A Metric for Gradient RG Flow of the Worldsheet Sigma Model Beyond First Order

Tseytlin has recently proposed that an action functional exists whose gradient generates to all orders in perturbation theory the Renormalization Group (RG) flow of the target space metric in the worldsheet sigma model. The gradient is defined with respect to a metric on the space of coupling constants which is explicitly known only to leading order in perturbation theory, but at that order is positive semi-definite, as follows from Perelman's work on the Ricci flow. This gives rise to a monotonicity formula for the flow which is expected to fail only if the beta function perturbation series fails to converge, which can happen if curvatures or their derivatives grow large. We test the validity of the monotonicity formula at next-to-leading order in perturbation theory by explicitly computing the second-order terms in the metric on the space of coupling constants. At this order, this metric is found not to be positive semi-definite. In situations where this might spoil monotonicity, derivatives of curvature become large enough for higher order perturbative corrections to be significant.Comment: 15 pages; Erroneous sentence in footnote 14 removed; this version therefore supersedes the published version (our thanks to Dezhong Chen for the correction

A Bakry-\'Emery Almost Splitting Result With Applications to the Topology of Black Holes

The almost splitting theorem of Cheeger-Colding is established in the setting of almost nonnegative generalized $m$-Bakry-\'{E}mery Ricci curvature, in which $m$ is positive and the associated vector field is not necessarily required to be the gradient of a function. In this context it is shown that with a diameter upper bound and volume lower bound the fundamental group of such manifolds is almost abelian. Furthermore, extensions of well-known results concerning Ricci curvature lower bounds are given for generalized $m$-Bakry-\'{E}mery Ricci curvature. These include: the first Betti number bound of Gromov and Gallot, Anderson's finiteness of fundamental group isomorphism types, volume comparison, the Abresch-Gromoll inequality, and a Cheng-Yau gradient estimate. Finally, this analysis is applied to stationary vacuum black holes in higher dimensions to find that low temperature horizons must have limited topology, similar to the restrictions exhibited by (extreme) horizons of zero temperature.Comment: Comm. Math. Phys., to appea

A uniqueness theorem for the adS soliton

The stability of physical systems depends on the existence of a state of least energy. In gravity, this is guaranteed by the positive energy theorem. For topological reasons this fails for nonsupersymmetric Kaluza-Klein compactifications, which can decay to arbitrarily negative energy. For related reasons, this also fails for the AdS soliton, a globally static, asymptotically toroidal $\Lambda<0$ spacetime with negative mass. Nonetheless, arguing from the AdS/CFT correspondence, Horowitz and Myers (hep-th/9808079) proposed a new positive energy conjecture, which asserts that the AdS soliton is the unique state of least energy in its asymptotic class. We give a new structure theorem for static $\Lambda<0$ spacetimes and use it to prove uniqueness of the AdS soliton. Our results offer significant support for the new positive energy conjecture and add to the body of rigorous results inspired by the AdS/CFT correspondence.Comment: Revtex, 4 pages; Matches published version. More detail in Abstract and one equation corrected. For details of proofs and further results, see hep-th/020408

The Positivity of Energy for Asymptotically Anti-de Sitter Spacetimes

We use the formulation of asymptotically anti-de Sitter boundary conditions given by Ashtekar and Magnon to obtain a coordinate expression for the general asymptotically AdeS metric in a neighbourhood of infinity. From this, we are able to compute the time delay of null curves propagating near infinity. If the gravitational mass is negative, so will be the time delay (relative to null geodesics at infinity) for certain null geodesics in the spacetime. Following closely an argument given by Penrose, Sorkin, and Woolgar, who treated the asymptotically flat case, we are then able to argue that a negative time delay is inconsistent with non-negative matter-energies in spacetimes having good causal properties. We thereby obtain a new positive mass theorem for these spacetimes. The theorem may be applied even when the matter flux near the boundary-at-infinity falls off so slowly that the mass changes, provided the theorem is applied in a time-averaged sense. The theorem also applies in certain spacetimes having local matter-energy that is sometimes negative, as can be the case in semi-classical gravity.Comment: (Plain TeX - figures not included

