The Singularity in Generic Gravitational Collapse Is Spacelike, Local, and Oscillatory

A longstanding conjecture by Belinskii, Khalatnikov, and Lifshitz that the singularity in generic gravitational collapse is spacelike, local, and oscillatory is explored analytically and numerically in spatially inhomogeneous cosmological spacetimes. With a convenient choice of variables, it can be seen analytically how nonlinear terms in Einstein's equations control the approach to the singularity and cause oscillatory behavior. The analytic picture requires the drastic assumption that each spatial point evolves toward the singularity as an independent spatially homogeneous universe. In every case, detailed numerical simulations of the full Einstein evolution equations support this assumption.Comment: 7 pages includes 4 figures. Uses Revtex and psfig. Received "honorable mention" in 1998 Gravity Research Foundation essay contest. Submitted to Mod. Phys. Lett.

Hunting Local Mixmaster Dynamics in Spatially Inhomogeneous Cosmologies

Heuristic arguments and numerical simulations support the Belinskii et al (BKL) claim that the approach to the singularity in generic gravitational collapse is characterized by local Mixmaster dynamics (LMD). Here, one way to identify LMD in collapsing spatially inhomogeneous cosmologies is explored. By writing the metric of one spacetime in the standard variables of another, signatures for LMD may be found. Such signatures for the dynamics of spatially homogeneous Mixmaster models in the variables of U(1)-symmetric cosmologies are reviewed. Similar constructions for U(1)-symmetric spacetimes in terms of the dynamics of generic $T^2$-symmetric spacetime are presented.Comment: 17 pages, 5 figures. Contribution to CQG Special Issue "A Spacetime Safari: Essays in Honour of Vincent Moncrief

Harmonic coordinate method for simulating generic singularities

This paper presents both a numerical method for general relativity and an application of that method. The method involves the use of harmonic coordinates in a 3+1 code to evolve the Einstein equations with scalar field matter. In such coordinates, the terms in Einstein's equations with the highest number of derivatives take a form similar to that of the wave equation. The application is an exploration of the generic approach to the singularity for this type of matter. The preliminary results indicate that the dynamics as one approaches the singularity is locally the dynamics of the Kasner spacetimes.Comment: 5 pages, 4 figures, Revtex, discussion expanded, references adde

Top quark physics at muon and other future colliders

The top quark will be extensively studied at future muon colliders. The threshold cross section can be measured precisely, and the small beam energy spread is especially effective at making the measurement useful. We report on all the activities of the top quark working group, including talks on top quark physics at other future colliders.Comment: 16 pages, 9 figures, Summary report of the Top Quark Working Group at the Workshop on Physics at the First Muon Collider and at the Front End of a Muon Collider, November 6-9, 1997, Fermi National Accelerator Laborator

Oscillatory approach to the singularity in vacuum T2T^2 symmetric spacetimes

A combination of qualitative analysis and numerical study indicates that vacuum $T^2$ symmetric spacetimes are, generically, oscillatory.Comment: 2 pages submitted to the Ninth Marcel Grossmann Proceedings; v2, "all known cases" changed to "various known cases" in the first paragrap

Timelike Compton scattering: exclusive photoproduction of lepton pairs

We investigate the exclusive photoproduction of a heavy timelike photon which decays into a lepton pair, gamma p -> l+ l- p. This can be seen as the analog of deeply virtual Compton scattering, and we argue that the two processes are complementary for studying generalized parton distributions in the nucleon. In an unpolarized experiment the angular distribution of the leptons readily provides access to the real part of the Compton amplitude. We estimate the possible size of this effect in kinematics where the Compton process should be dominated by quark exchange.Comment: 31 pages, 17 figure

On the area of the symmetry orbits in T2T^2 symmetric spacetimes

We obtain a global existence result for the Einstein equations. We show that in the maximal Cauchy development of vacuum $T^2$ symmetric initial data with nonvanishing twist constant, except for the special case of flat Kasner initial data, the area of the $T^2$ group orbits takes on all positive values. This result shows that the areal time coordinate $R$ which covers these spacetimes runs from zero to infinity, with the singularity occurring at R=0.Comment: The appendix which appears in version 1 has a technical problem (the inequality appearing as the first stage of (52) is not necessarily true), and since the appendix is unnecessary for the proof of our results, we leave it out. version 2 -- clarifications added, version 3 -- reference correcte

Evidence for an oscillatory singularity in generic U(1) symmetric cosmologies on T3×RT^3 \times R

A longstanding conjecture by Belinskii, Lifshitz, and Khalatnikov that the singularity in generic gravitational collapse is locally oscillatory is tested numerically in vacuum, U(1) symmetric cosmological spacetimes on $T^3 \times R$. If the velocity term dominated (VTD) solution to Einstein's equations is substituted into the Hamiltonian for the full Einstein evolution equations, one term is found to grow exponentially. This generates a prediction that oscillatory behavior involving this term and another (which the VTD solution causes to decay exponentially) should be observed in the approach to the singularity. Numerical simulations strongly support this prediction.Comment: 15 pages, Revtex, includes 12 figures, psfig. High resolution versions of figures 7, 8, 9, and 11 may be obtained from anonymous ftp to ftp://vela.acs.oakland.edu/pub/berger/u1genfig

Shintani functions, real spherical manifolds, and symmetry breaking operators

For a pair of reductive groups $G \supset G'$, we prove a geometric criterion for the space $Sh(\lambda, \nu)$ of Shintani functions to be finite-dimensional in the Archimedean case. This criterion leads us to a complete classification of the symmetric pairs $(G,G')$ having finite-dimensional Shintani spaces. A geometric criterion for uniform boundedness of $dim Sh(\lambda, \nu)$ is also obtained. Furthermore, we prove that symmetry breaking operators of the restriction of smooth admissible representations yield Shintani functions of moderate growth, of which the dimension is determined for $(G, G') = (O(n+1,1), O(n,1))$.Comment: to appear in Progress in Mathematics, Birkhause