J. Astrophys. Astr. (2021) 42:48 Ó Indian Academy of Sciences

https://doi.org/10.1007/s12036-021-09741-3Sadhana(0123456789().,-volV)FT3](0123456789().,-volV)

Trapped region in Kerr–Vaidya space–time

PRAVIN KUMAR DAHAL

Department of Physics and Astronomy, Macquarie University, Sydney, Australia. E-mail: [email protected]

MS received 18 November 2020; accepted 6 January 2021

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 does not exist in general for axisymmetric space–times. The reason being the surface at which the null tangent vectors are geodesics and the surface at which the expansion of such vectors vanishes do not coincide. The 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, seems to remove this non-uniqueness. We also propose a new definition of the boundary.

Keywords. —Kerr–Vaidya metric—black hole boundary—apparent horizon—geometric trapped surface—geometric horizon.

1. Introduction to define a black hole boundary such that the definition reproduces that answer. Such simple cases are station- The notion of the trapped surface was first introduced by ary space–times of spherical and axial symmetry for Penrose (1965), and the concept later emerged as a which all the horizons (event, apparent, killing, and quasi-local technique to characterize a black hole. They trapping horizons) coincide. The notion of trapped are seen as a replacement to the to define surfaces introduced in this way to define a black hole the black hole boundary, which depends on the global boundary poses no problem at all for stationary space– knowledge of space–time. The teleological nature times, even though trapped surfaces depend on the (Ashtekar & Krishnan 2004; Ashtekar & Galloway choice of foliations for these space–times (Krishnan 2005) and the unobservability (Visser 2014) of the 2014). Then comes the second stage of making use of event horizon have been discussed numerously in the the definition of the trapped surface to identify the black past. It has thus been regarded as unsuitable for char- hole boundary of the space–time for which the concept acterizing astrophysical black holes, which are dynamic of an event/killing horizon fails. The trapped surface as in nature and have to be studied by a finite-sized labo- a black hole boundary has celebrated its success in ratory for the finite time. However, even though the application to the general spherically symmetric space– trapped surfaces are quasi-local making them observ- times. The problem of foliation dependence, in that able in principle, they are not a well-defined quantity to case, can be eliminated by making the particular choice characterize black holes. The principal reason behind of foliation that respects the symmetry of space–time this is the foliation dependence of the trapped surface. (Faraoni et al. 2017). All spherically symmetric folia- As the choice of the foliation depends on the choice of tions give the same trapped surface for such null vectors, the trapped surface depends on the choice space–times. This problem, however, is more visible of null vectors (Faraoni et al. 2017). and profound if we move to the more general case of Defining a black hole boundary has been like a axially symmetric space–times, and there do not reverse engineering process. We somehow know the exist simple solutions to this problem in axial answer for some simple cases based on which we want symmetry. 48 Page 2 of 12 J. Astrophys. Astr. (2021) 42:48

