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Chapter 13: Einstein Field Equations
Solutions and Applications
This final page explores the weak field limit where the Einstein equations reduce to Newtonian gravity, the linearized equations for gravitational waves, the stress-energy tensors for common matter types, and an overview of the classification of exact solutions.
Weak Field Limit: Recovering Newtonian Gravity
General relativity must reduce to Newtonian gravity in the appropriate limit: weak gravitational fields and slow-moving particles. We write the metric as a small perturbation of flat spacetime:
$$g_{\mu\nu} = \eta_{\mu\nu} + h_{\mu\nu}, \qquad |h_{\mu\nu}| \ll 1$$
where \( \eta_{\mu\nu} = \text{diag}(-1, +1, +1, +1) \) is the Minkowski metric
For a static, weak field, the dominant metric perturbation is:
$$g_{00} \approx -\left(1 + \frac{2\Phi}{c^2}\right)$$
where \( \Phi \) is the Newtonian gravitational potential
This identification comes from requiring that the geodesic equation for slow particles (\( v \ll c \)) reproduces Newton's second law \( \ddot{x}^i = -\partial_i \Phi \). For a slow particle with \( u^\mu \approx (c, \vec{v}) \) where \( |\vec{v}| \ll c \):
$$\frac{d^2 x^i}{dt^2} \approx -\Gamma^i_{00}\, c^2 = -\frac{c^2}{2}\, \partial_i h_{00} = -\partial_i \Phi$$
This confirms that \( h_{00} = -2\Phi/c^2 \), connecting the metric perturbation to the Newtonian potential.
The Poisson Equation from Einstein's Equations
In the Newtonian limit, the \( (0,0) \) component of the Einstein equations gives:
$$R_{00} \approx \frac{1}{2}\nabla^2 h_{00} = -\frac{\nabla^2 \Phi}{c^2}$$
For a non-relativistic source with \( T_{00} \approx \rho c^2 \) and \( T \approx -\rho c^2 \), the trace-reversed Einstein equation \( R_{00} = 8\pi G(T_{00} - \frac{1}{2}g_{00}T)/c^4 \) gives:
$$R_{00} \approx \frac{8\pi G}{c^4}\left(\rho c^2 - \frac{1}{2}(-1)(-\rho c^2)\right) = \frac{4\pi G \rho}{c^2}$$
Combining:
$$\boxed{\nabla^2 \Phi = 4\pi G \rho}$$
The Poisson equation - Newton's law of gravity recovered from Einstein's equations
This is exactly the Poisson equation of Newtonian gravity! The factor of \( 8\pi G/c^4 \) in the Einstein equations was chosen precisely so that this limit works out correctly.
Linearized Einstein Equations
Beyond the static Newtonian limit, we can linearize the full Einstein equations to first order in \( h_{\mu\nu} \). Defining the trace-reversed perturbation:
$$\bar{h}_{\mu\nu} = h_{\mu\nu} - \frac{1}{2}\eta_{\mu\nu}\, h, \qquad h = \eta^{\mu\nu}h_{\mu\nu}$$
In the Lorenz gauge \( \partial^\mu \bar{h}_{\mu\nu} = 0 \) (analogous to the Lorenz gauge in electromagnetism), the linearized Einstein equations become:
$$\Box\, \bar{h}_{\mu\nu} = -\frac{16\pi G}{c^4}\, T_{\mu\nu}$$
where \( \Box = -\partial_t^2/c^2 + \nabla^2 \) is the d'Alembertian operator
This is a wave equation with source! In vacuum (\( T_{\mu\nu} = 0 \)):
$$\Box\, \bar{h}_{\mu\nu} = 0$$
Gravitational waves propagate at the speed of light
This predicts gravitational waves - ripples in spacetime that propagate at \( c \). The analogy with electromagnetism (\( \Box A_\mu = -\mu_0 J_\mu \)) is exact at the linearized level. The Lorenz gauge leaves residual gauge freedom that can be used to impose the transverse-traceless (TT) gauge, where only two independent polarizations (\( h_+ \) and \( h_\times \)) remain.
Stress-Energy: Perfect Fluid
The most common matter model in general relativity is the perfect fluid, which has no viscosity or heat conduction. Its stress-energy tensor is:
$$T^{\mu\nu} = (\rho + p/c^2)\, u^\mu u^\nu + p\, g^{\mu\nu}$$
Perfect fluid stress-energy tensor
where \( \rho \) is the energy density (including rest mass), \( p \) is the isotropic pressure, and \( u^\mu \) is the fluid 4-velocity. In the rest frame of the fluid:
$$T^{\mu\nu} = \begin{pmatrix} \rho c^2 & 0 & 0 & 0 \\ 0 & p & 0 & 0 \\ 0 & 0 & p & 0 \\ 0 & 0 & 0 & p \end{pmatrix}$$
The trace is \( T = g_{\mu\nu}T^{\mu\nu} = -\rho c^2 + 3p \). The conservation equation \( \nabla_\mu T^{\mu\nu} = 0 \) gives:
$$u^\mu\nabla_\mu \rho + (\rho + p/c^2)\nabla_\mu u^\mu = 0 \quad \text{(energy conservation)}$$
$$(\rho + p/c^2)\, u^\mu\nabla_\mu u^\nu + (g^{\mu\nu} + u^\mu u^\nu/c^2)\nabla_\mu p = 0 \quad \text{(Euler equation)}$$
Common equations of state include: dust (\( p = 0 \)), radiation (\( p = \rho c^2/3 \)), stiff matter (\( p = \rho c^2 \)), and dark energy (\( p = -\rho c^2 \)).
