# The Tomimatsu-Sato metric reloaded

D. Batic<EMAIL_ADDRESS>Department of Mathematics,
Khalifa University of Science and Technology,
Main Campus, Abu Dhabi,
United Arab Emirates

###### Abstract

In this work, we derive exact analytic formulae for the inner and outer
surfaces representing the boundary of the ergoregion appearing in the
Tomimatsu-Sato (TS) metric. Exact expressions for the radii of the ergoregion
in prolate spheroidal coordinates and in Boyer-Lindquist coordinates are
obtained. We also found that in addition to the ring-shaped naked singularity
there is an event horizon placed in the inner region inside the aforementioned
curvature singularity. In comparing our results with previous studies, we also
uncovered and corrected several errors in the literature. Finally, we provide
tables of numerical values for the inner and outer boundaries of the
ergoregion for different values of the rotational parameter. We hope this
study will be a useful resource for all researchers interested in the
Tomimatsu-Sato metric.

Tomimatsu-Sato metric, black hole, naked singularity

##  I Introduction

In 1972, nine years after Kerr obtained the metric associated to a spinning
mass, Tomimatsu and Sato were able to derive a new solution to the Einstein
field equations representing a stationary axisymmetric and asymptotically flat
metric describing the geometry around a deformed spinning mass with
deformation parameter $\delta=2$ tomsato ; tomsatop . Such a solution was
gradually generalized to the case of an arbitrary positive integer distortion
parameter in the series of papers tomsatop ; Yam01 ; Yam02 ; Hori ; Tani . In
the present work, we are interested in the case $\delta=2$ for which gary
showed that the corresponding space-time is characterized by four distinct
principal null directions and therefore, it is of type I according to Petrov’s
classification. We recall that the Kerr metric is instead of type D and as
such it admits a Carter constant which plays a fundamental role in the
separation process and reduction of order for the geodesic equation while the
TS metric being of general Petrov type will not allow for Carter’s integral of
motion CH . Another striking difference between the aforementioned manifolds
is that differently as in the Kerr metric where an event horizon shields the
ring-like curvature singularity, the TS space-time has a naked spinning
singularity gary  in vicinity of which closed time-like curves appear gary .
If from one side this feature is at odd with the Cosmic Censorship Conjecture
(CCC) Penrose  and we may be induced to dismiss the TS solution as unphysical,
on the other side, we should be aware that CCC is still a conjecture and up to
now, several counterexamples are known Papa1 ; Hollier ; Singh ; Gundlach ;
Harada ; Boyda ; Israel ; Drukker .

There has been a long debate in the literature on whether the TS metric may
admits an event horizon. Both Kerr and TS have an ergoregion but TS has a
ring-like curvature singularity at the intersection of the inner surface of
the ergoregion with the equatorial plane. Everybody uses prolate spheroidal
coordinates while we use Boyer-Lindquist coordinates adapted to the present
problem since Kerr metric also usually written in terms of BL-like coordinates
and this makes easier to higlight similarities and differences between these
two metrics.

One of the reasons leading us to study the TS metric is that the literature on
this topic is characterized by several mistakes, misprints happening from the
derivation of the TS metric and extending until Bambi where the shadow of a TS
gravitational object with $\delta=2$ is studied and instead of using the BL
coordinates corresponding to that case which can be found in Yamazaki Bambi
took the BL coordinates for $\delta=1$, i.e. the Kerr metric. This led us to
go over the derivation of the TS metric in Section 2 where we pointed out some
mistakes and typos occurring in the literature. In section 3, we focused our
attention on the study of null surfaces. We found that in addition to the
ring-shaped naked singularity there is an event horizon placed in the inner
region inside the aforementioned curvature singularity. In Section 4, we draw
our conclusions and discuss future research directions related to the TS
metric and its possible extension in the presence of a positive cosmological
constant.

##  II Derivation of the Tomimatsu-Sato metric with deformation parameter
$\delta=2$

The general form of the Lewis-Papapetrou line element describing cylindrical
solutions to the vacuum Einstein field equations in cylindrical coordinates
$(x^{0},x^{1},x^{2},x^{3})=(t,\rho,z,\varphi)$ is Lewis ; Papa

$ds^{2}=fdt^{2}-2\kappa dtd\varphi-\ell
d\varphi^{2}-e^{\mu}(d\rho^{2}+dz^{2}),$ (2.1)

where the four unknown functions appearing in (2.1) depend on the variables
$\rho$ and $z$ only. In the reminder of this section, we will assume that the
metric coefficients satisfy the condition
$\partial_{\beta\alpha}g_{\mu\nu}=\partial_{\alpha\beta}g_{\mu\nu}$. Note that
if we want the line element (2.1) to go over to the Minkowski metric in
cylindrical coordinates as $\rho,z\to\infty$, i.e.

$ds^{2}=dt^{2}-\rho^{2}d\varphi^{2}-d\rho^{2}-dz^{2},$ (2.2)

we need to require in such limit that $f\to 1$, $\kappa,\mu\to 0$ and
$\ell\to\rho^{2}$. If we impose that the Ricci tensor $R_{\alpha\beta}$
vanishes, we find by means of Maple that the only components that do not
identically vanish are $R_{00}$, $R_{03}$, $R_{11}$, $R_{12}$, $R_{22}$ and
$R_{33}$. More precisely, we end up with the following overdetermined coupled
system of PDEs

$\displaystyle\partial_{\rho}\left(\frac{\partial_{\rho}f}{D}\right)+\partial_{z}\left(\frac{\partial_{z}f}{D}\right)+\frac{f}{D^{3}}\left[\partial_{\rho}f\partial_{\rho}\ell+\partial_{z}f\partial_{z}\ell+(\partial_{\rho}\kappa)^{2}+(\partial_{z}\kappa)^{2}\right]$
$\displaystyle=$ $\displaystyle 0,$ (2.3)
$\displaystyle\partial_{\rho}\left(\frac{\partial_{\rho}\kappa}{D}\right)+\partial_{z}\left(\frac{\partial_{z}\kappa}{D}\right)+\frac{\kappa}{D^{3}}\left[\partial_{\rho}f\partial_{\rho}\ell+\partial_{z}f\partial_{z}\ell+(\partial_{\rho}\kappa)^{2}+(\partial_{z}\kappa)^{2}\right]$
$\displaystyle=$ $\displaystyle 0,$ (2.4)
$\displaystyle\partial_{\rho}\left(\frac{\partial_{\rho}\ell}{D}\right)+\partial_{z}\left(\frac{\partial_{z}\ell}{D}\right)+\frac{\ell}{D^{3}}\left[\partial_{\rho}f\partial_{\rho}\ell+\partial_{z}f\partial_{z}\ell+(\partial_{\rho}\kappa)^{2}+(\partial_{z}\kappa)^{2}\right]$
$\displaystyle=$ $\displaystyle 0,$ (2.5)
$\displaystyle\partial_{\rho\rho}\mu+\partial_{zz}\mu+2\frac{\partial_{\rho\rho}D}{D}-\frac{1}{D}\left(\partial_{\rho}\mu\partial_{\rho}D-\partial_{z}\mu\partial_{z}D\right)-\frac{1}{D^{2}}\left[\partial_{\rho}f\partial_{\rho}\ell+\left(\partial_{\rho}\kappa\right)^{2}\right]$
$\displaystyle=$ $\displaystyle 0,$ (2.6)
$\displaystyle\partial_{\rho\rho}\mu+\partial_{zz}\mu+2\frac{\partial_{zz}D}{D}+\frac{1}{D}\left(\partial_{\rho}\mu\partial_{\rho}D-\partial_{z}\mu\partial_{z}D\right)-\frac{1}{D^{2}}\left[\partial_{z}f\partial_{z}\ell+\left(\partial_{z}\kappa\right)^{2}\right]$
$\displaystyle=$ $\displaystyle 0,$ (2.7)
$\displaystyle\partial_{z}\left(\frac{\partial_{\rho}D}{D}\right)-\frac{1}{2D}\left(\partial_{z}\mu\partial_{\rho}D+\partial_{\rho}\mu\partial_{z}D\right)+\frac{\Psi}{4D^{4}}$
$\displaystyle=$ $\displaystyle 0,$ (2.8)

with

$\displaystyle D^{2}$ $\displaystyle=$ $\displaystyle\kappa^{2}+f\ell,$ (2.9)
$\displaystyle\Psi$ $\displaystyle=$
$\displaystyle\kappa^{2}\left(2\partial_{\rho}\kappa\partial_{z}\kappa-\partial_{\rho}f\partial_{z}\ell-\partial_{z}f\partial_{\rho}\ell\right)+2\kappa\ell\left(\partial_{z}\kappa\partial_{\rho}f+\partial_{\rho}\kappa\partial_{z}f\right)+2\kappa
f\left(\partial_{\rho}\kappa\partial_{z}\ell+\partial_{z}\kappa\partial_{\rho}\ell\right)$
(2.10) $\displaystyle-$ $\displaystyle
2f\ell\partial_{\rho}\kappa\partial_{z}\kappa+f^{2}\partial_{\rho}\ell\partial_{z}\ell+\ell^{2}\partial_{\rho}f\partial_{z}f.$

