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(Edited)

  1. Let $M$ be a smooth (pseudo-)Riemannian manifold and let $T(M)$ be the smooth tensor bundle on it. Is there a subspace $S \subset T(M)$ closed under (possibly infinitely many) finitary (possibly partially-defined) algebraic operations and finitely generated containing exactly the tensor fields on $M$ which are invariant under some fixed subgroup $G \le \mathrm{Diff}(M)$?
  2. In particular, the above is an open question because e.g. the $G$-invariant tensor fields have not been classified for general $G$, how about asking that $S$ contains only isometry-invariant tensor fields and in particular contains the metric tensor, inverse metric tensor, the Riemann tensor, the Ricci tensor, and the Ricci scalar?
  3. Suppose there is such an algebraic structure, and I adjoin an arbitrary tensor field to it. Will there be a non-trivial subgroup of $\mathrm{Diff}(M)$ under which the extended structure is invariant? Conversely, suppose I fix a non-trivial subgroup of $H < G$. Is there an algebraic structure of the same signature invariant under $H$ containing $S$?

Motivation

We can detect, using purely algebraic means in a certain sense, whether an arbitrary complex number $\alpha \in \mathbb{C}$ is an algebraic number: Look for the smallest subfield $\mathbb{Q}(\alpha) \le \mathbb{C}$ containing both $\mathbb{Q}$ and $\alpha$. It is a vector space over $\mathbb{Q}$, and $\alpha$ is algebraic if and only if this vector space is finite-dimensional. Moreover, we can even detect whether $\alpha$ can be expressed in terms of radicals simply by examining the automorphism group of the field extension $\mathbb{Q}(\alpha) / \mathbb{Q}$.

Here, I'm interested in whether or not something similar can be done for tensor fields on a manifold. The first question is analogous the inverse Galois problem — we have a symmetry group, and we are looking for an algebraic structure which is invariant under it. Having found such an algebraic structure, we can ask whether or not there is an analogue of the fundamental theorem of Galois theory, which establishes a bijective correspondence between subextensions of a Galois field extension and the subgroups of its automorphism group.

I'm aware of at least one result in this area, broadly interpreted as an algebraic approach to differential geometry — namely Lovelock's theorem, which classifies all the symmetric divergence-free second-order natural (0, 2)-tensors on a manifold. A corollary of this theorem, I'm told, tells us that the Einstein field equations are essentially unique (in 4 dimensions).

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Is there something wrong with the standard "invariant" characterization of a tensor, i.e. as a map from $T^1(M) \times \cdots T^1(M) \times T(M) \times \cdots \times T(M) \to C^{\infty}(M)$ which is multilinear over $C^{\infty}(M)$? $T(M)$ denotes the space of vector fields and $T^1(M)$ the space of $1$-forms. –  Yakov Shlapentokh-Rothman Dec 29 '10 at 8:40
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The confusion is that the $\Gamma$ you have defined is not a tensor. As Yakov mentions in his comment, the modern differential-geometric definition of a tensor is a section of a tensor bundle $\bigotimes^r TM \otimes \bigotimes^s T^*M$. In particular it has to be $C^\infty(M)$-linear in all entries. Your $\Gamma$, from its very definition, is not. –  José Figueroa-O'Farrill Dec 29 '10 at 11:04
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Could you rewrite your question and explicitly define what you mean by a "tensor bundle", a "tensor field", what it means for a "tensor field to be contained in a subspace", and, most importantly, what it means for tensor field to be "diffeomorphism-invariant"? It appears that you are using these terms in ways unfamiliar to many of us (perhaps because you are using definitions from a rather old treatise on general relativity?), so many of us are completely confused. –  Deane Yang Dec 30 '10 at 3:50
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How can we answer your question if you do not know the meaning of the words you are using? –  Deane Yang Dec 30 '10 at 16:03
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I don't want to hear your ideas. I want to know the definitions of the words you are using. I presume you are not using your own definitions. –  Deane Yang Dec 30 '10 at 16:04
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3 Answers 3

Hi, I'm not sure I understand what your question really is - frankly, I don't think that there is anything left to formalize about the principle of covariance in GR - but I hope that I can be of help nevertheless. Let me just state some remarks:

  • "I've read that the principle of general covariance in general relativity is best understood as a gauge symmetry with respect to the diffeomorphism group".

