Uniqueness of the Kerr-de Sitter spacetime as an algebraically special solution in five dimensions

We determine the most general solution of the five-dimensional vacuum Einstein equation, allowing for a cosmological constant, with (i) a Weyl tensor that is type II or more special in the classification of Coley et al., (ii) a non-degenerate"optical matrix"encoding the expansion, rotation and shear of the aligned null direction. The solution is specified by three parameters. It is locally isometric to the 5d Kerr-de Sitter solution, or related to this solution by analytic continuation or taking a limit. This is in contrast with four dimensions, where there exist infinitely many solutions with properties (i) and (ii).


Introduction
There is a long tradition of classifying solutions of the four-dimensional Einstein equation according to the algebraic type of the Weyl tensor. If the Weyl tensor is everywhere type II, or more special, then the solution is called algebraically special. In an algebraically special spacetime, the Einstein equation simplifies considerably and one can determine the explicit dependence of the metric on one of the coordinates. The Einstein equation then reduces to PDEs in 3 dimensions. The general solution of these PDEs is not known. However, it is clear from various special cases that the general solution involves arbitrary functions [1].
If one makes the stronger assumption that the Weyl tensor is (everywhere) of type D then much more progress can be made. The vacuum Einstein equation can be solved explicitly [2]. There are several families of solutions, each specified by a few parameters. These solutions include the Kerr solution. Performing this classification led to the discovery of a new vacuum solution: the spinning C-metric, describing a pair of rotating black holes being accelerated by cosmic strings.
In this paper we will consider higher-dimensional solutions of the vacuum Einstein equation, allowing for a cosmological constant: The algebraic classification of the Weyl tensor has been extended to d spacetime dimensions by Coley et al. [3]. A solution is type II or more special in this classification if it admits a multiple Weyl aligned null direction (multiple WAND) ℓ. The condition for ℓ to be a multiple WAND can be written [4] where C abcd is the Weyl tensor. In four dimensions, this is the same as the condition for ℓ to be a repeated principal null direction. The Myers-Perry black hole solution [5] ("higher dimensional Kerr") is known to have a Weyl tensor of type D [6,7,8], which means that it admits two distinct multiple WANDs. The Myers-Perry solution has been generalized to include a cosmological constant in five [9] and higher [10] dimensions. For simplicity, we will refer to these as "Kerr-de Sitter" solutions (for any value of Λ). These also have a Weyl tensor of type D [7]. By analogy with the four-dimensional case, one might expect there to exist a small number of families of solutions of (1) with a Weyl tensor of type D. Surprisingly, we will establish a stronger result in five dimensions: subject to one extra assumption, the Kerr-de Sitter solution is essentially the only solution with a Weyl tensor of type II.
To explain the extra assumption, we note that any solution of (1) admitting a multiple WAND must also admit a geodesic multiple WAND [11] hence there is no loss of generality in assuming ℓ to be geodesic. Recall that the expansion, rotation and shear of the null geodesic congruence tangent to ℓ are defined as the trace, antisymmetric part, and traceless symmetric part of the (d − 2) × (d − 2) "optical matrix" where m i are a set of (d − 2) orthonormal spacelike vectors orthogonal to ℓ. Our assumption is that ρ ij is non-degenerate. 1 We will prove: Theorem. Let (M, g) be a solution of the five-dimensional vacuum Einstein equation (1). Assume that (M, g) admits a geodesic multiple WAND ℓ for which the 3 × 3 matrix ρ ij is non-degenerate. Then one can define an affine parameter r along the null geodesics tangent to ℓ such that the eigenvalues of ρ ij are 1/r and 1/(r ± iχ) for some real function χ constant along each geodesic. Furthermore: 1. If χ = 0 and dχ = 0 then one can define local coordinates (u, r, χ, x, y) such that the metric is ds 2 = −2(du + χ 2 dy) dr + H(r, χ)(du + χ 2 dy) − E 0 dx − E 0 χ 2 dy + P dy +r 2 χ 2 dx − E 0 χ 2 dy 2 + (r 2 + χ 2 ) dχ 2 P (χ) + P (χ)dy 2 , where (A 0 , µ 0 , C 0 , E 0 ) are arbitrary real constants, and P (χ) > 0.
2. If χ = 0 and dχ ≡ 0 then one can define local coordinates (u, r, x, y 1 , y 2 ) such that the metric is where h αβ (y) (1 ≤ α, β ≤ 2) is the metric on a 2d Riemannian manifold of constant curvature, A = A α (y)dy α is a 1-form such that dA is a volume form for this 2d manifold, 1 Note that this does not depend on how the mi are chosen.
We will now make some remarks on the above theorem. For any of the above metrics ∂/∂r is a geodesic multiple WAND. The Weyl tensor vanishes if, and only if, µ 0 = 0. For µ 0 = 0 each solution is type D, i.e., there exists a second multiple WAND.
The metric (4), has a scaling symmetry: for λ = 0 one can perform a coordinate transformation and the metric in the primed coordinates takes the same form as (4) but with rescaled constants: This shows that the solution is really a 3-parameter family. 2 We will show (in section 4) that the solution (4) is locally isometric to the 5d Kerr-de Sitter solution [9] with two unequal rotation parameters. 3 The coordinates of (4) are closely related to the coordinates for the Kerr-de Sitter solution defined in Ref. [12]. Only when the parameters (A 0 , µ 0 , C 0 , E 0 ) lie within a certain set does the metric (4) describe a regular black hole solution. Values of the parameters outside this set can result in local metrics such as the "Kaluza-Klein bubble" spacetime of Ref. [13] which was shown to be algebraically special in Ref. [14]. 4 The metric (7) has a scaling symmetry analogous to (12). Hence this solution is really a 2parameter family. If R (2) > 0, one can perform a coordinate transformation to show that the solution is locally isometric to the Kerr-de Sitter solution with two equal, non-zero, rotation parameters. Solutions with R (2) ≤ 0 can be regarded as analytically continued versions of the Kerr-de Sitter solution.
The metric (10) is a generalized Schwarzschild metric written in outgoing Eddington-Finkelstein coordinates. Of course, this corresponds to the Kerr-de Sitter metric with vanishing rotation parameters.
