- #1

binbagsss

- 1,266

- 11

## Homework Statement

The most general form:

##ds^2=e^{2A(r)}dt^2-e^{2B(r)}dr^2-r^2(d\theta_1^2 +sin^2\theta_1(d\theta^2_2+sin^2\theta_2d\phi^2))##

Ricci tensors:

##R_{tt}=e^{2(A-B)}(\frac{3A'}{r}+A'^2-B'A'+A'')##

##R_{rr}=-A''+\frac{3B'}{r}+A'(B'-A')##

##R_{\theta_1 \theta_1}=2+e^{-2B}(-2+r(B'-A'))##

##R_{\theta_2 \theta_2} = sin^2 \theta_1 R_{\theta_1 \theta_1}##

##R_{\phi \phi} = sin^2 \theta_1 sin^2 \theta_2 R_{\theta_1 \theta_1}##

To solve the Ricci tensors for ##A## and ##B##?

## Homework Equations

## The Attempt at a Solution

I have tried preceeding as you do in the 4-d case. This is to do

##e^{-2(A-B)} R_{tt} + R_{rr} =0 ##

which gives ##A=-B##

and then to solve

##R_{\theta_1 \theta_1}=0##

However doing this i get:

##1+e^{2A}(1-A') = 0##

multiply though by ##e^{-3A}##

to get

##d/dr(e^{-A}r)=-e^{-A}##

##e^{-A}r=\int -e^{-A} dr ##and I can't solve for ##A(r)## explicitly

I can't see a better way to proceed as with the other components there are terms including ##A'' , A'^2## things.

Many thanks