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

  • Chethan N. GowdigereEmail author
  • Abhass Kumar
  • Himanshu Raj
  • Yogesh K. Srivastava
Research Article


We study the differentiability of the metric and other fields at any of the horizons of multi center Reissner–Nordstrom black hole solutions in \(d \ge 5\) and of multi center M2 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.


Smoothness of horizons Multi centre black holes Multi centre membranes 



Chethan N. Gowdigere and Yogesh K. Srivastava would like to thank Siddharth Satpathy for initial collaborations. Chethan N. Gowdigere would like to thank the very friendly staff at the various Cafe Coffee Day outlets in Bhubaneshwar, where quite a bit of his contribution to this work was done, for their warm hospitality.


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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  • Chethan N. Gowdigere
    • 1
    • 2
    Email author
  • Abhass Kumar
    • 1
    • 2
  • Himanshu Raj
    • 1
    • 2
  • Yogesh K. Srivastava
    • 1
    • 2
  1. 1.National Institute of Science Education and Research BhubaneshwarKhurdhaIndia
  2. 2.Homi Bhabha National InstituteMumbaiIndia

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