Systems Of Spherical Symmetry Research Articles

Spherically symmetric evolution of self-gravitating massive fields

We are interested in the global dynamics of a massive scalar field evolving under its own gravitational field and, in this paper, we study spherically symmetric solutions to Einstein's field equations coupled with a Klein-Gordon equation with quadratic potential. For the initial value problem we establish a global existence theory when initial data are prescribed on a future light cone with vertex at the center of symmetry. A suitably generalized solution in Bondi coordinates is sought which has low regularity and possibly large but finite Bondi mass. A similar result was established first by Christodoulou for massless fields. In order to deal with massive fields, we must overcome several challenges and significantly modify Christodoulou's original method. First of all, we formulate the Einstein-Klein-Gordon system in spherical symmetry as a non-local and nonlinear hyperbolic equation and, by carefully investigating the global dynamical behavior of the massive field, we establish various estimates concerning the Einstein operator, the Hawking mass, and the Bondi mass, including novel positivity and monotonicity properties. Importantly, in addition to a regularization at the center of symmetry we find it necessary to also introduce a regularization at null infinity. We also establish new energy and decay estimates for, both, regularized and generalized solutions.

Improved decay estimates and C2-asymptotic stability of solutions to the Einstein-scalar field system in spherical symmetry

We investigate the asymptotic stability of solutions to the characteristic initial value problem for the Einstein (massless) scalar field system with a positive cosmological constant. We prescribe spherically symmetric initial data on a future null cone with a wider range of decaying profiles than previously considered. New estimates are then derived in order to prove that, for small data, the system has a unique global classical solution. We also show that the solution decays exponentially in (Bondi) time and that the radial decay is essentially polynomial, although containing logarithmic factors in some special cases. This improved asymptotic analysis allows us to show that, under appropriate and natural decaying conditions on the initial data, the future asymptotic solution is differentiable, up to and including spatial null-infinity, and approaches the de Sitter solution, uniformly, in a neighborhood of infinity. Moreover, we analyze the decay of derivatives of the solution up to second order showing the (uniform) [Formula: see text]-asymptotic stability of the de Sitter attractor in this setting. This corresponds to a surprisingly strong realization of the cosmic no-hair conjecture.

A proof of the instability of AdS for the Einstein-massless Vlasov system

In recent years, the conjecture on the instability of Anti-de Sitter spacetime, put forward by Dafermos–Holzegel (Dynamic instability of solitons in 4 + 1 dimesnional gravity with negative cosmological constant, 2006. https://www.dpmms.cam.ac.uk/~md384/ADSinstability.pdf) and Dafermos (The Black Hole Stability problem, Talk at the Newton Institute, Cambridge, 2006. http://www-old.newton.ac.uk/webseminars/pg+ws/2006/gmx/1010/dafermos/) in 2006, has attracted a substantial amount of numerical and heuristic studies. Following the pioneering work (Phys Rev Lett 107(3):031102, 2011) of Bizon–Rostworowski, research efforts have been mainly focused on the study of the spherically symmetric Einstein-scalar field system. The first rigorous proof of the instability of AdS in the simplest spherically symmetric setting, namely for the Einstein-null dust system, was obtained in Moschidis (A proof of the instability of AdS for the Einstein-null dust system with an inner mirror, 2017. arXiv:1704.08681). In order to circumvent problems associated with the trivial break down of the Einstein-null dust system occuring at the center r=0, Moschidis (2017) studied the evolution of the system in the exterior of an inner mirror placed at r=r_{0}, r_{0}>0. However, in view of additional considerations on the nature of the instability, it was necessary for Moschidis (2017) to allow the mirror radius r_{0} to shrink to 0 with the size of the initial perturbation; well-posedness in the resulting complicated setup (involving low-regularity estimates of uniform modulus with respect to r_{0}) was obtained in Moschidis (The Einstein-null dust system in spherical symmetry with an inner mirror: structure of the maximal development and Cauchy stability, 2017. arXiv:1704.08685). In this paper, we establish the instability of AdS for the Einstein-massless Vlasov system in spherical symmetry; this will be the first proof of the AdS instability conjecture for an Einstein-matter system which is well-posed for regular initial data in the standard sense, without the addition of an inner mirror. The necessary well-posedness results for this system are obtained in our companion paper (Moschidis in The characteristic initial-boundary value problem for the Einstein-massless Vlasov system in spherical symmetry, 2018. arXiv:1812.04274). Our proof utilises an instability mechanism based on beam interactions which is superficially similar to the one appearing in Moschidis (A proof of the instability of AdS for the Einstein-null dust system with an inner mirror, 2017. arXiv:1704.08681). However, new difficulties associated with the Einstein-massless Vlasov system (such as the need for control on the paths of non-radial geodesics in a large curvature regime) will force us to develop a different strategy of proof involving a novel configuration of beam interactions. One of the main novelties of our construction is the introduction of a multi-scale hierarchy of domains in phase space, on which the initial support of the Vlasov field f is localised. The propagation of this hierarchical structure of the support of f along the evolution will be crucial both for controlling the geodesic flow under minimal regularity assumptions and for guaranteeing the existence of the solution until the time of trapped surface formation.

