#  Attractive voids

Raymond Isichei  João Magueijo<EMAIL_ADDRESS>Theoretical Physics
Group, The Blackett Laboratory, Imperial College, Prince Consort Rd., London,
SW7 2BZ, United Kingdom

###### Abstract

We explore the well know mass deficit/surplus phenomenon in General Relativity
to suggest that it could play a part in the dark matter conundrum.
Specifically in collapses and condensations of matter associated with negative
intrinsic curvature of the foliation associated with the asymptotic boundary
conditions, the external (ADM) mass can vastly exceed the integrated local
energy over the internal volume. This can be phrased in terms of a deficit of
volume for a given surface area (with respect to zero curvature). We explore
the phenomenon in the context of generalizations of the Oppenheimer-Snyder
models and other “cut and paste” models, the Lemaitre-Bondi-Tolman metric and
several others. We produce constructions where the internal object is
contracting or expanding, has a life time different from the asymptotic
Universe, as well as a volume different than the escavated volume from the
Universe. These are purely relativistic constructions and they could play a
role in the puzzle of dark matter: attraction without visible or indeed any
matter.

##  I Introduction

It has for long been noted that, with suitable definitions and conditions on
the space-time, the asymptotic mass of an extended object may differ from the
local mass integrated over its volume elements. This “mass deficit” pervades
classics of literature such as Zeldovich and Novikov Zelbook , and can be
interpreted in a quasi-Newtonian setting as the result of the “gravitational
binding energy” of the object. It applies to stable astrophysical objects,
such as stars, but also to collapsing objects. This can be illustrated by the
Oppenheimer-Snyder (OS) toy model of black hole collapse OS ; MTW  where an
internal piece of a collapsing closed FRW model is collated onto an outer
Schwarzchild solution. The matching conditions reveal that the integrated mass
in the FRW interior is larger than the Schwarzchild mass parameter (see e.g.
Misnerstudent ; Stephani ; Guth ). As this example shows, this effect can
transcend quasi-Newtonian physics and be purely relativistic. In the extreme
case we could construct an OS model using a near complete FRW 3-sphere for the
internal section, generating an infinitesimal outside mass, making the system
elusive to detection by the outside world. This would be a frustrated baby
Universe BabyU .

The point of this paper is two-fold. First we aim to generalize the formalism
to more complicated settings, specifically making use of the Lemaitre-Tolman-
Bondi (LTB) metric exactslns ; enqvist ; Goncalves . This will allow us to
consider a multitude of other scenarios where this phenomenon is present,
relevant to gravitational collapse but also to the primordial initial
conditions of our Universe. Second, we wish to highlight the fact that in the
same way that there can be a mass deficit, there can also be a mass surplus.
In extreme situations (converse to the frustrated baby Universe example BabyU
) it is possible to generate asymptotic mass even without any local energy
density. It is tempting to relate this feature to the puzzle of dark matter:
gravitational pull without visible matter. Perhaps some of the “dark” matter
is no matter at all. We provide examples of such “attractive voids”, as in the
title of this paper.

In addition to these two core motivations, we present several examples (and
even a general class of solutions) with the property that not only the
inferred outside mass, but also its inferred outside volume and age are
different from the internal ones. In fact the mass deficit/surplus and the
volume mismatches are related. All of these structures show that extrapolation
from the outside to the inside may miss the mark. This is a simple point,
related to the well know and accepted relativity of time, space and mass, but
we believe it has not been properly digested in the context of dark matter
studies.

Could these objects exist? There are several contexts in which these
structures could come to exist in the real world. They could have an
astrophysical origin in collapses dominated by kinetic energy in a Newtonian
setting or, perhaps more importantly, their relativistic equivalent. Such
collapses might be better characterized as ”implosions”, and complement the
usual picture of gravitational collapse. In a different context, these
structures could also be part of the primordial Universe, forming counterparts
to primordial black holes PBH ; PBHreview . They could even be a component of
the initial conditions of our Universe.

##  II Mass mismatch in Openheimer-Snyder collapse

The collapsing phase of a $k=1$ FRW dust model can be used as a toy model for
collapse into a black hole. This is the Oppenheimer-Snyder (OS) model OS ; MTW
, and it can be generalized Stephani ; Guth  to inside FRW metrics with the
full set of options $k=0,\pm 1$, as we now review. For $\chi\leq\chi_{0}$ we
take:

$ds^{2}=-d\tau^{2}+a^{2}(\tau)(d\chi^{2}+f^{2}(\chi)d\Omega^{2}_{2})$ (1)

where:

$\displaystyle f(\chi)=$ $\displaystyle\sin\chi$ $\displaystyle{\rm for}\;k=1$
$\displaystyle=$ $\displaystyle\chi$ $\displaystyle{\rm for}\;k=0$
$\displaystyle=$ $\displaystyle\sinh\chi$ $\displaystyle{\rm for}\;k=-1.$

The solutions to the Friedmann equations for dust are well known. For $k=0,-1$
one takes the branch containing contracting solutions (unlike for $k=1$, these
are disconnected from expanding solutions, a crucial difference). The outside
metric for all $k$ is the Schwarzchild metric:

$ds^{2}=-\left(1-\frac{R_{S}}{R}\right)dt^{2}+\frac{dR^{2}}{1-\frac{R_{S}}{R}}+R^{2}d\Omega^{2}_{2}$
(2)

(where $R_{S}=2GM$). The matching is performed on a radial geodesic surface,
so we consider the radial geodesics of (2). These satisfy MTW ; Stephani ;
Guth :

$\left(\frac{dR}{d\tilde{\tau}}\right)^{2}+\left(1-\frac{R_{S}}{R}\right)=E^{2},$
(3)

with $\tilde{\tau}$ the proper time and $E$ an integration constant. We must
select the appropriate $E$ to obtain the correct match to the various $k=0,\pm
1$ cases. The $\chi=\chi_{0}$ internal geodesic must match a radial outside
geodesic for some $R=R(\tilde{\tau})$ (with $\tau=\tilde{\tau}$, since $\tau$
in (1) is also the proper time of the $\chi=\chi_{0}$ geodesic). From the form
of the solutions for these geodesics MTW ; Stephani ; Guth  one sees that the
correct matching is achieved with:

$\displaystyle k=1$ $\displaystyle\longleftrightarrow$ $\displaystyle E^{2}<1$
(4) $\displaystyle k=-1$ $\displaystyle\longleftrightarrow$ $\displaystyle
E^{2}>1$ (5) $\displaystyle k=0$ $\displaystyle\longleftrightarrow$
$\displaystyle E^{2}=1.$ (6)

Hence, the internal closed, open and flat FRW models must match bounded,
unbounded and borderline outside geodesics, respectively. More generally we
can check all the jumping conditions, reviewed in Appendix ***, but these do
not add any new conditions.

