# Einstein's first postulate implies the second?

Einstein's two postulates of special relativity are as follows,

1. The principle of relativity: The laws of physics are the same in all inertial systems. There is no way to detect absolute motion, and no preferred inertial system exists.

2. The constancy of the speed of light: Observers in all inertial systems mea- sure the same value for the speed of light in a vacuum.

Now for an exercise in my book, I am requested the following:

'Explain why Einstein argued that the constancy of the speed of light (postulate 2) actually follows from the principle of relativity (postulate 1).'

I have been trying to figure this out for a very long time now with no luck. I thought of identifying a physical law that if not the same in two different inertial systems leads to a contradiction, but I can't think of anything. Could you push me in the right direction?

Further to Timaeus's Answer, the second postulate follows from the first postulate if we know about light. Otherwise, the second cannot follow from the first in a strict sense.

However, even if you don't know about light, there is still a way whereby the second postulate can be strongly motivated by the first, as follows.

The first postulate is essentially Galileo's notion of relativity as explained by his Allegory of Salviati's Ship.

If you assume:

1. The first relativity postulate; and
2. A concept of absolute time, i.e. that the time delay between two events will be measure to be the same for all inertial observers; and
3. Homogeneity of space and time so that linear transformation laws between inertial frames are implied (see footnote)

Then these three assumptions alone uniquely define Galilean Relativity.

However, if you ask yourself "what happens to Galileo's relativity if we relax the assumption of absolute time" but we keep 1. and 3. above, then instead we find that a whole family of Lorentz transformations, each parametrised by a parameter $c$, are possible. Galilean relativity is the limiting member of this family as $c\to\infty$. The study of this question was essentially Einstein's contribution to special relativity. You can think of it as Galileo's relativity with the added possibility of an observer-dependent time. I say more about this approach to special relativity in my answer to the Physics SE Question "What's so special about the speed of light?".

It follows from this analysis that if our Universe has a finite value of $c$, then something moving at this speed will be measured to have this speed by all inertial observers. However, there is nothing in the above argument to suggest that there actually is something that moves at this speed, although we could still measure $c$ if we can have two inertial frames moving relative to each other at an appreciable fraction of $c$. It becomes a purely experimental question as to whether there is anything whose speed transforms in this striking way.

Of course, the Michelson Morley experiment did find something with this striking transformation law.

Footnote: The homogeneity of space postulate implies the transformations act linearly on spacetime co-ordinates, as discussed by Joshphysic's answer to the Physics SE question "Homogeneity of space implies linearity of Lorentz transformations". Another beautiful write-up of the fact of linearity's following from homogeneity assumptions is Mark H's answer to the Physics SE question "Why do we write the lengths in the following way? Question about Lorentz transformation".

• Oh this is wonderful! Thanks! Also, I can see that special relativity follows directly from the relaxation of the second Galilean postulate and the acceptance of Maxwell's laws, but what arises when you also relax the third Galilean postulate (homogeneity of space & time)? Anything of interest? I would definitely doubt the homogeneity of space and time within our universe. – Arturo don Juan Jan 22 '15 at 17:56
• @ArturoDonJuan Inhomogeneous space and time implies the transformations are linear. Relaxation of this postulate means that almost anything could happen: in particular, nonlinear transformation laws. "I would definitely doubt the homogeneity of space and time": very insightful of you, and this is one of the problems that General Relativity addresses. However, now we need to add a great deal more physical insight to find the right transformation laws. It turns out that we still have a "local" homogeneity: as long as we look at a region of spacetime that is small enough, the same homogeneity ... – WetSavannaAnimal Jan 22 '15 at 23:17
• @ArturoDonJuan ... postulate still applies, if one is "freefalling", which is the GR equivalent of inertial frame. – WetSavannaAnimal Jan 22 '15 at 23:17

The first principle and Maxwell's Equation(s) together imply the second principle. If you'd never heard of light, or the speed of light and you had no laws predicting it, then the first principle would not imply the second.

For instance, the first principle was accepted in Newton's day, but Special Relativity was a long time coming.

