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THE THEORY OF GRAVITATIONAL WAVES

IN GENERAL RELATIVITY - A CRITICAL ANALYSIS

Professor Hasan Sehitoglu

ABSTRACT

The existence of gravitational waves has been a controversial subject from the inception

of Einstein’s geometric theory of gravity. In view of the recent announcements regarding the

real-time detection of gravitational wave effects attributed to the merger of black holes and

neutron stars, it is worthwhile to have a critical review of the theory of gravitational waves. This

paper details several internal contradictions and fundamental flaws within the said relativistic

theory. We discuss the consequences of these mathematical shortcomings and show that the

claims about direct observation of gravitational waves lack the extraordinary experimental

support as well as the necessary theoretical rigor they need.

Key words: Gravitational waves – radiation: dynamics

1.0 INTRODUCTION

Scientists have been debating the idea of gravitational waves from the very beginning. In

his lectures and published papers, Einstein himself questioned the physical existence of

gravitational waves. He was unsure whether his geometric theory of gravity predicted or ruled

out such waves. A historical discussion of this controversy can be found in Kennefick (2007) and

the references therein.

Before the installation of the advanced version of LIGO, the proponents of the

gravitational waves predicted that the binary neutron star detection rate would average about 40

events per year with a range between 0.4 and 400 per year (Abadie et al, 2010). Similarly, the

binary black hole detection rate was estimated to average 20 events per year with a range

between 0.4 and 1000 per year. However, these rate predictions have not materialized. So far,

there have been only 5 announcements regarding the detection of wave effects attributed to the

merger of black holes and one additional announcement concerning the merger of neutron stars.

From the details of an exceedingly noise corrupted signal, the LIGO collaboration claims to

extract not only the history of the in-spiral, coalescence, and the ring down, but also the masses

and spin rates of the initial black holes and the final black hole. These are extraordinary results.

The late astronomer Carl Sagan once famously stated that ‘extraordinary claims require

extraordinary evidence.’ A corollary of this rule is the requirement that extraordinary scientific

claims must be based on extraordinary mathematical rigor. Since the relativistic theory of

gravitational waves has been developed by using heuristic conjectures and phenomenological

arguments, it is worthwhile to scrutinize the essential elements of this important theory.

In the following sections of this paper, we shall show that the theory of gravitational

waves is rife with mathematical inconsistencies and internal contradictions. Regrettably, our

findings cast serious doubts on the recent claims regarding the direct detection of these waves.

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As pointed out in Creswell et al. (2017), Raman (2017), and Liu et al. ( 2018), it appears that

some unaccounted sources generate burst-like signals with correlated noise at the two LIGO

locations almost simultaneously.

This paper is organized as follows. For the sake of completeness, Section 2 gives a brief

overview of the relativistic theory of gravitational waves. In Section 3, we highlight the

disagreement between the theory and astronomical observations. Section 4 covers the

gravitational time dilation phenomenon in the context of gravitational waves. We itemize the key

differences between Newton’s and Einstein’s theories in Section 5 and show that, if these

theories are intermingled, inconsistent results become inevitable. Section 6 discusses various

internal contradictions arising from the use of the famous quadrupole formula. Section 7 lists

further inconsistencies in the theory of gravitational waves. An important property of the empty

space is the vacuum energy field, which has been ignored by wave theorists. In Section 8, we

obtain analytical solutions to the relativistic wave equation in the presence of the cosmological

constant and expose further internal disagreements within the theory. Section 9 deals with the

flaws in the signal analysis methods used by the LIGO collaboration. We present our final

remarks in Section 10.

2.0 The THEORY of GRAVITATIONAL WAVES

We begin by recalling that General Relativity regards gravity not as a force in the

physical sense but rather as a manifestation of the curvature of the union of space and time. In

other words, gravity is geometry. The union is called the spacetime and represented by a four-

dimensional vector x . General Relativity is based on a gauge-invariant line element constructed

by using spatial and temporal separations as shown below

2 Tds d d g dx dxx Q x (1)

where [ ]gQ is a sign indefinite matrix and is known as the metric tensor. Since the line

element can assume both positive and negative values, the geometry is said to be non-Euclidean.

It is important to keep in mind that the spacetime vector is a human construct and as such it is not

a physically observable quantity. In this paper, we adopt the ( , , , ) metric signature.

The field equations of General Relativity relates the spacetime curvature, described by

the Einstein tensor [ ]GG , to the totality of the local energy and momentum fields,

collectively represented by the tensor [ ]TT . Using the index notation, the field equations are

written as follows

G T (2)

where is a proportionality constant. Einstein proposed the above relationship as the

generalization of the Poisson’s equation of the Newtonian theory of gravity. The Einstein tensor

is composed of two terms

3

1

2G R Rq (3)

where R is the Ricci tensor and R is the curvature scalar. Similarly, the total energy-

momentum tensor is expressed in terms of two distinct components

matter vacuum

1( ) ( )T T T (4)

The term matter( )T represents the energy-momentum density of the local matter distribution as

well as all the other energy fields, for example, the electromagnetic field. The second term is the

energy-momentum tensor of the vacuum expressed by vacuum( )T g . In the literature,

is known as the cosmological constant.

Immediately after formulating General Relativity, Einstein linearized the following

version of his field equations

matter

1( )

2R Rq T (5)

He assumed a weak gravitational field and considered tiny fluctuations in the flat spacetime

metric of Minkowski

g h (6)

where [1, 1, 1, 1]diag . [ ]hH is a symmetric tensor field defined in Cartesian

coordinates on the background independent spacetime of the special relativity and 1h .

After some tedious algebra, the final equations of the linearized theory are

2

matter2 ( )h T (7)

where h is the trace reverse perturbation and the symbol 2 represents the 4-vector

D’Alembertian operator. The two perturbations are related by 12

h h h , in which h is

the trace of h . Moreover, Eq.7 is true provided that the perturbations satisfy the Lorenz gauge

condition 0vh , which implies that 0vT . Using this gauge and the geodesic equation

of General Relativity, one quickly finds that the acceleration of a test particle is governed by

0x , that is, the gravitational field has no effect whatsoever on the test particle. This clearly

contradicts the geodesic hypothesis of General Relativity. So, from the very beginning, we see

that the gravitational wave theory is internally inconsistent.

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In empty space, the linearized field equations constitute a set of decoupled wave

equations

2 0h (8)

for each component of h . Thus, Einstein concluded that the linearized field equations have

solutions in which the perturbations of Minkowski spacetime are plane waves traveling at the

speed of light. He called them ‘gravitational waves.’ Most textbooks consider a wave

propagating in the positive z-direction and employ a special choice of coordinates known as the

transverse-traceless gauge. Furthermore, solutions are obtained by using the 3+1 formalism,

namely, a three-dimensional space completely isolated from a one-dimensional time axis. This

coordinate separation is evident in the following solution of the wave equation in empty space

( )

0 0 0 0

0 0

0 0

0 0 0 0

ik z cth h

h eh h

(9)

where h and h are known as the ‘plus’ and ‘cross’ amplitudes. This completes our summary

of the theory of gravitational waves.

3.0 EXPERIMENTAL DISAGREEMENT

The true test of a theory is its success in accounting for experimental results. In the

present context, the theory of gravitational waves must agree with the astronomical observations.

The idea of a gravitational wave depends exclusively on the hypothesis that the three-

dimensional space is curved. General relativists have embraced this assumption as if it were a

scientific fact. For example, on page 188 of Einstein (1952), we read the following: ‘The

curvature of space is variable in time and place, according to the distribution of matter. … It is

to be emphasized, however, that a positive curvature of space is given by our results…’

Obviously, whether the physical space is curved or not has significant implications for

Einstein’s theory. If there is no curvature, according to General Relativity, there can be no

gravitational field and therefore no gravitational waves. It is important to note that the golden

years in the development of the theory of gravitational waves were in the 1970s and the 1980s,

way before the availability of technologically advanced astronomical observations by space-

based instruments. In those early years, the curvature of space was taken for granted by the

proponents of gravitational waves. In order to answer this vital question, several satellite-based

experiments have been performed in the last two decades. So, it was a little of a surprise to the

theorists, when the data collected by the Hubble telescope, COBE, WMAP, and Planck satellite

missions have shown unequivocally that the space is flat, not curved. In other words, the

Universe obeys the rules of Euclidean geometry, independent of the properties of matter and

energy filling the space. Mathematically speaking, if one slices up spacetime by introducing

three-dimensional spacelike hypersurfaces in connection with a series of time parameter, then

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each of these hypersurfaces at a particular time is flat. Hence, the core argument of the

linearization, namely, the space is slightly curved, is in conflict with the experimental findings.

Since the relativistic wave theory is based on the 3+1 formalism, the observation that the

physical space is flat translates into 0h everywhere in the Universe at any given time.

Clearly, it is illogical to talk about gravitational waves if metric perturbations with respect to the

flat Minkowski spacetime don’t exist. This simple disagreement with experiment is sufficient

enough to rule out the existence of relativistic gravitational waves.

Another important point is that experiments can only measure gauge-invariant quantities.

More specifically, only coordinate independent variables have physical meaning. Measuring all

ten functions of the perturbation tensor h at a certain event in the Universe requires a choice of

coordinates and that choice resides in the experimenter who studies the gravitational waves. In

order for the field equations of gravity to be relativistic, we must have them expressed in terms

of tensors so that the equations remain unchanged under coordinate transformations. However, a

well-known feature of the linearized wave theory is the fact that h does not transform as a

tensor under general coordinate transformations but only under the very restricted class of

Lorentz transformations of special relativity. For this reason, h and tensors derived from it are

called pseudo-tensors. In an actual physical experiment, different individuals using different

coordinate systems would disagree on the observed values of the variables represented by

pseudo-tensors. Recall that eigenvalues of a true tensor are gauge-invariant quantities. But,

different observers would measure completely different eigenvalues for the strain pseudo-tensor

h . This additional flaw shows that the real-time detection of relativistic gravitational waves

lacks the necessary experimental support.

The most recent astronomical observations confirm that the morphology of cosmic space

is akin to a porous medium with inhomogeneous and anisotropic physical properties. See, for

example, the three-dimensional cosmic velocity web and cartography by Pomarede et al. (2017).

On the other hand, the relativistic wave theory assumes an empty space with no material

boundaries. But, there are thousands of galaxies of all kinds and shapes between a source and the

Earth. For example, the Milky Way is part of a large number of galaxies, all moving towards the

Great Attractor under the influence of a cosmic flow. In addition, there is a ring of dwarf

galaxies and gas clouds known to orbit our galaxy in the same direction. Furthermore, the solar

system is located in the Orion Spur between two large spiral arms of the Milky Way. Thus, far

from being a void, the space between a gravitational wave source and a detector is filled with

intergalactic and interstellar media. According to the wave theory, the intervening space does not

absorb gravitational radiation. Since the length of a typical gravitational wave is much larger

than most solid objects, such as the Sun and the planets etc., absorption of waves by matter is

considered to be negligible (Thorne, 1982). But, this conclusion is based on the electromagnetic

wave analogy, which employs a lumped-parameter model (ordinary differential equation) and

ignores the field characteristics of gravitational phenomena (partial differential equation). For

example, intergalactic ionized gas formations of gigantic sizes are known to exist and there are

molecular clouds of astronomical proportions in every galaxy. The frequency of a passing

gravitational wave can match and excite one of the natural frequencies of such a distribution of

mass and energy field. The resulting fluctuations will cause thermal radiation and increase the

temperature of the cloud. Observe that the response of a molecular gas cloud to a gravitational

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wave in space is similar to the suspended tests particles of a laser interferometer. The cloud as a

whole will absorb some energy and oscillate, thereby reducing the luminosity of a wave.

In general, the absorbance of intergalactic and interstellar media is extremely small but

non-zero. The propagation velocity v of a gravitational wave in an energy dissipating medium

can be related to the speed of light by way of a complex-valued refractive index as follows

s

ci

v (10)

where is the usual index of refraction and s is the absorption coefficient of the medium.

Now, the 11h component of the plane wave solution can be rewritten as

( )

11( , )z

i tvh z t h e (11)

After substituting the propagation velocity into the above equation, we get

( )

11( , )s z

z i tc ch z t h e e (12)

Observe that the traveling wave is now damped because its amplitude is decreasing

exponentially with distance in the direction of propagation. The frequency-dependent term

sH has the dimensions of 1 1(km s Mpc ) . In cosmology, H is known as the Hubble

parameter, which is related to the redshift in galactic light due to the dissipation of optical

energy. To gain physical insight, let us assume that the absorbance of the empty space is the

same for both gravitational and electromagnetic waves. Astronomers estimate the Hubble

parameter to be 1 167 km s Mpc . Then, since the GW150914 event took place at a distance of

1.3 billion light-years (410 Mpc), one obtains

0.092 0.91

Hz

ce e (13)

Thus, there will be about 9 percent path-loss in the amplitude of the gravitational wave. Clearly,

energy dissipating nature of the cosmic medium has a substantial impact over cosmological

distances. We note in passing that the preceding discussion reveals the physical origin of the

Hubble parameter.

In summary, this section has demonstrated that astronomical observations and

experiments do not support the empty space assumptions made by the relativistic wave theory.

There is another important hypothesis of the said theory, namely, the physical effect of the

vacuum energy is negligible. We shall show later in Section 8 that this assumption is also false,

that is, the vacuum energy plays a significant role at cosmological distances.

4.0 INFINITE REDSHIFT

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Most textbooks point out that gravitational waves travel in the same manner as

electromagnetic waves. Indeed, the following is written on page 45 in Maggiore (2008): ‘Since

gravitons propagate along null geodesics, just as photons, their propagation through curved

space-time is the same as the propagation of photons, as long as geometric optics applies. For

instance, they suffer gravitational deflection when passing near a massive body, with the same

deflection angle as photons, and they undergo the same redshift in a gravitational potential.’

Gravitational redshift is a direct consequence of the difference introduced by gravity in

the local proper time relative to the proper time of a distant observer. In order to study the

redshift in the present context, let’s consider two events that take place in the Schwarzschild

spacetime surrounding a static, spherically symmetric body of mass M . Suppose that the events

are the emissions of two crests of a gravitational wave at a fixed radial location emr near a black

hole. Recall that black holes are said to be the most dense objects in the Universe, so they create

extreme curvature in spacetime. From the Schwarzschild line element, the interval associated

with these timelike events is given by

2 2 2

2

21em em

em

GMds c dt

c r (14)

where G is Newton’s gravitational constant and c is the speed of light in vacuum. The symbol

emdt represents the difference in coordinate time between the events. The proper time measured

by an inertial clock at the location of emission is em emd ds c . It follows then that the proper

time is less than the coordinate time as shown below

1

2

2

21em em

em

GMd dt

c r (15)

Next, consider an inertial observer outside the gravitational potential of the massive body

at some distant radial location obr . Such an observer will find that the coordinate time separating

two incoming wave crests will be the same as the coordinate time between the emission of these

crests, that is, ob emdt dt . Then, the invariant interval recorded by the distant observer is

2 2 2

2

21ob em

ob

GMds c dt

c r (16)

The proper time ob obd ds c of the distant observer becomes

1

2

2

21ob em

ob

GMd dt

c r (17)

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Let the observer be located at a sufficiently large distance, obr . Then, we have

emd dt .

After some algebra, one obtains the following relationship

1

2

2

21 em

em

GMd d

c r (18)

In words, the proper time between observation of the two gravitational wave crests at infinity is

greater than the proper time between their emission as measured at the site of the emission.

Therefore, the clock at emr r is running slower than the clock of a distant observer. Suppose

that the stationary clock at the source of gravitational waves is moved closer to the surface of a

black hole. How would its rate of ticking be measured by the distant observer? The distant

observer would find that the clock ticked even more slowly. This effect, the slowing of an

inertial clock in a gravitational field, is referred to as gravitational time dilation. Note, however,

that there is a conceptual difference between this effect and the cosmological time dilation

caused by the spatial expansion of the Universe.

Since the two events are related to the successive peaks of a gravitational wave, emd

represents the period of a wave at its site of emission and obd is the period of that same wave as

measured by the distant observer. The reciprocal of the period is equal to the frequency. Thus,

the distant observer measures the frequency of the radiation as

1

2

2

21em

em

GM

c r (19)

We see that the observed proper frequency is less than the emitted proper frequency. In other

words, gravitational waves are redshifted because they must overcome a gravitational field on

their way from a source to a distant observer. This phenomenon is known as the gravitational

redshift. The above formula predicts that the observed redshift will increase as the site of

emission approaches the event horizon. Indeed, as 22r GM c , we have 0 and the

redshift becomes infinite. Therefore, the luminosity of the radiation falls to zero and the distant

observer never detects the plunge and merging phases of the two coalescing black holes.

The foregoing discussion involves a static case. In reality, the exterior gravitational field

of the first black hole will be distorted in the presence of the matter distribution of the second

black hole and vice versa. Since the combined gravitational field is dynamic, the coordinate time

between the two gravitational wave crests is no longer the same at the sites of the emitter and the

observer. That is, we have ob emdt dt . Consequently, the actual redshift phenomenon is

physically more complex. However, even though the locally measured frequency increases as a

function of time, the radiation frequency recorded by a distant observer is still going to approach

zero as the black holes merge. This contradicts the claim that LIGO detects exactly the same

“chirping” signal as measured by an inertial observer located near the binary system.

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It needs to be mentioned that the gravitational redshift plays an important role in the

calculation of the time-of-arrival of signals from binary pulsars. This is because gravitational

field around a pulsar is so strong that the application of relativistic gravity becomes essential. For

example, we have 2( ) 0.2GM c R at the surface of a neutron star compared to 610 for the

Sun. Ironically, while wave theorists disregard the gravitational redshift phenomenon in the case

of black hole mergers, they pay attention to it when they study the orbital decay of pulsars

(Taylor & Weisberg 1989).

Note that the preceding results are entirely based on the theory of General Relativity. In

the literature, it is said that a collapsed black hole leaves a gravitational imprint frozen in the

space surrounding it. But, as two black holes coalesce, the gravitational field around them is

continuously changing. This shows that the relativistic wave theory is internally inconsistent.

Black holes get their name because they are causally disconnected from the rest of the Universe.

In other words, it is impossible to transmit outwardly any signal, even in the form of

gravitational waves. If no information can leave a black hole, how do then two black holes

gravitationally attract each other resulting in their merger? Unfortunately, the geometric theory

of gravity does not provide an answer to this important question.

5.0 MIXING of TWO DISJOINTED THEORIES

General Relativity, unlike the Newtonian gravity, is unable to describe the dynamical

evolution of a multi-body system. This is true because, according to the theory of relativity, each

object must have its own proper time. Recall that Einstein’s theory is based on a single line

element giving rise to a specific proper time and distance associated with a body. Consider, once

again, the case of two coalescing black holes with spacetime vectors 1x and

2x . If we find two

solutions 1( )A x and

2( )B x of the Einstein field equations, then aA bB (where a and

b are scalars) is not a valid solution because of the highly non-linear nature of General

Relativity. In words, one cannot take the overall gravitational field of two bodies to be the

summation of the individual gravitational fields of each body. Thus, the line element of the

binary system must be expressed as a function of two different spacetime vectors. Accordingly,

the metric tensor representing the combined gravitational field becomes 1 2( , )g x x . However,

in order to calculate the Christoffel symbols and the Riemann tensor, one must adopt a spacetime

vector for the system and take covariant derivatives with respect to systemx . But, General

Relativity is silent about how to define a single set of coordinates for a dynamic system in the

presence of multiple spacetimes. Equally important, in order to ensure the existence and

uniqueness of a solution to Einstein’s field equations, boundary and initial conditions of the

system must be specified. For example, expressions are needed for the temporal and spatial

variation of the gravitational field on the surface of each black hole. Since system( )g x is a priori

unknown, the boundary conditions remain unspecified and no physically meaningful solution can

be found. We see that relativistic multi-body problem is ill-posed.

On the other hand, Newton’s theory of gravity, while capable of solving the orbital

motion of a binary system, does not predict the existence of gravitational waves in any form or

shape. Faced with this conundrum, the relativistic wave theorists have decided to combine ideas

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and equations from these two disjointed theories. The conceptual differences between Newton’s

and Einstein’s theories are so profound that, if intermingled, inconsistent results become

inevitable. Before we discuss these internal contradictions, let us highlight some of the key

differences.

General Relativity asserts that the gravity is not a physical force but is a manifestation of

the curvature of spacetime. According to Newton’s theory, gravity is a mutually attractive

force.

General Relativity is a field theory. Thus, the speed of gravity is finite and is equal to the

speed of light in vacuum. Newton’s theory gives rise to an action-at-a-distance law,

which Einstein himself derided as ‘spooky’. The speed of Newtonian gravity is infinite.

In General Relativity, time is relative between two observers. In Newtonian mechanics,

time is absolute, that is, the passage of time is the same in all coordinates.

As mentioned earlier, General Relativity is predicated on the idea of a non-Euclidean

space. Newtonian mechanics considers the space to be Euclidean.

In Newtonian mechanics, the Lagrangian includes two terms; one to account for the

kinetic energy and the second for the potential energy storage capability of the physical

system. The Lagrangian used to determine the geodesic equation of General Relativity

does not include a potential energy term.

Newton’s theory solves a binary system by assuming point masses without any internal

structure. Using this assumption, the system is reduced to an Effective One Body (EOB)

problem. Thus, there is no spin-orbit interaction. But, in General Relativity, material

objects are represented by extended bodies with internal structure. For example, black

holes are said to vibrate during the ring-down phase of a merger. Moreover, objects

experience spin-orbit interaction.

