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Chapter 13: Einstein Field Equations
The Einstein Field Equations
The Einstein field equations are the core of general relativity. They relate the curvature of spacetime (geometry) to the distribution of matter and energy. "Matter tells spacetime how to curve; spacetime tells matter how to move." These ten coupled, nonlinear partial differential equations encode all of gravitational physics.
The Field Equations
The Einstein field equations, in their most general form with a cosmological constant, are:
$$G_{\mu\nu} + \Lambda g_{\mu\nu} = \frac{8\pi G}{c^4}\, T_{\mu\nu}$$
Geometry = Matter (with cosmological constant \( \Lambda \))
Each side of this equation is a symmetric \( 4 \times 4 \) tensor, giving 10 independent component equations. Let us examine each piece carefully.
Left Side: Geometry
The left side encodes the curvature of spacetime through the Einstein tensor:
$$G_{\mu\nu} = R_{\mu\nu} - \frac{1}{2}\, g_{\mu\nu}\, R$$
where \( R_{\mu\nu} \) is the Ricci tensor (contraction of the Riemann curvature tensor) and \( R = g^{\mu\nu}R_{\mu\nu} \) is the Ricci scalar. The Einstein tensor involves second derivatives of the metric tensor \( g_{\mu\nu} \) through the Christoffel symbols.
The cosmological constant term \( \Lambda g_{\mu\nu} \) was originally introduced by Einstein to allow a static universe. After Hubble's discovery of cosmic expansion, Einstein famously called it his "greatest blunder." However, the 1998 discovery of accelerated expansion has restored\( \Lambda \) as a critical part of modern cosmology, where it represents dark energy with \( \Lambda \approx 1.1 \times 10^{-52}\, \text{m}^{-2} \).
Geometric Content
The Einstein tensor encodes 10 of the 20 independent components of the Riemann tensor - specifically, those related to the Ricci curvature. The remaining 10 components (the Weyl tensor) represent "free" gravitational field that propagates even in vacuum. The Einstein equations determine the Ricci part from the matter content; the Weyl part is determined by boundary conditions and the propagation of gravitational degrees of freedom.
Right Side: Matter
The right side contains the stress-energy tensor \( T_{\mu\nu} \), which encodes all information about the matter and energy content of spacetime:
\( T^{00} \) = energy density \( \rho c^2 \)
\( T^{0i} \) = energy flux = momentum density \( \times\, c \)
\( T^{i0} \) = momentum density \( \times\, c \)
\( T^{ij} \) = stress (pressure and shear)
The coupling constant \( 8\pi G/c^4 \approx 2.08 \times 10^{-43}\, \text{N}^{-1} \) is extraordinarily small, which is why enormous amounts of energy are needed to produce detectable spacetime curvature. This is determined by requiring that the equations reduce to Newton's law of gravity in the weak-field, slow-motion limit.
In natural units where \( G = c = 1 \), the equations simplify to:
$$G_{\mu\nu} + \Lambda g_{\mu\nu} = 8\pi\, T_{\mu\nu}$$
10 Coupled Nonlinear PDEs
The Einstein equations constitute a system of 10 coupled, nonlinear, second-order partial differential equations for the 10 independent components of the metric tensor \( g_{\mu\nu} \). They are nonlinear because the Einstein tensor contains products of Christoffel symbols:
$$G_{\mu\nu} \sim \partial^2 g + (\partial g)^2$$
Schematic structure: linear part + nonlinear part
The nonlinearity has profound consequences:
No Superposition
The sum of two solutions is generally not a solution. You cannot simply add two black hole metrics to get a binary black hole. The gravitational field itself carries energy and creates more gravity.
Self-Interaction
Gravitational waves carry energy, and that energy itself gravitates. This makes the theory fundamentally different from electromagnetism, where photons do not interact with each other (at tree level).
Exact Solutions are Rare
The nonlinearity makes exact solutions extremely difficult to find. Only highly symmetric spacetimes (like Schwarzschild, Kerr, FLRW) admit closed-form solutions. Generic problems require numerical relativity.
Key Properties
Diffeomorphism Invariance
The equations are covariant: they take the same form in any coordinate system. This is the mathematical expression of the principle of general covariance - the laws of physics should not depend on coordinate choice.
This gauge freedom means that 4 of the 10 metric components are pure gauge (can be chosen freely), leaving 6 physical degrees of freedom. Combined with 4 constraint equations from \( \nabla^\mu G_{\mu\nu} = 0 \), only 2 degrees of freedom propagate - the two polarizations of gravitational waves.
Automatic Conservation
The contracted Bianchi identity \( \nabla^\mu G_{\mu\nu} = 0 \) (a geometric identity) automatically ensures \( \nabla^\mu T_{\mu\nu} = 0 \) (energy-momentum conservation). This is not an additional assumption but a mathematical consequence of the structure of the field equations.
Constraint + Evolution Structure
The 10 equations decompose into 4 constraint equations (involving only first time derivatives, analogous to Gauss's law in electromagnetism) and 6 evolution equations (involving second time derivatives). The constraints must be satisfied on the initial data surface; the Bianchi identity guarantees they remain satisfied under evolution.
Alternative Forms
Original Form (without \( \Lambda \))
$$R_{\mu\nu} - \frac{1}{2}\, g_{\mu\nu}\, R = \frac{8\pi G}{c^4}\, T_{\mu\nu}$$
Einstein's original 1915 form, before the cosmological constant
Trace-Reversed Form
$$R_{\mu\nu} = \frac{8\pi G}{c^4}\left(T_{\mu\nu} - \frac{1}{2}\, g_{\mu\nu}\, T\right)$$
where \( T = g^{\mu\nu}T_{\mu\nu} \). Obtained by taking the trace (\( R = -8\pi G T/c^4 \)) and substituting back. Often more convenient for calculations.
Vacuum Equations
$$R_{\mu\nu} = 0 \qquad \text{(when } T_{\mu\nu} = 0, \Lambda = 0 \text{)}$$
In vacuum, the trace-reversed form reduces to Ricci-flatness. The Schwarzschild and Kerr solutions satisfy these vacuum equations.
With \( \Lambda \) as Effective Stress-Energy
$$G_{\mu\nu} = 8\pi G\left(T_{\mu\nu} + T^{(\Lambda)}_{\mu\nu}\right) \quad \text{where} \quad T^{(\Lambda)}_{\mu\nu} = -\frac{\Lambda}{8\pi G}\, g_{\mu\nu}$$
The cosmological constant can be interpreted as a constant vacuum energy density\( \rho_\Lambda = \Lambda c^2/(8\pi G) \) with equation of state \( w = p/\rho = -1 \).