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

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].

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 g_μν and the extrinsic curvature K^μ_ν be continuous across the shell wall. While these conditions are common in the literature, doubt is raised about their application to certain physical scenarios [41] or in modified theories of gravity [42]. Some authors derive new junction conditions that specify jumps in curvature [43], jumps in the tangential metric components to account for domain wall spin currents [44], or other field behavior [45]. Others avoid junction conditions by use of a confining potential [46].

It may be significant that Israel’s original derivation was based on properties of electromagnetic fields rather than on general relativity (GR), although recent derivations, in contexts such as cosmological brane-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 n_i normal to the domain wall; the surface stress-energy tensor S^μ_ν for the transient domain; the extrinsic curvature K^μ_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 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 g_μν(r) are uniformly defined over all space, is that only two boundary conditions are needed to fix the metric:

1. g_μν must be nonsingular at r=0; and
2. g_μν must match Minkowski space as r─>∞,

where Minkowski space is here defined by the metric

g_μν = 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 g_μν(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 δ_c(r-r₀), will be derived in Section II. According to this model, the mass density distribution is

ρ(r) = μ₀ δ_c(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 T_μν 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 δ_c(r-r₀) with similar properties. One such function can be defined as follows:

1) Let δ_c(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)2/ε2}.                               (2)

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

∫₀^∞dr G(r) = ∫₀^∞dr (ε√π)^─1 e^{─(r─r₀)2/ε2} ≈ 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

∫₀^∞dx (ε√π)^─1 e^{─(x─x₀)2/ε2} ≈ ∫_─∞^∞dx (ε√π)^─1 e^{─(x─x₀)2/ε2} = 1.

The delta function may thus be written

δ(r-r₀) = lim_[ε─>0](ε√π)^─1 e^{─(r─r₀)2/ε2}.                (3)

2) The continuous or broadened delta function δ_c(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 δ_c(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 δ_c obeys, to any desired accuracy, the defining properties of the Dirac delta function:

a) ∫₀^∞dr δ_c(r-r₀) ≈ 1

b) ∫₀^∞dr f(r) δ_c(r-r₀) ≈ f(r₀)

provided that f(r) is slowly varying over the transient layer r₀─nε<r<r₀+nε.

5) The integral ∫₀^rdrδ_c, or the inverse derivative of the broadened delta function δ_c, is a continuous or broadened step function S_c(r;r₀) such that

∫₀^rdr f(r) δ_c(r-r₀) ≈ f(r₀) S_c(r;r₀),                         (4)

where f(r) varies slowly over the transient layer, and S_c(r;r₀) has the properties

S_c(r;r₀) ≈ 0                             r < r₀─nε 
S_c(r;r₀) ≈ 1/2                          r = r₀          ‘
S_c(r;r₀) ≈ 1                              r > r₀+nε.

(For convenience, the symbol r represents both the dummy variable and the integral limit.) That S_c≈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 S_c, 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 g_αβ of the form

ds² = g₀₀(r) dt² + g₁₁(r) dr² ─ r²dΩ²
= e^ν dt² ─ e^λ dr² ─ r²dΩ²

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

R^μ_ν ─ (1/2) g^μ_ν R = κ T^μ_ν                                  (5)

where R^μ_ν 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 T^μ_ν=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 Γ^μ_αβ and curvatures R and R^μ_ν, 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 e^λ, we obtain

e^λ = [1 + k₀/r + (κ/r)∫_r dr ρ(r) r²] ^{─ 1}.             (7)

where k₀ is a constant of integration. Substituting ρ(r)=μ₀δ_c(r─r₀) and μ₀=m₀/4πr₀², and applying Eq. (4), this becomes

e^λ = (1 + k₀/r ─ 2m₀S_c/r) ^{─ 1}.                        (8)

For an empty shell, the boundary condition that e^λ 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 = ─ e^λ = ─ (1 ─ 2m₀S_c/r) ^{─ 1}.                     (9)

Outside the shell, where S_c≈1, we see that g₁₁ matches the radial component of the Schwarzschild metric g^S_μν, as given by

ds² = (1 ─ 2m/r)dt² ─ (1 ─ 2m/r)^─1dr² ─ r²dΩ²        (10)

for m the central mass. In the interior of the shell, where S_c≈0, it is clear that g₁₁ matches the Minkowski metric.

