Lorentz transformation ↩

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created: 2022-02-22 16:59:02
modified: 2022-03-01 21:33:09

Let

be a Minkowski spacetime. The

linear transformation is a Lorentz transformation if:

Is true for .

Matrix of the transformation

We can see that is a "basis transformation" on :

Properties

Lorentz transformations leave the metric invariant:
The matrix of a L-transformation has the following property:

Where is the metric tensor

From this, the determinant of :

Motivation

Motivation for change of bases in Minkowski space time When changing between reference frames in special relativity, we do that by changing bases in Minkowski spacetime. These are linear transformations on .

Since these are just "change of base" transformations, they should not change the "dot product" of vectors in . (This property is analogous to unitary transformations.)

Where is the transformation.
We call the transformations satisfying this Lorentz transformations.