# Tipler Naked Singularities in $N$ Dimensions

Kharanshu N. Solanki♭ 0000-0002-6909-4370 Karim Mosani† 0000-0001-5682-1033
Omkar Deshpande♯ and Pankaj S. Joshi⋆ 0009-0006-5745-8081 0000-0003-0463-7554
♭ Department of Physics, Universität Wien, Boltzmanngasse 5, 1090 Wien  †
Fachbereich Mathematik, Universität Tübingen, Auf der Morgenstelle 10, 72076
Tübingen 00footnotetext: † Now at Fakultät für Mathematik, Ruhr-Universität
Bochum, Universitätsstr. 150, 44780 Bochum. ♯ Mathematisches Institut, Ludwig-
Maximilians-Universität, Theresienstr. 39, 80333 München  ♭,⋆ International
Center for Space and Cosmology, Ahmedabad University, 380009 Ahmedabad
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

A spacetime singularity, identified by the existence of incomplete causal
geodesics in the spacetime, is called a (Tipler) strong curvature singularity
if the volume form acting on independent Jacobi fields along causal geodesics
vanishes in the approach of the singularity. It is called naked if it is the
past endpoint of causal curves in spacetime. Here, we study the formation of
strong curvature naked singularities arising from spherically symmetric
gravitational collapse of general type-I matter fields in an arbitrarily
finite number of dimensions. In the spirit of Joshi and Dwivedi (1993 Phys.
Rev. D 47 5357), and Goswami and Joshi (2007 Phys. Rev. D 76 084026),
beginning with regular initial data, we derive two distinct (but not mutually
exclusive) conditions, which we call the positive root condition (PRC) and the
simple positive root condition (SPRC), that serve as necessary and sufficient
conditions, respectively, for the existence of naked singularities. In doing
so, we generalize the results of both the aforementioned works. We further
restrict the PRC and the SPRC by imposing the curvature growth condition (CGC)
of Clarke and Krolak (1985 J. Geom. Phys. 2(2) 127) on all causal curves that
satisfy the causal convergence condition. The CGC then gives a sufficient
condition ensuring that the naked singularities implying the PRC and implied
by the SPRC, are of strong curvature type and hence correspond to the
inextendibility of the spacetime. Using the CGC, we extend the results of
(2020 Phys. Rev. D 101 044052) (that hold for dimension $N=4$) to the case
$N=5$, showing that strong curvature naked singularities can occur in this
case. However, for the case $N\geq 6$, we show that past-incomplete causal
curves that identify naked singularities do not satisfy the CGC. These results
shed light on the validity of the cosmic censorship conjectures in arbitrary
dimensions.

###### keywords:

Naked singularity; Gravitational collapse; Cosmic Censorship; Higher
Dimensions.

###### Contents

  1. 1 Introduction
  2. 2 The Mathematical Model of a Collapsing Star
    1. 2.1 The Interior Spacetime
    2. 2.2 The Exterior Spacetime
    3. 2.3 Construction of Regular Initial Data
    4. 2.4 Singular Solutions from Regular Initial Data
  3. 3 Classification of Singular End States
    1. 3.1 Conditions for the formation of Black Holes
    2. 3.2 Conditions for the Formation of Naked Singularities
  4. 4 Gravitational Strength of the Naked Singularity
  5. 5 Concluding Remarks

##  1 Introduction

The modern formulations of the cosmic censorship conjectures are built on the
dynamical formulation of general relativity, indicating that these conjectures
(in a broad sense) speak about the nature of singularities that form
dynamically in general relativity. Fundamental to the idea of dynamics in
general relativity is the notion of initial data sets. These are $5$-tuples
$(\Sigma,\leftidx{{}^{\Sigma}}{\mathbf{g}}{},\leftidx{{}^{\Sigma}}{\mathbf{K}}{},\mu,J)$,
where $\Sigma$ is the initial spacelike hypersurface with first and second
fundamental form fields $\leftidx{{}^{\Sigma}}{\mathbf{g}}{}$ and
$\leftidx{{}^{\Sigma}}{\mathbf{K}}{}$; and $\mu\in\mathcal{C}^{k}(\Sigma)$ and
$J\in\Gamma(\mathcal{T}\Sigma)$ are the energy density and momentum density of
the matter field considered respectively, that are to be defined on $\Sigma$
with an appropriate regularity class. This initial data set must satisfy a
certain set of constraint equations [1], subject to which it is to be evolved.

Employing this formalism, Choquet-Bruhat [2] developed the fundamental theorem
of dynamics in general relativity in 1952. The theorem says that given a
vacuum initial data set
$(\Sigma,\leftidx{{}^{\Sigma}}{\mathbf{g}}{},\leftidx{{}^{\Sigma}}{\mathbf{K}}{},\mu=0,J=0)$
solving the vacuum constraint equations, one can always find a spacetime
$(M,\mathbf{g})$ that (i) solves the vacuum Einstein equations and (ii) admits
$\Sigma\hookrightarrow M$ as a Cauchy hypersurface. In other words, general
relativity can be posed as a well-defined Cauchy problem. Later, in 1969,
Choquet-Bruhat and Geroch [3] proved that111This proof appeals to Zorn’s
lemma, which is equivalent to the set-theoretic axiom of choice. This may seem
to indicate that the dynamics of general relativity rely on the axiom of
choice. However, in 2016, Sbierski [4] carried out an alternative proof of the
uniqueness of globally hyperbolic solutions without employing Zorn’s lemma. a
vacuum initial data set
$(\Sigma,\leftidx{{}^{\Sigma}}{\mathbf{g}}{},\leftidx{{}^{\Sigma}}{\mathbf{K}}{},\mu=0,J=0)$
solving the vacuum constraint equations admits a unique vacuum maximal
globally hyperbolic development $M$.

Before the advent of the dynamical formulation of relativity, the general
consensus about singular solutions of the Einstein equations was to discard
them as “bad” solutions. However, in the context of the dynamical formulation
of relativity, one is not allowed to discard a solution, but can only make
judgements about the initial data (which is regular in the first place) [5].
For instance, one may argue that perhaps the singular behaviour of the
Schwarzschild geometry is only an artefact of the spherically symmetric
initial data that it evolves from. However, Hawking and Penrose [6] showed
that such arguments must be considered invalid in the context of their
singularity or incompleteness theorems. These theorems show the geodesic
incompleteness of spacetime under the satisfaction of sufficiently general
constraints such as certain mild causality conditions, energy conditions and
appropriate initial or boundary conditions [7, 8].

At the root of the Hawking-Penrose singularity theorems of general relativity
is the notion of causal geodesic incompleteness, and not that of unbounded
curvatures along the geodesic or even the inextendibility of the spacetime in
question. This leads to a rather improper terminology for the notion of
singularities, which in the context of the singularity theorems, may also
serve as endpoints of geodesics, through which the spacetime itself may be
extended. Additionally, the singularity theorems make no claim whatsoever
about the existence of black holes. In other words, the theorems do not negate
the existence of naked singularities [9]. However, concerns regarding the
nature of naked singularities and their relation with the breakdown of
classical predictability, led Penrose [10] to introduce the cosmic censorship
conjecture (CCC). In present literature [11, 12, 5, 13], this conjecture is
often expressed as two separate statements. In the context of dynamical
relativity, these conjectures take the following forms:

######  Conjecture 1.

(Weak cosmic censorship - Choquet-Bruhat [14])
The maximal globally hyperbolic development (MGHD) $(M,\mathbf{g})$ of generic
asymptotically flat initial data
$(\Sigma,\leftidx{{}^{\Sigma}}{\mathbf{g}}{},\leftidx{{}^{\Sigma}}{\mathbf{K}}{},\mu,J)$
possesses a complete222Christodolou [15] defined that for an asymptotically
flat MGHD to possess a complete null infinity $\mathscr{I}^{+}$, the null
geodesic generators of $\mathscr{I}^{+}$ must be complete with respect to the
conformal metric of the maximal development. future null infinity.

######  Conjecture 2.

(Strong cosmic censorship - Choquet-Bruhat [14])
The MGHD of generic asymptotically flat initial data
$(\Sigma,\leftidx{{}^{\Sigma}}{\mathbf{g}}{},\leftidx{{}^{\Sigma}}{\mathbf{K}}{},\mu,J)$
is inextendible333Wald [11] adds to this saying that if the MGHD is
extendible, then for each point $p$ on the Cauchy horizon $H^{+}(\Sigma)$ in
the extension, either strong causality is violated, or
$\overline{I^{-}(p)\cap\Sigma}$ is non-compact. as a suitably regular
Lorentzian manifold.

The terminology “weak” and “strong” is rather unfortunate, since strong cosmic
censorship (SCC) does not imply weak cosmic censorship (WCC). That being said,
WCC is (in some sense) weaker than SCC because the former allows the
singularity to be visible to nearby observers, while the latter does not allow
the singularity to be visible to any observer. Singularities that violate WCC
are termed globally naked, while those that violate SCC are termed locally
naked. Fig. 1 illustrates these notions for the Lemaitre-Tolman-Bondi (LTB)
inhomogeneous dust collapse scenario [16].

The current form of the CCCs indicate that they have been introduced to
resolve two issues. The first issue concerns the existence of naked
singularities and their relation to the breakdown of predictability [17];
while the second one concerns the confusion regarding the meaning of the word
singularity as demanded by the Hawking-Penrose singularity theorems. Indeed,
in their current form, the cosmic censorship conjectures deal with the
existence of maximal globally hyperbolic inextendible developments of generic
initial data. In other words, the endpoints of all incomplete causal geodesics
must be genuine singularities through which the spacetime is inextendible.
Additionally, these singularities must be covered by a horizon.

Figure 1:  On the LHS, we have the conformal diagram for the well known
Oppenheimer-Synder-Datt homogeneous dust collapse, which results in the
formation of a black hole. Moreover, in the center we have the conformal
diagram for the LTB inhomogeneous dust collapse scenario with a locally naked
singularity, whereas on the RHS we have the conformal diagram for the LTB
inhomogeneous dust collapse scenario with a globally naked singularity [18,
19]. The shaded regions represent the MGHDs of some initial data hypersurface
$\Sigma$. Further, $H^{+}(\Sigma)$ denotes the future Cauchy horizon of
$\Sigma$, $\mathscr{E}$ denotes the event horizon, and $\mathscr{N}$ denotes
the naked singularity. Note that for the locally naked scenario,
$\mathscr{I}^{+}$ is complete and
$H^{+}(\Sigma)\cap\mathscr{I}^{+}=\emptyset$, whereas for the globally naked
scenario, $\mathscr{I}^{+}$ is incomplete and
$H^{+}(\Sigma)\cap\mathscr{I}^{+}\neq\emptyset$. Additionally for both the
scenarios, the MGHDs can be extended (to the unshaded region) which then does
not admit $\Sigma$ as a Cauchy surface, and hence is not globally hyperbolic.
Hence, globally naked singularities violate both WCC and SCC, whereas locally
naked singularities only violate the SCC.

The aforementioned precise formulations of cosmic censorship conjectures are
built on the assumption that the underlying spacetime is asymptotically flat
or that it is the MGHD of asymptotically flat initial data. We hold that such
an assumption is rather strong. An alternate characterization of the cosmic
censorship conjectures can be developed by incorporating the following
requirements on physical grounds [20]: (i) A suitable energy condition must be
obeyed, (ii) the collapse must develop from generic regular initial data,
(iii) any singularity arising from a realistic collapse must be
gravitationally strong, and (iv) the matter fields must be sufficiently
general.

A limitation of the cosmic censorship conjectures is the lack of a unique
definition of the term generic initial data. It is therefore imperative to
consider specific solutions to the Einstein equations that admit naked
singularities. These solutions must be studied in order to construct a
relevant notion of generic initial data for the CCCs, and then make
appropriate judgements about them. From the physical point of view, one may
consider the formation of spacetime singularities via the unhindered
gravitational collapse of a massive star that has used up its nuclear energy
[20]. Such studies have, by and large, been conducted by considering various
forms of the matter that constitute the collapsing star. Oppenheimer and
Snyder [21] studied the unhindered collapse of homogeneous dust in 1939,
wherein they showed that such a collapse inevitably leads to the formation of
black holes, i.e., singularities that are covered by an apparent horizon. In
1984, Christodoulou [22] showed the existence of naked singularities in
inhomogeneous dust collapse. However, Newman [23] showed that these
singularities do not satisfy the strong limiting focussing condition (SLFC)
[24], which is a sufficient condition for the singularity to be strong (in
Tipler’s sense) for spherically symmetric spacetimes [25]. The strength of a
singularity characterizes the continuous inextendibility of the spacetime
through it, and hence makes sure that the end point of the inextendible curve
in question is indeed a genuine spacetime singularity. In 1993, Joshi and
Dwivedi [26] showed the existence of naked singularities in the same settings
as Christodoulou, which satisfies Tipler’s criterion of strength. Recently,
Mosani et al. [27] extended these results to singularities arising in perfect
fluid collapse.

