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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?

Qmechanic
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Arturo don Juan
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4 Answers4

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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.

Timaeus
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    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 04:54
  • 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 05:08
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    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 05:18
  • "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 06:12
  • @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 06:15
  • @robertbristow-johnson Two different frames don't move "the same translation through the medium" they move relative to each other. The point of the second law is to explicitly say that Maxwell holds for every frame that they are physical laws for a frame not physical laws just for the frame of a medium. And as I already said the first postulate is ancient history, you don't get SR unless you explicitly say that Maxwell holds in all frames, bit just the frame of a medium. – Timaeus Nov 27 '15 at 06:21
  • i understand that the 1st postulate is ancient history. the insight of Einstein was simply to insist on its consistent application (and then to derive the implication of that consistent application on time, length, momentum). i agree that, without some kind of wind sheer, the two frames cannot translate identically through the medium if they move relative to each other. but my point continues to stand as stated. – robert bristow-johnson Nov 27 '15 at 06:25
  • @robertbristow-johnson What point is it that stands? I don't see a point. I pointed out that when a law says a wave travels relative to a medium this doesn't give you special relativity. Einstein is being rather explicit about rejecting a medium and then taking a frame independent speed and seeing how that changes relativity. Hence he sees what relativity is like when you add a frame invariant speed. And that's exactly what SR is. – Timaeus Nov 27 '15 at 06:37
  • @robertbristow-johnson Are you now claiming sound happens in a vacuum? Some people thought Maxwell was like sound in that it merely described waves relative to the medium so the laws and the constants $\epsilon_0$ and $\mu_0$ only show up in the frame of the medium in the same way equations for sound waves hold in their form only in the frame of the medium. Think of SR as telling you to use the modern interpretation of Maxwell rather than the older one. Since the equations allowed both, you have to pick one. Modern experiments favor the one SR tells us to use but SR had to tell us to use it. – Timaeus Nov 27 '15 at 07:03
  • of course not, @Timaeus. whoops, i see i made an error that one would say is the same as in the Wicked Bible. i forgot the word "not". i will repost it, since i cannot edit that comment. – robert bristow-johnson Nov 28 '15 at 05:10
  • @Timaeus i'm gonna bring it over to the other question that i just posted a half hour ago. again the point is the 1st postulate says both observers (moving relative to each other) experience the same laws of physics. that includes Maxwell's Eqs. that means they have the same $\epsilon_0$ and $\mu_0$. that means they have the same $c$ in a vacuum. bringing up the propagation of sound (which does not happen in a vacuum) does not speak to this. – robert bristow-johnson Nov 28 '15 at 05:11
  • @robertbristow-johnson There is no perfect vacuum. And never will be. And Maxwell is wrong anyway. If you want to replace the second principle with the statement that light travels through a vacuum, that there is a vacuum, and that Maxwell is a physical law that holds in every frame, then you can get the same predictions. Except now any problem with Maxwell (such as photon photon scattering from quantum effects or an exceedingly small but nonzero mass for the photon) destroys SR. Whereas SR actually holds ($c$ could be for any invariant speed) even though QED, for instance, supersedes Maxwell. – Timaeus Nov 28 '15 at 17:28
  • @Timaeus, I agree with you. Could you help me out with https://physics.stackexchange.com/q/754187/288306. This question is touching on exactly this subject. It has been closed while I think it's perfectly legitimate question. – Real Pattern Mar 10 '23 at 08:17
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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".

Community
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Selene Routley
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  • 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
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    @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 ... – Selene Routley Jan 22 '15 at 23:17
  • @ArturoDonJuan ... postulate still applies, if one is "freefalling", which is the GR equivalent of inertial frame. – Selene Routley Jan 22 '15 at 23:17
  • Nice answer @SeleneRoutley. However, you seem to suggest that special relativity is galilean relativity without assumption 2. I don't think that is correct. Strictly speaking, with only assumptions 1 and 3 the allowed transformations between inertial frames are 3 fold: euclidean, galilean and lorentzian. Special relativity is one "specific flavor" of relativity, one with a finite invariant speed. Both classical mechanics and special relativity share a version of the first assumption (RP), but they have a different second postulate, used to uniquely identify their transformation laws. – Real Pattern Mar 10 '23 at 08:01
  • Also, I think we need to be careful with the MM experiment. Taken in isolation, it does not conclude that the speed of light is finite and independent of the source. – Real Pattern Mar 10 '23 at 08:20
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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.

John Rennie
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  • 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 06:58
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    @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 07:17
  • 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 05:06
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    @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
  • Nice answer @JohnRennie. I have a question very much related to your point. That might be something for you: https://physics.stackexchange.com/q/754187/288306. I get the impression that my question is wildly misinterpreted. – Real Pattern Mar 08 '23 at 23:23
  • @JohnRennie, is the second postulate not simply stating that the invariant speed is finite? I'm trying to understand why an infinite light speed violates the second postulate. – Real Pattern Mar 11 '23 at 12:33
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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?:

Galilena vs. Special Relativity

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.

Markoul11
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  • I think you conflate galilean relativity and the relativity principle. Both SR and Newtonian mechanics are consistent with RP, but SR and galilean relativity are incompatible. – Real Pattern Mar 10 '23 at 08:13
  • @RealPattern These may be just general arguments which I am not even sure are stated clearly in the bibliography and general consensus. Please only specific arguments that invalidate my given answer herein. – Markoul11 Mar 10 '23 at 09:50
  • @RealPattern You surely don't want to argue that if light at $c$ from a stationary light source within the lab frame with the observer, has not reached yet an observer moving towards the light from opposite direction at $υ$ non-relativistic speed (i.e. Lorentz factor ignored) given the initial distance between them as χ will reach the observer at time $χ/c$ instead of $χ/(c+υ)$ ? – Markoul11 Mar 10 '23 at 10:09
  • I was referring to the first paragraph of your answer: "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." SR is not constistent with Galilean relativity. The rest may or may not be correct or insightful. – Real Pattern Mar 10 '23 at 10:32
  • @RealPattern "SR is not constistent with Galilean relativity." I highly debate this. If you swim in a wavy sea you get the exact same SR behaviour. The rate where you receive the water waves Doppler shifts according to your relative motion but the propagation speed of the waves relative to you does not change since you are engulfed by the waves. – Markoul11 Mar 10 '23 at 11:37
  • "I highly debate this", I don't. In Einsteinian relativity the invariant speed is finite, in Galilean relativity it's not. These are incompatible assertions. – Real Pattern Mar 10 '23 at 13:28
  • Is the speed of the propagation of the waves relative to you the same no matter in what direction you swim? You couldn't swim faster that the waves either? – Real Pattern Mar 10 '23 at 13:42
  • The speed is fixed, the frequecny and therefore also wavelength (or else wave distance relative to the observer) only changes of the water waves crushing into the observer similar to light $c=λf$. – Markoul11 Mar 10 '23 at 20:13
  • Let's stop here. – Real Pattern Mar 11 '23 at 12:28