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Special Relativity

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• David Stiff

At speeds that are a substantial fraction of the speed of light, the framework of Newtonian mechanics no longer suffices to describe many physical phenomena. Instead, one must start to take into account Einstein's theory of special relativity , which deals with the "special" case of physics in the absence of gravity. The more "general" case of general relativity takes into account gravitational effects.

Introduction

History of special relativity, einstein's postulates, time dilation, length contraction, loss of simultaneity, lorentz transformation, relativistic dynamics.

Given what is known about modern physics today, it is quite easy to take for granted that the speed of light, approximately \( 3 \cdot 10^8\: \text{m}/\text{s} \), is the unwavering speed limit of the universe. However, just a little over a century ago, this seemingly simple fact shook to the ground the Newtonian laws of physics that had stood for over two centuries and became the basis of Einstein's revolutionary 1905 paper On the Electrodynamics of Moving Bodies , in which Einstein posited the fundamental ideas of special relativity [1].

While Newtonian physics is adequate to describe the motion of most macroscopic objects, which usually move at speeds much slower than the speed of light, many macroscopic phenomena are fundamentally microscopic in nature. Due to their low mass-to-charge ratio , microscopic particles are relatively easy to accelerate to high speeds and must often be described relativistically. As it turns out, special relativity is crucial to explaining the correct chemical activity of the elements predicted by chemistry, the behavior of magnetic materials, and the behavior of subatomic particles in particle accelerators, all of which involve particles such as electrons or protons moving at relativistic speeds.

Relativity is also often used on a very macroscopic scale when an extremely precise measurement is required. For instance, GPS satellites, which orbit Earth tens of thousands of kilometers above its surface, must account for the effects of general relativity on the passage of time in order to provide directions accurate enough for daily use. Another well-known case is correctly describing the particularly strange orbit of the planet Mercury, as well as the dynamics of black holes and other large astronomical bodies.

The marriage of special relativity and quantum mechanics in the 1930s allowed for the prediction of antimatter and soon led to the birth of particle physics via quantum field theory .

The beginnings of special relativity came well before Einstein's 1905 paper introducing special relativity. In the decades after Maxwell fleshed out a framework for describing electricity and magnetism in the mid-nineteenth century, physicists started to become aware of possible holes in the laws of physics.

One of the direct implications of what became known as Maxwell's equations is that a propagating electromagnetic wave —that is, light—must travel at a fixed speed of approximately \( 3 \cdot 10^8 \: \text{m}/\text{s} \). This is a result that stands somewhat at odds with classical physics, which inherently requires that no speeds be absolute.

Recall that Newtonian physics assumes that relative velocities add directly according to the so-called Galilean transformation . Suppose an observer in an inertial frame of reference moves at velocity \( \mathbf{v} \) relative to some lab frame. If this observer measures an object to move with velocity \( \mathbf{w} \) in his or her frame, then one measures the velocity of the object in the lab frame to be \( \mathbf{v} + \mathbf{w} \). As a result, there always exists a frame in which an object's velocity is greater than in its rest frame. If a car driving at \( 40 \text{ m}/\text{s} \) with respect to observers on the side of the road flashes its headlights in the direction of its motion, then classical physics predicts that the observers on the side of the road should measure the speed of the light emitted from the headlights to be \( 40 \text{ m}/\text{s} + 3 \cdot 10^8 \text{ m}/\text{s} \). However, this is not what is predicted by Maxwell's equations as we currently understand them.

Similarly troubling was the fact that the velocity dependence of Maxwell's equations led to apparent asymmetries in the electric and magnetic forces. Consider, for instance, Einstein's example of a stationary loop of wire and moving magnet. According to Lenz's law , the moving magnet changes the magnetic flux across the wire loop and leads to an induced current, or an electric field , in the wire. However, the loop feels no magnetic force since it is stationary. Equivalently, one can also view the magnet as stationary and the loop as moving, in which case the loop experiences a magnetic force but no electric force. In this simplified analysis, the forces calculated in both frames turn out to be equal, but in one frame the loop perceives a magnetic field, while in the other it perceives an electric field.

Worse yet, when one considers Ampère's law (with Maxwell's modifications), one obtains an additional magnetic field term resulting from a changing electric field in the case of the moving magnet. This additional term is small but nonetheless cannot be accounted for classically.

Einstein's breakthrough in addressing these paradoxes in electromagnetism was to apply the mathematical framework of Lorentz and Poincaré from decades earlier to modify Galilean relativity. Confronted with these puzzles plus experimental observations in the late \(19^\text{th}\) century that suggested that the speed of light was indeed constant in all frames, Einstein realized Lorentz and Poincaré's mathematics could be perfectly adapted to account for a velocity bound in nature. Einstein's seminal paper on special relativity in electrodynamics incorporated their so-called Lorentz transformation to explain why propagating electromagnetic waves were consistent with the existing laws of physics. Einstein's genius was to make the bold statement that it was not Maxwell's equations but, rather, the centuries-old Newtonian framework that had to be modified. On this basis, Einstein was able to elegantly derive the Lorentz transformation from first principles in physics, leading to the modern theory of special relativity as we know it today.

Einstein proposed two simple postulates as the basis for special relativity, which were justified by previous experimental results and fundamental ideals for physical theories:

Postulate 1. All inertial frames are equivalent with respect to physical laws. There is no "preferred" frame of reference.

Postulate 2. The speed of light is measured to be the same value, \( c \approx 3 \cdot 10^8 \: \text{m}/\text{s} \), by observers in all inertial frames.

Recall that an inertial frame is one that is not accelerating. It is generally assumed that a frame is inertial if and only if no fictitious forces like the centrifugal force appear in the frame.

It can be shown that these two seemingly innocuous postulates have the following counterintuitive consequences, among many others:

• Time dilation. A time interval measured by an observer moving with respect to a stationary observer may be measured to be longer in the frame of the stationary observer.
• Length contraction. A length measured by an observer moving with respect to a stationary observer may be measured to be shorter in the frame of the stationary observer.
• Loss of simultaneity. Events measured to be simultaneous by an observer moving with respect to a stationary observer may not be simultaneous in the frame of the stationary observer.

Note that Postulate 1 demands that the consequences be symmetric between two frames. For example, a time interval measured in the frame of a stationary observer must also be longer in the frame of the moving observer, to whom his or her own frame appears stationary and the "stationary" observer appears to be moving. As a result, there is no longer any sense of giving absolute quantities of time or length to a single event, since even the relative lengths of distance and time are relative to the frame of measurement. The relativity of such physical quantities gave rise to the name of the theory.

An immediate consequence of Postulate 2 is the fact that measurements of time will depend on the frame of reference. In general, a measurement of the duration of some event (for instance, the tick of a clock) measured by an observer moving with respect to a stationary observer will be longer in the frame of the stationary observer. If an intergalactic spaceship flying by Earth measures each pulse of its laser to take one millisecond, observers on Earth measure each pulse to take longer than one millisecond—far slower (and thus "dilated") compared to the time measured by the inhabitants of the spaceship.

Time dilation is a simple consequence of the fact that if the speed of light is fixed to be the same value for two observers in different frames, either length or time will change from the usual Newtonian value when transforming between frames.

The lengths of time measured in both frames, however, are indeed different. Indeed, the ratio of time measured in the stationary observer's frame \( t \) to the time measured in the train observer's frame \( t' \) is given by

\[ \frac{t}{t'} = \frac{c}{\sqrt{c^2 - v^2}}. \]

Since lengths perpendicular to the direction of motion of a frame are measured to be the same by observers in all frames (as argued below), it makes sense to consider a thought experiment containing a "light clock" set up in the perpendicular direction. An open train car, moving to the right at speed \(v\) with a light source pulsing a beam of light off a mirror at the top of the train. Suppose an observer in an open car of a train of height \( l \) moving to the right at a speed of \( v \) keeps track of time by pulsing light from the bottom of the train directly toward a mirror at the top of the train and measuring the time elapsed between the initial pulse and the return of the light back to the source after being reflected at the top of the train. Clearly, to the observer in the train, the time \( t' \) elapsed for one such pulse is \[ t' = \frac{2l}{c}, \] where \( c \) is the speed of light. An observer outside the train, however, cannot measure the same \( t' \) for the duration of the pulse, or else the total speed of the light pulse would be \[ v_{\text{total}} = \sqrt{v_x^2 + v_y^2} = \sqrt{v^2 + c^2}, \] which is larger than \( c \) and thus violates Postulate 2. Instead, it must be the case that \( v_{\text{total}} = c \), in which case \[ c = \sqrt{v^2 + v_y^2}, \] so \[ v_y = \sqrt{c^2 - v^2}. \] Therefore, the time \( t \) measured by an observer outside the train is \[ t = \frac{2l}{\sqrt{c^2 - v^2}}, \] and the ratio \( \frac{t}{t'} \) of time measured in the stationary observer's frame to the time measured in the train observer's frame is \[ \frac{t}{t'} = \frac{c}{\sqrt{c^2 - v^2}}.\ _\square \]

It is customary in special relativity to name this ratio \( \gamma \) (the Greek letter gamma ) and rewrite the expression in the form

\[ \gamma = \frac{c}{\sqrt{c^2 - v^2}} = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}, \]

in which case one may write simply

\[ \frac{t}{t'} = \gamma. \]

In this form, the asymptotic behavior of \( \gamma \) as a function of \( v \) is clear. As \( v \rightarrow 0 \), \( \gamma \rightarrow 1\), and as \( v \rightarrow c \), \(\gamma \rightarrow \infty \). In other words, as the speed of the train goes to zero, the time measured in the stationary observer's frame approaches that measured in the train observer's frame. However, as the speed of the train approaches the speed of light, the ratio between the two times increases without bound.

In all cases, \( \gamma \geq 1 \), so \( t \geq t' \), which proves the earlier statement that a time interval measured by an observer moving with respect to a stationary observer may be measured to be longer in the frame of the stationary observer.

Muon decay 1. Elementary particles called muons are constantly produced in the upper atmosphere due to collisions from cosmic rays. Because muons are relatively light (about a few hundred times heavier than an electron), they travel at nearly the speed of light. For the sake of simplicity, consider that all of the muons produced travel at \( 0.998 c \). Given an average distance of \( 15 \, \text{km} \) from the upper atmosphere to the earth's surface, classically one might believe that no muons should reach the surface because muons have a half-life of \( 1.56 \cdot 10^{-6} \, \text{s} \). Even though the muons travel at nearly the speed of light \( \big(3 \cdot 10^8 \, \text{m}/\text{s}\big), \) they exist for such a short time that the average muon travels no more than a few hundred meters. However, note that the decay time is the time measured in the muon's frame. As measured by an observer on the earth, the half-life is longer by a factor of \( \gamma \approx 15.8 \). As a result, according to special relativity, the muon travels almost \( 16 \) times farther in the stationary observer's frame than the classical prediction. This extra distance traveled (on the order of kilometers ) allows an appreciable number of muons to reach the earth's surface. Indeed, the relativistic prediction matches the experimental observation of muon detection on the earth's surface.

With the time dilation result in hand, one can show that lengths parallel to the direction of travel are shorter in the frame of the stationary observer. Lengths perpendicular to the direction of travel are the same in both frames, however, as argued below.

Analogous to the time dilation result, one can express the ratio of the length measured in the stationary observer's frame \( l \) to the time measured in the train observer's frame \( l' \). The result is

\[ \frac{l}{l'} = \frac{1}{\gamma}. \]

Consider a thought experiment containing a "light clock" set up in the parallel direction. Consider a mirror placed at the right end of the train, with a light source placed at the left end. Let the length of the train in the train observer's frame be \( l' \), and let the length of the train measured in the stationary observer's frame outside of the train be \( l. \) \((\)Note that a priori it is not necessarily true that \( l \neq l'.) \) Suppose that the train moves to the right at a speed of \( v \) and that the observer in the train keeps track of time by pulsing light from the left end of the train and measuring the time elapsed between the initial pulse and the return of the light back to the source after being reflected at the right end. To the observer in the train, the time \( t' \) elapsed for one such pulse is \[ t' = \frac{2l'}{c}, \] where \( c \) is the speed of light. However, an observer outside the train measures a different result. The time to reach the mirror from the source is no longer just the length of the train divided by the speed of the light but rather \[ t_\text{right} = \frac{l}{c - v}. \] To obtain this result, note that the train travels distance \(v t_\text{right}\) in time \(t_\text{right}\), so \[ t_\text{right} = \frac{l+v t_\text{right}}{c}\] and solving for \( t_\text{right} \) gives \( t_\text{right} = \frac{l}{c - v}\). By the same argument with a sign change, the time needed for light to reach the source from the right end is \[ t_\text{left} = \frac{l}{c + v}. \] Therefore, the total time \( t \) for one tick of the clock in the stationary observer's frame is \[ t = t_\text{right} + t_\text{left} = \frac{2lc}{c^2 - v^2}. \] Since \( \frac{t}{t}' = \gamma \), it follows that \[ \frac{2lc}{c^2 - v^2} = \frac{2 \gamma l'}{c}. \] Straightforward algebra leads to the desired result: \[ \gamma = \frac{l'}{l}.\ _\square \]

As the speed of the moving frame approaches the speed of light, the length measured in the stationary observer's frame therefore becomes arbitrarily small, whereas it remains unchanged for \( v = 0 \implies \gamma = 1 \).

Muon decay 2. Postulate 1 suggests that the previous muon decay example should be equivalent when viewed in the frame of the muon, which travels toward the earth at a speed \( 0.998 c \). In this frame, the half-life is properly \( 1.56 \cdot 10^{-6} \, \text{s} \). Do muons still reach the earth's surface? In the frame of the muon, the earth travels toward the muon at almost the speed of light, which requires that lengths measured in the earth's frame must be shorter by a factor of \( \gamma = 15.8 \). As a result, the atmosphere-earth distance of \( 15 \, \text{km} \) is reduced by a factor of \( \gamma \), which produces results equivalent to the calculation performed in the frame of the earth.

A simple thought experiment, discussed below, shows that lengths perpendicular to the direction of motion of a frame must be measured to be the same by observers in all frames.

There are two sticks of equal length aligned parallel to each other as shown. If the stick on the right moves toward the other, show that the length measured in the frame of either stick is the same. Suppose that the stick on the right has globs of paint on both ends. For the sake of contradiction, suppose that lengths measured by an observer in a moving frame are measured to be shorter in the frame of a stationary observer. In that case, in the frame of the stick on the left, the stick on the right is shorter, and therefore the right stick paints the left stick as the right stick passes through the left sticks. However, in the frame of the stick on the right, the stick on the left is moving toward the right stick, which means that the left stick is shorter in the frame of the right stick. Therefore, the right stick does not point the left stick as the left stick passes through the right stick. Because Postulate 1 is violated, our assumption must be false, and both sticks must be the same length in both frames. Similarly, a contradiction results if one supposes that lengths are longer to observers in a stationary frame.

In the derivation for the length contraction result, there was a certain asymmetry between the pulse of light moving toward the right end of the train (in the direction of travel of the train) and the light moving back toward the left end of the train (opposite to the direction of travel of the train). This observation is the basis for the fact that events simultaneous in the train frame may not be simultaneous in the frame of the stationary observer.

Suppose events at the left and right ends of a train are simultaneous in the train frame. It turns out that these events are not simultaneous in a stationary observer's frame outside the train. Furthermore, the time elapsed between the events in the stationary observer's frame is

\[ \Delta t = \frac{\gamma l' v}{c^2}, \]

with the event on the side of the left end of the train occurring first, \( l' \) the length of the train in the train frame, and \( v \) the speed of the train.

