Separability, plane wave limits and rotating black holes

A detailed description of the construction of PCs on a spacetime associated to a null geodesic can be found in [11, 12]. Here for completeness we shall repeat some essential steps in the construction and make the computation a bit more explicit. To begin consider the null HJ equation

$$\begin{equation}{g}^{\mu \nu }{\partial }_{\mu }S{\partial }_{\nu }S=0,\end{equation} \tag{ 1 }$$

where S is the HJ function. This is a first order partial differential equation and so locally there is a unique solution provided that a boundary condition for S is specified on a spatial hyper-surface F(x^μ ) = 0 in a spacetime. In practice though such boundary value problems are hard to solve. Instead the aim is to find the 'complete' solution to the null HJ equation ³ . Such a solution is characterized by the presence of n integration constants α, where n is the dimension of the spacetime. Two of these constants are forgetful because if S is a solution, then pS + c will also be a solution for any constants p and c. Thus the complete solutions of the null HJ equation is specified by n − 2 independent integration constants.

Next observe that the solutions of

$$\begin{equation}{\dot {x}}^{\mu }={g}^{\mu \nu }{\partial }_{\nu }S\end{equation} \tag{ 2 }$$

are null geodesics, where ${\dot {x}}^{\mu }=\frac{\mathrm{d}{x}^{\mu }}{\mathrm{d}\lambda }$ and λ is the affine parameter. Indeed ${g}_{\mu \nu }{\dot {x}}^{\mu }{\dot {x}}^{\nu }=0$ as a consequence of (1) and

$$\begin{equation}{\nabla }_{\lambda }{\dot {x}}^{\mu }={g}^{\mu \nu }{\dot {x}}^{\rho }{\nabla }_{\rho }{\partial }_{\nu }S=\frac{1}{2}{g}^{\mu \nu }{\partial }_{\nu }\left({g}^{\rho \sigma }{\partial }_{\rho }S\enspace {\partial }_{\sigma }S\right)=0.\end{equation} \tag{ 3 }$$

The system of first order equations conditions (2) has a unique solution ${x}^{\mu }(\lambda ,{x}_{0}^{\nu },\alpha )$ for small λ subject to a boundary condition at λ = 0, ${x}^{\mu }(0,{x}_{0}^{\nu },\alpha )={x}_{0}^{\mu }$, where α are the integration constants of a complete solution to the HJ equation. Next consider the solutions of (2) for different boundary conditions ${x}_{0}^{\mu }$ and assume that they define a null geodesic congruence on the spacetime such that they intersect a hypersurface $\mathcal{H}$ only once. In such a case after a possible redefinition of the affine parameter of each geodesic, the boundary conditions for the solutions of (2) at λ = 0 can be taken to be the points of $\mathcal{H}$.

Furthermore observe that given a solution of (2), one has that

$$\begin{equation}S({x}^{\mu }(\lambda ,{x}_{0}^{\nu },\alpha ),\alpha )=S({x}_{0}^{\nu },\alpha ),\end{equation} \tag{ 4 }$$

i.e. S does not depend on the affine parameter λ. Indeed

$$\begin{equation}\dot {S}={\dot {x}}^{\mu }{\partial }_{\mu }S={g}^{\mu \nu }{\partial }_{\mu }S{\partial }_{\nu }S=0.\end{equation} \tag{ 5 }$$

Given a null geodesic congruence as described above, the definition of a PC system (λ, u, wⁱ ), i = 1, ..., n − 2, on a spacetime is as follows. First λ is the affine parameter of the null geodesic congruence. The coordinate u is identified with the HJ function as $u=S({x}_{0}^{\mu },\alpha )$. Assuming that the level sets of $S({x}_{0}^{\mu },\alpha )$ are transversal to the initial value hypersurface $\mathcal{H}$ given by $H({x}_{0}^{\mu })=0$, the coordinates wⁱ are identified with independent coordinates ${x}_{0}^{\mu }$ which remain after solving both $H({x}_{0}^{\mu })=0$ and $u=S({x}_{0}^{\mu },\alpha )$ for ${x}_{0}^{\mu }$. Formally, one has

$$\begin{equation}\left\{u,{w}^{i}\right\}=\left\{{x}_{0}^{\mu }\right\}/\left\{H({x}_{0}^{\mu }),u-S({x}_{0}^{\mu },\alpha )\right\}.\end{equation} \tag{ 6 }$$

An explicit computation that utilises the above procedure for constructing PCs can be found in section 2.2. The choice of H is not essential as a different choice will give rise to a different parameterisation of the same null geodesic congruence. Note however that a different choice of integration constants α may lead to a different choice of PCs on a spacetime. This is because a choice of different set of integration constants α may lead to a different null geodesic congruence.

