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

By Kathleen A. Rosser Ka [email protected] (Published 3 October 2019) Interest in general relativistic treatments of thin matter shells has flourished over recent decades, most notably in connection with astrophysical and cosmological applications such as 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

1 It may be significant that Israel's original derivation was as continuous approximation to the spherical Dirac delta based on properties of electromagnetic fields rather than function δ(r ─ r0), where the continuous or broadened on general relativity (GR), although recent derivations, in version of the delta function, to be written δc(r─ r0), will contexts such as cosmological brane-worlds, address the be derived in Section II. According to this model, the junction by adding a Gibbons-Hawking term to the mass density distribution is standard Einstein-Hilbert action of GR [47]. However, ρ(r) = μ δ (r ─ r ). (1) some authors point to contradictions in this method, 0 c 0 particularly when applied to infinitely thin shells [48]. Here, μ0 is the surface density of the shell and has dimensions [m/r2]. Recalling that the δ function has While procedures for deriving the Israel junction dimensions [1/r], it is clear that the volume density ρ(r) conditions are well established, their implementation has dimensions [m/r3], or [1/r2] in the units G=c=1. relies on concepts outside the core formalism of GR and This density distribution may be substituted into the other metric gravities, including the notion of induced energy-momentum tensor T on the right-hand side of metric, or the D─n dimensional metric in the transient μν EFE. The equations are then solved using a unique domain; the vector n normal to the domain wall; the i change of variable that allows integration to arbitrary surface stress-energy tensor Sμ for the transient domain; ν accuracy. The result is an asymptotically exact the extrinsic curvature Kμ ; the Gibbons-Hawking action ν continuous metric for an empty ultra-thin shell. term, and so forth. A treatment of thin shells that obviates the need for junction conditions may therefore The metric signature (+ - - -) and units c=G=1 will be be useful for its simplicity. Cosmic inhomogeneities used throughout this paper. Small Greek letters stand for using cubic lattices that avoid junction conditions have spacetime indices 0,1,2,3. The symbol ≈ denotes been studied by some authors [49,50]. Nevertheless, asymptotic equality, or equality in limit as thickness examples in the literature of continuous spherical thin- parameter ε approaches zero, although the formalism is shell solutions to the gravitational field equations have undefined at ε=0. An equation of state (EoS) of the form proven difficult to find. p(r)=wρ (r) for w a constant will be assumed. While the method here applies to static shells, it can in principle be The purpose of this paper is to derive an asymptotically generalized to account for expansion or contraction. This exact continuous solution to Einstein's field equations for is a topic for future research. The presentation is static, spherical, ultra-thin massive shells without the organized as follows. In Section II, the broadened need for junction conditions, employing a uniform set of spherical delta function will be derived. Section III coordinates defined over all space, with equation of state shows how to solve EFE for a thin shell using the p=wρ . Here, asymptotically exact means exact in the continuous solution method. In Section IV, the novel limit of vanishing thickness (although the solution is properties of black shells (those of radius less than or undefined for zero thickness), and ultra-thin denotes equal to the Schwarzschild radius) will be examined. arbitrarily thin but non-vanishing. One advantage to the Section V discusses how the galactic rotation curve continuous solution method, in which density ρ(r), might be explained by a supermassive black shell at the pressure p(r), and the metric g (r) are uniformly defined μν galactic core, and Section VI presents the concentric over all space, is that only two boundary conditions are shell solution as a method for discrete sampling. needed to fix the metric: Concluding remarks are found in Section VI. 1. g must be nonsingular at r=0, μν II. MASS DENSITY: DEFINING THE CONTINUOUS DELTA FUNCTION 2. gμν must match Minkowski space as r─ >∞, Spherical Dirac delta functions as models for mass or where Minkowski space is here defined by the metric charge distributions have appeared in the literature for 2 2 2 gμν=diag(1,─ 1, ─ r ,─ r sin (Θ )). The first condition many decades. Use of the delta function for thin shell follows from the absence of matter in the interior [51]. solutions to EFE is frequently encountered in such This is relaxed in the case of a central mass. The second applications as bubble universes and cosmic domain dictates that space be asymptotically flat, assuming walls. However, the discontinuities in the delta function g00─>1 as r─ >∞, or that the standard laboratory clock and its integral, the step function, necessitate piecewise rate is the same as that at infinity. solutions and attendant junction conditions, as noted To obtain a continuous solution to Einstein's field above. To apply a delta function model uniformly over δ ─ equations (EFE), i.e. a metric composed of continuous all space requires that the Dirac delta function (r r0) be analytic functions g (r) defined over all space, one must replaced by a continuous or broadened delta function μν δ ─ first define a continuous density ρ(r) spanning the range c(r r0) with similar properties. One such function can 0≤r≤∞. For an ultra-thin shell, ρ(r) will be modeled here be defined as follows:

2 1) Let δc(r─ r0) be an approximation to a spherical Dirac where f(r) varies slowly over the transient layer, and delta function δ(r ─ r0), where the latter is expressed in Sc(r;r0) has the properties terms of the normalized spherical Gaussian Sc(r;r0) ≈ 0 r < r0─n ε ─1 2 2 G(r) := (ε√ π) exp[─(r─r0) /ε ]. (2) Sc(r;r0) ≈ 1/2 r = r0 ' Here G(r) is defined over the domain r≥0, with a peak Sc(r;r0) ≈ 1 r > r0+nε . centered at r=r of height 1/(π1/2ε) and width 0 (For convenience, the symbol r represents both the proportional to ε. For ε<0] walls enclosing the known cosmos, or for a discrete sampling of any continuous spherical mass distribution. 2) The continuous or broadened delta function δc(r─ r0) III. SOLVING EINSTEIN'S FIELD EQUATIONS is obtained as an approximation to δ(r ─ r0) by taking an incomplete limit in Eq. (3), that is, by letting ε become FOR A THIN SHELL: THE CONTINUOUS arbitrarily small but nonzero. SOLUTION METHOD 3) For n a small integer such that 2nε approximates the We will now derive a locally continuous ultra-thin shell peak width to some selected accuracy, the broadened solution to EFE, assuming a static spherically symmetric metric g of the form delta function δc(r─ r0) nearly vanishes in the domains αβ rr +nε. Therefore mass density ρ(r) 0 0 2 2 2 2 2 approaches that of a near-vacuum in these regions. By ds = g00(r) dt + g11(r) dr ─ r dΩ increasing n and decreasing ε, the vacuum can be = eν dt2 ─ eλ dr2 ─ r2dΩ2 achieved as closely as desired. The appropriate gravitational field equations may be δ 4) The broadened delta function c obeys, to any desired found by substituting this metric into Einstein's field accuracy, the defining properties of the Dirac delta equations, given by function: μ μ μ R ν ─ (1/2) g ν R = κ T ν (5) a) ∫ ∞dr δ (r─ r ) ≈ 1 μ 0 c 0 where R ν is the curvature or Ricci tensor, R is the scalar curvature, κ is a constant with the value κ=─8 π G/c2 ∞ b) ∫0 dr f(r) δc(r─ r0) ≈ f(r0) (using Dirac's sign convention [54]), or κ = ─ 8 π for G=c=1, and Tμ =diag(ρ , ─ p, ─ p, ─ p) is the stress energy f(r) ν provided that is slowly varying over the transient tensor, with ρ(r) the mass-energy density and p(r) the layer r0─n ε

3 where primes denote derivatives with respect to r. Eq. Since δc(r─ r0)≈ 0 in the near-vacuum domains rr0+nε, the integrand vanishes to any desired pure differential (see Appendix for details of derivations accuracy in these domains. (An exception is the case in this section): r0=2m0, where the integrand approaches 0/0 rather than 0 for r>>r +nε , as will be discussed in Section IV.) κρr2 + 1 = (re─λ)′. 0 Hence in general, r changes by a near infinitesimal Integrating and solving for eλ, we obtain amount 2nε across the non-vanishing domain of the transient layer and may be treated as a constant r≈ r . λ κ ∫ ρ 2 ─ 1 0 e = [1 + k0 /r + ( /r) r dr (r) r ] . (7) Thus we have, where k0 is a constant of integration. Substituting 2 I(r) ≈ r ∫ r dr δ / (1 ─ 2m S /r ) r ≠2m . (13) ρ(r)=μ0δc(r─ r0) and μ0=m0/4π r0 , and applying Eq. (4), 0 0 c 0 c 0 0 0 this becomes (Here as elsewhere, the symbol ≈ denotes asymptotic λ ─ 1 e = (1 + k0 /r ─ 2m0Sc/r) . (8) equality, for which precision increases as ε decreases.) I(r) can now be integrated to asymptotic precision by a For an empty shell, the boundary condition that eλ be unique change of variable. Recalling from Eq. (4) that non-singular at r=0 requires that k =0. (If the shell 0 S =∫ δ dr and therefore dS =δ dr, the continuous contains a central mass M, an integration constant c r c c c monotonic function S can be used as the variable of k =─ 2M is generally assumed.) The rr component of the c 0 integration. The limits of integration become 0 and S (r), ultra-thin shell metric is therefore c and the integral may be written g = ─ eλ = ─ (1 ─ 2m S /r) ─ 1. (9) 11 0 c Sc(r) I(r) ≈ r0 ∫0 dSc / (1 ─ 2m0Sc / r0 ) Outside the shell, where Sc≈1, we see that g11 matches S 2 the radial component of the Schwarzschild metric g μν, ≈ ─ (r /2m ) ln |(1 ─ 2m S / r )|, as given by 0 0 0 c 0 where the absolute value, arising from the standard 2 ─ 2 ─ ─ ─1 2 ─ 2 Ω2 ds = (1 2m/r)dt (1 2m/r) dr r d (10) integral formula ∫(dx/x)=ln|x|, will impact later analysis. for m the central mass. In the interior of the shell, where Substituting I(r) back into Eq. (11) and evaluating the constants κ and μ yields Sc≈0, it is clear that g11 matches the Minkowski metric. 0 ν ν ≈ ─ λ + k ─ (1+w) ln |1 ─ 2m S /r )|. Next, the tt component g00=e can be evaluated by 1 0 c 0 subtracting Eq. (6b) from Eq. (6a) to obtain Upon substitution of eλ from Eq. (8), the result is κ( ρ +p) = ─ e─λ λ′/r ─ e─λ ν′/r. ν k1 ─ (1+w) e ≈ (1 ─ 2m0Sc/r) e |1 ─ 2m0Sc/r0| . Solving for ν′, substituting eλ from Eq. (9) and ρ(r) from ν ν Eq. (1), and using equation of state p=wρ, the result is Since e must obey the Minkowski condition e ─>1 as r─ >∞, the integration constant ek1 must cancel the right- ν′ = ─ λ′ ─ κ (1+w)μ δ r / (1 ─ 2m S /r), 0 c 0 c hand factor in the outer region where Sc─>1, leaving only the left-hand factor, which is asymptotically where δc and Sc are abbreviated notations for the broadened delta and step functions. Upon integrating, Minkowski. Hence the integration constant is this becomes k1 (1+w) e = |1 ─ 2m0/r0| ν ─ λ ─ κ ∫ δ ─ = + k1 (1+w)μ0 r dr [ cr / (1 2m0Sc/r)] (11) and the final result for the tt component of the ultra-thin shell metric is with k1 a constant of integration. Eq. (11) represents an exact solution to EFE for the tt metric component g =eν 00 (1+w) ─(1+w) of an ultra-thin shell. The integrand, however, contains g00≈(1 ─ 2m0Sc/r) |1─ 2m0/r0| |1─ 2m0Sc/r0| the spherical Gaussian G(r) and may be difficult to evaluate analytically. For the present, an arbitrarily close r0 ≠ 2m0. (14) approximation can be found using the properties of the broadened step and delta functions. This procedure To analyze this result, we evaluate g00 for the interior and exterior, obtaining requires care due to the rapid variation of Sc(r;r0) in the transient layer r ─n ε

