# Is 2nd postulate of SR necessary?

#### meopemuk

I'm not sure I understand what you're saying here. Are the "relativistic" "dynamical" theories you're talking about formulated in the framework of special relativity or not? (See my previous post for a clarification of what I mean by the "framework"). If they are, then this is all very trivial. One of the things that such a theory of inter-atomic forces would tell us is that the length of the moving rod can be calculated by calculating its rest length and doing a Lorentz transformation.
I am not sure that things are so simple. Let me take as an example the hydrogen molecule H2. Suppose that in the reference frame at rest I have a Hamiltonian H which describes interaction between 2 protons and 2 electrons. I can find the ground state of this Hamiltonian and calculate the equilibrium distance R between two protons. If my theory is fully relativistic, I should be able also to write the electron-proton Hamiltonian H', which is valid in the moving reference frame. (Note that H and H' must be different.) I can also find the ground state for H', and thus determine the equilibrium distance R' in the hydrogen molecule from the point of view of the moving observer. Now, your claim is that R and R' will be related exactly by the Einstein's length contraction formula. I am not convinced.

#### JesseM

I am not sure that things are so simple. Let me take as an example the hydrogen molecule H2. Suppose that in the reference frame at rest I have a Hamiltonian H which describes interaction between 2 protons and 2 electrons. I can find the ground state of this Hamiltonian and calculate the equilibrium distance R between two protons. If my theory is fully relativistic, I should be able also to write the electron-proton Hamiltonian H', which is valid in the moving reference frame. (Note that H and H' must be different.) I can also find the ground state for H', and thus determine the equilibrium distance R' in the hydrogen molecule from the point of view of the moving observer. Now, your claim is that R and R' will be related exactly by the Einstein's length contraction formula. I am not convinced.
Quantum field theories (the relativistic version of QM) are all Lorentz-invariant, which should mathematically guarantee that the predictions of different inertial frames are related by the Lorentz transformation.

#### meopemuk

Quantum field theories (the relativistic version of QM) are all Lorentz-invariant, which should mathematically guarantee that the predictions of different inertial frames are related by the Lorentz transformation.
If there is such a guarantee, then it should be not difficult to verify the transformations by direct calculations, as I described. Or perhaps, there should be a rigorous mathematical proof that direct calculations of equilibrium distances in different frames would yield exactly the length contraction formula. I haven't heard about such a proof.

I agree with you that Lorentz transformations are part of QFT. So, one possible approach is, as you said, just simply apply the length contraction formula. However, relativistic quantum theory gives us another approach: to find the ground state of the Hamiltonian H' in the moving frame. If the theory is self-consistent, then both approaches should yield the same result. If we, indeed, find that both results agree, then everything is great. The theory passed an important consistency check. But, what if the two results disagree?

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#### JesseM

If there is such a guarantee, then it should be not difficult to verify the transformations by direct calculations, as I described. Or perhaps, there should be a rigorous mathematical proof that direct calculations of equilibrium distances in different frames would yield exactly the length contraction formula. I haven't heard about such a proof.

I agree with you that Lorentz transformations are part of QFT. So, one possible approach is, as you said, just simply apply the length contraction formula. However, relativistic quantum theory gives us another approach: to find the ground state of the Hamiltonian H' in the moving frame. If the theory is self-consistent, then both approaches should yield the same result. If we, indeed, find that both results agree, then everything is great. The theory passed an important consistency check. But, what if the two results disagree?
I don't know how to do quantum field theory calculations myself, but just based on the fact that the theory is Lorentz-symmetric, it's mathematically impossible that they could disagree. If you know the dynamical laws in one frame, all you have to do is perform a Lorentz transformation on these laws to find equations that give physically identical predictions in a different frame. If we don't assume Lorentz-symmetry, in general the equations might be different in different frames, but this is just a way of relabeling events so it can't give physically different predictions. And the very definition of Lorentz-symmetry implies that if you perform a Lorentz transformation on equations which are Lorentz-symmetric, the equations will be unchanged in the new frame, so you can use the same equations in different frames and get physically identical predictions.

