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

By Kathleen A. Rosser

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

Interest in general relativistic treatments of thin matter shells has flourished over recent decades, most notably in connection with astrophysical and cosmological applications such as black hole matter accretion, spherical wormholes, bubble universes, and cosmic domain walls. In the present paper, an asymptotically exact solution to Einstein’s field equations for static ultra-thin spherical shells is derived using a continuous matter density distribution ρ(r) defined over all space. The matter density is modeled as a product of surface density μ0 and a continuous or broadened spherical delta function. Continuity over the full domain 0<r<∞ ensures unambiguous determination of both the metric and coordinates across the shell wall, obviating the need to patch interior and exterior solutions using junction conditions. A unique change of variable allows integration with asymptotic precision. It is found that ultra-thin shells smaller than the Schwarzschild radius can be used to model supermassive black holes believed to lie at the centers of galaxies, possibly accounting for the flattening of the galactic rotation curve as described by Modified Newtonian Dynamics (MOND). Concentric ultra-thin shells may also be used for discrete sampling of arbitrary spherical mass distributions with applications in cosmology. Ultra-thin shells are shown to exhibit constant interior time dilation proportional to mass m0 and inversely proportional to shell radius r0. The exterior solution matches the Schwarzschild metric. General black shell solutions, horizons, and singularities are also discussed. Various questions are finally listed as topics for future research.

I. INTRODUCTION

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

A long-standing unsolved problem in astrophysics is the observed discrepancy in the orbital velocity v(r) of the luminous matter of galaxies. This discrepancy, often called the flattening of the galactic rotation curve, has been ascertained from Doppler shift measurements that indicate the outlying stars and hydrogen clouds of galaxies orbit too fast to be gravitationally bound by baryonic matter alone. In regions outside the luminous disk, v(r) does not fall off as r─1/2 as predicted by Newtonian dynamics, but tends toward a constant as r increases. The discrepancy is generally attributed to the presence of dark matter, a hypothetical transparent nonradiating material that has never been independently detected nor reconciled with the standard model of particle physics. The failure to identify this elusive substance has given rise to modified gravity theories that obviate the need for dark matter, such as Mordehai Milgrom’s Modified Newtonian mechanics (MOND) [1,2] and others [3,4]. Here, a static spherical thin shell solution to Einstein’s field equations is derived that may suggest a new explanation for the galactic rotation curve. A solution for concentric shells is also presented that may be useful for the discrete sampling of arbitrary spherical mass distributions, with possible applications in cosmology.

Investigation into the gravitational properties of thin matter shells has flourished over the past few decades, most notably in studies of astrophysical and cosmological structures such as spherical wormholes [5-7], black hole accretion shells, bubble universes as models of cosmic inflation [8,9], false vacuum bubbles [10,11], and cosmic membranes or domain walls that split the universe into distinct spacetime regions [12-14].

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

The structures may be static, as in the case of spherical wormholes; contracting, as in the case of matter accretion shells around black holes [15] and shells collapsing into wormholes [16,17]; rotating and collapsing [18,19]; or expanding, as in the case of cosmic brane worlds [20], inflationary bubbles or bubble universes [21]. Such shells may split the universe into two domains, an interior and exterior joined by an infinitesimally thin wall of singular mass or pressure [22-26]; or into three domains [27], where the wall of finite thickness is sometimes called the transient layer [28]. Various interior and exterior metrics are assumed, including the Friedman-Robertson-Walker [29,30], Schwarzschild, de Sitter [31], anti-de Sitter [32], Minkowski, and Reissner-Nordstrom [33,34] metrics. The metrics are often selected a priori, their parameters later fixed by junction conditions that specify continuity or jumps in the metric at the inner and outer surfaces of the wall, or at the shell radius [35]. Common techniques frequently require patching solutions for inner, outer, and possible transient domains, using separate coordinate systems and metrics for each domain [36,37]. The most widely applied junction conditions, attributed to Israel [38,39], or Darmois and Israel [40], require that both the metric gμν and the extrinsic curvature Kμν be continuous across the shell wall. While these conditions are common in the literature, doubt is raised about their application to certain physical scenarios [41] or in modified theories of gravity [42]. Some authors derive new junction conditions that specify jumps in curvature [43], jumps in the tangential metric components to account for domain wall spin currents [44], or other field behavior [45]. Others avoid junction conditions by use of a confining potential [46].

It may be significant that Israel’s original derivation was based on properties of electromagnetic fields rather than on general relativity (GR), although recent derivations, in contexts such as cosmological brane-world scenarios, accommodate the junction by adding a Gibbons-Hawking term to the standard Einstein-Hilbert action of GR [47]. However, some authors point to contradictions in this method, particularly in applications involving infinitely thin shells [48].

While procedures for deriving the Israel junction conditions are well established, their implementation relies on concepts outside the core formalism of GR and other metric gravities, including the notion of induced metric, or the D─n dimensional metric in the transient domain; the vector ni normal to the domain wall; the surface stress-energy tensor Sμν for the transient domain; the extrinsic curvature Kμν; the Gibbons-Hawking action term, and so forth. A treatment of thin shells that obviates the need for junction conditions may therefore be useful for its simplicity. Cosmic inhomogeneities using cubic lattices that avoid junction conditions have been studied by some authors [49,50]. Nevertheless, examples in the literature of continuous spherical thin-shell solutions to the gravitational field equations have proven elusive.

The purpose of this paper is to derive an asymptotically exact continuous solution to Einstein’s field equations for static, spherical, ultra-thin massive shells without the need for junction conditions, employing a uniform set of coordinates defined over all space, with equation of state p=wρ. Here, asymptotically exact means exact in the limit of vanishing thickness (although the solution is undefined for zero thickness), and ultra-thin denotes arbitrarily thin but nonvanishing. One advantage to the continuous solution method, in which density ρ(r), pressure p(r), and the metric gμν(r) are uniformly defined over all space, is that only two boundary conditions are needed to fix the metric:

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

where Minkowski space is here defined by the metric

gμν = diag( 1, ─1, ─r2, ─r2sin2(Θ)).

The first boundary condition follows logically from the absence of matter in the interior [51]. This condition is relaxed in the case of a central mass. The second boundary condition dictates that space be asymptotically flat, with the assumption that g00─>1 as r─>∞, or equivalently, that the standard laboratory clock rate is the same as that at infinity.

To obtain a continuous solution to Einstein’s field equations (EFE), i.e. a metric composed of continuous analytic functions gμν(r) defined over all space, one must first define a continuous density distribution ρ(r) spanning the range 0≤r≤∞, with r the radial coordinate. For an ultra-thin shell, ρ(r) will be modeled here as continuous approximation to the spherical Dirac delta function δ(r-r0), where the continuous or broadened version of the delta function, to be written δc(r-r0), will be derived in Section II. According to this model, the mass density distribution is

ρ(r) = μ0 δc(r ─ r0).                                 (1)

Here, μ0 is the surface density of the shell and has dimensions [m/r2]. Recalling that the δ function has dimensions [1/r], it is clear that the volume density ρ(r) has dimensions [m/r3], or [1/r2] in the units G=c=1. This density distribution may be substituted into the energy-momentum tensor Tμν on the right-hand side of EFE. The equations are then solved using a unique change of variable that allows integration to arbitrary accuracy. The result is an asymptotically exact continuous metric for an empty ultra-thin shell.

The metric signature (+ – – -) and units c=G=1 will be used throughout this paper. Small Greek letters stand for spacetime indices 0,1,2,3. The symbol denotes asymptotic equality, or equality in limit as thickness parameter ε approaches zero, although the formalism is undefined at ε=0. An equation of state (EoS) of the form p(r)=wρ(r) for w a constant will be assumed. While the method here applies to static shells, it can in principle be generalized to account for expansion or contraction. This is a topic for future research.

The presentation is organized as follows. In Section II, the broadened spherical delta function will be derived. Section III shows how to solve EFE for a thin shell using the continuous solution method. In Section IV, the novel properties of black shells (those of radius less than or equal to the Schwarzschild radius) will be examined. Section V discusses how the galactic rotation curve might be explained by a supermassive black shell at the galactic core, and Section VI presents the concentric shell solution as a method for discrete sampling. Concluding remarks are found in Section VI.

II. MASS DENSITY: DEFINING THE CONTINUOUS DELTA FUNCTION

Spherical Dirac delta functions as models for mass or charge distributions have appeared in the literature for many decades. Use of the delta function for thin shell solutions to EFE is frequently encountered in such applications as bubble universes and cosmic domain walls. However, the discontinuities in the delta function and its integral, the step function, necessitate piecewise solutions and attendant junction conditions, as noted above. To apply the delta function technique uniformly over all space requires that the discontinuous Dirac delta function δ(r-r0) be replaced by a continuous or broadened delta function δc(r-r0) with similar properties. One such function can be defined as follows:

1) Let δc(r-r0) be an approximation to a spherical Dirac delta function δ(r-r0), where the latter is expressed in terms of the normalized spherical Gaussian

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

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

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

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

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

The delta function may thus be written

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

2) The continuous or broadened delta function δc(r-r0) is obtained as an approximation to δ(r-r0) by taking an incomplete limit in Eq. (3), that is, by letting ε become arbitrarily small but nonzero.

3) For n a small integer such that 2nε approximates the peak width to some selected accuracy, the broadened delta function δc(r-r0) nearly vanishes in the domains r<r0─nε and r>r0+nε. Therefore mass density ρ(r) approaches that of a near-vacuum in these regions. By increasing n and decreasing ε, the vacuum can be achieved as closely as desired.

4) The broadened delta function δc obeys, to any desired accuracy, the defining properties of the Dirac delta function:

a) 0dr δc(r-r0) ≈ 1

b) 0dr f(r) δc(r-r0) ≈ f(r0)

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

5) The integral 0rdrδc, or the inverse derivative of the broadened delta function δc, is a continuous or broadened step function Sc(r;r0) such that

0rdr f(r) δc(r-r0) ≈ f(r0) Sc(r;r0),                         (4)

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

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

(For convenience, the symbol r represents both the dummy variable and the integral limit.) That Sc≈1/2 for r=r0 can be seen by integrating G(r) from 0 to r0, and recalling that the integral over all space of a normalized Gaussian is unity. The function Sc, while locally continuous, appears globally discontinuous in that its value changes rapidly over the thickness 2nε of the transient layer.

