Dimensionally Challenged Gravities

It is a pleasure to dedicate this little travelogue/catalogue of exotic gravity models to John Stachel, whose loyalty to the D=4 Einstein cause is too steadfast to be subverted by reading it. Once unleashed by general relativity, dynamical geometry has become a fertile playground for generalization in many directions beyond Einstein’s D=4 Ricci-flat choice. This trend has intensified with string theory, where D=10 is normal as are (higher curvature power) corrections to the Einstein action. There are many other reasons to study different dimensions; here is one: As I became aware, thanks to John, Einstein already foresaw [1] the potential danger of letting geometry be at the mercy of field equations, in particular worrying about spaces with closed timelike curves, but also optimistically hoping that they would be forbidden in “physically acceptable” matter contexts (this is not a tautology since acceptable means having decent stress tensor). Although the best-known examples, such as Gödel’s universes [2], fall in this class, it is in the simpler setting of D=3 (“planar”) gravity that they have recently been studied on an industrial scale [3], and have yielded Einstein’s hoped-for taboo in a clear way. More generally, one can learn about D=4 Einstein’s virtues from studying different D’s, and the different sorts of models they support. What is more, we are very likely to be embedded in a world, which, if it has any classical geometry at all, is likely to have as many as eleven dimensions! In this short excursion, I can only point out some recent examples of theories that I have been involved with directly; equations and further references will have to be found in the citations. Let us begin with some remarks about ordinary Einstein theory in the smaller worlds of D<4. One does not normally think of the curvature components as being dimension-dependent, but we all know that in D=3, Einstein and Riemann tensors have the same number of components and indeed are equivalent, since Gν = 1 4 μαβ νλσRαβ λσ. Strangely, it was a long time before the import of this was appreciated: that outside sources, spacetime is flat! More precisely it is locally, but not globally, flat. Philosophically, D=3 Einstein theory presents the Machian dream in its purest form: there are no gravitational excitations, so geometry is entirely – and locally – determined by matter. There is a field-current identity: Riemann (being Einstein) equals stress tensor. So the picture that emerges is that this planar world consists of patches of Minkowski space glued together at the sources (most simply discrete point particles, representing parallel strings in a D=4 Einstein world). The 1-particle conical space solution is amusing enough [4] but things really get to be fun for two or more stationary or, better still, moving ones [5]. If a cosmological constant is present, it’s even more fun as the patches are constant curvature spacetimes [6]. In that case (for negative cosmological constant) it is also possible to have black holes by suitable identifications of points, [7]. Time-helical structures, requiring identification of times in a periodic way (as well as the space gluings)arise for stationary, rotating, solutions and lead to the whole gamut of possible closed timelike curves and, as mentioned, a clear arena to examine whether they can be physically generated. But D=3 can be more amusing still, for it permits (as does any odd-dimensional space) the construction of different invariants, the Chern–Simons (CS) terms. These are the gravitational analogs of the simple electrodynamics (or Yang–Mills) ∫ A∧F structures that in turn arise from the next higher dimensional topological invariants such as Fμν ?Fμν ≡ ∂μ( AνFαβ) in the abelian