Deformed and twisted black holes with NUTs

In what follows, we concentrate on analyzing the even-dimensional Kerr-NUT-(A)dS spacetimes⁵ with the spacetime dimensions parameterized as $D=2N$. The metric of such spacetime takes the following form [16] ⁶ :

$$\begin{eqnarray}&&{\boldsymbol{g}}=\displaystyle \sum _{\mu }\left[\;\displaystyle \frac{{U}_{\mu }}{{X}_{\mu }}\;{\boldsymbol{d}}{x}_{\mu }^{2}+\displaystyle \frac{{X}_{\mu }}{{U}_{\mu }}{\left(\displaystyle \sum _{k}{A}_{\mu }^{(k)}{\boldsymbol{d}}{\psi }_{k}\right)}^{2}\;\right].\end{eqnarray} \tag{ 2.1 }$$

A variety of the off-shell metrics is hidden in the metric functions, X_μ, each of which depends just on the corresponding coordinate x_μ,

$$\begin{eqnarray}&&{X}_{\mu }={X}_{\mu }({x}_{\mu }).\end{eqnarray} \tag{ 2.2 }$$

The vacuum Einstein equations restrict these metric functions to be particular polynomials ${X}_{\mu }={{ \mathcal X }}_{\mu }^{(0)}$, where

$$\begin{eqnarray}&&{{ \mathcal X }}_{\mu }^{(p)}=\lambda \left[{ \mathcal J }({x}_{\mu }^{2})-2{\beta }_{\mu }{a}_{\mu }^{2}\;{{ \mathcal U }}_{\mu }\;{\left(\displaystyle \frac{{x}_{\mu }}{{a}_{\mu }}\right)}^{1-2p}\right].\end{eqnarray} \tag{ 2.3 }$$

With this choice of X_μ we call the Kerr-NUT-(A)dS metric (2.1) on-shell, highlighting the fact that it solves the Einstein equations. (Polynomials ${{ \mathcal X }}_{\mu }^{(p)}$ with general p will emerge in the limiting process studied in the next section.)

The remaining metric functions ${A}_{\mu }^{(k)}$, ${U}_{\mu }$ and auxiliary functions ${J}_{\mu }({a}^{2})$ are explicit polynomials of coordinates x_μ, defined by the following relations⁷ :

$$\begin{eqnarray}&&{J}_{\mu }({a}^{2})=\displaystyle \prod _{\displaystyle \genfrac{}{}{0em}{}{\nu }{\nu \ne \mu }}({x}_{\nu }^{2}-{a}^{2})=\displaystyle \sum _{k}{A}_{\mu }^{(k)}{(-{a}^{2})}^{N-1-k},\end{eqnarray} \tag{ 2.4 }$$

$$\begin{eqnarray}&&{U}_{\mu }={J}_{\mu }({x}_{\mu }^{2}).\end{eqnarray} \tag{ 2.5 }$$

We also define auxiliary functions $J({a}^{2})$ and ${A}^{(k)}$,

$$\begin{eqnarray}&&J({a}^{2})=\displaystyle \prod _{\nu }({x}_{\nu }^{2}-{a}^{2})=\displaystyle \sum _{k=0,\ldots ,N}{A}^{(k)}{(-{a}^{2})}^{N-k}.\end{eqnarray} \tag{ 2.6 }$$

Properties of these functions are listed in appendix B.1.

Inspecting the polynomials (2.3), we see that the on-shell Kerr-NUT-(A)dS metric contains N parameters a_μ, these are related to rotations. They can be combined into polynomials ${{ \mathcal J }}_{\mu }({x}^{2})$, ${{ \mathcal A }}_{\mu }^{(k)}$, ${{ \mathcal U }}_{\mu }$, ${ \mathcal J }({x}^{2})$, and ${{ \mathcal A }}^{(k)}$ defined in a manner analogous to functions ${J}_{\mu }({a}^{2})$, ${A}_{\mu }^{(k)}$, ${U}_{\mu }$, $J({a}^{2})$, and ${A}^{(k)}$, (2.4)–(2.6), with x_μ and a_μ 'interchanged'. For example,

$$\begin{eqnarray}&&{{ \mathcal J }}_{\mu }({x}^{2})=\displaystyle \prod _{\displaystyle \genfrac{}{}{0em}{}{\nu }{\nu \ne \mu }}({a}_{\nu }^{2}-{x}^{2})=\displaystyle \sum _{k}{{ \mathcal A }}_{\mu }^{(k)}{(-{x}^{2})}^{N-1-k},\end{eqnarray} \tag{ 2.7 }$$

$$\begin{eqnarray}&&{{ \mathcal U }}_{\mu }={{ \mathcal J }}_{\mu }({a}_{\mu }^{2}),\end{eqnarray} \tag{ 2.8 }$$

see appendix B.1.

The on-shell metric also contains the parameter λ related to the cosmological constant according to

$$\begin{eqnarray}&&{\rm{\Lambda }}=(2N-1)(N-1)\lambda ,\end{eqnarray} \tag{ 2.9 }$$

and parameters ${\beta }_{\mu }$ which encode mass and what is usually vaguely called NUT parameters or NUT charges.

Let us note here that we took advantage of the 'vagueness' of NUT charges ${\beta }_{\mu }$ and introduced a particular combination of parameters ${\beta }_{\mu }{{ \mathcal U }}_{\mu }{a}_{\mu }^{2p+1}$ in (2.3) instead of writing just a simple parameter b_μ, which is common practice. The reason is that parameters ${\beta }_{\mu }$ introduced in this way will have a trivial scaling behavior under the limit performed below whereas parameters b_μ would have to be nontrivially rescaled to obtain the desired result.

The metric (2.1) can also be expressed using a different set of angular coordinates ${\phi }_{\alpha }$,

$$\begin{eqnarray}&&{\phi }_{\alpha }=\lambda {a}_{\alpha }\displaystyle \sum _{k}{{ \mathcal A }}_{\alpha }^{(k)}{\psi }_{k},\quad {\psi }_{k}=\displaystyle \sum _{\alpha }\displaystyle \frac{{(-{a}_{\alpha }^{2})}^{N-1-k}}{{{ \mathcal U }}_{\alpha }}\displaystyle \frac{{\phi }_{\alpha }}{\lambda {a}_{\alpha }}.\end{eqnarray} \tag{ 2.10 }$$

Since ϕ's are just constant linear combinations of ψ's, they are also Killing coordinates. In the maximally symmetric case, they are related to N independent planes of rotations. Using these angles, the metric takes the form

$$\begin{eqnarray}&&{\boldsymbol{g}}=\displaystyle \sum _{\mu }\left[\;\displaystyle \frac{{U}_{\mu }}{{X}_{\mu }}\;{\boldsymbol{d}}{x}_{\mu }^{2}+\displaystyle \frac{{X}_{\mu }}{{U}_{\mu }}{\left(\displaystyle \sum _{\alpha }\displaystyle \frac{{J}_{\mu }({a}_{\alpha }^{2})}{{{ \mathcal U }}_{\alpha }}\displaystyle \frac{1}{\lambda {a}_{\alpha }}{\boldsymbol{d}}{\phi }_{\alpha }\right)}^{2}\;\right].\end{eqnarray} \tag{ 2.11 }$$

Summarizing the previous subsection, the on-shell Kerr-NUT-(A)dS geometry is given by the metric (2.1) or (2.11) with X_μ given by ${{ \mathcal X }}_{\mu }^{(0)}$, (2.3). The geometry is labeled by parameters a_μ and ${\beta }_{\mu }$. The remaining parameter λ is fixed by the cosmological constant through (2.9).

However, the parameters a_μ and ${\beta }_{\mu }$ are not independent. There exist a one-parametric gauge freedom in rescaling coordinates, metric functions, and parameters which leaves the metric in the same form. Namely, it reads

$$\begin{eqnarray}&&{x}_{\mu }\to {{sx}}_{\mu },\quad {\phi }_{\alpha }\to {\phi }_{\alpha },\quad {\psi }_{k}\to {s}^{-(k+1)}{\psi }_{k},\\ &&{a}_{\mu }\to {{sa}}_{\mu },\quad {\beta }_{\mu }\to {\beta }_{\mu },\\ &&{X}_{\mu }\to {s}^{2N}{X}_{\mu },\quad {U}_{\mu }\to {s}^{2(N-1)}{U}_{\mu },\quad {A}_{\mu }^{(k)}\to {s}^{2k}{A}^{(k)}.\end{eqnarray} \tag{ 2.12 }$$

One of the parameters a_μ can thus be set to a suitable value using this gauge freedom. We will use this freedom in the Lorentzian case, requiring gauge condition (2.28) below. Thus, for a fixed cosmological constant the on-shell Kerr-NUT-(A)dS metrics form ($2N-1)$-parametric family of solutions of the Einstein equations.

Inspecting metric (2.11) with ${X}_{\mu }={{ \mathcal X }}_{\mu }^{(0)}$, we can also observe that the dependence on the cosmological parameter λ is just a global scaling. Indeed, one can easily see that

$$\begin{eqnarray}&&\lambda {\boldsymbol{g}}={\boldsymbol{g}}{| }_{\lambda =1}.\end{eqnarray} \tag{ 2.13 }$$

We will use this property mainly in the Euclidean case to eliminate a global scale in one part of the warped metric, see (3.16).

It is useful to introduce the following orthonormal frame of 1-forms:

$$\begin{eqnarray}&&{{\boldsymbol{e}}}^{\mu }={\left(\displaystyle \frac{{U}_{\mu }}{{X}_{\mu }}\right)}^{\displaystyle \frac{1}{2}}{\boldsymbol{d}}{x}_{\mu },\\ &&{\hat{{\boldsymbol{e}}}}^{\mu }={\left(\displaystyle \frac{{X}_{\mu }}{{U}_{\mu }}\right)}^{\displaystyle \frac{1}{2}}\displaystyle \sum _{k}{A}_{\mu }^{(k)}{\boldsymbol{d}}{\psi }_{k}={\left(\displaystyle \frac{{X}_{\mu }}{{U}_{\mu }}\right)}^{\displaystyle \frac{1}{2}}\displaystyle \sum _{\alpha }\displaystyle \frac{{J}_{\mu }({a}_{\alpha }^{2})}{{{ \mathcal U }}_{\alpha }}\displaystyle \frac{{\boldsymbol{d}}{\phi }_{\alpha }}{\lambda {a}_{\alpha }},\end{eqnarray} \tag{ 2.14 }$$

and the dual frame of vectors:

$$\begin{eqnarray}&&{{\boldsymbol{e}}}_{\mu }={\left(\displaystyle \frac{{X}_{\mu }}{{U}_{\mu }}\right)}^{\displaystyle \frac{1}{2}}{{\boldsymbol{\partial }}}_{{x}_{\mu }},{\hat{{\boldsymbol{e}}}}_{\mu }={\left(\displaystyle \frac{{U}_{\mu }}{{X}_{\mu }}\right)}^{\displaystyle \frac{1}{2}}\displaystyle \sum _{k}\displaystyle \frac{{(-{x}_{\mu }^{2})}^{N-1-k}}{{U}_{\mu }},\;{{\boldsymbol{\partial }}}_{{\psi }_{k}}={\left(\displaystyle \frac{{U}_{\mu }}{{X}_{\mu }}\right)}^{\displaystyle \frac{1}{2}}\displaystyle \sum _{\alpha }\displaystyle \frac{{{ \mathcal J }}_{\alpha }({x}_{\mu }^{2})}{{U}_{\mu }}\;\lambda {a}_{\alpha }\;{{\boldsymbol{\partial }}}_{{\phi }_{\alpha }}.\end{eqnarray} \tag{ 2.15 }$$

The form (2.1) of the metric indicates that there is a natural splitting of the tangent space into 2-planes spanned on ${{\boldsymbol{e}}}_{\mu }$ and ${\hat{{\boldsymbol{e}}}}_{\mu }$. These planes, which we call the μ-planes, are characterized by antisymmetric 2-forms

$$\begin{eqnarray}&&{{\boldsymbol{\omega }}}^{\mu }={{\boldsymbol{e}}}^{\mu }\wedge {\hat{{\boldsymbol{e}}}}^{\mu }.\end{eqnarray} \tag{ 2.16 }$$

The metric and the corresponding Levi-Civita tensor are then written as

$$\begin{eqnarray}&&{\boldsymbol{g}}=\displaystyle \sum _{\mu }({{\boldsymbol{e}}}^{\mu }{{\boldsymbol{e}}}^{\mu }+{\hat{{\boldsymbol{e}}}}^{\mu }{\hat{{\boldsymbol{e}}}}^{\mu }),\quad {\boldsymbol{\varepsilon }}={{\boldsymbol{\omega }}}^{1}\wedge \cdots \wedge {{\boldsymbol{\omega }}}^{N}.\end{eqnarray} \tag{ 2.17 }$$

Although the expression (2.17) suggests that the (off-shell) Kerr-NUT-(A)dS metric is positive definite, it can actually describe geometries of various signatures. The normalization in (2.14) and (2.15) can be only 'artificial' and the positively normalized vectors and 1-forms can be, in fact, imaginary. The real signature is given by signs of the metric functions ${X}_{\mu }/{U}_{\mu }$ and these depend on a particular choice of coordinate ranges. The signature can also be affected if we choose some of the coordinates to be Wick-rotated, i.e. purely imaginary.

We will be mainly interested in (i) geometries with the Euclidean signature and positive curvature and (ii) physical geometries of the Lorentzian signature and arbitrary sign of the curvature. Before specifying the coordinate ranges and Wick rotations in these cases, let us first identify the maximally symmetric geometry.

It has been already observed in [47] that the class of Kerr-NUT-(A)dS geometries contains also a trivial case of the maximally symmetric spaces. It is obtained by setting the NUT and mass parameters equal to zero, ${\beta }_{\mu }=0$, while the parameters a_μ remain unrestricted. The metric functions X_μ then simplify into a common polynomial $\lambda { \mathcal J }({x}^{2})$ in the corresponding variable, ${X}_{\mu }=\lambda { \mathcal J }({x}_{\mu }^{2})$. The roots of this polynomial are exactly the parameters ${a}_{\mu }^{2}$.

With this choice and employing the orthogonality relations (B23), the metric (2.11) can be transformed into

$$\begin{eqnarray}&&{\boldsymbol{g}}=\displaystyle \sum _{\mu }\left[\;\displaystyle \frac{{U}_{\mu }}{\lambda { \mathcal J }({x}_{\mu }^{2})}\;{\boldsymbol{d}}{x}_{\mu }^{2}-\displaystyle \frac{J({a}_{\mu }^{2})}{{{ \mathcal U }}_{\mu }}\displaystyle \frac{1}{\lambda {a}_{\mu }^{2}}{\boldsymbol{d}}{\phi }_{\mu }^{2}\;\right].\end{eqnarray} \tag{ 2.18 }$$

Introducing $N+1$ coordinates ${\rho }_{\mu }$, $\mu =0,1,\ldots ,N$, instead of N coordinates x_μ, and employing the Jacobi transformation⁸

$$\begin{eqnarray}&&\lambda {\rho }_{\mu }^{2}=\displaystyle \frac{J({a}_{\mu }^{2})}{-{a}_{\mu }^{2}\;{{ \mathcal U }}_{\mu }},\quad \lambda {\rho }_{0}^{2}=\displaystyle \frac{J(0)}{{ \mathcal J }(0)}=\displaystyle \frac{{A}^{(N)}}{{{ \mathcal A }}^{(N)}},\end{eqnarray} \tag{ 2.19 }$$

one can show that the new coordinates are restricted by the following constraint:

$$\begin{eqnarray}&&\lambda \displaystyle \sum _{\mu =0}^{N}{\rho }_{\mu }^{2}=1,\end{eqnarray} \tag{ 2.20 }$$

while the metric takes a simple form

$$\begin{eqnarray}&&{\boldsymbol{g}}={\boldsymbol{d}}{\rho }_{0}^{2}+\displaystyle \sum _{\mu }[{\boldsymbol{d}}{\rho }_{\mu }^{2}+{\rho }_{\mu }^{2}\;{\boldsymbol{d}}{\phi }_{\mu }^{2}].\end{eqnarray} \tag{ 2.21 }$$

Clearly, ${\rho }_{0}$, ${\rho }_{\mu }$, ${\phi }_{\mu }$ are multi-cylindrical coordinates on a $2N$-dimensional (pseudo-)sphere, given by the constraint (2.20), embedded into a $(2N+1)$-dimensional flat space.

Assuming that all coordinates x_μ, ${\psi }_{k}$ are real and parameters a_μ, ${\beta }_{\mu }$, λ are positive, the signature of the metric (2.1) is determined by signs of the metric functions ${X}_{\mu }/{U}_{\mu }$. To determine these signs we need to specify ranges of coordinates x_μ. All these coordinates, as well as corresponding parameters a_μ and ${\beta }_{\mu }$, enter our metric in an equivalent way. However, it will be useful to relabel these quantities in such a way that parameters a_μ are ordered as follows,

$$\begin{eqnarray}&&0\lt {a}_{1}\lt {a}_{2}\lt \ldots \;\lt \;{a}_{N}.\end{eqnarray} \tag{ 2.22 }$$

In the maximally symmetric case ${\beta }_{\mu }=0$, parameters a_μ coincide with the roots of the polynomials X_μ, and determine axes as well as restrict the ranges of coordinates x_μ. Zeros in functions ${U}_{\mu }$ can be avoided by choosing

$$\begin{eqnarray}&&-{a}_{1}\lt {x}_{1}\lt {a}_{1}\lt {x}_{2}\lt {a}_{2}\lt \ldots \;\lt \;{x}_{N}\lt {a}_{N}.\end{eqnarray} \tag{ 2.23 }$$

With angular coordinates restricted to a circle

$$\begin{eqnarray}&&-\pi \lt {\phi }_{\mu }\lt \pi ,\end{eqnarray} \tag{ 2.24 }$$

the metric (2.11) describes a homogeneous metric on the Euclidean sphere with radius ${\ell }=1/\sqrt{\lambda }$, that is written in 'multi-elliptic' coordinates parameterized by constants a_μ.

