# Metric Tensor

A metric tensor g on M assigns to each point p of M a metric gp in the tangent space at p in a way that varies smoothly with p. More precisely, given any open subset U of manifold M and any (smooth) vector fields X and Y on U, the real function

is a smooth function of p.

Read more about Metric Tensor:  Components of The Metric, Intrinsic Definitions of A Metric, Arclength and The Line Element, Canonical Measure and Volume Form

### Other articles related to "metric tensor, metric, tensor":

History Of Gravitational Theory - Modern Era (Origin of Gravitation) - General Relativity
... The solutions of the field equations are the components of the metric tensor of spacetime ... A metric tensor describes the geometry of spacetime ... of the universe (predicted by the Robertson-Walker metric) was confirmed by Edwin Hubble in 1929 ...
Metric Tensor - Examples - Lorentzian Metrics From Relativity
... special relativity), with coordinates the metric is For a curve with—for example—constant time coordinate, the length formula with this metric reduces to the usual ... The Schwarzschild metric describes the spacetime around a spherically symmetric body, such as a planet, or a black hole ... With coordinates, we can write the metric as where G (inside the matrix) is the gravitational constant and M the mass of the body ...
Metric Signature
... The signature of a metric tensor (or more generally a symmetric bilinear form, thought of as a quadratic form) is the number of positive, negative and zero eigenvalues of the metric ... If the matrix of the metric tensor is n × n, then the number of positive, negative and zero eigenvalues p, q and r may take values from 0 to n with p + q + r = n ... A Riemannian metric is a metric with a (positive) definite signature ...
Coordinate Conditions - Other Coordinates
... the harmonic and synchronous coordinate conditions, would be satisfied by a metric tensor that equals the Minkowski tensor everywhere ... However, since the Riemann and hence the Ricci tensor for Minkowski coordinates is identically zero, the Einstein equations give zero energy/matter for Minkowski ... is the algebraic statement that the determinant of the metric tensor is −1, which still leaves considerable gauge freedom ...
Bimetric Theory - Explanation
... assumed that the distance between two points in spacetime is given by the metric tensor ... Einstein's field equation is then used to calculate the form of the metric based on the distribution of energy and momentum ... Rosen (1940) has proposed at each point of space-time, a Euclidean metric tensor in addition to the Riemannian metric tensor ...