To write explicitly the change of coordinate transformation consider a solution ${x}^{\mu }(\lambda ,{x}_{0}^{\mu },\alpha )$ to (2), take the differential and express it in the new coordinates as

$$\begin{align}\mathrm{d}{x}^{\mu }& ={\dot {x}}^{\mu }\mathrm{d}\lambda +\frac{\partial {x}^{\mu }}{\partial {x}_{0}^{\nu }}(\frac{\partial {x}_{0}^{\nu }}{\partial u}\enspace \mathrm{d}u+\frac{\partial {x}_{0}^{\nu }}{\partial {w}^{i}}\enspace \mathrm{d}{w}^{i})\\ & ={g}^{\mu \nu }{\partial }_{\nu }S\enspace \mathrm{d}\lambda +\frac{\partial {x}^{\mu }}{\partial {x}_{0}^{\nu }}(\frac{\partial {x}_{0}^{\nu }}{\partial u}\enspace \mathrm{d}u+\frac{\partial {x}_{0}^{\nu }}{\partial {w}^{i}}\enspace \mathrm{d}{w}^{i}),\end{align} \tag{ 7 }$$

where the integration constants α are considered fixed and they will be suppressed from now on. In these new coordinates ⁴ the metric reads as

$$\begin{equation}g=2\mathrm{d}u\enspace \left(\enspace \mathrm{d}\lambda +{h}_{i}(\lambda ,u,w)\enspace \mathrm{d}{w}^{i}+\frac{1}{2}{\Delta}(\lambda ,u,w)\enspace \mathrm{d}u\right)+{\gamma }_{ij}(\lambda ,u,w)\enspace \mathrm{d}{w}^{i}\enspace \mathrm{d}{w}^{j},\end{equation} \tag{ 8 }$$

where we have used

$$\begin{equation}{g}_{\lambda \lambda }={g}_{\mu \nu }{\dot {x}}^{\mu }{\dot {x}}^{\nu }={g}_{\mu \nu }{g}^{\mu \rho }{\partial }_{\rho }S{g}^{\nu \sigma }{\partial }_{\sigma }S=0,\end{equation} \tag{ 9 }$$

as a consequence of (1) and

$$\begin{align}{g}_{\mu \nu }{g}^{\mu \rho }{\partial }_{\rho }S\frac{\partial {x}^{\nu }}{\partial {x}_{0}^{\sigma }}\left(\frac{\partial {x}_{0}^{\sigma }}{\partial u}\mathrm{d}u+\frac{\partial {x}_{0}^{\sigma }}{\partial {w}^{i}}\mathrm{d}{w}^{i}\right)\mathrm{d}\lambda & ={\partial }_{\nu }S\frac{\partial {x}^{\nu }}{\partial {x}_{0}^{\sigma }}\left(\frac{\partial {x}_{0}^{\sigma }}{\partial u}\mathrm{d}u+\frac{\partial {x}_{0}^{\sigma }}{\partial {w}^{i}}\mathrm{d}{w}^{i}\right)\mathrm{d}\lambda \\ & ={\partial }_{{x}_{0}^{\nu }}S({x}_{0}^{\rho })\frac{\partial {x}_{0}^{\nu }}{\partial u}\mathrm{d}\lambda \enspace \mathrm{d}u\\ & \quad +{\partial }_{{x}_{0}^{\nu }}S({x}_{0}^{\rho })\frac{\partial {x}_{0}^{\nu }}{\partial {w}^{i}}\mathrm{d}\lambda \enspace \mathrm{d}{w}^{i}\\ & =\frac{\partial u}{\partial u}\mathrm{d}\lambda \enspace \mathrm{d}u+\frac{\partial u}{\partial {w}^{i}}\mathrm{d}{w}^{i}\mathrm{d}\lambda \\ & =\mathrm{d}u\enspace \mathrm{d}\lambda .\end{align} \tag{ 10 }$$

The remaining components of the metric can be easily evaluated leading to (8).

The plane wave limit of a spacetime is defined as follows. First one adapts a set of PCs on a spacetime as in (8), scales the spacetime metric as g → ℓ⁻² g and simultaneously performs the coordinate transformation u → ℓ² u, wⁱ → ℓwⁱ and λ → λ, where ℓ is a constant parameter. Then one takes the limit ℓ → 0. The metric at the limit takes the form

$$\begin{equation}{g}_{pw}=2\mathrm{d}u\enspace \mathrm{d}\lambda +{\gamma }_{ij}(\lambda )\mathrm{d}{w}^{i}\mathrm{d}{w}^{j}.\end{equation} \tag{ 11 }$$

This is a plane-wave metric written in Rosen coordinates. The limit ℓ → 0 restricts the non-vanishing components of the metric at the null geodesic with boundary conditions u = wⁱ = 0. So the limit depends on the choice of this null geodesic.

