A large spherical asteroid of uniform density has a spherical cavity at its centre

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

A large spherical asteroid of uniform density has a spherical cavity at its centre, completely empty of matter and radiation. Take the cosmological constant to be exactly zero for this problem (its effect is negligible at solar-system scales). ((a)) Write the Einstein field equations with the cosmological constant in the standard form, and say in words what each symbol stands for. ((b)) State the value of

T_{\mu\nu}

inside the cavity, and use the trace-reversed field equations

R_{\mu\nu} = \kappa\left(T_{\mu\nu} - \tfrac12 Tg_{\mu\nu}\right)

to show that the Ricci tensor vanishes there.

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