Search results
Results from the WOW.Com Content Network
A metric tensor at p is a function g p (X p, Y p) which takes as inputs a pair of tangent vectors X p and Y p at p, and produces as an output a real number , so that the following conditions are satisfied:
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.
Here is the inverse matrix to the metric tensor . In other words, = and thus = = = is the dimension of ... The covariant derivative of a function (scalar) ...
In general relativity, the connection plays the role of the gravitational force field with the corresponding gravitational potential being the metric tensor. When the coordinate system and the metric tensor share some symmetry, many of the Γ i jk are zero. The Christoffel symbols are named for Elwin Bruno Christoffel (1829–1900). [7]
In mathematics, a metric space is a set together with a notion of distance between its elements, usually called points. The distance is measured by a function called a metric or distance function. [1] Metric spaces are the most general setting for studying many of the concepts of mathematical analysis and geometry.
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'.
The Minkowski metric η is the metric tensor of Minkowski space. It is a pseudo-Euclidean metric, or more generally, a constant pseudo-Riemannian metric in Cartesian coordinates. As such, it is a nondegenerate symmetric bilinear form, a type (0, 2) tensor.
In mathematics, metric may refer to one of two related, but distinct concepts: A function which measures distance between two points in a metric space A metric tensor , in differential geometry, which allows defining lengths of curves, angles, and distances in a manifold