Next, the tt component g₀₀=e^ν 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 e^λ from Eq. (9) and ρ(r) from Eq. (1), and using equation of state p=wρ, the result is

ν′ = ─ λ′ ─ κ(1+w)μ₀δ_cr / (1 ─ 2m₀S_c/r),

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

ν = ─ λ + k₁ ─ κ(1+w)μ₀ ∫_r dr [δ_cr / (1 ─ 2m₀S_c/r)]         (11)

with k₁ a constant of integration. Eq. (11) represents an exact solution to EFE for the tt metric component g₀₀=e^ν 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 S_c(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) = ∫₀^r dr δ_c r/(1 ─ 2m₀S_c/r).                       (12)

Since δ_c(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) ≈ r₀ ∫₀^r dr δ_c/(1 ─ 2m₀S_c/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 S_c=∫_rδ_cdr and therefore dS_c=δ_cdr, the continuous monotonic function S_c can be used as the variable of integration. The limits of integration become 0 and S_c(r), and the integral may be written

I(r) ≈ r₀ ∫₀^S_c(r) dS_c/(1 ─ 2m₀S_c/r₀ )

              ≈ ─ (r₀²/2m₀) ln| (1 ─ 2m₀S_c/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 ─ 2m₀S_c/r₀)|.

Upon substitution of e^λ from Eq. (8), the result is

e^ν ≈ (1 ─ 2m₀S_c/r) e^k₁ |1 ─ 2m₀S_c/r₀|^{─ (1+w)}.

Since e^ν must obey the Minkowski condition e^ν─>1 as r─>∞, the integration constant e^k₁ must cancel the right-hand factor in the outer region where S_c─>1, leaving only the left-hand factor, which is asymptotically Minkowski. Hence the integration constant is

e^k₁ = |1 ─ 2m₀/r₀| ^(1+w)

and the final result for the tt component of the ultra-thin shell metric is

g₀₀≈(1─2m₀S_c/r)|1─2m₀/r₀|^(1+w)|1─2m₀S_c/r₀|^─(1+w)

r₀ ≠ 2m₀.                                     (14)

To analyze this result, we evaluate g₀₀ for the interior and exterior, obtaining

g_00int ≈ |1 ─ 2m₀/r₀|^(1+w)                              (15a)

g_00ext ≈ (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 T^μ_ν=0, but also the massive equations, where T^μ_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 g_00int is a constant not equal to unity, while the exterior metric g_00ext 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 g_00int<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 g_00int depends on the EoS of the shell, while the exterior metric g_00ext 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 r_o≤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 g^S_μν as given by Eq. (10):

1. A coordinate singularity, or horizon, exists at r=2m, where g^S₀₀=0 and g^S₁₁─>─∞.
2. Inside the horizon, squared proper time intervals dτ²=g^S₀₀dt² are negative, and thus proper time is spacelike, while squared proper radial intervals dR²=g^S₀₀dr² are positive, and proper radial distance is timelike.
3. A physical singularity is generally assumed to exist at r=0, where g^S₀₀─>-∞ and g^S₁₁=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:

g_00int ≈ (1 ─ 2m₀/r₀)^(1+w)            r₀ ≠ 2m₀    (16a)
g_00ext ≈ 1 ─ 2m₀/r                           r₀ ≠ 2m₀    (16b)
g_11int ≈ ─1                                                             (16c)
g_11ext ≈ ─ (1 ─ 2m₀/r) ^─1.                                  (16d)

In the case of ordinary shells (r₀>2m₀), the Schwarzschild radius r_s=2m₀ lies inside the shell where the metric is constant. Thus there is no horizon at r=r_s. 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 r_s=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─2m₀S_c/r)|1─2m₀/r₀|^(1+w)|1─2m₀S_c/r₀|^─(1+w)    (17)

g₁₁ ≈ ─ 1/(1 ─ 2m₀S_c/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 S_c(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 δ_c 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 2m₀S_c(r)/r=1, or r/r₀=S_c(r). Since it is always true that S_c(r)<1, any such singularity can only reside at r<r₀. It will be stated without proof that since S_c(r)≈1/2 when r/r₀=1, and since S_c(r) falls to zero more rapidly than r/r₀, there can be no r>0 such that 2m₀S_c(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 S_c(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─2m₀S_c/r₀| in the right-hand factor (denominator) of Eq. (17). To locate this singularity, recall that S_c(r) increases monotonically over the range 0<S_c<1. Thus g₀₀ becomes singular at some unique r such that S_c(r)=r₀/2m₀. Since S_c(r) traverses nearly all of its range within a distance nε 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 S_c(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 S_c(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─2r₀S_c(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─2r₀S_c/r] / ∂_r |1─2S_c|

              = (2r₀S_c/r² ─ 2r₀δ_c/r) / (─ ⁄ + 2δ_c)