Note that, in four dimensions, there are many solutions satisfying the assumptions of the above theorem. In 4d, the Goldberg-Sachs theorem implies that ℓ is shear-free, which implies that ρ ij is degenerate if, and only if, ρ ij = 0 (this defines the Kundt family of solutions). So any algebraically special solution with ρ ij = 0 satisfies the assumptions of the theorem. As noted above, there is no simple explicit form for such solutions, and such solutions are known to involve free functions.
The non-degeneracy assumption on ρ ij cannot be eliminated from our theorem. To see this, note that one can take the product of a 4d Ricci flat algebraically special solution with a flat direction to obtain a 5d Ricci flat solution admitting a geodesic multiple WAND. This solution will have degenerate ρ ij because ℓ does not expand along the flat direction. Obviously there are as many such solutions as there are 4d algebraically special solutions. For Λ = 0 one can take a warped product to reach the same conclusion.
Our result can be viewed as a new kind of uniqueness theorem for the 5d Kerr-de Sitter solution. It should be contrasted with the usual uniqueness theorem [15,16] for the 5d Myers-Perry black hole, which assumes the existence of a R × U (1) 2 isometry group, asymptotic flatness, a regular horizon of spherical topology, and no topology outside the horizon. Our result assumes nothing about isometries or global structure and allows for a cosmological constant.
The local nature of our result is similar to the uniqueness result for spacetimes with certain "hidden symmetries", i.e., symmetries associated to Killing tensors rather than Killing vectors. Refs. [17,18] proved that the Kerr-de Sitter solution (generalized to allow for a NUT charge [12]) is the most general d-dimensional solution of (1) admitting a "principal conformal Killing-Yano 2form". Note that this result applies even for d = 4, where it yields a subset of the type D solutions. This is in contrast with our result, for which there exist infinitely many d = 4 solutions satisfying the assumptions of the theorem.
Our result is also reminscent of the theorem that asserts that the Kerr solution is the unique stationary solution of the 4d vacuum Einstein equation with vanishing Mars-Simon tensor [19]. However our theorem does not assume stationarity.
There have been several hints that higher-dimensional algebraically special solutions with nondegenerate ρ ij might be more rigid than their 4d counterparts. First, Ref. [20] considered higher dimensional Robinson-Trautman solutions, defined by the existence of a null geodesic ℓ with 0 = ρ ij ∝ δ ij (such ℓ must be a multiple WAND). It was found that these solutions are considerably simpler for d > 4 than for d = 4. For d = 5, the only such solution is the generalized Schwarzschild metric (10).
Second, Ref. [21] investigated the possible existence of families of algebraically special solutions that contain the Schwarzschild solution. A solution "close" to the Schwarzschild solution in such a family would have non-degenerate ρ ij . The approach of Ref. [21] was to consider linear perturbations of the Schwarzschild solution that preserve the algebraically special property. For d = 4 there are infinite families of such perturbations. But for d > 4 it was found that the only such perturbations that are regular on the orbits of spherical symmetry are perturbations corresponding to a linearisation of the Myers-Perry solution around the Schwarzschild solution.
Third, Ref. [22] studied algebraically special solutions in d > 4 dimensions that (i) have nondegenerate ρ ij and (ii) are asymptotically flat. It was assumed that the curvature components can be expanded in inverse powers of an affine parameter r along the null geodesics tangent to ℓ. It was found that such solutions are non-radiative, in contrast with the d = 4 case.
Ref. [14] showed that 5d algebraically special solutions can be classified according to the rank of ρ ij . Our theorem determines all solutions for which ρ ij has rank 3. Rank 0 defines the Kundt class of solutions which was studied in Refs [23,24]. For rank 2 or rank 1, all solutions for which ℓ is hypersurface-orthogonal (ρ [ij] = 0) were determined in Ref. [25]. We intend to return to the general (non-hypersurface-orthogonal) rank 2 and rank 1 cases in future work.
We end this introduction with an outline of the proof of our theorem. The starting point for the proof is the recent demonstration [14] that the optical matrix of a geodesic multiple WAND in 5d can be brought to a certain canonical form by an appropriate choice of the basis vectors m i . This is the 5d analogue of the "shearfree" property that holds in 4d because of the Goldberg-Sachs theorem. As noted in Ref. [14], case 3 of the theorem follows immediately from combining this canonical form with the results of Ref. [20].
For non-degenerate ρ ij , the canonical form involves two unknown functions. We show how the evolution equation for ρ ij can be integrated to determine the dependence of these functions on an affine parameter r along the geodesics tangent to ℓ. This determines the form of the eigenvalues of ρ ij as stated in the theorem. Next we complete {ℓ, m i } to a null basis {ℓ, n, m i } where n is null and orthogonal to m i . After exploiting a residual freedom in the choice of m i , we show how the "Newman-Penrose" and Bianchi equations can be integrated to determine the r-dependence of the basis vectors and hence the rdependence of the metric. The r-dependence of the connection and curvature components is also fully determined. This calculation reveals that the Weyl tensor is necessarily of type D.
The vanishing of certain connection components enables us to introduce local coordinates in a canonical way. After expressing our basis vectors in terms of these coordinates and using the results obtained previously we obtain a set of equations that can be integrated. At this stage, it becomes convenient to divide the analysis into two cases depending on whether χ is constant or not. If dχ = 0 then we use χ as a coordinate and show that residual coordinate freedom can be exploited to make the solution independent of three of the remaining coordinates. Finally we solve for the dependence on χ to obtain the solution (4). If dχ ≡ 0 then a similar procedure leads to the metric (7). This paper is organized as follows: section 2 contains the first part of the proof of the theorem in which we determine the connection and curvature components in a null basis. In section 3, we introduce coordinates and complete the proof of the theorem. Section 4 demonstrates how the metrics (4) and (7) are related to the Kerr-de Sitter solution.