Suppose a light source is placed in the center of a train, which in the stationary observer's frame is of length \( l \). In the train frame, the light reaches both ends of the train simultaneously, but this is not the case in the stationary observer's frame. If the train moves to the right, the amount of time required for light to reach the right end as measured by the stationary observer is \[ t_\text{right} = \frac{l}{2(c - v)}, \] whereas the time for light to reach the left end is \[ t_\text{left} = \frac{l}{2(c + v)}, \] via the same reasoning as before. Using the expression for length contraction \( l/l' = 1/\gamma \) yields \[ \Delta t = t_\text{right} - t_\text{left} = \frac{l'}{\gamma} \left[\frac{1}{2(c - v)} - \frac{1}{2(c + v)}\right], \] which can be rearranged to \[ \Delta t = \frac{\gamma l' v}{c^2}. \]

While the expressions for time dilation and length contraction involved only time or length coordinates separately, here there is clearly "mixing" of the length and time coordinates in different frames. For this reason, in relativistic physics, one often refers to "spacetime" as a single entity, in which time sits on equal footing with the three spatial dimensions.

As they have been presented, the expressions for the key relativistic effects (time dilation, length contraction, and loss of simultaneity) only apply directly to each of the effects taken separately. In general, all of the effects can be systematically dealt with in one fell swoop. One can show that the so-called Lorentz transformation , which relates the stationary observer's coordinates with that of an observer in a moving frame, is consistent with all of the expressions previously obtained:

\[ \begin{align} ct' &= \gamma \left( ct - \frac{vx}{c}\right)\\ x' &= \gamma\left(x - \frac{v}{c} ct\right) \\ y' &= y \\ z' &= z. \end{align} \]

Here, the moving frame travels along the \( x \)-axis with velocity \( v \), with the coordinates of the moving frame denoted by primes. Usually, the Lorentz transformation is written as a matrix transformation between vectors of coordinates, defining \(\beta = \frac{v}{c}\):

\[\begin{pmatrix} ct' \\ x' \\ y' \\ z' \end{pmatrix} = \begin{pmatrix} \gamma & -\gamma \beta & 0 & 0 \\ -\gamma \beta & \gamma & 0 &0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 &1 \end{pmatrix} \begin{pmatrix} ct \\ x \\ y \\ z \end{pmatrix} .\]

If special relativity is true, why is it that one never encounters relativistic phenomena in everyday experience? The key lies in the size of \( \gamma \) at everyday velocities. For \( v \ll c \), \(1 - v^2/c^2 \approx 1\), and therefore \( \gamma \approx 1 \). Thus, at speeds much smaller than the speed of light, relativistic effects are quite minimal, since the \(\gamma\) factor modifying equations of motion effectively does not contribute any correction.

To illustrate how \(\gamma\) changes with \(v\), consider the below tables of values of \( \gamma \) as a function of \( v \) for several given speeds \( v \) (in terms of fractions of the speed of light \( c \)). For comparison, the speed of light is a factor of ten thousand times that of the escape velocity from the earth:

\[ \begin{array}{cc} \\ v & \gamma \\[.01cm] \hline \\[.01cm] 0.1 c & 1.005 \\ 0.25 c & 1.033 \\ 0.5 c & 1.155 \\ 0.9 c & 2.294 \\ 0.99 c & 7.089 \end{array} \]

Clearly, the value of \( \gamma \) is barely larger than \( 1 \) for any speeds that are not an appreciable fraction of \( c \).

One of the clear implications of special relativity is the fact that no object with mass can travel at the speed of light or faster. This presents a clear problem with the Newtonian expressions of various dynamical quantities such as the kinetic energy \( \frac{1}{2} mv^2 \) and the momentum \( m \mathbf{v} \). In both cases, both quantities are constrained to be finite even though there are no physical laws that prevent an arbitrarily large amount (but finite) amount of energy to be added to a system.

It turns out that there exist proper relativistic expressions for the energy and momentum that have the proper asymptotic behavior as \( v \) approaches \( c \). The total energy of a particle of mass \( m \) and speed \( v \) is given by

\[ E = \gamma mc^2. \]

One peculiar outcome of this expression is that a particle at rest must have some rest energy \(E_0 = mc^2 \), which is a powerful implication in and of itself.

Similarly, the relativistic momentum is

\[ \mathbf{p} = \gamma m \mathbf{v}, \]

which, like the relativistic energy, has the correct limiting behavior.

These equations, like all others in special relativity, can be derived by demanding consistency with Einstein's two postulates or equivalently enforcing the Lorentz transformation in a variety of physical circumstances.

As in classical physics, the total energy and momentum of a system is always conserved. \((\)However, mass is no longer conserved in general. Einstein's equation \(E = mc^2\) shows that energy may be converted to mass and vice versa, and only the total energy including the mass will be conserved.\()\)

[1] Einstein, A. " On the Electrodynamics of Moving Bodies ." Annalen der Physik , 1905.

[2] Morin, D.J. Introduction to Classical Mechanics . Cambridge University Press, 2007.

[3] Taylor, J.R. Classical Mechanics . University Science Books, 2005.

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Introduction to Special Relativity

Chapter outline.

Have you ever looked up at the night sky and dreamed of traveling to other planets in faraway star systems? Would there be other life forms? What would other worlds look like? You might imagine that such an amazing trip would be possible if we could just travel fast enough, but you will read in this chapter why this is not true. In 1905 Albert Einstein developed the theory of special relativity. This theory explains the limit on an object’s speed and describes the consequences.

Relativity does not only apply to far-reaching and (as yet) unrealized activities like human interstellar travel. It affects everyday life in the form of communication, global trade, and even medicine. For example, Global Positioning Systems, which drive everything from airplane navigation to smart phone maps, rely on signals captured by multiple orbiting satellites and highly accurate measurements of time. Every signal passing between satellites, towers, and devices must be precisely measured and account for the relativistic effect of curved space and time dilation (discussed below). Variations in Earth’s landscape, its non-spherical shape, and the effects of gravity must also be considered in order to obtain accurate measurements. One of the most important contributors to these systems was Gladys West, a computer scientist and mathematician working at the Naval Proving Ground, where GPS and related technologies were advanced. West had previously developed altimeter models and managed the world’s first satellite-based ocean mapping project (Seastat). She then developed and programmed the algorithms capable of calculating positions and Earth’s shape to sufficient precisions to enable the existence of GPS. In these calculations, she accounted for the impacts of relativity and other complex principles related to it.

Relativity . The word relativity might conjure an image of Einstein, but the idea did not begin with him. People have been exploring relativity for many centuries. Relativity is the study of how different observers measure the same event. Galileo and Newton developed the first correct version of classical relativity. Einstein developed the modern theory of relativity. Modern relativity is divided into two parts. Special relativity deals with observers who are moving at constant velocity. General relativity deals with observers who are undergoing acceleration. Einstein is famous because his theories of relativity made revolutionary predictions. Most importantly, his theories have been verified to great precision in a vast range of experiments, altering forever our concept of space and time.

It is important to note that although classical mechanics, in general, and classical relativity, in particular, are limited, they are extremely good approximations for large, slow-moving objects. Otherwise, we could not use classical physics to launch satellites or build bridges. In the classical limit (objects larger than submicroscopic and moving slower than about 1% of the speed of light), relativistic mechanics becomes the same as classical mechanics. This fact will be noted at appropriate places throughout this chapter.

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Description: Professor David Kaiser teaches in the physics department at MIT and is also a historian of science. In this guest lecture, he describes how the most accomplished physicists of the mid-to-late 19 th century were thinking about motion of bodies through space and time, and how, at the end of that century, a rather young and very little-known person named Albert Einstein began asking similar questions but often in very different ways. (01:13:40)

Space, Time, and Spacetime (PDF - 5.8MB) (Courtesy of Prof. David Kaiser. Used with permission.)

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Course: physics library   >   unit 16.

• Starting to set up a Newtonian path–time diagram
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• Galilean transformation and contradictions with light

Introduction to special relativity and Minkowski spacetime diagrams

• Measuring time in meters in Minkowski spacetime
• Angle of x' axis in Minkowski spacetime

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Video transcript

Space Travel Calculator

Table of contents

Ever since the dawn of civilization, the idea of space travel has fascinated humans! Haven't we all looked up into the night sky and dreamed about space?

With the successful return of the first all-civilian crew of SpaceX's Inspiration4 mission after orbiting the Earth for three days, the dream of space travel looks more and more realistic now.

While traveling deep into space is still something out of science fiction movies like Star Trek and Star Wars, the tremendous progress made by private space companies so far seems very promising. Someday, space travel (or even interstellar travel) might be accessible to everyone!

It's never too early to start planning for a trip of a lifetime (or several lifetimes). You can also plan your own space trip and celebrate World Space Week in your own special way!

This space travel calculator is a comprehensive tool that allows you to estimate many essential parameters in theoretical interstellar space travel . Have you ever wondered how fast we can travel in space, how much time it will take to get to the nearest star or galaxy, or how much fuel it requires? In the following article, using a relativistic rocket equation, we'll try to answer questions like "Is interstellar travel possible?" , and "Can humans travel at the speed of light?"

Explore the world of light-speed travel of (hopefully) future spaceships with our relativistic space travel calculator!

If you're interested in astrophysics, check out our other calculators. Find out the speed required to leave the surface of any planet with the escape velocity calculator or estimate the parameters of the orbital motion of planets using the orbital velocity calculator .

One small step for man, one giant leap for humanity

Although human beings have been dreaming about space travel forever, the first landmark in the history of space travel is Russia's launch of Sputnik 2 into space in November 1957. The spacecraft carried the first earthling, the Russian dog Laika , into space.

Four years later, on 12 April 1961, Soviet cosmonaut Yuri A. Gagarin became the first human in space when his spacecraft, the Vostok 1, completed one orbit of Earth.

The first American astronaut to enter space was Alan Shepard (May 1961). During the Apollo 11 mission in July 1969, Neil Armstrong and Buzz Aldrin became the first men to land on the moon. Between 1969 and 1972, a total of 12 astronauts walked the moon, marking one of the most outstanding achievements for NASA.

In recent decades, space travel technology has seen some incredible advancements. Especially with the advent of private space companies like SpaceX, Virgin Galactic, and Blue Origin, the dream of space tourism is looking more and more realistic for everyone!

However, when it comes to including women, we are yet to make great strides. So far, 566 people have traveled to space. Only 65 of them were women .

Although the first woman in space, a Soviet astronaut Valentina Tereshkova , who orbited Earth 48 times, went into orbit in June 1963. It was only in October 2019 that the first all-female spacewalk was completed by NASA astronauts Jessica Meir and Christina Koch.

Women's access to space is still far from equal, but there are signs of progress, like NASA planning to land the first woman and first person of color on the moon by 2024 with its Artemis missions. World Space Week is also celebrating the achievements and contributions of women in space this year!

In the following sections, we will explore the feasibility of space travel and its associated challenges.

How fast can we travel in space? Is interstellar travel possible?

Interstellar space is a rather empty place. Its temperature is not much more than the coldest possible temperature, i.e., an absolute zero. It equals about 3 kelvins – minus 270 °C or minus 455 °F. You can't find air there, and therefore there is no drag or friction. On the one hand, humans can't survive in such a hostile place without expensive equipment like a spacesuit or a spaceship, but on the other hand, we can make use of space conditions and its emptiness.

The main advantage of future spaceships is that, since they are moving through a vacuum, they can theoretically accelerate to infinite speeds! However, this is only possible in the classical world of relatively low speeds, where Newtonian physics can be applied. Even if it's true, let's imagine, just for a moment, that we live in a world where any speed is allowed. How long will it take to visit the Andromeda Galaxy, the nearest galaxy to the Milky Way?

We will begin our intergalactic travel with a constant acceleration of 1 g (9.81 m/s² or 32.17 ft/s²) because it ensures that the crew experiences the same comfortable gravitational field as the one on Earth. By using this space travel calculator in Newton's universe mode, you can find out that you need about 2200 years to arrive at the nearest galaxy! And, if you want to stop there, you need an additional 1000 years . Nobody lives for 3000 years! Is intergalactic travel impossible for us, then? Luckily, we have good news. We live in a world of relativistic effects, where unusual phenomena readily occur.

Can humans travel at the speed of light? – relativistic space travel

In the previous example, where we traveled to Andromeda Galaxy, the maximum velocity was almost 3000 times greater than the speed of light c = 299,792,458 m/s , or about c = 3 × 10 8 m/s using scientific notation.

However, as velocity increases, relativistic effects start to play an essential role. According to special relativity proposed by Albert Einstein, nothing can exceed the speed of light. How can it help us with interstellar space travel? Doesn't it mean we will travel at a much lower speed? Yes, it does, but there are also a few new relativistic phenomena, including time dilation and length contraction, to name a few. The former is crucial in relativistic space travel.

Time dilation is a difference of time measured by two observers, one being in motion and the second at rest (relative to each other). It is something we are not used to on Earth. Clocks in a moving spaceship tick slower than the same clocks on Earth ! Time passing in a moving spaceship T T T and equivalent time observed on Earth t t t are related by the following formula:

where γ \gamma γ is the Lorentz factor that comprises the speed of the spaceship v v v and the speed of light c c c :

where β = v / c \beta = v/c β = v / c .

For example, if γ = 10 \gamma = 10 γ = 10 ( v = 0.995 c v = 0.995c v = 0.995 c ), then every second passing on Earth corresponds to ten seconds passing in the spaceship. Inside the spacecraft, events take place 90 percent slower; the difference can be even greater for higher velocities. Note that both observers can be in motion, too. In that case, to calculate the relative relativistic velocity, you can use our velocity addition calculator .

Let's go back to our example again, but this time we're in Einstein's universe of relativistic effects trying to reach Andromeda. The time needed to get there, measured by the crew of the spaceship, equals only 15 years ! Well, this is still a long time, but it is more achievable in a practical sense. If you would like to stop at the destination, you should start decelerating halfway through. In this situation, the time passed in the spaceship will be extended by about 13 additional years .

Unfortunately, this is only a one-way journey. You can, of course, go back to Earth, but nothing will be the same. During your interstellar space travel to the Andromeda Galaxy, about 2,500,000 years have passed on Earth. It would be a completely different planet, and nobody could foresee the fate of our civilization.

A similar problem was considered in the first Planet of the Apes movie, where astronauts crash-landed back on Earth. While these astronauts had only aged by 18 months, 2000 years had passed on Earth (sorry for the spoilers, but the film is over 50 years old at this point, you should have seen it by now). How about you? Would you be able to leave everything you know and love about our galaxy forever and begin a life of space exploration?

Space travel calculator – relativistic rocket equation

Now that you know whether interstellar travel is possible and how fast we can travel in space, it's time for some formulas. In this section, you can find the "classical" and relativistic rocket equations that are included in the relativistic space travel calculator.

There could be four combinations since we want to estimate how long it takes to arrive at the destination point at full speed as well as arrive at the destination point and stop. Every set contains distance, time passing on Earth and in the spaceship (only relativity approach), expected maximum velocity and corresponding kinetic energy (on the additional parameters section), and the required fuel mass (see Intergalactic travel — fuel problem section for more information). The notation is:

• a a a — Spaceship acceleration (by default 1   g 1\rm\, g 1 g ). We assume it is positive a > 0 a > 0 a > 0 (at least until halfway) and constant.
• m m m — Spaceship mass. It is required to calculate kinetic energy (and fuel).
• d d d — Distance to the destination. Note that you can select it from the list or type in any other distance to the desired object.
• T T T — Time that passed in a spaceship, or, in other words, how much the crew has aged.
• t t t — Time that passed in a resting frame of reference, e.g., on Earth.
• v v v — Maximum velocity reached by the spaceship.
• K E \rm KE KE — Maximum kinetic energy reached by the spaceship.

The relativistic space travel calculator is dedicated to very long journeys, interstellar or even intergalactic, in which we can neglect the influence of the gravitational field, e.g., from Earth. We didn't include our closest celestial bodies, like the Moon or Mars, in the destination list because it would be pointless. For them, we need different equations that also take into consideration gravitational force.