Stress-Energy: Electromagnetic Field
The electromagnetic field is described by the antisymmetric field tensor \( F_{\mu\nu} \). Its stress-energy tensor is:
$$T_{\mu\nu} = \frac{1}{\mu_0}\left(F_{\mu\alpha}F_\nu^{\ \alpha} - \frac{1}{4}g_{\mu\nu}F_{\alpha\beta}F^{\alpha\beta}\right)$$
This tensor is traceless (\( T = 0 \)), which means the trace-reversed Einstein equations reduce to:
$$R_{\mu\nu} = 8\pi G\, T_{\mu\nu} \qquad \text{(since } T = 0 \text{, Ricci scalar also vanishes: } R = 0 \text{)}$$
The Reissner-Nordstrom solution (charged black hole) and the Kerr-Newman solution (rotating charged black hole) are solutions of the Einstein-Maxwell equations, where the electromagnetic stress-energy serves as the source of curvature.
Classification of Exact Solutions
Despite the nonlinearity of the Einstein equations, many exact solutions have been found, typically by exploiting symmetries. Here is an overview of the major classes:
Vacuum Solutions (\( R_{\mu\nu} = 0 \))
Schwarzschild (1916): Unique spherically symmetric vacuum solution (Birkhoff's theorem). Describes non-rotating black holes and the exterior of spherical stars.
Kerr (1963): Axially symmetric, stationary vacuum solution. Describes rotating black holes. Uniqueness theorems show that Kerr is the most general stationary vacuum black hole.
Gravitational waves: pp-waves, impulsive waves, and sandwich waves are exact vacuum solutions representing gravitational radiation.
Electrovacuum Solutions (\( T_{\mu\nu} = T_{\mu\nu}^{\text{EM}} \))
Reissner-Nordstrom: Spherically symmetric charged black hole. Has two horizons for\( Q < M \) (in geometrized units).
Kerr-Newman: The most general stationary black hole with mass, charge, and angular momentum. "Black holes have no hair" - they are completely characterized by \( (M, Q, J) \).
Cosmological Solutions
FLRW: Homogeneous, isotropic universe filled with perfect fluid. The standard cosmological model.
de Sitter / anti-de Sitter: Maximally symmetric solutions with positive/negative cosmological constant. de Sitter approximates the late-time accelerating universe.
Bianchi models: Homogeneous but anisotropic cosmologies. Important for understanding the approach to the initial singularity (BKL/Mixmaster behavior).
Interior Solutions
Schwarzschild interior: Uniform density star with \( p = p(r) \), matched to the exterior Schwarzschild vacuum at the stellar surface.
Tolman-Oppenheimer-Volkoff: The relativistic structure equation for spherically symmetric stars in hydrostatic equilibrium.
Key Concepts Summary
Einstein field equations: \( G_{\mu\nu} + \Lambda g_{\mu\nu} = 8\pi G\, T_{\mu\nu}/c^4 \) - 10 coupled nonlinear PDEs relating geometry to matter.
Variational derivation: \( \delta S_{\text{EH}}/\delta g^{\mu\nu} = 0 \) with\( S_{\text{EH}} = \frac{1}{16\pi G}\int\sqrt{-g}\,(R - 2\Lambda)\,d^4x \).
Key variations: \( \delta\sqrt{-g} = -\frac{1}{2}\sqrt{-g}\,g_{\mu\nu}\,\delta g^{\mu\nu} \) and\( \delta R_{\mu\nu} = \nabla_\rho\delta\Gamma^\rho_{\mu\nu} - \nabla_\nu\delta\Gamma^\rho_{\mu\rho} \) (Palatini).
Newtonian limit: \( g_{00} \approx -(1 + 2\Phi/c^2) \) and the Einstein equations reduce to Poisson's equation \( \nabla^2\Phi = 4\pi G\rho \).
Linearized equations: \( \Box\bar{h}_{\mu\nu} = -16\pi G\,T_{\mu\nu}/c^4 \) - predicts gravitational waves propagating at speed \( c \).
Perfect fluid: \( T^{\mu\nu} = (\rho + p/c^2)u^\mu u^\nu + p\,g^{\mu\nu} \) - the standard matter model for stars and cosmology.
Exact solutions: Schwarzschild (static BH), Kerr (rotating BH), FLRW (cosmology), de Sitter (accelerating expansion) - found by exploiting symmetry.