Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 2 / 38D / b

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- Solution b

Solution

Contracting the stated Einstein field equations with

g^{α β}

in

n

dimensions gives

T = R - \frac{n}{2} R = - \frac{n - 2}{2} R .

Thus

R = - 2 T / (n - 2)

, and substitution back yields the Trace-reversed Einstein field equations in D dimensions

R_{α β} = T_{α β} - \frac{1}{n - 2} g_{α β} T, n > 2.

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