# Trapped region in Kerr-Vaidya space-time

Pravin Kumar Dahal<EMAIL_ADDRESS>Department of Physics and
Astronomy, Macquarie University.

###### Abstract

We review the basic definitions and properties of trapped surfaces and discuss
them in the context of Kerr-Vaidya line-element. Our study shows that the
apparent horizon does not exist in general for axisymmetric space-times. The
reason being the surface at which the null tangent vectors are geodesic and
the surface at which the expansion of such vectors vanishes do not coincide.
Calculation of an approximate apparent horizon for space-times that ensure its
existence seems to be the only way to get away with this problem. The
approximate apparent horizon, however, turned out to be non-unique. The choice
of the shear free null geodesics, at least in the leading order, seem to
remove this non-uniqueness. We also propose a new definition of the black hole
boundary.

##  I Introduction

The notion of the trapped surface was first introduced by Penrose31 , and the
concept later emerged as a quasi-local technique to characterize a black hole.
They are seen as a replacement to the event horizon to define the black hole
boundary, which depends on the global knowledge of the space-time. The
teleological nature32 ; 19  and the unobservability30  of the event horizon
have been discussed numerously in the past. It has thus been regarded as
unsuitable for characterizing astrophysical black holes, which are dynamical
in nature and has to be studied by a finite-sized laboratory for the finite
time. However, even though the trapped surfaces are quasi-local making them
observable in principle, they are not a well-defined quantity to characterize
black holes. The principal reason behind this is the foliation dependence of
the trapped surface. As the choice of the foliation depends on the choice of
null vectors, the trapped surface depends on the choice of null vectors33 .

Defining a black hole boundary has been like a reverse engineering process. We
somehow know the answer for some simple cases based on which we want to define
a black hole boundary such that the definition reproduces that answer. Such
simple cases are stationary space-times of spherical and axial symmetry for
which all the horizons (event, apparent, killing, trapping horizons) coincide.
The notion of trapped surfaces, introduced in this way to define a black hole
boundary poses no problem at all for stationary space-times, even though
trapped surfaces depend on the choice of foliations for these space-times18 .
Then comes the second stage of making the use of the definition of the trapped
surface to identify the black hole boundary of the space-time for which the
concept of an event/ killing horizon fails down. Trapped surface as a black
hole boundary has celebrated its success in application to the general
spherically symmetric space-times. The problem of foliation dependence, in
that case, can be eliminated by making the particular choice of foliation that
respects the symmetry of space-time33 . All spherically symmetric foliations
give the same trapped surface for such space-times. This problem, however, is
more visible and profound if we move to the more general case of axially
symmetric space-times, and there do not exist simple solutions of this problem
in axial symmetry.

This article is the introduction of the trapped surface and its application to
the axially symmetric space-time by taking the reference of the Kerr-Vaidya
metric. We introduce trapped surface, define various types of trapped
surfaces, and discuss, in brief, the notion of the geometric horizon as a
black hole boundary in Sec.I.1. Recently, the geometric horizon has been
considered as an alternative over the trapped surface in locating the black
hole boundary because of its invariant characterization, quasi-local nature,
and foliation independence. However, its applicability has been demonstrated
only for limited classes of space-times until now. In Sec.I.2, we present some
of the properties of the trapped surfaces based on their metric, and in
Sec.I.3, we introduce the slowly evolving horizon. The concept of a slowly
evolving horizon is useful in the sense that it does not mingle with the
problem of foliation dependence, provided that the condition for its existence
satisfies. Sec.II is the application of the trapped surface to the
axisymmetric space-time, and for this, we chose one of the simplest example of
Kerr-Vaidya geometry in advanced coordinates. We also discuss our results in
the context of the geometric horizon. Finally, in Sec.III, we propose a new
black hole boundary.

###  I.1 Trapped surfaces and Geometrical horizons

We begin here with the notion of an event horizon, which is believed to be the
true boundary of a black hole by some authors despite its teleological nature.

Asymptotically flat space-time containing black holes can be separated into
two regions: 1) the region from which all null curves can reach the future
null infinity (where future-directed null curves end in an infinite time in
the Minkowskian space-time). 2) The region from which no null curve reaches
the future null infinity. A boundary separating these two regions of space-
time is defined as an event horizon. This definition of the event horizon
depends on the complete knowledge of future-directed null curves, which is
impossible to achieve in a finite time making it physically unobservable30 .
This global nature of the event horizon also makes it a less useful entity for
dynamical space-times. Similarly, the definition of the event horizon could
not be applied to the space-times which are not asymptotically flat. Thus, the
notion of the trapped surfaces that depends on the quasi-local nature of
space-time is handy in describing black holes evolving in time. In principle,
it is not possible to define a horizon that can be detected by completely
local measurements. The reason behind this is rooted in the equivalence
principle, which implies an impossibility of measuring space-time curvature by
local measurements.

In the orientable space-time manifold ${\cal M}$, lets take future-directed
field of null vectors $l^{\alpha}$ and $n^{\alpha}$ satisfying
$l^{\alpha}n_{\alpha}=-1$. We assume that $l^{\alpha}$ is outgoing in the
sense it is pointed away from the trapped region, and $n^{\alpha}$ is ingoing.
The divergence (or the convergence) of the field of null congruence $l^{\mu}$
is given by its covariant derivative $l^{\mu}_{;\mu}$. If the vector field
$l^{\mu}$ is arbitrarily parametrized, then we affinely parametrize it before
the calculation of divergence, and this introduces the additional correction
term $\kappa=-n_{\mu}l^{\nu}l^{\mu}_{;\nu}$, called the surface gravity. Thus,
the divergence (or convergence) of the affinely parametrized null congruence
is given as2

$\theta_{l}=l^{\mu}_{;\mu}+n_{\mu}l^{\nu}l^{\mu}_{;\nu}\qquad\textrm{and}\qquad\theta_{n}=n^{\mu}_{;\mu}+n_{\mu}l^{\nu}n^{\mu}_{;\nu},$
(1)

where $\theta_{l}$ denotes the outgoing expansion, and $\theta_{n}$ denotes
the ingoing expansion (for details on the expansion of congruence see textbook
by Poisson3 ). Among these, the condition for the vanishing of outgoing
expansion $\theta_{l}=0$ is pivotal for the definition of trapped surfaces. A
surface is said to be trapped if $\theta_{l}<0$ and $\theta_{n}<0$. Thus on
the trapped surface, both the ingoing and outgoing null geodesics are
converging. A marginally outer trapped surface (MOTS) has $\theta_{l}=0$ and
$\theta_{n}<0$. This surface outlines the boundary from which the outgoing
null geodesics start to converge. The three-dimensional surface which can be
foliated entirely by the MOTS is called a marginally outer trapped tube
(MOTT). This MOTT is defined as the future apparent horizon in recent
literature27 ; 12 , and we will adopt this definition here. The definition of
an apparent horizon and, in general, of the trapped surfaces varies over the
literature. The Hayward’s trapping horizon is defined as an apparent horizon
satisfying $n^{\mu}(\theta_{l})_{;\mu}\neq 0$Hayward (1994). This additional
condition imposed implies that the space-time surface is trapped inside the
future outer trapping horizon (FOTH) and normal outside it. Similarly, a
spacelike apparent horizon is defined as a dynamical horizon32 .