One advantage to modeling mass density ρ(r) in terms of a broadened delta function is the ease of integration when solving EFE. Many integrals can be read off by simply applying Eq. (4). This technique can be extended to concentric shells, such as those discussed in reference [52], and may be useful for modeling astrophysical objects such as spherical dust accretion clouds surrounding dirty black holes [53], spherical domain walls enclosing the known cosmos, or for a discrete sampling of any continuous spherical mass distribution.

III. SOLVING EINSTEIN’S FIELD EQUATIONS FOR A THIN SHELL: THE CONTINUOUS SOLUTION METHOD

We will now derive a locally continuous ultra-thin shell solution to EFE, assuming a static spherically symmetric metric gαβ of the form

ds2 = g00(r) dt2 + g11(r) dr2 ─ r22
= eν dt2 ─ eλ dr2 ─ r22

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

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

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

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

κT11 = ─ κp(r) = e─λ/r2 ─ 1/r2 + e─λν/r,           (6b)

where primes denote derivatives with respect to r. Eq. (6a) can be solved by rearranging terms to produce a pure differential (see Appendix for details of derivations in this section):

κρr2 + 1 = (re─λ).

Integrating and solving for eλ, we obtain

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

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

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

For an empty shell, the boundary condition that eλ be nonsingular at r=0 requires that k0=0. (If the shell contains a central mass M, an integration constant k0=─2M is generally assumed.) The rr component of the ultra-thin shell metric is therefore

g11 = ─ eλ = ─ (1 ─ 2m0Sc/r) ─ 1.                     (9)

Outside the shell, where Sc≈1, we see that g11 matches the radial component of the Schwarzschild metric gSμν, as given by

ds2 = (1 ─ 2m/r)dt2 ─ (1 ─ 2m/r)─1dr2 ─ r22        (10)

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

Next, the tt component g00=eν can be evaluated by subtracting Eq. (6b) from Eq. (6a) to obtain

κ(ρ+p) = ─ e─λ λ/r ─ e─λ ν/r.

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

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

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

ν = ─ λ + k1 ─ κ(1+w)μ0r dr [δcr / (1 ─ 2m0Sc/r)]         (11)

with k1 a constant of integration. Eq. (11) represents an exact solution to EFE for the tt metric component g00=eν of an ultra-thin shell. The integrand, however, contains the spherical Gaussian G(r) and may be difficult to evaluate analytically. For the present, an arbitrarily close approximation can be found using the properties of the broadened step and delta functions. This procedure requires care due to the rapid variation of Sc(r;r0) in the transient layer r0─nε<r<r0+nε. We proceed by writing the integral in Eq. (11) as a function of the upper limit r

I(r) = ∫0r dr δc r/(1 ─ 2m0Sc/r).                       (12)

Since δc(r─r0)≈0 in the near-vacuum domains r<r0─nε and r>r0+nε, the integrand vanishes to any desired accuracy in these domains. (An exception is the case r0=2m0, where the integrand approaches 0/0 rather than 0 for r>>r0+nε, as will be discussed in Section IV.) Hence in general, r changes by a near infinitesimal amount 2nε across the non-vanishing domain of the transient layer and may be treated as a constant r≈r0. Thus we have,

I(r) ≈ r00r dr δc/(1 ─ 2m0Sc/r0)        r0≠2m0.       (13)

(Here as elsewhere, the symbol denotes asymptotic equality, for which precision increases as ε decreases.) I(r) can now be integrated to asymptotic precision by a unique change of variable. Recalling from Eq. (4) that Sc=∫rδcdr and therefore dSccdr, the continuous monotonic function Sc can be used as the variable of integration. The limits of integration become 0 and Sc(r), and the integral may be written

I(r) ≈ r00Sc(r) dSc/(1 ─ 2m0Sc/r0 )

              ≈ ─ (r02/2m0) ln| (1 ─ 2m0Sc/r0)|,

where the absolute value, arising from the standard integral formula ∫(dx/x)=ln|x|, will impact later analysis. Substituting I(r) back into Eq. (11) and evaluating the constants κ and μ0  yields

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

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

eν ≈ (1 ─ 2m0Sc/r) ek1 |1 ─ 2m0Sc/r0|─ (1+w).

Since eν must obey the Minkowski condition eν─>1 as r─>∞, the integration constant ek1 must cancel the right-hand factor in the outer region where Sc─>1, leaving only the left-hand factor, which is asymptotically Minkowski. Hence the integration constant is

ek1 = |1 ─ 2m0/r0| (1+w)

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

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

r0 ≠ 2m0.                                     (14)

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

g00int ≈ |1 ─ 2m0/r0|(1+w)                              (15a)

g00ext ≈ (1 ─ 2m0/r).                                        (15b)

The exterior component g00, like the exterior component g11, matches the Schwarzschild solution as expected. Note that the quantity ─2m0 in the exterior metric arises automatically from the field equations and, unlike for the case of Schwarzschild metric, is not put in as an integration constant. That this quantity is predetermined by EFE further confirms the consistency of general relativity, in that vacuum and non-vacuum solutions agree for regions surrounding a central mass. Thus, solar system tests confirm not just the vacuum equations, where Tμν=0, but also the massive equations, where Tμν≠0, insofar as a thin shell serves as well as a point mass for modeling a star or planet.

Regarding time dilation, it is significant that the interior metric g00int is a constant not equal to unity, while the exterior metric g00ext asymptotically approaches unity, indicating clocks inside the shell run at different rates than those at infinity. For so-called non-phantom matter, which has an EoS p(r)=wρ(r) with w>─1, we note that g00int<1, indicating time inside the shell is dilated with respect to infinity. This result may seem at odds with occasional claims that time does not dilate inside an empty shell. Such claims may arise from piecewise solutions and are often based on two arguments: 1) Minkowski spacetime, with g00=1, prevails inside a hollow shell; or 2) according to Birkhoff’s theorem, the Schwarzschild metric governs the vacuum in an empty shell, leading to g00=1 [51]. These arguments, however, depend on a rescaling of the time coordinate inside the shell. The continuous solution method, in contrast, assumes a uniform time coordinate over the whole space domain 0≤r<∞. It is clear, nevertheless, that no apparent gravitational forces exist inside an empty shell due to the constant value of the interior metric.

For a shell composed of dust, the EoS parameter is w=0, and the interior and exterior solutions match at r=r0. Therefore g00 and the corresponding clock rates are continuous across the shell wall. The tt component for a thin dust shell thus satisfies the first Israel junction condition.

For a shell composed of stiff matter, which has an EoS of w=1, we see that g00 changes abruptly across the shell wall, allowing interior time dilation up to twice that at the outer surface. Thus the continuous solution method predicts time dilation measurements using real non-dust shells would show a violation of the Israel conditions.

It seems interesting that the interior metric g00int depends on the EoS of the shell, while the exterior metric g00ext like the Schwarzschild metric, is independent of the EoS. This curious distinction resolves the seeming paradox, mentioned in a previous paper [56], that while non-vacuum solutions to EFE require an EoS, Schwarzschild vacuum solutions do not, even though mass appears in the metric.

IV. BLACK HOLES AND BLACK SHELLS

The ultra-thin shell metric of Eqs. (9) and (11) may be applied to shells of radius equal to or less than the Schwarzschild radius, or shells such that ro≤2m0. To be called black shells, these exotic objects would generally appear to a distant observer as a Schwarzschild black hole (although unexpected singularities may occur). At close range, black shells display unique properties with respect to horizons and singularities. To compare black holes and black shells, first recall the properties of the Schwarzschild black hole with metric gSμν as given by Eq. (10):

  1. A coordinate singularity, or horizon, exists at r=2m, where gS00=0 and gS11─>─∞.
  2. Inside the horizon, squared proper time intervals 2=gS00dt2 are negative, and thus proper time is spacelike, while squared proper radial intervals dR2=gS00dr2 are positive, and proper radial distance is timelike.
  3. A physical singularity is generally assumed to exist at r=0, where gS00─>-∞ and gS11=0.
  4. There are no finite discontinuities in the domain r>0.

To compare the properties of black shells, we consider the metrics for four shell types: ordinary shells with r0>2m0; horizon black shells with r0=2m0subhorizon black shells with r0<2m0, and semi-horizon black shells with r0=m0. First, recall the interior and exterior thin shell metrics of Section III:

g00int ≈ (1 ─ 2m0/r0)(1+w)            r0 ≠ 2m0    (16a)
g00ext ≈ 1 ─ 2m0/r                           r0 ≠ 2m0    (16b)
g11int ≈ ─1                                                             (16c)
g11ext ≈ ─ (1 ─ 2m0/r) ─1.                                  (16d)

In the case of ordinary shells (r0>2m0), the Schwarzschild radius rs=2m0 lies inside the shell where the metric is constant. Thus there is no horizon at r=rs. In addition, no singularity exists at r=0. Although the metric is locally continuous everywhere, comparison of Eqs. (16c) and (16d) reveals a global discontinuity or jump across r0 in the component g11. For non-dust models, for which w≠0, there is also a jump across r0 in the component g00, in apparent violation of the Israel junction conditions. However when w=0 as in the case of dust, g00 remains unchanged across r0, in agreement with the Israel conditions.

For horizon black shells (r0=2m0), the shell radius is equal to the Schwarzschild radius rs=2m0, and as noted earlier, the approximation r─>r0 in the integrand of I(r) of Eq. (13) is no longer valid. Deriving the properties of g00 would require computing the exact integral of Eq. (12) using the spherical Gaussian. Such a calculation is not attempted here. If, however, we naively allow the approximation r─>r0 and apply Eq. (14), the apparent properties of horizon black shells suggest such objects may be nonphysical. To illustrate, recall the full equations for the metric:

g00≈(1─2m0Sc/r)|1─2m0/r0|(1+w)|1─2m0Sc/r0|─(1+w)    (17)

g11 ≈ ─ 1/(1 ─ 2m0Sc/r)                                 (18)

Setting r0=2m0 in the first equation and assuming w>─1, it is clear that g00(r)=0 for 0<r<∞. This can be seen by noting that the middle factor in g00 vanishes identically, while the right-hand factor (denominator) is nonvanishing for all finite r due to the property Sc(r)<1, and the left-hand factor is finite for all r>0. The vanishing of g00 suggests that a horizon black shell would stop all clocks in the universe, a physical impossibility and a violation of the asymptotic Minkowski condition. Whether this nonphysical result can be avoided by evaluating g00 analytically using the function G(r), by applying numerical methods, or by redefining δc in terms of a function other than G(r), is a question for future research.