4 The exterior component g00, like the exterior component IV. BLACK HOLES AND BLACK SHELLS g , matches the Schwarzschild solution as expected. 11 The ultra-thin shell metric of Eqs. (9) and (11) may be Note that the quantity ─2m in the exterior metric arises 0 applied to shells of radius equal to or less than the automatically from the field equations and, unlike for the Schwarzschild radius, or shells such that r ≤2m . To be case of Schwarzschild metric, is not put in as an o 0 called black shells, these exotic objects would generally integration constant. That this quantity is predetermined appear to a distant observer as a Schwarzschild black by EFE further confirms the consistency of general hole (although unexpected singularities may occur). At relativity, in that vacuum and non-vacuum solutions close range, black shells display unique properties with agree for regions surrounding a central mass. Thus, solar respect to horizons and singularities. To compare black system tests confirm not just the vacuum equations, holes and black shells, first recall the properties of the where Tμ =0, but also the massive equations, where ν Schwarzschild black hole with metric gS as given by Tμ ≠0, insofar as a thin shell serves as well as a point μν ν Eq. (10): mass for modeling a star or planet. 1. A coordinate singularity, or horizon, exists at Regarding time dilation, it is significant that the interior r=2m, where gS =0 and gS ─> ─∞. metric g is a constant not equal to unity, while the 00 11 00int 2. Inside the horizon, squared proper time intervals exterior metric g asymptotically approaches unity, 00ext dτ2=gS dt2 are negative, and thus proper time is indicating clocks inside the shell run at different rates 00 spacelike, while squared proper radial intervals than those at infinity. For so-called non-phantom matter, dR2=gS dr2 are positive, and proper radial which has an EoS p(r)=wρ (r) with w>─ 1, we note that 00 distance is timelike. g <1, indicating time inside the shell is dilated with 00int 3. A physical singularity is generally assumed to respect to infinity. This result may seem at odds with exist at r=0, where gS ─>-∞ and gS =0. occasional claims that time does not dilate inside an 00 11 4. There are no finite discontinuities in the domain empty shell. Such claims may arise from piecewise r>0. solutions and are often based on two arguments: 1) Minkowski spacetime, with g00=1, prevails inside a To compare the properties of black shells, we consider hollow shell; or 2) according to Birkhoff's theorem, the the metrics for four shell types: ordinary shells with Schwarzschild metric governs the vacuum in an empty r0>2m0; horizon black shells with r0=2m0; subhorizon shell, leading to g =1 [51]. These arguments, however, 00 black shells with r0<2m0, and semi-horizon black shells depend on a rescaling of the time coordinate inside the with r0=m0. First, recall the interior and exterior thin shell. The continuous solution method, in contrast, shell metrics of Section III: assumes a uniform time coordinate over the whole space (1+w) domain 0≤r<∞. It is clear, nevertheless, that no apparent g00int ≈ (1 ─ 2m0 /r0) r0 ≠ 2m0 (16a) gravitational forces exist inside an empty shell due to the g ≈ 1 ─ 2m /r r ≠ 2m (16b) constant value of the interior metric. 00ext 0 0 0 g ≈ ─ 1 (16c) For a shell composed of dust, the EoS parameter is w=0, 11int g ≈ ─ (1 ─ 2m /r) ─1. (16d) and the interior and exterior solutions match at r=r0. 11ext 0 Therefore g00 and the corresponding clock rates are In the case of ordinary shells (r0>2m0), the continuous across the shell wall. The tt component for a Schwarzschild radius r =2m lies inside the shell where thin dust shell thus satisfies the first Israel junction s 0 the metric is constant. Thus there is no horizon at r=rs. In condition. addition, no singularity exists at r=0. Although the For a shell composed of stiff matter, which has an EoS of metric is locally continuous everywhere, comparison of Eqs. (16c) and (16d) reveals a global discontinuity or w=1, we see that g00 changes abruptly across the shell wall, allowing interior time dilation up to twice that at jump across r0 in the component g11. For non-dust the outer surface. Thus the continuous solution method models, for which w≠0, there is also a jump across r0 in predicts time dilation measurements using real non-dust the component g00, in apparent violation of the Israel shells would show a violation of the Israel conditions. It junction conditions. However when w=0 as in the case of dust, g remains unchanged across r , in agreement with seems interesting that the interior metric g00int depends 00 0 the Israel conditions. on the EoS of the shell, while the exterior metric g00ext like the Schwarzschild metric, is independent of the EoS. For horizon black shells (r0=2m0), the shell radius is This curious distinction resolves the seeming paradox, equal to the Schwarzschild radius r =2m , and as noted mentioned in a previous paper [56], that while non- s 0 earlier, the approximation r─ >r0 in the integrand of I(r) vacuum solutions to EFE require an EoS, Schwarzschild of Eq. (13) is no longer valid. Deriving the properties of vacuum solutions do not, even though mass appears in g would require computing the exact integral of Eq. the metric. 00

