# What does the first postulate of specially relativity really say?

I know these two versions of the same postulate is saying the same thing. But I failed to connect them. Please help me understand the links between them.

version1

The laws of physics are the same in all inertial reference frames.

version2

The speed light is traveling at is the same no matter what the constant velocity of the observer (inertial frame reference) looking at the light is.

If I am not mistaken, the observer, whether stationary or traveling at constant velocity, is the inertial reference frame, to which the speed that light is traveling at is relative.

If that is correct, the version 1 is contradicting what Einstein said about speed of light being always constant. Because the "traditional" law of physics clearly says that there has to be some change in the observed speed of light as the speed of the observer varies.

• actually they are TWO different postulates. Aug 10, 2015 at 1:23
• These postulates can't be equivalent because "Version 1" holds in Galilean relativity while "Version 2" does not. Aug 10, 2015 at 1:30
• I concur with Shing Lau; 'version1' is the Principle of Relativity while 'version2' is the Principle of Invariant Light Speed $c$ Aug 10, 2015 at 1:31

## 2 Answers

That motion was relative was realized by Galileo, so there was a theory of relativity -- Galilean relativity -- long before Einstein. That the speed of light should be the same according to all observers is indeed inconsistent with the Galilean relativity. This is because in Galilean relativity time is absolute. But it is not mathematically inconsistent to have a universal speed of light. We "just" have to give up the idea that time is absolute. (That's a very big "just" - a Nobel prize "just"!) From On the Electrodynamics of Moving Bodies:

Examples of this sort, together with the unsuccessful attempts to discover any motion of the earth relatively to the “light medium,” suggest that the phenomena of electrodynamics as well as of mechanics possess no properties corresponding to the idea of absolute rest. They suggest rather that ... the same laws of electrodynamics and optics will be valid for all frames of reference for which the equations of mechanics hold good. We will raise this conjecture (the purport of which will hereafter be called the “Principle of Relativity”) to the status of a postulate, and also introduce another postulate, which is only apparently irreconcilable with the former, namely, that light is always propagated in empty space with a definite velocity $c$ which is independent of the state of motion of the emitting body.

The Galilean relativity says that if Alice moves at a velocity $v$ relative to Bob, their space and time coordinates are related by \begin{align} t_A & = t_B \\ x_A & = x_B + vt_B \end{align} and it follows from this that if Alice emits a signal with velocity $w$, Bob will observe its velocity to be $v+w$. This is why Einstein says that the postulates are apparently irreconcilable, as you have found.

But in the Einsteinian relativity, Alice's and Bob's space and time coordinates are instead related by \begin{align} t_A & = \gamma(t_B - v x_B/c^2) \\ x_A & = \gamma(x_B + \frac{v}{c} t_B) \end{align} where $$\gamma = 1/\sqrt{1 - \frac{v^2}{c^2}}$$ and $c$ is some constant with the units of velocity -- at this point we haven't related $c$ to physical phenomena like light yet. Now it turns out that when space and time are related like this, because Alice does not agree with Bob what time is, velocities add differently. If Alice emits a signal with velocity $w$, the velocity measured by Bob is $$v\oplus w = \frac{v + w}{1 + vw/c^2}.$$

Now there are two interesting things about this formula. First of all, if $v, w < c$ then $v\oplus w$, and -- as the resolution to the conundrum -- if $w = c$, then regardless of $v$, $v\oplus w = c$. So under this theory of relativity, signals moving at $c$ are special: everyone will agree that they move at $c$.

To relate this to light, one of Maxwell's equations -- the equations that govern electricity and magnetism -- is $$\nabla\times\mathbf B = \mu_0 \mathbf J + C^{-2} \frac{\partial \mathbf E}{\partial t}$$ where $C$ is a constant with units of velocity -- it is the speed of electromagnetic waves, that is, light. It turns out that Maxwell's equations are never consistent with Galilean relativity, but they are consistent with Einsteinian relativity, under the condition that $c = C$. So the $c$ that enters into the coordinate transformation is indeed the speed of light. (You can do the math to make absolutely certain that it is like this, but you can also think of it like this: under Einsteinian relativity $c$ is the only speed that everyone agrees on. So if everyone is to agree on the form of the equation, the only velocity that can show up is $c$.)

