Written by Dr Georgios V Kraniotis, a theoretical physicist at the University of
Ioannina in the physics department.
Solving in closed form the Klein-Gordon-Fock equation on curved black hole spacetimes
A new exciting era in the exploration of spacetime
The investigation of the interaction of a scalar particle with the gravitational field is of importance in the attempts to construct quantum theories on curved spacetime backgrounds. The general relativistic form that models such interaction is the so called Klein-Gordon-Fock (KGF) wave equation named after its three independent inventors. The discovery of a Higgs-like scalar particle at CERN in conjuction with the recent spectacular observation of gravitational waves (GW) from the binary black hole mergers GW150914 and GW151226 by LIGO collaboration, adds a further impetus for probing the interaction of scalar degrees of freedom with the strong gravitational field of a black hole.
Kerr black hole perturbations and the separation of the Dirac’s equations was a central theme in the investigations of Teukolsky and Chandrasekhar .
All the above motivated our research recently published in CQG on the scalar charged massive field perturbations for the most general four dimensional curved spacetime background of a rotating, charged black hole, in the presence of the cosmological constant .
Where interesting physics meets profound mathematics
The KGF equation is the relativistic version of the Schrödinger equation and thus is one of the fundamental equations in physics.
In our recent CQG paper, we examined the KGF equation for a charged massive particle in the Kerr-Newman-(anti) de Sitter (KN(a)dS) and the Kerr-Newman spacetimes which describe the rotating solution of the Einstein-Maxwell equations . We solved the Klein-Gordon-Fock equation in these spacetime backgrounds. In the absence of the cosmological constant the solutions obtained are given in terms of confluent Heun functions.
In the presence of the cosmological term the solutions obtained were more involved and were expressed in terms of Heun functions. For particular values of the mass of the scalar field in terms of the cosmological constant we succeeded in transforming both angular and radial equations that result from the separation of the KGF equation in the KNdS background into Heun equations.
A bit of history of Heun functions
Heun’s differential equation (HE) belong to the class of Fuchsian differential equations (FDE), since it is the most general linear differential equation with four regular singular points. It was introduced by Karl Heun ( a Privatdozent in Munich at that period) as a generalisation of the hypergeometric equation studied by Gauß . Despite the phenomenal simplicity of the equation the theory of its solutions (the so called Heun functions) is far from being complete. A lot of research effort has been invested on the so called (still open) connection problem, i.e. the problem of finding relations between local solutions of HE about two different singularities. A series of open issues require further insight, among others we mention, the details of the monodromy group and finding integral representations for the Heun solutions.
The confluence of two regular singular points of the HE results in the confluent Heun equation (CHE) which still has regular singularities at the points and and an irregular singularity of rank at .
Solutions of KGF equation in KNdS background in terms of Heun functions and the connection problem
For the value of the inverse Compton wavelength of the scalar particle, , both radial and angular parts of the separated KGF equation were transformed into Heun equations . In our CQG paper we solved both Heun equations in terms of an infinite series of hypergeometric functions using the idea of augmented convergence . In this setup, the solution converges inside the ellipse with foci at two of the finite regular singularities and passing through the third finite regular singular point with possible exception of the line connecting the two foci. This method of constructing a Heun function offers a perspective for the connection problem. In particular, the solution obtained for the massive radial Heun equation in KNdS spacetime converges in the ellipse with foci at the event and Cauchy horizons of the black hole (BH). For more details of the construction of the solutions we refer the reader to our CQG paper .
An interesting research path for the future will be to construct radial solutions for the massive charged scalar particle which are valid not only inside an ellipse with foci at the event and Cauchy horizons but also at the cosmological horizon of the KNdS black hole.
