Metric. Part 1.

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A (semi-Riemannian) metric on a manifold M is a smooth field gab on M that is symmetric and invertible; i.e., there exists an (inverse) field gbc on M such that gabgbc = δac.

The inverse field gbc of a metric gab is symmetric and unique. It is symmetric since

gcb = gnb δnc = gnb(gnm gmc) = (gmn gnb)gmc = δmb gmc = gbc

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

g′bc = g′nc δnb = g′nc(gnm gmb) = (gmn g′nc) gmb = δmc gmb = gcb = gbc

(Here again we use the symmetry of gnm for the third equality; and we use the symmetry of gcb for the final equality.) The inverse field gbc of a metric gab 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 gab on the manifold M (not necessarily symmetric and not necessarily smooth), the following conditions are equivalent.

(1) There is a tensor field gbc on M such that gabgbc = δac.

(2) ∀ p in M, and all vectors ξa at p, if gabξa = 0, then ξa =0.

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

Untitled

In the presence of a metric gab, it is customary to adopt a notation convention for “lowering and raising indices.” Consider first the case of vectors. Given a contravariant vector ξa at some point, we write gab ξa as ξb; and given a covariant vector ηb, we write gbc ηb as ηc. 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 αcab at a point, for example, how should we write gmc αcab? As αmab? Or as αamb? Or as αabm? 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 αabc or αacb or αcab. (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 gmc αabc as αabm; gmc αacb as αamb; 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 αabc and αacb (at some point) need not be equal. Here is an example. Suppose ξ1a, ξ2a, … , ξna is a basis for the tangent space at a point p. Further suppose αabc = ξia ξjb ξkc at the point. Then αacb = ξia ξjc ξkb. Hence, lowering indices, we have αabc =ξia ξjb ξkc but αacb =ξia ξjc ξib at p. These two will not be equal unless j = k.

We have reserved special notation for two tensor fields: the index substiution field δba and the Riemann curvature field Rabcd (associated with some derivative operator). Our convention will be to write these as δab and Rabcd – i.e., with contravariant indices before covariant ones. As it turns out, the order does not matter in the case of the first since δab = δba. (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 αb:

δbaαb = (gbngamδnmb = gbnganαb = gbngnaαb = δabαb

Now suppose gab 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 ξ1a, ξ2a,…, ξna for the tangent space at p such that

gabξia ξib = +1 if 1≤i≤m

gabξiaξib = −1 if m<i≤n

gabξiaξjb = 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 gab 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 gab

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 gab has signature (m, n − m), and ξ1a, ξ2a, . . . , ξna is an orthonormal basis at a point. Further, suppose μa and νa are vectors there. If

μa = ∑ni=1 μi ξia and νa = ∑ni=1 νi ξia, then it follows from the linearity of gab that

gabμa νb = μ1ν1 +…+ μmνm − μ(m+1)ν(m+1) −…−μnνn.

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

gabμaνb = μ1ν1 +…+ μnνn

And where it is Lorentzian,

gab μaνb = 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 gab and ∇ are both defined on the manifold M. Further suppose

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

Disjointed Regularity in Open Classes of Elementary Topology

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Let x, y, … denote first-order structures in St𝜏, x ≈ y will denote isomorphism.

x ∼n,𝜏 y means that there is a sequence 0 ≠ I0 ⊆ …. ⊆ In of sets of 𝜏-partial isomorphism of finite domain so that, for i < j ≤ n, f ∈ Ii and a ∈ x (respectively, b ∈ y), there is g ∈ Ij such that g ⊇ f and a ∈ Dom(g) (respectively, b ∈ Im(g)). The later is called the extension property.

x ∼𝜏 y means the above holds for an infinite chain 0 ≠ I0 ⊆ …. ⊆ In ⊆ …

Fraïssé’s characterization of elementary equivalence says that for finite relational vocabularies: x ≡ y iff x ∼n,𝜏 y. To have it available for vocabularies containing function symbols add the complexity of terms in atomic formulas to the quantifier rank. It is well known that for countable x, y : x ∼𝜏 y implies x ≈ y.

Given a vocabulary 𝜏 let 𝜏 be a disjoint renaming of 𝜏. If x, y ∈ St𝜏 have the same power, let y be an isomorphic copy of y sharing the universe with x and renamed to be of type 𝜏. In this context, (x, y) will denote the 𝜏 ∪ 𝜏-structure that results of expanding x with the relations of y.

Lemma: There is a vocabulary 𝜏+ ⊇ 𝜏 ∪ 𝜏 such that for each finite vocabulary 𝜏0 ⊆ 𝜏 there is a sequence of elementary classes 𝛥1 ⊇ 𝛥2 ⊇ 𝛥3 ⊇ …. in St𝜏+ such that if 𝜋 = 𝜋𝜏+,𝜏∪𝜏 then (1) 𝜋(𝛥𝑛) = {(x,y) : |x| = |y| ≥ 𝜔, x ≡n,𝜏0 y}, (2) 𝜋(⋂n 𝛥n) = {(x, y) : |x| = |y| ≥ 𝜔, x ∼𝜏0 y}. Moreover, ⋂n𝛥n is the reduct of an elementary class.