Going beyond general relativity, one may identify the following motivations
for studying gravitational collapse in theories assuming a higher dimensional
spacetime: (i) It has been suggested [28] that we may be living in a lower
dimensional surface (called a brane) of a higher dimensional space. All the
non-gravitational forces are confined to this brane, but gravity is not. This
requires a higher dimensional generalization of Einstein gravity [29]. (ii)
Having knowledge of certain properties in higher dimensions could help better
understand those properties in four dimensions, for which general relativity
was originally constructed. Goswami and Joshi developed the formalism to study
the gravitational collapse of spherically symmetric type-I matter field in
higher dimensions [30] and showed the existence of naked singularities.
However, the strength of such singularities (in the sense of Tipler) has not
been examined [31, 32, 33, 34, 35]. (iii) It may be interesting to assess the
validity of the cosmic censorship conjectures in higher dimensions. In fact,
it has been shown that in some special scenarios, even though cosmic
censorship may be violated in four dimensional spacetimes, it is restored
beyond five dimensions [20].

In the context of the discussion thus far, our aim here is to probe the
conditions for the occurrence of strong curvature naked singularities from
spherically symmetric gravitational collapse in an arbitrary, albeit finite
number of spacetime dimensions. In particular, we take the matter fields to
belong to the general class of type-I matter fields. Choosing this class
allows for a unified treatment of previous studies, since dust, tangential
pressure, perfect fluid, scalar fields and Chaplygin gases [36], among other
kinds of matter, are included in this class. Moreover, we only impose mild
energy conditions (namely, the dominant and the null energy conditions) on the
spacetime from physical considerations. No further assumptions, such as the
existence of Killing fields (apart from those induced by spherical symmetry),
or self-similarity [37] are imposed. In section 2, we describe the formalism
employed to study spherically symmetric gravitational collapse. This includes
a discussion of the interior and exterior spacetimes for the collapsing matter
field. Here, we also construct regular initial data in terms of the matter and
geometric variables under consideration. In section 3, we discuss necessary
and sufficient condition for the formation of a naked singularity (in any
dimensions). In section 4, we give a sufficient condition for the naked
singularity to be gravitationally strong. Implications for the cosmic
censorship conjectures and future directions are discussed in section 5.
Throughout the paper (unless mentioned otherwise) we consider a spacetime to
be a $N$ dimensional Lorentzian manifold, endowed with an at least
$\mathcal{C}^{2}$ Lorentzian metric with signature $(1,N-1)$.

##  2 The Mathematical Model of a Collapsing Star

We model the gravitational collapse of a spherically symmetric matter cloud by
a collection of concentric spherical shells $\mathcal{S}^{N-2}_{r}$ of co-
dimension two, identified by its radial coordinate $r$. As time commences,
either all these shells contract to a point, or only shells identified by
radial coordinates less than or equal to $r_{e}<r_{b}$ (where $r_{b}$ is the
radial coordinate of the largest shell) contract to a point, or none of them
contract to a point. We refer to the resultant phenomenon in the first case as
unhindered gravitational collapse, and in the remaining two cases as hindered
gravitational collapse. The point to which these shells collapse (if they do)
is referred to as the singularity. The hindered collapse of matter clouds may
lead to either a singular equilibrium configuration [38], or a regular
equilibrium configuration [39].

In fact, we require two separate spacetimes: one that describes the interior
collapsing cloud, and the other that describes the surrounding exterior
region. These are then matched on a timelike hypersurface representing the
boundary of the collapsing cloud, via an appropriate procedure [40]. In this
fashion, we construct a “larger” spacetime by taking the union of the interior
and the exterior spacetime, and the matching hypersurface. In what follows, we
briefly review the interior and exterior spacetimes for a spherically
symmetric unhindered gravitational collapse. We then proceed to construct and
evolve the regular initial data.

###  2.1 The Interior Spacetime

Let $(M,\mathbf{g})$ be a spherically symmetric $N$ dimensional spacetime with
$\partial M\eqcolon B\hookrightarrow M$. We call this the interior spacetime.
Let $\gamma:\mathbb{R}\supseteq I\to M$ be the worldline of a timelike
particle constituting the matter field $\mathbf{T}:M\to\mathcal{T}^{0}_{\ 2}M$
over $M$. For each curve parameter $\lambda\in I$, let
$\textbf{v}_{\lambda}\in\mathcal{T}_{\gamma(\lambda)}M$ be the tangent vector
to $\gamma$ at the point $\gamma(\lambda)\in M$. Choose a basis
$\\{\mathbf{e}_{a}\\}=\\{\mathbf{e}_{1},\dots,\mathbf{e}_{N}\\}$ of each
tangent space $\mathcal{T}_{\gamma(\lambda)}M$, such that (i) $\forall\
\lambda\in
I:\mathbf{g}_{\gamma(\lambda)}(\mathbf{e}_{a},\mathbf{e}_{b})=\eta_{ab}$, and
(ii) $\mathbf{e}_{1}\coloneqq\textbf{v}_{\lambda}$. Here, $\eta_{ab}$ are the
components of the $N$ dimensional Minkowski metric. Condition (i) implies that
this frame is orthonormal (in the Lorentzian sense), and condition (ii)
stipulates that the particle does not move spatially with respect to itself.
Such a choice of basis is called a comoving frame, in the sense that the frame
moves with the matter. The corresponding choice of basis for the respective
cotangent spaces is called the comoving coframe.

By definition, a type-I matter field admits one timelike and $N-1$ spacelike
eigenvectors in a comoving frame. Note that $\mathbf{e}_{1}$ is timelike,
since it is the tangent to a timelike geodesic. Moreover, condition (i)
implies that $\mathbf{e}_{2},\dots,\mathbf{e}_{N}$ are all spacelike. We now
introduce a chart map $\mathbf{x}:M\to\mathbb{R}^{N}$ such that $\forall\ p\in
M:\mathbf{x}(p)=(\mathbf{x}^{1}(p),\dots,\mathbf{x}^{N}(p))\coloneqq(t,r,\theta^{1},\dots,\theta^{N-2})\in\mathbb{R}^{N}$.
Further, we require that for every $1\leq\mu\leq N$, the inverse mapping
$(\mathbf{x}^{\mu})^{-1}:\mathbb{R}\supseteq I\to M$ is the integral curve of
the vector field generated by $\mathbf{e}_{\mu}$ (as defined above). We call
these comoving coordinates. For future use, we define that $\forall\ 1\leq
i\leq N:\mathbf{X}_{i}\coloneqq\text{range}(\mathbf{x}^{i})\subset\mathbb{R}$,
and
$\mathbf{X}\coloneqq\mathbf{X}_{1}\times\mathbf{X}_{2}\subset\mathbb{R}^{2}$.

The comoving frame may now be denoted by the set $\\{\partial_{a}:1\leq a\leq
N\\}\subset\mathcal{T}_{p}M$, and the comoving coframe may be denoted by the
set $\\{d\mathbf{x}^{b}:1\leq a\leq N\\}\subset\mathcal{T}_{p}^{*}M$.
Consequently, one can construct a basis for $\mathcal{T}^{0}_{\ 2,p}M$ given
by $\\{d\mathbf{x}^{a}\otimes d\mathbf{x}^{b}:1\leq a,b\leq N\\}$. Then, the
spherically symmetric metric $\mathbf{g}:M\to\mathcal{T}^{0}_{\ 2}M$ can be
expanded in this basis in terms of three arbitrary functions
$\phi:\mathbf{X}\to\mathbb{R}$, $\psi:\mathbf{X}\to\mathbb{R}$ and
$R:\mathbf{X}\to\mathbb{R}$ as follows:

$\displaystyle\mathbf{g}_{p}$ $\displaystyle=-e^{2\phi(t,r)}\ dt\otimes
dt+e^{2\psi(t,r)}\ dr\otimes dr+\mathbf{h}_{p},~{}\textrm{where}$ (1)
$\displaystyle\mathbf{h}_{p}$
$\displaystyle=R^{2}(t,r)\sum_{i=1}^{N-2}\left(\prod_{j=1}^{i-1}\sin^{2}{\theta^{j}}\right)d\theta^{i}\otimes
d\theta^{i}.$

Here the notation $\mathbf{g}_{p}$ is used to denote the metric evaluated at a
point $p\in M$. Similar notation shall be adhered to for any other tensor
field henceforth. Likewise, one can construct a basis $\\{\partial_{a}\otimes
d\mathbf{x}^{b}:1\leq a,b\leq N\\}$ for $\mathcal{T}^{1}_{\ 1,p}M$. The
energy-momentum tensor for a type-I matter field can then be expanded in this
basis as follows:

$\mathbf{T}_{p}=-\rho(t,r)\ \partial_{t}\otimes dt+p_{r}(t,r)\
\partial_{r}\otimes
dr+p_{\theta}(t,r)\sum_{i=1}^{N-2}\partial_{\theta^{i}}\otimes d\theta^{i}.$
(2)

Here, the maps $\rho:\mathbf{X}\to\mathbb{R}$,
$p_{r}:\mathbf{X}\to\mathbb{R}$, and $p_{\theta}:\mathbf{X}\to\mathbb{R}$ are
called the energy density, radial pressure, and tangential pressure of the
cloud undergoing gravitational collapse. Additionally, in equation (1), the
function $R:\mathbf{X}\to\mathbb{R}$ is interpreted as the physical radius of
the matter cloud. The physical significance of the metric functions $\phi$ and
$\psi$ will be clarified later.

###  2.2 The Exterior Spacetime

Let $(\tilde{M},\mathbf{\tilde{g}})$ be a spherically symmetric $N$
dimensional spacetime with
$\partial\tilde{M}\eqcolon\tilde{B}\hookrightarrow\tilde{M}$. We call this the
exterior spacetime. This spacetime models the region surrounding the
collapsing cloud, which may be done by considering
$(\tilde{M},\mathbf{\tilde{g}})$ to be the generalized radiating Vaidya
spacetime [41].

We introduce a different chart $\mathbf{y}:\tilde{M}\to\mathbb{R}^{N}$, such
that $\forall\
p\in\tilde{M}:\mathbf{y}(p)=(\mathbf{y}^{1},\dots,\mathbf{y}^{N})(p)=(t_{v},r_{v},\theta_{v}^{1},\dots,\theta_{v}^{N-2})\in\mathbb{R}^{N}$,
where $t_{v}$ is called the retarded (or exploding) time, $r_{v}$ is called
generalized Vaidya radius, and for each $1\leq i\leq N-2$, the numbers
$\theta^{i}_{v}$ are the angular coordinates for the generalized Vaidya
spacetime. For future use, we define that $\forall 1\leq i\leq
D:\mathbf{Y}_{i}\coloneqq\text{range}(\mathbf{y}^{i})\subset\mathbb{R}$, and
$\mathbf{Y}\coloneqq\mathbf{Y}_{1}\times\mathbf{Y}_{2}\subset\mathbb{R}^{2}$.
In the corresponding coordinate basis, the exterior generalized Vaidya metric
$\tilde{\mathbf{g}}:\tilde{M}\to\mathcal{T}^{0}_{\ 2}\tilde{M}$ is written as
[42],

$\displaystyle\tilde{\mathbf{g}}_{p}$
$\displaystyle=-\left(1-\frac{2\mathfrak{M}(t_{v},r_{v})}{r_{v}^{N-3}}\right)\
dt_{v}\otimes dt_{v}-2\ dt_{v}\otimes
dr_{v}+\tilde{\mathbf{h}}_{p},~{}\textrm{where}$ (3)
$\displaystyle\tilde{\mathbf{h}}_{p}$
$\displaystyle=r_{v}^{2}\sum_{i=1}^{N-2}\left(\prod_{j=1}^{i-1}\sin^{2}{\theta_{v}^{j}}\right)d\theta_{v}^{i}\otimes
d\theta_{v}^{i}.$

Here, $\mathfrak{M}:\mathbf{Y}\to\mathbb{R}^{+}_{0}$ is called the generalized
Vaidya mass. The exterior spacetime must be “glued” to the interior spacetime
in order to form one total spacetime describing the phenomenon of
gravitational collapse.