This article is the introduction of the trapped sur- describing black holes evolving in time. In principle, face and its application to the axially symmetric it is not possible to define a horizon that can be space–time by taking the reference of the Kerr–Vai- detected by completely local measurements. The dya metric. We introduce trapped surfaces, define reason behind this is rooted in the equivalence prin- various types of trapped surfaces, and discuss, in brief, ciple, which implies an impossibility of measuring the notion of the geometric horizon as a black hole space–time curvature by local measurements. boundary in Section 1.1. Recently, the geometric In the orientable space–time manifold M, let us horizon has been considered as an alternative over the take a future-directed field of null vectors la and na a a trapped surface in locating the black hole boundary satisfying l na ¼À1. We assume that l is outgoing in because of its invariant characterization, quasi-local the sense it is pointed away from the trapped region, nature, and foliation independence. However, its and na is ingoing. The divergence (or the conver- applicability has been demonstrated only for limited gence) of the field of null congruence ll is given by its l l classes of space–times until now. In Section 1.2,we covariant derivative l;l. If the vector field l is arbi- present some of the properties of the trapped surfaces trarily parameterized, then we affinely parameterize it based on their metric, and in Section 1.3, we introduce before the calculation of divergence, and this intro- the slowly evolving horizon. The concept of a slowly m l duces the additional correction term j ¼Ànll l;m, evolving horizon is useful in the sense that it does not called the surface gravity. Thus, the divergence (or mingle with the problem of foliation dependence, convergence) of the affinely parameterized null con- provided that the condition for its existence satisfies. gruence is given as (Faraoni 2015) Section 2 describes the application of the trapped h ll n lmll and h nl n lmnl ; 1 surface to the axisymmetric space–time, and for this, l ¼ ;l þ l ;m n ¼ ;l þ l ;m ð Þ we chose one of the simplest examples of Kerr–Vai- where hl denotes the outgoing expansion and hn dya geometry in advanced coordinates. We also dis- denotes the ingoing expansion (for details on the cuss our results in the context of the geometric expansion of congruence see textbook by Poisson horizon. Finally, in Section 3, we propose a new black (2004)). Among these, the condition for the vanishing hole boundary. of outgoing expansion hl ¼ 0 is pivotal for the defi- nition of trapped surfaces. A surface is said to be trapped if hl \ 0 and hn \ 0. Thus, on the trapped 1.1 Trapped surfaces and geometrical horizons surface, both the ingoing and outgoing null geodesics are converging. A marginally outer trapped surface We begin here with the notion of an event horizon, (MOTS) has hl ¼ 0 and hn \ 0. This surface outlines which is believed to be the true boundary of a black the boundary from which the outgoing null geodesics hole by some authors despite its teleological nature. start to converge. The three-dimensional surface, Asymptotically flat space–time containing black which can be foliated entirely by the MOTS, is called holes can be separated into two regions: (1) the region a marginally outer trapped tube (MOTT). This MOTT from which all null curves can reach the future null is defined as the future apparent horizon in recent infinity (where future-directed null curves end in an literature (Barbado et al. 2016; Levi & Ori 2016), and infinite time in the Minkowskian space–time) and (2) we will adopt this definition here. The definition of an the region from which no null curve reaches the future apparent horizon and, in general, of the trapped sur- null infinity. A boundary separating these two regions faces varies over the literature. The Hayward’s trap- of space–time is defined as an event horizon. This ping horizon is defined as an apparent horizon l definition of the event horizon depends on the com- satisfying n ðhlÞ;l 6¼ 0 (Hayward 1994). This addi- plete knowledge of future-directed null curves, which tional condition imposed implies that the space–time is impossible to achieve in a finite time making it surface is trapped inside the future outer trapping physically unobservable (Visser 2014). This global horizon and normal outside it. Similarly, a space like nature of the event horizon also makes it a less useful apparent horizon is defined as a dynamical horizon entity for dynamical space–times. Similarly, the defi- (Ashtekar & Krishnan 2004). nition of the event horizon could not be applied to The definitions of trapped surfaces given here pro- space–times that are not asymptotically flat. Thus, the vide an easy way of identifying black holes and these notion of the trapped surfaces that depends on the are just by the calculation of null expansions hl and hn. quasi-local nature of space–time is handy in We thus do not have to wait until the end of time to J. Astrophys. Astr. (2021) 42:48 Page 3 of 12 48 know the formation of an event horizon to identify scalar polynomial curvature invariants could be cal- black holes. Trapped surfaces indeed are convenient culated for stationary space–times and space–times and unique for the numerical and analytical study of conformal to stationary space–times (McNutt 2017). spherically symmetric space–times (for spherically At the event horizon of such space–times, it has been symmetric foliation, which is an obvious choice of shown that the squared norm of the wedge products of foliation because of the symmetry of space–time). the n-linearly independent gradients of scalar poly- However, even for the trapped surfaces not being nomial curvature invariants vanishes, n being the local entirely local entities, their identification depends on co-homogeneity of space–time (Page & Shoom 2015; the closed hypersurface R of the space–time whose McNutt & Page 2017). This method of locating the foliation spans the whole space–time. Expansions black hole boundary is restricted to the stationary should be calculated over all the points of the closed space–times as this method, in principle, is looking for hypersurface R, not just at a single point for their the null surface where the magnitude of some partic- identification. Hence, trapped surfaces are quasi-local ular combinations of invariants vanishes. However, in nature and depend on the complete knowledge of the horizons of dynamical space–times are not null in the hypersurface R, if not of space–time. general. The extension of this method to the dynamic As mentioned above, with any choice of spherically space–times conformal to the stationary space–times symmetric foliations, a trapped surface in spherically lies in the fact that the event horizon is invariant under symmetric space–time is unique. However, the com- conformal transformation. plication starts to appear when we consider foliations Another procedure for calculating the geometric that are not spherically symmetric, even to calculate horizon is by finding the zeros of the certain combina- the trapped surface in spherically symmetric space– tions of Cartan invariants (McNutt et al. 2018; Coley times. The foliation dependence of the trapped surface et al. 2019). This method is exactly similar, in princi- in Vaidya space–time has been demonstrated by ple, to the scalar polynomial invariants for finding the numerical calculations with some special choice of geometric horizon. Thus, this method is also limited to non-spherically symmetric foliation in Schnetter & stationary space–times and any space–times confor- Krishnan (2006) and Nielsen et al. (2011). The dis- mal to them. However, Cartan invariants are regarded cussion of the foliation dependence of the trapped as the improvement over the scalar polynomial surface in Schwarzschild and Vaidya space–time can invariants as this method involves the linear also be found in the review by Krishnan (2014). We combinations of the components of a curvature tensor, will describe the trapped surface in Kerr–Vaidya and they are possible to construct from Cartan space–time and its foliation dependence. invariants. Another type of horizon that is commonly used to The third procedure for the calculation of the geo- characterize the black hole boundary is the notion of metric horizon is similar to the calculation of the geometric horizon (Coley et al. 2017; Coley & McNutt trapped surfaces, and thus the black hole boundary 2018). The existence of the geometric horizon is calculated using this procedure can be called a geo- assured by the conjecture that the non-stationary black metric trapped surface. The calculation is based on the holes contain the hypersurface, which is more alge- conjecture that the hypersurface constituting geomet- braically special (Coley & McNutt 2018; McNutt & ric trapped surface is more algebraically special (Page Coley 2018) and such a hypersurface is identified as & Shoom 2015; McNutt & Page 2017). The procedure the geometric horizon. This method of locating the is to look for the geometrically preferred outgoing null black hole does not suffer the problem of foliation vectors and then find the hypersurface where the dependence. The property of being more special can expansion of such vectors vanishes. The null vector is be quantified invariantly by making particular com- called the geometrically preferred null vector because binations of curvature invariants zero on the horizon. the congruence of the null vectors thus chosen is such Three different sets of invariants derived from cur- that the covariant derivatives of the curvature tensor vature tensors are commonly used in the literature to are more algebraically special there (Coley et al. calculate the geometric horizon in general. The first 2017; McNutt & Coley 2018). The applicability of approach relies on finding the suitable Killing vector this method extends beyond the above two procedures field that becomes the null generator on the horizon. of finding the geometric horizon as this method can be Then, the hypersurface where the square norm of the extended to the dynamical space–times also. How- Killing vector vanishes is the event horizon. If it is not ever, until now, it has not been used to locate the possible to find such a field of Killing vector, then black hole boundary of any axisymmetric dynamical 48 Page 4 of 12 J. Astrophys. Astr. (2021) 42:48 space–times. The equivalence/correspondence the horizon is receding and C ¼ 0or1 implies that between these three procedures for calculating the the horizon is isolated. Thus, the space like apparent geometrical horizon lies in the fact that they all horizon is expanding and the time like apparent involve the calculation of invariants derived from the horizon is receding, while isolated apparent horizons curvature tensor (see, e.g., McNutt & Coley 2018; are null. Coley et al. 2019 for details).