Notation
We will perform most calculations in a null basis. Refs. [26,27] developed a higher-dimensional analogue of the Newman-Penrose formalism used for calculations in such a basis. This was repackaged into a higher-dimensional analogue of the Geroch-Held-Penrose (GHP) formalism in Ref. [28]. We will follow the notation of Ref. [28] for the connection components and Weyl tensor components. In particular, we refer the reader to eqns. NP1-NP4, B1-B8 and C1-C3 of Ref. [28], which lists all the Newman-Penrose and Bianchi equations satisfied by the connection and curvature components, as well as equations for the commutator of derivatives.
2 Integration of GHP equations 2.1 Canonical form for ρ ij Consider an Einstein spacetime, i.e. a solution of (1), admitting a multiple WAND ℓ. Introduce a null basis {ℓ, n, m i }, i = 2, 3, 4. The multiple WAND condition is equivalent to the vanishing of the Weyl components of boost weights +2 and +1: Without loss of generality, ℓ can be assumed to be geodesic [11], i.e.
Ref. [14] showed that the spatial basis vectors m a i (i = 2, 3, 4) can be chosen so that the optical matrix ρ ij of ℓ takes one of the following forms: for functions a, b. For b = 0, these matrices have rank 3,2,1 respectively. Our assumption is that ρ ij is non-degenerate so We know all the GHP scalars with positive boost weight, both from the connection and the curvature: Ω ij , Ψ ijk , κ i , and we know the structure of ρ ij , which has boost weight +1. Using the Newman-Penrose equations and Bianchi identities, we can determine the GHP scalars of boost weight negative and zero by systematically examining these equations from higher to lower boost weight. We now proceed to indicate the steps involved in this calculation.

Choice of basis
The form of the optical matrix above does not fix the basis uniquely because this form is preserved by null rotations about ℓ and spins in the 2-3 directions. We can use this freedom to make some of the GHP scalars vanish. Consider a null rotation about ℓ with parameters z i [28] ℓ → ℓ, where z 2 = z i z i . This leaves ρ ij unchanged (as κ i = 0) but τ i changes according to [28] Since ρ ij is non-degenerate, we can choose z i to set τ i = 0. Equation NP2 of Ref. [28] then gives τ ′ i = 0 [28]. In summary, we choose our basis so that: It is convenient to combine the spatial vectors m 2 , m 3 into complex null vectors: In this frame, we have 5 while the optical matrix is written as Now consider a spin in the 2-3 directions, which can be phrased in terms of the null complex frame as for some function λ. This will induce changes in i M j0 as follows (recall that D ≡ ℓ · ∂): The last equation, in particular, implies that we can choose λ to set 5 M5 0 = 0 (note that the LHS is imaginary). Moreover, eqn. NP1 of Ref. [28] for ij = 45 gives 4 M 50 = 0 which, from the above, is preserved under such spins. Hence we have Finally, we are free to rescale ℓ so that the geodesics with tangent ℓ are affinely parameterized, which implies L 10 = 0. Note that the conditions mean that we have chosen our basis to be parallelly transported along the geodesics with tangent ℓ.