For sufficiently small positive values of parameters ${\beta }_{\mu }$, the roots of metric functions X_μ change only slightly; they determine a more narrow range, now different for each latitude coordinate x_μ. Namely, we require

$$\begin{eqnarray}&&{}^{(-)}{x}_{\mu }\lt {x}_{\mu }\lt {}^{(+)}{x}_{\mu },\end{eqnarray} \tag{ 2.25 }$$

where ${}^{(\pm )}{x}_{\mu }$ are roots of X_μ such that ${a}_{\mu -1}\lt {}^{(-)}{x}_{\mu }\lt {}^{(+)}{x}_{\mu }\lt {a}_{\mu }$ (with ${}^{(-)}{x}_{1}\lt -{a}_{1}\lt {}^{(+)}{x}_{1}\lt {a}_{1}$ for $\mu =1$). With this choice the metric (2.1) represents the Euclidean instanton of signature $(+\;+\;\cdots \;+)$.

Parameters ${\beta }_{\mu }$ encode deformations of the geometry, namely how it deviates from the geometry of the sphere. Parameters a_μ are for non-vanishing ${\beta }_{\mu }$ also non-trivial—they do not label just a choice of coordinates, as in the maximally symmetric case, but they also change the geometry. For non-trivial values of ${\beta }_{\mu }$ the geometry does not have a curvature singularity—the singularity of the Riemann tensor occurs outside the coordinate range (2.23) of the Euclidean instanton.

The global definition and regularity of the geometry described by metrics (2.1) or (2.11) has to be concluded by specifying which Killing angles should be identified. Any linear combination of Killing coordinates (with constant coefficients) forms again a Killing coordinate and it is not a priori clear which of the Killing coordinates should be periodic. Typically, angles ${\psi }_{k}$ are not those which should be periodic. Since Killing coordinates are non-trivially coupled in the metric (the metric is not diagonal in these directions), a particular choice of the periodicity of Killing coordinates can introduce a non-trivial twisting of the geometry, as well as possible irregularities on the axes.

We will not discuss these characteristics in more detail and for our purposes we will simply remember that the Euclidean instanton describes a deformed and twisted spherical-like geometry. (For more details we refer the interested reader to a vast literature on the subject of gravitational instantons, e.g. [20–23, 48–55].)

The Lorentzian signature can be achieved by Wick-rotating some of the coordinates and parameters. Different choices can lead to different interpretations of the metric. We restrict ourselves to the case where the coordinate x_N is Wick-rotated to a radial coordinate r and the angular coordinate ${\phi }_{N}$ to the time coordinate t. The remaining coordinates retain their original character. Namely, we set

$$\begin{eqnarray}&&{x}_{N}^{2}=-{r}^{2},\quad {\phi }_{N}=\lambda {a}_{N}t,\end{eqnarray} \tag{ 2.26 }$$

with r and t real, and introduce a real mass parameter m by

$$\begin{eqnarray}&&{\beta }_{N}=m{\lambda }^{N-\displaystyle \frac{3}{2}}.\end{eqnarray} \tag{ 2.27 }$$

Notice that all coordinates ${\psi }_{k}$ remain real. We also use the invariance (2.12) of the metric to correlate a_N with λ,

$$\begin{eqnarray}&&{a}_{N}^{2}=-\displaystyle \frac{1}{\lambda }.\end{eqnarray} \tag{ 2.28 }$$

Other parameters a_μ remain restricted as in the Euclidean case

$$\begin{eqnarray}&&0\lt {a}_{1}\lt {a}_{2}\lt \cdots \;\lt \;{a}_{N-1}\lt {a}_{N},\end{eqnarray} \tag{ 2.29 }$$

where the last inequality applies only for a_N real (i.e. for $\lambda \lt 0$). The ranges of coordinates ${x}_{\bar{\mu }}$, $\bar{\mu }=1,\ldots ,N-1$, are given again by roots of functions ${X}_{\bar{\mu }}$, as in (2.25).

With these choices metric (2.11) has the physical signature $(-+\;\cdots \;+)$. More explicitly, it reads

$$\begin{eqnarray}{\boldsymbol{g}} & = & -\quad \displaystyle \frac{{\rm{\Delta }}}{{\rm{\Sigma }}}{\left(\displaystyle \prod _{\bar{\nu }}\displaystyle \frac{1+\lambda {x}_{\bar{\nu }}^{2}}{1+\lambda {a}_{\bar{\nu }}^{2}}{\boldsymbol{d}}t-\displaystyle \sum _{\bar{\nu }}\displaystyle \frac{\bar{J}({a}_{\bar{\nu }}^{2})}{{a}_{\bar{\nu }}(1+\lambda {a}_{\bar{\nu }}^{2}){\bar{{ \mathcal U }}}_{\bar{\nu }}}{\boldsymbol{d}}{\phi }_{\bar{\nu }}\right)}^{2}+\displaystyle \frac{{\rm{\Sigma }}}{{\rm{\Delta }}}\;{\boldsymbol{d}}{r}^{2}+\displaystyle \sum _{\bar{\mu }}\displaystyle \frac{({r}^{2}+{x}_{\bar{\mu }}^{2}){\bar{U}}_{\bar{\mu }}}{-{{ \mathcal X }}_{\bar{\mu }}^{(0)}}\;{\boldsymbol{d}}{x}_{\bar{\mu }}^{2}\\ & & +\displaystyle \sum _{\bar{\mu }}\displaystyle \frac{-{{ \mathcal X }}_{\bar{\mu }}^{(0)}}{({r}^{2}+{x}_{\bar{\mu }}^{2}){\bar{U}}_{\bar{\mu }}}{\left(\displaystyle \frac{1-\lambda {r}^{2}}{1+\lambda {x}_{\bar{\mu }}^{2}}\displaystyle \prod _{\bar{\nu }}\displaystyle \frac{1+\lambda {x}_{\bar{\nu }}^{2}}{1+\lambda {a}_{\bar{\nu }}^{2}}{\boldsymbol{d}}t+\displaystyle \sum _{\bar{\nu }}\displaystyle \frac{({r}^{2}+{a}_{\bar{\nu }}^{2}){\bar{J}}_{\bar{\mu }}({a}_{\bar{\nu }}^{2})}{{a}_{\bar{\nu }}(1+\lambda {a}_{\bar{\nu }}^{2}){\bar{{ \mathcal U }}}_{\bar{\nu }}}{\boldsymbol{d}}{\phi }_{\bar{\nu }}\right)}^{2},\end{eqnarray} \tag{ 2.30 }$$

with

$$\begin{eqnarray}&&{\rm{\Delta }}=-{{ \mathcal X }}_{N}^{(0)}=(1-\lambda {r}^{2})\displaystyle \prod _{\bar{\nu }}({r}^{2}+{a}_{\bar{\nu }}^{2})-2m\;r\displaystyle \prod _{\bar{\nu }}(1+\lambda {a}_{\bar{\nu }}^{2}),\\ &&-{{ \mathcal X }}_{\bar{\mu }}^{(0)}=(1+\lambda {x}_{\bar{\mu }}^{2})\bar{{ \mathcal J }}({x}_{\bar{\mu }}^{2})-2{\beta }_{\bar{\mu }}{a}_{\bar{\mu }}(1+\lambda {a}_{\bar{\mu }}^{2}){\bar{{ \mathcal U }}}_{\bar{\mu }}\;{x}_{\bar{\mu }},\\ &&\phantom{\rule{4em}{0ex}}{\rm{\Sigma }}={U}_{N}=\displaystyle \prod _{\bar{\nu }}({r}^{2}+{x}_{\bar{\nu }}^{2}).\end{eqnarray} \tag{ 2.31 }$$

Here, the barred indices take values $\bar{\mu },\bar{\nu }=1,\ldots ,N-1$ and functions ${\bar{J}}_{\bar{\mu }}$, ${\bar{U}}_{\bar{\mu }}$, ${\bar{{ \mathcal U }}}_{\bar{\mu }}$ and $\bar{{ \mathcal J }}$ are defined by expressions (2.4) with restricted sets of coordinates ${x}_{\bar{\mu }}$ and parameters ${a}_{\bar{\mu }}$ (without x_N and a_N).

The roots of the metric function ${\rm{\Delta }}$ determine horizons. For $\lambda \leqslant 0$ there can be two roots (outer and inner horizons), one double root (extremal horizon) or no roots (the case of naked singularity). For $\lambda \gt 0$ there is always one additional root representing the cosmological horizon. See figure 1.

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For $m\ne 0$ and ${\beta }_{\bar{\mu }}=0$, $\bar{\mu }=1,\ldots ,N-1$, the parameters ${a}_{\bar{\mu }}$, $\bar{\mu }=1,\ldots ,N-1$, can be identified with the rotation parameters of the black hole. However, for non-trivial NUT parameters ${\beta }_{\bar{\mu }}$ the interpretation is more unclear. Similar to the Euclidean instanton, both sets of parameters ${\beta }_{\bar{\mu }}$ and ${a}_{\bar{\mu }}$, as well as the specification of the periodicity of the Killing coordinates, affect the deformation of the geometry. Since the Lorentzian signature is involved, the geometry has in general NUT-like irregular behavior including the existence of closed time-like curves around the axes.

Let us conclude this section by a brief review of the symmetry structure of the (off-shell) Kerr-NUT-(A)dS spacetimes; see [36, 37, 39, 40] for more details. As mentioned in the introduction, the canonical form of Kerr-NUT-(A)dS spacetimes (2.1) is uniquely determined by the existence of a non-degenerate CCKY 2-form ${\boldsymbol{h}}$ [24, 25, 27] ⁹ . This 2-form reads

$$\begin{eqnarray}&&{\boldsymbol{h}}=\displaystyle \sum _{\mu }\;{x}_{\mu }{{\boldsymbol{\omega }}}^{\mu }=\displaystyle \sum _{\mu }\;{x}_{\mu }{\boldsymbol{d}}{x}_{\mu }\wedge \displaystyle \sum _{k}\;{A}_{\mu }^{(k)}{\boldsymbol{d}}{\psi }_{k},\end{eqnarray} \tag{ 2.32 }$$

and satisfies the CCKY equation

$$\begin{eqnarray}&&{{\rm{\nabla }}}^{a}{h}_{{bc}}={g}_{{ab}}{\xi }_{c}-{g}_{{ac}}{\xi }_{b},\end{eqnarray} \tag{ 2.33 }$$

with the primary Killing–Yano 1-form ${\boldsymbol{\xi }}$ given by

$$\begin{eqnarray}&&{\boldsymbol{\xi }}=\displaystyle \frac{1}{D-1}{\boldsymbol{\nabla }}\cdot {\boldsymbol{h}}=\displaystyle \sum _{\mu }\;\displaystyle \frac{{X}_{\mu }}{{U}_{\mu }}\displaystyle \sum _{k}\;{A}_{\mu }^{(k)}{\boldsymbol{d}}{\psi }_{k}.\end{eqnarray} \tag{ 2.34 }$$

Raising the index, ${\boldsymbol{\xi }}$ turns into a primary Killing vector

$$\begin{eqnarray}&&{\boldsymbol{\xi }}={{\boldsymbol{l}}}_{(0)}={{\boldsymbol{\partial }}}_{{\psi }_{0}}.\end{eqnarray} \tag{ 2.35 }$$

The non-degenerate CCKY 2-form ${\boldsymbol{h}}$ generates a full Killing tower of symmetries [36, 56]. Namely, it defines N CCKY forms ${{\boldsymbol{h}}}^{(k)}$, $k=0,\ldots ,N-1$, of rank $2k$,

$$\begin{eqnarray}&&{{\boldsymbol{h}}}^{(k)}=\displaystyle \frac{1}{k!}\displaystyle \frac{1}{\sqrt{{{ \mathcal A }}^{(k)}}}\;{{\boldsymbol{h}}}^{\wedge k}.\end{eqnarray} \tag{ 2.36 }$$

Each of these CCKY forms then defines the following Killing–Yano (KY) forms ${{\boldsymbol{f}}}^{(k)}$ of rank $2(N-k)$:

$$\begin{eqnarray}&&{{\boldsymbol{f}}}^{(k)}=\ast {{\boldsymbol{h}}}^{(k)},\end{eqnarray} \tag{ 2.37 }$$

and the following rank-2 Killing tensors ${{\boldsymbol{k}}}^{(k)}$:

$$\begin{eqnarray}&&{{\boldsymbol{k}}}^{(k)}={{\boldsymbol{f}}}^{(k)}\underset{2N-2k-1}{\bullet }\;{{\boldsymbol{f}}}^{(k)}.\end{eqnarray} \tag{ 2.38 }$$

Here $\underset{p}{\bullet }$ denotes the partial contraction of two antisymmetric forms in the first p indices divided by $p!$,

$$\begin{eqnarray}&&{({\boldsymbol{\alpha }}\underset{p}{\bullet }{\boldsymbol{\beta }})}_{a\ldots b\ldots }=\displaystyle \frac{1}{p!}\;{{\boldsymbol{\alpha }}}_{{n}_{1}\ldots {n}_{p}a\ldots }\;{{\boldsymbol{\beta }}}^{{n}_{1}\ldots {n}_{p}}{}_{b\ldots },\end{eqnarray} \tag{ 2.39 }$$

where the upper indices were raised using the metric. If no index p is indicated, the full contraction in all indices is assumed. The primary Killing vector ${\boldsymbol{\xi }}$ and the above Killing tensors then define N Killing vectors ${{\boldsymbol{l}}}_{(j)}$ according to

$$\begin{eqnarray}&&{{\boldsymbol{l}}}_{(j)}={{\boldsymbol{k}}}_{(j)}\cdot {\boldsymbol{\xi }}.\end{eqnarray} \tag{ 2.40 }$$

In particular, we find the following expressions for the Killing vectors and Killing tensors:

$$\begin{eqnarray}&&{{\boldsymbol{l}}}_{(j)}=\displaystyle \frac{1}{{{ \mathcal A }}^{(j)}}\;{{\boldsymbol{\partial }}}_{{\psi }_{j}},\end{eqnarray} \tag{ 2.41 }$$

$$\begin{eqnarray}{{\boldsymbol{k}}}_{(j)} & = & \displaystyle \frac{1}{{{ \mathcal A }}^{(j)}}\displaystyle \sum _{\mu }\;{A}_{\mu }^{(j)}({{\boldsymbol{e}}}_{\mu }{{\boldsymbol{e}}}_{\mu }+{\hat{{\boldsymbol{e}}}}_{\mu }{\hat{{\boldsymbol{e}}}}_{\mu })\\ & = & \displaystyle \sum _{\mu }\;\displaystyle \frac{{A}_{\mu }^{(j)}}{{{ \mathcal A }}^{(j)}}\left[\;\displaystyle \frac{{X}_{\mu }}{{U}_{\mu }}\;{{\boldsymbol{\partial }}}_{{x}_{\mu }}^{\ 2}+\displaystyle \frac{{U}_{\mu }}{{X}_{\mu }}{\left(\displaystyle \sum _{k}\displaystyle \frac{{(-{x}_{\mu }^{2})}^{N-1-k}}{{U}_{\mu }}{{\boldsymbol{\partial }}}_{{\psi }_{k}}\right)}^{2}\;\right].\end{eqnarray} \tag{ 2.42 }$$

Note that, in expressions (2.41) and (2.42) we have a normalization prefactor $1/{{ \mathcal A }}^{(j)}$ that originates from a definition (2.36). Such normalization is not used in the literature. Since this prefactor is just a constant (a combination of parameters a_μ), it does not influence the Killing property of the defined objects. However, it will be useful in the limiting procedure below.

The above Killing tower of symmetries stands behind many of the remarkable integrability properties of Kerr-NUT-(A)dS spacetimes, see [28–39] for more details. In particular, the N Killing vectors ${{\boldsymbol{l}}}_{(j)}$ together with the N rank-2 Killing tensors ${{\boldsymbol{k}}}_{(j)}$ define $2N$ integrals for geodesic motion that are independent and in involution, making this motion completely integrable in the Liouville sense [35–39].

This property remains true for the twisted and deformed black holes obtained by a limiting procedure in the next section. Namely, as we shall see, all the objects from the Killing tower do survive the limit and correspondingly obey the same mutual commutation relations that guarantee the above integrability.

Let us now concentrate on a situation when the Kerr-NUT-(A)dS spacetimes describe the Lorentzian black hole solution, studied in section 2.6. In this case we have the following ordering of coordinates x_μ and parameters a_μ:

$$\begin{eqnarray}&&{x}_{1}\lt {a}_{1}\lt {x}_{2}\lt {a}_{2}\lt \cdots \;\lt \;{x}_{N-1}\lt {a}_{N-1}.\end{eqnarray} \tag{ 3.1 }$$

Parameters a_μ are closely related to rotations: for vanishing NUT parameters they are exact rotations [18, 19]—in the presence of NUTs a proper identification is more delicate. Nevertheless, in both cases it seems that 'unspinning' the rotations is equivalent to sending these parameters to zero.

However, it is obvious from (3.1) that our metrics (2.1) or (2.30) are not suitable for a straightforward limit of vanishing parameters a_μ, or for a limit when such parameters coincide. If such limits were performed naively, some of the coordinates x_μ would degenerate. Since ${x}_{\mu }$ are eigen-values of the principal CCKY tensor ${\bf{h}}$, the latter becomes degenerate in this limit. But what is worse, the metric itself would not be well defined. In the following we show that a well defined limit ${a}_{\mu }\to 0$ can be achieved by a proper simultaneous scaling of corresponding coordinates x_μ. We first perform the limit for the canonical metric (2.1), then specialize to the black hole case.

Namely, let us consider a multiple-spin-zero limit in which the first $\bar{N}$ parameters ${a}_{\bar{\mu }}$ and the corresponding coordinates ${x}_{\bar{\mu }}$ are homogeneously rescaled to zero, while the remaining $\tilde{N}=N-\bar{N}$ parameters and coordinates are left unscaled but renamed to be indexed from 1 to $\tilde{N}$. The Killing coordinates ${\phi }_{\mu }$, NUT parameters ${\beta }_{\mu }$, and λ remain unscaled but they also get appropriately renamed. We denote by $\bar{{ \mathcal M }}$ a $2\bar{N}$-dimensional submanifold which emerges in the limiting procedure—it is spanned on the first $\bar{N}$ μ-planes. Since we are sending the corresponding rotations to zero, we call it the unspinned sector. The complementary $2\tilde{N}$-dimensional submanifold $\tilde{{ \mathcal M }}$ emerging as a space spanned on the unscaled μ-planes will be called the regular sector. Similarly, we say that the scaled (unscaled) coordinates and parameters belong to the unspinned (regular) sector, respectively.

