metric

A metric is a mathematical generalization of distance, capturing some of its characteristics. A metric is a function that maps pairs of points in a space to non-negative numbers such that the metric from a point to itself is zero, that applying it to the reverse (i.e., B to A instead of A to B) yields the same result, and that the metric from A to C is always less than or equal to the metric from A to B plus that of B to C. Ordinary distance adheres to these conditions, and a mathematical term for it is Euclidean metric. In physics, such metrics of non-straight lines are of interest, e.g., specifying "the metric of A to C through B", and for curved lines in continuous spaces, using a line integral.

Within relativity and modern cosmology, a slightly-more general variant of this mathematical metric concept is used (and referred to as a metric, i.e., within such physics, the term is used slightly differently) to describe spacetimes: the same conditions hold except that there can be some particular pairs of points that have a zero metric between them even though the two points are distinct. (These metrics are for a distance-like quantity that is not normal spatial distance, but how far apart two points in spacetime, a distance-like equality that applies points differing in both space and time.) Examples of metrics used in astrophysics:

• Minkowski metric - for special relativity (relativistically-invariant, i.e., invariant under the Lorentz transformation).
• Schwarzschild metric - for general relativity given the influence of a mass.
• Kerr metric - a generalization of the Schwarzschild metric including the effects of rotation, used for the Kerr black hole model.
• Reissner-Nordström metric - generalization including mass and electric charge.
• Kerr-Newman metric - generalization including all three: mass, rotation, and charge.
• Robertson-Walker metric incorporating GR and the expansion/contraction of a model universe.

Special cases or variations on GR (modified GR) are often defined by a formula for its particular metric (see theoretical modified GR metric).