Spatial Infinity Research Articles

Asymptotics of linear differential systems and application to quasinormal modes of nonrotating black holes

The traditional approach to perturbations of nonrotating black holes in General Relativity uses the reformulation of the equations of motion into a radial second-order Schr\"odinger-like equation, whose asymptotic solutions are elementary. Imposing specific boundary conditions at spatial infinity and near the horizon defines, in particular, the quasi-normal modes of black holes. For more complicated equations of motion, as encountered for instance in modified gravity models with different background solutions and/or additional degrees of freedom, we present a new approach that analyses directly the first-order differential system in its original form and extracts the asymptotic behaviour of perturbations, without resorting to a second-order reformulation. As a pedagogical illustration, we apply this treatment to the perturbations of Schwarzschild black holes and then show that the standard quasi-normal modes can be obtained numerically by solving this first-order system with a spectral method. This new approach paves the way for a generic treatment of the asymptotic behaviour of black hole perturbations and the identification of quasi-normal modes in theories of modified gravity.

Black hole perturbations in modified gravity

We study the linear perturbations about nonrotating black holes in the context of degenerate higher-order scalar-tensor (DHOST) theories, using a systematic approach that extracts the asymptotic behaviour of perturbations (at spatial infinity and near the horizon) directly from the first-order radial differential system governing these perturbations. For axial (odd-parity) modes, this provides an alternative to the traditional approach based on a second-order Schr\"odinger-like equation with an effective potential, which we also discuss for completeness. For polar (even-parity) modes, which contain an additional degree of freedom in DHOST theories, and are thus more complex, we use a direct treatment of the four-dimensional first-order differential system (without resorting to a second order reformulation). We illustrate our study with two specific types of black hole solutions: 'stealth' Schwarzschild black holes, with a non trivial scalar hair, as well as a class of non-stealth black holes whose metric is distinct from Schwarzschild. The knowledge of the asymptotic behaviours of the perturbations enables us to compute numerically quasi-normal modes, as we show explicitly for the non-stealth solutions. Finally, the asymptotic form of the modes also signals some pathologies in the stealth and non-stealth solutions considered here.

Staticity and regularity for zero rest-mass fields near spatial infinity on flat spacetime

Linear zero-rest-mass fields generically develop logarithmic singularities at the critical sets where spatial infinity meets null infinity. Friedrich’s representation of spatial infinity is ideally suited to study this phenomenon. These logarithmic singularities are an obstruction to the smoothness of the zero-rest-mass field at null infinity and, in particular, to peeling. In the case of the spin-2 field it has been shown that these logarithmic singularities can be precluded if the initial data for the field satisfies a certain regularity condition involving the vanishing, at spatial infinity, of a certain spinor (the linearised cotton spinor) and its totally symmetrised derivatives. In this article we investigate the relation between this regularity condition and the staticity of the spin-2 field. It is shown that while any static spin-2 field satisfies the regularity condition, not every solution satisfying the regularity condition is static. This result is in contrast with what happens in the case of general relativity where staticity in a neighbourhood of spatial infinity and the smoothness of the field at future and past null infinities are much more closely related.

Thin accretion discs around spherically symmetric configurations with nonlinear scalar fields

We study stable circular orbits (SCO) around static spherically symmetric configuration of general relativity with a nonlinear scalar field (SF). The configurations are described by solutions of the Einstein-SF equations with monomial SF potential $V(\ensuremath{\phi})=|\ensuremath{\phi}{|}^{2n}$, $n>2$, under the conditions of the asymptotic flatness and behavior of SF $\ensuremath{\phi}\ensuremath{\sim}1/r$ at spatial infinity. We proved that under these conditions, the solution exists and is uniquely defined by the configuration mass $M>0$ and scalar ``charge'' $Q$. The solutions and the space-time geodesics have been investigated numerically in the range $n\ensuremath{\le}40$, $|Q|\ensuremath{\le}60$, $M\ensuremath{\le}60$. We focus on how nonlinearity of the field affects properties of SCO distributions (SCOD), which, in turn, affect topological form of the thin accretion disk around the configuration. Maps are presented showing the location of possible SCOD types for different $M$, $Q$, $n$. We found a lot of differences from the Fisher-Janis-Newman-Winicour metric (FJNW) dealing with the linear SF, though basic qualitative properties of the configurations have much in common with the FJNW case. For some values of $n$, a topologically new SCOD type was discovered that is not available for the FJNW metric. Like FJNW, all images of accretion disks have a dark spot in the center (mimicking the ordinary black hole), either because there is no SCO near the center or due to the strong deflection of photons near the singularity, although fine features other than a black hole can appear with a special choice of $M$, $Q$.

