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A metric tensor field g on M assigns to each point p of M a metric tensor 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 g ( X , Y ) ( p ) = g p ( X p , Y p ) {\displaystyle g(X,Y)(p)=g_{p}(X_{p},Y_{p})} is ...
In general relativity, the metric tensor (in this context often abbreviated to simply the metric) is the fundamental object of study.The metric captures all the geometric and causal structure of spacetime, being used to define notions such as time, distance, volume, curvature, angle, and separation of the future and the past.
A tensor whose components in an orthonormal basis are given by the Levi-Civita symbol (a tensor of covariant rank n) is sometimes called a permutation tensor. Under the ordinary transformation rules for tensors the Levi-Civita symbol is unchanged under pure rotations, consistent with that it is (by definition) the same in all coordinate systems ...
In these coordinates, the Euclidean metric tensor is given by = +. This can be seen via the change of variables formula for the metric tensor, or by computing the differential forms dx , dy via the exterior derivative of the 0-forms x = r cos( θ ) , y = r sin( θ ) and substituting them in the Euclidean metric tensor ds 2 = dx 2 + dy 2 .
Tensor [4] is a tensor package written for the Mathematica system. It provides many functions relevant for General Relativity calculations in general Riemann–Cartan geometries. Ricci [5] is a system for Mathematica 2.x and later for doing basic tensor analysis, available for free.
The metric tensor is a central object in general relativity that describes the local geometry of spacetime (as a result of solving the Einstein field equations). Using the weak-field approximation, the metric tensor can also be thought of as representing the 'gravitational potential'. The metric tensor is often just called 'the metric'.
In mathematics, the Poincaré metric, named after Henri Poincaré, is the metric tensor describing a two-dimensional surface of constant negative curvature. It is the natural metric commonly used in a variety of calculations in hyperbolic geometry or Riemann surfaces .
The signature of a metric tensor is defined as the signature of the corresponding quadratic form. [2] It is the number (v, p, r) of positive, negative and zero eigenvalues of any matrix (i.e. in any basis for the underlying vector space) representing the form, counted with their algebraic multiplicities.