The matching conditions translate into a condition between the Schwarzchild
(here also ADM) mass $M$ and $\rho$, $a$ and $\chi_{0}$ OS ; MTW ; Stephani ;
Guth :

$\displaystyle M=\frac{4\pi}{3}\rho
a^{3}f^{3}(\chi_{0})=M_{\star}f^{3}(\chi_{0})$ (7)

where we defined

$M_{\star}\equiv\frac{4\pi}{3}\rho a^{3}$ (8)

for later convenience. This is to be contrasted with the integrated local mass
of the dust inside the FRW model:

$M_{F}=4\pi\rho
a^{3}\int_{0}^{\chi_{0}}d\tilde{\chi}\,f^{2}(\chi)=3M_{\star}g(\chi_{0})$ (9)

where:

$\displaystyle g(\chi)=$ $\displaystyle\frac{\chi}{2}-\frac{\sin 2\chi}{4}$
$\displaystyle{\rm for}\;k=1$ $\displaystyle=$
$\displaystyle\frac{\chi^{3}}{3}$ $\displaystyle{\rm for}\;k=0$
$\displaystyle=$ $\displaystyle-\frac{\chi}{2}+\frac{\sinh 2\chi}{4}$
$\displaystyle{\rm for}\;k=-1.$

Except for $k=0$, we have $g\neq f^{3}/3$, leading to the advertised mismatch
between $M$ and $M_{F}$. We see that for $k=1$ ($E^{2}<1$, i.e. bound external
geodesics) we have a mass deficit, $M<M_{F}$. In a Newtonian picture this is
the case where we hide mass in a cloak of binding energy. For $k=-1$
($E^{2}>1$, i.e. unbound external geodesics), we have instead a surplus of
mass, $M>M_{F}$. In a Newtonian picture this can be seen as the effect of the
kinetic energy present, in addition to the rest mass and gravitational binding
energy. But the effect is present even when $\chi_{0}$ is not small, indeed
the larger the $\chi_{0}$, the bigger the mass mismatch. In extreme cases we
therefore have a purely relativistic effect, rendering the Newtonian
interpretation irrelevant.

An extreme case is that of a near-baby Universe BabyU , resulting from $k=1$
and $\chi_{0}=\pi^{-}$. Then, $M=0^{+}$, whilst $M_{F}=3/2M_{\star}$. This is
a hidden world, with its matter and volume cloaked from the outside Universe.
For $k=-1$ and $\chi_{0}\gg 1$ we have the opposite situation (and there is no
limit on $\chi_{0}$). In the extreme case, the internal world is empty: a
Milne Universe; yet the outside world sees matter.

###  II.1 Quantifying the effect

To quantify this effect in terms of more tangible quantities, we must work out
how much of the FRW we must cover in a given OS model, that is the value of
$\chi_{0}$ needed for a given set of parameters defining the object. For the
$k=1$ model this is usually done by examining the relation between mass and
density at the standard initial condition MTW  of turnaround $\dot{a}=0$. Such
a point does not exist for $k=-1$. As a generalization we can extract
$\chi_{0}$ from:

$f^{3}(\chi_{0})=\frac{M}{M_{\star}}.$ (10)

Note that both $M$ and $M_{\star}$ are constants. Up to a numerical factor,
$M_{\star}$ is the first integral of the conservation equation $m=\rho a^{3}$.
We can work it out from $\rho$ and $\Omega$, where as usual
$\Omega=\rho/\rho_{c}$, and $\rho_{c}$ is the value of $\rho$ obtained by
setting $k=0$ in the Friedman equation, keeping the other variables ($a$ and
$\dot{a}$) fixed. If we set $k=\pm 1$, then $a$ has units of length and
satisfies:

$a^{2}=\frac{\alpha}{\rho}\frac{\Omega}{|\Omega-1|},$ (11)

where $\alpha=3/(8\pi G)$ is proportional to the reduced Planck mass squared.
After some manipulations we arrive at:

$M_{\star}(\rho,\Omega)=\frac{4\pi}{3}\frac{\alpha^{3/2}}{\sqrt{\rho}}\left(\frac{\Omega}{|\Omega-1|}\right)^{3/2}$
(12)

With this formula we can evaluate $\chi_{0}$ from the mass, density and the
value of $\Omega$ at any point, that is:

$\chi_{0}=\chi_{0}(M,\rho,\Omega).$ (13)

For the standard OS ($k=1$) at the turnaround point we have $\Omega=\infty$
and:

$M_{\star}(\rho,\infty)=\frac{4\pi}{3}\frac{\alpha^{3/2}}{\sqrt{\rho}}.$ (14)

###  II.2 Geometrical interpretation

The phenomenon we illustrated within OS follows the pattern The outside
gravitational mass of a homogeneous spherical dust region with a given surface
area equals the integrated stress-energy mass over a volume with the same
surface area but flat.  The mass deficit and excess can then be understood
geometrically as the result of this volume being larger or smaller depending
on whether we have positive or negative intrinsic curvature.

---
Figure 1: The mass deficit for a $k=1$ collapse can be understood as a volume
excess (red region) for the same surface area (red line) with respect to
euclidean space (yellow region).
---
Figure 2: The equivalent situation for $k=-1$ can be illustrated using the
fact that geodesics shooting off from the pole diverge, instead of converging
as in the $k=1$ case. Filling in the space between them in red (right picture)
leaves a volume deficit (the regions in white) with respect to an euclidean
space with the same boundary area (yellow region). This translates into a mass
surplus.

This is illustrated in Figs. 1/2 for positive/negative curvature. In the
second case we use the fact that geodesics shooting away from the “pole”
converge for positive curvature, but diverge for negative curvature, in the
sense that the angle between them increases. This allows us to represent the
volume deficit (translating into a mass surplus) for the $k=-1$ case. These
figures are the converse of those in carroll .

We could quantify these interpretation in terms of the dimensionless ratio
between volume and area:

$\displaystyle\frac{V}{A^{3/2}}=\frac{1}{2\sqrt{\pi}r^{3}}\int^{r}_{0}\frac{d\tilde{r}\tilde{r}^{2}}{1-k\tilde{r}^{2}}$
(15)

with a reference flat value of $\frac{1}{6\sqrt{\pi}}$. The Milne limit is
interesting in that in areal coordinates $V\sim 4\pi r$, so this ratio goes
like $1/r$. Asymptotically it can be thought of as a finite area enclosing a
zero volume.

This has an observational meaning. Suppose that we take a near baby Universe
BabyU  full of stars. The brightness for the surface area will be very large.
The Keplerian mass will be negligible. Same in reverse with a $k=-1$ void.
Perfect darkness (zero brightness) but large Keplerian mass. Brightness is
flux, so energy per unit area. The energy comes form the volume but passes
through the surface area. So it is sensitive to this area/volume
deficit/surplus.

##  III Critical review of the foundations of LTB

The Lemaitre-Tolman-Bondi (LTB) formalism exactslns ; enqvist ; Goncalves
starts from the metric111Of the large number of notations used in the
literature we elected enqvist .:

$ds^{2}=-dt^{2}+X^{2}(r,t)dr^{2}+A^{2}(r,t)d\Omega^{2},$ (16)

before it is brought to the LTB form. At this stage only spherical symmetry
has been assumed.

