Separation of variables in Maxwell equations in Plebański-Demiański spacetime

Abstract

A new method for separating variables in the Maxwell equations in four-and higher-dimensional Kerr-(A)dS spacetimes proposed recently by Lunin is generalized to any off-shell metric that admits a principal Killing-Yano tensor. The key observation is that Lunin's ansatz for the vector potential can be formulated in a covariant form-in terms of the principal tensor. In particular, focusing on the fourdimensional case we demonstrate separability of Maxwell's equations in the Kerr-NUT-(A)dS and the Plebański-Demiański family of spacetimes. The new method of separation of variables is quite different from the standard approach based on the Newman-Penrose formalism.

On symmetry operators for the Maxwell equation on the Kerr-NUT-(A)dS spacetime

“…where its components are given bỹ This transformation does not affect the other metric components, hence the metric after this transformation manifestly fits into Benenti's canonical form, and then the separability of geodesic equations for the metric is guaranteed. 6 In terms of the Stäckel matrix for metric (4.1), Eq. (4.25) is given bỹ…”

“…where ω ab are the connection 1-forms on (M, g) and F ab are the components of the field strength F = dA on (M, g). Using the formula∇ẽ Aẽ B = −ω B C (ẽ A )ẽ C , we havẽ 6) and the others are zero. The curvature 2-formsR AB =dω AB +ω A C ∧ω CB on (M ,g) are then given bỹ…”

Massive vector fields in Kerr-Newman and Kerr-Sen black hole spacetimes

“…As argued in Sec. In order to separate these equations, we employ the LFKK ansatz [28][29][30][31], P a = B ab ∇ b Z , B ab (g bc + iµh bc ) = δ a c , (A. 14) where µ is a complex parameter, h bc in the generalized principal tensor (A.8), and the potential function Z is written in the multiplicative separated form Z = R 1 (x 1 )R 2 (x 2 )e iL 0 ψ 0 e iL 1 ψ 1 .…”

“…As a consequence the problem was investigated either using approximations [9,24,25] or employing serious numerical analysis [10,26,27].However, a separability renaissance for vector fields has begun in the last couple of years due to a new ansatz by Lunin [28]. Simplified and written in covariant form by Frolov-Krtouš-Kubizňák [29,30], the new ansatz works for the massive vector field case [31] and can be applied in the Kerr-NUT-AdS spacetimes for all dimensions. Importantly the Lunin-Frolov-Krtouš-Kubizňák (LFKK) ansatz exploits the existence of hidden symmetries in these spacetimes which are encoded in the principal tensor [18] and allows the Proca equations to be decoupled and separated into ordinary differential equations [31].…”

Principal tensor strikes again: Separability of vector equations with torsion

“…Lunin's approach was novel in that it provided a separable ansatz for the vector potential rather than the field strength, a method that had previously seen success in D = 4 dimensions [7,8]. In 2018, Frolov, Krtous, and Kubiznak showed that Lunin's ansatz can be written in a covariant form, in terms of the principal tensor [15,16], allowing them * rcayuso@perimeterinstitute.ca † fgray@perimeterinstitute.ca ‡ dkubiznak@perimeterinstitute.ca § amargalit@perimeterinstitute.ca ¶ rsouza@perimeterinstitute.ca * * lthiele@perimeterinstitute.ca to extend Lunin's result to general (possibly off-shell) Kerr-NUT-AdS spacetimes [17]. The separation of massive vector (Proca) field perturbations in these spacetimes (an achievement previously absent even for the four-dimensional Kerr geometry) followed shortly after that [18], see also [19][20][21].…”