Proof. Let 𝛥 be the class of structures (x, y, <, a, I) where < is a discrete linear order with minimum but no maximum and I codes for each c ≤ a a family Ic = {I(c, i, −, −)}i∈x of partial 𝜏0-𝜏0–isomorphisms from x into y, such that for c < c’ ≤ a : Ic ⊆ Ic and the extension property holds. Describe this by a first-order sentence 𝜃𝛥 of type 𝜏+ ⊇ 𝜏0 ∪ 𝜏0 and set 𝛥𝑛 = ModL(𝜃𝛥 ∧ ∃≥n x(x ≤ a)}. Then condition (1) in the Lemma is granted by Fraïssé’s characterization and the fact that x being (2) is granted because (x, y, <, a, I) ∈ ⋂n𝛥n iff < contains an infinite increasing 𝜔-chain below a, a ∑11 condition.

A topology on St𝜏 is invariant if its open (closed) classes are closed under isomorphic structures. Of course, it is superfluous if we identify isomorphic structures.

Theorem: Let Γ be a regular compact topology finer than the elementary topology on each class St𝜏 such that the countable structures are dense in St𝜏 and reducts and renamings are continuous for these topologies. Then Γ𝜏 is the elementary topology ∀ 𝜏.

Proof: We show that any pair of disjoint closed classes C1, C2 of Γ𝜏 may be separated by an elementary class. Assume this is not the case since Ci are compact in the topology Γ𝜏 then they are compact for the elementary topology and, by regularity of the latter, ∃ xi ∈ Ci such that x1 ≡ x2 in L𝜔𝜔(𝜏). The xi must be infinite, otherwise they would be isomorphic contradicting the disjointedness of the Ci. By normality of Γ𝜏, there are towers Ui ⊆ Ci ⊆ Ui ⊆ Ci, i = 1,2, separating the Ci with Ui, Ui open and Ci, Ci closed in Γ𝜏 and disjoint. Let I be a first-order sentence of type 𝜏 ⊇ 𝜏 such that (z, ..) |= I ⇔ z is infinite, and let π be the corresponding reduct operation. For fixed n ∈ ω and the finite 𝜏0  ⊆ 𝜏, let t be a first-order sentence describing the common ≡n,𝜏0 – equivalence class of x1, x2. As,

(xi,..) ∈ Mod𝜏(I) ∩ π-1 Mod(t) ∩ π-1Ui, i = 1, 2,..

and this class is open in Γ𝜏‘ by continuity of π, then by the density hypothesis there are countable xi ∈ Ui , i = 1, 2, such that x1n,𝜏 x2. Thus for some expansion of (x1, x2),

(x, x,..) ∈ 𝛥n,𝜏0 ∩ 𝜋1−1(𝐶1) ∩ (𝜌𝜋2)−1(C2) —– (1)

where 𝛥𝑛,𝜏0 is the class of Lemma, 𝜋1, 𝜋2 are reducts, and 𝜌 is a renaming:

𝜋1(x1, x2, …) = x1 𝜋1 : St𝜏+ → St𝜏∪𝜏 → St𝜏

𝜋2(x1, x2, …) = x2 𝜋2 : St𝜏+ → St𝜏∪𝜏 → St𝜏

𝜌(x2) = x2 𝜌 : St𝜏 → St𝜏

Since the classes (1) are closed by continuity of the above functors then ⋂n𝛥n,𝜏0 ∩ 𝜋1−1(C1) ∩ (𝜌𝜋2)−1(C2) is non-emtpy by compactness of Γ𝜏+. But ⋂n𝛥n,𝜏0 = 𝜋(V) with V elementary of type 𝜏++ ⊇ 𝜏+. Then

V ∩ π-1π1-1(U1) ∩ π-1(ρπ2)-1 (U2) ≠ 0

is open for ΓL++ and the density condition it must contain a countable structure (x1, x*2, ..). Thus (x1, x*2, ..) ∈ ∩n 𝛥𝑛,𝜏0, with xi ∈ Ui ⊆ Ci. It follows that x1 ~𝜏0 x2 and thus x1 |𝜏0 ≈ x2 |𝜏0. Let δ𝜏0 be a first-order sentence of type 𝜏 ∪ 𝜏* ∪{h} such that (x, y*, h) |= δ𝜏0 ⇔ h : x |𝜏0 ≈ y|𝜏0. By compactness,

(∩𝜏0fin𝜏 Mod𝜏∪𝜏*∪{f}𝜏0)) ∩ π1-1(C1) ∩ (ρπ2)-1(C2) ≠ 0

and we have h : x1 ≈ x2, xi ∈ Ci, contradicting the disjointedness of Ci. Finally, if C is a closed class of Γ𝜏 and x ∉ C, clΓ𝜏{x} is disjoint from C by regularity of Γ𝜏. Then clΓ𝜏{x} and C may be separated by open classes of elementary topology, which implies C is closed in this topology.