54 research outputs found
Curvature blow up in Bianchi VIII and IX vacuum spacetimes
The maximal globally hyperbolic development of non-Taub-NUT Bianchi IX vacuum
initial data and of non-NUT Bianchi VIII vacuum initial data is C2
inextendible. Furthermore, a curvature invariant is unbounded in the incomplete
directions of inextendible causal geodesics.Comment: 20 pages, no figures. Submitted to Classical and Quantum Gravit
Chaotic dynamics around astrophysical objects with nonisotropic stresses
The existence of chaotic behavior for the geodesics of the test particles
orbiting compact objects is a subject of much current research. Some years ago,
Gu\'eron and Letelier [Phys. Rev. E \textbf{66}, 046611 (2002)] reported the
existence of chaotic behavior for the geodesics of the test particles orbiting
compact objects like black holes induced by specific values of the quadrupolar
deformation of the source using as models the Erez--Rosen solution and the Kerr
black hole deformed by an internal multipole term. In this work, we are
interesting in the study of the dynamic behavior of geodesics around
astrophysical objects with intrinsic quadrupolar deformation or nonisotropic
stresses, which induces nonvanishing quadrupolar deformation for the
nonrotating limit. For our purpose, we use the Tomimatsu-Sato spacetime [Phys.
Rev. Lett. \textbf{29} 1344 (1972)] and its arbitrary deformed generalization
obtained as the particular vacuum case of the five parametric solution of Manko
et al [Phys. Rev. D 62, 044048 (2000)], characterizing the geodesic dynamics
throughout the Poincar\'e sections method. In contrast to the results by
Gu\'eron and Letelier we find chaotic motion for oblate deformations instead of
prolate deformations. It opens the possibility that the particles forming the
accretion disk around a large variety of different astrophysical bodies
(nonprolate, e.g., neutron stars) could exhibit chaotic dynamics. We also
conjecture that the existence of an arbitrary deformation parameter is
necessary for the existence of chaotic dynamics.Comment: 7 pages, 5 figure
Kolmogorov-Sinai entropy and black holes
It is shown that stringy matter near the event horizon of a Schwarzschild
black hole exhibits chaotic behavior (the spreading effect) which can be
characterized by the Kolmogorov-Sinai entropy. It is found that the
Kolmogorov-Sinai entropy of a spreading string equals to the half of the
inverse gravitational radius of the black hole. But the KS entropy is the same
for all objects collapsing into the black hole. The nature of this universality
is that the KS entropy possesses the main property of temperature: it is the
same for all bodies in thermal equilibrium with the black hole. The
Kolmogorov-Sinai entropy measures the rate at which information about the
string is lost as it spreads over the horizon. It is argued that it is the
maximum rate allowed by quantum theory. A possible relation between the
Kolmogorov-Sinai and Bekenstein-Hawking entropies is discussed.Comment: 10 pages, no figures; this is an extended version of my paper
arXiv:0711.313
Spike Oscillations
According to Belinskii, Khalatnikov and Lifshitz (BKL), a generic spacelike
singularity is characterized by asymptotic locality: Asymptotically, toward the
singularity, each spatial point evolves independently from its neighbors, in an
oscillatory manner that is represented by a sequence of Bianchi type I and II
vacuum models. Recent investigations support a modified conjecture: The
formation of spatial structures (`spikes') breaks asymptotic locality. The
complete description of a generic spacelike singularity involves spike
oscillations, which are described by sequences of Bianchi type I and certain
inhomogeneous vacuum models. In this paper we describe how BKL and spike
oscillations arise from concatenations of exact solutions in a
Hubble-normalized state space setting, suggesting the existence of hidden
symmetries and showing that the results of BKL are part of a greater picture.Comment: 38 pages, 14 figure
Notes on the integration of numerical relativity waveforms
A primary goal of numerical relativity is to provide estimates of the wave
strain, , from strong gravitational wave sources, to be used in detector
templates. The simulations, however, typically measure waves in terms of the
Weyl curvature component, . Assuming Bondi gauge, transforming to the
strain reduces to integration of twice in time. Integrations
performed in either the time or frequency domain, however, lead to secular
non-linear drifts in the resulting strain . These non-linear drifts are not
explained by the two unknown integration constants which can at most result in
linear drifts. We identify a number of fundamental difficulties which can arise
from integrating finite length, discretely sampled and noisy data streams.