If $\sigma$ is a scaling parameter which goes to zero, $\sigma \to 0$, the limit is characterized by

$$\begin{eqnarray}\mathrm{unspinned} & \quad \mathrm{regular}\\ {a}_{\bar{\mu }}=\sigma {\bar{a}}_{\mu }, & \quad {a}_{\bar{N}+\tilde{\mu }}={\tilde{a}}_{\tilde{\mu }},\\ {\beta }_{\bar{\mu }}={\bar{\beta }}_{\mu }, & \quad {\beta }_{\bar{N}+\tilde{\mu }}={\tilde{\beta }}_{\tilde{\mu }},\end{eqnarray} \tag{ 3.2 }$$

$$\begin{eqnarray}{x}_{\bar{\mu }}=\sigma {\bar{x}}_{\mu }, & \quad {x}_{\bar{N}+\tilde{\mu }}={\tilde{x}}_{\tilde{\mu }},\\ {\phi }_{\bar{\mu }}={\bar{\phi }}_{\mu }, & \quad {\phi }_{\bar{N}+\tilde{\mu }}={\tilde{\phi }}_{\tilde{\mu }}.\end{eqnarray} \tag{ 3.3 }$$

We use the barred indices $\bar{\mu },\bar{\nu },\ldots$ and $\bar{k},\bar{l},\ldots$ for the unspinned quantities, and the tilded indices $\tilde{\mu },\tilde{\nu },\ldots$ and $\tilde{k},\tilde{l}\;,\ldots$ for regular quantities. They take the following values:

$$\begin{eqnarray}&&\bar{\mu },\bar{\nu },\ldots \;=\;1,\ldots ,\bar{N},\tilde{\mu },\tilde{\nu },\ldots \;=\;1,\ldots \tilde{N},\\ &&\bar{k},\bar{l},\ldots \;=\;0,\ldots ,\bar{N}-1,\tilde{k},\tilde{l},\ldots \;=\;0,\ldots ,\tilde{N}-1.\end{eqnarray} \tag{ 3.4 }$$

Let $\approx$ denotes the equality valid up to the terms of the higher order in $\sigma$. Employing relations (2.10) between angular coordinates ${\phi }_{\mu }$ and ${\psi }_{k}$, we find

$$\begin{eqnarray}&&{\psi }_{\tilde{k}}\approx {\tilde{\psi }}_{\tilde{k}},\quad {\psi }_{\tilde{N}+\bar{k}}\approx {\sigma }^{-(2\bar{k}+1)}\displaystyle \frac{1}{{\tilde{{ \mathcal A }}}^{(\tilde{N})}}{\bar{\psi }}_{\bar{k}}.\end{eqnarray} \tag{ 3.5 }$$

Note that, comparing equations (3.3) and (3.5), the new coordinates are 'enumerated' differently for Latin and Greek indices. Namely, for Latin indices the unspinned sector corresponds to 'lower' values and the regular sector to 'upper' values of indices, whereas this is opposite for the Greek indices.

At the same time, metric functions (2.4) and (2.6) give

$$\begin{eqnarray}&&{U}_{\bar{\mu }}\approx {\sigma }^{2(\bar{N}-1)}{\tilde{A}}^{(\tilde{N})}{\bar{U}}_{\bar{\mu }},\quad {U}_{\bar{N}+\tilde{\mu }}\approx {(-{\tilde{x}}_{\tilde{\mu }}^{2})}^{\bar{N}}{\tilde{U}}_{\tilde{\mu }},\end{eqnarray} \tag{ 3.6 }$$

$$\begin{eqnarray}&&{A}_{\bar{\mu }}^{(\tilde{k})}\approx {\tilde{A}}^{(\tilde{k})},\phantom{\rule{4em}{0ex}}{A}_{\bar{\mu }}^{(\tilde{N}+\bar{k})}\approx {\sigma }^{2\bar{k}}{\tilde{A}}^{(\tilde{N})}{\bar{A}}_{\bar{\mu }}^{(\bar{k})},\\ &&{A}_{\bar{N}+\tilde{\mu }}^{(\tilde{k})}\approx {\tilde{A}}_{\tilde{\mu }}^{(\tilde{k})},\quad {A}_{\bar{N}+\tilde{\mu }}^{(\tilde{N}+\bar{k})}\approx {\sigma }^{2(\bar{k}+1)}{\tilde{A}}_{\tilde{\mu }}^{(\tilde{N}-1)}{\bar{A}}^{(\bar{k}+1)},\\ &&\quad {A}^{(\tilde{k})}\approx {\tilde{A}}^{(\tilde{k})},\phantom{\rule{4em}{0ex}}{A}^{(\tilde{N}+\bar{k})}\approx {\sigma }^{2\bar{k}}{\tilde{A}}^{(\tilde{N})}{\bar{A}}^{(\bar{k})},\end{eqnarray} \tag{ 3.7 }$$

and analogous relations hold for ${{ \mathcal U }}_{\mu }$, ${{ \mathcal A }}_{\mu }^{(k)}$, and ${{ \mathcal A }}^{(k)}$. Starting from (2.3), we also have

$$\begin{eqnarray}&&{{ \mathcal X }}_{\bar{\mu }}^{(p)}\approx {\sigma }^{2\bar{N}}{\tilde{A}}^{(\tilde{N})}{\bar{{ \mathcal X }}}_{\bar{\mu }}^{(p)},\quad {{ \mathcal X }}_{\bar{N}+\tilde{\mu }}^{(p)}\approx {(-{\tilde{x}}_{\tilde{\mu }}^{2})}^{\bar{N}}{\tilde{{ \mathcal X }}}_{\tilde{\mu }}^{(p+\bar{N})}.\end{eqnarray} \tag{ 3.8 }$$

Hence, we have obtained the following relations for various subterms of the metric (2.1):

$$\begin{eqnarray}&&\qquad \quad \displaystyle \frac{{{ \mathcal X }}_{\bar{\mu }}^{(p)}}{{U}_{\bar{\mu }}}\approx {\sigma }^{2}\displaystyle \frac{{\tilde{{ \mathcal A }}}^{(\tilde{N})}}{{\tilde{A}}^{(\tilde{N})}}\displaystyle \frac{{\bar{{ \mathcal X }}}_{\bar{\mu }}^{(p)}}{{\bar{U}}_{\bar{\mu }}},\quad \displaystyle \frac{{{ \mathcal X }}_{\bar{N}+\tilde{\mu }}^{(p)}}{{U}_{\bar{N}+\tilde{\mu }}}\approx \displaystyle \frac{{\tilde{{ \mathcal X }}}_{\tilde{\mu }}^{(p+\bar{N})}}{{\tilde{U}}_{\tilde{\mu }}},\\ &&\displaystyle \sum _{k}{A}_{\bar{\mu }}^{(k)}{\boldsymbol{d}}{\psi }_{k}\approx \displaystyle \frac{1}{\sigma }\displaystyle \frac{{\tilde{A}}^{(\tilde{N})}}{{\tilde{{ \mathcal A }}}^{(\tilde{N})}}\displaystyle \sum _{\bar{k}}{\bar{A}}_{\bar{\mu }}^{(\bar{k})}{\boldsymbol{d}}{\bar{\psi }}_{\bar{k}},\quad \displaystyle \sum _{k}{A}_{\bar{N}+\tilde{\mu }}^{(k)}{\boldsymbol{d}}{\psi }_{k}\approx \displaystyle \sum _{\tilde{k}}{\tilde{A}}_{\tilde{\mu }}^{(\tilde{k})}{\boldsymbol{d}}{\tilde{\psi }}_{\tilde{k}}.\end{eqnarray} \tag{ 3.9 }$$

In all these expressions barred (or tilded) functions are defined in terms of ${\bar{a}}_{\bar{\mu }}$ and ${\bar{x}}_{\bar{\mu }}$ (or ${\tilde{a}}_{\tilde{\mu }}$ and ${\tilde{x}}_{\tilde{\mu }}$) in a way analogous to how unbarred functions are defined in terms of a_μ and x_μ. In other words, in both regular and unspinned sectors we use the same functional definitions, using the appropriate coordinates and parameters for a given sector.

Putting everything together, we find that in the leading order of σ expansion the on-shell Kerr-NUT-(A)dS metric decouples to a warped product of the base and seed metrics in the regular and unspinned sectors,

$$\begin{eqnarray}&&{\boldsymbol{g}}\approx \tilde{{\boldsymbol{g}}}+{\tilde{w}}^{2}\bar{{\boldsymbol{g}}}.\end{eqnarray} \tag{ 3.10 }$$

In the regular sector, the base metric $\tilde{{\boldsymbol{g}}}$ is the off-shell $2\tilde{N}$-dimensional Kerr-NUT-(A)dS metric spanned on coordinates ${\tilde{x}}_{\tilde{\mu }}$, ${\tilde{\psi }}_{\tilde{k}}$, characterized by parameters ${\tilde{a}}_{\tilde{\mu }}$, ${\tilde{\beta }}_{\tilde{\mu }}$, and metric functions

$$\begin{eqnarray}&&{\tilde{X}}_{\tilde{\mu }}={\tilde{{ \mathcal X }}}_{\tilde{\mu }}^{(\tilde{N})}.\end{eqnarray} \tag{ 3.11 }$$

In the unspinned sector, the seed metric $\bar{{\boldsymbol{g}}}$ is the on-shell $2\bar{N}$-dimensional Kerr-NUT-(A)dS metric spanned on ${\bar{x}}_{\bar{\mu }}$, ${\bar{\psi }}_{\bar{k}}$, characterized by parameters ${\bar{a}}_{\bar{\mu }}$, ${\bar{\beta }}_{\bar{\mu }}$, and metric functions

$$\begin{eqnarray}&&{\bar{X}}_{\bar{\mu }}={\bar{{ \mathcal X }}}_{\bar{\mu }}^{(0)}.\end{eqnarray} \tag{ 3.12 }$$

The warp-factor $\tilde{w}$ is

$$\begin{eqnarray}&&{\tilde{w}}^{2}=\displaystyle \frac{{\tilde{A}}^{(\tilde{N})}}{{\tilde{{ \mathcal A }}}^{(\tilde{N})}}=\displaystyle \frac{{\tilde{x}}_{1}^{2}\ldots {\tilde{x}}_{\tilde{N}}^{2}}{{\tilde{a}}_{1}^{2}\ldots {\tilde{a}}_{\tilde{N}}^{2}}.\end{eqnarray} \tag{ 3.13 }$$

It depends only on the tilded variables, i.e. on the regular sector. The Levi-Civita tensor associated with the obtained metric reads

$$\begin{eqnarray}&&{\boldsymbol{\varepsilon }}\approx {\tilde{w}}^{2\bar{N}}\;\tilde{{\boldsymbol{\varepsilon }}}\wedge \bar{{\boldsymbol{\varepsilon }}}.\end{eqnarray} \tag{ 3.14 }$$

One can easily show that the obtained warped geometry (3.10) still satisfies the vacuum Einstein equations with the cosmological constant given by (2.9); for more details see [46]. One can also calculate the Riemann tensor and the Kretschmann scalar for such metrics, as described in appendix B.2.

For the Lorentzian signature, it is useful to eliminate the dependence of the unspinned metric $\bar{{\boldsymbol{g}}}$ on the curvature scale λ. After the Wick rotation (2.26) and normalization (2.28), the warp-factor $\tilde{w}$ becomes

$$\begin{eqnarray}&&{\tilde{w}}^{2}=\lambda \;{\tilde{r}}^{2}\;\displaystyle \frac{{\tilde{x}}_{1}^{2}\ldots {\tilde{x}}_{\tilde{N}-1}^{2}}{{\tilde{a}}_{1}^{2}\ldots {\tilde{a}}_{\tilde{N}-1}^{2}}.\end{eqnarray} \tag{ 3.15 }$$

As we discussed in (2.13), metric $\lambda \bar{{\boldsymbol{g}}}=\bar{{\boldsymbol{g}}}{| }_{\lambda =1}$ is independent of λ. Thus,

$$\begin{eqnarray}&&{\boldsymbol{g}}\approx \tilde{{\boldsymbol{g}}}+{\tilde{r}}^{2}\;\displaystyle \frac{{\tilde{x}}_{1}^{2}\ldots {\tilde{x}}_{\tilde{N}-1}^{2}}{{\tilde{a}}_{1}^{2}\ldots {\tilde{a}}_{\tilde{N}-1}^{2}}\ \bar{{\boldsymbol{g}}}{| }_{\lambda =1}.\end{eqnarray} \tag{ 3.16 }$$

The unspinned metric $\bar{{\boldsymbol{g}}}$ describes the on-shell Euclidean instanton, i.e. a deformed twisted sphere. In the black hole case, the regular metric $\tilde{{\boldsymbol{g}}}$ has a Lorentzian signature and describes a part of the metric which includes a time coordinate, radial coordinate, and rotating directions. Notice that the regular metric belongs only to the off-shell class of Kerr-NUT-(A)dS metrics since the metric functions ${\tilde{X}}_{\mu }$ are modified and their form depends on the dimensionality of the unspinned sector, see (3.11).

Inspecting metric (3.10), we observe that although we took the limit of vanishing ${a}_{\bar{\mu }}=\sigma {\bar{a}}_{\bar{\mu }}\to 0$, the unspinned parameters ${\bar{a}}_{\bar{\mu }}$ remain in the resulting metric. The limit only 'decouples' both sets of regular and unspinned variables and parameters. Counting parameters in the obtained metric, we have $2\tilde{N}$ parameters ${\tilde{a}}_{\tilde{\mu }}$, ${\tilde{\beta }}_{\tilde{\mu }}$ labeling regular metric $\tilde{{\boldsymbol{g}}}$ and $2\bar{N}$ parameters ${\bar{a}}_{\bar{\mu }}$, ${\bar{\beta }}_{\bar{\mu }}$ labeling unspinned metric $\bar{{\boldsymbol{g}}}$. However, now we can use the rescaling freedom (2.12) for both parts of the metric independently. Two of the parameters are therefore superfluous and they can be fixed by gauge fixing conditions. The limiting procedure therefore eliminated only one parameter from the original metric.

We have thus obtained a new rich class of higher-dimensional black hole solutions that solve the vacuum Einstein equations with the cosmological constant and contain one parameter less than the original Kerr-NUT-(A)dS spacetimes. Because of the presence of NUT parameters the new spacetimes represent a non-trivial generalization of Kerr-AdS black holes obtained in [17–19] which no longer belongs, due to the singular character of the scaling limit, to the Kerr-NUT-(A)dS class.

An important special case of the limit occurs for $\tilde{N}=1$, $\bar{N}=N-1$. Since we are interested in the physical signature, we employ the normalization (2.28), and the Wick rotation (2.26),

$$\begin{eqnarray}\lambda {r}^{2} & = & \displaystyle \frac{{x}_{N}^{2}}{{a}_{N}^{2}}=\displaystyle \frac{{\tilde{x}}_{1}^{2}}{{\tilde{a}}_{1}^{2}}={\tilde{w}}^{2},\\ t & = & {a}_{N}{\phi }_{N}={a}_{N}{\tilde{\phi }}_{1}\approx {\psi }_{0}={\tilde{\psi }}_{0}.\end{eqnarray} \tag{ 3.17 }$$

Introducing the mass parameter (2.27), the metric function ${\tilde{X}}_{1}={\tilde{{ \mathcal X }}}_{1}^{(\bar{N})}=-f$ can be rewritten as

$$\begin{eqnarray}&&f=1-\lambda {r}^{2}-\displaystyle \frac{2m}{{r}^{2N-3}}.\end{eqnarray} \tag{ 3.18 }$$

With these definitions, the limiting metric takes the following form:

$$\begin{eqnarray}&&{\boldsymbol{g}}\approx -f{\boldsymbol{d}}{t}^{2}+\displaystyle \frac{1}{f}{\boldsymbol{d}}{r}^{2}+{r}^{2}\bar{{\boldsymbol{g}}}{| }_{\lambda =1}.\end{eqnarray} \tag{ 3.19 }$$

It describes a deformed and twisted static black hole with an angular part given by the Euclidean instanton $\bar{{\boldsymbol{g}}}{| }_{\lambda =1}$.

In particular for ${\bar{\beta }}_{\bar{\mu }}=0$, the unspinned metric simplifies to a maximally symmetric geometry, that is, $\bar{{\boldsymbol{g}}}{| }_{\lambda =1}$ describes the Euclidean $2\bar{N}$-dimensional sphere of unit radius. We thus recovered the Schwarzschild–Tangherlini solution describing a non-rotating higher-dimensional black hole. The metric $\bar{{\boldsymbol{g}}}$ on the sphere is, however, written in multi-eliptical coordinates that are characterized by constants ${\bar{a}}_{\bar{\mu }}$.

We now turn to the limit of the symmetry objects in the Killing tower. Obviously, since in the limit $\sigma \to 0$ the first $\bar{N}$ eigenvalues ${x}_{\bar{\mu }}$ of the CCKY tensor ${\boldsymbol{h}}$ vanish, this tensor becomes a degenerate CCKY tensor $\tilde{{\boldsymbol{h}}}$ and the resulting metric has to belong to the class of generalized Kerr-NUT-(A)dS spacetimes [44, 45], reviewed in appendix A. However, as we show now, the obtained metric is a very distinguished one in this class as it possesses an enhanced symmetry of the full Killing tower. In particular, we show that the limiting $\tilde{{\boldsymbol{h}}}$ can be understood as a non-degenerate CCKY on the submanifold $\tilde{{ \mathcal M }}$ and that another 2-form $\bar{{\boldsymbol{h}}}$ emerges in the limit and effectively plays the role of a non-degenerate CCKY tensor on the unspinned submanifold $\bar{{ \mathcal M }}$. (See appendix B.3 for more details.)

First, employing (3.3) and (3.9), we obtain the following expressions for 2-forms ${{\boldsymbol{\omega }}}^{\mu }$:

$$\begin{eqnarray}&&{{\boldsymbol{\omega }}}^{\bar{\mu }}\approx {\tilde{w}}^{2}\;{\bar{{\boldsymbol{\omega }}}}^{\bar{\mu }},\quad {{\boldsymbol{\omega }}}^{\bar{N}+\tilde{\mu }}\approx {\tilde{{\boldsymbol{\omega }}}}^{\tilde{\mu }}.\end{eqnarray} \tag{ 3.20 }$$

The unspinned 2-forms ${\bar{{\boldsymbol{\omega }}}}^{\bar{\mu }}$ are normalized in the sense of metric $\bar{{\boldsymbol{g}}}$. The warp factor guarantees the normalization of ${{\boldsymbol{\omega }}}^{\bar{\mu }}$ with respect to ${\boldsymbol{g}}$.

Starting from expression (2.32) for ${\boldsymbol{h}}$, only terms with regular coordinates ${x}_{\tilde{N}+\tilde{\mu }}={\tilde{x}}_{\tilde{\mu }}$ survive in the leading order and we get

$$\begin{eqnarray}&&{\boldsymbol{h}}\approx \tilde{{\boldsymbol{h}}}=\displaystyle \sum _{\tilde{\mu }}{\tilde{x}}_{\tilde{\mu }}{\tilde{{\boldsymbol{\omega }}}}^{\tilde{\mu }}.\end{eqnarray} \tag{ 3.21 }$$

The limit of the primary KY form ${\boldsymbol{\xi }}$ is straightforward, see (2.34) and (3.9),

$$\begin{eqnarray}&&{\boldsymbol{\xi }}\approx \tilde{{\boldsymbol{\xi }}}.\end{eqnarray} \tag{ 3.22 }$$

The limit (3.21) of the CCKY form ${\boldsymbol{h}}$ is degenerate: it vanishes in the unspinned sector. Therefore the limiting metric does not have the canonical form (2.1) and the obtained $\tilde{{\boldsymbol{h}}}$ cannot be used for a direct construction of the Killing tower. (This tensor can, however, be understood as a non-degenerate CCKY 2-form on the base manifold $\tilde{{ \mathcal M }}$, see (3.21)). Nevertheless, we can find the limits of all objects ${{\boldsymbol{h}}}^{(k)}$, ${{\boldsymbol{f}}}^{(k)}$, ${{\boldsymbol{k}}}_{(k)}$, and ${{\boldsymbol{l}}}_{(k)}$ directly.

Had we used the standard definitions of the Killing tower objects without a prefactor given by suitable powers of ${{ \mathcal A }}^{(k)}$ (see the discussion below expressions (2.41) and (2.42)), we would find that some of the objects in the limit vanished. However, it would be possible to extract the coefficient of the leading term in the $\sigma$ expansion. Instead, this can be simply achieved by including the discussed prefactors into the definitions of Killing objects—under the limit these prefactors supply necessary powers of the scaling parameter σ to obtain a finite result.

After the limit, the Killing tower naturally splits into two sets, one associated with the metric $\bar{{\boldsymbol{g}}}$ of the unspinned sector and the other associated with the regular metric $\tilde{{\boldsymbol{g}}}$. Here, we present only the resulting expressions; details of the limiting process can be found in appendix B.3. The limits are

$$\begin{eqnarray}{{\boldsymbol{h}}}^{(\tilde{k})} & \approx & {\tilde{{\boldsymbol{h}}}}^{(\tilde{k})},\\ {{\boldsymbol{h}}}^{(\tilde{N}+\bar{k})} & \approx & {\tilde{w}}^{2\bar{k}+1}\;\tilde{{\boldsymbol{\varepsilon }}}\wedge {\bar{{\boldsymbol{h}}}}^{(\bar{k})}.\end{eqnarray} \tag{ 3.23 }$$

Similarly, for the KY forms we obtain

$$\begin{eqnarray}{{\boldsymbol{f}}}^{(\tilde{k})} & \approx & {\tilde{w}}^{2\bar{N}}\;{\tilde{{\boldsymbol{f}}}}^{(\tilde{k})}\wedge \bar{{\boldsymbol{\varepsilon }}},\\ {{\boldsymbol{f}}}^{(\tilde{N}+\bar{k})} & \approx & {\tilde{w}}^{2(\tilde{N}-\bar{k})+1}\;{\bar{{\boldsymbol{f}}}}^{(\bar{k})}.\end{eqnarray} \tag{ 3.24 }$$

The Killing tensors are quadratic in KY forms according to (2.38). The limit gives

$$\begin{eqnarray}{{\boldsymbol{k}}}_{(\tilde{k})} & \approx & {\tilde{{\boldsymbol{k}}}}_{(\tilde{k})}+\tfrac{{\tilde{A}}^{(\tilde{k})}}{{\tilde{{ \mathcal A }}}^{(\tilde{k})}}\;{\tilde{w}}^{-2}\;{\bar{{\boldsymbol{g}}}}^{-1},\\ {{\boldsymbol{k}}}_{(\tilde{N}+\bar{k})} & \approx & {\bar{{\boldsymbol{k}}}}_{(\bar{k})}.\end{eqnarray} \tag{ 3.25 }$$

Finally, we perform the limit of the Killing vectors. In this case, however, even after employing the normalization factor ${{ \mathcal A }}^{(k)}$ in (2.41), the limit of the Killing vectors related to the unspinned coordinates is not finite¹⁰

$$\begin{eqnarray}{{\boldsymbol{l}}}_{(\tilde{k})} & \approx & {\tilde{{\boldsymbol{l}}}}_{(\tilde{k})}=\tfrac{1}{{\tilde{{ \mathcal A }}}^{(\tilde{k})}}{{\boldsymbol{\partial }}}_{{\tilde{\psi }}_{\tilde{k}}},\\ {{\boldsymbol{l}}}_{(\tilde{N}+\bar{k})} & \approx & \sigma {\bar{{\boldsymbol{l}}}}_{(\bar{k})}=\sigma \tfrac{1}{{\bar{{ \mathcal A }}}^{(\bar{k})}}{{\boldsymbol{\partial }}}_{{\bar{\psi }}_{\bar{k}}}.\end{eqnarray} \tag{ 3.26 }$$

Nevertheless, the leading term of the Killing vectors ${{\boldsymbol{l}}}_{(\tilde{N}+\bar{k})}$ is proportional to the Killing vector ${\bar{{\boldsymbol{l}}}}_{(\bar{k})}$ of the unspinned part $\bar{{\boldsymbol{g}}}$ of the metric. The limiting metric thus admits at least the same number of explicit Killing vectors as the original one, only their relation to the original Killing vectors is slightly more complicated.

It is possible to check that both expressions (2.37) and (2.38), relating CCKY forms ${{\boldsymbol{h}}}^{(j)}$, KY forms ${{\boldsymbol{f}}}^{(j)}$, and Killing tensors ${{\boldsymbol{k}}}_{(j)}$ remain valid after the limit. The relation (2.40) between the Killing vectors ${{\boldsymbol{l}}}_{(j)}$ and Killing tensors ${{\boldsymbol{k}}}_{(j)}$ holds also in the leading order. However, with the normalization (3.26) it remains non-degenerate only for vectors related to regular coordinates, i.e. for $j=\tilde{j}=0,\ldots ,\tilde{N}-1$. For $j=\tilde{N}+\bar{j}=\tilde{N},\ldots ,N-1$ this relation degenerates in the zeroth order and one has to study next order terms in ${\boldsymbol{\xi }}$ and ${{\boldsymbol{k}}}_{(\tilde{N}+\bar{k})}$ to establish its validity.

To summarize, we have found that although the limiting metric ${\boldsymbol{g}}$ does not admit a non-degenerate CCKY form, it possesses the full Killing tower of symmetries that is similar to the Killing tower of the original metric. It splits into two sectors which are closely related to the Killing towers of regular metric $\tilde{{\boldsymbol{g}}}$ and unspinned metric $\bar{{\boldsymbol{g}}}$; see relations (3.23)–(3.26). Since the resulting Killing tower is obtained by the limiting procedure, the surviving symmetries satisfy the same mutual relations as the original symmetries, namely, they mutually commute. This then guarantees integrability and separability properties similar to the those of the original Kerr-NUT-(A)dS spacetimes [28–40]. The integrability and separability properties based on the Killing tower of a warped space (3.10) has been also recently discussed in a slightly broader context of metrics that admit complete separability of the Klein–Gordon equation [30].

To write down explicitly the limiting metric $\mathring{{\boldsymbol{g}}}$ and its symmetries, we first introduce some auxiliary objects. Let ${}^{\alpha }{\boldsymbol{g}}$ denote the canonical Kerr-NUT-(A)dS metric (2.11) in $2\alpha$ number of dimensions, spanned on coordinates ${x}_{1},\ldots ,{x}_{\alpha }$ and ${\phi }_{1},\ldots ,{\phi }_{\alpha }$. Let ${}^{\alpha }{\boldsymbol{\varepsilon }}$ represent the corresponding Levi-Civita tensor and ${}^{\alpha }{{\boldsymbol{h}}}^{(j)}$, ${}^{\alpha }{{\boldsymbol{f}}}^{(j)}$, ${}^{\alpha }{{\boldsymbol{k}}}_{(j)}$, $j=0,\ldots ,\alpha$ the symmetry objects in the corresponding Killing tower. We introduce 'auxiliary' $2\alpha$-dimensional metrics ${}^{\alpha }\mathring{{\boldsymbol{g}}}$ and ${}^{\alpha }\overset{\bullet }{{\boldsymbol{g}}}$ and the corresponding Levi-Civita tensors ${}^{\alpha }\mathring{{\boldsymbol{\varepsilon }}}$ and ${}^{\alpha }\overset{\bullet }{{\boldsymbol{\varepsilon }}}$ by the following expressions:

$$\begin{eqnarray}{}^{\alpha }\mathring{{\boldsymbol{g}}} & = & {{\boldsymbol{q}}}_{\alpha }+{\xi }_{\alpha }^{2}{{\boldsymbol{q}}}_{\alpha -1}+{\xi }_{\alpha }^{2}{\xi }_{\alpha -1}^{2}{{\boldsymbol{q}}}_{\alpha -2}\;+\;\cdots \;+\;{\xi }_{\alpha }^{2}\;\ldots \;{\xi }_{2}^{2}{{\boldsymbol{q}}}_{1},\\ {}^{\alpha }\mathring{{\boldsymbol{\epsilon }}} & = & {{\boldsymbol{\epsilon }}}_{\alpha }\wedge {\xi }_{\alpha }^{2}{{\boldsymbol{\epsilon }}}_{\alpha -1}\wedge {\xi }_{\alpha }^{2}{\xi }_{\alpha -1}^{2}{{\boldsymbol{\epsilon }}}_{\alpha -2}\wedge \cdots \wedge {\xi }_{\alpha }^{2}\ldots {\xi }_{2}^{2}{{\boldsymbol{\epsilon }}}_{1},\end{eqnarray} \tag{ 4.1 }$$

$$\begin{eqnarray}{}^{\alpha }\overset{\bullet }{{\boldsymbol{g}}} & = & {{\boldsymbol{q}}}_{N}+{\xi }_{N}^{2}{{\boldsymbol{q}}}_{N-1}+{\xi }_{N}^{2}{\xi }_{N-1}^{2}{{\boldsymbol{q}}}_{N-2}\;+\;\cdots \\ & & +{\xi }_{N}^{2}\ldots {\xi }_{N-\alpha +2}^{2}{{\boldsymbol{q}}}_{N-\alpha +1},\\ {}^{\alpha }\overset{\bullet }{{\boldsymbol{\varepsilon }}} & = & {{\boldsymbol{\varepsilon }}}_{N}+{\xi }_{N}^{2}{{\boldsymbol{\epsilon }}}_{N-1}\wedge {\xi }_{N}^{2}{\xi }_{N-1}^{2}{{\boldsymbol{\epsilon }}}_{N-2}\wedge \ldots \\ & & \wedge {\xi }_{N}^{2}\ldots {\xi }_{N-\alpha +2}^{2}{{\boldsymbol{\epsilon }}}_{N-\alpha +1}.\end{eqnarray} \tag{ 4.2 }$$

The above definitions are expressed in terms of rescaled coordinates ${\xi }_{\mu }=\tfrac{{x}_{\mu }}{{a}_{\mu }}$ and ${\phi }_{\mu }$, through atomic 2-metrics ${{\boldsymbol{q}}}_{\mu }$ and 2-forms ${{\boldsymbol{\epsilon }}}_{\mu }$

$$\begin{eqnarray}{{\boldsymbol{q}}}_{\mu } & = & \displaystyle \frac{1}{\lambda }\left({\displaystyle \frac{1}{{\rm{\Delta }}}}_{\mu }{\boldsymbol{d}}{\xi }_{\mu }^{2}+{{\rm{\Delta }}}_{\mu }{\boldsymbol{d}}{\phi }_{\mu }^{2}\right),\\ {{\boldsymbol{\epsilon }}}_{\mu } & = & \displaystyle \frac{1}{\lambda }{\boldsymbol{d}}{\xi }_{\mu }\wedge {\boldsymbol{d}}{\phi }_{\mu },\end{eqnarray} \tag{ 4.3 }$$

where

$$\begin{eqnarray}&&{{\rm{\Delta }}}_{\mu }=1-{\xi }_{\mu }^{2}-2{\beta }_{\mu }{\xi }_{\mu }^{-2\mu +3}.\end{eqnarray} \tag{ 4.4 }$$

Now we summarize the results of the combined limiting process—the derivation is delegated to appendix B.4. The limiting metric is $\mathring{{\boldsymbol{g}}}={}^{N}\mathring{{\boldsymbol{g}}}={}^{N}\overset{\bullet }{{\boldsymbol{g}}}$. The explicit expressions for it and the Levi-Civita tensors are

$$\begin{eqnarray}\mathring{{\boldsymbol{g}}} & = & {{\boldsymbol{q}}}_{N}+{\xi }_{N}^{2}\;{{\boldsymbol{q}}}_{N-1}\;+\;\cdots \;+\;{\xi }_{N}^{2}\cdots {\xi }_{2}^{2}\;{{\boldsymbol{q}}}_{1},\\ \mathring{{\boldsymbol{\epsilon }}} & = & {{\boldsymbol{\epsilon }}}_{N}\wedge {\xi }_{N}^{2}\;{{\boldsymbol{\epsilon }}}_{N-1}\wedge \cdots \wedge {\xi }_{N}^{2}\ldots {\xi }_{2}^{2}\;{{\boldsymbol{\epsilon }}}_{1}.\end{eqnarray} \tag{ 4.5 }$$

If we perform the Wick rotation (2.26), and assume normalization (2.28), we can write the limit of the Lorentzian metric as

$$\begin{eqnarray}\mathring{{\boldsymbol{g}}} & = & -f{\boldsymbol{d}}{t}^{2}+\displaystyle \frac{1}{f}{\boldsymbol{d}}{r}^{2}\\ & & {+{r}^{2}[{{\boldsymbol{q}}}_{N-1}+{\xi }_{N-1}^{2}({{\boldsymbol{q}}}_{N-2}+\cdots +{\xi }_{3}^{2}({{\boldsymbol{q}}}_{2}+{\xi }_{2}^{2}{{\boldsymbol{q}}}_{1}))]| }_{\lambda =1},\end{eqnarray} \tag{ 4.6 }$$

with $f$ given by (3.18). Here, coordinates ${\xi }_{\mu }$, $\mu \gt 1$, run between roots of ${{\rm{\Delta }}}_{\mu }$, (4.4), which are for small positive ${\beta }_{\mu }$ inside interval $(0,1)$. This restriction does not apply to ${\xi }_{N}$ in the Lorentzian case, since in that case ${\xi }_{N}$ is Wick rotated to radial coordinate $r$. The range of ${\xi }_{1}$ is inside $(-1,1)$ and the metric ${{\boldsymbol{q}}}_{1}$ is trivial; see the discussion of metric $\bar{{\boldsymbol{g}}}$ in section 5.1.

The curvature of $\mathring{{\boldsymbol{g}}}$ can be obtained by repetitive application of the warped splitting while employing relations (B24)–(B27), to get the following expression for the Riemann tensor:

$$\begin{eqnarray}{\boldsymbol{R}} & = & -\quad \displaystyle \frac{\lambda }{4}\displaystyle \sum _{\mu =1}^{N}{{\rm{\Delta }}}_{\mu }^{\prime\prime }\left(\displaystyle \prod _{\nu =\mu +1}^{N}{\xi }_{\nu }^{2}\right){{\boldsymbol{q}}}_{\mu }\wedge \wedge \;{{\boldsymbol{q}}}_{\mu }\\ & & -\displaystyle \frac{\lambda }{2}\displaystyle \sum _{\mu =2}^{N}\displaystyle \frac{{{\rm{\Delta }}}_{\mu }^{\prime }}{{\xi }_{\mu }}\displaystyle \sum _{\nu =1}^{\mu -1}\left(\displaystyle \prod _{\alpha =\nu +1}^{N}{\xi }_{\alpha }^{2}\right){{\boldsymbol{q}}}_{\mu }\wedge \wedge \;{{\boldsymbol{q}}}_{\nu }\\ & & -\displaystyle \frac{\lambda }{2}\displaystyle \sum _{\mu =2}^{N}{{\rm{\Delta }}}_{\mu }\displaystyle \sum _{\alpha ,\beta =1}^{\mu -1}\left(\displaystyle \prod _{\kappa =\alpha +1}^{N}{\xi }_{\kappa }^{2}\right)\left(\displaystyle \prod _{\lambda =\beta +1}^{\mu -1}{\xi }_{\lambda }^{2}\right){{\boldsymbol{q}}}_{\alpha }\wedge \wedge \;{{\boldsymbol{q}}}_{\beta },\end{eqnarray} \tag{ 4.7 }$$

where for two 2-forms ${\boldsymbol{p}}$, ${\boldsymbol{q}}$ we defined

$$\begin{eqnarray}&&{({\boldsymbol{p}}\wedge \wedge {\boldsymbol{q}})}_{{abcd}}=4\;{{\boldsymbol{p}}}_{[a| [c}{{\boldsymbol{q}}}_{d]| b]}.\end{eqnarray} \tag{ 4.8 }$$

Substituting the expressions for ${{\rm{\Delta }}}_{\mu }$, (4.4), we can analyze the singular behavior of the curvature. In our case, it is sufficiently represented by the singularities of the Kretschmann scalar, which simplifies to

$$\begin{eqnarray}&&{ \mathcal K }=4N(2N-1){\lambda }^{2}+{\lambda }^{2}\displaystyle \sum _{\mu =2}^{N}\displaystyle \frac{16{(\mu -1)}^{2}(2\mu -1)(2\mu -3){\beta }_{\mu }^{2}}{{\xi }_{\mu }^{4\mu -2}{({\displaystyle \prod }_{\alpha =\mu +1}^{N}{\xi }_{\alpha }^{2})}^{2}}.\end{eqnarray} \tag{ 4.9 }$$

For non-trivial parameters ${\beta }_{\mu }$ we thus obtained a deformed geometry which is, however, not twisted as the angular Killing coordinates ${\phi }_{\mu }$ do not mix among themselves (the metric is diagonal). For the Lorentzian signature the geometry has a curvature singularity at $r=0$. Inspecting the expression for the Kretschmann scalar (4.9), we see that the singularity occurs for ${\xi }_{\mu }=0$, $\mu \gt 1$, which is out of the coordinate ranges, except for the radial coordinate $r$.

For vanishing NUT parameters, ${\beta }_{\mu }=0$, $\mu =1,\ldots ,N-1$, one can introduce latitude angles ${\theta }_{\mu }$,

$$\begin{eqnarray}&&{\xi }_{\mu }=\mathrm{cos}{\theta }_{\mu }\end{eqnarray} \tag{ 4.10 }$$

and the atomic 2-metric (4.3) simplifies to the homogeneous metric on a 2-sphere

$$\begin{eqnarray}&&{{\boldsymbol{q}}}_{\mu }={\boldsymbol{d}}{\theta }_{\mu }^{2}+{\mathrm{sin}}^{2}{\theta }_{\mu }\;{\boldsymbol{d}}{\phi }_{\mu }^{2}.\end{eqnarray} \tag{ 4.11 }$$

The limits of the CCKY forms, KY forms and Killing tensors are

$$\begin{eqnarray}&&{\mathring{{\boldsymbol{h}}}}^{(j)}={}^{j}w\ {}^{j}\overset{\bullet }{{\boldsymbol{\varepsilon }}},\end{eqnarray} \tag{ 4.12 }$$

$$\begin{eqnarray}&&{\mathring{{\boldsymbol{f}}}}^{(j)}={}^{j}{w}^{2(N-j)+1}\ {}^{N-j}\mathring{{\boldsymbol{\epsilon }}},\end{eqnarray} \tag{ 4.13 }$$

$$\begin{eqnarray}&&{\mathring{{\boldsymbol{k}}}}_{(j)}={}^{N-j}{\mathring{{\boldsymbol{g}}}}^{-1}.\end{eqnarray} \tag{ 4.14 }$$

They satisfy relations (2.37) and (2.38). Relation (2.40) between Killing vectors and Killing tensors degenerates since in the limit Killing vectors ${{\boldsymbol{l}}}_{(j)}$ vanish except for j = 0. However, we can identify immediately another set of N explicit Killing vectors of the limiting metric $\mathring{{\boldsymbol{g}}}$, namely vectors ${{\boldsymbol{\partial }}}_{{\phi }_{\mu }}$.

Our starting point is the four-dimensional on-shell Kerr-NUT-(A)dS metric with Lorentzian signature $(-+++)$. Since in lower dimensions it is more natural to label coordinates by letters instead of indexing, we use the following notation: ${x}_{1}\to x$, ${a}_{1}\to a$, ${\psi }_{0}\to t,{\psi }_{1}\to \psi$. The metric (2.30) then reads

$$\begin{eqnarray}&&{\boldsymbol{g}}=-\displaystyle \frac{{\rm{\Delta }}}{{\rm{\Sigma }}}{({\boldsymbol{d}}t+{x}^{2}{\boldsymbol{d}}\psi )}^{2}+\displaystyle \frac{{\rm{\Sigma }}}{{\rm{\Delta }}}\;{\boldsymbol{d}}{r}^{2}+\displaystyle \frac{{\rm{\Sigma }}}{{ \mathcal S }}\;{\boldsymbol{d}}{x}^{2}+\displaystyle \frac{{ \mathcal S }}{{\rm{\Sigma }}}{({\boldsymbol{d}}t-{r}^{2}{\boldsymbol{d}}\psi )}^{2},\end{eqnarray} \tag{ 5.1 }$$

where

$$\begin{eqnarray}{\rm{\Delta }} & = & -{X}_{2}=(1-\lambda {r}^{2})({a}^{2}+{r}^{2})-2m(1+\lambda {a}^{2})\;r,\\ { \mathcal S } & = & -{X}_{1}=(1+\lambda {x}^{2})({a}^{2}-{x}^{2})-2\alpha a(1+\lambda {a}^{2})\;x,\\ {\rm{\Sigma }} & = & {U}_{2}=-{U}_{1}={r}^{2}+{x}^{2}.\end{eqnarray} \tag{ 5.2 }$$

This is the standard Kerr-NUT-(A)dS solution written in Carter's coordinates [2, 3]. It is labeled by parameter $\lambda$ fixed by the cosmological constant, mass $m$, NUT parameter $\alpha$, and rotation parameter $a$. The exact interpretration of parameters for nonvanishing $\alpha$ is, however, nontrivial, see [10, 11].

We perform the limit $N=2\to \bar{N}=1,\tilde{N}=1$ which leads to the warped metric

$$\begin{eqnarray}&&{\boldsymbol{g}}=\tilde{{\boldsymbol{g}}}+{\tilde{r}}^{2}\bar{{\boldsymbol{g}}}{| }_{\lambda =1}.\end{eqnarray} \tag{ 5.3 }$$

The regular sector is characterized by $\tilde{N}=1$ and signature $(-+)$. Coordinates $\tilde{r}$, $\tilde{t}$ and mass parameter $\tilde{m}$ are trivially related to the original ones: $\tilde{r}=r$, $\tilde{t}=t$, and $\tilde{m}=m$. The regular metric $\tilde{{\boldsymbol{g}}}$ is given by

$$\begin{eqnarray}&&\tilde{{\boldsymbol{g}}}=-f\quad {\boldsymbol{d}}{\tilde{t}}^{2}+\displaystyle \frac{1}{f}{\boldsymbol{d}}{\tilde{r}}^{2},\end{eqnarray} \tag{ 5.4 }$$

$$\begin{eqnarray}&&f=1-\lambda {r}^{2}-\displaystyle \frac{2m}{r}.\end{eqnarray} \tag{ 5.5 }$$

The unspinned sector is characterized by $\bar{N}=1$, signature $(++)$, and the following notation dictionary: ${\bar{x}}_{1}\to \bar{x}$, ${\bar{\psi }}_{0}\to \bar{\psi }$, ${\bar{a}}_{1}\to \bar{a}$, ${\bar{\beta }}_{1}\to \bar{\alpha }$. Coordinates and parameters are related to those before the limit as $\bar{x}\approx \tfrac{1}{\sigma }x$, $\bar{\psi }\approx -\sigma \tfrac{1}{\lambda }\psi$, $\bar{a}\approx \tfrac{1}{\sigma }a$, $\bar{\alpha }=\alpha$. The metric reads

$$\begin{eqnarray}&&\bar{{\boldsymbol{g}}}{| }_{\lambda =1}=\displaystyle \frac{1}{\bar{X}}{\boldsymbol{d}}{\bar{x}}^{2}+\bar{X}{\boldsymbol{d}}{\bar{\psi }}^{2},\end{eqnarray} \tag{ 5.6 }$$

$$\begin{eqnarray}&&\bar{X}={\bar{a}}^{2}-{\bar{x}}^{2}-2\bar{\alpha }\bar{a}\bar{x}.\end{eqnarray} \tag{ 5.7 }$$

By rescaling coordinates,

$$\begin{eqnarray}&&\xi =\displaystyle \frac{\bar{x}}{\bar{a}},\quad \phi =\bar{a}\bar{\psi },\end{eqnarray} \tag{ 5.8 }$$

it is possible to eliminate parameter $\bar{a}$, obtaining the following metric (setting $\bar{\alpha }=\alpha$):

$$\begin{eqnarray}&&\bar{{\boldsymbol{g}}}{| }_{\lambda =1}=\displaystyle \frac{1}{X}{\boldsymbol{d}}{\xi }^{2}+X{\boldsymbol{d}}{\phi }^{2},\end{eqnarray} \tag{ 5.9 }$$

$$\begin{eqnarray}&&X=1-{\xi }^{2}-2\alpha \xi .\end{eqnarray} \tag{ 5.10 }$$

This metric is an Euclidean version of the two-dimensional black hole solution of the $D\to 2$ Einstein–dilaton gravity with a cosmological constant, constructed in [57]; see also [58] for the interpretation of parameters and its thermodynamics.

Coordinate $\xi$ runs between two roots of X, $-\alpha$ being the center of this interval. An introduction of the angular coordinate $\vartheta \in (0,\pi )$,

$$\begin{eqnarray}&&\xi =-\alpha +\sqrt{1+{\alpha }^{2}}\mathrm{cos}\vartheta ,\end{eqnarray} \tag{ 5.11 }$$

brings the metric to its spherical form

$$\begin{eqnarray}&&\bar{{\boldsymbol{g}}}{| }_{\lambda =1}={\boldsymbol{d}}{\vartheta }^{2}+(1+{\alpha }^{2}){\mathrm{sin}}^{2}\vartheta \;{\boldsymbol{d}}{\phi }^{2},\end{eqnarray} \tag{ 5.12 }$$

with $\alpha$ affecting the angular deficit at poles of the sphere. We see that in this case, the NUT parameter $\alpha$ is rather trivial: it can be absorbed into conicity of the symmetry axis of the geometry.

Putting everything together, the obtained metric reads

$$\begin{eqnarray}&&{\boldsymbol{g}}=-f\;{\boldsymbol{d}}{\tilde{t}}^{2}+\displaystyle \frac{1}{f}{\boldsymbol{d}}{\tilde{r}}^{2}+{\tilde{r}}^{2}({\boldsymbol{d}}{\vartheta }^{2}+(1+{\alpha }^{2}){\mathrm{sin}}^{2}\vartheta \;{\boldsymbol{d}}{\phi }^{2}),\end{eqnarray} \tag{ 5.13 }$$

with the metric function $f$ given by (5.5). By switching-off the rotation we have recovered the Schwarzschild-(A)dS solution.

This examples illustrates that although the NUT parameter α survived in this case, it belongs to the unspinned sector and hence completely decouples from temporal and radial coordinates. Therefore, after the limit it does not play a role of real NUT charge—we have not recovered the Taub–NUT solution [5, 6]. Since our limiting procedure completely decouples regular and unspinned sectors, to maintain a nontrivial Lorentzian NUT parameter, we would need to have $\tilde{N}\geqslant 2$. Even in the regular sector the surviving parameter α is rather trivial. For $\bar{N}=1$, it contributes only to the conicity of the geometry. This is not the case for all surviving NUT parameters in the unspinned sector in higher dimensions $\bar{N}\gt 1$ as we are going to see in the next subsection.

For the $N=3$ Lorentzian on-shell Kerr-NUT-(A)dS metric we employ the following notation dictionary: ${x}_{1},{x}_{2}\to x,y$, ${\psi }_{0},{\psi }_{1},{\psi }_{2}\to t,\varphi ,\psi$, ${a}_{1},{a}_{2}\to a,b$, and ${\beta }_{1},{\beta }_{2}\to \alpha ,\beta$. The metric (2.30) then reads

$$\begin{eqnarray}{\boldsymbol{g}} & = & -\displaystyle \frac{{\rm{\Delta }}}{{\rm{\Sigma }}}{({\boldsymbol{d}}t+({x}^{2}+{y}^{2}){\boldsymbol{d}}\varphi +{x}^{2}{y}^{2}{\boldsymbol{d}}\psi )}^{2}+\displaystyle \frac{{\rm{\Sigma }}}{{\rm{\Delta }}}{\boldsymbol{d}}{r}^{2}\\ & & +\displaystyle \frac{U}{X}{\boldsymbol{d}}{x}^{2}+\displaystyle \frac{X}{U}{({\boldsymbol{d}}t+({y}^{2}-{r}^{2}){\boldsymbol{d}}\varphi -{y}^{2}{r}^{2}{\boldsymbol{d}}\psi )}^{2}\\ & & +\displaystyle \frac{V}{Y}{\boldsymbol{d}}{y}^{2}+\displaystyle \frac{Y}{V}{({\boldsymbol{d}}t+({x}^{2}-{r}^{2}){\boldsymbol{d}}\varphi -{x}^{2}{r}^{2}{\boldsymbol{d}}\psi )}^{2},\end{eqnarray} \tag{ 5.14 }$$

where

$$\begin{eqnarray*}{\rm{\Delta }} & = & (1-\lambda {r}^{2})({a}^{2}+{r}^{2})({b}^{2}+{r}^{2})-2m(1+\lambda {a}^{2})(1+\lambda {b}^{2})\;r,\\ X & = & -(1+\lambda {x}^{2})({a}^{2}-{x}^{2})({b}^{2}-{x}^{2})+2\alpha a(1+\lambda {a}^{2})({b}^{2}-{a}^{2})\;x,\\ Y & = & -(1+\lambda {y}^{2})({a}^{2}-{y}^{2})({b}^{2}-{y}^{2})+2\beta b(1+\lambda {b}^{2})({a}^{2}-{b}^{2})\;y,\end{eqnarray*}$$

$$\begin{eqnarray}{\rm{\Sigma }} & = & {U}_{3}=({x}^{2}+{r}^{2})({y}^{2}+{r}^{2}),\\ U & = & {U}_{1}=({x}^{2}-{y}^{2})({x}^{2}+{r}^{2}),\\ V & = & {U}_{2}=({y}^{2}-{x}^{2})({y}^{2}+{r}^{2}).\end{eqnarray} \tag{ 5.15 }$$

5.2.1. Black hole with one rotation

Let us first perform a single-spin-zero limit $N=3\to \bar{N}=1,\tilde{N}=2$, switching-off one rotation parameter, ${a}_{1}\to 0$. We obtain the warped metric, see (3.16),

$$\begin{eqnarray}&&{\boldsymbol{g}}=\tilde{{\boldsymbol{g}}}+{\tilde{r}}^{2}\displaystyle \frac{{\tilde{y}}^{2}}{{\tilde{b}}^{2}}\bar{{\boldsymbol{g}}}{| }_{\lambda =1}.\end{eqnarray} \tag{ 5.16 }$$

In the regular sector we have a four-dimensional metric $\tilde{{\boldsymbol{g}}}$ with signature $(-+++)$ for which we employ the following notation dictionary: ${\tilde{x}}_{1}\to \tilde{y}$, ${\tilde{\psi }}_{1}\to \tilde{\varphi }$, ${\tilde{a}}_{1}\to \tilde{b}$, ${\tilde{\beta }}_{1}\to \tilde{\beta }$. As usual, in the regular sector coordinates and parameters after the limit are trivially related to those before the limit: $\tilde{y}=y$, $\tilde{r}=r$, $\tilde{t}=t$, $\tilde{\varphi }=\varphi$, $\tilde{m}=m$, $\tilde{\beta }=\beta$, and $\tilde{b}=b$. The metric resembles the Kerr-NUT-(A)dS solution (5.1) in four dimensions,

$$\begin{eqnarray}&&\tilde{{\boldsymbol{g}}}=-\displaystyle \frac{{\rm{\Delta }}}{\tilde{{\rm{\Sigma }}}}{({\boldsymbol{d}}\tilde{t}+{\tilde{y}}^{2}{\boldsymbol{d}}\tilde{\varphi })}^{2}+\displaystyle \frac{\tilde{{\rm{\Sigma }}}}{{\rm{\Delta }}}\;{\boldsymbol{d}}{\tilde{r}}^{2}+\displaystyle \frac{\tilde{{\rm{\Sigma }}}}{\tilde{{\rm{\Delta }}}}\;{\boldsymbol{d}}{\tilde{y}}^{2}+\displaystyle \frac{\tilde{{\rm{\Delta }}}}{\tilde{{\rm{\Sigma }}}}{({\boldsymbol{d}}\tilde{t}-{\tilde{r}}^{2}{\boldsymbol{d}}\tilde{\varphi })}^{2},\end{eqnarray} \tag{ 5.17 }$$

However, the metric functions have the $\bar{N}=1$ off-shell form (3.11) instead of the usual (5.2),

$$\begin{eqnarray}{\rm{\Delta }} & = & -{\tilde{X}}_{2}=(1-\lambda {\tilde{r}}^{2})({\tilde{b}}^{2}+{\tilde{r}}^{2})-2\tilde{m}(1+\lambda {\tilde{b}}^{2})\displaystyle \frac{1}{\tilde{r}},\\ \tilde{{\rm{\Delta }}} & = & -{\tilde{X}}_{1}=(1+\lambda {\tilde{y}}^{2})({\tilde{b}}^{2}-{\tilde{y}}^{2})-2\tilde{\beta }{\tilde{b}}^{3}(1+\lambda {\tilde{b}}^{2})\displaystyle \frac{1}{\tilde{y}},\\ \tilde{{\rm{\Sigma }}} & = & {\tilde{U}}_{2}=-{\tilde{U}}_{1}={\tilde{r}}^{2}+{\tilde{y}}^{2}.\end{eqnarray} \tag{ 5.18 }$$

The unspinned sector leads to the $\tilde{N}=1$ metric $\bar{{\boldsymbol{g}}}{| }_{\lambda =1}$ with signature $(++)$. With the notation dictionary ${\bar{x}}_{1}\to \bar{x}$, ${\bar{\psi }}_{0}\to \bar{\psi }$, ${\bar{a}}_{1}\to \bar{a}$, and ${\bar{\beta }}_{1}\to \bar{\alpha }$, it takes exactly the form (5.6) studied in the previous section; a discussion leading to (5.12) applies again. New coordinates and parameters are related to those before the limit as follows: $\bar{x}\approx \tfrac{1}{\sigma }x$, $\bar{\psi }\approx -\sigma \tfrac{{b}^{2}}{\lambda }\psi$, $\bar{a}\approx \tfrac{1}{\sigma }a$, and $\bar{\alpha }=\alpha$.

