#  Kleinian black holes

Damien A<EMAIL_ADDRESS>Department of Physics, Arizona State
University, Tempe, AZ 85287-1504, USA  Max W.
<EMAIL_ADDRESS>Brown Theoretical Physics Center, Department
of Physics, Brown University, Providence, RI 02912, USA

###### Abstract

We prove there is a unique vacuum solution in split-signature spacetimes with
Kleinian $SO(2,1)$ spherical symmetry. We extend our analysis to accommodate a
positive or negative cosmological constant and we prove the Kleinian
spherically symmetric solutions to Einstein’s equation are locally isomorphic
to the split-signature analogues of Schwarzschild-(Anti)-de Sitter or Nariai
spacetimes. Our analysis provides a Kleinian extension of Birkhoff’s theorem
to metrics with split-signature. Axisymmetric vacuum solutions are also
considered, including $(2,2)$ signature formulations of the Kerr and Taub-NUT
metrics.

###### Contents

  1. I Introduction
  2. II Klein Space
  3. III Kleinian vacuum solution
    1. III.1 Kleinian Schwarzschild
      1. III.1.1 Kleinian Kantowski-Sachs
    2. III.2 (A)dS Generalization
    3. III.3 Kleinian Nariai
    4. III.4 (2,2) signature Kerr
      1. III.4.1 Kleinian Kerr-Schild coordinates
      2. III.4.2  Kleinian Kerr solution
    5. III.5 (2,2) signature Taub-NUT
  4. IV Conclusions
  5. IV.1 Alternative analytic continuation

##  I Introduction

The Birkhoff theorem stands as a cornerstone in the realm of general
relativity, affirming that any spherically symmetric vacuum solution derived
from Einstein’s equations can be locally equated to a corresponding segment of
the Schwarzschild spacetime. The ubiquitous use of the theorem, along with its
multiple origin history are a testament to its profound importance in
contemporary general relativity and cosmology, and naturally invite
contemplation regarding the theorem’s potential extension to diverse scenarios
within the realm of gravitational physics and theories of modified gravity.

Birkhoff’s theorem states that the Schwarzschild metric:

$\mathrm{d}s^{2}=-\Big{(}1-\frac{2m}{r}\Big{)}\mathrm{d}t^{2}+\Big{(}1-\frac{2m}{r}\Big{)}^{-1}\mathrm{d}r^{2}+r^{2}\mathrm{d}\theta^{2}+r^{2}\sin^{2}\theta\mathrm{d}\phi^{2}\,,$
(1.1)

is the unique vacuum solution with spherical symmetry to the Einstein
equations and there are no time-dependent solutions of this form, as long as
$t$ remains the timelike coordinate and $r,\theta,\phi$ remain spacelike. The
maximal analytic extension of the Schwarzschild solution has a non-static
interior with spacelike (rather than timelike) Killing vector field. The
theorem generalizes in the case of a non-vanishing cosmological constant
$\Lambda$, to the Schwarzschild-(A)dS solutions. The caveat concerning the
coordinate $t$ remaining timelike now extends to the region outside the de
Sitter cosmological horizon which is time-dependent. The theorem is
ubiquitously attributed to Birkhoff Birkhoff (1923), although it was
discovered earlier by Jebsen Jebsen (1921) and allegedly independently by
Alexandrow Alexandrow (1923) and Eiesland Eiesland (1921, 1925), in various
forms (see e.g. Deser and Franklin (2005); Schleich and Witt (2010) for
historical accounts).

The analytic continuation from Lorentzian $(1,3)$ signature to Kleinian
$(2,2)$ split-signature has proved useful in the study of quantum field theory
and quantum gravity. At the level of empty space, this translates into the
analytic continuation from Minkowski space, $\mathbb{M}^{3,1}$ to Klein space,
$\mathbb{K}^{2,2}$. Recently, studies of black holes written in Kleinian
signature have appeared in the literature, making a complete investigation of
their emergence from the vacuum theory desirable. For example, it has been
shown that linearized black hole geometries are captured by (2,2) three-point
scattering amplitudes of a graviton and a massive spinning particle. The
solutions include Kerr-Taub-NUT spacetimes which naturally encompass the
Schwarzschild solution Arkani-Hamed  _et al._ (2020); Crawley  _et al._
(2022).

In addition, two-dimensional quantum states associated to four-dimensional
linearized rotating self-dual black holes in split-signature have been shown
to be comprised of global conformal primaries circulating on the celestial
torus, the Kleinian analog of the celestial sphere. This allows for a direct
connection to the S-matrix Crawley  _et al._ (2023).

In some cases, it was shown that a spin parameter $a$ can be added to existing
solutions via a simple coordinate change, including Self-dual Taub-NUT (SDTN)
Crawley  _et al._ (2022) and the self-dual analog of Kerr Easson  _et al._
(2023). Furthermore, a double copy relation between the gluon amplitude on a
self-dual dyon and graviton amplitude on a SDTN spacetime was discussed in
Adamo  _et al._ (2023). And a possible explanation of hidden symmetries of
rotating black holes via analytic continuation to Kleinian signature was
explored in Guevara and Kol (2023).

Given these advances in Kleinian geometry it is therefore natural to consider
complete studies of black hole spacetimes with split signature, including
properties of their uniqueness. In this paper we initiate this effort, and
show that the Kleinian analog of Schwarzschild is the unique vacuum solution
to the Einstein equations with Kleinian spherical symmetry and there are no
time dependent solutions of this form, thus generalizing the Birkhoff theorem
to split signature spacetimes. Spacetimes with a cosmological constant are
also considered and we show the unique solutions are split-signature versions
of Schwarzschild-(A)dS or Nariai Klein spaces. Finally, expanding our
tolerance to allow for axially symmetric vacuum solutions, we conjure the
Kleinian analogue of the Kerr solution applying the Newman-Janis algorithm to
split-signature Schwarzschild written in Kerr-Schild coordinates, and discuss
the Kleinian Taub-NUT solution.

