Relativity Theory

Relativity theory was developed by Albert Einstein and others in the first half of the twentieth century and was a paradigm shift in our understanding of the physical laws of motion and energy that govern our universe.

Einstein’s special theory of relativity replaces Newton’s laws of motion at velocities comparable to the speed of light, and leads to the fundamental principal that mass and energy are equivalent – expressed in perhaps the most famous of all mathematical equations, E = mc2. Special relativity treats time as if it were equivalent to a fourth spacial dimension, introducing the unified concept of spacetime; recognises the speed of light as the ultimate speed limit; and deals with unintuitive concepts such as relativistic time dilation.

The general theory of relativity extends these principles to include the effects of a gravitational field and describes gravity as a distortion of the very fabric of spacetime.

Special Theory of Relativity

The special theory of relativity is so called because it deals with the special case of relativity in the absence of acceleration or a gravitational field. (The general theory of relativity, however, includes gravity and says that acceleration is mathematically equivalent to being stationary in a gravitational field.)

It is a fundamental principle of relativity theory that an observer will always measure the speed of light as 3×108 m/s, however fast they are travelling. This is because speed is a relative concept, and there is no fundamental frame of reference we can use to say that an object is either moving or at rest. In other words, it would feel no different to us if we were sitting in an armchair on a spacecraft floating in deep space or sitting in the same armchair on board the same spacecraft after having accelerated to a constant speed of one million miles an hour. Since we are no longer accelerating, we would feel no force acting upon us, and the laws of physics must appear to be the same to all non-accelerating observers outside of the influence of a gravitational field. If light did appear to travel at different speeds, depending on the speed of the observer, then it would be possible to identify which observer is at rest and which observer is moving.

The frame of reference of any observer who is not accelerating and is not under the influence of a gravitational field is known as an “inertial reference frame”. The fundamental laws of physics must appear the same in all inertial reference frames as there is no preferred inertial reference frame that can be claimed to be at absolute rest. Any inertial reference frame can always be considered to be in motion relative to any other inertial reference frame. Conversely, any inertial reference frame can be considered to be at rest, with all other inertial reference frames moving relative to it.

This absence of a fundamental resting frame of reference was suggested by the results of an experiment conducted by American physicists Michelson and Morley, in 1887, using a device known as an interferometer. This interferometer showed that a light beam took the same time to travel the same distance no matter in which direction it was travelling. Until this time, it had been presumed that we were surrounded by a mysterious medium referred to as the luminiferous aether, though which the Earth was moving as it travelled through space, and that light waves travelled through this aether like ripples in a pond. This would mean that light travelling against the direction of motion of the Earth through the aether would be measured as moving faster than light travelling in the same direction as the Earth. Michelson Morley’s experiment showed, however, that light always appeared to travel at the same speed in any direction, disproving the existence of this aether, abolishing the concept of fundamental rest and leading to the conclusion that the speed of any object is not an absolute quantity but can only be said to have a value relative to the speed of another object.

James Clerk Maxwell had also shown mathematically, in a paper of 1864, that light could be considered to consist of an electric and a magnetic field oscillating at right angles to each other. This electromagnetic wave is self-perpetuating and, unlike ripples on a pond, doesn’t require a medium though which to propagate. Maxwell also calculated the speed of this wave from the properties of electromagnetism, which agreed with measurements of the speed of light.

If the speed of light varied with the speed of the observer, this electromagnetic waveform would appear to be frozen to an observer moving alongside the beam of light. Since the observer would be in an inertial reference frame, however, the laws of physics must appear the same and the light must instead appear to be moving at the speed described by Maxwell’s equations.

Time Dilation

Imagine that you are sitting on board a spaceship, floating in deep space, which is not accelerating and which you consider to be at rest. Imagine, also, that you have a clock on board that consists of a beam of light bouncing up and down in a box. Each time the light beam hits a mirror at the top or the bottom of the box, the hands of the clock are moved on by a small amount.

Now imagine a friend of yours has another of these light clocks and is travelling past you on a second spaceship that, from your frame of reference, you consider to be moving at close to the speed of light.

As you watch the beam of light bouncing up and down in your friend’s moving light clock, the path travelled by the beam of light as it moves from the bottom of their light clock to the top, will appear to you to trace a diagonal line, because of the movement of the spaceship as it passes by.

So, to you, it seems that for each tick of your friend’s light clock, the beam of light has travelled further than the beam of light in your own clock sitting next to you.

This might not seem to be a big deal until you remember that the speed of light always appears to be 3×108 m/s to every observer. So, since the beam of light in your friend’s light clock has travelled further between ticks than the beam in your own light clock, it must seem to you that it has taken longer to tick than your own. Each tick of your friend’s light clock will take longer than the tick of your own clock, making your friend’s moving light clock appear to run slow.