We can observe that the Killing coordinate $\bar{\psi }$ is decoupled from the time coordinate $\tilde{t}$ (there are no off-diagonal terms ${g}_{\tilde{t}\bar{\psi }}$), i.e. there is no rotation in this direction. However, there remains a nontrivial rotation in the $\tilde{\varphi }$ direction. The surviving NUT parameter $\tilde{\beta }$ belongs to the regular sector and as such it is coupled to the temporal coordinate. Consequently, its presence has familiar NUT-like consequences on the geometry, including the existence of closed time-like curves around the axis. The case with vanishing $\tilde{\beta }$ has been discussed in [59].

5.2.2. Twisted black hole

Another alternative in six dimensions is to perform the multiple-spin-zero limit $N=3\to \bar{N}=2,\tilde{N}=1$. The resulting metric reads (3.19),

$$\begin{eqnarray}&&{\boldsymbol{g}}=-f{\boldsymbol{d}}{t}^{2}+\displaystyle \frac{1}{f}{\boldsymbol{d}}{r}^{2}+{r}^{2}\bar{{\boldsymbol{g}}}{| }_{\lambda =1}\end{eqnarray} \tag{ 5.19 }$$

with the metric function¹¹

$$\begin{eqnarray}&&f=1-\lambda {r}^{2}-\displaystyle \frac{2m}{{r}^{3}}.\end{eqnarray} \tag{ 5.20 }$$

We recognize the Schwarzschild–Tangherlini-like dependence on the radial coordinate in six dimensions. However, the geometry reduces to the Schwarzschild–Tangherlini solution only for the vanishing NUT parameters in the unspinned sector.

Indeed, an unspinned metric $\bar{{\boldsymbol{g}}}{| }_{\lambda =1}$ is the $\bar{N}=2$ Euclidean instanton with signature $(++++)$. With the notation dictionary ${\bar{x}}_{1},{\bar{x}}_{2}\to \bar{x},\bar{y}$, ${\bar{\psi }}_{0},{\bar{\psi }}_{1}\to \bar{\varphi },\bar{\psi }$, ${\bar{\beta }}_{1},{\bar{\beta }}_{2}\to \alpha ,\beta$, and ${\bar{a}}_{1},{\bar{a}}_{2}\to \bar{a},\bar{b}$, it reads

$$\begin{eqnarray}\bar{{\boldsymbol{g}}}{| }_{\lambda =1} & = & \displaystyle \frac{{\bar{y}}^{2}-{\bar{x}}^{2}}{\bar{X}}{\boldsymbol{d}}{\bar{x}}^{2}+\displaystyle \frac{\bar{X}}{{\bar{y}}^{2}-{\bar{x}}^{2}}{({\boldsymbol{d}}\bar{\varphi }+{\bar{y}}^{2}{\boldsymbol{d}}\bar{\psi })}^{2}\\ & & +\displaystyle \frac{{\bar{y}}^{2}-{\bar{x}}^{2}}{-\bar{Y}}{\boldsymbol{d}}{\bar{y}}^{2}+\displaystyle \frac{-\bar{Y}}{{\bar{y}}^{2}-{\bar{x}}^{2}}{({\boldsymbol{d}}\bar{\varphi }+{\bar{x}}^{2}{\boldsymbol{d}}\bar{\psi })}^{2},\end{eqnarray} \tag{ 5.21 }$$

where

$$\begin{eqnarray}\bar{X} & = & ({\bar{a}}^{2}-{\bar{x}}^{2})({\bar{b}}^{2}-{\bar{x}}^{2})-2\bar{\alpha }\bar{a}({\bar{b}}^{2}-{\bar{a}}^{2})\;\bar{x},\\ \bar{Y} & = & ({\bar{a}}^{2}-{\bar{y}}^{2})({\bar{b}}^{2}-{\bar{y}}^{2})-2\bar{\beta }\bar{b}({\bar{a}}^{2}-{\bar{b}}^{2})\;\bar{y}.\end{eqnarray} \tag{ 5.22 }$$

The ranges of $\bar{x}$ and $\bar{y}$ are given by intervals between the roots of $\bar{X}$ and $\bar{Y}$, respectively, such that $-\bar{a}\lt \bar{x}\lt \bar{a}\lt \bar{y}\lt \bar{b}$; see the general discussion near (2.25). The metric is not diagonal, it mixes Killing directions (we have a non-vanishing term ${g}_{\bar{\varphi }\bar{\psi }}$ ), which we refer to as local twisting of the geometry. For $\bar{\alpha },\bar{\beta }\gt 0$, the curvature is singular only at $r=0$. For demonstration, the Kretschmann scalar is

$$\begin{eqnarray}{ \mathcal K } & = & 100{\lambda }^{2}+\displaystyle \frac{1600{m}^{2}}{{r}^{10}}+\displaystyle \frac{80{({\bar{a}}^{2}-{\bar{b}}^{2})}^{2}}{{r}^{4}{({\bar{y}}^{2}-{\bar{x}}^{2})}^{6}}\\ & & \times (({\bar{a}}^{2}{\bar{\alpha }}^{2}+{\bar{b}}^{2}{\bar{\beta }}^{2})({\bar{x}}^{2}+{\bar{y}}^{2})({\bar{x}}^{4}+14{\bar{x}}^{2}{\bar{y}}^{2}+{\bar{y}}^{4})+4\bar{a}\bar{b}\bar{x}\bar{y}(3{\bar{x}}^{2}+{\bar{y}}^{2})({\bar{x}}^{2}+3{\bar{y}}^{2})),\end{eqnarray} \tag{ 5.23 }$$

$\bar{x}=\bar{y}$ is out of the allowed coordinate range. There can exist additional non-regularities on the axes of symmetry. They depend on a choice of identification of Killing coordinates, which also introduces global twisting of geometry in these directions.

The limiting metric (3.19) thus represents a generalization of the static Schwarzschild–Tangherlini solution with an angular part given by the twisted four-dimensional Euclidean instanton (5.21).

5.2.3. Deformed black hole

After 'decoupling' the four-dimensional Euclidean instanton, by 'unspinning' it from the radial and temporal directions, we can also 'untwist' its Killing angular direction. Performing so the second single-spin-zero limit $\bar{N}=2\to \bar{\bar{N}}=1,\tilde{\bar{N}}=1$, we arrive at the 'combined metric' (4.5). Using the following notations: ${\xi }_{1},{\xi }_{2}\to \xi ,\upsilon$, ${\phi }_{1},{\phi }_{2}\to \phi ,\psi$, ${\beta }_{1},{\beta }_{2}\to \alpha ,\beta$, the metric reads

$$\begin{eqnarray}{\boldsymbol{g}} & = & {{\boldsymbol{q}}}_{r}+{r}^{2}{{\boldsymbol{q}}}_{\upsilon }+{r}^{2}{\upsilon }^{2}{{\boldsymbol{q}}}_{\xi }\\ & = & -f{\boldsymbol{d}}{t}^{2}+\displaystyle \frac{1}{f}{\boldsymbol{d}}{r}^{2}+{r}^{2}\left(\displaystyle \frac{1}{{{\rm{\Delta }}}_{\upsilon }}{\boldsymbol{d}}{\upsilon }^{2}+{{\rm{\Delta }}}_{\upsilon }{\boldsymbol{d}}{\psi }^{2}\right)+{r}^{2}{\upsilon }^{2}\left(\displaystyle \frac{1}{{{\rm{\Delta }}}_{\xi }}{\boldsymbol{d}}{\xi }^{2}+{{\rm{\Delta }}}_{\xi }{\boldsymbol{d}}{\phi }^{2}\right),\end{eqnarray} \tag{ 5.24 }$$

where

$$\begin{eqnarray}f & = & 1-\lambda {r}^{2}-\displaystyle \frac{2m}{{r}^{3}},\\ {{\rm{\Delta }}}_{\upsilon } & = & 1-{\upsilon }^{2}-\displaystyle \frac{2\beta }{\upsilon },\\ {{\rm{\Delta }}}_{\xi } & = & 1-{\xi }^{2}-2\alpha \xi .\end{eqnarray} \tag{ 5.25 }$$

Metric ${{\boldsymbol{q}}}_{r}=-f{\boldsymbol{d}}{t}^{2}+{f}^{-1}{\boldsymbol{d}}{r}^{2}$ describes a geometry spanning the radial and temporal directions and has the familiar Schwarzschild–Tangherlini form d. Metric ${{\boldsymbol{q}}}_{\xi }={{\rm{\Delta }}}_{\xi }^{-1}{\boldsymbol{d}}{\xi }^{2}+{{\rm{\Delta }}}_{\xi }{\boldsymbol{d}}{\phi }^{2}$ is spherical, (5.9). The middle part ${{\boldsymbol{q}}}_{\upsilon }={{\rm{\Delta }}}_{\upsilon }^{-1}{\boldsymbol{d}}{\upsilon }^{2}+{{\rm{\Delta }}}_{\upsilon }{\psi }^{2}$ has the Euclidean signature for $\tfrac{1}{\sqrt{27}}\gt \beta \geqslant 0$, when ${{\rm{\Delta }}}_{\upsilon }$ has two roots which specify the allowed range of $\upsilon$. For $\beta \ne 0$ this metric describes a non-spherical geometry.

The solution (5.24) thus describes a static deformed Schwarzschild–Tangherlini black hole. Since the metric is diagonal, its Killing directions are not twisted. The curvature is singular for $r=0$, as can be seen, e.g. from the Kretschmann scalar

$$\begin{eqnarray}&&{ \mathcal K }=60{\lambda }^{2}+\displaystyle \frac{960{m}^{2}}{{r}^{10}}+\displaystyle \frac{48{\beta }^{2}}{{r}^{4}{\upsilon }^{6}}.\end{eqnarray} \tag{ 5.26 }$$

Remember that for $\beta \gt 0$, value $\upsilon =0$ is outside the allowed coordinate range.

In section 2 we have defined auxiliary functions $J({a}^{2})$ and ${A}^{(k)}$ as follows¹²

$$\begin{eqnarray}&&J({a}^{2})=\displaystyle \prod _{\nu }({x}_{\nu }^{2}-{a}^{2})=\displaystyle \sum _{k=0,\ldots ,N}{A}^{(k)}{(-{a}^{2})}^{N-k},\end{eqnarray} \tag{ B1 }$$

and their complementary functions

$$\begin{eqnarray}&&{ \mathcal J }({x}^{2})=\displaystyle \prod _{\nu }({a}_{\nu }^{2}-{x}^{2})=\displaystyle \sum _{k=0,\ldots ,N}{{ \mathcal A }}^{(k)}{(-{x}^{2})}^{N-k}.\end{eqnarray} \tag{ B2 }$$

The above definitions imply that

$$\begin{eqnarray}&&{A}^{(k)}=\displaystyle \sum _{\displaystyle \genfrac{}{}{0em}{}{{\mu }_{1},\ldots ,{\mu }_{k}}{{\mu }_{1}\lt \ldots \lt {\mu }_{k}}}{x}_{{\mu }_{1}}^{2}\ldots {x}_{{\mu }_{k}}^{2},\end{eqnarray} \tag{ B3 }$$

$$\begin{eqnarray}&&{{ \mathcal A }}^{(k)}=\displaystyle \sum _{\displaystyle \genfrac{}{}{0em}{}{{\mu }_{1},\ldots ,{\mu }_{k}}{{\mu }_{1}\lt \ldots \lt {\mu }_{k}}}{a}_{{\mu }_{1}}^{2}\ldots {a}_{{\mu }_{k}}^{2}.\end{eqnarray} \tag{ B4 }$$

Similarly, we define functions ${J}_{\mu }({a}^{2})$, ${A}_{\mu }^{(j)}$, ${{ \mathcal J }}_{\mu }({x}^{2})$, and ${{ \mathcal A }}_{\mu }^{(j)}$, which skip the μth variables x_μ and a_μ as follows

$$\begin{eqnarray}&&{J}_{\mu }({a}^{2})=\displaystyle \prod _{\displaystyle \genfrac{}{}{0em}{}{\nu }{\nu \ne \mu }}({x}_{\nu }^{2}-{a}^{2})=\displaystyle \sum _{k}{A}_{\mu }^{(k)}{(-{a}^{2})}^{N-1-k},\end{eqnarray} \tag{ B5 }$$

$$\begin{eqnarray}&&{{ \mathcal J }}_{\mu }({x}^{2})=\displaystyle \prod _{\displaystyle \genfrac{}{}{0em}{}{\nu }{\nu \ne \mu }}({a}_{\nu }^{2}-{x}^{2})=\displaystyle \sum _{k}{{ \mathcal A }}_{\mu }^{(k)}{(-{x}^{2})}^{N-1-k},\end{eqnarray} \tag{ B6 }$$

with

$$\begin{eqnarray}&&{A}_{\mu }^{(k)}=\displaystyle \sum _{\begin{array}{c}{\nu }_{1},\ldots ,{\nu }_{k}\\ {\nu }_{1}\lt \ldots \lt {\nu }_{k}\\ {\nu }_{i}\ne \mu \end{array}}{x}_{{\nu }_{1}}^{2}\ldots {x}_{{\nu }_{k}}^{2},\end{eqnarray} \tag{ B7 }$$

$$\begin{eqnarray}&&{{ \mathcal A }}_{\mu }^{(k)}=\displaystyle \sum _{\begin{array}{c}{\nu }_{1},\ldots ,{\nu }_{k}\\ {\nu }_{1}\lt \ldots \lt {\nu }_{k}\\ {\nu }_{i}\ne \mu \end{array}}{a}_{{\nu }_{1}}^{2}\ldots {a}_{{\nu }_{k}}^{2}.\end{eqnarray} \tag{ B8 }$$

These functions satisfy

$$\begin{eqnarray}&&J({x}_{\mu }^{2})=0,\qquad { \mathcal J }({a}_{\mu }^{2})=0,\\ &&{J}_{\mu }({x}_{\nu }^{2})=0,\qquad {{ \mathcal J }}_{\mu }({a}_{\nu }^{2})=0,\qquad {\rm{for}}\ \nu \ne \mu .\end{eqnarray} \tag{ B9 }$$

Finally, we define

$$\begin{eqnarray}&&{U}_{\mu }={J}_{\mu }({x}_{\mu }^{2})=\displaystyle \prod _{\displaystyle \genfrac{}{}{0em}{}{\nu }{\nu \ne \mu }}({x}_{\nu }^{2}-{x}_{\mu }^{2}),\end{eqnarray} \tag{ B10 }$$

$$\begin{eqnarray}&&{{ \mathcal U }}_{\mu }={{ \mathcal J }}_{\mu }({a}_{\mu }^{2})=\displaystyle \prod _{\displaystyle \genfrac{}{}{0em}{}{\nu }{\nu \ne \mu }}({a}_{\nu }^{2}-{a}_{\mu }^{2}).\end{eqnarray} \tag{ B11 }$$

The polynomials ${A}^{(k)}$ and ${A}_{\mu }^{(k)}$ satisfy the following identities:

$$\begin{eqnarray}&&{A}^{(k)}={A}_{\mu }^{(k)}+{x}_{\mu }^{2}{A}_{\mu }^{(k-1)},\end{eqnarray} \tag{ B12 }$$

$$\begin{eqnarray}&&\displaystyle \sum _{k}{A}_{\mu }^{(k)}\displaystyle \frac{{(-{x}_{\nu }^{2})}^{N-1-k}}{{U}_{\nu }}={\delta }_{\mu }^{\nu },\end{eqnarray} \tag{ B13 }$$

$$\begin{eqnarray}&&\displaystyle \sum _{\mu }{A}_{\mu }^{(k)}\displaystyle \frac{{(-{x}_{\mu }^{2})}^{N-1-l}}{{U}_{\mu }}={\delta }_{l}^{k},\end{eqnarray} \tag{ B14 }$$

$$\begin{eqnarray}&&\displaystyle \sum _{\mu }{A}_{\mu }^{(k)}\displaystyle \frac{{(-{x}_{\mu }^{2})}^{N}}{{U}_{\mu }}=-{A}^{(k+1)},\end{eqnarray} \tag{ B15 }$$

$$\begin{eqnarray}&&\displaystyle \sum _{\mu }{A}_{\mu }^{(k)}=(N-k){A}^{(k)},\end{eqnarray} \tag{ B16 }$$

$$\begin{eqnarray}&&\displaystyle \sum _{\mu }(N-k){A}^{(k)}\displaystyle \frac{{(-{x}_{\nu }^{2})}^{N-1-k}}{{U}_{\nu }}=1,\end{eqnarray} \tag{ B17 }$$

$$\begin{eqnarray}&&\displaystyle \sum _{k=0,\ldots ,N}{A}^{(k)}{(-{x}_{\nu }^{2})}^{N-k}=0,\end{eqnarray} \tag{ B18 }$$

$$\begin{eqnarray}&&\displaystyle \sum _{l=0,\ldots ,k}{A}^{(l)}{(-{x}_{\nu }^{2})}^{k-l}={A}_{\mu }^{(k)}.\end{eqnarray} \tag{ B19 }$$

Analogous identities hold also for the complementary polynomials ${{ \mathcal A }}^{(k)}$ and ${{ \mathcal A }}_{\mu }^{(k)}$.