The problem with this statement is that "gauge transformation" in a gauge theory means "a transformation of the mathematical model that does not have any measurable/observable effect". In this sense the existence of gauge transformations means that the mathematical model is redundant, there are degrees of freedom that are not observable. In general relativity a diffeomorphism represents a change of the reference frame. This is of course "observable" in the sense that observers living in different reference frames report different observations of the same event. So I'd say that this analogy is at least as misleading as it is helpful.

You can find more about this on the webpage of Ray Streater here: Diff M as a gauge group.

  • "the link between this and the notion of manifest covariance is not obvious to me."

"Manifest" simply means that the covariance of an equation is easy to see (for an educated human), it does not have any deeper meaning.

  • "Physicists have a "principle of general covariance" which basically states that physical laws (and in particular, physical quantities) can be stated in a form which is somehow coordinate-independent. The paradox is, such "coordinate-independent" quantities and equations are frequently stated in terms of (admittedly, arbitrary) coordinates!"

I'm not sure I understand what the paradox is. Let's say you sit in a train and pour yourself a cup of coffee. You report: "both the cup and the coffeepot don't move, therefore the coffee ends up in the cup". Let's say I observe this standing at a railroad crossing , I'll report "both the cup and the coffeepot move with 30 km/h to the east, therefore the coffee ends up in the cup". The principle of general covariance says that the event "the coffee ends up in the cup" needs to be a scalar, since all observers will agree on the fact. And the velocity of the cup and the coffeepot in the west-east direction needs to be a vector because all observers will disagree about it according to the relative velocities of their frames of reference.

So, in general relativity, we say that

a) every specific choice of coordinates of (a patch of) spacetime corresponds to an observer, who can observe events and tell us about his observations,

b) given these observations we can predict what every other observer will report by applying the diffeomorphism that takes one set of coordinates to the other set of coordinates to the mathematical gadgeds that represent observable entities/effects.

The principle of general covariance says that every physical quantity has to transform in a way that we don't get an inconsistency between what we predict what another observer will report and what he actually reports. If you report that the coffee ends up in the cup, and I report that the coffee ends up on you because in my reference frame the cup moves with a different velocity than the coffepot, the theory is in trouble.

More about general relativity, the principle of general covariance and the "hole argument" of Einstein can be found on the page spacetime on the nLab.

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Hmmm. I think the point of disagreement is that "manifest covariance" cannot be given deeper meaning. Thank you for the links, however. As an analogy, I could say that a number is "manifestly algebraic" if I can give it terms of radicals, and then the question is how to detect such numbers amongst all algebraic numbers. The answer, of course, is Galois theory, and here I'm hoping that there might be some sort of Galois theory of geometric/physical invariants. –  Zhen Lin Dec 29 '10 at 10:23
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The principle of general covariance has been explicitly described by J.-M. Souriau in his 1974 paper "Modèle de particule à spin dans le champ électromagnétique et gravitationnel"

http://www.jmsouriau.com/Publications/JMSouriau-ModPartSpin1974.pdf

It is in french, I don't know if there exists an english translation?

You can find a related paper in english, by Shlomo Sternberg, here:

http://www.academy.ac.il/data/books/539/Sternberg-einstein-lecture.pdf

Hope that helps.

Comment: The principle of general covariance gives you the so-called passive field equations, that is for example ${\rm div}(T) = 0$, e.g. $\nabla_\mu T^{\mu\nu} = 0$ (or the equations of geodesics, or more complex equations if you input more fields or data).

It works essentially this way. Let us say that a geometrical object is a space with a natural action of the group ${\rm Diff}(M)$, the diffeomorphisms of $M$. For example the space ${\frak M}$ of metrics with signature $(+,-,-,-)$, and ${\rm Diff}(M)$ acts by pullback, $(\varphi,g) \mapsto \varphi^*(g)$, for $\varphi \in {\rm Diff}(M)$ and $g \in {\frak M}$. Now, the principle of general covariance (its active interpretation, not with charts or frames) says that any physical object (submitted to the field $g$) belongs to the quotient ${\frak Q} = {\frak M}/{\rm Diff}(M)$. Actually it is not completely exact, we must restrict the group to ${\rm Diff}_\bullet(M)$, the group of compact supported diffeomorphisms.