Determining a, b
We introduce local coordinates as follows. Pick a hypersurface Σ transverse to ℓ and introduce coordinates x µ on Σ. Now assign coordinates (r, x µ ) to the point parameter distance r along the integral curve of ℓ through the point on Σ with coordinates x µ . Note that r is an affine parameter along the geodesics. We now have Consider eqn. NP1 of Ref. [28], which in our parallelly transported basis reads Taking the ij = 55 component gives: The solution is then for some complex function χ that does not depend on r. There is freedom in defining r in the sense that we can shift it by a function of the other coordinates, r → r + α(x µ ), which corresponds to moving the surface Σ used to define r. This freedom can be used to set Im(χ) = 0. With χ real, we take the real and imaginary parts of the above equation to find The r-dependence of ρ ij is then given by In the real basis, the eigenvalues of ρ ij are b(1 + a 2 ) and b(1 ± ia). Using (32) we see that these are 1/r and 1/(r ± iχ), as asserted in our Theorem.

The case χ ≡ 0
If χ ≡ 0 then we have ρ ij = (1/r)δ ij , so ℓ is free of rotation and shear with non-vanishing expansion. This defines the Robinson-Trautman class of solutions. These solutions were studied in Ref. [20], where is was proved that the only solution of this type (in 5d) is the generalized Schwarzschild solution (10). This establishes case 3 of the Theorem. Henceforth we will assume χ = 0.

Boost weight 0 components of the Weyl tensor
The only GHP scalars with boost weight 0 that we do not know yet are such components of the Weyl tensor: Φ ijkl , Φ ij . Their r-dependence can be determined completely using the Bianchi identities, as we will now explain. Notice that, in five dimensions, all the information regarding the boost weight 0 components of the Weyl tensor is encoded in Φ ij , for one can write in terms of the symmetric part Φ S ij = Φ (ij) . The relevant Bianchi identities to determine Φ ij are equations B2, B3 and B4 of Ref. [28], which become, in our basis, and respectively. Take first the ijkl = 5455 component of (37), which gives Taking the ijkl = 4555 component of (36) and substituting the expression for Φ A 45 then gives Comparing the previous two equations with the one obtained from the ij = 54 component of (35) gives Φ S 45 = 0, and hence Φ 45 = Φ 54 = 0. Now we compare the two equations that can be obtained for Φ 55 . The first comes from setting ij = 55 in (35), The second is obtained by putting ijkl = 4545 in (36), One then immediately sees that consistency requires Φ 55 = 0. Thus, the nontrivial components of Φ ij can only be Φ 44 , Φ 55 . Setting ijkl = 4455 in (37) gives the relation If we now substitute this into the ijkl = 4545 component of (36), we find On the other hand, the ij = 44 component of (35) gives Comparing these we find and hence Substituting this back into (44), we have for which the solution is simply for some function µ independent of r. Therefore, the form of Φ ij is One can then verify that any other component of equations (35), (36), (37) is trivially satisfied. Now recall that we have chosen the normalization of ℓ such that it is tangent to affinely parameterized null geodesics. Choosing the affine parameter as one of the coordinates, as above, allows ℓ to be written as ℓ = ∂/∂r. This, however, does not determine r uniquely. Previously we have used a shift in r by the other coordinates to make χ real. Now consider the effect of a boostl = λℓ for some non-zero function λ. The new vector fieldl will also be tangent to affinely parameterized geodesics provided that λ is independent of r. Thus, we can definê wherer = r/λ. If the analysis above were repeated usingr instead of r, the optical matrixρ ij ofl would have the same form as ρ ij if we definedχ = χ/λ. Note that this is consistent with the fact that ρ ij transforms with boost weight +1,ρ ij = λρ ij . On the other hand, Φ ij has boost weight 0 and so is invariant under boosts. Therefore it would retain the same form as in eqn. (49) by defininĝ µ = µ/λ 4 . But this shows that we can choose λ so as to makeμ constant (but we can't choose its sign). Dropping the hats, we can assume, without any loss of generality, that where µ 0 is a constant. Note that Φ ij vanishes if, and only if, µ 0 = 0.
Having determined the r-dependence of Φ ij , the r-dependence of ρ ′ ij can now be completely determined by using information from eqns. NP4, NP4 ′ of Ref. [28]. In our basis, these read (∆ ≡ n · ∂) and respectively.
We start by taking the ij = 44 component of (53), which gives The solution to this equation is where A is some function independent of r. Similarly, the ij = 45 component of the same equation yields to which the solution is for some complex function B 5 that does not depend on r. Substituting this into (52) for ij = 45, one finds and then the ij = 54 component of (52) determines In addition, (52) with ij = 55 immediately gives Next, putting ij = 55 in eqn. (53) gives the equation This can be integrated to give where A 5 is a complex function independent of r. No further information can then be extracted from eqn. (53). Consider now the ij = 44 component of (52), where n r is the r-component of n a in the coordinate basis defined by (r, x µ ). Similarly the ij = 55 component gives where we have used ∆r = n r . Subtracting (63) from (64) gives the relation Taking the real part yields simply and hence one can write where we have redefined Im(A 5 ) = −F . Thus the r-dependence of ρ ′ ij is now known. There is still some information left from eqns. (63) and (64), which will be exhausted in the next calculation.

