Einstein's theory of Special Relativity

1. 19th Century problems

In 1905, German-Swiss-American physicist Albert Einstein combined the idea that experiments performed at a constant velocity will give the same results as experiments that are stationery, with the idea that the speed of light will remain constant from both perspectives[1].

The first assumption is a result of Galileo's relativity. In 1632, Italian natural philosopher Galileo Galilei showed that there's no experiment that can distinguish if you are moving at a constant rate or if you are stationary[2].

The second assumption comes from the Michelson-Morley experiment of 1887[3].

Galileo's theory shows that if someone runs at velocity v across the deck of a ship moving at velocity u, then the speed measured by someone on the shore would be v + u. If a beam of light were to move across the ship at velocity c, the person on the shore should measure the speed to be c + u.

The Michelson-Morley experiment showed that this cannot be true. The speed of light was measured to be the same whether we are moving towards or away from it. This disproved the idea that absolute space is at rest relative to the aether.

2. Special relativity

In order to accommodate both views, Einstein realised that either of the two properties that velocity relies upon - time or space - must differ between observers. He then showed that both of these parameters can vary according to perspective.

2.1 The relativity of simultaneity

The first consequence of special relativity is known as the relativity of simultaneity. This shows that events that appear simultaneous in one reference frame, may not do so in another.

This means that what we perceive as the present only corresponds to what is occurring simultaneously to us, in our reference frame.

2.2 Length contraction

The second consequence of special relativity is length contraction. Einstein showed that an object will appear to be shorter, in its direction of motion, if we measure it while it's moving relative to us than if it's stationary.

The length that we measure when we are stationary with respect to an object is known as the 'proper' length, or rest length.

The difference between length and proper length depends on velocity v via:

ℓ =
0/γ
.
Here is length, 0 is the proper length, and γ =
1/√(1 - v2/c2)
, where c is the speed of light.

γ is known as the Lorentz factor, and is derived from the work of Dutch physicist Hendrik Lorentz.

Lorentz factor:
γ =
1/√(1- v2/c2)
Length contraction: ℓ =
0/γ
Time dilation: t = γt0
Mass increase: mrel = γm
Kinetic energy increase:
e - e0 = (γ - 1) mc2

2.3 Time dilation

The third consequence of special relativity is time dilation. This means that the time that a clock takes between ticks appears to be longer when the clock is moving relative to us.

This can be illustrated by imagining that we bounce a beam of light from two mirrors and observe it from both perspectives, one where the mirrors are stationary with respect to the observer and one where the mirrors are moving.

When the mirrors are stationary, the light will travel vertically, up and down.

When the mirrors are moving, the light will have to travel further.

Diagram showing a stationary light clock. The time light takes to move between two stationary panels and back is t= twice the distance between the panels divided by the speed of light.

Stationary light-clock. Image credit: Mdd4696/Public domain.

Diagram showing a moving light clock. The position of the panels changes during the time it takes for the light to travel between them, and so the light must travel a longer distance in the same time.

Moving light-clock. Image credit: Mdd4696/Public domain.

The speed of light is equal to the distance the light travels divided by the time it takes and so, in order for the speed to remain the same in both cases, time has to appear to pass more slowly for the moving clock, from the perspective of the stationary observer.

This effect cannot be due to length contraction because the contraction is parallel to the direction of motion. The time we measure between events that are stationary with respect us is known as 'proper' time, or rest time (t0). Time (t) and proper time are related via t = γt0.

2.4 The twin paradox

Time dilation and the relativity of simultaneity lead to the twin paradox. The twin paradox shows that if one twin travelled away from the other at near to the speed of light, and then turned around and came back again, they would be younger than their sibling on Earth.

This appears to be a paradox because Galileo's relativity states that there is no absolute state of rest. This means that the twin on Earth could equally consider themselves to be moving away from their sibling at this speed.

The paradox is resolved by the fact that the moving twin must accelerate when they leave Earth and when they turn around and some back, and this breaks the symmetry.

How much younger are astronauts when they return from orbit?

If a person spends 6 months on the International Space Station, which travels at about 8 km/s, then they will be about 0.0056 seconds younger when they return than they would be if they had remained on Earth.

Here,

t0 = 6 months = 15,768,000 s,
v = 8 km/s = 8000 m/s, and
c = 299,800,000 m/s.

t =
1/√(1 - v2/c2)
t0,

and so:

t =
1/√(1 - 80002/299,800,0002)
× 15,768,000,

t = 15768000.0056 s,

t - t0 = 0.0056 s.

2.5 Energy and mass

Einstein’s theory of special relativity also alters our understanding of mass. Einstein imagined a mass at rest (m) that emits two photons. One photon moves to the right and one to the left. Since we are at rest relative to the mass, the two photons appear to have equal and opposite momentums.

If we are moving relative to the mass, however, then one photon will be moving towards us, and one will be moving away. When objects move towards us, they appear to have more momentum than when they move away, due to the Doppler effect.

