# Special relativity vs Lorentz invariance

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## Summary:

Is there more about special relativity than claiming that the laws of physics are Lorentz invariant/covariant?
The Lorentz covariance of Maxwell equations was known before Einstein formulated special relativity. So what exactly special relativity brought new with respect to mere Lorentz covariance? Is special relativity just an interpretation of Lorentz invariance, in a sense in which Copenhagen interpretation interprets quantum mechanics? Or if there is something more about special relativity, then does it make measurable predictions that cannot be made by mere Lorentz covariance?

(To not attract the crackpots, I am giving my best to not mention the forbidden dirty "ae" word. )

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Dale
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Summary: Is there more about special relativity than claiming that the laws of physics are Lorentz invariant/covariant?

The Lorentz covariance of Maxwell equations was known before Einstein formulated special relativity. So what exactly special relativity brought new with respect to mere Lorentz covariance?
Personally, I don’t think that there is any difference between Lorentz covariant and relativity. However, I think that your characterization is a little off. First, the covariance of Maxwell’s equations was known, but many believed that Maxwell’s equations were approximations rather than laws of EM. Also, nobody believed that the laws of mechanics were Lorentz covariance.

Einstein’s work was not something more than Lorentz covariance, but it was a dramatic elevation of Lorentz covariance to a central principle of physics.

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Personally, I don’t think that there is any difference between Lorentz covariant and relativity.
So when the sound wave equation satisfies a Lorentz covariant wave equation (with "Lorentz" tranformations defined with the speed of sound instead of speed of light), would you say that this wave equation is "relativistic"? If not, why not?

Dale
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So when the sound wave equation satisfies a Lorentz covariant wave equation (with "Lorentz" tranformations defined with the speed of sound instead of speed of light), would you say that this wave equation is "relativistic"? If not, why not?
The equations of sound are not Lorentz covariant with the speed of sound in place of the speed of light.

martinbn
What if you want to consider non-inertial reference frames?

What is your asnswer to the question: Is Euclidean geometry the same as the study of properties preserved under transformations that preserve distance? What if I want to solve problems involving projections or homoteties, am I no longer doing Euclidean geometry?

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The equations of sound are not Lorentz covariant with the speed of sound in place of the speed of light.
$$\frac{1}{c_s^2}\frac{\partial^2\phi}{\partial t^2}-\nabla^2\phi=0$$
Are you saying that it is not Lorentz covariant or that it is not a wave equation for sound?

PeroK
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So what exactly special relativity brought new with respect to mere Lorentz covariance?
Perhaps what SR did was to take things away. Prior to Einstein, additional assumptions were needed to explain the phenomena where the reference frame was not the preferred one.

In your sound analogy, as far as I can understand it, if you transform the coordinates, using the equivalent of a LT with the speed of sound, then you get the same speed of sound for a frame moving wrt to the original. But, that is not what is observed. The observed speed of sound in the second frame is ##c_s + v## and not ##c_s##.

When you apply a LT in the case of light and vacuum, you get the same observed speed of light in the second frame. Prior to Einstein, additional assumptions about physical changes in all but the preferred frame were needed. Einstein, essentially, took these away and said that the invariance of the speed of light was sufficient to formulate a consistent theory.

Bruce Liu and martinbn
martinbn
The equations of sound are not Lorentz covariant with the speed of sound in place of the speed of light.
$$\frac{1}{c_s^2}\frac{\partial^2\phi}{\partial t^2}-\nabla^2\phi=0$$
Are you saying that it is not Lorentz covariant or that it is not a wave equation for sound?
I think he means the speed of sound instead of speed of light in the equation will make it not invariant under Lorentz transformation, in which you have the speed of light.

Dale
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1c2s∂2ϕ∂t2−∇2ϕ=01cs2∂2ϕ∂t2−∇2ϕ=0​

Are you saying that it is not Lorentz covariant or that it is not a wave equation for sound?
It is not the equation for sound. Specifically, that equation does not describe sound in any frame where the medium is moving.

The corrections to that equation which are needed to describe sound when the medium is moving are roughly the corrections that Einstein’s contemporaries believed were needed for Maxwell’s equations.

Gold Member
I think he means the speed of sound instead of speed of light in the equation will make it not invariant under Lorentz transformation, in which you have the speed of light.
Perhaps. To avoid any misunderstanding, the wave equation for sound is covariant under the Lorentz boosts such as
$$x'=\frac{x-vt}{\sqrt{1-v^2/c_s^2}}$$
$$t'=\frac{t-vx/c_s^2}{\sqrt{1-v^2/c_s^2}}$$
where ##c_s## is the speed of sound. So does this Lorentz covariance make sound relativistic? If not, why not?

PeroK
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Perhaps. To avoid any misunderstanding, the wave equation for sound is covariant under the Lorentz boosts such as
$$x'=\frac{x-vt}{\sqrt{1-v^2/c_s^2}}$$
$$t'=\frac{t-vx/c_s^2}{\sqrt{1-v^2/c_s^2}}$$
where ##c_s## is the speed of sound. So does this Lorentz covariance make sound relativistic? If not, why not?
As above, because the observed speed of sound does not agree with the predicted speed of sound - assuming that your Lorentz boost is supposed to represent a physical transformation between IRF's.

Dale
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the wave equation for sound is covariant under the Lorentz boosts
No. The wave equation is covariant under boosts, but when boosted it is not the equation for sound. The wave equation only describes sound in the frame where the medium is at rest.

