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

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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.
Thin shells in general relativity without junction conditions: A model for galactic
rotation and the discrete sampling of fields
By Kathleen A. Rosser
(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.
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].
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, r2sin2( )). 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
0r≤∞. 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 r0), will
be derived in Section II. According to this model, the
mass density distribution is
(r) = μρ0 δc(r r0). (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.
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 r0) with similar properties. One such function can
be defined as follows:
1) Let δc(r r0) 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)22]. (2)
Here G(r) is defined over the domain r0, 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
0dr G(r) = 0dr ( )ε π 1 exp[─(r─r0)22] 1,
This relation may be verified by evaluating the integral
of a normalized rectangular Gaussian G(x), which for
<<xε0 has the property
0dx ( )ε π 1 exp [─(x─x0)22 ]
≈ ∫
dx ( )ε π 1 exp [─(x─x0)22] = 1.
The delta function may thus be written
(r rδ 0 ) = lim[ >0] ε─ ()ε π 1 exp [─(r─r0)22]. (3)
2) The continuous or broadened delta function δc(r r0)
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 r0) 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
a) 0dr δc(r r0) 1
b) 0dr f(r) δc(r r0) f(r0)
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 r0) f(r0) 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.
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 r2dΩ2
= eν dt2 e λdr2 r2dΩ2
The appropriate gravitational field equations may be
found by substituting this metric into Einstein's field
equations, given by
ν (1/2) gμ 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
and curvatures R and Rμ
ν, EFE of Eq. (5) simplify to a
pair of simultaneous equations [55]
Tκ00 = (r) = eκρ ─λ/r2 1/r2 e─λλ/r (6a)
Tκ11 = p(r) = e─ κ ─λ/r2 1/r2 + e─λν/r, (6b)
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 r0) and μ0=m0/4 rπ02, and applying Eq. (4),
this becomes
eλ = (1 + k0 /r 2m0Sc/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 = eλ = (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)dt2 (1 2m/r)─ ─ 1dr2 r2dΩ2 (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 2m0Sc/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 2m0Sc/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 2m0Sc/r). (12)
Since δc(r r0) 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 r0.
Thus we have,
I(r) r0 0r dr δc / (1 2m0Sc/r0) r02m0. (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) r0 0Sc(r) dSc / (1 2m0Sc / r0 )
(r≈ ─ 02/2m0) ln |(1 2m0Sc / 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 2m0Sc/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 2m0/r0| (1+w)
and the final result for the tt component of the ultra-thin
shell metric is
g00 (1 2m≈ ─ 0Sc/r) |1 2m0/r0|(1+w) |1 2m0Sc/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)
The exterior component g00, like the exterior component
g11, matches the Schwarzschild solution as expected.
Note that the quantity 2m0 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
ν0, 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 0r<. 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
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.
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 ro2m0. 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
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 w0, 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 >r0 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.
(12) using the spherical Gaussian. Such a calculation is
not attempted here. If, however, we naively allow the
approximation r >r0 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 2m0/r0|(1+w)|1 2m0Sc/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
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 2m0. 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 2m0Sc/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 2Sc(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 2r0Sc/r] / r |1 2Sc|
= (2r0Sc/r2 2r0δc/r) / ( ⁄ + 2─ δc)
= ⁄ + (r0/r) (Sc/rδc 1)
where the sign ambiguity springs from the absolute
value. Taking the limit r >r0, 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.
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
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 8cm/sec2 H0 /2 = cπ2/R = c2/( /3)Λ1/2
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/2r2) [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/Rmr. (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/c2Rm) 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
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 2m0Sc(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,
g00SBS [1 r s(1 )/r] / | (r) |─ η σ η ─ σ
(1 r s/r) / | (r) |.σ η ─ σ (25)
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 + (rs/Rm) ln |r/(R0r)|. (26)
This approximation can be checked by calculating
acceleration a from potential φm
a ─ φm GM/R≅ ─ mr GM/Rm(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,
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.
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 rj) denotes a broadened delta function at
radius rj. Substituting (r)ρ into Eq. (7) and setting the
integration constant to zero yields
g11 = eλ = [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 = eλ = [1 2m─ ─ 0S0/r 2m1S1/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
= + kν ─ λ 1 ─ κ∫r dr (r) eρλ r
= + k─ λ 1 ─ κ∫rdr(μ0δ0+μ1δ1)r / [1 2(m0S0+m1S1)/r].