If $A^{\prime}=0$ at isolated points222We do not consider the case where
$A^{\prime}=0$ in an open set; see Theorem 15.7 of exactslns . None of what
follows applies to that case. it may happen that several $r$ have the same
area $A$. In the literature this is usually associated with physical “shell
crossing” exactslns ; Goncalves , implying that points with the same $A$ are
at the same location. We contend that this is the wrong interpretation within
LTB (it leads to a different formalism). Metric (16) implies a foliation with
a normal $n_{\mu}=(1,\mathbf{0})$ which is a geodesic 4-vector333This can be
proved for any metric where $N=N(t)$ (the fact that $N^{i}=0$, valid here, is
not needed). If $n_{\mu}=-Ndt$, then $n_{\mu}=\partial_{\mu}\tilde{t}$ is
locally equivalent to $\partial_{i}N=0$ (so that $dn=0$). But then (since
partials and nablas are equivalent in the exterior derivative)
$\nabla_{\mu}n_{\nu}=\nabla_{\nu}n_{\mu}$, so combined with
$n_{\mu}n^{\mu}=-1$ we have $n^{\mu}\nabla_{\mu}n^{\alpha}=0$.. It is then
assumed that $n^{\mu}$ is also the (geodesic) velocity 4-vector of the total
dust matter at each point: this is why these coordinates are dubbed “comoving”
coordinates. Then the stress energy tensor in these coordinates only has a
$00$ component, since:

$\displaystyle T_{\mu\nu}$ $\displaystyle=\rho\,n_{\mu}n_{\nu}.$ (17)

If shells were to cross at the same physical point, then the stress energy
tensor would instead have the form:

$T^{\mu}_{\;\nu}=\rho_{1}\,n_{\mu}n_{\nu}+\rho_{2}u_{\mu}u_{\nu}$ (18)

with $u^{\mu}\neq n^{\mu}$ in general, so that $T_{\mu\nu}$ would have
components other than 00. In such a situation, the Einstein equations would
not be the ones found in the LTB formalism before the metric is brought to its
final form, viz enqvist :

$\displaystyle-2\frac{A^{\prime\prime}}{AX^{2}}+2\frac{A^{\prime}X^{\prime}}{AX^{3}}+2\frac{\dot{X}\dot{A}}{AX}+\frac{1}{A^{2}}+\left(\frac{\dot{A}}{A}\right)^{2}-\left(\frac{A^{\prime}}{AX}\right)^{2}$
$\displaystyle=8\pi G\rho$ (19)
$\displaystyle\dot{A}^{\prime}-A^{\prime}\frac{\dot{X}}{X}$ $\displaystyle=0$
(20) $\displaystyle
2\frac{\ddot{A}}{A}+\frac{1}{A^{2}}+\left(\frac{\dot{A}}{A}\right)^{2}-\left(\frac{A^{\prime}}{AX}\right)^{2}$
$\displaystyle=0$ (21)
$\displaystyle-\frac{A^{\prime\prime}}{AX^{2}}+\frac{\ddot{A}}{A}+\frac{\dot{A}}{A}\frac{\dot{X}}{X}+\frac{A^{\prime}X^{\prime}}{AX^{3}}+\frac{\ddot{X}}{X}$
$\displaystyle=0.$ (22)

The Hamiltonian content of (19)-(22) can be read off easily. Eq.(19) is the
$00$ component of Einstein’s equations, and so represents the Hamiltonian
constraint. Eq.(20) is their $0i$ component, and so represents the
diffeomorphism or momentum constraint. The last two equations are the $rr$ and
$\theta\theta$ (or $\phi\phi$) components, making up the dynamical or
Hamilton’s equations.

If there were physical shell crossing, these equations would contain an extra
source with non-$00$ components. Specifically, they would have a source term
in the momentum constraint, essential for bringing the metric to the LTB form,
as we will presently show. For LTB to follow, we must assume no matter
currents contribute to the momentum constraint. Thus, the LTB formalism
requires more than spherical symmetry. It requires a specific type of dust
source which precludes shell crossing. It requires that $r$ be a good
coordinate, so that points with different $r$ and the same $A$ are different
points in space and no shell-crossing occurs. This is not that abstruse: in
fact the FRW $k=1$ solution expressed in LTB coordinates offers the simplest
example (as we will see in Section VI.1): as the radial coordinate increases
the area first increases, then past the equator it decreases.

##  IV LTB metric, equations and solutions

With this matter clarified the usual LTB derivations do follow through. In
points where $A^{\prime}\neq 0$, the source-free momentum constraint, Eq.(20),
implies $X=C(r)A^{\prime}$. Writing $C(r)=1/\sqrt{1-K(r)}$ with $K(r)<1$,
reduces the metric to the LTB form:

$ds^{2}=-dt^{2}+\frac{A^{\prime 2}}{1-K(r)}dr^{2}+A^{2}d\Omega^{2}.$ (23)

The remaining Einstein equations can then be brought to:

$\displaystyle\frac{\dot{A}^{2}}{A^{2}}$ $\displaystyle=$
$\displaystyle\frac{F(r)}{A^{3}}-\frac{K}{A^{2}}$ (24)
$\displaystyle\frac{2}{3}\frac{\ddot{A}}{A}+\frac{1}{3}\frac{\ddot{A}^{\prime}}{A^{\prime}}$
$\displaystyle=$ $\displaystyle-\frac{4\pi G}{3}\rho,$ (25)

where

$\frac{F^{\prime}}{A^{\prime}A^{2}}=8\pi G\rho.$ (26)

These can be condensed further into:

$\displaystyle\frac{\dot{A}^{2}}{A^{2}}$ $\displaystyle=$
$\displaystyle\frac{F(r)}{A^{3}}-\frac{K}{A^{2}}$ (27) $\displaystyle\dot{F}$
$\displaystyle=$ $\displaystyle 0$ (28) $\displaystyle\dot{K}$
$\displaystyle=$ $\displaystyle 0$ (29)

with constraint:

$F^{\prime}=8\pi G\rho A^{\prime}A^{2}.$ (30)

Hence the time independent functions $F(r)$ and $K(r)$ can be seen as the
inputs. If we want to read off the density from the constant function $F(r)$
(which parameterizes the solutions) we can reqrite (30) as:

$\rho=\frac{F^{\prime}}{8\pi GA^{\prime}A^{2}}.$ (31)

It implies the conservation law:

$\dot{\rho}+\left(2\frac{\dot{A}}{A}+\frac{\dot{A}^{\prime}}{A^{\prime}}\right)\rho=0.$
(32)

Obviously if $\rho\geq 0$ and finite, we must have that
$F^{\prime}/A^{\prime}>0$, that is $F^{\prime}$ and $A^{\prime}$ must have the
same sign. If $F^{\prime}=0$ then we are in vacuum and $A^{\prime}$ can have
any sign. In points where $A^{\prime}=0$ we must also have $F^{\prime}=0$.

