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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 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.
, is Minkowski space (or rather, Minkowski space in a choice of orthonormal basis), a model for spacetime with weak curvature. It is common convention to use greek indices when writing expressions involving tensors in Minkowski space, while Latin indices are reserved for Euclidean space.
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, ... In flat Minkowski space (special relativity), ...
(Some authors alternatively use the negative metric signature of (− + + +), with =, = = =.) Lorentz transformations leave the Minkowski metric invariant, so the d'Alembertian yields a Lorentz scalar. The above coordinate expressions remain valid for the standard coordinates in every inertial frame.
As a Jacobian matrix for the SR Minkowski metric tensor. The Jacobian matrix is the matrix of all first-order partial derivatives of a vector-valued function.
Rindler chart, for = in equation (), plotted on a Minkowski diagram.The dashed lines are the Rindler horizons. The worldline of a body in hyperbolic motion having constant proper acceleration in the -direction as a function of proper time and rapidity can be given by [16]
Because the Minkowski tensor is involved here, it becomes necessary to introduce something called the metric tensor in General Relativity. The metric tensor g is symmetric, and locally reduces to the Minkowski tensor in free fall.