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# Current conflicts in general relativity: Is Einstein's theory incomplete? (V2)

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## 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.
1
Current conflicts in general relativity: Is Einstein’s theory
incomplete?
By Kathleen A. Rosser
Kathleen.A.Rosser@ieee.org
v1: August 12, 2018
v2: September 20, 2019
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and unpublished communications.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 common reasons for refuting GR is
presented below. These topics will be discussed in
detail in later sections.
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(
x
), 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(
x
) 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
2
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
g

, which
contains curvature information, and the differentials
dx
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
2
ds g dx dx


, with repeated
indices indicating summation from 0 to 3.
The computational pipeline of the general relativistic
formalism for orthogonal energy-momentum tensors
( , , , )T diag p p p

is straightforward. One must
first select a coordinate system for the spacetime
region to be studied. Next, the mass-energy density ρ
as a function of the coordinates
x
must be specified
for the region. The function ρ(
x
) is then substituted
into the energy-momentum tensor
T

on the right
hand side of Einstein's Field Equations (EFE):
1
2
R g R T
  

where
R

is the Ricci tensor and depends on
derivatives of the metric,
g

is the metric to be solved
for,
the scalar curvature obtained by contracting the
Ricci tensor, and.
2
8Gc


is a constant. After
that, one must specify the momentum density p, or for
static configurations, the pressure density, also denoted
p, the latter determined by a selected equation of state
that relates mass-energy density to pressure, and
substitute the resulting function
()
i ii
p x T
into
T

.
The field equations are then solved to obtain the metric
g

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
/L mds dt
[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
24G
   
 
,
to argue the plausibility of his relativistic field
equations, in which the derivatives of φ are represented
by curvature
R

and mass density ρ by the energy
momentum tensor
T

[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 r is not, as can be
seen from Dirac's derivation of the Schwarzschild
3
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 does not
explicitly occur in the formalism of general relativity.
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 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].)
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
4
galaxies show a surprising uniformity in their would-be
DM distributions, as manifest in the universal
constant
82
01.2 10a x cm sec
, which accurately
specifies for most spiral galaxies the radial acceleration
at that distance 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. 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 criticize his simplified
galactic model, or claim that he has ignored other
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 substance 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:
2 2 2 2 2 2 2 2
( )( sin )ds dt a t dr r d r d
 
 
Using a metric of the above form, Einstein’s field
equations reduce to the following two simultaneous
equations in terms the time-dependent scale factor a(t):
2
283
aG
a

2
2
28
aa p
aa
 
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
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. Using
standard forms of a(t) which increase monotonically
with time, FLRW predicts that redshift increases with
distance for 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]. Calculations
based on GR 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
5
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
T

[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
11
g
but not the t component
00
g
. This
can be seen by examining Einstein’s field equations for
a static spherical non-zero mass distribution, which
reduce to the simultaneous equations:
11
2 2 2
11 11
00
2200 11
11
11
()
11
()
g
rg r r g r
g
pr g g r
g r r

 
 
where primes denote differentiation with respect to r. It
is clear from the first equation that
11
g
is fully
determined by mass-energy density ρ(r). To solve for
00
g
however, an auxiliary Equation of State (EoS)
relating ρ to pressure p is needed. In general
applications, 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 peer-reviewed literature, models using
an EoS of seeming unlimited complexity are
sometimes assumed [56-58]. This leads to the awkward
circumstance that in many cases the EoS yields more
information about gravitational effects than do
Einstein's equations themselves. In fact, almost any
desired gravitational effect can be induced 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
6
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 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],
2 2 1 2
(1 2 / ) (1 2 / )ds m r dt m r dr
   
2 2 2 2
( sin )r d d
 

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 as described previously. Another peculiar
fact is that the Schwarzschild metric has the form
00 11
1gg
, as if an EoS of p=ρ had been implicitly
assumed. Was it? In a sense, yes, in that both ρ and p
vanish for the vacuum and hence are equal. But this is
a trivial application of 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
( ,0,0,0)T diag

,
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
momentum, which vanishes for static configurations.
Such an interpretation seems reasonable to this author
in that the motions comprising pressure are random
rather than unidirectional, suggesting pressure should
not appear in the spatial components, but only in the
mass-energy density component
00
T
. The pressure
terms
ii
Tp
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
T

. 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
g
c
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
2
v c 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,
2
v c 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
7
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
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
g
c
is assumed to be c, and therefore the
lack of aberration does not imply
g
cc
[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 Carlips 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
g
c
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]. More precisely,
g
cc
to an accuracy of
15
10
[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
g
cc
[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
GW
c
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
8
that the results are 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
speed of light as seen from infinity. But time dilation
and the slowing of the speed of light are formally
equivalent. They are two different descriptions of a
single property of the metric, namely that
00 1g
. 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, there is no way in principle to directly
measure cosmic time dilation, which may not even
exist given that
00 1g
in the FLRW metric, although
cosmic redshift is certainly observed. 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
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. Indeed, they
may have misled 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
00 1g
. 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
g

.
The ultimate arbiter is the metric. When redshift is
calculated from
g

, 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
9
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
2 2 2
ds dt dr
.
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
10
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 proof of the incompleteness of
the theory, though to the knowledge of this author,
such a deficiency is never acknowledged in the
literature. Physicists invariably select an EoS as a
matter of course. The EoS is usually chosen either ad
11
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
11
g
component
of the metric. The
00
g
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
T

is replaced
by a function f(T) of the scalar T, the trace of
T

. But
such a theory is unlikely to explain the galactic rotation
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.
__________________________________________
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... It seems interesting that the interior metric g 00int depends on the EoS of the shell, while the exterior metric g 00ext like the Schwarzschild metric, is independent of the EoS. This curious distinction resolves the seeming paradox, mentioned in a previous paper [56], that while nonvacuum solutions to EFE require an EoS, Schwarzschild vacuum solutions do not, even though mass appears in the metric. ...
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