A Scattering Theory Approach to Cauchy Horizon Instability and Applications to Mass Inflation

Motivated by the strong cosmic censorship conjecture, we study the linear scalar wave equation in the interior of subextremal strictly charged Reissner–Nordström black holes by analyzing a suitably defined “scattering map” at 0 frequency. The method can already be demonstrated in the case of spherically symmetric scalar waves on Reissner–Nordström: we show that assuming suitable (\(L^2\)-averaged) upper and lower bounds on the event horizon, one can prove (\(L^2\)-averaged) polynomial lower bound for the solution (1) on any radial null hypersurface transversally intersecting the Cauchy horizon, and (2) along the Cauchy horizon toward timelike infinity. Taken together with known results regarding solutions to the wave equation in the exterior, (1) above in particular provides yet another proof of the linear instability of the Reissner–Nordström Cauchy horizon. As an application of (2) above, we prove a conditional mass inflation result for a nonlinear system, namely the Einstein–Maxwell-(real)-scalar field system in spherical symmetry. For this model, it is known that for a generic class of Cauchy data \(\mathcal G\), the maximal globally hyperbolic future developments are \(C^2\)-future-inextendible. We prove that if a (conjectural) improved decay result holds in the exterior region, then for the maximal globally hyperbolic developments arising from initial data in \(\mathcal G\), the Hawking mass blows up identically on the Cauchy horizon.

Global Non-linearly Stable Large Data Solutions to the Einstein Scalar Field System

I study a class of global, causal geodesically complete solutions to the spherically symmetric Einstein scalar field (SSESF) system . Extending results of Luk-Oh (Quantitative Decay Rates for Dispersive Solutions to the Einstein-Scalar Field System in Spherical Symmetry, arXiv:1402.2984), Luk-Oh-Yang (Solutions to the Einstein-Scalar-Field System in Spherical Symmetry with Large Bounded Variation Norms, arXiv:1605.03893), I provide new bounds controlling higher derivatives of both the metric components of the solution and the scalar field itself for large data solutions to SSESF. Moreover, by constructing a particular set of generalized wave-coordinates, I show that, assuming sufficient regularity of the data, these solutions are globally non-linearly stable to non-spherically symmetric perturbations by recent results of Luk and Oh. In particular, I demonstrate the existence of a large collection of non-trivial examples of large data, globally nonlinearly stable, dispersive solutions to the Einstein scalar field system.