Newton's universe — arrive at the destination at full speed

It's the simplest case because here, T T T equals t t t for any speed. To calculate the distance covered at constant acceleration during a certain time, you can use the following classical formula:

Since acceleration is constant, and we assume that the initial velocity equals zero, you can estimate the maximum velocity using this equation:

and the corresponding kinetic energy:

Newton's universe — arrive at the destination and stop

In this situation, we accelerate to the halfway point, reach maximum velocity, and then decelerate to stop at the destination point. Distance covered during the same time is, as you may expect, smaller than before:

Acceleration remains positive until we're halfway there (then it is negative – deceleration), so the maximum velocity is:

and the kinetic energy equation is the same as the previous one.

Einstein's universe — arrive at the destination at full speed

The relativistic rocket equation has to consider the effects of light-speed travel. These are not only speed limitations and time dilation but also how every length becomes shorter for a moving observer, which is a phenomenon of special relativity called length contraction. If l l l is the proper length observed in the rest frame and L L L is the length observed by a crew in a spaceship, then:

What does it mean? If a spaceship moves with the velocity of v = 0.995 c v = 0.995c v = 0.995 c , then γ = 10 \gamma = 10 γ = 10 , and the length observed by a moving object is ten times smaller than the real length. For example, the distance to the Andromeda Galaxy equals about 2,520,000 light years with Earth as the frame of reference. For a spaceship moving with v = 0.995 c v = 0.995c v = 0.995 c , it will be "only" 252,200 light years away. That's a 90 percent decrease or a 164 percent difference!

Now you probably understand why special relativity allows us to intergalactic travel. Below you can find the relativistic rocket equation for the case in which you want to arrive at the destination point at full speed (without stopping). You can find its derivation in the book by Messrs Misner, Thorne ( Co-Winner of the 2017 Nobel Prize in Physics ) and Wheller titled Gravitation , section §6.2. Hyperbolic motion. More accessible formulas are in the mathematical physicist John Baez's article The Relativistic Rocket :

• Time passed on Earth:
• Time passed in the spaceship:
• Maximum velocity:
• Relativistic kinetic energy remains the same:

The symbols sh ⁡ \sh sh , ch ⁡ \ch ch , and th ⁡ \th th are, respectively, sine, cosine, and tangent hyperbolic functions, which are analogs of the ordinary trigonometric functions. In turn, sh ⁡ − 1 \sh^{-1} sh − 1 and ch ⁡ − 1 \ch^{-1} ch − 1 are the inverse hyperbolic functions that can be expressed with natural logarithms and square roots, according to the article Inverse hyperbolic functions on Wikipedia.

Einstein's universe – arrive at destination point and stop

Most websites with relativistic rocket equations consider only arriving at the desired place at full speed. If you want to stop there, you should start decelerating at the halfway point. Below, you can find a set of equations estimating interstellar space travel parameters in the situation when you want to stop at the destination point :

Intergalactic travel – fuel problem

So, after all of these considerations, can humans travel at the speed of light, or at least at a speed close to it? Jet-rocket engines need a lot of fuel per unit of weight of the rocket. You can use our rocket equation calculator to see how much fuel you need to obtain a certain velocity (e.g., with an effective exhaust velocity of 4500 m/s).

Hopefully, future spaceships will be able to produce energy from matter-antimatter annihilation. This process releases energy from two particles that have mass (e.g., electron and positron) into photons. These photons may then be shot out at the back of the spaceship and accelerate the spaceship due to the conservation of momentum. If you want to know how much energy is contained in matter, check out our E = mc² calculator , which is about the famous Albert Einstein equation.

Now that you know the maximum amount of energy you can acquire from matter, it's time to estimate how much of it you need for intergalactic travel. Appropriate formulas are derived from the conservation of momentum and energy principles. For the relativistic case:

where e x e^x e x is an exponential function, and for classical case:

Remember that it assumes 100% efficiency! One of the promising future spaceships' power sources is the fusion of hydrogen into helium, which provides energy of 0.008 mc² . As you can see, in this reaction, efficiency equals only 0.8%.

Let's check whether the fuel mass amount is reasonable for sending a mass of 1 kg to the nearest galaxy. With a space travel calculator, you can find out that, even with 100% efficiency, you would need 5,200 tons of fuel to send only 1 kilogram of your spaceship . That's a lot!

So can humans travel at the speed of light? Right now, it seems impossible, but technology is still developing. For example, a photonic laser thruster is a good candidate since it doesn't require any matter to work, only photons. Infinity and beyond is actually within our reach!

How do I calculate the travel time to other planets?

To calculate the time it takes to travel to a specific star or galaxy using the space travel calculator, follow these steps:

• Choose the acceleration : the default mode is 1 g (gravitational field similar to Earth's).
• Enter the spaceship mass , excluding fuel.
• Select the destination : pick the star, planet, or galaxy you want to travel to from the dropdown menu.
• The distance between the Earth and your chosen stars will automatically appear. You can also input the distance in light-years directly if you select the Custom distance option in the previous dropdown.
• Define the aim : select whether you aim to " Arrive at destination and stop " or “ Arrive at destination at full speed ”.
• Pick the calculation mode : opt for either " Einstein's universe " mode for relativistic effects or " Newton's universe " for simpler calculations.
• Time passed in spaceship : estimated time experienced by the crew during the journey. (" Einstein's universe " mode)
• Time passed on Earth : estimated time elapsed on Earth during the trip. (" Einstein's universe " mode)
• Time passed : depends on the frame of reference, e.g., on Earth. (" Newton's universe " mode)
• Required fuel mass : estimated fuel quantity needed for the journey.
• Maximum velocity : maximum speed achieved by the spaceship.

How long does it take to get to space?

It takes about 8.5 minutes for a space shuttle or spacecraft to reach Earth's orbit, i.e., the limit of space where the Earth's atmosphere ends. This dividing line between the Earth's atmosphere and space is called the Kármán line . It happens so quickly because the shuttle goes from zero to around 17,500 miles per hour in those 8.5 minutes .

How fast does the space station travel?

The International Space Station travels at an average speed of 28,000 km/h or 17,500 mph . In a single day, the ISS can make several complete revolutions as it circumnavigates the globe in just 90 minutes . Placed in orbit at an altitude of 350 km , the station is visible to the naked eye, looking like a dot crossing the sky due to its very bright solar panels.

How do I reach the speed of light?

To reach the speed of light, you would have to overcome several obstacles, including:

Mass limit : traveling at the speed of light would mean traveling at 299,792,458 meters per second. But, thanks to Einstein's theory of relativity, we know that an object with non-zero mass cannot reach this speed.

Energy : accelerating to the speed of light would require infinite energy.

Effects of relativity : from the outside, time would slow down, and you would shrink.

Why can't sound travel in space?

Sound can’t travel in space because it is a mechanical wave that requires a medium to propagate — this medium can be solid, liquid, or gas. In space, there is no matter, or at least not enough for sound to propagate. The density of matter in space is of the order 1 particle per cubic centimeter . While on Earth , it's much denser at around 10 20 particles per cubic centimeter .

Dreaming of traveling into space? 🌌 Plan your interstellar travel (even to a Star Trek destination) using this calculator 👨‍🚀! Estimate how fast you can reach your destination and how much fuel you would need 🚀

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ol{padding-top:0px;}.css-4okk7a ul:not(:first-child),.css-4okk7a ol:not(:first-child){padding-top:4px;} Spaceship and destination 👩‍🚀👨‍🚀

Spaceship acceleration

Spaceship mass

Mass of spaceship excluding fuel.

Destination

Select a destination from the list or type in distance by hand.

Which star/galaxy?

If you want to input your own distance, select the 'Custom destination' option in the 'Which star/galaxy?' field.

Calculation options

Do you want to stop at destination point? If yes, the spaceship will start decelerating once it reaches the halfway point.

Calculations mode

You can compare Einstein's special relativity with non-relativistic Newton's physics. Remember that at near-light speeds only the former is correct!

Travel details 🚀

Time passed in spaceship

Time passed on Earth

Time passed in the resting frame of reference. It could be an observer on Earth.

Required fuel mass

Assuming 100% efficiency.

Maximum velocity

Note that our calculator may round velocity to the speed of light if it is really close to it.

Additional parameters

Relativistic Flight Mechanics and Space Travel

• © 2007
• Richard F. Tinder 0

Washington State University, USA

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Part of the book series: Synthesis Lectures on Engineering, Science, and Technology (SLEST)

Part of the book sub series: Synthesis Lectures on Engineering (SLE)

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Table of contents (6 chapters)

Front matter, introduction.

Richard F. Tinder

Relativistic Rocket Mechanics

Space travel and the photon rocket, minkowski diagrams, k-calculus, and relativistic effects, other prospective transport systems for relativistic space travel, back matter, about this book, authors and affiliations, about the author, bibliographic information.

Book Title : Relativistic Flight Mechanics and Space Travel

Authors : Richard F. Tinder

Series Title : Synthesis Lectures on Engineering, Science, and Technology

DOI : https://doi.org/10.1007/978-3-031-79297-7

Publisher : Springer Cham

eBook Packages : Synthesis Collection of Technology (R0) , eBColl Synthesis Collection 1

Copyright Information : Springer Nature Switzerland AG 2007

Softcover ISBN : 978-3-031-79296-0 Published: 31 December 2007

eBook ISBN : 978-3-031-79297-7 Published: 01 June 2022

Series ISSN : 2690-0300

Series E-ISSN : 2690-0327

Edition Number : 1

Number of Pages : XXII, 117

Topics : Engineering Design , Materials Engineering , Professional & Vocational Education

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Special Theory of Relativity

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• Page ID 3401

INTRODUCTION

In this document we discuss Einstein's Special Theory of Relativity. The treatment is non-mathematical, except for a brief use of Pythagoras' theorem about right triangles. We concentrate on the implications of the theory. The document is based on a discussion of the the theory for an upper-year liberal arts course in Physics without mathematics; in the context of that course the material here takes about 4 or 5 one-hour classes.

Einstein published this theory in 1905. The word special here means that we restrict ourselves to observers in uniform relative motion. This is as opposed the his General Theory of Relativity of 1916; this theory considers observers in any state of uniform motion including relative acceleration. It turns out that the general theory is also a theory of gravitation.

Sometimes one hears that the Special Theory of Relativity says that all motion is relative. This is not quite true. Galileo and Newton had a similar conception. Crucial to Newton's thinking is that there is an absolute space, independent of the things in that space:

"Absolute space, in its own nature, without relation to anything external, remains always similar and immovable. Relative space is some movable dimension or measure of the absolute spaces; which our senses determine by its position to bodies .. because the parts of space cannot be seen, or distinguished from one another by our senses, there in their stead we use sensible [i.e. perceptible by the sense] measures of them ... but in philosophical disquisition, we ought to abstract from our senses, and consider things themselves, distinct from what are only sensible measures of them." -- Principia I , Motte trans.

For Newton, the laws of physics, such as the principle of inertia, are true in any frame of reference either at rest relative to absolute space or in uniform motion in a straight line relative to absolute space. Such reference frames are called inertial . Notice there is a bit of a circular argument here: the laws of physics are true in inertial frames, and inertial frames are ones in which the laws of physics are true.

In any case, from the standpoint of any such inertial frame of reference all motion can be described as being relative. If you are standing by the highway watching a bus go by you at 100 km/hr, then relative to somebody on the bus you are traveling in the opposite direction at 100 km/hr.

This principle, called Galilean relativity , is kept in Einstein's Theory of Relativity.

Many of the consequences of the Special Theory of Relativity are counter-intuitive and violate common sense . Einstein correctly defined common sense as those prejudices that we acquire at an early age.

THE CONSTANCY OF THE SPEED OF LIGHT

Once we realize that light is some sort of a wave, a natural question is "what is waving?" One answer to this question is that it is the luminiferous ether . The idea behind this word is that there is an all-pervading homogenous massless substance everywhere in the universe, and it is this ether that is the medium through which light propagates. Note that this ether could define Newton's absolute space.

A rough analogy is to a sound wave traveling through the air. The air is the medium and oscillations of the molecules of the air are what is "waving." The speed of sound is about 1193 km/hr with respect to the air, depending on the temperature and pressure. Thus if I am traveling through the air at 1193 km/hr in the same direction as a sound wave, the speed of the wave relative to me will be zero.

The speed of light is measured to be about 1,079,253,000 km/hr, and presumably this is its speed relative to the ether. Presumably the ether is stationary with respect to the fixed stars. This section investigates these two presumptions.

Galileo attempted to measure the speed of light around 1600. He and a colleague each had a lantern with a shutter, and they went up on neighboring mountains. Galileo opened the shutter on his lantern and when his colleague saw the light from Galileo's lantern he opened the shutter on this lantern. The time delay between when Galileo opened the shutter on his lantern and when he saw the answering light from his colleague's lantern would allow him to calculate the speed of light. This is absolutely correct experimental procedure in principle. However, because of our human reaction times the lag between when the colleague saw the light from Galileo's lantern until when he could get the shutter of his lantern open is so long that the light could have circled the globe many many times.

In 1676 Römer successfully measured the speed of light, although his results differed from the accepted value today by about 30%

The Michelson-Morley Experiment

In this sub-section we discuss a famous experiment done in the late nineteenth century by Michelson and Morley. Some knowledge of the fact that light is a wave and can undergo interference is assumed. A discussion of this occurs in the the first two sections of the document http://www.upscale.utoronto.ca/GeneralInterest/Harrison/DoubleSlit/DoubleSlit.html .

In is ironic that Michelson himself wrote in 1899, "The more important fundamental laws and facts of physical reality have all been discovered and they are now so firmly established that the possibility of their ever being supplanted in consequence of new discoveries is exceedingly remote .... Our future discoveries must be looked for in the 6th place of decimals." At this time there were a couple of small clouds on the horizon. One of those clouds was his own experiment with Morley that we describe in this sub-section. As we shall see, the experiment played a part in the development of the Special Theory of Relativity, a profound advance.

Recently some people, especially John Horgan in his book The End of Science (1996), have been making similar claims about how the enterprise of science is complete. My opinion is that they are no more correct than was Michelson. I certainly hope they are wrong, because if they are correct all the fun goes out of physics. In fact, as we shall see, I think there are already a couple of clouds on the horizon. One cloud is the failure of our theories of cosmology to account for recent observations of the universe. The other is the failure of the quark model to produce any truly useful results.

Before we turn to the experiment itself we will consider a "race" between two swimmers.

We have two identical swimmers, 1 and 2, who each swim the same distance away from the raft, to the markers, and then swim back to the raft. The "race" ends in a tie.

Now the raft and markers are being towed to the left. In this case the race will no longer be a tie. In fact, it is not too hard to show that swimmer 2 wins this race.

A small Flash animation illustrating the above race may be found here .

These notes are intended to be non-mathematical, with the exception of a brief use of Pythagoras theorem about right triangles. However, some people would like to see a little bit of the math. Thus, a proof that swimmer 2 above wins the race may be found here . Below, a further small amount of math will appear, but will always be labelled as a Technical note .

One of the difficulties that students experience in learning about the theories of relativity is that it is easy to ask questions of themselves and/or others that are not well formed. Insisting on complete statements often makes the problems disappear. One common case of sloppy language leading to poorly formed questions involves the concept of speed. If we say, for example, that the swimmers in the above examples swim at 5 km/hr we have not made a complete statement; we should say that the swimmers swim at 5 km/hr with respect to the water . If we are stationary with respect to the water then they swim at 5 km/hr with respect to us. But if we are moving at, say, 5 km/hr with respect to the water in the direction that one of the swimmers is swimming, that swimmer will be stationary relative to us.