To your friend on the spaceship, however, the same thing seems to be happening. Your friend watches you and your light clock race past the window of their spaceship, with the beam of light in your clock appearing to move diagonally rather than straight up and down, in exactly the same way that your friend’s clock appeared to you. From your friend’s point of view, or reference frame, it is your beam of light that appears to travel further between each tick, and so, since light always appears to travel at the same speed, it takes longer for your clock to tick than the clock on board your friend’s spaceship.

This leads to the inescapable conclusion that you and your friend will both observe the other’s clock to be running slow.

This change in measurement and perception of the flow of time, depending on the observer’s relative speed, is known as relativistic time dilation.

If your friend’s spaceship and light clock were travelling at an incredibly high speed approaching the speed of light (N.B. it’s not possible for an object with mass to travel at the speed of light – see below), then the path of the beam of light travelling upwards from the bottom of their light clock would have to travel almost parallel to the direction of the spaceship in order to keep up. This means that it would take an extremely long time to reach the top of the light clock. The clock could take an indefinitely long time to tick, as the spaceship approached the speed of light, and time would appear to be almost frozen on board your friend’s spaceship.

Relativistic Length Contraction

For your friend on board their spaceship, travelling at almost the speed of light, time would still appear to pass normally, however, except that space in the direction of travel would become compressed, and it would seem to them that they had reached their destination almost instantaneously. This effect is known as relativistic length contraction. As with time dilation, both you and your friend will observe the other’s spaceship (and everything else that is moving relative to you) to be length contracted. However, to both you and your friend, your own spaceships will not appear length contracted, since they are moving along with you and are at rest with respect to your own inertial reference frames.

The degree of length contraction depends on the how fast an object is travelling. Length contraction is only observed in the direction of motion and, like time dilation, is only noticeable when travelling at speeds that are a substantial proportion of the speed of light.

Because of the effect of length contraction, If your friend’s spaceship started out travelling at the same speed as your own but accelerated at a constant rate of 10 metres per second every second – roughly equivalent to the acceleration of an object free-falling under gravity on Earth – it would be possible for them to travel around the entire observable universe within an average human lifetime. To you, waiting for their return, however, your friend’s journey would seem to have taken several billion years. This is because of the equivalent time dilation effect from the perspective of the observer who hasn’t undergone an acceleration – or, in the terminology of relativity theory, has stayed in the same inertial reference frame. See the twin paradox below.

The Twin Paradox

At first glance, it might be difficult to image how two moving clocks can both appear to run slow relative to each other. If both you and your friend observe the other’s clock to run slow, because of your relative motion, what would happen if you were to meet back up again and compare the time on your two clocks? Who’s clock would appear to have slowed down? This is known as the twin paradox, as the question is usually posed for the case of two twins. One travels on board a spaceship at near light speed and then returns to Earth. If they measure each other’s clock to have run slow, how can each twin have simultaneously aged more than the other?

Of course there is really no paradox, and the two twins will agree on who has aged the most when they are finally reunited. The situation isn’t symmetrical, as one twin stays in the same inertial reference frame, while the other must undergo acceleration in order to leave the Earth and then return to it. An in-depth analysis of the situation shows that it is the twin who stays behind, and has not undergone an acceleration, who will have aged the most.

The reason for this can be seen by considering the change in reference frame of the twin who leaves the Earth.

Experimental Confirmation of Special Relativity

Much of special relativity’s predictions are difficult to test directly, as they are only noticeable for objects moving at ‘relativistic’ speeds, i.e. at a significant proportion of the speed of light. One way by which the effects of time dilation and length contraction have been confirmed experimentally is by observing subatomic particles called muons (see particle physics) travelling at close to the speed of light. The muon’s halflife – how long it takes for half of any number of these particles to decay into other particles – is known, for when they are at rest with respect to our own reference frame. However, the halflife of these particles is extended if they are travelling at close to the speed of light, due to this relativistic time dilation effect. Hence a beam of these particle, moving at close to the speed of light, is observed to travel much further than would otherwise be expected before half the particles in the beam have decayed.

From the reference frame of the muon beam, however, which could also be considered to be at rest, the particles’ halflife is not extended. To an observer in this reference frame, i.e. travelling at the same relative speed as the muon beam, it is the distance that the muons have travelled before half of them have decayed that appears to be shorter, due to the equivalent effect of length contraction.