This paper is organized as follows. In section II, we introduce flat Klein
space and our conventions used to transition from Lorentzian to split
signature Kleinian metrics. In section III, we study Kleinian vacuum solutions
to the Einstein equations, uncovering the Kleinian Schwarzschild solution. In
a brief discussion of black hole horizon crossing we find the Kleinian
Kantowski-Sachs metric. We then consider Schwarzschild-(Anti)-de Sitter and
Nariai generalizations in the presence of non-vanishing cosmological constant.
We then produce the Kleinian Schwarzschild solution in Kerr-Schild coordinates
and using the Newman-Janis trick we construct a $(2,2)$ signature Kerr
solution. In this family of metrics with axial symmetry we also discuss
$(2,2)$ signature Taub-NUT space. We conclude in section IV. Appendix A is
included to demonstrate alternative analytic continuations from Lorentzian to
Klein space.

##  II Klein Space

Let us begin with local Cartesian coordinates $x^{\alpha}=(t,x,y,z)$, in which
the Lorentzian flat spacetime metric is

$\mathrm{d}s^{2}=-\mathrm{d}t^{2}+\mathrm{d}x^{2}+\mathrm{d}y^{2}+\mathrm{d}z^{2}\,.$
(2.2)

Transforming to spherical coordinates, the euclidean line element
$\mathrm{d}s^{2}=g_{ij}\mathrm{d}x^{i}\mathrm{d}x^{j}$ is then fixed by
invariance of the metric on the manifold $\mathcal{M}$ under the $SO(3)$
rotation group: $SO(3)\times\mathcal{M}\rightarrow\mathcal{M}$ and we find

$\mathrm{d}s^{2}=-\mathrm{d}t^{2}+\mathrm{d}r^{2}+r^{2}\mathrm{d}\theta^{2}+r^{2}\sin^{2}\theta\mathrm{d}\phi^{2}\,.$
(2.3)

This coordinate transformation is given by

$x=r\cos\phi\sin\theta\,,\qquad y=r\sin\phi\sin\theta\,,\qquad
z=r\cos\theta\,,$ (2.4)

with inverse transformation

$r=\sqrt{x^{2}+y^{2}+z^{2}}\,,\qquad\theta=\arccos\left(\frac{z}{r}\right)\,,\qquad\varphi=\arctan\left(\frac{y}{x}\right)$
(2.5)

and the most general spherically symmetric spacetime metric is

$\mathrm{d}s^{2}=-N(t,r)\mathrm{d}t^{2}+P(t,r)\mathrm{d}r^{2}+Q(r,t)\mathrm{d}\Omega_{L}^{2}\,,$
(2.6)

where
$\mathrm{d}\Omega_{L}^{2}=\mathrm{d}\theta^{2}+\sin^{2}\theta\mathrm{d}\phi^{2}$
is the metric on the unit sphere $\mathbf{S}^{2}$ and $N$, $P$, $Q$ are
arbitrary functions of the time and radial coordinate.

It is possible to move from Lorentzian signature metrics to split-signature
(Kleinian) metrics via analytic continuation. We may transition to Klein space
in multiple ways. 333An alternative choice of continuation to the one
presented here is provided in Appendix A. We choose to move to split signature
metric via the analytic continuation

$t\rightarrow it\,,\qquad x\rightarrow ix\,,\qquad y\rightarrow iy\,,$ (2.7)

so that in Kleinian signature the flat metric (2.2) becomes

$\mathrm{d}s^{2}=\mathrm{d}t^{2}-\mathrm{d}x^{2}-\mathrm{d}y^{2}+\mathrm{d}z^{2}\,.$
(2.8)

The analytic continuation (2.7) is equivalent to the spherical coordinate
euclideanization $\theta\rightarrow i\theta$ in (2.4), so that
$\cos{\theta}\rightarrow\cosh{\theta}$ and $\sin{\theta}\rightarrow
i\sinh{\theta}$. We can transform into coordinates where $SO(2,1)$ invariance
is manifest via

$x=r\cos\phi\sinh\theta\,,\qquad\qquad y=r\sin\phi\sinh\theta\,,\qquad\qquad
z=r\cosh\theta\,.$ (2.9)

The inverse coordinate transformation is

$r=\sqrt{z^{2}-x^{2}-y^{2}}\,,\qquad\theta=\mathrm{arccosh}\left(\frac{z}{\sqrt{z^{2}-x^{2}-y^{2}}}\right)\,,\qquad\phi=\arctan\left(\frac{y}{x}\right).$
(2.10)

The metric (2.8) becomes

$\mathrm{d}s^{2}=\mathrm{d}t^{2}+\mathrm{d}r^{2}-r^{2}\mathrm{d}\theta^{2}-r^{2}\sinh^{2}\theta\mathrm{d}\phi^{2}.$
(2.11)

Invariance under the action of the Kleinian rotation symmetry group on the
spacetime manifold yields corresponding infinitesimal motion generated by the
complex Killing vector fields

$\displaystyle\xi_{x}$ $\displaystyle=$ $\displaystyle
i\sin{\phi}\,\partial_{\theta}+i\coth{\theta}\cos{\phi}\,\partial_{\phi}$
$\displaystyle\xi_{y}$ $\displaystyle=$
$\displaystyle-i\cos{\phi}\,\partial_{\theta}+i\coth{\theta}\sin{\phi}\,\partial_{\phi}$
$\displaystyle\xi_{z}$ $\displaystyle=$ $\displaystyle\partial_{\phi}$ (2.12)

