Mathematics of General Relativity — Mathematics Knowledge System

Overview

General relativity (GR), formulated by Einstein in 1915, describes gravity not as a force but as the curvature of a Lorentzian manifold (spacetime) caused by mass and energy. The mathematical framework requires pseudo-Riemannian geometry: a smooth 4-manifold equipped with a metric of signature (-,+,+,+), the Levi-Civita connection, and curvature tensors. The Einstein field equations relate the Einstein tensor (encoding spacetime curvature) to the stress-energy tensor (encoding matter and energy content).

Intuition

Imagine spacetime as a rubber sheet. Massive objects like the Sun cause the sheet to curve. Other objects (planets, photons) then follow the straightest possible paths (geodesics) on this curved surface. What we perceive as the gravitational force is simply inertial motion on a curved background. The deeper insight is that there is no global 'flat' reference frame: curvature is an intrinsic property of the manifold, detectable without reference to an external space.

Formal Definition

Definition

Spacetime is a smooth 4-manifold M equipped with a Lorentzian metric g of signature (-,+,+,+). The Einstein field equations couple the geometry of (M,g) to the matter content.

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

Einstein field equations

R^\rho{}_{\sigma\mu\nu} = \partial_\mu \Gamma^\rho_{\nu\sigma} - \partial_\nu \Gamma^\rho_{\mu\sigma} + \Gamma^\rho_{\mu\lambda}\Gamma^\lambda_{\nu\sigma} - \Gamma^\rho_{\nu\lambda}\Gamma^\lambda_{\mu\sigma}

Riemann curvature tensor

\Gamma^\lambda_{\mu\nu} = \tfrac{1}{2} g^{\lambda\sigma}(\partial_\mu g_{\nu\sigma} + \partial_\nu g_{\mu\sigma} - \partial_\sigma g_{\mu\nu})

Christoffel symbols (Levi-Civita connection coefficients)

\frac{d^2 x^\mu}{d\tau^2} + \Gamma^\mu_{\alpha\beta}\frac{dx^\alpha}{d\tau}\frac{dx^\beta}{d\tau} = 0

Geodesic equation (free-fall trajectories)

S_{\mathrm{EH}} = \frac{c^4}{16\pi G}\int R\,\sqrt{-g}\;d^4x

Einstein-Hilbert action

Notation

Notation	Meaning
$g_{\mu\nu}$	Lorentzian metric tensor on spacetime
$\Gamma^\lambda_{\mu\nu}$	Christoffel symbols of the Levi-Civita connection
$R^\rho{}_{\sigma\mu\nu}$	Riemann curvature tensor
$R_{\mu\nu} = R^\rho{}_{\mu\rho\nu}$	Ricci tensor (contraction of Riemann)
$R = g^{\mu\nu}R_{\mu\nu}$	Ricci scalar curvature
$G_{\mu\nu}$	Einstein tensor
$T_{\mu\nu}$	Stress-energy tensor

Properties

Bianchi identity

\nabla_\lambda R_{\rho\sigma\mu\nu} + \nabla_\rho R_{\sigma\lambda\mu\nu} + \nabla_\sigma R_{\lambda\rho\mu\nu} = 0

Conservation of stress-energy

\nabla^\mu T_{\mu\nu} = 0

Equivalence principle (mathematical form)

\text{At any point } p \in M,\; \exists \text{ local coordinates such that } g_{\mu\nu}(p) = \eta_{\mu\nu},\; \Gamma^\lambda_{\mu\nu}(p) = 0

Worked Examples

1
Assume a static spherically symmetric metric in the form ds² = -e^{2α(r)}c²dt² + e^{2β(r)}dr² + r²dΩ².
$ds^2 = -e^{2\alpha(r)}c^2 dt^2 + e^{2\beta(r)}dr^2 + r^2 d\Omega^2$
2
Compute the Christoffel symbols and then the Ricci tensor components Rμν in these coordinates.
$R_{tt} = 0,\quad R_{rr} = 0,\quad R_{\theta\theta} = 0$
3
The vacuum equations Rμν = 0 and boundary condition gμν → ημν as r → ∞ uniquely yield e^{2α} = e^{-2β} = 1 - r_s/r.
$e^{2\alpha} = 1 - \frac{r_s}{r}, \quad r_s = \frac{2GM}{c^2}$

✓ Answer

The Schwarzschild metric is the unique (Birkhoff's theorem) spherically symmetric vacuum solution with Schwarzschild radius rs = 2GM/c².

Practice Problems

Hardfree response

State and explain the geodesic equation. What does it mean for a massive particle? For a photon?

Hardproof writing

Prove that the contracted Bianchi identity ∇^μ Gμν = 0 follows from the full Bianchi identity.

Common Mistakes

Common Mistake

Treating the Christoffel symbols as tensors

Christoffel symbols Γλμν are NOT tensors — they transform inhomogeneously under coordinate changes. The covariant derivative and curvature tensor are the properly tensorial objects.

Common Mistake

Confusing the Ricci tensor Rμν with the full Riemann tensor Rρσμν

The Riemann tensor has 4 indices and encodes all curvature information. The Ricci tensor is a contraction Rμν = Rρμρν with only 2 independent indices. Vacuum equations require Rμν = 0, not Rρσμν = 0 (flat spacetime).

Quiz

The Einstein field equations state that:

The Riemann tensor equals the stress-energy tensorThe Einstein tensor (Gμν = Rμν - gμν R/2) is proportional to the stress-energy tensorThe Ricci scalar equals the trace of the stress-energy tensorThe metric tensor is proportional to the stress-energy tensor

The Schwarzschild radius rs = 2GM/c² is the radius at which:

The escape velocity equals the speed of lightTime dilation becomes infinite (event horizon of a Schwarzschild black hole)Both of the aboveThe gravitational field vanishes

Historical Background

Einstein began developing GR in 1907, motivated by the incompatibility of Newtonian gravity with special relativity. After years of struggle with the mathematics of curved spacetime (guided significantly by Marcel Grossmann), he published the final field equations in November 1915, just days before Hilbert independently derived them from a variational principle. Experimental confirmation came with the 1919 solar eclipse measurement of light bending by Eddington.

1907
Einstein formulates the equivalence principle
Albert Einstein
1913
Einstein and Grossmann publish the Entwurf theory using Ricci calculus
Albert Einstein, Marcel Grossmann
1915
Einstein publishes the final field equations; Hilbert derives them from a variational principle
Albert Einstein, David Hilbert
1916
Schwarzschild finds the first exact solution (black hole metric)
Karl Schwarzschild
1919
Eddington's solar eclipse expedition confirms gravitational light deflection
Arthur Eddington

Summary

GR models spacetime as a pseudo-Riemannian 4-manifold (M, g) with Lorentzian signature.
The Einstein field equations Gμν = (8πG/c⁴) Tμν couple spacetime curvature (left side) to matter-energy (right side).
Free particles and photons follow geodesics; gravity is not a force but the geometry of curved spacetime.
The Einstein-Hilbert action provides a variational derivation of the field equations.
The contracted Bianchi identity implies ∇^μ Tμν = 0, encoding local energy-momentum conservation.

References

BookEinstein, A. — Die Feldgleichungen der Gravitation, Sitzungsberichte der Preussischen Akademie der Wissenschaften (1915)
BookWald, R.M. — General Relativity (1984), University of Chicago Press
BookMisner, C.W., Thorne, K.S. & Wheeler, J.A. — Gravitation (1973), W.H. Freeman
WebsiteWikipedia — Mathematics of general relativity
WebsiteMathWorld — Einstein Field Equations