36,300 research outputs found

### 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

### Oscillatory approach to the singularity in vacuum $T^2$ 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

### On the area of the symmetry orbits in $T^2$ 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 $T^3 \times R$

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

### On the push&pull protocol for rumour spreading

The asynchronous push&pull protocol, a randomized distributed algorithm for
spreading a rumour in a graph $G$, works as follows. Independent Poisson clocks
of rate 1 are associated with the vertices of $G$. Initially, one vertex of $G$
knows the rumour. Whenever the clock of a vertex $x$ rings, it calls a random
neighbour $y$: if $x$ knows the rumour and $y$ does not, then $x$ tells $y$ the
rumour (a push operation), and if $x$ does not know the rumour and $y$ knows
it, $y$ tells $x$ the rumour (a pull operation). The average spread time of $G$
is the expected time it takes for all vertices to know the rumour, and the
guaranteed spread time of $G$ is the smallest time $t$ such that with
probability at least $1-1/n$, after time $t$ all vertices know the rumour. The
synchronous variant of this protocol, in which each clock rings precisely at
times $1,2,\dots$, has been studied extensively. We prove the following results
for any $n$-vertex graph: In either version, the average spread time is at most
linear even if only the pull operation is used, and the guaranteed spread time
is within a logarithmic factor of the average spread time, so it is $O(n\log
n)$. In the asynchronous version, both the average and guaranteed spread times
are $\Omega(\log n)$. We give examples of graphs illustrating that these bounds
are best possible up to constant factors. We also prove theoretical
relationships between the guaranteed spread times in the two versions. Firstly,
in all graphs the guaranteed spread time in the asynchronous version is within
an $O(\log n)$ factor of that in the synchronous version, and this is tight.
Next, we find examples of graphs whose asynchronous spread times are
logarithmic, but the synchronous versions are polynomially large. Finally, we
show for any graph that the ratio of the synchronous spread time to the
asynchronous spread time is $O(n^{2/3})$.Comment: 25 page

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