5 (12) using the spherical Gaussian. Such a calculation is some extremely small quantity, a singularity may occur not attempted here. If, however, we naively allow the at some large radius r=R0 where Sc(R0)=r0/2m0≈1. This approximation r─ >r0 and apply Eq. (14), the apparent means subhorizon black shells could in principle cause properties of horizon black shells suggest such objects singularities in g00 at cosmological distances. Such may be nonphysical. To illustrate, recall the full models may have astrophysical applications related to equations for the metric: the composition of galactic cores (the topic of Section V), or cosmological interpretations with respect to (1+w) ─(1+w) g00≈(1 ─ 2m0Sc/r)|1─ 2m0/r0| |1─ 2m0Sc/r0| (17) Hubble redshift, bubble universes or spherical domain walls, to be addressed in a later paper. g ≈ ─ 1 / (1 ─ 2m S /r) (18) 11 0 c In the unique case of a semi-horizon black shell for ─ Setting r0=2m0 in the first equation and assuming w> 1, which the radius r0=m0 is half the Schwarzschild radius, it is clear that g00(r)=0 for 00. The vanishing of simplifies to g suggests that a horizon black shell would stop all 00 g (r) ≈ [1 ─ 2r S (r)/r] / |1─ 2S (r)|, clocks in the universe, a physical impossibility and a 00 0 c c violation of the asymptotic Minkowski condition. which, as r tends to r0, approaches the improper limit Whether this nonphysical result can be avoided by 0/0. Applying L'Hopital's rule yields the ratio H of the evaluating g00 analytically using the function G(r), by derivatives of numerator and denominator: applying numerical methods, or by redefining δ in terms c ─ ─ of a function other than G(r), is a question for future H = ∂r [1 2r0Sc/r] / ∂r |1 2Sc| research. 2 = (2r0Sc/r ─ 2r0δc/r) / (─ ⁄ + 2 δc) Concerning the rr metric component, we see from Eq. = ─ ⁄ + (r0/r) (Sc/rδc ─ 1) (16c) that g11≈─1 inside the shell, implying no interior singularities exist. To check this result, note that by Eq. where the sign ambiguity springs from the absolute (18), no singularity can exist unless there is an r such that value. Taking the limit r─ >r , the term S /rδ approaches 1/2 0 c c 2m S (r)/r=1, or r/r =S (r). Since it is always true that π ε/2r0<<1, and H tends to positive or negative unity, 0 c 0 c with the positive case corresponding to approach from Sc(r)<1, any such singularity can only reside at rr0 and the negative to r0 such that 2m S (r)/r=1, and semi-horizon black shell discontinuity arises as an 0 0 c artifact of the approximation is not known. hence no singularity in the domain 00] 2m0 Sc(r)/r = 0, ruling out a singularity at the origin. Thus a horizon Can supermassive black shells in the cores of galaxies black shell, unlike a Schwarzschild black hole, manifests explain the discrepancy in the galactic rotation curve? If no singularities in g . so, it would obviate the need for postulating a dark 11 matter halo. The discrepancy in orbital velocity v(r), as Subhorizon black shells (r0<2m0), in contrast, appear at noted earlier, arises from observations of differential close range like Schwarzschild black holes, with a Doppler shift, which indicate the outer stars and horizon at r≈ 2m0. Subhorizon black shells also have hydrogen clouds of galaxies orbit too fast to be approximate Schwarzschild behavior for r>2m0. gravitationally bound by luminous or baryonic matter However, a new singularity in g00 may arise due to the alone. Thus, outside the bright galactic disk, v(r) does not ─1/2 vanishing of 1─ 2m0Sc/r0 in the right-hand factor fall off as r , as would be expected from Newtonian (denominator) of Eq. (17). To locate this singularity, dynamics [57], but tends toward a constant as r recall that Sc(r) increases monotonically over the range increases. This anomaly was noted by Fritz Zwicky in 0

6 distances. The potential φm, for reasons evident below, φm = ─ ∫ a(r) dr ≅ ─ GM/2r + (GM/Rm) ln (r/Rm). will be called the MOND potential. The goal is to show The factor 1/2 in the first term on the right does not that a subhorizon black shell (SBS), or similar exotic appear in some presentations of MOND, where different black object, located in the galactic core, could interpolating functions apply and where the potential theoretically account for the observed excess orbital covers all space [66]. However, since the second term velocities, or equivalently, that an SBS potential φ (r) SBS increases with r and becomes dominant near R , we can can be made consistent with the MOND potential φ (r) m m neglect the first term and construct an effective metric in outlying regions. The MOND potential will be derived for the deep MOND region [67] first, followed by the SBS potential. The two will then be equated to show, by a redefinition of the broadened delta 2 2 g00 ≅ 1 + 2 φm/c ≅ 1 + (2GM/c Rm) ln (r/Rm), (22) function, a close correspondence in the metrics. which is accurate in the domain nRm∞ as Newtonian Dynamics (MOND), a formalism developed r─ >∞. Hence the effective metric violates the asymptotic in 1983 by Mordehai Milgrom [60] to account for the Minkowski condition and cannot, in the form of Eq. (22), discrepancy in the rotation velocity of galaxies. Although be consistent with a black shell metric. Consistency will the excess velocity is usually attributed to the presence of be attained through a later approximation. an unseen dark matter halo, the MOND formalism, φ (r) relying on baryonic matter alone, has proven accurate in Next, to calculate the SBS potential SBS , we assume predicting orbital motion [61], and thus provides a means the galaxy is centered on a supermassive ultra-thin SBS for testing theories. of radius ─ ς ─ σ The MOND formalism is based on the empirical relation r0 = 2m0 = (1 ) rs, (23)

where ς> ε is a small distance on the order of meters, rs is μ(a/a0)a = aN (19) the Schwarzschild radius 2m0, shell mass m0 is a large which connects observed radial acceleration a to fraction of galactic mass M, and parameter σ=ς/r 2 s predicted Newtonian acceleration aN=GM/r using an measures the small difference between shell size and interpolating function μ(a/a0), where Schwarzschild radius. Such an SBS would induce a singularity in g at some cosmic-scale radius R , at a = 1.2x10─8cm/sec2 H /2π = c2/R = c2/(Λ /3)−1/2 00 0 0 ≅ 0 which clocks would theoretically run at an infinite rate. is a universal constant with dimensions of acceleration, In realistic scenarios, no remote singularity can occur H0 is the Hubble parameter [62], and R is roughly 2π due to disturbance of the mass density by other fields. times the radius of the visible universe or the de Sitter Nevertheless, a remote virtual singularity implies a radius corresponding to cosmological constant Λ [63]. modification of the field in the neighborhood of the The interpolating function runs smoothly from the inner galaxy. galaxy, where the field falls off as roughly 1/r2, to the The distance to the singularity at R is inversely related region outside the bright galactic disk, called the deep 0 to ς and increases with step width 2nε. More specifically, MOND region, where the field tends to fall off as 1/r. from Eq. (17), R must satisfy Using the simple interpolating function 0 1 ─ 2m0Sc(R0)/r0 = 0, μ(a/a0) = (a/a0) / (1+a/a0) or, upon substiting 2m =r +ς and rearranging, proposed by Zhao, Famaey and Binney [64, 65], the 0 0 MOND relation of Eq. (19) becomes a quadratic Sc(R0) = 1 / (1 + ς / rs) ≅ 1 ─ ς / rs. (24) equation with solution, To calculate the impact of the distant singularity on the 2 2 2 a = ─ (GM/2r ) [1 + √(1+4r /Rm )]. (20) field in the galactic neighborhood, we start by introducing a new function η(r) and expressing the The radius R , to be called the MOND radius, lies near m broadened step function as S (r)=1─η (r), where η(r)<<1 the edge of the bright galactic disk and has the value c in the deep MOND region. This and Eq. (23) are then 2 Rm = √(GMR/c ) = √(GM/a0). substituted into the thin shell metric of Eq. (17), and the result is evaluated for the shell's far exterior r <