We construct the total spacetime $(\mathscr{M},\mathfrak{g})$ from
$(M,\mathbf{g})$ and $(\tilde{M},\tilde{\mathbf{g}})$ as follows [43]:

  1. 1.

$\mathscr{M}\cong M\sqcup\tilde{M}$ as Lorentzian manifolds. Moreover, for
every $p\in M$, we have $\mathfrak{g}_{p}\coloneqq\mathbf{g}_{p}$ and for
every $q\in\tilde{M}$, we have $\mathfrak{g}_{q}\coloneqq\mathbf{g}_{p}$.

  2. 2.

Identify points in $B$ with points in $\tilde{B}$ such that $B\cong\tilde{B}$
as differentiable manifolds. We may then say that these boundaries are
diffeomorphic to a $(N-1)$ dimensional manifold $\mathscr{S}$.

  3. 3.

The following junction conditions given by Israel [40] must hold:

$\displaystyle\leftidx{{}^{B}}{\mathbf{g}}{}$
$\displaystyle\equiv\leftidx{{}^{\tilde{B}}}{\tilde{\mathbf{g}}}{},~{}\textrm{and}$
(4) $\displaystyle\mathbf{K}$ $\displaystyle\equiv\tilde{\mathbf{K}}.$

Here $\leftidx{{}^{B}}{\mathbf{g}}{}$ denotes the metric induced on
$B\hookrightarrow M$ and $\leftidx{{}^{\tilde{B}}}{\tilde{\mathbf{g}}}{}$
denotes the metric induced on $\tilde{B}\hookrightarrow\tilde{M}$.
Additionally, the maps $\mathbf{K}:B\to\mathcal{T}^{0}_{\ 2}B$ and
$\tilde{\mathbf{K}}:\tilde{B}\to\mathcal{T}^{0}_{\ 2}\tilde{B}$ are defined
pointwise in the following manner:

$\displaystyle\forall\ p\in B;\ \mathbf{K}_{p}:$
$\displaystyle\mathcal{T}_{p}B\times\mathcal{T}_{p}B\to\mathbb{R}$ (5)
$\displaystyle(X,Y)\mapsto\mathbf{g}_{p}\left(\nabla_{X}Z,Y\right),~{}\textrm{and}$
$\displaystyle\forall\ q\in\tilde{B};\ \tilde{\mathbf{K}}_{q}:$
$\displaystyle\mathcal{T}_{q}\tilde{B}\times\mathcal{T}_{q}\tilde{B}\to\mathbb{R}$
$\displaystyle(X,Y)\mapsto\tilde{\mathbf{g}}_{q}\left(\nabla_{X}Z,Y\right).$

These are the extrinsic curvatures (or second fundamental forms) of
$B\hookrightarrow M$ and $\tilde{B}\hookrightarrow\tilde{M}$ respectively.
Here $Z\in\Gamma(TN)$ is the global unit normal vector field to $\mathscr{S}$.

Israel’s conditions were used in $N=4$ case to match interior perfect fluid
seeded spacetime with the exterior generalized Vaidya spacetimes in [30].
Generalizing the procedure to $N$ dimensions is fairly straightforward, and we
shall not focus on this aspect here.

###  2.3 Construction of Regular Initial Data

From the dynamical point of view, the problem of gravitational collapse of a
spherically symmetric matter cloud can be treated as an initial value problem,
wherein regular initial data defined on an initial spacelike hypersurface
$\Sigma_{t_{i}}\hookrightarrow M$ characterized by $t=t_{i}$, is evolved in
the future subject to the Einstein equations. Moreover, one would like to
impose some additional constraints on the initial data in terms of some
physically motivated conditions. One of these restrictions is that the metric
components $g_{ab}:\mathbb{R}^{N}\to\mathbb{R}$ in the chosen basis must be at
least $\mathcal{C}^{2}$ everywhere. This is required to solve the Einstein
equations in the first place.

Additionally, we require the stress-energy tensor to satisfy the weak energy
condition (WEC) and the null energy condition (NEC), which are together given
by the following pointwise defined condition,

$\forall\ \mathsf{V}\in\mathcal{T}_{p}M:~{}\lVert\mathsf{V}\rVert_{p}\leq
0\implies\mathbf{T}_{p}(\mathsf{V},\mathsf{V})\geq 0.$ (6)

Here,
$\lVert\mathsf{V}\rVert_{p}\coloneqq\mathbf{g}_{p}(\mathsf{V},\mathsf{V})$.
Moreover, we also ensure the satisfaction of dominant energy condition (DEC),
which requires that the WEC holds in addition to the following pointwise
defined condition:

$\forall\
\mathsf{V}\in\mathcal{T}_{p}M:~{}\lVert\mathsf{V}\rVert_{p}<0\implies\lVert(\mathbf{T}_{p}(\mathsf{V},\cdot))^{\sharp_{p}}\rVert_{p}\leq
0.$ (7)

Here, $(\cdot)^{\sharp_{p}}$ denotes the pointwise evaluation of the globally
defined musical isomorphism [44] $\sharp:\mathcal{T}^{*}M\to\mathcal{T}M$.
These energy conditions are rather mild, physically motivated conditions,
conveying the idea that the energy density along light trajectories, and as
measured by an observer locally, is non-negative, in addition to the notion of
causal matter flow. Furthermore, no other energy conditions are imposed. For
the specific form of energy-momentum tensor given in equation (2), the DEC and
NEC are together equivalent to the satisfaction of the following condition:

$\displaystyle\forall$
$\displaystyle(t,r)\in\mathbf{X}:\Delta_{(t,r)}\subset\mathbb{R}^{+}_{0},~{}\textrm{where}$
(8) $\displaystyle\Delta_{(t,r)}$
$\displaystyle\coloneqq\\{\rho(t,r),~{}(\rho+p_{r})(t,r),~{}(\rho+p_{\theta})(t,r),~{}(\rho-|p_{r}|)(t,r),~{}(\rho-|p_{\theta}|)(t,r)\\}.$

Now we define a particular notion of mass for the spacetimes under
consideration. As we shall see later, this mass function can be used to
characterize the apparent horizon.

######  Definition 1.

(Misner-Sharp mass function, Giambo et. al. [45])
For a $N-$dimensional spherically symmetric spacetime $(M,\mathbf{g})$, with
$\mathbf{g}$ given by equation (1), the Misner-Sharp mass function is a map
$F:\mathbf{X}\to\mathbb{R}$ defined by,

$F(t,r)\coloneqq R^{N-3}(t,r)(1-\mathbf{g}^{\sharp}_{\ p}(dR,dR))(t,r).$ (9)

Here, $\forall\ p\in M:\mathbf{g}^{\sharp}_{\
p}:\mathcal{T}^{*}_{p}M\times\mathcal{T}^{*}_{p}M\to\mathbb{R}$ is the scalar
product on each cotangent space, and will be referred to as the inverse
metric.

######  Remark 1.

In the context of the Hamiltonian formulation of relativity, one can define
various notions of mass [46]. For instance, one has the Hawking quasi-local
mass [47], the Arnowitt-Deser-Misner (ADM) mass [48] and the Bondi-Sachs (BS)
mass [49]. The Misner-Sharp mass is equivalent to the Hawking quasi-local mass
under the assumption of spherical symmetry. Moreover, the ADM mass and the BS
mass are equivalent to the Misner-Sharp mass in the limit of approach to
spatial infinity and null infinity respectively [50].

With these definitions in mind, for the metric given in equation (1), the
Einstein equations can be written in the following form:

$\displaystyle\rho(t,r)$
$\displaystyle=\frac{(N-2)F^{\prime}(t,r)}{2R^{N-2}(t,r)R^{\prime}(t,r)},$
(10a) $\displaystyle p_{r}(t,r)$
$\displaystyle=-\frac{(N-2)\dot{F}(t,r)}{2R^{N-2}(t,r)\dot{R}(t,r)},$ (10b)
$\displaystyle\phi^{\prime}(t,r)$
$\displaystyle=\frac{(N-2)(p_{\theta}(t,r)-p_{r}(t,r))}{\rho(t,r)+p_{r}(t,r)}\frac{R^{\prime}(t,r)}{R(t,r)}-\frac{p^{\prime}_{r}(t,r)}{\rho(t,r)+p_{r}(t,r)},$
(10c) $\displaystyle\dot{R}^{\prime}(t,r)$
$\displaystyle=\frac{1}{2}\left(\dot{R}(t,r)\frac{H^{\prime}(t,r)}{H(t,r)}+R^{\prime}(t,r)\frac{\dot{G}(t,r)}{G(t,r)}\right).$
(10d)

Here, we have used the notations,

$F^{\prime}(t,r)=\frac{\partial F}{\partial r}\bigg{|}_{t}(t,r);\ \ \
\dot{F}(t,r)=\frac{\partial F}{\partial t}\bigg{|}_{r}(t,r).$ (11)

The same notations applies to the physical radius $R$. Moreover, the functions
$G:\mathbf{X}\to\mathbb{R}$ and $H:\mathbf{X}\to\mathbb{R}$ are defined as,

$G(t,r)\coloneqq e^{-2\psi(t,r)}(R^{\prime})^{2}(t,r);\ \ \ H(t,r)\coloneqq
e^{-2\phi(t,r)}(\dot{R})^{2}(t,r).$ (12)

In terms of these functions, equation (9) gives,

$F(t,r)=R^{N-3}(t,r)(1-G(t,r)+H(t,r)).$ (13)

In the context of gravitational collapse of a dust cloud, the Misner sharp
mass function is to be interpreted as the quasi-local mass contained inside a
spherical matter shell of radius $r$ at time $t$. In fact, from equation
(10a), we find that at a given time $t$,

$F(t,r)=\frac{2}{\mathsf{D-2}}\int_{0}^{R}\ d\bar{R}\
\bar{R}^{\mathsf{D-2}}\rho.$ (14)

It is possible for the Misner-Sharp mass function to assume negative values a
priori. However, in the context of the positive energy theorem(s) of classical
relativity [51, 52], the Misner-Sharp mass corresponds to the mass of the
spacetime. As a consequence, it is necessary that the Misner-Sharp mass
function assume only non-negative values, i.e.,

$\text{range}(F)\subseteq\mathbb{R}^{+}_{0}.$ (15)

As we shall see later, the Misner-Sharp mass function can be used to
characterize the apparent horizon, much like the Schwarzschild mass can be
used to characterize the event horizon, which is equivalent to the apparent
horizon for the Schwarzschild spacetime. In fact, the Schwarzschild mass is
equivalent to the Misner-Sharp mass function for the Schwarzschild spacetime.
Using equations (6) and (10a), the WEC and NEC can be formulated along any
causal curve as,

$\forall\
(t,r)\in\mathbf{X}:~{}\\{F^{\prime}(t,r),~{}R^{\prime}(t,r)\\}\subset\mathbb{R}^{+}_{0}.$
(16)

Some restrictions on the behaviour of the physical radius $R(t,r)$ are in
order. The behaviour of $\dot{R}:\mathbf{X}\to\mathbb{R}$ determines whether
the matter cloud undergoes unhindered gravitational collapse, hindered
gravitational collapse, or expands. Clearly, for the matter field to undergo
unhindered gravitational collapse, we must have,

$\text{range}(\dot{R})\subseteq\mathbb{R}^{-}.$ (17)

If one has $\dot{R}(t_{f},r)\geq 0$ at some later time $t=t_{f}$, the
collapsing matter would either halt or bounce back. We do not concern
ourselves with such cases. Instead, we focus on the scenario wherein all the
matter shells completely collapse to a singularity. Now, for reasons to be
specified later, we introduce a scaling function for the physical radius.

