Future Null Infinity Research Articles

The hyperboloidal numerical evolution of a good-bad-ugly wave equation

One method for the numerical treatment of future null-infinity is to decouple coordinates from the tensor basis and choose each in a careful manner. This dual-frame approach is hampered by logarithmically divergent terms that appear in a naive choice of evolved variables. Here we consider a system of wave equations that satisfy the weak-null condition and serve as a model system with similar nonlinearities to those present in the Einstein field equations in generalized harmonic gauge. We show that these equations can be explicitly regularized by a nonlinear change of variables. Working in spherical symmetry, a numerical implementation of this model using compactified hyperboloidal slices is then presented. Clean convergence is found for the regularized system. Although more complicated, it is expected that general relativity can be treated similarly.

Repeated ringing of black holes: Quasinormal bursts from highly eccentric, extreme mass-ratio binaries

Recent studies of scalar and gravitational waveforms from high-eccentricity, extreme mass-ratio black hole binaries show the presence of quasi-normal bursts (QNB) -- lingering high frequency decaying oscillations (also known as ``wiggles'') -- soon after each periapsis passage. One puzzle associated with these QNB is that in the case of a nearly-extreme rotating central black hole the frequency of the QNB has been found to be in a range which is lower than the corresponding range of relevant quasi-normal modes. We reproduce these results using a different approach and perform a detailed analysis to find evidence for the resolution of the puzzle and for the origin of the QNB. We find that the QNB frequency as measured at future null infinity evolves in (retarded) time and approaches the dominant quasi-normal frequency exponentially in time. We also show that the QNB amplitude decays inversely in (retarded) time. We discuss the time dependence of both the QNB waveform frequency and its amplitude and argue that this behavior arises as a result of the excitation of many quasi-normal overtones and the summation thereof.

Conformal scattering of the Maxwell-scalar field system on de Sitter space

We prove small data energy estimates of all orders of differentiability between past null infinity and future null infinity of de Sitter space for the conformally invariant Maxwell-scalar field system. Using these, we construct bounded and invertible, but nonlinear, scattering operators taking past asymptotic data to future asymptotic data. We deduce exponential decay rates for solutions with data having at least two derivatives, and for more regular solutions discover an asymptotic decoupling of the scalar field from the charge. The construction involves a carefully chosen complete gauge fixing condition which allows us to control all components of the Maxwell potential, and a nonlinear Grönwall inequality for higher-order estimates.

Numerical investigation of the dynamics of linear spin s fields on a Kerr background: Late-time tails of spin s=±1,±2 fields

The time evolution of linear fields of spin $s = \pm 1$ and $s = \pm 2$ on Kerr black hole spacetimes are investigated by solving the homogeneous Teukolsky equation numerically. The applied numerical setup is based on a combination of conformal compactification and the hyperbolic initial value problem. The evolved basic variables are expanded in terms of spin-weighted spherical harmonics which allows us to evaluate all the angular derivatives analytically, whereas the evolution of the expansion coefficients, in the time-radial section, is determined by applying the method of lines implemented in a fourth order accurate finite differencing stencil. Concerning the initialization, in all of our investigations single mode excitations---either static or purely dynamical type initial data---are applied. Within this setup the late time tail behavior is investigated. Due to the applied conformal compactification the asymptotic decay rates are determined at three characteristic locations---in the domain of outer communication, at the event horizon and at future null infinity---simultaneously. Recently introduced new type of `energy' and `angular momentum' balance relations are also applied in order to demonstrate the feasibility and robustness of the developed numerical schema, and also to verify the proper implementation of the underlying mathematical model.

Transient scalar hair for nearly extreme black holes

It has been shown recently that extreme Reissner-Nordstr\"{o}m black holes perturbed by a minimally coupled, free, massless scalar field have permanent scalar hair. The hair - a conserved charge calculated at the black hole's event horizon - can be measured by a certain expression at future null infinity: the latter approaches the hair inversely in time. We generalize this newly discovered hair also for extreme Kerr black holes. We study the behavior of nearly extreme black hole hair and its measurement at future null infinity as a transient phenomenon. For nearly extreme black holes the measurement at future null infinity of the length of the newly grown hair decreases quadratically in time at intermediate times until its length becomes short and the rate at which the length shortens further slows down. Eventually, the nearly extreme BH becomes bald again like non-extreme BHs.

