General relativity: constant curvature, characterizing equation

1. May 18, 2010

Derivator

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

Show, that a three-dimensional space with constant curvature K is charaterized by the following equation for the Riemann curvature tensor:

$$R_{abcd} = K \cdot \left(g_{ac}g_{bd}-g_{ad}g_{bc}\right)$$

2. Relevant equations

3. The attempt at a solution

Hi folks,

I would like to give an own attempt, but I have no Idea how to start.

We haven't defined the curvature K in lecture. How is it defined?
Has anybody an idea, how to start?

--
derivator

2. May 19, 2010

Derivator

anybody an idea?

3. May 19, 2010

gabbagabbahey

Wald defines the Ricci tensor as

$$R_{ac}=R_{abc}{}^{b}$$

And then the scalar curvature is the trace of the Ricci tensor

$$R=R_{a}{}^{a}$$

Your text should have similar definitions.

So, I think want you want to do is to show that if $R_{abcd} = K\left(g_{ac}g_{bd}-g_{ad}g_{bc}\right)$, where $K$ is a constant, then the scalar curvature is $R=K$.

4. May 20, 2010

Derivator

ah, thx for your input.

i haven't seen this exercise from this point of view, but it makes sense.

I have found: R=6K, thus R is constant, thus we have a constant curvature.

Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook