Kerr-Schild Structure and Harmonic 2-forms on (A)dS-Kerr-NUT Metrics

We demonstrate that the general (A)dS-Kerr-NUT solutions in D dimensions with ([D/2], [(D+1)/2]) signature admit [D/2] linearly-independent, mutually-orthogonal and affinely-parameterised null geodesic congruences. This enables us to write the metrics in a multi-Kerr-Schild form, where the mass and all of the NUT parameters enter the metrics linearly. In the case of D=2n, we also obtain n harmonic 2-forms, which can be viewed as charged (A)dS-Kerr-NUT solution at the linear level of small-charge expansion, for the higher-dimensional Einstein-Maxwell theories. In the BPS limit, these 2-forms reduce to n-1 linearly-independent ones, whilst the resulting Calabi-Yau metric acquires a Kahler 2-form, leaving the total number the same.


Introduction
One intriguing feature of General Relativity is that, despite its high degree of non-linearity, many exact solutions can be cast into a Kerr-Schild form [1] where non-trivial parameters such as mass, charge, or cosmological constant enter the metrics as a linear perturbation of flat spacetime. A simple example is the (A)dS metric, which can be written as where the first three terms describe the (n + 2)-dimensional Minkowski spacetime and the cosmological constant enters the last term linearly. More complicated examples include the Plebanski metric [2]; in (2,2) signature, the Plebanski metric can have a double Kerr-Schild form where both the mass and the NUT charge enter the metric linearly [3].
The most general higher-dimensional (A)dS-Kerr-NUT solutions, which can be viewed as higher-dimensional generalisations of the Plebanski metric, were recently obtained in [4]. The solutions are parameterised by the mass, multiple NUT charges and arbitrary orthogonal rotations. The metrics have U (1) n isometries, where n = [(D + 1)/2]. They are demonstrated [5] to be of type D in the higher-dimensional generalisation [6] of the Petrov classification.
Many further interesting properties of the metrics were obtained, such as the separability of the Hamiltonian-Jacobi and Klein-Gordon equations [7], and the existence of Killing-Yano tensors [8]. The metrics also admit BPS limits where the Killing spinors can emerge [4]. In the odd 2n + 1 dimensions, this leads to a large class of Einstein-Sasaki metrics with U (1) n isometry, generalising the previously known Y p,q [9] and L pqr [10] spaces. In the even 2n dimensions, this leads to the non-compact Calabi-Yau metrics that can provide a resolution of the cone over the Einstein-Sasaki metrics constructed in the odd dimensions [11,12].
In this letter, we demonstrate in section 2 that the D-dimensional This enables us to cast the metric into the multi-Kerr-Schild form, where the mass and all of the NUT parameters enter the metric linearly. In section 3, we obtain n harmonic 2-forms on the (A)dS-Kerr-NUT metrics in D = 2n dimensions. In the BPS limit, these n harmonic 2-forms becomes linearly dependent, and the number of linearly-independent ones becomes n − 1. However, a Kähler 2-form emerges under the BPS limit, and hence the total number of harmonic 2-forms remains n. We conclude the letter in section 4.

