Symmetry operators and separability of massive Klein-Gordon and Dirac equations in the general 5-dimensional Kerr-(anti-)de Sitter black hole background

It is shown that the Dirac equation is separable by variables in a five-dimensional rotating Kerr-(anti-)de Sitter black hole with two independent angular momenta. A first order symmetry operator that commutes with the Dirac operator is constructed in terms of a rank-three Killing-Yano tensor whose square is a second order symmetric Stackel-Killing tensor admitted by the five-dimensional Kerr-(anti-)de Sitter spacetime. We highlight the construction procedure of such a symmetry operator. In addition, the first law of black hole thermodynamics has been extended to the case that the cosmological constant can be viewed as a thermodynamical variable.

💡 Research Summary

The paper investigates the separability of both the massive Klein‑Gordon and Dirac equations in the background of a five‑dimensional rotating Kerr‑(anti‑)de Sitter black hole that possesses two independent angular momenta. The authors begin by presenting the metric in a Boyer‑Lindquist‑type coordinate system, explicitly displaying the dependence on the rotation parameters (a_{1}) and (a_{2}) as well as the cosmological constant (\Lambda). They then turn to the scalar field equation (\Box\Phi - \mu^{2}\Phi = 0). By constructing a second‑order symmetric Stäckel‑Killing tensor (K^{\mu\nu}) that satisfies (\nabla_{(\alpha}K_{\beta\gamma)}=0), they demonstrate that the Klein‑Gordon equation admits a complete separation of variables. The separated ansatz (\Phi = e^{-i\omega t}e^{im_{1}\phi_{1}}e^{im_{2}\phi_{2}}R(r)\Theta(\theta)) leads to ordinary differential equations for the radial and angular parts, each containing a Carter‑type constant derived from the hidden symmetry encoded in (K^{\mu\nu}).

The central technical achievement concerns the Dirac equation (i\gamma^{\mu}\nabla_{\mu}\Psi - m\Psi = 0). The authors identify a rank‑three Killing‑Yano tensor (f_{\mu\nu\rho}) that obeys (\nabla_{(\mu}f_{\nu)\rho\sigma}=0). This antisymmetric tensor is non‑trivial in five dimensions and its square reproduces the previously found Stäckel‑Killing tensor: (K_{\mu\nu}=f_{\mu\alpha\beta}f_{\nu}{}^{\alpha\beta}). Using (f_{\mu\nu\rho}) they construct a first‑order symmetry operator \