Rindler horizons in a Schwarzschild spacetime

We investigate the past and future Rindler horizons for radial Rindler trajectories in the Schwarzschild spacetime. We assume the Rindler trajectory to be linearly uniformly accelerated (LUA) throughout its motion, in the sense of the curved spacetime generalisation of the Letaw-Frenet equations. The analytical solution for the radial LUA trajectories along with its past and future intercepts ${\cal C}$ with the past null infinity ${\cal J^-}$ and future null infinity ${\cal J^+}$ are presented. The Rindler horizons, in the presence of the black hole, are found to depend on both the magnitude of acceleration $|a|$ and the asymptotic initial data $h$, unlike in the flat Rindler spacetime case wherein they are only a function of the global translational shift $h$. The horizon features are discussed. The Rindler quadrant structure provides an alternate perspective to interpret the acceleration bounds, $|a| \leq |a|_b$ found earlier in arXiv:1901.04674.

The weak null condition in free-evolution schemes for numerical relativity: dual foliation GHG with constraint damping

All strategies for the treatment of future null-infinity in numerical relativity involve some form of regularization of the field equations. In a recent proposal that relies on the dual foliation formalism this is achieved by the use of an asymptotically Minkowskian generalized harmonic tensor basis. For the scheme to work however, derivatives of certain coordinate light-speeds must decay fast enough. Presently, we generalize the method of asymptotic expansions for nonlinear wave equations to treat first order symmetric hyperbolic systems. We then use this heuristic tool to extract the expected rates of decay of the metric near null-infinity in a free-evolution setting. We show, within the asymptotic expansion, that by carefully modifying the non-principal part of the field equations by the addition of constraints, we are able to obtain optimal decay rates even when the constraints are violated. The light-speed condition can hence be satisfied, which paves the way for the explicit numerical treatment of future null-infinity. We then study the behavior of the Trautman–Bondi mass under the decay results predicted by the asymptotic expansion. Naively the mass seems to be unbounded, but we see first that the divergent terms can be replaced with a combination of the constraints and the Einstein field equations, and second that the Bondi mass loss formula is recovered within the framework. Both of the latter results hold in the presence of small constraint violations.

Hausdorff closed limits and the c-boundary II: null infinity and black holes

We have shown in a companion paper (e Silva et al (2019)) that when a spacetime is globally hyperbolic with a (possibly empty) smooth timelike boundary , then a metrizable topology, the closed limit topology (CLT), can be advantageously adopted on the Geroch–Kronheimer–Penrose causal completion (or c-completion for short) of . The CLT retains essentially all the good properties of other topologies previously defined in the literature, such as the well-known chronological topology. In this paper we prove that if a globally hyperbolic spacetime admits a conformal boundary, defined in such broad terms as to include all the standard examples in the literature, then the latter is homeomorphic to the causal boundary endowed with the CLT. Then, we revisit the new definitions of (future) null infinity and black hole as given in terms of the c-boundaries in another recent work Costa e Silva et al (2018 J. High Energy Phys. JHEP(2018)123). We show that a number of the attending technicalities can be simplified for globally hyperbolic spacetimes when the CLT is considered.

Hawking radiation as instantons

There have been various interpretations of Hawking radiation proposed based on the perturbative approach, and all have confirmed Hawking’s original finding. One major conceptual challenge of Hawking evaporation is the associated black hole information loss paradox, which remains unresolved. A key factor to the issue is the end-stage of the black hole evaporation. Unfortunately by then the evaporation process becomes non-perturbative. Aspired to provide a tool for the eventual solution to this problem, here we introduce a new interpretation of Hawking radiation as the tunneling of instantons. We study instantons of a massless scalar field in Einstein gravity. We consider a complex-valued instanton that connects an initial pure black hole state to a black hole with a scalar field that represents the Hawking radiation at future null infinity, where its action depends only on the areal entropy difference. By comparing it with several independent approaches to Hawking radiation in the perturbative limit, we conclude that Hawking radiation may indeed be described by a family of instantons. Since the instanton approach can describe non-perturbative processes, we hope that our new interpretation and holistic method may shed lights on the information loss problem.