######  Definition 2.

(Scaling function)
The scaling function is a map $v:\mathbf{X}\to(0,1]$ defined by,

$v(t,r)\coloneqq\frac{R(t,r)}{r}.$ (18)

######  Remark 2.

The scaling function $v(t,r)$ is a monotone decreasing function of $t$
throughout the gravitational collapse. This allows us to use $v$ as a time-
coordinate. In other words, we employ the coordinate transformation $t\mapsto
v(t,r)$ for a fixed $r$. For the remainder of the paper, while referring to
$v$ as a coordinate, we shall (abusively) drop the argument $(t,r)$. Moreover,
a function $f(t,r)$ in the $(t,r)$ plane will be denoted in the $(v,r)$ plane
as a corresponding transformed function $\tilde{f}(v,r)$.

We now recall some useful definitions from the theory of spherically symmetric
gravitational collapse.

######  Definition 3.

(Epoch)
Let $(M,\mathbf{g})$ be the interior spacetime. A spacelike hypersurface
$\Sigma\subset M$ with normal $\partial_{t}$, defined by some
$t=t_{1}\in\mathbf{X}_{1}$, or equivalently by some $v=v_{1}\in(0,1]$ is
called a (regular) epoch.

######  Definition 4.

(Time curve)
A map $\tilde{t}:(0,1]\times\mathbf{X}_{2}\to\mathbb{R}$ describing the time
$\tilde{t}(v,r)$ for a matter shell identified by a radial coordinate $r$ to
reach the epoch defined by a constant $v\in(0,1]$ is called the time curve.

######  Definition 5.

(Singularity curve)
A singularity curve is a map $t_{s}:\mathbf{X}_{2}\to\mathbb{R}$ defined in
terms of the time curve $\tilde{t}$, as,

$\forall r\in\mathbf{X}_{2}:t_{s}(r)\coloneqq\lim_{v\to 0}\tilde{t}.$ (19)

Clearly, the singularity curve describes the time taken by a matter shell
identified by a radial coordinate $r$ to approach the singular epoch, i.e.,
the epoch defined in the limit $v\to 0$, or equivalently $t\to t_{s}$.

######  Definition 6.

(Shell-crossing singularity, Szekeres and Lun [53])
A spacetime $(M,\mathbf{g})$ is said to admit a shell-crossing singularity if
either of the following conditions hold:

  1. (i)

Let $\gamma:[\lambda_{0},0)\to M$ be an incomplete causal curve which
approaches the singularity as the curve parameter $\lambda\to 0$. If
$J:[\lambda_{0},0)\to\mathcal{T}M$ such that for every
$\lambda\in[\lambda_{0},0)$, $J(\lambda)\in\mathcal{T}_{\gamma(\lambda)}M$
constitutes a Jacobi field along $\gamma$, then in a parallely propagated (pp)
orthonormal frame along $\gamma$, its components $J^{a}\in\mathcal{C}^{k}(M)$
have finite non-zero values as $\lambda\to 0$.

  2. (ii)

It can be covered by a set of $\mathcal{C}^{1}$ regular boundary points, i.e.,
it is removable by a $\mathcal{C}^{1}$ chart transition.

The second part of the above definition employs the abstract boundary
formulation of Scott and Szekeres [54] to identify singularities as boundary
points attached to spacetime.

######  Definition 7.

(Shell-focussing singularity)
A spherical matter shell identified by a radial coordinate
$r\in\mathbf{X}_{2}$ is said to terminate at a shell-focussing singularity if
its physical radius $R$ vanishes in the limit of approach to the singular
epoch, i.e., if,

$\lim_{t\to t_{s}}R=0.$ (20)

From equation (10a), it is seen that the density $\rho$ diverges in the limit
$R(t,r)\to 0$ and/or $R^{\prime}(t,r)\to 0$. Clearly, $R(t,r)=0$ indicates the
vanishing of all matter shells, and $R^{\prime}(t,r)\geq 0$ implies a crossing
or intersection of distinct matter shells. It has been shown previously [26,
31] that the spacetime can be extended through the singularities characterized
by $R^{\prime}(t,r)\to 0$ and $R(t,r)\to c\in\mathbb{R}$. Hence, such
singularities can be classified as shell-crossing following definition 6. On
the other hand, the singularities characterized by $R(t,r)\to 0$ and
$R^{\prime}(t,r)\to 0$ are simultaneously shell-crossing and shell-focussing
following definitions 6 and 7 respectively. In a physically sound
gravitational collapse scenario, one would expect that matter shells
identified by smaller values of $r$ collapse prior to those identified by
larger values of $r$. Hence we would like to avoid the formation of shell-
crossing singularities before the final shell-focussing singularity by
construction. The behaviour of $R^{\prime}$ is thereby restricted as,

$\text{range}(R^{\prime})\subseteq\mathbb{R}^{+}_{0}.$ (21)

This condition ensures that the physical radius cannot exhibit a non-linear
response to the comoving radius, on the initial hypersurface.

######  Lemma 1.

Let $\beta\in\mathbb{R}$. For a complete gravitational collapse without
initial shell-crossings, from a regular initial hypersurface $\Sigma_{t_{i}}$
subject to the Einstein equations, we have,

$R(t_{i},r)=r^{\beta}\implies\beta=1.$ (22)

###### Proof.

We have $R^{\prime}(t_{i},r)=\beta r^{\beta-1}$. Suppose $\beta\neq 1$. Then,
at $t=t_{i}$, either $R^{\prime}(t_{i},0)=0$ or
$R^{\prime}(t_{i},0)\to\infty$. The former contradicts the assumption that
there are no initial shell-crossings, and the latter contradicts the
requirement that the metric functions must be at least $\mathcal{C}^{2}$
everywhere initially in order to solve the Einstein equations. Hence we must
have $\beta=1$.
∎

Using this lemma and the initial scaling freedom for the physical radius, we
make the choice $R(t_{i},r)\coloneqq r$. We now recall a useful theorem
regarding the occurrence of shell-crossing singularities in the context of
spherically symmetric gravitational collapse.

######  Theorem 1.

(Joshi and Saryakar [55])
Given $\mathbb{R}\ni\epsilon>0$, there exists a number
$r_{1}\in\mathbb{R}^{+}$ such that for every $r\in[0,r_{1}]$, we have
$R^{\prime}(t,r)>0$ upto $v(t,r)=\epsilon$.

This theorem says that given any spherically symmetric unhindered
gravitational collapse model, there always exists a $r_{1}\in\mathbb{R}^{+}$
such that there are no shell-crossings present for every $r\in[0,r_{1}]$,
throughout the evolution of collapse upto any arbitrarily close epoch to the
final singularity. With this in hindsight, we see that lemma 1 is a corollary
of theorem 1.

As an upshot of our discussion thus far, we find that the following conditions
on the scaling function describe the spherically symmetric gravitational
collapse of matter fields.

$v(t_{i},r)=1;\ \ \ \ \lim_{t\to t_{s}(r)}v=0;\ \ \ \ \dot{v}(t,r)<0.$ (23)

Clearly, $R(t,r)$ vanishes at $r=0$ on all regular epochs. However, $R(t,r)$
also vanishes in the limit of approach to the singular epoch $t\to t_{s}(r)$.
These cases are to be distinguished by choosing a suitable form of the Misner-
Sharp mass function $F(t,r)$, such that it vanishes sufficiently fast in the
limit $r\to 0$, so that even if $R(t,r)\to 0$ at the center of all regular
epochs, the density and curvatures remain finite. On the other hand, by
introducing $v(t,r)$ as given by equation (18) and (23), we get a better
characterization of the shell-focussing singularity, namely $v\to 0$ as $t\to
t_{s}(r)$. Note, however that unlike $R(t_{i},r)$, we have $v(t_{i},r)=1$.
Hence the regular and singular epochs are appropriately distinguished.

Moving further, from equation (10a), it is seen that the regularity and
finiteness of $\rho$ requires that
$F^{\prime}(t_{i},r)\sim\mathcal{O}(r^{N-2})$. Hence, the regularity of the
initial density profile requires that the Misner-Sharp mass function must have
the following general form:

$F(t,r)=r^{N-1}\tilde{\mathcal{M}}(v,r).$ (24)

Here, ${\tilde{\mathcal{M}}}:(0,1]\times\mathbf{X}_{2}\to\mathbb{R}^{+}$ is at
least $\mathcal{C}^{1}$ for $r=0$ and at least $\mathcal{C}^{2}$ for
$r\in\mathbb{R}^{+}$. Such construction implies that the Misner-Sharp mass
vanishes at the centre of the collapsing cloud on all regular epochs, as well
as in the limit of approach to the singular epoch, i.e.,

$\forall\ t\in\mathbb{R}\setminus\\{t_{s_{0}}\\}:~{}F(t,0)=0\ ;\ \ \ \ \
\lim_{(t,r)\to(t_{s_{0}},0)}F=0.$ (25)

Here, $t_{s_{0}}\coloneqq t_{s}(0)$ and hence
$(t,r)\to(t_{s_{0}},0)\iff(v,r)\to(0,0)$ represents the shell-focussing
singularity at the centre, which we shall call the central singularity. Since
the Misner-Sharp mass vanishes at such singularities, they are sometimes also
referred to as massless singularities [26]. As we shall see later, for
spherically symmetric gravitational collapse of type-I matter fields only the
class of central shell-focussing singularities can be naked. On the other
hand, the class of non-central shell-focussing singularities will turn out to
be covered by an apparent horizon.

We have now defined on the initial hypersurface $\Sigma_{t_{i}}$, seven
functions as initial data, given by,

$\displaystyle\phi(t_{i},r)$ $\displaystyle=\phi_{0}(r);\ \ \
\psi(t_{i},r)=\psi_{0}(r);\ \ \ R(t_{i},r)=r;\ \ \
F(t_{i},r)=r^{N-1}\tilde{\mathcal{M}}(1,r);$ (26) $\displaystyle\rho(t_{i},r)$
$\displaystyle=\rho_{0}(r);\ \ \ p_{r}(t_{i},r)=p_{r_{0}}(r);\ \ \
p_{\theta}(t,r)=p_{\theta_{0}}(r).$

Sufficient conditions to maintain the regularity of $\phi_{0}(r)$ are given in
[30]. These sufficient conditions are motivated from physical considerations
such as the vanishing of pressure gradient forces and the vanishing of the
difference between radial and tangential pressures near the center of the
collapsing matter cloud (i.e., the matter behaves like a perfect fluid near
its center). These conditions, in addition to those discussed in this
subsection, guarantee the regularity of the metric components and the matter
density and pressures at the initial epoch.

###  2.4 Singular Solutions from Regular Initial Data

Having prescribed regular initial data on the spacelike hypersurface
$\Sigma_{t_{i}}\subset M$, we now wish to investigate its future development
$(M,\mathbf{g})$. As aforementioned, the complete gravitational collapse of a
matter cloud may result in either a black hole or a naked singularity. Our
approach here will be to investigate the existence and finiteness of tangents
to a non-zero measure set of outgoing causal geodesics in the limit of
approach to the shell-focussing singularity. As we shall see, the polarity of
their tangents will decide whether the matter cloud collapses into a black
hole or a naked singularity.