Now we consider the Michelson interferometer, shown schematically to the right. The light source is the red star to the left of the figure. The light from it is incident on a half-silvered mirror, which is drawn as a blue line; this is a "crummy" mirror that only reflects one-half of the light incident on it, transmitting the other half. The two light beams then go to good mirrors, drawn as green rectangles, which reflect the light. The reflected light actually follows the same path as the incident beam, although I have drawn them slightly offset. When beam 1 returns to the half-silvered mirror, one half is reflected down; the other half is transmitted back toward the light source but I haven't bothered to draw that ray. Similarly, when beam 2 returns to the half-silvered mirror, one half is transmitted; the other half is reflected towards the source although I haven't drawn that ray either. The two combined beams go from the half-silvered mirror to the detector, which is the yellow object at the bottom of the figure.

If the distance from the half-silvered mirror to mirror 1 is equal to the distance to mirror 2, then when the two rays are re-combined they will have travelled identical distances. Thus, they will be "in phase" and will constructively interfere and we will get a strong signal at the detector. If we slowly move mirror 1 to the right, that ray will be travelling a longer total distance than ray 2; at some point the two rays will be "out of phase" and destructively interfere. Moving mirror 1 a bit further to the right, at some point the two rays will be "in phase" again, giving constructive interference.

Say we have the interferometer adjusted so we are getting constructive interference at the detector. Then the "race" between the two beams of light is essentially a tie. This may remind you of the race of the swimmers above.

Except that if we have the apparatus sitting on the earth, we have to remember that the speed of the earth in its orbit around the sun is on the order of 108,000 km/hr relative to the ether, depending on the season and time of day. So the situation is more like the second race above when the raft is being towed through the water. The interferometer is being "towed" through the ether.

Michelson and Morley did this experiment in the 1880's. The arms of the interferometer were about 1.2 meters long. The apparatus was mounted on a block of marble floating in a pool of mercury to reduce vibrations. They adjusted the interferometer for constructive interference, and then gently rotated the interferometer by 90 degrees.

Given the speed of light as 1,079,253,000 km/hr relative to the ether and the speed of the earth equal to some number like 108,000 km/hr relative to the ether, they calculated that they should easily see the combined beams going through maxima and minima in the interference pattern as they rotated the apparatus.

Except that when they did the experiment, they got no result. The interference pattern did not change!

It was suggested that maybe the speed of the earth due to its rotation on its axis was cancelling its speed due to its orbit around the sun. So they waited 12 hours and repeated the experiment. Again they got no result.

It was suggested that the Earth's motion in orbit around the Sun canceled the other motions. So they waited six months and tried the experiment again. And again they got no result.

It was suggested that maybe the mass of the earth "dragged" the ether along with it. So they hauled the apparatus up on top of a mountain, hoping that the mountain would be sticking up into the ether that was not being dragged by the earth. And again they got no result.

Thus, this attempt to measure the motion of the earth relative to the ether failed.

Lorentz was among many who were very puzzled by this result. He proposed that when an object was moving relative to the ether, its length along its direction of motion would be contracted by just the right amount needed to explain the experimental result. If the length of the object when it is at rest with respect to the ether is L 0 , then if is is moving at speed v through the ether its length becomes L given by:

\[L=\sqrt{1-v^2/c^2}L_0\]

where c is the speed of light relative to the ether. If you chose to look at the brief mathematical supplement above, the structure of this equation may look familiar to you.

Einstein "Explains" the Michelson-Morley Experiment

When Einstein was 16, in 1895, he asked himself an interesting question:

"If I pursue a beam of light with the velocity c I should observe such a beam of light as a spatially oscillatory electromagnetic field at rest. However, there seems to be no such thing, whether on the basis of experience or according to [the theory of electricity and magnetism]. From the very beginning it appeared to me intuitively clear that, judged from the standpoint of such an observer, everything would have to happen according to the same laws as for an observer who, relative to the earth, was at rest. For how, otherwise, should the first observer know, i.e.. be able to determine, that he is in a state of uniform motion?" -- As later written by Einstein in "Autobiographical Notes", in Schilpp, ed., Albert Einstein: Philosopher-Scientist .

He continued to work on this question for 10 years with the mixture of concentration and determination that characterized much of his work. He published his answer in 1905:

"... light is always propagated in empty space with a definite velocity c which is independent of the state of [relative] motion of the emitting body .... The introduction of a `luminiferous ether' will be superfluous inasmuch as the view here to be developed will not require an `absolutely stationary space' provided with special properties." -- Annalen Physik 17 (1905).

Put another way, the speed of light is 1,079,253,000 km/hr with respect to all observers.

As we shall see, this one statement is equivalent to all of the Special Theory of Relativity, and everything else is just a consequence.

Notice that the statement also explains the null result of the Michelson-Morley experiment. However, although the evidence is not certain it seems quite likely that in 1905 Einstein was unaware of the experiment (cf. Gerald Holton, "Einstein, Michelson and the 'Crucial' Experiment," which has appeared in Thematic Origins of Scientific Thought , pg. 261. and also in Isis 60 , 1969, pg. 133.).

EXPLORING THE CONSEQUENCES OF EINSTEIN'S "EXPLANATION"

Here we will begin to see why Einstein's statement about the constancy of the speed of light leads to all of the strange consequences such as time dilation, length contraction, etc. But first we should take a few moments to carefully explore just what we mean when we say some event occurred at some particular place at some particular time

We synchronise the clocks to the "Reference Clock." To do this correctly requires taking into account that if we are standing by one of the clocks looking at the Reference Clock, the time that we see on the Reference is not the current time, but is the time it was reading when the light we see left the clock. Thus we have to account for the small but finite time it takes light to travel from the Reference Clock to us standing beside another clock. A bit tedious, but fairly straightforward.

We imagine some event occurs. We define its position by where it happened relative to the lattice of meter sticks and we define the time when it happened as the time read by the nearest clock.

Of course, in practice nobody ever does this sort of thing.

Usually we don't bother to draw the whole lattice, but rather represent it by a set of coordinate axes, x and y , and a single clock measuring time t , as shown below. We have also put an observer, whom we shall name Lou , at rest in his coordinate system.

Next we imagine that Lou has a light bulb at the "origin" of his coordinate system. At some time t which we shall call zero he turns on the light. The light moves away from the light bulb at 1,079,253,000 km/hr as measured by Lou's system of rods and clocks. At some time t later the light will form a sphere with the light bulb right at the center.

There are two animations of this situation. One is a "simple" animated gif with a file size of 22k; it may be accessed by clicking here . The other is a Flash animation with a file size of 16k; it may be accessed by clicking here .

Now, Lou has a twin sister Sue , whom we shall assume was born at the same time as Lou (a biological impossibility). Sue has her own lattice of meter sticks and clocks and she is at rest relative to them. Just as for Lou, we represent Sue's rods and clocks as shown below.

Sue is an astronaut, and is in her rocket ship which is traveling at one-half the speed of light to the right relative to Lou. Of course, relative to Sue, Lou is travelling at half the speed of light to the left.

Let us imagine that Sue, traveling at half the speed of light relative to Lou, goes by Lou and he turned on the light bulb just at the moment that Sue passed by it. Sue will call this time zero as measured by her clocks.

Relative to Sue, the light bulb is traveling to the left at half the speed of light. However, because of Einstein's "explanation", the speed of light relative to her is exactly 1,079,253,000 km/hr. Thus, at some later time she will measure that the outer edge of the light forms a perfect sphere with her at the middle.

There are both a animated gif and Flash animation of the above. To access the 18k gif animation click here . To access the 18k Flash animation click here .

There is also a Flash animation of both Sue and Lou. To access the 22k animation click here .

If we think about the above a moment, it is clear that something weird is going on. Lou claims that the light forms a sphere with the light bulb at the center. Sue claims the light forms a sphere with her at the center. But except for the moment when the light bulb was first turned on, the light bulb and Sue are at nowhere near the same place. Evidently the position and time of the outer edge of the sphere as measured by Lou's system of rods and clocks and as measured by Sue's system of rods and clocks are not as our common sense would predict.

Note that the only assumption we have made here is the constancy of the speed of light. Thus, to avoid this sort of weirdness one must come up with another explanation of the null result of the Michelson-Morley experiment.

This theorem says that for any right triangle such as the one shown to the right:

x 2 + y 2 = h 2

Now, when Lou measures the position of the outer edge of the sphere of light he can use Pythagoras' Theorem to calculate the radius r of the sphere:

x 2 Lou + y 2 Lou = r 2 Lou

But the radius at time t is just the speed of the light, c , times the time:

r Lou = c t Lou

x 2 Lou + y 2 Lou = (c t) 2 Lou

Notice that we don't need to label c as being the speed of light relative to Lou, since it is the same number for all observers, including Sue.

Now, Sue measures the position of the outer edge of the sphere of light with her rods and clocks and will conclude that:

x 2 Sue + y 2 Sue = (c t) 2 Sue

I will write the relations for Sue and Lou in a form which will be useful later:

x 2 Lou + y 2 Lou - (c t) 2 Lou = 0 = x 2 Sue + y 2 Sue - (c t) 2 Sue

THE PARABLE OF THE SURVEYORS

Once upon a time there was a kingdom in which all positions were measured relative to the town square of the capitol.

This kingdom had a sort of strange religion that dictated that all North-South distances were to be measured in sacred units of feet ; East-West distances were measured in everyday units of meters .

Despite this religious requirement all positions in the kingdom could be uniquely specified.

There were two schools or surveying in operation. One, the daytime school, used a compass to determine the direction of North. The other, the nighttime school, used the North star to determine the direction of North.

As the sophistication of the measuring instruments increased, people began to notice that the daytime and nighttime measurements didn't quite agree. This is because magnetic North as determined by a compass is not in exactly the same direction as the North star. The figure to the right illustrates, although the actual difference is much less than in the diagram.

Finally, a young fellow named Albert attended both schools of surveying. He was also an irreligious person so he did not take the religious requirement of measuring North-South distances in feet seriously. He converted those North-South distances to everyday units by multiplying by k , the number of meters in a foot. He then discovered that although the daytime and nighttime numbers for the position of a particular place differed slightly, there was a constant:

E 2 night + (k N) 2 night = E 2 day + (k N) 2 day

What he is calculating, of course, is the distance squared between the town square and a particular location using Pythagoras' Theorem.

The original source for the above story is E.F. Taylor and J.A. Wheeler, Spacetime Physics (Freeman, 1966), pg. 1.

In the parable of the surveyors, we converted North-South distances from sacred units of feet to everyday units of meters, and found that for the two rotated reference frames, the daytime and nighttime frames, there was a constant for the position of a particular place in the kingdom relative to the town square:

In the section before that Sue and Lou were observing the same sphere of light expanding outwards and saw that here too there was a constant:

x 2 Lou + y 2 Lou - (c t) 2 Lou = x 2 Sue + y 2 Sue - (c t) 2 Sue

Notice the similarity to the surveyor system. Take time, measured in sacred units of seconds , and convert to everyday units of meters by multiplying the time by the speed of light. Take the normal position coordinates x and y plus the time coordinate, square them and combine them: the result is the same number for both Sue and Lou.

Thus we are led to the idea the time is just another coordinate, i.e. that time is the fourth dimension. The fact that there is a minus sign between the square of the normal spatial coordinates and the square of the time coordinates indicates that there is some difference between space and time, but it is not a large difference.

Thus, we tend to write spacetime as a single word as a mnemonic to remind us of all this.

Note that the speed of light, c , is now only a conversion factor for units. If we had started out measuring time in everyday units of meters instead of sacred units of seconds, the speed of light would just be one.

Spacetime Diagrams

The spacetime diagram is a useful visualisation technique.

The time axis is vertical, and of course we have multiplied t by c so we are measuring time in meters, the same as the other coordinates.

An object that is stationary does not have its position change with time: on a spacetime diagram this would be represented by a worldline that is vertical.

If an object is moving, its worldline is not vertical.

For something moving at the speed of light, it moves a distance of, say, 1 meter in a time of 1 meter. Thus the worldline makes an angle of 45 degrees with both the x and ct axes. In the diagram, we have drawn the light cone, representing rays of light that go through the point x=0 and ct=0.

The point x=0 and ct=0 is called the present. Coordinates in spacetime that are inside the light cone and have time coordinates greater than zero are in the future; locations inside the light cone with negative time are in the past.

A Spacetime Diagram

Consider that we are located at the present. We know that, for example, we can not know what happened at the star Alpha Centauri yesterday; it is about 4 light years away and since no information can travel faster than the speed of light we will have to wait four years to find out what happened there. Thus the coordinate of Alpha Centauri yesterday, which is outside the light cone, is inaccessible to us. Similarly, we can not get a signal to Alpha Centauri that will arrive tomorrow. Thus the entire region of spacetime outside the light cone is called elsewhere .

The Dimensions of Spacetime

There is a problem with the spacetime diagram: it only has one explicit spatial coordinate x . The way the light cone is drawn suggests, properly, that there is a second spatial coordinate, say y , that points out of the plane of the figure. But what about the third spatial coordinate? It has to be perpendicular to the ct axis and the x axis and the y axis. There is no simple way to draw such a circumstance.

The following figures indicates one way to approach a representation of such a four-dimensional object.

We begin with a zero-dimensional object, a point.

We move the point one unit to the right to generate a one-dimensional line.

Moving the line one unit perpendicular to itself generates a two-dimensional square.

We move the square one unit perpendicular to itself, and we represent the three dimensional cube as shown.

Finally, if the moving of the square down and to the left was used to get from a square to a cube, then we represent moving the cube perpendicular to itself as moving it down and to the right. The result is called a tesseract .

In about 1884 Edwin Abbott wrote a lovely little book called Flatland: a Romance of Many Dimensions ; the book has been reprinted many times and is readily available. In it he imagines a world with only two spatial dimensions. One of Flatland's inhabitants, named A. Square , became aware of the existence of a third spatial dimension through an interaction with a higher dimensional being, a Sphere . He attempts to explain this third dimension to the other inhabitants of Flatland, which of course promptly got him put in jail. The difficulties A. Square had in visualising the third spatial dimension is analogous to the difficulties we have in visualising a four-dimensional spacetime. An Flash animation of the interaction of the Sphere with Flatland may be seen here .

More Spacetime Diagrams and Some Discussion

The above spacetime diagram was drawn by Claude Bragdon in 1913 for his book A Primer of Higher Space . In this figure the time axis is horizontal.

Bragdon's "day job" was as an architect. He, along with Abbott, also believed that learning to comprehend a fourth dimension was in some sense equivalent to enlightenment. Bragdon designed many buildings in Rochester New York on which the tesseract can be found.

Einstein wrote when his friend Besso died, "For us believing physicists, the distinction between past, present, and future is illusion, however persistent."

Here is another spacetime diagram, this time from D. Postle, Fabric of the Universe , pg. 106:

We imagine our worldline in this spacetime diagram. Then, as David Park wrote, "our consciousness crawls along our worldline as a spark burns along a fuse" (in J.T. Fraser et al., eds., The Study of Time , pg. 113). As it crawls up our worldline we discover new slices of spacetime.

Postle included a continuous block of spacetime between the two different ways of slicing it. Quantum Mechanics calls into question whether such a concept is valid.

Imagine we take one of the piles of frames of the movie and shuffle it. The correlation between our consciousness and what it perceives remains the same. So -- would we notice any difference? I don't have any good way to approach a discussion of this question, but it is one that has fascinated me for years.

Louis de Broglie wrote a famous commentary on the worldview of the theory of relativity:

"In space-time, everything which for each of us constitutes the past, the present, and the future is given in block, and the entire collection of events, successive for us, which form the existence of a material particle is represented by a line, the world-line of the particle .... Each observer, as his time passes, discovers, so to speak, new slices of space-time which appear to him as successive aspects of the material world, though in reality this ensemble of events constituting space-time exist prior to his knowledge of them." -- in Albert Einstein: Philosopher-Scientist , pg. 114.

