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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.
Moreover, the metric is required to be nondegenerate with signature (− + + +). A manifold equipped with such a metric is a type of Lorentzian manifold. Explicitly, the metric tensor is a symmetric bilinear form on each tangent space of that varies
The signature of g is the pair of integers (p, n − p), signifying that there are p positive signs and n − p negative signs in any such expression. Equivalently, the metric has signature (p, n − p) if the matrix g ij of the metric has p positive and n − p negative eigenvalues. Certain metric signatures which arise frequently in ...
A pseudo-Riemannian manifold (M, g) is a differentiable manifold M that is equipped with an everywhere non-degenerate, smooth, symmetric metric tensor g. Such a metric is called a pseudo-Riemannian metric. Applied to a vector field, the resulting scalar field value at any point of the manifold can be positive, negative or zero.
For an overview, Minkowski space is a 4-dimensional real vector space equipped with a non-degenerate, symmetric bilinear form on the tangent space at each point in spacetime, here simply called the Minkowski inner product, with metric signature either (+ − − −) or (− + + +).
In physics, a sign convention is a choice of the physical significance of signs (plus or minus) for a set of quantities, in a case where the choice of sign is arbitrary. . "Arbitrary" here means that the same physical system can be correctly described using different choices for the signs, as long as one set of definitions is used consiste
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The value of the magnitude of an object's four-velocity, i.e. the quantity obtained by applying the metric tensor g to the four-velocity U, that is ‖ U ‖ 2 = U ⋅ U = g μν U ν U μ, is always equal to ±c 2, where c is the speed of light. Whether the plus or minus sign applies depends on the choice of metric signature.