The solutions to this system in regions where $A^{\prime}\neq 0$ are well
known, and separate for each $r$. For $K(r)>0$:

$\displaystyle A$ $\displaystyle=\frac{F}{2K}(1-\cos\eta)$ (33) $\displaystyle
t-t_{0}$ $\displaystyle=\pm\frac{F}{2K^{3/2}}(\eta-\sin\eta);$ (34)

for $K(r)<0$:

$\displaystyle A$ $\displaystyle=\frac{F}{2|K|}(\cosh\eta-1)$ (35)
$\displaystyle t-t_{0}$
$\displaystyle=\pm\frac{F}{2|K|^{3/2}}(\sinh\eta-\eta);$ (36)

and for $K(r)=0$:

$t-t_{0}=\pm\frac{2}{3}\frac{A^{3/2}}{F^{1/2}}$ (37)

or

$A=\left(\frac{9F}{4}\right)^{1/3}|t-t_{0}|^{2/3}$ (38)

where $t_{0}=t_{0}(r)$ is a free function. The $K\neq 0$ solutions reduce to
the $K=0$ ones when $\eta\ll 1$. The limit $\eta\rightarrow\infty$ of $K=-1$
is:

$A=\sqrt{|K|}(t-t_{0}(r))$ (39)

for any $F(r)$. This is equivalent to setting to zero the mass term in (24)
(or to assuming $F\ll|K|A$), so this is a good model for void. If $t_{0}(r)$
is constant this limit reduces to the Milne Universe.

Some comments on these solutions:

  * •

It may seem that there are 3 independent free functions ($F$, $K$ and
$t_{0}$), but $r$ can be redefined by an arbitrary monotonic function, so in
fact there are only 2 independent ones.

  * •

Beware if initial conditions are given in terms of the density, since $A$
appears in its expression (cf. (31)) which in turn contains $F$ within these
solutions. Reading off the appropriate solution may then be non-trivial.

##  V Generating solutions by cutting an pasting

The fact that these solutions seemingly decouple in $r$ does not mean that the
different shells are entirely decoupled, since the functions $F(r)$ and $K(r)$
cannot be completely arbitrary given the continuity of space. In turn, this
affects the process of cutting and pasting solutions, and can be phrased as
jumping conditions. These can be derived directly from the Einstein equations,
or appealing to well-known formalisms for gluing solutions across
hypersurfaces.

We backtrack to (19)-(22) and examine the impact of jumping conditions on
their left hand side, seeking the implications of avoiding a delta function at
a given gluing $r$ (and so the need to introduce a shell of matter in their
right hand side). Assuming first $A^{\prime}\neq 0$ (and also obviously $A\neq
0$), we see that absence of delta functions in Eq.(21) (or (20)) implies that
around any point $r$:

$\displaystyle A(r_{-})=A(r_{+}).$ (40)

Eq.(19) (or (22)) then shows that $A^{\prime}$ and $X$ can jump, but only
keeping their time independent proportionality factor $C(r)$ fixed. This
implies that

$\displaystyle K(r_{-})=K(r_{+}).$ (41)

Other than this, $K$ can change sign or approach 1, its upper bound, at a
gluing point. These are the two conditions for gluing at points where
$A^{\prime}\neq 0$.

In contrast, at isolated points $r=r_{\star}$ where $A^{\prime}=0$, Eq. (20)
only implies $\dot{A}^{\prime}=0$, so such points remain extremes of $A(r)$ at
all times. Of the other jumping conditions:

$A(r_{\star+})=A(r_{\star-})$ (42)

remains the same, but (19) (or (22)) now only implies that $A^{\prime}$ cannot
jump

$A^{\prime}(r_{\star+})=A^{\prime}(r_{\star-})=0$ (43)

whilst $X$ can, so $K$ could jump at these points. This is the opposite of
what happens for points with $A^{\prime}\neq 0$.

We add two important comments to this conclusion. First, we note that, as can
be seen in Appendix A, these conditions can be replicated appealing to the
Israel formalism for cutting and pasting. Second, we stress that there are
different variations for these jumping conditions, as critically reviewed in
bonner . The stronger versions are deemed to be physically surplus to
requirement, and will not be used in this paper.

(Parenthetically, we note that our cutting and pasting approach is almost
orthogonal to that seeking a smooth transition between astrophysical and
cosmological structures; eg baker ; Bolejko ; rivers .)

##  VI Critical review of basic solutions

We first illustrate some of the points made with well known solutions.

###  VI.1 FRW within LTB

The FRW solutions can be obtained by choosing:

$\displaystyle F$ $\displaystyle=\frac{8\pi G}{3}mr^{3}$ (44) $\displaystyle
K$ $\displaystyle=kr^{2}$ (45) $\displaystyle t_{0}$ $\displaystyle=0$ (46)

where $k=0,\pm 1$. Then we can recognize the usual FRW metric (and its
solutions in (33)-(38)) written in terms of a comoving areal coordinate, with
identifications:

$\displaystyle A$ $\displaystyle=a(t)r$ (47) $\displaystyle\rho(t)$
$\displaystyle=\frac{m}{a^{3}}.$ (48)

In terms of these variables, (24) and (25) reduce to the Friedman and
Raychaudhuri equation, and (32) to the FRW conservation equation for dust. For
latter convenience we can define the “Schwarzchild radius” of the FRW Universe
as $r_{S}=2Gm$ (in analogy with $R_{S}=2GM$). Then

$F=r_{S}V_{0}(r)$ (49)

with $V_{0}$ the comoving Euclidean ($k=0$) volume for a given area indexed by
$r$.

We can locally redefine $r$ at will, and a possibility is:

$r=\sin\chi$ (50)

where $\chi$ is the proper distance to the center/pole and also a latitude
angle in the case $k=1$. Indeed this coordinate is needed to cover the whole
sphere in the $k=1$ case. We have $K=r^{2}$ so $K$ touches 1. To cover the
whole sphere we need to use $\chi$. Then $A^{\prime}=0$ at $\chi=\pi/2$, and
there are points with different $\chi$ and the same $A$. This clearly does not
signal shell crossing, but simply a manifold where the area does not increase
monotonically with the radius, as obviously is the case of the 3-sphere. This
is an obvious point, illustrated in Fig. 3, but it helps to understand more
complicated solutions later.

---
Figure 3: Simplest example of an LTB solution where $A^{\prime}=0$ at an
isolated point, with $A^{\prime}$ changing sign, without shell-crossing.