These issues are an artifact of post-processing data. They are independent of
the characteristics of the original simulation, such as gauge or numerical
method used. We suggest, however, a simple procedure for integrating numerical
waveforms in the frequency domain, which is effective at strongly reducing
spurious secular non-linear drifts in the resulting strain.Comment: 23 pages, 10 figures, matches final published versio
New Algorithm for Mixmaster Dynamics
We present a new numerical algorithm for evolving the Mixmaster spacetimes.
By using symplectic integration techniques to take advantage of the exact Taub
solution for the scattering between asymptotic Kasner regimes, we evolve these
spacetimes with higher accuracy using much larger time steps than previously
possible. The longer Mixmaster evolution thus allowed enables detailed
comparison with the Belinskii, Khalatnikov, Lifshitz (BKL) approximate
Mixmaster dynamics. In particular, we show that errors between the BKL
prediction and the measured parameters early in the simulation can be
eliminated by relaxing the BKL assumptions to yield an improved map. The
improved map has different predictions for vacuum Bianchi Type IX and magnetic
Bianchi Type VI Mixmaster models which are clearly matched in the
simulation.Comment: 12 pages, Revtex, 4 eps figure
Electrocardiogram of the Mixmaster Universe
The Mixmaster dynamics is revisited in a new light as revealing a series of
transitions in the complex scale invariant scalar invariant of the Weyl
curvature tensor best represented by the speciality index , which
gives a 4-dimensional measure of the evolution of the spacetime independent of
all the 3-dimensional gauge-dependent variables except for the time used to
parametrize it. Its graph versus time characterized by correlated isolated
pulses in its real and imaginary parts corresponding to curvature wall
collisions serves as a sort of electrocardiogram of the Mixmaster universe,
with each such pulse pair arising from a single circuit or ``complex pulse''
around the origin in the complex plane. These pulses in the speciality index
and their limiting points on the real axis seem to invariantly characterize
some of the so called spike solutions in inhomogeneous cosmology and should
play an important role as a gauge invariant lens through which to view current
investigations of inhomogeneous Mixmaster dynamics.Comment: version 3: 20 pages iopart style, 19 eps figure files for 8 latex
figures; added example of a transient true spike to contrast with the
permanent true spike example from the Lim family of true spike solutions;
remarks in introduction and conclusion adjusted and toned down; minor
adjustments to the remaining tex
Bianchi VIII Empty Futures
Using a qualitative analysis based on the Hamiltonian formalism and the
orthonormal frame representation we investigate whether the chaotic behaviour
which occurs close to the initial singularity is still present in the far
future of Bianchi VIII models. We describe some features of the vacuum Bianchi
VIII models at late times which might be relevant for studying the nature of
the future asymptote of the general vacuum inhomogeneous solution to the
Einstein field equations.Comment: 22 pages, no figures, Latex fil
Global dynamics of the mixmaster model
The asymptotic behaviour of vacuum Bianchi models of class A near the initial
singularity is studied, in an effort to confirm the standard picture arising
from heuristic and numerical approaches by mathematical proofs. It is shown
that for solutions of types other than VIII and IX the singularity is velocity
dominated and that the Kretschmann scalar is unbounded there, except in the
explicitly known cases where the spacetime can be smoothly extended through a
Cauchy horizon. For types VIII and IX it is shown that there are at most two
possibilities for the evolution. When the first possibility is realized, and if
the spacetime is not one of the explicitly known solutions which can be
smoothly extended through a Cauchy horizon, then there are infinitely many
oscillations near the singularity and the Kretschmann scalar is unbounded
there. The second possibility remains mysterious and it is left open whether it
ever occurs. It is also shown that any finite sequence of distinct points
generated by iterating the Belinskii-Khalatnikov-Lifschitz mapping can be
realized approximately by a solution of the vacuum Einstein equations of
Bianchi type IX.Comment: 16 page
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