Separability of Maxwell equation in rotating black hole spacetime and its geometric aspects

TL;DR

This paper explores the geometric origins of symmetry operators that enable the separation of Maxwell's equations in rotating black hole spacetimes, advancing understanding of their mathematical structure.

Contribution

It introduces a method to derive symmetry operators from the geometric properties linked to hidden symmetries of the Kerr-NUT-(A)dS spacetime.

Findings

Symmetry operators can be constructed from geometric quantities.

The formalism simplifies Maxwell's equations in complex spacetimes.

The approach enhances understanding of separability in black hole backgrounds.

Abstract

Recently a new formalism for perturbations of Maxwell's equations on the background of the Kerr-NUT-(A)dS spacetime was proposed, with which the equations are reduced to a equation of motion of a scalar field that can be solved by separation of variables. In this formalism the differential operators that commute with the operators of the equations of motion, called symmetry operators, played a key role to establish the separable structure. In this work we propose a method to reproduce these commuting symmetry operators in terms of the geometric quantities associated to the hidden symmetry of the background spacetime.

Equations

$$g_{ab}d{x}^{a}d{x}^b=\sum _{\mu =1}^n\frac{U_{\mu }}{X_{\mu }}d{x}_{\mu }^2+\sum _{\mu =1}^n\frac{X_{\mu }}{U_{\mu }}{\left(\sum _{k=0}^{n-1}{A}_{\mu }^{(k)}d{\psi }_k\right)}^2,$$

$${\nabla }_c{h}_{ab}=g_{ca}{\xi }_b-g_{cb}{\xi }_a,{\xi }_a=\frac{1}{2n-1}{\nabla }^b{h}_{ba},$$

$${\mathsf{A}}^a=B^{ab}{\nabla }_{b}Z,B^{ac}\left(g_{cb}-\beta h_{cb}\right)={\delta }_b^a,$$

$$\square Z+2\beta {\xi }_c{B}^{cd}{\nabla }_{d}Z=0,$$

$$d{s}^2={\tilde{g}}_{AB}d{x}^{A}d{x}^B=g_{ab}d{x}^{a}d{x}^b+2q{A}_{a}d{x}^{a}du+2dudv-2Vd{u}^2,$$

$$H=\frac{1}{2}{g}^{ab}\left(p_a-q{A}_a\right)\left(p_b-q{A}_b\right)+V$$

$$d{\tilde{s}}^2={\tilde{g}}_{AB}d{x}^{A}d{x}^B=g_{ab}d{x}^{a}d{x}^b+2q{A}_{a}d{x}^{a}du+2dudv-2Vd{u}^2,$$

$$\tilde{H}=\frac{1}{2}{\tilde{g}}^{AB}{p}_A{p}_B=\frac{1}{2}{g}^{ab}(p_a-q{A}_a{p}_v)(p_b-q{A}_b{p}_v)+V{p}_v^2+p_u{p}_v,$$

$${\mathcal{H}}\psi\equiv\Big{[}g^{ab}(\nabla_{a}-iqA_{a})(\nabla_{b}-iqA_{b})-2V\Big{]}\psi=-m^{2}\psi\,,$$

$$\tilde{\square }\Phi =0,$$

$${\tilde{\square }}_{\tilde{g}}={\square }_g-2q{A}^a\frac{\partial }{\partial x^a}\frac{\partial }{\partial v}-q({\nabla }_a{A}^a)\frac{\partial }{\partial v}+(q^2{A}_a{A}^a+2V){\left(\frac{\partial }{\partial v}\right)}^2+2\frac{\partial }{\partial u}\frac{\partial }{\partial v},$$

$$\tilde{\square }\Phi =e^{-i\frac{m^2}{2}u+iv}\left(\mathcal{H}+m^2\right)\psi ,$$

