Thin shells in general relativity without junction conditions: A model for galactic rotation and the discrete sampling of fields

Abstract

Interest in general relativistic treatments of thin matter shells has flourished over recent decades, most notably in connection with astrophysical and cosmological applications such as black hole matter accretion, spherical wormholes, bubble universes, and cosmic domain walls. In the present paper, an asymptotically exact solution to Einstein's field equations for static ultra-thin spherical shells is derived using a continuous matter density distribution (r) ρ defined over all space. The matter density is modeled as a product of surface density μ 0 and a continuous or broadened spherical delta function. Continuity over the full domain 0<r<∞ ensures unambiguous determination of both the metric and coordinates across the shell wall, obviating the need to patch interior and exterior solutions using junction conditions. A unique change of variable allows integration with asymptotic precision. It is found that ultra-thin shells smaller than the Schwarzschild radius can be used to model supermassive black holes believed to lie at the centers of galaxies, possibly accounting for the flattening of the galactic rotation curve as described by Modified Newtonian Dynamics (MOND). Concentric ultra-thin shells may also be used for discrete sampling of arbitrary spherical mass distributions with applications in cosmology. Ultra-thin shells are shown to exhibit constant interior time dilation. The exterior solution matches the Schwarzschild metric. General black shell horizons, and singularities are also discussed.

Full text

Thin shells in general relativity without junction conditions: A model for galactic
rotation and the discrete sampling of fields
By Kathleen A. Rosser
Ka thleen.A.Rosser@ieee.org
(Published 3 October 2019)
Interest in general relativistic treatments of thin matter shells has flourished over recent decades, most notably in
connection with astrophysical and cosmological applications such as black hole matter accretion, spherical wormholes,
bubble universes, and cosmic domain walls. In the present paper, an asymptotically exact solution to Einstein's field
equations for static ultra-thin spherical shells is derived using a continuous matter density distribution (r) ρdefined over
all space. The matter density is modeled as a product of surface density μ0 and a continuous or broadened spherical delta
function. Continuity over the full domain 0<r<∞ ensures unambiguous determination of both the metric and coordinates
across the shell wall, obviating the need to patch interior and exterior solutions using junction conditions. A unique
change of variable allows integration with asymptotic precision. It is found that ultra-thin shells smaller than the
Schwarzschild radius can be used to model supermassive black holes believed to lie at the centers of galaxies, possibly
accounting for the flattening of the galactic rotation curve as described by Modified Newtonian Dynamics (MOND).
Concentric ultra-thin shells may also be used for discrete sampling of arbitrary spherical mass distributions with
applications in cosmology. Ultra-thin shells are shown to exhibit constant interior time dilation. The exterior solution
matches the Schwarzschild metric. General black shell horizons, and singularities are also discussed.
I. INTRODUCTION
A long-standing unsolved problem in astrophysics is the
observed discrepancy in the orbital velocity v(r) of the
luminous matter of galaxies. This discrepancy, often
called the flattening of the galactic rotation curve, has
been ascertained from Doppler shift measurements that
indicate the outlying stars and hydrogen clouds of
galaxies orbit too fast to be gravitationally bound by
baryonic matter alone. In regions outside the luminous
disk, v(r) does not fall off as r1/2─ as predicted by
Newtonian dynamics, but tends toward a constant as r
increases. The discrepancy is generally attributed to the
presence of dark matter, a hypothetical transparent
nonradiating material that has never been independently
detected nor reconciled with the standard model of
particle physics. The failure to identify this elusive
substance has given rise to modified gravity theories that
obviate the need for dark matter, such as Mordehai
Milgrom's Modified Newtonian mechanics (MOND)
[1,2] and others [3,4]. Here, a static spherical thin shell
solution to Einstein's field equations is derived that may
suggest a new explanation for the galactic rotation curve.
A solution for concentric shells is also presented that
may be useful for discrete sampling of arbitrary spherical
mass distributions with applications in cosmology.
Investigation into the gravitational properties of thin
matter shells has flourished over the past few decades,
most notably in studies of astrophysical and
cosmological structures such as spherical wormholes [5-
7], black hole accretion shells, bubble universes as
models of cosmic inflation [8,9], false vacuum bubbles
[10,11], and cosmic membranes or domain walls that
split the universe into distinct spacetime regions [12-14].
The structures may be static, as in the case of spherical
wormholes; contracting, as in the case of matter
accretion shells around black holes [15] and shells
collapsing into wormholes [16,17]; rotating and
collapsing [18,19]; or expanding, as in the case of cosmic
brane worlds [20], inflationary bubbles or bubble
universes [21]. Such shells may split the universe into
two domains, an interior and exterior joined by an
infinitesimally thin wall of singular mass or pressure [22-
26]; or into three domains [27], where the wall of finite
thickness is sometimes called the transient layer [28].
Various interior and exterior metrics are assumed,
including the Friedman-Robertson-Walker [29,30],
Schwarzschild, de Sitter [31], anti-de Sitter [32],
Minkowski, and Reissner-Nordstrom [33,34] metrics.
The metrics are often selected a priori, their parameters
later fixed by junction conditions at the inner and outer
surfaces of the wall, or at the shell radius [35]. Common
techniques frequently require patching solutions for
inner, outer, and possible transient domains, using
separate coordinate systems and metrics for each domain
[36,37]. The most widely applied junction conditions,
attributed to Israel [38,39], or Darmois and Israel [40],
require that both the metric gμν and the extrinsic
curvature Kμ
ν be continuous across the shell wall. While
these conditions are common in the literature, doubt is
raised about their application to certain physical
scenarios [41] or in modified theories of gravity [42].
Some authors derive new junction conditions that specify
jumps in curvature [43], jumps in the tangential metric
components to account for domain wall spin currents
[44], or other field behavior [45]. Others avoid junction
conditions by use of a confining potential [46].
1