Kink solutions in logarithmic scalar field theory: Excitation spectra, scattering, and decay of bions

We consider the (1+1)-dimensional Lorentz-symmetric field-theoretic model with logarithmic potential having a Mexican-hat form with two local minima similar to that of the quartic Higgs potential in conventional electroweak theory with spontaneous symmetry breaking and mass generation. We demonstrate that this model allows topological solutions — kinks. We analyze the kink excitation spectrum, and show that it does not contain any vibrational modes. We also study the scattering dynamics of kinks for a wide range of initial velocities. The critical value of the initial velocity occurs in kink-antikink collisions, which thus differentiates two regimes. Below this value, we observe the capture of kinks and their fast annihilation; while above this value, the kinks bounce off and escape to spatial infinities. Numerical studies show no resonance phenomena in the kink-antikink scattering.

Extended geometry of Gambini-Olmedo-Pullin polymer black hole and its quasinormal spectrum

In this paper we systematically study a model of spherically symmetric polymer black holes recently proposed by Gambini, Olmedo, and Pullin (GOP). Within the framework of loop quantum gravity, the quantum parameters in the GOP model depend on the minimal area gap and the size of the discretization of the physical states. In this model, a spacelike transition surface takes place of the classical singularity. By means of coordinate transformations, we first extend the metric to the white hole region, and find that the geometric structure of the quantum black hole is similar to the wormhole structure, and the radius of the most quantum region is equal to the wormhole radius. In addition, we show that the energy conditions are violated not only at throat, but also at horizons and the spatial infinities. In order to show how the quantum effects affect the spacetimes, we calculate the Ricci and Kretschmann scalars at different places. It turns out that, as expected, the most quantum region is at the throat. Finally, we consider the quasinormal modes (QNMs) of massless scalar field perturbations, electromagnetic field perturbations, and axial gravitational perturbations. QNMs in the Eikonal limits are also considered. As anticipated, the spectrum of QNMs deviates from that of the classical case due to quantum effects. Interestingly, our results show that the quasinormal frequencies of the perturbations share the same qualitative tendency while setting quantum parameters with various values in this effective model, even if the potential deviations are different with different spins.

Relativistic scattering of a fast spinning neutron star by a massive black hole

ABSTRACT The orbital dynamics of fast spinning neutron stars encountering a massive black hole (BH) with unbounded orbits are investigated using the quadratic-in-spin Mathisson–Papapetrou–Dixon (MPD) formulation. We consider the motion of the spinning neutron stars with astrophysically relevant speed in the gravity field of the BH. For such slow-speed scattering, the hyperbolic orbits followed by these neutron stars all have near the e = 1 eccentricity, and have distinct properties compared with those of e ≫ 1. We have found that, compared with geodesic motion, the spin–orbit and spin–spin coupling will lead to a variation of scattering angles at spatial infinity, and this variation is more prominent for slow-speed scattering than fast-speed scattering. Such a variation leads to an observable difference in pulse-arrival-time within a few hours of observation, and up to a few days or months for larger BH masses or longer spinning periods. Such a relativistic pulsar-BH system also emits a burst of gravitational waves (GWs) in the sensitivity band of Laser Interferometric Space Antenna, and for optimal settings, can be seen up to $100\, {\rm Mpc}$ away. A radio follow up of such a GW burst with SKA or FAST will allow for measuring the orbital parameters with high accuracy and testing the predictions of general relativity.

