Finsler Space as a Locally Minkowskian Space: Caught Between Curvature and Torsion Tensors.

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The extension of Riemannian “point”-space {xi} into a “line-space” {xi, dxi} make things clearer but not easier: how do you explain to a physicist a geometry supporting at least 3 curvature tensors and five torsion tensors? Not to speak of its usefulness for physics! Fortunately, the “impenetrable forest” by now has become a real, enjoyable park: through the application of the concepts of fibre bundle and non-linear connection. The different curvatures and torsion tensors result from vertical and horizontal parts of geometric objects in the tangent bundle, or in the Finsler bundle of the underlying manifold.

In essence, Finsler geometry is analogous to Riemannian geometry: there, the tangent space in a point p is euclidean space; here, the tangent space is just a normed space, i.e., Minkowski Space. Put differently: A Finsler metric for a differentiable manifold M is a map that assigns to each point x ∈ M a norm on the tangent space TxM. When referred to the almost exclusive use of methods from Riemannian geometry it means that this norm is demanded to derive from the length of a smooth path γ : [a, b] → M defined by ∫ab ∥ dγ(t)/dt ∥ dt. Then Finsler space becomes an example for the class of length spaces.

Starting from the length of the curve,

dγ(p, q):= ∫pq Lx(t), dx(t)/dt dt

the variational principle δdγ(p, q) = 0 leads to the Euler-Lagrange equation

d/dt(∂L/∂x ̇i) – ∂L/∂xi = 0

which may be rewritten into

d2xi/dt2 + 2Gi(xl, x ̇m) = 0

with Gi(xl, x ̇m) = 1/4gkl(-∂L/∂xl + ∂2L/∂xl∂x ̇m), and 2gik = ∂2L/∂x ̇l∂x ̇m, gilgjl = δij. The theory then is developed from the Lagrangian defined in this way. This involves an important object Nil := ∂Gi/∂yl, the geometrical meaning of which is a non-linear connection.

In general, a Finsler structure L(x, y) with y := dx(t))/dt = x ̇ and homogeneous degree 1 in y is introduced, from which the Finsler metric follows as:

fij = fji = ∂(1/2L2)/∂yi∂yj, fijyiyj = L2, yl∂L/∂yl = L, fijyj = L∂L/∂yi

A further totally symmetric tensor Cijk ensues:

Cijk := ∂(1/2L2)/∂yi∂yj∂yk

which will be interpreted as a torsion tensor. As an example of a Finsler metric is the Randers metric.

L(x.y) = bi(x)yi + √(aij(x)yiyj)

The Finsler metric following is

fik = bibk + aik + 2b(iak)lyˆl − aillakmm(bnn)

with yˆk := yk(alm(x)ylym)−1/2. Setting aij = ηij, yk = x ̇k, and identifying bi with the electromagnetic 4-potential eAi leads back to the Lagrangian for the motion of a charged particle.

In this context, a Finsler space thus is called a locally Minkowskian space if there exists a coordinate system, in which the Finsler structure is a function of yi alone. The use of the “element of support” (xi, dxk ≡ yk) essentially amounts to a step towards working in the tangent bundle TM of the manifold M.

Is General Theory of Relativity a Gauge Theory? Trajectories of Diffeomorphism.

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Historically the problem of observables in classical and quantum gravity is closely related to the so-called Einstein hole problem, i.e. to some of the consequences of general covariance in general relativity (GTR).

The central question is the physical meaning of the points of the event manifold underlying GTR. In contrast to pure mathematics this is a non-trivial point in physics. While in pure differential geometry one simply decrees the existence of, for example, a (pseudo-) Riemannian manifold with a differentiable structure (i.e., an appropriate cover with coordinate patches) plus a (pseudo-) Riemannian metric, g, the relation to physics is not simply one-one. In popular textbooks about GTR, it is frequently stated that all diffeomorphic (space-time) manifolds, M are physically indistinguishable. Put differently:

S − T = Riem/Diff —– (1)

This becomes particularly virulent in the Einstein hole problem. i.e., assuming that we have a region of space-time, free of matter, we can apply a local diffeomorphism which only acts within this hole, letting the exterior invariant. We get thus in general two different metric tensors

g(x) , g′(x) := Φ ◦ g(x) —– (2)

in the hole while certain inital conditions lying outside of the hole are unchanged, thus yielding two different solutions of the Einstein field equations.

Many physicists consider this to be a violation of determinism (which it is not!) and hence argue that the class of observable quantities have to be drastically reduced in (quantum) gravity theory. They follow the line of reasoning developed by Dirac in the context of gauge theory, thus implying that GTR is essentially also a gauge theory. This then winds up to the conclusion:

Dirac observables in quantum gravity are quantities which are diffeomorphism invariant with the diffeomorphism group, Diff acting from M to M, i.e.

