Gamma as a Jacobian of Lorentz transformations

[wil3]

Hello. When one is converting between coordinate systems, the Jacobian arises as a necessary consequence of the conversion. Does this occur with transformations between relativistic systems, and, if so, is this manifested through the prevalence of gamma in the transforms?

Any guidance would be appreciated. Thanks!

 

[haushofer]

No. The Jacobian of a Lorentz transformation is 1. That's why in special relativistic field theories you don't need to consider the subtlety that the measure is actually a scalar density., and you can define

$$S[\phi] = \int d^4 x L$$

For general coordinate transformations the measure is NOT invariant, and you would obtain the action

$$S[\phi] = \int \sqrt{|g|}d^4 x L$$

The squareroot becomes 1 for the Minkowski metric.

The gamma is part of the Lorentz transformation itself, NOT of the corresponding Jacobian.

 

[bcrowell]

For a relatively lowbrow discussion, see p. 629 of this book: http://www.lightandmatter.com/lm.pdf