General relativity §

1. Weak equivalence principle. Inertial mass equals gravitational mass, so all bodies fall with the same acceleration. Eötvös-type experiments verify this to $\sim 10^{-13}$. Newtonian gravity assumes it; GR builds the theory on it.
2. Einstein equivalence principle. In a freely falling frame, all non-gravitational physics reduces to special relativity. Gravity is therefore not a force in the usual sense: it is the geometry that remains after you remove every local effect.
3. Consequences. Gravitational time dilation: clocks deeper in a potential well run slow by $\Delta f/f \approx \Delta\Phi/c^2$. GPS satellites gain about 38 μs/day net (GR slows them by 45 μs/day at altitude, special relativity speeds them up by 7 μs/day from orbital speed); without correction, GPS positions would drift by kilometers per day.

Example. An astronaut in a windowless capsule in orbit floats because the capsule and the astronaut are both in free fall — the equivalence principle in action, not the absence of gravity.

1. Left side: geometry. $g_{\mu\nu}$ is the metric (the machine that converts coordinate separations into proper times and distances); $R_{\mu\nu}$ and $R$ are its curvatures. The Einstein tensor $G_{\mu\nu}$ is divergence-free, automatically conserving energy-momentum.
2. Right side: matter. $T_{\mu\nu}$ packages density, momentum flux, pressure, and shear. Crucially, pressure gravitates: radiation-dominated matter ($p = \rho c^2/3$) pulls three times harder than dust. This is why the early universe’s expansion decelerated faster, and why pressure can make collapse inevitable inside Compact objects.
3. Nonlinearity. Gravity gravitates: the field’s own energy is a source. This makes exact solutions rare and is the technical root of why quantizing gravity is hard (see the Gravity overview).
4. Weak-field limit. With $g_{00} \approx -(1 + 2\Phi/c^2)$, the field equations reduce to Poisson’s equation $\nabla^2\Phi = 4\pi G\rho$ — Newton’s theory reappears, which is why the Newtonian page works so well on Earth.

Example. The Schwarzschild solution ($T_{\mu\nu}=0$ outside a spherical mass) yields light deflection of $4GM/(c^2 b)$ at impact parameter $b$ — twice the Newtonian prediction, confirmed by Eddington in 1919.

1. Geodesic equation. $\dfrac{d^2x^\mu}{d\tau^2} + \Gamma^\mu_{\alpha\beta}\dfrac{dx^\alpha}{d\tau}\dfrac{dx^\beta}{d\tau} = 0$, where $\Gamma^\mu_{\alpha\beta}$ (Christoffel symbols) encode how the coordinate basis twists and stretches. In the weak field this reduces to Newton’s $\ddot{\mathbf{r}} = -\nabla\Phi$ plus small corrections.
2. Perihelion precession. The relativistic correction to the orbit equation is $\Delta\varphi \approx 6\pi GM/(c^2 a(1-e^2))$ per orbit — 43″/century for Mercury, and minutes of arc per orbit for binary pulsars.
3. Gravitational redshift. Light climbing out of a potential well loses frequency: $\Delta\lambda/\lambda \approx GM/(c^2 r)$. Pound–Rebka (1959) measured it over a 22.5 m tower.
4. Modern tests. Shapiro time delay, Gravity Probe B frame dragging, and Event Horizon Telescope shadow imaging all confirm GR in regimes spanning 15 orders of magnitude in field strength.

Example problem. How much slower does a clock run at Earth’s surface compared to far away?

Satellite clocks tick faster because they are higher in Earth’s gravitational potential (GR effect: $+\Delta\Phi/c^2$, about +45 μs/day) but slower because they move at ~3.9 km/s (special relativity: $-v^2/2c^2$, about −7 μs/day). The net is +38 μs/day, and since light travels ~300 m per μs, ignoring it would corrupt positions by ~11 km/day. Both effects are of the same order of magnitude ($v^2/c^2 \sim GM/(c^2 r)$), so neither can be neglected.