Metric Tensor


Also found in: Wikipedia.

metric tensor

[′me·trik ′ten·sər]
(mathematics)
A second rank tensor of a Riemannian space whose components are functions which help define magnitude and direction of vectors about a point. Also known as fundamental tensor.

Metric Tensor

 

the set of quantities that define the geometric properties of a space (the metric of the space). In the general case of an n -dimensional Riemannian space, the metric is defined as the square of the distance ds2Pointe (x1,x2,.......,xn) and (x1 + dx1, x2 + dx2,...., xn)+ dxn)

where xl, x2, . . . , xn are coordinates and the gik are certain functions of the coordinates. The set of the quantities gik forms a second-rank tensor, which is called the metric tensor. This tensor is symmetric, that is, gik = gki The form of the components of the metric tensor gik depends on the choice of the coordinate system, but ds2 does not change in changing from one coordinate system to another, that is, it is invariant with respect to transformations of coordinates. If the metric tensor can be reduced to the form

between two infinitesimally close by selection of the coordinate system, then the space is a plane, Euclidean space. (For a three-dimensional space, ds2 = dx2 + dy2 + dz2, where x1 = x, x2 = y, and x3 = z are the rectangular Cartesian coordinates.) If a metric tensor cannot be reduced to the form (2) by any transformation of coordinates, then the space is curved and the curvature of the space is defined by the metric tensor. In the theory of relativity, the space-time metric is defined by a metric tensor.

G. A. ZISMAN

References in periodicals archive ?
The metric tensor for the metric above, equation (3), is
mu]] of electromagnetic field and on metric tensor [g.
The components of the corresponding metric tensor h and the Christoffel symbols on the manifold N will be denoted by [h.
Thus the metric tensor of the manifold S it will be transformed under the law
In the embedding problem, let us recall, the intrinsic geometry of the spacetime (mainly determined through the metric tensor [G.
Extensive use of components of the Eucidean metric tensor enabled formulation in a material coordinate system.
In [1], the covariant metric tensor exterior to a homogeneous time varying distribution of mass within regions of spherical geometry is defined as:
where [DELTA] denotes the operator of covariant differentiation with respect to the metric tensor g and A, B are nowhere vanishing 1-forms such that g(X, [rho]) = A(X) and g(X, [mu]) = B(X) for all X and [rho], [mu] are called the basic vector fields of the manifold.
ab] is symmetric and is only function of the metric tensor components [g.