It may be significant that Israel's original derivation was
based on properties of electromagnetic fields rather than
on general relativity (GR), although recent derivations, in
contexts such as cosmological brane-worlds, address the
junction by adding a Gibbons-Hawking term to the
standard Einstein-Hilbert action of GR [47]. However,
some authors point to contradictions in this method,
particularly when applied to infinitely thin shells [48].
While procedures for deriving the Israel junction
conditions are well established, their implementation
relies on concepts outside the core formalism of GR and
other metric gravities, including the notion of induced
metric, or the D n─ dimensional metric in the transient
domain; the vector ni normal to the domain wall; the
surface stress-energy tensor Sμ
ν for the transient domain;
the extrinsic curvature Kμ
ν; the Gibbons-Hawking action
term, and so forth. A treatment of thin shells that
obviates the need for junction conditions may therefore
be useful for its simplicity. Cosmic inhomogeneities
using cubic lattices that avoid junction conditions have
been studied by some authors [49,50]. Nevertheless,
examples in the literature of continuous spherical thin-
shell solutions to the gravitational field equations have
proven difficult to find.
The purpose of this paper is to derive an asymptotically
exact continuous solution to Einstein's field equations for
static, spherical, ultra-thin massive shells without the
need for junction conditions, employing a uniform set of
coordinates defined over all space, with equation of state
p=w .ρ Here, asymptotically exact means exact in the
limit of vanishing thickness (although the solution is
undefined for zero thickness), and ultra-thin denotes
arbitrarily thin but non-vanishing. One advantage to the
continuous solution method, in which density (r)ρ,
pressure p(r), and the metric gμν(r) are uniformly defined
over all space, is that only two boundary conditions are
needed to fix the metric:
1. gμν must be nonsingular at r=0,
2. gμν must match Minkowski space as r >∞─,
where Minkowski space is here defined by the metric
gμν=diag(1, 1, r─ ─ 2, r─2sin2( )). ΘThe first condition
follows from the absence of matter in the interior [51].
This is relaxed in the case of a central mass. The second
dictates that space be asymptotically flat, assuming
g00 >1─ as r >∞─, or that the standard laboratory clock
rate is the same as that at infinity.
To obtain a continuous solution to Einstein's field
equations (EFE), i.e. a metric composed of continuous
analytic functions gμν(r) defined over all space, one must
first define a continuous density (r)ρ spanning the range
0≤r≤∞. For an ultra-thin shell, (r)ρ will be modeled here
as continuous approximation to the spherical Dirac delta
function (r rδ ─ 0), where the continuous or broadened
version of the delta function, to be written δc(r r─0), will
be derived in Section II. According to this model, the
mass density distribution is
(r) = μρ0 δc(r r─0). (1)
Here, μ0 is the surface density of the shell and has
dimensions [m/r2]. Recalling that the δ function has
dimensions [1/r], it is clear that the volume density (r)ρ
has dimensions [m/r3], or [1/r2] in the units G=c=1.
This density distribution may be substituted into the
energy-momentum tensor Tμν on the right-hand side of
EFE. The equations are then solved using a unique
change of variable that allows integration to arbitrary
accuracy. The result is an asymptotically exact
continuous metric for an empty ultra-thin shell.
The metric signature (+ - - -) and units c=G=1 will be
used throughout this paper. Small Greek letters stand for
spacetime indices 0,1,2,3. The symbol ≈ denotes
asymptotic equality, or equality in limit as thickness
parameter ε approaches zero, although the formalism is
undefined at =0ε. An equation of state (EoS) of the form
p(r)=w (r)ρ for w a constant will be assumed. While the
method here applies to static shells, it can in principle be
generalized to account for expansion or contraction. This
is a topic for future research. The presentation is
organized as follows. In Section II, the broadened
spherical delta function will be derived. Section III
shows how to solve EFE for a thin shell using the
continuous solution method. In Section IV, the novel
properties of black shells (those of radius less than or
equal to the Schwarzschild radius) will be examined.
Section V discusses how the galactic rotation curve
might be explained by a supermassive black shell at the
galactic core, and Section VI presents the concentric
shell solution as a method for discrete sampling.
Concluding remarks are found in Section VI.
II. MASS DENSITY: DEFINING THE
CONTINUOUS DELTA FUNCTION
Spherical Dirac delta functions as models for mass or
charge distributions have appeared in the literature for
many decades. Use of the delta function for thin shell
solutions to EFE is frequently encountered in such
applications as bubble universes and cosmic domain
walls. However, the discontinuities in the delta function
and its integral, the step function, necessitate piecewise
solutions and attendant junction conditions, as noted
above. To apply a delta function model uniformly over
all space requires that the Dirac delta function (r rδ ─ 0) be
replaced by a continuous or broadened delta function
δc(r r─0) with similar properties. One such function can
be defined as follows:
2

1) Let δc(r r─0) be an approximation to a spherical Dirac
delta function (r rδ ─ 0), where the latter is expressed in
terms of the normalized spherical Gaussian
G(r) := ( √)ε π 1─ exp[─(r─r0)2/ε2]. (2)
Here G(r) is defined over the domain r≥0, with a peak
centered at r=r0 of height 1/(π1/2 )ε and width
proportional to ε. For <<rε0, G(r) obeys the relation
∫0∞dr G(r) = ∫0∞dr ( √)ε π 1─ exp[─(r─r0)2/ε2] 1, ≈
<<rε0.
This relation may be verified by evaluating the integral
of a normalized rectangular Gaussian G(x), which for
<<xε0 has the property
∫0∞dx ( √)ε π 1─ exp [─(x─x0)2/ε2 ]
≈ ∫ ∞─
∞dx ( √)ε π 1─ exp [─(x─x0)2/ε2] = 1.
The delta function may thus be written
(r rδ ─ 0 ) = lim[ >0] ε─ (√)ε π 1─ exp [─(r─r0)2/ε2]. (3)
2) The continuous or broadened delta function δc(r r─0)
is obtained as an approximation to (r rδ ─ 0) by taking an
incomplete limit in Eq. (3), that is, by letting ε become
arbitrarily small but nonzero.
3) For n a small integer such that 2nε approximates the
peak width to some selected accuracy, the broadened
delta function δc(r r─0) nearly vanishes in the domains
r<r0n─ ε and r>r0+nε. Therefore mass density (r)ρ
approaches that of a near-vacuum in these regions. By
increasing n and decreasing ε, the vacuum can be
achieved as closely as desired.
4) The broadened delta function δc obeys, to any desired
accuracy, the defining properties of the Dirac delta
function:
a) ∫0∞dr δc(r r─0) 1≈
b) ∫0∞dr f(r) δc(r r─0) f(r≈0)
provided that f(r) is slowly varying over the transient
layer r0n <r<r─ ε 0+nε.
5) The integral ∫0rdrδc, or the inverse derivative of the
broadened delta function δc, is a continuous or
broadened step function Sc(r;r0) such that
∫0rdr f(r) δc(r r─0) f(r≈0) Sc(r;r0), (4)
where f(r) varies slowly over the transient layer, and
Sc(r;r0) has the properties
Sc(r;r0) 0 ≈ r < r0n─ ε
Sc(r;r0) 1/2≈ r = r0 '
Sc(r;r0) 1≈ r > r0+n .ε
(For convenience, the symbol r represents both the
dummy variable and the integral limit.) That Sc1/2≈ for
r=r0 can be seen by integrating G(r) from 0 to r0, and
recalling that the integral over all space of a normalized
Gaussian is unity. The function Sc, while locally
continuous, appears globally discontinuous in that its
value changes rapidly over the thickness 2nε of the
transient layer.
One advantage to modeling mass density (r) ρin terms of
a broadened delta function is the ease of integration
when solving EFE. Many integrals can be read off by
simply applying Eq. (4). This technique can be extended
to concentric shells, such as those discussed in reference
[52], and may be useful for modeling astrophysical
objects such as spherical dust accretion clouds
surrounding dirty black holes [53], spherical domain
walls enclosing the known cosmos, or for a discrete
sampling of any continuous spherical mass distribution.
III. SOLVING EINSTEIN'S FIELD EQUATIONS
FOR A THIN SHELL: THE CONTINUOUS
SOLUTION METHOD
We will now derive a locally continuous ultra-thin shell
solution to EFE, assuming a static spherically symmetric
metric gαβ of the form
ds2 = g00(r) dt2 + g11(r) dr2 r─2dΩ2
= eν dt2 e─ λdr2 r─2dΩ2
The appropriate gravitational field equations may be
found by substituting this metric into Einstein's field
equations, given by
Rμ
ν (1/2) g─μ νR = Tκμ
ν (5)
where Rμ
ν is the curvature or Ricci tensor, R is the scalar
curvature, κ is a constant with the value κ=8 G/c─ π 2
(using Dirac's sign convention [54]), or = 8 κ ─ π for
G=c=1, and Tμ
ν=diag( , p, p, p)ρ ─ ─ ─ is the stress energy
tensor, with (r)ρ the mass-energy density and p(r) the
pressure. After calculating the Christoffel symbols Γμ
αβ
and curvatures R and Rμ
ν, EFE of Eq. (5) simplify to a
pair of simultaneous equations [55]
Tκ00 = (r) = eκρ ─λ/r2 1/r─2 e──λλ′/r (6a)
Tκ11 = p(r) = e─ κ ─λ/r2 1/r─2 + e─λν′/r, (6b)
3

