Unique Derivative Operator: Reparametrization. Metric Part 2.

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Moving on from first part.

Suppose ∇ is a derivative operator, and gab is a metric, on the manifold M. Then ∇ is compatible with gab iff ∇a gbc = 0.

Suppose γ is an arbitrary smooth curve with tangent field ξa and λa is an arbitrary smooth field on γ satisfying ξnnλa = 0. Then

ξnn(gabλaλb) = gabλaξnnλb + gabλbξnnλa + λaλbξnngab

= λaλbξnngab

Suppose first that ∇ngab = 0. Then it follows immediately that ξnngabλaλb = 0. So ∇ is compatible with gab. Suppose next that ∇ is compatible with gab. Then ∀ choices of γ and λa (satisfying ξnnλa =0), we have λaλbξnngab = 0. Since the choice of λa (at any particular point) is arbitrary and gab is symmetric, it follows that ξnngab = 0. But this must be true for arbitrary ξa (at any particular point), and so we have ∇ngab = 0.

Note that the condition of compatibility is also equivalent to ∇agbc = 0. Hence,

0 = gbnaδcn = gbna(gnrgrc) = gbngnragrc + gbngrcagnr

= δbragrc + gbngrcagnr = ∇agbc + gbngrcagnr.

So if ∇agbc = 0,it follows immediately that ∇agbc = 0. Conversely, if ∇agbc =0, then gbngrcagnr = 0. And therefore,

0 = gpbgscgbngrcagnr = δnpδrsagnr = ∇agps

The basic fact about compatible derivative operators is the following.

Suppose gab is a metric on the manifold M. Then there is a unique derivative operator on M that is compatible with gab.

It turns out that if a manifold admits a metric, then it necessarily satisfies the countable cover condition. And then it guarantees the existence of a derivative operator.) We do prove that if M admits a derivative operator ∇, then it admits exactly one ∇′ that is compatible with gab.

Every derivative operator ∇′ on M can be realized as ∇′ = (∇, Cabc), where Cabc is a smooth, symmetric field on M. Now

∇′agbc = ∇agbc + gnc Cnab + gbn Cnac = ∇agbc + Ccab + Cbac. So ∇′ will be compatible with gab (i.e., ∇′agbc = 0) iff

agbc = −Ccab − Cbac —– (1)

Thus it suffices for us to prove that there exists a unique smooth, symmetric field Cabc on M satisfying equation (1). To do so, we write equation (1) twice more after permuting the indices:

cgab = −Cbca − Cacb,

bgac = −Ccba − Cabc

If we subtract these two from the first equation, and use the fact that Cabc is symmetric in (b, c), we get

Cabc = 1/2 (∇agbc − ∇bgac − ∇cgab) —– (2)

and, therefore,

Cabc = 1/2 gan (∇ngbc − ∇bgnc − ∇cgnb) —– (3)

This establishes uniqueness. But clearly the field Cabc defined by equation (3) is smooth, symmetric, and satisfies equation (1). So we have existence as well.

In the case of positive definite metrics, there is another way to capture the significance of compatibility of derivative operators with metrics. Suppose the metric gab on M is positive definite and γ : [s1, s2] → M is a smooth curve on M. We associate with γ a length

|γ| = ∫s1s2 gabξaξb ds,

where ξa is the tangent field to γ. This assigned length is invariant under reparametrization. For suppose σ : [t1, t2] → [s1, s2] is a diffeomorphism we shall write s = σ(t) and ξ′a is the tangent field of γ′ = γ ◦ σ : [t1, t2] → M. Then

ξ′a = ξads/dt

We may as well require that the reparametrization preserve the orientation of the original curve – i.e., require that σ (t1) = s1 and σ (t2) = s2. In this case, ds/dt > 0 everywhere. (Only small changes are needed if we allow the reparametrization to reverse the orientation of the curve. In that case, ds/dt < 0 everywhere.) It

follows that

|γ’| = ∫t1t2 (gabξ′aξ′b)1/2 dt = ∫t1t2 (gabξaξb)1/2 ds/dt

= ∫s1s2 (gabξaξb)1/2 ds = |γ|

Let us say that γ : I → M is a curve from p to q if I is of the form [s1, s2], p = γ(s1), and q = γ(s2). In this (positive definite) case, we take the distance from p to q to be

d(p,q)=g.l.b. |γ|:γ is a smooth curve from p to q.

Further, we say that a curve γ : I → M is minimal if, for all s ∈ I, ∃ an ε > 0 such that, for all s1, s2 ∈ I with s1 ≤ s ≤ s2, if s2 − s1 < ε and if γ′ = γ|[s1, s2] (the restriction of γ to [s1, s2]), then |γ′| = d(γ(s1), γ(s2)) . Intuitively, minimal curves are “locally shortest curves.” Certainly they need not be “shortest curves” outright. (Consider, for example, two points on the “equator” of a two-sphere that are not antipodal to one another. An equatorial curve running from one to the other the “long way” qualifies as a minimal curve.)

One can characterize the unique derivative operator compatible with a positive definite metric gab in terms of the latter’s associated minimal curves. But in doing so, one has to pay attention to parametrization.

Let us say that a smooth curve γ : I → M with tangent field ξa is parametrized by arc length if ∀ ξa, gabξaξb = 1. In this case, if I = [s1, s2], then

|γ| = ∫s1s2 (gabξaξb)1/2 ds = ∫s1s2 1.ds = s2 – s1

Any non-trivial smooth curve can always be reparametrized by arc length.

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 ∇.