Kerr Backgrounds Research Articles

The weak null condition on Kerr backgrounds

The weak null condition on Kerr backgrounds

Physical-space estimates for axisymmetric waves on extremal Kerr spacetime

We study axisymmetric solutions to the wave equation on extremal Kerr backgrounds and obtain integrated local energy decay (or Morawetz estimates) through an analysis exclusively in physical-space. Boundedness of the energy and Morawetz estimates for axisymmetric waves in extremal Kerr were first obtained by Aretakis [13] through the construction of frequency-localized currents used in particular to express the trapping degeneracy. Here we extend to extremal Kerr a method introduced by Stogin [63] in the sub-extremal case, simplifying Aretakis' derivation of Morawetz estimates through purely classical currents.

Canonical quantization of Teukolsky fields on Kerr backgrounds

Canonical quantization of Teukolsky fields on Kerr backgrounds

Scattering in black hole backgrounds and higher-spin amplitudes. Part I

The scattering of massless waves of helicity mid hmid =0,frac{1}{2},1 in Schwarzschild and Kerr backgrounds is revisited in the long-wavelength regime. Using a novel description of such backgrounds in terms of gravitating massive particles, we compute classical wave scattering in terms of 2 → 2 QFT amplitudes in flat space, to all orders in spin. The results are Newman-Penrose amplitudes which are in direct correspondence with solutions of the Regge-Wheeler/Teukolsky equation. By introducing a precise prescription for the point-particle limit, in Part I of this work we show how both agree for h = 0 at finite values of the scattering angle and arbitrary spin orientation.Associated classical observables such as the scattering cross sections, wave polarizations and time delay are studied at all orders in spin. The effect of the spin of the black hole on the polarization and helicity of the waves is found in agreement with previous analysis at linear order in spin. In the particular limit of small scattering angle, we argue that wave scattering admits a universal, point-particle description determined by the eikonal approximation. We show how our results recover the scattering eikonal phase with spin up to second post-Minkowskian order, and match it to the effective action of null geodesics in a Kerr background. Using this correspondence we derive classical observables such as polar and equatorial scattering angles.This study serves as a preceding analysis to Part II, where the Gravitational Wave (h = 2) case will be studied in detail.

Late-time tails and mode coupling of linear waves on Kerr spacetimes

We provide a rigorous derivation of the precise late-time asymptotics for solutions to the scalar wave equation on subextremal Kerr backgrounds, including the asymptotics for projections to angular frequencies ℓ≥1 and ℓ≥2. The ℓ-dependent asymptotics on Kerr spacetimes differ significantly from the non-rotating Schwarzschild setting (“Price's law”). The main differences with Schwarzschild are slower decay rates for higher angular frequencies and oscillations along the null generators of the event horizon. We introduce a physical space-based method that resolves the following two main difficulties for establishing ℓ-dependent asymptotics in the Kerr setting: 1) the coupling of angular modes and 2) a loss of ellipticity in the ergoregion. Our mechanism identifies and exploits the existence of conserved charges along null infinity via a time invertibility theory, which in turn relies on new elliptic estimates in the full black hole exterior. This framework is suitable for addressing conflicting results on the numerology of Kerr late-time asymptotics appearing in the numerics literature and definitively determining the correct numerology.

Pointwise decay for semilinear wave equations on Kerr spacetimes

In this article we prove optimal pointwise bounds for solutions to the semilinear wave equation with integer powers $p\geq 3$ on Kerr backgrounds with small angular momentum and small initial data.

Boundedness of Massless Scalar Waves on Kerr Interior Backgrounds

We consider solutions of the massless scalar wave equation $\Box_g\psi=0$, without symmetry, on fixed subextremal Kerr backgrounds $(\mathcal M, g)$. It follows from previous analyses in the Kerr exterior that for solutions $\psi$ arising from sufficiently regular data on a two ended Cauchy hypersurface, the solution and its derivatives decay suitably fast along the event horizon $\mathcal H^+$. Using the derived decay rate, we show that $\psi$ is in fact uniformly bounded, $|\psi|\leq C$, in the black hole interior up to and including the bifurcate Cauchy horizon $\mathcal C\mathcal H^+$, to which $\psi$ in fact extends continuously. In analogy to our previous paper, [30], on boundedness of solutions to the massless scalar wave equation on fixed subextremal Reissner--Nordstr\"om backgrounds, the analysis depends on weighted energy estimates, commutation by angular momentum operators and application of Sobolev embedding. In contrast to the Reissner--Nordstr\"om case the commutation leads to additional error terms that have to be controlled.