where primes denote derivatives with respect to r. Eq.
(6a) can be solved by rearranging terms to produce a
pure differential (see Appendix for details of derivations
in this section):
rκρ 2 + 1 = (re─λ)′.
Integrating and solving for eλ, we obtain
eλ = [1 + k0 /r + ( /r)κ ∫r dr (r) rρ2] 1─. (7)
where k0 is a constant of integration. Substituting
(r)=μρ0δc(r r─0) and μ0=m0/4 rπ02, and applying Eq. (4),
this becomes
eλ = (1 + k0 /r 2m─0Sc/r) 1─. (8)
For an empty shell, the boundary condition that eλ be
non-singular at r=0 requires that k0=0. (If the shell
contains a central mass M, an integration constant
k0= 2M─ is generally assumed.) The rr component of the
ultra-thin shell metric is therefore
g11 = e─λ = (1 2m─ ─ 0Sc /r) 1─. (9)
Outside the shell, where Sc1≈, we see that g11 matches
the radial component of the Schwarzschild metric gSμν,
as given by
ds2 = (1 2m/r)dt─2 (1 2m/r)─ ─ 1─dr2 r─2dΩ2 (10)
for m the central mass. In the interior of the shell, where
Sc0≈, it is clear that g11 matches the Minkowski metric.
Next, the tt component g00=eν can be evaluated by
subtracting Eq. (6b) from Eq. (6a) to obtain
( +p) = eκ ρ ─ ─λ λ′/r e─ ─λ ν′/r.
Solving for ν′, substituting eλ from Eq. (9) and (r) ρfrom
Eq. (1), and using equation of state p=wρ, the result is
ν′ = ─ λ′ (1+w)μ─ κ 0δcr / (1 2m─0Sc/r),
where δc and Sc are abbreviated notations for the
broadened delta and step functions. Upon integrating,
this becomes
= + kν ─ λ 1 (1+w)μ─ κ 0 ∫r dr [δcr / (1 2m─0Sc/r)] (11)
with k1 a constant of integration. Eq. (11) represents an
exact solution to EFE for the tt metric component g00=eν
of an ultra-thin shell. The integrand, however, contains
the spherical Gaussian G(r) and may be difficult to
evaluate analytically. For the present, an arbitrarily close
approximation can be found using the properties of the
broadened step and delta functions. This procedure
requires care due to the rapid variation of Sc(r;r0) in the
transient layer r0n <r<r─ ε 0+nε. We proceed by writing
the integral in Eq. (11) as a function of the upper limit r
I(r) = ∫0r dr δc r/(1 2m─0Sc/r). (12)
Since δc(r r─0) 0≈ in the near-vacuum domains r<r0n─ ε
and r>r0+nε, the integrand vanishes to any desired
accuracy in these domains. (An exception is the case
r0=2m0, where the integrand approaches 0/0 rather than
0 for r>>r0+n , εas will be discussed in Section IV.)
Hence in general, r changes by a near infinitesimal
amount 2nε across the non-vanishing domain of the
transient layer and may be treated as a constant r r≈0.
Thus we have,
I(r) r≈0 ∫0r dr δc / (1 2m─0Sc/r0) r0≠2m0. (13)
(Here as elsewhere, the symbol ≈ denotes asymptotic
equality, for which precision increases as εdecreases.)
I(r) can now be integrated to asymptotic precision by a
unique change of variable. Recalling from Eq. (4) that
Sc=∫rδcdr and therefore dSc=δcdr, the continuous
monotonic function Sc can be used as the variable of
integration. The limits of integration become 0 and Sc(r),
and the integral may be written
I(r) r≈0 ∫0Sc(r) dSc / (1 2m─0Sc / r0 )
(r≈ ─ 02/2m0) ln |(1 2m─0Sc / r0)|,
where the absolute value, arising from the standard
integral formula (dx/x)=ln|x|∫, will impact later analysis.
Substituting I(r) back into Eq. (11) and evaluating the
constants κ and μ0 yields
+ kν ≈ ─ λ 1 (1+w) ln |1 2m─ ─ 0Sc/r0)|.
Upon substitution of eλ from Eq. (8), the result is
eν (1 2m≈ ─ 0Sc/r) ek1 |1 2m─0Sc/r0| (1+w)─.
Since eν must obey the Minkowski condition eν>1─ as
r >∞─, the integration constant ek1 must cancel the right-
hand factor in the outer region where Sc>1─, leaving
only the left-hand factor, which is asymptotically
Minkowski. Hence the integration constant is
ek1 = |1 2m─0/r0| (1+w)
and the final result for the tt component of the ultra-thin
shell metric is
g00 (1 2m≈ ─ 0Sc/r) |1 2m─0/r0|(1+w) |1 2m─0Sc/r0|(1+w)─
r0 ≠ 2m0. (14)
To analyze this result, we evaluate g00 for the interior
and exterior, obtaining
g00int |1 2m≈ ─ 0/r0|(1+w) (15a)
g00ext (1 2m≈ ─ 0/r). (15b)
4

The exterior component g00, like the exterior component
g11, matches the Schwarzschild solution as expected.
Note that the quantity 2m─0 in the exterior metric arises
automatically from the field equations and, unlike for the
case of Schwarzschild metric, is not put in as an
integration constant. That this quantity is predetermined
by EFE further confirms the consistency of general
relativity, in that vacuum and non-vacuum solutions
agree for regions surrounding a central mass. Thus, solar
system tests confirm not just the vacuum equations,
where Tμ
ν=0, but also the massive equations, where
Tμ
ν≠0, insofar as a thin shell serves as well as a point
mass for modeling a star or planet.
Regarding time dilation, it is significant that the interior
metric g00int is a constant not equal to unity, while the
exterior metric g00ext asymptotically approaches unity,
indicating clocks inside the shell run at different rates
than those at infinity. For so-called non-phantom matter,
which has an EoS p(r)=w (r)ρ with w> 1─, we note that
g00int<1, indicating time inside the shell is dilated with
respect to infinity. This result may seem at odds with
occasional claims that time does not dilate inside an
empty shell. Such claims may arise from piecewise
solutions and are often based on two arguments: 1)
Minkowski spacetime, with g00=1, prevails inside a
hollow shell; or 2) according to Birkhoff's theorem, the
Schwarzschild metric governs the vacuum in an empty
shell, leading to g00=1 [51]. These arguments, however,
depend on a rescaling of the time coordinate inside the
shell. The continuous solution method, in contrast,
assumes a uniform time coordinate over the whole space
domain 0≤r<∞. It is clear, nevertheless, that no apparent
gravitational forces exist inside an empty shell due to the
constant value of the interior metric.
For a shell composed of dust, the EoS parameter is w=0,
and the interior and exterior solutions match at r=r0.
Therefore g00 and the corresponding clock rates are
continuous across the shell wall. The tt component for a
thin dust shell thus satisfies the first Israel junction
condition.
For a shell composed of stiff matter, which has an EoS of
w=1, we see that g00 changes abruptly across the shell
wall, allowing interior time dilation up to twice that at
the outer surface. Thus the continuous solution method
predicts time dilation measurements using real non-dust
shells would show a violation of the Israel conditions. It
seems interesting that the interior metric g00int depends
on the EoS of the shell, while the exterior metric g00ext
like the Schwarzschild metric, is independent of the EoS.
This curious distinction resolves the seeming paradox,
mentioned in a previous paper [56], that while non-
vacuum solutions to EFE require an EoS, Schwarzschild
vacuum solutions do not, even though mass appears in
the metric.
IV. BLACK HOLES AND BLACK SHELLS
The ultra-thin shell metric of Eqs. (9) and (11) may be
applied to shells of radius equal to or less than the
Schwarzschild radius, or shells such that ro≤2m0. To be
called black shells, these exotic objects would generally
appear to a distant observer as a Schwarzschild black
hole (although unexpected singularities may occur). At
close range, black shells display unique properties with
respect to horizons and singularities. To compare black
holes and black shells, first recall the properties of the
Schwarzschild black hole with metric gSμν as given by
Eq. (10):
1. A coordinate singularity, or horizon, exists at
r=2m, where gS00=0 and gS11 >∞.─ ─
2. Inside the horizon, squared proper time intervals
dτ2=gS00dt2 are negative, and thus proper time is
spacelike, while squared proper radial intervals
dR2=gS00dr2 are positive, and proper radial
distance is timelike.
3. A physical singularity is generally assumed to
exist at r=0, where gS00 >-∞ and ─gS11=0.
4. There are no finite discontinuities in the domain
r>0.
To compare the properties of black shells, we consider
the metrics for four shell types: ordinary shells with
r0>2m0; horizon black shells with r0=2m0; subhorizon
black shells with r0<2m0, and semi-horizon black shells
with r0=m0. First, recall the interior and exterior thin
shell metrics of Section III:
g00int (1 2m≈ ─ 0 /r0)(1+w) r0 ≠ 2m0 (16a)
g00ext 1 2m≈ ─ 0 /r r0 ≠ 2m0 (16b)
g11int 1≈ ─ (16c)
g11ext (1 2m≈ ─ ─ 0 /r) 1─. (16d)
In the case of ordinary shells (r0>2m0), the
Schwarzschild radius rs=2m0 lies inside the shell where
the metric is constant. Thus there is no horizon at r=rs. In
addition, no singularity exists at r=0. Although the
metric is locally continuous everywhere, comparison of
Eqs. (16c) and (16d) reveals a global discontinuity or
jump across r0 in the component g11. For non-dust
models, for which w≠0, there is also a jump across r0 in
the component g00, in apparent violation of the Israel
junction conditions. However when w=0 as in the case of
dust, g00 remains unchanged across r0, in agreement with
the Israel conditions.
For horizon black shells (r0=2m0), the shell radius is
equal to the Schwarzschild radius rs=2m0, and as noted
earlier, the approximation r >r─0 in the integrand of I(r)
of Eq. (13) is no longer valid. Deriving the properties of
g00 would require computing the exact integral of Eq.
5