Let $g \in {\frak M}$ and $\gamma = [g] \in {\frak Q}$, the "tangent space" at $\gamma$ identifies to the tangent space at $g \in {\frak M}$ (that is the space of any symmetric tensor field on $M$) modulo the tangent space to the orbit of $g$, but the tangent space to the orbit of $g$ identifies with the space of Lie derivative of $g$ by compact supported fields.

Example: Let us look for a "covector" of $\frak Q$ at the point $\gamma$, and let us assume that it is given by a smooth distribution $T^{\mu\nu}$ of contravariant symmetric tensor on $M$, according to $$\tau(\delta g) = \int_M T^{\mu\nu}\delta g_{\mu\nu}$$ This is a linear form defined on the compact supported tensor fields $\delta g$ on $M$. But to be defined on $T_\gamma \frak Q$, $\tau$ must satisfy the (Eulerian) condition $$\tau(\epsilon) = \int_M T^{\mu\nu}\epsilon_{\mu\nu} = 0 \quad \mbox{for all} \quad \epsilon = {\frak L}_\xi(g)$$ with $\xi$, any compact supported vector field. And then, you can check that this is equivalent to ${\rm div}(T) = 0$.

This may give you a taste of what contains the Souriau's paper above. So, the principle of general covariance is just a principle of invariance with respect to the action of the diffeomorphisms with compact support. It is possible to give a very precise meaning to all these heuristic considerations. It has still not been done completely.

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Here is an outline of an answer to one of your earlier versions of your question: What tensor fields are invariant under the group of diffeomorphisms? Here, a tensor field is assumed to be a section of a tensor bundle over a fixed manifold. A tensor bundle is defined to be the tensor product of a finite number of copies of the tangent bundle and a finite number of copies of the cotangent bundle. First, since you can map any point to any other point in the manifold with any given differential, it follows that given any two points, you can find local co-ordinates near each point such that the components of the tensor field at the two points are equal. It therefore suffices to study the tensor field at a single point.

We can also now restrict to diffeomorphisms that fix the given point. The tensor field at the point is just an element of $ V \otimes \cdots \otimes V\otimes V^*\otimes\cdots\otimes V^ *$, where $V$ is the tangent space at that point. Moreover the action of a diffeomorphism corresponds simply to the action of $GL(V)$ on this space. We therefore want to find all tensors that are fixed by the action of $GL(V)$. Once we have identified these tensors, then these will define natural sections of the tensor bundle that are invariant under all diffeomorphisms.

At this point, I need to defer to someone who knows $GL(n)$ representation theory a lot better than me to explain the known classification. Let me just point out that there are indeed examples, the simplest nonzero one being the identity map $ \delta \in V\otimes V^ * $. One thing that is worth noting though is that such nontrivial $GL(n)$-invariant elements exist only if there are the same number of $V$ factors as $V^*$ factors in the tensor product. And I believe that any such invariant element is simply a linear combination of tensor products of the $\delta$ tensor but where you use different "slots" for each $\delta$. So the symmetric group plays a crucial role here.

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MathJax remains an inscrutable mystery to me. –  Deane Yang Dec 30 '10 at 20:07
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To give a reference, much of the last paragraph is explained in H. Weyl, The classical groups, their invariants and representations. –  Willie Wong Dec 30 '10 at 20:34
    
I would add that I have no idea how to deal with an arbitrary subgroup $G$ of the group of diffeomorphisms. A more reasonable and more useful question is to ask about tensors or subspaces of tensors (at a fixed point in the manifold) that are invariant under a subgroup of $GL(v)$ (for example, $SL(v)$, $O(n)$, $U(n/2)$ for $n$ even, etc.). –  Deane Yang Dec 30 '10 at 23:53
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