Multi-Kerr-Schild structure
Let us first consider the case of D = 2n+1 dimensions, for which the metric was given in [4].
In order to put the metric in a Kerr-Schild form, it is necessary to Wick rotate to (n, n + 1) signature. This can be easily achieved by Wick rotating all the spatial U (1) coordinates.
The corresponding metric is then given by where The prime on the summation and product symbols in the definition of A (k) µ and U µ indicates that the index value µ is omitted in the summations of the ν indices over the range [1, n].
Note that ψ 0 was denoted as t in [4], playing the rôle of the time like coordinate in the (1, 2n) spacetime signature. In this way of writing the metric, all of the integration constants of the solution enter only in the functions X µ . The constant c n = (−1) n Λ is fixed by the value of the cosmological constant, with R µν = 2nΛ g µν . The other 2n constants c k , c and b µ are arbitrary. These are related to the n rotation parameters, the mass and the (n − 1) NUT parameters, with one parameter being trivial and removable through a scaling symmetry [4]. Note that in (n, n + 1) signature, the NUT charges are really masses with respect to different time-like Killing vectors. However, we shall continue to refer them as NUT charges.
We now re-arrange the metric (2) into the form If we perform the following coordinate transformation, the metric can then be cast into the n-Kerr-Schild form, namely where It is straightforward to verify that the metric ds 2 is that of pure (A)dS spacetime. The mass and NUT parameters b µ appear linearly in the metric ds 2 . It should be emphasised that although the constants c and c k with k < n are trivial in the metric ds 2 , they provide non-trivial angular momentum parameters in the metric ds 2 . It is interesting to note that all of the constants c k , including c n that is related to the cosmological constant, appear linearly in the metric, and can all be extracted from ds 2 and grouped in the second term of (6). This implies that all the parameters, the mass, NUTs and angular momenta and cosmological constant can enter the metric linearly as a perturbation of flat spacetime. In this letter, we shall consider in detail only the Kerr-Schild form where the mass and NUT parameters enter the metric linearly as a perturbation of pure (A)dS spacetime.
The (A)dS metric (7) can be diagonalised, in a way that the second term of (6) remains simple. To do so, let us first rewrite theX µ as follows Then we complete the square in ds 2 : and make the coordinate transformation, The metric can be put into a new form, where Performing a recombination of the U (1) coordinates, namely the odd dimensional (A)dS-Kerr-NUT metrics can be expressed as where If we set all but one of the b µ to zero, the result reduces to the Kerr-Schild form for rotating (A)dS black holes obtained previously in [13].
We now turn our attention to the the case of D = 2n dimensions. The corresponding (A)dS-Kerr-NUT metrics were obtained in [4]. After performing Wick rotations, the metric with (n, n) signature is given by where we Q µ , U µ and A (k) µ have the same form as those in the even dimensions, given in (3). The functions X µ are given by The constants c k and b µ are arbitrary, except for c n = (−1) n Λ, which is fixed by the value of the cosmological constant, R µν = (2n − 1)Λ g µν . The metric can be re-arranged into the form After performing the coordinate transformation the metric can be cast into the n-Kerr-Schild form, where It is straightforward to verify that ds 2 is the metric for pure (A)dS spacetime. As in the odd dimensions, this metric can be put into a diagonal form, while keeping the second term of (23) simple. To do that, we first reparameterise X µ as We then complete the square in ds 2 , i.e.
and make the coordinate transformation The metric (23) can then be put into a new form: where The ds 2 metric can now straightforwardly be diagonalised by means of the coordinate The even dimensional (A)-dS Kerr-NUT metrics can now be expressed as where where Ξ i , γ i and W have the same structure as that in the even dimensions, given by (18).
When all but one of the b µ vanishes, the metric reduces to the Kerr-Schild form of the rotating (A)dS black hole obtained in [13].
To summarise, we find that in both even and odd dimensions, the (A)dS-Kerr-NUT solution can be cast into the following multi-Kerr-Schild form: where f (x µ ) = 1 for odd dimensions and f (x µ ) = x µ for even dimensions. The vectors k (µ)α are n linearly-independent, mutually-orthogonal and affinely-parameterised null geodesic congruences, satisfying Note that the index α in k α(µ) can be raised with either g αβ orḡ αβ for the above conditions to be satisfied.
The potentials have a rather simple form, given by The metric (19) admits a natural vielbein basis, namely In this vielbein basis, the harmonic 2-forms G (µ) (2) are given by where the coefficients are We verify with low-lying examples that all of the G (µ) (2) are harmonic, i.e. dG (µ) (2) . It is worth observing that these 2-forms are harmonic regardless of the detailed structure of the functions X µ .

It was shown in [4] that the BPS limit of the metric (19) gives rise to the non-compact
Calabi-Yau metric that can provide a resolutions of the cone over the Einstein-Sasaki spaces.
Under suitable coordinate transformation, the metric is given by where we define Note that we have Wick rotated the metric to have Euclidean signature. We can choose the same form of the vielbein basis as in (38). The Kähler 2-form is then given by The 1-form potentials for the harmonic 2-forms are given by The corresponding harmonic 2-forms G (µ) (2) have the same form as in (39), with the functions f (µ) ν are given by Note that G (µ) (2) satisfy the linear relation n µ=1 G (µ) (2) = 0. Thus, in the BPS limit, there are (n − 1) linearly independent such harmonic 2-forms. Together with the Kähler 2-form, the total number of harmonic 2-forms is n again.

Conclusion
In this letter, we explicitly express the general (A)dS-Kerr-NUT metrics in Kerr-Schild form for both even and odd dimensions. We demonstrate that, in a suitable coordinate system the mass, NUT and angular momentum parameters enter linearly in the metric, and hence they can be viewed as a linear perturbation of pure (A)dS spacetime.
We also obtain n harmonic 2-forms on the 2n-dimensional (A)dS-Kerr-NUT metrics.
An interesting property of these harmonic 2-forms is that the closure and co-closure do not depend on the detailed structure of the functions X µ . This provides a potential ansatz for charged (A)dS-Kerr-NUT solutions for pure Einstein-Maxwell theories in higher dimensions, whose explicit analytical solutions remain elusive. In the case of four dimensions, the backreaction of the gauge field to the Einstein equations gives precisely the charged Plebanski metric [2], where only the functions X µ in the metric have extra contributions from the electric and magnetic charges. However, the same phenomenon does not occur in higher dimensions; nevertheless, the harmonic 2-forms we constructed can be viewed as charged (A)dS-Kerr-NUT solutions at the linear level for small-charge expansion. Together with the charged slowly-rotating black holes obtained in [14,15], our results may lead to the general charged (A)dS-Kerr-NUT solutions.