$$\tilde{\mathcal{K}}={\tilde{\nabla }}_A{\tilde{K}}^{AB}{\tilde{\nabla }}_B,$$

$$[\tilde{\square },\tilde{\mathcal{K}}]=0,$$

$${\tilde{\nabla }}^A\left({\tilde{K}}_{[A}{}^C{\tilde{R}}_{B]C}\right)=0,$$

$$\tilde{\mathcal{K}}\Phi =e^{iv}\mathcal{K}\psi .$$

$$({\tilde{K}}^{AB})=\left(\begin{array}{ccc}{\tilde{K}}^{ab}& {\tilde{K}}^{au}& {\tilde{K}}^{av}\\ {\tilde{K}}^{ub}& {\tilde{K}}^{uu}& {\tilde{K}}^{uv}\\ {\tilde{K}}^{vb}& {\tilde{K}}^{vu}& {\tilde{K}}^{vv}\end{array}\right)$$

$$=\left(\begin{array}{ccc}{K}^{ab}& L^a& -q{K}^{ac}{A}_c+M^a\\ L^b& W& -q{L}^c{A}_c+T+2WV\\ -q{K}^{bc}{A}_c+M^b& -q{L}^c{A}_c+T+2WV& q^2{K}^{cd}{A}_c{A}_d-2q{M}^c{A}_c+F\end{array}\right).$$

$$\mathcal{K}\psi$$

$$d{s}^2=-\frac{\Delta }{\Sigma }{\left(dt-a{\sin }^2\theta d\varphi \right)}^2+\frac{\Sigma d{r}^2}{\Delta }+\Sigma d{\theta }^2+\frac{{\sin }^2\theta }{\Sigma }{\left(adt-(r^2+a^2)d\varphi \right)}^2,$$

$$d{s}^2=-\frac{\mathcal{Q}(r)}{r^2+p^2}(d\tau -p^{2}d\sigma {)}^2+\frac{r^2+p^2}{\mathcal{Q}(r)}d{r}^2+\frac{r^2+p^2}{\mathcal{P}(p)}d{p}^2+\frac{\mathcal{P}(p)}{r^2+p^2}(d\tau +r^{2}d\sigma {)}^2,$$

$$g^{rr}=\frac{\mathcal{Q}}{r^2+p^2},g^{pp}=\frac{\mathcal{P}}{r^2+p^2},g^{\tau \tau }=-g^{rr}\frac{r^4}{{\mathcal{Q}}^2}+g^{pp}\frac{p^4}{{\mathcal{P}}^2},$$

$$g^{\tau \sigma }=g^{rr}\frac{r^2}{{\mathcal{Q}}^2}+g^{pp}\frac{p^2}{{\mathcal{P}}^2},g^{\sigma \sigma }=-g^{rr}\frac{1}{{\mathcal{Q}}^2}+g^{pp}\frac{1}{{\mathcal{P}}^2},$$

$$K^{rr}=\frac{p^2\mathcal{Q}}{r^2+p^2},K^{pp}=-\frac{r^2\mathcal{P}}{r^2+p^2}.$$

$${\mathcal{H}}^{(L)}\psi\equiv\Big{[}g^{ab}(\nabla_{a}-iqA^{(L)}_{a})(\nabla_{b}-iqA^{(L)}_{b})-2V^{(L)}\Big{]}\psi=0\,,$$

$$q{A}_a^{(L)}d{x}^a=$$

$$V^{(L)}=$$

$${\Xi }_r\mathcal{Q}\frac{d}{dr}\left(\frac{\mathcal{Q}}{{\Xi }_r}\frac{dR}{dr}\right)+\left[(\omega r^2+m{)}^2-\frac{2i\mathcal{Q}}{\beta {\Xi }_r}(\omega +{\beta }^{2}m)-{\kappa }^{(L)}\mathcal{Q}\right]R$$

$${\Xi }_p\mathcal{P}\frac{d}{dp}\left(\frac{\mathcal{P}}{{\Xi }_p}\frac{d\Theta }{dp}\right)+\left[-(\omega p^2-m{)}^2+\frac{2i\mathcal{P}}{\beta {\Xi }_p}(\omega +{\beta }^{2}m)+{\kappa }^{(L)}\mathcal{P}\right]\Theta$$

$${\tilde{g}}^{ab}=g^{ab},{\tilde{g}}^{rv}=-g^{rr}\frac{i{\beta }^{2}r}{{\Xi }_r},{\tilde{g}}^{pv}=g^{pp}\frac{i{\beta }^{2}p}{{\Xi }_p},$$

