_{1}

The resemblance between the equation for a characteristic hypersurface through which wavefronts of light rays pass and optical metrics of general relativity has long been known. Discontinuities in the hypersurface are due to refraction involving Snell’s law, as opposed to discontinuities in time that would involve the Doppler effect. The presence of a static gravitational potential in the metric coefficients is accounted by an index of refraction that is entirely dependent on the space coordinates. The two-time Einstein metric must be reinterpreted as a two-space scale metric because of the two different speeds of light. It is shown that the Schwarzschild metric is incompatible with the laws of classical physics. Gravitational waves are identified with the transverse-trans-verse plane wave solutions to Einstein’s equations in vacuum, which propagate at the speed of light. Yet, when energy loss is evaluated, his equations acquire, surprisingly, a source term. Poynting’s vector, which is not a true vector, is defined in terms of the pseudo-gravitational tensor, and hence energy is neither localizable nor conserved. The solutions to the equations of motion are geodesics and, by definition, do not radiate. A finite speed of propagation implies that gravitational waves should aberrate, like their electromagnetic wave counterparts, and if they do not aberrate they cannot radiate.

The classical tests of general relativity involving the bending of light in a gravitational field, and the advance of the perihelion of Mercury still form the cornerstone upon which the theory rests. The former has been interpreted by the apparent slowing down of clocks in a gravitational potential, while the latter makes use of the equivalence between geodesics of light-rays propagating in a static gravitational field and geodesic paths of general relativity [

One the one hand, there is remarkable numerical equivalence between predicted and observed phenomena of the two classical tests, yet on the other hand, there are huge discrepancies between the Schwarzschild metric and the laws of classical physics. Moreover, the phenomena have been given inaccurate physical explanations. This is not surprising insofar as the laws of Newtonian physics are independent of location whereas those of general relativity are apparently location dependent. Newtonian theory has no upper bound on the propagation speed, while general relativity has, since it is a generalization of the special theory.

There is even no clear meaning of the line element in general relativity. Einstein generalized the hyperbolic line element of special relativity by allowing the metric coefficients to become space dependent, and, therefore, to depend on the gravitational field. Yet, there are no known solutions to the two-body problem in general relativity, so the generalized line element describes a fictitious mass, or what is commonly designated as a “test” particle. We don’t even know if Newtonian attraction applies in general relativity.

In this regard, an anecdote of Levi-Civita [

During a conversation that I had with Einstein a few years ago, I asked him … if it was possible to give any concrete interpretation … to the elementary chromotopic interval

The saving grace of general relativity, as exemplified in the calculation of the perihelion advance of Mercury, is that geodesic motion can be described by the propagation of wavefronts traveling at the speed of light that happen to coincide with the null geodesics of electromagnetic waves [

Weyl [

Studying the exact solution, Baldwin and Jeffrey [

Later, Bonner [

Weyl [

The thesis of this paper is that general relativity describes the propagation of electromagnetic disturbances through an inhomogeneous medium caused by the presence of a static gravitational potential. We will show that the only unambiguous interpretation applies to null geodesics which are derived from an eikonal equation in which the index of refraction is space dependent. Time, or times―whether local or coordinate―have no role in the theory. Rather, the metric has to be interpreted as a measure of the different arc lengths in vacuo and in a medium with an index of refraction. Doppler’s principle, in the form of a second-order frequency contraction, which is the result of setting the space part of the metric equal to zero [

It is commonly accepted that clocks run slower in a gravitational field. Yet, Einstein’s equivalence principle allows one to replace gravitational acceleration by any other generic one, and it has been emphasized by Einstein himself that acceleration does not affect the rate at which clocks tick. The distinction between local and coordinate times in the metric are interpreted as differences in the paths due to discontinuities in the wave surfaces as light traverses media of different indices of refraction.

In §2, we will review the reasoning that led Einstein to originally determine the bending of light in a static gravitational field. In a series of mathematical manipulations, that maintain the same units but not the same physics, Einstein goes from a system of uniform velocity, whose frequencies experience a Doppler effect, to one of constant gravitational acceleration to, finally, to a static system in gravitational field. More recent authors have followed suit, thereby creating a great deal of confusion. Black body radiation is not something that is experienced by an accelerating observer that vanishes for a uniformly moving observer, or one that is at rest [

In §3 we will repeat Ives’ incomplete derivation of the Schwarzschild metric from the analysis of an interferometer in a gravitational field. We will show that the propagation of a light beam in an interferometer placed in a static gravitational field is analogous to that in a birefringent material whose index of refraction is related to the ratio of distances in Euclidean and hyperbolic spaces. Local time cannot be equated with the length of the arm of the interferometer, and instead of a two-time metric, Ives should have obtained a two-space metric due to the presence of a static gravitational field.