Concerning the rr metric component, we see from Eq. (16c) that g11≈─1 inside the shell, implying no interior singularities exist. To check this result, note that by Eq. (18), no singularity can exist unless there is an r such that 2m0Sc(r)/r=1, or r/r0=Sc(r). Since it is always true that Sc(r)<1, any such singularity can only reside at r<r0. It will be stated without proof that since Sc(r)≈1/2 when r/r0=1, and since Sc(r) falls to zero more rapidly than r/r0, there can be no r>0 such that 2m0Sc(r)/r=1, and hence no singularity in the domain 0<r<r0. Moreover, by L’Hopital’s rule it is found that

lim[r─>0] 2m0 Sc(r)/r = 0,

ruling out a singularity at the origin. Thus a horizon black shell, unlike a Schwarzschild black hole, manifests no singularities in g11.

Subhorizon black shells (r0<2m0), in contrast, appear at close range like Schwarzschild black holes, with a horizon at r≈2m0. Subhorizon black shells also have approximate Schwarzschild behavior for r>2m0. However, a new singularity in g00 may arise due to the vanishing of |1─2m0Sc/r0| in the right-hand factor (denominator) of Eq. (17). To locate this singularity, recall that Sc(r) increases monotonically over the range 0<Sc<1. Thus g00 becomes singular at some unique r such that Sc(r)=r0/2m0. Since Sc(r) traverses nearly all of its range within a distance of r0, such singularities usually fall within r0─nε<r<r0+nε, or in the transient layer of the wall itself. However if r0=2m0─ς, where ς is some extremely small quantity, a singularity may occur at some large radius r=R0 where Sc(R0)=r0/2m0≈1. This means subhorizon black shells could in principle cause singularities in g00 at cosmological distances. Such models may have astrophysical applications related to the composition of galactic cores (the topic of Section V), or cosmological interpretations with respect to Hubble redshift, bubble universes or spherical domain walls, to be addressed in a later paper.

In the unique case of a semi-horizon black shell for which the radius r0=m0 is half the Schwarzschild radius, one might expect a singularity in g00(r) at r=r0, where Sc(r)≈1/2. However, it turns out that g00(r) has a finite discontinuity rather than a singularity at r=r0. This can be shown as follows. Setting w=0 and r0=m0, Eq. (17) simplifies to

g00 (r) ≈ [1─2r0Sc(r)/r] / |1─2Sc(r)|,

which, as r tends to r0, approaches the improper limit 0/0. Applying L’Hopital’s rule yields the ratio H of the derivatives of numerator and denominator:

H = ∂r [1─2r0Sc/r] / ∂r |1─2Sc|

              = (2r0Sc/r2 ─ 2r0δc/r) / (─ ⁄ + 2δc)

= ─ ⁄ + (r0/r) (Sc/rδc ─ 1)

where the sign ambiguity springs from the absolute value. Taking the limit r─>r0, the term Sc/rδc approaches π1/2ε/2r0<<1, and H tends to positive or negative unity, with the positive case corresponding to approach from r>r0 and the negative to r<r0. Thus for r0=m0, the limit is not unique, leaving g00 undefined at r=r0. Whether the semi-horizon black shell discontinuity arises as an artifact of the approximation is not known.

V. BLACK SHELLS, MOND, AND THE GALACTIC ROTATION CURVE

Can supermassive black shells in the cores of galaxies explain the discrepancy in the galactic rotation curve? If so, it would obviate the need for postulating a dark matter halo. The discrepancy in orbital velocity v(r), as noted earlier, arises from observations of differential Doppler shift, which indicate the outer stars and hydrogen clouds of galaxies orbit too fast to be gravitationally bound by luminous or baryonic matter alone. Thus, outside the bright galactic disk, v(r) does not fall off as r─1/2, as would be expected from Newtonian dynamics [57], but tends toward a constant as r increases. This anomaly was noted by Fritz Zwicky in 1933 [58] and first quantified observationally by Vera Rubin [59].

The flattening of the galactic rotation curve can be described by an effective potential φm(r) that depends only baryonic mass and increases with r at large distances. The potential φm, for reasons evident below, will be called the MOND potential. The goal is to show that a subhorizon black shell (SBS), or similar exotic black object, located in the galactic core, could theoretically account for the observed excess orbital velocities, or equivalently, that an SBS potential φSBS(r) can be made consistent with the MOND potential φm(r) in outlying regions. The MOND potential will be derived first, followed by the SBS potential. The two will then be equated to show, by a redefinition of the broadened delta function, a close correspondence in the metrics.

The MOND potential can be calculated from Modified Newtonian Dynamics (MOND), a formalism developed in 1983 by Mordehai Milgrom [60] to account for the discrepancy in the rotation velocity of galaxies. Although the excess velocity is usually attributed to the presence of an unseen dark matter halo, the MOND formalism, relying on baryonic matter alone, has proven accurate in predicting orbital motion [61], and thus provides a means for testing theories.

The MOND formalism is based on the empirical relation

μ(a/a0)a = aN                                           (19)

which connects observed radial acceleration a to predicted Newtonian acceleration aN=GM/r2 using an interpolating function μ(a/a0), where

a0 = 1.2×10─8cm/sec2 ≅ H0/2π = c2/R = c2/(Λ/3)−1/2

is a universal constant with dimensions of acceleration, H0 is the Hubble parameter [62], and R is roughly times the radius of the visible universe or the de Sitter radius corresponding to cosmological constant Λ [63]. The interpolating function runs smoothly from the inner galaxy, where the field falls off as roughly 1/r2, to the region outside the bright galactic disk, called the deep MOND region, where the field tends to fall off as 1/r. Using the simple interpolating function

μ(a/a0) = (a/a0) / (1+a/a0)

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

a = ─ (GM/2r2) [1 + √(1+4r2/Rm2)].                   (20)

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

Rm = √(GMR/c2) = √(GM/a0).

In the domain of interest 2Rm<r<<R, which is roughly the region outside the luminous disk, the observed radial acceleration a of Eq. (20) can be approximated as

a ≅ ─ GM/2r2 ─ GM/Rmr.                           (21)

The potential in this domain can be expressed as

φm = ─ ∫a(r) dr ≅ ─ GM/2r + (GM/Rm) ln (r/Rm).

The factor 1/2 in the first term on the right does not appear in some presentations of MOND, where different interpolating functions apply and where the potential covers all space [66]. However, since the second term increases with r and becomes dominant near Rm, we can neglect the first term and construct an effective metric for the deep MOND region [67]

g00 ≅ 1 + 2φm/c2 ≅ 1 + (2GM/c2Rm) ln (r/Rm),          (22)

which is accurate in the domain nRm<r<<R, for n a small integer on the order of 4 or 5. Note that g00─>∞ as r─>∞. Hence the effective metric violates the asymptotic Minkowski condition and cannot, in the form of Eq. (22), be consistent with a black shell metric. Consistency will be attained through a later approximation.

Next, to calculate the SBS potential φSBS(r), we assume the galaxy is centered on a supermassive ultra-thin SBS of radius

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

where ς>ε is a small distance on the order of meters, rs is the Schwarzschild radius 2m0, shell mass m0 is a large fraction of galactic mass M, and parameter σ=ς/rs measures the small difference between shell size and Schwarzschild radius. Such an SBS would induce a singularity in g00 at some cosmic-scale radius R0, at which clocks would theoretically run at an infinite rate. In realistic scenarios, no remote singularity can occur due to disturbance of the mass density by other fields. Nevertheless, a remote virtual singularity implies a modification of the field in the neighborhood of the galaxy.

The distance to the singularity at R0 is inversely related to ς and increases with step width 2nε. More specifically, from Eq. (17), R0 must satisfy

1 ─ 2m0Sc(R0)/r0 = 0,

or, upon substiting 2m0=r0 and rearranging,

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

To calculate the impact of the distant singularity on the field in the galactic neighborhood, we start by introducing a new function η(r) and expressing the broadened step function as Sc(r)=1─η(r), where η(r)<<1 in the deep MOND region. This and Eq. (23)  are then substituted into the thin shell metric of Eq. (17), and the result is evaluated for the shell’s far exterior r0<<r<R0, yielding

g00SBS ≈ [1 ─ rs(1 ─ η)/r] σ/ |η(r) ─ σ|

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

From Eq. (24), we see that η(R0)≅σ=ς/rs, and the denominator of g00SBS vanishes near R0 as expected.

To match g00SBS to the MOND metric of Eq. (22), we first write a an approximation to the latter which repositions the singularity at infinity to a remote finite distance r=R0 as follows:

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

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

a ≅ ─ φm ≅ ─ GM/Rmr ─ GM/Rm(R0─r).

It is clear that for r in the neighborhood of the galaxy, (R0─r) is large enough that the right-hand term can be neglected. The remaining term matches the MOND acceleration of Eq. (21). Hence we see that g00MOND of Eq. (26) adequately approximates the MOND metric in the deep MOND region.

The MOND and SBS metrics may now be equated, giving

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

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

Sm(r) = 1 ─ η(r) = 1 ─ σ / [1 + (rs/Rm) ln |r/(R0─r)|]  ─ σ.

Simple calculation shows that Sm(r), while different from the broadened step function Sc(r) derived in Section II, has like properties in the domain r0<<r<R0. To wit, Sm(r) is slightly less than one and increases monotonically to the near-unity value 1─σ as r approaches the near-infinite distance R0. Thus, it is possible to derive a MOND-compatible step function Sm(r) by replacing the Gaussian G(r) with some appropriate function F(r) in the definition of the broadened delta function δc, thus obtaining a new delta function δm. An SBS modeled on δm, embedded in the galactic core, would then account for the anomalous orbital velocities. The exact function F(r) is unknown. Questions also remain about SBS formation and stability. What is important is the implication that an exotic black object, possibly a subhorizon black shell, could in principle cause the observed galactic rotation curve without the need for a dark matter halo.