Singularities in Static Spherically Symmetric Configurations of General Relativity with Strongly Nonlinear Scalar Fields

There are a number of publications on relativistic objects dealing either with black holes or naked singularities in the center. Here we show that there exist static spherically symmetric solutions of Einstein equations with a strongly nonlinear scalar field, which allow the appearance of singularities of a new type (“spherical singularities”) outside the center of curvature coordinates. As the example, we consider a scalar field potential ∼sinh(ϕ2n),n>2, which grows rapidly for large field values. The space-time is assumed to be asymptotically flat. We fulfill a numerical investigation of solutions with different n for different parameters, which define asymptotic properties at spatial infinity. Depending on the configuration parameters, we show that the distribution of the stable circular orbits of test bodies around the configuration is either similar to that in the case of the Schwarzschild solution (thus mimicking an ordinary black hole), or it contains additional rings of unstable orbits.

Non-Rindler horizons and radiating black holes

In this work we construct metrics corresponding to radiating black holes whose near horizon regions cannot be approximated by Rindler space–time. We first construct infinite parameter coordinate transformations from Minkowski coordinates, such that an observer using these coordinates to describe space–time events measures the Minkowski vacuum to be Planckian. Utilizing these results, we construct a family of black holes that radiate at spatial infinity. As an illustration, we study a subset of the black hole solutions and show that they satisfy the null energy condition.

Hawking radiation of massive fields in 2D

We extend the classic results of the paper P. C. W. Davies, S. A. Fulling, and W. G. Unruh "Energy-momentum tensor near an evaporating black hole" by considering a massive scalar field in a two dimensions in the presence of a thin shell collapse. We show that outside the shell the WKB approximation is valid for any value of $r$ if $mr_{g} \gg 1$, where $m$ is the mass of the field, and $r_{g}$ is the Schwarzschild radius. Thus, we use semiclassical modes to calculate the flux in the vicinity of the shell, and at spatial infinity, $r \rightarrow +\infty$ at the final stage of the collapse, $t \rightarrow +\infty$ with the use of the covariant point-splitting regularization. We get that near the shell and at the spatial infinity the radiation is thermal with Hawking temperature. We obtain the negative flux $T_{vv}$ in the vicinity of the shell, which is similar to the classic result in the massless case.

A Fuchsian viewpoint on the weak null condition

We analyze systems of semilinear wave equations in 3 + 1 dimensions whose associated asymptotic equation admit bounded solutions for suitably small choices of initial data. Under this special case of the weak null condition, which we refer to as the bounded weak null condition , we prove the existence of solutions to these systems of wave equations on neighborhoods of spatial infinity under a small initial data assumption. Existence is established using the Fuchsian method. This method involves transforming the wave equations into a Fuchsian equation defined on a bounded spacetime region. The existence of solutions to the Fuchsian equation then follows from an application of the existence theory developed in [11] . This, in turn, yields, by construction, solutions to the original system of wave equations on a neighborhood of spatial infinity.

Celestial IR divergences and the effective action of supertranslation modes

Infrared divergences in perturbative gravitational scattering amplitudes have been recently argued to be governed by the two-point function of the supertranslation Goldstone mode on the celestial sphere. We show that the form of this celestial two-point function simply derives from an effective action that also controls infrared divergences in the symplectic structure of General Relativity with asymptotically flat boundary conditions. This effective action finds its natural place in a path integral formulation of a celestial conformal field theory, as we illustrate by re-deriving the infrared soft factors in terms of celestial correlators. Our analysis relies on a well-posed action principle close to spatial infinity introduced by Compère and Dehouck.

On center of mass and foliations by constant spacetime mean curvature surfaces for isolated systems in General Relativity

We propose a new foliation of asymptotically Euclidean initial data sets by 2-spheres of constant spacetime mean curvature (STCMC). The leaves of the foliation have the STCMC-property regardless of the initial data set in which the foliation is constructed which asserts that there is a plethora of STCMC 2-spheres in a neighborhood of spatial infinity of any asymptotically flat spacetime. The STCMC-foliation can be understood as a equivariant relativistic generalization of the CMC-foliation suggested by Huisken and Yau (Invent Math 124:281–311, 1996). We show that a unique STCMC-foliation exists near infinity of any asymptotically Euclidean initial data set with non-vanishing energy which allows for the definition of a new notion of total center of mass for isolated systems. This STCMC-center of mass transforms equivariantly under the asymptotic Poincaré group of the ambient spacetime and in particular evolves under the Einstein evolution equations like a point particle in Special Relativity. The new definition also remedies subtle deficiencies in the CMC-approach to defining the total center of mass suggested by Huisken and Yau (Invent Math 124:281–311, 1996) which were described by Cederbaum and Nerz (Ann Henri Poincaré 16:1609–1631, 2015).