The plane wave metric g_pw can be rewritten in Brinkmann coordinates as

$$\begin{equation}{g}_{pw}=2\mathrm{d}\lambda \left(\mathrm{d}v+\frac{1}{2}{A}_{ab}(\lambda ){x}^{a}{x}^{b}\mathrm{d}\lambda \right)+{\delta }_{ab}\mathrm{d}{x}^{a}\mathrm{d}{x}^{b},\end{equation} \tag{ 12 }$$

where ${w}^{i}={\text{e}}_{a}^{\text{i}}(\lambda ){x}^{a}$, $u=v-\frac{1}{2}{\gamma }_{ij}{\partial }_{\lambda }{\text{e}}_{\left(\right.a}^{\text{i}}{e}_{b\left.\right)}^{j}{x}^{a}{x}^{b}$, with ${\gamma }_{ij}{\text{e}}_{a}^{\text{i}}{e}_{b}^{j}={\delta }_{ab}$ and

$$\begin{equation}{A}_{ab}={\gamma }_{ij}{\partial }_{\lambda }{\text{e}}_{a}^{\text{i}}{\partial }_{\lambda }{e}_{b}^{j}-{\partial }_{\lambda }\left({\gamma }_{ij}{\partial }_{\lambda }{\text{e}}_{\left(\right.a}^{\text{i}}{e}_{b\left.\right)}^{j}\right),\end{equation} \tag{ 13 }$$

is the plane wave profile mentioned in the introduction.

In gravitational systems that both the null geodesic and HJ equations are separable, there is a simplification in the description of PCs of the spacetime. The separability of the HJ equation means that there is a coordinate system {x^μ } on the spacetime such that the HJ function S can be written as

$$\begin{equation}S=\sum\limits _{\mu }{S}_{(\mu )}({x}^{\mu })\end{equation} \tag{ 14 }$$

where each function S_(μ) depends only on the coordinate x^μ , and the HJ equation reduces to a set of first order ordinary differential equations one for each S_(μ). Such differential equations can be integrated to determine S_(μ) in terms of x^μ up to an integration constant α.

The integrability of the geodesic equations means that among the coordinates {x^μ } that separate the null HJ equation, there is a coordinate, say x¹ ∈ {x^μ }, such that the geodesic equation ${\dot {x}}^{1}={g}^{1\nu }{\partial }_{\nu }S$ is an equation that involves only x¹ and the affine parameter λ, i.e. it does not depend on the remaining spacetime coordinates. As a result this can be integrated to express x¹ in terms of λ as ${x}^{1}={x}^{1}(\lambda ,{x}_{0}^{1})$. Then the remaining geodesic equations can be solved sequentially. This means that if the solution for the first k-coordinates is known, say ${x}^{\mu }={x}^{\mu }(\lambda ,{x}_{0}^{1},\dots ,{x}_{0}^{k})$, μ = 1, ..., k, then after a substitution of these solutions into the geodesic equation for the k + 1 coordinate, x^k+1 will lead to an equation which will involve only the coordinate x^k+1 and λ. As a result, it can also be integrated to give x^k+1 in terms of λ, and so on.

It is clear that the separability of the geodesic equations will lead to a solution of the geodesic equations which can be arranged, after a possible permutation of the coordinates, to be of the form

$$\begin{equation}{x}^{\mu }={x}^{\mu }(\lambda ,{x}_{0}^{\rho }),\quad \rho {\leqslant}\mu ,\enspace \mu =1,2,3,\dots ,n.\end{equation} \tag{ 15 }$$

Note that x^μ depends only on the first μ integration constants, ${x}_{0}^{\nu }$, ν = 1, ..., μ. To proceed, it is convenient to take $H({x}_{0}^{\nu })={x}_{0}^{1}=0$. In such a case, the change of coordinate transformation to the PCs reads

$$\begin{equation}\begin{aligned}& \mathrm{d}{x}^{1}={\dot {x}}^{1}\mathrm{d}\lambda ;\quad \mathrm{d}{x}^{\mu }={\dot {x}}^{\mu }\mathrm{d}\lambda +\sum\limits _{\nu =2}^{\mu }\frac{\partial {x}^{\mu }}{\partial {x}_{0}^{\nu }}\mathrm{d}{x}_{0}^{\nu },\mu { >}1;\quad u=S({x}_{0}^{\nu })=\sum\limits _{\mu { >}1}{S}_{(\mu )}({x}_{0}^{\mu }).\end{aligned}\end{equation} \tag{ 16 }$$