Determining the basis vectors
In this section we determine the r-dependence of the basis vectors. It is sufficient to consider the commutators together with the remaining information from eqn. (52) and also eqn. NP3 of Ref. [28], which in our basis is The µ-components of (68) reads Dn µ = 0. Hence where n 0µ is independent of r. In order to determine n r , we use (63) to write Substituting into (64) then gives immediately .
The r-component of (68) now gives All the information contained in eqn. (52) has now been used. Next consider the µ-components of (69) with i = 4: The solution is simply where m 0 4 µ are functions independent of r. Similarly, the r-component gives the relation Substituting this into eqn. (70) with ijk = 545 yields where m 0 4 (χ) ≡ m 0µ 4 ∂ µ χ. Now setting ijk = 455 in (70), gives showing that (r − iχ) 4 M 55 is purely imaginary. The last term on the RHS of eqn. (79) is then real. Taking the imaginary part of (79) gives a simple differential equation for m r 4 , which integrates to for some function E 4 of the x µ -coordinates only. Going back to (78), one then finds The real part of (79) then determines Now following the same procedure as above determines the form of m 5 . The µ-components of (69) with i = 5 imply for complex functions m 0 5 µ of the x µ -coordinates only. Then, using the r-component to write and substituting into (70) with ijk = 555 gives where m 0 5 (χ) ≡ m 0µ 5 ∂ µ χ. Integrating this equation one finds that for some complex function E 5 independent of r. Eqn. (86) then determines L 15 : In summary, the coordinate basis components of n a , m a 4 and m a 5 are given by (72) using previous results. The second is the ijk = 545 component, which gives Now consider the commutator C1 of Ref. [28] applied to a GHP scalar V i of arbitrary boost weight b and spin weight s = 1, [þ, þ ′ ] V i . This gives two relations: a boost part (i.e. the coefficient of b) which is automatically satisfied, and a boost-independent part, Notice that this implies that for some unknown, real function C of the x µ -coordinates only. One can follow the same procedure by applying the commutator C2 of [28], [þ, i ], to a GHP scalar V i of spin weight 1. This gives the equations and The former is automatically satisfied, while the latter gives the additional information where D 4 and D 5 are real and complex r-independent functions, respectively.

Boost weight −1 components of the Weyl tensor
We are now in a position to take þ ′ -and i -derivatives in full. In particular, we can now consider the boost weight 0 components of the Bianchi equation, eqns. B5, B5 ′ , B6 and B7 of Ref. [28], and determine completely Ψ ′ ijk . Start with eqn. B5 of [28], which reads The ijk = 545 component of this equation immediately gives that If one takes the ijk = 555 component, one finds that while setting ijk = 545 gives Substituting this result into the ijk = 455 component, one finds that and hence The only remaining independent component of (99) is then ijk = 445, giving Now consider eqn. B5 ′ of [28], Using the previous results, the ijk = 445 component gives the condition Eqns. (101) and (105) then imply and respectively. Of course, the symmetries of Ψ ′ ijk determine automatically One can then verify that all other components of (106) are automatically satisfied, as well as eqns. B6 and B7 of Ref. [28].

Equation NP2 ′
Eqn. NP2 ′ of Ref. [28], which reduces to This can be integrated to determine κ ′ i . The result is where G 4 (real) and G 5 (complex) are functions depending on x µ only.

Differential and algebraic constraints
The r-independent integration functions appearing in the various expressions in the previous sections are not completely independent. There is still information contained in the commutators [n, m i ] , [m i , m j ], as well as in the commutators of GHP derivatives [þ ′ , i ] and [ i , j ] applied to some GHP scalar V i of spin weight 1, and this information can be used to place algebraic and differential constraints on the r-independent functions found above. Later on, when we introduce coordinates, we will be interested in the symmetries and Killing fields admitted by the solutions considered here. In order to study these symmetries, we will need to know the derivatives of those functions along all basis vectors. Consider first the commutator for i = 4, j = 5. The µ-components of this equation give On the other hand, the r-component can be brought to the form p(r) = 0, where p(r) is a polynomial in r with coefficients depending on the x µ -coordinates. It is clear that the coefficient of each power of r must then vanish, resulting in the independent equations Since the action of m 0 4 and m 0 5 on χ is known, eqns. (103), (107), one can apply their commutator (115) to χ to find Comparison with (116) yields immediately Hence the only non-vanishing components of ρ ′ ij are ρ ′ 44 , ρ ′ 55 (and its complex conjugate). Using the above result, eqn. (117) reduces to An additional algebraic constraint can be obtained from eqn. NP3 ′ of [28], By taking the ijk = 455 component and using B 5 = 0, one finds If we act with m 0 5 ,m 0 5 on χ and use (124), we find n 0 (χ) = 2F (125) and hence we know all derivatives of χ. Also, we have the commutators From the i = 4 component one finds the relations and while the i = 5 component gives and Apart from (127), which gives directly the derivative of F along m 0 4 , the other relations determine only a combination of derivatives. One can, however, gain more information from other equations.
We start by going back to eqn. NP3 ′ , (121). Taking the ijk = 445 component one finds Substituting into (131) gives Now using this in (130) gives Setting ijk = 545 in (121) and using (127), one finds which can now be used in eqn. (128) to obtain No additional information can then be obtained from eqn. NP3 ′ , (121). Another source of information is the commutator of GHP derivatives [ i , j ]. Applying this to a GHP scalar V i with spin weight 1, one obtains two equations, just as in the study of [þ, þ ′ ] V i and [þ, i ] V j carried out before. The boost part gives which is automatically satisfied using previous results. The boost-independent part is while the ijkl = 4545 gives m 0 4 (E 4 ) = −3F χ.
There is now enough information to apply the commutators n 0 , m 0 i , m 0 i , m 0 j to the functions considered here. Before we used m 0 5 ,m 0 5 (χ) to find eqn. (125). Another non-trivial, algebraic relation can then be obtained by considering n 0 , m 0 5 (χ), namely Applying the same ideas to A, one finds that m 0 4 , m 0 5 (A) gives It turns out that the term in brackets always vanishes, for suppose that E 4 = 0. Then (141) implies that F = 0, which in turn, from (135), implies Thus, the latter is true irrespective of whether E 4 = 0 or E 4 = 0. Using this in (142), one finds another important algebraic constraint, In turn, m 0 5 ,m 0 5 (A) gives Finally, n 0 , m 0 4 (A) implies either E 4 = 0 or However, as just discussed above, E 4 = 0 implies F = 0, which is consistent with (147). Therefore, one can safely take (147) to be always true.

