Special relativity

Special relativity

Einstein, 1905

The first page, as it appeared in the Annalen der Physik for 1905. It opens on a magnet and a conductor, remarks in passing on the unsuccessful attempts to detect a motion of the Earth relative to the light medium, and proposes to raise the conjecture to a principle. There is no abstract and nothing is cited.

Published	26 September 1905, Annalen der Physik[1]
Postulates	Two
Adjustable parameters	None
Applies to	Uniform motion in a straight line
Does not apply to	Rotation; acceleration[2]
Absolute in the theory	The speed of light
Relative in the theory	Nearly everything else
Named after	The second of those
Empirically equivalent to	Lorentz's ether theory
Overturned	Not yet

Special relativity is the theory of space and time set out by Albert Einstein in 1905, in a paper of about thirty pages which reports no experiments and cites almost nothing: two footnotes, one crediting Planck and one observing that Lorentz's preceding memoir had not reached the author.^[1] It rests on two assumptions, and everything else in it follows by algebra a competent sixth-former can check.

Its name records the wrong half. The theory is not the claim that everything is relative; it is the claim that one thing is not. The speed of light in vacuum is the same for every observer in uniform motion, whatever they are doing and whatever the source is doing, and space and time are obliged to give way to keep it so. Almost every misreading of the theory, including all of those below, begins by taking the title as a summary.^[3]

The two postulates

1. The laws of physics take the same form in every inertial frame – that is, in every frame moving uniformly in a straight line.
2. A ray of light travels through empty space at the fixed speed $c$, whether it was emitted by a body at rest or by a body in motion.

The second is usually quoted in a stronger form, that light travels at $c$ for every observer whatever the observer is doing. That is true, and it is what the theory says, but it is not what Einstein postulated: he assumed only that the speed does not depend on the source, and the independence from the observer falls out when the first postulate is applied to the second.^[10]

The first is old and was not controversial; Galileo had it for mechanics. The second is the one that costs something. Taken together they are inconsistent with the assumption everyone had been making without noticing it – that two events either are or are not simultaneous, and that this is a fact about the events.

What follows

Set $\gamma = 1/\sqrt{1 - v^{2}/c^{2}}$. Then, for a frame moving at $v$ relative to another:

Moving clocks run slow, moving lengths contract along the direction of motion, and each observer says it of the other. Neither is mistaken and there is no fact of the matter about which is really moving, because uniform motion is not a property a thing has by itself.

At everyday speeds $\gamma$ is 1 to a dozen decimal places, which is why none of this was noticed for three hundred years. At 0.995 of the speed of light it is 10.

The relativity of simultaneity

This is the part that is rejected, and it is worth setting out properly, because the usual objection is to something the theory does not claim.

Einstein's own illustration, from the popular exposition he wrote twelve years later rather than from the paper, is a train. Lightning strikes the track at two points, one ahead of the train and one behind. An observer on the embankment, standing exactly midway between the two marks, receives both flashes together and calls the strikes simultaneous. An observer at the midpoint of the moving train receives the forward flash first.

The objection is that this is merely an artefact of the second observer's motion towards one flash – a delay in the signal, not a fact about time. If he simply subtracted the travel time, it is said, he would recover the simultaneity and the paradox would evaporate.

He does subtract it. That is the whole point and it is where the argument turns. The traveller knows he stands midway between the two ends of his own train; he measures the light from each end to have travelled at $c$, because the second postulate obliges him to; he therefore computes equal travel times over equal distances and concludes that the forward strike happened first. The correction has been applied and the disagreement survives it.^[4]

It is worth noting how ordinary the question was when Einstein reached it. Telling two distant clocks to agree was a commercial problem of the 1900s, and the world's timekeeping still rests on his answer to it, which is a definition rather than a discovery.

There is exactly one way to get absolute simultaneity back, and it is to declare that one particular frame's answer is the true one and that everybody else's clocks are simply mis-set. That move is available, it is consistent, and it costs a frame nobody can find. It has a name, and it is the subject of the next section but one.

What the theory does not say

The theory is narrower than its reputation, and the difference matters here more than anywhere.

Only the first line is relative, and the theory is named after the first line. Rotation is detectable from inside a sealed room, which was Newton's point in 1687 and remains true; it is measured optically by the Sagnac effect and mechanically by any gyroscope on any aircraft.

This is the whole of the reply to the argument that relativity licenses a stationary Earth. It licenses no such thing. It says that if the Earth moved uniformly in a straight line, no experiment aboard would reveal it – and the Earth does not move uniformly in a straight line. It turns, and it goes round something, and both of those are accelerations, and both are measured.^[5]

Lorentz's version

There is a rival account, and honesty requires that it be stated at full strength rather than dismissed.

Hendrik Lorentz had the transformations before Einstein and derived them differently: there is a stationary ether, bodies moving through it physically contract, and clocks moving through it physically run slow, by exactly the amounts required. In that theory simultaneity is absolute, there is a preferred frame, and the traveller on the train really is moving.