For spacetimes with isometries generated by Killing vectors ξ, there is a further simplification. Typically the separability coordinates {x^μ } are adapted to the commuting isometries of the spacetime. If there is such an isometry ξ which is adapted to any coordinate x^μ , μ > 1, say for simplicity x², i.e. $\xi ={\partial }_{{x}^{2}}$, then the HJ function reads as

$$\begin{equation}S={L}_{2}\enspace {x}^{2}+\sum\limits _{\mu \ne 2}{S}_{(\mu )}({x}^{\mu })\enspace ,\end{equation} \tag{ 17 }$$

where L₂ is a constant. L₂ is one of the integration constants α of the HJ equation. For null geodesic congruences for which L₂ ≠ 0, the coordinate transformation $u=S({x}_{0}^{\nu })$ can be solved as

$$\begin{equation}{x}_{0}^{2}=\frac{1}{{L}_{2}}\left(u-\sum\limits _{\mu \ne 2}{S}_{(\mu )}({x}_{0}^{\mu })\right).\end{equation} \tag{ 18 }$$

Combining the equation above with $H({x}_{0}^{\nu })={x}_{0}^{1}=0$, one concludes that the coordinates wⁱ in (6) can be identified with ${x}_{0}^{\mu }$, μ > 2.

There may be several commuting Killing vector fields ξ on a spacetime. In such a case the HJ function can be written as

$$\begin{equation}S=\sum\limits _{p=2}^{k+1}{L}_{p}\enspace {x}^{p}+\sum\limits _{\mu \ne 2,\dots ,k+1}{S}_{(\mu )}({x}^{\mu }),\end{equation} \tag{ 19 }$$

where x^p are the coordinates adapted to the k commuting Killing vector fields and L_p are integration constants of the HJ equation. Therefore for each L_p ≠ 0, one can solve $u=S({x}_{0}^{\nu })$ as in (18). This allows one to explore different plane wave limits of a spacetime. The constants L_p are conserved charges of the geodesic equation which label different kinematic regimes of the system. So different choices for the constants L_p may lead to different null geodesic congruences which in turn may lead to different plane wave limits for the spacetime.

The metric [26] of the near horizon geometry of the extreme Kerr black hole is

$$\begin{equation}g=(1+{z}^{2})(-{\rho }^{2}\mathrm{d}{\tau }^{2}+{\rho }^{-2}\mathrm{d}{\rho }^{2})+\frac{1+{z}^{2}}{1-{z}^{2}}\mathrm{d}{z}^{2}+\frac{1-{z}^{2}}{1+{z}^{2}}{(2\rho \mathrm{d}\tau +\mathrm{d}\varphi )}^{2}.\end{equation} \tag{ 35 }$$

Next introduce the coordinates

$$\begin{equation}v=\tau -{\rho }^{-1},\qquad \chi =\varphi -2\mathrm{d}\enspace \mathrm{log}\enspace \rho ,\end{equation} \tag{ 36 }$$

to express the above metric as

$$\begin{equation}g=(1+{z}^{2})(-{\rho }^{2}\mathrm{d}{v}^{2}+2\mathrm{d}v\enspace \mathrm{d}\rho )+\frac{1+{z}^{2}}{1-{z}^{2}}\mathrm{d}{z}^{2}+\frac{1-{z}^{2}}{1+{z}^{2}}{(2\rho \mathrm{d}v+\mathrm{d}\chi )}^{2}.\end{equation} \tag{ 37 }$$

In these coordinates the HJ equation is separable. In particular set S = Ev + Lχ + R(ρ) + Z(z), E and L constants, to find that

$$\begin{equation}{\rho }^{2}{\left(\frac{\mathrm{d}R}{\mathrm{d}\rho }\right)}^{2}+(2E-4L\rho )\frac{\mathrm{d}R}{\mathrm{d}\rho }=-{Q}^{2},\qquad {\left(\frac{\mathrm{d}Z}{\mathrm{d}z}\right)}^{2}+\frac{{(1+{z}^{2})}^{2}}{{(1-{z}^{2})}^{2}}{L}^{2}=\frac{{Q}^{2}}{1-{z}^{2}},\end{equation} \tag{ 38 }$$

where Q is a separation constant. We have also used that

$$\begin{equation}{g}^{-1}=\frac{2}{1+{z}^{2}}{\partial }_{v}{\partial }_{\rho }+\frac{{\rho }^{2}}{1+{x}^{2}}{\partial }_{\rho }^{2}-\frac{4\rho }{1+{z}^{2}}{\partial }_{\rho }{\partial }_{\chi }+\frac{1-{z}^{2}}{1+{z}^{2}}{\partial }_{z}^{2}+\frac{1+{z}^{2}}{1-{z}^{2}}{\partial }_{\chi }^{2}.\end{equation} \tag{ 39 }$$

Further progress depends on exploring (38) and the geodesic equations.

