Lorentz transformations leave the metric invariant

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created: 2022-02-22 17:14:19
modified: 2022-02-22 17:47:04

Let be a Minkowski spacetime equipped with the Minkowski metric .
Statement: A Lorentz transformation leaves the metric invariant and the converse is true as well:

Proof

This follows immediately, just use :

The basic notion is that is an arbitrary vector and it can be expressed as the sum of other two vectors, so use instead: