Metric. Part 1.

A (semi-Riemannian) metric on a manifold M is a smooth field g_ab on M that is symmetric and invertible; i.e., there exists an (inverse) field g^bc on M such that g_abg^bc = δ_a^c.

The inverse field g^bc of a metric g_ab is symmetric and unique. It is symmetric since

g^cb = g^nb δⁿ_c = g^nb(g_nm g^mc) = (g_mn g^nb)g^mc = δ_m^b g^mc = g^bc

(Here we use the symmetry of g^nm for the third equality.) It is unique because if g^′bc is also an inverse field, then

g^′bc = g^′nc δ_n^b = g^′nc(g_nm g^mb) = (g_mn g^′nc) g^mb = δ_m^c g^mb = g^cb = g^bc

(Here again we use the symmetry of g_nm for the third equality; and we use the symmetry of g^cb for the final equality.) The inverse field g^bc of a metric g_ab is smooth. This follows, essentially, because given any invertible square matrix A (over R), the components of the inverse matrix A⁻¹ depend smoothly on the components of A.

The requirement that a metric be invertible can be given a second formulation. Indeed, given any field g_ab on the manifold M (not necessarily symmetric and not necessarily smooth), the following conditions are equivalent.

(1) There is a tensor field g^bc on M such that g_abg^bc = δ_a^c.

(2) ∀ p in M, and all vectors ξ^a at p, if g_abξ^a = 0, then ξ^a =0.

(When the conditions obtain, we say that g_ab is non-degenerate.) To see this, assume first that (1) holds. Then given any vector ξ^a at any point p, if g_ab ξ^a = 0, it follows that ξ^c = δ_a^c ξ^a = g^bc g_ab ξ^a = 0. Conversely, suppose that (2) holds. Then at any point p, the map from (M_p)^a to (M_p)^b defined by ξ^a → g_ab ξ^a is an injective linear map. Since (M_p)^a and (M_p)^b have the same dimension, it must be surjective as well. So the map must have an inverse g^bc defined by g^bc(g_ab ξ^a) = ξ^c or g^bc g_ab = δ_a^c.

In the presence of a metric g_ab, it is customary to adopt a notation convention for “lowering and raising indices.” Consider first the case of vectors. Given a contravariant vector ξ^a at some point, we write g_ab ξ^a as ξ^b; and given a covariant vector η_b, we write g^bc η_b as η^c. The notation is evidently consistent in the sense that first lowering and then raising the index of a vector (or vice versa) leaves the vector intact.

One would like to extend this notational convention to tensors with more complex index structure. But now one confronts a problem. Given a tensor α_c^ab at a point, for example, how should we write g^mc α_c^ab? As α^mab? Or as α^amb? Or as α^abm? In general, these three tensors will not be equal. To get around the problem, we introduce a new convention. In any context where we may want to lower or raise indices, we shall write indices, whether contravariant or covariant, in a particular sequence. So, for example, we shall write α^ab_c or α^a_c^b or α_c^ab. (These tensors may be equal – they belong to the same vector space – but they need not be.) Clearly this convention solves our problem. We write g^mc α^abc as α^abm; g^mc α^a_c^b as α^amb; and so forth. No ambiguity arises. (And it is still the case that if we first lower an index on a tensor and then raise it (or vice versa), the result is to leave the tensor intact.)

We claimed in the preceding paragraph that the tensors α^ab_c and α^a_c^b (at some point) need not be equal. Here is an example. Suppose ξ1^a, ξ2^a, … , ξn^a is a basis for the tangent space at a point p. Further suppose α^abc = ξi^a ξj^b ξk^c at the point. Then α^acb = ξi^a ξj^c ξk^b. Hence, lowering indices, we have α^ab_c =ξi^a ξj^b ξk^c but α^a_c^b =ξi^a ξj_c ξi^b at p. These two will not be equal unless j = k.

We have reserved special notation for two tensor fields: the index substiution field δ_b^a and the Riemann curvature field R^a_bcd (associated with some derivative operator). Our convention will be to write these as δ^a_b and R^a_bcd – i.e., with contravariant indices before covariant ones. As it turns out, the order does not matter in the case of the first since δ^a_b = δ_b^a. (It does matter with the second.) To verify the equality, it suffices to observe that the two fields have the same action on an arbitrary field α^b:

δ_b^aα^b = (g_bng^amδⁿ_m)α^b = g_bng^anα^b = g_bng^naα^b = δ^a_bα^b

Now suppose g_ab is a metric on the n-dimensional manifold M and p is a point in M. Then there exists an m, with 0 ≤ m ≤ n, and a basis ξ1^a, ξ2^a,…, ξn^a for the tangent space at p such that

g_abξi^a ξi^b = +1 if 1≤i≤m

g_abξi^aξi^b = −1 if m<i≤n

g_abξi^aξj^b = 0 if i ≠ j

Such a basis is called orthonormal. Orthonormal bases at p are not unique, but all have the same associated number m. We call the pair (m, n − m) the signature of g_ab at p. (The existence of orthonormal bases and the invariance of the associated number m are basic facts of linear algebraic life.) A simple continuity argument shows that any connected manifold must have the same signature at each point. We shall henceforth restrict attention to connected manifolds and refer simply to the “signature of g_ab”

A metric with signature (n, 0) is said to be positive definite. With signature (0, n), it is said to be negative definite. With any other signature it is said to be indefinite. A Lorentzian metric is a metric with signature (1, n − 1). The mathematics of relativity theory is, to some degree, just a chapter in the theory of four-dimensional manifolds with Lorentzian metrics.

Suppose g_ab has signature (m, n − m), and ξ1^a, ξ2^a, . . . , ξn^a is an orthonormal basis at a point. Further, suppose μ^a and ν^a are vectors there. If

μ^a = ∑ⁿ_i=1 μⁱ ξi^a and ν^a = ∑ⁿ_i=1 νⁱ ξi^a, then it follows from the linearity of g_ab that

g_abμ^a ν^b = μ1ν1 +…+ μmνm − μ(m+1)ν(m+1) −…−μnνn.

In the special case where the metric is positive definite, this comes to

g_abμ^aν^b = μ1ν1 +…+ μnνn

And where it is Lorentzian,

g_ab μ^aν^b = μ1ν1 − μ2ν2 −…− μnνn

Metrics and derivative operators are not just independent objects, but, in a quite natural sense, a metric determines a unique derivative operator.

Suppose g_ab and ∇ are both defined on the manifold M. Further suppose

γ : I → M is a smooth curve on M with tangent field ξ^a and λ^a is a smooth field on γ. Both ∇ and g_ab determine a criterion of “constancy” for λ^a. λ^a is constant with respect to ∇ if ξⁿ∇_nλ^a = 0 and is constant with respect to g_ab if g_ab λ^a λ^b is constant along γ – i.e., if ξⁿ ∇_n (g_ab λ^a λ^b = 0. It seems natural to consider pairs g_ab and ∇ for which the first condition of constancy implies the second. Let us say that ∇ is compatible with g_ab if, for all γ and λ^a as above, λ^a is constant w.r.t. g_ab whenever it is constant with respect to ∇.

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