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353 Paperpresented at the 13th Int. Conf. on and Gravitation Cordoba, Argentina, 1992: Part 2, Workshop Summaries

Mathematical studies of Einstein’s and other relativistic equations/alternative theories

Rafael D, Sorlcin

Department of Physics, Syracuse University, Syracuse NY 13244—1130

Abstract. The topics represented in this workshop spanned a wide range, reflecting the diversity of the very large number of abstracts submitted (nearly 100). In View of this breadth, it would be impossible to organize the present report according to subject. Instead I will just summarize the talks of the three sessions individually, pointing out now and then some connections which exist between them. 354 General Relativity and Gravitalion 1992

The first session began with a presentation by Daniel Sudarsky (co-author Robert Wald) devoted mainly to the understanding of some old exact solutions of pure gravity and some new ones of gravity plus gauge fields. The basic ideas involve the extremal properties of the total energy-functional E on the space of field configurations. It is known in general that, when the field equations derive from an -principle, a so- lution extremizes E if it is stationary. Now for a , stationarity represents a kind of thermal equilibrium, and this strongly suggests the following analogous assertion, special cases of which have long been known: the region outside a horizon of area A is stationary 4=> one has dE = TdA—l—QdJ for appropriate constants T and Q and for arbi- trary variations of the exterior field configuration. Under the assumption of a ”bifurcate Killing horizon”, Sudarsky quoted a theorem which is essentially the assertion’s “=>” part, a strong form of the so-called first law of black hole thermodynamics. Under the additional assumption of a maximal slicing, he quoted a theorem which is a strengthened form of a special case of the assertion’s “¢” part, namely that initial data which extrem— ize E at fixed A are actually static. Together these results show that stationary implies static for non-rotating holes, thereby filling a gap in the black—hole uniqueness theorems. Moreover the theorems still apply when gauge—fields are present, and for non—abelian gauge groups, there exist disconnected minima of the energy differing from each other by "large” gauge transformations (ones disconnected from the identity). A differential topology argument then implies the existence between these minima of saddle points of the energy, and therefore of unstable static solutions. In this way several types of recently found “colored” black hole solutions can be understood, and analogous “colored excitations” of the Kerr metric can also be predicted. The next pair of presentations, by David Meyer and Alan Daughton, reported on work inspired by the “causal set" approach to . In his brief summary of this approach, which is based on the hypothesis that the Lorentzian manifold of General Relativity is only an approximation to a discrete substratum whose basic structure is that of a causal set (: locally finite partial order), Meyer pointed out that, although significant progress has been made in understanding the kinematics of causal sets, there is as yet no convincing candidate for the quantum amplitude [the discrete analog of exp(i5)] on which the theory of causal set dynamics would be based. Invoking recent string— theoretic work on discretized “Euclidean” two—dimensional gravity coupled to statistical mechanical models, Meyer introduced the analogous study of Ising models on causal sets. Remarkably, one is able to solve these models exactly in several different cases. In the most interesting of these, both the Ising “spins” and the causal set itself are summed over (the latter sum being restricted to causal sets generated by random sprinkling into 2— dimensional ) and the “partition function” turns out, for certain critical values of the parameters, to be a modified Bessel function. In this model (because 2-dimensional metrics are conformally fiat) one is effectively summing over all (1+1)- dimensional causal sets, and in that sense is dealing with a full quantum gravity coupled to a particular kind of “matter”. However the amplitude being used is not very realistic, in part because of its likely non-locality, a general difficulty highlighted in Daughton’s talk. In this connection, Meyer suggested that putting the Ising model at a critical point might lead to a more satisfactory form of “”, though perhaps one still limited to the very special subclass of two-dimensional causal sets. Daughton’s presentation (co-authors Jorge Pullin, and Eric Woolgar) also was concerned with a kind of “matter” living on “flat” causal sets, specifically with Mathematical studies os‘nstein’s and other relativistic equations 355