For the functions $J({a}^{2})$ and ${ \mathcal J }({x}^{2})$ we can write

$$\begin{eqnarray}&&\displaystyle \prod _{\mu }J({a}_{\nu }^{2})={(-1)}^{N}\displaystyle \prod _{\nu }{ \mathcal J }({x}_{\mu }^{2}),\\ &&\displaystyle \prod _{\displaystyle \genfrac{}{}{0em}{}{\mu }{\mu \ne \kappa }}{J}_{\kappa }({a}_{\nu }^{2})={(-1)}^{N-1}\displaystyle \prod _{\displaystyle \genfrac{}{}{0em}{}{\nu }{\nu \ne \kappa }}{{ \mathcal J }}_{\kappa }({x}_{\mu }^{2}).\end{eqnarray} \tag{ B20 }$$

These functions satisfy important orthogonality relations

$$\begin{eqnarray}&&\displaystyle \sum _{\alpha }\displaystyle \frac{{J}_{\nu }({a}_{\alpha }^{2})}{{{ \mathcal U }}_{\alpha }}\displaystyle \frac{{{ \mathcal J }}_{\alpha }({x}_{\mu }^{2})}{{U}_{\mu }}={\delta }_{\nu }^{\mu },\end{eqnarray} \tag{ B21 }$$

$$\begin{eqnarray}&&\displaystyle \sum _{\alpha }\displaystyle \frac{{J}_{\mu }({a}_{\alpha }^{2}){J}_{\nu }({a}_{\alpha }^{2})}{J({a}_{\alpha }^{2}){{ \mathcal U }}_{\alpha }}=-\displaystyle \frac{{U}_{\mu }}{{ \mathcal J }({x}_{\mu }^{2})}{\delta }_{\mu \nu },\end{eqnarray} \tag{ B22 }$$

$$\begin{eqnarray}&&\displaystyle \sum _{\mu }{J}_{\mu }({a}_{\alpha }^{2}){J}_{\mu }({a}_{\beta }^{2})\displaystyle \frac{{ \mathcal J }({x}_{\mu }^{2})}{{U}_{\mu }}=-J({a}_{\alpha }^{2}){{ \mathcal U }}_{\alpha }{\delta }^{\alpha \beta }.\end{eqnarray} \tag{ B23 }$$

The Riemann tensor of the off-shell Kerr-NUT-(A)dS metric has been calculated in [47]. The obtained limiting metric (3.10) has a warped-product structure with both components belonging to the Kerr-NUT-(A)dS class. The curvature of this metric can thus be obtained by using general formulae relating the curvature of the warped metric to curvatures of its components and derivatives of the warped factor. Following [46, 63], the Riemann and Ricci tensors read

$$\begin{eqnarray}&&{{\boldsymbol{R}}}_{{ab}}{}^{c}{}_{d}={\tilde{{\boldsymbol{R}}}}_{{ab}}{}^{c}{}_{d}+\displaystyle \frac{2}{\tilde{w}}{\tilde{{\boldsymbol{H}}}}_{d[a}{\bar{{\boldsymbol{\delta }}}}_{b]}^{c}-2\tilde{w}\;{\tilde{{\boldsymbol{g}}}}^{{ce}}{\tilde{{\boldsymbol{H}}}}_{e[a}{\bar{{\boldsymbol{g}}}}_{b]d}+{\bar{{\boldsymbol{R}}}}_{{ab}}{}^{c}{}_{d}-2{\tilde{w}}^{2}{\tilde{\lambda }}^{2}{\bar{{\boldsymbol{\delta }}}}_{[a}^{c}{\bar{{\boldsymbol{g}}}}_{b]d},\end{eqnarray} \tag{ B24 }$$

$$\begin{eqnarray}&&{{\bf{Ric}}}_{{ab}}={\tilde{{\bf{Ric}}}}_{{ab}}-\displaystyle \frac{\bar{N}}{\tilde{w}}{\tilde{{\boldsymbol{H}}}}_{{ab}}+{\bar{{\bf{Ric}}}}_{{ab}}-{\tilde{w}}^{2}\left(\displaystyle \frac{\tilde{{ \mathcal H }}}{\tilde{w}}+(\bar{N}-1){\tilde{\lambda }}^{2}\right){\bar{{\boldsymbol{g}}}}_{{ab}}.\end{eqnarray} \tag{ B25 }$$

Scalar curvature ${ \mathcal R }$ and the Kretschmann scalar ${ \mathcal K }$ are

$$\begin{eqnarray}&&{ \mathcal R }=\tilde{{ \mathcal R }}+\displaystyle \frac{\bar{{ \mathcal R }}}{{\tilde{w}}^{2}}-\displaystyle \frac{2\bar{N}}{\tilde{w}}\tilde{{ \mathcal H }}-\bar{N}(\bar{N}-1){\tilde{\lambda }}^{2},\end{eqnarray} \tag{ B26 }$$

$$\begin{eqnarray}&&{ \mathcal K }=\tilde{{ \mathcal K }}+\displaystyle \frac{4\bar{N}}{{\tilde{w}}^{2}}{\tilde{{\boldsymbol{H}}}}^{{ab}}{\tilde{{\boldsymbol{H}}}}_{{ab}}+\displaystyle \frac{1}{{\tilde{w}}^{4}}\bar{{ \mathcal K }}-4\displaystyle \frac{{\tilde{\lambda }}^{2}}{{\tilde{w}}^{2}}\bar{{ \mathcal R }}+2(\bar{N}-1)\bar{N}{\tilde{\lambda }}^{4}.\end{eqnarray} \tag{ B27 }$$

Here, the Hessian tensor $\tilde{{\boldsymbol{H}}}$ and scalar $\tilde{{ \mathcal H }}$ are defined as

$$\begin{eqnarray}&&{\tilde{{\boldsymbol{H}}}}_{{ab}}={\tilde{{\boldsymbol{\nabla }}}}_{a}{\tilde{{\boldsymbol{\nabla }}}}_{b}\tilde{w},\quad \tilde{{ \mathcal H }}={\tilde{{\boldsymbol{g}}}}^{{ab}}\;{\tilde{{\boldsymbol{H}}}}_{{ab}},\end{eqnarray} \tag{ B28 }$$

and

$$\begin{eqnarray}&&{\tilde{\lambda }}^{2}={\tilde{w}}^{-2}\;{\tilde{{\boldsymbol{g}}}}^{{ab}}\;{{\boldsymbol{d}}}_{a}\tilde{w}{{\boldsymbol{d}}}_{b}\tilde{w}.\end{eqnarray} \tag{ B29 }$$

In particular, for the warp factor (3.13) the Hessian tensor and scalars $\tilde{{ \mathcal H }}$, ${\tilde{\lambda }}^{2}$ can be expressed as

$$\begin{eqnarray}\tilde{{\boldsymbol{H}}} & = & \displaystyle \sum _{\tilde{\mu }}\displaystyle \frac{\tilde{w}}{2{\tilde{x}}_{\tilde{\mu }}}{\left(\displaystyle \frac{{\tilde{X}}_{\tilde{\mu }}}{{\tilde{U}}_{\tilde{\mu }}}\right)}_{,\tilde{\mu }}({\tilde{{\boldsymbol{e}}}}^{\tilde{\mu }}{\tilde{{\boldsymbol{e}}}}^{\tilde{\mu }}+{\hat{\tilde{{\boldsymbol{e}}}}}^{\tilde{\mu }}{\hat{\tilde{{\boldsymbol{e}}}}}^{\tilde{\mu }}),\\ \tilde{{ \mathcal H }} & = & \displaystyle \sum _{\tilde{\mu }}\displaystyle \frac{\tilde{w}{\tilde{X}}_{\tilde{\mu }}^{\prime }}{{\tilde{x}}_{\tilde{\mu }}{\tilde{U}}_{\tilde{\mu }}},\quad {\tilde{\lambda }}^{2}=\displaystyle \sum _{\tilde{\mu }}\displaystyle \frac{{\tilde{X}}_{\tilde{\mu }}}{{\tilde{x}}_{\tilde{\mu }}^{2}{\tilde{U}}_{\tilde{\mu }}}.\end{eqnarray} \tag{ B30 }$$

Let us now provide more details about the multiple-spin-zero limit of the Killing tower discussed briefly in section 3.4. We mentioned there that although the limit of the CCKY 2-form ${\boldsymbol{h}}\approx \tilde{{\boldsymbol{h}}}$ becomes degenerate, the full Killing tower of symmetries nevertheless survives. In some sense this is caused by the emergence of a new 2-form $\bar{{\boldsymbol{h}}}$ that is non-degenerate on manifold $\bar{{ \mathcal M }}$. Let us first briefly explain how this additional 2-form emerges.

We start from the CCKY 2-form ${\boldsymbol{h}}$. In the limit $\sigma \to 0$ its exterior power ${{\boldsymbol{h}}}^{\wedge \tilde{N}}$ becomes a rank $2\tilde{N}$ antisymmetric tensor that vanishes in the directions tangent to $\bar{{ \mathcal M }}$. It can be also understood as a tensor on the submanifold $\tilde{{ \mathcal M }}$. On this submanifold we can define the Levi-Civita tensor $\tilde{{\boldsymbol{\varepsilon }}}$ and, obviously, ${{\boldsymbol{h}}}^{\wedge \tilde{N}}$ must be proportional to it with some suitable prefactor,

$$\begin{eqnarray}&&\displaystyle \frac{1}{\tilde{N}!}{{\boldsymbol{h}}}^{\wedge \tilde{N}}\approx \displaystyle \frac{1}{\tilde{N}!}{\tilde{{\boldsymbol{h}}}}^{\wedge \tilde{N}}=\sqrt{{\tilde{A}}^{(\tilde{N})}}\;\tilde{{\boldsymbol{\varepsilon }}}.\end{eqnarray} \tag{ B31 }$$

One also has

$$\begin{eqnarray}&&\displaystyle \frac{1}{(\tilde{N}+1)!}{{\boldsymbol{h}}}^{\wedge (\tilde{N}+1)}\approx \sqrt{{\tilde{A}}^{(\tilde{N})}}\;\tilde{{\boldsymbol{\varepsilon }}}\wedge {\boldsymbol{h}}\approx { \mathcal O }(\sigma ).\end{eqnarray} \tag{ B32 }$$

As we will show, if this object is rescaled by ${\sigma }^{-1}$, its limit has a structure of the wedge-product of tensors from $\tilde{{ \mathcal M }}$ and $\bar{{ \mathcal M }}$

$$\begin{eqnarray}&&{{\boldsymbol{h}}}^{\wedge (\tilde{N}+1)}\propto \tilde{{\boldsymbol{\varepsilon }}}\wedge \bar{{\boldsymbol{h}}},\end{eqnarray} \tag{ B33 }$$

see relations (3.23) for $\bar{k}=1$. This determines a new 2-form $\bar{{\boldsymbol{h}}}$ on ${ \mathcal M }$ which is non-degenerate on the unspinned manifold $\bar{{ \mathcal M }}$. In some sense this 2-form is responsible for the survival of the Killing tower.

More concretely, as a result of the limiting procedure of the original CCKY tensor ${\boldsymbol{h}}$ one obtains a CCKY tensor $\tilde{{\boldsymbol{h}}}$ and a new rank two antisymmetric tensor $\bar{{\boldsymbol{h}}}$. Although $\tilde{{\boldsymbol{h}}}$ is degenerate on the full manifold, it is non-degenerate on $\tilde{{ \mathcal M }}$ where it plays a role similar to ${\boldsymbol{h}}$ on ${ \mathcal M }$. The other tensor, $\bar{{\boldsymbol{h}}}$, is not a CCKY 2-form of the limiting metric (since we factorized out some nontrivial prefactors); however we will show that it can effectively play a role of a non-degenerate CCKY tensor on $\bar{{ \mathcal M }}$. Indeed, as discussed in the main text, the limiting geometry has a structure of the warped product (3.10) and $\tilde{{\boldsymbol{h}}}$ is the CCKY form of metric $\tilde{{\boldsymbol{g}}}$ and $\bar{{\boldsymbol{h}}}$ is the CCKY form of metric $\bar{{\boldsymbol{g}}};$ both metrics belonging to the off-shell Kerr-NUT-(A)dS class. This guarantees the survival of the Killing tower.

With the above intuitive explanation in mind let us proceed to the actual limit of the Killing objects. The CCKY forms ${{\boldsymbol{h}}}^{(k)}$ defined in (2.36) can be written as

$$\begin{eqnarray}&&{{\boldsymbol{h}}}^{(k)}=\displaystyle \frac{1}{\sqrt{{{ \mathcal A }}^{(k)}}}\displaystyle \sum _{{\mu }_{1}\lt \ldots \lt {\mu }_{k}}{x}_{{\mu }_{1}}\ldots {x}_{{\mu }_{k}}\ {{\boldsymbol{\omega }}}^{{\mu }_{1}}\wedge \ldots \wedge {{\boldsymbol{\omega }}}^{{\mu }_{k}}.\end{eqnarray} \tag{ B34 }$$

For $k=\tilde{k}\leqslant \tilde{N}$, the leading order contains just unscaled terms ${x}_{{\mu }_{i}}$ with ${\mu }_{i}=\bar{N}+{\tilde{\mu }}_{i}\gt \bar{N}$,

$$\begin{eqnarray}&&{{\boldsymbol{h}}}^{(\tilde{k})}=\displaystyle \frac{1}{\sqrt{{{ \mathcal A }}^{(k)}}}\displaystyle \sum _{{\tilde{\mu }}_{1}\lt \ldots \lt {\tilde{\mu }}_{\tilde{k}}}{\tilde{x}}_{{\tilde{\mu }}_{1}}\ldots {\tilde{x}}_{{\tilde{\mu }}_{k}}\ {\tilde{{\boldsymbol{\omega }}}}^{{\tilde{\mu }}_{1}}\wedge \ldots \wedge {\tilde{{\boldsymbol{\omega }}}}^{{\tilde{\mu }}_{k}},\end{eqnarray} \tag{ B35 }$$

which together with (3.7) proves the first relation (3.23).

For $k=\tilde{N}+\bar{k}\gt \tilde{N}$ the leading term in (B34) must contain $\bar{k}$ rescaled terms, since there are just $\tilde{N}$ unscaled coordinates ${x}_{\bar{N}+1},\ldots ,{x}_{N}$. The unscaled terms can be factorized in front of the sum and with the help of (3.7), (3.20), and, in the second equality, with the help of (3.13), (2.17), and again (B34), we obtain

$$\begin{eqnarray}{{\boldsymbol{h}}}^{(\tilde{N}+\bar{k})} & \approx & {\sigma }^{-\bar{k}}{\left(\tfrac{1}{{\tilde{{ \mathcal A }}}^{(\tilde{N})}{\bar{{ \mathcal A }}}^{(\bar{k})}}\right)}^{\displaystyle \frac{1}{2}}{\sigma }^{\bar{k}}{\tilde{w}}^{2\bar{k}}\ {\tilde{x}}_{1}\ldots {\tilde{x}}_{\tilde{N}}\ {\tilde{{\boldsymbol{\omega }}}}^{1}\wedge \ldots \wedge {\tilde{{\boldsymbol{\omega }}}}^{\tilde{N}}\\ & & \quad \qquad \qquad \wedge \displaystyle \sum _{{\bar{\mu }}_{1}\lt \ldots \lt {\bar{\mu }}_{\bar{k}}}{\bar{x}}_{{\bar{\mu }}_{1}}\ldots {\bar{x}}_{{\bar{\mu }}_{\bar{k}}}\ {\bar{{\boldsymbol{\omega }}}}^{{\bar{\mu }}_{1}}\wedge \ldots \wedge {\bar{{\boldsymbol{\omega }}}}^{{\bar{\mu }}_{\bar{k}}}\\ & = & {\tilde{w}}^{2\bar{k}+1}\;\tilde{{\boldsymbol{\varepsilon }}}\wedge {\bar{{\boldsymbol{h}}}}^{(\bar{k})},\end{eqnarray} \tag{ B36 }$$

which concludes the proof of relations (3.23).

The limit of KY forms ${{\boldsymbol{f}}}^{(k)}$ can be obtained employing definition (2.37). The Hodge dual is given by the contraction with the Levi-Civita tensor ${\boldsymbol{\varepsilon }}$ which splits into unscaled and scaled parts (3.14). For $k=\tilde{k}\leqslant \tilde{N}$ we use the first relation of (3.23) and obtain

$$\begin{eqnarray}&&{{\boldsymbol{f}}}^{(\tilde{k})}={{\boldsymbol{h}}}^{(\tilde{k})}\;\bullet \;{\boldsymbol{\varepsilon }}\approx {\tilde{w}}^{2\bar{N}}{\tilde{{\boldsymbol{h}}}}^{(\tilde{k})}\;\bullet \;(\tilde{{\boldsymbol{\varepsilon }}}\wedge \bar{{\boldsymbol{\varepsilon }}}).\end{eqnarray} \tag{ B37 }$$

Since ${\tilde{{\boldsymbol{h}}}}^{(\tilde{k})}$ is nontrivial only in the unscaled directions, we can write

$$\begin{eqnarray}&&{{\boldsymbol{f}}}^{(\tilde{k})}\approx {\tilde{w}}^{2\bar{N}}({\tilde{{\boldsymbol{h}}}}^{(\tilde{k})}\;\tilde{\bullet }\;\tilde{{\boldsymbol{\varepsilon }}})\wedge \bar{{\boldsymbol{\varepsilon }}}={\tilde{w}}^{2\bar{N}}\;{\tilde{{\boldsymbol{f}}}}^{(\tilde{k})}\wedge \bar{{\boldsymbol{\varepsilon }}},\end{eqnarray} \tag{ B38 }$$

which is the first relation (3.24).

