Einstein Field Equations

The Einstein field equations G_μν = 8πG/c⁴ T_μν equate the geometry of spacetime (through the Einstein tensor G_μν built from the Riemann curvature tensor) with the distribution of energy and momentum (through the stress-energy tensor T_μν).

Key Concepts

Riemann tensor: R^ρ_μσν encodes spacetime curvature
Ricci tensor: R_μν = R^ρ_μρν (trace of Riemann)
Ricci scalar: R = g^μν R_μν
Einstein tensor: G_μν = R_μν - ½g_μν R
Cosmological constant Λ: G_μν + Λg_μν = 8πG/c⁴ T_μν

Key Equations

Einstein field equations

G_{\mu\nu} = R_{\mu\nu} - \frac{1}{2}g_{\mu\nu}R = \frac{8\pi G}{c^4}T_{\mu\nu}

Bianchi identity

\nabla_\mu G^{\mu\nu} = 0

Weak-field limit

\nabla^2\Phi = 4\pi G\rho

Trace of EFE

R = -\frac{8\pi G}{c^4}T

Worked Example

Example Problem

Problem

In the weak-field limit, the 00 component of the EFE gives the Poisson equation ∇²Φ = 4πGρ. For a uniform sphere of density ρ=10³ kg/m³ and radius R=100 m, find ∇²Φ inside.

Solution

∇²Φ = 4πGρ = 4π×6.674×10⁻¹¹×10³ = 8.39×10⁻⁷ s⁻².

Practice

Exercises

7 problems

1 of 7

The constant 8πG/c⁴ in SI units. Find it in units of s²/(kg·m).

Your answer

s²/(kg·m)

2 of 7

For a perfect fluid with T^μν = diag(ρc²,P,P,P), find the trace T = g_μν T^μν for the Minkowski metric (signs: -ρc²+3P). What is T for pressureless dust (P=0, ρ=1 kg/m³) in J/m³?

Your answer

J/m³

3 of 7

The Schwarzschild radius r_s = 2GM/c². For the Earth (M=5.97×10²⁴ kg), find r_s in mm.

Unlock Exercise 3