In §4 we will analyze the incongruities when Newtonian forces that act in Euclidean spaces are assumed valid in a hyperbolic space of non-constant curvature given by the Schwarzschild metric. In §5, we will contrast the two-time interpretation of the metric with one of two-space scales, arguing that it is not a Doppler effect, but, rather, one described by Snell’s law of refraction that is caused by the presence of a static gravitational field.

The motion described by general relativity is geodesic, and, by definition, cannot radiate. The coincidence between Einstein’s equations of motion and those of null geodesics of electromagnetic waves has led to the incorrect conclusion that if electromagnetic waves radiate so, too, must gravitational waves. There is no self-energy of a particle in a gravitational field like there is in an electromagnetic field.

A well-defined energy stress tensor for electromagnetic waves does not carry over to their gravitational analogue; as Einstein [

In §6 we claim that since gravitational waves do not aberrate they cannot radiate; radiation occurs in the direction of the Euler force. A fortiori, they would need a Poynting vector to determine the direction of energy flow, and this would require the analog of a magnetic field, the so-called gravitomagnetic field [

By replacing the velocity in the Doppler expression for the frequency shift by the product of gravitational acceleration with time, and then replacing the latter by the ratio of the distance that light traverses in the given time interval, and the speed at which it propagates, Einstein was able to convert a uniform velocity into a completely static expression. The only common thread in the three expressions is the same units. However, it implied that a static gravitational field can cause a frequency shift [

Although it could have been derived simply by appealing to the viral theorem, it points to the fact that a medley of different physical situations can be used to transform the Doppler effect into something completely foreign to it. This has led to many completely erroneous claims like accelerations can cause Doppler shifts [

When we realize that the sole generator of the Doppler effect is the uniform velocity of a body then any reference to acceleration, or to a static gravitational field, as the cause of the frequency shift is patently false. The Doppler effect maintains a constant speed of light so that any change in the frequency must be compensated by a corresponding change in the wavelength of light in order to keep their product constant. Other optical phenomena, like refraction, permit either frequency, or wavelength, to change at the expense of a constant speed of light, c.

When c is not a constant, we can have a stationary, but inhomogeneous medium, or a homogenous but non-stationary one. The gravitational field,

more slowly in a material medium than in vacuum, the velocity being inversely proportional to the refractive index of the medium. The phenomenon of refraction is in fact caused by the slewing of the wave-front in passing into a region of smaller velocity. We can thus imitate the gravitational effect on light precisely, if we imagine the space round the sun filled with a refracting medium which gives the appropriate velocity of light [my italics]. To give a velocity of

Any problem on the paths of rays near the sun can … be solved by the methods of geometrical optics applied to the equivalent refracting medium … the total deflection of light passing at a distance r from the centre of the sun is

whereas the deflection of the same ray calculated on the basis of Newtonian theory would be

[in units where

The factor of 2 is precisely that required to convert the angular velocity of a Keplerian orbit into an escape velocity.

We will argue that the tortuous derivation of Einstein [^{4}-law for a black body [

However, Einstein [

from what just has been said we must use clocks of unlike constitution, for measuring time at places of different gravitational potential. For measuring time at a place which, relative to the origin of the coordinates, has a gravitational potential

The principle of the constancy of the velocity of light holds good according to this theory in a different form from that which usually underlies the ordinary theory of relativity.

By writing down (1), Einstein was admitting that the velocity of light has been decreased by an amount

However, (1) has nothing whatsoever to do with a frequency shift given by the classical Doppler effect,

where g is the gravitational acceleration at the surface of the earth,

This is not not Kepler’s third law. The seeming analogy with the Doppler effect has allowed Einstein to introduce the discrepancy in the rates of clock motion in a gravitational field and in its absence. The Doppler effect has absolutely no place in the discussion, that can simply be treated as the propagation of light from the vacuum to a medium of index of refraction (2).

If it were the Doppler effect, the conservation of energy and momentum would have to be satisfied. Einstein writes the conservation of energy as

where the right-hand side represents “the increase in gravitational mass…, and therefore equal to the increase in inertia mass as given by the theory of relativity”.