###  VI.2 Schwarzchild space time variations

Lines of constant LTB spatial coordinates are geodesics, so setting $F$ to
constant

$F=2GM=R_{S}$ (51)

in (33)-(38) must produce portions of Schwarzchild space-time in timelike
geodesic coordinates, each radial geodesic indexed by constant (comoving)
$r,\theta,\phi$. These can be incoming or outgoing, depending on the sign
choice in (33)-(38). They can be bounded, unbounded or borderline,
corresponding to the various $K(r)$. $t_{0}(r)$ selects the initial condition
for each geodesic, and can be $r$ dependent. Different choices cover different
partial patches of the Kruskal manifold and these coordinates can be seen as
the time-like equivalent of the Eddington-Finkelstein null coordinates.

For $K=0$, $t_{0}$ must depend on $r$, or else $A^{\prime}=0$ everywhere. Once
this is satisfied any monotonic $t_{0}(r)$ will do. Setting $t_{0}=r$
corresponds to Lemaitre coordinates exactslns . The radial Schwarzchild
coordinate, $R$, being an areal coordinate, relates to the LTB $r$ and $t$
via:

$R=A(r,t)$ (52)

(since $A^{2}$ is the areal factor of (23)). Hence for $K=0$ the horizons are
at:

$\pm(t-t_{0}(r))=\frac{2}{3}R_{S}$ (53)

and the singularity is at $t=t_{0}(r)$.

For $K\neq 0$ we have significantly more choice (this is sometime called
Novikov coordinates). Comparison with the solutions to (3), bearing in mind
(52), implies:

$\displaystyle t-t_{0}(r)$ $\displaystyle=\tau$ (54) $\displaystyle K(r)$
$\displaystyle=1-E^{2}$ (55)

Hence $t_{0}(r)$ is the conventional zero of the proper time of these
geodesics, and $K(r)$ the choice for their “total energy”. Considerable
ambiguity is now present.

The choice $K(r)=k/r$ does not lead to the Hubble radial gauge, which in
general does not exist (cf. Kopinski ). One exception is $k=-1$ and $\eta\gg
1$ which naturally touches Milne (both being empty spaces), so that $K=kr^{2}$
with $t_{0}=0$ still is a Hubble gauge. Better find the equivalent of the
Lemaitre gauge for when $\eta\ll 1$ and the solutions reduce to $K=0$. This is
achieved with:

$\displaystyle K(r)$ $\displaystyle=K_{0}$ (56) $\displaystyle t_{0}$
$\displaystyle=r.$ (57)

We can also change the sign of $K$ as long as the geodesics do not cross: for
outgoing charts this means we can go from $K>0$ to $K<0$, but not the other
way, and the opposite for incoming charts.

###  VI.3 OS and its variations, and vacuoles

The OS solution and its variations, as well as the similar vacuole solutions
(see, e.g. Stephani ) can be found within (33)-(38) subject to constraints
(40) and (41), and exceptions at $A^{\prime}=0$ points at the end of Section
III.

For OS-like solutions, at $r<r_{0}$ we pick an FRW solution, for example with
choices (44)-(46). Traditionally one takes contracting solutions (with any
$k$) but we can also consider expanding ones (modelling explosions rather than
collapses). For $r>r_{0}$ we need a Schwarzchild representation (among those
in Section VI.2) satisfying:

$\displaystyle F(r)=F(r_{0})$ $\displaystyle=\frac{8\pi G}{3}mr_{0}^{3}$ (58)
$\displaystyle K(r_{0})$ $\displaystyle=kr_{0}^{2}$ (59) $\displaystyle
t(r_{0})$ $\displaystyle=0$ (60)

and with the same of sign $t$ at $r_{0}$ (i.e. the same expanding/outgoing or
contracting/incoming nature). With these conditions, $A_{+}=A_{-}$ and
$K_{+}=K_{-}$ at all times.

This still leaves plenty of choice (which can be exploited to link to outer
solutions). A possibility is:

$\displaystyle K$ $\displaystyle=kr_{0}^{2}$ (61) $\displaystyle t_{0}$
$\displaystyle=r-r_{0}$ (62)

which reduces to the Lemaitre coordinates when $\eta\ll 1$, i.e. curvature can
be ignored. Other possibilities will be explored later.

The same applies in reverse (Schwarzchild for $r<r_{0}$, FRW for $r>r_{0}$),
as is the case of a vacuole.

Thus the matching conditions found in Sec. II (as well as those for the
similar vacuole solution; see Stephani ) appear naturally, together with the
concept of mass deficit/surplus: (58) implies

$M=\frac{4\pi}{3}\rho a^{3}r_{0}^{3}=\rho a^{3}V_{0}(r_{0})$ (63)

with $V_{0}$ the comoving Euclidean ($k=0$) volume for the area indexed by
areal coordinate $r_{0}$. The same issues happen in vacuoles where the mass
must be the escavated FRW mass corresponding to a $k=0$ FRW Universe,
regardless of the actual value of $k$.

##  VII Attractive dark voids

We are now in position to present examples of a large class of solutions in
which the mass surplus effect is in evidence.

###  VII.1  OS-FRW combinations

A vacuole within a FRW Universe containing an OS-like structure in the middle
can be obtained combining the matching conditions of the previous subsections.
It then all boils down to finding a Schwarzchild chart that links the inner
and the outer solutions. We assume there are no mass shells, so $F$ cannot
jump, implying:

$\displaystyle r<r_{1}\implies$ $\displaystyle F=\frac{4\pi}{3}r^{3}r_{S1}$
$\displaystyle r_{1}<r<r_{2}\implies$ $\displaystyle F=r_{S2}$ $\displaystyle
r>r_{2}\implies$ $\displaystyle F=\frac{4\pi}{3}r^{3}r_{S3}$ (64)

with:

$r_{S2}=\frac{4\pi}{3}r_{1}^{3}r_{S1}=\frac{4\pi}{3}r_{2}^{3}r_{S3}$ (65)

which requires:

$\frac{r_{S1}}{r_{S3}}=\frac{m_{in}}{m_{out}}=\left(\frac{r_{2}}{r_{1}}\right)^{3}.$
(66)

In addition, $K(r)$ cannot jump, but we can choose any function interpolating
between $K(r_{1})=k_{1}r_{1}^{2}$ and $K(r_{2})=k_{2}r_{2}^{2}$ in the region
$r_{1}<r<r_{2}$, where $k_{1}$ and $k_{2}$ are the normalized curvatures in
the inner and outer FRW models (and they all lead to equivalent models). If
the inner FRW is contracting and the outer one is expanding, we may need a
designer $K(r)$ and $t_{0}(r)$ to allow the geodesics in $r_{1}<r<r_{2}$ to
change from incoming to outgoing.

A large class of solutions therefore follows, with different values of $k_{1}$
and $k_{2}$, and with the inner FRW contracting or expanding (we assume the
outer FRW is expanding). In addition, we can assume the same bang time $t_{0}$
for both FRW solutions (so that they have the same age), or not.

