Homework Help: Need to find the Ricci scalar curvature of this metric

1. May 27, 2012

chinared

1. The problem statement, all variables and given/known data
Need to find the Ricci scalar curvature of this metric:
ds2 = e2a(z)(dx2 + dy2) + dz2 − e2b(z)dt2

2. Relevant equations

3. The attempt at a solution

I tried to find the solution, but failed to pass the calculation of Riemann curvature tensor:

<The Christoffel connection> Here a'(z) denotes the first derivative of a(z) respect to z.
$\Gamma\stackrel{x}{xz}$=$\Gamma\stackrel{x}{zx}$=a'(z)
$\Gamma\stackrel{y}{yz}$=$\Gamma\stackrel{y}{zy}$=a'(z)
$\Gamma\stackrel{z}{tt}$=b'(z)e2b(z)
$\Gamma\stackrel{z}{xx}$=$\Gamma\stackrel{z}{yy}$=-a'(z)e2a(z)
$\Gamma\stackrel{t}{tz}$=$\Gamma\stackrel{t}{zt}$=b'(z)
$\Gamma\stackrel{}{either}$=0

<The Riemann curvature tensor>
$\R\stackrel{x}{zxz}$=$\R\stackrel{y}{zyz}$=-a''(z)-[a'(z)]2
$\R\stackrel{z}{tzt}$=b''(z)+[b'(z)]2

I tried to find the Ricci scalar curvature(R) from current result, but it gave a function depend on z. Is there any problem in my calculation?

1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

2. May 28, 2012

clamtrox

Why wouldn't it depend on z when your metric does. You can compare this to the FRW-metric. In that case, the metric and the scalar curvature depend on time.

The results look reasonable but I'm a bit too lazy to check it explicitly.