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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.

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