The integral may be expressed as a sum of two terms:
I(r) = μ0rdr δ0r / [1 2(m0S0+m1S1)/r]
+ μ1rdr δ1r / [1 2(m0S0+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) μ0r0rdr δ0 / [1 2(m0S0+m1S1)/r0]
+ μ1r1rdr δ1 / [1 2(m0S0+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
slowly over the nonzero domain of δ1 and may be set to a
constant S01. The total integral then simplifies to
I(r) μ0r0rdr δ0 / (1 2m0S0/r0)
+ μ1r1rdr δ1 / (1 2m0/r1 2m1S1/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) (μ0r02/2m0) ln |1 2m0S0/r0|
+ (μ1r12/2m1) ln |1 2m0/r0 2m1S1/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 2m1S1 /r1|.
and hence
g00 = e ν= [1 2(m0S0+m1S1)/r] ek1 |1 2m0S0/r0|1
X |1 2m0/r0 2m1S1/r1|1 (30)
where X denotes multiplication. Again, to meet the
asymptotic Minkowski condition, the integration
constant must be
ek1 = |1 2m0/r0| |1 2m0/r0 2m1/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 ri). 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.
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,
3. Comparison of ultra-thin shell boundary
properties to Israel junction conditions under a
general EoS,
4. Collapsing ultra-thin shells and black shell
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.
Using the line element
ds2 = g00(r) dt2 + g11(r) dr2 r2dΩ2
= eν dt2 e λdr2 r2dΩ2
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/r2 e─λλ/r (a1)
Tκ11 = p(r) = e─ κ ─λ/r2 1/r2 + 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 ─ λr)
= (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 r0) and application of Eq.
(4) gives
eλ = (1 + μκ0r02Sc/r + k0/r) 1.
Using surface density μ0=m0/4 rπ02, this becomes
eλ = (1 2m0Sc/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 = eλ = (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 2m0Sc/r).
Upon integrating, this becomes
= + kν ─ λ 1 (1+w)μ─ κ 0 r dr δcr / (1 2m0Sc/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
I(r) = 0r dr δc r/(1 2m0Sc/r).
Since r changes by the near infinitesimal amount 2nε
across the transient layer, it may be treated as a constant
r r0, hence
I(r) r0 0r dr δc/(1 2m0Sc/r0) r02m0.
I(r) can be integrated by a change of variable dSc=δcdr,
with limits of integration 0 and Sc(r):
I(r) r0 0Sc(r) dSc/(1 2m0Sc/r0)
(r≈ ─ 02/2m0) ln| (1 2m0Sc/r0)|0Sc(r)
(r≈ ─ 02/2m0) ln| (1 2m0Sc/r0)|,
Substituting I(r) into Eq. (a5) yields,
+ kν ≈ ─ λ 1 + [ (1+w)μκ0r02/2m0] ln |1 2m0Sc/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 2m0Sc/r0| (1+w)
(1 2m≈ ─ 0Sc/r) ek1 |1 2m0Sc/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 2m0/r0| (1+w)
and the tt component of the ultra-thin shell metric
g00 (1 2m≈ ─ 0Sc/r)|1 2m0/r0|(1+w)|1 2m0Sc/r0|(1+w)
r0 2m0.
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A review of refutations of general relativity commonly found in today's literature is presented, with comments on the status of Einstein's theory and brief analyses of the arguments for modified gravity. Topics include dark matter and the galactic rotation curve, dark energy and cosmic acceleration, completeness and the equation of state, the speed of gravity, the singularity problem, redshift, gravitational time dilation, localized energy, and the gravitational potential. It is conjectured that the contemporary formalism of general relativity offers an incomplete description of gravitational effects, which may be the most compelling reason for seeking new theories of gravity.
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We consider the Einstein-scalar-Gauss-Bonnet theory in the presence of a cosmological constant Λ, either positive or negative, and look for novel, regular black-hole solutions with a nontrivial scalar hair. We first perform an analytic study in the near-horizon asymptotic regime and demonstrate that a regular black-hole horizon with a nontrivial hair may always be formed, for either sign of Λ and for arbitrary choices of the coupling function between the scalar field and the Gauss-Bonnet term. At the faraway regime, the sign of Λ determines the form of the asymptotic gravitational background leading to either a Schwarzschild–anti-de Sitter–type background (Λ<0) or a regular cosmological horizon (Λ>0), with a nontrivial scalar field in both cases. We demonstrate that families of novel black-hole solutions with scalar hair emerge for Λ<0, for every choice of the coupling function between the scalar field and the Gauss-Bonnet term, whereas for Λ>0, no such solutions may be found. In the former case, we perform a comprehensive study of the physical properties of the solutions found such as the temperature, entropy, horizon area, and asymptotic behavior of the scalar field.