The conservation of energy, according to Einstein’s Doppler principle is

where h is Planck’s constant. In addition, we must also have the conservation of momentum,

where m must be the peripheral mass that does not appear in Kepler’s law. Dividing (5) by (6),

where we imposed Einstein’s stationary condition, (4). There is no way that (7) can be satisfied so that we come out with

―even as an approximation. Einstein’s entire analogy with the Doppler effect, and, consequently, with it the different rates of ticking of clocks, is vacuous.

Rather, by writing down (1), Einstein has implicitly admitted that he is treating an inhomogenous, static medium that is amenable to Snell’s law of refraction. All what he said previously is completely superfluous to what follows, and by using

Snell’s law reads

where the angles of incidence,

Einstein’s relation is completely fortuitous, and his result from his general theory of relativity cannot be related to the difference between proper and world times since, as we have shown, time and the rate of ticking of clocks do not enter into the physics of the problem.

Ives’ [

The length of the interferometer supposedly undergoes a Lorentz contraction by a factor,

where c is the speed of light, and V is the “escape” velocity

Reference to motion is completely superfluous, what Ives needs is the static contraction factor in (10).

The length of the stationary interferometer, L, will only be contracted in the direction of the field,

but not in the direction normal to the field. Frequencies will be shortened by the same factor while the period of the interferometer in a stationary field,

Ives first considered the interferometer in a “stationary” gravitational field where one arm is in the radial direction, while the other arm is normal to it. Light sent out and reflected from a mirror in the normal direction will take a time

In constrast, the time taken for light to make an out-and-back journey in the radial direction is

From this it follows that

Everything can be explained without recourse to contracted motion simply by considering a birefringent crystal with two indices of refraction: the index of refraction of the ordinary wave,

Ives was also under the mistaken impression that the “ticks” on a clock should be altered when placed in a static gravitational field. Based on the result he wants to derive, namely, the Schwarzschild metric, he claims that the frequency of a light ray in the presence of a large mass will be “red-shifted”.

Ives then imagined that there is a hollow spherical sphere encompassing a diagonal mirror of the interferometer. When the gravitational field is turned on, the sphere of radius a becomes an oblate ellipsoid, with radii a, a, b, where

Ives went on to consider the speed of light along any radius r making an angle

The velocity of light in any direction is

It is evident that if the radius is in the direction of the field,

In terms of the polar coordinates,

so that its velocity will be

so that

Ives then claimed that the velocity of light,

Why he has changed nomenclature is that

The generic speed (14) can now be expressed as:

The ratio,

is an index of refraction measuring the difference between Euclidean,

For (16) is c times the ratio of the Euclidean distance

and

is the hyperbolic distance of the well-known Beltrami metric, although it will not be of constant curvature unless

Instead of (15), the ratio of the velocity (16) to the radial coordinate r should be the same ratio as c to arm of the interferometer, L, viz.,

Thus, relative speed is

The time taken to complete the out-and-back journey in any arbitrary direction is

on the strength of (19). However, Ives uses (15) which replaces

If somehow

Ives clearly realized this is the weak point of his derivation. As he openly admitted:

The identification of

So what does (21) describe? According to Levi-Civita [

Even the two-body problem, solved long ago by Newton, has very little chance of being solved successfully in general relativity. This is because, in the relativistic scheme, the reaction principle no longer holds, and we do not even know how to begin going about the reduction of the partial differential equations to ordinary differential equations―let alone integrate them.

Moreover, if the correct relation, (19), were used, the indefinite metric would be:

However, as far as the optical metric,

It should be clear from Ives’ derivation of the Schwarzschild metric that two times, local,

It is truly amazing that the metrics of general relatively can yield such precise values for the advance of the perihelion of Mercury, and the gravitational deflection of light by a celestial body, when the Schwarzschild metric cannot even distinguish between proper and coordinate time in deriving Newton’s law and Kepler’s second and third laws, predicting gravitational repulsion for velocities surpassing a critical value, and the existence of a critical radius where the centrifugal force becomes attractive. The interpretation of r as a radial coordinate is also troublesome since

We will see that no matter how hard authors have tried to unite the Schwarzschild metric with the dynamical laws of physics it is impossible. Since discontinuities appear, the two-time interpretation of the metric is at fault and one must consider different measures of path lengths when different indices of refraction are present.