Φ : M → M —– (3)

One should note that with respect to physical observations there is no violation of determinism. An observer can never really observe two different metric fields on one and the same space-time manifold. This can only happen on the mathematical paper. He will use a fixed measurement protocol, using rods and clocks in e.g. a local inertial frame where special relativity locally applies and then extend the results to general coordinate frames.

We get a certain orbit under Diff if we start from a particular manifold M with a metric tensor g and take the orbit

{M, Φ ◦g} —– (4)

In general we have additional fields and matter distributions on M which are transformd accordingly.

Note that not even scalars are invariant in general in the above sense, i.e., not even the Ricci scalar is observable in the Dirac sense:

R(x) ≠ Φ ◦ R(x) —– (5)

in the generic case. Thus, this would imply that the class of admissible observables can be pretty small (even empty!). Furthermore, it follows that points of M are not a priori distinguishable. On the other hand, many consider the Ricci scalar at a point to be an observable quantity.

This winds up to the question whether GTR is a true gauge theory or perhaps only apparently so at a first glance, while on a more fundamental level it is something different. In the words of Kuchar (What is observable..),

Quantities non-invariant under the full diffeomorphism group are observable in gravity.

The reason for these apparently diverging opinions stems from the role reference systems are assumed to play in GTR with some arguing that the gauge property of general coordinate invariance is only of a formal nature.

In the hole argument it is for example argued that it is important to add some particle trajectories which cross each other, thus generating concrete events on M. As these point events transform accordingly under a diffeomorphism, the distance between the corresponding coordinates x, y equals the distance between the transformed points Φ(x), Φ(y), thus being a Dirac observable. On the other hand, the coordinates x or y are not observable.

One should note that this observation is somewhat tautological in the realm of Riemannian geometry as the metric is an absolute quantity, put differently (and somewhat sloppily), ds2 is invariant under passive and by the same token active coordinate transformation (diffeomorphisms) because, while conceptually different, the transformation properties under the latter operations are defined as in the passive case. In the case of GTR this absolute quantity enters via the equivalence principle i.e., distances are measured for example in a local inertial frame (LIF) where special relativity holds and are then generalized to arbitrary coordinate systems.

Sobolev Spaces

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For any integer n ≥ 0, the Sobolev space Hn(R) is defined to be the set of functions f which are square-integrable together with all their derivatives of order up to n:

f ∈ Hn(R) ⇐⇒ ∫-∞ [f2 + ∑k=1n (dkf/dxk)2 dx ≤ ∞

This is a linear space, and in fact a Hilbert space with norm given by:

∥f∥Hn = ∫-∞ [f2 + ∑k=1n (dkf/dxk)2) dx]1/2

It is a standard fact that this norm of f can be expressed in terms of the Fourier transform fˆ (appropriately normalized) of f by:

∥f∥2Hn = ∫-∞ [(1 + y2)n |fˆ(y)|2 dy

The advantage of that new definition is that it can be extended to non-integral and non-positive values. For any real number s, not necessarily an integer nor positive, we define the Sobolev space Hs(R) to be the Hilbert space of functions associated with the following norm:

∥f∥2Hs = ∫-∞ [(1 + y2)s |fˆ(y)|2 dy —– (1)

Clearly, H0(R) = L2(R) and Hs(R) ⊂ Hs′(R) for s ≥ s′ and in particular Hs(R) ⊂ L2(R) ⊂ H−s(R), for s ≥ 0. Hs(R) is, for general s ∈ R, a space of (tempered) distributions. For example δ(k), the k-th derivative of a delta Dirac distribution, is in H−k−1/2</sup−ε(R) for ε > 0.

In the case when s > 1/2, there are two classical results.

Continuity of Multiplicity:

If s > 1/2, if f and g belong to Hs(R), then fg belongs to Hs(R), and the map (f,g) → fg from Hs × Hs to Hs is continuous.

Denote by Cbn(R) the space of n times continuously differentiable real-valued functions which are bounded together with all their n first derivatives. Let Cnb0(R) be the closed subspace of Cbn(R) of functions which converges to 0 at ±∞ together with all their n first derivatives. These are Banach spaces for the norm:

∥f∥Cbn = max0≤k≤n supx |f(k)(x)| = max0≤k≤n ∥f(k)∥ C0b

Sobolev embedding:

If s > n + 1/2 and if f ∈ Hs(R), then there is a function g in Cnb0(R) which is equal to f almost everywhere. In addition, there is a constant cs, depending only on s, such that:

∥g∥Cbn ≤ c∥f∥Hs

From now on we shall always take the continuous representative of any function in Hs(R). As a consequence of the Sobolev embedding theorem, if s > 1/2, then any function f in Hs(R) is continuous and bounded on the real line and converges to zero at ±∞, so that its value is defined everywhere.