Invariance under Kleinian rotations imply the vanishing of the Lie derivative
$L_{\xi}g_{ij}=0$ along these vector fields. The commutators of the above
$[\xi_{i},\xi_{j}]=\epsilon_{ijk}\xi_{k}$ form the Lie algebra of the Kleinian
rotation group $SO(2,1)$, and thus we have invariance of the metric under the
action $\xi:SO(2,1)\times\mathcal{M}\rightarrow\mathcal{M}$, of the group
$SO(2,1)$ on the spacetime manifold $\mathcal{M}$. This suggests the most
general Kleinian spherically symmetric metric may be written

$\mathrm{d}s^{2}=N(T,R)\mathrm{d}T^{2}+P(T,R)\mathrm{d}R^{2}-Q(T,R)(\mathrm{d}\theta^{2}+\sinh^{2}\theta\mathrm{d}\phi^{2})\,,$
(2.13)

where $N$, $P$, and $Q$ are scalar functions of $T$ and $R$ only. We will also
abbreviate the angular term in parentheses as
$\mathrm{d}\theta^{2}+\sinh^{2}\theta\mathrm{d}\phi^{2}\equiv\mathrm{d}\Omega^{2}$,
the unit radius hyperbolic space with constant Ricci scalar curvature $R=-2$.
We can make a diffeomorphism to coordinates where $Q=r^{2}$ for some
coordinate $r$: this transforms the metric into

$\mathrm{d}s^{2}=\Bigg{(}N+P\Big{(}\frac{\partial R}{\partial
T}\Big{)}^{2}\Bigg{)}\mathrm{d}T^{2}+P\frac{\partial R}{\partial
T}\frac{\partial R}{\partial
r}(\mathrm{d}T\mathrm{d}r+\mathrm{d}r\mathrm{d}T)+P\Big{(}\frac{\partial
R}{\partial r}\Big{)}^{2}\mathrm{d}r^{2}-r^{2}\mathrm{d}\Omega^{2}.$ (2.14)

Defining new metric profile functions $g_{TT}$, $g_{Tr}$, and $g_{rr}$, we see
that a Kleinian spherically symmetric metric can also be written as

$\mathrm{d}s^{2}=g_{TT}\mathrm{d}T^{2}+g_{Tr}(\mathrm{d}T\mathrm{d}r+\mathrm{d}r\mathrm{d}T)+g_{rr}\mathrm{d}r^{2}-r^{2}\mathrm{d}\Omega^{2}.$
(2.15)

Next, we show that this metric can be written in the form

$\mathrm{d}s^{2}=n(t,r)\mathrm{d}t^{2}+p(t,r)\mathrm{d}r^{2}-r^{2}\mathrm{d}\Omega^{2}.$
(2.16)

For if we make the coordinate transformation $\mathrm{d}t=\frac{\partial
t}{\partial T}\mathrm{d}T+\frac{\partial t}{\partial r}\mathrm{d}r$ on the
above line element, we have that

$\mathrm{d}s^{2}=n\Big{(}\frac{\partial t}{\partial
T}\Big{)}^{2}\mathrm{d}T^{2}+n\frac{\partial t}{\partial T}\frac{\partial
t}{\partial
r}(\mathrm{d}T\mathrm{d}r+\mathrm{d}r\mathrm{d}T)+\Bigg{(}n\Big{(}\frac{\partial
t}{\partial r}\Big{)}^{2}+p\Bigg{)}\mathrm{d}r^{2}-r^{2}\mathrm{d}\Omega^{2}.$
(2.17)

Matching the above line element to (2.15), we obtain

$n\Big{(}\frac{\partial t}{\partial T}\Big{)}^{2}=g_{TT}\mathrm{,\
}n\frac{\partial t}{\partial T}\frac{\partial t}{\partial r}=g_{Tr}\mathrm{,\
and\ }n\Big{(}\frac{\partial t}{\partial r}\Big{)}^{2}+p=g_{rr}.$ (2.18)

These are three differential equations for three unknown functions $t(T,r)$,
$n(t,r)$, and $p(t,r)$, which is enough to specify each along with supplying
initial conditions on $t$. The most general Kleinian spherically symmetric
metric is given by (2.16).

##  III Kleinian vacuum solution

We now begin an exploration of Kleinian solutions to the vacuum Einstein
equations $G_{\mu\nu}=0$, where as usual $G_{\mu\nu}$ is the Einstein tensor
defined in terms of the Ricci tensor and scalar,
$G_{\mu\nu}=R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R$.

###  III.1 Kleinian Schwarzschild

Here we produce the slit-signature generalization of the Birkhoff theorem, by
searching for spherically symmetric solutions to the vaccum Einstein
equations. We substitute (2.16) into the vacuum Einstein equations
$G_{\mu\nu}=0$ to find that

$G_{tr}=\frac{\partial_{t}p}{pr}=0\,,$ (3.19)

which implies that $p$ is a function independent of time. With this condition,
no time derivatives of $n$ appear in the other equations, and this implies
that $n$ is independent of time as well. $G_{tt}=0$ implies that

$p(p-1)+r\partial_{r}p=0,$ (3.20)

which has the solution $p=(1+A/r)^{-1}$ with $A$ a constant. Additionally
$G_{rr}=0$ implies that

$1-p+\frac{r\partial_{r}n}{n}=0.$ (3.21)

