An experiment to test the isotropy of the one-way speed of light
An experimental setup capable, in principle, to test the isotropy of the {\em one way} propagation of light to 1 part in $10^8$ (or better), is suggested.
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arXiv subjects
Reinaldo J. Gleiser
Publications and source records attributed to Reinaldo J. Gleiser.
The Linet-Tian metrics are a restricted set of Krasiński's solutions of Einstein's field equations for a rotating perfect fluid
In this note we show that the Linet-Tian family of solutions of the vacuum Einstein equations with a cosmological constant are a restricted set of the solutions of the Einstein field equations for a rotating perfect fluid previously found by A. Krasiński.
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Linear perturbations of the Linet-Tian metrics with a positive cosmological constant
The Linet-Tian metrics are solutions of the Einstein equations with a cosmological constant, $Λ$, that can be positive or negative. The linear instability of these metrics in the case $Λ<0$, has already been established. In the case $Λ>0$, it was found in a recent analysis that the perturbation equations admit unstable modes. The analysis was based on the construction of a gauge invariant function of the metric perturbation coefficients, called here $W(y)$. This function satisfied a linear second order equation that could be used to set up a boundary value problem determining the allowed, real or purely imaginary frequencies for the perturbations. Nevertheless, the relation of these solutions to the full spectrum of perturbations, and, therefore, to the evolution of arbitrary perturbations, remained open. In this paper we consider again the perturbations of the Linet-Tian metric with $Λ>0$, and show, using a form of the Darboux transformation, that one can associate with the perturbation equations a self adjoint problem that provides a solution to the completeness and spectrum of the perturbations. This is also used to construct the explicit relation between the solutions of the gauge invariant equation for $W(y)$, and the evolution of arbitrary initial data, thus solving the problem that remained open in the previous study. Numerical methods are then used to confirm the existence of unstable modes as a part of the complete spectrum of the perturbations, thus establishing the linear gravitational instability of the Linet-Tian metrics with $Λ>0$.
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Linear stability of the Linet - Tian solution with positive cosmological constant
The purpose of this paper is to extend the analysis of gravitational instability of the Linet - Tian solution to the case $Λ> 0$. A fundamental difference brought about by $Λ>0$, as compared to $Λ<0$ is in the structure of the resulting space time. Associated with each of the two commuting Killing vectors $\partial_ϕ$, and $\partial_{z}$, there is a curvature singularity that has the same characteristics as that associated to $\partial_ϕ$ in the Levi - Civita metric, and we show that there is an isometry relating these singularities that reduces the effective parameter space of the metrics. In attempting to set up and solve the linearized perturbation equations we are confronted with the problem of a gauge ambiguity that leads to the introduction of a gauge invariant function, $W_1$, that is shown to be also a {\em master function}, that satisfies a second order ODE, and in terms of which one can express all the perturbation functions. Unfortunately, the equation satisfied by $W_1$ contains singular coefficients, and, although {\em all} its solutions are regular, because of the presence of these singularities one cannot, as in the case of negative $Λ$, set up an associated self adjoint problem that provides a complete set of solutions for $W_1$. We are thus restricted to solving numerically the perturbation equations, and using those solutions for constructing $W_1$, for particular values of the parameters. In all the cases analyzed we find unstable modes, which strongly suggests that all the Linet - Tian space times with $Λ> 0$ are linearly unstable under gravitational perturbations. The problem of determining the time evolution of arbitrary initial data in terms of the $W_1$, or something equivalent, remains open.
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Linear stability of the Linet - Tian solution with negative cosmological constant
In this paper we analyze the linear stability of the Linet - Tian solution with negative cosmological constant. In the limit of vanishing cosmological constant the Linet - Tian metric reduces to a form of the Levi - Civita metric, and, therefore, it can be considered as a generalization of the former to include a cosmological constant. The gravitational instability of the Levi - Civita metric was recently established, and the purpose of this paper is to investigate what changes result from the introduction of a cosmological constant. A fundamental difference brought about by a (negative) cosmological constant is in the structure at infinity. This introduces an added problem in attempting to define an evolution for the perturbations because the constant time hypersurfaces are not Cauchy surfaces. In this paper we show that under a large set of boundary conditions that lead to a unique evolution of the perturbations, we always find unstable modes, that would generically be present in the evolution of arbitrary initial data, leading to the conclusion that the Linet - Tian space times with negative cosmological constant are linearly unstable under gravitational perturbations.