The Classical Doppler Effect Applied to Light

The spectrum of visible light is one small part of the electromagnetic spectrum, which includes all wavelengths of electromagnetic radiation, from radio waves at the longest wavelengths, to gamma rays at the shortest wavelengths.

The frequency of any wave corresponds to the number of wave fronts passing an observer within a given time period. The frequency can be calculated by dividing the speed of the wave by the wavelength. E.g. visible light has a frequency of between 430 terahertz, at the red end of the spectrum (where one terahertz (THz) is 1012 wave fronts per second) and 790 THz, at the violet (deep blue) end of the spectrum. This corresponds to a wavelength of between 700 nanometres at the red end (where one nanometre (nm) is 10 -9 metres) and 390 nm at the violet end.

A boat will meet waves at a faster rate, and hence higher frequency, when sailing against the waves on a lake than when sailing along with them. The absolute speed of water waves can be measured relative to the water through which they are travelling, and the speed of the waves, as measured by an observer, varies with their relative motion across the water.

For light, their is no such propagation medium, equivalent to the water by which to measure its speed, and the speed of light remains constant to any observer, moving at any speed.

However, imagine your friend’s spaceship is floating in space some distance away but moving at the same relative velocity as you, so that you consider your friend to be at rest with respect to your reference frame. Your friend then turns on a source of very long wavelength radio waves. Radio waves are part of the electromagnetic spectrum and, therefore, behave in the same way and travel at the same speed as light, except they have much longer wavelengths and therefore much lower frequencies. (N.B. It’s easier to think of longer wavelength in this and the following scenarios, which is why this analogy uses radio waves rather than visible light waves.)

The wavelength of the radio waves transmitted by your friend happens to be exactly equal to the distance between you and his spaceship, so that the first wave crest hits your radio receiver, which makes a ping noise, just as the second wave crest is released from your friend’s transmitter. The second wave crest travels towards you and you again hear a second ping just as the third wave crest is leaving your friend’s transmitter.

Suppose, instead, that your friend had been travelling towards you at a significant fraction of the speed of light and, at the same distance they were from you in the first scenario, suddenly turned on the their radio wave transmitter. In this scenario, from your perspective, by the time the first wave crest arrives at your detector and makes its ping sound, your friend’s spaceship will have moved a good proportion of the distance that the radio wave has covered. So the second wave crest has to travel a shorter distance to reach you than it did in the first scenario. This means its journey time is also shorter, since the speed of light is constant and it has less distance to traverse. Therefore, you will measure the time interval between the pings that you hear in this second scenario as shorter than in the first scenario when your friend was stationary relative to you.

This time interval is a direct measurement of the frequency of the wave, and consequently the radio waves appears to have a higher frequency (i.e. more waves passing by per unit time) in this scenario. Since the second wave crest started its journey towards you at a closer distance than in the first scenario, the whole waveform of the radio wave is compressed and, therefore the wavelength is shorter. This allows the speed of the wave, which is calculated as the frequency multiplied by the wavelength, to still equal the speed of light, as it must always do.

To your friend, however, they consider you to be moving and that their own spaceship is at rest. To them, it appears that the radio waves they emit have the same frequency and wavelength in both scenarios.

Conversely, if your friend was, instead, moving away from you at a significant fraction of the speed of light, the frequency of their radio wave transmission would appear lower and the wavelength longer as the second wavecrest would have further to travel than the first.

If your friend had been emitting visible light, instead of radiowaves, as they moved towards you, this increase in frequency and shortening of wavelength would cause you to perceive the light as having shifted towards the blue end of the spectrum, compared to the colour of the light as you perceived it when your friend’s spaceship was at rest. If your friend had been moving towards you, the light would have shifted towards the red end of the spectrum.

The magnitude of this “blueshift”, for electromagnetic waves emitted by a source travelling towards you and “redshift” for electromagnetic waves emitted by a source moving away from you, increases with the increasing relative speed of the source to the observer.

This existence of this effect was first proposed by Christian Doppler in 1842 to explain the colours of stars in binary systems, which change depending on whether a star is moving away from us or towards us in its orbit.

The Doppler effect is also what causes the redshift in the spectral lines of light from distant galaxies, which led Edwin Hubble to propose that the universe is expanding (see Hubble’s Law).

The Relativistic and Transverse Doppler Effect

The classical Doppler effect, as described above, applies also to sound waves, which causes the change in pitch of the sound that we hear when an ambulance siren or a speeding car passes by.

For the classical Doppler effect, at the point where the direction of travel is exactly perpendicular to the line between the source and the observer – i.e. the point of closest approach – the pitch of the sound will be exactly the same as it would be when the source of the sound is stationery (ignoring any effects of aberration due to the finite speed of the wave). However, the theory of relativity predicts an extra redshift in the frequency of light, which is present even when the source of light is at this point of closest approach, moving in a transverse direction to the observer.