VI. CONCENTRIC SHELLS AND DISCRETE DENSITY SAMPLING

The continuous solution method is easily generalized to n concentric shells of arbitrary mass and radius. This technique provides a formalism for solving EFE for any continuous static spherical density distribution ρ(r), where ρ(r) is modeled by a discrete sampling at r={r0, r1 … rn-1}.  The method for concentric shell solutions will be illustrated for the simple case of two shells with EoS parameter w=0. Assuming surface densities μ0 and μ1, radii r0 and r1, and masses m0=4πμ0r02 and m1=4πμ1r12, the mass density can be expressed in terms of broadened delta functions as

ρ(r) = μ0δ0 + μ1δ1

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

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

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

Upon integration, the double thin-shell solution becomes

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

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

g11int  ≈ ─1                                                            (28a)
g11mid ≈ ─ (1 ─ 2m0/r) ─1                                  (28b)
g11ext  ≈ ─ [1 ─ 2(m0 + m1)/r] ─1,                    (28c)

displaying Minkowski properties inside the smaller shell, Schwarzschild behavior between shells, and combined Schwarzschild behavior outside the larger shell.

To solve for the time component g00 = eν, the method of Section III will be applied. From Eqs. (11) and (27), we have

    ν = ─ λ + k1 ─ κ∫r dr ρ(r) eλ r

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

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

I(r) = μ0rdr δ0r / [1─ 2(m0S0+m1S1)/r]
           + μ1rdr δ1r / [1─ 2(m0S0+m1S1)/r].

Since r is slowly varying over the two transient layers, it can be approximated by r0 and r1 in the two respective integrands, yielding

I(r) ≈ μ0r0rdr δ0 / [1─ 2(m0S0+m1S1)/r0]
         + μ1r1rdr δ1 / [1─ 2(m0S0+m1S1)/r1]

Note that in the first integral, the outer step function S1(r) varies slowly over the nonzero domain of the inner delta function δ0, and hence may be set to a constant S1≈0. Analogously, in the second integral, S0(r) varies slowly over the nonzero domain of δ1 and may be set to a constant S0≈1. The total integral then simplifies to

I(r) ≈ μ0r0rdr δ0 / (1─ 2m0S0/r0)        
                       + μ1r1rdr δ1 / (1─ 2m0/r1 ─ 2m1S1/r1).

Following the method of Section III, a change of variable from r to S0(r) and S1(r) in the respective integrals gives, upon integration,

I(r) ≈ (μ0r02/2m0) ln|1─ 2m0S0/r0|          
                  + (μ1r12/2m1) ln|1─ 2m0/r0 ─ 2m1S1/r1|.

Multiplying I(r) by κ and substituting back into Eq. (29) then yields

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

and hence

g00 = eν = [1 ─ 2(m0S0+m1S1)/r]  ek|1─2m0S0/r0|─1

X  |1 ─ 2m0/r0 ─ 2m1S1/r1|─1                           (30)

where X denotes multiplication. Again, to meet the asymptotic Minkowski condition, the integration constant must be

ek1 = |1 ─ 2m0/r0| |1 ─ 2m0/r0 ─ 2m1/r1|             (31)

The constant ek1 is then substituted back into Eq. (30), yielding the g00 component of the double concentric shell metric. Taken together, Eqs. (27), (30) and (31) represent a complete continuous asymptotically exact solution to EFE for two concentric ultra-thin dust shells of arbitrary mass and radius.

It is straightforward to extend this result to n concentric shells of mass mi, radius ri, and thickness εi, as long as εi<<(ri+1─ri). Such a set of locally continuous thin shells may be viewed as a discrete sampling, at arbitrary radii ri, of a globally continuous mass density distribution ρ(r). The concentric shell formalism thus provides a discrete method for approximating the solution to EFE for any static, spherically symmetric mass-energy density. Hence Einstein’s equations can be readily solved for complicated scenarios such as a star surrounded by spherical dust clouds embedded in cosmic bubbles, and so forth. The impact of discreteness is a topic for future discussion.

VII. CONCLUSION

We have derived an asymptotically exact solution to Einstein’s field equations for individual and multiple concentric ultra-thin shells of arbitrary mass and radius using a continuous solution method that does not require junction conditions. The single shell solution is given by Eqs. (9) and (14), and the double shell solution by Eqs. (27), (30) and (31). These solutions are fixed by two boundary conditions: asymptotic flatness at infinity and non-singularity at the origin. The interior of a thin shell is found to manifest no effective gravitational forces. However, interior clocks run at different rates from those at infinity. For non-phantom matter (w>─1), time in the interior of the shell is dilated with respect to infinity, while for phantom matter, time is contracted.

Exterior to the shell, the field generally matches that of the Schwarzschild metric. Exceptions are found for black shells, i.e. shells of radius less than or equal to the Schwarzschild radius. The method breaks down for equal radii, and an asymptotically exact solution was not attempted. However, approximations suggest such objects may be unphysical. Subhorizon black shells, which have a radius smaller than the Schwarzschild radius, are more easily analyzed, and were shown in general to appear as Schwarzschild black holes everywhere outside the shell. This holds with one key exception. When the radius of a supermassive black shell is less than its Schwarzschild radius by a very small distance on the order of meters, a singularity may occur in the time component of the metric at cosmological distances. It was then shown that this singular metric approximates an effective MOND metric, where the latter is expressed in terms of an effective potential that accounts for the observed galactic orbital velocities. Thus, a supermassive subhorizon ultra-thin black shell or similar exotic black object, located at the center of a galaxy, could theoretically explain the flattening of the galactic rotation curve without the need for dark matter.

It was also shown that the solution for a series of concentric shells provides a discrete sampling method for calculating the approximate gravitational field of any spherical static mass distribution. Applications might include detailed scenarios such as spherical accretion shells around black holes embedded in a constant background density enclosed by a cosmic bubble.

The method developed here applies to static scenarios. It can in principle be generalized to dynamic configurations such as colliding shells in anti-deSitter spacetime [68] or black holes embedded in expanding bubble universes described by the Friedman-Robertson-Walker metric. These are topics for future research. Other questions also remain concerning

  1. Multiple concentric shell techniques for discrete sampling of cosmological mass distributions,
  2. The impact of discreteness on accuracy,
  3. Comparison of ultra-thin shell boundary properties to Israel junction conditions under a general EoS,
  4. Collapsing ultra-thin shells and black shell formation,
  5. Whether possible nonphysical features of horizon black shells interfere with black shell formation,
  6. The nature and stability of rotating or charged ultra-thin shells,
  7. Stability of ultra-thin shells, particularly of subhorizon black shells in galactic cores, and
  8. The mathematical properties of functions F(r) and δm compatible with MOND and the galactic rotation curve.

APPENDIX

Using the line element

ds2 = g00(r) dt2 + g11(r) dr2 ─ r22
= eν dt2 ─ eλ dr2 ─ r22

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

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

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

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

κρr2 + 1 = e─λ (1 ─ λr)
= (re─λ).

Integration then yields

re─λ = k0 +  ∫r dr (κρr2 + 1) .

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

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

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

eλ = (1 + κμ0r02Sc/r + k0/r) ─ 1.

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

eλ = (1 ─ 2m0Sc/r + k0/r) ─ 1.                             (a3)

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

g11 = ─ eλ = ─ (1 ─ 2m0Sc/r) ─ 1.                      (a4)

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

κ(ρ+p) = ─ e─λ λ/r ─ e─λ ν/r.

Solving for ν yields

ν = ─ λ ─ κ(ρ+p) eλ r.

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

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

Upon integrating, this becomes

ν = ─ λ + k1 ─ κ(1+w)μ0r dr δcr / (1 ─ 2m0Sc/r)        (a5)

with k1 a constant of integration. Eq. (a5) represents an exact solution to Einstein’s field equations for the tt metric component g00=eν of an ultra-thin shell. To approximate the integral, we use the properties of the broadened step and delta functions. The integral may be written

I(r) = ∫0r dr δc r/(1 ─ 2m0Sc/r).

Since r changes by the near infinitesimal amount 2nε across the transient layer, it may be treated as a constant r≈r0, hence

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

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

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

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

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

Evaluating the constants κ and μ0, this simplifies to

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

with the result

eν ≈ e─λ ek1 |1 ─ 2m0Sc/r0|─ (1+w)

≈ (1 ─ 2m0Sc/r) ek1 |1 ─ 2m0Sc/r0|─ (1+w).

Since eν must obey the Minkowski condition eν─>1 as r─>∞, the integration constant ek1 must cancel the right-hand factor in the outer region where Sc≈1,. Hence the integration constant is

ek1 = |1 ─ 2m0/r0| (1+w)

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

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

r0 ≠ 2m0.

_____________

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Current conflicts in general relativity: Is Einstein’s theory incomplete?

By Kathleen A. Rosser
Kathleen.A.Rosser@ieee.org

Version 2
September 20, 2019
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[Also available at ResearchGate and viXra]

ABSTRACT: A review of refutations of general relativity commonly found in today’s literature is presented, with comments on the status of Einstein’s theory and brief analyses of the arguments for modified gravity. Topics include dark matter and the galactic rotation curve, dark energy and cosmic acceleration, completeness and the equation of state, the speed of gravity, the singularity problem, redshift, gravitational time dilation, localized energy, and the gravitational potential. It is conjectured that the contemporary formalism of general relativity offers an incomplete description of gravitational effects, which may be the most compelling reason for seeking new theories of gravity.

I. INTRODUCTION

Researchers both inside and outside the established physics community are currently questioning the theory of General Relativity (GR) for a number of reasons. The present review article is intended to catalogue some of these objections and lend perspective on their validity. It is hoped this effort will help reduce the growing confusion that has permeated the literature at all levels, from strict peer-reviewed journals, to publications with little or no peer review, technical books, educational websites, physics forums, and unpublished communications. Also proposed here is the hypothesis that incompleteness is the most critical flaw in the current general relativistic formalism, along with the conjecture that for some physical systems, GR offers no independent information about such observables as gravitational redshift and time dilation.

A list of the common reasons for refuting GR is presented below. These topics will be discussed in detail in later sections.

(Click to enlarge.) Rotation curve of spiral galaxy M 33 (yellow and blue points), and curve predicted from distribution visible matter (white line). The discrepancy can be accounted for by adding a dark matter halo around the galaxy. [Wikipedia, Galaxy rotation curve]
1) Galactic rotation curve (dark matter): Many physicists and astronomers believe that general relativity fails to explain the unexpectedly rapid orbital motion of the outer regions of galaxies except through the introduction of dark matter, a supposed non-radiating transparent material that has never been directly observed astronomically, nor verified to exist in particle accelerators, despite over half a century of searching.