The definitions of trapped surfaces given here provide an easy way of
identifying black holes and these are just by the calculation of null
expansions $\theta_{l}$ and $\theta_{n}$. We thus do not have to wait till the
end of time to know the formation of an event horizon to identify black holes.
Trapped surfaces indeed are convenient and unique for the numerical and
analytical study of spherically symmetric space-times (for spherically
symmetric foliation, which is an obvious choice of foliation because of the
symmetry of space-time). However, even the trapped surfaces not being entirely
local entities, their identification depends on the closed hyper-surface
$\Sigma$ of the space-time whose foliation spans the whole space-time.
Expansions should be calculated over all the points of the closed hyper-
surface $\Sigma$, not just at a single point for their identification. Hence,
trapped surfaces are quasi-local in nature and depend on the complete
knowledge of the hypersurface $\Sigma$, if not of the space-time.

As mentioned above, with any choice of spherically symmetric foliations,
trapped surface in spherically symmetric space-time is unique. However, the
complication starts to appear when we consider foliations that are not
spherically symmetric, even to calculate the trapped surface in spherically
symmetric space-times. The foliation dependence of trapped surface in Vaidya
space-time has been demonstrated by numerical calculations with some special
choice of non-spherically symmetric foliation in Refs.16 ; 17 . Discussion of
the foliation dependence of the trapped surface in Schwarzschild and Vaidya
space-time can also be found in the review by Krishnan18 . We will describe
the trapped surface in Kerr-Vaidya space-time and its foliation dependence.

Another type of horizon that is commonly used to characterize the black hole
boundary is the notion of geometric horizon15g ; 16g . The existence of the
geometric horizon is assured by the conjecture that the non-stationary black
holes contain the hypersurface which is more algebraically special14g ; 15g
and such a hypersurface is identified as the geometrical horizon. This method
of locating the black hole does not suffer the problem of foliation
dependence. The property of being more special can be quantified invariantly
by making particular combinations of curvature invariants zero on the horizon.
Three different sets of invariants derived from curvature tensors are commonly
used in the literature to calculate the geometric horizon in general. The
first approach relies on finding the suitable Killing vector field that
becomes the null generator on the horizon. Then, the hypersurface where the
square norm of the Killing vector vanishes is the event horizon. If it is not
possible to find such a field of Killing vector, then scalar polynomial
curvature invariants could be calculated for stationary space-times and space-
times conformal to stationary space-times21g . At the event horizon of such
space-times, it has been shown that the squared norm of the wedge products of
the n-linearly independent gradients of scalar polynomial curvature invariants
vanishes, n being the local cohomogenity of space-time17g ; 18g . This method
of locating the black hole boundary is restricted to the stationary space-
times as this method, in principle, is looking for the null surface where the
magnitude of some particular combinations of invariants vanishes. However, the
horizons of dynamical space-times are not null in general. The extension of
this method to the dynamic space-times conformal to the stationary space-times
lies on the fact that the event horizon is invariant under conformal
transformation.

Another procedure for calculating the geometric horizon is by finding the
zeroes of the certain combinations of Cartan invariants19g ; 20g . This method
is exactly similar, in principle, to the scalar polynomial invariants for
finding the geometrical horizon. Thus, this method is also limited to
stationary space-times and any space-times conformal to them. However, Cartan
invariants are regarded as the improvement over the scalar polynomial
invariants as this method involves the linear combinations of the components
of curvature tensor, and they are possible to construct from Cartan
invariants.

The third procedure for the calculation of the geometric horizon is similar to
the calculation of the trapped surfaces, and thus the black hole boundary
calculated using this procedure can be called as geometrically trapped
surface. The calculation is based on the conjecture that the hypersurface
constituting geometrically trapped surface is more algebraically special17g ;
18g . The procedure is to look for the geometrically preferred outgoing null
vectors and then find the hypersurface where the expansion of such vectors
vanishes. The null vector is called the geometrically preferred null vector
because the congruence of the null vectors thus chosen is such that the
covariant derivatives of the curvature tensor are more algebraically special
there14g ; 16g . The applicability of this method extends beyond the above two
procedures of finding the geometric horizon as this method can be extended to
the dynamical space-times also. However, until now, it has not been used to
locate the black hole boundary of any axisymmetric dynamical space-times. The
equivalence/correspondence between these three procedures for calculating the
geometrical horizon lies in the fact that they all involve the calculation of
invariants derived from the curvature tensor (see, for example, 14g , 20g  for
details).

###  I.2 Properties of trapped surfaces

Refs.22 ; 23  presented a simple formalism to determine the metric signature
of an apparent horizon and demonstrated that for spherically symmetric space-
times. We will here discuss it for the purpose to determine some properties of
the MOTT of the Kerr-Vaidya geometry. Let, $\chi^{\mu}$ be the vector field
tangential to the MOTT and orthogonal to the MOTSs that foliate the MOTT. Also
assume that the vector field $\chi^{\mu}$ generates the flow that preserves
the foliation. Then, ${\cal L}_{\chi}v=f(v)$, where ${\cal L}_{\chi}$ denotes
the Lie derivative along $\chi$ and $f(v)$ denotes some function on the
foliation $v$. We similarly denote by $\bar{\chi}^{\mu}$ the vector field
normal to the MOTT which satisfies $\bar{\chi}_{\mu}\chi^{\mu}=0$. If we
assume both $\chi^{\mu}$ and $\bar{\chi}^{\mu}$ are future directed, then one
of them is spacelike and another is timelike. Now, in terms of the null
vectors $l^{\mu}$ and $n^{\mu}$ given above, we can write: & χ^μ=l^μ-C n^μ,
¯χ^μ=l^μ+C n^μ, for some function $C$. Along $\chi^{\mu}$, the Lie derivative
of an outgoing expansion is zero thereby implying

${\cal L}_{\chi}\theta_{l}={\cal L}_{l}\theta_{l}-C{\cal L}_{n}\theta_{l}=0,$
(2)

or,

$C=\frac{{\cal L}_{l}\theta_{l}}{{\cal L}_{n}\theta_{l}}.$ (3)

Eq.e16 can be used to calculate $C$ and as $C\propto\chi^{\mu}\chi_{\mu}$, the
metric signature of the MOTT is determined by the sign of $C$. $C>0$ means
that an apparent horizon is spacelike, $C<0$ means that an apparent horizon is
timelike and $C=0$ or $\infty$ means that an apparent horizon is null. The
sign of $C$ also determines whether an apparent horizon is expanding or
contracting. To see this, lets take an area element of the two-surface be
$\tilde{q}$ and evaluate its Lie derivative along $\chi^{\mu}$:

${\cal L}_{\chi}\tilde{q}=-C\theta_{n}\tilde{q}$ (4)

As $\theta_{n}<0$ on trapped surface, $C>0$ implies that an apparent horizon
is expanding, $C<0$ implies that the horizon is receding and $C=0$ or $\infty$
implies that the horizon is isolated. Thus, the spacelike apparent horizon is
expanding and the timelike apparent horizon is receding, while an isolated
apparent horizons are null.