Equation NP1 ′
Now that we have the information about all derivatives of the functions involved, we can finally consider eqn. NP1 ′ of Ref. [28] to calculate Ω ′ ij , the Weyl tensor components with boost weight −2. In our basis, eqn. NP1 ′ of [28] reads Taking the trace of this equation and recalling that Ω ′ ii = 0, one finds Then, the independent components of Ω ′ ij can be obtained by putting ij = 44, 45, 55 in eqn. (148), giving with Ω ′ 55 = −Ω ′ 44 /2 following from the traceless condition. No further information can be obtained from eqn. (148).

Further algebraic constraints
Notice that, with F = 0, we have With this information, we find When computing ∇ a A, one then finds showing that the term in brackets must be constant, and hence for some constant E 0 .
We have now finished calculating all the components of the connection, including non-GHP scalars, and the components of the Weyl tensor. Furthermore, we have found the r-dependence of the basis vectors and several algebraic and differential constraints involving the r-independent functions.

Null rotations and type D
Consider a null rotation (17) about ℓ. Choosing one finds that the following GHP scalars transform according to and where We have thus found a basis in which only the boost weight 0 components of the Weyl tensor are non-zero, and where both ℓ and n are geodesic with corresponding optical matrices in their canonical form. This shows that all solutions considered here are of type D. If b ′ = 0 then the optical matrix of the second multiple WAND, has rank 3 (in agreement with an argument of Ref. [14]). If b ′ ≡ 0 then the optical matrix of the second multiple WAND vanishes identically in which case the solution belongs to the Kundt family. The condition for this is We will study this special case in more detail in section 4.

Integrable submanifolds
So far we have kept the coordinates x µ arbitrary. We will now show how the results derived above lead to a canonical way of choosing these coordinates. From one finds, using B 5 = 0, that Thus, m 5 ∧ dm 5 = 0. This is the integrability condition for the existence of a complex function z such that m 5 = m5 ∝ dz. Thus, we write for some function M.
Thus {n 0 , m 0 4 } is integrable and we can choose coordinates (r, u, x, z,z) so that for real functions N = 0, L x independent of r. From (m 5 ) 2 = 0 we have mz 5 = 0 and hence for complex functions M = 0, Y , L z independent of r. We now have where depends on r. Now define V x (real) and V z (complex), both independent of r, by . We then have From the inner products we then find from which it is easy to write down the metric. Recall that n 0 (χ) = m 0 4 (χ) = 0. These imply that hence χ = χ(z,z). Now, m 0 5 (χ) = −E 5 reduces to while the commutator n 0 , m 0 4 = 0 gives From (158) we then have ∂ u V x = 0. The result above implies that L x = L x (x, z,z). Note that we have the residual coordinate freedom u → u ′ = u + h(x, z,z), which has the effect We can therefore choose h appropriately to set L ′ x = 0, and then drop the primes. Henceforth, The expression n 0 , m 0 reduces to The latter equation implies and where we used the fact that ∂ u M/M is purely imaginary to write C in a form that is manifestly real. If we now recall that n 0 (E 5 ) = 0, we find where we make use of (145). Hence, ∂ u V z = 0. Thus, we conclude that all functions appearing in the metric are independent of u, and hence ∂/∂u is a Killing vector field. Notice, in particular, that the metric depends on M only through |M | 2 . The commutator gives The latter equation can be used to show that as well as to write D 4 as Moreover, there is still freedom in redefining x by transforming x → x ′ (x, z,z). This changes N to N ′ = N ∂x ′ /∂x. Since N = N (x, z,z), we can use this to impose the condition N ′ = 1/χ. Dropping the primes, N = 1/χ, which implies that the LHS of (195) vanishes, as well as ∂ x N . Thus (195) simply reduces to Notice also that then reduces to Therefore, similarly to the argument above for ∂/∂u, we have shown that ∂/∂x is also a Killing vector field for these solutions. The function M depends on u and x only through a phase, which may be eliminated with an r-independent spin transformation of the form (23). Consider now the expression for m 0 5 ,m 0 5 , which reduces to With the above equation for D 5 , the equation can be written as But if we recall that E 5 = −M ∂ z χ, see (183), the quantity in brackets is −|M | 2 ∂ z χ, which is a function of z,z only. The equation above then shows that it must actually be a function ofz only: for some analytic function k = k(z). If we now consider a holomorphic transformation z → z ′ (z), the effect is to change M → M ′ = M dz ′ /dz and hence k → k ′ = kdz ′ /dz. We can therefore use this transformation to set k ′ to −1 or 0. Dropping the primes, we have shown that we can write We now see that ∂ z χ = ∂zχ. If we now write z = z 1 + iz 2 , we see that hence χ = χ(z 1 ). We can now distinguish between two different cases.