In Newtonian mechanics, coordinate separation is the same as the physical spatial

separation. Thus, in the EOB formulation, the radial and angular variables represent

physical distances. On the other hand, in General Relativity, coordinates are arbitrary and

have no intrinsic (physical) significance. One is free to choose a coordinate system with

two timelike dimensions in addition to two spacelike dimensions. For example, the

Schwarzschild radial coordinate is not a true measure of the proper radius. This means

that the radial and angular velocities determined by the EOB model have nothing to do

with their general relativistic counterparts.

In General Relativity, a spacelike coordinate may acquire timelike behavior. For

example, the Schwarzschild radial coordinate becomes timelike inside a black hole.

Newtonian mechanics comes with a precise definition of gravitational energy. In General

Relativity, the energy contained in a gravitational field cannot be described in a gauge-

invariant way.

The ordinary derivative of a vector is not a tensor. Therefore, General Relativity employs

the concept of covariant derivative. In Newtonian mechanics, one uses the usual partial

derivatives.

Vectors of Newtonian mechanics must always obey the triangle inequality. In General

Relativity, spacetime vectors can satisfy the reverse triangle inequality.

General Relativity deals with only conservative systems by demanding that the covariant

divergence of the metric tensor must be zero. This particular requirement corresponds to

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the energy-momentum conservation relation 0T . Therefore, the loss of energy due

to dissipative effects is anathema to the theory. Moreover, the presence of damping terms

destroys the time-reversal invariance property of General Relativity. On the other hand,

Newtonian mechanics has no problem dealing with dissipative phenomena.

As a typical example of the mixing of results from these two disjoint theories, consider

the perihelion advance of Mercury. Astronomers have determined that the unresolved advance of

Mercury’s perihelion is about 600 arcsec per century. General Relativity predicts an advance of

43 arcsec per century, a relatively minor value. The remaining amount (more than 90 per cent of

the unaccounted total) is attributed to the Newtonian action-at-a-distance forces due to the outer

planets within the solar system. Recall that Newton’s gravitational law and Einstein’s field

equations are highly non-linear differential equations. Therefore, it is mathematically incorrect to

superpose these two solutions. It is amazing to see that general relativists casually sum the

Newtonian and Einsteinian contributions without any further thought. Even when a system has

linear dynamics, the amplitudes and phases due to multiple simultaneous inputs do not add up

directly. The late Prof. J. L. Synge, an eminent general relativist of his time, called this fusion of

the two conceptually different theories ‘intellectually repellant’ in his book Synge (1960).

Einstein’s theory asserts that a binary system with a time-varying mass quadrupole

moment emits gravitational waves. The energy and momentum carried away in these waves is

removed from the binary system. As a result, the orbit decays and the stars spiral toward each

other. Since a two-body problem is analytically unsolvable in General Relativity, the theory of

gravitational waves has no choice but to borrow results from the Newtonian gravity. Recall that

the EOB solution reduces the two-body problem to one fictitious body in a central potential. This

solution is obtained by using Newton’s third law, namely, the gravitational forces are equal in

magnitude but opposite in direction. However, there is no such relationship in General Relativity.

Since the total angular momentum is conserved in a central potential, the motion is restricted to a

plane. Thus, the motion of one body relative to the other can be expressed in terms of polar

coordinates. The conservation of angular momentum provides an equation for the evolution of

the azimuthal angle . Similarly, conservation of the total energy yields an equation for the

evolution of the radial separation r . By combining these equations, wave theorists find that the

possible solutions for the relative motion are conic sections. From this result, Kepler’s laws

follow.

Adoption of the Newtonian EOB model implies that c . In this case, there can be no

gravitational waves because the right hand side of Einstein’s field equations drops out, that is,

the matter distribution has lost its influence. We have just identified one of the many theoretical

inconsistencies. In order to find the other flaws, let us briefly outline the strategy employed in

many textbooks (e.g., Misner et al. 1973; Hubson et al. 2006). Consider a coalescing binary

system of mass 1m and

2m in a nearly circular orbit. The virial theorem states that the total

energy is equal to half the potential energy or to the negative of the kinetic energy:

1 2orbit

2KE PE

Gm mE E E

r (20)

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For a nearly circular orbit, the semi-major axis is a r and Kepler’s third law becomes

2

3

GM

r (21)

where 1 2M m m and the symbol represents the orbital angular velocity, that is, .

Eliminating r in favor of in the energy equation, one finds

132 2

1 2orbit

2

m m GE

M (22)

Next, wave theorists turn to General Relativity for the rate of energy loss from the system. For a

nearly circular orbit, the quadrupole formula (which we discuss in the next section) gives

7 103 3

23

2

GR 1 2

5

32 ( )

5

dE G m m

dt c M (23)

At this point, Newton’s action-at-a-distance and Einstein’s geometric theories are fused together

by assuming

orbit GRdE dE

dt dt (24)

There are major problems associated with the above equation. To begin with, the right-

hand side represents time-averaged luminosity over several wavelengths whereas the left-hand

side is an instantaneous time-rate of change of energy. Also, Eq.24 is incomplete because there

are other physical mechanisms that can cause a binary system to lose energy. For example, stars

are known to eject substantial fraction of their mass via stellar winds. Furthermore, in the

modern implementations of the post-Newtonian approximation of General Relativity, one

considers a binary system as a fluid distribution that breaks up into two components, calling each

component a body. See, for example, chapter 9 of Poisson & Will (2014). Mechanical interaction

of these bodies will decrease the orbital energy via viscous heating of the interstellar medium. If

there is a high amount of turbulence, then the damping coefficient will be large. When the total

energy of the system decreases due to drag forces, it must move to a smaller radii to lose

potential energy and hence total energy. According to the virial theorem, as the potential energy

becomes more negative, the kinetic energy must increase. The increase in KEE is equal to only

one-half the loss of PEE . So, the total energy of the system indeed decreases. Paradoxically,

while damping forces try to slow down the system, the stars fall to a lower orbit and speed up.

This explains why the angular velocity increases as a function of time.

Despite the doubts about its validity, wave theorists use Eq.24 to determine the orbital

angular acceleration as a function of the angular velocity. For a nearly circular motion, their

result is

13

53 11

3

13

1 2GR 5

96

5

G m m

c M (25)

Using the time derivative of Kepler’s third law, the following radial velocity is obtained

3

1 2GR 5 3

64

5

G m m Mr

c r (26)

If the above equation is correct, then the their radial acceleration is given by

23

GR 1 25 7

64 13

5

Gr m m M

c r (27)

It must be emphasized that the preceding developments are entirely kinematical in nature.

Since the system dynamics is not taken into account, the resulting equations show unusual

characteristics. According to the EOB model, the radial dynamics must be governed by

2

EOB 2

GMr r

r (28)

In comparison, the result obtained by wave theorists does not contain and therefore totally

ignores the centrifugal acceleration term. Furthermore, the relativistic wave theory result is

proportional to the inverse of the seventh (!) power of the radial distance. Likewise, the

azimuthal equation motion of the EOB model is

EOB

2 r

r (29)

But, the wave theory result claims that the orbital angular acceleration is proportional to 11 3 .

More importantly, Eq.25 is independent of the radial velocity, ignoring the presence of the

Coriolis acceleration term. In summary, the mixing of the Newtonian and Einsteinian energy

equations gives rise to the loss of interaction between the radial and angular dynamics. This

decoupling is, of course, unphysical. Note that the above developments did not include the post-

Newtonian terms and the necessary corrections for an elliptical orbit. However, these additional

modifications do not affect the conclusions presented here. For example, the angular acceleration

still remains decoupled from the radial dynamics (Blanchet et al. 1995).