= ─ ⁄ + (r₀/r) (S_c/rδ_c ─ 1)

where the sign ambiguity springs from the absolute value. Taking the limit r─>r₀, the term S_c/rδ_c approaches π^1/2ε/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 φ_m(r) that depends only baryonic mass and increases with r at large distances. The potential φ_m, 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 φ_SBS(r) can be made consistent with the MOND potential φ_m(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 = a_N                                           (19)

which connects observed radial acceleration a to predicted Newtonian acceleration a_N=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 2π times the radius of the visible universe or the de Sitter radius corresponding to cosmological constant Λ [63]. The interpolating function runs smoothly from the inner galaxy, where the field falls off as roughly 1/r², to the region outside the bright galactic disk, called the deep MOND region, where the field tends to fall off as 1/r. Using the simple interpolating function

μ(a/a₀) = (a/a₀) / (1+a/a₀)

proposed by Zhao, Famaey and Binney [64, 65], the MOND relation of Eq. (19) becomes a quadratic equation with solution,

a = ─ (GM/2r²) [1 + √(1+4r²/R_m²)].                   (20)

The radius R_m, to be called the MOND radius, lies near the edge of the bright galactic disk and has the value

R_m = √(GMR/c²) = √(GM/a₀).

In the domain of interest 2R_m<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/R_mr.                           (21)

The potential in this domain can be expressed as

φ_m = ─ ∫a(r) dr ≅ ─ GM/2r + (GM/R_m) ln (r/R_m).

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 R_m, we can neglect the first term and construct an effective metric for the deep MOND region [67]

g₀₀ ≅ 1 + 2φ_m/c² ≅ 1 + (2GM/c²R_m) ln (r/R_m),          (22)

which is accurate in the domain nR_m<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 φ_SBS(r), we assume the galaxy is centered on a supermassive ultra-thin SBS of radius

r₀ = 2m₀ ─ ς = (1 ─ σ) r_s,                                  (23)

where ς>ε is a small distance on the order of meters, r_s is the Schwarzschild radius 2m₀, shell mass m₀ is a large fraction of galactic mass M, and parameter σ=ς/r_s 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 ─ 2m₀S_c(R₀)/r₀ = 0,

or, upon substiting 2m₀=r₀+ς and rearranging,

            S_c(R₀) = 1/(1+ς/r_s) ≅ 1 ─ ς/r_s.                          (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 S_c(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

g_00SBS ≈ [1 ─ r_s(1 ─ η)/r] σ/ |η(r) ─ σ|

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

From Eq. (24), we see that η(R₀)≅σ=ς/r_s, and the denominator of g_00SBS vanishes near R₀ as expected.

To match g_00SBS 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:

g_00MOND ≅ 1 + 2φ_m/c²≅ 1 + (r_s/R_m) ln |r/(R₀─r)|.      (26)

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

a ≅ ─ φ_m′ ≅ ─ GM/R_mr ─ GM/R_m(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 g_00MOND of Eq. (26) adequately approximates the MOND metric in the deep MOND region.

The MOND and SBS metrics may now be equated, giving

1 + (r_s/R_m) ln |r/(R₀─r)| = (1 ─ r_s/r) σ / [η(r) ─ σ].

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

S_m(r) = 1 ─ η(r) = 1 ─ σ / [1 + (r_s/R_m) ln |r/(R₀─r)|]  ─ σ.

Simple calculation shows that S_m(r), while different from the broadened step function S_c(r) derived in Section II, has like properties in the domain r₀<<r<R₀. To wit, S_m(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 S_m(r) by replacing the Gaussian G(r) with some appropriate function F(r) in the definition of the broadened delta function δ_c, thus obtaining a new delta function δ_m. An SBS modeled on δ_m, 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 … r_n-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 δ_j=δ_c(r-r_j) denotes a broadened delta function at radius r_j. Substituting ρ(r) into Eq. (7) and setting the integration constant to zero yields

g₁₁ = ─ e^λ = ─ [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 = ─ e^λ = ─ [1─ 2m₀S₀/r ─ 2m₁S₁/r] ^─1            (27)

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

g_11int  ≈ ─1                                                            (28a)
g_11mid ≈ ─ (1 ─ 2m₀/r) ^─1                                  (28b)
g_11ext  ≈ ─ [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 = e^ν_, the method of Section III will be applied. From Eqs. (11) and (27), we have

    ν = ─ λ + k₁ ─ κ∫_r dr ρ(r) e^λ r

= ─ λ + k₁ ─ κ∫_rdr(μ₀δ₀+μ₁δ₁)r / [1─ 2(m₀S₀+m₁S₁)/r].   (29)

