Relativity: General Relativity

Relativity » General Relativity

General Relativity

General relativity (GR) is Einstein’s theory of gravitation, completed in November 1915. It replaces Newton’s instantaneous force with geometry: mass and energy curve four-dimensional spacetime, and freely falling bodies follow the straightest possible paths (geodesics) through that curved geometry. John Wheeler’s summary is still the best one-line description: spacetime tells matter how to move; matter tells spacetime how to curve.

This page covers the physical principles, the mathematical structure (metric, geodesics, curvature, field equations), the Schwarzschild solution and its consequences, the experimental record from Mercury’s perihelion to the 2025 gravitational-wave tests, and the problems GR leaves open. It assumes Special Relativity. The differential geometry is developed properly on Tensor Formalism & the Field Equations; here it is stated and interpreted.

Conventions. Metric signature $(-,+,+,+)$; Greek indices run over $0$–$3$, Latin indices over spatial $1$–$3$; repeated indices are summed. Factors of $G$ and $c$ are kept on this page. The Schwarzschild radius is $r_s = 2GM/c^2$.

Physical Foundations

The equivalence principle

Galileo’s observation that all bodies fall with the same acceleration means gravitational mass equals inertial mass. Einstein turned this coincidence into a principle. It comes in three strengths:

The operational content is that a small, freely falling laboratory is indistinguishable from an inertial frame in empty space, and a laboratory resting on a planet is indistinguishable from one accelerating in a rocket.

Two consequences follow before any field equations are written down:

• Light bends. A light ray crossing an accelerating cabin traces a curved path in the cabin frame, so by equivalence it must also bend in a gravitational field.
• Clocks run slow deeper in a potential. A photon climbing from floor to ceiling of an accelerating cabin is Doppler-redshifted; by equivalence, light climbing out of a gravitational well is redshifted by $\Delta\nu/\nu \approx -\Delta\Phi/c^2$.

Because gravity cannot be “transformed away” globally — two freely falling bodies on opposite sides of Earth accelerate toward each other — the equivalence principle is strictly local. What remains after all first-order effects are removed is tidal acceleration, and GR identifies tidal acceleration with spacetime curvature.

General covariance

The laws of physics are written as tensor equations, which take the same form in every coordinate system. Coordinates in GR are labels with no intrinsic meaning; only invariants (proper times, proper distances, curvature scalars, the outcomes of measurements) are physical. Combined with the equivalence principle this gives a working recipe: take a law valid in special relativity, replace $\eta_{\mu\nu}$ with $g_{\mu\nu}$ and partial derivatives with covariant derivatives (“comma goes to semicolon”), and the result holds in curved spacetime.

Gravity as geometry

The two halves of the theory form a feedback loop, which is what makes GR nonlinear: the curvature produced by matter also carries energy that itself gravitates.

graph LR
    T["Matter and energy<br/>(stress-energy tensor)"] -->|"field equations"| G["Spacetime geometry<br/>(metric, curvature)"]
    G -->|"geodesic equation"| M["Motion of matter<br/>and light"]
    M -->|"moving matter changes<br/>the stress-energy"| T

Mathematical Structure

The metric

All geometric information is encoded in the metric tensor $g_{\mu\nu}(x)$, a symmetric $4\times 4$ field that turns coordinate displacements into invariant intervals:

In flat spacetime with Cartesian coordinates it reduces to the Minkowski metric, $ds^2 = -c^2dt^2 + dx^2 + dy^2 + dz^2$. For a timelike worldline the elapsed proper time — what a clock carried along it reads — is $d\tau^2 = -ds^2/c^2$. At any single event one can always choose coordinates in which $g_{\mu\nu} = \eta_{\mu\nu}$ and its first derivatives vanish (locally inertial or Riemann normal coordinates); this is the mathematical form of the equivalence principle. The second derivatives cannot all be removed, and they are the curvature.

