# Deriving lorentz transformation equations using linear transformation

While I was deriving the Lorentz transformation equations using linear transformation and eigenvectors i noticed that i get the correct equation only when determinant of the transformation matrix is $$\pm 1$$.What is the physical meaning behind this result?

• The determinant of a rotation is also equal to 1. The determinant of the Galilean transformation is also equal to 1. In your Lorentz transformation, hopefully you noticed that your eigenvalues are reciprocals of each other with eigenvectors along the lightcone. So, areas are preserved (the "volume scaling factor" mentioned by @Charles Francis)... in particular, the areas of causal diamonds. Possibly useful: physics.stackexchange.com/questions/527472/… – robphy Apr 19 '20 at 23:19
• what eigenvectors are you referring to? – magma May 1 '20 at 22:04

## 1 Answer

The determinant of a transformation matrix is generally interpreted as the volume scaling factor of the transformation, but this does not give a direct physical interpretation for the determinant being $$\pm 1$$. It results from the fact that the inverse of a Lorentz boost $$v$$ is a Lorentz boost $$-v$$, and that these boosts are identical in magnitude, so must have the same determinant, and that transformations of space coordinates (e.g. space rotation) make no scaling changes.