Spacetime: Difference between revisions

In special relativity and general relativity, time and three-dimensional space are treated together as a single four-dimensional pseudo-Riemannian manifold called spacetime. A point in spacetime may be referred to as an event. Each event has four coordinates (t, x, y, z); or, in angular coordinates, t, r, θ, and φ.

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 ${\displaystyle (3,1)}$.

Just as the x, y, z spatial coordinates of a point depend on the axes one is using, in special relativity all x, y, z, t coordinates depend on selection of axis, including orientation of time axis. Two observers moving at different speeds have time axis with different orientation. What is pure "space" in one reference frame is mixed up with time in the other. That's one of the main points of special theory of relativity.

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 in special relativity by:

${\displaystyle ds^{2}=c^{2}dt^{2}-dx^{2}-dy^{2}-dz^{2}}$

where c is the speed of light (see sign convention). 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 into spatial coordinates plus temporal coordinates, which is not possible in the general case.

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 led 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.

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.

• Galilean transformation
• Lorentz transformation
• Minkowski space
• Lorentz invariance
• Simultaneity
• Manifold
• Metric space
• Gravity
• Four-vector
• Space-time theories of consciousness