###  I.3 Slowly evolving horizon

We are working in the semi-classical domain for which the quasi-static
evaporation law given in 26 ; 27  is valid. In this domain, an apparent
horizon of the Kerr-Vaidya metric can be approximated as a slowly evolving
horizon. So, we present some characteristics of the slowly evolving horizon
here. An almost-isolated trapping horizon is defined as a slowly evolving
horizon and the invariant characterizations of such horizon are made by Booth
and Fairhurst24 ; 25 . We have mentioned above that an apparent horizon is
null and isolated if $C=0$. It thus seems that a slowly evolving horizon can
be characterized by the spacelike or timelike horizon for which $C$ is small.
$C$, however, can be arbitrarily varied by rescaling
$n^{\mu}\to\frac{n^{\mu}}{\alpha}$ and $C\to\frac{C}{\alpha^{2}}$, for some
$\alpha$. The slowly evolving horizon should thus be identified in a scaling
independent manner and for this, we write the area evolution law of Eq.e17 as:

${\cal
L}_{\chi}\tilde{q}=-||\chi||\bigg{(}\sqrt{\frac{C}{2}}\theta_{n}\tilde{q}\bigg{)}$
(5)

such that, the term in parentheses is independent of rescaling. This term
gives an invariant area evolution and if this term is small, then the horizon
is slowly evolving. Thus, one of the conditions proposed by Booth and
Fairhurst24  for the horizon to be slowly evolving over some MOTS in a given
foliation is

$\sqrt{C}\theta_{n}<\frac{\epsilon}{R},$ (6)

for arbitrary small $\epsilon$ where $R$ is the areal radius of the MOTS.
Another condition is the choice of scaling of the null vectors, such that,
$||\chi||\sim\epsilon$. There are other conditions given in 24 ; 25  requiring
that the horizon evolves smoothly over a long period of time. These conditions
will be automatically satisfied for the slowly evolving Kerr-Vaidya metric
where the horizon offsets from the isolated horizon, at most, by an order of
$M_{v}$. The reason behind this is, every parameter calculated at the horizon,
in this case, differs from the stationary Kerr metric by an order of $M_{v}$.

##  II Trapped region in Advanced Kerr-Vaidya space-time

###  II.1 Some Examples

We have, the line-element for Kerr-Vaidya space-time in advanced coordinates
$(v,r,\theta,\tilde{\phi})$ is given as1 :

$ds^{2}=-\bigg{(}1-\frac{2M(v)r}{\rho^{2}}\bigg{)}dv^{2}+2dvdr+\rho^{2}d\theta^{2}-\frac{4aM(v)r\sin^{2}\theta}{\rho^{2}}d\tilde{\phi}dv-2a\sin^{2}\theta
d\tilde{\phi}dr+\frac{(r^{2}+a^{2})^{2}-a^{2}\Delta\sin^{2}\theta}{\rho^{2}}\sin^{2}\theta
d\tilde{\phi}^{2},$ (7)

where $\rho^{2}=r^{2}+a^{2}\cos^{2}\theta$ and $\Delta=r^{2}-2M(v)r+a^{2}$.
This metric reduces to the familiar Kerr metric for some advanced time $v$ if
$M(v)$ is constant6 . The trapped region associated with this metric can be
determined by the calculation of the expansion scalars $\theta_{l}$ and
$\theta_{n}$. To find the trapped surface for this metric, we use the future-
directed null vectors of the form1

$\split
l_{\mu}=\frac{\rho^{2}}{2\Omega^{2}}\bigg{(}-\Delta,\;r^{2}+a^{2}+\Omega,\;0,\;0\bigg{)},\\\
n_{\mu}=\bigg{(}-1,\;\frac{r^{2}+a^{2}-\Omega}{\Delta},\;0,\;0\bigg{)},$ (8)

where $\Omega=\sqrt{(r^{2}+a^{2})\rho^{2}+2M(v)ra^{2}\sin^{2}\theta}$. These
null vectors also being normalized, satisfy the relations

$l_{\mu}l^{\mu}=n_{\mu}n^{\mu}=0\;\text{and}\;l_{\mu}n^{\mu}=-1.$ (9)

Moreover, calculation of the geodesic equation $l^{\mu}l_{\nu;\mu}=\lambda
l_{\nu}$ for some parameter $\lambda$ shows that the vector $l^{\mu}$ is
tangent to the null geodesic only on $\Delta=0$ surface. We now use Eq.e1 to
calculate the expansion scalars using these null vectors. The expansions thus
obtained are given as

$\split\theta_{l}=\frac{1}{2\Omega^{3}}\bigg{(}\Delta(2r^{3}+2a^{2}r-a^{2}r\sin^{2}\theta+a^{2}M\sin^{2}\theta)+a^{2}rM_{v}\sin^{2}\theta(r^{2}+a^{2}+\Omega)\bigg{)},\\\
\theta_{n}=-\frac{1}{2\rho^{2}\Delta\Omega}\bigg{(}\Delta(2r^{3}+2a^{2}r-a^{2}r\sin^{2}\theta+a^{2}M\sin^{2}\theta)+a^{2}rM_{v}\sin^{2}\theta(r^{2}+a^{2}-\Omega)\bigg{)},$
(10)

where $M_{v}=\frac{\partial{M}}{\partial{v}}$. Now, a MOTS is the two-surface
where outgoing null expansion ($\theta_{l}$) vanishes and this marks the
boundary of the trapped region. Senovilla and Torres1  have argued that the
solution for $\theta_{l}=0$ does not exist in general for the $v=constant$
foliation considered. This is indeed true and the reason behind is that the
null vectors are geodesic only on $\Delta=0$ surface and on this surface,
$\theta_{l}=0$ only when $M_{v}=0$. They have thus considered the surface of
intersection between $\Delta=0$ and $M_{v}=0$ where
$r_{g}=M\pm\sqrt{M^{2}-a^{2}}$ are the two real solutions of $\theta_{l}=0$.
For dynamical space-time considered, $M_{v}=0$ is not true in general and
might occur only for some $v=v_{0}$. Thus, $v=constant$ surface does not
foliate an apparent horizon in general.