(12) using the spherical Gaussian. Such a calculation is
not attempted here. If, however, we naively allow the
approximation r >r─0 and apply Eq. (14), the apparent
properties of horizon black shells suggest such objects
may be nonphysical. To illustrate, recall the full
equations for the metric:
g00 (1 2m≈ ─ 0Sc/r)|1 2m─0/r0|(1+w)|1 2m─0Sc/r0|(1+w) ─(17)
g11 1 / (1 2m≈ ─ ─ 0Sc/r) (18)
Setting r0=2m0 in the first equation and assuming w> 1─,
it is clear that g00(r)=0 for 0<r<∞. This can be seen by
noting that the middle factor in g00 vanishes identically,
while the right-hand factor (denominator) is non-
vanishing for all finite r due to the property Sc(r)<1, and
the left-hand factor is finite for all r>0. The vanishing of
g00 suggests that a horizon black shell would stop all
clocks in the universe, a physical impossibility and a
violation of the asymptotic Minkowski condition.
Whether this nonphysical result can be avoided by
evaluating g00 analytically using the function G(r), by
applying numerical methods, or by redefining δc in terms
of a function other than G(r), is a question for future
research.
Concerning the rr metric component, we see from Eq.
(16c) that g11 1≈─ inside the shell, implying no interior
singularities exist. To check this result, note that by Eq.
(18), no singularity can exist unless there is an r such that
2m0Sc(r)/r=1, or r/r0=Sc(r). Since it is always true that
Sc(r)<1, any such singularity can only reside at r<r0. It
will be stated without proof that since Sc(r) 1/2≈ when
r/r0=1, and since Sc(r) falls to zero more rapidly than
r/r0, there can be no r>0 such that 2m0Sc(r)/r=1, and
hence no singularity in the domain 0<r<r0. Moreover,
by L'Hopital's rule it is found that
lim[r >0]─ 2m0 Sc(r)/r = 0,
ruling out a singularity at the origin. Thus a horizon
black shell, unlike a Schwarzschild black hole, manifests
no singularities in g11.
Subhorizon black shells (r0<2m0), in contrast, appear at
close range like Schwarzschild black holes, with a
horizon at r 2m≈0. Subhorizon black shells also have
approximate Schwarzschild behavior for r>2m0.
However, a new singularity in g00 may arise due to the
vanishing of 1 2m─0Sc/r0 in the right-hand factor
(denominator) of Eq. (17). To locate this singularity,
recall that Sc(r) increases monotonically over the range
0<Sc<1. Thus g00 becomes singular at some unique r
such that Sc(r)=r0/2m0. Since Sc(r) traverses nearly all of
its range within a distance n εof r0, such singularities
usually fall within r0n <r<r─ ε 0+nε, or in the transient
layer of the wall itself. However if r0=2m0─ς, where ς is
some extremely small quantity, a singularity may occur
at some large radius r=R0 where Sc(R0)=r0/2m01≈. This
means subhorizon black shells could in principle cause
singularities in g00 at cosmological distances. Such
models may have astrophysical applications related to
the composition of galactic cores (the topic of Section
V), or cosmological interpretations with respect to
Hubble redshift, bubble universes or spherical domain
walls, to be addressed in a later paper.
In the unique case of a semi-horizon black shell for
which the radius r0=m0 is half the Schwarzschild radius,
one might expect a singularity in g00(r) at r=r0, where
Sc(r) 1/2≈. However, it turns out that g00(r) has a finite
discontinuity rather than a singularity at r=r0. This can
be shown as follows. Setting w=0 and r0=m0, Eq. (17)
simplifies to
g00 (r) [1 2r≈ ─ 0Sc(r)/r] / |1 2S─c(r)|,
which, as r tends to r0, approaches the improper limit
0/0. Applying L'Hopital's rule yields the ratio H of the
derivatives of numerator and denominator:
H = ∂r [1 2r─0Sc/r] / ∂r |1 2S─c|
= (2r0Sc/r2 2r─0δc/r) / ( ⁄ + 2─ δc)
= ⁄ + (r─0/r) (Sc/rδc 1)─
where the sign ambiguity springs from the absolute
value. Taking the limit r >r─0, the term Sc/rδc approaches
π1/2 /2rε0<<1, and H tends to positive or negative unity,
with the positive case corresponding to approach from
r>r0 and the negative to r<r0. Thus for r0=m0, the limit
is not unique, leaving g00 undefined at r=r0. Whether the
semi-horizon black shell discontinuity arises as an
artifact of the approximation is not known.
V. BLACK SHELLS, MOND, AND THE
GALACTIC ROTATION CURVE
Can supermassive black shells in the cores of galaxies
explain the discrepancy in the galactic rotation curve? If
so, it would obviate the need for postulating a dark
matter halo. The discrepancy in orbital velocity v(r), as
noted earlier, arises from observations of differential
Doppler shift, which indicate the outer stars and
hydrogen clouds of galaxies orbit too fast to be
gravitationally bound by luminous or baryonic matter
alone. Thus, outside the bright galactic disk, v(r) does not
fall off as r1/2─, as would be expected from Newtonian
dynamics [57], but tends toward a constant as r
increases. This anomaly was noted by Fritz Zwicky in
1933 [58] and first quantified observationally by Vera
Rubin [59].
The flattening of the galactic rotation curve can be
described by an effective potential φm(r) that depends
only baryonic mass and increases with r at large
6