randomly into a scalar field on a background causal set generated by sprinkling N points work, the a so-called interval-subset of Minkowski space. As contrasted with Meyer’s couple matter to additional motivation here — beyond the same ones of learning how to gravity” — was a causal set, and the possibility of thereby producing a theory of “induced based on causal to investigate whether it is possible to recover effective locality in theories discreteness, sets. (That this is not easy, is due to the inherent contradiction among any successful discrete locality, and Lorentz invariance, these being three properties which easy to implement theory must combine. For regular “lattices” locality proves causal set, the but Lorentz invariance difficult; for random “lattices” such as a sprinkled be great however, situation is reversed. The promise of a successful combination would leading almost as locality in the presence of local Lorentz invariance is very restrictive, presented inevitably to an effective Lagrangian of the Einstein-Hilbert form.) Daughton to the the usual several approaches to defining a scalar field Action which would reduce is based one in the (naive) continuum limit, but he concentrated on a scheme which than vice on thinking of the d’Alembertian as the inverse of a Green’s function rather discussed versa. This scheme is most natural in two and four , and Daughton for its mainly the lower—dimensional case, offering both analytic and numerical evidence function feasibility there. Specifically, he showed that the inverse of the retarded Green’s and on the (perfectly regular) “trellis-causal—set” is precisely a discretized d’Alembertian; ! on he described preliminary computer simulaticns which indicate a similar relationship sprinkled average — for the half—retarded half—advanced Green’s function on a randomly causal set. Jona- The presentation by John Friedman (co—authors Nicolas Papastamatiou and the two pre- than Simon) concerned a possibility in a sense opposite to that animating violations in vious talks, namely the possibility that spacetime might contain Thorne’s ple- the form of closed timelike curves. Such a possibility (reviewed in Kip but because it nary lecture) is of interest, not only for its own intrinsic fascination, the globally provides one way for the spacetime topology to change without forsaking there is regular Lorentzian metric mandated by the “equivalence principle”. However in particular grave doubt whether closed timelike curves are physically consistent, and them. The whether quantum fields can consistently propagate in containing was difficulty raised by Friedman (citing also similar conclusions by David Boulware) a compact that of unitarity, in a situation where the causality violation is confined to problem, it spacetime region. He explained that, even though free fields experience no reduces to is quite otherwise when interactions are present. Then perturbative unitarity propagator a set of spacetime “cutting identities” whose validity relies on the Feynman its sense in taking the form of a “time-ordered two-point function”, a form which loses of unitarity the absence of a consistent causal ordering of events. Given this destruction inter- by interactions, Friedman considered whether a more general sum-over-histories does pretation could salvage quantum field theory. He argued that such an interpretation results of allow a consistent assignment of probabilities to histories, but at a price: the future. present experiments would depend on whether closed timelike curves form in the to be If it is only recently that the possibility of causality violations has begun as old as General taken seriously, then the desire to localize gravitational energy is a true energy- Relativity itself. Having pretty much abandoned the hope of defining giving the momentum current, people would now settle for a “quasi-local” expression A particularly energy, angular momentum, etcetera, contained within some 2-surface. 356 General Relau‘vity and Gravitation [992