$${\tilde{g}}^{\tau v}=-g^{rr}\frac{i}{2\beta }\frac{1+{\beta }^2{r}^2}{{\Xi }_r\mathcal{Q}}+g^{pp}\frac{i}{2\beta }\frac{1-{\beta }^2{p}^2}{{\Xi }_p\mathcal{P}},{\tilde{g}}^{\sigma v}=g^{rr}\frac{i\beta }{2}\frac{1+{\beta }^2{r}^2}{{\Xi }_r\mathcal{Q}}-g^{pp}\frac{i\beta }{2}\frac{1-{\beta }^2{p}^2}{{\Xi }_p\mathcal{P}},$$

$${\tilde{g}}^{uv}=1=g^{rr}\frac{r^2}{\mathcal{Q}}+g^{pp}\frac{p^2}{\mathcal{P}},{\tilde{g}}^{vv}=\frac{g^{rr}}{\mathcal{Q}}\frac{d}{dr}\left(\frac{{\beta }^{2}r}{{\Xi }_r}\mathcal{Q}\right)-\frac{g^{pp}}{\mathcal{P}}\frac{d}{dp}\left(\frac{{\beta }^{2}p}{{\Xi }_p}\mathcal{P}\right),$$

$${\mathcal{K}}^{(L)}\psi =$$

Full text

1. Introduction

Black holes are basic objects in the gravitational theory, and their various aspects have been studied intensively. Particularly perturbative dynamics of fields in black hole spacetime has attracted much interest since it governs astrophysical phenomena around black holes and also shows intriguing mathematical properties. In this context, the separability of the perturbation equations plays a key role since it reduces the perturbation equations given as partial differential equations to to a set of ordinary differential equations that may be solved analytically.

In this work we focus on spacetimes with rotating black holes [1, 2], in which the perturbation equations become complicated due to the rotation. For the Kerr black holes in four dimensions, Teukolsky made a significant progress in his seminal work, which showed that the equations of motion for the gravitational and Maxwell field can be separated if they are expressed in the so-called Newman-Penrose formalism [3, 4]. However, this formalism works only in the four dimensions and generalization to higher dimensions remained unaccomplished for a long time.

Recently a breakthrough about this problem was made by Lunin [5] by introducing new variables to express the vector potential, which is the dynamical variables of the Maxwell field. In terms of these new variables, the perturbation equations can be separated for all the propagating modes, and also this formalism can be generalized to the higher-dimensional case straightforwardly. In this work symmetries of the rotating black hole spacetime played an important role. It is known that the Kerr spacetime and its higher-dimensional generalization have hidden symmetries signified by the existence of nontrivial symmetric tensor satisfying $\nabla_{(a}K_{bc)}=0$. This tensor is called the Killing tensor, and it yields a constant of motion $K^{ab}p_{a}p_{b}$, where $p^{a}$ is the momentum vector, that is preserved on a geodesic. In Lunin’s work, this tensor was used to introduce special coordinates that helped separating the variables of the perturbation equations.

Krtouš, Frolov and Kubizňák made a step forward about clarifying the significance of the hidden symmetries [6] in this problem. They focused on the off-shell Kerr-NUT-AdS metric in $2n$ dimensions

[TABLE]

where $U_{\mu}$, $X_{\mu}$ and $A^{(k)}_{\mu}$ are certain functions of $x^{\mu}$. This metric is the most general one admitting the principal tensor $h_{ab}$ which is an anti-symmetric generalization of the Killing tensor defined by

[TABLE]

where $\nabla_{a}$ is the covariant derivative (Levi-Civita connection) of $g_{ab}$. Then they showed that, by expressing the vector potential expressed with a scalar $Z$ and a constant $\beta$ as

[TABLE]

Maxwell’s equation in the Lorenz gauge $\nabla_{a}{\mathsf{A}}^{a}=\nabla_{a}\left(B^{ab}\nabla_{b}Z\right)=0$ can be reduced to a single scalar equation given by

[TABLE]

where $\square\equiv g^{ab}\nabla_{a}\nabla_{b}$ is the d’Alembertian operator. In our work, we call it the Lunin-Krtouš-Frolov-Kubizňák (LKFK) equation. They also showed that these equations can be expressed using differential operators whose commutators vanish, which makes it possible to solve the equations by separation of variables. This result suggests that the separability of the perturbation equation and the hidden symmetry of the spacetime are intimately related with each other, although their precise correspondence still remains unclear.

