- #51

WannabeNewton

Science Advisor

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You can also express it as [itex]\tilde{g}_{\mu \nu} = \Omega^2 g_{\mu \nu}[/itex] for some smooth scalar field ##\Omega##, which is how you will usually see it. It's the same thing in the end anyways.Okay, I saw a definition on the internet, that two metrics [itex]g_{\mu \nu}[/itex] and [itex]\tilde{g}_{\mu \nu}[/itex] are "conformally equivalent" if you can get from one to the other by a scaling factor (in general, position-dependent). In other words:

[itex]\tilde{g}_{\mu \nu} = e^\phi g_{\mu \nu}[/itex]

But what's the significance of that? Are the mass/energy distributions that give rise to those two metrics related in some simple way?

Geometrically, conformally equivalent metrics preserve angles but not lengths. I have already proven one significance of them, which is that conformally equivalent metrics agree on null geodesics (post #45). The converse is true as well: two metrics which agree on null geodesics are conformally equivalent. Also, Maxwell's equations in curved space-time (as well as certain other physical equations in curved space-time) remain invariant under conformal transformations.

The most important application of conformal transformations, in the context of general relativity, is probably in the definition of asymptotic flatness which involves a conformal isometry of a certain space-time onto an open subset of another space-time satisfying certain conditions (for example a conformal isometry of Minkowski space-time onto an open subset of the Einstein static universe).

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