34;spin down" (loss of angular momentum) of a spinning black hole caused by a stationary non-axisymmetric perturbation. a) Perturbation problem for black holes (b) Newman-Penrose formalism. One interesting application of the perturbation equations concerns the "spin down" effect , which occurs when a rotating black hole is subjected to a stationary, non-axisymmetric perturbation.
PART II
BACKGROUND MATERIAL
For gravitational perturbations (e.g. when φ is the mass of an incident particle), this is the complete set of equations. The quantity g describes the unperturbed Kerr solution, while g B is the perturbation due to .
10- gravitational field
Since most readers are probably more familiar with tetrads than spinors, I will briefly describe the NP formalism in the language of tetrads without following the spinor path; a good description of the spinor approach was given by Pirani (1964). Note that equation (2.5) is invariant under Lorentz transformations of the tetrad:. 2.9) Since the Lorentz group has six parameters, there are six levels.
2.9) Since the Lorentz group has six parameters, there are six degrees of
The Riemann tensor can be decomposed into the Weyl tensor, the Ricci tensor and the scalar curvature: The vectors £ and n are real numbers, while m is complex with real and imaginary parts, each of which is space-like.).
2.28) They reduce to four complex equations
PART III
SCALAR-FIELD CALCULATIONS OF ROTATING BLACK-HOLE PHENOMENA
ROTATING BLACK HOLES: LOCALLY NONROTATING FRAMES, ENERGY EXTRACTION, AND SCALAR
SYNCHROTRON RADIATION*
INTRODUCTION
Phenomena near a rotating black hole are considerably more complicated than in the non-rotating case. The Boyer-Lindquist (1967) coordinates are the natural generalization of Schwarzschild curvature coordinates and are the best for many purposes, but sufficiently close to the hole - in the 'ergosphere' - they are somewhat unphysical.
BASIC PROPERTIES OF THE KERR METRIC AND ITS ORBITS
2.2) Here M is the mass of the black hole, a is its angular momentum per unit mass
THE SCALAR WAVE EQUATION AND SCALAR SYNCHROTRON RADIATION
At radius r, the direction cosine with respect to the ^-direction in the LNRF is for the trajectory of a photon with energy E and axial angular momentum L. A photon with energy E and axial angular momentum L has a locally measured energy (frequency) in the LNRF. < - e -∖E - ωL).
CONCLUSION
In particular, the frequency of a photon emitted tangentially to the circle of the speed of light at r = r1 for which E∣L = Ω varies with radius in the same way as the frequency of scalar synchrotron radiation.
APPENDIX A
BOUNDS ON ENERGIES AND RELATIVE VELOCITIES OF PARTICLE ORBITS
Therefore, of all physically plausible diving trajectories, this class is always separated from the negative energy region by at least half the speed of light. To achieve energy extraction, hydrodynamic forces or super-strong radiation reactions would have to accelerate particle fragments to more than half the speed of light in the 'short' environment.
APPENDIX B
Full. 178 . around any rotating black hole) and E2 = 0 (the boundary of the negative energy region), we get. Even for a ≈ M, however, the highly relativistic orbits at r ~ rpb cannot extract energy from the black hole.
APPENDIX C
BOUNDS ON EIGENVALUES OF SPHEROIDAL HARMONICS Define the following differential operator L on the closed interval [—1, 1]
1 shows the results of our numerical integrations of equation (3) for the most favorable case, a maximally rotating black hole with a=M. At this 'floating' beam, the particle gradually radiates the rotational energy away from the black hole; as a decreases, the beam of the lowest stable orbit moves outward with respect to the floating beam.
KERR PERTURBATION EQUATIONS
DECOUPLED GRAVITATIONAL EQUATIONS The derivation in this section applies to
Now consider the following three non-vacuum NP equations, taken from Pirani (l96⅛)z. 2.U) The Ricci tensor terms on the right-hand side of equations (2.2) and (2.3) are given by the Einstein field equations:. and so on, where R is the Ricci tensor and T is the stress energy tensor. This relationship holds for any Type D metric, where equations (2.1) hold, and can be proven using equations. This is the decoupled equation for ψθ. The entire set of NP equations is invariant under the exchange l* n, m<÷ m* (Geroch et al. 1972).
DECOUPLED ELECTROMAGNETIC EQUATIONS
A proves that ψθ and are invariant under gauge transfor. matations and infinitesimal tetrad rotations, and are therefore completely ma surahle physical quantities. 3.6) By interchanging I and n, and m and m*, we get the equation for [which is also directly derivable from eq. 3.8) Fackerell and Ipser (1972) derived an analogous decoupled equation for 0^, but this equation does not appear to be separable in the Kerr case.
SEPARATION OF THE EQUATIONS
In the general case, we will call the eigenfunctions "spin-weighted spheroidal harmonics". The numerical calculation of these functions and the corresponding eigenvalues is described in Paper II of this series. When the sources are present (T ∕ θ), we can use the eigenfunctions of equation (U.lθ) to separate equation (U.7) by expansion.