Gold Member
Specifically, that equation does not describe sound in any frame where the medium is moving.
Well, it depends on how do you describe the moving frame. If you describe the moving frame by the Lorentz transformations in post #10, then it does describe the sound in the moving frame. And if now you tell me that this is not how the moving frame should be described, then my question is: How can one know that, unless special relativity is something more than mere Lorentz covariance?

Dale
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Well, it depends on how do you describe the moving frame.
With standard rulers and clocks. Otherwise the laws of mechanics don’t work.

PeroK
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Well, it depends on how do you describe the moving frame. If you describe the moving frame by the Lorentz transformations in post #10, then it does describe the sound in the moving frame. And if now you tell me that this is not how the moving frame should be described, then my question is: How can one know that, unless special relativity is something more than mere Lorentz covariance?
The critical thing is to identify the LT with the coordinate transformation between inertial reference frames. That, ultimately, has to be decided by experiment. (Although, in fact, from considerations of isotropy and homogeneity of spacetime, the LT and the Galilean Transformation are the only possible candidates.)

Gold Member
The wave equation is covariant under boosts, but when boosted it is not the equation for sound.
That's a very strange answer. Are you saying that it is an equation for something else? For what? Or are you saying that this transformed equation does not have any physical interpretation? I claim that it has. For instance, if you define simultaneity through Einstein synchronization but with using sound (rather than light) for the purpose of communication, then ##t'## given by the boost above can be interpreted as the physical time.

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PeroK
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That's a very strange answer. Are you saying that is in equation for something else? For what? Or are you saying that this transformed equation does not have any physical interpretation? I claim that it has. For instance, if you define simultaneity through Einstein synchronization but with using sound (rather than light) for the purpose of communication, then ##t'## given by the boost above can be interpreted as the physical time.
And then we would all have experienced this time dilation and the impossiblity of accelerating an object up to the speed of sound!

Dale
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With standard rulers and clocks. Otherwise the laws of mechanics don’t work.
Fine, but why do I must use standard rulers and clocks? Why am I not allowed to use rulers and clocks based on sound? Is it a principle derived from Lorentz covariance or a separate principle?

PeroK
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Fine, but why do I must use standard rulers and clocks? Why am I not allowed to use rulers and clocks based on sound? Is it a principle derived from Lorentz covariance or a separate principle?
Because then you would need different clocks. Or, more to the point, no clock would do the job. E.g. a quartz crystal clock in two reference frames moving at almost the speed of sound relative to each other measure the same elapsed time for an experiment. Your theory would require extreme time dilation in this case.

You could only achieve this by demanding that clocks are mechanically altered in each reference frame to keep your "sound time".

The point about SR is that, experimentally, the same clocks (without mechanical alteration!) moving wrt each other do indeed show the effects of time dilation and differential ageing in agreement with the theory.

In your "sound" universe, effectively, you would lose the second postulate of SR that the laws of physics are the same in all IRF's.

You would have a preferred frame where the known laws of physics hold and in every other (subsonic) frame you would have the invariance of the speed of sound but nothing else!

And you'd have to figure something out for supersonic frames.

PeterDonis
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And then we would all have experienced this time dilation and the impossiblity of accelerating an object up to the speed of sound!
Well, if the objects (which are not made of sound) also satisfied sound-Lorentz covariant equations, then this indeed would be true.

Dale
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Fine, but why do I must use standard rulers and clocks?
I already answered that. Because otherwise the laws of mechanics wouldn’t work.

Why am I not allowed to use rulers and clocks based on sound?
Such clocks could fairly easily be shown to not keep proper time in frames where the medium is moving.

Frankly, I don’t see your point, but it seems pretty obvious that insofar as the wave equation is Lorentz covariant it does not describe sound. Please do not push personal speculation here. This is not the QM forum.

Gold Member
Because then you would need different clocks. Or, more to the point, no clock would do the job. E.g. a quartz crystal clock in two reference frames moving at almost the speed of sound relative to each other measure the same elapsed time for an experiment. Your theory would require extreme time dilation in this case.

You could only achieve this by demanding that clocks are mechanically altered in each reference frame to keep your "sound time".

The point about SR is that, experimentally, the same clocks (without mechanical alteration!) moving wrt each other do indeed show the effects of time dilation and differential ageing in agreement with the theory.

In your "sound" universe, effectively, you would lose the second postulate of SR that the laws of physics are the same in all IRF's.

You would have a preferred frame where the known laws of physics hold and in every other (subsonic) frame you would have the invariance of the speed of sound but nothing else!

And you'd have to figure something out for supersonic frames.
All this can boiled down to the fact that the equations for matter (of which real clocks are made) are not sound-Lorentz covariant. But hypothetically, if we had some exotic state of matter that was sound-Lorentz covariant, then it would behave according to the principles of SR, except with the speed of sound instead of the speed of light. Do you agree?

PeroK
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All this can boiled down to the fact that the equations for matter (of which real clocks are made) are not sound-Lorentz covariant. But hypothetically, if had some exotic state of matter that was sound-Lorentz covariant, then it would behave according to the principles of SR, except with the speed of sound instead of the speed of light. Do you agree?
I don't know. Sound can't travel through a vacuum.

I'd say the alternative is to have a universe filled with sound-permitting aether and Newtonian physics.

Gold Member
Sound can't travel through a vacuum.
It depends on what one means by "vacuum". In condensed matter physics (which is a relevant branch of physics when we talk about sound), by vacuum one often means a crystal or liquid without quasiparticles. Sound can travel through this.