It is clear from the Einstein equations (10a) to (10d) and equation (11), that
not all of the initial data is independent of each other. Indeed, we have five
equations and seven unknowns. One possible approach in such a scenario is to
introduce an appropriate equation of state relating the density and pressure
profiles [56, 57, 58]. However, in order not to bring in undue assumptions, we
will not fix any equation of state for our analysis. Instead, the formalism
employed here allows one to work with and construct general classes of
functions that are required to have a certain form by the regularity
conditions, much like the form taken by the Misner-Sharp mass function, as
dictated by the regularity of the energy density. To begin with, we define a
suitably differentiable function
$\tilde{A}:(0,1]\times\mathbf{X}_{2}\to\mathbb{R}$ such that,

$\tilde{A},_{v}(v,r)\coloneqq\frac{\partial\tilde{A}}{\partial
v}\bigg{|}_{r}(t,r)=r\frac{\phi^{\prime}(t,r)}{R^{\prime}(t,r)}.$ (27)

The regularity of $\phi^{\prime}:\mathbf{X}\to\mathbb{R}$ will decide the
regularity of $A_{,v}:(0,1]\times\mathbf{X}_{2}\to\mathbb{R}$. For any
function in the $(v,r)$ plane, ($,_{v}$) denotes its partial derivative with
respect to $v$ along an $r$ constant hypersurface and ($,_{r}$) denotes its
partial derivative with respect to $r$ along a $v$ constant hypersurface.
Using equation (27) in equation (10d) and integrating, yields the expression
for $\tilde{G}:(0,1]\times\mathbf{X}_{2}$ as,

$\tilde{G}(v,r)=b(r)e^{2\tilde{A}(v,r)}.$ (28)

Here $b(r)$ is the integration constant and is called the velocity profile
[27] of the collapsing matter shell identified by $r$. The regularity of
$\dot{v}:\mathbf{X}\to\mathbb{R}$ at $r=0$ implies that $b(r)$ must have the
following form:

$b(r)=1+r^{2}b_{0}(r).$ (29)

$b_{0}(r)$ is called the energy distribution [27] within a shell of radius
$r$. Equations (21), (27), (28) and (13) yield,

$v^{(N-3)/2}(t,r)\
\dot{v}(t,r)=-e^{\phi(t,r)}\sqrt{\tilde{\mathcal{M}}(v,r)+\frac{v^{N-3}(t,r)(b(r)e^{2\tilde{A}(v,r)}-1)}{r^{2}}}.$
(30)

Integrating equation (30) with respect to $v$ yields the following expression
for the time curve:

$\tilde{t}(v,r)=\int_{v}^{1}d\bar{v}\frac{e^{-\phi(t,r)}}{\sqrt{\dfrac{\tilde{\mathcal{M}}(\bar{v},r)}{\bar{v}^{N-3}}+\dfrac{b(r)e^{2\tilde{A}(\bar{v},r)}-1}{r^{2}}}}.$
(31)

The time curve describes the time taken by a shell labelled by the radial
coordinate $r$ to arrive at some epoch defined by constant $v$ from the
initial epoch $v=1$. Hence, the time taken by the shell to collapse into the
shell-focussing singularity defined by $v\to 0$, i.e., the expression for the
singularity curve $t_{s}:\mathbf{X}_{2}\to\mathbb{R}$, is given by,

$t_{s}(r)\coloneqq\lim_{v\to
0}\tilde{t}=\int_{0}^{1}d\bar{v}\frac{e^{-\phi(t,r)}}{\sqrt{\dfrac{\tilde{\mathcal{M}}(\bar{v},r)}{\bar{v}^{N-3}}+\dfrac{b(r)e^{2\tilde{A}(\bar{v},r)}-1}{r^{2}}}}.$
(32)

The existence and polarity of finite tangents $(dt_{s}/dr)(r)$ of the
singularity curve will decide the existence or otherwise of ingoing and
outgoing causal geodesics approaching or emanating from the shell-focussing
singularity. The existence of such causal curves in turn play a crucial role
in deciding whether their singular end points are naked or not. A sufficient
condition for such a singularity curve to exist is the existence of an at
least $\mathcal{C}^{2}$ time curve. From equations (31) and (32), it can be
seen that the (at least) $\mathcal{C}^{2}$ regularity of the functions
$\tilde{A}$, $\tilde{\mathcal{M}}$ and $\phi$ is enough to show that the time
curve and hence the singularity curve terminate at the shell-focussing
singularity with a finite tangent. On the other hand, even if these functions
blow up in the limit of the singularity, the time curve remains finite
nonetheless. Therefore, regardless of the regularity of the integrand in
equation (31), the time curve is finite. The reader is referred to the
monograph [20] for a detailed discussion regarding the conditions for the
existence of a $\mathcal{C}^{2}$ time curve.

The existence of a regular time curve with a finite tangent in the limit of
approach to the singularity implies that the collapse terminates into a
singularity in a finite amount of time, which is expected from any physically
reasonable collapse scenario. Equations (10a)-(10c) along with equations (24)
and (27) yield

$\displaystyle\tilde{\rho}(v,r)$
$\displaystyle=\frac{N-2}{2}\left(\frac{(N-1)\tilde{\mathcal{M}}(v,r)+r(\tilde{\mathcal{M}}_{,r}(v,r)+\tilde{\mathcal{M}}_{,v}(v,r)v^{\prime}(t,r))}{v^{N-2}(v+rv^{\prime}(t,r))}\right),$
(33a) $\displaystyle\tilde{p}_{r}(v,r)$
$\displaystyle=-\frac{N-2}{2}\left(\frac{\tilde{\mathcal{M}}_{,v}(v,r)}{v^{N-2}}\right),$
(33b) $\displaystyle\tilde{p}_{\theta}(v,r)$
$\displaystyle=p_{r}(v,r)+\frac{R(t,r)}{N-2}\left(\frac{p^{\prime}_{r}(t,r)}{R^{\prime}(t,r)}+\tilde{A}_{,v}(v,r)(\rho(v,r)+p_{r}(v,r))\right).$
(33c)

It is clear from these equations that the densities and pressures diverge in
the limit of approach to the shell-focussing singularity, i.e.,

$\noindent\lim_{v\to 0}\tilde{\rho}=\infty;\ \ \ \lim_{v\to
0}\tilde{p}_{r}=\infty;\ \ \ \lim_{v\to 0}\tilde{p}_{\theta}=\infty.$ (34)

Hence we have the following well-known result which we write for the sake of
completion: Given spherically symmetric regular initial data for a collapsing
type-I matter field, on a spacelike hypersurface $\Sigma_{t_{i}}$ subject to
the DEC, NEC and the Einstein equations, its future development444Note that we
are not using the usual notion of Cauchy developments here, since we do not
require the developements to be globally hyperbolic. See [59] for a definition
of general developments. $(M,\mathbf{g})$ is singular, i.e., there exists at
least one incomplete and inextendible curve $\gamma:\mathbb{R}\supseteq I\to
M$, whose end point is a spacetime singularity. Hence the interior spacetime
$(M,\mathbf{g})$ is singular. No comments have been made on the nakedness (or
otherwise) of the singularity yet.

##  3 Classification of Singular End States

Now that it is guaranteed that the spherically symmetric gravitational
collapse of a general type-I matter field must result in either a black hole
or a naked singularity, we wish to tackle the problem of finding the
conditions for the existence of each of these possible end states. More
importantly, we wish to investigate the conditions under which the singularity
will be naked. This involves studying trapped surfaces and the apparent
horizon.

For the sake of completeness, we review some relevant notions from Lorentzian
geometry that aid the study of the causal structure of a spacetime.

######  Definition 8.

(Geodesic congruence)
Let $(M,\mathbf{g})$ be a spacetime, and $U\subseteq M$ be open in $M$. Then a
geodesic congruence through $U$ is a set of geodesics
$\Gamma\coloneqq\\{\gamma\ |\ \gamma:I\subseteq\mathbb{R}\to U\subseteq M\\}$
with the property that for every $\ p\in U$, there exists a
$~{}\gamma\in\Gamma$ such that $\gamma(\lambda)=p$ for some $\lambda\in I$.

In other words, there exists precisely one curve through each point $p\in U$.
Given a spacelike hypersurface $\Sigma\subset M$, we can define two kinds of
geodesic congruences through it.

######  Definition 9.

(Ingoing and outgoing null geodesic congruence)
Given a spacetime $(M,\mathbf{g})$, let $U\subseteq M$ be open in $M$, and let
$\Sigma\subset M$ be a spacelike hypersurface. Let $\mathsf{T}$ be a ($N-2$)
dimensional closed (compact and without boundary) submanifold of $M$ such that
$\mathsf{T}\subset\Sigma$. Let $\mathcal{N}\in\Gamma(\mathcal{T}\Sigma)$ be
the globally defined unit normal vector field to $\mathsf{T}$. Further, define
a set,

$\mathcal{T}Q\coloneqq\bigcup_{p\in\mathsf{T}}\mathcal{T}_{p}M.$ (35)

Then, a future-directed vector field $\mathcal{N}^{-}\in\Gamma(\mathcal{T}Q)$
satisfying $\forall
p\in\mathsf{T}:\mathbf{g}_{p}(\mathcal{N}(p),\mathcal{N}^{-}(p))=-1$ is called
the ingoing null normal field. Similarly, a future-directed vector field
$\mathcal{N}^{+}\in\Gamma(\mathcal{T}Q)$ satisfying $\forall
p\in\mathsf{T}:\mathbf{g}_{p}(\mathcal{N}(p),\mathcal{N}^{+}(p))=1$ is called
the outgoing null normal field. The integral curves of the ingoing null normal
field form the ingoing null geodesic congruence (INGC), and the integral
curves of the outgoing null normal field form the outgoing null geodesic
congruence (ONGC).

The reader is referred to fig. 2 for a visualization of this concept. Once we
have defined a geodesic congruence, we can discuss its behaviour in terms of
three parameters - expansion, vorticity and shear [11]. For the purpose of
this paper, we consider only hypersurface orthogonal congruences, which means
their vorticity vanishes [11]. The expansion of a geodesic congruence can be
used to define trapped surfaces and the apparent horizon.

Figure 2:  On the LHS, we have cartoon depicting the concepts of ingoing and
outgoing geodesic congruences. At any point $p\in M$, we may define two
vectors $\mathcal{N}^{-}(p)$ and $\mathcal{N}^{+}(p)$, that form the tangents
to ingoing and outgoing curves $\gamma_{i}$ and $\gamma_{o}$ through that
point. In the theory of gravitational collapse, the terminology “ingoing” and
“outgoing” is often used for geodesics running into or emanating from a
singularity. In the latter case, we refer to the singularity as being naked.
On the RHS, we have a cartoon depicting a trapped surface. For such surfaces,
ingoing and outgoing geodesics converge in the future (or past).

######  Definition 10.

(Null mean curvature/ expansion)
Let the conditions in definition 9 hold. Let
$\leftidx{{}^{\mathsf{T}}}{\mathbf{g}}{}$ be the metric induced on
$\mathsf{T}$ from $M$, and for each $p\in\mathsf{T}$, let
$\\{\mathbf{e}_{1},\dots\mathbf{e}_{N-2}\\}$ be an orthonormal basis for
$\mathcal{T}_{p}\mathsf{T}$. Then, the ingoing and outgoing null mean
curvatures $\Theta^{-}$ and $\Theta^{+}$ of $\mathsf{T}$ at $p\in\mathsf{T}$
are defined as,

$\Theta^{\mp}(p)\coloneqq\mathsf{tr}_{\leftidx{{}^{\mathsf{T}}}{\mathbf{g}}{}}(\nabla\mathcal{N}^{\mp})(p)\coloneqq\sum_{i=1}^{N-2}\leftidx{{}^{\mathsf{T}}}{\mathbf{g}}{}_{p}(\nabla_{\mathbf{e}_{i}}\mathcal{N}^{\mp}(p),\mathbf{e}_{i}).$
(36)

Moreover, since $N^{-}$ and $N^{+}$ are the tangent fields to null geodesics,
this is also called the expansion of a null geodesic congruence through
$\mathsf{T}$.

######  Definition 11.

(Trapped surface, marginally trapped surface, MOTS)
Let $\mathsf{T}$ be defined as in definition 9. Then,

  1. (i)

$\mathsf{T}$ is called a trapped surface if $\ \forall\
p\in\mathsf{T}:\Theta^{\mp}(p)<0$ (see fig. 2).

  2. (ii)

$\mathsf{T}$ is called a marginally trapped surface if $\ \forall\
p\in\mathsf{T}:\Theta^{\mp}(p)\leq 0$.

  3. (iii)

$\mathsf{T}$ is called a marginally outer trapped surface (MOTS) if $\
\forall\ p\in\mathsf{T}:\Theta^{+}(p)\leq 0.$

######  Definition 12.

(Edge, partial Cauchy surface, trapped region, apparent horizon)
The edge of a spacelike hypersurface $\Sigma\subset M$ is the set of points
$p\in\bar{\Sigma}=\Sigma\cup\partial\Sigma$ such that for every open
neighbourhood $U\subset\bar{\Sigma}$ of $p$, there exists a causal curve
$\gamma:J^{-}(p)\to J^{+}(p)$ that does not intersect $\Sigma$. Further,
$\Sigma$ is called a partial Cauchy surface if its edge is empty. Moreover, if
$\Sigma$ is a partial Cauchy surface, then a trapped region is a $(N-1)$
dimensional submanifold $\mathscr{T}\subset\Sigma$ whose boundary
$\mathscr{A}=\partial\mathscr{T}$ is a MOTS with $\Theta^{+}(p)=0$ for every
$p\in\mathscr{A}$. The boundary $\mathscr{A}$ is called an apparent horizon.