It is well known that all regular (extreme) Killing horizons can be expressed in null Gaussian coordinates, see [27, 28]. For the near horizon Kerr geometry, these is a special case of PCs which are associated with null geodesics with Q = L = 0; for a different derivation see [29]. In this kinematic regime the HJ equation can be integrated to yield

$$\begin{equation}S=Ev+2E{\rho }^{-1}.\end{equation} \tag{ 40 }$$

Moreover, the associated null geodesic equations can be solved to find

$$\begin{equation}z={z}_{0},\qquad \rho =-\frac{E}{1+{z}_{0}^{2}}\lambda ,\qquad u=\frac{2}{E}(1+{z}_{0}^{2}){\lambda }^{-1}+{u}_{0},\qquad \chi =-4\enspace \mathrm{log}\enspace \lambda +{\chi }_{0},\end{equation} \tag{ 41 }$$

where λ is the affine parameter of the geodesics and no integration constant has been introduced for ρ. Adapting PCs to this null geodesic congruence, one finds that

$$\begin{equation}\begin{aligned}& h=\lambda \left(\frac{4(1-{x}^{2})}{{(1+{x}^{2})}^{2}}\mathrm{d}\phi -\frac{2x}{1+{x}^{2}}\mathrm{d}x\right),\\ & {\Delta}={\lambda }^{2}\left(\frac{3-6{x}^{2}-{x}^{4}}{{a}^{2}{(1+{x}^{2})}^{3}}\right),\\ & \gamma ={a}^{2}\frac{1+{x}^{2}}{1-{x}^{2}}\mathrm{d}{x}^{2}+4{a}^{2}\frac{1-{x}^{2}}{1+{x}^{2}}\mathrm{d}{\phi }^{2}\enspace ,\end{aligned}\end{equation} \tag{ 42 }$$

with a² = 1, where we have set x = z₀, $-2\phi ={\chi }_{0}-4\enspace \mathrm{log}(1+{z}_{0}^{2})$ and u = S = Ev₀. A non-trivial a² factor can be introduced after rescaling the whole metric with a² and changing coordinates as λ → a² λ. The metric of the near horizon geometry has been put in Gaussian null coordinates, i.e. it is of the form

$$\begin{equation}g=2\mathrm{d}u(\mathrm{d}\lambda +\lambda \enspace {~{h}}_{i}\mathrm{d}{x}^{i}+\frac{1}{2}{\lambda }^{2}\enspace ~{{\Delta}}\mathrm{d}u)+{\gamma }_{ij}\mathrm{d}{x}^{i}\mathrm{d}{x}^{j},\end{equation} \tag{ 43 }$$

where $~{h}$, $~{{\Delta}}$ and γ depend only on the coordinate x, and where $h=\lambda ~{h}$ and ${\Delta}={\lambda }^{2}~{{\Delta}}$.

It has already been mentioned in the beginning of this section that all near horizon geometries of extreme regular Killing horizons can be written in null Gaussian coordinates. This means that the metric takes the form (43) and its components $~{h}$, $~{{\Delta}}$ and γ do not depend on the coordinates u and λ but they may depend on all the rest of the coordinates. As γ does not depend of λ, the plane wave limit is the Minkowski spacetime. Therefore all near horizon geometries of extreme Killing horizons admit Minkowski spacetime as a plane wave limit. In particular, the near horizon geometry of the Kerr black hole admits Minkowski space as plane wave limit. Typically near horizon geometries of extreme regular Killing horizons may also admit other plane wave limits adapted to a different set of PCs. For example the AdS₂ × S² near horizon geometry of the extreme Reissner–Nordström black hole admits two different plane wave limits [20] and only one of them is the Minkowski spacetime.

