All About the Einstein Field Equations

[Greg Bernhardt]

Table of Contents
ToggleDefinition/SummaryFast FactsEquationsShort Version (using Einstein tensor:Simplified in Cosmological UnitsLong Version (using Ricci curvature tensor and scalar...

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[Aaron Paulo]

The Einstein Field Equations are a set of ten differential equations which express the general theory of relativity mathematically. These equations relate the curvature of spacetime to the energy/matter content of spacetime, and can be written in two ways. The short version is expressed as a single symmetric tensor equation, with G_{\mu\nu} equaling 8\pi G/c^4 multiplied by T_{\mu\nu}. The long version is expressed as two equations: one scalar equation, R = -8\piT, and one traceless symmetric tensor equation, R_{\mu\nu} - (1/4)Rg_{\mu\nu} = 8\pi(T_{\mu\nu} - (1/4)Tg_{\mu\nu}). Cosmological units are used in cosmology, where G = c = 1. The trace of a symmetric tensor is a scalar invariant, and by splitting the equation into scalar and traceless parts, we can see that the trace of the Ricci curvature equals minus the trace of the stress-energy, while the traceless Ricci curvature equals the traceless stress-energy. The factor 8\pi is ultimately related to the weak-field limit giving the inverse-square law of Newtonian gravity.