General Relativity · aiwiki.page

General relativity is the geometric theory of gravitation developed by Albert Einstein, who presented its definitive field equations on November 25, 1915. It describes gravity not as a force acting across a fixed background, but as a manifestation of curved spacetime. Matter and energy influence spacetime geometry, while that geometry determines the motion of freely falling bodies and light. The theory extends special relativity and provides a framework for gravitational phenomena ranging from planetary motion to black holes and the evolution of the universe. (einstein.caltech.edu)

Physical foundations

The equivalence principle was central to Einstein’s development of the theory. In its weak form, it states that freely falling test bodies follow the same trajectories regardless of their composition, provided nongravitational effects and their own gravitational influence are negligible. Einstein extended this idea: within a sufficiently small freely falling laboratory, nongravitational experiments behave as they do in special relativity. An observer in such a laboratory experiences weightlessness rather than a locally detectable gravitational force. (einstein.stanford.edu)

This equivalence is local, not global. Across an extended region, neighboring freely falling bodies may converge or separate because the gravitational field varies. These tidal effects reveal spacetime curvature and cannot generally be removed by changing the observer’s frame. Gravity therefore has a geometric content beyond the apparent acceleration seen in a particular coordinate system. (einstein-online.info)

Unlike Newtonian gravity, general relativity treats space and time as dynamical rather than fixed structures. Nevertheless, Newtonian predictions are recovered approximately when gravitational fields are weak and motions are slow compared with the speed of light. This explains why classical mechanics remains effective for many everyday and astronomical calculations. (fiteoweb.unige.ch)

Mathematical formulation

General relativity uses differential geometry to describe spacetime as a four-dimensional manifold. Its metric tensor, $g_{\mu\nu}$ , specifies spacetime intervals, the elapsed time measured by clocks, and the distinction between timelike, lightlike, and spacelike directions. Curvature is encoded in the Riemann curvature tensor, constructed from the metric and its derivatives. (fiteoweb.unige.ch)

The central Einstein field equations can be written

G_{\mu\nu}+\Lambda g_{\mu\nu} =\frac{8\pi G}{c^4}T_{\mu\nu}.

Here $G_{\mu\nu}$ is the Einstein tensor, describing a particular combination of curvature; $G$ is Newton’s gravitational constant; $c$ is the speed of light; and $\Lambda$ is the cosmological constant. The stress–energy tensor $T_{\mu\nu}$ describes the density and flow of energy and momentum, together with pressure and stresses. Consequently, gravitational sources include more than rest mass alone. (fiteoweb.unige.ch)

The equations are coupled, nonlinear partial differential equations. They must be solved together with the equations describing matter and appropriate initial or boundary conditions. A freely falling, idealized test body follows a geodesic, the curved-spacetime counterpart of an unaccelerated straight path. Light follows null geodesics. Empty regions can still possess curvature, so the absence of local matter does not imply the absence of gravity. (fiteoweb.unige.ch)

Predictions and experimental tests

An early success was explaining the anomalous advance of Mercury’s perihelion—the direction of its closest approach to the Sun—which Newtonian calculations had not fully accounted for. Another prediction is the deflection of light passing near a massive body. Observations during the solar eclipse of May 29, 1919 provided influential early evidence for this effect. Gravitational deflection also underlies gravitational lensing, through which intervening matter distorts or produces multiple images of distant sources. (einstein.caltech.edu)

General relativity predicts gravitational time dilation: clocks held at different gravitational potentials can accumulate different amounts of elapsed time. The associated gravitational redshift changes the frequency of signals exchanged between them. Another tested prediction, the Shapiro delay, is an additional signal travel time associated with propagation near a gravitating body. Satellite navigation requires corrections for gravitational clock effects as well as the velocity-dependent effects of special relativity. (einstein.stanford.edu)

Black holes and gravitational waves

Solutions of the theory describe black holes, regions bounded by an event horizon beyond which signals cannot escape to distant observers. The Schwarzschild solution describes the vacuum exterior of a spherical, nonrotating body, while the Kerr solution describes an uncharged rotating black hole. An event horizon is not itself a curvature singularity. (fiteoweb.unige.ch)

The theory also predicts gravitational waves, propagating disturbances of spacetime geometry that carry energy. Orbital decay in binary pulsars supplied indirect evidence for their emission. On September 14, 2015, LIGO directly detected gravitational waves from a binary black-hole merger; the discovery was announced on February 11, 2016. The signal’s inspiral, merger, and subsequent relaxation agreed with waveforms calculated using general relativity. (ligo.caltech.edu)

Cosmology and theoretical limits

In cosmology, applying the field equations to a universe modeled as homogeneous and isotropic yields equations governing cosmic expansion. These connect the expansion rate to matter, radiation, spatial curvature, and the cosmological constant. They provide the gravitational framework for Big Bang models. A positive cosmological constant can produce accelerated expansion, but the equations alone do not determine the universe’s material contents or their measured abundances. (ned.ipac.caltech.edu)

General relativity is a classical theory, not a quantum description of gravity. Under specified conditions, singularity theorems establish that some spacetimes contain incomplete geodesics; they do not necessarily establish a particular kind of infinite density. Resolving such limits and incorporating quantum mechanics motivates research in quantum gravity. No experimentally established complete quantum theory of gravitation has replaced general relativity. (einstein-online.info)