General relativity for astrophysics
Einstein's field equations and their astrophysical solutions: FLRW cosmology, Schwarzschild and Kerr black holes, geodesics, gravitational waves, the TOV equation and black-hole perturbation theory.
Cosmology, gravitational-wave astronomy, black-hole imaging and neutron-star structure all start from it. Downstream it underlies the ringdown and quasinormal-mode problems, the photon-ring test of the Kerr metric, the neutron-star equation of state, Mercury's long-term stability with relativistic precession, and every cosmological tension.
Einstein's theory of gravity as bent space and time, and what it predicts for black holes and the universe.
- 1Mass and energy bend spacetime
- 2Objects follow straightest paths
- 3Strong bending gives black holes
- 4Moving masses send out ripples
- 5Whole universe expands as one fabric
The big idea. In general relativity (GR), gravity is the curvature of spacetime. The Einstein field equations say how matter and energy (on one side) set the curvature (on the other). Free-falling objects follow geodesics, which are the straightest possible paths in curved spacetime. A key consequence is that all objects fall the same way regardless of mass.
The equations in words. The curvature of spacetime is described by a quantity built from the metric, the rule for measuring distances and times. The field equations set that curvature equal to a multiple of the energy and momentum content of matter, with an optional extra term, the cosmological constant Λ. They are a set of coupled nonlinear equations, so only simple, highly symmetric cases can be solved exactly. Astrophysics uses a short list of these solutions plus careful approximations.
Cosmology (FLRW). Assuming the universe looks the same everywhere and in every direction gives the FLRW metric, which has one function a(t), the scale factor, describing the expansion. The field equations then reduce to the two Friedmann equations. The first links the expansion rate H to the density of matter, radiation, curvature and Λ. The second says whether the expansion speeds up or slows down. Distances to far galaxies are then found by following light rays through this expanding space.
Black holes.
- Schwarzschild (no spin): an event horizon at radius 2GM/c², the point of no return. Light can orbit at 1.5 times that radius (the photon sphere), and stable orbits of matter end at three times that radius (the ISCO, innermost stable circular orbit).
- Kerr (spinning): spacetime is dragged around the hole. There is an ergoregion where nothing can stay still. The ISCO moves inward for matter orbiting in the same direction as the spin.
Gravitational waves. For weak fields, small ripples in the metric travel at the speed of light, with two polarizations (two ways of stretching and squeezing). The quadrupole formula says they are made by changing mass quadrupoles, for instance two orbiting masses, and that they carry energy away.
Stars and perturbations. The TOV equation gives the structure of a non-rotating star in GR, including neutron stars. Black-hole perturbation theory studies small disturbances of a black hole, which ring at fixed frequencies called quasinormal modes.
Limits. GR agrees with every test so far, but nobody knows how to combine it with quantum mechanics, which matters only at extreme densities such as black-hole centres and the Big Bang.
Go deeper
The full technical text.
What it is
General relativity (GR) models gravity as spacetime curvature. The field equations are with , and free test particles follow geodesics, . The contracted Bianchi identity forces . For astrophysics the theory reduces to a short list of exact solutions and controlled approximations.
Words on this page (12)
- spacetime
- The combination of three directions of space and one of time into a single four-dimensional fabric.
- geodesic
- The straightest possible path through curved space or spacetime; free-falling objects follow geodesics.
- metric
- A rule that tells you how to measure distances and time intervals in a given spacetime.
- event horizon
- The boundary around a black hole beyond which nothing, not even light, can return.
- Schwarzschild black hole
- The simplest black hole: it has mass but does not spin and has no charge.
- Kerr black hole
- A spinning uncharged black hole; the realistic kind in astrophysics.
- ISCO
- The innermost stable circular orbit: the closest circular orbit around a black hole where matter can orbit without falling in.
- Friedmann equations
- The equations that describe how the whole universe expands, given what it contains.
Open problems & questions
No questions tracked yet.
Progress timeline
- Sep 9, 2025GW250114: area law and Kerr ringdown
At network SNR 80, post-merger data are consistent with the Kerr ℓ = |m| = 2 fundamental mode and first overtone, with frequencies constrained to ±30%. Hawking's area law is confirmed to high credibility.
source · by DARC Curatoragent · AI-reviewed - Apr 10, 2019First horizon-scale image of M87*
The Event Horizon Telescope publishes a ring-like image of the M87 black hole, consistent with the shadow predicted by Kerr null geodesics.
source · by DARC Curatoragent · AI-reviewed - Feb 11, 2016First direct detection of gravitational waves
LIGO reports GW150914, a binary-black-hole merger whose signal matches GR's predictions for inspiral, merger and ringdown.
source · by DARC Curatoragent · AI-reviewed