Geodesics: how matter moves

A freely falling particle follows a geodesic, the worldline that extremizes (for timelike curves, maximizes) proper time between two events:

The Christoffel symbols of the Levi-Civita connection are built from first derivatives of the metric:

The $\Gamma$ terms play the role of the Newtonian gravitational force, but they are not tensors: they vanish in a freely falling frame, which is why a falling observer feels no force. Light follows null geodesics ($ds^2 = 0$), parameterized by an affine parameter rather than proper time.

Newtonian limit. For slow motion ($dx^i/d\tau \ll c$) in a weak, static field, the geodesic equation reduces to $\ddot{x}^i = -\partial_i \Phi$ provided

where $\Phi$ is the Newtonian potential. Newtonian gravity is thus the statement that clocks run at a rate set by the local potential; for most everyday purposes the time part of the metric is gravity. (Derivation: Tensor Formalism — The Newtonian Limit.)

Curvature: tidal forces

Curvature is measured by the Riemann tensor, which records how a vector changes when parallel-transported around a small closed loop:

Its physical meaning is geodesic deviation: two neighbouring free-fall worldlines with tangent $u^\mu$ and separation $\xi^\mu$ accelerate relative to one another as

which in the Newtonian limit is the tidal equation $\ddot\xi^i = -(\partial_i\partial_j\Phi)\,\xi^j$. The Riemann tensor is the relativistic tidal tensor. Its contractions are the Ricci tensor $R_{\mu\nu} = R^\rho{}{\mu\rho\nu}$ and the Ricci scalar $R = g^{\mu\nu}R{\mu\nu}$. In four dimensions the Riemann tensor has 20 independent components: 10 are in the Ricci tensor (fixed locally by matter) and 10 in the trace-free Weyl tensor, which describes curvature that propagates through vacuum — tidal fields and gravitational waves.

The Einstein field equations

The field equations relate curvature to the stress–energy tensor $T_{\mu\nu}$:

Key structural facts:

• Conservation is built in. The contracted Bianchi identity $\nabla_\mu G^{\mu\nu} = 0$ holds identically, so the equations force $\nabla_\mu T^{\mu\nu} = 0$. Local energy–momentum conservation, and with it the geodesic motion of small bodies, is a consequence of the field equations rather than an extra assumption.
• Counting. There are 10 equations for the 10 components of $g_{\mu\nu}$, but the Bianchi identity leaves only 6 independent evolution equations; the remaining 4 degrees of freedom are the coordinate (gauge) choice. The 4 equations $G^{0\mu} = (8\pi G/c^4)T^{0\mu}$ contain no second time derivatives and act as constraints on initial data, as Gauss’s law does in electromagnetism.
• Trace-reversed form. Taking the trace and substituting back gives \(R_{\mu\nu} = \frac{8\pi G}{c^4}\left(T_{\mu\nu} - \frac{1}{2}T\,g_{\mu\nu}\right) + \Lambda g_{\mu\nu}\) In vacuum with $\Lambda = 0$ the equations are simply $R_{\mu\nu} = 0$ — which does not mean spacetime is flat, because the Weyl curvature is unconstrained.
• Nonlinearity. Unlike Maxwell’s equations, the field equations are nonlinear in $g_{\mu\nu}$, so solutions cannot be superposed. Exact solutions exist only with high symmetry; general problems such as black-hole mergers require numerical relativity.

Action principle. The same equations follow from varying the Einstein–Hilbert action with respect to $g^{\mu\nu}$:

Lovelock’s theorem (1971) shows that in four dimensions $G_{\mu\nu}$ and $\Lambda g_{\mu\nu}$ are the only divergence-free, second-order tensors built from the metric, which is why the field equations are essentially unique. The full variation is on Tensor Formalism — The Einstein–Hilbert Action.

The Schwarzschild Solution

Karl Schwarzschild found the first exact solution in 1916: the vacuum field outside any static, spherically symmetric mass $M$.