Returning to the example of the light clock ticking aboard a friend’s spaceship travelling past you at close to the speed of light, the ticking of this clock will appear slower to you than it does to your friend on board, due to time dilation, as described above. Imagine that the light clock is instead an oscillating electromagnetic field that produces a beam of electromagnetic radiation in the wavelength range of visible light. Since the electromagnetic field will appear to be oscillating more slowly to you than it does to your friend travelling with the source, the light produced will seem to you to be of a lower frequency and longer wavelength than it would appear to your friend aboard their spaceship.

Hence, an extra relativistic component to the Doppler shift is predicted by the special theory of relativity. This relativistic redshift is detectable even when the source is moving at right angles to the line joining the source and observer, when it is known as the transverse Doppler effect. This extra relativistic redshift has been observed experimentally, providing another useful test of the theory of special relativity.

Equivalence of Mass and Energy, E = mc2

E = mc2 is considered to be the most famous of all mathematical equations.

The E represents energy, m represents mass and c is the speed of light in a vacuum, so the equation is actually saying that energy is equivalent to mass times the speed of light squared.
(Note that in algebra the multiplication symbol is omitted as it could be confused with the letter x, which is often used as an algebraic symbol. However, the equation could also be written as E = m x c2.)

In other words if you add energy to an object, e.g. by heating it up or accelerating it, you will actually increase the object’s mass. Since the speed of light squared is a very large number, 1 kilogram in mass is approximately equal to 9×1016  Joules (or 90 petajoules) of energy.

So, how did Einstein reach this conclusion?

Light can be considered to behave as a particle as well as wave (see wave-particle duality under quantum mechanics). A particle of light is also known as a photon and the equations of James Clerke Maxwell had shown, in 1862, that a photon of light can be considered to have momentum but no mass.

This is unusual in itself, since the momentum of an object is usually defined as its mass times its velocity. However, in the case of light, this momentum means that a beam of light will exert a “radiation pressure” on a surface that it illuminates. For example this radiation pressure can be measured by a device known as a Nichols radiometer and was first proposed by Johannes Kepler in 1619 to explain why the tails of comets always point away from the Sun. These effects can be explained as a consequence of the light transferring its energy to the object whose surface it is absorbed by or the reactive force caused by the change in direction of the photon if it is reflected by an object. The photon’s momentum is directly proportional to its energy and hence it’s frequency. The higher the energy and frequency, and hence the longer the wavelength, the higher the momentum of the photon.

Now imagine a box floating in space and that a photon is emitted from one side of the box towards the other side. Because the photon has momentum, this means that the box must recoil by an imperceptible amount when the photon is released. When the photon is absorbed by the other side, the box stops moving, due to conservation of momentum.

Since there are no external forces acting on the box, the centre of mass of the system must not move throughout the process. However, if the photon has no mass then this would not be true.

Einstein suggested that the the photon must possess a relativistic mass, related to its energy, in order that the centre of mass of the box and photon system remains stationary. The equation E = mc2 then follows from a mathematical consideration of the laws of conservation of momentum.

An alternative thought experiment, based on the non-relativistic Doppler Effect, leading to the same conclusion, was proposed by Fritz Rohrlich in 1990. Imagine a light source produces two photons in opposite directions. In the rest frame of the source, both photons are of the same wavelength, i.e. not red or blue-shifted. For an observer in a reference frame moving towards the source object, however, a photon emitted towards the observer will appear blue shifted and a photon moving in the opposite direction will be red-shifted. This means that to the observer moving relative to the source, the blue-shifted photon moving towards them must impart more momentum to the source than the red-shifted photon. From the rest frame of the observer this should cause a deceleration in the motion of the source as it travels towards the observer. However, accelerations should be measured to be the same in all inertial reference frames, so this contradicts what appears to be happening in the reference frame of the source, where the momentum of the two photons balance out.

This difference between reference frames does not occur, however, if the photons actually carry away some of the mass of the source object, if this mass is equivalent to the energy of the two photons by E = mc2. The source will then simultaneously increase its speed towards the observer due to this loss of mass, since it’s kinetic energy needs to remain constant, while also decreasing its speed relative to the observer due to the difference between the momentum of the two photons. This leads to no overall acceleration of the source object in any other inertial reference frame, since the two factors exactly cancel each other out if the mass reduction of the source object is equivalent to the energy of the photons via E = mc2. Explained another way, from the reference frame of the observer, the photons are carrying away a net momentum from the source but the source cannot appear to change its velocity in any reference frame. Since momentum is equal to mass times velocity, and the velocity of the source object doesn’t change, it must be its mass that is reduced as viewed from the reference frame of the observer.

