Einstein's first postulate implies the second?

Question

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?

Answer 1

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.

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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".

Answer 2

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.

Answer 3

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.

Answer 4

If you regard Galilean relativity as a law of nature then Einstein's 2nd postulate confirms his 1st postulate and vice versa since SR as strange initially this may sound, is consistent with Galilean relativity and actually depends on Galilean relativity.

Even today this is by many not fully understood and cause for misinterpretations and misunderstanding thus non-intuitive. However, this is because this 2nd postulate comes with a condition that is little know or made clear. As soon this condition described herein is understood everything becomes clear in Einstein's Special Relativity (SR) concerning this 2nd postulate.

The illustration below of Fig. 1 explains to what Einstein's 2nd postulate that speed of light being absolute and not depending of the velocity and speed of the observer (receiver) really refers to, and which is IMHO the main root of confusion and difficulty most people have to intuitively understand what Einstein is actually saying?:

Base Graphics Credits: Steffen Kühn

In the first case where the EM wave did not reach yet the receiver (i.e. Broadcast station just transmitted its first wave transmission) propagation times tA and tB from the station to the receiver are different and are depending on the velocity and speed v of the receiver therefore subject of Galilean relativity.

In the second illustration the observer (receiver) has been already encompassed by the transmitted EM wave from the station. This is the case which Einstein SR refers to! The observer (receiver) is already inside the transmitted EM wave and fully surrounded by it. Therefore, tB=tA and independent the speed v of the receiver as long as the receiver is engulfed by the wave. The receiver (observer) is now subject of Einstein's Special Relativity (SR) and not Galilean relativity.

The speed v only generates a frequency Doppler shift of the received frequency of the EM waves but does not affect the propagation speed of the received waves which is always fixed at speed c in a vacuum.

That's it. Nothing really non-intuitive.

That is the key point for understanding Einstein's absolute c speed of light postulate, independent by the velocity of the observer. Einstein assumes that when you see the light on the train coming towards you, since you see the light that means it already has reached you and you are already inside it and engulfed by it.

An observer located within the wavelength of wave does not experience different propagation delays with his motion inside it therefore perceives the wave at constant velocity c in a vaccum independend the observer's relative motion.

So, all laws of nature hold and are the same independent of inertial frame in Einstein's SR even including Galilean relativity. This is the beauty of Einstein's theory.