In our work, we shed new light on this problem based on studies about the hidden symmetry of the spacetime and the separability. One technique useful here is the Eisenhart-Duval lift [7, 8], in which the original spacetime is supplemented with additional dimensions as

[TABLE]

where $g_{ab}$ is the original spacetime metric and $u,v$ are the coordinates for the additional spacetime dimensions. By suitably choosing the metric functions $A^{a},V$, the derivative operator $\square+2\beta\xi_{c}B^{cd}\nabla_{d}$ in (4) can be expressed simply as the d’Alembertian $\tilde{\square}=\tilde{g}^{AB}\tilde{\nabla}_{A}\tilde{\nabla}_{B}$ in terms of the covariant derivative $\tilde{\nabla}_{A}$ associated with $\tilde{g}_{AB}$. Hence, with this technique we can reduce the perturbation equation (4) to a massless Klein-Gordon equation by absorbing the additional term $2\beta\xi_{c}B^{cd}\nabla_{d}$ into the structure of the higher-dimensional spacetime.

It turns out that Eq. (5) falls into the standard form of the metric whose geodesics is completely integrable [9], hence it is associated with the Killing tensors $\tilde{K}_{AB}^{(i)}$ corresponding to the constants of motion on those geodesics. Using these Killing tensors we can construct differential operators $\tilde{\mathcal{K}}^{(i)}\equiv\tilde{\nabla}_{A}\tilde{K}_{(i)}^{AB}\tilde{\nabla}_{B}$, and it can be shown that these operators commute with the d’Alembertian \bigl{[}\tilde{\square},\tilde{\mathcal{K}}^{(i)}\bigr{]}=0 and also that they commutes with each other \bigl{[}\tilde{\mathcal{K}}^{(i)},\tilde{\mathcal{K}}^{(j)}\bigr{]}=0. It implies that $\tilde{\mathcal{K}}^{(i)}$ act as commuting symmetry operators that maps a solution of (4) to another one, which plays a key role in separation of variables. It can be further confirmed that the differential operators $\tilde{\mathcal{K}}^{(i)}$ coincide with the commuting derivative operators found in [6], where only the explicit form of the operators in terms of the coordinates were given. Hence we found a way to give a geometric interpretation to the commuting symmetry operators found by [6].

2. Construction of commuting symmetry operators

2.1. Eisenhart-Duval lift of a classical mechanical system

In the Hamiltonian formulation, the motion of a relativistic particle with mass $m$ and charge $q$ in the presence of vector and scalar potentials $A_{a}$ and $V$ on a $D$-dimensional spacetime $(M,g_{ab})$, on which the line element is given by $ds^{2}=g_{ab}dx^{a}dx^{b}$, is governed by the Hamiltonian

[TABLE]

with the constraint $H=-m^{2}$. For such a system, we may take the Eisenhart-Duval lift by considering the $(D+2)$-dimensional manifold $\tilde{M}=M\times R^{2}$ with the metric

[TABLE]

where ${\tilde{g}}_{ab}=g_{ab}(x^{a})\,,{\tilde{g}}_{au}=qA_{a}(x^{a})\,,{\tilde{g}}_{uv}=1\,,{\tilde{g}}_{uu}=-2V(x^{a})$. The geodesic Hamiltonian on the uplifted spacetime $(\tilde{M},{\tilde{g}}_{AB})$ is written as

[TABLE]

where ${\tilde{g}}^{ab}=g^{ab}\,,{\tilde{g}}^{av}=-qg^{ab}A_{b}\,,{\tilde{g}}^{uv}=1\,,{\tilde{g}}^{vv}=q^{2}A_{a}A^{a}+2V$. Since $g_{ab}$, $A_{a}$ and $V$ are functions of $x^{a}$ only, $\partial/\partial u$ and $\partial/\partial v$ are Killing vector fields; hence, the corresponding momenta $p_{u}$ and $p_{v}$ are constants. Putting $\tilde{H}=0$, $p_{u}=m^{2}$ and $p_{v}=1$, we obtain the equations of motion for the Hamiltonian (6) satisfying $H=-m^{2}$ from (8). Thus null geodesics on the uplifted spacetime ($\tilde{M},{\tilde{g}}_{AB}$) project onto the solutions of the original system in $M$.

