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So I'm trying to prove that the light cone [math] \eta _{ \mu \nu } dx ^{ \mu } dx ^{ \nu } = 0 [/math] is invariant under a couple of transformations. The way I'm thinking of doing this is to act on the metric with a couple of Jacobians, so given a discrete transformation [math] x _{ \mu } \to x _{ \mu } / x^2 [/math] I'd have: [eqn] \eta _{ \mu \nu } \Lambda ^{ \mu } _{ \rho } \Lambda ^{ \nu } _{ \sigma } dx ^{ \rho } dx ^{ \sigma } \\ \text { Where } ~ \Lambda ^{ \mu } _{ \rho } = \frac { \delta ^{ \mu } _{ \rho } } { x^2 } - 2 \frac { x ^{ \mu } x _{ \nu } } { x^4 } [/eqn] So after some algebra [eqn] \eta _{ \mu \nu } \Lambda ^{ \mu } _{ \rho } \Lambda ^{ \nu } _{ \sigma } dx ^{ \rho } dx ^{ \sigma } = \frac { \eta _{ \rho \sigma } } { x^2 } dx ^{ \rho } dx ^{ \sigma } =0 [/eqn] Is that it? Does that imply that the interval is invariant? And if it does, is that a general method for solving these types of problems? I'm sure I've done this before in the past but that would have been some years ago.
Pic mostly unrelated
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>>8493788
Bump.
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Go to mathoverflow man, good mathematicians don't post here.
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I think you need to act on the metric and on the differentials, then the interval is automatically an invariant quantity
[eqn]\bar{\eta}_{\mu\nu}d\bar{x}^{\mu}d\bar{x}^{\nu}={\eta}_{\mu\nu}dx^{\alpha}dx^{\beta}\Lambda^{\mu}_{\bar{\mu}}\Lambda^{\nu}_{\bar{\nu}}\Lambda^{\bar{\mu}}_{\alpha}\Lambda^{\bar{\nu}}_{\beta}={\eta}_{\mu\nu}dx^{\alpha}dx^{\beta}\delta^{\mu}_{\alpha}\delta^{\nu}_{\beta}={\eta}_{\mu\nu}dx^{\mu}dx^{\nu}[/eqn]
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>>8494594
Thanks, that was actually my argument.
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