Are Lorentz transforms unique?

[Naty1]

Apparently Poincare suggested and Einstein adopted/proposed that all physical laws should remain unchanged in form after a Lorentz transformation.

Do we know these are the only transforms that work or does the possibility exist that other transforms might also??

For example, if I ask the question "What transforms retain the form of Maxwell's equations in different frames?" Lorentz transforms is one answer; but is it the only one??

If any particular transform worked in one physical law and not another, would that hint at anything...is that even possible??

 

[Fredrik]

If you assume that the set of functions that represent a coordinate change from one inertial frame (an undefined concept at this point) to another, has the structure of a group (with composition of functions as the group multiplication), and that each such function takes straight lines to straight lines (an inertial observer should describe another inertial observer as moving with constant velocity), we're already down to just two possibilities: The Galilei group and the Poincaré group (and their subgroups of course).

As you know, the Galilei group isn't consistent with an invariant speed of light, so we have to drop that too.

Oh yeah, you obviously also have to assume that we're talking about functions that take $\mathbb R^4$ to itself.

 

[Naty1]

Fredrik: thanks for the reply.

we're already down to just two possibilities: The Galilei group and the Poincaré group (and their subgroups of course).

It's been waaaaay too long since I studied groups...I can accept that group theory applies to coordinate changes/ transforms...but how do we know the combination applies to the physical world...

because "so far observation matches theory"?

Maybe that's as far as my question takes goes. Or maybe I should ask if the Lorentz transforms can be derived from group theory...I assume not; so how do we know another suitable transform is not hidden in there?

 

[Fredrik]

The assumption that they form a group is not a strong assumption. Essentially it's saying this: Suppose that S, S' and S'' are three physical observers, that x is the coordinate change from S to S' and that y the coordinate change from S' to S''. Then $x^{-1}$ is the coordinate change from S' to S and $y\circ x$ is the coordinate change from S to S''. This assumption seems even more natural to me than the one about straight lines.

..but how do we know the combination applies to the physical world...

because "so far observation matches theory"?

Not sure if I understand the question. You kind of answered it yourself, I think. I only have these two things to add:

1. We never know if a theory really describes the world. The best we can do is to find out how accurate the theory's predictions are.

2. The type of argument we're discussing here is supposed to be used at the stage where we're trying to find a mathematical model of space and time that might be useful in a new theory. (Of course SR isn't really new anymore, but you get the idea). There really is no need to know that something useful will come out of it, when we're doing this. It would actually be absurd to require that, since we don't even have a theory yet.

so how do we know another suitable transform is not hidden in there?

Isn't that the question I answered in #2? Or are you asking for the details of the derivation? I did most of it here (for the 1+1-dimensional case), but I don't think I figured out the minimal set of assumptions that we need. This paper has more on that.

 

[samalkhaiat]

Do we know these are the only transforms that work or does the possibility exist that other transforms might also??

Einstein's equations are invariant under ARBITRARY coordinates transformations; for example, they are invariant under non-linear coordinates transformations known by the name "conformal" coordinates transformations.

For example, if I ask the question "What transforms retain the form of Maxwell's equations in different frames?" Lorentz transforms is one answer; but is it the only one??

Maxwell's equations are also invariant under the conformal coordinates transformations.

If any particular transform worked in one physical law and not another, would that hint at anything...

Yes, it would LIMIT the DOMAIN of INVARIANCE to a certain CLASS of theories.

is that even possible??

Yes it is possible; if we introduce a mass term into Maxwell's equations, the resulting theory will no longer be conformally-invariant. This mass term will also spoil the gauge invariance of Maxwell's theory.

regards

sam

 

[Naty1]

Fredrik: yes, I asked essentially the same question again...you caught me!...While the paper you referenced was a bit exotic for me mathematically, it did provide some perspective:

The theoretical results currently available fall into two categories:
rigorous results on approximate models and approximate
results in realistic models. ...To me this seems to be the generic situation in theoretical physics. In that respect, Minkowski space is certainly an approximate model, but to a very
good approximation indeed: as global model of spacetime if gravity plays
no dynamical role, and as local model of spacetime in far more general situ-
ations.

This seems a natural extension of the idea I was trying to explore...

I also checked Einstein's own book RELATIVITY (The special and the general theory) and found this in Appendix V :

The question now arises...What kind of nonlinear transformations are to be permitted, or how is the Lorentz Transformation to be generalised.

Sam,
thanks for your reply...Your comments relate to others that appear in Einstein's appendix V and putting the pieces together makes me realize I am mathematically limited for this issue...

What's the significance (anything physical?) of invariance under conformal transformations? I know too little of conformal geometry to draw any conclusions.

And rereading my own original post I realized that my last question:

If any particular transform worked in one physical law and not another, would that hint at anything?

had at least one obvious example I should have remembered : Galilean transform works in Newton's Theory, not in Maxwell's. According to Feynman's lectures, (SIX Not So Easy Pieces) when that was discovered it initially cast doubt on Maxwell's equations (!) and appears to have led to the widespread adoption of Lorentz Tranforms...and final confirmation of Mawell's 20 year old equations!

 

[Fredrik]

Maxwell's equations are also invariant under the conformal coordinates transformations.

Here's my thoughts on this. Let me know if you agree or disagree. (I know almost nothing about conformal transformations. I still haven't read that thread you started).