Note that equations (2.3), (2.4) and (2.5) agree with the corresponding
equations (2.2a), (2.2b) and (2.2c) in Islam  where the corresponding
equations arising from the Ricci components $R_{11}$, $R_{12}$ and $R_{22}$
have not been given. As already observed by Lewis , the combination $\ell
R_{00}-2\kappa R_{03}-fR_{33}$ gives rise to the Laplace equation
$\partial_{\rho\rho}D+\partial_{zz}D=0$. This is a remarkable property
satisfied by the function $D$ which leads to a further simplification of the
line element (2.1). More precisely, $D$ is the real part of some complex-
valued function $\Sigma(\rho+iz)=D(\rho,z)+iE(\rho,z)$ where the functions $D$
and $E$ are intertwined through the Cauchy-Riemann equations
$\partial_{\rho}D=\partial_{z}E$ and $\partial_{z}D=-\partial_{\rho}E$. If we
introduce the coordinate transformation

$\overline{\rho}=D(\rho,z),\quad\overline{z}=E(\rho,z)$ (2.11)

together with the invertibility condition
$\partial_{\rho}D\partial_{z}E-\partial_{\rho}E\partial_{z}D=(\partial_{\rho}D)^{2}+(\partial_{z}D)^{2}\neq
0$, we can rewrite the line element (2.1) as

$ds^{2}=fdt^{2}-2\kappa dtd\varphi-\ell
d\varphi^{2}-e^{\overline{\mu}}(d\overline{\rho}^{2}+d\overline{z}^{2}),\quad
e^{\overline{\mu}}=\frac{e^{\mu}}{(\partial_{\rho}D)^{2}+(\partial_{z}D)^{2}},$
(2.12)

where the functions $f$, $\kappa$ and $\ell$ depend on the new variables
$\overline{\rho}$ and $\overline{z}$. At this point, we can exploit the
freedom of choosing $D$ so that it satisfies the Laplace equation. The
simplest choice is $D=\rho$ which implies that $\overline{\rho}=\rho$ while
the Cauchy-Riemann equations lead to $E=z$ modulo an integration constant that
can be chosen to be zero. Hence, we have $\overline{z}=z$ and as a
consequence, the transformed metric (2.12) coincides again with (2.1) but the
unknown functions $f$, $\kappa$ and $\ell$ are not anymore independent because
they are connected through the relation

$D^{2}=\kappa^{2}+f\ell=\rho^{2}.$ (2.13)

If we implement (2.13) into (2.3)-(2.8), the resulting Einstein field
equations read

$\displaystyle\partial_{\rho\rho}f+\partial_{zz}f-\frac{\partial_{\rho}f}{\rho}+\frac{f}{\rho^{2}}\left[\partial_{\rho}f\partial_{\rho}\ell+\partial_{z}f\partial_{z}\ell+(\partial_{\rho}\kappa)^{2}+(\partial_{z}\kappa)^{2}\right]$
$\displaystyle=$ $\displaystyle 0,$ (2.14)
$\displaystyle\partial_{\rho\rho}\kappa+\partial_{zz}\kappa-\frac{\partial_{\rho}\kappa}{\rho}+\frac{\kappa}{\rho^{2}}\left[\partial_{\rho}f\partial_{\rho}\ell+\partial_{z}f\partial_{z}\ell+(\partial_{\rho}\kappa)^{2}+(\partial_{z}\kappa)^{2}\right]$
$\displaystyle=$ $\displaystyle 0,$ (2.15)
$\displaystyle\partial_{\rho\rho}\ell+\partial_{zz}\ell-\frac{\partial_{\rho}\ell}{\rho}+\frac{\ell}{\rho^{2}}\left[\partial_{\rho}f\partial_{\rho}\ell+\partial_{z}f\partial_{z}\ell+(\partial_{\rho}\kappa)^{2}+(\partial_{z}\kappa)^{2}\right]$
$\displaystyle=$ $\displaystyle 0,$ (2.16)
$\displaystyle\partial_{\rho\rho}\mu+\partial_{zz}\mu-\frac{\partial_{\rho}\mu}{\rho}-\frac{1}{\rho^{2}}\left[\partial_{\rho}f\partial_{\rho}\ell+\left(\partial_{\rho}\kappa\right)^{2}\right]$
$\displaystyle=$ $\displaystyle 0,$ (2.17)
$\displaystyle\partial_{z}\mu+\frac{1}{2\rho}\left(\partial_{\rho}f\partial_{z}\ell+\partial_{z}f\partial_{\rho}\ell+2\partial_{\rho}\kappa\partial_{z}\kappa\right)$
$\displaystyle=$ $\displaystyle 0,$ (2.18)
$\displaystyle\partial_{\rho\rho}\mu+\partial_{zz}\mu+\frac{\partial_{\rho}\mu}{\rho}-\frac{1}{\rho^{2}}\left[\partial_{z}f\partial_{z}\ell+\left(\partial_{z}\kappa\right)^{2}\right]$
$\displaystyle=$ $\displaystyle 0.$ (2.19)

As a consistency check, it is gratifying to observe that equations
(2.14)-(2.16) coincide with equations (A)-(C) in Papa . Moreover,
(2.17)-(2.19) agree with (2.10a)-(2.10b) in Islam , respectively. Finally, if
we consider the combinations (2.17)$\pm$(2.19), we obtain equations (2.2) and
(2.2a) in Papa , namely

$\displaystyle\partial_{\rho\rho}\mu+\partial_{zz}\mu$ $\displaystyle=$
$\displaystyle\frac{1}{2\rho^{2}}\left[\partial_{\rho}f\partial_{\rho}\ell+\partial_{z}f\partial_{z}\ell+(\partial_{\rho}\kappa)^{2}+(\partial_{z}\kappa)^{2}\right],$
(2.20) $\displaystyle\partial_{\rho}\mu$ $\displaystyle=$
$\displaystyle-\frac{1}{2\rho}\left[\partial_{\rho}f\partial_{\rho}\ell-\partial_{z}f\partial_{z}\ell+(\partial_{\rho}\kappa)^{2}-(\partial_{z}\kappa)^{2}\right].$
(2.21)

Note that equations (2.14)-(2.16) control the functions $k$, $f$ and $\ell$
and once they are found, we can compute $\mu$ by quadratures from (2.18) and
(2.21). In order to derive the Tomimatsu-Sato metric with deformation
parameter $\delta=2$, it is convenient to recast (2.1) in the so-called Weyl-
Lewis-Papapetrou form. This is achieved with the help of (2.13) and by
introducing the new function

$w=\frac{\kappa}{f}.$ (2.22)

As a result, the line element (2.1) becomes

$ds^{2}=f(dt-
wd\varphi)^{2}-\frac{\rho^{2}}{f}d\varphi^{2}-e^{\mu}(d\rho^{2}+dz^{2})$
(2.23)

The corresponding Einstein field equations can be readily obtained from
(2.14), (2.15), (2.18) and (2.21). They are

$\displaystyle
f\left(\partial_{\rho\rho}f+\partial_{zz}f+\frac{\partial_{\rho}f}{\rho}\right)-\left(\partial_{\rho}f\right)^{2}-\left(\partial_{z}f\right)^{2}+\frac{f^{4}}{\rho^{2}}\left[\left(\partial_{\rho}w\right)^{2}+\left(\partial_{z}w\right)^{2}\right]=0,$
(2.24) $\displaystyle
f\left(\partial_{\rho\rho}w+\partial_{zz}w-\frac{\partial_{\rho}w}{\rho}\right)+2\left(\partial_{\rho}w\partial_{\rho}f+\partial_{z}w\partial_{z}f\right)=0,$
(2.25)
$\displaystyle\partial_{\rho}\mu=-\frac{\partial_{\rho}f}{f}+\frac{\rho}{2f^{2}}\left[\left(\partial_{\rho}f\right)^{2}-\left(\partial_{z}f\right)^{2}\right]-\frac{f^{2}}{2\rho}\left[\left(\partial_{\rho}w\right)^{2}-\left(\partial_{z}w\right)^{2}\right],$
(2.26)
$\displaystyle\partial_{z}\mu=-\frac{\partial_{z}f}{f}+\frac{\rho}{f^{2}}\partial_{\rho}f\partial_{z}f-\frac{f^{2}}{\rho}\partial_{\rho}w\partial_{z}w.$
(2.27)

