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

Notes on the integration of numerical relativity waveforms

A primary goal of numerical relativity is to provide estimates of the wave strain, $h$, from strong gravitational wave sources, to be used in detector templates. The simulations, however, typically measure waves in terms of the Weyl curvature component, $\psi_4$. Assuming Bondi gauge, transforming to the strain $h$ reduces to integration of $\psi_4$ twice in time. Integrations performed in either the time or frequency domain, however, lead to secular non-linear drifts in the resulting strain $h$. 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

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 $\mathcal{S}$, 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

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$_0$ Mixmaster models which are clearly matched in the simulation.Comment: 12 pages, Revtex, 4 eps figure

Non-integrability of the mixmaster universe

We comment on an analysis by Contopoulos et al. which demonstrates that the governing six-dimensional Einstein equations for the mixmaster space-time metric pass the ARS or reduced Painlev\'{e} test. We note that this is the case irrespective of the value, $I$, of the generating Hamiltonian which is a constant of motion. For $I < 0$ we find numerous closed orbits with two unstable eigenvalues strongly indicating that there cannot exist two additional first integrals apart from the Hamiltonian and thus that the system, at least for this case, is very likely not integrable. In addition, we present numerical evidence that the average Lyapunov exponent nevertheless vanishes. The model is thus a very interesting example of a Hamiltonian dynamical system, which is likely non-integrable yet passes the reduced Painlev\'{e} test.Comment: 11 pages LaTeX in J.Phys.A style (ioplppt.sty) + 6 PostScript figures compressed and uuencoded with uufiles. Revised version to appear in J Phys.

Marginally trapped tubes and dynamical horizons

We investigate the generic behaviour of marginally trapped tubes (roughly time-evolved apparent horizons) using simple, spherically symmetric examples of dust and scalar field collapse/accretion onto pre-existing black holes. We find that given appropriate physical conditions the evolution of the marginally trapped tube may be either null, timelike, or spacelike and further that the marginally trapped two-sphere cross-sections may either expand or contract in area. Spacelike expansions occur when the matter falling into a black hole satisfies $\rho - P \leq 1/A$, where $A$ is the area of the horizon while $\rho$ and $P$ are respectively the density and pressure of the matter. Timelike evolutions occur when $(\rho - P)$ is greater than this cut-off and so would be expected to be more common for large black holes. Physically they correspond to horizon "jumps" as extreme conditions force the formation of new horizons outside of the old.Comment: 31 pages, many figures. Final Version to appear in CQG: improvements include more complete references, a discussion of those references, Penrose-Carter diagrams for several of the spacetimes, and improved numerics for the scalar field

Chaotic Friedmann-Robertson-Walker Cosmology

We show that the dynamics of a spatially closed Friedmann - Robertson - Walker Universe conformally coupled to a real, free, massive scalar field, is chaotic, for large enough field amplitudes. We do so by proving that this system is integrable under the adiabatic approximation, but that the corresponding KAM tori break up when non adiabatic terms are considered. This finding is confirmed by numerical evaluation of the Lyapunov exponents associated with the system, among other criteria. Chaos sets strong limitations to our ability to predict the value of the field at the Big Crunch, from its given value at the Big Bang. (Figures available on request)Comment: 28 pages, 11 figure

Chaos in Static Axisymmetric Spacetimes I : Vacuum Case

We study the motion of test particle in static axisymmetric vacuum spacetimes and discuss two criteria for strong chaos to occur: (1) a local instability measured by the Weyl curvature, and (2) a tangle of a homoclinic orbit, which is closely related to an unstable periodic orbit in general relativity. We analyze several static axisymmetric spacetimes and find that the first criterion is a sufficient condition for chaos, at least qualitatively. Although some test particles which do not satisfy the first criterion show chaotic behavior in some spacetimes, these can be accounted for the second criterion.Comment: More comments for the quantitative estimation of chaos are added, and some inappropriate terms are changed. This will appear on Class. Quant. Gra

(Non)Invariance of dynamical quantities for orbit equivalent flows

We study how dynamical quantities such as Lyapunov exponents, metric entropy, topological pressure, recurrence rates, and dimension-like characteristics change under a time reparameterization of a dynamical system. These quantities are shown to either remain invariant, transform according to a multiplicative factor or transform through a convoluted dependence that may take the form of an integral over the initial local values. We discuss the significance of these results for the apparent non-invariance of chaos in general relativity and explore applications to the synchronization of equilibrium states and the elimination of expansions

On the relation between mathematical and numerical relativity

The large scale binary black hole effort in numerical relativity has led to an increasing distinction between numerical and mathematical relativity. This note discusses this situation and gives some examples of succesful interactions between numerical and mathematical methods is general relativity.Comment: 12 page