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Axial gravitational perturbations of an infinite static line source
In this paper we study axial gravitational perturbations of an infinite static line source, represented by a form of the Levi-Civita metric. The perturbations are restricted to axial symmetry but break the cylindrical symmetry of the background metric. We analyze the gauge issues that arise in setting up the appropriate form of the perturbed metric and show that it is possible to restrict to diagonal terms, but that this does not fix the gauge completely. We derive the perturbation equations and show that they can be solved by solving a third order ordinary differential equation for an appropriately chosen function of the perturbed metric coefficients. The set of solutions of this equation contains gauge trivial parts, and we show how to extract the gauge non trivial components. We introduce appropriate boundary conditions on the solutions and show that these lead to a boundary value problem that determines the allowed functional forms of the perturbation modes. The associated eigenvalues determine a sort of "dispersion relation" for the frequencies and corresponding "wave vector" components. The central result of this analysis is that the spectrum of allowed frequencies contains one unstable (imaginary frequency) mode for every possible choice of the background metric. The completeness of the mode expansion in relation to the initial value problem and to the gauge problem is discussed in detail, and we show that the perturbations contain an unstable component for generic initial data, and, therefore, that the Levi-Civita space times are gravitationally unstable. We also include, for completeness, a set of approximate eigenvalues, and examples of the functional form of the solutions.
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On the evolution of the momentarily static radiation free data in the Apostolatos - Thorne cylindrical shell model
We study the evolution of the "Momentarily Static and Radiation Free" (MSRF) initial data for the Apostolatos - Thorne cylindrical shell model. We analyze the relation between the parameters characterizing the MSRF data those for the corresponding final static configuration, and show that there is a priori no conflict for any choice of initial MSRF data, in contrast with some recent results of Nakao, Ida and Kurita. We also consider the problem in the linear approximation, and show that the evolution is stable in all cases. We find that the approach to the final state is very slow, with an inverse logarithmic dependence on time at fixed radius. To complement these results we introduce a numerical computation procedure that allows us to visualize the explicit form of the evolution of the shell and of the gravitational field up to large times. The results are in agreement with the qualitative behaviour conjectured by Apostolatos and Thorne, with an initial damped oscillatory stage, but with oscillations about a position that approaches slowly that of the static final state, as indicated by our analysis. We also include an Appendix, where we prove the existence of solutions of the cylindrical wave equation with vanishing initial value for $r > R_0$, ($R_0 > 0$ some finite constant), that approach a constant value for large times. This result is crucial for the proof of compatibility of arbitrary MSRF initial data and a final static configuration for the system.
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Late time tails in the Kerr spacetime
Outside a black hole, perturbation fields die off in time as $1/t^n$. For spherical holes $n=2\ell+3$ where $\ell$ is the multipole index. In the nonspherical Kerr spacetime there is no coordinate-independent meaning of "multipole," and a common sense viewpoint is to set $\ell$ to the lowest radiatiable index, although theoretical studies have led to very different claims. Numerical results, to date, have been controversial. Here we show that expansion for small Kerr spin parameter $a$ leads to very definite numerical results confirming previous theoretical analyses.
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Perturbative evolution of the static configurations, quasinormal modes and quasi normal ringing in the Apostolatos - Thorne cylindrical shell model
We study the perturbative evolution of the static configurations, quasinormal modes and quasi normal ringing in the Apostolatos - Thorne cylindrical shell model. We consider first an expansion in harmonic modes and show that it provides a complete solution for the characteristic value problem for the finite perturbations of a static configuration. As a consequence of this completeness we obtain a proof of the stability of static solutions under this type of perturbations. The explicit expression for the mode expansion are then used to obtain numerical values for some of the quasi normal mode complex frequencies. Some examples involving the numerical evaluation of the integral mode expansions are described and analyzed, and the quasi normal ringing displayed by the solutions is found to be in agreement with quasi normal modes found previously. Going back to the full relativistic equations of motion we find their general linear form by expanding to first order about a static solution. We then show that the resulting set of coupled ordinary and partial differential equations for the dynamical variables of the system can be used to set an initial plus boundary values problem, and prove that there is an associated positive definite constant of the motion that puts absolute bounds on the dynamic variables of the system, establishing the stability of the motion of the shell under arbitrary, finite perturbations. We also show that the problem can be solved numerically, and provide some explicit examples that display the complete agreement between the purely numerical evolution and that obtained using the mode expansion, in particular regarding the quasi normal ringing that results in the evolution of the system. We also discuss the relation of the present work to some recent results on the same model that have appeared in the literature.
