Marginally
Trapped Surfaces in 4D Einstein-Gauss-Bonnet Theory
Suresh C. Jaryal #1, 2 , Ayan
Chatterjee†1
1Department of
Physics and Astronomical Science, Central University of Himachal Pradesh,
Dharamshala 176215, India
2Department of
Physics and Astronomical Sciences, Central University of Jammu,
Samba, J&K 181143, India
*Corresponding Author E-mail: suresh.fifthd@gmail.com,
ayan.theory@gmail.com
Abstract:
We investigate the
formation and evolution of horizons during gravitational collapse of dust
matter in 4D Einstein- Gauss- Bonnet gravity. The singularity forms as the end
state of the collapse and depending on time formation of these horizons/
marginally trapped surfaces (MTT), the collapse leads to either a black hole or
a massive naked singularity. The effects of the GB coupling parameter (
) and the Misner- Sharp mass function F(
) on the formation of MTT and time of formation of
central singularity are investigated. Our results show that the relationship between
GB coupling parameter (
) and the Misner- Sharp mass function F(
) determines the end state of collapse to be either
black hole or naked singularity. We find that, if
, there are no
trapped surfaces/ MTT on the initial Cauchy hypersurface and the central
singularity if massive and naked. However, for F(
) ≥ 2
, central
singularity is always hidden/ censored by marginally trapped surface of
topology
which eventually
becomes null and coincides with the event horizon at equilibrium. We validated
these results for various matter density profiles using analytic and numerical
techniques.
Keywords: Alternative theory of gravity, Classical black holes,
Naked singularity.
I. INTRODUCTION:
Einstein's general theory of relativity (GR) is the
foundation for our understanding of both small- and large-scale gravitational
phenomena [1, 2]. Although GR has been empirically proven to be accurate and
predictive, its modifications still draw attention [3, 6]. Nowadays, there are
many theories of gravity that are alternatives to Einstein’s theory [7, 8]. Among
these modified theories, we shall study a effective and natural generalization
of GR to
dimensions [9, 10]. This
extended theory has certain unique qualities with higher derivative terms among
the wide class of generic higher order curvature theories. Nevertheless, this alternate
theory's field equations are second-order, much like those in the general
theory of relativity. The EGB
gravity, also known as quadratic order Lovelock theory, allows us to
investigate how higher curvature corrections to blackhole physics significantly
alter the qualitative features available due to black holes in the context of general
relativity. Boulware and Deser's discovery of the spherically symmetric static
solution in the EGB theory [11] led to the generalisation of the
D-dimensional Scwarzschild- Tangherlini black hole [12].
In four dimensions, the GB Lagrangian is known to be a
total derivate and hence, it does not contribute to the equations of motion.
This implies that the usual limit to 4d does not exist at field equations level
and violates the Lovelock theorems. However, it was recently suggested that this
violation of the Lovelock theorems can be avoided, if one rescale the Gauss-
Bonnet coupling constant
then it is
possible that variation of the GB action can result into a non trivial
4-dimensional limit of GB theory [13]. There
have been numerous attempts with dimensional reductions and/ or field
re-definitions, and these efforts have produced some variants of the 4D model [14,
15]. Out of these, in this work, we have followed the dimensional reduction
formalism as given by Lu and Pang [14].
This version of the 4D EGB theory's spherically
symmetric horizon formation is of particular interest to us since it may shed
light on issues pertaining to the cosmic censorship concept. In this work we
will employ the formalism of trapped surfaces [16]. The fundamental of this
formalism is that, when the matter cloud starts to collapse it will eventually
led to the spacetime singularity and during this process trapped surfaces will
emerge. In these trapped surfaces, the null rays orthogonal to the closed
surface have
negative expansion. The boundary of these trapped regions is a
-dimensional surface foliated by
-dimensional spheres and these surfaces are called
trapped or untrapped corresponding to the respective value of expansion
to be less than
or greater than respectively. On the boundary of these regions have negative
ingoing expansion
and the
vanishing of the outgoing expansion
and are called
the marginally trapped spheres (MTS) and the cylinder foliated by MTS is called
a marginally trapped tube (MTT) [17, 18].
In this work, we employ the MTT formalism to study the
collapsing phenomena in the four dimensional EGB gravity. Our study shows that for
the collapse of dustlike matter cloud, if the mass function
is less than
that GB coupling constant
i.e.
, the MTT does
not form on the initial Cauchy hypersurface, implying the spacetime singularity
is massive and naked. However, for F(
) ≥ 2
, the spacetime
singularity is always covered by non-central MTT, and at equilibrium it eventually coincides
with the event horizon. We validated our results for various matter density
profiles using analytic and numerical techniques [17, 18].