Solution of $\theta_{l}=0$ is simpler and more evident if we calculate the
expansion scalars for the null geodesics with the following tangent vectors:

$\split&l_{\mu}=\frac{\Delta}{2\rho^{2}}\bigg{(}-1,\;\frac{2\rho^{2}}{\Delta},\;0,\;a\sin^{2}\theta\bigg{)},\\\
n_{\mu}=\bigg{(}-1,\;0,\;0,\;a\sin^{2}\theta\bigg{)}.$ (11)

$l_{\mu}$ satisfy the geodesic equation $l^{\mu}l_{\nu;\mu}=\lambda l_{\nu}$,
for some parameter $\lambda$ defining the geodesic curve only when $M_{v}=0$.
The expansions using these pairs of null vectors are:

$\theta_{l}=\frac{r\Delta}{\rho^{4}},\;\;\;\;\;\theta_{n}=-\frac{2r}{\rho^{2}}.$
(12)

We thus get $\theta_{l}=0$ when $\Delta=0$. Hence, the MOTS should be the
intersection between $\Delta=0$ and $M_{v}=0$ leading us again to the
conclusion that $v=constant$ surface does not foliate an apparent horizon in
general.

###  II.2 MOTSs are not unique

Here, we will first show that depending on the choice of the null vectors, the
MOTS (the surface for which $\theta_{l}=0$) can extend all the way from the
singularity to the infinity. For this, we take the general null vector of the
form:

$\split l_{\mu}=(-\alpha,\;1,\;\beta,\;\gamma)\\\
n_{\mu}=(-1,\;0,\;0,\;a\sin^{2}\theta)$ (13)

where $\alpha=\alpha(v,r,\theta,M(v))$. The outgoing vector $l_{\mu}$ will be
future directed for $\alpha>0$ and it is null implies that $l_{\mu}l^{\mu}=0$.
This allows us to express $\gamma$ in terms of $\alpha$ and $\beta$ as:

$\gamma=(-a+\alpha
a\pm\sqrt{(1-2\alpha)a^{2}-(\Delta-2\alpha(a^{2}+r^{2})+\beta^{2})\csc^{2}\theta})\sin^{2}\theta$
(14)

With this value of $\gamma$, the vectors $l_{\mu}$ and $n_{\mu}$ satisfies all
the relations given in Eq.3. We can now calculate the outgoing expansion
$\theta_{l}$ and its value is obtained as:

$\theta_{l}=\theta_{Kerr}+\frac{2aM_{v}\bigg{(}r+\rho^{2}\frac{\partial\alpha}{\partial
M}-\beta\frac{\partial\beta}{\partial
M}\bigg{)}}{\rho^{2}\sqrt{(1-2\alpha)a^{2}-(\Delta-2\alpha(a^{2}+r^{2})+\beta^{2})\csc^{2}\theta}}$
(15)

where $\theta_{Kerr}$ is an expression containing terms that does not have
$M_{v}$. This corresponds to the outgoing expansion for the Kerr metric with
$M=constant$. But, for the Kerr geometry, we should have

$\theta_{Kerr}=\frac{\Delta
g(t,r,\theta)}{\rho^{2}\sqrt{(1-2\alpha)a^{2}-(\Delta-2\alpha(a^{2}+r^{2})+\beta^{2})\csc^{2}\theta}},$
(16)

where $g(t,r,\theta)$ is an arbitrary function. However, we should have
$g(t,r,\theta)>0$ so as to ensure that $\theta_{Kerr}>0$ for $\Delta>0$. We
thus get

$\theta_{l}=\frac{g\bigg{(}\Delta+\frac{2aM_{v}}{g}\bigg{(}r+\rho^{2}\frac{\partial\alpha}{\partial
M}-\beta\frac{\partial\beta}{\partial
M}\bigg{)}\bigg{)}}{\rho^{2}\sqrt{(1-2\alpha)a^{2}-(\Delta-2\alpha(a^{2}+r^{2})+\beta^{2})\csc^{2}\theta}}.$
(17)

Again, to ensure that $\theta_{l}>0$ outside the horizon, we should have

$\Delta+\frac{2aM_{v}}{g}\bigg{(}r+\rho^{2}\frac{\partial\alpha}{\partial
M}-\beta\frac{\partial\beta}{\partial M}\bigg{)}>0,$ (18)

as $g>0$. We will analyse the two cases of $\Delta>0$ (for the possibility of
horizon lying outside $M+\sqrt{M^{2}-a^{2}}$) and $\Delta<0$ (for the
possibility of horizon lying inside $M+\sqrt{M^{2}-a^{2}}$) individually.

First let us consider the case of $\Delta>0$ at the horizon. To achieve this,
that is, to get $\theta_{l}=0$, we should have from Eq.h15

$\split&r+\rho^{2}\frac{\partial\alpha}{\partial
M}-\beta\frac{\partial\beta}{\partial M}>0\qquad\text{for}\qquad
M_{v}<0\qquad\text{and}\qquad g>0,\\\ r+\rho^{2}\frac{\partial\alpha}{\partial
M}-\beta\frac{\partial\beta}{\partial M}<0\qquad\text{for}\qquad
M_{v}>0\qquad\text{and}\qquad g>0.$ (19)

Without risk of losing the generality of the the result, we will assume
$\beta=0$ for the rest of the analysis. This gives

$\split r+\rho^{2}\frac{\partial\alpha}{\partial M}>0\qquad\text{for}\qquad
M_{v}<0,\\\ r+\rho^{2}\frac{\partial\alpha}{\partial
M}<0\qquad\text{for}\qquad M_{v}>0.$ (20)

The lower/upper bound of this inequality gives
$\alpha=\frac{\Delta}{2\rho^{2}}$ with the choice of $(a^{2}+r^{2})/2$ as an
integration constant and this is the null vector given in Eq.o8 to obtain
$\Delta=0$ as the horizon. In general, the solution for $\theta_{l}=0$ from
Eq.h15 is given as

$\Delta+\frac{2aM_{v}}{g}\bigg{(}r+\rho^{2}\frac{\partial\alpha}{\partial
M}\bigg{)}=0,$ (21)

or

$\bigg{(}1+\frac{2aM_{v}}{g}\frac{\partial\alpha}{\partial
M}\bigg{)}r^{2}-2\bigg{(}M-\frac{aM_{v}}{g}\bigg{)}r+a^{2}\bigg{(}1+\frac{2aM_{v}}{g}\frac{\partial\alpha}{\partial
M}\cos^{2}\theta\bigg{)}=0.$ (22)

The solution of this equation will be diverging as
$\frac{\partial\alpha}{\partial M}\to-\frac{g}{2aM_{v}}$, which is the
reasonable possibility. We will here give an example to support this argument.
But, let us first assume $M_{vv}\approx 0$ (only for this example), which is
the reasonable approximation in the semi-classical limit. We then take

$\alpha=\frac{a^{2}Q^{2}+M^{2}\csc^{2}\theta}{2a^{2}Q^{2}},$ (23)

where $Q=Q(v)$ is some arbitrary function. With this choice of $\alpha$, we
get

$\gamma=\frac{M^{2}-a^{2}Q^{2}\sin^{2}\theta+2\sqrt{MQ^{2}(\rho^{2}M+2a^{2}rQ^{2}\sin^{2}\theta)}}{2aQ^{2}}.$
(24)