Substituting the solution for $p$ into the above equation allows us to solve
for $n$: we find that $n=B/A+B/r$ with $B$ a constant. If we impose that the
metric reduce to its flat form (2.11) as $r\rightarrow\infty$ then $A=B$. We
discover that the unique spherically symmetric solution to the vacuum Einstein
equations (2.16) is given by

$\mathrm{d}s^{2}=\Big{(}1+\frac{A}{r}\Big{)}\mathrm{d}t^{2}+\Big{(}1+\frac{A}{r}\Big{)}^{-1}\mathrm{d}r^{2}-r^{2}\mathrm{d}\theta^{2}-r^{2}\sinh^{2}\theta\mathrm{d}\phi^{2}.$
(3.22)

This is the Kleinian form of the familiar Schwarzschild metric, as identified
through analytic continuation $t\rightarrow it$, $\theta\rightarrow i\theta$
and by setting integration constant $A=-2m$, returning us to (1.1).

####  III.1.1 Kleinian Kantowski-Sachs

Here we consider moving across $r=2m$ in the Kleinien Schwarzschild black hole
solution. In Lorentzian Schwarzschild spacetime the metric function
$g_{tt}=(1-2m/r)$ changes sign while passing to the black hole interior
$r<2m$. This is related to the changing of the roles of space and time as seen
by the transition of $t$ and $r$ to become the spacelike and timelike
coordinates respectively. The interior Schwarzschild metric is not static but
rather a homogeneous and anisotropic Kantowski-Sachs metric

$\mathrm{d}s^{2}=-\nu(t)^{-1}\mathrm{d}t^{2}+\nu(t)\mathrm{d}r^{2}+h^{2}(t)(\mathrm{d}\theta^{2}+\sin^{2}\theta\mathrm{d}\phi^{2})\,,$
(3.23)

with two scale factors, $\nu(t)=(A/t-1)$ and $h(t)=t$. Under the analytic
continuation leading to (3.22), we see that crossing $r=2m=-A$ leads to an
Euclidean metric with signature $(-,-,-,-)$! This is in stark contrast to what
happens in ordinary Lorentzian Schwarzschild when crossing the horizon. We
note that this transition to a Euclidean metric does not occur if one chooses
to analytically continue Lorentzian Schwarzschild to Klein space via the
transformation leading to (4.62).

In Crawley  _et al._ (2022), split-signature linearized metrics were
constructed from purely on-shell radiative modes. Under the analytic
continuation used above, the Kleinian traced-reverse linearized metric has a
real integrand, and ultimately reduces to a convergent Gaussian integral (in
the region of interest). In contrast, under the continuation leading to
(4.62), the integrand for the metric is imaginary and oscillates, requiring an
$i\epsilon$ prescription for convergence. We leave further physical
interpretation of this phenomenon to future work.

###  III.2 (A)dS Generalization

If we allow for a nonzero cosmological constant, then the Einstein equations
state $G_{\mu\nu}+\Lambda g_{\mu\nu}=0$. We proceed as in the previous section
and substitute the Kleinian spherically symmetric metric (2.16) into the field
equations. The equation $G_{tr}=-\Lambda g_{tr}$ implies that
$\partial_{t}p=0$, as before; this further implies that $\partial_{t}n=0$ from
the other field equations. The equation $G_{tt}=-\Lambda g_{tt}$ demands that

$p(p-1)+r\partial_{r}p=\Lambda pr^{2}.$ (3.24)

This has the solution $p=(1+A/r-\Lambda r^{2}/3)^{-1}$. Then the
$G_{rr}=-\Lambda g_{rr}$ field equations imply that

$1-p+\frac{r\partial_{r}n}{n}=-\Lambda pr^{2}.$ (3.25)

After enforcing that $n\rightarrow 1$ as $r\rightarrow\infty$ as before, this
has the solution $n=1+A/r-\Lambda r^{2}/3=p^{-1}$. The unique solution with
metric asatz $\ref{so21metric})$ and nonzero cosmological constant is then

$\mathrm{d}s^{2}=\Big{(}1+\frac{A}{r}-\frac{\Lambda}{3}r^{2}\Big{)}\mathrm{d}t^{2}+\Big{(}1+\frac{A}{r}-\frac{\Lambda}{3}r^{2}\Big{)}^{-1}\mathrm{d}r^{2}-r^{2}\mathrm{d}\theta^{2}-r^{2}\sinh^{2}\theta\mathrm{d}\phi^{2}\,,$
(3.26)

which is the Kleinian Schwarzshild-(A)dS solution corresponding to
asymptotically Anti-de Sitter/de Sitter space for $\Lambda<0$ or $\Lambda>0$,
respectively.

###  III.3 Kleinian Nariai

There is one further possibility to consider. Returning to our general metric
(2.6) in Kleinian signature

$\mathrm{d}s^{2}=N(t,r)\mathrm{d}t^{2}+P(t,r)\mathrm{d}r^{2}-Q(r,t)\mathrm{d}\Omega^{2}\,,$
(3.27)

where again
$\mathrm{d}\Omega^{2}=\mathrm{d}\theta^{2}+\sinh^{2}{\theta}\mathrm{d}\phi^{2}$.
The Einstein equation for this metric along with vacuum energy support,
$G_{\mu\nu}+\Lambda g_{\mu\nu}=0$, gives

