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 below). A point in spacetime may
be referred to as an event. Each event has four coordinates (t, x, y, z).

Reference frame

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. This is the central lesson of special
relativity.

The central lesson of general relativity is that spacetime cannot be a fixed
background, but is rather a network of evolving relationships.

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

     ds2 = c2dt - 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.

The spacetime intervals on a manifold define a pseudo-metric called the
Lorentz metric. This metric is 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.

One of the simplest interesting examples of a spacetime is R4 with the
spacetime interval defined above. This is known as Minkowski space, and is
the usual geometric setting for Special Relativity. In contrast, General
Relativity says that the underlying manifold will not be flat, if gravity is
present, and thus it calls for the use of spacetime rather than Minkowski space.

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.

Some general facts about spacetimes

A compact manifold can be turned into a spacetime if and only if its Euler
characteristic is 0.

Any non-compact 4-manifold can be turned into a spacetime.

Many spacetimes have physical interpretations which most physicists would
consider bizarre or unsettling. For example, a compact spacetime has closed
timelike curves, which violate our usual ideas of causality. For this
reason, mathematical physicists usually consider only restricted subsets of
all the possible spacetimes. One way to do this is to study "realistic"
solutions of the equations of General Relativity. Another way is add some
additional "physically reasonable" but still fairly general geometric
restrictions, and try to prove interesting things about the resulting
spacetimes. The latter approach has lead to some important results, most
notably the Penrose-Hawking singularity theorems.

In mathematical physics it is also usual to restrict the manifold to be
connected and Hausdorff. A Hausdorff spacetime is always paracompact.

Is Spacetime Quantized?

Current research is focused on the nature of spacetime at the Planck scale.
Loop quantum gravity, string theory, and black hole thermodynamics all
predict a quantized spacetime with agreement on the order of magnitude. Loop
quantum gravity even makes precise predictions about the geometry of
spacetime at the Planck scale.