7 From Eq. (24), we see that η(R0)≅σ = ς /rs, and the r = {r0, r1 ... rn-1}. denominator of g vanishes near R as expected. 00SBS 0 The method for concentric shell solutions will be To match g00SBS to the MOND metric of Eq. (22), we illustrated for the simple case of two shells with EoS first write a an approximation to the latter which parameter w=0. Assuming surface densities μ and μ , 2 0 21 repositions the singularity from infinity to a remote finite radii r0 and r1, and masses m0=4πμ0r0 and m1=4πμ1r1 , distance r=R0 as follows: the mass density can be expressed in terms of broadened 2 delta functions as g00MOND ≅ 1 + 2 φm/c ≅ 1 + (rs/Rm) ln |r/(R0─r)|. (26) ρ(r) = μ δ + μ δ This approximation can be checked by calculating 0 0 1 1 acceleration a from potential φm where δj=δc(r─ rj) denotes a broadened delta function at radius r . Substituting ρ(r) into Eq. (7) and setting the a ′ GM/R r ─ GM/R (R ─r). j ≅ ─ φm ≅ ─ m m 0 integration constant to zero yields It is clear that for r in the neighborhood of the galaxy, g = ─ eλ = ─ [1 + ( κ /r) ∫ dr ρ r2] ─ 1 (R ─r) is large enough that the right-hand term can be 11 r 0 2 2 ─1 neglected. The remaining term matches the MOND = ─ [1 + ( κμ0/r)∫r dr r δ0 + (κμ1/r)∫r dr r δ1] . acceleration of Eq. (21). Hence we see that g of 00MOND Upon integration, the double thin-shell solution becomes Eq. (26) adequately approximates the MOND metric in the deep MOND region. λ ─ 1 g11 = ─ e = ─ [1 ─ 2m0S0/r ─ 2m1S1/r] (27) The MOND and SBS metrics may now be equated, where S0=Sc(r;r0) and S1=Sc(r;r1). The interior giving (rr1+nε ) solutions are therefore 1 + (rs/Rm) ln |r/(R0─r)| = (1 ─ rs/r) σ / [ η (r) ─ σ ]. g ≈ ─ 1 (28a) By solving for η(r), a new form Sm(r) of the broadened 11int step function is obtained that is consistent with the g ≈ ─ (1 ─ 2m /r) ─1 (28b) MOND metric as follows: 11mid 0 ─1 g11ext ≈ ─ [1 ─ 2(m0 + m1)/r] , (28c) Sm(r) = 1─η (r) = 1 ─ σ / [1+(rs/Rm) ln |r/(R0─r)|] ─ σ . displaying Minkowski properties inside the smaller shell, Simple calculation shows that Sm(r), while different from Schwarzschild behavior between shells, and combined the broadened step function Sc(r) derived in Section II, Schwarzschild behavior outside the larger shell. has like properties in the domain r0<

8 slowly over the nonzero domain of δ1 and may be set to a using a continuous solution method that does not require constant S0≈1. The total integral then simplifies to junction conditions. The single shell solution is given by Eqs. (9) and (14), and the double shell solution by Eqs. I(r) ≈ μ0r0∫rdr δ0 / (1─ 2m0S0/r0) (27), (30) and (31). These solutions are fixed by two + μ r ∫ dr δ / (1─ 2m /r ─ 2m S /r ). boundary conditions: asymptotic flatness at infinity and 1 1 r 1 0 1 1 1 1 non-singularity at the origin. The interior of a thin shell Following the method of Section III, a change of variable is found to manifest no effective gravitational forces. from r to S0(r) and S1(r) in the respective integrals gives, However, interior clocks run at different rates from those upon integration, at infinity. For non-phantom matter (w>─ 1), time in the interior of the shell is dilated with respect to infinity, 2 I(r) ≈ (μ0r0 /2m0) ln |1─ 2m0S0/r0| while for phantom matter, time is contracted. 2 + (μ1r1 /2m1) ln |1─ 2m0/r0 ─ 2m1S1/r1|. Exterior to the shell, the field generally matches that of Multiplying I(r) by κ and substituting back into Eq. (29) the Schwarzschild metric. Exceptions are found for black then yields shells, i.e. shells of radius less than or equal to the Schwarzschild radius. The method breaks down for equal ν = ─λ + k1 ─ ln |1 ─ 2m0S0 /r0| radii, and an asymptotically exact solution was not attempted. However, approximations suggest such ─ ln |1 ─ 2m /r ─ 2m S /r |. 0 0 1 1 1 objects may be unphysical. Subhorizon black shells, and hence which have a radius smaller than the Schwarzschild radius, are more easily analyzed, and were shown in ν k1 ─1 g00 = e = [1 ─ 2(m0S0+m1S1)/r] e |1─ 2m0S0/r0| general to appear as Schwarzschild black holes everywhere outside the shell. This holds with one key ─1 X |1 ─ 2m0/r0 ─ 2m1S1/r1| (30) exception. When the radius of a supermassive black shell is less than its Schwarzschild radius by a very small where X denotes multiplication. Again, to meet the distance on the order of meters, a singularity may occur asymptotic Minkowski condition, the integration in the time component of the metric at cosmological constant must be distances. It was then shown that this singular metric k1 approximates an effective MOND metric, where the e = |1 ─ 2m0/r0| |1 ─ 2m0/r0 ─ 2m1/r1| (31) k1 latter is expressed in terms of an effective potential that The constant e is then substituted back into Eq. (30), accounts for the observed galactic orbital velocities. g yielding the 00 component of the double concentric Thus, a supermassive subhorizon ultra-thin black shell or shell metric. Taken together, Eqs. (27), (30) and (31) similar exotic black object, located at the center of a represent a complete continuous asymptotically exact galaxy, could theoretically explain the flattening of the solution to EFE for two concentric ultra-thin dust shells galactic rotation curve without the need for dark matter. of arbitrary mass and radius. It was also shown that the solution for a series of It is straightforward to extend this result to n concentric concentric shells provides a discrete sampling method for m r ε shells of mass i, radius i, and thickness i, as long as calculating the approximate gravitational field of any ε <<(r ─r ). locally i i+1 i Such a set of continuous thin spherical static mass distribution. Applications might shells may be viewed as a discrete sampling, at arbitrary include detailed scenarios such as spherical accretion r globally radii i, of a continuous mass density shells around black holes embedded in a constant ρ distribution (r). The concentric shell formalism thus background density enclosed by a cosmic bubble. provides a discrete method for approximating the solution to EFE for any static, spherically symmetric The method developed here applies to static scenarios. It mass-energy density. Hence Einstein's equations can be can in principle be generalized to dynamic configurations readily solved for complicated scenarios such as a star such as colliding shells in anti-deSitter spacetime [68] or surrounded by spherical dust clouds embedded in cosmic black holes embedded in expanding bubble universes bubbles, and so forth. The impact of discreteness on described by the Friedman-Robertson-Walker metric. accuracy is a topic for future discussion. These are topics for future research. Other questions also remain concerning: 1. Multiple concentric shell techniques for discrete VII. CONCLUSION sampling of cosmological mass distributions, We have derived an asymptotically exact solution to 2. The impact of discreteness on accuracy, Einstein's field equations for individual and multiple concentric ultra-thin shells of arbitrary mass and radius

