Einstein field equation

The Einstein field equation is the extremal to the Einstein-Hilbert action. We have

𝐺 𝑎 𝑏 + Λ 𝑔 𝑎 𝑏 = 𝜅 𝑇 𝑎 𝑏

where $𝐺 𝑎 𝑏 : = 𝑅 𝑎 𝑏 - 12 𝑅 𝑔 𝑎 𝑏$ is the Einstein tensor, $𝑇 𝑎 𝑏$ is the Stress-energy tensor, $Λ$ is the Cosmological constant, and

𝜅 = 8 𝜋 𝐺 𝑐 4 \approx 2.076 65 \times 10 - 43 N - 1

is the Einstein gravitational constant.

Proof

proof

Einstein field equation in vacuum

In a vacuum $𝑇 𝑎 𝑏 = 0$ , so the Einstein field equation becomes $𝐺 𝑎 𝑏 = Λ 𝑔 𝑎 𝑏$ . On a manifold $𝑀$ of dimension $𝑚 \neq 2$ this becomes

𝑅 𝑎 𝑏 = 2 Λ 𝑛 - 2 𝑔 𝑎 𝑏 .

Proof

We have
$0 = 𝑅 𝑎 𝑏 - 12 𝑅 𝑔 𝑎 𝑏 + Λ 𝑔 𝑎 𝑏 .$
Contracting with $𝑔 𝑎 𝑏$ , we have
$0 = (1 - 𝑛 2) 𝑅 + 𝑛 Λ ⟹ 𝑅 = 2 𝑛 Λ 𝑛 - 2$
so the Einstein field equation becomes
$0 = 𝑅 𝑎 𝑏 - 𝑛 Λ 𝑛 - 2 𝑔 𝑎 𝑏 + Λ 𝑔 𝑎 𝑏 = 𝑅 𝑎 𝑏 - 2 Λ 𝑛 - 2 𝑔 𝑎 𝑏$
giving the claimed equation.

Thus solutions to the Einstein field equation in a vacuum for arbitrary $Λ$ are precisely those for which the Ricci curvature is proportional to the metric tensor. Such a manifold is called an Einstein manifold. In particular, for $Λ = 0$ , we see the solution is precisely a Ricci flat metric.

develop | en | SemBr

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Einstein field equation

Einstein field equation in vacuum

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