BOUNDARY CONDITIONS, ENERGY, AND POLARIZATION
Appendix A)*. This is the non-special, as you can see by writing it in the form e ^ωv e^mt^. So the correct precondition is. 5.10). However, you can find the energy flux in the two main cases: at infinity and at the horizon. See analyzes by Press (1972) (scalar perturbation) and Hartle (1973) (gravitational perturbation with a «M).] The calculation for arbi.
DISCUSSION
The shear gives the rate of change of the horizon surface, dA∕dt. We will not discuss the source terms here, nor the physical interpretations. When these equations are written in Boyer-Lindquist coordinates, they have the same form as the main equation (4.7), with ψ = χθ (s = l∕2) and.
STABILITY OF ROTATING BLACK HOLES
USE OF THE DECOUPLED, SEPARATED WAVE EQUATION
If ‡ , seen as a function of ω, contains no poles (we will assume that the w' branch cute is excluded by the shape of ^) in the region above the contour, then equation (2.4) remains a valid "reconstruction" of the complete field. First, for the equation with spin weight s = -2 (which for numerical reasons will turn out to be the actual case integral, see §IIl), the integral part of the solution. Here, however, one can use the Regge-Wħeeler (1957) and Zerilli (1970) radial equations instead of the more complicated equation (2.9) of the Kerr background.
NUMERICAL SOLUTION OF THE EQUATIONS a) Angular Eigenfunctions and Eigenvalues
Similarly, if we integrate at the horizon and try to establish boundary conditions, we are faced with finding a zero of the exponentially small irregular solution, which is lost in numerical truncation. The case s = -2 is pleasantly the reverse: integrating outwards, any contamination of the irregular solution vanishes exponentially; or, inte. A second method is the one we actually used for most of the results presented here.
DISCUSSION AND CONCLUSIONS
The important result is that the dominant behavior of the integral is still determined by the value of e 2^^ωr* at a saddle point;. Thus, a differential equation for χ can be integrated without loss of meaning for either solution. It is almost certain that any instability must occur via a real frequency (see text); furthermore, the smoothness of the Z function in a/M directly argues that there is no sudden appearance of a pole in the upper-half complex plane.
FURTHER APPLICATIONS
If there is an angular momentum flux along the 2-surface element formed by the intersection of a horizon element with two surfaces of time constant V separated by dv 9, then the change in angular momentum of the hole is.
148- (iv) Source Terms
150- These enable us to write
This means that the lowest source multipole contributing to the spin-down effect is the quadrupole 9 in contrast to the dipole contribution to the scalar and electromagnetic perturbation. This is as expected - gravitational rotation is a tidal effect. v) Solving the radial equation. The first of these is the exact boundary condition; it represents an ongoing flow of energy to a local observer (see Section 5 of Paper IV).
152- such that
5.6) (5.2) are
5.8) Here the hypergeometric functions F are polynomials which have the
154- dü
Here we see that as the source of disturbance approaches the ergo limit. This is because stationary charge moves at a speed approaching the speed of light as seen by a physical observer at the boundary of the ergosphere. The above calculation does not enable us to calculate the time evolution of the angular momentum of the hole: we need to know the velocity e.
7.5) This is completely negligible for an interstellar gravitational field
The only known theoretical limit on the amount of energy that can in principle be obtained is Hawking's (1972) theorem that the surface of a black hole cannot shrink. However, we saw in Article V that a rotating black hole is stable for all values of a. A particle with rest mass mθ and total energy E θ travels from infinity through the ergosphere of a rotating black hole.
The Kerr metric is stationary and axisymmetric, so any wave can be decomposed into shape modes. The magnitude of the superradiation effect can be calculated using the separable equations of Paper IV. tromagnetic or gravitational) with amplitude ∑1∙ n is sent from infinity, scattered by the hole and exits with amplitude Zθ ut. The ratio of the outgoing to incoming energy flux at infinity can be found from the expression for the electromagnetic energy-voltage tensor [Paper IV, equation (5.12)].
166- and then
It is clear that more detailed astrophysical models of black hole processes are required before we can say anything definitive about the importance of exotic effects such as superradiation. The vertical scale is ( I zou √zi n ∣ corrected ’ 1} ’ where "cθr - corrected,* means converted to give energy rather than amplitude, as described in the text.
PÄRT VII APPENDIX
ON THE EVOLUTION OF THE SECULARLY UNSTABLE, VISCOUS MACLAURIN SPHEROIDS*
DISCUSSION OF PREVIOUS INVESTIGATIONS
Thus, although a boundary layer stress is carried over into the hydrodynamic equations, the anti-damped term appearing in the equation of motion of the unsteady mode is the same mode-projected term allowed by the tensor-virial approximation. . The point of confusion is that the anti-damped oscillations of the linearized treatments refer to Lagrangian displacements of fluid from a "fictitious" . undisturbed configuration; they are not overdamped oscillations in the physical shape of the ellipsoid. In Fujimoto's numerical solutions, the additional damped oscillations occur at early times because his initial perturbation is not a pure mode; it is a mixture of the two toroidal modes in the linearized theory;.
NEW RESULTS
If ξ is the Lagrangian displacement at position x, without time dependence e~iRe