Strong cosmic censorship in spherical symmetry for two-ended asymptotically flat initial data I. The interior of the black hole region

This is the first and main paper of a two-part series, in which we prove the $C^{2}$-formulation of the strong cosmic censorship conjecture for the Einstein-Maxwell-(real)-scalar-field system in spherical symmetry for two-ended asymptotically flat data. For this model, it is known through the works of Dafermos and Dafermos-Rodnianski that the maximal globally hyperbolic future development of any admissible two-ended asymptotically flat Cauchy initial data set possesses a non-empty Cauchy horizon, across which the spacetime is $C^{0}$-future-extendible (in particular, the $C^{0}$-formulation of the strong cosmic censorship conjecture is false). Nevertheless, the main conclusion of the present series of papers is that for a generic (in the sense of being open and dense relative to appropriate topologies) class of such data, the spacetime is future-inextendible with a Lorentzian metric of higher regularity (specifically, $C^{2}$). In this paper, we prove that the solution is $C^{2}$-future-inextendible under the condition that the scalar field obeys an $L^{2}$-averaged polynomial lower bound along each of the event horizons. This, in particular, improves upon a previous result of Dafermos, which required instead a pointwise lower bound. Key to the proof are appropriate stability and instability results in the interior of the black hole region, whose proofs are in turn based on ideas from the work of Dafermos-Luk on the stability of the Kerr Cauchy horizon (without symmetry) and from our previous paper on linear instability of Reissner-Nordstrom Cauchy horizon. In the second paper of the series (arXiv:1702.05716), which concerns analysis in the exterior of the black hole region, we show that the $L^2$-averaged polynomial lower bound needed for the instability result indeed holds for a generic class of admissible two-ended asymptotically flat Cauchy initial data.

Strong Cosmic Censorship in Spherical Symmetry for Two-Ended Asymptotically Flat Initial Data II: The Exterior of the Black Hole Region

This is the second and last paper of a two-part series in which we prove the $$C^2$$ -formulation of the strong cosmic censorship conjecture for the Einstein–Maxwell–(real)–scalar–field system in spherical symmetry for two-ended asymptotically flat data. In the first paper, we showed that the maximal globally hyperbolic future development of an admissible asymptotially flat Cauchy initial data set is $$C^2$$ -future-inextendible provided that an $$L^2$$ -averaged (inverse) polynomial lower bound for the derivative of the scalar field holds along each horizon. In this paper, we show that this lower bound is indeed satisfied for solutions arising from a generic set of Cauchy initial data. Roughly speaking, the generic set is open with respect to a (weighted) $$C^1$$ topology and is dense with respect to a (weighted) $$C^\infty $$ topology. The proof of the theorem is based on extensions of the ideas in our previous work on the linear instability of Reissner–Nordstrom Cauchy horizon, as well as a new large data asymptotic stability result which gives good decay estimates for the difference of the radiation fields for small perturbations of an arbitrary solution.

A new property of the electromagnetic/Yang–Mills-conformal gravity system in spherical symmetry

We find a new property in W2-conformal gravity in spherical symmetry. We demonstrate that the charge of the electromagnetic field varies with respect to the partial scaling symmetry (conformal transformations in subspaces of a spacetime) in the conformal gravity. We find that the electric and magnetic charges can vanish or appear via partial conformal transformations for the on-shell configurations in the conformal gravity. We find a solution of a Yang–Mills SU(2)-charged black hole in the conformal gravity, and further demonstrate that the SU(2) charge has similar properties with electromagnetic magnetic charges in the conformal gravity. We demonstrate how to imbed the electromagnetic and Yang–Mills SU(2) gauge field into the conformal gravity via such a symmetry. We make a brief discussion about the possibility to extend this argument to the Einstein gravity.

Solutions to the Einstein-Scalar-Field System in Spherical Symmetry with Large Bounded Variation Norms

It is well-known that small, regular, spherically symmetric characteristic initial data to the Einstein-scalar-field system which are decaying towards (future null) infinity give rise to solutions which are foward-in-time global (in the sense of future causal geodesic completeness). We construct a class of spherically symmetric solutions which are global but the initial norms are consistent with initial data not decaying towards infinity. This gives the following consequences: (1) We prove that there exist foward-in-time global solutions with arbitrarily large (and in fact infinite) initial bounded variation (BV) norms and initial Bondi masses. (2) While general solutions with non-decaying data do not approach Minkowski spacetime, we show using the results of Luk--Oh that if a sufficiently strong asymptotic flatness condition is imposed on the initial data, then the solutions we construct (with large BV norms) approach Minkowski spacetime with a sharp inverse polynomial rate. (3) Our construction can be easily extended so that data are posed at past null infinity and we obtain solutions with large BV norms which are causally geodesically complete both to the past and to the future. Finally, we discuss applications of our method to construct global solutions for other nonlinear wave equations with infinite critical norms.