natural approach to finding such an expression is to seek it as the variation of some corresponding quasilocal Action, and this is the approach adopted by David Brown, whose presentation inaugurated the second session. Defining an Action appropriate for a spatially bounded region, and varying the lapse-part of the metric on the timelike boundary led him to an energy given in terms of the extrinsic curvature of the 2-surface (with respect to the hypersurface in which it lies). This energy behaves favorably at spatial infinity and in the Newtonian limit, but has the drawback (in my opinion) that it appears difficult to generalize to a full lO-parameter set of conserved quantities. The contributions by Brian Edgar and Graham Hall (co-author B.M. Haddow) were in the realm of pure differential geometry, but have obvious implications for the possibility — in either standard Relativity or one of its generalizations like Weyl’s theory — of taking a connection (or possibly a curvature tensor) to be the fundamental physical field, rather than, for example, a metric. A series of related results was presented, but the focus was on the case of so—called Weyl connections, i.e. ones which preserve a compatible metric up to a conformal factor in the sense that cab : gab/\C. The main result of Edgar asserted that a given symmetric connection Va is Weyl iff there exists a putative curvature tensor and a putative metric tensor such that the former has the correct algebraic symmetries and fulfills the Bianchi identity, and the latter yields a symmetric index-pair when used to lower the raised index of the former. (Here a genericity condition has been assumed; in its absence conditions on the derivatives of the curvature enter as well.) The main result of Hall was somewhat more global in character, being couched in terms of the holonomy group of the connection rather than its curvature tensor. It asserts that a connection in an n—manifold is Weyl iff its holonomy group is a Lie subgroup of some conformal group (rotations plus scalings) in n—dimensions. A companion result characterizes the ambiguity in the pairs (gum/k) which “fit” a given connection. In addition to the conformal ambiguity from which “gauge transformations” in the original sense got their name, there can be further ambiguities only in the non—generic case where the holonomy group at a point (by definition a group of linear transformations of the tangent space) is reducible in the sense of admitting an invariant subspace; moreover there will be such ambiguities if the subspace is not null. However, even in these special cases, Weyl’s “electromagnetic field”, curl A, remains unique. Closely related to the Weyl “unified field theory” are the scalar—tensor gravity theories, and in particular the so-called Brans—Dicke alternative gravity theory, which is distinguished by a special choice for the coupling between ordinary matter and the scalar field. Although this theory allows in principle for an additional “scalar charge”, in addition to the mass, the electric charge, etcetera, it is known that, as with Einstein gravity, the only spherically symmetric “vacuum” solution with a horizon is again the Schwarzschild metric, this being especially evident when the scalar field is constant, in which case the field equations reduce to the vacuum Einstein equations. In his contri- bution to the second session, however, Carlos Lousto (co—author Manuela Campanelli) presented a family of non-Ricci-fiat spherically-symmetric solutions of the Brans-Dicke field equations, which remain non-Ricci-fiat even in a certain w —> oo limit in which the scalar field becomes constant. In a certain formal sense these solutions do have a horizon, but that “surface” has infinite area, and is perhaps better interpreted as a kind of internal infinity of the spacetime. For certain values of their parameters the solutions presented are supposed to be astrophysically viable, and What is also interesting, it is Mathematical studies of Einstein”: and other relativistic equations 357 claimed that their “surface-gravity” vanishes, apparently lending them a zero horizon temperature. What seems needed now is a better understanding of the global structure of these solutions. Another set of unified field theories in which extra scalar fields play a prominent role are those named for Kaluza and Klein. This approach to unification was represented in the workshop by the talk of Alfredo Macias (co-author H. Dehnen), who considered an 8-dimensional spacetime with “internal” dimensions in the form of a SU(2) >< U(1) group-manifold. In order to provide explicit fermionic degrees of freedom, a Dirac field is introduced to complement the 8-dimensional metric. With the Higgs-like modes (the , etc.) frozen out by hand, and with a prescribed dependence of the spinor field on the internal coordinates, one obtains a dimensionally reduced Lagrangian resembling the electro—weak sector of the . However, the fermions, including the neutrino, all have large masses, and of course, no Higgs fields are present. In addition the neutrino acquires an anomalous electromagnetic moment. The third and last session of the workshop commenced with a talk by James Isen- berg (co—authors Vincent Moncrief and Yvonne Choquet—Bruhat), who reported on a new theorem guaranteeing under certain conditions the existence of vacuum initial data with non—constant mean curvature. He began, however, by reviewing the rather satisfactory understanding which we have of those solutions of the initial-value constraints which do possess constant mean curvature. In that case, one specifies certain free hypersurface data (the conformal metric, Aab, the “conformal extrinsic curvature”, U“ (a symmetric tensor with zero trace and divergence), and a number, 7', giving the mean curvature) and solves an elliptic equation for the conformal factor relating this data to the true metric and extrinsic curvature. The question of when this equation has a solution is fully understood, the answer depending on which of 12 cases the free data falls into, as determined by the Yamabe class of the conformal metric and by the vanishing or not of 'r and 0‘“. Unfortunately, not all solutions of the Einstein equations admit constant mean curvature slices, and so one is forced to consider a more complicated scheme in which 7' is non-constant and one has a pair of coupled equations to solve, rather than a single one. As a first step in sorting out necessary and sufficient conditions for this set to be soluble, Isenberg presented a sufficient condition which requires roughly that An], be in the negative Yamabe class and that 7' be sufficiently close to a non—zero constant. The method of proof is one which he expects to yield existence for a much more general class of data, as well. The final two presentations to be discussed were connected more or less closely with . That by Robert Mann, dealt more generally with 2-dimcnsional gravity, or rather with what he called “dilaton gravity”, in which a scalar field multiplies the scalar-curvature term in the Lagrangian, and may couple also to the “matter fields” if such are present (there is of course no “Einstein-Hilbert gravity” in 2-dimensions, at least classically, since the unadorned scalar curvature is a total divergence there). Working with a rather general class of such theories, Mann introduced a certain vector field 5" for which he could show, in many cases, that J“ :2 Tt‘Vfiu is conserved, even though {“ itself is not in general a Killing vector. Interpreting the potential for this current as a “mass function”, he illustrated the relations found on some “black-hole” solutions, ending with the provocative conclusion that the same black-hole metric can be associated with two very different entropies, depending on which theory one interprets it within. Finally Tevian Dray (reporting on behalf of Corinne Manogue) described a group- 358 General Relativity and Gravitation 1992 theoretic result of interest for octonionic string theory, as well as more generally. The relevant mathematical coincidences here are that 50(9, 1) is isomorphic to SL(2,0) (0 being the octonions), and that correspondingly vectors in lO-dimensional Minkowski space can be represented in terms of octonions analogously to how 4-vectors can be represented as 2 X 2 complex matrices. Now, however, there is a puzzle. If X is the 2 x 2 octonionic-valued matrix representing a lO-vector, and M a matrix in SL(2, 0), then X —» MXMl does not yield all possible Lorentz transforms on X, despite the isomorphism we began with. The resolution is that one obtains the missing transfor- mations by iterating the simple ones, the former not collapsing to the latter precisely because octonions are non-associative. Dray showed explicitly how this works in a. simple example.