9 λ ─ 1 3. Comparison of ultra-thin shell boundary g11 = ─ e = ─ (1 ─ 2m0Sc/r) . (a4) properties to Israel junction conditions under a The tt component g =eν can be evaluated by subtracting general EoS, 00 Eq. (a2) from Eq. (a1) to obtain 4. Collapsing ultra-thin shells and black shell formation, κ( ρ +p) = ─ e─λ λ′/r ─ e─λ ν′/r. 5. Whether possible nonphysical features of horizon ν′ black shells interfere with black shell formation, Solving for yields 6. The nature and stability of rotating or charged ν′ = ─ λ′ ─ κ ( ρ +p) eλ r. ultra-thin shells, λ 7. Stability of ultra-thin shells, particularly of If we now substitute ρ(r) and e from Eq. (a4), and apply subhorizon black shells in galactic cores, and the equation of state p=wρ for w a constant, the result is 8. The mathematical properties of functions F(r) ν′ = ─ λ′ ─ κ (1+w)μ0δcr / (1 ─ 2m0Sc/r). and δm compatible with MOND and the galactic rotation curve. Upon integrating, this becomes

ν = ─ λ + k1 ─ κ (1+w)μ0 ∫r dr δcr / (1 ─ 2m0Sc/r) (a5)

with k1 a constant of integration. Eq. (a5) represents an exact solution to Einstein's field equations for the tt ν APPENDIX metric component g00=e of an ultra-thin shell. To approximate the integral, we use the properties of the Using the line element broadened step and delta functions. The integral may be 2 2 2 2 2 written ds = g00(r) dt + g11(r) dr ─ r dΩ I(r) = ∫ r dr δ r/(1 ─ 2m S /r). = eν dt2 ─ eλ dr2 ─ r2dΩ2 0 c 0 c Since r changes by the near infinitesimal amount 2nε with κ=─8 π and a diagonal stress-energy tensor of the across the transient layer, it may be treated as a constant μ ρ ─ ─ ─ form T ν=diag( , p, p, p), Einstein's field equations r≈ r hence simplify to 0, I(r) ≈ r ∫ r dr δ /(1 ─ 2m S /r ) r ≠2m . 0 ─λ 2 2 ─λ 0 0 c 0 c 0 0 0 κT 0 = κρ (r) = e /r ─ 1/r ─ e λ′/r (a1) I(r) can be integrated by a change of variable dSc=δcdr, 1 ─λ 2 2 ─λ with limits of integration 0 and Sc(r): κT 1 = ─ κ p(r) = e /r ─ 1/r + e ν′/r, (a2) Sc(r) where primes denote derivatives with respect to r. Eq. I(r) ≈ r0 ∫0 dSc/(1 ─ 2m0Sc/r0) (a1) can be solved by rearranging terms 2 Sc(r) ≈ ─ (r0 /2m0) ln| (1 ─ 2m0Sc/r0)|0 κρr2 + 1 = e─λ (1 ─ λ′r) 2 ≈ ─ (r0 /2m0) ln| (1 ─ 2m0Sc/r0)|, = (re─λ)′. Substituting I(r) into Eq. (a5) yields, Integration then yields ν ≈ ─ λ + k + [κ (1+w)μ r 2/2m ] ln |1 ─ 2m S /r |. ─λ 2 1 0 0 0 0 c 0 re = k0 + ∫r dr (κρ r + 1) . Evaluating the constants κ and μ , this simplifies to Here, ∫ denotes the inverse derivative and k is a 0 r λ 0 constant of integration. Solving for e , we obtain ν ≈ ─ λ + k1 ─ (1+w) ln |1 ─ 2m0Sc/r0)|, λ 2 ─ 1 with the result e = [1 + k0/r + (κ /r) ∫r dr ρ (r) r ] . Substitution of ρ(r)=μ δ (r─ r ) and application of Eq. ν ─λ k1 ─ (1+w) 0 c 0 e ≈ e e |1 ─ 2m0Sc/r0| (4) gives k1 ─ (1+w) λ 2 ─ 1 ≈ (1 ─ 2m0Sc/r) e |1 ─ 2m0Sc/r0| . e = (1 + κμ0r0 Sc/r + k0/r) . ν ν Using surface density μ =m /4π r 2, this becomes Since e must obey the Minkowski condition e ─>1 as 0 0 0 r─ >∞, the integration constant ek1 must cancel the right- eλ = (1 ─ 2m S /r + k /r) ─ 1. (a3) 0 c 0 hand factor in the outer region where Sc≈1, Hence the The boundary condition that eλ be nonsingular at r=0 integration constant is requires that k =0. The rr component of the ultra-thin k1 (1+w) 0 e = |1 ─ 2m0/r0| shell metric is therefore