• You said the first postulate is inconsistent with the second, but does that depends on what you call the "law of physics"? Aug 11, 2015 at 23:51
• So the first postulate of special relativity is just the result of Einstein simply restating what Galileo said? Aug 11, 2015 at 23:55
• Is my textbook then incorrect in stating as the first postulate that "the speed of light and all laws of physics are the same for any inertial reference frame"? If version 1 is the correct first postulate of special relativity, then which other postulate says that light speed is the same to all observers? Aug 11, 2015 at 23:59
• @user132522 - well, yes, it depends on the laws of physics in a certain sense. Both Einstein's and Galileo's theories are fine proposals for how to relate inertial observers, so we need something else to figure out which one to prefer. That "something else" is that some law of physics, or experimental fact, is consistent with one but not the other. That's where Postulate 2, that some speed is the same for everyone, comes in. Postulate 2 is consistent with Einstein's theory but not Galileo's. Aug 12, 2015 at 0:49
• It is not that the postulates are inconsistent but that Postulate 1 is not sufficient on its own. The existence of a universal speed you can take as Postulate 2 to derive the mathematical form of the theory, or you can take it as one of the laws of physics in Postulate 1. Aug 12, 2015 at 0:57

version1

The laws of physics are the same in all inertial reference frames.

This is a correct principle, but it is useless unless you also state which things are the laws of physics. If they the laws include Newton's laws as usually written then you get Galilean Relativity, if the laws include Maxwell's equation then you get special relativity.

version2

The speed light is traveling at is the same no matter what the constant velocity of the observer (inertial frame reference) looking at the light is.

This is a special case of the first one where the laws of physics include Maxwell's equation, hence the prediction that there are waves travelling g at speed c relative to the inertial frame (Maxwell predicts waves travelling at speed c in the frame in which Maxwell holds).

This is all we need to actually derive the rest of special relativity, so it's a good thing to have. Since it is a special case of the first one and the first one is the correct principle it is redundant to have both.

If I am not mistaken, the observer, whether stationary or traveling at constant velocity, is the inertial reference frame, to which the speed that light is traveling at is relative.

If Maxwell holds in every inertial frame and Maxwell predicts light travelling at speed c in the frame where Maxwell holds then you are wrong. It turns out your opinion about how velocity in one frame looks like to another frame, was just that. An opinion. Baggage in fact.

And you can hold onto the baggage, but if you do so you have to give up the principle of relativity (version 1) or you have to say that Maxwell doesn't hold in evey frame.

Or you can put your cards on the table and see what experiments go along with. If Maxwell only held in one frame then light travels at speed c in just one frame so we should be able to detect that. We try. We fail. Instead, experiments show that for every inertial frame, regardless of velocity, light moves at speed c.

If that is the case then everything else must change, every assumption about how velocity and space and time and relative velocity how hey all work together. Each one must be reexamined to see which ones only held approximately.

If that is correct, the version 1 is contradicting what Einstein said about speed of light being always constant.

Absolutely not. The contradiction comes from your baggage. And if you ignore the fact that you assumed things that are wrong, then contradictions will appear over and over again.

Because the "traditional" law of physics clearly says that there has to be some change in the observed speed of light as the speed of the observer varies.

Newton's Laws don't say that (and Newton's laws do have to be changed). Maxwell's don't say that (and Maxwell doesn't have to be changed). Basically start with Maxwell and trust nothing else and figure out what has to hold if you want Maxwell and version 1. The first thing you get from taking version 1 and taking Maxwell is you get version 2 and from that you can get the rest.

I was saying it would contradict Einstein's own theory if like you said, the "law of physics" were Galilean Relativity. I just wanted to make sure Einstein meant Maxwell when he said the "law of physics".

He called his paper " On the electrodynamics of moving bodies" so he clearly meant Maxwell and clearly wanted the principle of relativity not the specific Galilean frame transformations.

Are both parts of the first postulate of relativity, just that the second version is more specific than the first?

The second version is more what he used, the first is more what he meant and what we think is true. If for instance photons are massive then light goes at different speeds but the speed $c$ is still an invariant upper bound to all speeds that everyone agrees on so relativity wouldn't change. So relativity will be the same even if Maxwell is wrong. The first is correct. And the general transformation allows an invariant speed (for Galilean relativity that speed is infinite) and that speed can be determined by experiment. We've done that and the speed is $c.$ Even if Maxwell is wrong.

• Did Einstein proposed the first version or Galileo did? My book used Einstein said... I was wondering why would he include something to contradict his own thoery. Aug 12, 2015 at 12:12
• @user132522 If you think it contradicts his own theory, then either you have bias and baggage so strong you can never learn physics (because you can't listen) or you didn't even read my answer, or else I have no idea why you think that. The first two are your issue and the last one is mine. If it is mine, then I apologize for failing you. Aug 12, 2015 at 15:00
• I was saying it would contradict Einstein's own theory if like you said, the "law of physics" were Galilean Relativity. I just wanted to make sure Einstein meant Maxwell when he said the "law of physics". Aug 13, 2015 at 21:29
• Are both parts of the first postulate of relativity, just that the second version is more specific than the first? Aug 13, 2015 at 21:31
• @mostvenerablesir "I just wanted to make sure Einstein meant Maxwell when he said the "law of physics" - Einstein meant the laws of physics, whatever they are. Any laws of physics must be formulated in such a way that they are valid unchanged in any inertial reference frame. Feb 4 at 15:33