#### clem

I just came across some references that all dispense with the 2nd postulate:

[1] R. Weinstock: New approach to special relativity, Am. J. Phys. 33 (1965) 540-545.
[2] V. Mitavalsky: Special relativity without the postulate of constancy of light, Am. J. Phys. 34 (1966) 825.
[3] A. R. Lee, T. M. Kalotas: Lorentz transformations from the first postulate, Am. J. Phys. 43 (1975) 434-437.
[4] J.-M. Levy-Leblond: One more derivation of the Lorentz transformation, Am. J. Phys.
44 (1976) 271-277.
[5] A. M. Srivastava: Invariant speed in special relativity, Am. J. Phys. 49 (1981) 504-505.
[6] N. D. Mermin: Relativity without light, Am. J. Phys. 52 (1984) 119-124.
[7] H. M. Schwartz: Deduction of the general Lorentz transformations from a set of necessary assumptions, Am. J. Phys. 52 (1984) 346-350.
[8] H. M. Schwartz: A simple new approach to the deduction of the Lorentz transformations, Am. J. Phys. 53 (1985) 1007-1008.
[9] S. Singh: Lorentz transformations in Mermin’s relativity without light, Am. J. Phys. 54
(1986) 183-184.
[10] A. Sen: How Galileo could have derived the special theory of relativity, Am. J. Phys. 62 (1994) 157-162.
[11] M. J. Feigenbaum, N. D. Mermin: E = mc2, Am. J. Phys. 56 (1988

#### meopemuk

clem,

thanks for the references. I didn't read all of them, but I am familiar with some of them. They suffer from a common unjustified assumption, which, I think, is important.

For example, take Levy-Leblond's paper. In his eq. (1) he assumes that event's space-time coordinates (x', t') in the moving frame depend ONLY on its coordinates (x, t) in the frame at rest and on parameters {a}, which determine the relative velocity of the two frames. He doesn't entertain the possibility that x' and t' may also depend on the physical nature of the event.

Let me explain what I mean on the following example. Suppose that we are interested in the event of collision of two particles. This event is rather well localized in both space and time for both observers. So, coordinates (x, t) and (x', t') have clear meaning, and it makes sense to ask which transformation connects these two pairs of numbers. Consider two possibilities. In one case the particles are non-interacting. In the other case the particles interact with each other (for example, these are two opposite charges). Is there a good reason to assume a priori that the transformation laws (x, t) -> (x', t') will be exactly the same in these two cases? I don't think so.

I would be much happier if Levy-Leblond (and others) explicitly formulated their tacit postulate, which sounds roughly like this: "Transformations of space-time coordinates of events do not depend (i) on the physical nature of events; (ii) on the composition of the system in which the events occur; (iii) on the nature and strength of interactions in the system." This assumption lies in the foundation of all special relativity, but it is never (or rarely) mentioned.

#### Fredrik

Staff Emeritus
Gold Member
My opinion is that people should care a lot less on Einstein's postulates (and possible alternatives to them) and a lot more about the details of how to interpret the mathematics of Minkwski space as predictions about the results of experiments. But I probably say that too often in this forum.

I decided to look up the original version of the postulates. I found a translation here (page 140) and the original german http://www.pro-physik.de/Phy/pdfs/ger_890_921.pdf [Broken]. They are a bit different from what I said in #23. This is a better summary of what they say:

1. The laws of electrodynamics and optics are valid in any coordinate system where the equations of mechanics are valid.

2. In empty space, light always propagates at a definite velocity V which is independent of the motion of the light source.

These aren't exactly well-defined mathematical axioms either. Does the specification of what coordinate systems he's talking about in postulate 1 even make sense? (That's not a rhetorical question. I don't know what the answer is, but then I'm always a bit dumber the last hour before I go to sleep). Also, note that postulate 2 doesn't even mention coordinates.

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#### clem

Fred: I agree with your first paragraph, so I probably should stop right here but....
Neither postulate, as stated in that translation make much sense today.
For (1), the "equations of mechanics" as Einstein knew them at the time were mostly not valid.
For (2), isn't that completely compatible with an aether? That postulate holds for sound in air.
If anyone does want one or two basic postulates for SR, it can't be either one of those.

#### robphy

Homework Helper
Gold Member
Be fair. The only filled 15 pages with words that say very little. The rest are references. And besides, they are philosophers. It's their job to fill pages with words that say very little.
In defense of some "philosophers [of science]",
there are some more-technical ones...
like Malament ( http://www.lps.uci.edu/home/fac-staff/faculty/malament/ )
who is interested in foundational aspects that most physicists gloss over in their treatment of the subject. (Foundational aspects are important, of course, in any search for a more fundamental theory.)

a review article "Classical General Relativity" http://arxiv.org/abs/gr-qc/0506065

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#### DrGreg

Gold Member
I decided to look up the original version of the postulates. I found a translation here (page 140) and the original german http://www.pro-physik.de/Phy/pdfs/ger_890_921.pdf [Broken]. They are a bit different from what I said in #23. This is a better summary of what they say:

1. The laws of electrodynamics and optics are valid in any coordinate system where the equations of mechanics are valid.

2. In empty space, light always propagates at a definite velocity V which is independent of the motion of the light source.