1.3 Slowly evolving horizon 1.2 Properties of trapped surfaces We are working in the semi-classical domain for Booth et al. (2006) and Sherif et al. (2019) presented a which the quasi-static evaporation law given in Levi simple formalism to determine the metric signature of & Ori (1995) and Brout et al. (2016) is valid. In this an apparent horizon and demonstrated that for spheri- domain, an apparent horizon of the Kerr–Vaidya cally symmetric space–times. We will here discuss it for metric can be approximated as a slowly evolving the purpose to determine some properties of the MOTT horizon. So, we present some characteristics of the of the Kerr–Vaidya geometry. Let vl be the vector field slowly evolving horizon here. An almost-isolated tangential to the MOTT and orthogonal to the MOTSs trapping horizon is defined as a slowly evolving that foliate the MOTT. Also, assume that the vector field horizon and the invariant characterizations of such a vl generates the flow that preserves the foliation. Then horizon are made by Booth & Fairhurst (2004, 2007). We have mentioned above that an apparent horizon is Lvv ¼ f ðvÞ, where Lv denotes the Lie derivative along v and f(v) denotes some function on the foliation v.We null and isolated if C ¼ 0. It thus seems that a slowly similarly denote by vl the vector field normal to the evolving horizon can be characterized by the space- MOTT, which satisfies v vl ¼ 0. If we assume both vl like or time-like horizon for which C is small. C, l l and vl are future directed, then one of them is space like however, can be arbitrarily varied by rescaling n ! l 2 and another is time like. Now, in terms of the null n =a and C ! C=a , for some a. The slowly evolving vectors ll and nl given above, we can write horizon should thus be identified in a scaling-inde- l l l pendent manner and for this, we write the area evo- v ¼ l À Cn ; ð2Þ lution law of Equation (6)as l l l rffiffiffiffi  v ¼ l þ Cn ; ð3Þ C Lvq~ ¼Àkvk hnq~ ð7Þ for some function C. Along vl, the Lie derivative of 2 an outgoing expansion is zero thereby implying such that the term in parentheses is independent of Lvhl ¼Llhl À CLnhl ¼ 0 ð4Þ rescaling. This term gives an invariant area evolution and if this term is small, then the horizon is slowly or evolving. Thus, one of the conditions proposed by Llhl Booth & Fairhurst (2004) for the horizon to be slowly C ¼ : ð5Þ Lnhl evolving over some MOTS in a given foliation is pffiffiffiffi  Equation (5) can be used to calculate C and as Chn\ ; ð8Þ l R C / v vl, the metric signature of the MOTT is determined by the sign of C. C [ 0 means that an for arbitrary small ; where R is the areal radius of the apparent horizon is space like, C\0 means that an MOTS. Another condition is the choice of scaling of apparent horizon is time like, and C ¼ 0or1 means the null vectors, such that, kvk. There are other that an apparent horizon is null. The sign of C also conditions given in Booth & Fairhurst (2004, 2007) determines whether an apparent horizon is expanding requiring that the horizon evolves smoothly over a or contracting. To see this, let us take an area element long period of time. These conditions will be auto- of the two surface be q~ and evaluate its Lie derivative matically satisfied for the slowly evolving Kerr–Vai- along vl: dya metric where the horizon offsets from the isolated horizon, at most, by an order of Mv. The reason behind Lvq~ ¼ÀChnq~: ð6Þ this is every parameter calculated at the horizon, in As hn\0 on the trapped surface, C [ 0 implies that this case, differs from the stationary Kerr metric by an an apparent horizon is expanding, C\0 implies that order of Mv. J. Astrophys. Astr. (2021) 42:48 Page 5 of 12 48

2. Trapped region in advanced Kerr–Vaidya and this marks the boundary of the trapped region. space–time Senovilla & Torres (2015) have argued that the solution for hl ¼ 0 does not exist in general for the 2.1 Some examples v ¼ constant foliation considered. This is indeed true and the reason behind this is that the null We have the line element for Kerr–Vaidya space–time vectors are geodesic only on the D ¼ 0 surface and in advanced coordinates ðv; r; h; /~Þ given as (Sen- on this surface, hl ¼ 0 only when Mv ¼ 0. They ovilla & Torres 2015) have thus considered the surface of intersection   between D ¼ 0 and Mv ¼ 0; where rg ¼ M Æ 2 2MðvÞr 2 2 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ds ¼À 1 À dv þ 2dvdr þ q dh 2 2 q2 M À a are the two real solutions of hl ¼ 0. For dynamical space–time considered, Mv ¼ 0 is not 4aMðvÞr sin2 h À d/~dv À 2a sin2 hd/~dr true in general and might occur only for some q2 v ¼ v0. Thus, the v, constant surface does not foliate 2 2 2 2 2 ðr þ a Þ À a D sin h 2 an apparent horizon in general. þ sin2 hd/~ ; ð9Þ q2 The solution of hl ¼ 0 is simpler and more evident if we calculate the expansion scalars for the null where q2 ¼ r2 þ a2 cos2 h and D ¼ r2 À 2MðvÞr þ a2. geodesics with the following tangent vectors: This metric reduces to the familiar Kerr metric for v M v   some advanced time if ( ) is constant (Kerr 1963). D 2q2 The trapped region associated with this metric can be l ¼ À 1; ; 0; a sin2 h ; l 2q2 D determined by the calculation of the expansion scalars 2 hl and hn. To find the trapped surface for this metric, nl ¼ðÀ1; 0; 0; a sin hÞ: ð13Þ we use the future-directed null vectors of the form l satisfies the geodesic equation lll ¼ kl , for some (Senovilla & Torres 2015) l m;l m 2 parameter k defining the geodesic curve only when q 2 2 ll ¼ ðÀD; r þ a þ X; 0; 0Þ; Mv ¼ 0. The expansions using these pairs of null 2X2   vectors are as follows: r2 þ a2 À X n ¼ À 1; ; 0; 0 ; ð10Þ l D qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rD 2r hl ¼ 4 ; hn ¼À 2 : ð14Þ where X ¼ ðr2 þ a2Þq2 þ 2MðvÞra2 sin2 h. These q q null vectors also being normalized satisfy the relations We thus get hl ¼ 0 when D ¼ 0. Hence, the MOTS should be the intersection between D ¼ 0 and M ¼ 0 l ll ¼ n nl ¼ 0 and l nl ¼À1: ð11Þ v l l l leading us again to the conclusion that v, constant Moreover, the calculation of the geodesic equation surface does not foliate an apparent horizon in l l lm;l ¼ klm for some parameter k shows that the vector general. ll is tangent to the null geodesic only on D ¼ 0 sur- face. We now use Equation (1) to calculate the expansion scalars using these null vectors. The expansions thus obtained are given as 2.2 MOTSs are not unique