Finally, we must point out another glaring inconsistency. The EOB solution is based on

the assumption that radial and angular variables represent the actual distances. As mentioned

earlier, in General Relativity, r and have nothing to do with the physical positions. While

wave theorists ignore this important distinction between the coordinate and physical distances at

the emission site, they welcome it at the detection site. Consider a cloud of non-interacting test

particles. When a gravitational wave passes, the application of the geodesic differential equation

14

shows that the coordinate separation between any two particles remains constant. Thus, in order

to obtain a coordinate-independent effect, wave theorists use the spacetime line element to

calculate the proper spatial distance. The foregoing philosophy can be summed up as follows: If

the mixing of the Newtonian and Einsteinian concepts suits the needs of wave theorists, they will

embrace the results. Otherwise, the mixing will be rejected as an inappropriate idea.

6.0 The QUADRUPOLE FORMULA

We are now ready to discuss the theoretical shortcomings associated with the famous

quadrupole formula, details of which can be found in any textbook on General Relativity. The

solution to the relativistic wave equation in the presence of some non-zero matter distribution is

expressed in terms of the integral of a time-retarded scalar Green’s function multiplied by the

energy-momentum tensor T . However, in four dimensional spacetime, the directions of the

source and the field are different and related to each other in a complicated way. For example,

the spatial distribution of gravitational waves will be influenced by the energy-momentum

distribution of the coalescing binary black holes. It is, therefore, necessary to use a tensor-valued

Green’s function. We have just identified a mathematical flaw in the derivation of the

quadrupole formula.

Due to its analytical complexity, wave theorists convert their integral of the scalar

Green’s function multiplied by T into a second mass moment by applying various

approximations and more importantly by using the energy-momentum conservation relation

0vT . If r is the distance between the source of radiation and the field position of interest,

then their strain pseudo-tensor is given by

2

4 2

( )2( , )

ij

ij

d I tGh t

c r dtr (30)

where t t r c is the retarded time. The symbol ijI represents the tensor for the second

moment of the matter density distribution ( )x as defined below

3( ) ( )ij i jI t x x dx x (31)

One immediately recognizes that the quadrupole Eq.30 is unbalanced. The left hand side

has been calculated in the transverse-traceless gauge. But, the trace of the second mass moment

tensor is always non-zero. In order to circumvent this mathematical defect, wave theorists

conveniently convert ijI into a trace-free tensor. Clearly, this is an artificial step introduced by

hand without any physical justification.

The quadrupole formula of gravitational radiation holds true if and only if an

astrophysical system is isolated, that is, if the system has absolutely no dynamic interaction with

the rest of the Universe. Total energy and momentum are conserved in such a system.

Conservation of linear momentum means that the center of mass of the system does not

15

accelerate. Therefore, the dipolar component cannot contribute to the production of gravitational

waves. Likewise, there is no magnetic dipole radiation by conservation of angular momentum. In

short, gravitational radiation starts with the quadrupole term in General Relativity. But, this line

of reasoning contradicts the reality. Purely conservative systems represent mathematical

idealizations. All physical systems possess some degree of damping. When a dynamic system

loses energy through an irreversible process, its momentum is not conserved. Since energy

permanently leaves a binary star system as a result of gravitational radiation, the system is non-

conservative. Mathematically, this means that 0vT . The energy loss can be determined by

calculating the energy flux through a control volume, e.g., a sphere enclosing the system.

Furthermore, a pulsar system is always influenced by external forces. Firstly, there exists a force

vector arising from the gravitational potential of the host galaxy. Secondly, it is well-known that

galaxies flutter like a flag in the wind. So, there is a force transverse to the galactic plane. Hence,

contrary to the general relativistic claim, a pulsar system is not isolated and a non-zero dipole

radiation is physically possible. On dimensional grounds, we expect the rate of energy loss due

to dipole radiation to be

2 2 4

dipole

3

dE GM L

dt c (32)

where L is some length scale characteristic to the system. However, the dipolar radiation is

extremely rare because the system must be either a singleton or, in the case of a binary, mass of

the pulsar must be much greater than that of its companion. In the case of the binary pulsar PSR

1913+16, the two neutron stars have almost equal mass values (Hulse & Taylor 1974). The

system’s center of mass hardly moves as the stars orbit each other. Therefore, there is no

observable dipolar radiation.

A key experimental test of the quadrupolar formula involves its prediction for the decay

rate in orbital period bP . There are many pulsar systems where the general relativistic value of

bP does not agree with astronomical observations. In the literature, the offset of bP is usually

attributed to a number of ‘classical’ influences. A typical kinematic effect is a secular change due

to the presence of galactic gravitational field. Intrinsic variations include changes in the

quadrupole moment of the companion star and mass losses either from the pulsar or its

companion. For example, one can reduce the offset significantly by adjusting the distance to the

pulsar in the Skhlovskii (1970) effect. However, in some cases, even these additional fixes

cannot rescue the theory. Indeed, it is written in Stappers et al. (1998) that ‘… the orbital period

of PSR J2051-0827 is decreasing at a rate 12( 11 1) 10bP . This bP is some two orders of

magnitude greater than the contribution expected from general relativistic effects, 14( 3 1) 10bP , and the possible influence of the Shklovskii term is negligible.’ Regarding

the pulsar PSR J1756+2251, we read in Ferdman et al. (2014): ‘The observed and kinematic-bias

corrected orbital decay rates ( obs

bP and intr

bP , respectively) disagree with the GR prediction by

2 3 … It may be that the GR formulation for quadrupolar gravitational-wave radiation is

incorrect, or that GR itself has broken down in the case of this system.’ Furthermore, explosive

sources like supernovae offer another test of the quadrupolar formula. According to an order-of-

16

magnitude calculation done for a typical supernova, the formula gives an unrealistically low

radiation rate, more specifically, four orders of magnitude smaller than the Sun’s luminosity

(Narlikar 2010).

Because of the importance of the quadrupole formula, its use in the literature needs to be

critically examined. Most application papers follow the same logic. First, the fusion of General

Relativity and Newtonian gravity is assumed to be valid. Second, the system is considered to be

binary, rejecting the presence of extra objects or external rings of matter. Third, the system is

said to be clean, that is, gravitational radiation is the only energy loss mechanism. These papers

use the same software package to obtain a least-square fit for more than 20 orbital parameters,

leaving two major unknowns, namely, pulsar and companion mass values. If the relativistic wave

theory is correct, then the curves of the advance of the periastron p, the time dilation parameter

, the rate of change in orbital period bP , and the orbital inclination factor sins must all

meet at a single point in the pulsar-companion mass plane. The orbit parameters p and

indicate the existence of a strong gravitational field while bP and s are associated with the

gravitational radiation. In the case of the Hulse-Taylor pulsar, although the curves of the triad

p bP meet at a single point, the curve of the orbital inclination factor has a significant

uncertainty band (Weisberg & Huang 2016). On the other hand, the pulsars PSR J1141-6545

(Bhat et al. 2008) and PSR B1534+12 (Fonseca et al. 2014) don’t even pass the p bP test.

This failure exposes another defect in the relativistic wave theory as we explain below.