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We present spherically symmetric solutions to Einstein’s equations, which are equivalent to canonical Schwarzschild and Reissner-Nordstrom black holes on the exterior, but with singular (Planck-density) shells at their respective event and inner horizons. The locally measured mass of the shell and the singularity are much larger than the asymptotic Arnowitt-Deser-Misner mass. The area of the shell is equal to that of the corresponding canonical black hole, but the physical distance from the shell to the singularity is a Planck length, suggesting a natural explanation for the scaling of the black hole entropy with area. The existence of such singular shells enables solutions to the black hole information problem of Schwarzschild black holes and the Cauchy horizon problem of Reissner-Nordstrom black holes. While we cannot rigorously address the formation of these solutions, we suggest plausibility arguments for how “normal” black hole solutions may evolve into such states. We also comment on the possibility of negative-mass Schwarzschild solutions that could be constructed using our methods. Requirements for the nonexistence of negative-mass solutions may put restrictions on the types of singularities allowed in an ultraviolet theory of gravity.
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In a previous work we suggested a self-gravitating electromagnetic monopole solution in a string-inspired model involving global spontaneous breaking of a $SO(3)$ internal symmetry and Kalb-Ramond (KR) axions, stemming from an antisymmetric tensor field in the massless string multiplet. These axions carry a charge, which, in our model, also plays the r\^ole of the magnetic charge. The resulting geometry is close to that of a Reissner-Nordstr\"om (RN) black hole with charge proportional to the KR-axion charge. We proposed the existence of a thin shell structure inside a (large) core radius as the dominant mass contribution to the energy functional. The resulting energy was finite, and proportional to the KR-axion charge; however, the size of the shell was not determined and left as a phenomenological parameter. In the current article, we can calculate the mass-shell size, on proposing a regularisation of the black hole singularity via a matching procedure between the RN metric in the outer region and, in the inner region, a de Sitter space with a (positive) cosmological constant proportional to the scale of the spontaneous symmetry breaking of $SO(3)$ . The matching, which involves the Israel junction conditions for the metric and its first derivatives at the inner surface of the shell, determines the inner mass-shell radius. The axion charge plays an important r\^ole in guaranteeing positivity of the "mass coefficient" of the gravitational potential term appearing in the metric component; so the KR electromagnetic monopole shows normal attractive gravitational effects. This is to be contrasted with the global monopole case (in the absence of KR axions) where such a matching is known to yield a negative "mass coefficient" (and, hence, repulsive gravitational effects). The total energy of the monopole within the shell is calculated.
We constructed a simple cosmological model which approximates the Einstein-de Sitter background with periodically distributed dust inhomogeneities. By taking the metric as a power series up to the third order in some perturbative parameter λ, we are able to achieve large values of the density contrast. With a metric explicitly given, many model properties can be calculated in a straightforward way which is interesting in the context of the current discussion concerning the averaging of the inhomogeneities and their backreaction in cosmology. Although the Einstein-de Sitter model can be thought of as the model average, the light propagation differs from that of Einstein-de Sitter. The angular diameter distance-redshift relation is affected by the presence of inhomogeneities and depends on the observer’s position. The model construction scheme enables some generalizations in the future, so the present work is a step toward more realistic cosmological model described by a relatively simple analytical metric.
The scattering of light by water droplets can produce one of the most beautiful phenomena in nature: the rainbow. This optical phenomenon has analogues in molecular, atomic, and nuclear physics. Recently, rainbow scattering has been shown to arise from the gravitational interaction of a scalar field with a compact horizonless object. We show that rainbow scattering can also occur in the background of a black hole with surrounding matter. We study the scattering of null geodesics and planar massless scalar waves by Schwarzschild black holes surrounded by a thin spherical shell of matter. We explore various configurations of this system, analyzing changes in mass fraction and radius of the shell. We show that the deflection function can present stationary points, which leads to rainbow scattering. We analyze the large-angle scattered amplitude as a function of the shell’s parameters, showing that it presents a nonmonotonic behavior.