1 3 2 2 2 2 2 hl ¼ ðDð2r þ 2a r À a r sin h þ a M sin hÞ Here, we will first show that depending on the choice 2X3 of the null vectors, the MOTS (the surface for which þ a2rM sin2 hðr2 þ a2 þ XÞÞ; v hl ¼ 0) can extend all the way from the singularity to 1 the infinity. For this, we take the general null vector of h ¼À ðDð2r3 þ 2a2r À a2r sin2 h þ a2M sin2 hÞ n 2q2DX the form: 2 2 2 2 2 þ a rMv sin hðr þ a À XÞÞ; ð12Þ ll ¼ðÀa; 1; b; cÞ; nl ¼ðÀ1; 0; 0; a sin hÞ; ð15Þ where Mv ¼ oM=ov. Now, an MOTS is the two where a ¼ aðv; r; h; MðvÞÞ. The outgoing vector ll will surface where outgoing null expansion (hl) vanishes be future directed for a [ 0 and its null implies that 48 Page 6 of 12 J. Astrophys. Astr. (2021) 42:48

l lll ¼ 0. This allows us to express c in terms of a and b as Without the risk of losing the generality of the result, we will assume b ¼ 0 for the rest of the analysis. This c ¼ðÀa þ aa qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi gives Æ ð1 À 2aÞa2 ÀðD À 2aða2 þ r2Þþb2Þ csc2 hÞ sin2 h: oa r þ q2 [ 0 for M [ 0; ð16Þ oM v o With this value of c, the vectors l and n satisfy all 2 a l l r þ q \0 for Mv\0: ð22Þ the relations given in Equation (11). We can now oM calculate the outgoing expansion hl and its value is The lower/upper bound of this inequality gives a ¼ obtained as follows: D=2q2 with the choice of ða2 þ r2Þ=2asanintegration hl ¼ hKerr constant and this is the null vector given in Equation (13)   to obtain D ¼ 0 as the horizon. In general, the solution 2 oa ob 2aMv r þ q oM À b oM for hl ¼ 0 from Equation (19)isgivenas qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ;   þ 2aM oa q2 ð1 À 2aÞa2 ÀðD À 2aða2 þ r2Þþb2Þ csc2 h D þ v r þ q2 ¼ 0 ð23Þ g oM ð17Þ or where h is an expression containing terms that do     Kerr o not have Mv. This corresponds to the outgoing 2aMv a 2 aMv 1 þ o r À 2 M À r expansion for the Kerr metric with M, constant. g M  g However, for the Kerr geometry, we should have 2aM oa þ a2 1 þ v cos2 h ¼ 0: ð24Þ Dgðt; r; hÞ g oM hKerr ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; 2 2 2 2 2 2 q ð1 À 2aÞa ÀðD À 2aða þ r Þþb Þ csc h The solution to this equation will be diverging as ð18Þ oa=oM !Àðg=2aMvÞ, which is the reasonable possibility. We will here give an example to where gðt; r; hÞ is an arbitrary function. However, we support this argument. But, let us first assume should have gðt; r; hÞ [ 0 to ensure that hKerr [ 0 for Mvv  0 (only for this example), which is the D [ 0. We thus obtain reasonable approximation in the semi-classical 2aMv 2 oa ob gðD þ g ðr þ q oM À b oMÞÞ limit. We then take hl ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : q2 ð1 À 2aÞa2 ÀðD À 2aða2 þ r2Þþb2Þ csc2 h a2Q2 þ M2 csc2 h a ¼ 2 2 ; ð25Þ ð19Þ 2a Q Q Q v Again, to ensure that h [ 0 outside the horizon, we where ¼ ð Þ is some arbitrary function. With this l a should have choice of , we obtain   qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi o o 2 2 2 2 2 2 2 2 2 2aMv 2 a b M À a Q sin h þ 2 MQ ðq M þ 2a rQ sin hÞ D þ r þ q À b [ 0; ð20Þ c ¼ : g oM oM 2aQ2 as g [ 0. We will analyse the two cases of D [ 0 ð26Þ (for the possibility of horizon lying outside pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi We thus have the pair of null vectors given as 2 2 M þ M À a ) and D\0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (for the possibility of  p a2Q2 þ M2 csc2 h horizon lying inside M þ M2 À a2) individually. l ¼À ; 1; 0; l 2a2Q2 First, let us consider the case of D [ 0 at the qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 horizon. To achieve this, that is, to get h 0, we l ¼ M2 À a2Q2 sin2 h þ 2 MQ2ðq2M þ 2a2rQ2 sin2 hÞ should have from Equation (19) A; o o 2aQ2 2 a b r þ q À b [ 0 for Mv\0 and g [ 0; oM oM 2 nl ¼ðÀ1; 0; 0; a sin hÞ: ð27Þ oa ob r þ q2 À b \0 for M [ 0 and g [ 0: oM oM v They satisfy the relations given in Equation (11) ð21Þ and thus can be used for the calculation of J. Astrophys. Astr. (2021) 42:48 Page 7 of 12 48 expansion scalars. The value of hl is obtained as whether space–time inside it is trapped and outside it follows: is normal. So, the problem of locating the trapping

  1 MQðrM þ a2Q2 sin2 hÞþQM ðq2M þ a2rQ2 sin2 hÞÀq2M2Q h ¼ r À M þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiv v ; l q2 ð28Þ MQ4ðq2M þ 2a2rQ2 sin2 hÞ

where Qv ¼ oQ=ov. The solution of hl ¼ 0foran horizon extends further beyond the complexity of apparent horizon will be convenient when identifying the apparent horizon. 2 2 Q ! Mv , for which case, Qv  0. The solution for r in that regime would be

