Electromagnetic fields on Kerr spacetime, Hertz potentials, and Lorenz gauge

Abstract

We review two procedures for constructing the vector potential of the electromagnetic field on Kerr spacetime, namely, the classic method of Cohen & Kegeles, yielding A µ in a radiation gauge, and the newer method of Frolov et al., yielding A µ in Lorenz gauge. We demonstrate that the vector potentials are related by straightforward gauge transformations, which we give in closed form. We obtain a new result for a separable Hertz potential H µν such that A µ lor = ∇ ν H µν . * s.dolan@sheffield.ac.ukRecent results from gravitational wave detectors and the Event Horizon Telescope support the hypothesis that the universe is replete with rotating (Kerr) black holes, across a range of mass scales (∼ 10-10 9 M ⊙ ). These experimental breakthroughs are underpinned by a solid theoretical understanding of how fields propagate on rotating black hole spacetimes, developed over several decades.This paper returns once more to the venerable topic of massless (test) fields on rotating black hole spacetimes. This area of enquiry blossomed in the 1970s, after Teukolsky [1][2][3][4] showed that certain components of the electromagnetic and gravitational fields on Kerr spacetime satisfy decoupled scalar equations that admit a full separation of variables. Shortly thereafter, Cohen & Kegeles [5,6], Chrzanowski [7], Chandrasekhar [8,9], Wald [10], Stewart [11], and others [12,13] developed methods for reconstructing, from scalar potentials, both the vector potential A µ of the electromagnetic field, and the metric perturbation h µν of the gravitational field. Several applications rely on these methods, from the scattering of gravitational waves [14] to gravitational self-force calculations [15][16][17].The classic Hertz potential method [5][6][7] of the 1970s generates fields in a radiation gauge: l µ A µ = l µ h µν = 0, where l µ is a principal null direction of the spacetime. However, for certain applications, it is preferable to work with a field in Lorenz gauge 1 : ∇ µ A µ = 0 and ∇ µ h µν = 0, 1 The gauge condition takes the name of L. V. Lorenz (1829-1891) rather than H. A. Lorentz (1853A. Lorentz ( -1928.2 Similarly, the statement δF = 0 implies that F = δC for some three-form C.

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

“…They fit into Benenti's canonical form (3.8) when the Stäckel matrix φ and its inverseφ are given by 11) and ζ-matrices ζ (µ) are given by…”

Principal tensor strikes again: Separability of vector equations with torsion

“…where of the four free parameters A, B, C, c only three are physical (one can be scaled away) and are related to the mass and two rotations. As per usual, the separability property shown below does not need the special form (21) and occurs "off-shell", for arbitrary functions…”

“…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].…”

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“…In the shadowless gauge, the softened string singularity is confined to the middle region r min < r < r max . We can understand its form by analyzing the form of T R ab together with the equations ( 56), (57), and (58) for x R ab and the inversion relation…”