$\frac{{\dot{P}}Q^{\prime}}{{2PQ}}+\frac{{N^{\prime}\dot{Q}}}{{2NQ}}+\frac{{Q^{\prime}\dot{Q}}}{{2Q^{2}}}-\frac{{\dot{Q}^{\prime}}}{{Q}}=0\,,$
(3.28) $-\frac{N}{Q}-\frac{NP^{\prime}Q^{\prime}}{2P^{2}Q}-\frac{NQ^{\prime
2}}{4PQ^{2}}+\frac{NQ^{\prime\prime}}{PQ}+\frac{\dot{P}\dot{Q}}{2PQ}+\frac{\dot{Q}^{2}}{4Q^{2}}=-N\Lambda\,,$
(3.29) $-\frac{P}{Q}+\frac{N^{\prime}Q^{\prime}}{2NQ}+\frac{Q^{\prime
2}}{4Q^{2}}-\frac{P\dot{N}\dot{Q}}{2N^{2}Q}-\frac{P\dot{Q}^{2}}{4NQ^{2}}+\frac{P\ddot{Q}}{NQ}=-P\Lambda\,.$
(3.30)

The aforementioned case of interest occurs when the Kleinian sphere orbits of
the $SO(2,1)$ isometry have the same constant radius of $r=\sqrt{Q}$. A
detailed discussion may be found in Morrow-Jones and Witt (1993). In this case
from (3.29) with constant $Q$ we find

$Q=\frac{1}{\Lambda}\,.$ (3.31)

We further discover

$N=P^{-1}=1-\Lambda r^{2}\,,$ (3.32)

which gives the Kleinian analogue of the Nariai spacetime

$\mathrm{d}s^{2}=\Big{(}1-\Lambda r^{2}\Big{)}\mathrm{d}t^{2}+\Big{(}1-\Lambda
r^{2}\Big{)}^{-1}\mathrm{d}r^{2}-\Lambda^{-1}(\mathrm{d}\theta^{2}+\sinh^{2}\theta\mathrm{d}\phi^{2})\,,$
(3.33)

whose corresponding Lorentzian metric analogue is only well defined for
$\Lambda>0$.

###  III.4 (2,2) signature Kerr

We now discuss the case of nonzero angular momentum with spin parameter $a$.
We will arrive at our solution using arguments analogous to the Newman-Janis
algorithm Newman and Janis (1965) and it is instructive to reformulate our
Kleinian Schwarzschild Birkhoff argument of section III using Kerr-Schild
coordinates.

####  III.4.1 Kleinian Kerr-Schild coordinates

The metric (2.16) may be put into Kerr-Schild form Bini  _et al._ (2010)

$g_{\mu\nu}=\eta_{\mu\nu}+\Phi k_{\mu}k_{\nu}\,,$ (3.34)

where $k^{\sigma}$ is a null vector and for our purposes $\Phi$ is a function
of $r$. For this discussion we will move to split signature with the
continuation: $\theta\rightarrow i\theta$ and $\phi\rightarrow i\phi$, which
allows for a real Kerr-Schild null vector and a real metric. This is
equivalent to analytically continuing $z\rightarrow iz$ in Cartesian
coordinates leading to flat space metric (4.58). 444An alternative choice of
continuation leading to a complex Kerr-Schild null vector and complex metric
is presented in Appendix A. We will use the coordinate transformation
convention (2.9). Taking the null vector

$k^{\sigma}=(1,1,0,0)$ (3.35)

we see $k_{\mu}k^{\mu}=0$ with respect to both the flat metric

$\mathrm{d}s^{2}=-\mathrm{d}t^{2}+\mathrm{d}r^{2}-r^{2}\mathrm{d}\theta^{2}+r^{2}\sinh^{2}\theta\mathrm{d}\phi^{2}\,,$
(3.36)

and the full Kleinian Kerr-Schild metric (3.34), having line element

$ds^{2}=-\mathrm{d}t^{2}+\mathrm{d}r^{2}-r^{2}\mathrm{d}\theta^{2}+r^{2}\sinh^{2}\theta\mathrm{d}\phi^{2}+(\mathrm{d}r-\mathrm{d}t)^{2}\,\Phi(r)\,.$
(3.37)

Note the appearance of cross-terms $drdt$ in the metric remains compatible
with spherical $SO(3)$ symmetry. With this ansatz we compute the Ricci scalar
$R$ and set it to zero as is appropriate for a vacuum solution

$R=\Phi^{\prime\prime}(r)+\frac{2(\Phi(r)+2r\Phi^{\prime}(r))}{r^{2}}=0\,,$
(3.38)

where prime denotes differentiation with respect to $r$. We find the solution
for $\Phi$ is

$\Phi(r)=\frac{c_{1}}{r^{2}}+\frac{c_{2}}{r}\,.$ (3.39)

Further demanding the vacuum condition on the Ricci scalar $R_{\mu\nu}=0$
yields $c_{1}=0$. Hence setting $c_{2}=A$ we find the Schwarzschild solution
written in Kerr-Schild form with

$\Phi(r)=\frac{A}{r}\,,$ (3.40)

and the Kleinian Schwarzschild metric written in Kerr-Schild coordinates is
thus

$\mathrm{d}s^{2}=-\left(1-\frac{A}{r}\right)\mathrm{d}t^{2}+\left(1+\frac{A}{r}\right)\mathrm{d}r^{2}-r^{2}\mathrm{d}\theta^{2}+r^{2}\sinh^{2}\theta\mathrm{d}\phi^{2}-2\frac{A}{r}\mathrm{d}r\mathrm{d}t\,.$
(3.41)

The curvature invariants computed from (3.41) are identical to Eqs. (4.63).
The familiar Lorentzian Schwarzschild metric in Kerr-Schild form is recovered
from the above by taking

