# Special relativity vs Lorentz invariance

• I
• Demystifier
In summary: No, the Lorentz covariance of the wave equation for sound does not make sound relativistic. However, the wave equation for sound does have the property of being Lorentz covariant under a Lorentz boost.
Demystifier
Gold Member
TL;DR 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. )

Demystifier said:
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.

Dale said:
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?

Demystifier said:
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.

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?

Dale said:
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?

Demystifier said:
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
Dale said:
The equations of sound are not Lorentz covariant with the speed of sound in place of the speed of light.
Demystifier said:
$$\frac{1}{c_s^2}\frac{\partial^2\phi}{\partial t^2}-\nabla^2\phi=0$$
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.

Demystifier said:

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.

martinbn said:
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?

Demystifier said:
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.

Demystifier said:
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.

Dale said:
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?

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

Demystifier said:
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.)

Dale said:
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.

Last edited:
Demystifier said:
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
Dale said:
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?

Demystifier said:
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
PeroK said:
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.

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

Demystifier said:
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.

PeroK said:
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?

Demystifier said:
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.

PeroK said:
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.

Dale said:
Frankly, I don’t see your point
Then let me try to rephrase my point. When Einstein formulated his special relativity, he didn't base his arguments on actually performed experiments (such as Michelson-Morley) with actual clocks and rulers. The only established fact that he used was Lorentz covariance of Maxwell equations. All the rest was his speculation based on some thought experiments. Later actual experiments have shown that Einstein was right, but it was not obvious in the beginning when he formulated his theory. And Lorentz covariance of Maxwell equations alone is fully analogous to the sound-Lorentz covariance of the sound equation. So it seems that Einstein had to assume something that was not already encoded in Lorentz covariance of Maxwell equations. He had to assume something more (which later turned out to be true). So I want to better understand what exactly did he assumed in addition to the Lorentz covariance of Maxwell equations?

I think that his assumption was that matter too (not just the EM field) obeys Lorentz covariant equations. I think that it's enough to derive all the rules of SR. But I would like to see what others think, is it really enough (as I think it is) or is something more needed?

The case of sound serves for analogy, to guide thinking.

I would say, at least the following define what I would consider 'standard SR':

1) Lorentz covariance with c is a universal principle that must apply to all physical laws, and that light traveling at c is true only to the extent Maxwell equations are exact (that is Lorentz covariance is more fundamental, and there could be a Lorentz covariant alternative to Maxwell's equations in which light travels at c-epsilon, which is found to be true without abolishing SR).

2) No information (matter and energy are special cases) of any kind can travel > c (the Lorentz constant) in an inertial frame. This makes the null cone structure of spacetime fundamental, and is definitely a prediction in that it applies to all conceivable mechanisms (now and 'forever' unless dis-proven). This makes the causal structure of spacetime fundamental, allowing SR to be built from this starting point.

(Note that all predictions of these are still compatible with suitably generized LET, as far as I know).

Dale
PAllen said:
I would say, at least the following define what I would consider 'standard SR':

1) Lorentz covariance with c is a universal principle that must apply to all physical laws, and that light traveling at c is true only to the extent Maxwell equations are exact (that is Lorentz covariance is more fundamental, and there could be a Lorentz covariant alternative to Maxwell's equations in which light travels at c-epsilon, which is found to be true without abolishing SR).

2) No information (matter and energy are special cases) of any kind can travel > c (the Lorentz constant) in an inertial frame. This makes the null cone structure of spacetime fundamental, and is definitely a prediction in that it applies to all conceivable mechanisms (now and 'forever' unless dis-proven). This makes the causal structure of spacetime fundamental, allowing SR to be built from this starting point.

(Note that all predictions of these are still compatible with suitably generized LET, as far as I know).
Couldn't one just say that standatd SR says that space-time is Minkowski space?

Dale
Demystifier said:
I think that his assumption was that matter too (not just the EM field) obeys Lorentz covariant equations. I think that it's enough to derive all the rules of SR. But I would like to see what others think, is it really enough (as I think it is) or is something more needed?