In order to answer the question of how the gravitational radiation affects the orbital

evolution of an astrophysical system, the modern implementation of the relativistic wave theory

refers to the classical electromagnetic radiation. Since the theory assumes the absence of dipole

radiation, the rate of energy loss scales as 5c instead of 3c . This means that, in the post-

Newtonian approximation, the radiation-reaction force terms of the equation of motion must

come with odd powers of 1c starting at order 5c (Poisson & Will 2014) . Using this heuristic

argument, the formula for the instantaneous rate of change of the orbital period is found to be

53

3

192 2( )

5

b c

b

dP GMf e

dt c P (33)

where 3 5 1 5

1 2 1 2( ) ( )cM m m m m is known as the chirp mass. The symbol ( )f e represents the

correction for elliptical orbits. There are grave problems associated with the above formula.

Firstly, it is physically unrealistic because the right hand side does not include a contribution

from the radial dynamics. Secondly, the rate of change is always negative because ( ) 0bP t for

all time. Thirdly, bP is inversely proportional to ( )bP t . But, more importantly, the least-square fit

treats bP as a constant by assuming a secular drift of the form 0 0( ) ( )b b bP t P P t t . This also

does not reflect the reality. For example, since the Hulse-Taylor pulsar is highly eccentric with

0.617e , the rate of change of its orbital period oscillates widely between positive and negative

values like a sinusoidal signal.

17

As we discussed previously, in addition to gravitational radiation, there are other physical

mechanisms by which an astrophysical system can dissipate energy and thus experience orbital

decay. Since it is impossible to quantify the exact contribution of each of these mechanisms, the

physically sensible approach is to regard the overall effect as some kind of a drag force that

drives the system toward coalescence. More specifically, Lagrange’s equations of motion must

include non-conservative forces derivable from Rayleigh’s dissipation function. In view of the

experimental fact that the damping forces and torques are proportional to the velocity and

opposite in direction to the velocity, one arrives at

2

EOB 2 r

GMr r D r

r (34)

EOB

2 rD

r (35)

where rD and D represent the damping coefficients in the radial and azimuthal directions,

respectively. Now, using the definition 2bP , we obtain

EOB

2( )b

b b

P rP P D

r (36)

Note that the period time derivative is not only directly proportional to the orbital period but also

dependent on the radial dynamics.

The evolution of the Hulse-Taylor pulsar can be simulated by utilizing its latest orbital

parameters. Furthermore, by adjusting the damping coefficients, one can easily determine the

decay in the orbital period derivative as good as if not better than the degree of accuracy

achieved by the least-squares fitting algorithm. In fact, we have found the decay in bP averaged

over an orbital period to be 122.3953 10 for 16 16.432 10 sD and 16 16.0 10 srD .

This result is in excellent agreement with the fitted value of 12( 2.398 0.004) 10 as reported

in Weisberg & Huang (2016).

7.0 MORE INTERNAL INCONSISTENCIES

We have already drawn attention to many inconsistencies in the theory of gravitational

waves. Alas, the relativistic theory has even more internal conflicts as listed below:

The loss of linear and angular momenta in a radiating binary system has an important

consequence: the stars cannot be orbiting in the same plane as assumed by the EOB

model. Thus, one must include the polar dynamics in the direction and consider orbit-

orbit interaction. Since the orbital motion is governed by non-linear differential

equations, the trajectory will deviate considerably from a pure post-Keplerian ellipse.

The use of a retarded time in the quadrupole formula automatically assumes that a clock

at the source ticks at the same rate as a clock at the site of a detector. We have already

18

shown that this is manifestly not true. To a distant observer, clocks near a black hole

appear to tick more slowly due to the time-dilation effect of a gravitational field.

Since a binary star system is always influenced by external forces, the divergence of the

energy-momentum tensor is not zero, that is, v vT f where f represents the force

density field. This means that 0vh , which violates the key hypothesis of the wave

theory, namely, the Lorenz gauge condition.

Gravitational wave interferometers are designed to record the physical separation of two

massive particles (mirrors) in order to detect the oncoming distortion in space, expansion

in one direction and contraction in another. But, cosmologists assert that gravitationally

bound objects such as planets, atoms etc. do not experience spatial expansion. We see

that one group of general relativists contradicts a second group. Each LIGO installation

uses laser interference to detect changes in length on the order of 1810 m and strains on

the order of 2110 . Note that these measurements involve quantum-domain waves, which

are subject to Heisenberg’s Uncertainty Principle. Since every quantum-domain wave is a

complex-valued probability wave, the outcome of experiments has an intrinsic

indeterminacy. Zero-point energy excitations such as virtual particle-antiparticle

interactions interfere with the local radiation. Therefore, it is impossible to deduce if an

interference pattern is due to gravitational or electromagnetic radiation. More

importantly, General Relativity is a deterministic field theory whereas quantum field

theory is stochastic. That’s why there are numerous theoreticians working very hard to

combine quantum mechanics and gravity. But, neither the String Theory nor the Loop

Quantum Gravity offers physically meaningful explanation to the gravitational

phenomena at the subatomic level. In summary, in the absence of a correct quantum

theory of gravity, one simply cannot explain or interpret the highly noise corrupted data

collected by LIGO.

According to the quadrupole formula, the amplitude of the strain pseudo-tensor falls off

as 1 r in empty space, which is due to the assumption that gravitational waves are

generated by isotropic point sources, emitting energy equally in all directions. This

assumption leads to the use of a scalar-valued Green’s function in the solution of the

inhomogeneous wave equation. But, a binary star system with unequal mass values is not

spherically symmetric. Therefore, the correct solution must be expressed in terms of a

Green’s function which relates all components of the source tensor vT to all components

of the strain pseudo-tensor vh . In other words, Green’s function must be a tensor. As we

mentioned earlier, this is a fatal mathematical mistake in the relativistic wave theory.

In electromagnetism, an accelerating charged particle emits radiation. On the other hand,

in General Relativity, gravitational radiation is due to the time-rate of change of matter

distribution. An individual black hole, even if it is accelerating, cannot generate

gravitational waves by itself because its mass distribution is time-invariant. Only a system

of moving black holes or neutron stars can produce radiation. The LIGO collaboration

has published numerous videos illustrating gravitational wave emission from a coalescing

black hole pair by using numerical simulation of Einstein’s field equations. These videos

routinely show two spheres representing the black holes and two spirals trailing the

spheres. Watch, for example, the LIGO simulation (2015). The spirals are said to be the

outwardly propagating waves. This is grossly misleading picture of gravitational

radiation. First, since a single black hole is unable to radiate, a trailing spiral cannot

19

emerge. Second, according to the relativistic theory, gravitational waves emanating from

a binary system are transverse (in the z direction) to the plane of the coalescing black

holes. See, for example, fig. 3.7 in Maggiore (2008) for the angular distribution of

radiated power. But, the videos show gravitational waves traveling outward in the x-y

plane of the binary like ripples on a pond. It is ironic that even the LIGO simulations

don’t agree with the theory they are supposed to represent.

