General Relativity — Curved Spacetime & Gravity
Newton described gravity as a force acting at a distance. Einstein showed that gravity is not a force at all — it is the curvature of spacetime itself, produced by mass and energy, and experienced by everything that moves through it. General relativity (GR) predicted phenomena that seemed fantastical in 1915 — black holes, gravitational waves, the expanding universe, gravitational time dilation — and every single prediction has since been confirmed by experiment to extraordinary precision.
1. The Equivalence Principle
Einstein's path to general relativity began with a thought experiment he later called "the happiest thought of my life": a person falling freely in a gravitational field feels no gravity. A person in an accelerating rocket feels a force indistinguishable from gravity. This is the equivalence principle:
- Weak equivalence principle (WEP): inertial mass equals gravitational mass. All objects fall at the same rate regardless of composition. Tested by Eötvös (1889) and modern experiments to 1 part in 10¹⁵.
- Einstein equivalence principle (EEP): in a small enough freely falling laboratory, the laws of physics are those of special relativity — no local experiment can detect whether you are in free fall near a mass or floating in empty space.
- Strong equivalence principle (SEP): this equivalence holds even for self-gravitating bodies (e.g., neutron stars), tested by lunar laser ranging.
From EEP, Einstein deduced that gravity must bend light (since photons follow straight paths in freely falling frames, and such frames are accelerated relative to a distant observer), and that clocks at different gravitational potentials must tick at different rates. Both predictions preceded and were later confirmed by experiment.
2. The Metric Tensor and Spacetime Geometry
In special relativity, the spacetime interval between two events is:
In general relativity, spacetime is curved. The interval becomes:
The metric tensor plays the role that the gravitational potential played in Newtonian theory, but it contains 10 independent components (since g_{μν} = g_{νμ}) and encodes both the geometry of space and the flow of time. From g_{μν} one computes the Christoffel symbols Γ^λ_{μν} (the "connection"), which describe how vectors change as they are parallel-transported through curved spacetime.
The Riemann curvature tensor R^ρ_{σμν}, built from derivatives of the Christoffel symbols, measures the curvature. Its contractions give the Ricci tensor R_{μν} and Ricci scalar R, which appear in Einstein's field equations.
3. Geodesic Equation and Free Fall
In curved spacetime, freely falling particles (no non-gravitational forces) follow geodesics — the generalisation of straight lines to curved geometry. The geodesic equation is:
In the Newtonian limit (weak gravity, slow velocities), the geodesic equation reduces to Newton's second law with gravitational acceleration a = −∇Φ, where Φ is the Newtonian potential. The correspondence requires g₀₀ ≈ −(1 + 2Φ/c²), so the metric component g₀₀ generalises the Newtonian potential.
Massless particles (photons) follow null geodesics with ds² = 0. The geodesic equation governs how light bends around massive objects — a prediction confirmed by Eddington's 1919 solar eclipse measurement of starlight deflection, which made Einstein world-famous overnight.
4. Einstein Field Equations
The central dynamical equations of GR relate spacetime curvature to the distribution of energy and momentum:
These 10 coupled, non-linear partial differential equations are among the most complex in physics. The right-hand side encodes all forms of energy and momentum (mass, pressure, heat flux, stress); the left-hand side encodes how spacetime curves in response. Wheeler's famous summary: "Matter tells spacetime how to curve, and spacetime tells matter how to move."
5. The Schwarzschild Solution
Karl Schwarzschild found the first exact solution to Einstein's equations just weeks after GR was published (and while serving on the Russian front in World War I). It describes the spacetime geometry outside a spherically symmetric, non-rotating, uncharged mass M:
Two singularities appear at r = R_s and r = 0. The singularity at r = R_s is a coordinate singularity — it can be removed by changing coordinates (Eddington-Finkelstein or Kruskal-Szekeres). The singularity at r = 0 is a genuine curvature singularity where tidal forces diverge.