We define, for s ∈ R, a continuous bilinear form on H−s(R) × Hs(R) by:

〈f, g〉= ∫-∞ (fˆ(y))’ gˆ(y)dy —– (2)

where z’ is the complex conjugate of z. Schwarz inequality and (1) give that

|< f , g >| ≤ ∥f∥H−s∥g∥Hs —– (3)

which indeed shows that the bilinear form in (2) is continuous. We note that formally the bilinear form (2) can be written as

〈f, g〉= ∫-∞ f(x) g(x) dx

where, if s ≥ 0, f is in a space of distributions H−s(R) and g is in a space of “test functions” Hs(R).

Any continuous linear form g → u(g) on Hs(R) is, due to (1), of the form u(g) = 〈f, g〉 for some f ∈ H−s(R), with ∥f∥H−s = ∥u∥(Hs)′, so that henceforth we can identify the dual (Hs(R))′ of Hs(R) with H−s(R). In particular, if s > 1/2 then Hs(R) ⊂ C0b0 (R), so H−s(R) contains all bounded Radon measures.

Abelian Categories, or Injective Resolutions are Diagrammatic. Note Quote.

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Jean-Pierre Serre gave a more thoroughly cohomological turn to the conjectures than Weil had. Grothendieck says

Anyway Serre explained the Weil conjectures to me in cohomological terms around 1955 – and it was only in these terms that they could possibly ‘hook’ me …I am not sure anyone but Serre and I, not even Weil if that is possible, was deeply convinced such [a cohomology] must exist.

Specifically Serre approached the problem through sheaves, a new method in topology that he and others were exploring. Grothendieck would later describe each sheaf on a space T as a “meter stick” measuring T. The cohomology of a given sheaf gives a very coarse summary of the information in it – and in the best case it highlights just the information you want. Certain sheaves on T produced the Betti numbers. If you could put such “meter sticks” on Weil’s arithmetic spaces, and prove standard topological theorems in this form, the conjectures would follow.

By the nuts and bolts definition, a sheaf F on a topological space T is an assignment of Abelian groups to open subsets of T, plus group homomorphisms among them, all meeting a certain covering condition. Precisely these nuts and bolts were unavailable for the Weil conjectures because the arithmetic spaces had no useful topology in the then-existing sense.

At the École Normale Supérieure, Henri Cartan’s seminar spent 1948-49 and 1950-51 focussing on sheaf cohomology. As one motive, there was already de Rham cohomology on differentiable manifolds, which not only described their topology but also described differential analysis on manifolds. And during the time of the seminar Cartan saw how to modify sheaf cohomology as a tool in complex analysis. Given a complex analytic variety V Cartan could define sheaves that reflected not only the topology of V but also complex analysis on V.

These were promising for the Weil conjectures since Weil cohomology would need sheaves reflecting algebra on those spaces. But understand, this differential analysis and complex analysis used sheaves and cohomology in the usual topological sense. Their innovation was to find particular new sheaves which capture analytic or algebraic information that a pure topologist might not focus on.

The greater challenge to the Séminaire Cartan was, that along with the cohomology of topological spaces, the seminar looked at the cohomology of groups. Here sheaves are replaced by G-modules. This was formally quite different from topology yet it had grown from topology and was tightly tied to it. Indeed Eilenberg and Mac Lane created category theory in large part to explain both kinds of cohomology by clarifying the links between them. The seminar aimed to find what was common to the two kinds of cohomology and they found it in a pattern of functors.

The cohomology of a topological space X assigns to each sheaf F on X a series of Abelian groups HnF and to each sheaf map f : F → F′ a series of group homomorphisms Hnf : HnF → HnF′. The definition requires that each Hn is a functor, from sheaves on X to Abelian groups. A crucial property of these functors is:

HnF = 0 for n > 0

for any fine sheaf F where a sheaf is fine if it meets a certain condition borrowed from differential geometry by way of Cartan’s complex analytic geometry.