At this point a couple of remarks are necessary. First of all, it is not
difficult to check that the equations (2.26) and (2.27) are consistent, i.e.
$\partial_{\rho z}\mu=\partial_{z\rho}\mu$. Moreover, $\mu$ defined through
(2.26) and (2.27) satisfies the equations (2.17) and (2.19). This can be
verified by replacing $\kappa=wf$ and $\ell=\rho^{2}/f-fw^{2}$ therein.
Finally, equation (2.16) after the aforementioned substitution coincides with
(2.24). Hence, the new Einstein field equations are represented by
(2.24)-(2.27). It is worth mentioning that Papa  solved the above system of
equations under the additional assumption that the r.h.s. of (2.25) vanishes.
Such an approach leads to a solution describing a gravitational field where
either the mass or the angular momentum can be different from zero. Last but
not least, instead of working with the functions $f$ and $w$, Cosgrove1 ;
Cosgrove2  introduced a new function $\gamma=(\mu+\log{f})/2$ and derived a
fourth order quasi-linear PDE for $\gamma$. A less restrictive procedure than
the one adopted in Papa  relies on the observation that the divergence of the
vector field
$\mathcal{F}:\Omega\subseteq(0,\infty)\times\mathbb{R}\longrightarrow\mathbb{R}^{2}$
defined as follows

$\mathcal{F}(\rho,z)=\frac{f^{2}}{\rho}\left(\partial_{\rho}w,\partial_{z}w\right)$
(2.28)

coincides with the LHS of (2.25). This is easily done by showing that the
equation
$\partial_{\rho}(\mathcal{A}\partial_{\rho}w)+\partial_{z}(\mathcal{A}\partial_{z}w)=0$
coincides with (2.25) if $\mathcal{A}=f^{2}/\rho$. Hence, it is possible to
construct a function $u=u(\rho,z)$ such that

$\partial_{\rho}u=\frac{f^{2}}{\rho}\partial_{z}w,\quad\partial_{z}u=-\frac{f^{2}}{\rho}\partial_{\rho}w.$
(2.29)

By means of (2.29) it is not difficult to derive from (2.24)-(2.27) the
following PDEs governing the functions $f$ and $u$, namely

$\displaystyle f\nabla^{2}f$ $\displaystyle=$
$\displaystyle\left(\partial_{\rho}f\right)^{2}+\left(\partial_{z}f\right)^{2}-\left[\left(\partial_{\rho}u\right)^{2}+\left(\partial_{z}u\right)^{2}\right],$
(2.30) $\displaystyle f\nabla^{2}u$ $\displaystyle=$ $\displaystyle
2\left(\partial_{\rho}f\partial_{\rho}u+\partial_{z}f\partial_{z}u\right),$
(2.31) $\displaystyle\partial_{\rho}\left(\mu+\ln{f}\right)$ $\displaystyle=$
$\displaystyle\frac{\rho}{2f^{2}}\left[\left(\partial_{\rho}f\right)^{2}-\left(\partial_{z}f\right)^{2}\right]+\frac{\rho}{2f^{2}}\left[\left(\partial_{\rho}u\right)^{2}-\left(\partial_{z}u\right)^{2}\right],$
(2.32) $\displaystyle\partial_{z}\left(\mu+\ln{f}\right)$ $\displaystyle=$
$\displaystyle\frac{\rho}{f^{2}}\left(\partial_{\rho}f\partial_{z}f+\partial_{\rho}u\partial_{z}u\right),$
(2.33)

where
$\nabla^{2}=\rho^{-1}\partial_{\rho}(\rho\partial_{\rho}\cdot))+\rho^{-2}\partial_{\varphi\varphi}+\partial_{zz}$
is the Laplace operator in cylindrical coordinates. We draw the attention of
the reader to the fact that there is a typo in equation (3.3a) in Islam  where
the term $\partial_{z}u$ should appear squared. The Ernst equation Ernst
emerges from (2.30) and (2.31) by introducing some complex-valued function
$\mathcal{E}=f+iu$ and realizing that the aforementioned equations are the
real and imaginary parts of the complex PDE

$\Re{\left(\mathcal{E}\right)}\nabla^{2}\mathcal{E}=\left(\partial_{\rho}\mathcal{E}\right)^{2}+\left(\partial_{z}\mathcal{E}\right)^{2},\quad\Re{\mathcal{E}}=f.$
(2.34)

Hence, (2.34) is an equivalent form of the Einstein field equations (2.30) and
(2.31). In order to derive the Tomimatsu-Sato metric, it is convenient to
introduce the Ansatz

$\mathcal{E}=\frac{\Phi-1}{\Phi+1}$ (2.35)

with $\Phi$ a complex-valued function yet to be determined. The Ernst equation
becomes

$\left(|\Phi|^{2}-1\right)\nabla^{2}\Phi=2\Phi^{*}\left[\left(\partial_{\rho}\Phi\right)^{2}+\left(\partial_{z}\Phi\right)^{2}\right],$
(2.36)

where star denotes complex conjugation. Note that our notation departs from
that in Islam ; tomsato ; tomsatop  where the lowercase Greek letter $\xi$ is
used instead of $\Phi$. The reason behind our choice is that the letter $\xi$
will denote one of the prolate spheroidal coordinates. Finally, the Tomimatsu-
Sato line element can be constructed by searching for solutions to (2.36) in
prolate spheroidal coordinates $\xi=\xi(\rho,z)$, $\eta=\eta(\rho,z)$ with
$\xi\geq 1$ and $-1\leq\eta\leq 1$ such that

$\rho=\sigma\sqrt{(\xi^{2}-1)(1-\eta^{2})},\quad
z=\sigma\xi\eta,\quad\sigma>0.$ (2.37)

The corresponding inversion formulae are

$\xi=\frac{\sqrt{R_{+}}+\sqrt{R_{-}}}{2\sigma},\quad\eta=\frac{\sqrt{R_{+}}-\sqrt{R_{-}}}{2\sigma},\quad
R_{\pm}=\rho^{2}+(z\pm\sigma)^{2}.$ (2.38)

We alert the reader that our notation for the prolate spheroidal coordinates
$(\xi,\eta)$ differs from that employed in Islam ; tomsato ; tomsatop  where
the lowercase Latin letters $x$ and $y$ stand for $\xi$ and $\eta$,
respectively. If we apply the coordinate transformation (2.37) to (2.36), we
end up with the Ernst equation in prolate spheroidal coordinates, i.e.

$\left(|\Phi|^{2}-1\right)\left[(\xi^{2}-1)\partial_{\xi\xi}\Phi+(1-\eta^{2})\partial_{\eta\eta}\Phi+2\xi\partial_{\xi}\Phi-2\eta\partial_{\eta}\Phi\right]=$
$2\Phi^{*}\left[(\xi^{2}-1)\left(\partial_{\xi}\Phi\right)^{2}+(1-\eta^{2})\left(\partial_{\eta}\Phi\right)^{2}\right].$
(2.39)

