nLab spacetime

Idea

General

A spacetime is a manifold that models space and time in physics.

This is formalized by saying that a spacetime is a smooth Lorentzian space $(X,\mu)$ equipped with a time orientation (see there).

Hence a point in a spacetime is called an event.

In the context of classical general relativity a spacetime is usually in addition assumed to be connected and four-dimensional. A connected Lorentzian manifold is either time orientable or it has a two-sheeted covering which is time orientable.

In classical physics, notably in special relativity and general relativity points in $X$ model coordinates where events can take place from the viewpoint of an observer (“points in space and time”) while the metric $\mu$ models the field of gravity in general relativity.

Intermingling of space and time

The noun “spacetime” is used in both special relativity and general relativity, but is best motivated from the viewpoint of general relativity. Special relativity deals with the Minkowski spacetime only. The Minkowski spacetime allows a canonical choice of global coordinates such that the metric tensor has in every point the form diag(-1, 1, 1, 1), which identifies the first coordinate as representing the time coordinate and the others as representing space coordinates.

Given a general spacetime, there is not necessarily a globally defined coordinate system, and therefore not necessarily a globally defined canonical time coordinate. More specifically, there are spacetimes that admit coordinates defined on subsets where the physical interpretation of the coordinates as modelling time and space coordinates changes over the domain of definition.

(TODO: references and explanations).

Examples

• Minkowski spacetime

black hole spacetimes	vanishing angular momentum	positive angular momentum
vanishing charge	Schwarzschild spacetime	Kerr spacetime
positive charge	Reissner-Nordstrom spacetime	Kerr-Newman spacetime

• asymptotically flat spacetime

• Kerr-Schild spacetime

• piecewise flat spacetime

• gravitational wave

• anti de Sitter spacetime

• FRW spacetime

• Taub-NUT spacetime

• Einstein manifold

• pp-wave spacetime

• KK-monopole

• Malament–Hogarth spacetime

• super spacetime

  • super Minkowski spacetime

• black brane

Related concepts

• spacetime signature

• static spacetime, stationary spacetime

• universal spacetime

• general covariance

• gravity, cosmology, black hole

• particle physics, matter, sigma-model

• force, gauge theory

• mass

References

Books

• Stephen W. Hawking, George F. R. Ellis: The large scale structure of space-time, Cambridge Univ. Press (1973, 2010) [doi:10.1017/CBO9780511524646, Wikipedia entry]

• John Beem, Paul Ehrlich, Global Lorentzian geometry, Marcel Dekker 1981 (and Russian, updated translation, Mir 1985)

Articles

• L. Markus, Line element fields and Lorentz structures on differentiable manifolds, Ann. of Math. (2) 62 (1955), 411–417, MR0073169 jstor

• Roger Penrose, Gravitational collapse and space-time singularities, Phys. Rev. Lett. 14, 57–59

• Paul Ernest, The Mathematics of Space-Time and the Limits of Human Understanding, Philosophy of Mathematics Education Journal, No. 30 (October 2016) ISSN:1465-2978

category: physics, geometry