The cohomology of a group G assigns to each G-module M a series of Abelian groups HnM and to each homomorphism f : M →M′ a series of homomorphisms HnF : HnM → HnM′. Each Hn is a functor, from G-modules to Abelian groups. These functors have the same properties as topological cohomology except that:

HnM = 0 for n > 0

for any injective module M. A G-module I is injective if: For every G-module inclusion N M and homomorphism f : N → I there is at least one g : M → I making this commute

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Cartan could treat the cohomology of several different algebraic structures: groups, Lie groups, associative algebras. These all rest on injective resolutions. But, he could not include topological spaces, the source of the whole, and still one of the main motives for pursuing the other cohomologies. Topological cohomology rested on the completely different apparatus of fine resolutions. As to the search for a Weil cohomology, this left two questions: What would Weil cohomology use in place of topological sheaves or G-modules? And what resolutions would give their cohomology? Specifically, Cartan & Eilenberg defines group cohomology (like several other constructions) as a derived functor, which in turn is defined using injective resolutions. So the cohomology of a topological space was not a derived functor in their technical sense. But a looser sense was apparently current.

Grothendieck wrote to Serre:

I have realized that by formulating the theory of derived functors for categories more general than modules, one gets the cohomology of spaces at the same time at small cost. The existence follows from a general criterion, and fine sheaves will play the role of injective modules. One gets the fundamental spectral sequences as special cases of delectable and useful general spectral sequences. But I am not yet sure if it all works as well for non-separated spaces and I recall your doubts on the existence of an exact sequence in cohomology for dimensions ≥ 2. Besides this is probably all more or less explicit in Cartan-Eilenberg’s book which I have not yet had the pleasure to see.

Here he lays out the whole paper, commonly cited as Tôhoku for the journal that published it. There are several issues. For one thing, fine resolutions do not work for all topological spaces but only for the paracompact – that is, Hausdorff spaces where every open cover has a locally finite refinement. The Séminaire Cartan called these separated spaces. The limitation was no problem for differential geometry. All differential manifolds are paracompact. Nor was it a problem for most of analysis. But it was discouraging from the viewpoint of the Weil conjectures since non-trivial algebraic varieties are never Hausdorff.

Serre replied using the same loose sense of derived functor:

The fact that sheaf cohomology is a special case of derived func- tors (at least for the paracompact case) is not in Cartan-Sammy. Cartan was aware of it and told [David] Buchsbaum to work on it, but he seems not to have done it. The interest of it would be to show just which properties of fine sheaves we need to use; and so one might be able to figure out whether or not there are enough fine sheaves in the non-separated case (I think the answer is no but I am not at all sure!).

So Grothendieck began rewriting Cartan-Eilenberg before he had seen it. Among other things he preempted the question of resolutions for Weil cohomology. Before anyone knew what “sheaves” it would use, Grothendieck knew it would use injective resolutions. He did this by asking not what sheaves “are” but how they relate to one another. As he later put it, he set out to:

consider the set13 of all sheaves on a given topological space or, if you like, the prodigious arsenal of all the “meter sticks” that measure it. We consider this “set” or “arsenal” as equipped with its most evident structure, the way it appears so to speak “right in front of your nose”; that is what we call the structure of a “category”…From here on, this kind of “measuring superstructure” called the “category of sheaves” will be taken as “incarnating” what is most essential to that space.

The Séminaire Cartan had shown this structure in front of your nose suffices for much of cohomology. Definitions and proofs can be given in terms of commutative diagrams and exact sequences without asking, most of the time, what these are diagrams of.  Grothendieck went farther than any other, insisting that the “formal analogy” between sheaf cohomology and group cohomology should become “a common framework including these theories and others”. To start with, injectives have a nice categorical sense: An object I in any category is injective if, for every monic N → M and arrow f : N → I there is at least one g : M → I such that

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Fine sheaves are not so diagrammatic.

Grothendieck saw that Reinhold Baer’s original proof that modules have injective resolutions was largely diagrammatic itself. So Grothendieck gave diagrammatic axioms for the basic properties used in cohomology, and called any category that satisfies them an Abelian category. He gave further diagrammatic axioms tailored to Baer’s proof: Every category satisfying these axioms has injective resolutions. Such a category is called an AB5 category, and sometimes around the 1960s a Grothendieck category though that term has been used in several senses.

So sheaves on any topological space have injective resolutions and thus have derived functor cohomology in the strict sense. For paracompact spaces this agrees with cohomology from fine, flabby, or soft resolutions. So you can still use those, if you want them, and you will. But Grothendieck treats paracompactness as a “restrictive condition”, well removed from the basic theory, and he specifically mentions the Weil conjectures.

Beyond that, Grothendieck’s approach works for topology the same way it does for all cohomology. And, much further, the axioms apply to many categories other than categories of sheaves on topological spaces or categories of modules. They go far beyond topological and group cohomology, in principle, though in fact there were few if any known examples outside that framework when they were given.