The integral may be expressed as a sum of two terms:

I(r) = μ₀∫_rdr δ₀r / [1─ 2(m₀S₀+m₁S₁)/r]
           + μ₁∫_rdr 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) ≈ μ₀r₀∫_rdr δ₀ / [1─ 2(m₀S₀+m₁S₁)/r₀]
         + μ₁r₁∫_rdr 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) ≈ μ₀r₀∫_rdr δ₀ / (1─ 2m₀S₀/r₀)        
                       + μ₁r₁∫_rdr 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 = e^ν = [1 ─ 2(m₀S₀+m₁S₁)/r]  e^k₁  |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

e^k₁ = |1 ─ 2m₀/r₀| |1 ─ 2m₀/r₀ ─ 2m₁/r₁|             (31)

The constant e^k₁ 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 m_i, radius r_i, and thickness ε_i, as long as ε_i<<(r_i+1─r_i). Such a set of locally continuous thin shells may be viewed as a discrete sampling, at arbitrary radii r_i, 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 δ_m compatible with MOND and the galactic rotation curve.

APPENDIX

Using the line element

ds² = g₀₀(r) dt² + g₁₁(r) dr² ─ r²dΩ²
= e^ν dt² ─ e^λ dr² ─ r²dΩ²

with κ=─8π and stress-energy tensor T^μ_ν=diag(ρ,─p,─p,─p),  Einstein’s field equations simplify to

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

κT¹₁  = ─ 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 ─ λ′r)
= (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 e^λ, we obtain

e^λ = [1 + k₀/r + (κ/r)∫_r dr ρ(r) r²] ^{─ 1}.

Substitution of ρ(r)=μ₀δ_c(r─r₀)  and application of Eq. (4) gives

e^λ = (1 + κμ₀r₀²S_c/r + k₀/r) ^{─ 1}.

Using surface density μ₀=m₀/4πr₀², this becomes

e^λ = (1 ─ 2m₀S_c/r + k₀/r) ^{─ 1}.                             (a3)

The boundary condition that e^λ be nonsingular at r=0 requires that k₀=0. The rr component of the ultra-thin shell metric is therefore

g₁₁ = ─ e^λ = ─ (1 ─ 2m₀S_c/r) ^{─ 1}.                      (a4)

The tt component g₀₀=e^ν 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) e^λ r.

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

ν′ = ─ λ′ ─ κ(1+w)μ₀δ_cr / (1 ─ 2m₀S_c/r).

Upon integrating, this becomes

ν = ─ λ + k₁ ─ κ(1+w)μ₀ ∫_r dr δ_cr / (1 ─ 2m₀S_c/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=e^ν 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) = ∫₀^r dr δ_c r/(1 ─ 2m₀S_c/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) ≈ r₀ ∫₀^r dr δ_c/(1 ─ 2m₀S_c/r₀)              r₀≠2m₀.

I(r) can be integrated by a change of variable dS_c=δ_cdr, with limits of integration 0 and S_c(r):

I(r) ≈ r₀ ∫₀^S_c(r) dS_c/(1 ─ 2m₀S_c/r₀)
                       ≈ ─ (r₀²/2m₀)ln| (1 ─ 2m₀S_c/r₀)|₀^S_c(r)
                 ≈ ─ (r₀²/2m₀) ln| (1 ─ 2m₀S_c/r₀)|,

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

ν ≈ ─ λ + k₁ + [κ(1+w)μ₀r₀²/2m₀] ln |1 ─ 2m₀S_c/r₀|.

Evaluating the constants κ and μ₀, this simplifies to

ν ≈ ─ λ + k₁ ─ (1+w) ln |1 ─ 2m₀S_c/r₀)|,

with the result

e^ν ≈ e^─λ e^k₁ |1 ─ 2m₀S_c/r₀|^{─ (1+w)}

≈ (1 ─ 2m₀S_c/r) e^k₁ |1 ─ 2m₀S_c/r₀|^{─ (1+w)}.

Since e^ν must obey the Minkowski condition e^ν─>1 as r─>∞, the integration constant e^k₁ must cancel the right-hand factor in the outer region where S_c≈1,. Hence the integration constant is

e^k₁ = |1 ─ 2m₀/r₀| ^(1+w)

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

g₀₀≈(1─2m₀S_c/r)|1─2m₀/r₀|^(1+w)|1─2m₀S_c/r₀|^─(1+w)

r₀ ≠ 2m₀.

_____________

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