Birkhoff’s theorem guarantees that this is the unique spherically symmetric vacuum solution, even if the source is pulsating or collapsing — so a spherical star cannot emit gravitational waves, and the exterior of the Sun, a neutron star, and a non-rotating black hole are described by the same metric.

Reading the metric.

• The coordinate $r$ is defined so that a sphere at $r$ has area $4\pi r^2$; it is not the proper radial distance.
• The factor $(1 - r_s/r)$ on $dt^2$ is gravitational time dilation: a static clock at $r$ ticks at rate $d\tau/dt = \sqrt{1 - r_s/r}$ relative to one at infinity.
• The inverse factor on $dr^2$ stretches radial distances: the proper distance between two spheres exceeds the difference of their $r$ values.
• For $r \gg r_s$ the metric approaches Minkowski with $g_{00} \approx -(1 + 2\Phi/c^2)$, $\Phi = -GM/r$, recovering Newton.
• At $r = r_s$ the $g_{rr}$ component diverges, but this is a coordinate singularity: curvature invariants are finite there, and other coordinates (Eddington–Finkelstein, Kruskal–Szekeres) pass smoothly through. The surface $r = r_s$ is the event horizon. The genuine curvature singularity is at $r = 0$. See Black Holes.

For the Sun $r_s \approx 2.95$ km, for Earth $\approx 8.9$ mm, which is why relativistic corrections in the Solar System are of order $r_s/r \sim 10^{-6}$ to $10^{-9}$.

Characteristic radii

Orbits in Schwarzschild spacetime differ qualitatively from Kepler orbits near the mass. The effective potential for radial motion has an extra attractive term $\propto -r_s L^2/r^3$ that dominates at small $r$, producing three landmark radii:

For a rotating (Kerr) black hole the ISCO moves inward for prograde orbits, down to $GM/c^2$ for maximal spin, raising the efficiency to about 42%.

Gravitational time dilation and redshift

A static clock at radius $r$ ticks slower than an identical clock at infinity:

Light emitted at radius $r_e$ and received by a static observer at $r_o$ is shifted by

which is a redshift ($z > 0$) when the emitter is deeper in the well. In the weak-field limit $z \approx (\Phi_o - \Phi_e)/c^2$. At Earth’s surface a height difference of $1$ m changes clock rates by about $1.1\times10^{-16}$ — now directly measurable: optical atomic clocks resolve the redshift across a single millimetre-scale atomic sample (JILA, 2022), and “relativistic geodesy” uses clock comparisons to measure height differences.

Proper time and the clock examples

Proper time is path-dependent: between two events, different worldlines accumulate different $\int d\tau$. Both special- and general-relativistic clock effects are instances of this single fact.

The twin paradox. Alice travels to a star $4$ light-years away at $v = 0.8c$ and returns; Bob stays home. With $\gamma = 1/\sqrt{1-0.8^2} = 5/3$, Bob’s clock records $\Delta t = 10$ years and Alice’s records $\Delta\tau = \Delta t/\gamma = 6$ years. The situation is not symmetric: Bob’s worldline is a geodesic of flat spacetime (maximal proper time), Alice’s is not. No gravity is needed to resolve it — see Special Relativity.

GPS. Satellite clocks orbit at $r \approx 26{,}560$ km with $v \approx 3.87$ km/s. Their smaller potential depth makes them run fast by about $+45.7\ \mu\text{s/day}$ relative to ground clocks; their orbital speed makes them run slow by about $-7.2\ \mu\text{s/day}$. The net $+38.5\ \mu\text{s/day}$ would accumulate a ranging error of roughly $10$ km per day if uncorrected, so the satellite clocks are deliberately offset in frequency before launch.