Einstein-Vlasov system in spherical symmetry. II. Spherical perturbations of static solutions

We reduce the equations governing the spherically symmetric perturbations of static spherically symmetric solutions of the Einstein-Vlasov system (with either massive or massless particles) to a single stratified wave equation $-\psi_{,tt}=H\psi$, with $H$ containing second derivatives in radius, and integrals over energy and angular momentum. We identify an inner product with respect to which $H$ is symmetric, and use the Ritz method to approximate the lowest eigenvalues of $H$ numerically. For two representative background solutions with massless particles we find a single unstable mode with a growth rate consistent with the universal one found by Akbarian and Choptuik in nonlinear numerical time evolutions.

Einstein-Vlasov system in spherical symmetry: Reduction of the equations of motion and classification of single-shell static solutions in the limit of massless particles

We express the Einstein-Vlasov system in spherical symmetry in terms of a dimensionless momentum variable $z$ (radial over angular momentum). This regularises the limit of massless particles, and in that limit allows us to obtain a reduced system in independent variables $(t,r,z)$ only. Similarly, in this limit the Vlasov density function $f$ for static solutions depends on a single variable $Q$ (energy over angular momentum). This reduction allows us to show that any given static metric which has vanishing Ricci scalar, is vacuum at the centre and for $r>3M$ and obeys certain energy conditions uniquely determines a consistent $f=\bar k(Q)$ (in closed form). Vice versa, any $\bar k(Q)$ within a certain class uniquely determines a static metric (as the solution of a system of two first-order quasilinear ODEs). Hence the space of static spherically symmetric solutions of Einstein-Vlasov is locally a space of functions of one variable. For a simple 2-parameter family of functions $\bar k(Q)$, we construct the corresponding static spherically symmetric solutions, finding that their compactness is in the interval $0.7\lesssim \rm max_r(2M/r)\le 8/9$. This class of static solutions includes one that agrees with the approximately universal type-I critical solution recently found by Akbarian and Choptuik (AC) in numerical time evolutions. We speculate on what singles it out as the critical solution found by fine-tuning generic data to the collapse threshold, given that AC also found that {\em all} static solutions are one-parameter unstable and sit on the threshold of collapse.

Quantitative decay rates for dispersive solutions to the Einstein-scalar field system in spherical symmetry

In this paper, we study the future causally geodesically complete solutions of the spherically symmetric Einstein-scalar field system. Under the a priori assumption that the scalar field $\phi$ scatters locally in the scale-invariant bounded-variation (BV) norm, we prove that $\phi$ and its derivatives decay polynomially. Moreover, we show that the decay rates are sharp. In particular, we obtain sharp quantitative decay for the class of global solutions with small BV norms constructed by Christodoulou. As a consequence of our results, for every future causally geodesically complete solution with sufficiently regular initial data, we show the dichotomy that either the sharp power law tail holds or that the spacetime blows up at infinity in the sense that some scale invariant spacetime norms blow up.

Critical collapse in the spherically symmetric Einstein-Vlasov model

We solve the coupled Einstein-Vlasov system in spherical symmetry using direct numerical integration of the Vlasov equation in phase space. Focusing on the case of massless particles we study critical phenomena in the model, finding strong evidence for generic type I behaviour at the black hole threshold that parallels what has previously been observed in the massive sector. For differing families of initial data we find distinct critical solutions, so there is no universality of the critical configuration itself. However we find indications of at least a weak universality in the lifetime scaling exponent, which is yet to be understood. Additionally, we clarify the role that angular momentum plays in the critical behaviour in the massless case.