Boosted Kerr black holes in general relativity

A solution of Einstein's vacuum field equation is derived that describes a general boosted Kerr black hole relative to a Lorentz frame at future null infinity. The metric contains five independent parameters -- mass $m$, rotation $\omega$, boost parameter $v/c$ and the boost direction defined by $(n_1,n_2,n_3)$ satisfying $(n_1)^2+(n_2)^2+(n_3)^2=1$ -- and reduces to the Kerr black hole when the boost parameter is zero and $n_1=1$. The solution describes the most general configuration that an astrophysical black hole must have. The black hole rotates about the $z$-axis with angular momentum proportional to $m \omega$ and the geometry has just one Killing vector $\partial/\partial{u}$, where $u$ is the retarded time coordinate. The boost turns the ergosphere asymmetric, with dominant lobes in the direction opposite to the boost. The event and Cauchy horizons, defined for the case $\omega < m$, are specified respectively by the radii $r_{\pm}=m \pm \sqrt{m^2-\omega^2}$. The horizons are topologically spherical and the singularity has the topology of a circle on planes that are orthogonal to the boost direction. We argue that this black hole geometry is the natural set to describe the remnants of the recently observed gravitational wave events $GW150914$, $GW151226$, $GW170814$ and $GW170817$\cite{gw1,gw2,gw3,gw4}. In the conclusions we discuss possible astrophysical processes in the asymmetric ergosphere and the electromagnetic dynamical effects that may result from the rotating black hole moving at relativistic speeds together with the precession of the boost axis about the rotation axis of the black hole.

Asymptotic behavior of massless fields and the memory effect

We investigate the behavior of massless scalar, electromagnetic, and linearized gravitational perturbations near null infinity in d \geq 4 dimensional Minkowski spacetime (of both even and odd dimension) under the assumption that these fields admit a suitable expansion in 1/r. We also investigate the behavior of asymptotically flat, nonlinear gravitational perturbations near null infinity in all dimensions d\geq 4. We then consider the memory effect in fully nonlinear general relativity. We show that in even dimensions, the memory effect first arises at Coulombic order--i.e., order 1/r^{d-3}--and can naturally be decomposed into `null memory' and `ordinary memory.' Null memory is associated with an energy flux to null infinity. We show that ordinary memory is associated with the metric failing to be stationary at one order faster fall-off than Coulombic in the past and/or future. In odd dimensions, we show that the total memory effect at Coulombic order and slower fall-off always vanishes. Null memory is always of `scalar type', but the ordinary memory can be of any (i.e., scalar, vector, or tensor) type. In 4-spacetime dimensions, we give an explicit example in linearized gravity which gives rise to a nontrivial vector (i.e., magnetic parity) ordinary memory effect at order 1/r. We show that scalar memory is described by a diffeomorphism; vector and tensor memory cannot be. In d=4 dimensions, we show that there is a close relationship between memory and the charge and flux expressions associated with supertranslations. We analyze the behavior of solutions that are stationary at Coulombic order and show how these suggest `antipodal matching' between future and past null infinity, which gives rise to conservation laws. The relationship between memory and infrared divergences of the `out' state in quantum gravity is analyzed, and the nature of the `soft theorems' is explained.

Flux-balance equations for linear momentum and center-of-mass position of self-gravitating post-Newtonian systems

We revisit the problem of flux-balance equations for isolated post-Newtonian matter systems due to the emission of gravitational waves. In particular we show by a local derivation confined to the system, using the expression of radiation-reaction forces up to the 3.5 PN order, that not only the energy, angular momentum and linear momentum of the system, but also the position of its center of mass, obey some (non-trivial) flux-balance equations. The balance equation for the center-of-mass position completes the description of the secular evolution by gravitational waves of relativistic post-Newtonian isolated matter systems. We then confirm this result by a direct computation of the gravitational-wave fluxes at future null infinity, obtaining the full multipole-moment expansion of the flux associated with the center-of-mass position (probably new with this paper), and rederiving as well the known multipole-moment expansions of the fluxes of energy, angular momentum and linear momentum. We also check our analysis by a direct calculation of radiation-reaction effects in the case of compact binary systems.

