Someone argues: "The equivalence principle already tells us that free fall is…

PHYSICS · Subjective · Fermi University Physics · Relativity & Gravitation · Differential geometry of curved spacetime · Einstein's field equations

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

Problem

Someone argues: "The equivalence principle already tells us that free fall is geodesic motion in curved spacetime. So the equivalence principle by itself must also fix the field equation to be

G_{\mu\nu} = \kappa T_{\mu\nu}

; there is no need for a separate list of assumptions about the left-hand side." As a test of this claim, consider the alternative candidate equation

R_{\mu\nu} = \kappa T_{\mu\nu}

(the Ricci tensor alone, not the Einstein tensor). This is still built from the metric, is symmetric, has two derivatives of the metric, and matter still couples to the one metric

g_{\mu\nu}

, so it still respects the equivalence principle. Explain why this candidate is nonetheless not an acceptable field equation, and say which of the assumptions listed for the left-hand side it fails.

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