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Unstable fields in Kerr spacetimes
We show that both the interior region $r M^2$ Kerr naked singularity admit unstable solutions of the Teukolsky equation for any value of the spin weight. For every harmonic number there is at least one axially symmetric mode that grows exponentially in time and decays properly in the radial directions. These can be used as Debye potentials to generate solutions for the scalar, Weyl spinor, Maxwell and linearized gravity field equations on these backgrounds, satisfying appropriate spatial boundary conditions and growing exponentially in time, as shown in detail for the Maxwell case. It is suggested that the existence of the unstable modes is related to the so called "time machine" region, where the axial Killing vector field is time-like, and the Teukolsky equation, restricted to axially symmetric fields, changes its character from hyperbolic to elliptic.
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Instabilities in Kerr Spacetimes
We present a generalization of previous results regarding the stability under gravitational perturbations of nakedly singular super extreme Kerr spacetime and Kerr black hole interior beyond the Cauchy horizon. To do so we study solutions to the radial and angular Teukolsky's equations with different spin weights, particulary $s=\pm 1$ representing electromagnetic perturbations, $s=\pm 1/2$ representing a perturbation by a Dirac field and $s=0$ representing perturbations by a scalar field. By analizing the properties of radial and angular eigenvalues we prove the existence of an infinite family of unstable modes.
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The effect of radiative gravitational modes on the dynamics of a cylindrical shell of counter rotating particles
In this paper we consider some aspects of the relativistic dynamics of a cylindrical shell of counter rotating particles. In some sense these are the simplest systems with a physically acceptable matter content that display in a well defined sense an interaction with the radiative modes of the gravitational field. These systems have been analyzed previously, but in most cases resorting to approximations, or considering a particular form for the initial value data. Here we show that there exists a family of solutions where the space time inside the shell is flat and the equation of motion of the shell decouples completely from the gravitational modes. The motion of the shell is governed by an equation of the same form as that of a particle in a time independent one dimensional potential. We find that under appropriate initial conditions one can have collapsing, bounded periodic, and unbounded motions. We analyze and solve also the linearized equations that describe the dynamics of the system near a stable static solutions, keeping a regular interior. The surprising result here is that the motion of the shell is completely determined by the configuration of the radiative modes of the gravitational field. In particular, there are oscillating solutions for any chosen period, in contrast with the "approximately Newtonian plus small radiative corrections" motion expectation. We comment on the physical meaning of these results and provide some explicit examples. We also discuss the relation of our results to the initial value problem for the linearized dynamics of the shell.
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Gravitational instability of the inner static region of a Reissner-Nordstrom black hole
Reissner--Nordström black holes have two static regions: $r > \ro$ and $0 < r < \ri$, where $\ri$ and $\ro$ are the inner and outer horizon radii. The stability of the exterior static region has been established long time ago. In this work we prove that the interior static region is unstable under linear gravitational perturbations, by showing that field perturbations compactly supported within this region will generically excite a mode that grows exponentially in time. This result gives an alternative reason to mass inflation to consider the space time extension beyond the Cauchy horizon as physically irrelevant, and thus provides support to the strong cosmic censorship conjecture, which is also backed by recent evidence of a linear gravitational instability in the interior region of Kerr black holes found by the authors. The use of intertwiners to solve for the evolution of initial data plays a key role, and adapts without change to the case of super-extremal \rn black holes, allowing to complete the proof of the linear instability of this naked singularity. A particular intertwiner is found such that the intertwined Zerilli field has a geometrical meaning -it is the first order variation of a particular Riemann tensor invariant-. Using this, calculations can be carried out explicitely for every harmonic number.