Moreover, the equations (2.29), (2.32) and (2.33) become

$\displaystyle\partial_{\xi}w$ $\displaystyle=$
$\displaystyle-\frac{\sigma(1-\eta^{2})}{f^{2}}\partial_{\eta}u,$ (2.40)
$\displaystyle\partial_{\eta}w$ $\displaystyle=$
$\displaystyle\frac{\sigma(\xi^{2}-1)}{f^{2}}\partial_{\xi}u,,$ (2.41)
$\displaystyle\partial_{\xi}\left(\mu+\ln{f}\right)$ $\displaystyle=$
$\displaystyle\frac{1-\eta^{2}}{2f^{2}(\xi^{2}-\eta^{2})}\left\\{\xi(\xi^{2}-1)\left[\left(\partial_{\xi}f\right)^{2}+\left(\partial_{\xi}u\right)^{2}\right]-\xi(1-\eta^{2})\left[\left(\partial_{\eta}f\right)^{2}+\left(\partial_{\eta}u\right)^{2}\right]\right.$
(2.42) $\displaystyle-$
$\displaystyle\left.2\eta(\xi^{2}-1)\left(\partial_{\xi}f\partial_{\eta}f+\partial_{\xi}u\partial_{\eta}u\right)\right\\},$
$\displaystyle\partial_{\eta}\left(\mu+\ln{f}\right)$ $\displaystyle=$
$\displaystyle\frac{\xi^{2}-1}{2f^{2}(\xi^{2}-\eta^{2})}\left\\{\eta(\xi^{2}-1)\left[\left(\partial_{\xi}f\right)^{2}+\left(\partial_{\xi}u\right)^{2}\right]-\eta(1-\eta^{2})\left[\left(\partial_{\eta}f\right)^{2}+\left(\partial_{\eta}u\right)^{2}\right]\right.$
(2.43) $\displaystyle+$
$\displaystyle\left.2\xi(1-\eta^{2})\left(\partial_{\xi}f\partial_{\eta}f+\partial_{\xi}u\partial_{\eta}u\right)\right\\}.$

It is well-known that the vacuum stationary axisymmetric solutions to
Einstein’s field equations can be generated from the Ernst equation (2.34),
(2.36) or equivalently from (2.39). For instance, the Kerr metric emerges from
the case

$\Phi_{K}=p\xi-iq\eta,\quad q=\frac{J}{M^{2}},\quad p=\sqrt{1-q^{2}},$ (2.44)

where $M$ is the mass of the gravitational source and $J$ its total angular
momentum. A further solution to (2.39) was found by tomsato ; tomsatop  in the
form

$\Phi_{TS}=\frac{\mathfrak{u}+i\mathfrak{v}}{\mathfrak{m}+i\mathfrak{n}}$
(2.45)

with

$\mathfrak{u}=p^{2}\xi^{4}+q^{2}\eta^{4}-1,\quad\mathfrak{v}=-2pq\xi\eta(\xi^{2}-\eta^{2}),\quad\mathfrak{m}=2p\xi(\xi^{2}-1),\quad\mathfrak{n}=-2q\eta(1-\eta^{2}).$
(2.46)

It is a straightforward exercise with Maple to verify that (2.45) indeed
satisfies (2.39). Since $\Phi_{TS}\to(\xi^{2}+1)/2\xi$ for $q\to 0$, in this
limit the Tomimatsu-Sato solution coincides with the class of Weyl’s metrics
generated by

$\Phi_{W}=\frac{(\xi+1)^{\delta}+(\xi-1)^{\delta}}{(\xi+1)^{\delta}-(\xi-1)^{\delta}}$
(2.47)

when $\delta=2$. In the case $\delta=1$, the function $\Phi_{W}$ leads to the
Schwarzschild metric Voor . Therefore, $\delta$ and $q$ can be viewed as
positive parameters measuring the deviation from spherical symmetry. If we
insert (2.45) into (2.35) and recall that $\mathcal{E}=f+iu$, we can easily
compute the real and imaginary parts of $\mathcal{E}$ with the help of Maple.
More precisely, we find that

$f=\Re{\left(\mathcal{E}\right)}=\frac{A}{B}~{},\quad
u=\Im{\left(\mathcal{E}\right)}=\frac{2I}{B}$ (2.48)

with

$\displaystyle A$ $\displaystyle=$
$\displaystyle\mathfrak{u}^{2}+\mathfrak{v}^{2}-(\mathfrak{m}^{2}+\mathfrak{n}^{2}),$
(2.49) $\displaystyle=$
$\displaystyle\left[p^{2}(\xi^{2}-1)^{2}+q^{2}(1-\eta^{2})^{2}\right]^{2}-4p^{2}q^{2}(\xi^{2}-1)(1-\eta^{2})(\xi^{2}-\eta^{2})^{2},$
(2.50) $\displaystyle B$ $\displaystyle=$
$\displaystyle(\mathfrak{u}+\mathfrak{m})^{2}+(\mathfrak{v}+\mathfrak{n})^{2},$
(2.51) $\displaystyle=$
$\displaystyle(p^{2}\xi^{4}+q^{2}\eta^{4}-1+2p\xi^{3}-2p\xi)^{2}+4q^{2}\eta^{2}(p\xi^{3}-p\xi\eta^{2}+1-\eta^{2})^{2},$
(2.52) $\displaystyle I$ $\displaystyle=$
$\displaystyle\mathfrak{m}\mathfrak{v}-\mathfrak{n}\mathfrak{u},$ (2.53)
$\displaystyle=$
$\displaystyle-2q\eta\left[(1-\eta^{2})(1-q^{2}\eta^{4})+p^{2}\xi^{2}\eta^{2}(2-\xi^{2})+p^{2}\xi^{4}(2\xi^{2}-3)\right].$
(2.54)

It is gratifying to observe that the expressions for the coefficients $A$ and
$B$ represented by (2.50) and (2.52) agree with the corresponding ones offered
inIslam ; tomsato ; tomsatop . Moreover, $f$ exhibits the desired behaviour at
space-like infinity, that is $f\to 1$ as $\xi\to\infty$. Concerning the
function $w$, it can derived from the equations (2.40) and (2.41). In
particular, we try as in tomsato ; tomsatop ; Islam  the Ansatz

$w=\frac{2Mq(1-\eta^{2})}{A}C,\quad C=\sum_{\begin{subarray}{c}k,l=0\\\ 0\leq
k+l\leq 7\end{subarray}}^{7}a_{kl}\xi^{k}\eta^{l},$ (2.55)

where $M$ is the total mass of the gravitational source and $C$ is some
polynomial in $\xi$ and $\eta$ of maximum degree seven. This requirement
together with the fact that $A$ is a polynomial of degree eight in $\xi$ is
necessary in order to ensure that $w\to 0$ as $\xi\to\infty$. If we substitute
(2.55) into (2.40) and (2.41), we obtain the following system

$\displaystyle A\partial_{\xi}C-C\partial_{\xi}A$ $\displaystyle=$
$\displaystyle\frac{\sigma}{Mq}\left(I\partial_{\eta}B-B\partial_{\eta}I\right),$
(2.56) $\displaystyle(1-\eta^{2})(A\partial_{\eta}C-C\partial_{\eta}A)-2\eta
AC$ $\displaystyle=$
$\displaystyle\frac{\sigma}{Mq}(\xi^{2}-1)\left(B\partial_{\xi}I-I\partial_{\xi}B\right)$
(2.57)

in agreement with equations (7) and (8) in Yam01 . Setting $\sigma=Mp/2$ and
substituting (2.55) into (2.56) leads to a linear system for the unknown
coefficients $a_{kl}$ which is easily handled by Maple. The final result is

$C=q^{2}(1+p\xi)(1-\eta^{2})^{3}-p^{2}(\xi^{2}-1)(1-\eta^{2})(p\xi^{3}+3\xi^{2}+3p\xi+1)$
$-2p^{2}\xi(\xi^{2}-1)^{2}(p\xi^{2}+2\xi+p).$ (2.58)

As a consistency check, we used Maple to verify that (2.58) is indeed a
solution to both PDEs (2.56) and (2.57). It is gratifying to see that (2.58)
agrees with the corresponding expressions given in tomsatop ; Yam01 . However,
a comment is in order. tomsatop  also derived a system of first order PDEs for
$C$ represented by equations (3.3) and (3.4) therein. Even though the l.h.s.’s
of (2.56) and (3.3) in tomsatop  coincide, the same cannot be said for the
r.h.s.’s. Despite the fact that the expression for $C$ given in tomsatop
agrees with the corresponding expressions presented here and in Yam01 , it
does not satisfy equation (3.3) in tomsatop  as it can be easily verified by
means of Maple. The only conclusion is that such a discrepancy is due to a
typo in the aforementioned equation. Finally, Islam  instead of deriving a
formula for $C$ gives the same result as in tomsatop  but with a typo in the
last term. It should also be mentioned that Islam  makes an opposite choice
for the signs entering in front of the l.h.s. of the equations represented by
(2.29) in the present work. It is not difficult to check that such a choice
would alter the sign in front of the l.h.s.’s in (2.56) and (2.57) and send
$C$ into $-C$. A formula for the metric coefficient $e^{2\gamma}/f$ was
derived in tomsatop  while Islam  gives an expression for $e^{\mu}$ which is
linked to the aforementioned metric coefficient in tomsatop  by the relation
$e^{\mu}=e^{2\gamma}/f$. We checked with Maple that