2.3 An approximate apparent horizon 2MMv r ¼ : ð29Þ M2 À Q2 v Besides the problem of foliation dependence, there is 2 2 another more serious problem in general dynamical We thus have diverging r ¼ rg as Q ! Mv . Now, we consider the case of D\0 at the horizon. axially symmetric space–times in the calculation of Then, to ensure that hl ¼ 0, we should have the trapped surfaces. In general, for time-dependent o axial symmetry, it is observed that the future-directed 2 a r þ q o \0 for Mv\0; outgoing null vector is not tangent to the geodesic M everywhere. This is because, the general future-di- oa r þ q2 [ 0 for M [ 0: ð30Þ rected outgoing null vector has three independent oM v parameters and can be written as ll ¼ðÀa; 1; b; cÞ, Again, the upper/lower bound of this inequality gives where a 6¼ 0. However, there are four constraints a ¼ D=2q2 with the choice of ða2 þ r2Þ=2asan altogether to satisfy by this vector: three independent l integration constant, which is the null vector given in constraints from the geodesic equation l lm;l ¼ klm, for l Equation (13). The solution for hl ¼ 0 from Equa- some parameter k (one of the equation l lr;l ¼ klr is tion (19) is given as satisfied identically) and one constraint from the null     condition lll ¼ 0. There are thus four equations to 2aM oa aM l 1 þ v r2 À 2 M À v r satisfy by three variables and this will hold only on g oM g   some region/hypersurface of the space–time. This is in 2aM oa contrast to the situation in both the time-dependent þ a2 1 þ v cos2 h ¼ 0: ð31Þ g oM spherical symmetry and the stationary axial symme- try, where an arbitrary null vector can be made geo- Both for Mv\0 and Mv [ 0, a proper choice of the desic by the proper choice of a parameter called an value of oa=oM could make the value of r ¼ rg recede affine parameter defining the curve. Thus, an outgoing up to the singularity. This can be seen by minimizing r null vector is a tangent to the geodesic only on some with respect to oa=oM in this equation, whose solution surface in axial symmetry in general, and on that 2 2 2 is r þ a cos h ¼ 0. Thus, depending on the choice surface, the outgoing expansion hl does not vanish of null vectors (which in fact determines the choice of usually. A way to address this problem is to calculate the foliation), the value of the MOTS can extend from an approximate apparent horizon. the singularity to the infinity. As we have seen in the To approximate an apparent horizon for the Kerr– previous two examples, all the MOTSs do not foliate Vaidya geometry, let us take the null vectors of an apparent horizon. However, it is not clear which Equation (13) for which the solution of hl ¼ 0is MOTSs dynamically evolve to foliate the apparent clearly given by D ¼ 0. However, as concluded above, horizon and if the apparent horizon is unique the intersection of v is constant and D ¼ 0 surface (Ashtekar & Galloway 2005; Hawking & Ellis 2011). does not foliate the apparent horizon in general. The Moreover, to find the trapping horizon, we have to reason behind this is that the vectors of Equation (13) check each apparent horizon hypersurfaces to confirm are not tangent to the null geodesics in general. To see 48 Page 8 of 12 J. Astrophys. Astr. (2021) 42:48

l this, the geodesic deviation equation l lm;l À jlm for Again, solving the geodesic deviation equation at 2 2 some parameter j is given as follows: M ¼ða þ r þ f ðt; r; hÞMvÞ=2r, with the value of f a2r sin2 hM given by Equation (38), we get lll À jl ¼ v ; ð32Þ v;l v q4 ÀÁ M ðÞr2 À a2 k þ 4a2r2 sin2 h l l v l l À jl ¼ 0; ð33Þ l lv;l À jlv ¼ r;l r ÀÁ4rq4 l 2 l lh;l À jlh ¼ 0; ð34Þ þO Mv ; ð39Þ 2 2 2 lll jl 0; 40 l arða þ r Þ sin hMv r;l À r ¼ ð Þ l l/;l À jl/ ¼À : ð35Þ q4 2 2 ÀÁ l ðÞa À r Mvf 2 l lh;l À jlh ¼ þO M ; ð41Þ Thus, the null vectors are offset from being the geo- 2rq2 v desics by an order of M , which is obviously small in ÀÁ v ðÞa2 þ r2 M ðÞa2 À r2 k À 4a2r2 sin2 h the semi-classical limit as pointed out in Section 1.3. l v l l/;l À jl/ ¼ 4 So in the semi-classical region, we can approximate ÀÁ 4aq r 2 D ¼ 0 as an apparent horizon, and we will explain þO Mv ; ð42Þ below that this satisfies all the properties for being the ÀÁ 2 2 slowly evolving horizon. However, before that, we will where O Mv represents the terms of order Mv and present a technique that is similar to the perturbation higher. Now, requiring the geodesicÀÁ deviation of the null 2 expansion to approximate an apparent horizon. Possi- tangent vectors to be of at least O Mv we should have bly, the validity of this approximation method would be for all jM j\1 and not only for very small M . 4a2r2 sin2 h v v k ¼ and f ¼ 0: ð43Þ For this, we proceed forward by making a slight a2 À r2 modification on the null vectors of Equation (13)  Substituting this in Equation (38), we get D þ kðr; hÞM l ¼ À v ; 1; fðr; hÞM ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi l 2 v 2 q 2 2 4 2 2 2 4 4 4 4r Mv 2 2a sin h 2a cos h þ 2a r À ðÞa cos 2h þ a þ 2r a2Àr2 þ 1  f ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 aD sin h þ mðv; r; hÞ 2 2 4r Mv ða À rÞ ða þ rÞ a2Àr2 þ 1 2 ; 2q 4a2r2 sin2 h ¼ þOðÞMv : ð44Þ 2 a2 À r2 nl ¼ðÀ1; 0; 0; a sin hÞ: ð36Þ

