Thin shells in general relativity 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 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. The exterior solution matches the Schwarzschild metric. General black shell horizons, and singularities are also discussed. split the universe into distinct spacetime regions [12-14]. The structures may be static, as in the case of spherical I. INTRODUCTION wormholes; contracting, as in the case of matter A long-standing unsolved problem in astrophysics is the accretion shells around black holes [15] and shells observed discrepancy in the orbital velocity v(r) of the collapsing into wormholes [16,17]; rotating and luminous matter of galaxies. This discrepancy, often collapsing [18,19]; or expanding, as in the case of cosmic called the flattening of the galactic rotation curve, has brane worlds [20], inflationary bubbles or bubble been ascertained from Doppler shift measurements that universes [21]. Such shells may split the universe into indicate the outlying stars and hydrogen clouds of two domains, an interior and exterior joined by an galaxies orbit too fast to be gravitationally bound by infinitesimally thin wall of singular mass or pressure [22- baryonic matter alone. In regions outside the luminous 26]; or into three domains [27], where the wall of finite disk, v(r) does not fall off as r─1/2 as predicted by thickness is sometimes called the transient layer [28]. Newtonian dynamics, but tends toward a constant as r Various interior and exterior metrics are assumed, increases. The discrepancy is generally attributed to the including the Friedman-Robertson-Walker [29,30], presence of dark matter, a hypothetical transparent Schwarzschild, de Sitter [31], anti-de Sitter [32], nonradiating material that has never been independently Minkowski, and Reissner-Nordstrom [33,34] metrics. detected nor reconciled with the standard model of The metrics are often selected a priori, their parameters particle physics. The failure to identify this elusive later fixed by junction conditions at the inner and outer substance has given rise to modified gravity theories that surfaces of the wall, or at the shell radius [35]. Common obviate the need for dark matter, such as Mordehai techniques frequently require patching solutions for Milgrom©s Modified Newtonian mechanics (MOND) inner, outer, and possible transient domains, using [1,2] and others [3,4]. Here, a static spherical thin shell separate coordinate systems and metrics for each domain solution to Einstein©s field equations is derived that may [36,37]. The most widely applied junction conditions, suggest a new explanation for the galactic rotation curve. attributed to Israel [38,39], or Darmois and Israel [40], A solution for concentric shells is also presented that require that both the metric g ν and the extrinsic μ μ may be useful for discrete sampling of arbitrary spherical curvature K ν be continuous across the shell wall. While mass distributions with applications in cosmology. these conditions are common in the literature, doubt is raised about their application to certain physical Investigation into the gravitational properties of thin scenarios [41] or in modified theories of gravity [42]. matter shells has flourished over the past few decades, Some authors derive new junction conditions that specify most notably in studies of astrophysical and jumps in curvature [43], jumps in the tangential metric cosmological structures such as spherical wormholes [5- components to account for domain wall spin currents 7], black hole accretion shells, bubble universes as [44], or other field behavior [45]. Others avoid junction models of cosmic inflation [8,9], false vacuum bubbles conditions by use of a confining potential [46]. [10,11], and cosmic membranes or domain walls that 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.