$e^{\mu}=\frac{B}{p^{4}(\xi^{2}-\eta^{2})^{4}}$ (2.59)

is indeed a solution of the system (2.42) and (2.43). Finally, the line
element (2.23) can be written in prolate spheoridal coordinates as follows

$\displaystyle ds^{2}=\frac{A}{B}dt^{2}$ $\displaystyle-$
$\displaystyle\frac{4Mq(1-\eta^{2})C}{B}dtd\varphi-\frac{M^{2}B}{4p^{2}(\xi^{2}-\eta^{2})^{3}}\left(\frac{d\xi^{2}}{\xi^{2}-1}+\frac{d\eta^{2}}{1-\eta^{2}}\right)$
(2.60) $\displaystyle-$
$\displaystyle\frac{M^{2}(1-\eta^{2})}{A}\left[\frac{p^{2}}{4}(\xi^{2}-1)B-4q^{2}(1-\eta^{2})\frac{C^{2}}{B}\right]d\varphi^{2},$

where $A$, $B$ and $C$ are given by (2.50), (2.52) and (2.58), respectively.
If, instead, we use the standard definition of prolate spheoridal coordinates
$(u,v,\varphi)$ with $\xi=\cosh{u}$ and $\eta=\cos{v}$ where $u\geq 0$ and
$0\leq v\leq\pi$, (2.60) can be cast into the form

$ds^{2}=\frac{A}{B}dt^{2}-4Mq\sin^{2}{v}\frac{C}{B}dtd\varphi-\frac{M^{2}B}{4p^{2}(\cosh^{2}{u}-\cos^{2}{v})^{3}}\left(du^{2}+dv^{2}\right)$
$-\frac{M^{2}\sin^{2}{v}}{4AB}\left(p^{2}B^{2}\sinh^{2}{u}-16q^{2}C^{2}\sin^{2}{v}\right)d\varphi^{2},$
(2.61)

where

$\displaystyle A$ $\displaystyle=$
$\displaystyle\left(p^{2}\sinh^{4}{u}+q^{2}\sin^{4}{v}\right)^{2}-4p^{2}q^{2}\sinh^{2}{u}\sin^{2}{v}\left(\cosh^{2}{u}-\cos^{2}{v}\right)^{2},$
(2.62) $\displaystyle B$ $\displaystyle=$
$\displaystyle\left(q^{2}\cos^{4}{v}+p^{2}\cosh^{4}{u}+2p\cosh{u}\sinh^{2}{u}-1\right)^{2}$
(2.63) $\displaystyle+$ $\displaystyle
4q^{2}\cos^{2}{v}\left(p\cosh^{3}{u}-p\cosh{u}\cos^{2}{v}+\sin^{2}{v}\right)^{2},$
(2.64) $\displaystyle C$ $\displaystyle=$ $\displaystyle
q^{2}\sin^{6}{v}(1+p\cosh{u})-p^{2}\sinh^{2}{u}\sin^{2}{v}(p\cosh^{3}{u}+3\cosh^{2}{u}+3p\cosh{u}+1)$
(2.65) $\displaystyle-$ $\displaystyle
2p^{2}\cosh{u}\sinh^{4}{u}\left(p\cosh^{2}{u}+2\cosh{u}+p\right).$

##  III Analysis of the metric

In this section, we improve the singularity and ergoregion analysis provided
in gary ; Kodama . First of all, a direct inspection of (2.50), (2.52), (2.58)
and (2.59) shows that all metric coefficients are invariant under the
transformation $\eta\to-\eta$. The fact that $B$ is always nonnegative implies
that the sign of the metric coefficient $f$ is uniquely controlled by the sign
of the polynomial function $A$. Moreover, it can be immediately seen that the
functions (2.50), (2.52) and (2.58) have common roots at $\xi=1$ and $\eta=\pm
1$. If we Taylor expand $B$ around $\xi=\pm 1$ and $\eta=\pm 1$ as follows

$\displaystyle B$ $\displaystyle=$ $\displaystyle
p^{2}(\xi^{2}-1)^{2}a_{1}(\xi)-4p^{2}q^{2}(\xi^{2}-1)^{3}(1-\eta^{2})+a_{2}(\xi)(1-\eta^{2})^{2}$
(3.1) $\displaystyle+$ $\displaystyle
a_{3}(\xi)(\eta^{2}-1)^{3}+q^{4}(\eta^{2}-1)^{4},$ (3.2)

with

$\displaystyle a_{1}(\xi)$ $\displaystyle=$ $\displaystyle
p^{2}\xi^{4}+4p\xi^{3}+2(1+q^{2})\xi^{2}+4p\xi+p^{2},$ (3.4) $\displaystyle
a_{2}(\xi)$ $\displaystyle=$
$\displaystyle-6p^{2}q^{2}\xi^{4}-4pq^{2}\xi^{3}+12p^{2}q^{2}\xi^{2}+12pq^{2}\xi+2q^{2}(1+3q^{2}),$
(3.5) $\displaystyle a_{3}(\xi)$ $\displaystyle=$ $\displaystyle
4p^{2}q^{2}\xi^{2}+8pq^{2}\xi+4q^{2}(1+q^{2})$ (3.6)

and we recall that $f=A/B$, we conclude that the metric coefficient $g_{tt}$
has a regular behaviour on the line segment described by $\xi=1$ and
$-1<\eta<1$. The points $(\xi,\eta)=(1,\pm 1)$ making $f$ singular have the
interpretation of quasi-regular singularities Kodama . At this step, it is not
obvious whether the polynomial functions $A$ and $B$ have some other common
zeroes in addition to those mentioned above. Inspired by the parametric
surface representation of the ergosphere of a Kerr black hole, we introduce
the ansatz gary

$\xi^{2}=1+\lambda^{2}(1-\eta^{2})$ (3.7)

with $\lambda\in\mathbb{R}$. This allows to cast $A$ into the form

$A=(1-\eta^{2})^{4}\mathfrak{p}_{\pm},\quad\mathfrak{p}_{\pm}=p^{2}\lambda^{4}+q^{2}\pm
2pq\lambda(\lambda^{2}+1).$ (3.8)

First of all, we observe that a quartic polynomial similar to
$\mathfrak{p}_{-}$ has been also given in gary . However, there is a typo
there and the factor 4 multiplying the term $\lambda(\lambda+1)$ should be
replaced by 2. The polynomial $\mathfrak{p}_{+}$ does not admit any real root
for $0<q<1$ and therefore, if there are some other real roots of $A$ distinct
from those already identified above, they must come from the quartic
$\mathfrak{p}_{-}$. This observation is important because it signalizes that
$g_{tt}$ may not be everywhere positive definite for $\xi>1$ and
$-1\leq\eta\leq 1$. In other words, for decreasing values of $\xi$ the Killing
vector field $\partial_{t}$ may already loose its property of being time-like
before $\xi$ reaches the value one and an ergoregion may arise as pointed out
by gary ; Kodama . Concerning the zeroes of the polynomial $\mathfrak{p}_{-}$,
there are two sign changes and Descartes’ rule of signs predicts that the
number of positive roots is either equal two or zero. Moreover, after the
transformation $\lambda\to-\lambda$ the corresponding polynomial has no sign
changes and therefore, $\mathfrak{p}_{-}$ does not have negative real roots.
Finally, by applying (1.163a) and (1.163b) in Bron  the two positive real
roots are

$\lambda_{\pm}=\frac{q}{2p}+\frac{\sqrt{\sqrt[3]{4p^{2}q^{2}}+q^{2}}}{2p}\pm\sqrt{2q^{2}-\sqrt[3]{4p^{2}q^{2}}+\frac{2\sqrt[3]{q^{2}}(2p^{2}+q^{2})}{\sqrt{\sqrt[3]{4p^{2}}+q\sqrt[3]{q}}}},$
(3.9)

to which there correspond the following values of $\xi$ determined with the
help of (3.7), i.e.