Tidal effects away from the equatorial plane in Kerr backgrounds

Tidal effects on self-gravitating Newtonian stars rotating around a Kerr black hole in stable spherical orbits away from the equatorial plane are exemplified by a non-vanishing Carter's constant, and are substantially different from those on the equatorial plane. In this paper, we calculate the tidal disruption limit of the star numerically, in Fermi normal coordinates on such non-equatorial orbits, and find significant differences in the nature of tidal forces compared to equatorial orbits.

Constrained field theories on Kerr backgrounds

We analyze the constraints of gauge theories on Kerr and Kerr-de Sitter spacetimes, which contain one or more horizons. We find that the constraints are modified on such backgrounds through the presence of additional surface terms at the horizons. As a concrete example, we consider the Maxwell field and find that the Gauss law constraint involves surface corrections at the horizons. These surface contributions correspond to induced surface charges and currents on the horizons, which agree with those found within the membrane paradigm. The modification of the Gauss law constraint also influences the gauge fixing and Dirac brackets of the theory.

Planar domain walls in black hole spacetimes

We investigate the behaviour of low-mass, planar domain walls in the so-called $\phi^4$ model of the scalar field on the Schwarzschild and Kerr backgrounds. We focus on a transit of a domain wall through a black hole and solve numerically the equations of motion for a range of parameters of the domain wall and the black hole. We observe a behavior resembling an occurrence of ringing modes. Perturbations of domain walls vanish during latter evolution, suggesting their stability against a passage through the black hole. The results obtained for Kerr and Reissner-Nordstr\"om black holes are also compared.

Decay for solutions of the wave equation on Kerr exterior spacetimes III: The full subextremal case |a| < M

This paper concludes the series begun in [M. Dafermos and I. Rodnianski, Decay for solutions of the wave equation on Kerr exterior spacetimes I{II: the cases jaj M or axisymmetry, arXiv:1010.5132], providing the complete proof of denitive boundedness and decay results for the scalar wave equation on Kerr backgrounds in the general subextremal jaj < M case without symmetry assumptions. The essential ideas of the proof (together with explicit constructions of the most dicult mul

Boundedness of Massless Scalar Waves on Reissner-Nordström Interior Backgrounds

We consider solutions of the scalar wave equation \({\Box_g\phi=0}\), without symmetry, on fixed subextremal Reissner-Nordstrom backgrounds \({(\mathcal{M}, g)}\) with nonvanishing charge. Previously, it has been shown that for ϕ arising from sufficiently regular data on a two ended Cauchy hypersurface, the solution and its derivatives decay suitably fast on the event horizon \({\mathcal{H}^+}\). Using this, we show here that ϕ is in fact uniformly bounded, \({|\phi| \leq C}\), in the black hole interior up to and including the bifurcate Cauchy horizon \({\mathcal{C}\mathcal{H}^+}\), to which ϕ in fact extends continuously. The proof depends on novel weighted energy estimates in the black hole interior which, in combination with commutation by angular momentum operators and application of Sobolev embedding, yield uniform pointwise estimates. In a forthcoming companion paper we will extend the result to subextremal Kerr backgrounds with nonvanishing rotation.

Static- and stationary-complete spacetimes: algebraic and causal structures

This is intended as an analysis of the global properties of static and stationary spacetimes with complete (timelike) Killing field, with particular attention to quotients by group actions. This is presented in terms of algebraic structures which are fairly simple for the static case and more involved for the stationary case; the most important tool, the fundamental cocycle, is a cohomological class for static spacetimes but of somewhat looser structure in the stationary case. In particular: (1) A new measurement, similar to the spacetime interval in Minkowski space, is devised for detecting whether two points are causally related in a stationary spacetime; this proves very useful for analysis. (2) All stationary spacetimes are categorized by how they behave with respect to the fundamental cocycle; this enables a complete characterization of global causality properties. (3) It is shown how these tools determine whether global hyperbolicity of a stationary spacetime is inherited by its quotients. (4) Examples are examined in detail, a large range including both ones of mathematical curiosity and ones of physical interest, such as cosmic strings in flat, accelerated, Schwarzschild, Kerr, and other backgrounds.