Next consider the kinematic regime for which E, Q ≠ 0 and L = 0. The HJ function is S = Ev + R(ρ) + Z(z), where

$$\begin{equation}R(\rho )=\frac{Q}{\mathrm{cos}\enspace \theta }-{{\epsilon}}_{\rho }Q\enspace \mathrm{tan}\enspace \theta +{{\epsilon}}_{\rho }Q\theta ,\quad \rho =\frac{E}{Q}\enspace \mathrm{cos}\enspace \theta ,\quad {{\epsilon}}_{\rho }={\pm}1,\end{equation} \tag{ 44 }$$

and

$$\begin{equation}Z(z)=-{{\epsilon}}_{z}Q\psi ,\quad z=\mathrm{cos}\psi ,\quad {{\epsilon}}_{z}={\pm}1,\end{equation} \tag{ 45 }$$

which follow from (38). In turn, the solution to the geodesic equation can be expressed as

$$\begin{align}& \frac{3}{2}\psi +\frac{1}{4}\mathrm{sin}(2\psi )=-{{\epsilon}}_{z}Q\enspace \lambda +{\psi }_{0},\quad \theta ={{\epsilon}}_{\rho }{{\epsilon}}_{z}\psi +{\theta }_{0},\\ v& =-\frac{Q}{E}\left[\mathrm{tan}(-{{\epsilon}}_{z}\psi -{{\epsilon}}_{\rho }{\theta }_{0})+{\mathrm{cos}}^{-1}(-{\epsilon}\psi -{{\epsilon}}_{\rho }{\theta }_{0})\right]+{v}_{0},\\ \chi & =2\left[\mathrm{log}\enspace \mathrm{tan}\left(\frac{-{{\epsilon}}_{z}\psi -{{\epsilon}}_{\rho }{\theta }_{0}}{2}+\frac{\pi }{4}\right)-\mathrm{log}\enspace \mathrm{cos}(-{{\epsilon}}_{z}\psi -{{\epsilon}}_{\rho }{\theta }_{0})\right]+{\chi }_{0}.\end{align} \tag{ 46 }$$

The first equation above can be used to determine ψ in terms of the affine parameter λ. Following the general description of the PCs for systems with separable HJ and geodesic equations, one writes

$$\begin{equation}u=S={{\epsilon}}_{\rho }\enspace Q\enspace {\theta }_{0}+E{v}_{0},\quad \lambda =\lambda ,\quad \phi ={\chi }_{0},\quad x={\theta }_{0},\end{equation} \tag{ 47 }$$

where ψ₀ is fixed, i.e. ψ₀ is set to zero. The metric in these coordinates reads as

$$\begin{align}& g=\mathrm{d}u\left[2\mathrm{d}\lambda +{{\epsilon}}_{\rho }\frac{2}{Q}\left((1+{\mathrm{cos}}^{2}\enspace \psi )+{Q}^{2}{Y}_{Q}\enspace {\mathrm{cos}}^{2}({{\epsilon}}_{z}\psi +{{\epsilon}}_{\rho }x)\right)\mathrm{d}x\right.\\ & \quad \left.+\frac{4}{Q}\frac{{\mathrm{sin}}^{2}\psi }{1+{\mathrm{cos}}^{2}\psi }\mathrm{cos}({{\epsilon}}_{z}\psi +{{\epsilon}}_{\rho }x)\mathrm{d}\phi +{Y}_{Q}\enspace {\mathrm{cos}}^{2}({{\epsilon}}_{z}+{{\epsilon}}_{\rho }x)\mathrm{d}u\right]\\ & \quad +\frac{{\mathrm{sin}}^{2}\enspace \psi }{1+{\mathrm{cos}}^{2}\enspace \psi }{\left(\mathrm{d}\phi -2{{\epsilon}}_{\rho }\enspace \mathrm{cos}({{\epsilon}}_{z}\psi +{{\epsilon}}_{\rho }x)\mathrm{d}x\right)}^{2}\\ & \quad +(1+{\mathrm{cos}}^{2}\enspace \psi ){\mathrm{sin}}^{2}({{\epsilon}}_{z}\psi +{{\epsilon}}_{\rho }x)\enspace \mathrm{d}{x}^{2},\end{align} \tag{ 48 }$$

where ψ = ψ(λ) is a solution of the first equation in (46) and

$$\begin{equation}{Y}_{Q}=\frac{3-6\enspace {\mathrm{cos}}^{2}\enspace \psi -{\mathrm{cos}}^{4}\enspace \psi }{{Q}^{2}(1+{\mathrm{cos}}^{2}\enspace \psi )}.\end{equation} \tag{ 49 }$$

The plane wave limit metric in Rosen coordinates is

$$\begin{equation}{g}_{pw}=2\mathrm{d}u\enspace \mathrm{d}\lambda +\frac{{\mathrm{sin}}^{2}\enspace \psi }{1+{\mathrm{cos}}^{2}\enspace \psi }{\left(\mathrm{d}\phi -2{{\epsilon}}_{\rho }\enspace \mathrm{cos}(\psi )\mathrm{d}x\right)}^{2}+(1+{\mathrm{cos}}^{2}\enspace \psi ){\mathrm{sin}}^{2}\enspace (\psi )\enspace \mathrm{d}{x}^{2}.\end{equation} \tag{ 50 }$$

This can be written in (u, ψ, x, ϕ) coordinates using (1 + cos²  ψ)dψ = −_z Qdλ.

