Chandrasekhar's algebraically special enigmas: the common origin of the Starobinsky constant's three manifestations, the conjugation identity for $P_{+2}$, and the special class of black-hole potential barriers

Chandrasekhar's 1984 analysis of algebraically special perturbations of black holes found that the Starobinsky constant $|\mathscr{C}|^2$ emerges in three apparently independent ways in Kerr perturbation theory — from the Teukolsky–Starobinsky identities, from the condition admitting polynomial radial solutions at the purely imaginary algebraically special frequencies, and as the factored product $K^{(+\sigma)}K^{(-\sigma)}$ of the transformation-theory operators — and posed the question of their common origin, together with two companion puzzles: why complex conjugation of the spin-weight $+2$ radial function $P_{+2}$ acts as the fourth-order operator $\Delta^2(\mathscr{D}_0^\dagger)^4$, and why the potential barriers of the Schwarzschild and Reissner–Nordström black holes belong to a very special class producing backscatter-free solutions. That common origin remains unidentified.

Background

Black-hole perturbation theory studies the linearized Einstein (and Einstein–Maxwell) equations about an exact black-hole solution: the Schwarzschild metric (mass $M$ only), the Reissner–Nordström or RN metric (mass $M$ and electric charge $Q$), and the Kerr metric (mass $M$ and angular momentum $a$). Instead of perturbing metric components, the standard formalism of Newman and Penrose perturbs the Weyl scalars $\psi_0,\dots,\psi_4$ — five complex scalars that encode the curvature relative to a tetrad of null vectors. Teukolsky showed that on Kerr a single master equation, the Teukolsky equation, governs massless fields of arbitrary spin weight $s$; throughout, $\Delta = r^2 - 2Mr + a^2$ is the Kerr horizon function, $\mathscr{D}_n$ is the standard first-order radial derivative operator of this formalism, and $\mathscr{D}_0^\dagger$ its complex-conjugate (adjoint) operator. By $P_{+2}$ one denotes the separated radial function of spin weight $+2$; in the non-rotating limit this is the radial variable of the Bardeen–Press equation, the Schwarzschild-limit forerunner of the Teukolsky equation.

A first structural fact is that the separated radial functions of spin $+s$ and $-s$ determine one another: applying suitable differential operators to one produces the other, and consistency of the round trip forces a spin-dependent constant, the Starobinsky constant $\mathscr{C}$, into what are called the Teukolsky–Starobinsky identities. A second structure is Chandrasekhar's transformation theory, which maps solutions of one master equation to those of another by paired differential operators — most famously between the Regge–Wheeler and Zerilli equations, the odd- and even-parity wave equations governing Schwarzschild perturbations in the tortoise coordinate $r_*$ (the radial coordinate in which the perturbation waves obey short-range equations with the horizon at $r_*\to-\infty$) — thereby establishing that the two parities are isospectral: they have identical quasinormal modes (the damped resonant frequencies at which the black hole rings after perturbation) and identical scattering spectra. In this theory paired operators denoted $K^{(+\sigma)}$ and $K^{(-\sigma)}$ appear, and their factored product reproduces the Starobinsky constant.

The third ingredient is algebraic specialness. In Petrov's classification a spacetime is algebraically special when principal null directions of its Weyl tensor coincide; Schwarzschild, RN and Kerr are of Petrov type D. Algebraically special perturbations are those for which the perturbed spacetime is again algebraically special; physically they are purely ingoing or purely outgoing gravitational waves that suffer no backscattering off the curvature potential barrier, and they exist only at discrete, purely imaginary frequencies. Couch and Newman (1973) found them for Schwarzschild, Wald (1973) characterized the type-D sector for Kerr, and Chandrasekhar (1984) treated Kerr, Schwarzschild and RN in one framework, observing that for Kerr the frequency condition is precisely $\mathscr{C}=0$: the algebraically special frequencies are the roots of the Starobinsky constant.

In that 1984 paper the same constant $|\mathscr{C}|^2$ surfaced in three different ways: (i) in the Teukolsky–Starobinsky identities themselves; (ii) as the condition under which the radial equations admit polynomial (closed-form) solutions, namely the algebraically special modes; and (iii) as the factored product $K^{(+\sigma)}K^{(-\sigma)}$ in the transformation theory. Chandrasekhar also recorded that, for these solutions, taking the complex conjugate of $P_{+2}$ is equivalent to applying the fourth-order operator $\Delta^2(\mathscr{D}_0^\dagger)^4$, and that the potential barriers surrounding the Schwarzschild and RN holes belong to a very special class of potentials that yields these backscatter-free solutions. All the separated equations of the theory are confluent Heun equations (a class of second-order ordinary differential equations with two regular singular points and one irregular singular point, generalizing the hypergeometric equation), and later work has clarified parts of this web: the transformation theory is an instance of the Darboux transformation (a nineteenth-century technique for generating isospectral Schrödinger operators, underlying supersymmetric quantum mechanics), and the Teukolsky–Starobinsky identities follow from a special parameter condition (the $\delta_N$ condition) on confluent Heun functions. What has not been achieved is a single account from which all of these facts flow at once — which is the question posed here.

Problem Statement

Identify the common origin of the three simultaneous manifestations of the Starobinsky constant $|\mathscr{C}|^2$ in the perturbation theory of the Kerr black hole, as posed in Chandrasekhar (1984):

(i) $|\mathscr{C}|^2$ appears in the Teukolsky–Starobinsky identities relating the separated radial functions of spin weight $+s$ and $-s$;

(ii) the vanishing of $\mathscr{C}$ is the condition under which the radial Teukolsky equations admit polynomial solutions — the algebraically special modes, purely ingoing or purely outgoing waves that suffer no backscatter and have purely imaginary frequencies;

(iii) the same $|\mathscr{C}|^2$ arises as the factored product $K^{(+\sigma)}K^{(-\sigma)}$ of the paired operators of Chandrasekhar's transformation theory.

Two further facts to be accounted for within the same framework are:

(iv) why, for these solutions, taking the complex conjugate of the spin-weight $+2$ radial function $P_{+2}$ is equivalent to applying the fourth-order differential operator $\Delta^2(\mathscr{D}_0^\dagger)^4$; and

(v) why the potential barriers surrounding the Schwarzschild and Reissner–Nordström black holes belong to the very special class of potentials that yields these algebraically special solutions — i.e. characterize that class precisely and give the structural reason why membership in it produces backscatter-free solutions.

The question is resolved by exhibiting a common origin for (i)–(v) in the Kerr, Schwarzschild and Reissner–Nordström cases — a framework from which (i)–(v) follow as theorems — or by demonstrating rigorously that no such common origin exists.

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