For the general spherically symmetric metric $\mathbf{g}$ given by equation
(1), the expansion for an ONGC can be written (in the comoving coordinates) in
terms of the metric components [19], using an orthonormal null tetrad as,

$\forall\
(t,r)\in\mathbf{X}:~{}\Theta^{+}(t,r)=\frac{N-2}{R^{N-3}(t,r)}\left(1-\sqrt{\frac{F(t,r)}{R^{N-3}(t,r)}}\right).$
(37)

Hence for a general spherically symmetric metric, the apparent horizon is
characterized by the following condition:

$\forall\ (t,r)\in\mathbf{X}:~{}\Theta^{+}(t,r)=0\iff F(t,r)=R^{N-3}(t,r).$
(38)

Here $(t,r)$ represents the image of some point $p\in M$ under the comoving
chart maps.

###  3.1 Conditions for the formation of Black Holes

The apparent horizon has a direct relation with geometric wave optics. This
can be seen by realizing that equations (9) and (38) together require that the
eikonal equation $\mathbf{g}^{\sharp}_{\ p}(dR,dR)=0$, for the general
spherically symmetric metric [60, 61] given by equation (1) holds. Hence, at
the apparent horizon, the gradient of the physical radius is a null covector.
If the shell-focussing singularity is formed after the apparent horizon, it
ends up being covered by the apparent horizon, and hence in this case, the
matter cloud collapses (in finite time) to a black hole. Using equation (24),
we find the epoch $v=v_{\text{ah}}(r)$ corresponding to the formation of the
apparent horizon, is a root of the following equation:

$v^{N-3}(t,r)=r^{2}\tilde{\mathcal{M}}(v,r).$ (39)

Using the time curve given by equation (31), we get the expression for the
apparent horizon curve $t_{\text{ah}}:\mathbf{X}_{2}\to\mathbb{R}$ as,

$t_{\text{ah}}(r)\coloneqq
t_{s}(r)-\int_{0}^{v_{\text{ah}}(r)}d\bar{v}\frac{e^{-\phi(t,r)}}{\sqrt{\dfrac{\tilde{\mathcal{M}}(\bar{v},r)}{\bar{v}^{N-3}}+\dfrac{b(r)e^{2\tilde{A}(\bar{v},r)}-1}{r^{2}}}}.$
(40)

The apparent horizon curve describes the time taken by a shell labelled by
comoving radius $r$ to cross the apparent horizon. Hence, a sufficient
condition for formation of a black hole as the end state of spherically
symmetric gravitational collapse is that,

$\forall\ r\in\mathbf{X}_{2}:t_{\text{ah}}(r)\leq t_{s}(r).$ (41)

On the other hand, if $t_{\text{ah}}(r)>t_{s}(r)$, then there may exist
outgoing causal curves terminating in the past at the singularity, that may
escape to future infinity. This corresponds to a naked singular end state. In
light of the cosmic censorship conjectures of general relativity, these are of
immediate interest, and we shall now turn towards an investigation of the
formation of naked singularities as end states of gravitational collapse.

###  3.2 Conditions for the Formation of Naked Singularities

A naked singularity is (in principle) characterized by the existence of at
least one past incomplete causal geodesic, i.e., we define a naked singularity
by the existence of such a geodesic. However, for an observer, a single (null)
geodesic emanating from the singularity corresponds to a single wavefront
reaching them. Hence, to have observational and physical relevance, we
investigate the conditions for the existence of a family of outgoing radial
null geodesics (ORNGs) that constitute a set of non-zero measure. Moreover, it
can be shown [20] using the causal boundary constructions of Geroch,
Kronheimer and Penrose [62], that the existence of ORNGs emanating from a
naked singularity implies the existence of non-radial null, as well as
timelike geodesics emanating from the naked singularity. Hence, it is
sufficient to investigate only ORNGs, i.e., the existence of at least one
past-incomplete ORNG is sufficient to say that the spacetime is naked
singular. Firstly, we note that since the time curve is at least
$\mathcal{C}^{2}$, it may be expanded around $r=0$, using Taylor’s theorem as
follows:

$t(v,r)=t(v,0)+\sum_{k=1}^{j\geq 1}r^{k}\chi_{k}(v)+\mathscr{O}(r^{j+1});\ \ \
\noindent\chi_{k}(v)\coloneqq\frac{1}{k!}\frac{d^{k}t}{dv^{k}}\bigg{|}_{r=0}.$
(42)

For ORNGs, the relation between the tangents is given by,

$\frac{K^{t}}{K^{r}}\coloneqq\frac{dt/d\lambda}{dr/d\lambda}=e^{\psi(t,r)-\phi(t,r)}>0.$
(43)

Here $\lambda\in[\lambda_{0},0)\subset\mathbb{R}$ is the affine parameter. We
parameterize the geodesics such that $(v,r)\to(0,0)$ implies $\lambda\to 0$,
i.e., the central singularity is identified by the vanishing of the affine
parameter. In the limit of approach to the singularity, we must have $t\to
t_{s}(r)$ and $R(t,r)\to 0$. We now introduce a new variable $u\coloneqq
r^{\alpha}$, where $\alpha\geq 1$. Then the ORNG equation may be rewritten as,

$\frac{dR}{du}(t,r)=\frac{1}{\alpha}\frac{R^{\prime}(t,r)}{r^{\alpha-1}}\left(1+\frac{\dot{R}(t,r)}{R^{\prime}(t,r)}\frac{dt}{dr}\right).$
(44)

It is easy to see that provided $\dot{R}(t,r)<0$ and $R^{\prime}(t,r)>0$
(which they are by construction), we have,

$\forall(t,r)\in\mathbf{X}:\frac{dR}{du}(t,r)>0\implies\frac{dt}{dr}>0.$ (45)

Defining $X(t,r)\coloneqq R(t,r)/u$, the tangents to ORNGs are described by
$dR/du$ in the $(R,u)$ plane. Using equations (12), (13) and (18) in equation
(44) yields,

$\frac{dR}{du}=\frac{1}{\alpha}\left(X+\dfrac{v^{\frac{N-3}{2}}v^{\prime}r^{\frac{N-3}{2}+2-\alpha\left(\frac{N-1}{2}\right)}}{X^{\frac{N-3}{2}}}\right)\left(\frac{1-(F/R^{N-3})}{\sqrt{G}(\sqrt{G}+\sqrt{H})}\right)\eqcolon\mathcal{U}(X,u).$
(46)

Using the Pfaffian differential equation $dv=\dot{v}dt+v^{\prime}dr=0$, along
a constant $v$ hypersurface, the expression for the term
$v^{(N-3)/2}v^{\prime}$ featuring in equation (46) is obtained as,

$\displaystyle(v^{\frac{N-3}{2}}v^{\prime})(t,r)$
$\displaystyle=e^{\psi(t,r)}\sqrt{v^{N-3}(b_{0}(r)e^{2\tilde{A}(v,r)}+\tilde{h}(v,r))+\tilde{\mathcal{M}}(v,r)})$
(47) $\displaystyle\tilde{h}(v,r)$
$\displaystyle\coloneqq\frac{e^{2\tilde{A}(v,r)}-1}{r^{2}}.$

Note that from equation (46), we find that,

$\lim_{(v,r)\to(0,r)}\frac{F}{R^{N-3}}=\infty\implies\lim_{(v,r)\to(0,r)}\frac{dR}{du}=-\infty.$
(48)

If the naked singularly can be characterized by the existence of positive
$dR/du$ (in the limit $v\to 0$) as a simple root of the ORNG equation in the
limit of approach to the singularity, the following statement holds: every
non-central singularity is not naked; since in the neighbourhood of the
singularity, there do not exist any ORNGs. This is because the tangents to the
geodesics are negative. Owing to the results of theorem 3 (to be discussed
soon), we limit the scope of our paper to the cases where there exist positive
values of $dR/du$ that constitute simple roots of the ORNG equation in the
limit of approach to the singularity. In other words, we restrict ourselves to
central singularities. Since $X(t,r)\coloneqq R(t,r)/u$, the tangents to the
ORNGs may be expressed near the central singularity, using L’Hospital’s rule
as,

$X_{0}\coloneqq\lim_{(v,r)\to(0,0)}\frac{R}{u}=\lim_{(v,r)\to(0,0)}\frac{dR}{du}.$
(49)

In what follows, we give a criterion for the central singularity to be naked.
This generalises the work done in [26] for the case of spherically symmetric
inhomogeneous dust collapse, and the work done in [30] for the specific cases
of spherically symmetric type-I matter collapse with $\chi_{1}\neq 0$.

######  Definition 13.

(Positive root condition)
The future development $(M,\mathbf{g})$ resulting from the $N(>3)$ dimensional
spherically symmetric gravitational collapse of a general type-I matter field
$($from regular initial data subject to the DEC, NEC and the Einstein
equations$)$ is said to satisfy the positive root condition (PRC) if the
function $V:\mathbb{R}\supset I\to\mathbb{R}$ defined by,

$V(X)=X-\mathcal{U}(X,0),$ (50)

admits at least one real positive root $X_{0}\in\mathbb{R}^{+}$. Here,

$\displaystyle\mathcal{U}(X,0)$
$\displaystyle\coloneqq\frac{1}{\alpha}\left(X+\frac{A}{X^{\frac{N-3}{2}}}\right)\left(1+\frac{B}{X^{\frac{N-3}{2}}}\right)~{}\textrm{where}$
(51) $\displaystyle\mathbb{R}\ni A$
$\displaystyle\coloneqq\sqrt{\tilde{\mathcal{M}}_{00}}\left(\lim_{r\to
0}\left(\sum_{k=1}^{j\geq 1}k\
r^{(k+1)+\frac{N-3}{2}-\alpha\left(\frac{N-1}{2}\right)}\chi_{k}(0)+\mathscr{O}\left(r^{(j+3)+\frac{N-3}{2}-\alpha\left(\frac{N-1}{2}\right)}\right)\right)\right);$
$\displaystyle\mathbb{R}\ni B$
$\displaystyle\coloneqq-\sqrt{\tilde{\mathcal{M}}_{00}}\lim_{r\to
0}r^{\frac{N-1-\alpha(N-3)}{2}};\ \ \
\tilde{\mathcal{M}}_{00}\coloneqq\tilde{\mathcal{M}}(0,0).$

######  Remark 3.

In definition 13, for $V$ to be well defined we must have,

$\alpha\in\left\\{\frac{2k+N-1}{N-1}:\mathbb{N}\ni
k\leq\frac{N-1}{N-3}\right\\}$ (52)

If (52) does not hold, either $A$ or $B$ blows up to $\pm\infty$, thereby
resulting in an ill-definition of $V$.

######  Definition 14.

(Simple positive root condition)
The future development $(M,\mathbf{g})$ is said to satisfy the simple positive
root condition (SPRC) if the function $V$ in definition 13 admits a simple
positive real root.

######  Theorem 2.

(Positive root theorem - PRT)
If the future development $(M,\mathbf{g})$ resulting from the $N$ dimensional
spherically symmetric gravitational collapse of a general type-I matter field
$($from regular initial data subject to the DEC, NEC and the Einstein
equations$)$ contains at least one past-incomplete ORNG, then $(M,\mathbf{g})$
satisfies the positive root condition.

