Einstein Equation

Prerequisites: Levi-Civita Connection, Curvature, Newton's law of Gravity. Full path (11 pages)

Best explanations

Intuitive How to Invent General Relativity
Concrete The Meaning of Einstein’s Equation

Intuitive
Concrete
Abstract
Why is it interesting?
Definitions

$G_{\mu \nu} = 8 \pi G T_{\mu \nu}$

Intuitive

Einstein's equation describes how spacetime gets curved through mass and energy.

In Newton's theory, gravity is a force that a mass exerts on other masses. In general relativity there is no gravitational force. A body on which no other forces act moves along the straightest possible path through spacetime, and what we see as the pull of gravity is the curvature of spacetime making those paths bend towards each other.

A good picture is a small cloud of dust grains that floats freely and starts out at rest relative to each other. In empty space far from everything, the cloud keeps its size. Near the Earth the cloud gets stretched along the direction towards the Earth and squeezed sideways, because the grains closer to the Earth fall a bit faster and the grains to the left and right fall towards the same center. These tidal effects are what curvature means physically. If the cloud contains matter, for example if the grains float inside a cloud of gas, then the cloud also starts to shrink in volume.

Einstein's equation says how much it shrinks. The rate at which the volume of such a freely falling cloud starts to shrink is proportional to the energy density inside it plus the sum of the pressures in the three space directions. So mass and energy act as sources of curvature, in the same way that electric charges act as sources of the electric field. Pressure also counts as a source, which has no analogue in Newton's theory.

The equation is local: it relates the curvature at a point to the energy and momentum at that same point. Changes in the curvature spread at the speed of light and can travel through empty space as gravitational waves.

How to Invent General Relativity by J. Schwichtenberg

Concrete

We use units with $c = 1$ (as in the formula above) and the metric signature $(-,+,+,+)$ . Spacetime is described by a metric $g_{\mu\nu}(x)$ , a symmetric $4\times 4$ matrix at every point. The Einstein equation

$R_{\mu\nu} - \frac12 R\, g_{\mu\nu} = 8\pi G\, T_{\mu\nu}$

is a set of ten coupled nonlinear second-order partial differential equations for the ten components of $g_{\mu\nu}$ , one for each independent pair $(\mu,\nu)$ . The left side is built from the metric and its first and second derivatives (see the Definitions below). The right side is the stress-energy tensor: $T_{00}$ is the energy density, $T_{0i}$ the momentum density and $T_{ij}$ the stress, which for a perfect fluid at rest is $p\,\delta_{ij}$ with the pressure $p$ .

Trace-reversed form. Contracting with $g^{\mu\nu}$ and using $g^{\mu\nu}g_{\mu\nu} = 4$ gives $R - 2R = 8\pi G\, T$ , so $R = -8\pi G\, T$ with $T = g^{\mu\nu}T_{\mu\nu}$ . Inserting this back,

$R_{\mu\nu} = 8\pi G\left( T_{\mu\nu} - \frac12 T\, g_{\mu\nu}\right).$

In vacuum, $T_{\mu\nu} = 0$ , the equation reduces to $R_{\mu\nu} = 0$ . This does not mean spacetime is flat: the Schwarzschild metric outside a star and gravitational waves both have $R_{\mu\nu} = 0$ but nonzero Riemann tensor.

Newtonian limit. Take a static, weak field with $g_{00} = -(1 + 2\Phi)$ , $|\Phi| \ll 1$ , and slowly moving dust with $T_{00} = \rho$ and all other components negligible. Then $T = -\rho$ and the $00$ component of the trace-reversed equation reads $R_{00} = 8\pi G(\rho - \frac12\rho) = 4\pi G\rho$ . To first order in $\Phi$ and for a static field, $\Gamma^i_{00} = -\frac12 \partial_i g_{00} = \partial_i \Phi$ and $R_{00} \approx \partial_i \Gamma^i_{00} = \nabla^2\Phi$ . So we get

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

which is the Poisson equation of Newtonian gravity. The geodesic equation of a slow particle gives $\ddot x^i = -\Gamma^i_{00} = -\partial_i\Phi$ , which is Newton's law of motion in the potential $\Phi$ .

Conservation. The Einstein tensor obeys the contracted Bianchi identity $\nabla^\mu G_{\mu\nu} = 0$ for every metric. The equation therefore forces $\nabla^\mu T_{\mu\nu} = 0$ , the local conservation of energy and momentum of matter. The same identity means that only six of the ten equations determine the evolution, while four are constraints on the initial data, matching the freedom to choose four coordinates.

Cosmological constant. One can add a term $\Lambda g_{\mu\nu}$ to the left side, $G_{\mu\nu} + \Lambda g_{\mu\nu} = 8\pi G\, T_{\mu\nu}$ , without spoiling the conservation law, since $\nabla^\mu g_{\mu\nu} = 0$ . See the cosmological constant problem.

The Meaning of Einstein’s Equation by John C. Baez and Emory F. Bunn explains the Einstein equations perfectly.