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The initial value problem for linearized gravitational perturbations of the Schwarzschild naked singularity
The coupled equations for the scalar modes of the linearized Einstein equations around Schwarzschild's spacetime were reduced by Zerilli to a 1+1 wave equation with a potential $V$, on a field $Ψ_z$. For smooth metric perturbations $Ψ_z$ is singular at $r_s=-6M/(\ell-1)(\ell+2)$, $\ell$ the mode harmonic number, and $V$ has a second order pole at $r_s$. This is irrelevant to the black hole exterior stability problem, where $r>2M>0$, and $r_s <0$, but it introduces a non trivial problem in the naked singular case where $M<0$, then $r_s>0$, and the singularity appears in the relevant range of $r$. We solve this problem by developing a new approach to the evolution of the even mode, based on a {\em new gauge invariant function}, $\hat Ψ$ -related to $Ψ_z$ by an intertwiner operator- that is a regular function of the metric perturbation {\em for any value of $M$}. This allows to address the issue of evolution of gravitational perturbations in this non globally hyperbolic background, and to complete the proof of the linear instability of the Schwarzschild naked singularity, by showing that a previously found unstable mode is excitable by generic initial data. This is further illustrated by numerically solving the linearized equations for suitably chosen initial data.
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Static spherically symmetric Einstein-Vlasov shells made up of particles with a discrete set of values of their angular momentum
In this paper we study static spherically symmetric Einstein-Vlasov shells, made up of equal mass particles, where the angular momentum L of particles takes values only on a discrete finite set. We consider first the case where there is only one value of L, and prove their existence by constructing explicit examples. Shells with either hollow or black hole interiors have finite thickness. Of particular interest is the thin shell limit of these systems and we study its properties using both numerical and analytic arguments to compare with known results. The general case of a set of values of L is also considered and the particular case where L takes only two values is analyzed, and compared with the corresponding thin shell limit already given in the literature, finding good agreement in all cases.
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Instabilities of naked singularities and black hole interiors in General Relativity
Metrics representing black holes in General Relativity may exhibit naked singularities for certain values of their parameters. This is the case for super-extremal ($J^2 > M>0$) Kerr and super-extremal ($|Q|>M>0$) Reissner-Nördstrom spacetimes, and also for the negative mass Schwarzschild spacetime. We review our recent work where we show that these nakedly singular spacetimes are unstable under linear gravitational perturbations, a result that supports the cosmic censorship conjecture, and also that the inner stationary region beyond the inner horizon of a Kerr black hole ($J^2 <M$) is linearly unstable.
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Gravitational instabilities in Kerr space-times
In this paper we consider the possible existence of unstable axisymmetric modes in Kerr space times, resulting from exponentially growing solutions of the Teukolsky equation. We describe a transformation that casts the radial equation that results upon separation of variables in the Teukolsky equation, in the form of a Schrödinger equation, and combine the properties of the solutions of this equations with some recent results on the asymptotic behaviour of spin weighted spheroidal harmonics to prove the existence of an infinite family of unstable modes. Thus we prove that the stationary region beyond a Kerr black hole inner horizon is unstable under gravitational linear perturbations. We also prove that Kerr space-time with angular momentum larger than its square mass, which has a naked singularity, is unstable.
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On the dynamics of thin shells of counter rotating particles
In this paper we study the dynamics of self gravitating spherically symmetric thin shells of counter rotating particles. We consider all possible velocity distributions for the particles, and show that the equations of motion by themselves do not constrain this distribution. We therefore consider the dynamical stability of the resulting configurations under several possible processes. This include the stability of static configurations as a whole, where we find a lower bound for the compactness of the shells. We analyze also the stability of the single particle orbits, and find conditions for "single particle evaporation". Finally, in the case of a shell with particles whose angular momenta are restricted to two values, we consider the conditions for stability under splitting into two separate shells. This analysis leads to the conclusion that under certain conditions, that are given explicitly, an evolving shell may split into one or more separate shells. We provide explicit examples to illustrate this phenomenon. We also include a derivation of the thick to thin shell limit for an Einstein shell that shows that the limiting distribution of angular momenta is unique, covering continuously a finite range of values.
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