Introduction
In the theories of special relativity and general relativity, time and three-dimensional space are treated together as a single four-dimensional object called a spacetime. Spacetimes are the arenas in which all physical events take place. For example, the motion of planets around the Sun may be described in a particular type of spacetime, or the motion of light around a rotating star may be described in another type of spacetime. A point in spacetime is usually referred to as an event. An example of an event may be the explosion of a star, for example. The worldline of a particle or light beam is the path that this particle or beam takes in the spacetime, this path being a geodesic. In classical physics, geodesics are timelike or null (lightlike).
Mathematics of spacetimes
For physical reasons, a spacetime is mathematically defined as a 4-dimensional, smooth, connected pseudo-Riemannian manifold together with a smooth, Lorentz metric of signature . About each point (event), , on this manifold, we have coordinate charts. Usually, we use Cartesian coordinates . Examples of spacetimes include Minkowski space.
Spacetime topology
The definition of a pseudo-Riemannian manifold includes the Hausdorff condition, and an important property of a Hausdorff space is that it is paracompact. This is useful in relativity.
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.
Reference frames in spacetime
A reference frame is identified with one of these coordinate charts. Physically, any observer in a reference frame can describe the event . Another reference frame may be identified by a second coordinate chart about . Two observers (one in each reference frame) may describe the same event but obtain different descriptions. For example, two observers, one of whom is on Earth, but the other one who is on a fast rocket to Jupiter, may observe a comet crashing into Jupiter (this is the event ). In general, they will disagree about the exact location and timing of this impact, i.e., they will have different 4-tuples (as they are using different coordinate systems). However, the two sets of coordinates are related by a non-singular coordinate transformation (on the intersection of the two charts). An important feature of relativity theory is it's local nature (observers measure physical quantities in their own locality). The local nature of the coordinate charts and the coordinate transformations between them serve to emphasise this fact. In general, global coordinate charts are seldom used in realistic physical situations. In the comet example above, although the kinematic descriptions will, in general, be different, dynamical (physical) laws, such as momentum conservation and the first law of thermodynamics, will still hold. In fact, relativity theory requires more than this in the sense that it stipulates these laws must take the same form in all coordinate systems. This introduces the tensor formalism into relativity.
Distance in spacetime
A spacetime interval between two events is the frame-invariant quantity analogous to distance in Euclidean space. The spacetime interval s along a curve (geodesic) is defined in special relativity by:
where c is the speed of light (see 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 into spatial coordinates plus temporal coordinates, which is not possible in the general case.
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 led to some important results, most notably the Penrose-Hawking singularity theorems.
In general relativity, spacetime is assumed to be 'smooth' and 'continuous' (not just in the mathematical sense). In the theory of quantum mechanics, there is an inherent 'discreteness' present in physics. In attempting to reconcile these two theories, it is sometimes postulated that spacetime should be quantized at the very smallest scales. Current theory 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.
