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Dec 1998

Volume 39, Issue 12, pp. 6247-6756

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Timelike infinity and asymptotic symmetry

Uchida Gen and Tetsuya Shiromizu

J. Math. Phys. 39, 6573 (1998); http://dx.doi.org/10.1063/1.532666 (20 pages) | Cited 2 times

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By extending Ashtekar and Romano’s definition of spacelike infinity to the timelike direction, a new definition of asymptotic flatness at timelike infinity for an isolated system with a source is proposed. The treatment provides unit spacelike three-hyperboloid timelike infinity and avoids the introduction of the troublesome differentiability conditions which were necessary in the previous works on asymptotically flat space–times at timelike infinity. Asymptotic flatness is characterized by the falloff rate of the energy-momentum tensor at timelike infinity, which makes it easier to understand physically what space–times are investigated. The notion of the order of the asymptotic flatness is naturally introduced from the rate. The definition gives a systematized picture of hierarchy in the asymptotic structure, which was not clear in the previous works. It is found that if the energy-momentum tensor falls off at a rate faster than ∼ t−2, the space–time is asymptotically flat and asymptotically stationary in the sense that the Lie derivative of the metric with respect to ∂t falls off at the rate ∼ t−2. It also admits an asymptotic symmetry group similar to the Poincaré group. If the energy-momentum tensor falls off at a rate faster than ∼ t−3, the four-momentum of a space–time may be defined. On the other hand, angular momentum is defined only for space–times in which the energy-momentum tensor falls off at a rate faster than ∼ t−4. © 1998 American Institute of Physics.
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04.20.Gz Spacetime topology, causal structure, spinor structure
95.30.Sf Relativity and gravitation
97.60.-s Late stages of stellar evolution (including black holes)

Simplicial quantum gravity in the elongated phase

Gabriele Gionti

J. Math. Phys. 39, 6593 (1998); http://dx.doi.org/10.1063/1.532667 (10 pages)

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We discuss the elongated phase of four-dimensional simplicial quantum gravity by exploiting recent analytical results. In particular using Walkup’s theorem we prove that the dominating configurations in the elongated phase are treelike structures called “stacked spheres.” Such configurations can be mapped into branched polymers and baby universes arguments are used in order to analyze the critical behavior of theory in the weak coupling regime. © 1998 American Institute of Physics.
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04.60.-m Quantum gravity
98.80.Jk Mathematical and relativistic aspects of cosmology
98.80.Qc Quantum cosmology
95.30.Sf Relativity and gravitation

Is there a general area theorem for black holes?

Domenico Giulini

J. Math. Phys. 39, 6603 (1998); http://dx.doi.org/10.1063/1.532668 (4 pages) | Cited 1 time

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The general validity of the area law for black holes is still an open problem. We first show in detail how to complete the usually incompletely stated textbook proofs under the assumption of piecewise C2-smoothness for the surface of the black hole. Then we prove that a black hole surface necessarily contains points where it is not C1 (called “cusps”) at any time before caustics of the horizon generators show up, like, e.g., in merging processes. This implies that caustics never disappear in the past and that black holes without initial cusps will never develop such. Hence black holes which will undergo any nontrivial processes anywhere in the future will always show cusps. Although this does not yet imply a strict incompatibility with piecewise C2 structures, it indicates that the latter are likely to be physically unnatural. We conclude by calling for a purely measure theoretic proof of the area theorem. © 1998 American Institute of Physics.
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97.60.Lf Black holes

Time-independent bounds on solutions with shocks to the relativistic Euler equations on conformally flat space–times with curvature singularities

Jeffrey M. Groah

J. Math. Phys. 39, 6607 (1998); http://dx.doi.org/10.1063/1.532669 (24 pages)

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We consider the initial value problem of the relativistic Euler equations when the underlying space–time is not flat but conformally flat, and demonstrate existence of solutions with shocks, with time-independent bounds, in some cases from big bang till big crunch. Our theorem requires that the space–time metric satisfy certain bounds, but these do not constrain the curvature of space–time, and hence our theorem may guarantee uniform bounds on solutions until the formation of curvature singularities. © 1998 American Institute of Physics.
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43.28.Mw Shock and blast waves, sonic boom
47.40.Nm Shock wave interactions and shock effects

Rest frame system for asymptotically flat space–times

Osvaldo M. Moreschi and Sergio Dain

J. Math. Phys. 39, 6631 (1998); http://dx.doi.org/10.1063/1.532646 (20 pages) | Cited 3 times

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See Also: Erratum

Show Abstract
The notion of center of mass for an isolated system has been previously encoded in the definition of the so-called nice sections. In this article we present a generalization of the proof of existence of solutions to the linearized equation for nice sections, and formalize a local existence proof of nice sections relaxing the radiation condition. We report on the differentiable and non-self-crossing properties of this family of solutions. We also give a proof of the global existence of nice sections. © 1998 American Institute of Physics.
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04.20.Gz Spacetime topology, causal structure, spinor structure
02.40.-k Geometry, differential geometry, and topology

On stationary black holes of the Einstein conformally invariant scalar system

T. Zannias

J. Math. Phys. 39, 6651 (1998); http://dx.doi.org/10.1063/1.532647 (17 pages) | Cited 5 times

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The circularity property of stationary-axisymmetric asymptotically flat solutions of the coupled Einstein conformally invariant scalar field equations is investigated. It is proven that as a result of the coupling of the scalar field to the background scalar curvature, in general such solutions fail to be circular. Consequently, it is argued that the space of stationary, black hole equilibrium states may contain states which are not circular. However, the circular stationary-axisymmetric sector is nonempty and in fact it is proven that: All circular, stationary-axisymmetric asymptotically flat black hole solutions are those and only those black holes states described by the two parameter family of Kerr black holes. © 1998 American Institute of Physics.
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04.70.-s Physics of black holes
04.20.-q Classical general relativity
97.60.Lf Black holes
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