spacetimes

Spacetimes are the four-dimensional continua that fuse the three spatial dimensions with time into a single manifold endowed with a metric that defines distances and causal relations. In relativity, the geometry of spacetimes governs the motion of matter and the behavior of clocks and light.

Mathematically, spacetime is modeled as a smooth four-dimensional manifold equipped with a Lorentzian metric g. The invariant interval between events is ds^2 = g_mu nu dx^mu dx^nu. The metric has signature typically -+++, though other conventions exist. In special relativity, spacetime is flat (Minkowski space); in general relativity it can be curved, with curvature described by the Riemann tensor.

The Einstein field equations relate spacetime curvature to energy and momentum: G_mu nu = 8πG/c^4 T_mu nu. Free-falling trajectories are geodesics, representing what inertial observers would measure as the path of a freely falling particle.

Causality is encoded by light cones; intervals are classified as timelike, null (lightlike), or spacelike, determining possible causal influence between events. Observers traversing spacetime trace worldlines that reflect their history of events.

Examples include Minkowski spacetime (special relativity), Schwarzschild spacetime around a static spherical mass, Kerr spacetime for rotating black holes, and Friedmann–Lemaître–Robertson–Walker spacetimes used in cosmology.

Consequences of spacetime geometry include gravitational time dilation, gravitational redshift, lensing of light, black holes, gravitational waves, and the expansion of the universe. Observational evidence comes from experiments and observations such as gravitational waves, precise clock timing on Earth and in satellites, and measurements of cosmic expansion.