Localized Energy Estimates for Wave Equations on (1+4)-dimensional Myers--Perry Space-times

Localized energy estimates for the wave equation have been increasingly used to prove various other dispersive estimates. This article focuses on proving such localized energy estimates on (1+4)-dimensional Myers--Perry black hole backgrounds with small angular momenta. The Myers--Perry space-times are generalizations of higher-dimensional Kerr backgrounds where additional planes of rotation are available while still maintaining axial symmetry. Once it is determined that all trapped geodesics have constant $r$, the method developed by Tataru and the fourth author, which perturbs off of the Schwarzschild case by using a pseudodifferential multiplier, can be adapted.

Quantitative Mode Stability for the Wave Equation on the Kerr Spacetime

We give a quantitative refinement and simple proofs of mode stability type statements for the wave equation on Kerr backgrounds in the full sub-extremal range (|a| < M). As an application, we are able to quantitatively control the energy flux along the horizon and null infinity and establish integrated local energy decay for solutions to the wave equation in any bounded-frequency regime.

Local decay of waves on asymptotically flat stationary space-times

In this article we study the pointwise decay properties of solutions to the wave equation on a class of stationary asymptotically flat backgrounds in three space dimensions. Under the assumption that uniform energy bounds and a weak form of local energy decay hold forward in time we establish a $t^{-3}$ local uniform decay rate for linear waves. This work was motivated by open problems concerning decay rates for linear waves on Schwarzschild and Kerr backgrounds. In the Schwarzschild case, such a decay rate has been heuristically derived by Price. Our results apply to both of these cases.

Decay of axisymmetric solutions of the wave equation on extreme Kerr backgrounds

We study the Cauchy problem for the wave equation □gψ=0 on extreme Kerr backgrounds. Specifically, we consider regular axisymmetric initial data prescribed on a Cauchy hypersurface Σ0 which connects the future event horizon with spacelike or null infinity, and we solve the linear wave equation on the domain of dependence of Σ0. We show that the spacetime integral of an energy-type density is bounded by the initial conserved flux corresponding to the stationary Killing field T, and we derive boundedness of the non-degenerate energy flux corresponding to a globally timelike vector field N. Finally, we prove uniform pointwise boundedness and power-law decay for ψ up to and including the event horizon H+.

Oscillations of relativistic axisymmetric tori and implications for modelling kHz-QPOs in neutron star X-ray binaries

We perform a global linear perturbative analysis, and investigate the oscillation properties of relativistic, non-selfgravitating tori orbiting around neutron stars in the slow rotation limit approximation. Extending the work done in Schwarzschild and Kerr backgrounds, we consider the axisymmetric oscillations of vertically integrated tori in the Hartle-Thorne spacetime. The equilibrium models are constructed by selecting a number of different non-Keplerian distributions of specific angular momentum, allowing for disc sizes $L \sim 0.5 - 600$ gravitational radii. Our results, obtained after solving a global eigenvalue problem to compute the xisymmetric $p$-modes, indicate that such oscillation modes could account with most observed lower ($\nu_L$) and upper ($\nu_U$) high frequency quasi-periodic oscillations for Sco X-1, and for some Z sources and Atoll sources with $\nu_L\ gtrsim 500$ Hz. However, when $\nu_L \lesssim 500$ Hz, $p$-modes oscillations do not account for the linear relation $\nu_U = A \nu_L + B$, $B \neq 0$ between the upper and the lower high frequency quasi-periodic oscillations that are observed in neutron star low-mass X-ray binaries.

Mathisson–Papapetrou–Dixon equations in the Schwarzschild and Kerr backgrounds

A new representation, which does not contain the third-order derivatives of the coordinates, of the exact Mathisson–Papapetrou–Dixon equations, describing the motion of a spinning test particle, is obtained under the assumption of the Mathisson–Pirani condition in a Kerr background. For this purpose the integrals of energy and angular momentum of the spinning particle as well as a differential relationship following from the Mathisson–Papapetrou–Dixon equations are used. The form of these equations is adapted for their computer integration with the aim to investigate the influence of the spin–curvature interaction on the particle's behavior in the gravitational field without restrictions on its velocity and spin orientation. Some numerical examples for a Schwarzschild background are presented.

The general relativistic Poynting–Robertson effect: II. A photon flux with nonzero angular momentum

We study the motion of a test particle in a stationary, axially and reflection-symmetric spacetime of a central compact object, as affected by interaction with a test radiation field of the same symmetries. Considering the radiation flux with fixed but arbitrary (nonzero) angular momentum, we extend previous results limited to an equatorial plane motion within a zero angular-momentum photon flux in the Kerr and Schwarzschild backgrounds. While a unique equilibrium circular orbit exists if the photon flux has zero angular momentum, multiples of such orbits appear if the photon angular momentum is sufficiently high.