Case 1: dχ = 0
We will now prove case 1 of our Theorem. dχ = 0 implies ∂ z χ = 0 so we must have k 0 = −1. The expressionm which is derived from eqn. (124) and previous results, reduces to Using ∂ z χ = ∂ 1 χ/2 = 1/|M | 2 , we can rewrite this as Since the quantity in brackets is a function of z 1 only, it follows that it must be a constant C 0 . Thus, We can therefore use χ as a coordinate rather than z 1 , with the transformation rule given by (214). Note that from which it follows that χ must lie in a range for which P (χ) > 0. The only quantities that we do not yet know are the functions L z and Y , which are determined by eqns. (203) and (204), respectively. Both have the same structure: where F denotes L z or Y and the RHS is a known function. Notice that and the quantity in brackets on the LHS is precisely the combination in which both L z and Y appear in the metric. If a particular solution Fdz + Fdz is found, any other solution will differ from this by a gradient dα, where α = α(z,z). In the case of L z , this can be absorbed by defining a new coordinate u ′ = u − α(z,z), which does not change any other quantity and, in particular, L x = 0 is maintained. Similarly, defining x ′ = x − α(z,z) eliminates the gradient dα from the expression for Y (notice that ∂x ′ /∂x = 1, and hence every other quantity is unchanged). Hence all we need to do is to find particular solutions for L z and Y . Consider first the equation for L z . If we search for a particular solution in which L z depends only on χ, (203) becomes This can be solved, in particular, for Similarly, if we look for a solution for Y such that Y = Y (χ), (204) reduces to Then, from the definition E 5 = −iM (V z − Y V x ) and from (209), (214), one finds Thus, all functions appearing in the metric (192) depend only on χ (equivalently z 1 ) and r. In addition to ∂/∂u and ∂/∂x, we now find that ∂/∂z 2 is also a Killing vector field for this metric. Using our definitions and results above, the metric becomes Making the definition y ≡ z 2 this gives the metric (4) in the statement of our Theorem.

Case 2: dχ ≡ 0
In this case, χ is constant. Eqn. (209) gives E 5 = k 0 = 0. The equation for m 0 5 (E 5 ) is then trivial, but the one form 0 5 (E 5 ), eqn. (211), becomes which gives an algebraic constraint involving these constants. There is a non-trivial component of (139) that has not yet been considered. The ijkl = 5555 component gives In general, using our expressions for C, D 4 , D 5 , this gives a second-order differential equation involving M,M which can be put in the form (using (224)) where R (2) is the Ricci scalar of the two-dimensional metric Hence this two-dimensional metric has constant curvature. A volume form for this two-dimensional metric is Eqns. (203) and (204) then become respectively. If we define a one-form A = A z (z,z)dz + Az(z,z)dz by then particular solutions to the equations above are As in case 1, any other solution would differ from these by some gradients dα(z,z) and dβ(z,z), respectively. These can be absorbed into u and x using the residual coordinate freedom u ′ = u + α(z,z), x ′ = x + β(z,z), which preserve all quantities fixed above. Then, from The metric is then (2) .
We now transform to arbitrary real coordinates y α (z,z) on the 2d part of the metric, so that g (2) = h αβ (y)dy α dy β , and this gives the metric (7) in the statement of our Theorem. The metric (234) has a scaling symmetry analogous to (12). For λ = 0 we can perform the coordinate transformation and the metric takes the same form as before but now with the constants rescaled as and g (2) replaced by g (2) ′ = λ 2 g (2) and A replaced by A ′ = λ 2 A.

Relation to Kerr-de Sitter
In this section we will perform coordinate transformations to demonstrate how the metrics (4) and (7) are related to the 5d Kerr-de Sitter solution.