Orbital Mechanics in Schwarzschild Geometry
Geodesics in the Schwarzschild metric predict three famous GR effects:
- Perihelion precession: Mercury's orbit precesses by an extra 43 arcseconds per century beyond Newtonian predictions. GR gives exactly 42.98 arcsec/century — a perfect match confirmed to 0.3%.
- Light deflection: a photon grazing the Sun is deflected by 1.75 arcseconds — twice the Newtonian prediction, confirmed since 1919.
- Shapiro delay: radar signals passing near the Sun arrive ~200 μs late due to the extra path length through curved spacetime; measured to 0.1% with the Cassini spacecraft.
6. Gravitational Time Dilation and Redshift
A clock at radius r in a Schwarzschild field ticks more slowly than a clock at infinity. Comparing proper time dτ at r to coordinate time dt (the rate at infinity):
The complementary effect — photons lose energy climbing out of a gravitational well — is gravitational redshift. A photon emitted at frequency f₀ from radius r₀ and received at infinity has frequency:
7. Black Holes
When mass is compressed within its Schwarzschild radius, a black hole forms. The surface r = R_s is the event horizon: no signal, no matter, no photon can escape from within it. To an outside observer, an object falling toward a black hole appears to slow down and redshift to invisibility as it approaches the horizon — it never quite crosses in finite coordinate time. Yet the infalling observer crosses the event horizon in finite proper time, experiencing nothing special at the horizon (for a large black hole, tidal forces there are weak).
Types of Black Holes
- Stellar black holes (M ~ 3–100 M_⊙): form from core collapse of massive stars. First detected via X-ray binaries (Cygnus X-1, 1971).
- Supermassive black holes (M ~ 10⁶–10¹⁰ M_⊙): reside at the centre of nearly every large galaxy. M87* (6.5 × 10⁹ M_⊙, R_s ≈ 19 billion km) was imaged by the Event Horizon Telescope in 2019. Sagittarius A* (4 × 10⁶ M_⊙) is at the Milky Way centre.
- Rotating black holes (Kerr metric): more realistic than Schwarzschild; the rotation introduces frame dragging (Lense-Thirring effect) and an ergosphere outside the horizon where spacetime itself is dragged along.
8. Gravitational Waves
Einstein's field equations allow wave-like solutions — ripples in spacetime curvature propagating at the speed of light. For a weak perturbation h_{μν} over flat spacetime (g_{μν} = η_{μν} + h_{μν}, with |h_{μν}| ≪ 1), the linearised field equations give:
Gravitational waves stretch and compress space transversely as they pass. The two polarisations h_+ and h_× distort a ring of test masses into alternating ellipses at 45° to each other. The amplitude is characterised by the strain h = ΔL/L — the fractional change in length between test masses.
On 14 September 2015, LIGO detected the first gravitational wave event GW150914: two black holes (36 M_⊙ and 29 M_⊙) merging 1.3 billion light-years away. The peak strain was h ≈ 10⁻²¹ — a change in length of 10⁻¹⁸ m over LIGO's 4 km arms, 1/1000 the diameter of a proton. The signal exactly matched GR predictions.
9. GPS: Relativity in Your Pocket
The Global Positioning System provides a striking everyday demonstration that both special and general relativity are real and necessary. GPS satellites orbit at altitude h ≈ 20,200 km with orbital speed v ≈ 3.87 km/s. Two relativistic corrections apply:
GPS clocks are pre-adjusted to tick at 10.23 MHz × (1 − 4.465 × 10⁻¹⁰) ≈ 10.22999999543 MHz on the ground, so that they tick at exactly 10.23 MHz in orbit after both relativistic corrections apply. Without this adjustment, GPS position errors would grow at ~11 km/day, making the system useless within minutes.
This is not an abstract thought experiment — it is an engineered relativistic correction running continuously in every satellite navigation system on Earth. General relativity is not just theoretically profound; it is a practical engineering requirement.