As you know, an arbitrary Lorentz transformation in 1+1 dimensions can be written as

$$\Lambda=\gamma\begin{pmatrix}1 & -v\\ -v & 1\end{pmatrix}$$

where

$$\gamma=\frac{1}{\sqrt{1-v^2}}$$

What's the corresponding expression for a conformal transformation? Is it

$$\Lambda=k\gamma\begin{pmatrix}1 & -v\\ -v & 1\end{pmatrix}$$

with k an arbitrary constant? Should I perhaps replace $k\gamma$ with an arbitrary function of v?

If this (either of the two possibilities I suggested) is right, the reason we don't consider the conformal group to be a candidate when we go through the argument that's supposed to help us find a mathematical model of space and time that's appropriate for a new (in 1905) theory of space and time, is that it doesn't satisfy the condition $\Lambda(v)^{-1}=\Lambda(-v)$.

 

[DrGreg]

Entirely from memory, I thought the conformal transformations we discussed a month or two ago were of the form

$$t' = \frac{At + Bx + C}{Pt + Qx + R}$$
$$x' = \frac{Dt + Ex + F}{Pt + Qx + R}$$​

which suffer from the defect that they're not actually defined when the denominator vanishes. I think when you do the maths with these, you can represent such a transform by the matrix

$$\begin{pmatrix}A & B & C\\ D & E & F\\P & Q & R\end{pmatrix}$$​

and a composition of transforms corresponds to a multiplication of the matrices.

 

[Fredrik]

Yeah, I might be way off. Somehow I got the idea a long time ago that a conformal transformation is kind of like a Lorentz transformation except that it also changes the scale involved, and I never tried to find out if that's even true. To be more specific, I think I heard the word "conformal" used in the following context:

Define $g(x,y)=x^T\eta y$. A Lorentz transformation leaves this quantity invariant, $g(\Lambda x,\Lambda y)=g(x,y)$, but what do you call the class of transformations that instead satisfies $g(\Lambda x,\Lambda y)=K^2g(x,y)$, were K is some real and positive constant?

 

[Naty1]

Regarding post 7,8,9: you might find some common source material for discussion at
http://en.wikipedia.org/wiki/Conformal_geometry#Minkowski_space

In Feynman's SIX NOT SO EASY PIECES he makes this comment in his lecture series (Page 77):
It has turned out
[QUOTE..to be of enormous utility in our study of other physical laws..to look at the symmetry of the laws...or more specifically to look for the ways in which the laws can be transformed and leave their form the same. So this idea of studying the patterns or operations under which the fundamental laws are not changed has proved to be a very useful one.[/QUOTE]

So I should have made the connection, I think, that the mathematical properties posted by others above actually relate to symmetries...How about Noether's Theorem's? (symmetry and conservation)

 

[DrGreg]

Yeah, I might be way off. Somehow I got the idea a long time ago that a conformal transformation is kind of like a Lorentz transformation except that it also changes the scale involved, and I never tried to find out if that's even true. To be more specific, I think I heard the word "conformal" used in the following context:

Define $g(x,y)=x^T\eta y$. A Lorentz transformation leaves this quantity invariant, $g(\Lambda x,\Lambda y)=g(x,y)$, but what do you call the class of transformations that instead satisfies $g(\Lambda x,\Lambda y)=K^2g(x,y)$, were K is some real and positive constant?

Actually, you may well be right over what "conformal" means. It's not my area of expertise but what you said sounds familiar.

But I suspect what samalkhaiat was referring to in post #5 was what I described in post #8. I could be wrong.

 

[A-wal]

How were the Lorentz transformations derived? I don't mean in a mathematical sense (I know it's a mathmatical construct so that doesn't really make sense. I just mean was it trail and error or a prediction of somthing else, or just maths), I just mean does it represent something more fundamental? Would it work with any self consistent transformation just as c can have any value? Would a different value for c change the transformations? No right?

 

[maverick280857]

How were the Lorentz transformations derived? I don't mean in a mathematical sense (I know it's a mathmatical construct so that doesn't really make sense. I just mean was it trail and error or a prediction of somthing else, or just maths), I just mean does it represent something more fundamental? Would it work with any self consistent transformation just as c can have any value? Would a different value for c change the transformations? No right?

From some fundamental constraints imposed by the requirement that laws of physics remain the same in all inertial frames (read: the transformations were historically derived by imposing the requirement that a spherical wavefront in one frame remain spherical in another inertial frame moving wrt the first one), and that the speed of light be the same in all inertial frames. It should be noted that Lorentz, who originally derived the transformation laws (and so did Poincare), subscribed to the ether theory and NOT to Einstein's radical viewpoint (which would only come much later). So while he used the same equations, he ascribed the effects to be dynamical and not kinematic.

The value of c was an experimental input, and yes, the form of the transformations is not dependent on the numerical value of c. To that extent, the transformation is "arbitrary". But then that is parametric arbitrariness, not form arbitrariness. I believe Naty was asking if some other functional form of the transformation law exists which is NOT the Lorentz transformation.

My answer to it is that Lorentz transformations as "derived" above are the high velocity versions of Galilean transformations which are the simplest linear transformations connecting two inertial frames of reference while enforcing homogeneity and isotropy of space and time. If you can find an alternative to Galilean transformations (which must be linear by the way, or else homogeneity and isotropy will break down) then you can enforce the form invariance of a spherical wavefront (as was done to get Lorentz transformations from Galilean transformations) and see what terms you need to add or multiply to get consistent results in all inertial frames. But since you will always start with something that is Galilean, the answer is no. Lorentz transformations are unique.

 

[A-wal]

Thankyou.

Okay, they took a light bubble from another frame, which would not be spherical due to the difference in velocities, then made it spherical again because of the constant speed of light and you have the Lorentz transformations.

I have another way. I'll post it when I've seen if I can take it any further. I'm trying to get the value of c from it.