- #1
Milsomonk
- 96
- 17
Homework Statement
In special relativity the metric is invariant under lorentz transformations and therefore so is the determinant of the metric. How does the metric determinant transform under a more general transformation $$x^{a\prime}=J^{a\prime}_{\quad a}x^{a}$$ where $$J^{a\prime}_{\quad a}$$ may not satisfy the conditions of the lorentz group.
Homework Equations
$$x^{a\prime}=J^{a\prime}_{\quad a}x^{a}$$
The Attempt at a Solution
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So I need to show how the determinant transforms, but in general I thought that the determinant was a scalar and thus did not transform, clearly this isn't correct so my next thought was to perform the above transformation on the metric as follows:
$$\eta_{a^{\prime}b^{\prime}}=J_{a\prime}^{\quad a}J_{b\prime}^{\quad b}\eta_{ab}$$
Then take the determinant of the result. But this doesn't appear to get me anywhere and doesn't make use of the fact that J is in general not a lorentz transform, any guidance as to where I should go next or whether I'm barking up the wrong tree entirely would be much appreciated :)