Dogen Zenji seemed to have a similar view 800 years ago. "It is believed by most that time passes; in actual fact it stays where it is. This idea of passing may be called time, but it is an incorrect idea, for since one only sees it as passing, one cannot understand that it stays just where it is. In a word, every being in the entire world is a separate time in one continuum." -- Shobogenzo .

Finally, Arthur I. Miller has argued that this new way of conceiving space and time in Special Relativity is mirrored by the cubist revolution in painting and especially Picasso's "Les Demoiselles d'Avignon" of 1907. He believes that both Einstein and Picasso were influenced by a statement by Poincaré in 1902 that "There is no absolute space … There is no absolute time." Reference: Einstein, Picasso: Space, Time, and the Beauty That Causes Havoc (Basic Books, 2001) ISBN: 0465018599.

Pablo Picasso, Les Demoiselles d'Avignon , 1907.

The Significance of the Minus Sign

In the parable of the surveyors, we saw that:

E 2 night + k N 2 night = E 2 day + k N 2 day

where the daytime and nighttime coordinate systems were rotated relative to each other.

For Sue and Lou we saw that:

This is similar to the surveyors, except that there is a minus sign between the spatial coordinates and the time coordinate.

The significance of the minus sign can be shown by drawing the spacetime diagram for Sue and Lou, as shown to the below. Sue's coordinate system is almost rotated relative to Lou's, except that the time axis and the position axis are rotating in opposite directions towards each other.

We saw a pre-cursor to this understanding in Postle's spacetime diagram made of movie frames above, where we saw that different observers slice spacetime in different ways.

FURTHER CONSEQUENCES OF EINSTEIN'S "EXPLANATION"

We have just seen that Sue's time and position axes point in different directions than Lou's. This seems to indicate that time is flowing at a different rate for Sue than for Lou, and that her measurements of the distance between two events will be different than his. This is correct, and in this section we shall explore this and other consequences of the Special Theory of Relativity.

Time Dilation

A Flash presentation similar to the discussion of this subsection has been prepared. It requires the Flash player of at least Version 6, and has a file size of 57k. To access the presentation click here .

We imagine that Lou has a light bulb, a mirror, and a light detector: the light bulb and detector are at nearly the same physical location. At some time t equal to zero he turns on the lightbulb. The light travels up to the mirror and is reflected back to the detector.

In the figure to the left, we show the light bulb emitting a light pulse which travels up to the mirror. The figure to the right shows the light traveling from the mirror back to a detector.

Lou measures the time between the two events, turning on the lightbulb and detecting the return ray with the detector.

We imagine the Sue is moving to the right relative to Lou at, say, half the speed of light. Relative to Sue the light bulb, mirror, and detector are moving to left at half the speed of light. She measures the time between the same two events that Lou measured; she will need two synchronized clocks to do this.

Clearly the light traveled a longer distance from the lightbulb to the detector for Sue than it did for Lou. But the speed of the light is the same for both Sue and Lou. Therefore, the time between the two events as measured by Sue's clocks is greater than the time between the same two events as measured by Lou's clock. We therefore conclude the Sue's clocks are running quickly compared to Lou's clock: Sue's clocks measure a greater elapsed time than Lou's.

This phenomenon is called time dilation : time is flowing at different rates for Sue and Lou.

If it were not for Einstein's "explanation," our common sense would say that if the speed of light relative to Lou is c , then the speed of that same light relative to Sue would have to be larger than c . In fact, if one does the mathematics ignoring Einstein's postulate, the time between the two events is the same for Sue and Lou.

Technical note: if Sue is traveling at a speed v relative to Lou, the mathematical relation between the time between the two events relative to Sue and relative to Lou is given by:

\[t_{Sue} = \frac{t_{Lou}}{\sqrt{1-v^2/c^2}}\]

The mathematics that derives this above relationship may be seen here .

Note that if the speed of light c is infinite, the denominator above becomes one, and the times as measured by Sue and Lou are the same. This is a general feature of Special Relativity: in the limit where the speed of light is effectively infinite these effects are unobservable and common sense prevails.

This prediction of Special Relativity has been experimentally confirmed many times. For example the muon is a type of cosmic ray formed in the upper atmosphere. It is unstable, decaying into an electron and an anti-neutrino. The lifetime of the muon when it is at rest relative to us is 2.196 micro-seconds. The distance from the surface of the Earth to the upper atmosphere where these cosmic rays are formed is about 25 kilometers.

These cosmic ray muons are traveling very close to but not quite at the speed of light. Even if they were traveling at the speed of light, in 2.196 micro-seconds they would only travel 660 meters before they decay. Since they are traveling somewhat less than this speed they will travel somewhat less than 660 meters.

However, when we look at the surface of the Earth we see many of these cosmic ray muons. How can they live long enough to travel 25 kilometers? Because their internal clocks are running slowly compared to our clocks so they are living longer than 2.196 micro-seconds.

Length Contraction

A Flash animation that covers the material of this sub-section has been prepared. It is very similar to this discussion, except for the technical notes and mathematical supplement at the end which are not included. It requires the Flash player of at least version 5 to be installed on your computer. The file size is 37k, and the animation will appear in a separate window. To access the animation click here .

In the previous sub-section we saw that when a muon is traveling at high speeds relative to us, its clock runs slowly compared to ours.

But imagine that we are moving at near-lightspeed towards the surface of the Earth and that a muon formed in the upper atmosphere is stationary relative to us.

Now the muon's clock is running at the same rate as our clocks, so it will live only 2.196 micro-seconds.

Meanwhile the Earth is rushing towards us at near-lightspeed. Further when the Earth's surface reaches us the muon will still not have decayed.

The only way that this is possible is that when the muon was formed, the Earth had to have been less than 660 meters away from us.

Thus we conclude that lengths are contracted when they are moving relative to us. The distance from the Earth to where the muons is formed is 25 km relative to a reference frame stationary on the Earth; the same distance is less than 660 meters in a frame in which the muons are stationary.

The length of an object when it at rest relative to us is called the rest length . If the object is moving relative to us, its length along its direction of motion will be less than the rest length.

Technical note: if we call L 0 the rest length, then the length when it is moving at a speed v relative to us is:

Note that this is the same equation Lorentz proposed for the contraction of objects in motion through the ether. Here, though, we interpret the effect quite differently. In any case, physicists sometimes call this Special Relativistic effect the Lorentz contraction .

A final mathematical supplementary document on time dilation and length contraction may be accessed here .

Simultaneity

A Flash animation that covers the material of this sub-section has been prepared. It is very similar to this discussion. It requires the Flash player of at least version 5 to be installed on your computer. The file size is 39k, and the animation will appear in a separate window. To access the animation click here .

Thus the two events, the right hand side of the platform passing the front of the locomotive and the left hand side of the platform passing the back of the locomotive, can not happen simultaneously.

Thus, two events that are simultaneous for one observer may not be simultaneous for some other observer. In terms of Postle's movie-frame spacetime diagram above, we would say that if two events are in the same frame of the movie for one observer they will not necessarily be in the same frame for some other observer.

A Little About Language

According to Whorf, the Hopi language cannot even express the idea of absolute simultaneity, that prediction of relativity that so badly tramples on our common sense. But, the Hopi language discusses reality quite differently from English:

"The SAE [Standard Average European language] microcosm has analyzed reality largely in terms of what it calls `things' (bodies and quasibodies) plus modes of extensional but formless existence that it calls `substances' or `matter' .... The Hopi microcosm seems to have analyzed reality largely in terms of EVENTS (or better `eventing'), referred to in two ways, objective and subjective. Objectively, and only if perceptible physical experience, events are expressed mainly as outlines, colors, movements and other perceptive reports. Subjectively, for both the physical and nonphysical, events are considered the expression of invisible intensity factors, on which depend their stability and persistence, or their fugitiveness and proclivities." -- B.L. Whorf, Language, Thought and Reality , pg. 147.

I have a great deal of sympathy with the view that the language with which we think has a close correlation with what we think; this tends to put me in opposition to Chomsky, Pinsky, et al. In any case, I find it interesting that the Hopi language analysis reality in a way so similar to the careful approach to measuring positions and times of events that we set up earlier to discuss relativity, and that the Hopi language and relativity agree on the absence of absolute simultaneity of events.

Relative Speeds

Imagine some object, say a bus, is moving from left to right at 120 km/hr relative to Lou. Also imagine that Sue is moving from left to right at 50 km/hr relative to Lou. Then our common sense tells us that the bus is moving from left to right at 120 - 50 = 70 km/hr relative to Sue.

However, we know from relativity that if instead of a bus moving at some speed less than the speed of light, we think about a light wave moving at c relative to Lou, then the same light wave will move at c relative to Sue.

You will probably not be surprised to learn that our common sense result for the speed of the bus relative to Sue is not quite correct. In fact, according to Special Relativity the speed of the bus relative to Sue is greater than the expected 70 km/hr by about 0.000,000,000,000,35 km/hr, which is 0.003 millimeters per year!

Imagine an unmanned rocket ship that is moving from left to right at three-quarters of the speed of light relative to Lou, and that Sue is moving from left to right at one-half the speed of light relative to Lou. Then relative to Sue the unmanned rocket is moving from left to right at 0.40 times the speed of light, which is noticeably larger than the common sense prediction of 0.75 - 0.50 = 0.25 times the speed of light.

Technical note: if we say that some object is moving at speed u Lou relative to Lou, and Sue is moving at speed v relative to Lou in the same direction as the object, then the speed of the object relative to Sue is as shown to the right. This equation has the property that if u Lou equals c , then so does u Sue regardless of the value of v .

\[u_{Sue}=\frac{u_{Lou}-v}{1-u_{Lou}v/c^2}\]

Mass-Energy Equivalence

Another prediction of Special Relativity is that:

Since we now know that the speed of light is just a conversion factor for units, we can "read" this equation to say that mass and energy are equivalent.

We call the mass of an object when it is at rest relative to us its rest mass . If the object is moving relative to us its mass will be greater than its rest mass.

The relation between the mass m and the speed v of an object is shown below.

Note that the mass approaches infinity as the speed approaches the speed of light. Thus, it would take infinite energy to accelerate a massive object to the speed of light; another way of saying this is no massive object can ever travel at the speed of light relative to us.

This prediction of Special Relativity has been experimentally confirmed many times. It forms the basis for nuclear energy.

Now we can answer Einstein's original question about what would happen if we pursue a beam of light at the speed of light. The answer is that we can't. No object with a non-zero rest mass can travel at the speed of light relative to any inertial frame of reference, although we can get as close to the speed of light as we wish by providing enough energy. But the light will still always be moving away from us at exactly 1,079,253,000 km/hr.

A supplement about how E = mc 2 arises in the theory has been prepared. The html version is here and the pdf version is here .

Technical note: if the rest mass is m 0 , then the mass m of the object when it is moving at speed v relative to us is:

\[m=\frac{m_0}{\sqrt{1-v^2/c^2}}\]

For light, the denominator is equal to zero since its speed v is equal to the speed of light. By convention, we say that the rest mass of light is zero, so we are dividing 0 by 0. The mathematicians say that this is impossible, but physicists tend to shrug off such pronouncements and say that in this case that the division of the two zeroes works out to be the finite mass-energy of the light.

Probably not Einstein's blackboard in 1905

For a long time people interpreted the material of the previous sub-section to mean that nothing can travel faster than the speed of light. In 1967 Feinberg showed that this is not correct. There is room in the theory for objects whose speed is always greater than c. Feinberg called these hypothetical objects tachyons ; the word has the same root as, say, tachometer.

If these objects exist, their properties include:

• It takes infinite energy to slow a tachyon down to the speed of light. Thus c is still a speed limit, but it is a limit from both sides. Ordinary matter always travels at less than the speed of light, light always travels at exactly the speed of light, and tachyons always travel at greater than the speed of light.
• If the tachyon has real energy, its rest mass must be imaginary, i.e. have a factor of the square root of minus 1. This is reasonable, since relativity says that there is no reference frame accessible to us in which the tachyon is at rest.
• If, say, Lou observes a tachyon produced at point A and then traveling to point B where it it detected, for certain states of motion of Sue relative to Lou she would see the tachyon traveling from B to A . Thus it is uncertain which event created the tachyon and which was its detection. Thus tachyons indicate some difficulty with causality .

Many attempts have been made to observe the existence of tachyons; so far all have failed.

One of the attempts to observe tachyons involves a phenomenon called Cerenkov radiation . In order to understand this, we must first realise that when we say that the speed of light is exactly c with respect to all observers, we are referring to the speed of light in a vacuum. When light travels through a medium such as glass, its speed is less than c ; for a typical glass the speed of light in it is only about two-thirds of the speed in a vacuum.

It turns out that when an electrically charged object travels through a medium at a speed greater than the speed of light in that medium, a characteristic electromagnetic radiation is emitted. This is Cerenkov radiation. The radiation is shaped roughly like the bow wave from a speedboat.

For the bow wave of a boat, it similarly arises when the speed of the boat through the water is greater than the speed of a water wave.

Nuclear reactors are sometimes encased in water to protect us from the radiation. Often there is a blue glow emitted by charged objects emitted from the reactor that are traveling through the water at a speed greater than the speed of light in the water. This is an example of Cerenkov radiation.

For an electrically charged tachyon traveling through a vacuum, its speed is greater than the speed of light in the vacuum and thus it should similarly emit Cerenkov radiation. Thus, some attempts to observe tachyons has been to look for anamolous Cerenkov radiation.

Superluminal Connections

The reason for the ambiguity in the direction of motion of a tachyon discussed in the previous sub-section arises from the way speeds add for different observers, as discussed in the Relative Speeds sub-section.

Say a tachyon is moving from left to right at 100 times the speed of light relative to Lou. Then if Sue is moving from left to right at a speed greater than 0.01 times the speed of light relative to Lou, the tachyon will be moving from right to left relative to her. If she is moving at 0.1 times the speed of light relative to Lou, the tachyon will be moving at a speed of -111 times the speed of light relative to Sue.

Imagine we wish to send a signal to Alpha Centauri, which is 4.35 light years away from us. If we send the signal at the speed of light, it will take 4.35 years relative to us until the signal gets there. If we could send a signal at, say, 100 times the speed of light then it would arrive in only 0.0435 years.

But if this signal is traveling at this "superluminal" speed relative to us, then for an observer moving towards Alpha Centauri at a speed greater than 0.01 times the speed of light relative to us the signal will be going from Alpha Centauri towards us. So if in our reference frame we say we have sent a signal to Alpha Centauri, there are other frames in which the observers would say that Alpha Centauri has sent a signal to us.

It is these considerations that lead us to say that according to relativity no signal or information can travel faster than the speed of light.

Recently, some controversy has re-ignited on this topic. Further information may be found here .

The "Speed" of Objects

Reference: Brian Greene, The Elegant Universe (Norton, 1999), pg. 47 ff.

In the figure to the right, we imagine a race between two identical cars. However, although the two cars travel at exactly the same speed relative to the ground, the car on the left wins the race since it travels the shortest path from the Start to the Finish line.

We will find it useful to state that the velocity of the car on the left is only in the North direction, while the velocity of the car on the right has a component in the North direction and another component in the East direction.

Although the example is fairly simple, we are about to use it to make a conceptual leap:

According to the Special Theory of Relativity, all objects travel at the speed of light at all times

We imagine an object that is stationary relative to us. Then its worldline on a spacetime diagram is vertical. We use the fact that time is another dimension of spacetime to say: the object is moving at the speed of light in the direction of the time axis. This is analogous to the car on the left in the above race.