II. Action, Field Equations and Solution:
Einstein- Gauss-
Bonnet (EGB) theory of gravity is a natural generalization of the GR. The D-dimensional
EGB gravity has the following the following Lagrangian functional [9,10,17]:
(1)
where
,
,
and
are Ricci
scalar, D- dimensional GB constant, cosmological constant and Lagrangian of the
matter field
, respectively,
is the GB
functional in D- dimensions with the tensor field
.
The line element of spherically symmetric distribution
of collapsing dust like matter cloud is
, (2)
The energy
momentum of the dustlike matter cloud is given by
, (3)
where
is the unit
timelike four velocity vector satisfying
= −1 and
is the energy
density. The set of non zero EGB field
equations takes the following form [17]:
, (4)
, (5)
, (6)
(7)
, (8)
where
is the Misner
Sharp mass function and the functions
and
are defined as
;
. The collapsing scenario requires
<0. By using these Eqns. (4)-(8), we find that
;
. The interior
metric Eqn (2) becomes
(9)
The
choice of
stands for the
marginally bund collapse when
and for bound
and unbound collapse, when
is greater or
less than zero, respectively. As stated earlier, MTT forms the boundary of
black hole region. So, we'll utilise the MTT formulation to find horizons. The
MTT formalism also has the advantage of allowing a direct examination of the
horizon's growth process. For instance, as matter passes through, the MTT's
radius grows. To study this formalism, let
is tangent to
the MTT and depending upon the sign/ nature of the quantity
, the corresponding
becomes null,
spacelike or timelike. For the dustlike matter in EGB gravity, the form of
is given by
[17]
,
(10)
where
represent the
area of the MTT,
and
can be
expressed in terms of the matter variables density and mass function [17, 18].
It is observed that
, when there is no matter shells are falling and it
shows that the MTT is null. When there are matter shells falling
becomes
positive, implies that the MTT is spacelike [17].
III.
Discussion of the Results:
In the present
work, we consider different matter- density profiles to study the gravitational
collapse of the dustlike matter cloud in 4D EGB theory of gravity. Our study
presents quite interesting results which does not emerges in its counterpart
theory i.e. 4D general relativity [17]. Our results shows that the formation of
the trapped surfaces/ MTT are delayed until the mass
of the
collapsing shell becomes equals to that of the
. The absence of the MTT/ trapped surfaces in the
region
implies that the
central spacetime singularity is massive and always naked in 4D EGB. As mass
equals
, the non
central MTT begins to form and when
, the MTT/ trapped surfaces covers the singularity. It
is interesting
Marginally
Trapped Surfaces in 4D Einstein-Gauss-Bonnet Theory
Suresh C. Jaryal #1, 2 , Ayan
Chatterjee†1
1Department of
Physics and Astronomical Science, Central University of Himachal Pradesh,
Dharamshala 176215, India
2Department of
Physics and Astronomical Sciences, Central University of Jammu,
Samba, J&K 181143, India
*Corresponding Author E-mail: suresh.fifthd@gmail.com,
ayan.theory@gmail.com
Abstract:
We investigate the
formation and evolution of horizons during gravitational collapse of dust
matter in 4D Einstein- Gauss- Bonnet gravity. The singularity forms as the end
state of the collapse and depending on time formation of these horizons/
marginally trapped surfaces (MTT), the collapse leads to either a black hole or
a massive naked singularity. The effects of the GB coupling parameter (
) and the Misner- Sharp mass function F(
) on the formation of MTT and time of formation of
central singularity are investigated. Our results show that the relationship between
GB coupling parameter (
) and the Misner- Sharp mass function F(
) determines the end state of collapse to be either
black hole or naked singularity. We find that, if
, there are no
trapped surfaces/ MTT on the initial Cauchy hypersurface and the central
singularity if massive and naked. However, for F(
) ≥ 2
, central
singularity is always hidden/ censored by marginally trapped surface of
topology
which eventually
becomes null and coincides with the event horizon at equilibrium. We validated
these results for various matter density profiles using analytic and numerical
techniques.
Keywords: Alternative theory of gravity, Classical black holes,
Naked singularity.
I. INTRODUCTION:
Einstein's general theory of relativity (GR) is the
foundation for our understanding of both small- and large-scale gravitational
phenomena [1, 2]. Although GR has been empirically proven to be accurate and
predictive, its modifications still draw attention [3, 6]. Nowadays, there are
many theories of gravity that are alternatives to Einstein’s theory [7, 8]. Among
these modified theories, we shall study a effective and natural generalization
of GR to
dimensions [9, 10]. This
extended theory has certain unique qualities with higher derivative terms among
the wide class of generic higher order curvature theories. Nevertheless, this alternate
theory's field equations are second-order, much like those in the general
theory of relativity. The EGB
gravity, also known as quadratic order Lovelock theory, allows us to
investigate how higher curvature corrections to blackhole physics significantly
alter the qualitative features available due to black holes in the context of general
relativity. Boulware and Deser's discovery of the spherically symmetric static
solution in the EGB theory [11] led to the generalisation of the
D-dimensional Scwarzschild- Tangherlini black hole [12].