2) Cosmic acceleration (dark energy): GR does not explain the apparent increasing expansion rate of the universe without the reintroduction of Einstein’s abandoned cosmological constant Λ, which must be fine-tuned in a seemingly improbable way, or the postulation of some form of phantom pressure called dark energy.

3) Incompleteness: Einstein’s field equations are possibly incomplete in that the gravitational mass-energy density ρ(xμ), which presumably comprises the source of the field, does not uniquely determine the metric, or equivalently, does not fully determine the geometry of spacetime, unless one selects an often ad hoc equation of state. Thus ρ(xμ) does not define such observables as time dilation, redshift, and certain properties of motion, except in special cases, which points to an inconsistency in the theory.

4) Speed of gravity: GR predicts that gravitational effects travel at the speed of light. However many independent researchers, as well as mainstream modified gravity theorists, postulate that the effects of gravity travel at higher or lower speeds.

5) Time dilation: Some researchers deny that time dilation, as predicted by GR, actually exists, asserting that redshift, which is often cited as proof of time dilation, is due to other causes such as motion of the photon through a potential.

6) Spacetime curvature: Some theorists doubt that the curvature of spacetime is the cause of gravitational effects, or even that 4-dimensional spacetime itself has physical meaning.

7) Energy: GR does not offer a definition of the localized energy of the field, which some researchers consider a flaw in the theory.

8) The singularity problem: The GR formalism leads to coordinate singularities as well to real singularities in the mass density. Yet the formalism is believed to break down at singularities, pointing to a contradiction.

This paper is organized as follows: In Section II, an overview of how the GR formalism is derived and applied will be presented. Sections III through X offer discussion of each of the refutations listed above. Section XI is a brief conclusion summarizing those objections to GR that may be the most valid.

II. PERSPECTIVES ON THE GENERAL RELATIVISTIC FORMALISM

General relativity, due to the subtlety and complexity of the mathematics, may rival only quantum mechanics as one of the most confusing theories ever developed. As a result, GR is sometimes improperly taught. Textbook authors and professors often rely on plausibility arguments rather than emphasizing the mathematical formalism. Plausibility arguments are however usually approximations and can be misleading. Heuristic analogies may compound the confusion and delay the tackling of Einstein’s field equations, which many graduate physics students never learn to solve.

To understand GR, one must grasp that it is one and only one thing: a theory of geometry. Whether GR is correct or not is another topic. But if one wishes to apply GR, either as a practical formalism or as a tentative description, it is necessary to realize that geometry is its total content. The geometry resulting from any specific mass, energy, momentum and pressure distribution in spacetime is uniquely and exhaustively described by the line element ds, which is the 4-dimensional differential distance along a path through space and time. The line element is constructed from the product of the metric gμν, which contains curvature information, and the differentials dxμ of the coordinates, where μ normally ranges from 0 to 3, with 0 corresponding to time, and 1 to 3 to the space coordinates. The line element is usually written in squared form as ds2=gμνdxμdxν, with repeated indices indicating summation from 0 to 3.

The computational pipeline of the general relativistic formalism for orthogonal energy-momentum tensors

Tμν = diag [ρ(u0)2, ρ(u1)2, ρ(u2)2, ρ(u3)2]

with 4-velocity uμ, is straightforward. One must first select a coordinate system for the spacetime region to be studied. Next, the mass-energy density ρ and the 4-velocity uμ a as a function of the coordinates must be specified for the region. The quantity ρuμuν is then substituted into the energy-momentum tensor Tμν on the right hand side of Einstein’s Field Equations (EFE):

Rμν − (1/2)gμνR = κTμν

where Rμν is the Ricci tensor and depends on derivatives of the metric, gμν is the metric to be solved for, R the scalar curvature obtained by contracting the Ricci tensor, and κ=-8πG/c2 is a constant. For static configurations, in which ui=0,  one must specify the pressure pi, which is determined by a selected equation of state that relates mass-energy density to pressure, and substitute the resulting quantity into the spatial diagonal elements Tii. The field equations are then solved to obtain the metric and hence the line element ds. Physical observables such as redshift, time dilation, and the motion of photons and test bodies are then calculated from the line element, which is proportional to the particle Lagrangian [1]. Thus, with the application of the Euler-Lagrange equation, the metric yields all test particle trajectories. (These are often calculated in a more general way using the geodesic equation, which can be derived by applying the Euler-Lagrange equation to the general line element.)

It is important to note that in GR, none of the physical observables are to be calculated from Newtonian quantities such as gravitational force or potential. Newtonian mechanics may provide guidelines for constructing the elements of the energy-momentum tensor, or boundary conditions on the solutions to EFE, but the concepts of force or potential play a role in plausibility arguments only. Indeed, Albert Einstein, in his original paper Cosmological Considerations in the General Theory of Relativity (1917) [2], used the Newtonian potential φ, along with a modified version of Laplace’s equation

2φ-λφ=4πGρ

to argue the plausibility of his relativistic field equations, in which the derivatives of φ are represented by curvature and mass density ρ by the energy momentum tensor [3].

One reason gravitational potential so often arises in heuristic arguments is that, for many spacetime geometries, the metric has terms proportional to the classical gravitational potential Gm/r. These potential-like terms emerge from solving EFE, however, and are not put in by hand. More specifically, while the dependence on mass m, usually entered as an integration constant, is borrowed from Newton’s law of gravity, the inverse dependence on coordinate r is not, as can be seen from Dirac’s derivation of the Schwarzschild solution [4]. Moreover, no concept of potential need be assumed in the derivation of Einstein’s equations. The only concept that must be assumed is that the energy, mass, momentum and pressure densities determine spacetime curvature, which in turn governs gravitational effects.

The above cautionary note is emphasized here because plausibility arguments, often based on gravitational force or potential, are frequently presented in textbooks [5,6] and on-line sources [7.8], as well as by independent researchers [9]. .For instance, Robert M. Wald in his scholarly text General Relativity, discusses for heuristic purposes the problem of measuring gravitational forces in the context of GR. Yet in the rigorous GR framework, such so-called forces do not exist. It would therefore be inappropriate to attempt to measure them, a fact that is not made clear. [10]. Further instances are found in James B. Hartle’s textbook Gravity, An Introduction to Einstein’s General Relativity, in which he says, “What is the difference between the rates at which signals are emitted and received at two different gravitational potentials?” [5]. Hartle continues by analyzing the effects of gravitational potential on clock rates. Yet the quantity called gravitational potential is not intrinsic to the formalism of general relativity, and to allude to it is misleading, as will be shown in Section VII. Similarly, Steven Weinberg, in his text Gravitation and Cosmology, uses a plausibility argument based on gravitational force to derive the general relativistic equation of motion for a freely falling body [6]. Later however, he discusses gravitational potential more accurately in the framework of the post-Newtonian approximation, making the Newtonian nature of the quantity unambiguous [11].

Other misleading plausibility arguments are found in the clearly written critique by Miles Mathis entitled The Speed of Gravity [12]. Mathis states, “The strong form [of the equivalence principle] says that gravity and acceleration are the same thing. [Therefore] asking what is the speed of gravity makes no sense [because] like acceleration, gravity is not a force, it is a motion.” What Mathis may be overlooking is the fact that spacetime curvature, not acceleration, constitutes the fundamental nature of gravity in GR. While it is true that test bodies accelerate in a gravitational field, and that accelerated reference frames mimic certain gravitational effects, it is also true that gravity can exist without acceleration, such as near an isolated black hole where no test bodies are present. Conversely, acceleration can exist without gravity, such as in a centrifuge rotating in free space. In view of these counterexamples, it is clear gravity is equivalent not to acceleration but to curvature. And it does after all make sense to ask at what speed changes in curvature propagate. Mathis seems further misled when he later claims that spatial curvature does not describe linear acceleration from rest. Indeed, spatial curvature does not, but spacetime curvature does. It is the time component of the metric that is important.

In the following sections, I will offer impressions of why the eight refutations of GR noted above arise and whether they are valid objections.

III. GALACTIC ROTATION CURVE (DARK MATTER)

A large body of precise galactic redshift data tabulated over the last century has shown that the outer stars and hydrogen clouds of galaxies orbit too fast to be explained by Newtonian gravitational attraction of visible or baryonic matter alone. The pattern of orbital velocities, called the galactic rotation curve, remains one of the most important unsolved problems in astrophysics. The data are extensive, accurate, and independent of any specific theory, yet the solution has remained mysterious for many decades. (See full historical summary at Ref. [13].)

(Click to enlarge.) Giant gravitational lens formed by large cluster of galaxies RCS2 032727 -132623. The gravitational field in which the (mostly yellow) cluster moves produces multiple, highly sheared images of a background (blue) galaxy (with redshift z = 1.7). The radius of the blue arcs is ~500,000 light years. Observations such as this are used to map dark matter. [–ScienceMag.org – Image at source courtesy: J. Rigby, K. Sharon, M. Gladders, and E. Wuyts]
Astronomers and physicists are somewhat divided on the issue of the galactic rotation curve anomaly. Astronomers generally accept the hypothesis that Dark Matter (DM), which supposedly comprises the majority of galactic material, fully explains the extra orbital velocity. Their research goals, however, are largely observational, and the DM hypothesis simplifies their theoretical framework. On the other hand, a significant minority of mainstream physicists doubt that DM exists [14]. This is because, after decades of theoretical, observational and experimental research seeking any type of particle or energy that exhibits the properties of dark matter, no direct evidence for this exotic substance has been found [15]. Astronomers might disagree, pointing to phenomena such as the gravitational lensing of light from distant objects by supposed excess matter in intervening galaxies [16]. (For extensive summary with images see Ref. [17].) But these arguments are theory dependent, and the observational data are less precise and abundant. Such arguments also do not take into account the possibly significant nonlinear effects that arise from a full general relativistic treatment [18].