Leray's plane stationary solutions at small Reynolds numbers

In the celebrated 1933 paper [22] by J. Leray, the invading domains method was proposed to construct D-solutions for the stationary Navier-Stokes flow around obstacle problem. In two dimensions, whether Leray's D-solution achieves the prescribed limiting velocity at spatial infinity became a major open problem since then. In this paper, we solve this problem at small Reynolds numbers. The proof builds on a novel blow-down argument which rescales the invading domains to the unit disc, as well as the ideas developed in the recent works [19,20].

A comparison of Ashtekar’s and Friedrich’s formalisms of spatial infinity

Penrose’s idea of asymptotic flatness provides a framework for understanding the asymptotic structure of gravitational fields of isolated systems at null infinity. However, the studies of the asymptotic behaviour of fields near spatial infinity are more challenging due to the singular nature of spatial infinity in a regular point compactification for spacetimes with non-vanishing ADM mass. Two different frameworks that address this challenge are Friedrich’s cylinder at spatial infinity and Ashtekar’s definition of asymptotically Minkowskian spacetimes at spatial infinity that give rise to the three-dimensional asymptote at spatial infinity . Both frameworks address the singularity at spatial infinity although the link between the two approaches had not been investigated in the literature. This article aims to show the relation between Friedrich’s cylinder and the asymptote as spatial infinity. To do so, we initially consider this relation for Minkowski spacetime. It can be shown that the solution to the conformal geodesic equations provides a conformal factor that links the cylinder and the asymptote. For general spacetimes satisfying Ashtekar’s definition, the conformal factor cannot be determined explicitly. However, proof of the existence of this conformal factor is provided in this article. Additionally, the conditions satisfied by physical fields on the asymptote are derived systematically using the conformal constraint equations. Finally, it is shown that a solution to the conformal geodesic equations on the asymptote can be extended to a small neighbourhood of spatial infinity by making use of the stability theorem for ordinary differential equations. This solution can be used to construct a conformal Gaussian system in a neighbourhood of .

First law of entanglement entropy in AdS black hole backgrounds

The first law for entanglement entropy in CFT in an odd-dimensional asymptotically AdS black hole is studied by using the AdS/CFT duality. The entropy of CFT considered here is due to the entanglement between two subsystems separated by the horizon of the AdS black hole, which itself is realized as the conformal boundary of a black droplet in even-dimensional global AdS bulk spacetime. In $(2+1)$-dimensional CFT, the first law is shown to be always satisfied by analyzing a class of metric perturbations of the exact solution of a $4$-dimensional black droplet. In $(4+1)$-dimensions, the first law for CFT is shown to hold under the Neumann boundary condition at a certain bulk hypersurface anchored to the conformal boundary of the boundary AdS black hole. From the boundary view point, this Neumann condition yields there being no energy flux across the boundary of the boundary AdS black hole. Furthermore, the asymptotic geometry of a $6$-dimensional small AdS black droplet is constructed as the gravity dual of our $(4+1)$-dimensional CFT, which exhibits a negative energy near the spatial infinity, as expected from vacuum polarization.

Spontaneous scalarization of charged stars

We study static and spherically symmetric charged stars with a nontrivial profile of the scalar field ϕ in Einstein-Maxwell-scalar theories. The scalar field is coupled to a U(1) gauge field Aμ with the form −α(ϕ)FμνFμν/4, where Fμν=∂μAν−∂νAμ is the field strength tensor. Analogous to the case of charged black holes, we show that this type of interaction can induce spontaneous scalarization of charged stars under the conditions (dα/dϕ)(0)=0 and (d2α/dϕ2)(0)>0. For the coupling α(ϕ)=exp⁡(−βϕ2/Mpl2), where β(<0) is a coupling constant and Mpl is a reduced Planck mass, there is a branch of charged star solutions with a nontrivial profile of ϕ approaching 0 toward spatial infinity, besides a branch of general relativistic solutions with a vanishing scalar field, i.e., solutions in the Einstein-Maxwell model. As the ratio ρc/ρm between charge density ρc and matter density ρm increases toward its maximum value, the mass M of charged stars in general relativity tends to be enhanced due to the increase of repulsive Coulomb force against gravity. In this regime, the appearance of nontrivial branches induced by negative β of order −1 effectively reduces the Coulomb force for a wide range of central matter densities, leading to charged stars with smaller masses and radii in comparison to those in the general relativistic branch. Our analysis indicates that spontaneous scalarization of stars can be induced not only by the coupling to curvature invariants but also by the scalar-gauge coupling in Einstein gravity.