In four dimensions, the GB Lagrangian is known to be a
total derivate and hence, it does not contribute to the equations of motion.
This implies that the usual limit to 4d does not exist at field equations level
and violates the Lovelock theorems. However, it was recently suggested that this
violation of the Lovelock theorems can be avoided, if one rescale the Gauss-
Bonnet coupling constant
then it is
possible that variation of the GB action can result into a non trivial
4-dimensional limit of GB theory [13]. There
have been numerous attempts with dimensional reductions and/ or field
re-definitions, and these efforts have produced some variants of the 4D model [14,
15]. Out of these, in this work, we have followed the dimensional reduction
formalism as given by Lu and Pang [14].
This version of the 4D EGB theory's spherically
symmetric horizon formation is of particular interest to us since it may shed
light on issues pertaining to the cosmic censorship concept. In this work we
will employ the formalism of trapped surfaces [16]. The fundamental of this
formalism is that, when the matter cloud starts to collapse it will eventually
led to the spacetime singularity and during this process trapped surfaces will
emerge. In these trapped surfaces, the null rays orthogonal to the closed
surface have
negative expansion. The boundary of these trapped regions is a
-dimensional surface foliated by
-dimensional spheres and these surfaces are called
trapped or untrapped corresponding to the respective value of expansion
to be less than
or greater than respectively. On the boundary of these regions have negative
ingoing expansion
and the
vanishing of the outgoing expansion
and are called
the marginally trapped spheres (MTS) and the cylinder foliated by MTS is called
a marginally trapped tube (MTT) [17, 18].
In this work, we employ the MTT formalism to study the
collapsing phenomena in the four dimensional EGB gravity. Our study shows that for
the collapse of dustlike matter cloud, if the mass function
is less than
that GB coupling constant
i.e.
, the MTT does
not form on the initial Cauchy hypersurface, implying the spacetime singularity
is massive and naked. However, for F(
) ≥ 2
, the spacetime
singularity is always covered by non-central MTT, and at equilibrium it eventually coincides
with the event horizon. We validated our results for various matter density
profiles using analytic and numerical techniques [17, 18].
II. Action, Field Equations and Solution:
Einstein- Gauss-
Bonnet (EGB) theory of gravity is a natural generalization of the GR. The D-dimensional
EGB gravity has the following the following Lagrangian functional [9,10,17]:
(1)
where
,
,
and
are Ricci
scalar, D- dimensional GB constant, cosmological constant and Lagrangian of the
matter field
, respectively,
is the GB
functional in D- dimensions with the tensor field
.
The line element of spherically symmetric distribution
of collapsing dust like matter cloud is
, (2)
The energy
momentum of the dustlike matter cloud is given by
, (3)
where
is the unit
timelike four velocity vector satisfying
= −1 and
is the energy
density. The set of non zero EGB field
equations takes the following form [17]:
, (4)
, (5)
, (6)
(7)
, (8)
where
is the Misner
Sharp mass function and the functions
and
are defined as
;
. The collapsing scenario requires
<0. By using these Eqns. (4)-(8), we find that
;
. The interior
metric Eqn (2) becomes
(9)
The
choice of
stands for the
marginally bund collapse when
and for bound
and unbound collapse, when
is greater or
less than zero, respectively. As stated earlier, MTT forms the boundary of
black hole region. So, we'll utilise the MTT formulation to find horizons. The
MTT formalism also has the advantage of allowing a direct examination of the
horizon's growth process. For instance, as matter passes through, the MTT's
radius grows. To study this formalism, let
is tangent to
the MTT and depending upon the sign/ nature of the quantity
, the corresponding
becomes null,
spacelike or timelike. For the dustlike matter in EGB gravity, the form of
is given by
[17]
,
(10)
where
represent the
area of the MTT,
and
can be
expressed in terms of the matter variables density and mass function [17, 18].
It is observed that
, when there is no matter shells are falling and it
shows that the MTT is null. When there are matter shells falling
becomes
positive, implies that the MTT is spacelike [17].
III.
Discussion of the Results:
In the present
work, we consider different matter- density profiles to study the gravitational
collapse of the dustlike matter cloud in 4D EGB theory of gravity. Our study
presents quite interesting results which does not emerges in its counterpart
theory i.e. 4D general relativity [17]. Our results shows that the formation of
the trapped surfaces/ MTT are delayed until the mass
of the
collapsing shell becomes equals to that of the
. The absence of the MTT/ trapped surfaces in the
region
implies that the
central spacetime singularity is massive and always naked in 4D EGB. As mass
equals
, the non
central MTT begins to form and when
, the MTT/ trapped surfaces covers the singularity. It
is interesting