A new final state of gravitational collapse is proposed. By extending the concept of Bose-Einstein condensation to gravitational systems, a cold, dark, compact object with an interior de Sitter condensate p v ‫؍‬ ؊ v and an exterior Schwarzschild geometry of arbitrary total mass M is constructed. These regions are separated by a shell with a small but finite proper thickness ഞ of fluid with equation of state p ‫؍‬ ؉ , replacing both the Schwarzschild and de Sitter classical horizons. The new solution has no singularities, no event horizons, and a global time. Its entropy is maximized under small fluctuations and is given by the standard hydrodynamic entropy of the thin shell, which is of the order k BഞMc͞ -h, instead of the Bekenstein-Hawking entropy formula, S BH ‫؍‬ 4 kBGM 2 ͞ -hc. Hence, unlike black holes, the new solution is thermodynamically stable and has no information paradox.C old superdense stars with a mass greater than some critical value undergo rapid gravitational collapse. Due to the impossibility of halting this collapse by any known equation of state for high-density matter, a kind of consensus has developed that a collapsing star must inevitably arrive in a finite proper time at a singular condition called a black hole.The characteristic feature of a black hole is its event horizon, the null surface of finite area at which outwardly directed light rays hover indefinitely. For simplicity, consider an uncharged, nonrotating Schwarzschild black hole with the static, spherically symmetric line element,The functions f(r) and h(r) are equal in this case, andAt the event horizon, r ϭ r s , the metric (Eq. 1) becomes singular. Since local curvature invariants remain regular at r ϭ r s , a test particle falling through the horizon experiences nothing catastrophic there (if M is large enough), and it is possible to find regular coordinates that analytically extend the exterior Schwarzschild geometry through the event horizon into the interior region.It is important to recognize that this mathematical procedure of analytic continuation through a null hypersurface involves a physical assumption, namely that the stress-energy tensor is vanishing there. Even in the classical theory, the hyperbolic character of the Einstein equations allows generically for sources and discontinuities on the horizon that would violate this assumption. Whether such analytic continuation is mandatory, or even permissable in a more complete theory taking quantum effects into account, is still less certain.Nonanalytic behavior is typical of quantum many-body systems at a phase transition. Quantum systems also exhibit macroscopic coherence effects that do not depend on local forces becoming large. Thus, the fact that the tidal forces on classical test bodies falling through the event horizon are arbitrarily weak (proportional to M͞r s 3 ϳ 1͞M 2 ) for an arbitrarily large black hole does not imply that quantum effects must be unimportant there. Electron waves restricted to the region outside an AharonovBohm solenoid, where the elec...

We present a general method of deriving the effective action for conformal anomalies in any even dimension, which satisfies the Wess-Zumino consistency condition by construction. The method relies on defining the coboundary operator of the local Weyl group, g ab →exp(2)g ab , and giving a cohomological interpretation to counterterms in the effective action in dimensional regularization with respect to this group. Nontrivial cocycles of the Weyl group arise from local functionals that are Weyl invariant in and only in the physical even integer dimension dϭ2k. In the physical dimension the nontrivial cocycles generate covariant nonlocal action functionals characterized by sensitivity to global Weyl rescalings. The nonlocal action so obtained is unique up to the addition of trivial cocycles and Weyl invariant terms, both of which are insensitive to global Weyl rescalings. These distinct behaviors under rigid dilations can be used to distinguish between infrared relevant and irrelevant operators in a generally covariant manner. Variation of the dϭ4 nonlocal effective action yields two new conserved geometric stress tensors with local traces equal to the square of the Weyl tensor and the Gauss-Bonnet-Euler density, respectively. The second of these conserved tensors becomes (3) H ab in conformally flat spaces, exposing the previously unsuspected origin of this tensor. The method may be extended to any even dimension by making use of the general construction of conformal invariants given by Fefferman and Graham. As a corollary, conformal field theory ͑CFT͒ behavior of correlators at the asymptotic infinity of either anti-de Sitter or de Sitter spacetimes follows, i.e., AdS dϩ1 -or deS dϩ1 -CFT d correspondence. The same construction naturally selects all infrared relevant terms ͑and only those terms͒ in the low energy effective action of gravity in any even integer dimension. The infrared relevant terms arising from the known anomalies in dϭ4 imply that the classical Einstein theory is modified at large distances.

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A new solution for the endpoint of gravitational collapse is proposed. By extending the concept of Bose-Einstein condensation to gravitational systems, a cold, compact object with an interior de Sitter condensate phase and an exterior Schwarzschild geometry of arbitrary total mass M is constructed. These are separated by a phase boundary with a small but finite thickness ℓ of fluid with eq. of state p = +ρc 2 , replacing both the Schwarzschild and de Sitter classical horizons. The new solution has no singularities, no event horizons, and a global time. Its entropy is maximized under small fluctuations and is given by the standard hydrodynamic entropy of the thin shell, which is of order k B ℓM c/h, instead of the Bekenstein-Hawking entropy, S BH = 4πk B GM 2 /hc. Unlike black holes, a collapsed star of this kind is thermodynamically stable and has no information paradox.