distances. The potential φm, for reasons evident below,
will be called the MOND potential. The goal is to show
that a subhorizon black shell (SBS), or similar exotic
black object, located in the galactic core, could
theoretically account for the observed excess orbital
velocities, or equivalently, that an SBS potential φSBS(r)
can be made consistent with the MOND potential φm(r)
in outlying regions. The MOND potential will be derived
first, followed by the SBS potential. The two will then be
equated to show, by a redefinition of the broadened delta
function, a close correspondence in the metrics.
The MOND potential can be calculated from Modified
Newtonian Dynamics (MOND), a formalism developed
in 1983 by Mordehai Milgrom [60] to account for the
discrepancy in the rotation velocity of galaxies. Although
the excess velocity is usually attributed to the presence of
an unseen dark matter halo, the MOND formalism,
relying on baryonic matter alone, has proven accurate in
predicting orbital motion [61], and thus provides a means
for testing theories.
The MOND formalism is based on the empirical relation
μ(a/a0)a = aN (19)
which connects observed radial acceleration a to
predicted Newtonian acceleration aN=GM/r2 using an
interpolating function μ(a/a0), where
a0 = 1.2x10 8─cm/sec2 H≅0 /2 = cπ2/R = c2/( /3)Λ−1/2
is a universal constant with dimensions of acceleration,
H0 is the Hubble parameter [62], and R is roughly 2π
times the radius of the visible universe or the de Sitter
radius corresponding to cosmological constant Λ [63].
The interpolating function runs smoothly from the inner
galaxy, where the field falls off as roughly 1/r2, to the
region outside the bright galactic disk, called the deep
MOND region, where the field tends to fall off as 1/r.
Using the simple interpolating function
μ(a/a0) = (a/a0) / (1+a/a0)
proposed by Zhao, Famaey and Binney [64, 65], the
MOND relation of Eq. (19) becomes a quadratic
equation with solution,
a = (GM/2r─2) [1 + √(1+4r2/Rm2)]. (20)
The radius Rm, to be called the MOND radius, lies near
the edge of the bright galactic disk and has the value
Rm = √(GMR/c2) = √(GM/a0).
In the domain of interest 2Rm<r<<R, which is roughly
the region outside the luminous disk, the observed radial
acceleration a of Eq. (20) can be approximated as
a GM/2r≅ ─ 2 GM/R─mr. (21)
The potential in this domain can be expressed as
φm = a(r) dr GM/2r + (GM/R─ ∫ ─≅m) ln (r/Rm).
The factor 1/2 in the first term on the right does not
appear in some presentations of MOND, where different
interpolating functions apply and where the potential
covers all space [66]. However, since the second term
increases with r and becomes dominant near Rm, we can
neglect the first term and construct an effective metric
for the deep MOND region [67]
g00 1 + 2≅ φm/c2 1 + (2GM/c≅2Rm) ln (r/Rm), (22)
which is accurate in the domain nRm<r<<R, for n a
small integer on the order of 4 or 5. Note that g00 >∞─ as
r >∞.─ Hence the effective metric violates the asymptotic
Minkowski condition and cannot, in the form of Eq. (22),
be consistent with a black shell metric. Consistency will
be attained through a later approximation.
Next, to calculate the SBS potential φSBS(r), we assume
the galaxy is centered on a supermassive ultra-thin SBS
of radius
r0 = 2m0 = (1 ) r─ ς ─ σ s, (23)
where >ς ε is a small distance on the order of meters, rs is
the Schwarzschild radius 2m0, shell mass m0 is a large
fraction of galactic mass M, and parameter σ=/rςs
measures the small difference between shell size and
Schwarzschild radius. Such an SBS would induce a
singularity in g00 at some cosmic-scale radius R0, at
which clocks would theoretically run at an infinite rate.
In realistic scenarios, no remote singularity can occur
due to disturbance of the mass density by other fields.
Nevertheless, a remote virtual singularity implies a
modification of the field in the neighborhood of the
galaxy.
The distance to the singularity at R0 is inversely related
to ς and increases with step width 2nε. More specifically,
from Eq. (17), R0 must satisfy
1 2m─0Sc(R0)/r0 = 0,
or, upon substiting 2m0=r0+ς and rearranging,
Sc(R0) = 1 / (1 + / rςs) 1 / r≅ ─ ς s. (24)
To calculate the impact of the distant singularity on the
field in the galactic neighborhood, we start by
introducing a new function (r)η and expressing the
broadened step function as Sc(r)=1 (r)─η , where (r)<<1η
in the deep MOND region. This and Eq. (23) are then
substituted into the thin shell metric of Eq. (17), and the
result is evaluated for the shell's far exterior r0<<r<R0,
yielding
g00SBS [1 r≈ ─ s(1 )/r] / | (r) |─ η σ η ─ σ
(1 r≅ ─ s/r) / | (r) |.σ η ─ σ (25)
7

From Eq. (24), we see that (Rη0) = /r≅σ ς s, and the
denominator of g00SBS vanishes near R0 as expected.
To match g00SBS to the MOND metric of Eq. (22), we
first write a an approximation to the latter which
repositions the singularity from infinity to a remote finite
distance r=R0 as follows:
g00MOND 1 + 2≅ φm/c2 1 + (r≅s/Rm) ln |r/(R0r)|.─ (26)
This approximation can be checked by calculating
acceleration a from potential φm
a ≅ ─ φm′ GM/R≅ ─ mr GM/R─m(R0r).─
It is clear that for r in the neighborhood of the galaxy,
(R0r)─ is large enough that the right-hand term can be
neglected. The remaining term matches the MOND
acceleration of Eq. (21). Hence we see that g00MOND of
Eq. (26) adequately approximates the MOND metric in
the deep MOND region.
The MOND and SBS metrics may now be equated,
giving
1 + (rs/Rm) ln |r/(R0r)| = (1 r─ ─ s/r) / [ (r) ].σ η ─ σ
By solving for (r), ηa new form Sm(r) of the broadened
step function is obtained that is consistent with the
MOND metric as follows:
Sm(r) = 1 (r) = 1 / [1+(r─η ─ σ s/Rm) ln |r/(R0r)|] .─ ─ σ
Simple calculation shows that Sm(r), while different from
the broadened step function Sc(r) derived in Section II,
has like properties in the domain r0<<r<R0. To wit,
Sm(r) is slightly less than one and increases
monotonically to the near-unity value 1─σ as r
approaches the near-infinite distance R0. Thus, it is
possible to derive a MOND-compatible step function
Sm(r) by replacing the Gaussian G(r) with some
appropriate function F(r) in the definition of the
broadened delta function δc, thus obtaining a new delta
function δm. An SBS modeled on δm, embedded in the
galactic core, would then account for the anomalous
orbital velocities. The exact function F(r) is unknown.
Questions also remain about SBS formation and stability.
What is important is the implication that an exotic black
object, possibly a subhorizon black shell, could in
principle cause the observed galactic rotation curve
without the need for a dark matter halo.
VI. CONCENTRIC SHELLS AND DISCRETE
DENSITY SAMPLING
The continuous solution method is easily generalized to
n concentric shells of arbitrary mass and radius. This
technique provides a formalism for solving EFE for any
continuous static spherical density distribution (r)ρ,
where (r)ρ is modeled by a discrete sampling at
r = {r0, r1 ... rn-1}.
The method for concentric shell solutions will be
illustrated for the simple case of two shells with EoS
parameter w=0. Assuming surface densities μ0 and μ1,
radii r0 and r1, and masses m0=4 μπ0r02 and m1=4 μπ1r12,
the mass density can be expressed in terms of broadened
delta functions as
(r) = μρ0δ0 + μ1δ1
where δj=δc(r r─j) denotes a broadened delta function at
radius rj. Substituting (r)ρ into Eq. (7) and setting the
integration constant to zero yields
g11 = e─λ = [1 + ( /r)─ κ ∫r dr rρ2] 1 ─
= [1 + ( μ─ κ 0/r)∫r dr r2δ0 + ( μκ1/r)∫r dr r2δ1] 1─.
Upon integration, the double thin-shell solution becomes
g11 = e─λ = [1 2m─ ─ 0S0/r 2m─1S1/r] 1─ (27)
where S0=Sc(r;r0) and S1=Sc(r;r1). The interior
(r<r0n )─ ε , middle (r0+n <r<rε1n )─ ε , and exterior
(r>r1+n )ε solutions are therefore
g11int 1≈ ─ (28a)
g11mid (1 2m≈ ─ ─ 0/r) 1─ (28b)
g11ext [1 2(m≈ ─ ─ 0 + m1)/r] 1─, (28c)
displaying Minkowski properties inside the smaller shell,
Schwarzschild behavior between shells, and combined
Schwarzschild behavior outside the larger shell.
To solve for the time component g00 = eν, the method of
Section III will be applied. From Eqs. (11) and (27), we
have
= + kν ─ λ 1 ─ κ∫r dr (r) eρλ r
= + k─ λ 1 ─ κ∫rdr(μ0δ0+μ1δ1)r / [1 2(m─0S0+m1S1)/r].
(29)
The integral may be expressed as a sum of two terms:
I(r) = μ0∫rdr δ0r / [1 2(m─0S0+m1S1)/r]
+ μ1∫rdr δ1r / [1 2(m─0S0+m1S1)/r].
Since r is slowly varying over the two transient layers, it
can be approximated by r0 and r1 in the two respective
integrands, yielding
I(r) μ≈0r0∫rdr δ0 / [1 2(m─0S0+m1S1)/r0]
+ μ1r1∫rdr δ1 / [1 2(m─0S0+m1S1)/r1]
Note that in the first integral, the outer step function
S1(r) varies slowly over the nonzero domain of the inner
delta function δ0, and hence may be set to a constant
S10≈. Analogously, in the second integral, S0(r) varies
8

