Spacetime

In special relativity and general relativity, time and three-dimensional space are treated together as a single four-dimensional manifold called spacetime (alternatively, space-time; see /Talk for details). A point in spacetime may be referred to as an event. Each event has four coordinates (t, x, y, z).

Just as the x, y, z coordinates of a point depend on the axes one is using, so distances and time intervals, invariant in Newtonian physics, may depend on the reference frame of an observer, in relativistic physics. See length contraction and time dilation.

A spacetime interval between two events is the frame-invariant quantity analagous to distance in Euclidean space. The spacetime interval s along a curve is defined by

ds² = c²dt² - dx² - dy² - dz²

where c is the speed of light (some people flip the signs of the equation). A basic assumption of relativity is that coordinate transformations have to leave intervals invariant. Intervals are invariant under Lorentz transformations.

They form a pseudo-metric very similar to distance in Euclidean space. However, note that whereas distances are always positive, intervals may be positive, zero, or negative. Events with a spacetime interval of zero are separated by the propagation of a light signal. Events with a positive spacetime interval are in each other's future or past, and the value of the interval defines the proper time measured by an observer travelling between them. Spacetime together with this pseudo-metric makes up a pseudo-Riemannian manifold.

Strictly speaking one can also consider events in Newtonian physics as a single spacetime. This is Galilean-Newtonian relativity, and the coordinate systems are related by Galilean transformations. However, since these preserve spatial and temporal distances independently, such a spacetime can be decomposed unarbitrarily, which is not possible in the general case.

See also:

• Galilean transformation

• Lorentz transformation

• Lorentz invariance

• Manifold

• Metric space

/Talk