Every experiment ever performed gives the same answer under both. This is not a coincidence but a construction: Lorentz built his contractions to reproduce the observations, and they do. No measurement distinguishes the two, and none can, because the preferred frame is undetectable by hypothesis.^[6]

The theories are therefore chosen between on other grounds. Einstein's carries no undetectable structure, derives the contractions rather than imposing them, and generalises to gravitation, which Lorentz's does not. Those are good reasons and they are not experimental ones. A person may prefer the Lorentzian formulation and be committed to no error at all – and equally, may not use it to claim that relativity has been refuted, since the two agree about every number.

The evidence

The theory has no free parameters, so every test is a test of the whole of it.

• Muons. A muon lives about 2.2 microseconds. Since nothing outruns $c$, a muon should manage no more than about 660 metres before decaying, and no amount of energy should improve that, the lifetime being the limit rather than the speed. Muons are made some fifteen kilometres up by cosmic rays and arrive at sea level in quantity. Their clocks run slow by the factor the theory specifies, and the number of survivors matches.^[7]
• The transverse Doppler shift. Ives and Stilwell measured it directly in 1938, using a beam of hydrogen ions, and found the shift relativity requires and classical physics does not predict at all. Ives spent the rest of his career arguing against relativity and read his own result as supporting the ether.^[8]
• Accelerators. Every particle accelerator built since the 1930s is designed around relativistic dynamics. They are not tests so much as a standing demonstration: the machines are engineered to the equations and the beams arrive where the equations say.
• GPS. The satellite clocks are detuned before launch by the amount the two relativities require, one of them this one.

The use made of it

The theory is unusual among the subjects of this encyclopedia in being attacked from two directions at once, by people who agree with each other about nothing else.

It is held to be false because it is counterintuitive, which is not an argument; because the relativity of simultaneity offends common sense, which is true and is not an argument either; and because a rotating interferometer, a satellite constellation and an orbiting muon each behave in ways the objector has been told they should not. In each of those cases the apparatus behaves exactly as the theory requires, and the requirement has been checked.

It is also, more often, misused rather than attacked: quoted for the proposition that all motion is relative and therefore that any body may be placed at the centre of anything. The theory says that of uniform straight-line motion and of nothing else, and the bodies in question are not moving that way.^[9]

A hundred and twenty years of attempts have not produced a discrepancy. This is not offered as proof that none exists; it is offered as the reason the working physicist keeps using it, which is a lower and more defensible claim.

See also

• Albert Einstein – who set it out, and later extended it to gravitation
• General relativity – that extension, and what it does not license either
• Hendrik Lorentz – whose version predicts the same numbers from an ether
• Michelson–Morley experiment – the null result it accounted for
• The Sagnac effect – the relativity of simultaneity, measured on a turntable
• GPS – where both relativities are applied before the satellite leaves the ground
• Newton's bucket – on rotation being absolute, agreed across two centuries
• Kinematic and dynamic equivalence – where the misuse above is examined at length
• Occam's razor – why the profession chose this account over Lorentz's, and on what grounds
• Time – what the theory did to the world's clocks, and what the clocks did back
• The speed of light – the one thing that is not relative, and what became of the number

References

1. ^ "Zur Elektrodynamik bewegter Körper", Annalen der Physik 17 (1905), 891–921, received 30 June and published 26 September. It cites nobody, which was remarked upon at the time and has been ever since.
2. ^ Both are handled, and neither of them needs general relativity to do it: an accelerating body has a perfectly good worldline in flat spacetime, and a rotating frame has a metric of its own. The row above says only that they are not relative. What actually requires general relativity is gravitation, and nothing else does. See the Sagnac effect for the rotating case worked through.
3. ^ A theory of invariants is what it is. The name that stuck describes the least important of its claims, and has been causing trouble ever since.
4. ^ The objection would be sound against a theory in which light travelled at c only in one frame. It is not sound against this one, and the difference between those two theories is the subject of the section on Lorentz.
5. ^ The Earth's orbital motion is an acceleration because its direction changes; its rotation likewise. Neither is uniform straight-line motion, and neither is hidden. See stellar aberration, stellar parallax and the CMB dipole for three independent measurements of the first.
6. ^ This is the point at which most arguments about relativity ought to stop, and almost none do. See GPS for the same observation applied to a modern advocate of the Lorentzian formulation.
7. ^ The muon's proper lifetime is 2.2 µs, so cτ is about 660 m and that figure does not improve with energy. What improves with energy is γ, and the observed survival rate follows it.
8. ^ H. Ives and G. Stilwell, 1938. Ives was the most determined American opponent of relativity of his generation, and regarded his own result as a confirmation of the Lorentzian ether rather than of Einstein. On the argument of the section above, he was not simply wrong to: the two accounts predict the same shift. What he could not claim, and did claim, was that the measurement told against relativity.
9. ^ See kinematic and dynamic equivalence, where the argument is set out in the form its advocates give it and answered on its own terms.
10. ^ Stated in his §2 as a property of the ray and the emitting body, in a paper whose whole method is to assume as little as possible and then be ruthless about it.

External links

• Einstein, "Zur Elektrodynamik bewegter Körper" – Annalen der Physik (1905)
• The paper as printed – Annalen der Physik 17 (1905), 891, via Wikimedia Commons
• Ives & Stilwell, "An Experimental Study of the Rate of a Moving Atomic Clock" – JOSA 28 (1938)
• What is the experimental basis of Special Relativity? – the Physics FAQ
• Special relativity – general reference