8.0 EFFECT of the VACUUM ENERGY

An observant reader will note that the most remarkable aspect of the formulation of the

relativistic wave theory is the omission of the vacuum energy term g . In the literature, the

exclusion of this term is attributed to the tiny magnitude of the cosmological constant, which

makes it unimportant for determining the motion of planets, stars, black holes, etc. However, the

LIGO collaboration reports that the GW150914 event occurred at a distance of 1.3 billion light-

years (410 Mpc). The GW170104 event was even further away at a distance of about 3 billion

light-years. When dealing with distances on cosmological scales, the intervening vacuum plays

an important role and its presence cannot be ignored. According to the LCDM model of the

contemporary cosmology, the energy density of the vacuum is associated with the accelerating

expansion of the Universe. Historically, the foundational research on gravitational waves and

numerical relativity took place mostly in the 1940s, 50s, and 60s well before the advent of the

LCDM model. It is therefore no surprise that the relativistic wave theory does not pay any

attention to the vacuum energy. In what follows, we obtain solutions to the linearized field

equations in the presence of the cosmological constant term. By substituting Eq.6 into the

energy-momentum tensor of the vacuum, one finds

12

( ) ( )g h h h (37)

Now, the linearized wave equation becomes

2 1matter 2

12 [( ) ( )]h T h h (38)

In empty space, we have

2 12

2 ( )h h h (39)

Observe that the partial differential equations are non-homogenous in the presence of the

vacuum energy. To gain insight, we shall first study the non-diagonal component 12h . In

Cartesian coordinates, it is governed by

2 2 2 2

12 12 12 12122 2 2 2 2

2h h h h

hc t x y z

(40)

20

In the case of a gravitational wave propagating in the positive z-direction, the wave is

independent of x and y coordinates and the above equation reduces to

2 2

2 212 12122 2

h hh c

t z (41)

where 2 22 c . This linear differential equation also occurs in quantum mechanics and is

called the Klein-Gordon equation. We see that the empty space with vacuum energy constitutes a

dispersive medium. Recall that a dispersive dynamic system transmits waves of differing

frequency at different speeds. Consider, for example, a harmonic wave of the form

( )

12( , ) i kz th z t Ae (42)

where A is a complex constant. Now, the wave 12( , )h z t is a solution provided that

2 2 2 2 0c k (43)

The above is the dispersion relation for the empty space with vacuum energy. In a dispersive

medium, waves of different frequencies can form a wave packet. Typically, a wave packet

contains a harmonic wave traveling with the phase velocity modulated by another harmonic

wave traveling with the group velocity. The phase velocity of an empty-space wave packet is

2

2 2 2

2( ) 1 1phasev k c c

k c k k (44)

Similarly, the group velocity is

2

22 2

( )2

11

group

d c cv k

dk

kc k

(45)

Observe that the phase velocity of the wave packet is greater than the speed of light in empty

space. However, the group velocity is physically more significant because the energy of the wave

packet is transmitted at groupv , which is always less than c . Waves of any length can travel, but

their frequencies must be at least . Hence, there are no traveling waves for , that is, no

signal can be transmitted. The standard gravitational wave theory claims that dispersion is totally

negligible (Thorne 1982). We see that this assertion is not true. Furthermore, according to the

announcement of the GW170817 event, the gravitational wave signal arrived 1.7 seconds earlier

than the electromagnetic waves. This is not possible in the presence of the vacuum energy.

21

Next, let’s discuss the diagonal components of the empty space differential equation.

Here, the terms h and h can be omitted because they are the product of two very small

quantities. In this case, one has

2

matter

12 [( ) ]h T (46)

and the empty space wave equation is

2 2h (47)

To gain insight, let’s consider the component 11( , )h z t representing a wave propagating in

the positive z-direction. Its differential equation is

2 2

11 11

2 2 22

h h

c t z (48)

Because the right-hand side is independent of time, the presence of the vacuum introduces a

steady-state effect. By keeping t constant, the steady-state solution can be determined from

2

11

22

d h

dz (49)

After integrating twice, one finds

2

11 1 2( )h z z C z C (50)

The parameters 1C and

2C are normally found by specifying two boundary conditions. However,

as two black holes merge, their boundaries will change with respect to time. Since the event

horizon of a black hole is not observable, 1C and

2C remain undetermined. After combining the

homogenous and non-homogenous solutions, we obtain

2

11 1 2( , ) sin( )h z t h kz t z C z C (51)

Note that small perturbations in the spacetime metric tensor tend to grow without a bound as

z . In other words, vacuum energy amplifies the perturbations as the waves travel. A

similar increase in magnitude also occurs in the other diagonal elements of the strain pseudo-

tensor. Thus, a tiny explosion in a distant galaxy will eventually give rise to enormous

gravitational waves which will destroy everything in their paths. However, these tsunami-like

activities don’t occur in our Universe. According to the relativistic wave theory, the signal

strength falls off in proportion to the inverse of the distance travelled. This assertion is in

disagreement with the result when vacuum energy is taken into account.

22

The LCDM model of standard cosmology maintains that the value of the cosmological

constant is about 52 210 m . Accepting this value, the difference between the quantum

mechanical estimate of the vacuum energy density and the observed density has been estimated

to be about 120 orders of magnitude, which is known as the worst prediction in the history of

physics. Referring to Eq.51, in order to have a strain amplitude on the order of 2110 resulting

from the merger of two black holes at a distance of 410 Mpc as in the GW150914 event, the

value of must be less than 72 210 m . This is even worst than the worst prediction in physics.

In short, either the LCDM model is wrong or the theory of gravitational waves is not correct. Yet

again, an application of the theory of General Relativity in one particular area contradicts the

results in another area.

9.0 FLAWS in SIGNAL PROCESSING

The purpose of this section is to show that the signal processing models and data

reduction algorithms used by the LIGO collaboration have significant flaws. According to the

collaboration, a traveling gravitational wave is observed if there is a near simultaneous signal

with consistent waveforms at their two detector locations. Among the announced six events, the

first GW150914 event is statistically the most significant because its signal, after whitening and

filtering, rises above the detector noise level while the remaining five are very weak events with

signal amplitudes considerably below the detector noise even after whitening and filtering.

The working assumption of the collaboration is that no other physical source can possibly

produce these waveforms. However, when a large blob of anti-matter encounters another large

blob of matter in the star forming regions of a neighboring galaxy, a similar quantum-domain

waveform emerges. That is, the emitted energy waves will continuously increase in strength,

followed by a ring-down period. Furthermore, merging of two strong magnetic fields exhibits the

same frequency signature. Also, nuclear fusion reactions deep inside stars can emit a burst of

energy with waveforms comparable to the ones observed by LIGO. Although these are high

frequency waves, due to the Doppler, cosmological, and gravitational redshifts, their frequencies

will occasionally fall into the finite bandwidth of the land-based wave detectors.

The signal processing model of the LIGO collaboration (Abbot et al. 2016) is the

superposition of the gravitational wave signal ( )h t and of the detector noise ( )n t

( ) ( ) ( )s t h t n t (52)

The detector is assumed to be recording an exact copy of the wave signal albeit with an additive

stationary Gaussian random noise with zero mean value. The goal of the collaboration is to

maximize the following signal-to-noise ratio (SNR)

2

2( ) | ( )

( )( ) | ( )

s t h tt

h t h t (53)

where the inner product of two time-domain signals is given in terms of their Fourier transforms

23

*

(2 )1 21 2

0

( ) ( )| 4 [ ]

( )

i f t

n

h f h fh h e df

S f (54)

The symbol ( )nS f represents the (supposedly known) power spectral density of the detector

noise.

