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atrahasis
#4
Mar10-12, 12:31 AM
P: 11
Induced metric on the brane

Thanks for the reply,
I checked on Poisson's book and also Gourgoulhon's review but I couldn't found the reason.

I finally understood my mistake, [itex]h_{\mu\nu}[/itex] is not the induced metric but only the projection tensor. For to have the induced metric we have to look to the tangential components of the tensor and not to [itex]h_{00}[/itex].
In fact the 3 vectors orthogonal to the normal vector and which define a basis on the hypersurface are
[itex]V1^\mu=(1,\dot a,0,0)[/itex]
[itex]V2^\mu=(0,0,1,0)[/itex]
[itex]V3^\mu=(0,0,0,1)[/itex]

so it is perfectly fine to look for [itex]h_{22}[/itex] and [itex]h_{33}[/itex]. But the last component is not [itex]h_{00}=h_{tt}[/itex] but [itex]h_{V1 V1}[/itex]

So now we have [itex]\partial_{V1}=\partial_t+\dot a \partial_\rho[/itex] which implies that

[itex]h_{V1V1}=h_{00}+2\dot a h_{01}+\dot a^2 h_{11}[/itex] which gives the correct result [itex]h_{V1V1}=-1+\dot a^2[/itex].

So it is a modification of the coordinates ...

Thanks