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

By Kathleen A. Rosser

Kathleen.A.Rosser@ieee.org
Published 27 September 2019
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Abstract

Interest in general relativistic treatments of thin matter shells has flourished over recent decades, most notably in connection with astrophysical and cosmological applications such as black hole matter accretion, spherical wormholes, bubble universes, and cosmic domain walls. In the present paper, an asymptotically exact solution to Einstein’s field equations for static ultra-thin spherical shells is derived using a continuous matter density distribution ρ(r) defined 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 proportional to mass m0 and inversely proportional to shell radius r0. The exterior solution matches the Schwarzschild metric. General black shell solutions, horizons, and singularities are also discussed. Various questions are finally listed as topics for future research.

I. INTRODUCTION

M81 Spiral Galaxy, 12 million light years away (–X-ray: NASA/CXC/SAO; Optical: Detlef Hartmann; Infrared: NASA/JPL-Caltech)

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 r─1/2 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 the discrete sampling of arbitrary spherical mass distributions, with possible 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].

Bubble Universe (–artist’s conception youtu.be/DkRYU1OLv7c)

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 that specify continuity or jumps in the metric 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 Kμν 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-world scenarios, accommodate 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 in applications involving 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 Sμν for the transient domain; the extrinsic curvature Kμν; 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 elusive.

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=wρ. 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 nonvanishing. One advantage to the continuous solution method, in which density ρ(r), 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; and
  2. gμν must match Minkowski space as r─>∞,

where Minkowski space is here defined by the metric

gμν = diag( 1, ─1, ─r2, ─r2sin2(Θ)).

The first boundary condition follows logically from the absence of matter in the interior [51]. This condition is relaxed in the case of a central mass. The second boundary condition dictates that space be asymptotically flat, with the assumption that g00─>1 as r─>∞, or equivalently, 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 distribution ρ(r) spanning the range 0≤r≤∞, with r the radial coordinate. For an ultra-thin shell, ρ(r) will be modeled here as continuous approximation to the spherical Dirac delta function δ(r-r0), 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

ρ(r) = μ0 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 ρ(r) 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 ε=0. An equation of state (EoS) of the form p(r)=wρ(r) 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 the delta function technique uniformly over all space requires that the discontinuous Dirac delta function δ(r-r0) 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 δ(r-r0), where the latter is expressed in terms of the normalized spherical Gaussian

G(r) := (ε√π)─1 e─(r─r0)22.                               (2)

Here G(r) is defined over the domain r≥0, with a peak centered at r=r0 of height 1/(π1/2ε) and width proportional to ε. For ε<<r0, G(r) obeys the relation

0dr G(r) = ∫0dr (ε√π)─1 e─(r─r0)22 ≈ 1,        ε<<r0.

This relation may be verified by evaluating the integral of a normalized rectangular Gaussian G(x), which for ε<<x0 has the property

0dx (ε√π)─1 e─(x─x0)22 ≈ ∫─∞dx (ε√π)─1 e─(x─x0)22 = 1.

The delta function may thus be written

δ(r-r0) = lim[ε─>0](ε√π)─1 e─(r─r0)22.                (3)

2) The continuous or broadened delta function δc(r-r0) is obtained as an approximation to δ(r-r0) 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<r0─nε and r>r0+nε. Therefore mass density ρ(r) 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) 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 r0─nε<r<r0+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 < r0─nε 
Sc(r;r0) ≈ 1/2                          r = r0         
Sc(r;r0) ≈ 1                              r > r0+nε.

(For convenience, the symbol r represents both the dummy variable and the integral limit.) That Sc≈1/2 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 ρ(r) in 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 ─ r22
= eν dt2 ─ eλ dr2 ─ r22

The appropriate gravitational field equations may be found by substituting this metric into Einstein’s field equations, given by

Rμν ─ (1/2) gμν R = κ Tμν                                  (5)

where Rμν is the curvature or Ricci tensor, R is the scalar curvature, κ is a constant with the value κ=─8πG/c2 (using Dirac’s sign convention [54]), or κ=─8π for G=c=1, and Tμν=diag(ρ,─p,─p,─p) is the stress energy tensor, with ρ(r) the mass-energy density and p(r) the pressure. After calculating the Christoffel symbols Γμαβ and curvatures R and Rμν, EFE of Eq. (5) simplify to a pair of simultaneous equations [55]

κT00 = κρ(r) = e─λ/r2 ─ 1/r2 ─ e─λλ/r                (6a)

κT11 = ─ κp(r) = e─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):

κρr2 + 1 = (re─λ).