I don't think that Einstein specifically assumed anything like this. His 1905 paper is available online, so you can read exactly what he published.

Naively, it was all based on a simple assumption about the invariance of the speed of light and, implicitly, some assumptions about isotropy and homogeneity of space and time.

A more modern approach might ignore EM radiation and from principles of isotropy and homogeneity of space and time alone demonstrate only the two candidates: Newtonian space and time with the Galilean transformation and no invariant speed; and, Minkowski spacetime with the Lorentz Transformation based on a universal constant ##c##, which can be seen to be an invariant speed.

I don't think you need any more than that.

Demystifier said:
Invariance under what?

Between inertial reference frames: "in which the equations of Newtonian mechanics hold good".

Demystifier said:
So I want to better understand what exactly did he assumed in addition to the Lorentz covariance of Maxwell equations?
This sounds like a historical or biographical question rather than a scientific question.

Demystifier said:
I think that his assumption was that matter too (not just the EM field) obeys Lorentz covariant equations.
I think that is clear when he referred to a frame where the equations of mechanics hold good, which is why I specifically mentioned it in post #2. That was one of the two key insights I mentioned that others were missing at the time.

Demystifier said:
He had to assume something more (which later turned out to be true). So I want to better understand what exactly did he assumed in addition to the Lorentz covariance of Maxwell equations?
I don't think that he did. He just had to not be silly in proposing that the then known laws of acoustics were Lorentz covariant. Frankly, your assertion to the contrary is ludicrous.

At this time I am closing this thread. If you find some professional scientific publications on this topic please PM me and we can reopen the thread. I am not interested in permitting a discussion based on personal speculation like claims that the laws of sound are Lorentz covariant.

Last edited:
martinbn and PeterDonis

## 1. What is the difference between special relativity and Lorentz invariance?

Special relativity is a theory proposed by Albert Einstein that describes the behavior of objects in space and time, particularly at high speeds. It is based on the principle of relativity, which states that the laws of physics should be the same for all observers in uniform motion. Lorentz invariance, on the other hand, is a more general concept that refers to the invariance of physical laws under transformations between different frames of reference. While special relativity is one specific theory that is Lorentz invariant, there are other theories that are also Lorentz invariant, such as general relativity.

## 2. How does special relativity explain the concept of time dilation?

Special relativity explains time dilation as a consequence of the speed of light being constant for all observers. This means that as an object approaches the speed of light, time slows down for that object relative to an observer at rest. This phenomenon has been experimentally verified and is a fundamental aspect of special relativity.

## 3. Does Lorentz invariance only apply to objects moving at high speeds?

No, Lorentz invariance applies to all objects and all speeds. It is a fundamental principle of physics that states that the laws of physics should be the same for all observers in different frames of reference. This means that the laws of physics should be the same whether an object is at rest or moving at a constant velocity.

## 4. Can special relativity and Lorentz invariance be reconciled with each other?

Yes, special relativity and Lorentz invariance are compatible with each other. In fact, special relativity is based on the principle of Lorentz invariance, and the two concepts are closely related. Special relativity is a specific theory that is Lorentz invariant, while Lorentz invariance is a more general concept that applies to a wider range of theories and phenomena.

## 5. Are there any experimental tests that support the concept of Lorentz invariance?

Yes, there have been numerous experimental tests that support the concept of Lorentz invariance. One of the most famous is the Michelson-Morley experiment, which showed that the speed of light is the same in all directions, regardless of the motion of the observer. Other experiments, such as the Kennedy-Thorndike experiment and the Ives-Stilwell experiment, have also confirmed the predictions of Lorentz invariance. Additionally, the predictions of special relativity, which is based on Lorentz invariance, have been extensively tested and found to be accurate.

Replies
7
Views
2K
Replies
4
Views
1K
Replies
32
Views
3K
Replies
6
Views
569
Replies
11
Views
2K
Replies
31
Views
8K
Replies
3
Views
2K
Replies
1
Views
870
Replies
22
Views
2K
Replies
7
Views
1K