S. DESER

DIMENSIONALLY CHALLENGED GRAVITIES*

Abstract. I review some ways through which spacetime dimensional ity enters explicitly in gravitation.In particular, I recall the unusual geometrical gravity models that are constructible in dimensions differentfrom four, espec ially in D=3 where even ordina ry Einstein theory is "different," e.g., fully Machian .

Once unleashed by general relativity, dynamical geometry has become a fertile play-ground 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 (highercurvature power) corrections to the Einstein action. There are many other reasons tostudy different dimensions; here is one: As I became aware, thanks to John, Einsteinalready foresaw (1914, 1079; 1957) the potential danger of letting geometry be atthe mercy of field equations , in particular worrying about spaces with closed time-like curves, but also optimistically hoping that they would be forbidden in "physicallyacceptable" matter contexts (this is not a tautology since acceptable means havingdecent stress tensor) . Although the best-known examples, such as Godel 's universes(G6deI1949; van Stockum 1937), fall in this class, it is in the simpler setting ofD=3("planar") gravity that they have recently been studied on an industrial scale, I and haveyielded Einstein's hoped-for taboo in a clear way. More generally, one can learn aboutD=4 Einstein's virtues from studying different D' s, and the different sorts of modelsthey support. What is more, we are very likely to be embedded in a world, which, ifit has any classical geometry at all, is likely to have as many as eleven dimensions! Inthis short excursion, I can only point out some recent examples of theories that I havebeen involved with directly; equations and further references will have to be found inthe citations.

Let us begin with some remarks about ordinary Einstein theory in the smallerworlds of D.aRQ(3 Aa . Strangely, it was a long time before the import of this,that outside sources, spacetime is flat was appreciated! Philosophically, D=3 Ein-stein theory presents the Machian dream in its purest form: there are no gravitationalexcitations , so geometry is entirely - and locally - determined by matter. There isa field-current identity : Riemann (being Einstein) equals stress tensor. So the pic-ture that emerges is that this planar world consists of patches of Minkowski space

397A. Ashtekar et al. (eds.), Revisiting the Foundations ofRelativistic Physics, 397-401. 2003 Kluwer Academic Publishers.

398 S. DESER

glued together at the sources (most simply discrete point particles, representing paral-lel strings in a D=4 Einstein world). The l-particle conical space solution is amusingenough (Staruszkiewicz 1963) but things really get to be fun for two or more station-ary or, better still , moving ones (Deser, Jackiw, and 't Hooft 1984). If a cosmologicalconstant is present , it's even more fun as the patches are constant curvature space-times (Deser and Jackiw 1984). In that case (for negative cosmological constant) it isalso possible to have black holes by suitable identifications of points (Banados et al.1993). Time-helical structures, requiring identification of times in a periodic way (aswell as the space gluings)arise for stationary, rotating, solutions and lead to the wholegamut of possible closed timelike curves and, as mentioned, a clear arena to examinewhether they can be physically generated . But D=3 can be more amusing still , forit permits (as does any odd-dimensional space) the construction of different invari-ants, the Chern-Simons (CS) terms. These are the gravitational analogs of the simpleelectrodynamics (or Yang-Mills) JA /\ F structures that in turn arise from the nexthigher dimensional topological invariants, such as F1w *Fr" == aIL ( ElLva(3AvFa(3) inthe abelian context. Here we have the Pontryagin invariant R*R instead. Varyingthese gravitational CS terms with respect to the metric leads to a tensor, because theintegral (if not the CS integrand) is gauge invariant. In D=3, this is the famous Cottontensor, CILV == ElLa(3Da (R~ - ~ 8~R), discovered long before general relativity (Cot-ton 1898); CILV is the conformal tensor in D=3, replacing the (identically vanishing)Weyl curvature. It is a symmetric , traceless, identically conserved quantity, althoughit superficially seems to be none of these. Its interest lies not so much for generating atheory of gravity in its own right (it could at best only couple to traceless sources) butas an added term to the Einstein one. Being of third derivative order, it has a coefficientwith relative inverse length or mass dimension (in Planck units) to that of the Einsteinaction . This mass is in fact that of small excitations (of helicity 2) of the metricabout flat space: adding CS has restored a degree offreedom absent in either R or CSalone. This combined theory (Deser, Jackiw, and Templeton 1982a, 1982b), calledtopologically massive gravity (TMG) for obvious reasons, has many other wondrousproperties and unsolved aspects. First, despite being a higher derivative theory, it hasno unitarity or ghost problems; it may even be finite as a quantum theory, althoughthat is still an open mathematical problem (Deser and Yang 1990). If so, it mightreally have some lessons for us, for it would be unique in this respect amongst trulydynamical gravity models without ghosts (unlike four-derivative theories) but witha dimensional coupling constant; pure Einstein D=3 theory is renormalizable (Witten1988; Deser, McCarthy, and Yang 1989) but that doesn't count - it is non-dynamical.Second, TMG, at least in its linearized guise (Deser 1990) acts to turn its sources intoanyons; that is, a particle can acquire any desired spin simply by coupling to TMG.But, thirdly, no-one has succeeded as yet in finding the simplest possible, "Schwarz-schild" solution to the nonlinear model, i.e., a circularly symmetric time-independent(we don 't even know if there's a Birkhofftheorem) exterior geometry that obeys theGILV + m- 1C ILV = 0 equations . Although CS-like terms can be constructed for higherodd-D spaces, they have not been studied much because they have no linearized , kin-ematical , effects beyond D=3 because they are ofhigher powers in an expansion aboutflat space. There are both strong similarities and differences between TMG and its

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spin 1 counterpart, topologically massive Yang-Mills theory. The most striking dif-ference is that in the quantum theory, the coefficient of the CS term in "TM- YM"must necessarily be quantized (Deser, Jackiw, and Templeton 1982a; Deser, Jackiw,and Templeton 1982b), but not that ofTMG (Percacci 1987).

