Tuesday, Oct 2nd.
Derek Wise, FAU Erlangen
Title: Lifting General Relativity to Observer Space
PDF of the talk (700k) Audio [.wav 34MB], Audio [.aif 3MB].
by Jeffrey Morton, University of Hamburg.
You can read a more technical and precise version of this post at Jeff's own blog.
This talk was based on a project of Steffen Gielen and Derek Wise, which has taken written form in a few papers (two shorter ones, "Spontaneously broken Lorentz symmetry for Hamiltonian gravity", "Linking Covariant and Canonical General Relativity via Local Observers", and a new, longer one called "Lifting General Relativity to Observer Space").
The key idea behind this project is the notion of "observer space": a space of all observers in a given universe. This is easiest to picture when one starts with a space-time. Mathematically, this is a manifold M with a Lorentzian metric, g, which among other things determines which directions are "timelike" at a given point. Then an observer can be specified by choosing two things. First, a particular point (x0,x1,x2,x3) = x, an event in space-time. Second, a future-directed timelike direction, which is the tangent to the space-time trajectory of a "physical" observer passing through the event x. The space of observers consists of all these choices: what is known as the "future unit tangent bundle of M". However, using the notion of a "Cartan geometry", one can give a general definition of observer space which makes sense even when there is no underlying space-time manifold.
The result is a surprising, relatively new physical intuition saying that "space-time" is a local and observer-dependent notion, which in some special cases can be extended so that all observers see the same space-time. This appears to be somewhat related to the idea of relativity of locality. More directly, it is geometrically similar to the fact that a slicing of space-time into space and time is not unique, and not respected by the full symmetries of the theory of relativity. Rather, the division between space and time depends on the observer.
So, how is this described mathematically? In particular, what did I mean up there by saying that space-time itself becomes observer-dependent? The answer uses Cartan geometry.
Cartan GeometryRoughly, Cartan geometry is to Klein geometry as Riemannian geometry is to Euclidean geometry.
Klein's Erlangen Program, carried out in the mid-19th-century, systematically brought abstract algebra, and specifically the theory of Lie groups, into geometry, by placing the idea of symmetry in the leading role. It describes "homogeneous spaces" X, which are geometries in which every point is indistinguishable from every other point. This is expressed by an action of some Lie group G, which consists of all transformations of an underlying space which preserve its geometric structure. For n-dimensional Euclidean space En, the symmetry group is precisely the group of transformations that leave the data of Euclidean geometry, namely lengths and angles, invariant. This is the Euclidean group, and is generated by rotations and translations.
But any point x will be fixed by some symmetries, and not others, so there is a subgroup H , the "stabilizer subgroup", consisting of all symmetries which leave x fixed.
The pair (G,H) is all we need to specify a homogeneous space, or Klein geometry. Thus, a point will be fixed by the group of rotations centered at that point. Klein's insight is to reverse this: we may may obtain Euclidean space from the group G itself, essentially by "ignoring" (or more technically, by "modding out") the subgroup H of transformations that leave a particular point fixed. Klein's program lets us do this in general, given a pair (G,H). The advantage of this program is that it gives a great many examples of geometries (including ones previously not known) treated in a unified way. But the most relevant ones for now are:
- n-dimensional Euclidean space, as we just described.
- n-dimensional Minkowski space. The Euclidean group gets replaced by the Poincaré group, which includes translations and rotations, but also the boosts of special relativity. This is the group of all transformations that fix the geometry determined by the Minkowski metric of flat space-time.
- de Sitter space and anti-de Sitter spaces, which are relevant for studying general relativity with a cosmological constant.
Since curvature is measured by the development of curves, we can think of each homogeneous space as a flat Cartan geometry with itself as a local model, just as the Minkowski space of special relativity is a particular example of a solution to general relativity.
The idea that the curvature of a manifold depends on the model geometry being used to measure it, shows up in the way we apply this geometry to physics.
