Verifying a solution to Einstein's vacuum field equations

Question

I need to verify a solution to Einsteins vacuum field equations. I have the solution as follows $$ds^2=a\,dt^2+b\,dr^2+\cdots$$

Is the following the right approach? Einsteins equation reduces to $R_{ab}=0$, which can be expressed in terms of Christoffel symbols which can be defined by the metric.

How does one obtain the metric from the line-element?

Once I have the metric, do I just plug it into $R_{ab}=0$ and check? This seems very tedious, am I on the right track?

Added:

If I change the coordinates, and show that this new metric satisfies Einsteins vacuum Equation, would it follow that the original coordiantes and metric does so as well?

Answer 1

The line element in terms of the metric $g_{\mu\nu}$ is given by,

$$ds^2=g_{\mu\nu}dx^\mu dx^\nu$$

As you haven't provided the entire line element in the post, I can only say $g_{tt} =a$ and $g_{rr}=b$. If it is the case that $a,b$ are constants, as well as any other components of the metric, then trivially $R_{\mu\nu}=R=0$.

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Otherwise, you pretty much have to compute the Ricci tensor $R_{ab}$ directly. However, there is a much faster method than using Christoffel symbols which is called the Cartan formalism. I have summarised it in other answers, see:

• Some hints for special case of metric tensor in GR
• Calculating the Riemann tensor for a 3-Sphere
• 5D Ricci Curvature

Very quickly: you choose an orthonormal basis $e^a_\mu$ such that $\eta_{ab}e^a_\mu e^b_\nu = g_{\mu\nu}$, and you can read off connection components $\omega^a_b$ from $de^a + \omega^a_b \wedge e^b = 0$. The curvature 2-form is $R=d\omega^a_b + \omega^a_c \wedge \omega^c_b$.

In addition, I recommend Ruth Gregory's lecture at perimeterscholars.org which provide an introduction to the method, and preliminaries introducing differential forms if you are not familiar with them.

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If you are given a metric $g_{\mu\nu}$, and change coordinates, you are still describing the same manifold; general relativity is invariant under diffeomorphisms.