2.2. Application to quantum mechanical systems

In quantum mechanics, via the Dirac quantization $p_{a}\to-i\nabla_{a}$, the equation of motion corresponding to the system of the Hamiltonian (6) is obtained as

[TABLE]

where $\nabla_{a}$ is the Levi-Civita connection on $(M,g_{ab})$. A remarkable fact is that solutions to the equation of motion (9) are reproduced from a particular class of the solutions to the massless Klein-Gordon equation on the uplifted spacetime $(\tilde{M},{\tilde{g}}_{AB})$,

[TABLE]

where $\tilde{\Box}\equiv{\tilde{g}}^{AB}\tilde{\nabla}_{A}\tilde{\nabla}_{B}$ and $\tilde{\nabla}_{A}$ are the d’Alembertian and Levi-Civita connection on $(\tilde{M},{\tilde{g}}_{AB})$. Actually $\tilde{\Box}$ is calculated from (8) as

[TABLE]

where $\Box\equiv g^{ab}\nabla_{a}\nabla_{b}$ and $\nabla_{a}$ are the d’Alembertian and Levi-Civita connection on $(M,g_{ab})$. Then restricting the solutions to the massless Klein-Gordon equation (10) to the specific ones of the form $\Phi=e^{-i\frac{m^{2}}{2}u+iv}\psi({x^{\rho}})$, we have

[TABLE]

where ${\mathcal{H}}$ is the one appeared in (9). For simplicity we set $m=0$ henceforth. Thus, instead of solving the equation of motion (9), we may solve the massless Klein-Gordon equation (10) on the uplifted spacetime $(\tilde{M},G_{AB})$. This fact is the key in constructing symmetry operators of the equation of motion (9), as shown in the subsequent sections.

2.3. Separability and commuting symmetry operators

Generally, the separability of the differential equation ${\mathcal{O}}\psi=0$ is related to the existence of first- and/or second-order differential operators ${\mathcal{K}}^{(i)}$ satisfying the commutation relations $[{\mathcal{O}},{\mathcal{K}}^{(i)}]=0\,,[{\mathcal{K}}^{(i)},{\mathcal{K}}^{(j)}]=0\,,$ where the square bracket $[\,\,,\,]$ denotes the commutator $[A,B]\equiv AB-BA$. The operators satisfying the first condition are called symmetry operators since they map a solution of ${\mathcal{O}}\psi=0$ to another solution, and the second condition implies that they commute with each other as differential operators. If we are able to solve the differential equation ${\mathcal{O}}\psi=0$ by separation of variables, we have separation constants $\kappa$ and they are related to second-order symmetry operators by ${\mathcal{K}}^{(i)}\psi=\kappa^{(i)}\psi\,.$

2.4. Construction of symmetry operators

In the Eisenhart-Duval lift, symmetry operators of the equation of motion (9) can be constructed from symmetry operators of the massless Klein-Gordon equation (10). To show it, we begin with the second-order differential operator $\tilde{{\mathcal{K}}}$ on $\tilde{M}$,

[TABLE]

where $\tilde{K}_{AB}$ is a rank-2 symmetric tensor on $\tilde{M}$. For $\tilde{{\mathcal{K}}}$ to be a symmetry operator of the massless Klein-Gordon equation (10), i.e.,

[TABLE]

$\tilde{K}_{AB}$ must be a Killing tensor on $(\tilde{M},{\tilde{g}}_{AB})$, which obeys the Killing equation $\tilde{\nabla}_{(A}\tilde{K}_{BC)}=0\,,$ and must satisfy, in addition, the anomaly-free condition

[TABLE]

where $\tilde{R}_{AB}$ is the Ricci tensor of $(\tilde{M},{\tilde{g}}_{AB})$ [10].

