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Content uploaded by Kathleen Rosser

Author content

All content in this area was uploaded by Kathleen Rosser on Sep 19, 2019

Content may be subject to copyright.

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and unpublished communications.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,

R

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

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

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 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

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

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. 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

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.

__________________________________________

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