Black hole S-matrix for a scalar field

We describe a unitary scattering process, as observed from spatial infinity, of massless scalar particles on an asymptotically flat Schwarzschild black hole background. In order to do so, we split the problem in two different regimes governing the dynamics of the scattering process. The first describes the evolution of the modes in the region away from the horizon and can be analysed in terms of the effective Regge-Wheeler potential. In the near horizon region, where the Regge-Wheeler potential becomes insignificant, the WKB geometric optics approximation of Hawking’s is replaced by the near-horizon gravitational scattering matrix that captures non-perturbative soft graviton exchanges near the horizon. We perform an appropriate matching for the scattering solutions of these two dynamical problems and compute the resulting Bogoliubov relations, that combines both dynamics. This allows us to formulate an S-matrix for the scattering process that is manifestly unitary. We discuss the analogue of the (quasi)-normal modes in this setup and the emergence of gravitational echoes that follow an original burst of radiation as the excited black hole relaxes to equilibrium.

TT¯/JT¯ -deformed WZW models from Chern-Simons AdS3 gravity with mixed boundary conditions

In this work we consider AdS$_3$ gravitational theory with certain mixed boundary conditions at spatial infinity. Using the Chern-Simons formalism of AdS$_3$ gravity, we find that these boundary conditions lead to non-trivial boundary terms, which, in turn, produce exactly the spectrum of the $T\bar{T}/J\bar{T}$-deformed CFTs. We then follow the procedure for constructing asymptotic boundary dynamics of AdS$_3$ to derive the constrained $T\bar{T}$-deformed WZW model from Chern-Simons gravity. The resulting theory turns out to be the $T\bar{T}$-deformed Alekseev-Shatashvili action after disentangling the constraints. Furthermore, by adding a $U(1)$ gauge field associated to the current $J$, we obtain one type of the $J\bar T$-deformed WZW model, and show that its action can be constructed from the gravity side. These results provide a check on the correspondence between the $T\bar{T}/J\bar{T}$-deformed CFTs and the deformations of boundary conditions of AdS$_3$, the latter of which may be regarded as coordinate transformations.

Fast diffusion on noncompact manifolds: Well-posedness theory and connections with semilinear elliptic equations

We investigate the well-posedness of the fast diffusion equation (FDE) on noncompact Riemannian manifolds. Existence and uniqueness of solutions for L 1 L^1 initial data was established in Bonforte, Grillo, and Vázquez [J. Evol. Equ. 8 (2008), pp. 99–128]. However, in the Euclidean space, it is known from Herrero and Pierre [Trans. Amer. Math. Soc. 291 (1985), pp. 145–158] that the Cauchy problem associated with the FDE is well posed for initial data that are merely in L l o c 1 L^1_{\mathrm {loc}} . We establish here that such data still give rise to global solutions on general manifolds. If, moreover, the radial Ricci curvature satisfies a suitable pointwise bound from below (possibly diverging to − ∞ -\infty at spatial infinity), we prove that also uniqueness holds, for the same type of data, in the class of strong solutions. Besides, assuming in addition that the initial datum is in L l o c 2 L^2_{\mathrm {loc}} and nonnegative, a minimal solution is shown to exist, and we establish uniqueness of purely (nonnegative) distributional solutions, a fact that to our knowledge was not known before even in the Euclidean space. The required curvature bound is sharp, since on model manifolds it is equivalent to stochastic completeness, and it was shown in Grillo, Ishige, and Muratori [J. Math. Pures Appl. (9) 139 (2020), pp. 63–82] that uniqueness for the FDE fails even in the class of bounded solutions when stochastic completeness does not hold. A crucial ingredient of the uniqueness result is the proof of nonexistence of nonnegative, nontrivial distributional subsolutions to certain semilinear elliptic equations with power nonlinearities, of independent interest.