# Four vectors and Lorentz invariance

1. May 9, 2004

### electronman

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

2. May 9, 2004

### 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

Last edited: May 9, 2004