If the object is moving relative to us, then its wordline is not vertical and looks more like the path of the car on the right. But, since it is moving relative to us, the internal clocks of the object run slowly compared to our clocks. Remember that the speed of the object is the "distance" it travels divided by the "time" for it to travel that distance. So, in a fixed amount of "time" the vertical "distance" is less for the moving object but the time for it to travel that distance also becomes less, so their ratio stays the same value, the speed of light.

For objects that travel at the speed of light relative to us, time dilation means their clocks have stopped: they have no component of their speed in the direction of the time axis.

The Lorentz Contraction is Invisible

We have discussed the fact that these relativistic effects violate our common sense because they are unobservable in our everyday life. The reason for the unobservability is that the speed of light is so large compared to everyday speeds that it is effectively infinite.

In 1940 physicist George Gamow published a book, Mr. Tompkins in Wonderland , that imagines a world where the speed of light is only 30 miles per hour. In this world these relativistic effects are readily observable. It has been collected with another of his works dealing with Quantum Mechanics by Cambridge University Press into a book titled Mr. Tompkins in Paperback.

In Wonderland, people observed length contractions, time dilations, etc. in their everyday life. In 1959 Terrell showed that this is not quite correct. When we see the length of a moving rod, we are seeing the light from the back and from the front of the rod that enters our eyes simultaneously. But if the rod is to our left and moving toward us, the light entering our eyes from the back left the rod before the light entering our eyes from the front. Thus it looks longer than it really is. It turns out that this effect cancels the length contraction. So we do not see the length contraction, although careful measurements of the simultaneous positions of the front and back of the rod will indicate that the length is in fact contracted.

In fact, the object will look like it is rotated but not contracted.

A Flash animation demonstrating this effect has been prepared. It requires the Flash 5 player on your computer, and has a file size of 92k. To access the animation click here .

THE TWIN PARADOX

Imagine that Sue blasts off from Earth, travels at high speed to Alpha Centauri, turns around and returns to Earth. Her twin Lou stays on Earth. According to relativity, since Sue's clocks are running slow compared to Lou's, when they rejoin Sue will be younger than Lou. Presumably all the clocks on Sue's rocket ship are running slow, including her internal clocks. So, for example, if she is listening to a CD during her trip the music will sound perfectly normal to her.

Now consider what Sue will observe. From her point of view she is, of course, stationary. But after blast-off Lou moves away from her at some high speed as Alpha Centauri approaches. Then, Alpha Centauri reaches her position and reverses it motion, starting to recede; now Lou and the Earth are getting closer. Throughout Lou has been moving relative to Sue, so his clocks should be running slow compared to Sue's, so he is the one that ends up being younger.

So we have argued that Sue ends up younger, and then have shown that Lou ends up younger. This is often called the twin paradox .

Resolving the paradox is fairly easy. Recall that we have said that we can only do physics in inertial reference frames, frames in which the principle of inertia is true. Although the Earth is in a circular orbit around the Sun and is also rotating on its axis, these accelerations are sufficiently small that we usually treat the Earth as an inertial reference frame.

However, Sue is very far away from being in an inertial reference frame. She experiences high g-forces when she blasts off, experiences yet others when reaches Alpha Centauri and turns around, and yet again when she decelerates and lands on the Earth at the end of her trip. During all these times the principle of inertia is not true. Thus we can not analyse the twin paradox from Sue's reference frame.

We analyse the twins in an inertial reference frame in which, say, Sue is stationary relative to us on her outbound trip. If she is traveling to Alpha Centuari at 99% of the speed of light, then on her outward trip Alpha Centuari is approaching us at 99% of the speed of the light and Lou is receding away from us at 99% of c. When Alpha Centuari reaches us, Sue decelerates, turns around, and chases after Lou. But we're not allowed to go with her: we have to stay in our inertial frame. So relative to us Lou is still receding away at 99% of the speed of light, and Sue is chasing him at an even faster speed than 0.99c. While Sue was stationary relative to us, Lou's clocks were running slow relative to us. But when Sue was chasing after him her clocks were running even slower than Lou's. If one does the math, it turns out that when Sue and Lou are reunited, Sue will end up younger than Lou.

Here is the spacetime diagram in a frame where Lou is stationary:

Here is the spacetime diagram in a frame where Sue is stationary on her outward trip.

Without doing any of the math, however, we do have what turns out to be a general principle. If we analyze the twins in any inertial reference frame and draw the spacetime diagram, the twin with the longer worldline ends up being the younger twin.

There are many approaches to this paradox. One uses the Relativistic Doppler Effect . A Flash animation of this approach has been prepared; it is somewhat more advanced than the discussion in this document. It requires the Flash 6 player on your computer, and has a file size of 92k. To access the animation, click here.

A FAVORITE PUZZLE

We have a 25 m long pole and a 20 m long barn, both as measured at rest relative to the pole and the barn. We will assume the back wall of the barn is very very strong.

If the pole is moving towards the barn at 70% of the speed of light, its length will be contracted to about 18 m. Thus it clearly fits in the barn, and we can slam the door shut (and run!).

But if we are riding along with the pole, its length is not contracted and is 25 m long. But the barn is contracted and is now about 14 m long. Clearly the pole does not fit in the barn.

Does the pole fit into the barn or not?

A famous Zen story: Two Zen monks were arguing about a flag waving in the breeze, and whether it was the flag or the wind that was moving. The Sixth Patriarch of Zen, Hui Neng, overheard; "I suggested it was neither, that what moved was their own mind."

A commentary by Mumon:

Wind, flag, mind moves. The same understanding. When the mouth opens All are wrong.

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Would you really age more slowly on a spaceship at close to light speed?

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Every week, the readers of our space newsletter, The Airlock , send in their questions for space reporter Neel V. Patel to answer. This week: time dilation during space travel. 

I heard that time dilation affects high-speed space travel and I am wondering the magnitude of that affect. If we were to launch a round-trip flight to a nearby exoplanet—let's say 10 or 50 light-years away––how would that affect time for humans on the spaceship versus humans on Earth? When the space travelers came back, will they be much younger or older relative to people who stayed on Earth? —Serge

Time dilation is a concept that pops up in lots of sci-fi, including Orson Scott Card’s Ender’s Game , where one character ages only eight years in space while 50 years pass on Earth. This is precisely the scenario outlined in the famous thought experiment the Twin Paradox : an astronaut with an identical twin at mission control makes a journey into space on a high-speed rocket and returns home to find that the twin has aged faster.

Time dilation goes back to Einstein’s theory of special relativity, which teaches us that motion through space actually creates alterations in the flow of time. The faster you move through the three dimensions that define physical space, the more slowly you’re moving through the fourth dimension, time––at least relative to another object. Time is measured differently for the twin who moved through space and the twin who stayed on Earth. The clock in motion will tick more slowly than the clocks we’re watching on Earth. If you’re able to travel near the speed of light, the effects are much more pronounced. 

Unlike the Twin Paradox, time dilation isn’t a thought experiment or a hypothetical concept––it’s real. The 1971 Hafele-Keating experiments proved as much, when two atomic clocks were flown on planes traveling in opposite directions. The relative motion actually had a measurable impact and created a time difference between the two clocks. This has also been confirmed in other physics experiments (e.g., fast-moving muon particles take longer to decay ). 

So in your question, an astronaut returning from a space journey at “relativistic speeds” (where the effects of relativity start to manifest—generally at least one-tenth the speed of light ) would, upon return, be younger than same-age friends and family who stayed on Earth. Exactly how much younger depends on exactly how fast the spacecraft had been moving and accelerating, so it’s not something we can readily answer. But if you’re trying to reach an exoplanet 10 to 50 light-years away and still make it home before you yourself die of old age, you’d have to be moving at close to light speed. 

There’s another wrinkle here worth mentioning: time dilation as a result of gravitational effects. You might have seen Christopher Nolan’s movie Interstellar , where the close proximity of a black hole causes time on another planet to slow down tremendously (one hour on that planet is seven Earth years).

This form of time dilation is also real, and it’s because in Einstein’s theory of general relativity, gravity can bend spacetime, and therefore time itself. The closer the clock is to the source of gravitation, the slower time passes; the farther away the clock is from gravity, the faster time will pass. (We can save the details of that explanation for a future Airlock.)

The search for extraterrestrial life is targeting Jupiter’s icy moon Europa

NASA’s Europa Clipper mission will travel to one of Jupiter's largest moons to look for evidence of conditions that could support life.

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Special Relativity/Spacetime

• 1.1 Interpreting space-time diagrams
• 1.2 Spacetime
• 2 The lightcone
• 3 The Lorentz transformation equations
• 4 A spacetime representation of the Lorentz Transformation

The modern approach to relativity [ edit | edit source ]

Although the special theory of relativity was first proposed by Einstein in 1905, the modern approach to the theory depends upon the concept of a four-dimensional universe, that was first proposed by Hermann Minkowski in 1908.

Minkowski's contribution appears complicated but is simply an extension of Pythagoras' Theorem:

The modern approach uses the concept of invariance to explore the types of coordinate systems that are required to provide a full physical description of the location and extent of things. The modern theory of special relativity begins with the concept of "length". In everyday experience, it seems that the length of objects remains the same no matter how they are rotated or moved from place to place. We think that the simple length of a thing is "invariant". However, as is shown in the illustrations below, what we are actually suggesting is that length seems to be invariant in a three-dimensional coordinate system.

The length of a thing in a two-dimensional coordinate system is given by Pythagoras's theorem:

This two-dimensional length is not invariant if the thing is tilted out of the two-dimensional plane. In everyday life, a three-dimensional coordinate system seems to describe the length fully. The length is given by the three-dimensional version of Pythagoras's theorem:

The derivation of this formula is shown in the illustration below.

It seems that, provided all the directions in which a thing can be tilted or arranged are represented within a coordinate system, then the coordinate system can fully represent the length of a thing. However, it is clear that things may also be changed over a period of time. Time is another direction in which things can be arranged. This is shown in the following diagram:

The length of a straight line between two events in space and time is called a "space-time interval".

In 1908 Hermann Minkowski pointed out that if things could be rearranged in time, then the universe might be four-dimensional. He boldly suggested that Einstein's recently-discovered theory of Special Relativity was a consequence of this four-dimensional universe. He proposed that the space-time interval might be related to space and time by Pythagoras' theorem in four dimensions:

Minkowski's use of the imaginary unit has been superseded by the use of advanced geometry that uses a tool known as the "metric tensor". The metric tensor permits the existence of "real" time and the negative sign in the expression for the square of the space-time interval originates in the way that distance changes with time when the curvature of spacetime is analysed (see advanced text). We now use real time but Minkowski's original equation for the square of the interval survives so that the space-time interval is still given by:

Space-time intervals are difficult to imagine; they extend between one place and time and another place and time, so the velocity of the thing that travels along the interval is already determined for a given observer.

If the universe is four-dimensional, then the space-time interval (rather than the spatial length) will be invariant. Whoever measures a particular space-time interval will get the same value, no matter how fast they are travelling. In physical terminology the invariance of the spacetime interval is a type of Lorentz Invariance . The invariance of the spacetime interval has some dramatic consequences.

The first consequence is the prediction that if a thing is travelling at a velocity of c metres per second, then all observers, no matter how fast they are travelling, will measure the same velocity for the thing. The velocity c will be a universal constant. This is explained below.

When an object is travelling at c , the space time interval is zero , this is shown below:

A space-time interval of zero only occurs when the velocity is c (if x>0). All observers observe the same space-time interval so when observers observe something with a space-time interval of zero, they all observe it to have a velocity of c , no matter how fast they are moving themselves.

The universal constant, c , is known for historical reasons as the "speed of light in a vacuum". In the first decade or two after the formulation of Minkowski's approach many physicists, although supporting Special Relativity, expected that light might not travel at exactly c , but might travel at very nearly c . There are now few physicists who believe that light in a vacuum does not propagate at c .

The second consequence of the invariance of the space-time interval is that clocks will appear to go slower on objects that are moving relative to you. Suppose there are two people, Bill and John, on separate planets that are moving away from each other. John draws a graph of Bill's motion through space and time. This is shown in the illustration below:

Being on planets, both Bill and John think they are stationary, and just moving through time. John spots that Bill is moving through what John calls space, as well as time, when Bill thinks he is moving through time alone. Bill would also draw the same conclusion about John's motion. To John, it is as if Bill's time axis is leaning over in the direction of travel and to Bill, it is as if John's time axis leans over.

This equation can be derived directly and validly from the time dilation result with the assumption that the speed of light is constant.

The last consequence is that clocks will appear to be out of phase with each other along the length of a moving object. This means that if one observer sets up a line of clocks that are all synchronised so they all read the same time, then another observer who is moving along the line at high speed will see the clocks all reading different times. In other words observers who are moving relative to each other see different events as simultaneous . This effect is known as Relativistic Phase or the Relativity of Simultaneity . Relativistic phase is often overlooked by students of Special Relativity, but if it is understood then phenomena such as the twin paradox are easier to understand.

The way that clocks go out of phase along the line of travel can be calculated from the concepts of the invariance of the space-time interval and length contraction.

In the diagram above John is conventionally stationary. Distances between two points according to Bill are simple lengths in space (x) all at t=0 whereas John sees Bill's measurement of distance as a combination of a distance (X) and a time interval (T):

Notice that the quantities represented by capital letters are proper lengths and times and in this example refer to John's measurements.

Bill's distance, x, is the length that he would obtain for things that John believes to be X metres in length. For Bill it is John who has rods that contract in the direction of motion so Bill's determination "x" of John's distance "X" is given from:

This relationship between proper and coordinate lengths was seen above to relate Bill's proper lengths to John's measurements. It also applies to how Bill observes John's proper lengths.

The net effect of the four-dimensional universe is that observers who are in motion relative to you seem to have time coordinates that lean over in the direction of motion and consider things to be simultaneous that are not simultaneous for you. Spatial lengths in the direction of travel are shortened, because they tip upwards and downwards, relative to the time axis in the direction of travel, akin to a rotation out of three-dimensional space.

Interpreting space-time diagrams [ edit | edit source ]

Great care is needed when interpreting space-time diagrams. Diagrams present data in two dimensions, and cannot show faithfully how, for instance, a zero length space-time interval appears.

When diagrams are used to show both space and time it is important to be alert to space and time being related by Minkowski's equation and not by simple Euclidean geometry. The diagrams are only aids to understanding the approximate relation between space and time and it must not be assumed, for instance, that simple trigonometric relationships can be used to relate lines that represent spatial displacements and lines that represent temporal displacements.

It is sometimes mistakenly held that the time dilation and length contraction results only apply for observers at x=0 and t=0. This is untrue. An inertial frame of reference is defined so that length and time comparisons can be made anywhere within a given reference frame.

Time differences in one inertial reference frame can be compared with time differences anywhere in another inertial reference frame provided it is remembered that these differences apply to corresponding pairs of lines or pairs of planes of simultaneous events.

Spacetime [ edit | edit source ]

In order to gain an understanding of both Galilean and Special Relativity it is important to begin thinking of space and time as being different dimensions of a four-dimensional vector space called spacetime. Actually, since we can't visualize four dimensions very well, it is easiest to start with only one space dimension and the time dimension. The figure shows a graph with time plotted on the vertical axis and the one space dimension plotted on the horizontal axis. An event is something that occurs at a particular time and a particular point in space. ("Julius X. wrecks his car in Lemitar, NM on 21 June at 6:17 PM.") A world line is a plot of the position of some object as a function of time (more properly, the time of the object as a function of position) on a spacetime diagram. Thus, a world line is really a line in spacetime, while an event is a point in spacetime. A horizontal line parallel to the position axis (x-axis) is a line of simultaneity ; in Galilean Relativity all events on this line occur simultaneously for all observers. It will be seen that the line of simultaneity differs between Galilean and Special Relativity; in Special Relativity the line of simultaneity depends on the state of motion of the observer.