The static limit of the Einstein equation is known as Newton's law.

Abstract

Let $(M,g)$ be a four-dimensional Lorentzian manifold, with signature $(-,+,+,+)$ and $c=1$ , and let $\nabla$ be the Levi-Civita connection of $g$ . The Einstein equation is the Euler-Lagrange equation of the action

$S[g,\phi] = \frac{1}{16\pi G}\int_M R\,\sqrt{-g}\;d^4x + S_{\text{m}}[g,\phi]$

under variations of the metric. Up to a total derivative, $\delta\left(\sqrt{-g}\,R\right) = \sqrt{-g}\,G_{\mu\nu}\,\delta g^{\mu\nu}$ , and the stress-energy tensor is defined by

$T_{\mu\nu} = -\frac{2}{\sqrt{-g}}\frac{\delta S_{\text{m}}}{\delta g^{\mu\nu}} .$

Setting $\delta S = 0$ gives $G_{\mu\nu} = 8\pi G\,T_{\mu\nu}$ . Because both terms of $S$ are invariant under diffeomorphisms, $T_{\mu\nu}$ is covariantly conserved on solutions of the matter equations, and $G_{\mu\nu}$ is divergence free identically. This is the Noether identity for diffeomorphism invariance, here the contracted Bianchi identity.

The form of the left-hand side is essentially unique. Lovelock's theorem states that in four dimensions the only symmetric, divergence-free tensors built locally from $g_{\mu\nu}$ and its first two derivatives are $a\,G_{\mu\nu} + b\,g_{\mu\nu}$ with constants $a, b$ . The constant $b$ is the cosmological constant.

The Riemann tensor splits into the Ricci tensor and the Weyl tensor. The Einstein equation fixes the Ricci part algebraically in terms of $T_{\mu\nu}$ , while the Weyl part is constrained only through the Bianchi identities. The Weyl part carries the gravitational degrees of freedom that exist in vacuum, such as gravitational waves.

As a system of PDEs the equation is not hyperbolic as it stands, because of the diffeomorphism freedom. After a choice of gauge, for example harmonic coordinates $\Box_g x^\mu = 0$ , it becomes a quasilinear hyperbolic system. Given a spacelike hypersurface $\Sigma$ with a Riemannian metric $h_{ij}$ and a symmetric tensor $K_{ij}$ (the second fundamental form) that satisfy the Hamiltonian and momentum constraints

${}^{(3)}\!R + K^2 - K_{ij}K^{ij} = 16\pi G\,\rho, \qquad D_j\left(K^{ij} - h^{ij}K\right) = 8\pi G\, j^i ,$

with the convention $K_{ij} = -\frac12 \mathcal{L}_n h_{ij}$ and where $\rho = T_{\mu\nu}n^\mu n^\nu$ and $j^i = -h^{i\mu}T_{\mu\nu}n^\nu$ are measured by the unit normal $n$ , there exists (in vacuum, or with suitable matter equations and matter data) a globally hyperbolic development of the data, unique up to diffeomorphism (Choquet-Bruhat, and Choquet-Bruhat and Geroch for the maximal development).

Why is it interesting?

The Einstein equation is the fundamental equation of general relativity. It describes how spacetime is curved through the presence of matter and energy.

Definitions

On the right-hand side, Newton's gravitational constant $G$ (not to be confused with the Einstein tensor $G_{\mu\nu}$ ) and the stress-energy tensor $T_{\mu \nu}$ , in units with $c=1$ .

On the left-hand side, the Einstein tensor $G_{\mu \nu}$ is defined as the Ricci Tensor $R_{\mu\nu}$ minus one half the metric times the trace of the Ricci tensor, called Ricci scalar $R =R_{\nu}^\nu$ $\begin{equation} G_{\mu \nu} = R_{\mu\nu}-\frac{1}{2}Rg_{\mu \nu} \end{equation}$ where the Ricci Tensor $R_{\mu\nu}$ is defined in terms of the Christoffel symbols $\Gamma^\mu_{\nu \rho}$

$\begin{equation} R_{\alpha\beta} = \partial_{\rho}{\Gamma^\rho_{\beta\alpha}} - \partial_{\beta}\Gamma^\rho_{\rho\alpha} + \Gamma^\rho_{\rho\lambda} \Gamma^\lambda_{\beta\alpha} - \Gamma^\rho_{\beta\lambda}\Gamma^\lambda_{\rho\alpha} \end{equation}$ and the Christoffel Symbols are defined in terms of the metric $\begin{equation} \Gamma_{\alpha \beta \rho} =\frac12 \left(\frac{\partial g_{\alpha \beta}}{\partial x^\rho} + \frac{\partial g_{\alpha \rho}}{\partial x^\beta} - \frac{\partial g_{\beta \rho}}{\partial x^\alpha} \right) = \frac12\, \left(\partial_{\rho}g_{\alpha \beta} + \partial_{\beta}g_{\alpha \rho} - \partial_{\alpha}g_{\beta \rho}\right). \end{equation}$

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