General Relativity

General relativity (GR) is Einstein’s theory of gravitation, published in 1915. Its central insight is breathtaking: gravity is not a force but the curvature of spacetime. Mass and energy warp the geometry of four-dimensional spacetime; freely falling bodies simply follow the straightest possible paths — geodesics — in that curved geometry.

The equivalence principle

Einstein’s key observation (1907) was the equivalence principle: the effects of gravity are locally indistinguishable from those of acceleration. An observer in a sealed box cannot tell, by any local experiment, whether they are in a uniform gravitational field or in an accelerating rocket. This elevates gravity from a force to a geometric property.

Weak equivalence principle: inertial mass equals gravitational mass — all bodies fall at the same rate (Galileo’s experiment). Tested to \(\sim 10^{-15}\).

Strong equivalence principle: locally (in a small enough region), spacetime looks flat and the laws of special relativity hold.

Riemannian geometry

Spacetime in GR is a four-dimensional pseudo-Riemannian manifold with metric \(g_{\mu\nu}\). The invariant interval is

\begin{equation} ds^2 = g_{\mu\nu}(x)\, dx^\mu dx^\nu. \end{equation}

The metric replaces the Newtonian gravitational potential. From it one constructs:

Christoffel symbols (connection coefficients): \begin{equation} \Gamma^\lambda_{\mu\nu} = \frac{1}{2}g^{\lambda\sigma}\left( \partial_\mu g_{\nu\sigma} + \partial_\nu g_{\mu\sigma} - \partial_\sigma g_{\mu\nu} \right). \end{equation}

Riemann curvature tensor: \begin{equation} 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}. \end{equation}

Ricci tensor and scalar: \begin{equation} R_{\mu\nu} = R^\lambda{}{\mu\lambda\nu}, \qquad R = g^{\mu\nu}R{\mu\nu}. \end{equation}

Einstein tensor: \begin{equation} G_{\mu\nu} = R_{\mu\nu} - \tfrac{1}{2}g_{\mu\nu} R. \end{equation}

The Einstein tensor satisfies \(\nabla^\mu G_{\mu\nu} = 0\) (Bianchi identity), which ensures conservation of energy-momentum.

Geodesics

A freely falling particle follows a geodesic — the curve that extremises proper time:

\begin{equation}\label{eq.geodesic} \frac{d^2 x^\mu}{d\tau^2} + \Gamma^\mu_{\alpha\beta}\frac{dx^\alpha}{d\tau}\frac{dx^\beta}{d\tau} = 0. \end{equation}

The Christoffel symbol terms play the role of the “gravitational force” in the new coordinates. In flat spacetime (\(\Gamma = 0\)) this reduces to \(d^2x^\mu/d\tau^2 = 0\): uniform motion.

Einstein field equations

The Einstein field equations (EFE) relate spacetime curvature to the energy-momentum content of matter:

\begin{equation}\label{eq.EFE} \boxed{G_{\mu\nu} + \Lambda g_{\mu\nu} = \frac{8\pi G}{c^4}\, T_{\mu\nu}.} \end{equation}

Here \(T_{\mu\nu}\) is the stress-energy tensor, \(G = 6.674\times10^{-11}\) N m² kg⁻² is Newton’s constant, and \(\Lambda\) is the cosmological constant (associated with dark energy). The prefactor \(8\pi G/c^4 \approx 2\times 10^{-43}\) N⁻¹ expresses the extraordinary stiffness of spacetime — enormous energy-momentum is needed to produce a measurable curvature.

Despite their compact appearance, the EFE are ten coupled, nonlinear partial differential equations for \(g_{\mu\nu}\). Exact solutions exist only in highly symmetric situations.

The Schwarzschild solution

Karl Schwarzschild found the exact solution for a static, spherically symmetric vacuum (\(T_{\mu\nu} = 0\), \(\Lambda = 0\)) in 1916 — just weeks after Einstein published the EFE:

\begin{equation}\label{eq.schwarz} ds^2 = -!\left(1 - \frac{r_s}{r}\right)c^2 dt^2 + \left(1 - \frac{r_s}{r}\right)^{-1} dr^2 + r^2\,d\Omega^2, \end{equation}

where \(d\Omega^2 = d\theta^2 + \sin^2\theta\, d\phi^2\) and the Schwarzschild radius is

\begin{equation} r_s = \frac{2GM}{c^2}. \end{equation}

For the Sun, \(r_s \approx 3\) km (the Sun’s radius is \(\sim 7\times10^5\) km); for the Earth, \(r_s \approx 9\) mm. The metric \eqref{eq.schwarz} is singular at \(r = r_s\) — not a physical singularity, but a coordinate singularity (the event horizon). The true singularity lies at \(r = 0\).

Predictions confirmed by observation

1. Perihelion precession of Mercury. GR predicts an extra 43 arcseconds/century, matching observation precisely — a long-standing discrepancy with Newtonian gravity.

2. Gravitational deflection of light. A ray passing the Sun is deflected by \(1.75''\), twice the Newtonian prediction. Confirmed by Eddington’s 1919 solar eclipse expedition.

3. Gravitational redshift. A photon climbing out of a gravitational well loses energy:

\begin{equation} \frac{\Delta\nu}{\nu} = -\frac{GM}{rc^2}. \end{equation}

Confirmed to high precision by Pound–Rebka (1959) and atomic clocks on GPS satellites (which require GR corrections of \(\sim 45\,\mu\)s/day).

4. Gravitational waves. The linearised EFE admit wave solutions that propagate at \(c\). Detected directly by LIGO in 2015 from a binary black hole merger — 100 years after the EFE.

The post-Newtonian limit

In the weak-field, slow-motion limit, the EFE reduce to Poisson’s equation

\begin{equation} \nabla^2\Phi = 4\pi G\rho, \end{equation}

where \(\Phi\) is the Newtonian gravitational potential and \(g_{00} \approx -(1 + 2\Phi/c^2)\). Newtonian gravity is thus the leading approximation to GR when both \(v/c \ll 1\) and \(\Phi/c^2 \ll 1\).

Cosmology and Friedmann equations

Applying the EFE to a homogeneous, isotropic universe (Friedmann–Lemaître–Robertson–Walker metric) gives the Friedmann equations:

\begin{equation} \left(\frac{\dot{a}}{a}\right)^2 = \frac{8\pi G}{3}\rho + \frac{\Lambda c^2}{3} - \frac{kc^2}{a^2}, \end{equation}

\begin{equation} \frac{\ddot{a}}{a} = -\frac{4\pi G}{3}!\left(\rho + \frac{3p}{c^2}\right) + \frac{\Lambda c^2}{3}, \end{equation}

where \(a(t)\) is the scale factor, \(k = -1, 0, +1\) is the spatial curvature, and \(p\) is pressure. The accelerating expansion of the universe (\(\ddot{a} > 0\)), driven by \(\Lambda > 0\), was discovered in 1998 — one of the deepest open problems in physics today.