Ansatz metric of the 4 dimensional spacetime:(adsbygoogle = window.adsbygoogle || []).push({});

[itex]ds^2=a^2 g_{ij}dx^i dx^j + du^2[/itex] (1)

where:

Signature: [itex]- + + +[/itex]

Metric [itex]g_{ij} \equiv g_{ij} (x^i) [/itex] describes 3 dimensional AdS spacetime

[itex]i,j = 0,1,2 = 3[/itex] dimensional curved spacetime indices

[itex]a(u)=[/itex] warped factor

[itex]u = x^D = x^3[/itex]

[itex]D = 3 =[/itex] number of spacial dimensional

Now I have to proof that

[itex]R_{ij} = [(\frac{a'}{a})' + 3 (\frac{a'}{a})^2 - \frac{\Lambda_3}{a^2}] a^2 g_{ij}[/itex]

[itex]R_{33} = -3 [(\frac{a'}{a})' + (\frac{a'}{a})^2][/itex]

[itex]R = 6 [(\frac{a'}{a})' + 2 (\frac{a'}{a})^2] - \frac{3\Lambda_3}{a^2}[/itex]

where

[itex]R_{ij} =[/itex] the Ricci curvature of metric (1)

[itex]R =[/itex] the Ricci scalar of metric (1)

[itex]a' = \frac{∂a}{∂u}[/itex]

My steps to calculate [itex]R_{ij}[/itex]:

- calculating [itex]R_{\mu\nu}[/itex], where [itex]\mu,\nu = 0,1,2,3 = 4[/itex] dimensional curved spacetime indices

- finding that [itex]R_{\mu\nu} = ... R_{ij} [/itex] (failed)

- Subtituting [itex]R_{\mu\nu} = ... R_{ij}[/itex] to [itex]R_{\mu\nu} = \Lambda_D g_{\mu\nu} [/itex] (failed)

Furthermore I can't find [itex]R_{33}[/itex] and [itex]R[/itex]

What are the right steps to find [itex]R_{ij}, R_{33}, R[/itex] ?

**Physics Forums | Science Articles, Homework Help, Discussion**

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Ricci curvature

Loading...

**Physics Forums | Science Articles, Homework Help, Discussion**