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1
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.
2
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.
4
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
5
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
6
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
7
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)
8
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.
9
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
10
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
11
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)
12
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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