$(\phi\rightarrow i\phi)\,,\qquad(\theta\rightarrow i\theta)\,,$ (3.42)

and is given by line element

$\mathrm{d}s^{2}=-\left(1-\frac{A}{r}\right)\mathrm{d}t^{2}+\left(1+\frac{A}{r}\right)\mathrm{d}r^{2}+r^{2}\mathrm{d}\theta^{2}+r^{2}\sin^{2}\theta\mathrm{d}\phi^{2}-2\frac{A}{r}\mathrm{d}r\mathrm{d}t\,.$
(3.43)

####  III.4.2  Kleinian Kerr solution

Having discovered Kleinian Schwarzschild in Kerr-Schild coordiantes, we may
now formulate the Kerr solution in $(2,2)$ signature, by employing the
Newmann-Janis trick Newman and Janis (1965) compexifying $z\rightarrow z-ia$,
so that the Kerr-Schild scalar for Kleinian Schwarzschild becomes

$\Phi=\frac{A}{\sqrt{-x^{2}-y^{2}+(z-ia)^{2}}}\,.$ (3.44)

Introducing oblate spheroidal coordinates

$x+iy=(r+ia)\sinh{\theta}e^{i\phi}\,,\qquad z=r\cosh{\theta}\,,$ (3.45)

the generating Kerr-Schild scalar is given by

$\Phi=\frac{A}{r+ia\cosh{\theta}}\,.$ (3.46)

We define

$\displaystyle\alpha$ $\displaystyle=$ $\displaystyle
Re\Phi=\frac{A}{2}\left(\frac{1}{r+ia\cosh{\theta}}+\frac{A}{r-ia\cosh{\theta}}\right)=\frac{r}{r^{2}+a^{2}\cosh^{2}{\theta}}\,,$
(3.47) $\displaystyle\beta$ $\displaystyle=$ $\displaystyle
Im\Phi=\frac{A}{2i}\left(\frac{1}{r+ia\cosh{\theta}}-\frac{A}{r-ia\cosh{\theta}}\right)=\frac{-a\cosh{\theta}}{r^{2}+a^{2}\cosh^{2}{\theta}}\,,$
(3.48)

which yields the Kerr space-time in Cartesian Kerr–Schild form with

$A\alpha=\frac{2mr}{r^{2}+a^{2}\cosh^{2}{\theta}}=\frac{2mr^{3}}{r^{4}+a^{2}z^{2}}\,,$
(3.49)

where $A=2m$ and Kerr-Schild metric

$g_{\mu\nu}=\eta_{\mu\nu}+\frac{2mr^{3}}{r^{4}+a^{2}z^{2}}k_{\mu}k_{\nu}$
(3.50)

with null vector in terms of the coordinates introduced in section II,

$k_{\sigma}=i\left(1,\frac{xr+ay}{r^{2}+a^{2}},\frac{yr-
ax}{r^{2}+a^{2}},\frac{-iz}{r}\right)\,.$ (3.51)

This concludes our derivation of the Kleinian Kerr solution.

###  III.5 (2,2) signature Taub-NUT

As a second example of a stationary axisymmetric spacetime we consider
Lorentzian Taub-NUT Taub (1951); Newman  _et al._ (1963) given by

$\mathrm{d}s^{2}=-f(r)(\mathrm{d}t-2N\cos{\theta}\mathrm{d}\phi)^{2}+\frac{\mathrm{d}r^{2}}{f(r)}+(r^{2}+N^{2})(\mathrm{d}\theta^{2}+\sin^{2}{\theta}\mathrm{d}\phi^{2})\,,$
(3.52)

where

$f(r)=\frac{r^{2}-2mr-N^{2}}{r^{2}+N^{2}}\,.$ (3.53)

In the above $m$ is the ADM mass and $N$ is the so-called NUT charge. We may
transition to Klein space via the complex rotations (4.67) along with
$N\rightarrow iN$ to arrive at the Kleinian Taub-NUT line element

$\mathrm{d}s^{2}=f(r)(\mathrm{d}t-2N\cosh{\theta}\mathrm{d}\phi)^{2}+\frac{\mathrm{d}r^{2}}{f(r)}-(r^{2}-N^{2})(\mathrm{d}\theta^{2}+\sinh^{2}{\theta}\mathrm{d}\phi^{2})\,.$
(3.54)

For an extensive analysis of this space see Crawley  _et al._ (2022). Let us
focus on the (anti)-self-dual case where $m=\pm N$, so that

$f(r)=\frac{r+m}{r-m}\,.$ (3.55)

The metric for the self-dual case $m=+N$ becomes

$\mathrm{d}s^{2}=\frac{r-m}{r+m}(\mathrm{d}t-2m\cosh{\theta}\mathrm{d}\phi)^{2}+\frac{r+m}{r-m}\mathrm{d}r^{2}-(r^{2}-m^{2})(\mathrm{d}\theta^{2}+\sinh^{2}{\theta}\mathrm{d}\phi^{2})\,.$
(3.56)

Interestingly, taking $r\rightarrow r-m$, puts the SDTN metric (3.56) in the
form

$\displaystyle\mathrm{d}s^{2}$ $\displaystyle=$
$\displaystyle\Big{(}1-\frac{2m}{r}\Big{)}\mathrm{d}t^{2}+\Big{(}1-\frac{2m}{r}\Big{)}^{-1}\mathrm{d}r^{2}-r^{2}\mathrm{d}\theta^{2}-r^{2}\sinh^{2}\theta\mathrm{d}\phi^{2}$
(3.57) $\displaystyle-$ $\displaystyle
4m\Big{(}1-\frac{2m}{r}\Big{)}\cosh\theta\,\mathrm{d}t\,\mathrm{d}\phi+2mr\,\mathrm{d}\theta^{2}$
$\displaystyle+$ $\displaystyle
2m\Big{(}2m\Big{(}1-\frac{2m}{r}\Big{)}\cosh^{2}\theta+r\sinh^{2}\theta\Big{)}\mathrm{d}\phi^{2}\,.$