10 and the tt component of the ultra-thin shell metric Kinoshita, What wormhole is traversable? A case of a becomes wormhole supported by a spherical thin shell, Phys Rev D 88, 044036 (2013) (1+w) ─(1+w) g00≈(1 ─ 2m0Sc/r)|1─ 2m0/r0| |1─ 2m0Sc/r0| [18] Térence Delsate, Jorge V. Rocha, and Raphael Santarelli, Collapsing thin shells with rotation, Phys Rev r0 ≠ 2m0. D 89, 121501(R) (2014) [19] Robert B. Mann John J. Oh and Mu-In Park, Role of ______angular momentum and cosmic censorship in (2+1)- [1] Mordehai Milgrom, MOND in galaxy groups: A dimensional rotating shell collapse, Phys Rev D 79, superior sample, Phys Rev D 99, 044041 (2019) 064005 (2009) [2] Mordehai Milgrom, Bimetric MOND gravity, Phys [20] Hideo Kodama, Akihiro Ishibashi, and Osamu Seto, Rev D 80, 123536 (2009) Brane world cosmology: Gauge-invariant formalism for [3] A. Stabile and S. Capozziell, Galaxy rotation curves perturbation, Phys Rev D 62, 064022 (2000) in f(R,φ ) gravity, Phys Rev D 87,064002 (2013) [21] Matthew C. Johnson, Hiranya V. Peiris, and Luis [4] Edmund A. Chadwick, Timothy F. Hodgkinson, and Lehner, Determining the outcome of cosmic bubble Graham S. McDonald, Gravitational theoretical collisions in full general relativity, Phys Rev D 85, development supporting MOND, Phys Rev D Phys. Rev. 083516 (2012) D 88, 024036 (2013) [22] Benjamin K. Tippett, Gravitational lensing as a [5] Naoki Tsukamoto and Takafumi Kokubu, Linear, mechanism for effective cloaking, Phys Rev D 84, stability analysis of a rotating thin-shell wormhole, Phys 104034 (2011) Rev D 98, 044026 (2018) [23] Kenji Tomita, Junction conditions of cosmological [6] Roberto Balbinot, Claude Barrabès, and Alessandro perturbations, Phys Rev D 70, 123502 (2004) Fabbri, Friedmann universes connected by Reissner- [24] Elias Gravanis and Steven Willison, ‘‘Mass without Nordström wormholes, Phys. Rev. D 49, 2801 (1994) mass’’ from thin shells in Gauss-Bonnet gravity, Phys [7] Z. Amirabi, M. Halilsoy, and S. Habib Rev D 75, 084025 (2007) Mazharimousavi, Effect of the Gauss-Bonnet parameter [25] Chih-Hung Wang, Hing-Tong Cho, and Yu-Huei in the stability of thin-shell wormholes, Phys Rev D 88, Wu, Domain wall space-times with a cosmological 124023 (2013) constant, Phys Rev D 83, 084014 (2011) [8] S. Shajidul Haque and Bret Underwood, Consistent [26] Donald Marolf and Sho Yaida, Energy conditions cosmic bubble embeddings, Phys Rev D 95, 103513 and junction conditions, Phys Rev D 72, 044016 (2005) (2017) [27] S. Khakshournia and R. Mansouri, Dynamics of [9] Anthony Aguirre and Matthew C. Johnson, Towards general relativistic spherically symmetric dust thick observable signatures of other bubble universes. II. shells, arXiv:gr-qc/0308025v1 (2003) Exact solutions for thin-wall bubble collisions, Phys Rev [28] V. A. Berezin, V. A. Kuzmin, and I. I. Tkachev, D 77, 123536 (2008) Dynamics of bubbles in general relativity, Phys Rev D [10] Eduardo Guendelman and Idan Shilon, 36, 2919 (1987) Gravitational trapping near domain walls and stable [29] Cédric Grenon and Kayll Lake, Generalized Swiss- solitons, Phys Rev D 76, 025021 (2007) cheese . II. Spherical dust, Phys Rev D 84, [11] Bum-Hoon Lee et. al., Dynamics of false vacuum 083506 (2011) bubbles with nonminimal coupling, Phys Rev D 77, [30] László Á. Gergely, No Swiss-cheese universe on the 063502 (2008) brane, Phys Rev D 71, 084017 (2005) [12] Debaprasad Maity, Dynamic domain walls in a [31] Shu Lin and Edward Shuryak, Toward the AdS/CFT Maxwell-dilaton background, Phys Rev D 78, 084008 gravity dual for high energy collisions. III. (2008) Gravitationally collapsing shell and quasiequilibrium, [13] Debaprasad Maity, Semirealistic bouncing domain Phys Rev D 78, 125018 (2008) wall cosmology, Phys Rev D 86, 084056 (2012) [32] Adam Balcerzak and Mariusz P. Dąbrowski, Brane [14] S. Habib Mazharimousavi and M. Halilsoy, Domain f(R) gravity cosmologies, Phys Rev D 81, 123527 (2010) walls in Einstein-Gauss-Bonnet bulk, Phys Rev D 82, [33] Nami Uchikata, Shijun Yoshida, and Toshifumi 087502 (2010) Futamase, New solutions of charged regular black holes [15] Shu Lin and Ho-Ung Yee, Out-of-equilibrium chiral and their stability, Phys Rev D 86, 084025 (2012) magnetic effect at strong coupling, Phys Rev D 88, [34] Nick E Mavromatos and Sarben Sarkar, Regularized 025030 (2013) Kalb-Ramond magnetic monopole with finite energy, [16] Yumi Akai and Ken-ichi Nakao, Nonlinear stability Phys Rev D 97, 125010 (2018) of a brane wormhole, Phys Rev D 96, 024033 (2017) [35] David E. Kaplan and Surjeet Rajendran, Firewalls [17] Ken-ichi Nakao, Tatsuya Uno, and Shunichiro in general relativity, Phys Rev D 99, 044033 (2019)