Gravity and Cartan GeometryThe MacDowell-Mansouri formulation of gravity can be understood as a theory in which general relativity is modeled by a Cartan geometry. Of course, a standard way of presenting general relativity is in terms of the geometry of a Lorentzian manifold. The Palatini formalism describes general relativity instead of in terms of a metric, in terms of a set of vector fields governed by the Palatini equations. This can be derived from a Cartan geometry through the theory of MacDowell-Mansouri, which "breaks the full symmetry" of the geometry at each point, generating the vector fields that arise in the Palatini formalism. So General Relativity can be written as the theory of a Cartan geometry modeled on a de Sitter space.
Observer SpaceThe idea in defining an observer space is to combine two symmetry reductions into one. One has a model Klein geometry, which reflects the "symmetry breaking" that happens when choosing one particular point in space-time, or event. The time directions are tangent vectors to the world-line (space-time trajectory) of a "physical" observer at the chosen event. So the model Klein geometry is the space of such possible observers at a fixed event. The stabilizer subgroup for a point in this space consists of just the rotations of space-time around the corresponding observer - the boosts in the Lorentz transformations that relate different observers. Locally, choosing an observer amounts to a splitting of the model space-time at the point into a product of space and time. If we combine both reductions at once, we get a 7-dimensional Klein geometry that is related to de Sitter space, which we think of as a homogeneous model for the "space of observers"
This may be intuitively surprising: it gives a perfectly concrete geometric model in which "space-time" is relative and observer-dependent, and perhaps only locally meaningful, in just the same way as the distinction between "space" and "time" in general relativity. That is, it may be impossible to determine objectively whether two observers are located at the same base event or not. This is a kind of "relativity of locality" which is geometrically much like the by-now more familiar relativity of simultaneity. Each observer will reach certain conclusions as to which observers share the same base event, but different observers may not agree. The coincident observers according to a given observer are those reached by a good class of geodesics in observer space moving only in directions that observer sees as boosts.
When one has a certain integrability condition, one can reconstruct a space-time from the observer space: two observers will agree whether or not they are at the same event. This is the familiar world of relativity, where simultaneity may be relative, but locality is absolute.
Lifting Gravity to Observer SpaceApart from describing this model of relative space-time, another motivation for describing observer space is that one can formulate canonical (Hamiltonian) general relativity locally near each point in such an observer space. The goal is to make a link between covariant and canonical quantization of gravity. Covariant quantization treats the geometry of space-time all at once, by means of of what is known as a Lagrangian. This is mathematically appealing, since it respects the symmetry of general relativity, namely its diffeomorphism-invariance (or, speaking more physically, that its laws take the same form for all observers). On the other hand, it is remote from the canonical (Hamiltonian) approach to quantization of physical systems, in which the concept of time is fundamental. In the canonical approach, one quantizes the space of states of a system at a given point in time, and the Hamiltonian for the theory describes its evolution. This is problematic for diffeomorphism-, or even Lorentz-invariance, since coordinate time depends on a choice of observer. The point of observer space is that we consider all these choices at once. Describing general relativity in observer space is both covariant, and based on (local) choices of time direction. Then a "field of observers" is a choice, at each base event in M, of an observer based at that event. A field of observers may or may not correspond to a particular decomposition of space-time into space evolving in time, but locally, at each point in observer space, it always looks like one. The resulting theory describes the dynamics of space-geometry over time, as seen locally by a given observer, in terms of a Cartan geometry.
This splitting, along the same lines as the one in MacDowell-Mansouri gravity described above, suggests that one could lift general relativity to a theory on an observer space. This amount to describing fields on observer space and a theory for them, so that the splitting of the fields gives back the usual fields of general relativity on space-time, and the equations give back the usual equations. This part of the project is still under development, but there is indeed a lifting of the equations of general relativity to observer space. This tells us that general relativity can be defined purely in terms of the space of all possible observers, and when there is an objective space-time, the resulting theory looks just like general relativity. In the case when there is no "objective" space-time, the result includes some surprising new fields: whether this is a good or a bad thing is not yet clear.