Curved Spacetime
Curved spacetime is general relativity's central concept: mass and energy curve the geometry of the four-dimensional spacetime they occupy, and this curvature — not a conventional force acting at a distance — is what objects experience and respond to as gravity.
Curious
Newton described gravity as an invisible force reaching out across empty space between two masses, pulling them together. Einstein's general relativity replaces this picture entirely: mass and energy don't reach out and pull on distant objects — instead, they curve the very geometry of spacetime around them, and other objects simply follow the straightest possible path through that curved geometry, which looks to us like being "pulled" toward the mass.
This is why physicist John Wheeler's famous summary of general relativity is so often quoted: "Spacetime tells matter how to move; matter tells spacetime how to curve." Gravity isn't a force reaching across space in this picture — it's the shape of space and time themselves, responding to whatever mass and energy is present.
Exploring
The mathematical machinery describing curved spacetime is a "metric" — essentially a rule for calculating distances (and time intervals) between nearby points, generalising the flat-spacetime Minkowski metric introduced by Hermann Minkowski for special relativity to allow for curvature. Objects moving under gravity alone (with no other forces acting) follow "geodesics" — paths that are the curved-spacetime generalisation of straight lines, the shortest (or, more precisely, extremal) path between two points given the local curvature.
Deep Dive
The precise mathematical relationship between spacetime curvature and the mass/energy causing it is given by the Einstein field equations, G_μν + Λg_μν = (8πG/c⁴)T_μν, where the left side (the Einstein tensor G_μν, plus a cosmological constant term) describes the geometry and curvature of spacetime, and the right side (the stress-energy tensor T_μν) describes the distribution of mass, energy, and momentum. These ten coupled, highly nonlinear differential equations are notoriously difficult to solve exactly except in situations with substantial symmetry — the Schwarzschild metric (for a spherically symmetric, non-rotating mass) being the first and most famous exact solution, found by Karl Schwarzschild within months of Einstein's 1915 publication.
Geodesics and the geodesic equation: In curved spacetime, free-falling objects (subject to gravity alone) follow geodesics — paths satisfying the geodesic equation, d²x^μ/dτ² + Γ^μ_νρ(dx^ν/dτ)(dx^ρ/dτ) = 0, where the Christoffel symbols Γ^μ_νρ encode the local curvature. This single equation replaces Newton's F = ma for gravitational motion: there is no separate "gravitational force" term at all in this formulation, only the geometry the object moves through.
The mathematics of curved spacetime relies on Riemannian (more precisely, pseudo-Riemannian or Lorentzian) differential geometry, developed in the 19th century by mathematicians including Bernhard Riemann, well before any physical application was known — one of the more striking examples in physics of pure mathematics developed independently later proving to be exactly the tool needed for a major physical theory.
Concept Relationship Map
Requires: equivalence principle, Minkowski spacetime
Enables: Schwarzschild metric, black holes, Einstein field equations [PENDING FULL ENTRY]
Commonly confused with: the rubber-sheet analogy taken too literally (it depicts only spatial curvature, missing the dominant role of time curvature)
Sources
- Einstein, A. (1916). Annalen der Physik, 49(7), 769–822.
- Dyson, F.W., Eddington, A.S., Davidson, C. (1920). "A Determination of the Deflection of Light by the Sun's Gravitational Field." Philosophical Transactions of the Royal Society A, 220, 291–333.