Despite the simplifications, the plane wave metric (50) is implicitly described. However notice that for small ψ, one has that 2ψ = −_z Qλ and the plane wave limit is Minkowski spacetime. Similarly for large ψ, one has $\frac{3}{2}\psi =-{{\epsilon}}_{z}Q\lambda$ and the plane wave metric (50) can be explicitly expressed in terms of λ as

$$\begin{align}\mathrm{d}{s}^{2}& =2\mathrm{d}u\enspace \mathrm{d}\lambda +\frac{{\mathrm{sin}}^{2}\left(\frac{2}{3}Q\lambda \right)}{1+{\mathrm{cos}}^{2}\left(\frac{2}{3}Q\lambda \right)}{\left(\mathrm{d}\phi -2{{\epsilon}}_{\rho }\mathrm{cos}\left(\frac{2}{3}Q\lambda \right)\mathrm{d}x\right)}^{2}\\ & \quad +\left(1+{\mathrm{cos}}^{2}\left(\frac{2}{3}Q\lambda \right)\right){\mathrm{sin}}^{2}\left(\frac{2}{3}Q\lambda \right)\mathrm{d}{x}^{2}.\end{align} \tag{ 51 }$$

This completes the description of the plane wave limits of the near horizon geometry of extreme Kerr black hole in the kinematic regime E, Q ≠ 0 and L = 0.

Next consider the general case with E, Q, L ≠ 0. The HJ equation (38) for R can be solved to yield

$$\begin{equation}R=2L\enspace \mathrm{log}\enspace \rho +\frac{E}{\rho }+~{R}(\rho ),\end{equation} \tag{ 52 }$$

where

$$\begin{equation}~{R}(\rho )={{\epsilon}}_{\rho }{\int }^{\rho }\mathrm{d}p\sqrt{{\left(\frac{E}{{p}^{2}}-\frac{2L}{p}\right)}^{2}-\frac{{Q}^{2}}{{p}^{2}}}.\end{equation} \tag{ 53 }$$

Furthermore setting z = cos ψ, one finds that

$$\begin{equation}Z(z)={\Psi}(\psi )={{\epsilon}}_{z}{\int }^{\psi }\mathrm{d}\beta \enspace \frac{1}{\mathrm{sin}\enspace \beta }\sqrt{{Q}^{2}{\mathrm{sin}}^{2}\enspace \beta -{L}^{2}{(1+{\mathrm{cos}}^{2}\enspace \beta )}^{2}}.\end{equation} \tag{ 54 }$$

The geodesic equations can be expressed as

$$\begin{equation}\begin{aligned}& \dot {v}=\frac{1}{{\mathrm{cos}}^{2}\enspace \psi }\frac{\mathrm{d}R}{\mathrm{d}\rho },\\ & \dot {\psi }=\frac{1}{1+{\mathrm{cos}}^{2}\enspace \psi }\frac{\mathrm{d}{\Psi}}{\mathrm{d}\psi },\\ & \dot {\chi }=\frac{1+{\mathrm{cos}}^{2}\enspace \psi }{{\mathrm{sin}}^{2}\enspace \psi }L-\frac{2\rho }{1+{\mathrm{cos}}^{2}\enspace \psi }\frac{\mathrm{d}R}{\mathrm{d}\rho },\\ & \dot {\rho }=\frac{E}{1+{\mathrm{cos}}^{2}\enspace \psi }+\frac{{\rho }^{2}}{1+{\mathrm{cos}}^{2}\enspace \psi }\frac{\mathrm{d}R}{\mathrm{d}\rho }-\frac{2L\rho }{1+{\mathrm{cos}}^{2}\enspace \psi }.\end{aligned}\end{equation} \tag{ 55 }$$