11 [36] Rui Andre´, Jose´ P. S. Lemos, and Gonçalo M. (2012). Quinta, Thermodynamics and entropy of self-gravitating [51] Ignazio Ciuffolini and John Archibald Wheeler, matter shells and black holes in d dimensions, Phys Rev Gravitataion and Intertia, Princeton University Press D 99, 125013 (2019) (1995), p40 [37] Aharon Davidson and Ilya Gurwich, Dirac [52] Metin Gürses and Burak Himmetoḡ lu, Multishell Relaxation of the Israel Junction Conditions: Unified model for Majumdar-Papapetrou spacetimes, Phys Rev Randall-Sundrum Brane Theory, arXiv:gr-qc/0606098v1 D 72, 024032 (2005) (2006) [53] Luiz C. S. Leite, Caio F. B. Macedo, and Luís C. B. [38] Chih-Hung Wang, Hing-Tong Cho, and Yu-Huei Crispino, Black holes with surrounding matter and Wu, Domain wall space-times with a cosmological rainbow scattering, Phys Rev D 99, 064020 (2019) constant, Phys Rev D 83, 084014 (2011) [54] P.A.M. Dirac, General Theory of Relativity, John [39] Kota Ogasawara1 and Naoki Tsukamoto, Effect of Wiley & Sons, 1975, p47 the self-gravity of shells on a high energy collision in a [55] A. Bakopoulos, G. Antoniou, and P. Kanti, Novel rotating Bañados-Teitelboim-Zanelli spacetime, Phys black-hole solutions in Einstein-scalar-Gauss-Bonnet Rev D 99, 024016 (2019) theories with a cosmological constant, Phys Rev D 99, [40] Peter Musgrave and Kayll Lake, Junctions and thin 064003 (2019) shells in general relativity using computer algebra: I. [56] Kathleen A. Rosser, Current conflicts in general The Darmois-Israel formalism, Classical and Quanum relativity: Is Einstein's theory incomplete?, ResearchGate Gravity, Volume 13, Number 7 DOI: 10.13140/RG.2.2.19056.40966 (2019); [41] Carlos Barceló and Matt Visser, Moduli fields and Quemadojournal.org brane tensions: Generalizing the junction conditions, [57] Jacob D. Bekenstein, Relativistic gravitation theory Phys Rev D 63, 024004 (2000) for the modified Newtonian dynamics paradigm, Phys [42] Jesse Velay-Vitow and Andrew DeBenedictis, Rev D 70, 083509 (2004) Junction conditions for FðTÞ gravity from a variational [58] F. Zwicky, Helv. Phys. Acta 6, 110 (1933) principle, Phys Rev D 96, 024055 (2017) [59] V. C. Rubin, N. Thonnard, and W. K. Ford, Jr., [43] Charles Hellaby and Tevian Dray, Failure of Astro-phys. J. 225, L107 (1978) standard conservation laws at a classical change of [60] Milgrom M., ApJ 270, 365 (1983) signature, Phys Rev D 49, 5096 (1994) [61] Robert H. Sanders and Stacy S. McGaugh, Modified [44] Alex Giacomini, Ricardo Troncoso, and Steven Newtonian Dynamics as a alternative to dark matter, Willison, Junction conditions in general relativity with arXiv:astro-ph/0204521v1 2002 spin sources, 73, 104014 (2006) [62] Lasha Berezhiani and Justin Khoury, Theory of dark [45] José M. M. Senovilla, Junction conditions for F(R) matter superfluidity, Phys Rev D 92, 103510 (2015) gravity and their consequences, Phys Rev D 88, 064015 [63] Milgrom, Mordehai, MOND in galaxy groups, Phys (2013) Rev D 98, 104036 (2018) [46] M. Heydari-Fard and H. R. Sepangi, Gauss-Bonnet [64] Zhao, H.S. and Famaey, B., Refining MOND brane gravity with a confining potential, Phys Rev D 75, interpolating function and TeVeS Lagrangian, 064010 (2007) arXiv:astro-ph/0512425v3 (2006) [47] Stephen C. Davis, Generalised Israel Junction [65] Benoit Famaey and James Binney, Modified Conditions for a Gauss-Bonnet Brane World, arXiv:hep- Newtonian Dynamics in the Milky Way, arXiv:astro- th/0208205v3 (2002) ph/0506723v2 (2005) [48] Adam Balcerzak and Mariusz P. Daa browski, [66]: HongSheng Zhao, Baojiu Li and Olivier Bienayme, Generalized Israel Junction Conditions for a Fourth- Modified Kepler’s law, escape speed, and two-body Order Brane World, arXiv:0710.3670v2; Phys.Rev.D 77, problem in modified Newtonian dynamics-like theories, 023524 (2008) Phys Rev D 82, 103001 (2010) [49] Szymon Sikora and Krzysztof Glod, Perturbatively [67] Benoit Famaey, Jean-Philippe Bruneton and constructed cosmological model with periodically HongSheng Zhao, Escaping from Modified Newtonian distributed dust inhomogeneities, Phys Rev D 99, Dynamics, arXiv:astro-ph/0702275 (2007) 083521 (2019) [68] Richard Brito, Vitor Cardos, Jorge V. Rocha, Two [50] Jean-Philippe Bruneton and Julien Larena, worlds collide: Interacting shells in AdS spacetime and Dynamics of a lattice Universe: the dust approximation chaos, arXiv:1602.03535v3 (2016) in cosmology, Classical Quantum Gravity 29, 155001

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