In a spacetime diagram the slope of a world line has a special meaning. Notice that a vertical world line means that the object it represents does not move -- the velocity is zero. If the object moves to the right, then the world line tilts to the right, and the faster it moves, the more the world line tilts. Quantitatively, we say that

Notice that this works for negative slopes and velocities as well as positive ones. If the object changes its velocity with time, then the world line is curved, and the instantaneous velocity at any time is the inverse of the slope of the tangent to the world line at that time.

The hardest thing to realize about spacetime diagrams is that they represent the past, present, and future all in one diagram. Thus, spacetime diagrams don't change with time -- the evolution of physical systems is represented by looking at successive horizontal slices in the diagram at successive times. Spacetime diagrams represent the evolution of events, but they don't evolve themselves.

The lightcone [ edit | edit source ]

Things that move at the speed of light in our four dimensional universe have surprising properties. If something travels at the speed of light along the x-axis and covers x meters from the origin in t seconds the space-time interval of its path is zero.

Extending this result to the general case, if something travels at the speed of light in any direction into or out from the origin it has a space-time interval of 0:

This equation is known as the Minkowski Light Cone Equation. If light were travelling towards the origin then the Light Cone Equation would describe the position and time of emission of all those photons that could be at the origin at a particular instant. If light were travelling away from the origin the equation would describe the position of the photons emitted at a particular instant at any future time 't'.

At the superficial level the light cone is easy to interpret. Its backward surface represents the path of light rays that strike a point observer at an instant and its forward surface represents the possible paths of rays emitted from the point observer. Things that travel along the surface of the light cone are said to be light- like and the path taken by such things is known as a null geodesic .

Events that lie outside the cones are said to be space-like or, better still space separated because their space time interval from the observer has the same sign as space (positive according to the convention used here). Events that lie within the cones are said to be time-like or time separated because their space-time interval has the same sign as time.

However, there is more to the light cone than the propagation of light. If the added assumption is made that the speed of light is the maximum possible velocity then events that are space separated cannot affect the observer directly. Events within the backward cone can have affected the observer so the backward cone is known as the "affective past" and the observer can affect events in the forward cone hence the forward cone is known as the "affective future".

The assumption that the speed of light is the maximum velocity for all communications is neither inherent in nor required by four dimensional geometry although the speed of light is indeed the maximum velocity for objects if the principle of causality is to be preserved by physical theories (ie: that causes precede effects).

The Lorentz transformation equations [ edit | edit source ]

The discussion so far has involved the comparison of interval measurements (time intervals and space intervals) between two observers. The observers might also want to compare more general sorts of measurement such as the time and position of a single event that is recorded by both of them. The equations that describe how each observer describes the other's recordings in this circumstance are known as the Lorentz Transformation Equations. (Note that the symbols below signify coordinates.)

The table below shows the Lorentz Transformation Equations.

See mathematical derivation of Lorentz transformation .

Notice how the phase ( (v/c 2 )x ) is important and how these formulae for absolute time and position of a joint event differ from the formulae for intervals.

A spacetime representation of the Lorentz Transformation [ edit | edit source ]

This relationship between the times of a common event between reference frames is known as the Lorentz Transformation Equation for time.

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Three ways to travel at (nearly) the speed of light.

Katy Mersmann

1) electromagnetic fields, 2) magnetic explosions, 3) wave-particle interactions.

One hundred years ago today, on May 29, 1919, measurements of a solar eclipse offered verification for Einstein’s theory of general relativity. Even before that, Einstein had developed the theory of special relativity, which revolutionized the way we understand light. To this day, it provides guidance on understanding how particles move through space — a key area of research to keep spacecraft and astronauts safe from radiation.

The theory of special relativity showed that particles of light, photons, travel through a vacuum at a constant pace of 670,616,629 miles per hour — a speed that’s immensely difficult to achieve and impossible to surpass in that environment. Yet all across space, from black holes to our near-Earth environment, particles are, in fact, being accelerated to incredible speeds, some even reaching 99.9% the speed of light.

One of NASA’s jobs is to better understand how these particles are accelerated. Studying these superfast, or relativistic, particles can ultimately help protect missions exploring the solar system, traveling to the Moon, and they can teach us more about our galactic neighborhood: A well-aimed near-light-speed particle can trip onboard electronics and too many at once could have negative radiation effects on space-faring astronauts as they travel to the Moon — or beyond.

Here are three ways that acceleration happens.

Most of the processes that accelerate particles to relativistic speeds work with electromagnetic fields — the same force that keeps magnets on your fridge. The two components, electric and magnetic fields, like two sides of the same coin, work together to whisk particles at relativistic speeds throughout the universe.

In essence, electromagnetic fields accelerate charged particles because the particles feel a force in an electromagnetic field that pushes them along, similar to how gravity pulls at objects with mass. In the right conditions, electromagnetic fields can accelerate particles at near-light-speed.

On Earth, electric fields are often specifically harnessed on smaller scales to speed up particles in laboratories. Particle accelerators, like the Large Hadron Collider and Fermilab, use pulsed electromagnetic fields to accelerate charged particles up to 99.99999896% the speed of light. At these speeds, the particles can be smashed together to produce collisions with immense amounts of energy. This allows scientists to look for elementary particles and understand what the universe was like in the very first fractions of a second after the Big Bang. 

Download related video from NASA Goddard’s Scientific Visualization Studio

Magnetic fields are everywhere in space, encircling Earth and spanning the solar system. They even guide charged particles moving through space, which spiral around the fields.

When these magnetic fields run into each other, they can become tangled. When the tension between the crossed lines becomes too great, the lines explosively snap and realign in a process known as magnetic reconnection. The rapid change in a region’s magnetic field creates electric fields, which causes all the attendant charged particles to be flung away at high speeds. Scientists suspect magnetic reconnection is one way that particles — for example, the solar wind, which is the constant stream of charged particles from the Sun — is accelerated to relativistic speeds.

Those speedy particles also create a variety of side-effects near planets.  Magnetic reconnection occurs close to us at points where the Sun’s magnetic field pushes against Earth’s magnetosphere — its protective magnetic environment. When magnetic reconnection occurs on the side of Earth facing away from the Sun, the particles can be hurled into Earth’s upper atmosphere where they spark the auroras. Magnetic reconnection is also thought to be responsible around other planets like Jupiter and Saturn, though in slightly different ways.

NASA’s Magnetospheric Multiscale spacecraft were designed and built to focus on understanding all aspects of magnetic reconnection. Using four identical spacecraft, the mission flies around Earth to catch magnetic reconnection in action. The results of the analyzed data can help scientists understand particle acceleration at relativistic speeds around Earth and across the universe.

Particles can be accelerated by interactions with electromagnetic waves, called wave-particle interactions. When electromagnetic waves collide, their fields can become compressed. Charged particles bouncing back and forth between the waves can gain energy similar to a ball bouncing between two merging walls.

These types of interactions are constantly occurring in near-Earth space and are responsible for accelerating particles to speeds that can damage electronics on spacecraft and satellites in space. NASA missions, like the Van Allen Probes , help scientists understand wave-particle interactions.

Wave-particle interactions are also thought to be responsible for accelerating some cosmic rays that originate outside our solar system. After a supernova explosion, a hot, dense shell of compressed gas called a blast wave is ejected away from the stellar core. Filled with magnetic fields and charged particles, wave-particle interactions in these bubbles can launch high-energy cosmic rays at 99.6% the speed of light. Wave-particle interactions may also be partially responsible for accelerating the solar wind and cosmic rays from the Sun.

Download this and related videos in HD formats from NASA Goddard’s Scientific Visualization Studio

By Mara Johnson-Groh NASA’s Goddard Space Flight Center , Greenbelt, Md.

Is Time Travel Possible?

We all travel in time! We travel one year in time between birthdays, for example. And we are all traveling in time at approximately the same speed: 1 second per second.

We typically experience time at one second per second. Credit: NASA/JPL-Caltech

NASA's space telescopes also give us a way to look back in time. Telescopes help us see stars and galaxies that are very far away . It takes a long time for the light from faraway galaxies to reach us. So, when we look into the sky with a telescope, we are seeing what those stars and galaxies looked like a very long time ago.

However, when we think of the phrase "time travel," we are usually thinking of traveling faster than 1 second per second. That kind of time travel sounds like something you'd only see in movies or science fiction books. Could it be real? Science says yes!

This image from the Hubble Space Telescope shows galaxies that are very far away as they existed a very long time ago. Credit: NASA, ESA and R. Thompson (Univ. Arizona)

How do we know that time travel is possible?

More than 100 years ago, a famous scientist named Albert Einstein came up with an idea about how time works. He called it relativity. This theory says that time and space are linked together. Einstein also said our universe has a speed limit: nothing can travel faster than the speed of light (186,000 miles per second).

Einstein's theory of relativity says that space and time are linked together. Credit: NASA/JPL-Caltech

What does this mean for time travel? Well, according to this theory, the faster you travel, the slower you experience time. Scientists have done some experiments to show that this is true.

For example, there was an experiment that used two clocks set to the exact same time. One clock stayed on Earth, while the other flew in an airplane (going in the same direction Earth rotates).

After the airplane flew around the world, scientists compared the two clocks. The clock on the fast-moving airplane was slightly behind the clock on the ground. So, the clock on the airplane was traveling slightly slower in time than 1 second per second.

Credit: NASA/JPL-Caltech

Can we use time travel in everyday life?

We can't use a time machine to travel hundreds of years into the past or future. That kind of time travel only happens in books and movies. But the math of time travel does affect the things we use every day.

For example, we use GPS satellites to help us figure out how to get to new places. (Check out our video about how GPS satellites work .) NASA scientists also use a high-accuracy version of GPS to keep track of where satellites are in space. But did you know that GPS relies on time-travel calculations to help you get around town?

GPS satellites orbit around Earth very quickly at about 8,700 miles (14,000 kilometers) per hour. This slows down GPS satellite clocks by a small fraction of a second (similar to the airplane example above).

GPS satellites orbit around Earth at about 8,700 miles (14,000 kilometers) per hour. Credit: GPS.gov

However, the satellites are also orbiting Earth about 12,550 miles (20,200 km) above the surface. This actually speeds up GPS satellite clocks by a slighter larger fraction of a second.

Here's how: Einstein's theory also says that gravity curves space and time, causing the passage of time to slow down. High up where the satellites orbit, Earth's gravity is much weaker. This causes the clocks on GPS satellites to run faster than clocks on the ground.

The combined result is that the clocks on GPS satellites experience time at a rate slightly faster than 1 second per second. Luckily, scientists can use math to correct these differences in time.

If scientists didn't correct the GPS clocks, there would be big problems. GPS satellites wouldn't be able to correctly calculate their position or yours. The errors would add up to a few miles each day, which is a big deal. GPS maps might think your home is nowhere near where it actually is!

In Summary:

Yes, time travel is indeed a real thing. But it's not quite what you've probably seen in the movies. Under certain conditions, it is possible to experience time passing at a different rate than 1 second per second. And there are important reasons why we need to understand this real-world form of time travel.

If you liked this, you may like:

Time travel: Is it possible?

Science says time travel is possible, but probably not in the way you're thinking.

Albert Einstein's theory

• General relativity and GPS
• Wormhole travel
• Alternate theories

Science fiction

Is time travel possible? Short answer: Yes, and you're doing it right now — hurtling into the future at the impressive rate of one second per second. 

You're pretty much always moving through time at the same speed, whether you're watching paint dry or wishing you had more hours to visit with a friend from out of town. 

But this isn't the kind of time travel that's captivated countless science fiction writers, or spurred a genre so extensive that Wikipedia lists over 400 titles in the category "Movies about Time Travel." In franchises like " Doctor Who ," " Star Trek ," and "Back to the Future" characters climb into some wild vehicle to blast into the past or spin into the future. Once the characters have traveled through time, they grapple with what happens if you change the past or present based on information from the future (which is where time travel stories intersect with the idea of parallel universes or alternate timelines). 

Related: The best sci-fi time machines ever

Although many people are fascinated by the idea of changing the past or seeing the future before it's due, no person has ever demonstrated the kind of back-and-forth time travel seen in science fiction or proposed a method of sending a person through significant periods of time that wouldn't destroy them on the way. And, as physicist Stephen Hawking pointed out in his book " Black Holes and Baby Universes" (Bantam, 1994), "The best evidence we have that time travel is not possible, and never will be, is that we have not been invaded by hordes of tourists from the future."

Science does support some amount of time-bending, though. For example, physicist Albert Einstein 's theory of special relativity proposes that time is an illusion that moves relative to an observer. An observer traveling near the speed of light will experience time, with all its aftereffects (boredom, aging, etc.) much more slowly than an observer at rest. That's why astronaut Scott Kelly aged ever so slightly less over the course of a year in orbit than his twin brother who stayed here on Earth. 

Related: Controversially, physicist argues that time is real

There are other scientific theories about time travel, including some weird physics that arise around wormholes , black holes and string theory . For the most part, though, time travel remains the domain of an ever-growing array of science fiction books, movies, television shows, comics, video games and more. 

Einstein developed his theory of special relativity in 1905. Along with his later expansion, the theory of general relativity , it has become one of the foundational tenets of modern physics. Special relativity describes the relationship between space and time for objects moving at constant speeds in a straight line. 

The short version of the theory is deceptively simple. First, all things are measured in relation to something else — that is to say, there is no "absolute" frame of reference. Second, the speed of light is constant. It stays the same no matter what, and no matter where it's measured from. And third, nothing can go faster than the speed of light.

From those simple tenets unfolds actual, real-life time travel. An observer traveling at high velocity will experience time at a slower rate than an observer who isn't speeding through space. 

While we don't accelerate humans to near-light-speed, we do send them swinging around the planet at 17,500 mph (28,160 km/h) aboard the International Space Station . Astronaut Scott Kelly was born after his twin brother, and fellow astronaut, Mark Kelly . Scott Kelly spent 520 days in orbit, while Mark logged 54 days in space. The difference in the speed at which they experienced time over the course of their lifetimes has actually widened the age gap between the two men.

"So, where[as] I used to be just 6 minutes older, now I am 6 minutes and 5 milliseconds older," Mark Kelly said in a panel discussion on July 12, 2020, Space.com previously reported . "Now I've got that over his head."

General relativity and GPS time travel

The difference that low earth orbit makes in an astronaut's life span may be negligible — better suited for jokes among siblings than actual life extension or visiting the distant future — but the dilation in time between people on Earth and GPS satellites flying through space does make a difference. 

Read more: Can we stop time?

The Global Positioning System , or GPS, helps us know exactly where we are by communicating with a network of a few dozen satellites positioned in a high Earth orbit. The satellites circle the planet from 12,500 miles (20,100 kilometers) away, moving at 8,700 mph (14,000 km/h). 

According to special relativity, the faster an object moves relative to another object, the slower that first object experiences time. For GPS satellites with atomic clocks, this effect cuts 7 microseconds, or 7 millionths of a second, off each day, according to the American Physical Society publication Physics Central .  

Read more: Could Star Trek's faster-than-light warp drive actually work?

Then, according to general relativity, clocks closer to the center of a large gravitational mass like Earth tick more slowly than those farther away. So, because the GPS satellites are much farther from the center of Earth compared to clocks on the surface, Physics Central added, that adds another 45 microseconds onto the GPS satellite clocks each day. Combined with the negative 7 microseconds from the special relativity calculation, the net result is an added 38 microseconds. 

This means that in order to maintain the accuracy needed to pinpoint your car or phone — or, since the system is run by the U.S. Department of Defense, a military drone — engineers must account for an extra 38 microseconds in each satellite's day. The atomic clocks onboard don’t tick over to the next day until they have run 38 microseconds longer than comparable clocks on Earth.

Given those numbers, it would take more than seven years for the atomic clock in a GPS satellite to un-sync itself from an Earth clock by more than a blink of an eye. (We did the math: If you estimate a blink to last at least 100,000 microseconds, as the Harvard Database of Useful Biological Numbers does, it would take thousands of days for those 38 microsecond shifts to add up.) 