l Thus, the solution of equation M ¼ The null condition l ll ¼ 0 gives 2 2  ða þ r þ f ðt; r; hÞMvÞ=2r with the value of f from Equation (44) gives the value of r. This r corresponds to m 2 h akM aq2 ¼ sin v À 2 the apparent horizon of the Kerr–Vaidya space–time up qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi to the first-order correction in Mv. Specifically, þ 2q2 a2 þ csc2 hkM À csc2 hf2M2 : ð37Þ v v pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r ¼ M þ M2 À a2  We now calculate the outgoing expansion hl at r ¼ rg pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi given by the solution of equation a2 sin2 h 2M2 À a2 þ 2M M2 À a2 2 2 þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi M : ð45Þ M ¼ða þ r þ f ðt; r; hÞMvÞ=2r, where f ðt; r; hÞ is p v ðÞM2 À a2 M þ M2 À a2 another arbitrary function. Assuming Mvv  0, we get from the solution of hl ¼ 0

ÀÁqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiÀÁ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiÀÁ  2 of ok 2 2 2 2 2 2 of ok 2q 2 oh À or csc hMv k À Mvf þ a þ 2f cot h csc hMv k À Mvf þ a À aMv or þ a or f ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiÀÁ þ k: 2 2 2 4r csc hMv k À Mvf þ a ð38Þ J. Astrophys. Astr. (2021) 42:48 Page 9 of 12 48

For the Kerr–Vaidya metric in advanced coordinates, times, we should choose the foliation that respects the when Mv\0, the apparent horizon is ellipsoid flat- symmetry of the space–time (Faraoni et al. 2017). For tened at the equator, provided that a is small. Simi- this spherical foliation of choice, the null geodesic larly, when Mv [ 0, for small a, the apparent horizon congruence normal to this surface is radial and shear is ellipsoid flattened at the pole. It can be shown that free. Some choices of non-spherical foliations, even in this approximate horizon is not the unique approxi- stationary spherically symmetric space–times, do not mate apparent horizon. There exist other surfaces contain trapped surfaces (Krishnan 2014). Obviously, where the null vectors satisfy geodesic equations up to choosing an arbitrary axially symmetric foliation does the second order in Mv and have the vanishing out- not work for axially symmetric space–times. Shear- going expansion hl (e.g. the exact procedures for the free geodesic congruence is the axially symmetric calculation of an approximate horizon applied to the analogy of radial geodesic in the spherical symmetric null vectors of Equation (10) give different surfaces as space–time (Chandrasekhar 1985). Thus, the foliation an approximate horizon). However, these surfaces associated with the shear-free geodesic in axially differ from each other and the stationary black hole symmetric space–time complements the symmetry horizon, at most, by an order of Mv. respecting foliation of the spherically symmetric The MOTSs foliating the apparent horizon is unique space–time. (Ashtekar & Galloway 2005). So, if an apparent horizon Furthermore, our calculation of an approximate is known a priori, then we can infer that this has been apparent horizon also follows from an assumption of the foliated by a dynamical evolution of the unique MOTS. validity of the geometric horizon conjecture stated in This is unlike the case of isolated horizons where foli- Section 1.1. The corollary of the Goldberg & Sachs ations are freely deformable. However, we could not (2009) theorem given in (Chandrasekhar (1985), p. 63) say anything about the uniqueness of the apparent implies that if the congruence formed by the two prin- horizon itself. (Remark: the assumption of the finite cipal null directions ll and nl are geodesic and shear time formation of an apparent horizon constrains its free then space–time is algebraically special. As the null location. However, it is not sufficient constraint to give geodesics of Equation (36) for the calculation of an the unique apparent horizon.) approximate horizon is shear free, at least in first-order The approximate apparent horizon calculated here has approximation in Mv, the congruence formed by them is promising features, which we list below, and this might more algebraically special. This implies that the provide a clue for the appropriate choice of null vectors approximate apparent horizon is also an approximate for the calculation of the trapped surface. The null vec- geometric-trapped surface. Moreover, the procedure for tors given in Equation (36) have the vanishing shear the calculation of an approximate apparent horizon is tensor in the leading order approximation in Mv.The the generalization of the third procedure for the calcu- shear tensor rlm can be calculated by using the relation lation of the geometric horizon, explained above, to the h more general case of axial symmetry. r ¼ B~ À h ; ð46Þ lm lm 2 lm ~ 1 where hab ¼ gab þ lanb þ nalb and Bab ¼ 2 hhab (see Poisson (2004) for details). Now, at the apparent 2.4 Features of the apparent horizon horizon, hl ¼ 0. The direct calculation using Equa- tion (46) gives We now take the pair of null vectors given in Equa- tion (36) with the value of respective parameters m 2 2 2 ÀÁ l lm 4a rg cos h 2 given in Equation (37) and k and l given in Equa- rlmrn ¼ 4 2 2 Mv þO Mv ; ð47Þ q ða À rgÞ tion (43). The direct calculation using these null vectors yields which is already the first order in M . The choice of v 2 ÀÁ null vectors with a vanishing shear tensor, at least, in l 2r Mv 2 Llhl ¼ l ðhlÞ;l ¼À þO Mv ; the leading order approximation is the natural choice q4 of null vectors (and hence the natural choice of foli- a2 À r2 L h ¼ nlðh Þ ¼ ation) for the calculation of the trapped surface. This n l l ;l q4 can be explained as follows. 2 2 2 2 2 ÀÁ 4a r ðÞa þ r sin h 2 It is undoubtedly true that for the calculation of À Mv þO M : ð48Þ 4 2 2 2 v trapped surfaces in spherically symmetric space– q ðÞa À r 48 Page 10 of 12 J. Astrophys. Astr. (2021) 42:48