The Schwarzschild metric is both diagonal and time independent. If we write the Schwarzschild metric in polar coordinates

The equations of motion for a free particle in general relativity are:

where

The equation of motion for the radial coordinate in the equatorial plane,

where the prime stands for differentiation with respect to r.

The pure radial part of the equation of motion is obtained by setting

where the dot denotes differentiation with respect to coordinate time, t. Inserting the Schwarzschild metric (21) for radial motion,

gives:

which resembles Newton’s law but is not Newton’s law because it is expressed in proper time,

Multiplying both sides of (27) by

apart from an arbitrary constant of integration we have set equal to zero. Hence, as a consequence of Newton’s law in proper time, (27), we get expression (28) for non-relativistic virial, also in proper time. This is indeed surprising since Newtonian mechanics knows no limit on the speed of particles, and, yet, general relativity predicts a critical speed, [cf., (35) below]!

Introducing this expression into the radial Schwarzschild metric,

gives:

after rearranging and taking the positive square root. This is not the same expression that would be obtained by considering only the temporial component of the Schwarzschild metric [

The velocity,

exists for all

is negative, and, hence, attractive, for

For

Hilbert [

where

It appears, however, that the first to introduce the notion of gravitational repulsion was Droste [

as the physical definition of the length

and derived the “renormalized” velocity:

and acceleration:

in proper time.

However, it is not the proper time (30), and has nothing to do with the Schwarzschild metric since it is defined by (37) having set

The distinction between local and coordinate times couldn’t be more paradoxical than in the case of purely rotational motion. When

Neglecting the minor point that the Schwarzschild radial coordinate is not the Newtonian radial coordinate, (40) resembles Kepler’s third law,

Consequently, the Schwarzschild metric, (21), yields two laws of classical physics, Newton’s, (27), and Kepler’s, (40), but with two different times!

Kepler’s third law, (40), can be extracted from the radial equation,

which reduces to

for a circular orbit,

Yet, for Kepler’s second law, (24) gives:

The condition for a central force to hold is that Kepler’s second law has to be obeyed. Notwithstanding the claim that (43) is Kepler’s third law, and not Kepler’s second, Birkhoff [

Birkhoff also sees no problem in determining the time coordinate t by sending out radial light signals from the center. Yet, the coordinate velocity, (31), goes to zero at

The equation for the geodesics, (24), also gives

where k can be considered as the conserved energy when it is associated with the fourth component of a 4-velocity. This constant scaling factor is usually set equal to one for radial motion [

If the left-hand side is interpreted as the “radial” kinetic energy per unit mass [

Even worse, the radial equation of motion, (41), can be written explicitly as:

The left-hand side of (46) is the Schwarzschild radial acceleration,

implying gravitational repulsion.

Ohanian [

where the critical speed

This would imply that light is bent backward away from the mass M which it grazes.

However, Einstein [

…a light ray passing along by a heavenly body suffers a deletion to the side directed toward the heavenly body.

It seems difficult then to accept that “light is repulsed everywhere, …increasing from zero at

The proper velocity is unity differing from the coordinate velocity by a factor

Another disturbing feature is that that the speed of light should be frame independent, and here it is clearly not. In the laboratory frame the speed is

Moreover, if light has mass equivalent, why should it be always repulsive in contrast to particle behavior? The deflection of light by a central mass would imply that light is attractive, and not repulsive. The fact that light is always repulsive and increases from 0 at

It would not be unreasonable to question the fact that since the index of refraction is the ratio of the coordinate time to the proper time, according to (44), that

For light waves, the proper time interval vanishes. Since t is a measure of the distance along a light ray in units of the speed of light, we may introduce a geometrical length s of the light ray by using the relation

where the only independent parameter measuring the length along the light rays is s which replaces the proper time

More recent controversies regarding the existence of gravitational repulsion have arisen [

the discrepancy between the signs of the accelerations of low-speed and high-speed particles is a perplexing violation of the equivalence principle. General Relativity attributes this discrepancy to a bad choice of coordi- nates―the coordinates r and t do not represent locally measured distances and times.

Reference to the equivalence principle is a red-herring, for we ask: equivalence of what? Reference to local coordinates is equally as bad: the Schwarzschild metric could not be more democratic for it gives us Newton’s and Kepler’s second law in proper time, while, simultaneously, it gives Kepler’s third law in coordinate time for circular motion!

