Tensor Calculus in a Riemann Manifold
An elementary introduction to the tensor representations of the diffeomorphism group and to the related covariant differential calculus. Includes, in particular, tensor densities, isometries, and a discussion of the properties of the affine connection. Of
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Tensor Calculus in a Riemann Manifold
Motivated by the discussion of the previous chapter let us then consider a space–time with a Riemann geometrical structure and a pseudo-Euclidean signature. Namely, let us assume that the physical space–time is a differentiable manifold1 equipped with a metric g which defines scalar products according to the postulates of Sect. 2.1, and which is represented by a real symmetric 4 × 4 matrix with time-like and space-like eigenvalues of opposite sign. Following our conventions, we will choose a positive time-like eigenvalue: gμν = diag(+, −, −, −).
(3.1)
We will assume, also, that the so-called “affine connection” of our Riemannian manifold is symmetric and compatible with the metric (see Sect. 3.5 below). We recall, first of all, that the metric eigenvalues—like any matrix eigenvalues— are left unchanged by the action of the so-called “similarity transformations”, represented by the map g → g = U −1 gU , where U is an arbitrary 4 × 4 matrix. However, the metric eigenvalues may change under a general coordinate transformation. In that case, in fact, the metric transformation is determined by Eq. (2.18) which, by introducing the Jacobian matrix J μ ν defined by J μν =
∂x μ , ∂x ν
J −1
μ ν
=
∂x μ , ∂x ν
(3.2)
can be rewritten as μ ν T ν = J −1 α gμν J −1 β ≡ J −1 α μ gμν J −1 β , gαβ
(3.3)
or, in compact matrix form: T g = J −1 g J −1 . 1 Differentiable
(3.4)
manifold: a topological Hausdorff space locally homeomorphic to Rn .
M. Gasperini, Theory of Gravitational Interactions, Undergraduate Lecture Notes in Physics, DOI 10.1007/978-88-470-2691-9_3, © Springer-Verlag Italia 2013
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Tensor Calculus in a Riemann Manifold
This type of transformation is called a “congruence”, and does not preserve in general the eigenvalues of the matrix g. However, it preserves the number of eigenvalues of a given sign, so that the 3 + 1 signature of the metric is left unchanged. This result is also known as the “Sylvester theorem”. In the context of the Riemann geometry the basic notion of inertial observer (or inertial reference system), typical of special relativity, is replaced by the notion of general coordinate system, also called “chart” in the language of differential geometry. The functional relations among different charts are no longer linear, unlike the case of Lorentz transformations. Also, a single chart could be not sufficiently extended to map all points of the given Riemann manifold: in that case we must introduce a collection of charts, called “atlas”. In the region where two charts are intersecting each space–time point is mapped by two different coordinate systems, {x} and {x }: it is thus possible, in that region, to define the coordinate transformation x → x . According to our assumptions on the space–time geometry such a transformation must correspond to a diffeomorphism, i.e. to a bijective, invertible map which is differentiable and has a differentiable inverse. Such a transformation, in particular, has to be characterize
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