Most astronomers believe the DM hypothesis is entirely compatible with GR. Thus by and large they uphold general relativity as the best theory of gravity. On the other hand (although some researchers disagree, as noted below), it is commonly assumed that if DM does not exist, a modified theory of gravity is needed to explain the galactic rotation curve. Another motivation for modifying gravity is the fact that galaxies show a surprising uniformity in their would-be DM distributions, as manifest in the universal constant a0 = 1.2 x 10-8 cm/sec2, which accurately specifies for most spiral galaxies the centripetal acceleration at that radius where the excess velocity becomes dominant. This suggests that the rotation anomaly is not due to invisible matter, which should vary from galaxy to galaxy, but to an extra gravitational attraction beyond that predicted by GR. This idea has given rise to a number of modified gravity theories, including Chameleon Bigravity [15], and Modified Newtonian Dynamics (MOND) [19-21].

A few theorists argue that if DM did not exist, it would still not be necessary to modify gravity, as the rotation curve is adequately described by a full general relativistic treatment. This argument refutes the common belief that general relativistic corrections to the galactic rotation curve are insignificant due to the non-relativistic velocities and weak fields of galaxies. This belief, added to the intractable nature of the dynamical formalism, has led most researchers to dismiss the need for applying EFE to galactic orbital motion. One exception is Fred L. Cooperstock, whose calculations show that the unexpected nonlinear effects of GR may account for most of the excess orbital velocity, and that only a small amount of unseen matter is needed to make up the difference [22]. This invisible substance could be ordinary non-radiating matter, rather than the exotic variety called dark matter.

If Cooperstock’s solution is correct, the galactic rotation curve would support rather than contradict GR, and the orbital motion of galaxies would no longer provide a compelling reason for modifying gravity. Furthermore, were Cooperstock’s results widely acknowledged, it would render moot the search for exotic dark matter. A full analysis of Cooperstock’s derivation, in which he solves EFE for a fluid disk using a cylindrical co-rotating coordinate system, would be required to settle the matter. Articles have appeared disputing Cooperstock’s results [23,24]. But the authors fail to rigorously analyze Cooperstock’s calculations, and instead attack his simplified galactic model, or the fact that he has ignored other supposed evidence for DM such as that found in galactic cluster data. The question of whether there is a need for exotic DM or modified theories of gravity to account for galactic motion thus remains open.

IV. COSMIC ACCELERATION AND DARK ENERGY

One commonly noted problem with GR is that it does not explain the apparent increasing expansion rate of the universe without the reintroduction of Einstein’s abandoned cosmological constant Λ, or without the postulation of some form of phantom pressure called dark energy [25]. To offer brief background, the idea that the cosmos is expanding is based on the big bang theory, a cornerstone of the standard or ΛCDM model of cosmology. This theory is governed by the Friedman-Lemaitre-Robertson-Walker (FLRW) metric, which for spherical co-moving coordinates in flat spacetime is written:

ds2 = dt2 – a(t)(dr2+ r22+ r2sin2Θdφ2)

Using a metric of the above form, Einstein’s field equations reduce to the following two simultaneous equations in terms of the time-dependent scale factor a(t):

a•2/a2 = 8πGρ/3

2a••/a + a•2/a2 = -8πp

where overdots mean derivatives with respect to the time coordinate t [26]. The first of these equations is called the Friedman equation. Note that a(t), which is critical in that it defines the cosmic expansion rate, is determined not just by mass density ρ, but also by pressure density p, which is fixed by an auxiliary equation of state specifying p as a function of ρ. Using the standard forms of a(t), which increase monotonically with time, FLRW predicts that redshift increases with distance for unbound galaxies beyond our local cluster. This redshift is considered to arise not at the galaxies themselves, which it would if it were a Doppler effect, but in the expanding space as photons traverse the cosmos on their way to the observer.

Assuming the universe is expanding, Supernovae Type 1a redshift versus distance data, among other evidence, suggest that the cosmic expansion rate is accelerating in the present epoch [27]. Preliminary GR calculations however predict the expansion should decelerate. This discrepancy is often resolved in one of two related ways. The first is the Dark Energy (DE) hypothesis. According to this, some unknown energy source, possibly related to the vacuum, pushes the universe apart. The existence of DE, however, seems implausible to many researchers. This phantom energy not only has a negative sign for pressure, it supposedly makes up most of the energy in the universe [28], despite that it has never been independently observed [29]. Thus, many astrophysicists propose instead the introduction of a cosmological constant Λ, which serves the same purpose. The cosmological constant is an ad hoc coefficient that can be put into Einstein’s field equations, and was first introduced by Einstein himself to counteract gravitational collapse in a universe he believed to be static. The constant was abandoned when the big bang theory obviated the need for cosmic repulsion, and was later reintroduced to account for cosmic acceleration. However, to match observation, Λ must be fine-tuned in a way that seems improbable [30-33]. Another problem relates to the odd coincidence that energy densities due to the cosmological constant and to matter are nearly the same in the present era [34]. Many researchers therefore reject the Λ and dark energy hypotheses.

Cosmic acceleration is arguably the phenomenon most frequently cited in peer-reviewed literature as a motivation for modified gravity [31,35-38]. Such theories are often published in mainstream journals, indicating the physics community provisionally accepts that modified gravity is relevant to current research. Among these theories are Horndeski-type scalar tensor models such as the Brans-Dicke theory [39], Born-Infeld gravity [40], Galileon theories, Gauss-Bonnet theories [41,42], f(R) theories where R is the Ricci scalar, such as the Starobinsky model [35,43,44], f(R,Q) gravity where Q is square of the Ricci tensor [45], unimodular f(R,T) gravity, where T the trace of the energy momentum tensor Tμν [46-48], and a recently proposed local antigravity model [49]. For discussions of modified gravities, see Refs [50,51].

But is cosmic acceleration really a valid reason for modifying or rejecting the well-tested theory of GR? Arguably not. First of all, astronomical evidence for cosmic acceleration is inconclusive. Analysis of the redshift data entails fitting a set of ideal curves to a comparatively small number of data points, where the curves to be fitted are close together relative to the size of the error bars. The data itself, moreover, is accurate only insofar as Supernovae Type Ia radiate as true standard candles, a question currently being debated in peer-reviewed journals [31]. Secondly, the interpretation of the redshift data is theory dependent. Modified gravities and alternate cosmologies suggest possible scenarios in which acceleration does not exist [28,52]. R. Monjo for example proposes an inhomogeneous cosmological metric with linear rather than accelerated expansion that fits SNIa data as well as the standard model [53]. Other researchers also note that apparent cosmic acceleration arises due to the assumption of a homogeneous universe. Hua Kai-Deng and Hao Wei say, “If the cosmological principle can be relaxed, it is possible to explain the apparent cosmic acceleration … without invoking dark energy or modified gravity. For instance, giving up the cosmic homogeneity, it is reasonable to imagine we are living in a locally underdense void.” [54] What is more, cosmic acceleration only makes sense in the context of an expanding universe, whose dynamics is usually assumed to be governed by the FLRW metric, itself a cornerstone of GR. Thus any such refutation of GR assumes GR at least in part, which may seem inconsistent.

Modified gravity theories have had some success in accounting for cosmic acceleration. However, insofar as observational evidence for accelerated expansion seems inconclusive, and can possibly be accounted for by alternate theories of cosmology, the apparent increase in universal expansion rate may not provide sufficient reason to modify or replace GR.

V. INCOMPLETENESS

Einstein’s field equations can be interpreted as incomplete in that mass-energy density ρ, presumably the source of gravity, does not uniquely determine all the components of the metric. For example, in the general spherical static non-vacuum case, ρ determines the r component g11 but not the t component g00. This can be seen by examining Einstein’s field equations for a static spherical non-zero mass distribution, which reduce to the simultaneous equations:

κT00 = κρ(r) = ─ (1/g11 + 1)/r2 + g11′/g112r

κT11 = ─κp(r) = ─ (1/g11 + 1)/r2 ─ g00′/g00g11r

where the index on T has been lowered by multiplication with  gμν, and primes denote differentiation with respect to r. It is clear from the first equation that g11 is fully determined by mass-energy density ρ(r). To solve for g00 however, an auxiliary Equation of State (EoS) relating mass-energy density ρ to pressure density p is needed. In general situations, the EoS as a practical matter is often chosen ad hoc. A commonly used EoS is p=wρ where w is a coefficient often set to 1 or 0. The coefficient w can also be negative, as is assumed in descriptions of dark energy, although this may seem unphysical [55]. Moreover, the EoS can in general vary with space and time. Indeed, in the peer-reviewed literature, models using an EoS of seeming unlimited complexity are assumed to be in principle rigorous [56-58]. This leads to the inconvenient circumstance that in many instances the EoS yields more information about gravitational effects than do Einstein’s equations themselves. In fact, almost any desired gravitational effect can be manufactured by tailoring the EoS, and since the EoS is derived not from gravitation theory but from the separate discipline of thermodynamics, this leads to the conjecture that EFE, and thus GR, provide no independent information at all about certain measurable gravitational effects. In particular, Einstein’s field equations provide no information about redshift and time dilation for static spherical non-zero mass distributions. (This conjecture will be proved in a later paper.)

One seeming contradiction arising from the requirement for an EoS is that, in the case of the static spherical vacuum solution, which by the Jebsen-Birkhoff theorem is uniquely the Schwarzschild metric [43],

ds2=(1-2m/r)dt2 – (1-2m/r)-1dr2 – r2(dΘ2+sin2Θdφ2)

no equation of state is needed. The Schwarzschild metric can be derived without one, and depends only on the central mass m. At the same time, this metric, which accurately describes gravity in the vicinity of stars and planets, is the only solution to EFE that has been extensively tested in a theory-independent way. The success of the Schwarzschild metric thus implies that gravitational effects are adequately determined by mass alone. But this contradicts the formalism for the non-vacuum. Another peculiar fact is that the Schwarzschild metric has the form g00=─1/g11, as if an EoS of p=─ρ had been implicitly assumed, as can be seen by solving the two simultaneous equations at the beginning of this section. Was it? In a sense, yes, in that both ρ and p vanish for the vacuum and hence satisfy the EoS. But this is a trivial application of the EoS. More relevant is the fact that no EoS is applied to the mass m itself, which is put into the metric by hand as a constant of integration.

It may be significant that Einstein’s original static energy-momentum tensor

Tμν= diag(ρ, 0,0,0),

as defined in his paper of 1917 [2], contained mass density ρ but not pressure p. This implies that Einstein interpreted the spatial components as strictly due to momentum, which vanishes for static configurations. Such an interpretation seems reasonable to this author in that the motions comprising pressure are random rather than directional, suggesting pressure should not appear in the spatial components, but only in the mass-energy component T00. The explicit pressure terms Tii =p were first suggested to Einstein in a letter from Erwin Schroedinger (1918) as a solution to the cosmological constant problem [3], and later became an established feature of GR. The history and impact of this development is a topic for future research.

