3.6 Riemannian and Pseudo-Riemannian Metrics: The Mathematical Definition |
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The two additional properties satisfied by a Riemannian metric state that it should be not only a symmetric non-degenerate form but also a positive definite one.
The definition of pseudo-Riemannian and Riemannian metrics can be summarized by saying that a metric is a section of the second symmetric tensor power of the cotangent bundle, namely we have:
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In a coordinate patch the metric is described by a symmetric rank two tensor gμν (x). Indeed we can write:
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X, Y X(M ) : g(X, Y) = gμν (x)Xμ(x)Y ν (x) |
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are the local expression of the two vector fields in the considered coordinate patch. The essential point in the definition of the metric is the statement that g(X, Y ) C∞(M ), namely that g(X, Y ) should be a scalar. This implies that the coefficients gμν (x) should transform from a coordinate patch to the next one with the inverse of the transition functions with which transform the vector field components. This is what implies (3.6.1). On the other hand, according to the viewpoint started by Gauss and developed by Riemann, Ricci and Levi Civita, the metric is the mathematical structure which allows to define infinitesimal lengths and, by integration, the length of any curve. Indeed given a curve C : [0, 1] −→ M let t(s) be its tangent vector at any point of the curve C , parameterized s [0, 1]: we define the length of the curve as:
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3.6.1 Signatures
At each point of the manifold p M , the coefficients gμν (p) of a metric g constitute a symmetric non-degenerate real matrix. Such a matrix can always be diagonalized by means of an orthogonal transformation O (p) SO(N ), which obviously varies from point to point namely we can write:
138 |
3 Connections and Metrics |
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where the real numbers λi (p) are the eigenvalues. Each of them depends on the point p M , but none of them can vanish, otherwise the determinant of the metric would be zero which contradicts one of the defining axioms. Consider next the diagonal matrix:
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The matrix M = O · L , where for brevity the dependence on the point p has been omitted, is such that:
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having denoted by sign[x] the function which takes the value 1 if x > 0 and the value −1 if x < 0. Hence we arrive at the following conclusion and definition.
Definition 3.6.3 Let M be a differentiable manifold of dimension m endowed with a metric g. At every point p M by means of a transformation g → ST · g · S, the metric tensor gμν (p) can be reduced to a diagonal matrix with p entries equal to 1 and m − p entries equal to −1. The pair of integers (p, m − p) is named the signature of the metric g
The rationale of the above definition is that the signature of a metric is an intrinsic property of g, independent both from the chosen coordinate patch and from the chosen point. The proof of this statement relies on a theorem proved in 1852 by the English Mathematician James Joseph Sylvester (see Fig. 3.18) and named by him the Law of Inertia of Quadratic Forms [23].
According to Sylvester’s theorem a symmetric non-degenerate matrix A can always be transformed into a diagonal one with ±1 entries by means of a substitution A → BT · A · B. On the other hand no such transformation can alter the signature