Furthermore, if you interpreted Maxwell's theory as being about the behaviour of a medium, then you might expect it to only hold in the frame of the medium. For instance, sometimes we derive equations for a fluid that are only designed for the frame of the fluid (which is sometimes a good reason to include non-inertial frames).

So you'd specifically have to say that Maxwell's Electromagnetism is something that you expect to hold equally well in every inertial frame. Which isn't really much different than the wording of the second principle.

• Wait, what if we just use this logic: Assume the propagation of light is isotropic. Suppose one measures the speed of light in an inertial reference frame in a vacuum. Then the speed of light is measured the same in every other inertial reference frame in a vacuum by postulate 1. This is postulate 2. – Arturo don Juan Jan 22 '15 at 4:54
• When invoking your logic, check first to make it clear that the speed of a sound wave is not the same to every inertial frame (the frame of the fluid/solid/etc. and another frame moving relative to the fluid/solid/etc.). If somehow you deduce that sound waves travel at the same speed to everyone, then (re)check your logic. – Timaeus Jan 22 '15 at 5:08
• Yes ok, I understand. So if instead we said: Assume that Maxwell's equations are physical laws in a particular inertial frame (say, our 'stationary' one). Then they are also the same physical laws in every other inertial frame, thus implying postulate 2. – Arturo don Juan Jan 22 '15 at 5:18
• "So you'd specifically have to say that Maxwell's Electromagnetism is something that you expect to hold equally well in every inertial frame. Which isn't really much different than the wording of the second principle." i must disagree. the 1st postulate already specifically says that the laws of physics are identical in every inertial frame. that means that Maxwell's Equations must be identical for every inertial frame. that means $\epsilon_0$ and $\mu_0$ must be identical for every inertial frame. that means $c=\frac{1}{\sqrt{\epsilon_0 \mu_0}}$ must be identical for every frame. – robert bristow-johnson Nov 27 '15 at 6:12
• @ArturodonJuan is correct. Timaeus response doesn't cut it, logically. sound propagates in a medium. if both inertial frames experienced the same translation through the medium, they would measure the speed of sound the same. – robert bristow-johnson Nov 27 '15 at 6:15

The first postulate is satisfied by Galilean relativity with an infinite speed of light, but this violates the second postulate. Therefore the second postulate does not follow from the first.

Of course experiment tells us that the speed of light isn't infinite, and if we combine the first postulate with a finite speed of light we find they are inconsistent unless further assumptions are made. This is where the second postulate comes in i.e. it is one way of reconciling the first postulate with a finite speed of light. The second postulate requires physical laws to be Lorentz covariant, which leads immediately to special relativity.

• the point is John that the 1st postulate says that all laws of physics are the same for inertial observers moving relative to each other, but otherwise having the same conditions. if someone comes up with a physical "law" that says $c < \infty$. that means $c$ = some number and the 1st postulate says that all inertial observers must have the same number. (there are other issues regarding units, but the 1st Postulate would say that all inertial observers must measure the same $\alpha$, too, which is even stronger than the same $c$.) – robert bristow-johnson Nov 27 '15 at 6:58
• @robertbristow-johnson: No, your claim in the comment above is false because you misquote the first postulate. The first postulate does not say all possible laws of phyics are the same for all observers. It says the laws of physics, whatever they are, are the same for all observers. The laws of physics could be Galilean relativity with infinite speed of light but for those annoying experimental scientists. – John Rennie Nov 27 '15 at 7:17
• neither i nor the 1st postulate is misconstrued to mean invalid "laws" of physics. the 1st postulate says that any law of physics that is applicable to one inertial observer is equally applicable to another inertial observer even if moving relative to the first. the onus is on you, John, to point out a law that applies to one and not to the other. – robert bristow-johnson Nov 28 '15 at 5:06
• @robertbristow-johnson Is it so hard to imagine that SR is a specific flavor of Relativity? One with a finite invariant speed (unlike Galilean Relativity)? And one where the invariant speed is $c$? One that is like that regardless of whether you think Maxwell is valid or invalid? (And Maxwell is invalid!) – Timaeus Nov 28 '15 at 19:56