Integrating and solving for eλ, we obtain

eλ = [1 + k0/r + (κ/r)∫r dr ρ(r) r2] ─ 1.             (7)

where k0 is a constant of integration. Substituting ρ(r)=μ0δc(r─r0) and μ0=m0/4πr02, and applying Eq. (4), this becomes

eλ = (1 + k0/r ─ 2m0Sc/r) ─ 1.                        (8)

For an empty shell, the boundary condition that eλ be nonsingular 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 ─ 2m0Sc/r) ─ 1.                     (9)

Outside the shell, where Sc≈1, 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 ─ r22        (10)

for m the central mass. In the interior of the shell, where Sc≈0, 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

κ(ρ+p) = ─ e─λ λ/r ─ e─λ ν/r.

Solving for ν,  substituting eλ from Eq. (9) and ρ(r) from Eq. (1), and using equation of state p=wρ, the result is

ν = ─ λ ─ κ(1+w)μ0δcr / (1 ─ 2m0Sc/r),

where δc and Sc are abbreviated notations for the broadened delta and step functions. Upon integrating, this becomes

ν = ─ λ + k1 ─ κ(1+w)μ0r 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 r0─nε<r<r0+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<r0─nε 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+nε, as 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) ≈ r00r dr δc/(1 ─ 2m0Sc/r0)        r0≠2m0.       (13)

(Here as elsewhere, the symbol denotes asymptotic equality, for which precision increases as ε decreases.) I(r) can now be integrated to asymptotic precision by a unique change of variable. Recalling from Eq. (4) that Sc=∫rδcdr and therefore dSccdr, 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) ≈ r00Sc(r) dSc/(1 ─ 2m0Sc/r0 )

              ≈ ─ (r02/2m0) 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

ν ≈ ─ λ + k1 ─ (1+w) ln |1 ─ 2m0Sc/r0)|.

Upon substitution of eλ from Eq. (8), the result is

eν ≈ (1 ─ 2m0Sc/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─2m0Sc/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 ─ 2m0/r0|(1+w)                              (15a)

g00ext ≈ (1 ─ 2m0/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 Tμν=0, but also the massive equations, where Tμν≠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)=wρ(r) 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 2=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=2m0subhorizon 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 ─ 2m0/r0)(1+w)            r0 ≠ 2m0    (16a)
g00ext ≈ 1 ─ 2m0/r                           r0 ≠ 2m0    (16b)
g11int ≈ ─1                                                             (16c)
g11ext ≈ ─ (1 ─ 2m0/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─>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─2m0Sc/r)|1─2m0/r0|(1+w)|1─2m0Sc/r0|─(1+w)    (17)

g11 ≈ ─ 1/(1 ─ 2m0Sc/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 nonvanishing 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≈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 of r0, such singularities usually fall within r0─nε<r<r0+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)=r0/2m0≈1. 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─2r0Sc(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) / (─ ⁄ + 2δc)

= ─ ⁄ + (r0/r) (Sc/rδc ─ 1)

where the sign ambiguity springs from the absolute value. Taking the limit r─>r0, the term Sc/rδc approaches π1/2ε/2r0<<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 r─1/2, 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 = 1.2×10─8cm/sec2 ≅ H0/2π = c2/R = c2/(Λ/3)−1/2

is a universal constant with dimensions of acceleration, H0 is the Hubble parameter [62], and R is roughly 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/2r2 ─ GM/Rmr.                           (21)

The potential in this domain can be expressed as

φm = ─ ∫a(r) dr ≅ ─ GM/2r + (GM/Rm) 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 + 2φm/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 ─ ς = (1 ─ σ) rs,                                  (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=ς/rs 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 ─ 2m0Sc(R0)/r0 = 0,

or, upon substiting 2m0=r0 and rearranging,

            Sc(R0) = 1/(1+ς/rs) ≅ 1 ─ ς/rs.                          (24)

To calculate the impact of the distant singularity on the field in the galactic neighborhood, we start by introducing a new function η(r) and expressing the broadened step function as Sc(r)=1─η(r), where η(r)<<1 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 ─ rs(1 ─ η)/r] σ/ |η(r) ─ σ|

≅  (1 ─ rs/r) σ / |η(r) ─ σ|.                           (25)

From Eq. (24), we see that η(R0)≅σ=ς/rs, 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 at infinity to a remote finite distance r=R0 as follows:

g00MOND ≅ 1 + 2φm/c2≅ 1 + (rs/Rm) ln |r/(R0─r)|.      (26)

This approximation can be checked by calculating acceleration a from potential φm

a ≅ ─ φm ≅ ─ GM/Rmr ─ GM/Rm(R0─r).

It is clear that for r in the neighborhood of the galaxy, (R0─r) 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/(R0─r)| = (1 ─ rs/r) σ / [η(r) ─ σ].