###### Proof.

Assume that $(M,\mathbf{g})$ contains at least one past-incomplete ORNG. It
will follow the ORNG equation given by equation (46). Using equation (30) and
(42) and the Pfaffian differential equation $dv=0$, in the limit of approach
to the central singularity, we have,

$\lim_{(v,r)\to(0,0)}v^{\frac{N-3}{2}}v^{\prime}=\sqrt{\tilde{\mathcal{M}}_{00}}\left(\left(\sum_{k=1}^{j\geq
1}k\ r^{k-1}\chi_{k}(0)\right)+\mathscr{O}(r^{j+1})\right).$ (53)

Using equations (27)-(29), we find that,

$\lim_{(t,r)\to(t_{s_{0}},0)}G=1.$ (54)

Using equation (54) in the limiting case of equation (13) yields,

$\lim_{(t,r)\to(t_{s_{0}},0)}\sqrt{\frac{F}{R^{N-3}}}=1-\frac{\sqrt{\tilde{\mathcal{M}}_{00}}}{X_{0}^{\frac{N-3}{2}}}\lim_{r\to
0}r^{\frac{N-1-\alpha(N-3)}{2}}\eqcolon 1+\frac{B}{X_{0}^{\frac{N-3}{2}}}.$
(55)

Using equations (53)-(55), the ORNG equation (46) can be evaluated in the
limit of approach to the central singularity, to yield the required equation,

$\displaystyle X_{0}$
$\displaystyle=\frac{1}{\alpha}\left(X_{0}+\frac{\sqrt{\tilde{\mathcal{M}}_{00}}}{X_{0}^{\frac{N-3}{2}}}\left(\lim_{r\to
0}\left(\sum_{k=1}^{j\geq 1}k\
r^{(k+1)+\frac{N-3}{2}-\alpha\left(\frac{N-1}{2}\right)}\chi_{k}(0)+\mathscr{O}\left(r^{(j+3)+\frac{N-3}{2}-\alpha\left(\frac{N-1}{2}\right)}\right)\right)\right)\right)$
(56) $\displaystyle\ \ \ \ \
\times\left(1-\frac{\sqrt{\tilde{\mathcal{M}}_{00}}}{X_{0}^{\frac{N-3}{2}}}\lim_{r\to
0}r^{\frac{N-1-\alpha(N-3)}{2}}\right)$
$\displaystyle\eqcolon\frac{1}{\alpha}\left(X_{0}+\frac{A}{X_{0}^{\frac{N-3}{2}}}\right)\left(1+\frac{B}{X_{0}^{\frac{N-3}{2}}}\right)\eqcolon\mathcal{U}(X_{0},0).$

Here, using (45), since we have considered ORNGs, we find that
$X_{0}\in\mathbb{R}^{+}$ provided equation (52) holds.
∎

It is important to note that there may exist a class of values of $X_{0}$ that
satisfy $V(X)=0$, however, it need not be true that each of these values of
$X_{0}$ correspond to a tangent to singular null geodesics terminating at the
central singularity in the past. In order to prove that the existence of a
positive real root of $V(X)$ is sufficient to declare the central singularity
naked, we must therefore assume first that there exists an
$X=X_{0}\in\mathbb{R}^{+}$ that satisfies $V(X)=0$, and then check if we can
construct outgoing singular null geodesics with $X_{0}$ as its tangent in
$(R,u)$ coordinates. We show this for the special case where $X_{0}$ is a
simple root of $V$.

######  Theorem 3.

(Simple positive root theorem - SPRT)
If $(M,\mathbf{g})$ satisfies the simple positive root condition, then it
contains at least one past-incomplete ORNG.

###### Proof.

Assume there exists a simple positive real root $X_{0}\in\mathbb{R}^{+}$ of
the function $V$ defined in theorem 2. Then we may write [26],

$V(X)=(X-X_{0})(h_{0}-1)+h(X).$ (57)

Here, $h_{0}\coloneqq h(X_{0})$ and $h(X)$ is to be chosen such that,

$\frac{dh}{dX}\bigg{|}_{X=X_{0}}=0.$ (58)

In other words, $h(X)$ must be chosen such that it contains only nonlinear
terms in $(X-X_{0})$. Differentiating equation (57) yields,

$h_{0}=\frac{dV}{dX}\bigg{|}_{X=X_{0}}+1.$ (59)

Differentiate equation (50) using equation (51) and evaluating at $X=X_{0}$.
Then substituting the result in equation (59), we get the following expression
for $h_{0}$:

$h_{0}=\frac{A(N-3)(2BX_{0}^{2}+X_{0}^{\frac{N+1}{2}})+B(N-5)X_{0}^{\frac{N+3}{2}}+2X_{0}^{N}(\alpha-1)}{2\alpha
X_{0}^{N}}.$ (60)

Now consider the following equation in the ($X,u$) plane:

$\frac{dX}{du}=\frac{1}{u}\left(\frac{dR}{du}-X\right)=\frac{\mathcal{U}(X,u)-X}{u}.$
(61)

The solutions of this equation (if they exist) are trajectories of RNGs in the
$(X,u)$ plane of the form $u=u(X)$. If these trajectories terminate (in the
past) at a singularity with tangents $X=X_{0}$, then as $u\to 0$, we have
$X\to X_{0}$. Substituting equations (50) and (60) in equation (61), and
defining $S(X,u)\coloneqq h(X)+\mathcal{U}(X,u)-\mathcal{U}(X,0)$, we find
that,

$\frac{dX}{du}=\frac{(X-X_{0})(h_{0}-1)}{u}+\frac{S(X,u)}{u}.$ (62)

Multiplying this equation by $u^{h_{0}-1}$ and integrating with respect to $u$
yields,

$X-X_{0}=Eu^{h_{0}-1}+u^{h_{0}-1}\int S(X,u)u^{-h_{0}}\ du.$ (63)

These represent a family of solutions to equation (61). Since
$X_{0}\in\mathbb{R}^{+}$, we find that in the limit of approach to the central
singularity, $dR/du>0$, and hence equation (63) represents trajectories of
past-incomplete ORNGs in the ($X,u$) plane. Note that $E$ is an integration
constant that labels different ORNGs in the $(X,u)$ plane. In the limit of
approach to the central singularity, i.e., in the limit $(X,u)\to(X_{0},0)$,
the second term in equation (63) vanishes as $S\to 0$ by construction.
However, the first term vanishes if either (i) $E=0$ and $h_{0}\leq 1$, or
(ii) $E=0$ and $h_{0}>1$, or (iii) $E\neq 0$ and $h_{0}>1$. In cases (i) and
(ii), we find that a single ORNG terminates in the past with tangent
$X=X_{0}$, and in case (iii), we find that an infinite family of ORNGs
terminate at the singularity in the past with tangents $X=X_{0}$, each ORNG
being labelled by a distinct value of the integration constant $E$. Using
equation (60), we find that if $V(X)$ is an increasing function of $X$ at
$X=X_{0}$, then a non-zero measure set of ORNGs emanate from the singularity.
In any case, if $X_{0}\in\mathbb{R}^{+}$ satisfies equation (50), then there
exists at least one past-incomplete ORNG in $(M,\mathbf{g})$ that terminates
in the past at the central singularity with tangent $X=X_{0}$.
∎

Theorems 2 and 3 provide necessary and sufficient conditions (respectively)
for the central singularity to be naked. Moreover, the restriction on $\alpha$
given by equation (52) also represents a necessary condition for the central
singularity to be naked, since it is required for the well-definition of $V$.
Note that substituting $k=1$ in equation (52) yields the choice
$\alpha=(N+1)/(N-1)$, which is the only value considered in [30]. Moreover,
substituting $N=4$ in equation (52) recovers the results of [27], however with
the correction that $k\leq 3$. Hence, we have generalized the results of [30]
for any value of the $\alpha$ parameter.

##  4 Gravitational Strength of the Naked Singularity

We have now established that there exist classes of solutions to the Einstein
field equations subject to the DEC and NEC, that admit a naked singularity. It
is often argued that singularities must only be taken seriously if they are of
the so-called strong curvature type. The idea of a strong curvature
singularity goes back to 1977, when Ellis and Schmidt [63] first defined the
gravitational strength of a singularity by referring to the destruction or
crushing of objects approaching the singularity. In the same year, Tipler [25]
provided a mathematically rigorous formulation of this idea, by utilising the
notion of the volume form defined by the Jacobi fields along causal geodesics
approaching the singularity.

######  Definition 15.

(Volume form along a geodesic)
Let $M$ be an $N$ dimensional Lorentzian manifold and
$\gamma:\mathbb{R}\supseteq I\to M$ be a causal geodesic. Then the volume form
is a $(N-m)$-form along $\gamma$, given by the wedge product of the $(N-m)$
one-forms corresponding to the $(N-m)$ independent vorticity-free Jacobi
fields orthogonal to $\gamma$ $($where $m=1$ for timelike and $m=2$ for null
geodesics$)$, i.e.,

$\forall\ \lambda\in
I:~{}\mathsf{vol}(\lambda)\coloneqq(J_{1}(\lambda))^{\flat}\wedge\dots\wedge(J_{N-m}(\lambda))^{\flat}\in\bigwedge^{N-m}\mathcal{T}^{*}_{\gamma(\lambda)}M.$
(64)

Here, $\flat:\mathcal{T}M\to\mathcal{T}^{*}M$ is the standard musical
isomorphism.

######  Definition 16.

(Strong curvature singularity, Tipler [25])
Let $(M,\mathbf{g})$ be singular, and let $\gamma:[\lambda_{0},0)\to M$ be an
incomplete and inextendible causal geodesic, such that its affine parameter
$\lambda\to 0$ in the limit of approach to the singularity. Then the curve
$\gamma$ is said to terminate strongly if,

$\lim_{\lambda\to 0}\mathsf{vol}(J_{1},\cdots,J_{N-m})=0.$ (65)

The singularity itself is called a strong curvature singularity if all causal
geodesics of $M$ terminate strongly.

In 1985, Clarke and Krolak [24] gave a sufficient condition for a curve to
terminate strongly, in terms of the rate of growth of the Ricci curvature in
the limit of approach to the singularity.

######  Definition 17.

(Strong limiting focussing condition, Clarke and Krolak [24])
Let $(M,\mathbf{g})$ be singular and satisfy the causal convergence condition,
i.e., if for all causal curves $\gamma:\mathbb{R}\supseteq I\to M$ with
parameter $\lambda\in I$ and tangents
$\mathsf{K}(\lambda)\in\mathcal{T}_{\gamma(\lambda)}M$, we have,

$\mathbf{Ric}_{\gamma(\lambda)}(\mathsf{K}(\lambda),\mathsf{K}(\lambda))\geq
0,$ (66)

then $\gamma$ is said to satisfy the strong limiting focussing condition
(SLFC) if the following integral is non-integrable on $I$:

$\int_{I}\mathbf{Ric}(\mathsf{K},\mathsf{K})\ d\lambda.$ (67)

######  Proposition 1.

(Curvature growth condition, Clarke and Krolak [24])
A causal geodesic $\gamma:[\lambda_{0},0)\to M$ with tangents
$\mathsf{K}(\lambda)\in\mathcal{T}_{\gamma(\lambda)}M$ and $\lambda\to 0$ at
the singularity, terminates strongly if the SLFC holds along $\gamma$. A
sufficient condition for this is as follows:

$\lim_{\lambda\to 0}\lambda^{2}\ \mathbf{Ric}(\mathsf{K},\mathsf{K})>0.$ (68)

We refer to this as the curvature growth condition (CGC), since it implies a
strong rate of growth of the Ricci curvature along $\gamma$ as it approaches
the singularity.

It follows that if the CGC is satisfied along all causal geodesics in $M$,
then the singularity is by definition strong-curvature type. Moreover, for the
purposes of this paper, the assumed spherical symmetry leads to the following
useful result, the clear proof of which, the authors could not find in
existing literature.

######  Proposition 2.

A spherically symmetric spacetime $(M,\mathbf{g})$ satisfying the DEC and NEC,
with $\mathbf{g}$ given by equation $($1$)$, contains a strong curvature
singularity if the CGC holds along at least one ORNG
$\gamma:[\lambda_{0},0)\to M$.