We thus have the pair of null vectors given as

$\split
l_{\mu}=&\left(-\frac{a^{2}Q^{2}+M^{2}\csc^{2}\theta}{2a^{2}Q^{2}},\;1,\;0,\;\frac{M^{2}-a^{2}Q^{2}\sin^{2}\theta+2\sqrt{MQ^{2}(\rho^{2}M+2a^{2}rQ^{2}\sin^{2}\theta)}}{2aQ^{2}}\right),\\\
n_{\mu}=(-1,\;0,\;0,\;a\sin^{2}\theta).$ (25)

They satisfy the relations given in Eq.3 and thus can be used for the
calculation of expansion scalars. The value of $\theta_{l}$ is obtained as:

$\theta_{l}=\frac{1}{\rho^{2}}\bigg{(}r-M+\frac{MQ(rM+a^{2}Q^{2}\sin^{2}\theta)+QM_{v}(\rho^{2}M+a^{2}rQ^{2}\sin^{2}\theta)-\rho^{2}M^{2}Q_{v}}{\sqrt{MQ^{4}(\rho^{2}M+2a^{2}rQ^{2}\sin^{2}\theta)}}\bigg{)},$
(26)

where $Q_{v}=\partial Q/\partial v$. The solution of $\theta_{l}=0$ for an
apparent horizon will be convenient when $Q^{2}\to M_{v}^{2}$, for which case,
$Q_{v}\sim 0$. The solution for $r$ in that regime would be

$r=\left|\frac{2MM_{v}}{M_{v}^{2}-Q^{2}}\right|.$ (27)

We thus have diverging $r=r_{g}$ as $Q^{2}\to M_{v}^{2}$.

Now, we consider the case of $\Delta<0$ at the horizon. Then, to ensure that
$\theta_{l}=0$, we should have

$\split r+\rho^{2}\frac{\partial\alpha}{\partial M}<0\qquad\text{for}\qquad
M_{v}<0,\\\ r+\rho^{2}\frac{\partial\alpha}{\partial
M}>0\qquad\text{for}\qquad M_{v}>0.$ (28)

Again, the upper/lower bound of this inequality gives
$\alpha=\frac{\Delta}{2\rho^{2}}$ with the choice of $(a^{2}+r^{2})/2$ as an
integration constant, which is the null vector given in Eq.o8. The solution
for $\theta_{l}=0$ from Eq.h15 is given as

$\bigg{(}1+\frac{2aM_{v}}{g}\frac{\partial\alpha}{\partial
M}\bigg{)}r^{2}-2\bigg{(}M-\frac{aM_{v}}{g}\bigg{)}r+a^{2}\bigg{(}1+\frac{2aM_{v}}{g}\frac{\partial\alpha}{\partial
M}\cos^{2}\theta\bigg{)}=0.$ (29)

Both for $M_{v}<0$ and $M_{v}>0$, proper choice of the value of
$\partial\alpha/\partial M$ could make the value of $r=r_{g}$ recede up to the
singularity. This can be seen by minimizing $r$ with respect to
$\partial\alpha/\partial M$ in this equation, whose solution is
$r^{2}+a^{2}\cos^{2}\theta=0$. Thus, depending on the choice of null vectors
(which in fact determines the choice of the foliation), the value of the MOTS
can extend from the singularity to the infinity. As we have seen in the
previous two examples, all the MOTSs do not foliate an apparent horizon.
However, it is not clear which MOTSs dynamically evolve to foliate the
apparent horizon and if the apparent horizon is unique19 ; 14 . Moreover, to
find the trapping horizon, we have to check each apparent horizon
hypersurfaces to confirm whether the space-time inside it is trapped and
outside it is normal. So, the problem of locating the trapping horizon extends
further beyond the complexity of identifying the apparent horizon.

###  II.3 An approximate apparent horizon

Besides the problem of foliation dependence, there is another more serious
problem in general dynamical axially symmetric space-times in the calculation
of the trapped surfaces. In general, for time-dependent axial symmetry, it is
observed that the future directed outgoing null vector is not tangent to the
geodesic everywhere. This is because, the general future-directed outgoing
null vector has three independent parameters and can be written as
$l_{\mu}=(-\alpha,\;1,\;\beta,\;\gamma)$, where $\alpha\neq 0$. However, there
are four constraints altogether to satisfy by this vector: three independent
constraints from the geodesic equation $l^{\mu}l_{\nu;\mu}=\lambda l_{\nu}$,
for some parameter $\lambda$ (one of the equation $l^{\mu}l_{r;\mu}=\lambda
l_{r}$ is satisfied identically) and one constraint from the null condition
$l^{\mu}l_{\mu}=0$. There are thus four equations to satisfy by three
variables and this will hold only on some region/hypersurface of the space-
time. This is in contrast to the situation in both the time-dependent
spherical symmetry and the stationary axial symmetry, where an arbitrary null
vector can be made geodesic by the proper choice of a parameter called an
affine parameter defining the curve. Thus, an outgoing null vector is a
tangent to the geodesic only on some surface in axial symmetry in general and
on that surface, the outgoing expansion $\theta_{l}$ does not vanish usually.
A way to address this problem is to calculate an approximate apparent horizon.

To approximate an apparent horizon for the Kerr-Vaidya geometry, let us take
the null vectors of Eq.o8 for which the solution of $\theta_{l}=0$ is clearly
given by $\Delta=0$. However, as concluded above, the intersection of
$v=constant$ and $\Delta=0$ surface does not foliate the apparent horizon in
general. The reason behind this is that the vectors of Eq.o8 is not tangent to
the null geodesics in general. To see this, the geodesic deviation equation
$l^{\mu}l_{\nu;\mu}-\kappa l_{\nu}$ for some parameter $\kappa$ is given as &
l^μl_v; μ-κl_v=a2r sin2θMvρ4,
l^μl_r; μ-κl_r=0,
l^μl_θ; μ-κl_θ=0,
l^μl_ϕ; μ-κl_ϕ=-a r (a2+r2) sin2θMvρ4. Thus the null vectors are offset from
being the geodesics by an order of $M_{v}$, which is obviously small in the
semi-classical limit as pointed out in Sec.I.3. So in the semi-classical
region, we can approximate $\Delta=0$ as an apparent horizon, and we will
explain below that this satisfies all the property for being the slowly
evolving horizon. But before that, we will present a technique that is similar
to the perturbation expansion to approximate an apparent horizon. Possibly,
the validity of this approximation method would be for all $|M_{v}|<1$ and not
only for very small $M_{v}$.