Intermediate-richness galaxy groups are an important testing ground for modified Newtonian dynamics (MOND). First, they constitute a distinct type of galactic systems, with their own evolution histories and underlying physical processes; second, they probe little-chartered regions of parameter space, as they have baryonic masses similar to massive galaxies, and similar velocity dispersions, but much larger sizes—similar to cluster cores (or even to clusters), but much lower dispersions. Importantly in the context of MOND, they have the lowest internal accelerations reachable inside galactic systems. Following my recent analysis of MOND in galaxy groups, I came across a much superior sample, which I analyze here. This extensive catalog permits strict quality cuts that still leave a large sample of 56 medium-richness groups, better suited for dynamical analysis—e.g., in having a large number (≥15) of members with measured velocities. I find that these groups obey the deep-MOND relation between baryonic mass, MM, and velocity dispersion, σ: MMGa0=(81/4)σ4, with individual, MOND, mass-to-light ratios, MM/LK of order 1 M⊙/LK,⊙, and a sample median value of (MM/LK)med=0.7 M⊙/LK,⊙. These compare well with stellar values deduced for single galaxies, and with values deduced from population-synthesis analyses. In contrast, the dynamical, Newtonian Md/LK values are much larger—several tens solar units, and (Md/LK)med=37 M⊙/LK,⊙. The same MOND relation describes (isolated) dwarf spheroidals—two-three orders smaller in size, and seven-eight orders lower in mass. The groups conformation to the MOND relation is equivalent to their lying on the deep-MOND branch of the “mass-discrepancy-acceleration relation,” g≈(gNa0)1/2, for g as low as a few percents of a0 (gN is the Newtonian, baryonic, gravitational acceleration, and g the actual one). This argues against systematic departure from MOND at extremely low accelerations (as has been suggested). This conformation also argues against the hypothesis that the remaining MOND conundrum in cluster cores bespeaks a breakdown of MOND on large-distance scales; our groups are as large as cluster cores, but do not show obvious disagreement with MOND. I also discuss the possible presence of the idiosyncratic, MOND external-field effect.
We consider a collision of two dust thin shells with a high center-of-mass (CM) energy including their self-gravity in a Bañados-Teitelboim-Zanelli (BTZ) spacetime. The shells divide the BTZ spacetime into three domains and the domains are matched by the Darmois-Israel junction conditions. We treat only the collision of two shells which corotate with a background BTZ spacetime because of the junction conditions. The counterpart of the corotating shell collision is a collision of two particles with vanishing angular momenta. We compare the dust thin shell collision and the particle collision in order to investigate the effects of the self-gravity of colliding objects on the high CM energy collision. We show that the self-gravity of the shells affects the position of an event horizon and it covers the high-energy collisional event. Therefore, we conclude that the self-gravity of colliding objects suppresses its CM energy and that any observer who stands outside of the event horizon cannot observe the collision with an arbitrary high CM energy.
Galaxy groups, which have hardly been looked at in modified Newtonian dynamics (MOND), afford probing the acceleration discrepancies in regions of system-parameter space that are not accessible in well-studied galactic systems: galaxies, galaxy clusters, and dwarf spheroidal satellites of galaxies. Groups are typically the size of galaxy-cluster cores, but have masses typically only a few times that of a single galaxy. Accelerations in groups get far below those in galaxies, and far below the MOND acceleration, so much so that many groups might be affected by the external-field effect, which is unique to MOND, due to background accelerations. Here, I analyze the MOND dynamics of 53 galaxy groups, recently catalogued in three lists. Their Newtonian, K-band, dynamical M/L ratios are a few tens to several hundreds solar units, with ⟨Md/LK⟩=(56,25,30) M⊙/L⊙, respectively, for the three lists, thus evincing very large acceleration discrepancies. I find here that MOND requires dynamical MM/LK values of order 1 M⊙/L⊙, with ⟨MM/LK⟩=(0.8,0.56,1.0) M⊙/L⊙, for the three lists, which are in good agreement with population-synthesis stellar values, and with those found in individual galaxies. MOND thus accounts for the observed dynamics in those groups with baryons alone, with no need for dark matter—an important extension of MOND analysis from galaxies to galactic systems, which, to boot, have characteristic sizes of several hundred Kpc, and accelerations much lower than probed before—only a few percent of MOND’s a0. The acceleration discrepancies evinced by these groups thus conform to the deep-MOND prediction: g≈(gNa0)1/2, down to these very low accelerations (g is the measured, and gN the baryonic, Newtonian acceleration).