We substitute this in Equation (5) to get these two classes of space–times. This is because of the two fundamental reasons: 2r2M ÀÁ C ¼ v þO M2 : ð49Þ r2 À a2 v • The problem of the foliation dependence of trapped surfaces: even in the simplest class of From this equation, M \0 gives C\0 thereby v space–times such as Schwarzschild, Vaidya implying that the apparent horizon of the Kerr–Vaidya (1951), the trapped surface depends on the line element in advanced coordinates is time like and choice of foliation on which the family of null receding in M \0 domain. These are important fea- v congruence is orthogonal. tures, as the M \0 regime of the Kerr–Vaidya line v • An arbitrary outgoing null vector l is not element in advanced coordinate is believed to be the l geodesic everywhere in general axisymmetric evaporating black hole solution of the Einstein equa- space–time. In general, the region where l is tion. Similarly, for M [ 0, the apparent horizon is l v geodesic is not where the outgoing null expan- space like and advancing. sion h vanishes. Because of this, even an exact Again, for the null vectors of Equation (36), we get l location of a geometrical trapped surface, which h ¼À2r=q2. We thus have the condition for the n is a foliation-independent entity is not possible. slowly evolving horizon sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi In an attempt to address these problems, we present an pffiffiffiffiffiffi 4 8r jMvj  alternative way of characterization of a black hole jCjhn ¼ \ ; ð50Þ q4ðr2 À a2Þ r2 þ a2 boundary. The common procedure for locating a black hole boundary is by looking for some identically non- that holds only whenpffiffiffiffiffiffiffiffiffi the space–time is slowly rotating, zero invariant quantity whose value vanishes on the that is, a  r and jMvj OðÞ. However, in the semi- horizon. For example, the expansion scalar of an classical limit, for a slowly rotating case near the outgoing null geodesic vanishes at the trapped surface. 0 À3 À4 apparent horizon rg, jMvjrg  10 À10 (Brout Similarly, the norm of the Killing vector field or some et al. 1995; Levi & Ori 2016). This gives   0:01  1. particular combinations of the scalar polynomial cur- Thus, the apparent horizon of an advanced Kerr–Vaidya vature invariants or Cartan invariants vanishes at the line element, in the semi-classical domain, satisfies the geometrical horizon. So, analogously, we are also condition to be called as the slowly evolving horizon looking for an invariant quantity that is identically provided a  r. zero at the horizon and non-zero elsewhere. However, Similarly, the calculation using the pair of null for us, that quantity should be foliation/observer-in- vectors of Equation (13) yields C  Mv for a  r. dependent and applicable to the more general case of This again leads to the conclusion that the surface dynamical axial symmetry. Such a quantity would be r D ffiffiffiffiffiffi¼ 0 can be called the slowly evolving horizon when an arbitrary radial trajectory l , which becomes null on p l h / Mv  . Thus, D ¼ 0 is the slowly evolving horizon the horizon, that is, l ll ¼ 0, given l and l are null of the Kerr–Vaidya metric in advanced coordinates (which can be constructed by choice). provided a  r. To demonstrate that all the radial trajectories indeed become null at the horizon, we here give an example of the general spherically symmetric space–time 3. Possible characterization of a black hole h t;r 1 2 boundary ds2 ¼Àe2 ð Þf ðt; rÞdt2 þ dr2 þ r2dX : ð51Þ f ðt; rÞ To begin with, we first summarize the complications Now, for this space–time, the equation of motion in we encounter in identifying trapped surfaces as a the h-direction is black hole boundary. Identification of trapped surfaces  d oL d oL d/ 2 as a black hole boundary is straightforward and poses ¼ ðr2h_Þ¼À ¼ r2 sin h cos h ; no problem in stationary space–times (where the event ds oh_ ds oh ds horizon can also be undoubtedly located) and in ð52Þ dynamical spherically symmetric space–times (where _ the location of the event horizon is problematic). where h ¼ dh=ds. So, if we initially choose h ¼ p=2 However, trapped surfaces turned out to be the ill- and h_ ¼ 0; then h€ ¼ 0. This implies that the geodesic defined concept when we try to explore them beyond motion can be described in an invariant plane and we J. Astrophys. Astr. (2021) 42:48 Page 11 of 12 48