The fact that one photon appears to have less momentum than the other means that this photon also appears to have less mass. This is because Momentum = Mass × Velocity, and the velocity of a photon is always the same: c. This mass would appear to be different if you moved, and so this mass is refered to as the relativistic mass (mrel). Einstein showed that the lost relativistic mass was equal to the energy of the photons divided by c2, and so:

e = mrelc2

The relativistic mass is related to mass (m) via:

mrel = γm

and so

E = γmc2,    γ =
1/√(1 - v2/c2)

giving

e2 = p2c2 + m2c4.

Here e refers to the total energy, p to momentum, and m to mass.

An object has no momentum if it is not moving, and so:

e = mc2.

An object that is moving, but has a rest mass of 0, like a photon, will have an energy of e2 = p2c2, or:

e = pc.

This shows that an object with no rest mass can have a momentum. Classically:

p = mv.

The momentum of a photon can be found by applying the Planck relation, which was derived by German physicist Max Planck in 1900. Planck showed that:

e = hν.

Here ν is the light's frequency, and h is a constant, known as Planck's constant.

This gives, p =
e/c
=
hν/c
, c = λν, where λ is the light's wavelength, and so p =
h/λ
.

It would take an infinite amount of energy to accelerate an object from rest to the speed of light. This is because an increase in kinetic energy (ek) is equal to the total energy minus the rest energy, which gives:

ek = e - e0,

ek = γmc2 - mc2,

ek =
mc2/√(1 - v2/c2)
- mc2.

v2/c2 = 1 when v = c, and so the formula becomes, (mc2/√0) - mc2.

This shows that as v approaches c, the energy requirement approaches infinity.

Some argue that special relativity implies that mass and energy are the same. British physicist Arthur Eddington stated that:

"it seems very probable that mass and energy are two ways of measuring what is essentially the same thing, in the same sense that the parallax and distance of a star are two ways of expressing the same property of location"[4].

This is because the speed of light is equal to one if the distance is measured in light-years, and so e = m, although there is still much debate about the meaning of the energy-mass relationship.

2.6 Spacetime

In 1908, Polish-German mathematician Hermann Minkowski showed that special relativity is best described using the concept of a four-dimensional spacetime[5]. This unites the time dimension with the three spatial dimensions we observe. Minkowski showed that events take place in a point in spacetime that can be specified by four numbers in any coordinate system.

In special relativity, unlike Newtonian physics, there's a universal speed limit; things cannot happen instantaneously. This means there are regions of spacetime that we can never affect, and that can never affect us. These things can be mapped in a 'light cone', which defines the observable universe.

An event in spacetime can only affect the areas it can physically reach, given that information cannot travel faster than the speed of light. The propagation of light from an event can be mapped in two dimensions. If time is added as the vertical dimension, this forms a cone shape, as more space is reached in time. Nothing can travel faster than light, and so nothing from the event can affect spacetime outside of this cone.

We can map the past in the same way. We can see distant objects, like stars, as they were thousands of years ago because their light has travelled across space for thousands of years before it reached us. The further we look outwards in space, the further we look back in time, but we cannot be affected by anything that is too far away for its light to ever reach us.

2-D animation showing an event in spacetime, light from the event propagates in all spatial directions at the speed of light.

After an event in spacetime, light from the event propagates in all spatial directions at the speed of light. Image credit: Helen Klus/CC-NC-SA.

2-D diagram showing light propagating from an event in spacetime.

The light from an event in spacetime can be mapped in 2-D. Image credit: Helen Klus/CC-NC-SA.

3-D diagram showing light propagating from an event in spacetime.

The light from an event in spacetime can be mapped in 3-D, with time as the vertical dimension. Image credit: Helen Klus/CC-NC-SA.

The concept of mass and its effect on spacetime were better understood when Einstein reconciled his theory of special relativity with Newton's law of universal gravitation. He finally achieved this in 1916, with his theory of general relativity[6].

3. References

  1. Einstein, A., 1905, 'On the electrodynamics of moving bodies', Annalen der Physik, 17, pp.891-921, reprinted in in 'The Principle of Relativity: A Collection of Original Memoirs on the Special and General Theory of Relativity', 1920, Courier Corporation.

  2. Galilei, G. and Drake, S. (trans), 1953 (1632), 'Dialogue Concerning the Two Chief World Systems, Ptolemaic and Copernican', Modern Library.

  3. Michelson, A. A. and Morley, E. W., 1887, 'On the Relative Motion of the Earth and of the Luminiferous Ether', Sidereal Messenger, 6, pp.306-310.

  4. Eddington, A., 2009 (1920), 'Space, Time, and Gravitation', Project Gutenberg.

  5. Minkowski, H., 1920 (1908), 'The Fundamental Equations for Electromagnetic Processes in Moving Bodies' in 'The Principle of Relativity: A Collection of Original Memoirs on the Special and General Theory of Relativity', Courier Corporation.

  6. Einstein, A., 1916, 'The foundation of the generalised theory of relativity', Annalen der Physik, 354, pp.769-822, reprinted in 'The Principle of Relativity: A Collection of Original Memoirs on the Special and General Theory of Relativity', 1920, Courier Corporation.

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