$\xi^{\pm}_{\mathcal{E}}=\sqrt{1+\lambda_{\pm}^{2}(1-\eta^{2})},$ (3.10)

where the subscript $\mathcal{E}$ stands for the ergoregion

$\mathcal{E}=\\{(\xi,\eta,\varphi)\in\mathbb{R}^{3}~{}|~{}\xi_{\mathcal{E}}^{-}<\xi<\xi_{\mathcal{E}}^{+},~{}-1<u<1,~{}0\leq\varphi<2\pi\\}.$
(3.11)

Note that on the equator, i.e. $\eta=0$, the inner and outer boundaries of the
ergoregion are located at

$\xi^{\pm}_{\mathcal{E},e}=\sqrt{1+\lambda_{\pm}^{2}}.$ (3.12)

In order to investigate whether or not the metric coefficient $f$ becomes
singular, it is necessary to study the zeroes of $B$. If we restrict our
attention to the equatorial plane as in gary , we end up with the problem of
finding the roots of the polynomial equation

$B(\xi,0)=\mathfrak{P}^{2}=0,\quad\mathfrak{P}=p^{2}\xi^{4}+2p\xi^{3}-2p\xi-1.$
(3.13)

Since $\mathfrak{P}$ exhibits only one sign change, there is only one positive
real root, here denoted as $\xi_{B}$, and it must be a zero of order two for
$B$. Moreover, the polynomial obtained by means of the transformation
$\xi\to-\xi$ has three sign changes meaning that there can be three or one
negative real roots. As it can be seen from Table 1, it turns out that
$\xi_{B}=\xi_{\mathcal{E}}^{-}$ in agreement with gary  where however no
numerical/analytic evidence was given for such a result. Since
$\xi_{\mathcal{E}}^{-}$ is a simple zero for $A$, while $\xi_{B}$ is a root of
order two for $B$, we can conclude that the metric coefficient $f$ exhibits a
singularity at $\xi=\xi_{B}$ on the equatorial plane. We checked numerically
that $A$ and $B$ have no common real roots away from the equatorial plane (see
Appendix A for typical values of $\xi_{\mathcal{E}}^{\pm}$ when $\eta\neq 0$).
More precisely, all zeroes of $B$ are complex whenever $\eta\neq 0$. This
indicates that the metric coefficient $f$ can only have a singularity at the
point $(\xi,\eta)=(\xi_{B},0)$ in addition to the singular points at $(1,\pm
1)$.

Table 1: Typical values of the zeroes $\lambda_{\pm}$ of the polynomial equation $\mathfrak{p}_{-}=0$ in (3.8) with the corresponding values $\xi_{\mathcal{E}}^{\pm}$ for the inner and outer boundaries of the ergoregion on the equatorial plane $\eta=0$. The last column represents the only positive real root of order two for the function $B$ restricted to the equatorial plane. $\eta=0$ | $\lambda_{-}$ | $\xi_{\mathcal{E}}^{-}$ | $\lambda_{+}$ | $\xi_{\mathcal{E}}^{+}$ | $\xi_{B}$
---|---|---|---|---|---
$q=0.1$ | 0.050157194 | 1.001257082 | 0.645657713 | 1.190325117 | 1.001257082
$q=0.2$ | 0.101280894 | 1.005115824 | 0.877134836 | 1.330174996 | 1.005115824
$q=0.3$ | 0.154462485 | 1.011859012 | 1.085464940 | 1.475884188 | 1.011859012
$q=0.4$ | 0.211086895 | 1.022036045 | 1.301672592 | 1.641448000 | 1.022036045
$q=0.5$ | 0.273120641 | 1.036626685 | 1.547108167 | 1.842157344 | 1.036626685
$q=0.6$ | 0.343701531 | 1.057417014 | 1.849576162 | 2.102601241 | 1.057417014
$q=0.7$ | 0.428584499 | 1.087972735 | 2.260794743 | 2.472082699 | 1.087972735
$q=0.8$ | 0.540664408 | 1.136801655 | 2.909501150 | 3.076556019 | 1.136801655
$q=0.9$ | 0.721912637 | 1.233352284 | 4.299249247 | 4.414016774 | 1.233352284
$q=0.99$ | 1.338695474 | 1.670959476 | 14.08894707 | 14.12439130 | 1.670959476

If we recall that prolate spheroidal and cylindrical coordinates are connected
by (2.37), it can be easily checked that $(\xi_{B},0)$ corresponds to a ring
singularity with radius

$\rho_{s}=\frac{Mp}{2}\lambda_{-}.$ (3.14)

This is where the inner boundary of the ergoregion intersects the equatorial
plane. Hence, such a singularity resides inside the ergoregion. The ring
singularity was proved to be a curvature singularity by analysing the
behaviour of the Weyl curvature invariants there (see Kodama  for a detailed
discussion). Kodama  also showed that such a ring singularity has zero Komar
mass: an unexpected result if we think that an axisymmetric space-time such as
the one described by a Kerr black hole has instead a positive Komar mass Wald
despite the fact it also exhibits a ring singularity. This aspect of the TS
metric and its physical implications seem not to have been addressed in the
related existing literature and they are definitely worth being studied in
more detail elsewhere. Finally, the ring singularity is naked because it is
not hidden by a Cauchy or event horizon as it has been already discussed in
Kodama  (see Fig. 10 therein).

It is interesting to understand how the aforementioned singularities are
mapped when we switch from prolate to Boyer-Lindquist (BL) coordinates. When
we reviewed the existing literature, we observed that there has been some
confusion regarding the transformations from cylindrical to BL coordinates in
the presence of an arbitrary deformation parameter. For instance, tomsato
gives without proof the formulae

$\displaystyle\rho$ $\displaystyle=$
$\displaystyle\frac{Mp}{\delta}\sqrt{\xi^{2}-1}\sqrt{1-\eta^{2}}=\sqrt{\widehat{\Delta}}\sin{\vartheta},$
(3.15) $\displaystyle z$ $\displaystyle=$
$\displaystyle\frac{Mp}{\delta}\xi\eta=(r-M)\cos{\vartheta},\quad\widehat{\Delta}=r^{2}-2Mr+M^{2}q^{2},$
(3.16)

where strangely enough the deformation parameter completely disappeared from
the expressions written in terms of the BL coordinates and the function
$\widehat{\Delta}$ is the same as the one entering in the Kerr metric.
Moreover, tomsato  refers to Voor  for more details but Voor  provides only
the transformations from prolate spheroidal to cylindrical coordinates.
Finally, the relations $\rho=\sqrt{\widehat{\Delta}}\sin{\vartheta}$ and
$z=(r-M)\cos{\vartheta}$ appear in the work of Ernst  where Ernst showed how
the metric derived therein, is equivalent to the Kerr solution if the
coordinate transformation $\xi=(r-M)/Mp$ and $\eta=\cos{\vartheta}$ is
introduced. Hence, those parts in the expressions (3.15) and (3.16) where the
$(r,\vartheta)$ coordinates enter, are correct only in the special case
$\delta=1$. At this point, we warn the reader about the fact that Bambi  when
analyzing the shadow of a TS manifold with $\delta=2$ made use of the
coordinate transformations (3.15) and (3.16) and therefore, the numerical
results presented there should be taken with some caution. The correct
transformation for arbitrary $\delta>0$ has been given by Yam02 , namely

$r=\frac{Mp}{\delta}\xi+M,\quad\cos{\vartheta}=\eta.$ (3.17)

Note that the condition $\xi\geq 1$ requires that

$r\geq r_{\delta}=M+\frac{Mp}{\delta}.$ (3.18)

Then, it is straightforward to verify that the cylindrical coordinates can be
expressed as

$\rho=\sqrt{(r-M)^{2}-\left(\frac{Mp}{\delta}\right)^{2}}\sin{\vartheta},\quad
z=(r-M)\cos{\vartheta}.$ (3.19)

Note that $r=r_{\delta}$ corresponds to $\rho=0$ and $-Mp/\delta\leq z\leq
Mp/\delta$, that is to an infinitesimally thin rod of length $2Mp/\delta$. In
the present work, we are interested in the case $\delta=2$. Hence, (3.19)
becomes

$\displaystyle\rho$ $\displaystyle=$
$\displaystyle\frac{Mp}{2}\sqrt{\xi^{2}-1}\sqrt{1-\eta^{2}}=\sqrt{\Delta}\sin{\vartheta},$
(3.20) $\displaystyle z$ $\displaystyle=$
$\displaystyle\frac{Mp}{2}\xi\eta=(r-M)\cos{\vartheta},\quad\Delta=(r-M)^{2}-\frac{M^{2}p^{2}}{4}.$
(3.21)

It is not difficult to verify that the prolate spheroidal coordinates can be
expressed in terms of the BL coordinates as

