General Relativity

Notes on the classical theory of gravitation — Einstein's 1915 geometrization of gravity, in which spacetime is a curved Lorentzian manifold whose geometry is sourced by matter and energy. The theory generalizes Special Relativity from the flat spacetime of inertial frames to a dynamical, curved spacetime, and reduces to Newtonian gravity in the weak-field, slow-motion limit.

We use natural units $c=1$ (keeping Newton's constant $G$ explicit), and adopt the mostly-plus metric signature $g_{\mu \nu }=\mathrm{diag}(-1,+1,+1,+1)$ standard in the general-relativity literature (Wald, MTW). This differs from the mostly-minus $\eta$ of the SR pages; the sign convention and its consequences are flagged on The Geometry of Gravity.

Contents

From the equivalence principle to the field equations

1. The Equivalence Principle — the universality of free fall, weak/Einstein/strong forms, and why gravity must be geometry.
2. The Geometry of Gravity — spacetime as a curved Lorentzian manifold; the metric as the gravitational potential.
3. Covariant Derivative and Parallel Transport — the Levi-Civita connection, Christoffel symbols, geodesics as free fall.
4. Curvature of Spacetime — the Riemann tensor, geodesic deviation, Ricci and Weyl.
5. The Einstein Field Equations — $G_{\mu \nu }+\Lambda g_{\mu \nu }=8\pi G{T}_{\mu \nu }$, its derivation, uniqueness, and Newtonian limit.

Matter and sources

6. The Stress–Energy Tensor — the source of gravity: dust, perfect fluids, fields, the Hilbert definition, and conservation.
7. Energy Conditions — positivity constraints on matter and the global theorems they underwrite.

Exact solutions

8. The Schwarzschild Solution — the field of a spherical mass, the classical tests, and the non-rotating black hole.
9. Cosmological Solutions (FLRW) — the homogeneous, isotropic universe: the Friedmann equations, expansion, and $\Lambda$CDM.
10. The Kerr and Charged Solutions — the rotating black hole, frame dragging, the ergosphere, and the no-hair theorem.

Black holes

11. Horizons and Singularities — crossing the horizon, Penrose diagrams, the singularity theorems, and cosmic censorship.
12. Black-Hole Thermodynamics — the four laws, Hawking radiation, entropy, and the information paradox.

Radiation, formulation, and frontiers

13. Linearized Gravity and Gravitational Waves — the wave equation, polarizations, the quadrupole formula, and LIGO.
14. The Variational Formulation — the Einstein–Hilbert action, the GHY boundary term, and conservation from diffeomorphism invariance.
15. The 3+1 Split and the Initial-Value Problem — the ADM formulation, constraints, the Cauchy problem, and numerical relativity.
16. QFT in Curved Spacetime and Quantum Gravity — Hawking/Unruh, non-renormalizability, the problem of time, and the main programs.
17. Experimental Status — the tests of general relativity from the solar system to cosmology.

Mathematical prerequisites

The GR pages assume the differential geometry collected in the Mathematics section, specialized here to Lorentzian signature:

• Topology and manifolds — manifolds, charts, tangent and cotangent spaces, tensor fields.
• Metric geometry — the metric tensor, the Levi-Civita connection, parallel transport.
• Curvature — the Riemann tensor and its contractions, geodesics.
• Differential forms — for the variational formulation.

Relation to other sections

General relativity takes the flat spacetime of Special Relativity — with its Lorentz/Poincaré symmetry and invariant interval — and promotes the metric to a dynamical field. The special-relativistic description is recovered locally, in a freely falling frame, expressing the equivalence principle. The union of GR with Quantum Field Theory remains unsolved; the tension surfaces in black-hole thermodynamics and the problem of time. The conceptual questions the theory raises — the reality of spacetime, the hole argument, the nature of the metric field — are treated in Philosophy of Space and Time, in particular General Relativity and Dynamical Spacetime.

Reading order

The five core pages are linear: equivalence principle → geometry of gravity → covariant derivative → curvature → field equations. A reader fluent in differential geometry can skip directly to the field equations after the equivalence principle.