###### Proof.

We work in the comoving coordinates as defined earlier. In these coordinates,
the components of the tangents to outgoing causal geodesics are related as
follows [27]:

$\displaystyle\left(\frac{dt}{d\lambda}\right)^{2}$
$\displaystyle=\frac{1}{e^{2\phi(t,r)}}\left[\left(\frac{dr}{d\lambda}\right)^{2}e^{2\psi(t,r)}+\frac{\ell^{2}}{R^{2}(t,r)}-\mathcal{B}(\lambda)\right];$
(69a) $\displaystyle\left(\frac{d\theta^{1}}{d\lambda}\right)^{2}$
$\displaystyle+\sum_{i=2}^{N-2}\left(\prod_{j=1}^{i-1}\sin^{2}\theta^{j}\right)\left(\frac{d\theta^{i}}{d\lambda}\right)^{2}=\frac{\ell^{2}}{R^{4}(t,r)}.$
(69b)

Here, $\ell$ is a conserved quantity called the impact parameter [27], which
vanishes for any radial curve. Moreover, we have defined,

$\forall\
\lambda\in[\lambda_{0},0):~{}\mathcal{B}(\lambda)\coloneqq\mathbf{g}_{\gamma(\lambda)}(\mathsf{K}(\lambda),\mathsf{K}(\lambda)).$
(70)

For outgoing causal geodesics, we have,

$\mathbf{Ric}(\mathsf{K},\mathsf{K})=\mathbf{T}(\mathsf{K},\mathsf{K})-\frac{\mathcal{B}(\lambda)}{N-2}\mathsf{tr}_{\mathbf{g}}\mathbf{T}.$
(71)

Here, $\mathbf{T}$ is the matter field over $M$. Using equations (2), (69a)
and (69b), and substituting $B(\lambda)=-1$, we find that for any non-radial
timelike geodesic,

$\displaystyle\lim_{\lambda\to 0}\lambda^{2}\
\mathbf{Ric}(\mathsf{K},\mathsf{K})=$ $\displaystyle\lim_{\lambda\to
0}\lambda^{2}(\rho+p_{r})e^{2\psi}\left(\frac{dr}{d\lambda}\right)^{2}$ (72)
$\displaystyle+\lim_{\lambda\to
0}\lambda^{2}(\rho+p_{\theta})\frac{\ell^{2}}{R^{2}}$
$\displaystyle+\lim_{\lambda\to 0}\frac{\lambda^{2}}{N-2}(\rho+p_{r})$
$\displaystyle+\lim_{\lambda\to 0}\lambda^{2}(\rho+p_{\theta}).$

For radial timelike geodesics, the second term in equation (72) vanishes
(taking $\ell=0$), and for non-radial null geodesics, the last two terms
vanish (taking $B(\lambda)=0$). Note that the first term on the RHS turns out
to be common among radial null geodesics and all other kinds of causal
geodesics. Moreover, from equation (8), it is clear that the NEC [64] implies
that the all the terms on the RHS of equation (72) are non-negative. For the
first term on the RHS of equation (70), since $r\to 0$ as $\lambda\to 0$,
using L’Hospital’s rule, we find that,

$\lim_{\lambda\to
0}\lambda^{2}(\rho+p_{r})e^{2\psi}\left(\frac{dr}{d\lambda}\right)^{2}=\lim_{\lambda\to
0}(\rho+p_{r})r^{2}e^{2\psi}.$ (73)

This limit is the only term featuring in the expression for the CGC for ORNGs.
Since all the other terms in equation (72) are non-negative, we have the
useful result that if the CGC holds along an ORNG, then it holds along all
causal curves, and hence, using proposition 1 and definitions 16 and 17, we
find that the singularity is of strong-curvature type.
∎

As before, it is useful to characterize the properties of the singularity in
terms of the parameter $\alpha$. The positivity (or otherwise) of the limit in
equation (73) depends on what values $\alpha$ takes. In other words, the
requirement that the central naked singularity be of strong-curvature type
leads to a more restriction on $\alpha$ that is already restricted as in (52).
Using the Einstein equations (10a) and (10b), and equations (12), (43) and
(73), the CGC for any ORNG $\gamma:[\lambda_{0},0)\to M$ may be rewritten by
evaluating the limit in the $(v,r)$ plane as,

$\lim_{\lambda\to 0}\lambda^{2}\
\mathbf{Ric}(\mathsf{K},\mathsf{K})=\frac{N-2}{2}\lim_{(t,r)\to(t_{s_{0}},0)}\left(\frac{F^{\prime}}{R^{\prime}}-\frac{\dot{F}}{\dot{R}}\right)\frac{(R^{\prime})^{2}}{R^{N-2}}.$
(74)

Using equation (41) and (46), the limit of $R^{\prime}$ can be written as,

$\lim_{\lambda\to 0}R^{\prime}=\alpha X_{0}\lim_{r\to 0}r^{\alpha-1}.$ (75)

Using this result, in addition to equation (24) and the requirement that
$\tilde{\mathcal{M}}$ must be least $\mathcal{C}^{1}$ in the limit of approach
to the singular epoch, the CGC for ORNGs is satisfied if

$\frac{\alpha(N-2)\tilde{\mathcal{M}}(0,0)}{2X_{0}^{N-3}}\left(\lim_{r\to
0}r^{(3-N)\alpha+N}\right)+\frac{\alpha(N-2)\tilde{\mathcal{M}}(0,0)}{2X_{0}^{N-3}}\left(\lim_{r\to
0}r^{(3-N)\alpha+(N-1)}\right)>0.$ (76)

This inequality restricts the values that $\alpha$ can take in order for the
central singularity to be of the strong curvature type. Indeed, from equation
(76) we find that for any ORNG $\gamma:[\lambda_{0},0)\to M$, we have,

$\alpha\in\left[\frac{N-1}{N-3},\infty\right)\iff\lim_{\lambda\to
0}\lambda^{2}\ \mathbf{Ric}(\mathsf{K},\mathsf{K})>0.$ (77)

In other words, the limit in (76) diverges, and hence the ORNG satisfies the
CGC. Hence, by propositions 1 and 2, all causal geodesics terminate strongly,
and hence the singularity is of the strong curvature type.

######  Theorem 4.

Central naked singularities for $N\in\\{4,5\\}$ are of strong-curvature type
if

$\alpha=\frac{N-1}{N-3}.$ (78)

###### Proof.

This result follows directly from equations (52) and (77), by employing
theorems 2 and 3, and propositions 1 and 2. For the singularity to be naked,
equation (51) must hold. For the singularity to be strong, equation (77) must
hold. We break down our proof into two cases:

  1. (i)

For naked singularities in $N=4$, it is necessary to have,

$\alpha\in\left\\{\frac{5}{3},\frac{7}{3},3\right\\},$ (79)

and for strong curvature singularities in $N=4$, it is sufficient to have
$\alpha=3$. Hence for $\alpha=(N-1)/(N-3)=3$ we get strong curvature naked
singularities in 4 dimensions.

  2. (ii)

For naked singularities in $N=5$, it is necessary to have,

$\alpha\in\left\\{\frac{3}{2},2\right\\},$ (80)

and for strong curvature singularities in $N=5$, it is sufficient to have
$\alpha=2$. Hence for $\alpha=(N-1)/(N-3)=2$ we get strong curvature naked
singularities in 5 dimensions.

In any case, we find that equation (78) must hold.
∎

######  Proposition 3.

Let $(M,\mathbf{g})$ be the future development arising from the $N\geq 6$
dimensional collapse of type-I matter fields subject to the WEC, DEC and
Einstein equations. Further, let $\gamma:[\lambda_{0},0)\to M$ be a past-
incomplete geodesic that terminates at a central naked singularity in the
limit $\lambda\to 0$. Then $\gamma$ does not satisfy the curvature growth
condition.

###### Proof.

It follows from equation (52) that for naked singularities in $N\geq 6$, it is
necessary to have,

$\alpha=\frac{N+1}{N-1},$ (81)

For strong curvature singularities in $N\geq 6$, it is sufficient to have
$\alpha\geq(N-1)/(N-3)$ (using equation (77)). Combining these values of
$\alpha$ with the value of $\alpha$ given by equation (81), we find that in
$N\geq 6$, we have strong curvature naked singularities provided,

$\frac{N+1}{N-1}\geq\frac{N-1}{N-3}.$ (82)

This inequality leads to a contradiction and hence is not realized. Hence if
these naked singularities constitute the end points of a past-incomplete
curves in the future development, then these curves do not satisfy the
curvature growth condition of Clark and Krolak.
∎

######  Remark 4.

Note that equation (51) does not apply in the case $N=3$; hence, in this case,
we cannot proceed using the methodology of theorem 4.

The cosmic censorship conjectures as discussed in section 1 require that all
genuine singularities (i.e., singularities through which the spacetime is
inextendible) must be covered by a horizon. However, through our analysis
here, we have shown that there exists a class of solutions to the Einstein
field equations evolving from spherically symmetric regular initial data (in
at least $N=4$ and $N=5$), that contain strong curvature naked singularities,
and hence the spacetime does not admit a $\mathcal{C}^{0}$ extension through
these naked singularities, i.e., the total spacetime manifold cannot be
isometrically embedded into some other manifold in a continuous fashion. In
particular this means that there exist maximal non-globally hyperbolic
inextendible solutions to the Einstein equations in at least four and five
dimensions. In other words, remark theorem 4 may be interpreted as a violation
of some form of the cosmic censorship conjectures of general relativity, since
these occur for a wide class of matter fields - namely type-I matter fields.
On the other hand, we may interpret proposition 3 as being indicative of a
possible restoration of cosmic censorship in higher dimensions, in the sense
that for the unhindered gravitational collapse of type-I matter fields in
$N\geq 6$, if there exists a subclass of the initial data which evolves to a
maximal development containing past incomplete ORNGs, then the naked
singularity that constitutes the end point of these ORNGs could possibly be
gravitationally weak.

##  5 Concluding Remarks

Here we summarize the results obtained, and thereby provide some future
directions and scope for further research.

  1. (i)

In order to precisely formulate the notion of generic initial data for cosmic
censorship, it is imperative to investigate the gravitational collapse of
specific forms of matter that leads to the formation of a naked singularity.
In this spirit, we constructed here the sufficient conditions for the
existence of naked singularities as the end state of the spherically symmetric
collapse of general type-I matter fields in an arbitrary and finite number of
dimensions $N$. This analysis was previously conducted in [30], but with a
fixed value of the parameter $\alpha=(N+1)/(N-1)$, or equivalently for the
specific case $\chi_{1}(v)\neq 0$. In the present paper, we drop this
specialisation and later restrict the range of $\alpha$ by imposing the
necessary and sufficient conditions for the formation of a naked singularity.

  2. (ii)

The analysis of the strength of the central naked singularity forming in
higher dimensions was not considered in [30]. As such, the physical relevance
of the singularities discussed therein is not accounted for. Here, we derived
sufficient conditions for the naked singularity to be of the strong curvature
type (Tipler strong [25]) in terms of the parameter $\alpha$. This leads to a
class of inextendible spacetimes that admit a strong curvature naked
singularity. This investigation shows that the naked singularities discussed
in [30] need not necessarily be strong curvature singularities.

  3. (iii)

An important point to consider here is the stability and genericity of the
naked singularities in question. One such notion of stability is dynamical
stability. To be considered significant, these naked singularities must not be
removable by a perturbation of the initial data, and they should form for a
sufficiently dense class of initial data. We have not conducted any such
analysis here. Following the general theme of dynamical relativity, one must
again emphasize that it is the initial data that must be judged and not the
final solution. Strictly speaking, one must check whether a generic set of
choices for the initial data leads to a central naked singularity or not; only
then can one speak about the genericity of the naked singularities. In other
words, it is crucial to formalize the notion of genericity of the naked
singularity, in order to make judgements about the cosmic censorship
conjectures. Hence, a possible future scope would be to construct an
appropriate notion of genericity for spherically symmetric gravitational
collapse. These definitions may be motivated, for instance, from the notion of
generic properties of vector fields in the study of dynamical systems [65], or
from topologically induced notion of genericity for some suitable function
space of initial data [20].

  4. (iv)

The naked singularities discussed herein may be either locally or globally
naked, i.e., we have not investigated whether ORNGs escape the collapsing
cloud before or after the apparent horizon does. Although the general
conditions under which a singularity will be globally naked are available in
the existing literature [26], one needs to further investigate under which
conditions do these general conditions hold.

  5. (v)

Although we did not discuss this explicitly for the general class of
spacetimes considered here, it is known that for a few specific subclasses
(like tangential pressure models and dust models), all the singularities for
$N\geq 6$ are black hole singularities [66, 67, 68]. This is due to the fact
that the Taylor expansion of the apparent horizon curve contains a term that
becomes significant beyond a critical dimension, after which the apparent
horizon curve has a decreasing behaviour. We find a somewhat similar result
here, where for $N\geq 6$, a sufficient criterion for checking the curvature
strength of the singularity fails to hold. The formation of strong curvature
naked singularities could be physically better understood in the context of
quantum gravitationally motivated regimes [69, 70, 71, 72, 73, 74, 75, 76, 77,
78, 79, 80], for instance by analyzing the physical effects of extra spatial
dimensions in such theories.

*Acknowledgments KNS would like to thank Oem Trivedi for discussions regarding the interpretation of the results obtained in terms of quantum gravitationally motivated concepts.

*Data availability Data sharing is not applicable to this article as no datasets were generated or analysed during the current study.

*Declarations

*Conflict of interest The authors have no competing interests to declare that are relevant to the content of this article.

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