[Paper Review] Exact solution of the wave equation of a scalar particle in the zero mass limit of Kerr and Kerr-(anti-)de-Sitter space-times

Tolga Birkandan, M. Hortaçsu|Istanbul Technical University Academic Open Archive (Istanbul Technical University)|Nov 18, 2017

Black Holes and Theoretical Physics17 references3 citations

TL;DR

This paper derives exact solutions to the Klein-Gordon equation for a massive scalar particle in the zero mass limit of Kerr and Kerr-(anti)-de Sitter spacetimes. Using spheroidal coordinates, it shows that both radial and angular equations reduce to confluent and general Heun-type differential equations, confirming the spacetime is not globally Minkowski due to non-analyticity and a z-axis excision, thus validating Gibbons and Volkov's wormhole interpretation.

ABSTRACT

Heun-type exact solutions emerge for both the radial and the angular equations for the case of a scalar particle coupled to the zero mass limit of both the Kerr and Kerr-(anti)de-Sitter spacetime. Since any type D metric has Heun-type solutions, it is interesting that this property is retained in the zero mass case. This work further refutes the claims that $M$ going to zero limit of the Kerr metric is both locally and globally the same as the Minkowski metric.

Motivation & Objective

• To resolve the long-standing debate on whether the zero mass limit of the Kerr metric is globally equivalent to Minkowski spacetime.
• To derive exact solutions of the scalar wave equation in the zero mass limit of Kerr and Kerr-(anti)-de Sitter spacetimes.
• To investigate whether the Heun-type structure of wave equations in type D spacetimes persists in the zero mass limit.
• To verify the topological and geometric claims of Gibbons and Volkov regarding a non-trivial wormhole structure in the zero mass limit.

Proposed method

• Transforming the metric into oblate spheroidal coordinates to enable separation of variables in the Klein-Gordon equation.
• Reducing the wave equation to radial and angular ordinary differential equations via separation of variables in time and azimuthal angle.
• Applying the substitution $ x = \cos^2\theta $ to convert the angular equation into a standard form solvable by confluent Heun functions.
• Deriving exact solutions in terms of confluent Heun functions $ H_C $ for the Kerr case and general Heun functions $ H_G $ for the Kerr-(anti)-de Sitter case.
• Using variable transformations such as $ u = -r^2/a^2 $ to reveal structural isomorphism between radial and angular equations.
• Analyzing singularities and analyticity, particularly the square root cut at $ r=0 $, to assess global geometry and topological structure.

Experimental results

Research questions

• RQ1 Does the zero mass limit of the Kerr metric yield a solution that is globally distinct from Minkowski spacetime?
• RQ2 Are the radial and angular equations in the zero mass limit of Kerr and Kerr-(anti)-de Sitter spacetimes solvable in terms of Heun functions?
• RQ3 What is the nature of the singularity at $ r=0 $, and how does it affect the global structure of the spacetime?
• RQ4 Can polynomial solutions exist for the Heun-type equations in this limit, and what does this imply for analyticity?
• RQ5 How does the behavior of the wave equation in the zero mass limit support or contradict the wormhole interpretation of Gibbons and Volkov?

Key findings

• The radial and angular equations for a scalar particle in the zero mass limit of Kerr spacetime yield exact solutions in terms of confluent Heun functions $ H_C $.
• The angular equation in the Kerr-(anti)-de Sitter case reduces to a general Heun equation with parameters dependent on $ \omega, \mu, a, \Lambda, m, \lambda $.
• The solution exhibits a square root branch cut at $ r=0 $, corresponding to a ring singularity at $ \rho = a $, which is not present in Minkowski spacetime.
• The z-axis is excised in the solution space, indicating a non-trivial global topology distinct from Minkowski space.
• The radial and angular equations are structurally identical under the transformation $ u = -r^2/a^2 $, revealing a deep symmetry in the wave equation.
• The event horizon singularity in the Kerr-(anti)-de Sitter case appears at $ u = -3/(a^2\Lambda) $, and the solution remains of Heun type beyond the horizon, confirming the presence of a non-trivial geometry.