###  VII.2 A simple example

A simple example results from the choice:

$\displaystyle r<r_{1}\implies$ $\displaystyle K=kr^{2}$ (67) $\displaystyle
r_{1}<r<r_{2}\implies$ $\displaystyle K=kr_{1}^{2}\frac{r_{2}-r}{r_{2}-r_{1}}$
(68) $\displaystyle r>r_{2}\implies$ $\displaystyle K=0$ (69)

picking expanding solutions for both FRWs, with the same $t_{0}$. We focus on
the $k=-1$ because it is the relevant one for a mass surplus. The two FRWs are
connected by a Schwarzchild buffer zone that can be read off from solutions
(33)-(38) choosing the + branch, and setting $t_{0}(r)=0$ throughout:

$\displaystyle A$
$\displaystyle=\frac{r_{S2}(r_{2}-r_{1})}{2r_{1}^{2}(r_{2}-r)}(\cosh\eta-1)$
(70) $\displaystyle t$
$\displaystyle=\frac{r_{S2}(r_{2}-r_{1})}{2r_{1}^{2}(r_{2}-r)}(\sinh\eta-\eta);$
(71)

(note that at $r\approx r_{2-}$ we can take the $\eta\rightarrow 0$ limit).
With this choice we satisfy the matching conditions (no jumps in $A$ and $K$)
at both gluing points.

The inside/outside solutions are just FRW spaces which touch as we approach
the Big Bang444Proof: find the $\eta\rightarrow 0$ limit of $K\neq 0$ and use
(66).. There is a mass surplus in the Schwarzchild region, with respect to the
integrated mass in the inside expanding region. As we have seen this results
from from a volume deficit inside the sphere bounded by $r=r_{1}$. We now find
a complementary effect to this on the outside. The area at $r=r_{2}$ is bigger
than the area $r=r_{1}$, with $r_{2}>r_{2}$. In the extreme case, these large
volumes could be completely hidden from the outside world except for their
mass.

Note that this strange property follows from that $A^{\prime}$ can switch
sign, just as it does for a pure $k=1$ FRW model. Here $A^{\prime}>0$ for the
inner and outer expandin FRW, but $A^{\prime}<0$ in the Schwarzchild buffer.
This is indeed what happens here, as depicted by the embedding in Fig. 4. The
fact that a jump in $A^{\prime}$ is possible goes back to the issues discussed
in bonner , and in Section V.

Could such objects be part of the initial conditions of our Universe, and be
about and be part of dark matter?

---
Figure 4: An expanding inner $k=-1$ Universe which has outgrown the cavity
size in the outer $k=0$ Universe. A large internal volume could therefore be
hidden in side this structure.

###  VII.3 Asynchronous Big Bang non-homogeneous generalizations

Finally there could also be a mismatch between the lifetime of the inner and
the outer structures, as measured in the synchronous gauge (i.e. by the proper
time of the geodesic observers attached to the LTB coordinate frame).

Note that in the solutions above we could make the Big Bang asynchronous in
either inner and outer solution, replacing (46) by $t_{0}=t_{0}(r)$ and
keeping the other 2 choices. Asynchronous Big Bangs have been discussed
($t_{0}$ is usually called the “Bang time”). We could also keep $t_{0}$
constant in the inner and outer FRW but they could take different values.

####  VII.3.1 Age mismatch effect: Asynchronous Big Bang with an empty buffer

In the simplest case we can have two homogeneous solutions with different Bang
times linked by vacuum. A large class of solutions can be generated this way;
here we present a simple example. The point is that generally the inner
solution may be larger than the excavated hole of the outer region, leading to
situations similar to that depicted in Fig. 4.

Consider, for example, the solution obtained from (33)-(38) by setting $K=0$
everywhere, choosing $F$ as in (64), and setting:

$\displaystyle r<r_{1}\implies$ $\displaystyle t_{0}^{in}=0$ (72)
$\displaystyle r_{1}<r<r_{2}\implies$ $\displaystyle
t_{0}(r)=t_{0}^{out}\frac{r-r_{1}}{r_{2}-r_{1}}$ (73) $\displaystyle
r>r_{2}\implies$ $\displaystyle t_{0}^{out}\neq 0.$ (74)

This solution satisfies the jumping conditions described in Section V.
Admittedly $A^{\prime}$ is discontinuous but that is OK, unless we take
seriously the physically unmotivated stronger condition of the 3 classes
described in bonner .

If $t_{0}^{out}=0$ we obviously would have $A_{-}=A_{+}$ and $A^{\prime}=0$ in
the open set $r_{1}<r<r_{2}$, disqualified by our assumptions. This is just a
standard FRW with a fictitious and unnecessary buffer region. If
$t_{0}^{out}>0$ then the inner region is older and larger than the the volume
escavated form the outer FRW Universe, and the Schwarzchild like buffer region
will be of the same shape as Fig. 4. A large volume and past life may then be
hidden inside a small hole in a much younger Universe. (The case
$t_{0}^{out}<0$ leads to a more standard vacuole.)

####  VII.3.2 More general solutions

More generally $t_{0}$ might vary with $r$ even within the inner and outer
non-vacuum solution, in which case they are no longer pieces of FRW models.
Indeed replacing (46) by $t_{0}=t_{0}(r)$ and keeping (44) and (45) implies
that $A$ is no longer proportional to $r$ and this implies inhomogeneous
solutions via (31). Homogeneity implies $F\propto A^{3}$ at constant time,
which implies $t_{0}^{\prime}=0$ (by inspection of solutions). For example,
for $K=0$, introducing a $t_{0}^{\prime}\neq 0$, we have:

$\rho=\frac{1}{6\pi
G(t-t_{0}(r))^{2}\left(1-\frac{2rt_{0}^{\prime}(r)}{3(t-t0(r))}\right)}$ (75)

We could also keep (44) (instead of introducing a vacuum buffer, as in the
previous subsection), make $t(r)$ constant except for a region linking
different Big Bang times $t_{0}$, thereby we linking two homogeneous FRW
models with different uniform densities with a buffer region which is not
uniform.

In general, the singularity time ($\rho\rightarrow\infty$) is the Bang time
$t_{0}$ even if $t_{0}^{\prime}(r)\neq 0$. However, near the singularity,
$\rho<0$ if $t_{0}^{\prime}>0$, that is, with positive energy we should
require the inner shells to have their BB retarded rather than advanced.

We can find Milne-like equivalents from (39). With a choice for $r$ leading
to(45) we have

$A=\pm r(t-t_{0}(r))$ (76)

so that (44) and (31) imply the profile for the density (assumed negligible
with respect to the curvature):

$\rho=\frac{m}{|t-t_{0}|^{3}\left(1-\frac{rt_{0}^{\prime}}{|t-t_{0}|}\right)}$
(77)

This solution could be used in variations of all the mass surplus situations
we have examined

##  VIII Generalization of the argument for mass surplus and deficit

With suitable adaptations, we can generalize the argument in Section II.2 to
non-homogeneous situations within the LTB framework. The mass felt outside a
shell with label $r$ (and not just the asymptotic mass, assuming asymptotic
flatness) is given by:

$M(r)=\frac{F}{2G}=4\pi\int dr\,A^{\prime}A^{2}\rho=\int
d^{3}x\,\sqrt{h_{0}}\rho,$ (78)

where $h_{0}$ is the determinant of the spatial metric assuming $K=0$, and
where we used $A^{\prime}>0$ in the last step. If $A^{\prime}$ changes sign
and $r(A)$ is multivalued, then we have to take this into account computing
the integral:

$M=4\pi\sum_{i}\int dAA^{2}\rho_{i}(A)=\sum_{i}{\rm sgn}(A^{\prime})\int
d^{3}x\,\sqrt{h_{0}}\rho,$

(with $i$ indexing all the branches). With these modifications, the mass felt
outside the body is still that obtained by filling the boundary with area $A$
with a surface with $K=0$ (taking the possibility that different branches may
connect $A=0$ and the given $A$.