Rest Mass

A photon can only be considered to possess a mass, as described above, due to its kinetic energy or momentum. In theory, it is possible to create a photon with an arbitrarily low energy, (i.e. an arbitrarily long wavelength) and hence an arbitrarily low relativistic mass.

However, all other particles (with the exception of the gluon) posses an intrinsic mass even when they are at rest relative to an observer (which isn’t possible for a photon). This intrinsic mass is known as the particle’s rest mass or invariant mass.

The full energy of a moving particle (with a non-zero rest mass), is a combination of the energy due to the particle’s rest mass and the energy due to its momentum.

This total energy is given by the energy-momentum relation:

E2 = m02c4 + p2c2

Where m0 denotes the particle’s rest mass and p denotes its momentum.

Note, however, that the total relativistic mass of the moving particle can still be calculated from the total energy, and vice-versa, using the equation E = mc2 and that the energy-momentum relation does not “replace” E = mc2, since the mass in this relation refers to the rest mass only (m0).

E=mc2 is often associated with the energy released by a nuclear bomb or in nuclear power production. It is true that the nuclei of the atoms that make up a nuclear bomb would collectively weight less after the explosion than they did before, with the mass difference carried away by photons. However, this mass difference can be attributed to the change in “binding energy” of protons and neutrons in the atomic nuclei. This binding energy is potential energy from the strong nuclear force, which also contributes to the total mass of the a system, via E = mc2, as does any form of energy. It is this energy that is released rather than any change in the rest mass of the particles. This is analogous to the binding energy released in a chemical reaction, except that it is electromagnetic binding energy rather than binding energy from the strong nuclear force that is released and the differences in relativistic mass are far less for chemical reactions.

However, when a particle of antimatter annihilates with a particle of matter, then the rest masses of the particles are converted entirely into energy, which is carried away by the gamma ray photons produced. This energy provides the photon’s with their relativistic mass, as described above.

In fact, the existence of antimatter was first predicted by Paul Dirac in 1928 from a relativistic version of the Schrödinger equation from quantum mechanics, which describes how the quantum state of a physical system changes with time. Put simply, since this relativistic Schrödinger equation makes use of the relativistic energy-momentum relation, it has two answers because the energy is given by the square root of m02c4 + p2c2. Any square root has a positive and a negative answer, e.g. the square root of 4 is either 2 or -2. The positive square root solution to Dirac’s relativistic version of the Schrödinger equation represents matter, while the negative version represents antimatter.

Light Speed – The Ultimate Speed Limit

Imagine you are watching your friend’s spaceship accelerating from being at rest with respect to your own inertial reference frame. The energy that is put in to accelerate the spaceship goes towards accelerating not only the rest mass of the spaceship but the total relativistic mass, which is a combination of the mass due to the spaceship’s kinetic energy and its rest mass. At first, when the spaceship is travelling at a very small fraction of the speed of light, the highest proportion of the total mass comes from its rest mass, since mass is equivalent to a very large amount of energy due to the large multiplication factor of c2. Compared to this, the kinetic energy of the spaceship is negligible at low, non-relativistic speeds. Most of the energy put in to accelerate the spaceship goes towards accelerating this initial rest mass, so the spaceship accelerates rapidly as we’d expect from a classical Newtonian consideration of acceleration. However as the spaceship reaches speeds that are a greater and greater proportions of the speed of light, the kinetic energy grows larger in comparison to the rest mass energy. To accelerate the spaceship you must accelerate its total mass, including the mass due to the kinetic energy. As the spaceship approaches light speed the proportion of mass due to kinetic energy becomes the dominant factor, and the energy put in to accelerate the spaceship goes mainly to accelerating this extra kinetic energy. To accelerate the spaceship to light speed itself would take an infinite amount of energy, which is why it is not possible to accelerate an object to the speed of light.

Light speed is, therefore, the ultimate speed limit in the universe.

However, to your friend, the mass of their spaceship does not appear to grow with acceleration, as the relative speed of the spaceship to the observer is still zero. It is the kinetic energy and hence the relativistic mass of your spaceship, and any other objects moving relative to you friend that appears to increase, from their perspective.

They would continue to accelerate, and any light they emitted would still be moving away from them at c. If they accelerated to a fast enough speed, length contraction and time dilation would make it possible for them to reach distant stars and galaxies in less time than you would calculate light would take to reach these distances from your reference frame. Even so, looking back towards your own spaceship, you would never appear to be receding away them at faster than the speed of light, because of length contraction.

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