The Cosmic Censor Forbids Naked Topology

For any asymptotically flat spacetime with a suitable causal structure obeying (a weak form of) Penrose's cosmic censorship conjecture and satisfying conditions guaranteeing focusing of complete null geodesics, we prove that active topological censorship holds. We do not assume global hyperbolicity, and therefore make no use of Cauchy surfaces and their topology. Instead, we replace this with two underlying assumptions concerning the causal structure: that no compact set can signal to arbitrarily small neighbourhoods of spatial infinity (``$i^0$-avoidance''), and that no future incomplete null geodesic is visible from future null infinity. We show that these and the focusing condition together imply that the domain of outer communications is simply connected. Furthermore, we prove lemmas which have as a consequence that if a future incomplete null geodesic were visible from infinity, then given our $i^0$-avoidance assumption, it would also be visible from points of spacetime that can communicate with infinity, and so would signify a true naked singularity.Comment: To appear in CQG, this improved version contains minor revisions to incorporate referee's suggestions. Two revised references. Plain TeX, 12 page

Theorems on gravitational time delay and related issues

Two theorems related to gravitational time delay are proven. Both theorems apply to spacetimes satisfying the null energy condition and the null generic condition. The first theorem states that if the spacetime is null geodesically complete, then given any compact set $K$, there exists another compact set $K'$ such that for any $p,q \not\in K'$, if there exists a ``fastest null geodesic'', $\gamma$, between $p$ and $q$, then $\gamma$ cannot enter $K$. As an application of this theorem, we show that if, in addition, the spacetime is globally hyperbolic with a compact Cauchy surface, then any observer at sufficiently late times cannot have a particle horizon. The second theorem states that if a timelike conformal boundary can be attached to the spacetime such that the spacetime with boundary satisfies strong causality as well as a compactness condition, then any ``fastest null geodesic'' connecting two points on the boundary must lie entirely within the boundary. It follows from this theorem that generic perturbations of anti-de Sitter spacetime always produce a time delay relative to anti-de Sitter spacetime itself.Comment: 15 pages, 1 figure. Example of gauge perturbation changed/corrected. Two footnotes added and one footnote remove

A Causal Order for Spacetimes with C0C^0 Lorentzian Metrics: Proof of Compactness of the Space of Causal Curves

We recast the tools of ``global causal analysis'' in accord with an approach to the subject animated by two distinctive features: a thoroughgoing reliance on order-theoretic concepts, and a utilization of the Vietoris topology for the space of closed subsets of a compact set. We are led to work with a new causal relation which we call $K^+$, and in terms of it we formulate extended definitions of concepts like causal curve and global hyperbolicity. In particular we prove that, in a spacetime \M which is free of causal cycles, one may define a causal curve simply as a compact connected subset of \M which is linearly ordered by $K^+$. Our definitions all make sense for arbitrary $C^0$ metrics (and even for certain metrics which fail to be invertible in places). Using this feature, we prove for a general $C^0$ metric, the familiar theorem that the space of causal curves between any two compact subsets of a globally hyperbolic spacetime is compact. We feel that our approach, in addition to yielding a more general theorem, simplifies and clarifies the reasoning involved. Our results have application in a recent positive energy theorem, and may also prove useful in the study of topology change. We have tried to make our treatment self-contained by including proofs of all the facts we use which are not widely available in reference works on topology and differential geometry.Comment: Two small revisions to accomodate errors brought to our attention by R.S. Garcia. No change to chief results. 33 page

The Generalized Second Law implies a Quantum Singularity Theorem

The generalized second law can be used to prove a singularity theorem, by generalizing the notion of a trapped surface to quantum situations. Like Penrose's original singularity theorem, it implies that spacetime is null geodesically incomplete inside black holes, and to the past of spatially infinite Friedmann--Robertson--Walker cosmologies. If space is finite instead, the generalized second law requires that there only be a finite amount of entropy producing processes in the past, unless there is a reversal of the arrow of time. In asymptotically flat spacetime, the generalized second law also rules out traversable wormholes, negative masses, and other forms of faster-than-light travel between asymptotic regions, as well as closed timelike curves. Furthermore it is impossible to form baby universes which eventually become independent of the mother universe, or to restart inflation. Since the semiclassical approximation is used only in regions with low curvature, it is argued that the results may hold in full quantum gravity. An introductory section describes the second law and its time-reverse, in ordinary and generalized thermodynamics, using either the fine-grained or the coarse-grained entropy. (The fine-grained version is used in all results except those relating to the arrow of time.) A proof of the coarse-grained ordinary second law is given.Comment: 46 pages, 8 figures. v2: discussion of global hyperbolicity revised (4.1, 5.2), more comments on AdS. v3: major revisions including change of title. v4: similar to published version, but with corrections to plan of paper (1) and definition of global hyperbolicity (3.2). v5: fixed proof of Thm. 1, changed wording of Thm. 3 & proof of Thm. 4, revised Sec. 5.2, new footnote