This kind of time travel may seem as negligible as the Kelly brothers' age gap, but given the hyper-accuracy of modern GPS technology, it actually does matter. If it can communicate with the satellites whizzing overhead, your phone can nail down your location in space and time with incredible accuracy. 

Can wormholes take us back in time?

General relativity might also provide scenarios that could allow travelers to go back in time, according to NASA . But the physical reality of those time-travel methods is no piece of cake. 

Wormholes are theoretical "tunnels" through the fabric of space-time that could connect different moments or locations in reality to others. Also known as Einstein-Rosen bridges or white holes, as opposed to black holes, speculation about wormholes abounds. But despite taking up a lot of space (or space-time) in science fiction, no wormholes of any kind have been identified in real life. 

Related: Best time travel movies

"The whole thing is very hypothetical at this point," Stephen Hsu, a professor of theoretical physics at the University of Oregon, told Space.com sister site Live Science . "No one thinks we're going to find a wormhole anytime soon."

Primordial wormholes are predicted to be just 10^-34 inches (10^-33 centimeters) at the tunnel's "mouth". Previously, they were expected to be too unstable for anything to be able to travel through them. However, a study claims that this is not the case, Live Science reported . 

The theory, which suggests that wormholes could work as viable space-time shortcuts, was described by physicist Pascal Koiran. As part of the study, Koiran used the Eddington-Finkelstein metric, as opposed to the Schwarzschild metric which has been used in the majority of previous analyses.

In the past, the path of a particle could not be traced through a hypothetical wormhole. However, using the Eddington-Finkelstein metric, the physicist was able to achieve just that.

Koiran's paper was described in October 2021, in the preprint database arXiv , before being published in the Journal of Modern Physics D.

Alternate time travel theories

While Einstein's theories appear to make time travel difficult, some researchers have proposed other solutions that could allow jumps back and forth in time. These alternate theories share one major flaw: As far as scientists can tell, there's no way a person could survive the kind of gravitational pulling and pushing that each solution requires.

Infinite cylinder theory

Astronomer Frank Tipler proposed a mechanism (sometimes known as a Tipler Cylinder ) where one could take matter that is 10 times the sun's mass, then roll it into a very long, but very dense cylinder. The Anderson Institute , a time travel research organization, described the cylinder as "a black hole that has passed through a spaghetti factory."

After spinning this black hole spaghetti a few billion revolutions per minute, a spaceship nearby — following a very precise spiral around the cylinder — could travel backward in time on a "closed, time-like curve," according to the Anderson Institute. 

The major problem is that in order for the Tipler Cylinder to become reality, the cylinder would need to be infinitely long or be made of some unknown kind of matter. At least for the foreseeable future, endless interstellar pasta is beyond our reach.

Time donuts

Theoretical physicist Amos Ori at the Technion-Israel Institute of Technology in Haifa, Israel, proposed a model for a time machine made out of curved space-time — a donut-shaped vacuum surrounded by a sphere of normal matter.

"The machine is space-time itself," Ori told Live Science . "If we were to create an area with a warp like this in space that would enable time lines to close on themselves, it might enable future generations to return to visit our time."

Amos Ori is a theoretical physicist at the Technion-Israel Institute of Technology in Haifa, Israel. His research interests and publications span the fields of general relativity, black holes, gravitational waves and closed time lines.

There are a few caveats to Ori's time machine. First, visitors to the past wouldn't be able to travel to times earlier than the invention and construction of the time donut. Second, and more importantly, the invention and construction of this machine would depend on our ability to manipulate gravitational fields at will — a feat that may be theoretically possible but is certainly beyond our immediate reach.

Time travel has long occupied a significant place in fiction. Since as early as the "Mahabharata," an ancient Sanskrit epic poem compiled around 400 B.C., humans have dreamed of warping time, Lisa Yaszek, a professor of science fiction studies at the Georgia Institute of Technology in Atlanta, told Live Science .  

Every work of time-travel fiction creates its own version of space-time, glossing over one or more scientific hurdles and paradoxes to achieve its plot requirements. 

Some make a nod to research and physics, like " Interstellar ," a 2014 film directed by Christopher Nolan. In the movie, a character played by Matthew McConaughey spends a few hours on a planet orbiting a supermassive black hole, but because of time dilation, observers on Earth experience those hours as a matter of decades. 

Others take a more whimsical approach, like the "Doctor Who" television series. The series features the Doctor, an extraterrestrial "Time Lord" who travels in a spaceship resembling a blue British police box. "People assume," the Doctor explained in the show, "that time is a strict progression from cause to effect, but actually from a non-linear, non-subjective viewpoint, it's more like a big ball of wibbly-wobbly, timey-wimey stuff." 

Long-standing franchises like the "Star Trek" movies and television series, as well as comic universes like DC and Marvel Comics, revisit the idea of time travel over and over. 

Related: Marvel movies in order: chronological & release order

Here is an incomplete (and deeply subjective) list of some influential or notable works of time travel fiction:

Books about time travel:

• Rip Van Winkle (Cornelius S. Van Winkle, 1819) by Washington Irving
• A Christmas Carol (Chapman & Hall, 1843) by Charles Dickens
• The Time Machine (William Heinemann, 1895) by H. G. Wells
• A Connecticut Yankee in King Arthur's Court (Charles L. Webster and Co., 1889) by Mark Twain
• The Restaurant at the End of the Universe (Pan Books, 1980) by Douglas Adams
• A Tale of Time City (Methuen, 1987) by Diana Wynn Jones
• The Outlander series (Delacorte Press, 1991-present) by Diana Gabaldon
• Harry Potter and the Prisoner of Azkaban (Bloomsbury/Scholastic, 1999) by J. K. Rowling
• Thief of Time (Doubleday, 2001) by Terry Pratchett
• The Time Traveler's Wife (MacAdam/Cage, 2003) by Audrey Niffenegger
• All You Need is Kill (Shueisha, 2004) by Hiroshi Sakurazaka

Movies about time travel:

• Planet of the Apes (1968)
• Superman (1978)
• Time Bandits (1981)
• The Terminator (1984)
• Back to the Future series (1985, 1989, 1990)
• Star Trek IV: The Voyage Home (1986)
• Bill & Ted's Excellent Adventure (1989)
• Groundhog Day (1993)
• Galaxy Quest (1999)
• The Butterfly Effect (2004)
• 13 Going on 30 (2004)
• The Lake House (2006)
• Meet the Robinsons (2007)
• Hot Tub Time Machine (2010)
• Midnight in Paris (2011)
• Looper (2012)
• X-Men: Days of Future Past (2014)
• Edge of Tomorrow (2014)
• Interstellar (2014)
• Doctor Strange (2016)
• A Wrinkle in Time (2018)
• The Last Sharknado: It's About Time (2018)
• Avengers: Endgame (2019)
• Tenet (2020)
• Palm Springs (2020)
• Zach Snyder's Justice League (2021)
• The Tomorrow War (2021)

Television about time travel:

• Doctor Who (1963-present)
• The Twilight Zone (1959-1964) (multiple episodes)
• Star Trek (multiple series, multiple episodes)
• Samurai Jack (2001-2004)
• Lost (2004-2010)
• Phil of the Future (2004-2006)
• Steins;Gate (2011)
• Outlander (2014-2023)
• Loki (2021-present)

Games about time travel:

• Chrono Trigger (1995)
• TimeSplitters (2000-2005)
• Kingdom Hearts (2002-2019)
• Prince of Persia: Sands of Time (2003)
• God of War II (2007)
• Ratchet and Clank Future: A Crack In Time (2009)
• Sly Cooper: Thieves in Time (2013)
• Dishonored 2 (2016)
• Titanfall 2 (2016)
• Outer Wilds (2019)

Additional resources

Explore physicist Peter Millington's thoughts about Stephen Hawking's time travel theories at The Conversation . Check out a kid-friendly explanation of real-world time travel from NASA's Space Place . For an overview of time travel in fiction and the collective consciousness, read " Time Travel: A History " (Pantheon, 2016) by James Gleik. 

Join our Space Forums to keep talking space on the latest missions, night sky and more! And if you have a news tip, correction or comment, let us know at: [email protected].

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Ailsa is a staff writer for How It Works magazine, where she writes science, technology, space, history and environment features. Based in the U.K., she graduated from the University of Stirling with a BA (Hons) journalism degree. Previously, Ailsa has written for Cardiff Times magazine, Psychology Now and numerous science bookazines. 

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Quantum Time Travel: Understanding the Physics Behind Time Jumps

Quantum mechanics, the cornerstone of modern physics, has revolutionized our understanding of the fundamental nature of reality, challenging our intuitive notions of time, space, and causality. Among its many intriguing implications, the concept of quantum time travel stands as one of the most captivating and enigmatic. This theoretical possibility, entwined with the fabric of quantum theory and general relativity, explores the potential for time jumps, time dilation, and the complex interplay of quantum phenomena that could allow for journeys through time. In this article, we will delve into the fascinating world of quantum time travel, unraveling its theoretical foundations, paradoxes, and the profound insights it offers into the nature of time, reality, and the universe.

Introduction to Quantum Time Travel

Quantum time travel delves into the quantum mechanical phenomena, spacetime geometry, and the theoretical frameworks that explore the potential for time travel, time loops, and time dilation within the context of quantum mechanics, general relativity, and the mysteries of the cosmos.

• Time Dilation and Relativistic Effects: Time dilation, a consequence of Einstein’s theory of special relativity, describes the slowing down of time for an observer moving relative to another inertial frame or experiencing gravitational effects, revealing the interconnectedness of time, space, and motion within the fabric of spacetime.
• Quantum Mechanics and Time Evolution: Quantum mechanics, governed by the Schrödinger equation and the principles of superposition, entanglement, and quantum states, offers insights into the non-deterministic, probabilistic nature of quantum systems, time evolution, and the quantum phenomena that challenge our classical understanding of time, reality, and the quantum landscape.

Wormholes, Black Holes, and Spacetime Geometry

Theoretical constructs, such as wormholes and black holes, serve as gateways to exploring the potential for time travel, time loops, and the intricate spacetime geometry that shapes the cosmic landscape and the possibilities for traversing through time and space.

• Wormholes and Einstein-Rosen Bridges: Wormholes, hypothetical passages connecting distant regions of spacetime, offer potential pathways for time travel, enabling journeys through time and space, linking distant cosmic regions, and revealing the interconnectedness of the universe through the fabric of spacetime and the curvature of reality.
• Black Holes, Event Horizons, and Cosmic Singularities: Black holes, regions of spacetime exhibiting strong gravitational effects that nothing can escape from, including light, represent cosmic laboratories for exploring extreme gravitational fields, time dilation, and the potential for time loops, revealing the mysterious nature of black holes, event horizons, and the cosmic singularities that define the cosmic landscape and the boundaries of our understanding of reality.

Quantum Paradoxes and Temporal Dilemmas

Theoretical paradoxes, conundrums, and thought experiments, such as the grandfather paradox, the twin paradox, and the Möbius strip of time, challenge our intuitive notions of causality, time directionality, and the consequences of time travel within the quantum realm.

• Grandfather Paradox and Causality Loops: The grandfather paradox, a classic time travel dilemma, explores the potential consequences, contradictions, and causal loops arising from changing the past, altering timelines, and the implications for causality, free will, and the interconnectedness of events within the fabric of time.
• Twin Paradox and Relativistic Effects: The twin paradox, a consequence of special relativity, illustrates the differential aging, time dilation, and the relativistic effects experienced by observers traveling at different velocities or experiencing different gravitational potentials, revealing the dynamic nature of time, spacetime geometry, and the interconnectedness of time, motion, and reality within the cosmic landscape.

Quantum Mechanics, Entanglement, and Time Symmetry

Quantum mechanics introduces the concepts of entanglement, superposition, and time symmetry, offering new perspectives on the nature of time, reality, and the quantum phenomena that challenge our classical understanding of time’s arrow, quantum states, and the mysterious quantum correlations linking distant events and particles within the quantum realm.

• Quantum Entanglement and Nonlocal Connections: Quantum entanglement, a phenomenon where particles become interconnected and the state of one particle instantly influences another, illustrates the nonlocality, quantum correlations, and the mysterious quantum connections that transcend classical boundaries, challenging our understanding of spacetime, locality, and the nature of reality within the quantum landscape.
• Time Symmetry, Quantum States, and Timeless Quantum Mechanics: Time symmetry, quantum superposition, and the principles of quantum mechanics challenge our classical perceptions of time’s arrow, temporal directionality, and the timeless nature of quantum states, revealing the interconnectedness of time, reality, and the quantum phenomena that shape the quantum landscape, cosmic dynamics, and the mysteries of the cosmos within the fabric of quantum time travel, quantum entanglement, and the quantum realm’s profound insights into the nature of time, reality, and the universe.

Quantum time travel, with its theoretical richness, paradoxes, and profound implications, stands as a testament to the intricate interplay of quantum mechanics, general relativity, and the mysteries of the cosmos, unveiling the potential pathways, cosmic connections, and the dynamic interplay of time, spacetime geometry, and the quantum phenomena shaping the cosmic landscape.

As we explore, investigate, and unravel the mysteries of quantum time travel through scientific inquiry, theoretical physics, and the pursuit of knowledge, we embark on a journey of discovery, exploration, and enlightenment that transcends boundaries, deepens our understanding of the universe’s complexity, beauty, and the mysterious interplay of quantum forces, spacetime curvature, and the cosmic dynamics shaping our cosmic journey, destiny, and the eternal quest for truth, meaning, and the timeless wonders that inspire wonder, curiosity, and a renewed appreciation for the grandeur, diversity, and interconnectedness of the cosmos, quantum phenomena, and the boundless realms of the universe and beyond.

Read More: The Goldilocks Zone: Finding Planets Just Right for Life

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How to destroy a black hole

A black hole would be tough to destroy, but in the season two premiere of Dead Planets Society our hosts are willing to go to extremes, from faster-than-light bombs to time travel

By Leah Crane

16 April 2024

Dead Planets Society is a podcast that takes outlandish ideas about how to tinker with the cosmos – from snapping the moon in half to causing a gravitational wave apocalypse – and subjects them to the laws of physics to see how they fare. Listen on Apple , Spotify or on our podcast page .

Dead Planets Society is back for season two, and our intrepid hosts Chelsea Whyte and Leah Crane are going after the toughest adversaries in the universe: black holes. These cosmic behemoths are so big and so sturdy that they can devour pretty much anything that is thrown at them without so much as flinching – so is it even possible to destroy one?

Black holes are expected to evaporate on their own thanks to Hawking radiation , a process by which they emit a slow leak of particles, but this would take much longer than the age of the universe to happen naturally. Just waiting isn’t really an option, so our hosts are joined by black hole astronomer Allison Kirkpatrick at the University of Kansas in an attempt to find a faster way.

Throwing anything at the black hole won’t really help either, whether it is a planet made of TNT or clumps of antimatter – the black hole will just swallow it up and get even more massive.

That doesn’t mean it is impossible to dream up something that would destroy a black hole by falling in . The escape velocity of a black hole – the speed at which one would have to fly away from its centre to escape its gravitational influence – is faster than the speed of light, so a ship that could travel beyond that physical limitation might be able to escape, or a bomb that could explode faster than the speed of light might be able to make a dent.

That is only the beginning of the outlandish ways to potentially wreck a black hole. Theoretical objects called white holes might work, but that could mean sending the black holes back in time, which wouldn’t be great for the past or the future.

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A black hole could perhaps be stretched out, but whether that works depends on the question of how quantum mechanics and general relativity mesh together, which may be the biggest unsolved question in physics. Our hosts find that giant magnets could help, with potentially horrifying results.

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• Dead Planets Society

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