As mentioned earlier, the EoS can vary with time. In the standard model of the expanding universe, for example, the EoS is assumed to change from epoch to epoch, depending on whether space is dominated by radiation, matter or the vacuum [59]. This epoch-dependent model is called the ΛCDM model, where CDM stands for Cold Dark Matter, and Λ is the cosmological constant. It is well known that if the standard EoS is assumed, the ΛCDM model accurately accounts for most astronomical observations. Thus, ΛCDM provides a useful framework for cataloguing astronomical data. However, the important point is that EFE, and hence GR, offer only partial information about how the universe evolves through time. An additional criterion for determining the cosmic scale factor a(t) is embodied in the EoS, and this auxiliary equation is chosen either after the fact by fitting observational data to redshift versus distance curves, or by applying thermodynamics, a separate branch of physics [60]. The above example again shows that the requirement for an EoS to determine the metric implies general relativity may be deficient. Incompleteness thus seems the most compelling reason to modify GR.

Some authors have proposed a type of modified gravity, called f(T) gravity (not to be confused with torsion or teleparallel gravities sometimes also called f(T)), in which the field equations contain only functions of the trace T of the energy-momentum tensor Tμν. This obviates the need for an EoS, and may be a start toward a more complete theory of gravity.

VI. SPEED OF GRAVITY

GR is widely believed to predict that gravitational effects travel at the speed of light c. If we assume the principles of Special Relativity (SR), a formalism confirmed in arguably millions of particle accelerator experiments, c is the speed at which the effects of gravity should be expected to travel. The speed of gravity cg  cannot be greater than c, insofar as messages can in principle be sent via gravity, and if messages could travel faster than c, they could be sent into the past in certain reference frames.

There is, however, a remote chance that non-oscillating gravitational effects could travel at a velocity greater than c. They might for example travel at v=c2/u, where u is the velocity of the source relative to the test particle. In that case, gravitational effects would be instantaneous in the rest frame of the source. Stated in terms of special relativistic spacetime diagrams, v=c2/u is the slope, in t-r coordinates, of the source’s plane of simultaneity, where u points in the direction r. This tachyonic value of v is of interest because it matches the phase velocity of de Broglie waves as defined by the relativistic single-particle Dirac and Klein-Gordon equations Nevertheless, it must remain true that oscillating effects such as gravitational waves, which carry energy and information, are confined to the limiting velocity c [61].

Whether a dual-velocity picture of gravitational propagation leads to contradictions is not yet known. However, the tachyonic speed of non-oscillating gravitational effects can be visualized in the following thought experiment. Imagine two stars of equal mass in circular orbits around their center of mass. First, it is known that in the framework of Newtonian celestial mechanics, which involves forces in absolute space and time, gravitational attraction must propagate instantaneously. Why? Were there any time delay, each star would feel a gravitational force pointing toward an earlier spot in the other star’s orbit [62]. If visualized correctly, the reader will see that this small offset, sometimes referred to as gravitational aberration, exerts a slight forward force on each star, making both stars orbit faster and faster, an instability which to Newtonian order is not observed. Thus, in real physical situations, each star accelerates toward the spot where the other star is now, and the gravitational force must therefore be instantaneous. Of course, this Newtonian scenario cannot tell us the speed of gravity in GR. It is a plausibility argument only. It does however present a paradox. How can the Newtonian infinite gravitational speed be reconciled with the supposed speed c predicted by GR?

One possible answer is suggested by the following treatment of the above thought experiment. Imagine a co-rotating coordinate system with respect to which the two orbiting stars described above are at rest (neglecting the small amount of radiative orbital decay.) The two stars can now be modeled by a static double-Schwarzschild metric. Such a metric has already been derived by other authors as an exact solution to Einstein’s field equations [63]. Since the metric is static in the co-rotating frame, the curvature and thus the mutual gravitational effects are also static in that frame. Defining the speed of gravity is now a matter of semantics. One might say that no effects at all are propagating in the co-rotating frame, or alternatively, that the effects of gravity propagate at infinite speed in that frame. In either case, the computed orbital motion, to Newtonian order, is the same as that of classical celestial mechanics. Again, it is important to stress that in the dual-velocity picture, these mutual gravitational effects cannot carry energy, since oscillating or energy-carrying effects must travel at c or less. (The small amount of gravitational radiation emitted from the rotating star system does of course propagate at c.)

The question of gravitational aberration has been a source of confusion in the literature. Some authors claim that the absence of gravitational aberration for orbiting bodies would constitute proof of an instantaneous gravitational interaction. Others, such as S. Carlip, argue that in a formal general relativistic treatment, aberration terms almost perfectly cancel even though cg is assumed to be c, and therefore the lack of aberration does not imply  cg>>c [64]. It is unclear, however, whether Carlip’s professed formal treatment, which employs a novel light-cone coordinate description of a mass-changing object called a photon rocket [65], is based on rigorous principles.

There remains in Carlip’s calculation a small higher-order residual gravitational aberration. Curiously, mathematical physicist Michal Krizek proposes that such an aberration is actually observed, and is the partial cause, along with tidal forces, of the increase in mean distance between the Earth and the Moon [66].

Can the speed of gravity be less than c? Some peer-reviewed theories of modified gravity, including quantized massive graviton theories, predict that it can (for extensive discussion see Ref. [67]). If true, the speed of gravity would not be the same in every reference frame. It might for example travel at a speed relative the source, much like Ritz’s old ballistic theory of light [68]. But to many theorists this seems implausible, especially in view of recent observations. Specifically, the reported near-simultaneous LIGO gravitational wave detection GW170817 and gamma ray burst GRB 170817a, received with a time lag of only 1.7 seconds from an event thought to be some 130 million light years away, seem to indicate gravity waves and electromagnetic waves travel at the same speed [69] for low redshift objects. More precisely, cg=c to an accuracy of 10-15  [70,71]. The small time lag is believed to be due to size of the source. Many astrophysicists have therefore concluded that these near-simultaneous GW and GRB detections disprove modified gravity theories in which cg ≠ c  [72-74], or that such theories must be strongly constrained [71,75]. For example, Crisostomi and Koyama say, [76] “The almost simultaneous detection of gravitational waves and gamma-ray bursts from the merging of a neutron stars binary system unequivocally fixed the speed of gravity to be the same as the speed of light c.” However, that this conclusion should be called unequivocal may be premature. Engineers and scientists familiar with large-scale government-funded research, especially involving extensive computer analysis, sometimes find that the results may be prone to error. Even if disparities rarely occurred, doubts might still be raised. Indeed, independent theorist and critic Miles Mathis doubts there is any truth at all to the professed LIGO gravitational wave detections, and while Mathis’s technical arguments have apparently not been peer-reviewed, his allegations of disregard for the scientific method on the part of the LIGO team may be justified [77]. It therefore seems reasonable that the raw data from the LIGO observations, as well as the experimental apparatus and its underlying assumptions, be analyzed by independent parties before conflicting theories are abandoned. To the knowledge of this author, an independent analysis has not been conducted. (See however James Creswell of the Niels Bohr Institute and associates, who perform an extensive analysis of LIGO detector noise and conclude that the gravity wave signals are questionable, stating, “A clear distinction between signal and noise therefore remains to be established in order to determine the contribution of gravitational waves to the detected signals.” [78]) Note that as recently as two decades ago, independent verification was the hallmark of physics. This standard should not be compromised. Meanwhile, it is still too early to call an end to all research into different speeds of gravity.

VII. GRAVITATIONAL TIME DILATION

Some theorists deny that time dilation, as predicted by GR, actually exists, claiming that redshift, which is often treated as equivalent to time dilation, is due to other causes such as photon motion through a gravitational potential. First, there seems to be confusion in the literature about the relation between time dilation and redshift, which will be discussed below. So the immediate question is, are there ways to measure time dilation without relying on redshift? One method is via the Shapiro time delay, which is the time delay of light as it traverses the field of the Sun [79]. This delay has been measured to a high degree of accuracy. The simplest explanation is that the delay is due in part to time dilation along the path of the photon as it passes close to the gravitational source, and in part to relativistic path length increase. Alternatively, the time delay might be attributed to a slowing of the coordinate speed of light, or the speed of light as seen from infinity. However, combined time dilation and path length increase are formally equivalent to the slowing of the coordinate speed of light; these are just two different interpretations of the same properties of the metric, namely that g00<1 and g11>1 . In any case, the Shapiro time delay does indeed verify time dilation independently of redshift.

Blurring of the distinction between gravitational time dilation and gravitational redshift is so prevalent, many authors use the terms almost interchangeably, even though they might be different phenomena. For example, although cosmic redshift is certainly observed, there is no way in principle to directly measure cosmic time dilation, which may not even exist given that g00= 1 in the FLRW metric. The confusion is further compounded by the fact that some authors contend that time dilation causes redshift, or that gravitational potential causes redshift. That such claims lead to contradictions has been demonstrated by Vasily Yanchilin [9]. In his paper entitled The Experiment with a Laser to Refute General Relativity, he points out that general relativists, in textbooks and peer-reviewed journals alike, contradict themselves by purporting on the one hand that gravitational redshift, for example in a Schwarzschild field, is caused by energy loss as photons climb through the gravitational potential, and on the other hand, by time dilation at the emitter. If both were true, Yanchilin explains, we would see twice the redshift we do. So it must be one or the other. This seems patently logical, and Yanchilin proposes an earth-based experiment to distinguish between the two purported causes. However there is a subtle point that Yanchilin and others may have missed. The notions that redshift is caused by energy loss in transit or by time dilation at the source are both plausibility arguments, put forth to help students visualize why redshift occurs in a gravitational field [80]. These arguments are misleading, and may have led Yanchilin into designing an experiment that will fail to prove what he seeks to prove, as will be discussed below.