For this, we proceed forward by making a slight modification on the null
vectors of Eq.o8

$\split&l_{\mu}=\bigg{(}-\frac{\Delta+\lambda(r,\theta)M_{v}}{2\rho^{2}},\;1,\;\zeta(r,\theta)M_{v},\;\frac{a\Delta\sin^{2}\theta+\nu(v,r,\theta)}{2\rho^{2}}\bigg{)},\\\
n_{\mu}=\bigg{(}-1,\;0,\;0,\;a\sin^{2}\theta\bigg{)}.$ (30)

The null condition $l^{\mu}l_{\mu}=0$ gives

$\nu=\sin^{2}\theta\bigg{(}a\lambda
M_{v}-2a\rho^{2}+2\rho^{2}\sqrt{a^{2}+\csc^{2}\theta\lambda
M_{v}-\csc^{2}\theta\zeta^{2}M_{v}^{2}}\bigg{)}.$ (31)

We now calculate the outgoing expansion $\theta_{l}$ at $r=r_{g}$ given by the
solution of the equation $M=\frac{a^{2}+r^{2}+f(t,r,\theta)M_{v}}{2r}$ where
$f(t,r,\theta)$ is another arbitrary function. Assuming $M_{vv}\sim 0$, we
get, from the solution of $\theta_{l}=0$

$f=\frac{2\rho^{2}\left(\left(2\frac{\partial\zeta}{\partial\theta}-\frac{\partial\lambda}{\partial
r}\right)\sqrt{\csc^{2}\theta M_{v}\left(\lambda-
M_{v}\zeta^{2}\right)+a^{2}}+2\zeta\left(\cot\theta\sqrt{\csc^{2}\theta
M_{v}\left(\lambda-
M_{v}\zeta^{2}\right)+a^{2}}-aM_{v}\frac{\partial\zeta}{\partial
r}\right)+a\frac{\partial\lambda}{\partial r}\right)}{4r\sqrt{\csc^{2}\theta
M_{v}\left(\lambda-M_{v}\zeta^{2}\right)+a^{2}}}+\lambda.$ (32)

Again, solving the geodesic deviation equation at
$M=\frac{a^{2}+r^{2}+f(t,r,\theta)M_{v}}{2r}$, with the value of $f$ given by
Eq.o27, we get & l^μl_v; μ-κl_v=Mv((r2-a2)λ+4 a2r2sin2θ)4 r ρ4+O(M_v^2),
l^μl_r; μ-κl_r=0,
l^μl_θ; μ-κl_θ=(a2-r2)Mvζ2r ρ2+O(M_v^2),
l^μl_ϕ; μ-κl_ϕ= (a2+r2)Mv((a2-r2)λ-4 a2r2sin2θ)4a ρ4r+O(M_v^2), where ${\cal
O}\left(M_{v}^{2}\right)$ represents the terms of order $M_{v}^{2}$ and
higher. Now, requiring the geodesic deviation of the null tangent vectors to
be of at least ${\cal O}\left(M_{v}^{2}\right)$ we should have

$\lambda=\frac{4a^{2}r^{2}\sin^{2}\theta}{a^{2}-r^{2}}\qquad\textrm{and}\qquad\zeta=0.$
(33)

Substituting this in Eq.o27, we get

$f=\frac{2a^{2}\sin^{2}\theta\left(2a^{4}\cos^{2}\theta+2a^{2}r^{2}-\left(a^{4}\cos
2\theta+a^{4}+2r^{4}\right)\sqrt{\frac{4r^{2}M_{v}}{a^{2}-r^{2}}+1}\right)}{(a-r)^{2}(a+r)^{2}\sqrt{\frac{4r^{2}M_{v}}{a^{2}-r^{2}}+1}}=\frac{4a^{2}r^{2}\sin^{2}\theta}{a^{2}-r^{2}}+{\cal
O}\left(M_{v}\right).$ (34)

Thus, the solution of the equation
$M=\frac{a^{2}+r^{2}+f(t,r,\theta)M_{v}}{2r}$ with the value of $f$ from
Eq.o33 gives the value of $r$. This $r$ corresponds to the apparent horizon of
the Kerr-Vaidya space-time up to the first order correction in $M_{v}$.
Specifically,

$r=M+\sqrt{M^{2}-a^{2}}+\frac{a^{2}\sin^{2}\theta\left(2M^{2}-a^{2}+2M\sqrt{M^{2}-a^{2}}\right)}{\left(M^{2}-a^{2}\right)\left(M+\sqrt{M^{2}-a^{2}}\right)}M_{v}.$
(35)

For the Kerr-Vaidya metric in advanced coordinates, when $M_{v}<0$, the
apparent horizon is ellipsoid flattened at the equator, provided that $a$ is
small. Similarly, when $M_{v}>0$, for small $a$, the apparent horizon is
ellipsoid flattened at the pole. It can be shown that this approximate horizon
is not the unique approximate apparent horizon. There exists other surfaces
where the null vectors satisfy geodesic equations up to the second order in
$M_{v}$ and has the vanishing outgoing expansion $\theta_{l}$ (for example,
the exact procedures for the calculation of an approximate horizon applied to
the null vectors of Eq.s9 give different surface as an approximate horizon).
However, these surfaces differs from each other and from the stationary black
hole horizon, at most, by an order of $M_{v}$.

The MOTSs foliating the apparent horizon is unique19 . So, if an apparent
horizon is known a priori, then we can infer that this has been foliated by a
dynamical evolution of the unique MOTS. This is unlike the case of isolated
horizons where foliations are freely deformable. However, we could not say
anything about the uniqueness of the apparent horizon itself. (Remark: the
assumption of the finite time formation of an apparent horizon constrains its
location. However, it is not sufficient constraint to give the unique apparent
horizon.)

The approximate apparent horizon calculated here has promising features which
we list below and this might provide a clue for the appropriate choice of null
vectors for the calculation of trapped surface. The null vectors given in
Eq.o25 have the vanishing shear tensor in the leading order approximation in
$M_{v}$. The shear tensor $\sigma_{\mu\nu}$ can be calculated by using the
relation

$\sigma_{\mu\nu}=\tilde{B}_{\mu\nu}-\frac{\theta}{2}h_{\mu\nu},$ (36)

where
$h_{\alpha\beta}=g_{\alpha\beta}+l_{\alpha}n_{\beta}+n_{\alpha}l_{\beta}$ and
$\tilde{B}_{\alpha\beta}=\frac{1}{2}\theta h_{\alpha\beta}$ (see 3  for
details). Now, at the apparent horizon, $\theta_{l}=0$. Direct calculation
using Eq.d37 gives

$\sigma^{l}_{\mu\nu}\sigma_{n}^{\mu\nu}=\frac{4a^{2}r_{g}^{2}\cos^{2}\theta}{\rho^{4}\left(a^{2}-r_{g}^{2}\right)}M_{v}+{\cal
O}\left(M_{v}^{2}\right),$ (37)

which is already first order in $M_{v}$. The choice of null vectors with
vanishing shear tensor, at least, in the leading order approximation is the
natural choice of null vectors (and hence the natural choice of foliation) for
the calculation of the trapped surface. This can be explained as follows:

It is undoubtedly true that for the calculation of trapped surfaces in
spherically symmetric space-times, we should choose the foliation that
respects the symmetry of the space-time33 . For this spherical foliation of
choice, the null geodesic congruence normal to this surface is radial and
shear free. Some choice of non-spherical foliations, even in stationary
spherically symmetric space-times, do not contain trapped surfaces18 .
Obviously, choosing an arbitrary axially symmetric foliation does not work for
axially symmetric space-times. Shear free geodesic congruence is the axially
symmetric analogy of radial geodesic in the spherical symmetric space-
timeChandrasekhar (1983). Thus, the foliation associated with the shear free
geodesic in axially symmetric space-time compliments the symmetry respecting
foliation of the spherically symmetric space-time.