As if this was not enough, physics of black holes points to a generalisation of Heun functions
The general case, however, is that the solution of the KGF equation with the method of separation of variables, for a rotating charged cosmological black hole, results in FDE for the radial and angular parts which for most of the parameter space contain more than three finite singularities and thereby generalise the Heun differential equations. We already mentioned regions for the parameter space (e.g. scalar mass) for which the FDE reduce to Heun equations. There are also other regions of the parameter space for which the extra singular points become false or apparent singular points. A singular point is called false if both of its exponents are non-negative integers and there are no logarithmic terms in the local expansion near the singular point. In our CQG paper, we derived a condition that guarantees the absence of logarithmic terms local to a singular point with exponents and proved that for , provided the coefficients of the angular equations satisfy the abovementioned condition, the extra singular point becomes a false point . One might even go a step further, by conjecturing that in the case of a Fuchsian equation with 5 singular points, as it is for example the case of the radial part of the KGF equation for a charged massive particle in the KNdS background for most of the parameter space, that if one of the singular points is false than the solution will be expressed in terms of general Heun functions. The investigation of such a conjecture will be examined in a future endeavour. Additional ramifications to be explored include the Riemann-Hilbert problem which in layman’s terms states: for a FDE in order that its isomonodromy problem is non-trivial additional degrees of freedom in the form of apparent singularities need to be introduced . If one considers the apparent singularities and the conjugate momenta as functions of the non-apparent regular singularities (other than and ) a system of Hamilton’s equations emerges. For one additional false singularity this Hamiltonian system is equivalent to a non-linear second order ordinary differential equation the celebrated sixth Painlevé equation . The interested reader is kindly invited to read our CQG paper for more details, where one can also consult a summary of known mathematical results on the connection of isomonodromy mappings and the solution of Fuchsian equations that possess false singularities.
Exact solutions of the KGF equation for the massive charged particle in the limit
The exact solutions of the radial and angular parts of the KGF equation in KN spacetime were expressed in terms of confluent Heun functions. The radial solutions are valid at all radii from the event horizon up to infinity. In our CQG paper we examined the physical importance of these solutions by investigating the asymptotic behaviour of the solutions and also obtaining the solutions in regions near the event horizon and far from the horizon of the black hole. For further details the reader should consult our CQG paper .
Possible applications: superradiance instabilities of a KN(a)dS black hole
As discussed in our CQG paper a possible application of our work would be to examine gravitational radiation from a hypothetical axion cloud around a rotating charged BH. Ultralight axion fields are ubiquitous in Calabi Yau compactifications of string theory. As was first shown by Ya. B. Zel’dovich rotating bodies amplify incident waves-this phenomenenon is called superradiance . A superradiant instability effectively takes place if the Compton wavelength of the axion mass has the order of the gravitational radius of the black hole. As the superradiant instability develops, an important effect is gradually taking place, namely the emission of GW . One can study using our exact solutions, GW emissions from a hypothetical axion bosonic cloud around the galactic centre Sagittarius A* (SgrA*) supermassive BH-assuming the latter provides a KN(a)dS BH background- and eventually constrain the mass of such ultralight (of order eV) axionic degrees of freedom. Such efforts can be aided by future precise measurements of the relativistic effects, associated with observed stellar orbits in the central arcsecond of SgrA* , such as periastron precession and frame dragging complemented by studies of strong gravitational lensing effects by the galactic centre BH. Another interesting research path would be to investigate superradiance for bosonic massive fields with spin and/or fermionic massive degrees of freedom by solving the corresponding differential equations. Studies of this kind will eventually offer the exciting possibility of testing the BH “no hair’” hypothesis.
 S. A. Teukolsky, The Kerr metric, Class. Quantum Grav. (2015) 32 124006, S. Chandrasekhar, The Mathematical Theory of Black Holes, (Oxford: Oxford University Press) 1998
 G V Kraniotis, The Klein-Gordon-Fock equation in the curved spacetime of the Kerr-Newman (anti) de Sitter black hole (2016) Class. Quantum Grav. 33 225011
 K. Heun, Zur Theorie der Riemann’schen Functionen zweiter Ordnung mit vier Verzweigungspunkten, Math.Ann. 33, (1889), 161-179
 A. Ronveaux (ed), Heun’s differential equations, Oxford University Press (1995)
 M. Yoshida, Fuchsian differential equations, Aspects of Mathematics, Springer 1987.
 Ya. B. Zel’dovich, Amplification of cylindrical electromagnetic waves reflected from a rotating body, Sov. Phys.-JETP 35, 1085-7
 A. Arvanitaki et al, String axiverse, Phys. Rev. D 81, 2010, 123530
Read the full article in Classical and Quantum Gravity:
The Klein-Gordon-Fock equation in the curved spacetime of the Kerr-Newman (anti) de Sitter black hole
G V Kraniotis 2016 Class. Quantum Grav. 33 225011
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