A rigorous analysis of the behavior of light as it climbs through a gravitational field shows that, while photon energy E=hν, where ν is the proper frequency measured along the photon’s path, is indeed lost during transit, and time, as viewed from infinity, is dilated at the emitter, these are two different descriptions of a single property of the metric, which in static cases is simply that g00 < 1. These phenomena do not cause redshift; spacetime curvature does. In fact, spacetime curvature causes all three phenomena: time dilation at the emitter, photon energy loss in transit, and redshift at the detector. And all three have the same value, obtained from g00.

The ultimate arbiter is the metric. When redshift is calculated from gμν, the result is unambiguous. There is one value of redshift, and it is not doubled. So if Yanchilin successfully conducts his experiment, in which light is to be emitted both upwards and downwards from a central height in a tall building, and the results tabulated by a frequency counter at that same central height, he will measure the correct GR redshift. However, believing the two plausibility arguments are mutually exclusive, he may misinterpret his results as a confirmation of photon energy loss, and hence as a repudiation of time dilation. Intending to disprove GR, he may find that many physicists will only claim he has proven it. Yet Yanchilin has simply carried to its logical conclusion a set of common misconceptions. I would venture that the fault lies in today’s education system, in which plausibility arguments are emphasized while mathematical formalism is neglected.

VIII. SPACETIME AND CURVATURE

Some researchers doubt that the curvature of spacetime, as embodied in the metric, is the origin of gravitational effects, or even that 4-dimensional Minkowski spacetime is a valid physical concept. In the latter case, they are refuting special relativity (See for example Ref [81]). A number of authors are currently investigating new physics beyond SR, and peer-reviewed articles state there is a consensus among physicists that the spacetime structure of SR will have to be modified in order to quantize gravity [82]. There is also renewed interest in Lorentz-violating theories such as Horava gravity, whose low energy limit is dynamically equivalent to the Einstein-aether theory [83,84]. Yet in a classical (non-quantum) context, a formalism describing time, space and linear motion more concise and accurate than SR has, to the knowledge of this author, never been derived. Occam’s razor alone says this validates SR.

It is true of course that time and space have very different properties. One such property is the signature in the line element, as can be seen from the 2D spherical Minkowski line element ds2=dt2-dr2. The sign of the temporal term is opposite that of the radial term, implying that if t is a dimension, it is in some sense an imaginary one. Another such property is the arrow of time. Space, in contrast, has no arrow. These disparities may make space and time hard to conceptualize as a homogeneous entity. Some critics thus reject spacetime altogether, and attempt to explain the constancy of the speed of light, which forms the mathematical basis of SR, by attributing the shortening of rulers and slowing of clocks to electromagnetic or mechanical processes [81]. However, since every moving clock and object slows and shortens, it might as well be said that time dilates and length contracts, as there is no way in principle to distinguish time and length from clocks and objects. In any event, refutations of SR are rarely mentioned in modern peer-reviewed journals except in the context of quantization. This does not mean, of course, that spacetime could not eventually be replaced by a simpler or more accurate construct, conceived perhaps as a product of brilliant intuition.

That gravity arises due to the curvature of spacetime is more frequently doubted. Some researchers accept Minkowski spacetime, yet reject the idea that pseudo-Riemannian geometry, which is defined by a (possibly) curved line element in which one term is of opposite sign, determines the properties of space, time and motion in a gravitational field. Among such theories are teleparallel gravity (TEGR) [85] or torsion-f(T) gravity [86,87].

The notion that gravity is caused by curved spacetime springs from the principle of equivalence. This principle may be paraphrased by saying that all point-like test particles, regardless of their mass or composition, follow the same trajectory in a gravitational field. So to doubt that gravity is geometry is to doubt the principle of equivalence. Yet the principle of equivalence has been demonstrated to a high degree of accuracy. In response to this fact, physicists who refute geometric gravity have proposed a hierarchy of equivalence principles, from strong to weak [88,89], claiming that only the weaker versions have been proven. This allows small deviations from pseudo-Riemannian geometry, which may be needed, for example, in attempts to quantize gravity.

As an aside, it can be argued that if gravity is geometry, then it cannot in principle be quantized. Geometric gravity does not involve any forces that might be mediated by gravitons. All apparent forces are pseudo forces. Thus, centrifugal force is as real or unreal as centripetal force. Both occur when an object deviates from a geodesic. (An example is found in the apparent forces at the near and far walls of an orbiting space station.) So if one wishes to quantize the attractive gravitational force, one should also quantize centrifugal force, which seems absurd. It is perhaps relevant that after almost a century of effort, no attempt to quantize gravity has been fully successful [84]. On the other hand, quantization efforts are justified insofar as GR does not tell us how spacetime curvature propagates outward from a massive body, only that it does so at the speed of light. To address this omission, it may be necessary to extend GR to include gravitons or some other mediating mechanism.

Whether gravity is or is not geometry is a separate question from whether GR is valid. GR of course requires that gravity be geometry. But there is an unlimited set of geometric gravity theories, often called metric theories, that differ from GR. These theories involve curved metrics, possibly in higher dimensions, but the metrics are not necessarily solutions to Einstein’s field equations. Examples include modified gravity theories such as f(R) gravities, in which the field equations contain higher order terms in the scalar curvature R [35,43,44], or f(R,T) theories, where T is the trace of the energy-momentum tensor [46-48]. The variations are endless.

Meanwhile, unless the equivalence principle can be disproved, there is no reason to reject curved spacetime as a description of how objects behave under the influence of gravity. Even if the metric is considered to be only a shorthand notation for gravitational effects, this does not change the fact that by Occam’s razor, curved spacetime provides the simplest and most accurate formalism for gravity known today.

IX. ENERGY AND THE GR FORMALISM

That GR does not offer a clear definition of the localized energy of the field is considered by some to be a defect in the theory. P.A.M. Dirac, in his concise textbook General Theory of Relativity [90], summarizes the situation as follows, “It is not possible to obtain an expression for the energy of the gravitational field satisfying both the conditions: (i) when added to other forms of energy the total energy is conserved, and (ii) the energy within a definite region at a certain time is independent of the coordinate system. Thus in general, gravitational energy cannot be localized.” Authors in peer-reviewed journals occasionally raise objections to the lack of local conserved energy, and suggest possible conserved quantities other than energy [91].

The absence in GR of a definite field energy meeting the requirements given by Dirac does not imply that Einstein’s theory is incomplete or should be modified. Conservation of energy is a classical law by virtue of the concept of potential energy, an arguably contrived quantity which is proportional to the potential. Yet potential, as explained before, is not intrinsic to GR. Therefore, GR should not be expected to comply with conservation of energy.

X. THE SINGULARITY PROBLEM

The formalism of GR predicts real physical singularities, such as those at t=0 in the FLRW metric (the time of the big bang) or r=0 in the Schwarzschild metric, as well as coordinate singularities such as that at r=2m, the horizon of a black hole. Yet the mathematical formalism is believed to break down at singularities [92]. Is this a contradiction in the theory? Some mainstream physicists contend that it is, citing for example a problem known as geodesic incompleteness, by which a photon traveling on a geodesic would cease to exist at a singularity [93,94]. Thus, there are ongoing efforts modify GR so that singularities do not arise [71].

Many researchers claim that a correct theory of quantized gravity will remove all singularities. These endeavors toward quantization are well documented in mainstream journals [95,96]. Yet the so-called singularity problem may not constitute a valid reason for rejecting or modifying GR. It could be true of course that singularities are unphysical. For example, it can be shown from the Schwarzschild metric that a black hole would take forever to form by gravitational attraction alone [97]. Therefore, unless black holes are primordial or created by other forces, they do not exist in a universe governed by GR. (Some astrophysicists ignore this result. As Naoki Tsukamoto says, in the introduction to an article on black hole shadows, “Recently, LIGO detected three gravitational wave events from binary black hole systems. The events showed stellar-mass black holes really exist in our universe.” [98]) Bouncing cosmological models have also been proposed that avoid the singularity at the big bang [48,92,99]. In any case, singularities do not seem to pose a problem from a mathematical standpoint. Coordinate singularities can be transformed away, while so-called real singularities can be handled as mathematical limits.

XI. CONCLUSION

Of the many reasons theorists refute general relativity, there are two that stand out as possibly the most compelling: 1) the galactic rotation curve anomaly, and 2) incompleteness, or the need for an equation of state. Finding a modified gravity theory that accounts for the galactic rotation curve has proven surprisingly difficult. One problem is that GR describes solar system observations to a high degree of accuracy, yet a naive scaling of the galactic rotation curve to fit the orbits of outer planets gives erroneous results. Thus, any modified gravity theory must employ some screening mechanism whereby GR holds at smaller scales, but not on the scale of galaxies or the cosmos. Many such mechanisms exist, but so far no modified gravity theory has gained acceptance as a replacement for GR. This problem is widely discussed in Physical Review D [73,100]. (For a summary of screening mechanisms see Ref. [36].)

More significantly, GR’s requirement for an equation of state seems to constitute proof of the incompleteness of the theory, though to the knowledge of this author, such a deficiency has never been acknowledged in the literature. Physicists invariably select an EoS as a matter of course. The EoS is usually chosen either ad hoc, or based on thermodynamic arguments. The EoS can be as complicated as desired, and in principle tailored to produce almost any physical result. For example, in the static spherical non-vacuum case, the mass density ρ(r) determines only the g11 component of the metric. The g00 component, which describes observables such as time dilation and redshift, depends on the EoS, and if the EoS is suitably varied, can in practice be anything conceivable. Thus, these time-coordinate observables do not in general depend on the mass distribution. This fact contradicts the common interpretation of the Schwarzschild metric, according to which such observables depend on mass alone. It is seldom if ever mentioned that the Schwarzschild metric, the only metric to have been observationally tested in a theory-independent way, does not require an EoS and therefore seems at odds with the rest of the theory.

Criticisms of general relativity abound, yet no suitable replacement has been proposed. It might be possible to derive a theory of gravity based on a field equation that does not require an EoS, for example in which the energy-momentum tensor Tμν is replaced by a function f(T) of the scalar T, the trace of Tμν. But such a theory is unlikely to explain the galactic rotation curve. Many of the questions raised in this article therefore remain open.

ACKNOWLEDGMENTS

I would like to thank Dale H. Fulton for providing valuable references on the speed of gravity, time dilation, and the LIGO gravitational wave detections, as well as for insightful discussions on those and other topics covered in this paper.
__________________________________________

[Many references at Physical Review D are publicly available on arXiv. If links are not provided below, they may be found by searching for author and title at Google Scholar.]

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