$\displaystyle\xi$ $\displaystyle=$
$\displaystyle\frac{\sqrt{R_{+}}+\sqrt{R_{-}}}{Mp},\quad\eta=\frac{(r-M)\cos{\vartheta}}{\sqrt{R_{+}}+\sqrt{R_{-}}},$
(3.22) $\displaystyle R_{\pm}$ $\displaystyle=$
$\displaystyle\Delta\sin^{2}{\vartheta}+\left[(r-M)\cos{\vartheta}\pm\frac{Mp}{2}\right]^{2}=\left(r-M\pm\frac{Mp}{2}\cos{\vartheta}\right)^{2}$
(3.23)

and as a consistency check, it can be easily shown that under the condition
$r\geq r_{2}=M+Mp/2$, formulae (3.22) and (3.23) give

$\xi=\frac{2}{Mp}(r-M),\quad\eta=\cos{\vartheta}$ (3.24)

in agreement with (3.17). Finally, the TS line element in BL coordinates is

$ds^{2}=\frac{A}{B}dt^{2}-4Mq\frac{C}{B}\sin^{2}{\vartheta}dtd\varphi-\Sigma_{1}\left(\frac{dr^{2}}{\Delta}+d\vartheta^{2}\right)-\Sigma_{2}\sin^{2}{\vartheta}d\varphi^{2}$
(3.25)

with

$\Sigma_{1}=\frac{M^{8}p^{4}B}{256\left[(r-M)^{2}-\frac{M^{2}p^{2}}{4}\cos^{2}{\vartheta}\right]^{3}},\quad\Sigma_{2}=\frac{1}{A}\left(B\Delta-4M^{2}q^{2}\frac{C^{2}}{B}\sin^{2}{\vartheta}\right)$
(3.26)

and $A$, $B$ and $C$ given by (2.50), (2.52) and (2.58) where $\xi$ and $\eta$
must be replaced according to (3.24). To find the radii of the ergoregion on
the equatorial plane, we can use (3.10) with $\eta=0$ and (3.22) with
$\vartheta=\pi/2$ together with the constraint $r\geq r_{2}$ where $r_{2}$ is
defined according to (3.18) to obtain the formulae

$r_{\mathcal{E},e}^{\pm}=M\left(1+\frac{p}{2}\xi^{\pm}_{\mathcal{E},e}\right)$
(3.27)

with $\xi^{\pm}_{\mathcal{E},e}$ given as in (3.12). Here, the subscript $e$
stands for equator. For typical numerical values of these radii we refer to
Table 2. In Fig. 1 we represented the ergoregion for the case $q=0.5$ where
cusp singularities emerge at the top and bottom of the inner surface of the
ergoregion.

At this point, some comments are in order. First of all,
$r=r_{\mathcal{E},e}^{-}$ remains a curvature singularity also after we
performed the coordinate transformation on the TS metric. Moreover, according
to the discussion here below, it turns out that switching to Boyer-Lindquist
coordinates has the effect of introducing an event horizon located in the
inner region of the ring singularity. This behaviour is not surprising if we
keep in mind the example of the Schwarzschild metric. In Kruskal-Szekeres
coordinates there is no horizon but if we rewrite the line element in
spherical coordinates an horizon appears at $r=2M$. In order to proceed
further, we recall that an event horizon can be characterized in terms of a
null surface $\mathcal{S}$, that is
$g^{\alpha\beta}\partial_{\alpha}\mathcal{S}\partial_{\beta}\mathcal{S}=0$
Islam . In the case of the TS metric in Boyer-Lindquist coordinates such an
equation reduces to

$\left[(r-M)^{2}-\frac{M^{2}p^{2}}{4}\cos^{2}{\vartheta}\right]^{3}\left[\Delta\left(\partial_{r}\mathcal{S}\right)^{2}+\left(\partial_{\vartheta}\mathcal{S}\right)^{2}\right]=0.$
(3.28)

Let us consider the surface $\Delta=0$, i.e. $r=M\pm Mp/2$. Call these
surfaces $\mathcal{S}_{+}$ and $\mathcal{S}_{-}$. It can be easily verified
that these surfaces indeed satisfy (3.28) so that they are null surfaces.
However, due to the condition $r\geq r_{2}$ only the surface $\mathcal{S}_{+}$
should be taken into account. Since $\xi^{-}_{\mathcal{E},e}>1$ when $q>0$
(see Table 1), it follows from (3.27) that $r_{\mathcal{E},e}^{-}>r_{2}$.
Hence, in addition to the ring-shaped naked singularity there is an event
horizon placed in the inner region inside the aforementioned curvature
singularity. However, such a null surface corresponds to the segment
$-Mp/2\leq z\leq Mp/2$. Finally, we observe that the term in the first bracket
of (3.28) vanishes at

$r_{\pm}=M\pm\frac{Mp}{2}\cos{\vartheta}.$ (3.29)

with $r_{+}$ and $r_{-}$ coinciding with $r_{2}$ for $\vartheta=0$ and
$\vartheta=\pi$, respectively, and otherwise, $r_{\pm}<r_{2}$. Since $\Delta$
becomes purely imaginary on the interval $r_{-}<r<r_{+}$, we can neglect this
case.

Figure 1:  Plot of of the ergoregion for $q=0.5$ and $M=1$. Cusp singularities
emerge at the top and bottom of the inner surface of the ergoregion.

##  IV Conclusions and Outlook

Even though the TS space-time is relatively unknown to the majority of
researchers in General Relativity and its application in astrophysics might be
questionable as pointed out by gary , we decided to focus the present work on
the TS metric due mainly to two reasons. The first one is related to the
observation that the literature on this topic is littered with mistakes and
misprints which have been able to propagate from the seminal work of tomsato ;
tomsatop  to the most recent result concerning the shadow of a gravitational
object described by the TS metric (see Bambi ). Hence, the first merit of the
present work is to have uncovered and corrected several mistakes in the
literature. In doing that, we were able to obtain an exact expression for the
ergoregion boundary of the TS space-time which in turn allowed to derive exact
formulae of the radii of the ergoregion in prolates spheroidal coordinates and
in Boyer-Lindquist coordinates. In addition, we also discovered that there is
an event horizon located at the inner region of a ring-shaped naked
singularity.

The second reason behind our work is related to the claims made in Bose  where
the authors, beside presenting the derivation of a weak lensing formula
recently corrected by BG , analyse the motion of a massive test particle
starting at space-like infinity with zero total angular momentum. It was found
by Bose  that for $q=0$ and $q=1$ the corresponding trajectories emerging from
the geodesic equations exhibit a perihelion. However, Bose  was not able to
show whether the same happens for all intermediate values of the parameter
$q$. Hence, Bose  conjectured that the conclusion should also hold for
$0<q<1$. In order to prove/disprove such a conjecture in a forthcoming paper,
it is desirable to have bulletproof formulae to rely on and therefore, the
present work represents a first step in this direction.

Moreover, in view of the fact that the capabilities of very long baseline
interferometry have improved in the recent years and are expected to further
increase in the next decade VB1 ; VB2  at the extent of allowing the
observation of the accreation disk around a candidate black hole, a strong
gravitational lensing analysis of the TS metric might provide a valuable
theoretical resource for the experimental search of such a gravitational
object in the observed universe. A first step in this direction would require
the (in)-stability analysis of null circular orbits for the TS metric. Should
they emerge from a saddle point in the effective potential, then extra care
needs to be applied in order to find out whether or not such orbits are
(un)-stable. Preliminary computations show that in principle this problem can
be reasonably solved using techniques similar to those adopted in Cm1 ; Cm2 .

Last but not least, the TS metric is asymptotically flat while the
cosmological data collected in the last decades seem to suggest that our
universe has a positive cosmological constant. On the other hand, to the best
of the author’s knowledge, no study has been conducted so far in the
literature to extend the TS metric in the presence of a positive cosmological
constant. Therefore,it would be interesting to address the problem of deriving
the deSitter-TS metric in a forthcoming paper. Excellent starting points for
such an endeavour are represented by Ch ; As .

