Local and gauge invariant observables in gravity

Read the full article for free* in Classical and Quantum Gravity:
Local and gauge invariant observables in gravity
Igor Khavkine 2015 Class. Quantum Grav. 32 185019
arXiv:1503.03754
*until 14/10/15

Generalized locality leads to lots of observables in gravity

Igor Khavkine is finishing up his term as a postdoctoral researcher at the University of Trento, Italy. His main interests are mathematical aspects of classical and quantum field theory, with an emphasis on gravity.

The problem of observables in general relativity is essentially as old as the theory itself. Einstein’s guiding principle of “general covariance”, that is, explicit tensorial transformation of basic physical fields and their equations under general coordinate transformations, leads to a formulation of the theory with “gauge” degrees of freedom. Those are degrees of freedom that, simply speaking, don’t contain any physical information and can be arbitrarily altered by the application of a coordinate transformation or, more abstractly, a diffeomorphism. Such a formulation is simple and elegant, but leads to the challenge of explicitly separating the gauge degrees of freedom from the remaining “physical” degrees of freedom in the analysis of (perhaps only hypothetical) physical measurements.

Historically, the difficulty in an effective separation of the gauge and physical degrees of freedom has lead to various confusions about the physical significance of ideas as varied as the hole argument, coordinate singularities, gravitation waves, the problem of time and the relation between general covariance and quantization.

Given its importance, the problem of observables has received quite a bit of attention. We now have a well-defined notion of a (“physical” or “diffeomorphism-invariant”) observable: a function on the (usually infinite dimensional) field configuration space (also phase space, when we restrict configurations to solutions of the equations of motion) that is invariant under the action of gauge transformations (diffeomorphisms, in this case) on the fundamental fields.

In more generality, the diffeomorphism group may be augmented to include other gauge symmetries, and also reduced to a subgroup that preserves some additional property, like asymptotic or boundary conditions. The persistent difficulty that has remained is an explicit and effective description of a sufficiently “nice” and “large” class of such observables.

The adjectives sufficiently “nice” and sufficiently “large” should be defined more precisely to investigate this question in more detail. While there might be more than one way to do so, we start with the following choice, keeping applications to classical and quantum field theory in mind. Under “nice” it is reasonable to ask for the observables to be “local” in the sense of being given by the integral over the spacetime of an expression with compact spacetime support that is written in terms of the metric field and its derivatives. Under “large” we ask for the requirement that the family of observables separates the points of the physical phase space (the space of gauge orbits of field configurations, in this case, isometry classes of metrics solving the Einstein equations). A set of functions separates the points of a space if there is no pair of points on which each function takes equal values. Such a separating set of functions is important because it is large enough to generate an algebra whose study is equivalent to the study of the underlying space.

Unfortunately, it is well-known that the above “nice” and “large” properties cannot be simultaneously satisfied for general relativity. This is usually stated as follows: gravity does not admit any local (gauge invariant) observables.

In my recent CQG paper, I consider slightly relaxed versions of this notion of “nice” and “large”, and show that, using ideas that already have an extended history in the relativity literature, there does exist a family of observables that simultaneously satisfies both of them. Of course, the goal in mathematical physics is to make such ideas as precise as possible and to prove interesting results about them. Here is a brief sketch of the technical content of my paper.

First, recall that a standard local observable is a functional of the dynamical fields $\phi$ (which could contain the metric, as well as other fields) of the form:

where $M$ is the spacetime, $\phi\colon M\to F$ is a section of the appropriate field bundle and $\tilde{\alpha}[\phi]$ is a top-degree differential form that can depend locally on $\phi$ and its derivatives. A crucial property asked of $\tilde{\alpha}[\phi]$ is that its support is contained in a given compact set for any $\phi$ . An equivalent, somewhat more geometrical description interprets $\tilde{\alpha}$ as a “horizontal” form on the total space of the jet bundle $J^k F$ . Jet are the geometric construction that allows us to express the dependence of $\tilde{\alpha}[\phi]$ on the derivatives of $\phi$ to order $k$ . The observable $A[\phi]$ is then simply the integral of $\tilde{\alpha}$ over the graph in $J^k F$ of the jet extension $j^k \phi$ of the field section $\phi$ , which is illustrated in the figure below.

Again, crucially, the support of $\tilde{\alpha}$ on $J^k F$ is required to have a compact projection to $M$ .

Geometrically, the invariance of $A[\phi]$ under gauge transformations follows from the invariance of $\tilde{\alpha}$ under the (jet-extended) action of any gauge transformation $g$ on $J^k F$ . Unfortunately, in the case of gravity, diffeomorphisms “move points”, meaning that their orbits in the jet space of metrics cover all of $M$ , as illustrated above, implying that no form $\tilde{\alpha}$ with support as above could define a diffeomorphism invariant observable $A[\phi]$ .

I call a functional of type (∗) “generalized local” provided the weaker condition, that the support of $\tilde{\alpha}$ intersects compactly the jet-extended graph $j^k \phi$ of each section $\phi\in \mathcal{U}$ in an open subset of the phase space, holds. This is illustrated in the figure below.

This notion is of course relative to the choice of $\mathcal{U}$ , which could be different for different observables or in different physical contexts. I argue in more detail in the paper why it is reasonable to restrict one’s attention to proper open subsets of the phase space, both classically and at least semi-classically in quantum theory. This simple generalization is sufficient to admit a much larger class of functionals including many that are gauge invariant. The forms $\tilde{\alpha}$ that have this property have been extensively studied in differential geometry and are known as differential invariants. In relativity, they are better known as curvature invariants, since they are built covariantly out of the curvature tensor and its derivatives. As illustrated above, differential invariants with desired support properties can be easily constructed by considering compactly supported forms $\tilde{\beta}$ on the quotient $\mathcal{M}^k = J^kF/\mathscr{G}$ by the group $\mathscr{G}$ of gauge transformations.

Another important property of these generalized local functionals is that they yield standard local observables upon linearization, as well as to any fixed perturbative order.

The question remains, given this generalized notion of “nice”, whether we could fulfill the sufficiently “large” requirement. The simple answer is unfortunately No. The problem is two-fold. Since our generalized local observables depend only on curvature, any sufficiently symmetric spacetime, where regions with identical or similar curvature profiles can fill at least a non-compact open set, may give rise to a divergent integral in (∗). The other problem also comes from symmetry. There exist non-isometric curvature profiles, necessarily associated with local existence of Killing vector fields, that cannot be distinguished by our curvature invariants. For example, we cannot distinguish Minkowski space from a pp-wave spacetime.

The good news is that the above obstacles are the only ones, which is the main result in my work. Namely, defining generic metrics to be those that avoid both of these problems gives us an open subset of the phase space whose points can be separated by gauge invariant generalized local observables. I invite the readers to consult the paper for further details.

One interesting question remains open. Is the set of generic metrics dense in the phase space of general relativity? I conjecture that the answer is Yes, perhaps under some additional physically reasonable hypothesis.

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