For $k=\tilde{N}+\bar{k}\gt \tilde{N}$ we employ the second relation (3.23), to get

$$\begin{eqnarray} & & {{\boldsymbol{f}}}^{(\tilde{N}+\bar{k})}={{\boldsymbol{h}}}^{(\tilde{N}+\bar{k})}\;\bullet \;{\boldsymbol{\varepsilon }}\\ & & \quad \approx ({\tilde{w}}^{2\bar{k}+1}\;\tilde{{\boldsymbol{\varepsilon }}}\wedge {\bar{{\boldsymbol{h}}}}^{(\bar{k})})\;\bullet \;({\tilde{w}}^{2\bar{N}}\;\tilde{{\boldsymbol{\varepsilon }}}\wedge \bar{{\boldsymbol{\varepsilon }}})\\ & & \quad \approx {\tilde{w}}^{2\bar{k}+1}{\tilde{w}}^{2\bar{N}}{\tilde{w}}^{-4\bar{k}}\;(\tilde{{\boldsymbol{\varepsilon }}}\;\tilde{\bullet }\;\tilde{{\boldsymbol{\varepsilon }}})({\bar{{\boldsymbol{h}}}}^{(\bar{k})}\;\bar{\bullet }\;\bar{{\boldsymbol{\varepsilon }}}).\end{eqnarray} \tag{ B39 }$$

The additional warp-factor ${\tilde{w}}^{-4\bar{k}}$ has appeared due to a conversion of $\bullet$ into $\bar{\bullet }$, since these operations involve the inverse metric ${{\boldsymbol{g}}}^{-1}$, ${\bar{{\boldsymbol{g}}}}^{-1}$, respectively, used for raising $\bar{k}$ indices before the contraction. Since $\tilde{{\boldsymbol{\varepsilon }}}\;\tilde{\bullet }\;\tilde{{\boldsymbol{\varepsilon }}}=1$, and recalling (2.37), we proved (3.24),

$$\begin{eqnarray}&&{{\boldsymbol{f}}}^{(\tilde{N}+\bar{k})}\approx {\tilde{w}}^{2(\tilde{N}-\bar{k})+1}\;{\bar{{\boldsymbol{f}}}}^{(\bar{k})}.\end{eqnarray} \tag{ B40 }$$

To find the limit of the Killing tensors (2.42), it is useful to write them with indices in the lower position,

$$\begin{eqnarray}&&{{\boldsymbol{k}}}^{(j)}=\displaystyle \sum _{\mu }\displaystyle \frac{{A}_{\mu }^{(j)}}{{{ \mathcal A }}^{(j)}}\left[\;\displaystyle \frac{{U}_{\mu }}{{X}_{\mu }}\;{\boldsymbol{d}}{x}_{\mu }^{2}+\displaystyle \frac{{X}_{\mu }}{{U}_{\mu }}{\left(\displaystyle \sum _{k}{A}_{\mu }^{(k)}{\boldsymbol{d}}{\psi }_{k}\right)}^{2}\;\right].\end{eqnarray} \tag{ B41 }$$

Employing relations (3.7) and (3.9), the sum splits into sums over scaled and unscaled coordinates. For $j=\tilde{j}\leqslant \tilde{N}$ both parts contribute to the leading order:

$$\begin{eqnarray}{{\boldsymbol{k}}}^{(\tilde{j})} & \approx & \displaystyle \frac{1}{{\tilde{{ \mathcal A }}}^{(\tilde{j})}}\displaystyle \sum _{\tilde{\mu }}{\tilde{A}}_{\tilde{\mu }}^{(\tilde{j})}\left[\;\displaystyle \frac{{\tilde{U}}_{\tilde{\mu }}}{{\tilde{X}}_{\tilde{\mu }}}\;{\boldsymbol{d}}{\tilde{x}}_{\tilde{\mu }}^{2}+\displaystyle \frac{{\tilde{X}}_{\tilde{\mu }}}{{\tilde{U}}_{\tilde{\mu }}}{\left(\displaystyle \sum _{\tilde{k}}{\tilde{A}}_{\tilde{\mu }}^{(\tilde{k})}{\boldsymbol{d}}{\tilde{\psi }}_{\tilde{k}}\right)}^{2}\;\right]\\ & & +\displaystyle \frac{{\tilde{A}}^{(\tilde{j})}}{{\tilde{{ \mathcal A }}}^{(\tilde{j})}}\displaystyle \sum _{\bar{\mu }}\left[\displaystyle \frac{1}{{\sigma }^{2}}\displaystyle \frac{{\tilde{A}}^{(\tilde{N})}}{{\tilde{{ \mathcal A }}}^{(\tilde{N})}}\displaystyle \frac{{\bar{U}}_{\bar{\mu }}}{{\bar{X}}_{\bar{\mu }}}\;{\sigma }^{2}{\boldsymbol{d}}{\bar{x}}_{\bar{\mu }}^{2}+{\sigma }^{2}\displaystyle \frac{{\tilde{{ \mathcal A }}}^{(\tilde{N})}}{{\tilde{A}}^{(\tilde{N})}}\displaystyle \frac{{\bar{X}}_{\bar{\mu }}}{{\bar{U}}_{\bar{\mu }}}\displaystyle \frac{1}{{\sigma }^{2}}{\left(\displaystyle \frac{{\tilde{A}}^{(\tilde{N})}}{{\tilde{{ \mathcal A }}}^{(\tilde{N})}}\displaystyle \sum _{\bar{k}}{\bar{A}}_{\bar{\mu }}^{(\bar{k})}{\boldsymbol{d}}{\bar{\psi }}_{\bar{k}}\right)}^{2}\;\right]\\ & = & {\tilde{{\boldsymbol{k}}}}^{(\tilde{j})}+\displaystyle \frac{{\tilde{A}}^{(\tilde{j})}}{{\tilde{{ \mathcal A }}}^{(\tilde{j})}}\;{\tilde{w}}^{2}\;\bar{{\boldsymbol{g}}}.\end{eqnarray} \tag{ B42 }$$

For $j=\tilde{N}+\bar{j}\gt \tilde{N}$ the sum over unscaled coordinates disappears in the leading order:

$$\begin{eqnarray}{{\boldsymbol{k}}}^{(\tilde{N}+\bar{j})} & \approx & {\sigma }^{-2\bar{j}}\displaystyle \frac{1}{{\tilde{{ \mathcal A }}}^{(\tilde{N})}{\bar{{ \mathcal A }}}^{(\bar{j})}}{\sigma }^{2(\bar{j}+1)}{\bar{A}}^{(\bar{j}+1)}\displaystyle \sum _{\tilde{\mu }}{\tilde{A}}_{\tilde{\mu }}^{(\tilde{N}-1)}\left[\;\displaystyle \frac{{\tilde{U}}_{\tilde{\mu }}}{{\tilde{X}}_{\tilde{\mu }}}\;{\boldsymbol{d}}{\tilde{x}}_{\tilde{\mu }}^{2}+\displaystyle \frac{{\tilde{X}}_{\tilde{\mu }}}{{\tilde{U}}_{\tilde{\mu }}}{\left(\displaystyle \sum _{\tilde{k}}{\tilde{A}}_{\tilde{\mu }}^{(\tilde{k})}{\boldsymbol{d}}{\tilde{\psi }}_{\tilde{k}}\right)}^{2}\;\right]\\ & & +{\sigma }^{-2\bar{j}}\displaystyle \frac{1}{{\tilde{{ \mathcal A }}}^{(\tilde{N})}{\bar{{ \mathcal A }}}^{(\bar{j})}}{\sigma }^{2\bar{j}}{\tilde{A}}^{(\tilde{N})}\displaystyle \sum _{\bar{\mu }}{\bar{A}}_{\bar{\mu }}^{(\bar{j})}\left[\displaystyle \frac{1}{{\sigma }^{2}}\displaystyle \frac{{\tilde{A}}^{(\tilde{N})}}{{\tilde{{ \mathcal A }}}^{(\tilde{N})}}\displaystyle \frac{{\bar{U}}_{\bar{\mu }}}{{\bar{X}}_{\bar{\mu }}}\;{\sigma }^{2}{\boldsymbol{d}}{\bar{x}}_{\bar{\mu }}^{2}\right.\\ & & \left.+{\sigma }^{2}\displaystyle \frac{{\tilde{{ \mathcal A }}}^{(\tilde{N})}}{{\tilde{A}}^{(\tilde{N})}}\displaystyle \frac{{\bar{X}}_{\bar{\mu }}}{{\bar{U}}_{\bar{\mu }}}\displaystyle \frac{1}{{\sigma }^{2}}{\left(\displaystyle \frac{{\tilde{A}}^{(\tilde{N})}}{{\tilde{{ \mathcal A }}}^{(\tilde{N})}}\displaystyle \sum _{\bar{k}}{\bar{A}}_{\bar{\mu }}^{(\bar{k})}{\boldsymbol{d}}{\bar{\psi }}_{\bar{k}}\right)}^{2}\;\right]\\ & \approx & {\tilde{w}}^{4}\;{{\boldsymbol{k}}}^{(\bar{j})}.\end{eqnarray} \tag{ B43 }$$

Here, the functions ${\tilde{X}}_{\tilde{\mu }}$ and ${\bar{X}}_{\bar{\mu }}$ are given by relations (3.11) and (3.12). Raising indices with the limiting metric (3.10) we obtain relations (3.25).

In this appendix, we provide some details about the limiting procedure performed in section 4. Namely, we will repeatedly apply the multiple-spin-zero limit to the Kerr-NUT-(A)dS metric and its Killing tower objects.

Let us first list some useful relations for the intermediate metrics introduced in (4.1) and (4.2). Metric $\mathring{{\boldsymbol{g}}}$ and its Levi-Civita tensor can be split into a warped product as follows

$$\begin{eqnarray}\mathring{{\boldsymbol{g}}} & = & {}^{\alpha }\overset{\bullet }{{\boldsymbol{g}}}+{}^{\alpha }{w}^{2}\ {}^{N-\alpha }\mathring{{\boldsymbol{g}}},\\ \mathring{{\boldsymbol{\varepsilon }}} & = & {}^{\alpha }\overset{\bullet }{{\boldsymbol{\varepsilon }}}\wedge {}^{\alpha }{w}^{2(N-\alpha )}\ {}^{N-\alpha }\mathring{{\boldsymbol{\varepsilon }}},\end{eqnarray} \tag{ B44 }$$

where, for arbitrary α, the warp factor reads

$$\begin{eqnarray}&&{}^{\alpha }w={\xi }_{N}\ldots {\xi }_{N-\alpha -1}.\end{eqnarray} \tag{ B45 }$$

Intermediate metrics (4.1) and (4.2) satisfy recurrence relations

$$\begin{eqnarray}{}^{\alpha +1}\mathring{{\boldsymbol{g}}} & = & {{\boldsymbol{q}}}_{\alpha +1}+{\xi }_{\alpha +1}^{2}\ {}^{\alpha }\mathring{{\boldsymbol{g}}},\quad {}^{\alpha +1}\mathring{{\boldsymbol{\varepsilon }}}={{\boldsymbol{\epsilon }}}_{\alpha +1}\wedge {\xi }_{\alpha +1}^{2\alpha }\ {}^{\alpha }\mathring{{\boldsymbol{\varepsilon }}},\\ {}^{\alpha +1}\overset{\bullet }{{\boldsymbol{g}}} & = & {}^{\alpha }\overset{\bullet }{{\boldsymbol{g}}}+{}^{\alpha }{w}^{2}\ {{\boldsymbol{q}}}_{N-\alpha },\quad {}^{\alpha +1}\overset{\bullet }{{\boldsymbol{\varepsilon }}}={}^{\alpha }\overset{\bullet }{{\boldsymbol{\varepsilon }}}\wedge {}^{\alpha }{w}^{2}\ {{\boldsymbol{\epsilon }}}_{N-\alpha }.\end{eqnarray} \tag{ B46 }$$

We want to take repeatedly the limit $\bar{N}=N-1$, $\tilde{N}=1$. To obtain the limit of the metric, it is enough to look at one 'inductive' step. We claim that after $N-\alpha$ limiting steps, the metric is

$$\begin{eqnarray}&&{\boldsymbol{g}}\approx {}^{N-\alpha }\overset{\bullet }{{\boldsymbol{g}}}+{}^{\alpha }{w}^{2}\ {}^{\alpha }{\boldsymbol{g}}.\end{eqnarray} \tag{ B47 }$$

Let us assume it for a particular $\alpha$. Performing the $\tilde{N}=1$ limit (3.10) of the metric ${}^{\alpha }{\boldsymbol{g}}$ we get

$$\begin{eqnarray}&&{\boldsymbol{g}}\approx {}^{N-\alpha }\overset{\bullet }{{\boldsymbol{g}}}+{}^{\alpha }{w}^{2}\ \left({}^{\alpha }\tilde{{\boldsymbol{g}}}+\displaystyle \frac{{\tilde{x}}_{1}^{2}}{{\tilde{a}}_{1}^{2}}\ {}^{\alpha }\bar{{\boldsymbol{g}}}\right).\end{eqnarray} \tag{ B48 }$$

Introducing the rescaled coordinate ${\xi }_{\alpha }=\tfrac{{\tilde{x}}_{1}}{{\tilde{a}}_{1}}=\tfrac{{x}_{\alpha }}{{a}_{\alpha }}$, the 2-metric ${}^{\alpha }\tilde{{\boldsymbol{g}}}$ can be rewritten as the atomic metric ${{\boldsymbol{q}}}_{\alpha }$ defined in (4.3). Relabeling back ${\bar{x}}_{\bar{\mu }}\to {x}_{\bar{\mu }}$, ${\bar{a}}_{\bar{\mu }}\to {a}_{\bar{\mu }}$ we get ${}^{\alpha }\bar{{\boldsymbol{g}}}\to {}^{\alpha -1}{\boldsymbol{g}}$. (Note that the relabeling of coordinates does not affect the definition of the rescaled coordinate ${\xi }_{\mu }$ in the following limiting steps, as ${\xi }_{\mu }=\tfrac{{\bar{x}}_{\mu }}{{\bar{a}}_{\mu }}=\tfrac{{x}_{\mu }}{{a}_{\mu }}$.) Taking into account relations (B46) and definition (B45), we prove the inductive step

$$\begin{eqnarray}{\boldsymbol{g}} & \approx & ({}^{N-\alpha }\overset{\bullet }{{\boldsymbol{g}}}+{}^{\alpha }{w}^{2}\ {{\boldsymbol{q}}}_{\alpha })+{}^{\alpha }{w}^{2}\;{\xi }_{\alpha }^{2}\ {}^{\alpha -1}{\boldsymbol{g}}\\ & = & {}^{N-\alpha +1}\overset{\bullet }{{\boldsymbol{g}}}+{}^{\alpha -1}{w}^{2}\ {}^{\alpha -1}{\boldsymbol{g}},\end{eqnarray} \tag{ B49 }$$

and thus the relation (B47). A straightforward check of the last step in the limit shows that after $N$ steps we obtain the desired relation ${\boldsymbol{g}}\approx {}^{N}\overset{\bullet }{{\boldsymbol{g}}}=\mathring{{\boldsymbol{g}}}$.

For the multiple single-spin-zero limit of the CCKY forms we indicate just a few first steps which elucidate the resulting expressions. The relation (3.23) for $\tilde{N}=1$, together with the same redefinitions as for the metric, allows us to write the first $k$ limits as follows

$$\begin{eqnarray}\phantom{\rule{4em}{0ex}}{{\boldsymbol{h}}}^{(k)} & \approx & {\xi }_{N}^{2(k-1)+1}\ {{\boldsymbol{\epsilon }}}_{N}\wedge {}^{N-1}{{\boldsymbol{h}}}^{(k-1)}\\ & \approx & {\xi }_{N}^{2(k-2)+1}{\xi }_{N-1}^{2(k-2)+1}\ {{\boldsymbol{\epsilon }}}_{N}\wedge {\xi }_{N}^{2}{{\boldsymbol{\epsilon }}}_{N-1}\wedge {}^{N-2}{{\boldsymbol{h}}}^{(k-2)}\\ & & \ldots \\ & \approx & {\xi }_{N}\ldots {\xi }_{N-k+1}\ {{\boldsymbol{\epsilon }}}_{N}\wedge {\xi }_{N}^{2}{{\boldsymbol{\epsilon }}}_{N-1}\wedge {\xi }_{N}^{2}{\xi }_{N-1}^{2}{{\boldsymbol{\epsilon }}}_{N-2}\wedge \ldots \wedge {}^{N-k}{{\boldsymbol{h}}}^{(0)}\\ & = & {}^{k}w\ {}^{k}\overset{\bullet }{{\boldsymbol{\varepsilon }}}.\end{eqnarray} \tag{ B50 }$$

The remaining limits do not affect the resulting object any more. To obtain the limit of KY forms ${{\boldsymbol{f}}}^{(k)}$ we employ the relation (3.24) for the first $k$ limits and the relation (3.14) for the remaining limits:

$$\begin{eqnarray}{{\boldsymbol{f}}}^{(k)} & \approx & {\xi }_{N}^{2(N-k)+1}\;{}^{N-1}{{\boldsymbol{f}}}^{(k-1)}\\ & & \ldots \\ & \approx & {({\xi }_{N}\ldots {\xi }_{N+1-k})}^{2(N-k)+1}\;{}^{N-k}{{\boldsymbol{f}}}^{(0)}={}^{k}{w}^{2(N-k)+1}\ \ {}^{N-k}{\boldsymbol{\varepsilon }}\\ & \approx & {}^{k}{w}^{2(N-k)+1}\ \ {{\boldsymbol{\epsilon }}}_{N-k}\wedge {\xi }_{N-k}^{2(N-k-1)}\ {}^{N-k-1}{\boldsymbol{\varepsilon }}\\ & & \ldots \\ & \approx & {}^{k}{w}^{2(N-k)+1}\ \ {{\boldsymbol{\epsilon }}}_{N-k}\wedge {\xi }_{N-k}^{2}\;{{\boldsymbol{\epsilon }}}_{N-k-1}\wedge \cdots \wedge {\xi }_{N-k}^{2}\ldots {\xi }_{2}^{2}\;{{\boldsymbol{\epsilon }}}_{1}\\ & = & {}^{k}{w}^{2(N-k)+1}\ \ {}^{N-k}\mathring{{\boldsymbol{\varepsilon }}}.\end{eqnarray} \tag{ B51 }$$

The first $j$ limits in the multiple single-spin-zero limit of the Killing tensor ${{\boldsymbol{k}}}_{(j)}$ is given by the second relation of (3.25), the remaining limits act on the lower-dimensional Kerr-NUT-(A)dS metric in a way analogous to (4.5),

$$\begin{eqnarray}&&{{\boldsymbol{k}}}_{(j)}\approx {}^{N-1}{{\boldsymbol{k}}}_{(j-1)}\approx \cdots \;\approx {}^{N-j}{{\boldsymbol{k}}}_{(0)}={}^{N-j}{{\boldsymbol{g}}}^{-1}\approx {}^{N-j}{\mathring{{\boldsymbol{g}}}}^{-1},\end{eqnarray} \tag{ B52 }$$

which completes this appendix.