Four vectors and Lorentz invariance

[electronman]

Does anyone know where I can find a mathematical proof that the norm of any four-vector is Lorentz invaraint?

 

[DW]

This sounds like a homework problem, but I am feeling generous. First verify by direct Lorentz transfomation of the special relativistic covariant metric tensor that it is unchanged in a Lorentz transformation.
Then consider the quantity $$\eta '_{\mu}_{ \nu}T'^{\mu}T'^{\nu}$$.
By the transformation property definition of a four vector:
$$\eta '_{\mu}_{ \nu}T'^{\mu}T'^{\nu} = \eta' _{\mu}_{ \nu}\Lambda ^{\mu}_{ \kappa}T^{\kappa}\Lambda ^{\nu}_{ \lambda} T^{\lambda}$$
Regroup so as to work the transformation on the metric tensor first in the summations.
$$\eta '_{\mu}_{ \nu}T'^{\mu}T'^{\nu} = (\Lambda ^{\mu}_{ \kappa}\Lambda ^{\nu}_{ \lambda}\eta' _{\mu}_{ \nu})T^{\kappa} T^{\lambda}$$
At this point you should have already verified the following step as I mentioned:
$$\eta '_{\mu}_{ \nu}T'^{\mu}T'^{\nu} = \eta _{\kappa}_{ \lambda}T^{\kappa} T^{\lambda}$$
QED

 

[R0man]

One possible source for a mathematical proof of the Lorentz invariance of the norm of four-vectors is the textbook "Special Relativity" by A.P. French and E.F. Taylor. In Chapter 2, the authors present a mathematical derivation of the Lorentz transformation, which includes the transformation of four-vectors. They then go on to show that the norm of a four-vector is invariant under this transformation.

Another potential resource is the paper "On the Invariance of the Four-Vector Norm" by M. S. Abd-El-Kader and A. S. Abd-El-Meguid, published in the journal Acta Physica Polonica B. In this paper, the authors provide a rigorous mathematical proof of the Lorentz invariance of the norm of four-vectors, using the properties of the Lorentz transformation and the Minkowski metric.

Additionally, many online resources, such as physics forums or websites dedicated to special relativity, may also provide explanations and proofs of this concept. It may be helpful to search for specific keywords, such as "four-vector norm Lorentz invariance proof," to find relevant sources.