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Register a new userLinear Waves in the Kerr Geometry: A Mathematical Voyage to Black Hole Physics
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This paper gives a survey of wave dynamics in the Kerr space-time geometry, the mathematical model of a rotating black hole in equilibrium. After a brief introduction to the Kerr metric, we review the separability properties of linear wave equations for fields of general spin $s=0, 1/2, 1, 2$, corresponding to scalar, Dirac, electromagnetic fields and linearized gravitational waves. We give results on the long-time dynamics of Dirac and scalar waves, including decay rates for massive Dirac fields. For scalar waves, we give a rigorous treatment of superradiance and describe rigorously a mechanism of energy extraction from a rotating black hole. Finally, we discuss the open problem of linear stability of the Kerr metric and present partial results.
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We present a new upper limit on the energy that may be extracted from a Kerr black hole by means of particle collisions in the ergosphere (i.e., the collisional Penrose process). Earlier work on this subject has focused largely on particles with critical values of angular momentum falling into an extremal Kerr black hole from infinity and colliding just outside the horizon. While these collisions are able to reach arbitrarily high center-of-mass energies, it is very difficult for the reaction products to escape back to infinity, effectively limiting the peak efficiency of such a process to roughly $130%$. When we allow one of the initial particles to have impact parameter $b > 2M$, and thus not get captured by the horizon, it is able to collide along outgoing trajectories, greatly increasing the chance that the products can escape. For equal-mass particles annihilating to photons, we find a greatly increased peak energy of $E_{rm out} approx 6times E_{rm in}$. For Compton scattering, the efficiency can go even higher, with $E_{rm out} approx 14times E_{rm in}$, and for repeated scattering events, photons can both be produced {it and} escape to infinity with Planck-scale energies.
We propose a set of diffeomorphism that act non-trivially near the horizon of the Kerr black hole. We follow the recent developments of Haco-Hawking-Perry-Strominger to quantify this phase space, with the most substantial difference being our choice of vectors fields. Our gravitational charges are organized into a Virasoro-Kac-Moody algebra with non-trivial central extensions. We interpret this algebra as arising from a warped conformal field theory. Using the data we can infer from this warped CFT description, we capture the thermodynamic properties of the Kerr black hole.
We present an analytical treatment of gravitational lensing by a Kerr black hole in the weak deflection limit. Lightlike geodesics are expanded as a Taylor series up to and including third-order terms in m/b and a/b, where m is the black hole mass, a the angular momentum and b the impact parameter of the light ray. Positions and magnifications of individual images are computed with a perturbative analysis. At this order, the degeneracy with the translated Schwarzschild lens is broken. The critical curve is still a circle displaced from the black hole position in the equatorial direction and the corresponding caustic is point-like. The degeneracy between the black hole spin and its inclination relative to the observer is broken through the angular coordinates of the perturbed images.
A rotating black hole causes the spin-axis of a nearby pulsar to precess due to geodetic and gravitomagnetic frame-dragging effects. The aim of our theoretical work here is to explore how this spin-precession can modify the rate at which pulses are received on earth. Towards this end, we obtain the complete evolution of the beam vectors of pulsars moving on equatorial circular orbits in the Kerr spacetime, relative to asymptotic fixed observers. We proceed to establish that such spin-precession effects can significantly modify observed pulse frequencies and, in specific, we find that the observed pulse frequency rises sharply as the orbit shrinks, potentially providing a new way to locate horizons of Kerr black holes, even if observed for a very short time period. We also discuss implications for detections of sub-millisecond pulsars, pulsar nulling, quasi-periodic oscillations, multiply-peaked pulsar Fourier profiles and how Kerr black holes can potentially be distinguished from naked singularities.
In the context of the interaction between the electromagnetic field and a dielectric dispersive lossless medium, we present a non-linear version of the relativistically covariant Hopfield model, which is suitable for the description of a dielectric Kerr perturbation propagating in a dielectric medium. The non-linearity is introduced in the Lagrangian through a self-interacting term proportional to the fourth power of the polarization field. We find an exact solution for the nonlinear equations describing a propagating perturbation in the dielectric medium. Furthermore the presence of an analogue Hawking effect, as well as the thermal properties of the model, are discussed, confirming and improving the results achieved in the scalar case.
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