A **metric tensor** *g* on *M* assigns to each point *p* of *M* a metric *g*_{p} 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

... 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

... 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

... 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

... 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**...