slowly over the nonzero domain of δ1 and may be set to a
constant S01≈. The total integral then simplifies to
I(r) μ≈0r0∫rdr δ0 / (1 2m─0S0/r0)
+ μ1r1∫rdr δ1 / (1 2m─0/r1 2m─1S1/r1).
Following the method of Section III, a change of variable
from r to S0(r) and S1(r) in the respective integrals gives,
upon integration,
I(r) (μ≈0r02/2m0) ln |1 2m─0S0/r0|
+ (μ1r12/2m1) ln |1 2m─0/r0 2m─1S1/r1|.
Multiplying I(r) by κ and substituting back into Eq. (29)
then yields
= + kν ─λ 1 ln |1 2m─ ─ 0S0 /r0|
ln |1 2m─ ─ 0 /r0 2m─1S1 /r1|.
and hence
g00 = e ν= [1 2(m─0S0+m1S1)/r] ek1 |1 2m─0S0/r0|1 ─
X |1 2m─0/r0 2m─1S1/r1|1─ (30)
where X denotes multiplication. Again, to meet the
asymptotic Minkowski condition, the integration
constant must be
ek1 = |1 2m─0/r0| |1 2m─0/r0 2m─1/r1| (31)
The constant ek1 is then substituted back into Eq. (30),
yielding the g00 component of the double concentric
shell metric. Taken together, Eqs. (27), (30) and (31)
represent a complete continuous asymptotically exact
solution to EFE for two concentric ultra-thin dust shells
of arbitrary mass and radius.
It is straightforward to extend this result to n concentric
shells of mass mi, radius ri, and thickness εi, as long as
εi<<(ri+1 r─i). Such a set of locally continuous thin
shells may be viewed as a discrete sampling, at arbitrary
radii ri, of a globally continuous mass density
distribution (r)ρ. The concentric shell formalism thus
provides a discrete method for approximating the
solution to EFE for any static, spherically symmetric
mass-energy density. Hence Einstein's equations can be
readily solved for complicated scenarios such as a star
surrounded by spherical dust clouds embedded in cosmic
bubbles, and so forth. The impact of discreteness on
accuracy is a topic for future discussion.
VII. CONCLUSION
We have derived an asymptotically exact solution to
Einstein's field equations for individual and multiple
concentric ultra-thin shells of arbitrary mass and radius
using a continuous solution method that does not require
junction conditions. The single shell solution is given by
Eqs. (9) and (14), and the double shell solution by Eqs.
(27), (30) and (31). These solutions are fixed by two
boundary conditions: asymptotic flatness at infinity and
non-singularity at the origin. The interior of a thin shell
is found to manifest no effective gravitational forces.
However, interior clocks run at different rates from those
at infinity. For non-phantom matter (w> 1)─, time in the
interior of the shell is dilated with respect to infinity,
while for phantom matter, time is contracted.
Exterior to the shell, the field generally matches that of
the Schwarzschild metric. Exceptions are found for black
shells, i.e. shells of radius less than or equal to the
Schwarzschild radius. The method breaks down for equal
radii, and an asymptotically exact solution was not
attempted. However, approximations suggest such
objects may be unphysical. Subhorizon black shells,
which have a radius smaller than the Schwarzschild
radius, are more easily analyzed, and were shown in
general to appear as Schwarzschild black holes
everywhere outside the shell. This holds with one key
exception. When the radius of a supermassive black shell
is less than its Schwarzschild radius by a very small
distance on the order of meters, a singularity may occur
in the time component of the metric at cosmological
distances. It was then shown that this singular metric
approximates an effective MOND metric, where the
latter is expressed in terms of an effective potential that
accounts for the observed galactic orbital velocities.
Thus, a supermassive subhorizon ultra-thin black shell or
similar exotic black object, located at the center of a
galaxy, could theoretically explain the flattening of the
galactic rotation curve without the need for dark matter.
It was also shown that the solution for a series of
concentric shells provides a discrete sampling method for
calculating the approximate gravitational field of any
spherical static mass distribution. Applications might
include detailed scenarios such as spherical accretion
shells around black holes embedded in a constant
background density enclosed by a cosmic bubble.
The method developed here applies to static scenarios. It
can in principle be generalized to dynamic configurations
such as colliding shells in anti-deSitter spacetime [68] or
black holes embedded in expanding bubble universes
described by the Friedman-Robertson-Walker metric.
These are topics for future research. Other questions also
remain concerning:
1. Multiple concentric shell techniques for discrete
sampling of cosmological mass distributions,
2. The impact of discreteness on accuracy,
9