Conservation of asymptotic charges from past to future null infinity: supermomentum in general relativity

We show that the BMS-supertranslations and their associated supermomenta on past null infinity can be related to those on future null infinity, proving the conjecture of Strominger for a class of spacetimes which are asymptotically-flat in the sense of Ashtekar and Hansen. Using a cylindrical 3-manifold of both null and spatial directions of approach towards spatial infinity, we impose appropriate regularity conditions on the Weyl tensor near spatial infinity along null directions. The asymptotic Einstein equations on this 3-manifold and the regularity conditions imply that the relevant Weyl tensor components on past null infinity are antipodally matched to those on future null infinity. The subalgebra of totally fluxless supertranslations near spatial infinity provides a natural isomorphism between the BMS-supertranslations on past and future null infinity. This proves that the flux of the supermomenta is conserved from past to future null infinity in a classical gravitational scattering process provided additional suitable conditions are satisfied at the timelike infinities.

A changed perspective concerning asymptotically flat Einstein/Einstein–Maxwell space–times

Approximately 60 years ago, Herman Bondi took the major step in the study of gravitational radiation of introducing the use of null surfaces as coordinates for the study and integration of the Einstein equations. This led to the well-known Bondi mass loss theorem, the basis of the recent observation by the Ligo team of gravitational radiation. In Bondi’s approach to the integration procedure, he used special null surfaces (now referred to as Bondi null surfaces) where the null generators possessed (in general) non-vanishing asymptotic shear—the free radiation data. The use of this Bondi strategy over the years has become almost sacrosanct—being the central approach in almost all discussions of gravitational radiation issues. It led to the idea of an asymptotic symmetry—the BMS group. Eventually Bondi’s description of null infinity became elegantly formalized via Penrose’s future null infinity and associated structures. However recently an alternative picture of null infinity has been developed—now based on the similarity of the null surfaces to those of flat-space near null infinity. The new null surfaces, that are now asymptotically shear-free, are very different from the Bondi surfaces. These surfaces are as similar as possible to flat-space light cones near infinity. Totally new structures—e.g., the geometric asymptotically shear-free null geodesic congruences and even the physical classical equations of motion, now appear in this new picture. The Bondi–Sachs energy–momentum conservations laws remain but are augmented by angular momentum (orbital and spin) conservation laws. The BMS group again reappears, not as an asymptotic symmetry group, but as a transformation group acting on these new structures. An unanswered question arises: with this new point of view, have we lost the asymptotic symmetries of the BMS group?

Black hole memory effect

We compute the memory effect produced at the black hole horizon by a transient gravitational shockwave. As shown by Hawking, Perry, and Strominger (HPS) such a gravitational wave produces a deformation of the black hole geometry which from future null infinity is seen as a Bondi-Metzner-Sachs (BMS) supertranslation. This results in a diffeomorphic but physically distinct geometry which differs from the original black hole by their charges at infinity. Here we give the complementary description of this physical process in the near-horizon region as seen by an observer hovering just outside the event horizon. From this perspective, in addition to a supertranslation the shockwave also induces a horizon superrotation. We compute the associated superrotation charge and show that its form agrees with the one obtained by HPS at infinity. In addition, there is a supertranslation contribution to the horizon charge, which measures the entropy change in the process. We then turn to electrically and magnetically charged black holes and generalize the near-horizon asymptotic symmetry analysis to Einstein-Maxwell theory. This reveals an additional infinite-dimensional current algebra that acts non-trivially on the horizon superrotations. Finally, we generalize the black hole memory effect to Reissner-Nordstr\"{o}m black holes.