Now, we suppose that the Killing tensor $\tilde{K}^{AB}$ are independent of the coordinates $u$ and $v$, $\partial_{u}\tilde{K}^{AB}=0\,,\partial_{v}\tilde{K}^{AB}=0\,.$ Then, applying the symmetry operator $\tilde{{\mathcal{K}}}$ on $\tilde{M}$ to the wave function of the form $\Phi=e^{-i\frac{m^{2}}{2}u+iv}\psi({x^{\rho}})$, we obtain the differential operator ${\mathcal{K}}$ on $M$ by

[TABLE]

Without loss of generality, we may express $\tilde{K}^{AB}$ as

[TABLE]

Here, using the fact that $\tilde{K}^{AB}$ is a Killing tensor on $\tilde{M}$, we can show that $K^{ab}$ and $L^{a}$ are Killing tensor and Killing vector on $M$, and also that $W$ is a constant and $\nabla_{a}M^{a}=0$ [11]. Using $\nabla_{a}M^{a}=0$ among them, we find

[TABLE]

where $\nabla^{(A)}_{a}\equiv\nabla_{a}-iqA_{a}$. Since $[\tilde{\Box},\tilde{K}]\Phi=e^{iv}[{\mathcal{H}},{\mathcal{K}}]\psi\,,$ we obtain $[{\mathcal{H}},{\mathcal{K}}]=0\,,$ which means that ${\mathcal{K}}$ is a symmetry operator ${\mathcal{K}}$ of the equation of motion (9).

For commutativity of two symmetry operators \bigl{[}\tilde{\mathcal{K}}^{(i)},\tilde{\mathcal{K}}^{(j)}\bigr{]}=0 additional conditions similar to (15) must be satisfied, which must be examined separately [12].

To summarize, the procedure for obtaining commuting symmetry operators ${\mathcal{K}}^{(i)}$ for the equation of motion (9) is given as follows:

1. (1)

Read out $g_{ab}$, $A_{a}$ and $V$ from the equation of motion (9). 2. (2)

Using these quantities, construct the uplifted metric ${\tilde{g}}_{AB}$ (7). 3. (3)

Find Killing tensors $\tilde{K}^{(i)}_{AB}$ on ($\tilde{M},{\tilde{g}}_{AB}$) such that the components are independent of $u$ and $v$. 4. (4)

Check if $\tilde{K}^{(i)}_{AB}$ satisfy the anomaly-free condition (15), and also the commutativity conditions. 5. (5)

If they hold, construct the commuting symmetry operators $\tilde{{\mathcal{K}}}^{(i)}$ by (13). Then it follows that $[\tilde{\Box},\tilde{{\mathcal{K}}}^{(i)}]=0=[\tilde{{\mathcal{K}}}^{(i)},\tilde{{\mathcal{K}}}^{(j)}]$. 6. (6)

Finally, obtain commuting symmetry operators ${\mathcal{K}}^{(i)}$ from $\tilde{{\mathcal{K}}}^{(i)}$ via (16) and (17). Then $[{\mathcal{H}},{\mathcal{K}}^{(i)}]=0=[{\mathcal{K}}^{(i)},{\mathcal{K}}^{(j)}]$.

3. Separability and symmetry operators of Maxwell’s equations

on the Kerr-NUT-(A)dS spacetime in four dimensions

To illustrate the procedure to construct the symmetry operators summarized in section 2.4, we apply it to the LKFK equation in four-dimensional case. The Teukolsky equation in four dimensions and the LKFK equations can be treated in a similar manner.

3.1. Carter form of the Kerr-NUT-(A)dS metric

The Kerr metric in the Boyer-Lindquist coordinates $(t,r,\theta,\phi)$ is

[TABLE]

where $\Delta=r^{2}-2mr+a^{2}\,,\Sigma=r^{2}+a^{2}\cos^{2}\theta\,.$ Performing the coordinate transformation $p^{2}=a^{2}\cos^{2}\theta\,,\tau=t-a\phi\,,\sigma=-\frac{\phi}{a}\,,$ we obtain the Kerr metric in the Carter form

