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There is something that has been bugging me for a long time about the meaning of Lorentz Transformations when looked at in the context of tensor analysis. I will try to be as clear as possible while at the same time remaining faithful to the train of thought that brought me here.

Most, if not all, derivations of Lorentz Transformations start by requiring a set of matrices (linear transformations) which leave the components of the Minkowski metric ##\eta## invariant. These would be a set of matrices ##\Lambda^\alpha_\beta## such that

$$ \Lambda^\alpha_\beta\Lambda^\mu_\nu\eta _{\alpha\mu}=\eta_{\beta\nu}.$$

Such derivations claim that this is a direct consequence of requiring the constancy of the speed of light, i.e., the constancy of the space-time interval.

Nevertheless, if one looks at it from the point of view of tensor analysis, this appears to be false (at least to me). Let us consider flat space-time. Suppose we find a reference system ##x^\mu## such that the components of the metric tensor, ##g##, do take the values of Minkowski space-time. This means that ##g_{\alpha\beta}=\eta_{\alpha\beta}##. Let us consider the four-vector ##P##, which describes the infinitesimal displacement of a ray of light. Then

$$g(P,P)=(g_{\alpha\beta} dx^\alpha\otimes dx^\beta)(P^\rho\partial_\rho)(P^\lambda\partial_\lambda)=g_{\alpha\beta}P^\rho P^\lambda\delta^\alpha_\rho\delta^\beta_\lambda=\eta_{\alpha\beta}P^\alpha P^\beta.$$

Let us now look for the most general transformation ##x\rightarrow x'(x)## such that the interval remains invariant. In the new reference system, we have

$$g(P,P)=(g'_{ab} dx'^a\otimes dx'^b)(P'^r\partial'_r)(P'^l\partial'_l)=g'_{ab}P'^a P'^b.$$

Since the interval is invariant, we must have

$$g'_{ab}P'^a P'^b=\eta_{\alpha\beta}P^\alpha P^\beta,$$

but we know how vectors transform, so

$$g'_{ab}P'^a P'^b=g'_{ab}\frac{\partial x^\alpha}{\partial x'^a}\frac{\partial x^\beta}{\partial x'^b}P^\alpha P^\beta,$$

$$g'_{ab}\frac{\partial x^\alpha}{\partial x'^a}\frac{\partial x^\beta}{\partial x'^b}=\eta_{\alpha\beta}.$$

This equation is the transformation law for the components of a tensor of type ##(0,2)##, and imposes no

restrictions on the allowed transformations. In other words, as far as tensor analysis is concerned, if one defines Lorentz Transformations to be those coordinate transformations which leave the speed of light invariant, then all coordinate transformations are Lorentz Transformations. This has to be true, since tensors are geometric objects that do not depend on the reference system.

It is my understanding that the actual restrictions on the transformations come from requiring that they be a symmetry of Maxwell's Equations. Indeed, this is how they were first derived.

Another thing that really gets to me is that the treatment of Lorentz Transformations is usually restricted to the use of cartesian systems. I think this looses generality.

What do you guys think? Can Lorentz Transformations be derived from the constancy of the speed of light, without reference to Maxwell's Equations, and in the language of tensor analysis?

Cheers!

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# I Lorentz Transformations in the context of tensor analysis

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