Furthermore, our calculation of an approximate apparent horizon also follows
from an assumption of the validity of the geometric horizon conjecture stated
in Sec.I.1. The corollary of the Goldberg and Sachs34  theorem given in
Chandrasekhar (1983) (p.63) implies that if the congruence formed by the two
principal null directions $l^{\mu}$ and $n^{\mu}$ are geodesic and shear free
then the space-time is algebraically special. As the null geodesics of Eq.o25
for the calculation of an approximate horizon is shear free, at least in
first-order approximation in $M_{v}$, the congruence formed by them is more
algebraically special. This implies that the approximate apparent horizon is
also an approximate geometric trapped surface. Moreover, the procedure for the
calculation of an approximate apparent horizon is the generalization of the
third procedure for the calculation of the geometric horizon, explained above,
to the more general case of axial symmetry.

###  II.4 Features of the apparent horizon

We now take the pair of null vectors given in Eq.o25 with the value of
respective parameters $\nu$ given in Eq.o44 and $\lambda$ and $\mu$ given in
Eq.o50. Direct calculation using these null vectors yield

$\split&{\cal
L}_{l}\theta_{l}=l^{\mu}(\theta_{l})_{;\mu}=-\frac{2r^{2}M_{v}}{\rho^{4}}+{\cal
O}\left(M_{v}^{2}\right),\\\ {\cal
L}_{n}\theta_{l}=n^{\mu}(\theta_{l})_{;\mu}=\frac{a^{2}-r^{2}}{\rho^{4}}-\frac{4a^{2}r^{2}\left(a^{2}+r^{2}\right)\sin^{2}\theta}{\rho^{4}\left(a^{2}-r^{2}\right)^{2}}M_{v}+{\cal
O}\left(M_{v}^{2}\right).$ (38)

We substitute this in Eq.e16 to get:

$C=\frac{2r^{2}M_{v}}{r^{2}-a^{2}}+{\cal O}\left(M_{v}^{2}\right).$ (39)

From this equation, $M_{v}<0$ gives $C<0$ thereby implying that the apparent
horizon of the Kerr-Vaidya line-element in advanced coordinates is timelike
and receding in $M_{v}<0$ domain. These are important features, as $M_{v}<0$
regime of the Kerr-Vaidya line-element in advanced coordinate is believed to
be the evaporating black hole solution of the Einstein equation. Similarly,
for $M_{v}>0$, the apparent horizon is spacelike and advancing.

Again, for the null vectors of Eq.o25, we get $\theta_{n}=-2r/\rho^{2}$. We
thus have the condition for the slowly evolving horizon

$\sqrt{|C|}\theta_{n}=\sqrt{\frac{8r^{4}|M_{v}|}{\rho^{4}(r^{2}-a^{2})}}<\frac{\epsilon}{r^{2}+a^{2}},$
(40)

that holds only when the space-time is slowly rotating, that is, $a\ll r$ and
$\sqrt{|M_{v}|}\sim{\cal O}(\epsilon)$. But, in the semi-classical limit, for
slowly rotating case near the apparent horizon $r_{g}$, $|M_{v}|\approx
r_{g}^{\prime}\sim 10^{-3}-10^{-4}$26 ; 27 . This gives $\epsilon\sim 0.01\ll
1$. Thus, the apparent horizon of an advanced Kerr-Vaidya line-element, in the
semi-classical domain, satisfies the condition to be called as the slowly
evolving horizon provided $a\ll r$.

Similarly, the calculation using the pair of null vectors of Eq.o8 yields
$C\sim M_{v}$ for $a\ll r$. This again leads to the conclusion that the
surface $\Delta=0$ can be called as the slowly evolving horizon when
$\sqrt{M_{v}}\sim\epsilon$. Thus, $\Delta=0$ is the slowly evolving horizon of
the Kerr-Vaidya metric in advanced coordinates provided $a\ll r$.

##  III Another possible characterization of a black hole boundary

To begin with, we first summarize the complications we encounter in
identifying trapped surfaces as a black hole boundary. Identification of
trapped surfaces as black hole boundary is straightforward and poses no
problem in stationary space-times (where the event horizon can also be
undoubtedly located) and in dynamical spherically symmetric space-times (where
the location of the event horizon is problematic). However, trapped surfaces
turned out to be the ill-defined concept when we try to explore them beyond
these two classes of space-times. This is because of the two fundamental
reasons:

  * •

The problem of the foliation dependence of trapped surfaces: Even in the
simplest class of space-times like Schwarzschild and Vaidya5 , trapped surface
depends on the choice of foliation on which the family of null congruence is
orthogonal.

  * •

An arbitrary outgoing null vector $l_{\mu}$ is not geodesic everywhere in
general axisymmetric space-time. In general, the region where $l_{\mu}$ is
geodesic is not where the outgoing null expansion $\theta_{l}$ vanishes.
Because of this, even an exact location of a geometrical trapped surface which
is a foliation independent entity is not possible.

In an attempt to address these problems, we present an alternative way of
characterization of a black hole boundary. The common procedure for locating a
black hole boundary is by looking for some identically non-zero invariant
quantity whose value vanishes on the horizon. For example, the expansion
scalar of an outgoing null geodesic vanishes at the trapped surface.
Similarly, the norm of the Killing vector field or some particular
combinations of the scalar polynomial curvature invariants or Cartan
invariants vanishes at the geometrical horizon. So, analogously, we are also
looking for an invariant quantity that is identically zero at the horizon and
non-zero elsewhere. However, for us, that quantity should be
foliation/observer-independent and applicable to the more general case of
dynamical axial symmetry. Such a quantity would be an arbitrary radial
trajectory $l^{r}$, which becomes null on the horizon, that is,
$l^{\mu}l_{\mu}=0$, given $l^{\theta}$ and $l^{\phi}$ are null (which can be
constructed by choice).