3. Comparison of ultra-thin shell boundary
properties to Israel junction conditions under a
general EoS,
4. Collapsing ultra-thin shells and black shell
formation,
5. Whether possible nonphysical features of horizon
black shells interfere with black shell formation,
6. The nature and stability of rotating or charged
ultra-thin shells,
7. Stability of ultra-thin shells, particularly of
subhorizon black shells in galactic cores, and
8. The mathematical properties of functions F(r)
and δm compatible with MOND and the galactic
rotation curve.
APPENDIX
Using the line element
ds2 = g00(r) dt2 + g11(r) dr2 r─2dΩ2
= eν dt2 e─ λdr2 r─2dΩ2
with κ=8─ π and a diagonal stress-energy tensor of the
form Tμ
ν=diag( , p, p, p)ρ ─ ─ ─ , Einstein's field equations
simplify to
Tκ00 = (r)κρ = e─λ/r2 1/r─2 e──λλ′/r (a1)
Tκ11 = p(r) = e─ κ ─λ/r2 1/r─2 + e─λν′/r, (a2)
where primes denote derivatives with respect to r. Eq.
(a1) can be solved by rearranging terms
rκρ 2 + 1 = e─λ (1 ─ λ′r)
= (re─λ)′.
Integration then yields
re─λ = k0 + ∫r dr ( rκρ 2 + 1) .
Here, ∫r denotes the inverse derivative and k0 is a
constant of integration. Solving for eλ, we obtain
eλ = [1 + k0/r + ( /r)κ ∫r dr (r) rρ2] 1─.
Substitution of (r)=μρ0δc(r r─0) and application of Eq.
(4) gives
eλ = (1 + μκ0r02Sc/r + k0/r) 1─.
Using surface density μ0=m0/4 rπ02, this becomes
eλ = (1 2m─0Sc/r + k0/r) 1─. (a3)
The boundary condition that eλ be nonsingular at r=0
requires that k0=0. The rr component of the ultra-thin
shell metric is therefore
g11 = e─λ = (1 2m─ ─ 0Sc/r) 1─. (a4)
The tt component g00=eν can be evaluated by subtracting
Eq. (a2) from Eq. (a1) to obtain
( +p) = eκ ρ ─ ─λ λ′/r e─ ─λ ν′/r.
Solving for ν′ yields
ν′ = ─ λ′ ( +p) e─ κ ρ λr.
If we now substitute (r)ρ and eλ from Eq. (a4), and apply
the equation of state p=wρ for w a constant, the result is
ν′ = ─ λ′ (1+w)μ─ κ 0δcr / (1 2m─0Sc/r).
Upon integrating, this becomes
= + kν ─ λ 1 (1+w)μ─ κ 0 ∫r dr δcr / (1 2m─0Sc/r) (a5)
with k1 a constant of integration. Eq. (a5) represents an
exact solution to Einstein's field equations for the tt
metric component g00=eν of an ultra-thin shell. To
approximate the integral, we use the properties of the
broadened step and delta functions. The integral may be
written
I(r) = ∫0r dr δc r/(1 2m─0Sc/r).
Since r changes by the near infinitesimal amount 2nε
across the transient layer, it may be treated as a constant
r r≈0, hence
I(r) r≈0 ∫0r dr δc/(1 2m─0Sc/r0) r0≠2m0.
I(r) can be integrated by a change of variable dSc=δcdr,
with limits of integration 0 and Sc(r):
I(r) r≈0 ∫0Sc(r) dSc/(1 2m─0Sc/r0)
(r≈ ─ 02/2m0) ln| (1 2m─0Sc/r0)|0Sc(r)
(r≈ ─ 02/2m0) ln| (1 2m─0Sc/r0)|,
Substituting I(r) into Eq. (a5) yields,
+ kν ≈ ─ λ 1 + [ (1+w)μκ0r02/2m0] ln |1 2m─0Sc/r0|.
Evaluating the constants κ and μ0, this simplifies to
+ kν ≈ ─ λ 1 (1+w) ln |1 2m─ ─ 0Sc/r0)|,
with the result
eν e≈─λ ek1 |1 2m─0Sc/r0| (1+w)─
(1 2m≈ ─ 0Sc/r) ek1 |1 2m─0Sc/r0| (1+w)─.
Since eν must obey the Minkowski condition eν>1─ as
r >∞─, the integration constant ek1 must cancel the right-
hand factor in the outer region where Sc1≈, Hence the
integration constant is
ek1 = |1 2m─0/r0| (1+w)
10

and the tt component of the ultra-thin shell metric
becomes
g00 (1 2m≈ ─ 0Sc/r)|1 2m─0/r0|(1+w)|1 2m─0Sc/r0|(1+w)─
r0 ≠ 2m0.
_____________
[1] Mordehai Milgrom, MOND in galaxy groups: A
superior sample, Phys Rev D 99, 044041 (2019)
[2] Mordehai Milgrom, Bimetric MOND gravity, Phys
Rev D 80, 123536 (2009)
[3] A. Stabile and S. Capozziell, Galaxy rotation curves
in f(R, ) gravityφ, Phys Rev D 87,064002 (2013)
[4] Edmund A. Chadwick, Timothy F. Hodgkinson, and
Graham S. McDonald, Gravitational theoretical
development supporting MOND, Phys Rev D Phys. Rev.
D 88, 024036 (2013)
[5] Naoki Tsukamoto and Takafumi Kokubu, Linear,
stability analysis of a rotating thin-shell wormhole, Phys
Rev D 98, 044026 (2018)
[6] Roberto Balbinot, Claude Barrabès, and Alessandro
Fabbri, Friedmann universes connected by Reissner-
Nordström wormholes, Phys. Rev. D 49, 2801 (1994)
[7] Z. Amirabi, M. Halilsoy, and S. Habib
Mazharimousavi, Effect of the Gauss-Bonnet parameter
in the stability of thin-shell wormholes, Phys Rev D 88,
124023 (2013)
[8] S. Shajidul Haque and Bret Underwood, Consistent
cosmic bubble embeddings, Phys Rev D 95, 103513
(2017)
[9] Anthony Aguirre and Matthew C. Johnson, Towards
observable signatures of other bubble universes. II.
Exact solutions for thin-wall bubble collisions, Phys Rev
D 77, 123536 (2008)
[10] Eduardo Guendelman and Idan Shilon,
Gravitational trapping near domain walls and stable
solitons, Phys Rev D 76, 025021 (2007)
[11] Bum-Hoon Lee et. al., Dynamics of false vacuum
bubbles with nonminimal coupling, Phys Rev D 77,
063502 (2008)
[12] Debaprasad Maity, Dynamic domain walls in a
Maxwell-dilaton background, Phys Rev D 78, 084008
(2008)
[13] Debaprasad Maity, Semirealistic bouncing domain
wall cosmology, Phys Rev D 86, 084056 (2012)
[14] S. Habib Mazharimousavi and M. Halilsoy, Domain
walls in Einstein-Gauss-Bonnet bulk, Phys Rev D 82,
087502 (2010)
[15] Shu Lin and Ho-Ung Yee, Out-of-equilibrium chiral
magnetic effect at strong coupling, Phys Rev D 88,
025030 (2013)
[16] Yumi Akai and Ken-ichi Nakao, Nonlinear stability
of a brane wormhole, Phys Rev D 96, 024033 (2017)
[17] Ken-ichi Nakao, Tatsuya Uno, and Shunichiro
Kinoshita, What wormhole is traversable? A case of a
wormhole supported by a spherical thin shell, Phys Rev
D 88, 044036 (2013)
[18] Térence Delsate, Jorge V. Rocha, and Raphael
Santarelli, Collapsing thin shells with rotation, Phys Rev
D 89, 121501(R) (2014)
[19] Robert B. Mann John J. Oh and Mu-In Park, Role of
angular momentum and cosmic censorship in (2+1)-
dimensional rotating shell collapse, Phys Rev D 79,
064005 (2009)
[20] Hideo Kodama, Akihiro Ishibashi, and Osamu Seto,
Brane world cosmology: Gauge-invariant formalism for
perturbation, Phys Rev D 62, 064022 (2000)
[21] Matthew C. Johnson, Hiranya V. Peiris, and Luis
Lehner, Determining the outcome of cosmic bubble
collisions in full general relativity, Phys Rev D 85,
083516 (2012)
[22] Benjamin K. Tippett, Gravitational lensing as a
mechanism for effective cloaking, Phys Rev D 84,
104034 (2011)
[23] Kenji Tomita, Junction conditions of cosmological
perturbations, Phys Rev D 70, 123502 (2004)
[24] Elias Gravanis and Steven Willison, ‘‘Mass without
mass’’ from thin shells in Gauss-Bonnet gravity, Phys
Rev D 75, 084025 (2007)
[25] Chih-Hung Wang, Hing-Tong Cho, and Yu-Huei
Wu, Domain wall space-times with a cosmological
constant, Phys Rev D 83, 084014 (2011)
[26] Donald Marolf and Sho Yaida, Energy conditions
and junction conditions, Phys Rev D 72, 044016 (2005)
[27] S. Khakshournia and R. Mansouri, Dynamics of
general relativistic spherically symmetric dust thick
shells, arXiv:gr-qc/0308025v1 (2003)
[28] V. A. Berezin, V. A. Kuzmin, and I. I. Tkachev,
Dynamics of bubbles in general relativity, Phys Rev D
36, 2919 (1987)
[29] Cédric Grenon and Kayll Lake, Generalized Swiss-
cheese cosmologies. II. Spherical dust, Phys Rev D 84,
083506 (2011)
[30] László Á. Gergely, No Swiss-cheese universe on the
brane, Phys Rev D 71, 084017 (2005)
[31] Shu Lin and Edward Shuryak, Toward the AdS/CFT
gravity dual for high energy collisions. III.
Gravitationally collapsing shell and quasiequilibrium,
Phys Rev D 78, 125018 (2008)
[32] Adam Balcerzak and Mariusz P. Dąbrowski, Brane
f(R) gravity cosmologies, Phys Rev D 81, 123527 (2010)
[33] Nami Uchikata, Shijun Yoshida, and Toshifumi
Futamase, New solutions of charged regular black holes
and their stability, Phys Rev D 86, 084025 (2012)
[34] Nick E Mavromatos and Sarben Sarkar, Regularized
Kalb-Ramond magnetic monopole with finite energy,
Phys Rev D 97, 125010 (2018)
[35] David E. Kaplan and Surjeet Rajendran, Firewalls
in general relativity, Phys Rev D 99, 044033 (2019)
11

