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Spacetime

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).

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 = 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.

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.

Image Caption: Artist concept of Gravity Probe B orbiting the Earth to measure space-time, a four-dimensional description of the universe including height, width, length, and time. Credit: NASA/Wikipedia

Spacetime


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