l choose that plane to be the equator h ¼ p=2. Thus, the trajectory, it should satisfy l ll ¼À1 and this gives vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi radial equation of motion for the space–time of u ÀÁ ! 2 u 2rM eh a sin2 hd t D 1 À 2h 2 2 2 2 2 2 Equation (51) can be written as at ¼ À À þ e À q c À ðÞa þ r sin hd : q2 b2 q2 L2 r_2 ¼Àdf ðt; rÞþðehðt;rÞf ðt; rÞt_Þ2 À f ðt; rÞ; ð53Þ ð56Þ r2 Now, this time like radial trajectory coincides identi- where d ¼ 0 for null geodesic, d ¼ 1 for time like cally with the null trajectory given by 2 geodesic and L ¼ r d/=ds is a constant. As t_does not vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u ÀÁ ! u 2 2 depend on the geodesic being null or time like and as t D 2rM eh À a sin hd a ¼ À þ e2h À q2c2 À ðÞa2 þ r2 sin2 hd2 ; L is arbitrary, the only surface where the radial tra- n q2 q2 jectory r_ is always null is f ðt; rÞ¼0. So, in dynamical ð57Þ spherically symmetric space–times, all the radial tra- jectories become null on some hypersurface, and this on the surface D ¼ 0 irrespective of the form of the three surface is uniquely characterized as the black parameters eh, c and d assuming that the normalization hole boundary by our prescription. Also, looking at factor b 6¼ 0 there (this is expected otherwise, the the geodesic equations for the Kerr space–time given trajectory is identically zero at D ¼ 0. Thus, for the in Chandrasekhar (1985), we can see that all of the general form of the axisymmetric metric given by time like radial geodesics approach null geodesic on Equation (54), adopting the definition presented here the r2 þ a2 À 2rM ¼ 0 surface. This surface is, in fact, gives grr ¼ 0 as the black hole boundary. both the event and the apparent horizon of the Kerr space–time. 4. Discussions and conclusion Hence, it might be reasonable to characterize a black hole boundary as the hypersurface where It seems that the quasi-local measures for determining every time like radial geodesics approaches null a black hole boundary are not suitable for the general geodesics. This is the asymptotic three surface, and axisymmetric space–times. In a spherical symmetry, the motivation for this is the classical picture of the there exists a preferred choice of foliation, which black hole as an asymptotic state of the gravita- obeys the symmetry of the space–time and gives a tional collapse. For most of the space–times, the unique MOTS. However, such a preferred choice of black hole boundary in this characterization is given foliations cannot be made in axisymmetric space– by the solution of grr ¼ 0. Some space–times, such times. We have shown that for axisymmetric space– as Vaidya, has grr ¼ 0 identically and non-zero gtr. times, MOTS is not unique and lies anywhere from Coordinate transformation is possible for such the singularity to the infinity depending on the choice space–times to make gtr ¼ 0, and then the solution of the null vectors. Although all these MOTSs do not of grr ¼ 0 gives the black hole boundary (otherwise, foliate apparent horizons, many of them could. the full geodesic equation should be solved). However, there is no preferred choice of MOTS We now take an example of the general axisym- foliating an apparent horizon that acts as a true black metric space–time with two parameters hole boundary.   Another problem in the calculation of an apparent 2Mr 4ehaMrsin2 h q2 2 2h 2 À dr2 horizon as explained in Section 2.2 is that the outgo- ds ¼e 1À 2 dt þ 2 dtd/ q q D ing null vectors are not geodesic everywhere, in general, in axisymmetric space–times. At least, for ðr2 þa2Þ2 Àa2Dsin2 h Kerr–Vaidya line element, it is found that the surface Àq2dh2 À sin2 hd/2; ð54Þ q2 where hl ¼ 0 does not coincide with the hypersurface where the null vectors are geodesic. We expect the where h¼hðt;r;hÞ and M¼Mðt;r;hÞ. To calculate the same situation to occur in most of the general boundary of the black hole for this space–time, we axisymmetric space–times. If this is the case, then the assume the trajectory of the form calculation of the trapped surface as a black hole boundary does not even make a sense for axisym- ll ¼ bðÞ1; a; c; d ; ð55Þ metric space–times. One way to get out of this prob- where b, a, c, and d are the functions of t, r, and h, lem was the calculation of an approximate apparent respectively. Now, assuming ll to be a time like horizon, explained in Section 2.3. The approximate 48 Page 12 of 12 J. Astrophys. Astr. (2021) 42:48 horizon exists only for small Mv, and when it exists, Booth I., Fairhurst S. 2004, Phys. Rev. Lett., 92, 011102 Section 2.4 shows that it has some promising features. Booth I., Fairhurst S. 2007, Phys. Rev. D 75, 084019 The problem, however, is that the approximate Booth I., Brits L., Gonzalez J. A., Van Den Broeck C. apparent horizon calculated in the way given in Sec- 2006, Class. Quantum Gravity, 23, 413 tion 2.3 is not unique. Brout R., Massar S., Parentani R., Spindel P. 1995, Phys. Rep., 260, 329 We have obtained an approximate apparent horizon Chandrasekhar S. 1985, The Mathematical Theory of Black that has some promising features. This encouraging Holes, Oxford University Press, Oxford result has been obtained by using the null vectors of Coley A. A., McNutt D. D., Shoom A. A. 2017, Phys. Lett. Equation (36) that has the vanishing shear tensor rlm B, 771, 131, 1710.08457 in the leading order. The null congruence used for the Coley A., McNutt D. 2018, Class. Quantum Gravity 35, calculation of an approximate horizon being more 025013, 1710.08773 algebraically special makes it eligible to be called the Coley A., Layden N., McNutt D. 2019, Gen. Relativ. geometric trapped surface. The null vectors with Gravit., 51, 164 vanishing shear tensor rlm, at least in the leading Faraoni V. 2015, Cosmological and Black Hole Apparent order, could thus be a natural choice for calculation of Horizons, vol. 907, Springer International Publishing, the trapped surface. The choice of null vectors deter- Switzerland mines the choice of foliation. Faraoni V., Ellis G. F. R., Firouzjaee J. T., Helou A., MuscoI.2017,Phys.Rev.D,95,024008, We have also proposed a new way to characterize a 1610.05822 black hole boundary. Although our procedure for Goldberg J. N., Sachs R. K. 2009, Gen. Relativ. Gravit., 41, locating the black hole boundary requires solving the 433 full radial geodesic equation of the space–time in gen- Hawking S., Ellis G. 2011, The Large Scale Structure of eral, for some forms of space–time it is just the solution Space–Time, Cambridge Monographs on Mathematical of grr ¼ 0. Equation (54) is an example of such types of Physics, Cambridge University Press, Cambridge space–times. The validity of our approach for locating Hayward S. A. 1994, Phys. Rev. D, 49, 6467, gr-qc/ the black hole boundary lies in the fact that such a 9303006. surface exists in the black hole solution of the Einstein Kerr R. P. 1963, Phys. Rev. Lett., 11, 237 equation (e.g. the Schwarzschild and the Kerr solution). Krishnan B. 2014, Springer Handbook of , The universality of our approach relies on whether such Springer, p. 527 surface where all the time like radial geodesics Levi A., Ori A. 2016, Phys. Rev. Lett., 117, 231101 McNutt D. D. 2017, Phys. Rev. D, 96, 104022 approaching the null geodesics exists in all black hole McNutt D. D., Page D. N. 2017, Phys. Rev. D, 95, solutions. From our study on the black hole boundary, 084044 we have noticed that the definition of the black hole McNutt D., Coley A. 2018, Phys. Rev. D, 98, 064043 boundary plays a crucial role in our understanding of McNutt D., MacCallum M., Gregoris D., et al. 2018, Gen. black holes. The new definition is therefore exciting and Relativ. Gravit., 50, 37 is believed to meet this expectation. Nielsen A. B., Jasiulek M., Krishnan B., Schnetter E. 2011, Phys. Rev. D, 83, 124022 Page D. N., Shoom A. A. 2015, Phys. Rev. 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