The entropy function for the extremal Kerr-(anti-)de Sitter black holes Original Research Article

Based on Senʹs entropy function formalism, we consider the Bekenstein–Hawking entropy of the extremal Kerr-(anti-)de Sitter black holes in 4-dimensions. Unlike the extremal Kerr black hole case with flat asymptotic geometry, where the Bekenstein–Hawking entropy S is proportional to the angular momentum J, we get a quartic algebraic relation between S and J by using the known solution to the Einstein equation. We recover the same relation in the entropy function formalism. Instead of full geometry, we write down an ansatz for the near horizon geometry only. The exact form of the unknown functions and parameters in the ansatz are obtained by solving the differential equations which extremize the entropy function. The results agree with the nontrivial relation between S and J.