These aren't exactly well-defined mathematical axioms either. Does the specification of what coordinate systems he's talking about in postulate 1 even make sense? (That's not a rhetorical question. I don't know what the answer is, but then I'm always a bit dumber the last hour before I go to sleep). Also, note that postulate 2 doesn't even mention coordinates.
I would first paraphrase (1) as

1a. An inertial coordinate system is defined as one in which the equations of mechanics are valid.
1b. The laws of electrodynamics and optics are valid in any inertial coordinate system.

Then I would simplify (1a) to say "Newton's first law" instead of "the equations of mechanics", that is:

1a. An inertial coordinate system is defined as one in which every free-falling particle has constant coordinate velocity.

I would then generalise (1b) as

1b. Every law of physics takes the same form in every inertial coordinate system.

(In fact, that isn't technically quite good enough, because it makes no distinction between orthogonal and skew coordinates, so a bit more is needed to resolve that ambiguity. There is also an implicit assumption that inertial coordinate systems do actually exist, otherwise (1b) would be trivially true on an empty set of coordinate systems.)

Einstein's first postulate essentially extends the principle of relativity from Newtonian mechanics (where it was already known to be true -- "Galilean relativity") to all branches of physics.

Although (2) doesn't explicitly mention coordinates, velocity is implicitly defined in terms of coordinates. It can be interpreted as valid in any single frame where V might vary from one frame to another. Subsequent application of (1) would imply all frames have the same V.

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#### Phrak

To those who have been arguing the point, I fail to see how the exclusion of postulate 2, specifying a finite invariant velocity, would obtain anything but Galilean relativity.

#### Fredrik

Staff Emeritus
Gold Member
In fact, that isn't technically quite good enough, because it makes no distinction between orthogonal and skew coordinates, so a bit more is needed to resolve that ambiguity.
The funny thing is that what we need to resolve that ambiguity is something like..."a coordinate transformation from one inertial frame to another maps the light cone at the origin onto itself", but that's a statement that sounds a lot like the second postulate. It just isn't possible to define "inertial frame" in a way that's appropriate for SR without including (some version of) both of the postulates in the definition.

Good post by the way. You noticed a couple of things that I hadn't.

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#### Fredrik

Staff Emeritus
Gold Member
To those who have been arguing the point, I fail to see how the exclusion of postulate 2, specifying a finite invariant velocity, would obtain anything but Galilean relativity.
The first postulate mentions the equations of electrodynamics explicitly, and Maxwell's equations predict that the speed of an electromagnetic wave is c in all inertial frames. So if we completely ignore that the first postulate actually fails to define what an inertial frame is (which actually seems to be what everyone is doing), then the second is implied by the first.

If we choose to interpret the first postulate as if it doesn't actually include Maxwell's equations (i.e. if we interpret "electrodynamics" as an unspecified theory of electricity and magnetism), then we still don't get Galilean relativity. To get Galilean relativity, we must at the very least postulate that the invariant speed is infinite.

#### xantox

Gold Member
To those who have been arguing the point, I fail to see how the exclusion of postulate 2, specifying a finite invariant velocity, would obtain anything but Galilean relativity.
One-postulate derivations such as Levy-Leblond's cited earlier will give a generic transformation law, where depending on the value of some undefined parameter you may obtain both Lorentzian, Galilean and Euclidean transformations.

#### Phrak

The first postulate mentions the equations of electrodynamics explicitly, and Maxwell's equations predict that the speed of an electromagnetic wave is c in all inertial frames. So if we completely ignore that the first postulate actually fails to define what an inertial frame is (which actually seems to be what everyone is doing), then the second is implied by the first.

If we choose to interpret the first postulate as if it doesn't actually include Maxwell's equations (i.e. if we interpret "electrodynamics" as an unspecified theory of electricity and magnetism), then we still don't get Galilean relativity. To get Galilean relativity, we must at the very least postulate that the invariant speed is infinite.
Then this is all historical nit-picking over axioms better stated. We could easily have Galilean relativity, compatible with Maxwell's equations within an aether where c is the velocity in the aether. But it Einstein relativity without a finite nonzero invariant velocity.

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