[TABLE]

where ${\mathcal{Q}}(r)=r^{2}-2mr+a^{2}\,,{\mathcal{P}}(p)=a^{2}-p^{2}$. The metric (19) solves the vacuum Einstein equation with cosmological constant $R_{ab}=\lambda g_{ab}$ if and only if ${\mathcal{Q}}(r)$ and ${\mathcal{P}}(p)$ are given by ${\mathcal{Q}}(r)=-\frac{\lambda}{3}r^{4}+\epsilon r^{2}-2mr+a^{2}\,,{\mathcal{P}}(p)=-\frac{\lambda}{3}p^{4}-\epsilon p^{2}+2lp+a^{2}$ with constants $\epsilon$, $a$, $m$ and $l$. This solution is called the Kerr-NUT-(A)dS metric. The metric (19) with ${\mathcal{Q}}(r)$ and ${\mathcal{P}}(p)$ replaced by arbitrary functions depending only on $r$ and $p$ is called the off-shell metric. In the following calculation, we will work with the off-shell metric without using imposing the Einstein equation.

It is important to mention that the off-shell metric in the Carter form fits into Benenti’s canonical form for metrics admitting the separability of the Hamilton-Jacobi equation for geodesics [9]. Actually, the components of the inverse metric in the coordinate basis are written in the canonical form

[TABLE]

and we can easily construct a Killing tensor on the Kerr spacetime. Following Benenti’s construction, the Killing tensor turns out to be given by (20) with $g^{ab},g^{rr},g^{pp}$ replaced to $K^{ab},K^{rr},K^{pp}$, where

[TABLE]

3.2. Symmetry operators of the

LKFK equations

In the LKFK formulation in four dimensions, Maxwell’s equation (4) can be written in the form

[TABLE]

where $g_{ab}$ is the off-shell metric (19), and $A^{(L)}$ and $V^{(L)}$ are given by

[TABLE]

where $\Xi_{r}=1-\beta^{2}r^{2}\,,\Xi_{p}=1+\beta^{2}p^{2}$. This can be solved by separation of variables by using the ansatz $\psi=e^{-i\omega\tau}e^{im\sigma}R(r)\Theta(p)$ and we obtain

[TABLE]

where $\kappa^{(L)}$ is a separation constant.

The uplifted metric ${\tilde{g}}_{AB}$ on $\tilde{M}$ is obtained from (7) with (19) and (23). As explained before, the uplifted metric ${\tilde{g}}_{AB}$ fits into Benenti’s canonical form

[TABLE]

and the Killing tensor is given by the expression same as the metric (25) with $\tilde{g}^{AB},g^{ab},g^{rr},g^{pp}$ replaced to $\tilde{K}^{AB},K^{ab},K^{rr},K^{pp}$, respectively. Then we obtain the symmetry operator of the LKFK equations according to (17) as

[TABLE]

Using the multiplicative separation ansatz $\psi=e^{-i\omega\tau}e^{im\sigma}R(r)\Theta(p)$ and the separated equations (24), we find

[TABLE]

It can be confirmed by direct calculations that the anomaly-free condition (15) is satisfied by the Killing tensor constructed above [11], hence we have the commutation relations

[TABLE]

4. Outlook

An obvious next step for this work is to inspect the uplifted metrics corresponding to different perturbations. In four dimensions Maxwell’s equations can be reduced to the Teukolsky equation and also to the LKFK equations, and these two equations lead to different uplifted metrics. Since these two equations should describe the same time evolution of the Maxwell fields, there would be intertwining operators connecting the solutions of these solutions. If they exist, it is expected that the uplifted metrics corresponding to the different equations of motion are related to each other by some transformations. It would be fruitful to investigate such possible connection between the uplifted spacetimes by clarifying their geometric properties such as the curvature invariants and Weyl tensors.

Also, for both equations the uplifted metric is intimately related to the symmetry of the base spacetime. For example, the vector field for the LKFK equation is given by $A_{a}\sim\xi^{b}B_{ba}$, where $\xi^{b}$ is a Killing vector and the polarization tensor $B_{ab}$ is constructed from the metric and the principal tensor corresponding to the hidden symmetry of the base spacetime. Geometric properties of this kind could provide useful information of properties of the uplifted metrics and also the vector potential ansatz that reduced the Maxwell equation to a scalar field equation. Pursuing this idea would be another future tasks for this study.