Table 2: Typical values of the horizon $r_{2}$ and the inner and outer boundaries of the ergoregion $r_{\mathcal{E},e}^{\pm}$ on the equatorial plane. $\eta=0$ | $r_{2}/M$ | $r^{-}_{E,e}/M$ | $r^{+}_{E,e}/M$
---|---|---|---
$q=0.1$ | 1.497493719 | 1.498119109 | 1.592179268
$q=0.2$ | 1.489897949 | 1.492404180 | 1.651650002
$q=0.3$ | 1.476969601 | 1.482625988 | 1.703951892
$q=0.4$ | 1.458257570 | 1.468355754 | 1.752205971
$q=0.5$ | 1.433012702 | 1.448872522 | 1.797677529
$q=0.6$ | 7/5 | 1.422966806 | 1.841040496
$q=0.7$ | 1.357071421 | 1.388483971 | 1.882710084
$q=0.8$ | 13/10 | 1.341040496 | 1.922966806
$q=0.9$ | 1.217944947 | 1.268802898 | 1.962012653
$q=0.99$ | 1.070533680 | 1.117858921 | 1.996245294

##  Appendix A Numerical tables

We present here the numerical values for the inner and outer boundaries
$\xi_{\mathcal{E}}^{\pm}$ of the ergoregion for different values of the
rotational parameter $q$ and of the angular variable $\eta$ (see Tables 3, 4
and 5). More precisely, we considered the intervals $0.1\leq q\leq 0.99$ and
$0.1\leq\eta\leq 0.99$. In these ranges, the roots of the polynomial function
$B$ entering the metric coefficient $f$ are always complex. This indicates
that modulo the quasi-regular singularities at $(1,\pm 1)$, $f$ can only
become singular on the equatorial plane.

Table 3: Typical values of $\xi_{\mathcal{E}}^{\pm}$ in prolate spheroidal
coordinates for $0.1\leq q\leq 0.99$ and $0.1\leq\eta\leq 0.3$.

$\eta=0.1$ | $\xi_{\mathcal{E}}^{-}$ | $\xi_{\mathcal{E}}^{+}$
---|---|---
$q=0.1$ | 1.001244519 | 1.188572734
$q=0.2$ | 1.005064795 | 1.327279875
$q=0.3$ | 1.011741110 | 1.471887154
$q=0.4$ | 1.021818037 | 1.636278712
$q=0.5$ | 1.036266826 | 1.835649270
$q=0.6$ | 1.056858285 | 2.094450443
$q=0.7$ | 1.087128247 | 2.461723165
$q=0.8$ | 1.135515223 | 3.062767535
$q=0.9$ | 1.231237701 | 4.393029551
$q=0.99$ | 1.665588339 | 14.05394768

$\eta=0.2$ | $\xi_{\mathcal{E}}^{-}$ | $\xi_{\mathcal{E}}^{+}$
---|---|---
$q=0.1$ | 1.001206829 | 1.183300016
$q=0.2$ | 1.004911691 | 1.318556370
$q=0.3$ | 1.011387321 | 1.459830391
$q=0.4$ | 1.021163733 | 1.620671921
$q=0.5$ | 1.035186500 | 1.815985114
$q=0.6$ | 1.055180322 | 2.069805474
$q=0.7$ | 1.084590838 | 2.430379632
$q=0.8$ | 1.131647154 | 3.021024505
$q=0.9$ | 1.224872051 | 4.329457508
$q=0.99$ | 1.649369985 | 13.84046576

$\eta=0.3$ | $\xi_{\mathcal{E}}^{-}$ | $\xi_{\mathcal{E}}^{+}$
---|---|---
$q=0.1$ | 1.001144009 | 1.174459550
$q=0.2$ | 1.004656467 | 1.303887504
$q=0.3$ | 1.010797398 | 1.439511398
$q=0.4$ | 1.020072295 | 1.594321140
$q=0.5$ | 1.033383445 | 1.782729578
$q=0.6$ | 1.052377772 | 2.028065112
$q=0.7$ | 1.080348579 | 2.377222226
$q=0.8$ | 1.125170823 | 2.950140542
$q=0.9$ | 1.214188473 | 4.221377158
$q=0.99$ | 1.621979060 | 13.47714996

Table 4: Typical values of $\xi_{\mathcal{E}}^{\pm}$ in prolate spheroidal
coordinates for $0.1\leq q\leq 0.99$ and $0.4\leq\eta\leq 0.6$.

$\eta=0.4$ | $\xi_{\mathcal{E}}^{-}$ | $\xi_{\mathcal{E}}^{+}$
---|---|---
$q=0.1$ | 1.001056055 | 1.161969906
$q=0.2$ | 1.004299043 | 1.283069381
$q=0.3$ | 1.009970927 | 1.410573173
$q=0.4$ | 1.018542316 | 1.556680857
$q=0.5$ | 1.030853871 | 1.735101349
$q=0.6$ | 1.048441617 | 1.968141982
$q=0.7$ | 1.074381275 | 2.300739449
$q=0.8$ | 1.116040824 | 2.847940559
$q=0.9$ | 1.199071557 | 4.065240094
$q=0.99$ | 1.582835645 | 12.95139687

$\eta=0.5$ | $\xi_{\mathcal{E}}^{-}$ | $\xi_{\mathcal{E}}^{+}$
---|---|---
$q=0.1$ | 1.000942959 | 1.145711749
$q=0.2$ | 1.003839312 | 1.255796218
$q=0.3$ | 1.008907327 | 1.372470620
$q=0.4$ | 1.016571816 | 1.506905323
$q=0.5$ | 1.027592411 | 1.671872531
$q=0.6$ | 1.043359026 | 1.888305850
$q=0.7$ | 1.066659976 | 2.198498272
$q=0.8$ | 1.104191334 | 2.710885041
$q=0.9$ | 1.179350835 | 3.855211805
$q=0.99$ | 1.531038595 | 12.24229644

$\eta=0.6$ | $\xi_{\mathcal{E}}^{-}$ | $\xi_{\mathcal{E}}^{+}$
---|---|---
$q=0.1$ | 1.000804714 | 1.125521784
$q=0.2$ | 1.003277132 | 1.221635761
$q=0.3$ | 1.007605846 | 1.324413020
$q=0.4$ | 1.014158229 | 1.443739929
$q=0.5$ | 1.023592070 | 1.591184450
$q=0.6$ | 1.037113145 | 1.785888145
$q=0.7$ | 1.057146248 | 2.066679326
$q=0.8$ | 1.089533626 | 2.533323122
$q=0.9$ | 1.154790469 | 3.581824704
$q=0.99$ | 1.465246589 | 11.31543172

Table 5: Typical values of $\xi_{\mathcal{E}}^{\pm}$ in prolate spheroidal
coordinates for $0.1\leq q\leq 0.99$ and $0.7\leq\eta\leq 0.9$.

$\eta=0.7$ | $\xi_{\mathcal{E}}^{-}$ | $\xi_{\mathcal{E}}^{+}$
---|---|---
$q=0.1$ | 1.000641309 | 1.101183763
$q=0.2$ | 1.002612332 | 1.179990007
$q=0.3$ | 1.006065563 | 1.265266537
$q=0.4$ | 1.011298381 | 1.365327537
$q=0.5$ | 1.018844145 | 1.490203770
$q=0.6$ | 1.029682805 | 1.656706163
$q=0.7$ | 1.045791176 | 1.899133583
$q=0.8$ | 1.071952509 | 2.305916399
$q=0.9$ | 1.125073556 | 3.229025779
$q=0.99$ | 1.383464435 | 10.11109287

$\eta=0.8$ | $\xi_{\mathcal{E}}^{-}$ | $\xi_{\mathcal{E}}^{+}$
---|---|---
$q=0.1$ | 1.000452731 | 1.072415310
$q=0.2$ | 1.001844706 | 1.130031675
$q=0.3$ | 1.004285376 | 1.193383547
$q=0.4$ | 1.007988474 | 1.268844574
$q=0.5$ | 1.013338126 | 1.364432382
$q=0.6$ | 1.021042148 | 1.493832491
$q=0.7$ | 1.032534010 | 1.685238687
$q=0.8$ | 1.051301327 | 2.011832721
$q=0.9$ | 1.089778339 | 2.766600056
$q=0.99$ | 1.282637129 | 8.512310772

$\eta=0.9$ | $\xi_{\mathcal{E}}^{-}$ | $\xi_{\mathcal{E}}^{+}$
---|---|---
$q=0.1$ | 1.000238967 | 1.038848419
$q=0.2$ | 1.000974018 | 1.070597706
$q=0.3$ | 1.002264010 | 1.106284089
$q=0.4$ | 1.004224058 | 1.149750752
$q=0.5$ | 1.007061581 | 1.206139834
$q=0.6$ | 1.011160146 | 1.284514335
$q=0.7$ | 1.017300392 | 1.403968178
$q=0.8$ | 1.027394968 | 1.615050284
$q=0.9$ | 1.048341544 | 2.124117081
$q=0.99$ | 1.157799663 | 6.222113920

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