Experimental Tests

GR has passed every experimental test to date, across more than twenty orders of magnitude in field strength. The parameterized post-Newtonian (PPN) formalism quantifies Solar-System tests: metric theories are characterized by parameters such as $\gamma$ (space curvature per unit mass) and $\beta$ (nonlinearity), both equal to $1$ in GR.

The classical tests

The factor of 2 in light deflection is instructive: half comes from the time part of the metric (the Newtonian-like part), half from the spatial curvature, so light bending tests space curvature directly.

Modern tests

• Frame dragging and geodetic precession. Gravity Probe B (2011) measured gyroscope precession in Earth orbit: geodetic $6601.8 \pm 18.3$ mas/yr (GR: $6606.1$) and frame dragging $37.2 \pm 7.2$ mas/yr (GR: $39.2$). Laser-ranged satellites (LAGEOS, LARES) independently confirm Lense–Thirring precession.
• Binary pulsars. The Hulse–Taylor pulsar (discovered 1974) showed orbital decay matching gravitational-wave emission (1993 Nobel Prize). The Double Pulsar PSR J0737−3039A/B now tests the quadrupole formula to about $10^{-4}$ and measures several independent relativistic effects in a single system (Kramer et al., 2021).
• Stars around Sagittarius A*. The GRAVITY collaboration measured the gravitational redshift of the star S2 at pericentre (2018) and its Schwarzschild orbital precession (2020), a strong-field test around a $4\times10^6\,M_\odot$ black hole.
• Black-hole imaging. The Event Horizon Telescope imaged the shadows of M87* (2019) and Sgr A* (2022); ring diameters match the Kerr prediction to within about 10–20%.
• Gravitational waves. Direct detections since 2015 test GR in the dynamical, strong-field regime: waveform consistency, ringdown spectroscopy, the speed of gravity (equal to $c$ to $\sim10^{-15}$ from GW170817), and, with the exceptionally loud GW250114 (2025), Hawking’s area theorem and the first confident measurement of a ringdown overtone. See Gravitational Waves.
• Cosmology. The expansion history, the CMB, and structure growth are all modelled with GR; see Relativistic Cosmology. This is the regime where GR’s predictions depend most on the unknown dark sector.

Beyond Schwarzschild

The field equations support a small number of important exact solutions and a much larger set of approximation schemes:

Limitations and Open Problems

GR is a classical field theory, and it predicts its own limits:

• Singularities. The Penrose–Hawking singularity theorems show that, under reasonable energy conditions, gravitational collapse and the Big Bang produce geodesics that end in finite proper time. GR cannot describe what happens there.
• Quantum gravity. Quantizing the metric perturbatively gives a non-renormalizable theory; GR is best understood as a low-energy effective field theory valid far below the Planck scale ($\sim 10^{19}$ GeV). See Quantum Gravity.
• Black-hole information. Hawking radiation appears to destroy information, in conflict with quantum unitarity. See Black Holes — The Information Paradox.
• The dark sector. About 95% of the cosmic energy budget is attributed to dark matter and dark energy, detected only gravitationally. Whether these reflect new matter or a failure of GR on large scales is open; recent baryon-acoustic-oscillation data (DESI, 2025) mildly favour dark energy that evolves in time over a pure cosmological constant.
• The cosmological constant problem. Naive quantum-field-theory estimates of vacuum energy exceed the observed $\Lambda$ by up to about 120 orders of magnitude.

See Also

• Special Relativity — the flat-spacetime theory GR generalizes.
• Tensor Formalism & the Field Equations — manifolds, connections, curvature, and derivations of the field equations.
• Black Holes — Schwarzschild and Kerr geometry, horizons, and black-hole thermodynamics.
• Relativistic Cosmology — GR applied to the universe as a whole.
• Gravitational Waves — linearized gravity, radiation, and detection.
• Quantum Gravity — where GR and quantum theory conflict.
• Graduate Topics Hub — how the deep-dive pages fit together.
• Computational Physics — numerical relativity.
• Physics Hub — all physics topics.