To demonstrate that all the radial trajectory indeed becomes null at the
horizon, we here given an example of the general spherically symmetric space-
time

$d\tau^{2}=-e^{2h(t,r)}f(t,r)dt^{2}+\frac{1}{f(t,r)}dr^{2}+r^{2}d\Omega^{2}.$
(41)

Now, for this space-time, the equation of motion in $\theta$ direction is

$\frac{d}{d\tau}\frac{\partial{\mathcal{L}}}{\partial\dot{\theta}}=\frac{d}{d\tau}(r^{2}\dot{\theta})=-\frac{\partial{\mathcal{L}}}{\partial\theta}=r^{2}\sin\theta\cos\theta\left(\frac{d\phi}{d\tau}\right)^{2},$
(42)

where $\dot{\theta}=d\theta/d\tau$. So, if we initially choose $\theta=\pi/2$
and $\dot{\theta}=0$ then, $\ddot{\theta}=0$. This implies that the geodesic
motion can be described in an invariant plane and we choose that plane to be
the equator $\theta=\pi/2$. Thus, the radial equation of motion for the space-
time of Eq.s47 can be written as

$\dot{r}^{2}=-\delta
f(t,r)+\left(e^{h(t,r)}f(t,r)\dot{t}\right)^{2}-\frac{L^{2}}{r^{2}}f(t,r),$
(43)

where $\delta=0$ for null geodesic, $\delta=1$ for timelike geodesic and
$L=r^{2}\frac{d\phi}{d\tau}$ is constant. As $\dot{t}$ does not depend on the
geodesic being null or timelike and as $L$ is arbitrary, the only surface
where the radial trajectory $\dot{r}$ is always null is $f(t,r)=0$. So, in
dynamical spherically symmetric space-times, all the radial trajectory becomes
null on some hypersurface, and this three-surface is uniquely characterized as
the black hole boundary by our prescription. Also, looking at the geodesic
equations for the Kerr space-time given in Chandrasekhar (1983), we can see
that all of the timelike radial geodesics approach null geodesic on
$r^{2}+a^{2}-2rM=0$ surface. This surface is, in fact, both the event and the
apparent horizon of the Kerr space-time.

Hence, it might be reasonable to characterize a black hole boundary as the
hypersurface where every timelike radial geodesics approaches null geodesics.
This is the asymptotic three surface, and the motivation for this is the
classical picture of the black hole as an asymptotic state of the
gravitational collapse. For most of the space-times, the black hole boundary
in this characterization is given by the solution of $g^{rr}=0$. Some space-
times, like Vaidya, has $g^{rr}=0$ identically and non-zero $g^{tr}$.
Coordinate transformation is possible for such space-times to make $g^{tr}=0$,
and then the solution of $g^{rr}=0$ gives the black hole boundary (otherwise,
the full geodesic equation should be solved).

We now take an example of the general axisymmetric space-time with two
parameters

$d\tau^{2}=e^{2h}\bigg{(}1-\frac{2Mr}{\rho^{2}}\bigg{)}dt^{2}+\frac{4e^{h}aMr\sin^{2}\theta}{\rho^{2}}dtd\phi-\frac{\rho^{2}}{\Delta}dr^{2}-\rho^{2}d\theta^{2}-\frac{(r^{2}+a^{2})^{2}-a^{2}\Delta\sin^{2}\theta}{\rho^{2}}\sin^{2}\theta
d\phi^{2},$ (44)

where $h=h(t,r,\theta)$ and $M=M(t,r,\theta)$. To calculate the boundary of
the black hole for this space-time, we assume the trajectory of the form

$l^{\mu}=\beta\left(1,\;\alpha,\;\gamma,\;\delta\right),$ (45)

where $\beta$, $\alpha$, $\gamma$ and $\delta$ are the functions of $t$, $r$
and $\theta$. Now, assuming $l^{\mu}$ to be a time like trajectory, it should
satisfy $l^{\mu}l_{\mu}=-1$ and this gives

$\alpha_{t}=\sqrt{\frac{\Delta}{\rho^{2}}\left(-\frac{1}{\beta^{2}}-\frac{2rM\left(e^{h}-a\sin^{2}\theta\delta\right)^{2}}{\rho^{2}}+e^{2h}-\rho^{2}\gamma^{2}-\left(a^{2}+r^{2}\right)\sin^{2}\theta\delta^{2}\right)}.$
(46)

Now, this time like radial trajectory coincides identically with the null
trajectory given by

$\alpha_{n}=\sqrt{\frac{\Delta}{\rho^{2}}\left(-\frac{2rM\left(e^{h}-a\sin^{2}\theta\delta\right)^{2}}{\rho^{2}}+e^{2h}-\rho^{2}\gamma^{2}-\left(a^{2}+r^{2}\right)\sin^{2}\theta\delta^{2}\right)},$
(47)

on the surface $\Delta=0$ irrespective of the form of the parameters $e^{h}$,
$\gamma$ and $\delta$ assuming that the normalization factor $\beta\neq 0$
there (this is expected otherwise, the trajectory is identically zero at
$\Delta=0$. Thus, for the general form of the axisymmetric metric given by
Eq.n69, adopting the definition presented here gives $g^{rr}=0$ as the black
hole boundary.

##  IV Conclusion and Discussions

It seems that the quasi-local measures for determining a black hole boundary
are not suitable for the general axisymmetric space-times. In spherical
symmetry, there exists a preferred choice of foliation which obeys the
symmetry of the space-time and gives a unique MOTS. However, such a preferred
choice of foliations can not be made in axisymmetric space-times. We have
shown that for axisymmetric space-times, MOTS is not unique and lies anywhere
from the singularity to the infinity depending on the choice of the null
vectors. Although all these MOTSs do not foliate apparent horizons, many of
them could. However, there is not a preferred choice of MOTS foliating an
apparent horizon that acts as a true black hole boundary.

Another problem in the calculation of an apparent horizon as explained in
Sec.II.2 is that the outgoing null vectors are not geodesic everywhere in
general in axisymmetric space-times. At least, for Kerr-Vaidya line-element,
it is found that the surface where $\theta_{l}=0$ does not coincide with the
hypersurface where the null vectors are geodesic. We expect the same situation
to occur in most of the general axisymmetric space-times. If this is the case,
then the calculation of the trapped surface as a black hole boundary does not
even make a sense for axisymmetric space-times. One way to get out of this
problem was the calculation of an approximate apparent horizon, explained in
Sec.II.3. The approximate horizon exists only for small $M_{v}$, and when it
exists, Sec.II.4 shows that it has some promising features. The problem,
however, is that the approximate apparent horizon calculated in the way given
in Sec.II.3 is not unique.

We have obtained an approximate apparent horizon that has some promising
features. This encouraging result has been obtained by using the null vectors
of Eq.o25 that has the vanishing shear tensor $\sigma_{\mu\nu}$ in the leading
order. The null congruence used for the calculation of an approximate horizon
being more algebraically special makes it eligible to be called as the
geometric trapped surface. The null vectors with vanishing shear tensor
$\sigma_{\mu\nu}$, at least in the leading order, could thus be a natural
choice for calculation of the trapped surface. The choice of null vectors
determines the choice of foliation.

We have also proposed a new way to characterize a black hole boundary.
Although our procedure for locating the black hole boundary requires solving
the full radial geodesic equation of the space-time in general, for some forms
of space-time it is just the solution of $g^{rr}=0$. Eq.n69 is an example of
such types of space-time. The validity of our approach for locating the black
hole boundary lies in the fact that such a surface exists in the black hole
solution of the Einstein equation (for example, the Schwarzschild and the Kerr
solution). The universality of our approach relies on whether such surface
where all the timelike radial geodesics approaching the null geodesics exists
in all black hole solutions. From our study on the black hole boundary, we
have noticed that the definition of the black hole boundary plays a crucial
role in our understanding of black holes. The new definition is therefore
exciting and is believed to meet this expectation.

###### Acknowledgements.

Pravin Kumar Dahal is supported by IMQRES.

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