Analytical distance distributions in systems of spherical symmetry with applications to double electron–electron resonance

Based on a simple geometrical approach, we derive analytical expression of the probability density functions (pdfs) of distance of probe molecules distributed homogeneously in spherical aggregates with shell structure. These distance distributions can be utilized in the investigation of double electron–electron resonance (DEER) data of disordered nanometer-sized spin clusters. Structural insights and geometrical parameters of the aggregates can be extracted by modeling the DEER time traces based on the analytical pdfs. This approach is efficient and avoids difficulties of the model-free solution of the inverse problem that are related to multi-spin effects, limited excitation bandwidth, bias introduced by the regularization scheme, or ambiguity resulting from broad distance distributions. The derived pdfs can serve as building blocks, from which the distance distributions in arbitrary spherically symmetric objects can be assembled. The scenario of the pumped species being chemically distinct from the observed species is covered as well as that of a single type of probe molecules. We demonstrate the merits of analytical distance distributions by studying the distribution of three different spin probes in SDS micelles. By simultaneously analyzing DEER data corresponding to different spin probe concentrations, the distribution of the spin probes over the micelle can be determined. Employing Bayesian inference it is found that for all probes studied, a spherical shell model is most appropriate among the studied models and by orders of magnitude more likely than a homogeneous distribution in a ball. This statement also applies to probes that are deemed nonpolar. We envisage that the spin probe distributions in disordered soft and hard matter systems can now be quantified using DEER spectroscopy with greater precision and reduced ambiguity.

Modified general relativity as a model for quantum gravitational collapse

We study a class of Hamiltonian deformations of the massless Einstein–Klein–Gordon system in spherical symmetry for which the Dirac constraint algebra closes. The system may be regarded as providing effective equations for quantum gravitational collapse. Guided by the observation that scalar field fluxes do not follow metric null directions due to the deformation, we find that the equations take a simple form in characteristic coordinates. We analyze these equations by a unique combination of numerical methods and find that Choptuik’s mass scaling law is modified by a mass gap as well as jagged oscillations. Furthermore, the results are universal with respect to different initial data profiles and robust under changes of the deformation.

Dynamics and entanglement in spherically symmetric quantum gravity

The gravity-scalar field system in spherical symmetry provides a natural setting for exploring gravitational collapse and its aftermath in quantum gravity. In a canonical approach, we give constructions of the Hamiltonian operator, and of semiclassical states peaked on constraint free data. Such states provide explicit examples of physical states. We also show that matter-gravity entanglement is an inherent feature of physical states, whether or not there is a black hole.

Numerical black hole interiors and string cosmology initial conditions

Recent work has proposed the principle of `asymptotic past triviality' to characterize the initial state in the pre-big bang scenario of string cosmology, that it is a generic perturbative solution of the low-energy effective action. Among the more generic sets of solutions which is simple enough to investigate thoroughly, yet complex enough to exhibit interesting behavior, is the gravity-dilaton system in spherical symmetry. Since, in the Einstein frame, this system reduces to a massless minimally coupled scalar, which has been target of a large body of previous investigation, we will draw on this and interpret it in the cosmological context. Since this scenario necessarily involves the transition from weak field initial data into the strong field regime, gravitational collapse, we have made numerical computations to answer some of the questions raised on the road to the proposal that `the pre-big bang is as generic as gravitational collapse'.

Trapped surfaces in vacuum arising dynamically from mild incoming radiation

In this paper, we study the minimal requirement on the incoming radiation that guarantees a trapped surface to form in vacuum. First, we extend the region of existence in Christodoulou's theorem on the formation of trapped surfaces and consequently show that the lower bound required to form a trapped surface can be relaxed. Second, we demonstrate that trapped surfaces form dynamically from a class of initial data which are large merely in a scaling-critical norm. This result is motivated in part by the scaling in Christodoulou's formation of trapped surfaces theorem for the Einstein-scalar field system in spherical symmetry.