[36] Rui Andre´, Jose´ P. S. Lemos, and Gonçalo M.
Quinta, Thermodynamics and entropy of self-gravitating
matter shells and black holes in d dimensions, Phys Rev
D 99, 125013 (2019)
[37] Aharon Davidson and Ilya Gurwich, Dirac
Relaxation of the Israel Junction Conditions: Unified
Randall-Sundrum Brane Theory, arXiv:gr-qc/0606098v1
(2006)
[38] Chih-Hung Wang, Hing-Tong Cho, and Yu-Huei
Wu, Domain wall space-times with a cosmological
constant, Phys Rev D 83, 084014 (2011)
[39] Kota Ogasawara1 and Naoki Tsukamoto, Effect of
the self-gravity of shells on a high energy collision in a
rotating Bañados-Teitelboim-Zanelli spacetime, Phys
Rev D 99, 024016 (2019)
[40] Peter Musgrave and Kayll Lake, Junctions and thin
shells in general relativity using computer algebra: I.
The Darmois-Israel formalism, Classical and Quanum
Gravity, Volume 13, Number 7
[41] Carlos Barceló and Matt Visser, Moduli fields and
brane tensions: Generalizing the junction conditions,
Phys Rev D 63, 024004 (2000)
[42] Jesse Velay-Vitow and Andrew DeBenedictis,
Junction conditions for FðTÞ gravity from a variational
principle, Phys Rev D 96, 024055 (2017)
[43] Charles Hellaby and Tevian Dray, Failure of
standard conservation laws at a classical change of
signature, Phys Rev D 49, 5096 (1994)
[44] Alex Giacomini, Ricardo Troncoso, and Steven
Willison, Junction conditions in general relativity with
spin sources, 73, 104014 (2006)
[45] José M. M. Senovilla, Junction conditions for F(R)
gravity and their consequences, Phys Rev D 88, 064015
(2013)
[46] M. Heydari-Fard and H. R. Sepangi, Gauss-Bonnet
brane gravity with a confining potential, Phys Rev D 75,
064010 (2007)
[47] Stephen C. Davis, Generalised Israel Junction
Conditions for a Gauss-Bonnet Brane World, arXiv:hep-
th/0208205v3 (2002)
[48] Adam Balcerzak and Mariusz P. D browski, 
Generalized Israel Junction Conditions for a Fourth-
Order Brane World, arXiv:0710.3670v2; Phys.Rev.D 77,
023524 (2008)
[49] Szymon Sikora and Krzysztof Glod, Perturbatively
constructed cosmological model with periodically
distributed dust inhomogeneities, Phys Rev D 99,
083521 (2019)
[50] Jean-Philippe Bruneton and Julien Larena,
Dynamics of a lattice Universe: the dust approximation
in cosmology, Classical Quantum Gravity 29, 155001
(2012).
[51] Ignazio Ciuffolini and John Archibald Wheeler,
Gravitataion and Intertia, Princeton University Press
(1995), p40
[52] Metin Gürses and Burak Himmeto lu, Multishell
model for Majumdar-Papapetrou spacetimes, Phys Rev
D 72, 024032 (2005)
[53] Luiz C. S. Leite, Caio F. B. Macedo, and Luís C. B.
Crispino, Black holes with surrounding matter and
rainbow scattering, Phys Rev D 99, 064020 (2019)
[54] P.A.M. Dirac, General Theory of Relativity, John
Wiley & Sons, 1975, p47
[55] A. Bakopoulos, G. Antoniou, and P. Kanti, Novel
black-hole solutions in Einstein-scalar-Gauss-Bonnet
theories with a cosmological constant, Phys Rev D 99,
064003 (2019)
[56] Kathleen A. Rosser, Current conflicts in general
relativity: Is Einstein's theory incomplete?, ResearchGate
DOI: 10.13140/RG.2.2.19056.40966 (2019);
Quemadojournal.org
[57] Jacob D. Bekenstein, Relativistic gravitation theory
for the modified Newtonian dynamics paradigm, Phys
Rev D 70, 083509 (2004)
[58] F. Zwicky, Helv. Phys. Acta 6, 110 (1933)
[59] V. C. Rubin, N. Thonnard, and W. K. Ford, Jr.,
Astro-phys. J. 225, L107 (1978)
[60] Milgrom M., ApJ 270, 365 (1983)
[61] Robert H. Sanders and Stacy S. McGaugh, Modified
Newtonian Dynamics as a alternative to dark matter,
arXiv:astro-ph/0204521v1 2002
[62] Lasha Berezhiani and Justin Khoury, Theory of dark
matter superfluidity, Phys Rev D 92, 103510 (2015)
[63] Milgrom, Mordehai, MOND in galaxy groups, Phys
Rev D 98, 104036 (2018)
[64] Zhao, H.S. and Famaey, B., Refining MOND
interpolating function and TeVeS Lagrangian,
arXiv:astro-ph/0512425v3 (2006)
[65] Benoit Famaey and James Binney, Modified
Newtonian Dynamics in the Milky Way, arXiv:astro-
ph/0506723v2 (2005)
[66]: HongSheng Zhao, Baojiu Li and Olivier Bienayme,
Modified Kepler’s law, escape speed, and two-body
problem in modified Newtonian dynamics-like theories,
Phys Rev D 82, 103001 (2010)
[67] Benoit Famaey, Jean-Philippe Bruneton and
HongSheng Zhao, Escaping from Modified Newtonian
Dynamics, arXiv:astro-ph/0702275 (2007)
[68] Richard Brito, Vitor Cardos, Jorge V. Rocha, Two
worlds collide: Interacting shells in AdS spacetime and
chaos, arXiv:1602.03535v3 (2016)
12