The signaling model adopted by the LIGO collaboration is very simplistic in the sense

that it ignores the effects of the cosmic medium between the wave source and the detector (the

channel in the language of the signal processing literature). First of all, it is impossible to have a

line of sight transfer of gravitational energy between the source and the detector because the

theory deals with a plane wave of infinite extent not a narrow stream of gravitons. This means

that an incoming plane wave will encounter multiple black holes, worm holes, globular star

clusters, molecular clouds, etc. on its way to a detector. Since a gravitational wave is distorted by

the material content of the Universe, the collaboration makes a mistake by totally ignoring the

refraction, reflection, diffraction, and focusing effects in its signal detection model. A

gravitational wave impinging on another cosmic medium will be partially transmitted and

reflected. Similarly, when a wave passes through a galactic gravitational potential well, the shape

of its wave-front will change (refraction). A black hole, for example, may reflect a gravitational

wave as well as increase its frequency by injecting energy. The theory of General Relativity

allows the existence of worm holes that can transmit waves from the future(!). Spiral arms, the

bar structure, and the spherical bulge of a galaxy can diffract a gravitational wave causing

multiple versions of the same wave to arrive at a detector in different directions and

polarizations. In fact, the space between the arms of a spiral galaxy can serve as a gravitational

waveguide. Furthermore, a spatial variation in the electric permittivity and magnetic

permeability values is unavoidable because the intergalactic, interstellar, and interplanetary

media all have non-uniform physical characteristics. There is empirical evidence that

electromagnetic properties of a medium are functions of the frequency of any wave propagating

within the medium. Therefore, different regions of a plane wave will be traveling at different

speeds, 1c , producing time-delayed versions before the wave arrives at a detector.

Taken as a whole, these diverse channel conditions will cause the incoming gravitational waves

to exhibit interference and beat phenomena. Since these phenomena are not being taken into

account by the relativistic theory, the overall noise in detection must have multiplicative, non-

stationary, and non-Gaussian characteristics.

A modern and more realistic signaling model would assume the presence of multiple

emitters and multiple detectors. This kind of spatial multiplexing is known as a Multiple-Input

Multiple-Output (MIMO) system in the signal processing literature. Let ( , )a t denote the time-

varying channel impulse response at time t to an impulse at time t . If ( )nh t is gravitational

signal radiated by the thn source, then the signal received by the thm detector is given by

1 1

( ) [ ( , ) ( ) ] ( , ) ( )N N

m mn n mn n

n n

s t a t h t d a t h t (55)

24

where the symbol denotes the convolution operator. Suppose that the LIGO predictions for the

rate of mergers are true. Then, at any given time, there must be several compact binaries in their

late stages of coalescence. Thus, N represents the number of astrophysical systems that are

generating strong gravitational waves. Currently, there are three detectors in operation, namely,

two LIGO locations and a VIRGO detector. Accordingly, the MIMO signal model becomes

1 11 1 1 1

2 21 2 2

3 31 3 3

( ) ( , ) ( , ) ( ) ( )

( ) ( , ) ( , ) ( )

( ) ( , ) ( , ) ( ) ( )

N

N

N N

s t a t a t h t n t

s t a t a t n t

s t a t a t h t n t

(56)

or

( ) ( , ) ( ) ( )t t t ts A h n (57)

where ( )tn is a vector of non-stationary and non-Gaussian noise signals. We shall not discuss

how to extract the desired wave signal because it is beyond the scope of this paper.

The routine practice of the LIGO collaboration is to fit highly noise corrupted data to

various templates by using a matched-filter algorithm. But, in data reduction with such a filter,

the transmitted waveform is known and the objective is the detection of this signal against a

background noise. In addition to the aforementioned flaws in signal modeling, data whitening

and cleaning algorithms used by the collaboration have major mistakes. For example, the article

by Raman (2018) provides evidence that the collaboration’s matched-filter misfires with high

SNR and cross-correlation function (CCF) peaks all the time. This erratic behavior of the filter is

due to the lack of cyclic prefix necessary to account for circular convolution and error in

whitening operations. The normalized CCF of the wave events is indistinguishable from

correlating the template vs bogus chirp templates. Regarding the GW170817 event, the same

article concludes that an external electromagnetic signal is the most likely candidate for the

coincident false detection. In a series of publications (Liu & Jackson 2016; Creswell et al. 2017;

Liu et al. 2018), the researchers at the Niels Bohr Institute have demonstrated that detection of a

gravitational wave signal using methods based on simulated templates can misidentify the

transients and/or systematic effects as part of the signal. In their latest paper (Creswell et al.

2018), the group investigates the degeneracy of simulated waveforms using the EOB model. For

the GW150914 event, they show that waveforms with greatly increased masses (e.g.,

1 70m M vs LIGO’s 1 36m M and 2 35m M vs LIGO’s 2 29m M ) yield almost the

same SNR in the strain data.

It is clear that, due to the experimental bias acquired from fitting data to their simulated

templates, the LIGO collaboration is inferring what they wish to see. This is typical cherry-

picking of data designed for confirmation of theory. Their false-alarm rate of 5-sigma is

associated with the probability of observing a certain type of waveform assuming the simulated

templates are correct. This statistics has no grounds because, in the present context, one is

interested in the probability of the relativistic wave theory being correct given the observations.

The accuracy and robustness of the data (recorded as well as simulated) are beset by major

problems. Simply put, the claims regarding the real-time observation of gravitational waves lack

the extraordinary experimental and theoretical rigor they need. Therefore, the evidence presented

by the LIGO collaboration does not pass the Sagan test.

25

10.0 FINAL REMARKS

The preceding sections pointed out the experimental (the space is flat, non-uniform

mass/energy distribution in the Universe, infinite redshift), theoretical (the use of a scalar rather

than tensor-valued Green’s function, omission of vacuum energy, etc.), illogical (fusion of two

disjoint theories of gravity, the use of the EOB model), and unnatural (no dipole radiation, no

refraction, diffraction etc.) reasons why the general relativistic wave theory is polluted with

unphysical assumptions and inappropriate approximations. Even though any one of these reasons

is sufficient, when one considers the totality of the mathematical inconsistencies and internal

contradictions, it becomes clear that one must rule out the existence of gravitational waves based

on the experimentally refuted hypothesis of General Relativity, that is, the supposedly non-

Euclidean geometry of the physical space.

It must be emphasized that the above conclusion does not exclude the gravitational waves

arising from the fluctuations of the interstellar and intergalactic media. Despite Einstein’s

conviction that ‘space is not a thing’ (Cheng 2005), the physical space, far from being a void, is

filled with fluid-like deformable plasma of electron-positron pairs, proton-antiproton pairs, all

kinds of neutrinos, quanta of electromagnetic and gravitational energy fields, etc. Thus,

gravitational waves generated by explosive sources, such as supernovae, can travel through this

physically active cosmic medium. Images of planetary nebulae provide observational evidence

for the existence of such waves. For instance, the image of Abell 39 is a beautiful example of a

spherically symmetric gravitational shock wave.

Finally, we expect that there will be considerable irritation among supporters of the

currently accepted theory. However, the validity of a scientific theory depends on its

mathematically rigorous foundation and experimentally verified predictions but not on its

heuristic arguments and inherently contradictory assumptions. As the famous British philosopher

Bertrand Russel once said: ‘Even when the experts all agree, they may well be mistaken.’ It is in

the best interest of the scientific community if research efforts are directed towards finding the

physically correct theory of gravitational waves based on the quantum and continuum dynamics

of the cosmic medium filling the Universe.

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