In a large cosmic void

PHYSICS · Multi-part · Fermi University Physics · Relativity & Gravitation · Einstein's field equations · Einstein's field equations

The Geometry of Gravity, Ch The Einstein field equations and the Newtonian limit

Problem

In a large cosmic void, far from any galaxy, the local matter density is taken as exactly zero, but the cosmological constant Λ=1.1×10−52 m−2\Lambda = 1.1\times10^{-52}\,\mathrm{m^{-2}} is not. ((a)) Write the field equations for this region with Tμν=0T_{\mu\nu}=0 exactly. ((b)) Take the trace (using gμνgμν=4g^{\mu\nu}g_{\mu\nu}=4) to show R=4ΛR=4\Lambda, then substitute back to show Rμν=ΛgμνR_{\mu\nu}=\Lambda g_{\mu\nu}. ((c)) Evaluate the Ricci scalar RR numerically for this void.

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