In contrast the integrated local mass is:

$M_{L}=\int d^{3}x\,\sqrt{h}\rho=4\pi\int
dr\,\frac{|A^{\prime}|A^{2}}{\sqrt{1-K}}\rho.$ (79)

which generalizes $M_{F}$. The mass mismatch formula is thus generalized.

If the space is asymptotically flat, then $M$ is the ADM mass.

##  IX Conclusions

In this paper we constructed a number of solutions which generalize the well
know mass deficit/surplus effect in various ways. In particular, we found that
not only the inferred outside mass may be different to the integrated inside
mass, but the volume and lifetime may be different. We can thus produce
objects converse to quasi-baby Universes BabyU  (which have no asymptotic
mass): attractive voids, which have asymptotic mass with minimal internal
integrated mass. Such objects, have a deficit of internal volume for a given
surface area. They can also have very different ages and that of the
asymptotic Universe.

More generally we can have mismatches leading to structures with near
vanishing surface area and very large internal volumes, such as an inner
expanding Universe that outgrew the outer one. The lifetimes of such
structures can also be different, using the proper time of geodesic observers
as a clock. We gave a number of concrete examples, but it is clear that an
even larger class of solutions can be found.

Are these solutions realistic? They certainly could be part of the initial
conditions of our Universe. But we can also speculate that they could be
purely astrophysical objects: who knows what lurks out there? We stress that
the effects can be purely relativistic, so that interpreting, for example, the
mass mismatch in a quasi-Newtonian framework (as the contribution of the
kinetic or the binding energy) may not be appropriate. If a collapsing $k=-1$
OS-like model could be seen as the result of a violent non-gravitational
implosion, it could also be a purely relativistic object.

##  X Acknowledgments

We thank Bernard Carr and Ray Rivers for discussions and Farbod Rassouli for
assistance with the pictures. This work was supported by a Bell-Burnell
Fellowship (RI) and partly supported by the STFC Consolidated Grants
ST/T000791/1 and ST/X00575/1 (JM).

##  Appendix A Israel Junction Conditions and the LTB Metric

As explained previously, the LTB metric reduces to either the Schwarzschild
metric in Lemaitre coordinates or the FRW metric. The outcome is contingent on
one’s choice of areal function $A(r,t)$, mass function $F(r,t)$ and curvature
function $K(r)$. We may consider a scenario in which the Schwarzschild and FRW
reductions of the LTB metric are glued at a common hypersurface. The two
metrics partitioned by the hypersurface are therefore

$ds^{2}_{\pm}=-dt^{2}+X_{\pm}^{2}(r,t)dr^{2}+A_{\pm}^{2}(r,t)d\Omega^{2}.$
(80)

We take the hypersurface to be a surface of constant radius which is defined
by the constraint $\Phi=0$ where $\Phi=r-r_{0}$. In general, a vector normal
to the hypersurface defined by a constraint $\Phi$ is determined by the
equation

$n_{\alpha}=\frac{\partial_{\alpha}\Phi}{|g^{\mu\nu}\partial_{\mu}\Phi\partial_{\nu}\Phi|^{\frac{1}{2}}}.$
(81)

For the LTB metric we arrive at

$\displaystyle\begin{split}n_{\alpha}dx^{\alpha}=X(r,t)dr\\\
n^{\alpha}\partial_{\alpha}=\frac{1}{X(r,t)}\partial_{r}\end{split}$ (82)

Generally, the projection tensors $e^{\mu}_{a}$ can be used to project out the
components of a tensor which are normal to the hypersurface. In this notation,
$x^{\mu}$ are coordinates associated with the full geometry while $y^{a}$ are
the hypersurface coordinates. For example, the induced metric $h_{ab}$ is
related to the full metric by the relation
$h_{ab}=g_{\mu\nu}e^{\mu}_{a}e^{\nu}_{b}$. When two LTB metrics are glued
together at a constant radius hypersurface, the induced metric on the
hypersurface as approached from either metric is given by

$d\Sigma_{\pm}=-dt^{2}+A_{\pm}^{2}(r_{0},t)d\Omega^{2},$ (83)

where $h_{tt}=-1$, $h_{\theta\theta}=A^{2}(r_{0},t)$ and
$h_{\phi\phi}=A^{2}(r_{0},t)\sin^{2}\theta$. When dealing with metrics which
are separated by hypersurfaces, we can define the following tensorial
quantities;

$\displaystyle\begin{split}&\big{[}h_{ab}\big{]}\equiv h_{ab(+)}-h_{ab(-)},\\\
&\big{[}n_{\alpha}\big{]}\equiv n_{\alpha(+)}-n_{\alpha(-)},\\\
&\big{[}k_{ab}\big{]}\equiv k_{ab(+)}-k_{ab(-),}\end{split}$ (84)

where $\big{[}h_{ab}\big{]}$ is the difference between the induced
hypersurface metric components across the hypersurface,
$\big{[}n_{\alpha}\big{]}$ corresponds to a jump in the normal vector across
the hypersurface and $\big{[}k_{ab}\big{]}$ is the difference in extrinsic
curvature across the hypersurface.

###  A.1 1st Israel Junction Condition

The 1st Israel junction condition requires the geometry on the hypersurface
should be the same when approached from both metrics. This corresponds to
enforcing $\big{[}h_{ab}\big{]}=0$ across the hypersurface. In the case of two
LTB metrics, enforcing the first Israel junction condition requires the
identification

$\big{[}A(r_{0},t)\big{]}\equiv A_{+}(r_{0},t)-A_{-}(r_{0},t)=0.$ (85)

As mentioned previously, the traditional Friedmann-Schwarzschild junction is
recovered under a specific choice of LTB free functions. The interior metric
corresponds to Friedmann geometry when we choose the areal function to be

$A_{-}(r_{0},t)=a(t)r_{0}$ (86)

where $a(t)$ is the usual scale factor for the internal Friedmann region. We
can identify the exterior geometry with the Schwarzschild metric in Lemaitre
coordinates by choosing the corresponding areal function to be

$A_{+}(r_{0},t)=\bigg{[}\frac{3}{2}(r_{0}-t)\bigg{]}^{\frac{2}{3}}r_{s}^{\frac{1}{3}},$
(87)

where the full LTB-Schwarzschild metric is given by

$ds^{2}=-dt^{2}+\frac{r_{s}}{A(r,t)}dr^{2}+A^{2}(r,t)d\Omega^{2}.$ (88)

In contrast to the usual curvature coordinates in which the Schwarzschild
metric is usually expressed, Lemaitre coordinates are synchronous.
Consequently, the time coordinate in the metric corresponds to to the proper-
time of time-like observers. This choice of areal function can be interpreted
as a surface which is initially at a maximum radius $r_{0}$ at initial proper-
time $t_{0}$. Then for proper-time $t>t_{0}$, the surface is in free-fall from
the maximum value $r_{0}$. In these coordinates, the analogy with the
Friedmann scale factor is obvious. Indeed, the 1st Israel junction condition
is simply the interpretation of this free-fall in terms of the behaviour of
the internal Friedmann scale factor $a(t)$.