But the twists in D=3 gravity do not stop there: there is yet another "CS-ness"present. Once it is noted that, in Einstein gravity, spacetime is flat outside sources,one realizes that this is just the same as what happens to abelian or nonabelian vectorfields in their pure CS models: the field equations are just *F /-I == ~ /-IVQ;FVQ; = 0,so that the field "curvature" also vanishes here [in both cases, the full "curvatures" areproportional to currents, *F /-I = J/-I ] . Indeed, there is an equivalence (except for someinteresting find print) between the two models and one can formally recast the Einsteinaction and equations into non-abelian vector field CS form in terms of the dreibein andspin connections. So this is yet another vast subject straddl ing two ostensibly differenttypes of theory?

We will not descend much to D=2, another well-studi ed subject (Jackiw and Teitel-boim 1984), because there is no Einstein gravity there at all: only the Ricci scalar isnon-vanishing, being the "double-dual" of Riemann, while the Einstein tensor van-ishes identically. As usual, D=2 is different from all other dimensions in this respect(it is also here that Maxwell theory ceases to have excitations); some sort of addi-tional scalar field is required to assure the Hilbert action from just being a dull Eulertopological invariant, and this departs from the realm of pure geometry.

What about dimensions beyond D=4? This becomes a generic area where thedifferences from D=4 are more quantitative than qualitative. Still, there are someamusing points to be noted. For example, consider the Gauss-Bonnet invariant R*R* ,defined in D=4. There, it is a total divergence and hence irrelevant to field equations.However, in higher dimensions, it can still be defined by writing it out in terms ofmetrics; for example we all know it is proportional to the combination (R~vQ;f3 -

4R~v + R 2 ) . As a Lagrangian, it is seemingly dangerous to unitarity of excitationsbecause of its fourth derivative order. In fact, there is no danger, because (say aboutflat space) this combinat ion is a total divergence in its leading quadratic order in h/-lv ==(9/-1v - TJ /-Iv ) in any D. Thus, this is a "safe" class ofalternative actions, say when addedto Einstein's. Some oftheir solutions, e.g., Schwarzschild-like ones, have been studiedto see whether they are better or still unique. For example, there can be cosmologicalsolutions without an explicit cosmological constant (Boulware and Deser 1985; Deserand Yang 1989), which is not necessarily a good thing, physically.

In quantum field theory, a powerful tool has been the "large N limit" of Yang-Mills theory in which the number of flavors (internal degrees of freedom: A~, a =1 ... N ) is sent to infinity. The equivalent in gravity, rather naturally, is the dimen-sionality D of spacetime, over which the "internal" index a of the vielbein e~ ranges.As far as I know, the literature here consists of but one brave paper (Strominger 1981),which however did not get far. This seems to me a worthy subjec t of study also byclassical relativists, who have instead mostly considered what is in some ways the op-posite, ultra-local, limit (Isham 1976; Teitelboim 1980). There must be some simpli-fying aspects as the number of degrees offreedom ~D(D -3), rises quadratically, andthe Newtonian potential that enters in the Schwarzschild metric behaves as r- (D -3 ) .

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There are also auxiliary quant ities that are interestingly dimension-dependent; weencountered some of them in current work on D= II supergravity (Deser and Semin-ara 1999, 2000). I will not transgress further into the superworld here, except to saythat it is absolutely amazing that a) Einstein gravity always has a "Dirac square root"for all D::;II , i.e., can always be consisten tly supersymmetrized without the need forhigher spins or more than one graviton, and b) that this possibility stops (Nahm 1978)at 0 =11 and c) that cosmologica l terms are allowed for all 0 . {3 = i gJlV C2in 0 =4, from antisymmetrizing expressions such as Ct::vC~gx;J ' where X is arbit-rary and bracket s indicate antisymmetizations over 5 indices. In the anomaly contextwe are actually interested, using so-called dimensional regularization, in spaces ofdi-mension differing from an integer by an infinitesimal parameter, still another unlikel ydeparture from D=4, but one that has its own unlikely set of geometrical rules.

In conclusion, I have tried to indicate that the list of interesting, useful and evenimportan t consequences to be drawn from excursions away from our favorite, Einstein,action in its D=4 world is substantial and by no means complete.

Brandeis University

NOTES

* 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 subve rted by reading it.This work was supported by NSF grants PHY93-15811 and PHY99-73935.1 am grateful to my collab-orators in our explorations of the areas discussed here.

I. (Got! 1991; Deser, Jackiw, and 't Hoot! 1992; Carroll, Farhi, and Guth 1992; 't Hoot! 1992; Deser andJackiw 1992).

2. For a review see e.g. (Achuc arro and Townsend 1986).

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