Case 1: dχ = 0
If we define the 1-forms then the metric (4) can be written where If we let x I = {u, x, y} then the metric can be written where Now consider a change of coordinates for some functions A I (r). We want to choose the functions A I to eliminate dy I dr terms from the resulting metric. This requires We can only do this if the RHS above is independent of χ. We find that 6 where So the RHS of (243) is indeed independent of χ and the coordinate transformation is permissible provided we work in a region where F (r) = 0. Note that The metric in the new coordinates is where ν i is defined by replacing x I with y I in σ i . Now using (245) to eliminate H we can write For E 0 = 0 we define and the metric is If we now define x 1 = χ and x 2 = ir then this is the Kerr-de Sitter solution [9] with two non-zero, unequal, spin parameters, as written in eqn. (22) of Ref. [12]. 7 The case E 0 = 0 corresponds to the Kerr-de Sitter metric with a single non-vanishing spin parameter. It can be obtained from the solution as written in Ref. [12] by defining ψ 2 =ψ 2 /E 0 and taking the limit E 0 → 0. Now we return to the special case F (r) ≡ 0, for which the above coordinate transformation no longer works. This condition can be understood geometrically as follows. The metric admits a 3d isometry group (associated to the Killing fields ∂/∂u, ∂/∂x, ∂/∂y) with 3d orbits. The condition F (r) ≡ 0 is the condition for these orbits to be null everywhere. 8 Note that This is precisely the condition (164) for the second multiple WAND to have vanishing expansion, rotation and shear, i.e., for the spacetime to be Kundt. 9 In this special case, the metric simplifies to ds 2 = −2dr du + χ 2 dy + µ 0 r 2 + χ 2 du − r 2 dy 2 + r 2 χ 2 dx 2 + r 2 + χ 2 µ 0 dχ 2 , where µ 0 > 0 follows from P (χ) > 0. (The scaling symmetry (12) could be used to set µ 0 = 1.) For this metric, the second multiple WAND is which obeys k a = −(dr) a so surfaces of constant r are all null. A surface r = r 0 is a Killing horizon of the vector field r 2 0 ∂/∂u + ∂/∂y. The surface gravity vanishes, so this spacetime is foliated by degenerate Killing horizons. The metric is smooth at χ = 0 (for r = 0) provided that x is identified with period 2π/ √ µ 0 , which implies that cross-sections of the Killing horizons have topology R 3 (assuming the coordinates u, y are non-compact). The curvature diverges at r = χ = 0 (see (49)). The curvature vanishes as r → ∞ so the spacetime is asymptotically locally flat. Obviously this special case is a limit of the generic case with F (r) = 0. One can obtain the solution as a limit of a regular Myers-Perry black hole solution. Start from a single spinning Myers-Perry black hole, which has Λ = E 0 = 0 and A 0 > 0 and C 0 > 0. Using the scaling freedom (12) we set A 0 = 1/2. Then the MP spin parameter is a = √ C 0 and the mass parameter is µ 0 . It has a regular horizon provided µ 0 > a 2 . The extremal solution with µ 0 = a 2 is nakedly singular. To take the limit, perform the rescaling (12) and take the limit λ → ∞ with µ 0 /λ 4 fixed and (µ 0 − a 2 )/λ 4 → 0. This corresponds to scaling the Myers-Perry black hole towards the extremal solution whilst simultaneously taking its mass to infinity. 10 4.2 Case 2: dχ ≡ 0 In the metric (7) written as in (234), set and let σ = du + 2χA.
The local symmetries of g (2) , which has constant curvature, extend to symmetries of the full metric (260), hence this metric is cohomogeneity-1, where the surfaces of (local) homogeneity are surfaces of constant r.
If E 0 = 0 then the Killing vector field ∂/∂q is hypersurface orthogonal. The scaling symmetry (236) can be used to eliminate one parameter, so these form a 1-parameter family. These solutions were discussed in Ref. [29].
To analyse the metric (260), consider the case for which R (2) > 0. Perform the coordinate transformation p = χψ − 8E 0 χR (2) q, where the coefficient of the second term is chosen to eliminate the O(r 2 ) terms in the g qψ component of the resulting metric. After defining new coordinates ρ = r 2 + χ 2 and t = χq, the metric is where and Ω(ρ) = 8µ 0 E 0 ρ 4 γ(ρ)R (2) .
We can use the scaling freedom (236) to set R (2) = 8 so (9) implies Λχ 2 /4 ≤ 1. If this inequality is strict (which is always the case for Λ ≤ 0) then the above metric is the Kerr-de Sitter solution with equal rotation parameters as written in [30], where the 2 parameters (spin and mass) are If the inequality is saturated, i.e., Λχ 2 /4 = 1 (only possible for Λ > 0) then we have E 0 = 0. We see that this solution can be obtained by taking a limit of the Kerr-de Sitter solution in which a → ∞, M → 0 with M a 2 approaching a finite limit. If Λ > 0 then R (2) > 0 is the only possibility. If Λ = 0 then it is also possible to have R (2) = 0, which requires E 0 = 0. If Λ < 0 then we can have R (2) = 0 (with E 0 = 0) or R (2) < 0. Note that the coordinate transformation (262) that brings the metric to the form (263) is valid also if R (2) < 0.