Glancing back at (3.22), we see that the SDTN may be written as ordinary
Kleinian Schwarzschild (identifying $A=-2m$) plus additional terms given by
the second and third lines of (3.57). The additional terms are made of the
cross-term $g_{t\phi}$ and an extra $g_{\theta\theta}$ and $g_{\phi\phi}$
piece, breaking the Kleinian spherical symmetry of (3.22). Remarkably, the
curvature invariants for (3.57) are identical to (4.63), and unlike the Kerr
metric, depend only on the $r$ coordinate. Of course, the above metric is
Ricci flat $R_{\mu\nu}=R=0$. All components of the Riemann tensor and
curvature invariants go to zero as $r\rightarrow\infty$.

As a final comment we note that unlike their electromagnetic linear
counterparts, Lorentzian (real) gravitational solutions are not in one-to-one
correspondence with self-dual solutions. However, for Kerr-Schild spacetimes
there is a simple correspondence between real vacuum solutions and self-dual
spacetimes due to the Einstein equations linearizing for Kerr-Schild metrics.
This fact was emphasized in Easson  _et al._ (2023), which also presented a
self-dual version of the Kerr solution.

##  IV Conclusions

We have completed a study of the uniqueness of Kleinian black hole solutions
with maximally symmetric vacua. A comprehensive study of Kleinian vacuum
solutions is essential, given that split-signature metrics and their
associated properties have emerged as a rich and prominent subject of study in
contemporary physics. In this paper we have demonstrated that there is a
uniqueness theorem for $SO(2,1)$ symmetric metrics in Kleinian signature
analogous to Birkhoff’s theorem in Lorentzian signature.

In vacuum, the unique Kleinian-spherically symmetric metric is a complexified
version of the Schwarzschild metric. In a spacetime with nonzero cosmological
constant, the unique solution is generalized to a complexified version of
Schwarzschild-(A)dS or Nariai space. Additionally, we have written down the
unique metric form in Kerr-Schild coordinates, and through the Newman-Janis
transform we were able to construct the Kleinian Kerr metric. Finally we
considered Kleinian Taub-NUT showing that it can be written as Kleinian
Schwarzschild with additional terms, further enriching our understanding of
axisymmetric solutions in split-signature. A natural continuation of this work
would be to extend the analysis to Kleinian electrovac solutions and we leave
such exploration to future research.

## Acknowledgements

We are delighted to thank Lars Aalsma, Samarth Chawla, Gabe Herczeg, Cindy
Keeler, Tucker Manton, Maulik Parikh and Andy Strominger for useful
discussions and correspondence. DAE is supported in part by the U.S.
Department of Energy, Office of High Energy Physics, under Award Number DE-
SC0019470.

## Appendix

###  IV.1 Alternative analytic continuation

An alternative choice of continuation from Lorentzian metric (2.2) to Kleinian
space is the minimal complexification on $z\rightarrow iz$ alone, so that flat
Klein space is given by metric

$\mathrm{d}s^{2}=-\mathrm{d}t^{2}-\mathrm{d}z^{2}+\mathrm{d}x^{2}+\mathrm{d}y^{2}.$
(4.58)

Transforming to spherical coordinates with

$x=r\cos\phi\cosh\theta\,,\qquad y=r\sin\phi\cosh\theta\,,\qquad
z=r\sinh\theta\,.$ (4.59)

The inverse coordinate transformation is

$r=\sqrt{x^{2}+y^{2}-z^{2}}\mathrm{,\ }\phi=\tan^{-1}\frac{y}{x}\mathrm{,\
}\theta=\sinh^{-1}\frac{z}{\sqrt{x^{2}+y^{2}-z^{2}}}.$ (4.60)

The flat space metric is then

$\mathrm{d}s^{2}=-\mathrm{d}t^{2}+\mathrm{d}r^{2}-r^{2}\mathrm{d}\theta^{2}+r^{2}\cosh^{2}\theta\mathrm{d}\phi^{2}.$
(4.61)

Under this continuation the Kleinian Schwarzschild metric is

$\mathrm{d}s^{2}=-\Big{(}1+\frac{A}{r}\Big{)}\mathrm{d}t^{2}+\Big{(}1+\frac{A}{r}\Big{)}^{-1}\mathrm{d}r^{2}-r^{2}\mathrm{d}\theta^{2}+r^{2}\cosh^{2}\theta\mathrm{d}\phi^{2}.$
(4.62)

Naturally, the choice of transformation does not matter for the spacetime
structure and corresponding curvature invariants for either analytic
continuation prescription are equivalent. Some quadratic, cubic and quartic
curvature invariants are found to be

$\displaystyle R_{\mu\nu\sigma\rho}R^{\mu\nu\sigma\rho}$ $\displaystyle=$
$\displaystyle\frac{12A^{2}}{r^{6}}\,,$ $\displaystyle
R_{\mu\nu}{}^{\alpha\beta}R_{\alpha\beta}{}^{\rho\sigma}R_{\rho\sigma}{}^{\mu\nu}$
$\displaystyle=$ $\displaystyle-\frac{12A^{3}}{r^{9}}\,,$
$\displaystyle\nabla_{\alpha}R_{\beta\gamma\sigma\rho}\nabla^{\alpha}R^{\beta\gamma\sigma\rho}$
$\displaystyle=$ $\displaystyle\frac{180A^{2}(A+r)}{r^{9}}\,,$ $\displaystyle
R_{\alpha\beta\gamma\rho}R^{\alpha\beta\gamma}{}_{\sigma}R_{\mu\nu\kappa}{}^{\rho}R^{\mu\nu\kappa\sigma}$
$\displaystyle=$ $\displaystyle\frac{36A^{4}}{r^{12}}\,.$ (4.63)