The second equation above can be solved to determine ψ in terms of the affine parameter λ as ψ = ψ(λ). As in the previous case, we do not introduce a boundary condition for ψ. Then the last geodesic equation can be solved as ρ = ρ(λ, ρ₀). Using this, one can solve the geodesic equation for v and χ as v = V(λ, ρ₀) + v₀ and χ = X(λ, ρ₀) + χ₀, respectively, where

$$\begin{equation}\begin{aligned}V& ={\int }^{\lambda }\mathrm{d}\mu \frac{1}{1+{\mathrm{cos}}^{2}\enspace \psi }\frac{\mathrm{d}R}{\mathrm{d}\rho },\quad X={\int }^{\lambda }\mathrm{d}\mu \left(\frac{1+{\mathrm{cos}}^{2}\enspace \psi }{{\mathrm{sin}}^{2}\psi }L-\frac{2\rho }{1+{\mathrm{cos}}^{2}\enspace \psi }\frac{\mathrm{d}R}{\mathrm{d}\rho }\right).\end{aligned}\end{equation} \tag{ 56 }$$

To express the metric in PCs, set x = ρ₀, ϕ = χ₀, λ = λ and u = S(ρ₀, χ₀, v₀). Eliminating v₀ in favour of the u coordinate, one finds that the components of the metric in the new coordinates can be expressed as

$$\begin{align}& {\Delta}=\frac{3-6\enspace {\mathrm{cos}}^{2}\enspace \phi -{\mathrm{cos}}^{4}\enspace \psi }{{E}^{2}(1+{\mathrm{cos}}^{2}\enspace \psi )},\quad {h}_{\phi }=-L\lambda \enspace {\Delta}+\frac{2\enspace {\in }^{2}\enspace \psi }{E(1+{\mathrm{cos}}^{2}\enspace \psi )}\\ & {h}_{x}=\frac{2\enspace {\mathrm{sin}}^{2}\enspace \psi }{E(1+{\mathrm{cos}}^{2}\enspace \psi )}{\partial }_{{\rho }_{0}}X+\frac{1}{E\lambda \enspace }(1+{\mathrm{cos}}^{2}\enspace \psi ){\partial }_{{\rho }_{0}}\rho -{\Delta}\lambda \enspace {\partial }_{{\rho }_{0}}R({\rho }_{0})+E{\Delta}\lambda \enspace {\partial }_{{\rho }_{0}}V,\\ & {\gamma }_{\phi \phi }=\frac{{\mathrm{sin}}^{2}\enspace \psi }{1+{\mathrm{cos}}^{2}\enspace \psi }-\frac{4\lambda \enspace L\enspace {\mathrm{sin}}^{2}\enspace \psi }{E(1+{\mathrm{cos}}^{2}\enspace \psi )}+{L}^{2}{\Delta}{\lambda }^{2},\\ & {\gamma }_{xx}={E}^{2}{\Delta}{({E}^{-1}{\partial }_{{\rho }_{0}}R(\rho )-{\partial }_{{\rho }_{0}}V)}^{2}-2\lambda \enspace \frac{{\mathrm{sin}}^{2}\enspace \psi }{1+{\mathrm{cos}}^{2}\enspace \psi }{\partial }_{{\rho }_{0}}X({E}^{-1}{\partial }_{{\rho }_{0}}R(\rho )-{\partial }_{{\rho }_{0}}V)\\ & \qquad -2{\partial }_{{\rho }_{0}}\rho (1+{\mathrm{cos}}^{2}\enspace \psi )({E}^{-1}{\partial }_{{\rho }_{0}}R(\rho )-{\partial }_{{\rho }_{0}}V)+\frac{{\mathrm{sin}}^{2}\enspace \psi }{1+{\mathrm{cos}}^{2}\enspace \psi }{({\partial }_{{\rho }_{0}}X)}^{2},\\ & {\gamma }_{x\phi }=-\frac{L}{E}(1+{\mathrm{cos}}^{2}\psi ){\partial }_{{\rho }_{0}}\rho +{E}^{2}{\Delta}{\lambda }^{2}({E}^{-1}{\partial }_{{\rho }_{0}}R(\rho )-{\partial }_{{\rho }_{0}}V)\\ & \qquad +\frac{{\mathrm{sin}}^{2}\enspace \psi }{1+{\mathrm{cos}}^{2}\enspace \psi }\left((1-2\lambda \enspace L{E}^{-1}){\partial }_{{\rho }_{0}}X+2\lambda \enspace {\partial }_{{\rho }_{0}}V-2\lambda \enspace {E}^{-1}{\partial }_{{\rho }_{0}}R({\rho }_{0})\right).\end{align} \tag{ 57 }$$

The plane wave limit of this metric can be easily taken and it will not stated here. The above expression of the metric can become explicit provided one can carry out the integrals in (56) and so integrate the null geodesic equation. It may not be possible to express such integrals in terms of 'simple' functions, see for example [30] for related studies.