By solving for η(r), a new form Sm(r) of the broadened step function is obtained that is consistent with the MOND metric as follows:

Sm(r) = 1 ─ η(r) = 1 ─ σ / [1 + (rs/Rm) ln |r/(R0─r)|]  ─ 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 ρ(r), where ρ(r) 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=4πμ0r02 and m1=4πμ1r12, the mass density can be expressed in terms of broadened delta functions as

ρ(r) = μ0δ0 + μ1δ1

where δjc(r-rj) denotes a broadened delta function at radius rj. Substituting ρ(r) into Eq. (7) and setting the integration constant to zero yields

g11 = ─ eλ = ─ [1 + (κ/r)∫r dr ρr2] ─1

                = ─ [1  +  (κμ0/r)∫r dr r2δ0  +  (κμ1/r)∫r dr r2δ1] ─1.

Upon integration, the double thin-shell solution becomes

g11 = ─ eλ = ─ [1─ 2m0S0/r ─ 2m1S1/r] ─1            (27)

where S0=Sc(r;r0) and S1=Sc(r;r1). The interior (r<r0─nε), middle (r0+nε<r<r1─nε), and exterior (r>r1+nε) solutions are therefore

g11int  ≈ ─1                                                            (28a)
g11mid ≈ ─ (1 ─ 2m0/r) ─1                                  (28b)
g11ext  ≈ ─ [1 ─ 2(m0 + 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

    ν = ─ λ + k1 ─ κ∫r dr ρ(r) eλ r

= ─ λ + k1 ─ κ∫rdr(μ0δ01δ1)r / [1─ 2(m0S0+m1S1)/r].   (29)

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 S1≈0. Analogously, in the second integral, S0(r) varies slowly over the nonzero domain of δ1 and may be set to a constant S0≈1. 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

ν = ─λ + k1 ─ ln|1─2m0S0/r0| ─ ln|1─2m0/r0─2m1S1/r1|.

and hence

g00 = eν = [1 ─ 2(m0S0+m1S1)/r]  ek|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 ρ(r). 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 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,
  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 ─ r22
= eν dt2 ─ eλ dr2 ─ r22

with κ=─8π and stress-energy tensor Tμν=diag(ρ,─p,─p,─p),  Einstein’s field equations simplify to

κT00 = κρ(r)  =  e─λ/r2 ─ 1/r2 ─ e─λλ/r                  (a1)

κT1 = ─ κp(r) = e─λ/r2 ─ 1/r2 + e─λν/r,              (a2)

where primes denote derivatives with respect to r. Eq. (a1) can be solved by rearranging terms

κρr2 + 1 = e─λ (1 ─ λr)
= (re─λ).

Integration then yields

re─λ = k0 +  ∫r dr (κρr2 + 1) .

Here, r denotes the inverse derivative and k0 is a constant of integration. Solving for eλ, we obtain

eλ = [1 + k0/r + (κ/r)∫r dr ρ(r) r2] ─ 1.

Substitution of ρ(r)=μ0δc(r─r0)  and application of Eq. (4) gives

eλ = (1 + κμ0r02Sc/r + k0/r) ─ 1.

Using surface density μ0=m0/4πr02, 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 ─ 2m0Sc/r) ─ 1.                      (a4)

The tt component g00=eν can be evaluated by subtracting Eq. (a2) from Eq. (a1) to obtain

κ(ρ+p) = ─ e─λ λ/r ─ e─λ ν/r.

Solving for ν yields

ν = ─ λ ─ κ(ρ+p) eλ r.

If we now substitute ρ(r) and eλ from Eq. (a4), and apply the equation of state p=wρ for w a constant, the result is

ν = ─ λ ─ κ(1+w)μ0δcr / (1 ─ 2m0Sc/r).

Upon integrating, this becomes

ν = ─ λ + k1 ─ κ(1+w)μ0r 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 written

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) ≈ r00r dr δc/(1 ─ 2m0Sc/r0)              r0≠2m0.

I(r) can be integrated by a change of variable dSccdr, with limits of integration 0 and Sc(r):

I(r) ≈ r00Sc(r) dSc/(1 ─ 2m0Sc/r0)
                       ≈ ─ (r02/2m0)ln| (1 ─ 2m0Sc/r0)|0Sc(r)
                 ≈ ─ (r02/2m0) ln| (1 ─ 2m0Sc/r0)|,

Substituting I(r) into Eq. (a5) yields,

ν ≈ ─ λ + k1 + [κ(1+w)μ0r02/2m0] ln |1 ─ 2m0Sc/r0|.

Evaluating the constants κ and μ0, this simplifies to

ν ≈ ─ λ + k1 ─ (1+w) ln |1 ─ 2m0Sc/r0)|,

with the result

eν ≈ e─λ ek1 |1 ─ 2m0Sc/r0|─ (1+w)

≈ (1 ─ 2m0Sc/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,. Hence the integration constant is

ek1 = |1 ─ 2m0/r0| (1+w)

and the tt component of the ultra-thin shell metric becomes

g00≈(1─2m0Sc/r)|1─2m0/r0|(1+w)|1─2m0Sc/r0|─(1+w)

r0 ≠ 2m0.

_____________

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