As another example of alternative continuations, the arguments leading to the
Kleinian Schwarzschild solution in Kerr-Schild coordinates of section III.4.1
are easily repeated using an alternative analytic continuation. One may
journey to Klein space via the continuation $t\rightarrow it$,
$\theta\rightarrow i\theta$, thusly accommodating a complex null vector:

$k^{\sigma}=(i,1,0,0)$ (4.64)

so that $k_{\mu}k^{\mu}=0$ with respect to both the flat metric (2.11) and the
full metric (4.65). Our Kleinian Kerr-Schild line element from (3.34) is now
the complex metric

$\mathrm{d}s^{2}=\mathrm{d}t^{2}+\mathrm{d}r^{2}-r^{2}\mathrm{d}\theta^{2}-r^{2}\sinh^{2}\theta\mathrm{d}\phi^{2}+(\mathrm{d}r+i\,\mathrm{d}t)^{2}\,\Phi(r)$
(4.65)

The Kleinian Schwarzschild metric written in Kerr-Schild form under the above
continuation reads

$\mathrm{d}s^{2}=\left(1-\frac{A}{r}\right)\mathrm{d}t^{2}+\left(1+\frac{A}{r}\right)\mathrm{d}r^{2}-r^{2}\mathrm{d}\theta^{2}-r^{2}\sinh^{2}\theta\mathrm{d}\phi^{2}+\frac{2iA\mathrm{d}r\mathrm{d}t}{r}\,.$
(4.66)

The curvature invariants computed from (4.66), Eqs. (4.63) remain identical.
The real Schwarzschild result (3.43) is recovered from the above by taking

$(t\rightarrow it)\,,\qquad(\theta\rightarrow i\theta)\,.$ (4.67)

## References

  * Birkhoff (1923) G. D. Birkhoff, Relativity and Modern Physics, Harvard University Press, , p. 253 (1923).
  * Jebsen (1921) J.T. Jebsen, Ark. Mat. Ast. Fys. 15, no. 18, 1 (1921).
  * Alexandrow (1923) W. Alexandrow, Ann. Physik 72, 141 (1923).
  * Eiesland (1921) J. Eiesland, Amer. Math. Soc. Bull. 27, 410 (1921).
  * Eiesland (1925) J. Eiesland, Trans. Amer. Math. Soc. 27, 213 (1925).
  * Deser and Franklin (2005) Stanley Deser and J. Franklin, “Schwarzschild and Birkhoff a la Weyl,” Am. J. Phys. 73, 261–264 (2005), arXiv:gr-qc/0408067 .
  * Schleich and Witt (2010) Kristin Schleich and Donald M. Witt, “A simple proof of Birkhoff’s theorem for cosmological constant,” J. Math. Phys. 51, 112502 (2010), arXiv:0908.4110 [gr-qc] .
  * Arkani-Hamed  _et al._ (2020) Nima Arkani-Hamed, Yu-tin Huang, and Donal O’Connell, “Kerr black holes as elementary particles,” JHEP 01, 046 (2020), arXiv:1906.10100 [hep-th] .
  * Crawley  _et al._ (2022) Erin Crawley, Alfredo Guevara, Noah Miller, and Andrew Strominger, “Black holes in Klein space,” JHEP 10, 135 (2022), arXiv:2112.03954 [hep-th] .
  * Crawley  _et al._ (2023) Erin Crawley, Alfredo Guevara, Elizabeth Himwich, and Andrew Strominger, “Self-dual black holes in celestial holography,” JHEP 09, 109 (2023), arXiv:2302.06661 [hep-th] .
  * Easson  _et al._ (2023) Damien A. Easson, Gabriel Herczeg, Tucker Manton, and Max Pezzelle, “Isometries and the double copy,” JHEP 09, 162 (2023), arXiv:2306.13687 [gr-qc] .
  * Adamo  _et al._ (2023) Tim Adamo, Giuseppe Bogna, Lionel Mason, and Atul Sharma, “Scattering on self-dual Taub-NUT,” (2023), arXiv:2309.03834 [hep-th] .
  * Guevara and Kol (2023) Alfredo Guevara and Uri Kol, “Black Hole Hidden Symmetries from the Self-Dual Point,” (2023), arXiv:2311.07933 [hep-th] .
  * Morrow-Jones and Witt (1993) J. Morrow-Jones and D. M. Witt, “Inflationary initial data for generic spatial topology,” Phys. Rev. D 48, 2516–2528 (1993).
  * Newman and Janis (1965) E. T. Newman and A. I. Janis, “Note on the Kerr spinning particle metric,” J. Math. Phys. 6, 915–917 (1965).
  * Bini  _et al._ (2010) Donato Bini, Andrea Geralico, and Roy P. Kerr, “The Kerr-Schild ansatz revised,” Int. J. Geom. Meth. Mod. Phys. 7, 693 (2010), arXiv:1408.4601 [gr-qc] .
  * Taub (1951) A. H. Taub, “Empty space-times admitting a three parameter group of motions,” Annals Math. 53, 472–490 (1951).
  * Newman  _et al._ (1963) E. Newman, L. Tamburino, and T. Unti, “Empty space generalization of the Schwarzschild metric,” J. Math. Phys. 4, 915 (1963).