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Dec 1998

Volume 39, Issue 12, pp. 6247-6756

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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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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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Integrable discretizations of the Euler top

A. I. Bobenko, B. Lorbeer, and Yu. B. Suris

J. Math. Phys. 39, 6668 (1998); http://dx.doi.org/10.1063/1.532648 (16 pages) | Cited 9 times

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Discretizations of the Euler top sharing the integrals of motion with the continuous time system are studied. Those of them which are also Poisson with respect to the invariant Poisson bracket of the Euler top are characterized. For all these Poisson discretizations a solution in terms of elliptic functions is found, allowing a direct comparison with the continuous time case. We demonstrate that the Veselov–Moser discretization also belongs to our family, and apply our methods to this particular example. © 1998 American Institute of Physics.
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45.05.+x General theory of classical mechanics of discrete systems
05.45.-a Nonlinear dynamics and chaos
02.30.Cj Measure and integration

On harmonic maps into gauge groups

Qing Ding

J. Math. Phys. 39, 6684 (1998); http://dx.doi.org/10.1063/1.532649 (12 pages)

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We continue the work of the paper [J. Math. Phys. 37, 4076 (1996)] on the existence of globally defined harmonic maps from the Minkowski plane R1,1 into an infinite-dimensional Hilbert Lie group. We prove that the Cauchy problem for harmonic maps from R1,1 into Hilbert loop groups Hs(LG) can be solved globally for all remaining cases ½<s ⩽ ¾ and we obtain similar results for harmonic maps from R1,1 into certain Hilbert Lie gauge groups Hs(Sn,G) (n ≥ 2). These answer the two questions remaining from the above paper. © 1998 American Institute of Physics.
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11.15.-q Gauge field theories
02.20.Qs General properties, structure, and representation of Lie groups

The topological structure of the space–time disclination

Yishi Duan and Sheng Li

J. Math. Phys. 39, 6696 (1998); http://dx.doi.org/10.1063/1.532650 (10 pages) | Cited 2 times

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The space–time disclination is studied by making use of the decomposition theory of gauge potential in terms of the antisymmetric tensor field and ϕ-mapping method. It is shown that the self-dual and anti-self-dual parts of the curvature compose the space–time disclinations which are classified in terms of topological invariants—winding number. The projection of space–time disclination density along an antisymmetric tensor field is quantized topologically and characterized by Brouwer degree and Hopf index. © 1998 American Institute of Physics.
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98.80.Jk Mathematical and relativistic aspects of cosmology
98.80.Qc Quantum cosmology
04.60.-m Quantum gravity

Maximum entropy in the generalized moment problem

M. Frontini and A. Tagliani

J. Math. Phys. 39, 6706 (1998); http://dx.doi.org/10.1063/1.532651 (9 pages) | Cited 3 times

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The generalized moment problem in the framework of the maximum entropy approach is considered. A proof for the existence conditions of the solution is provided. © 1998 American Institute of Physics.
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02.50.Cw Probability theory

On the use of a new integral theorem for the quantum mechanical treatment of electric circuits

W. Magnus and W. Schoenmaker

J. Math. Phys. 39, 6715 (1998); http://dx.doi.org/10.1063/1.532652 (5 pages) | Cited 6 times

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We present a new integral theorem based on the well-known Gauss and Stokes theorems which provides a straightforward physical interpretation of the quantum mechanical energy balance equation governing the steady-state propagation of charge carriers through closed electric circuits. © 1998 American Institute of Physics.
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84.30.Bv Circuit theory
85.35.Ds Quantum interference devices
73.23.-b Electronic transport in mesoscopic systems
02.30.Rz Integral equations

Solving Poisson’s equation with interior conditions

J. E. McCarthy, E. Yu. Backhaus, and J. Fajans

J. Math. Phys. 39, 6720 (1998); http://dx.doi.org/10.1063/1.532653 (10 pages) | Cited 2 times

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We consider the problem of extending the solution of a particular two-dimensional Poisson equation to a larger domain. This problem is related to the problem of putting a non-neutral plasma into equilibrium by applying a suitable wall potential, and to similar problems in two-dimensional fluid dynamics. While one cannot always find an exact solution, one can always find an approximate solution if the plasma has no holes. © 1998 American Institute of Physics.
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52.55.Dy General theory and basic studies of plasma lifetime, particle and heat loss, energy balance, field structure, etc.
52.35.Py Macroinstabilities (hydromagnetic, e.g., kink, fire-hose, mirror, ballooning, tearing, trapped-particle, flute, Rayleigh-Taylor, etc.)
52.35.Qz Microinstabilities (ion-acoustic, two-stream, loss-cone, beam-plasma, drift, ion- or electron-cyclotron, etc.)
47.10.-g General theory in fluid dynamics
52.40.Hf Plasma-material interactions; boundary layer effects

On the triple sum formula for Wigner 9j-symbols

Hjalmar Rosengren

J. Math. Phys. 39, 6730 (1998); http://dx.doi.org/10.1063/1.532634 (15 pages) | Cited 5 times

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We give a new proof of the triple sum formula for Wigner 9j-symbols due to Ališauskas and Jucys. The proof uses explicit expressions for the coupling kernels recently introduced by the author. Parts of our results generalize to general recoupling coefficients. © 1998 American Institute of Physics.
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03.65.Ca Formalism

Lie symmetry and integrability of ordinary differential equations

R. Z. Zhdanov

J. Math. Phys. 39, 6745 (1998); http://dx.doi.org/10.1063/1.532654 (12 pages)

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Combining a Lie algebraic approach that is due to Wei and Norman [J. Math. Phys. 4, 475 (1963)] and the ideas suggested by Drach [Compt. Rend. 168, 337 (1919)], we have constructed several classes of systems of linear ordinary differential equations that are integrable by quadratures. Their integrability is ensured by integrability of the corresponding stationary cubic Schrödinger, KdV, and Harry–Dym equations. Next, we obtain a hierarchy of integrable reductions of the Dirac equation of an electron moving in the external field. Their integrability is shown to be in correspondence with integrability of the stationary mKdV hierarchy. © 1998 American Institute of Physics.
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41.20.Jb Electromagnetic wave propagation; radiowave propagation
02.30.Hq Ordinary differential equations
02.10.Ud Linear algebra
02.10.Xm Multilinear algebra
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