It is worth noting that using the Schwarzschild metric in synchronous Lemaitre
coordinates offers significant advantages over the usual curvature
coordinates. In the usual gluing process, the internal Friedmann and external
Schwarzschild metrics are taken to be

$\displaystyle\begin{split}&ds^{2}_{-}=-dt^{2}+a(t)^{2}\bigg{[}\frac{dr^{2}}{1-kr^{2}}+r^{2}d\Omega^{2}\bigg{]},\\\
&ds^{2}_{+}=-f(r)dt^{2}+f(r)^{-1}dr^{2}+r^{2}d\Omega,\\\ \end{split}$ (89)

where

$f(r)=1-\frac{2M}{r}.$ (90)

However in this case we cannot simply use the constraint on the $r$ coordinate
for both metrics as we did in the LTB case. To match the internal Friedmann,
geometry with the external Schwarzschild geometry, the radius of the surface
as approached from the exterior Schwarzschild metric must be promoted to a
function $R(t)$ where $t$ is Friedmann synchronous time. One must also take
into account that the Schwarzschild time coordinate differs from synchronous
time. The Schwarzschild time coordinate on the surface of the star is
therefore promoted to $T(t)$, a function of the interior synchronous time. We
may then take an induced metric on a $r=r_{0}$ hypersurface from the Friedmann
perspective. This must be matched with the $r=R(t)$ surface from the
Schwarzschild perspective. The induced metrics on the hypersurface are
therefore given by

$\displaystyle\begin{split}&d\Sigma^{2}_{-}=-dt^{2}+a(t)^{2}r_{0}^{2}d\Omega^{2},\\\
&d\Sigma^{2}_{+}=-\bigg{[}f(R)\dot{T}^{2}+f(R)\dot{R}^{2}\bigg{]}dt^{2}+R^{2}d\Omega^{2}\\\
\end{split}$ (91)

Enforcing the 1st Israel junction condition leads to the identifications

$\displaystyle\begin{split}&-1=-\bigg{[}f(R)\dot{T}^{2}+f(R)\dot{R}^{2}\bigg{]},\\\
&R(t)=a(t)r_{0}.\\\ \end{split}$ (92)

The first identification may be interpreted as matching the time on the
surface of the dynamic Schwarzschild star with comoving Friedmann time. The
second identification is simply an expression for the radius of the dynamic
Schwarzschild star in terms of Friedmann variables.

###  A.2 2nd Israel Junction condition

To enforce the 2nd junction condition, we start with the definition of the
extrinsic curvature $k_{\mu\nu}$ projected onto the the hypersurface $\Sigma$
with coordinates $y^{a}$;

$\displaystyle\begin{split}k_{ab}=&e^{\mu}_{a}e^{\nu}_{b}k_{\mu\nu},\\\
=&e^{\mu}_{a}e^{\nu}_{b}(\nabla_{\nu}n_{\mu}-a_{\nu}n_{\mu}),\\\
=&\nabla_{b}n_{a},\end{split}$ (93)

where $e^{\mu}_{a}$ is a projection tensor which projects out components of
the extrinsic curvature normal to the surface $\Sigma$ as
$e^{\mu}_{a}n_{\mu}=0$ and $a_{\nu}$ is the acceleration vector defined in
terms of the normal as $a_{\nu}=n_{\lambda}\nabla^{\lambda}n_{\nu}$. As the
hypersurface coordinates are $y^{a}=(t,\theta,\phi)$ and the LTB metric is
diagonal, we’re only concerned with the $k_{tt}$, $k_{\theta\theta}$ and
$k_{\phi\phi}$ components of the extrinsic curvature. When considering the
$k_{tt}$ component, it is worth noting that the LTB metric is both diagonal
and expressed in a co-moving coordinate system. Therefore we would expect the
$k_{tt}$ component of the extrinsic curvature to vanish

$\displaystyle\begin{split}k_{tt}=\
&\partial_{t}n_{t}-n_{r}\Gamma^{r}_{tt},\\\ =\ &0,\end{split}$ (94)

since $\partial_{t}g_{at}=0$ for $a\neq t$ and $\partial_{r}g_{tt}=0$. The non
vanishing components of the extrinsic curvature are therefore the angular
components. Given that $\partial_{a}n_{r}=0$ for $a\neq r$ and
$\Gamma^{r}_{ab}=-\frac{1}{2}g^{rr}(\partial_{r}h_{ab})$,

$\displaystyle\begin{split}k_{ab}=&-n_{r}\Gamma^{r}_{ab}\\\
=&\frac{1}{2}n^{r}\big{(}\partial_{r}h_{ab}\big{)}.\end{split}$ (95)

The relevant components of the induced metric are
$h_{\theta\theta}=A(r_{0},t)^{2}$ and
$h_{\phi\phi}=h_{\theta\theta}\sin^{2}\theta$. Using the above formula for
extrinsic curvature, the $k_{\theta\theta}$ and $k_{\phi\phi}$ are explicitly
given by

$\displaystyle\begin{split}&k_{\theta\theta}=A(r_{0},t)\cdot\frac{A^{\prime}(r_{0},t)}{X(r,t)},\\\
&k_{\phi\phi}=k_{\theta\theta}\sin^{2}\theta.\end{split}$ (96)

Enforcing the 2nd Israel junction condition we have

$\displaystyle\begin{split}\big{[}k_{ab}\big{]}=&\bigg{[}A(r_{0},t)\frac{A^{\prime}(r_{0},t)}{X(r,t)}\bigg{]}\\\
=&\bigg{[}\frac{A^{\prime}(r_{0},t)}{X(r,t)}\bigg{]}\\\
=&\big{[}K(r)\big{]}.\end{split}$ (97)

The second line is derived from the first by using the fact that we’ve
enforced the first Israel junction condition that $\big{[}A\big{]}=0$. The
third line is then a consequence of the momentum constraint for the general
LTB metric leading to the identification $X=C(r)A^{\prime}$ where
$C(r)=1/\sqrt{1-K(r)}$ with $K(r)<1$.

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