Conservation of asymptotic charges from past to future null infinity: Maxwell fields

On any asymptotically-flat spacetime, we show that the asymptotic symmetries and charges of Maxwell fields on past null infinity can be related to those on future null infinity as recently proposed by Strominger. We extend the covariant formalism of Ashtekar and Hansen by constructing a 3-manifold of both null and spatial directions of approach to spatial infinity. This allows us to systematically impose appropriate regularity conditions on the Maxwell fields near spatial infinity along null directions. The Maxwell equations on this 3-manifold and the regularity conditions imply that the relevant field quantities on past null infinity are antipodally matched to those on future null infinity. Imposing the condition that in a scattering process the total flux of charges through spatial infinity vanishes, we isolate the subalgebra of totally fluxless symmetries near spatial infinity. This subalgebra provides a natural isomorphism between the asymptotic symmetry algebras on past and future null infinity, such that the corresponding charges are equal near spatial infinity. This proves that the flux of charges is conserved from past to future null infinity in a classical scattering process of Maxwell fields. We also comment on possible extensions of our method to scattering in general relativity.

On late time tails in an extreme Reissner–Nordström black hole: frequency domain analysis

In this paper, we revisit the study of the leading order late time decay tails of massless scalar perturbations outside an extreme Reissner–Nordström black hole. Previous authors have analysed this problem in the time domain; we analyse the problem in the frequency domain. We first consider initial perturbations with generic regular behaviour across the horizon on characteristic surfaces. For this set-up, we reproduce some of the previous results of Sela (2016 Phys. Rev. D 93 024054) using Fourier methods. Next we consider related initial data on hypersurfaces, and present decay results at timelike infinity, near future null infinity, and near the future horizon. Along the way, using the inversion symmetry of the extreme Reissner–Nordström spacetime, we relate the higher multipole Aretakis and Newman–Penrose constants for a massless scalar in this background.

On the local extension of the future null infinity

We consider a characteristic problem of the vacuum Einstein equations with part of the initial data given on a future asymptotically flat null cone, and show that the solution exists uniformly around the null cone for general such initial data. Therefore, the solution contains a piece of the future null infinity. The initial data are not required to be small and the decaying condition is consistent with those in the works of [8] and [11].

Quantum fields during black hole formation: how good an approximation is the Unruh state?

We study the quantum effects of a test Klein-Gordon field in a Vaidya space-time consisting of a collapsing null shell that forms a Schwazschild black hole, by explicitly obtaining, in a (1 + 1)-dimensional model, the Wightman function, the renormalised stress-energy tensor, and by analysing particle detector rates along stationary orbits in the exterior black hole region, and make a comparison with the folklore that the Unruh state is the state that emerges from black hole formation. In the causal future of the shell, we find a negative ingoing flux at the horizon that agrees precisely with the Unruh state calculation, and is the source of black hole radiation, while in the future null infinity we find that the radiation flux output in the Unruh state is an upper bound for the positive outgoing flux in the collapsing null shell spacetime. This indicates that back-reaction estimates based on Unruh state calculations over-estimate the energy output carried by so-called pre-Hawking radiation. The value of the output predicted by the Unruh state is however approached exponentially fast. Finally, we find that at late times, stationary observers in the exterior black hole region in the collapsing shell spacetime detect the local Hawking temperature, which is also well characterised by the Unruh state, coming from right-movers. Early-time discrepancies between the detector rates for the Unruh state and for the state in the collapsing shell spacetime are explored numerically.

On propagation of energy flux in de Sitter spacetime

In this paper, we explore propagation of energy flux in the future Poincar\'e patch of de Sitter spacetime. We present two results. First, we compute the flux integral of energy using the symplectic current density of the covariant phase space approach on hypersurfaces of constant radial physical distance. Using this computation we show that in the tt-projection, the integrand in the energy flux expression on the cosmological horizon is same as that on the future null infinity. This suggests that propagation of energy flux in de Sitter spacetime is sharp. Second, we relate our energy flux expression in tt-projection to a previously obtained expression using the Isaacson stress-tensor approach. We also comment on the energy flux computation in TT-gauge on hypersurfaces of constant radial physical distance.