On the smoothness of multi center coplanar black hole and membrane horizons

• Published in: General relativity and gravitation Vol. 51; no. 11; pp. 1 - 49
• Main Authors: Gowdigere, Chethan N., Kumar, Abhass, Raj, Himanshu, Srivastava, Yogesh K.
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.11.2019; Springer Nature B.V
• Subjects: Astronomy; Astrophysics and Cosmology; Black holes; Classical and Quantum Gravitation; Differential Geometry; Gravity; Harmonic functions; Mathematical and Computational Physics; Physics; Physics and Astronomy; Polynomials; Quantum Physics; Relativity Theory; Research Article; Smoothness; Theoretical; Smoothness of horizons; Multi centre membranes; Multi centre black holes
• ISSN: 0001-7701; 1572-9532
• DOI: 10.1007/s10714-019-2634-y

We study the differentiability of the metric and other fields at any of the horizons of multi center Reissner–Nordstrom black hole solutions in d ≥ 5 and of multi center M 2 brane solutions. The centers are distributed in a plane in transverse space, hence termed coplanar. We construct the Gaussian null co-ordinate system for the neighborhood of a horizon by solving the geodesic equations in expansions of (appropriate powers of) the affine parameter. Organizing the harmonic functions that appear in the solution in terms of what can be called generalized Gegenbauer polynomials is key to obtaining the solution to the geodesic equations in a compact and manageable form. We then compute the metric and other fields in the Gaussian null co-ordinate system and find that the differentiability of the coplanar solution is identical to the differentiability of the collinear solution (centers distributed on a line in transverse space). We end the paper with a conjecture on the degree of smoothness of the most general multi center solution, the one with centers distributed arbitrarily.