Paperpresented at the 13th Int. Conf. on General Relativity and Gravitation 353 Cordoba, Argentina, 1992: Part 2, Workshop Summaries Mathematical studies of Einstein’s and other relativistic equations/alternative gravity 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 Action-principle, a so- lution extremizes E if it is stationary. Now for a black hole, 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 quantum gravity. 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 Minkowski space) 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 “induced gravity”, though perhaps one still limited to the very special subclass of two-dimensional causal sets. Daughton’s presentation (co-authors Jorge Pullin, Rafael Sorkin 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 a scalar field on a background causal set generated by sprinkling N points randomly into a so-called interval-subset of Minkowski space. As contrasted with Meyer’s work, the additional motivation here — beyond the same ones of learning how to couple matter to a causal set, and the possibility of thereby producing a theory of “induced gravity” — was to investigate whether it is possible to recover effective locality in theories based on causal sets. (That this is not easy, is due to the inherent contradiction among discreteness, locality, and Lorentz invariance, these being three properties which any successful discrete theory must combine. For regular spacetime “lattices” locality proves easy to implement but Lorentz invariance difficult; for random “lattices” such as a sprinkled causal set, the situation is reversed. The promise of a successful combination would be great however, as locality in the presence of local Lorentz invariance is very restrictive, leading almost inevitably to an effective Lagrangian of the Einstein-Hilbert form.) Daughton presented several approaches to defining a scalar field Action which would reduce to the the usual one in the (naive) continuum limit, but he concentrated on a scheme which is based on thinking of the d’Alembertian as the inverse of a Green’s function rather than vice versa. This scheme is most natural in two and four dimensions, and Daughton discussed mainly the lower—dimensional case, offering both analytic and numerical evidence for its feasibility there. Specifically, he showed that the inverse of the retarded Green’s function on the (perfectly regular) “trellis-causal—set” is precisely a discretized d’Alembertian; and he described preliminary computer simulaticns which indicate a similar relationship ! on average — for the half—retarded half—advanced Green’s function on a randomly sprinkled causal set. The presentation by John Friedman (co—authors Nicolas Papastamatiou and Jona- than Simon) concerned a possibility in a sense opposite to that animating the two pre- vious talks, namely the possibility that spacetime might contain causality violations in the form of closed timelike curves. Such a possibility (reviewed in Kip Thorne’s ple- nary lecture) is of interest, not only for its own intrinsic fascination, but because it provides one way for the spacetime topology to change without forsaking the globally regular Lorentzian metric mandated by the “equivalence principle”. However there is grave doubt whether closed timelike curves are physically consistent, and in particular whether quantum fields can consistently propagate in spacetimes containing them. The difficulty raised by Friedman (citing also similar conclusions by David Boulware) was that of unitarity, in a situation where the causality violation is confined to a compact spacetime region. He explained that, even though free fields experience no problem, it is quite otherwise when interactions are present. Then perturbative unitarity reduces to a set of spacetime “cutting identities” whose validity relies on the Feynman propagator taking the form of a “time-ordered two-point function”, a form which loses its sense in the absence of a consistent causal ordering of events. Given this destruction of unitarity by interactions, Friedman considered whether a more general sum-over-histories inter- pretation could salvage quantum field theory. He argued that such an interpretation does allow a consistent assignment of probabilities to histories, but at a price: the results of present experiments would depend on whether closed timelike curves form in the future. If it is only recently that the possibility of causality violations has begun to be taken seriously, then the desire to localize gravitational energy is as old as General Relativity itself. Having pretty much abandoned the hope of defining a true energy- momentum current, people would now settle for a “quasi-local” expression giving the energy, angular momentum, etcetera, contained within some 2-surface. A particularly 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.