Ohanian also misconstrues attraction and repulsion with acceleration and deceleration. Newton’s law has the same form as Coulomb’s law (which was Coulomb’s impetus for his derivation). There is one sign to Coulomb’s law for unlike charges and the other for like charges. The signs refer to attraction and repulsion, respectively.

Ohanian further faults Hilbert, who, “being a better mathematican than a physicist”,

unfortunately misconstrued this deceleration as a repulsive gravitational force… Hilbert naively assumed that the force is in the direction of the coordinate acceleration

which is always negative. Therefore the direction of the force (and also the direction of the proper, or relativistic acceleration

Reference to the rate of change of the momentum (51) is also a red-herring: the peripheral mass m is velocity independent so that it reduces to the acceleration (27). In the law of gravitation the peripheral mass does not appear at all. It means that the acceleration at any point in the field depends only on the point where the particle is situated and not on its mass. Moreover, if the momentum is a function of the proper time, so too should the angular velocity, and from (40) we know that it isn’t since otherwise we would not get Kepler’s law.

The confusion of what is the angular velocity leads to other incongruous results. The Schwarzschild metric can be cast in the form

Assuming J is angular momentum where

In comparison with “Newton’s” law, (27), there is now the additional term which looks like centrifugal force, except for a coefficient which can be both positive or negative. Both Hilbert and Einstein claimed that the radial coordinate of a circular trajectory must satisfy

in order that the centrifugal force be repulsive. However, the left-hand side of (53) does not represent acceleration, just like J does not represent angular momentum per unit mass.

Hilbert’s inequality (54) follows directly from the Schwarzschild metric,

for circular motion in the equatorial plane by dividing through by

which clearly shows that inequality (54) must be strictly obeyed for circular orbits.

Although Kepler’s law does not condone restrictions on the radial coordinate, the Schwarzschild metric does! The inverse of the positive root of (56) can be considered as an index of refraction in the direction of rotation, just as the first expression in Ives’ expression (12). The times are proportional to the path lengths; there will be two of them: one for light in vacuum and the other in a medium of index of refraction,

The change in definition of the angular velocity, and the unwarranted conservation of the pseudo-angular momentum, (43), deform an otherwise circular orbit and, not unsurprisingly, predict the advance of the perihelion. Although the numerical result coincided with the advance of Mercury it did not fare as well with the other planetary orbits, which differed in all cases in magnitude, and, in some cases, even in sign [

This, supposedly, applies to circular orbits for which^{1}. It has even been claimed that the Rayleigh condition for orbital stability for

Inequality (54) has nothing to do with the centrifugal force changing sign and becoming attractive. Rather, it has to do with the curvature of the metric [^{2}, the condition for “Rayleigh” stability would be

where

Inequality (54) can be traced back to Hilbert [

where

Parenthetically, we mention the black hole “excision” technique has been suggested [

Notwithstanding how many “advances” in numerical relativity that have been made over the past two decades, and their supposed final experimental vindication in the LIGO claim of the observation of gravitational waves [

Instead of using the equation of motion (25), one [

and using the first integrals, (43) and (44). J would be the conserved angular momentum in local time, but not in coordinate time, and k is an arbitrary constant of integration in (44), which we shall appreciate is not so arbitrary insofar as it is related to the total energy of the system. This is in contrast to the arbitrary constant of integration that emerges from the integration of Einstein’s equations under the condition of emptiness which is identified as the mass of the “empty” space.

With the aid of the first integrals, (43) and (44), (58) can be written as:

where an index of refraction has been defined as:

Light rays in this medium, with refractive index (60), are identical to the paths of particles that move in a Newtonian potential

The energy is

provided the small perturbation

with its focus at the origin. The integration constant has been chosen so as to make r a minimum at

The Hamilton-Jacobi equation of geometrical optics is [

where the indices vary over the space coordinates, and the index of refraction, n, is given by (60). If the

For a relativistic system, the right-hand side of (61) would be equal to the square of the peripheral mass [

This resolves Ives’ conundrum of setting the length of the interferometer equal to a time interval. As far as gravity is concerned, the general relativistic metric does not relate proper time to coordinate time, but, rather, it relates two path lengths, or wavelengths, that occur when light traverses an inhomogeneous medium whose index of refraction is proportional to a static gravitational field.

In this case (60) reduces simply to:

or, equivalently,

upon taking the positive square root, where

A light signal travels through space over a set of wavefronts:

that propagate at the speed of light, c, where S appears as Hamilton’s charac- teristic function. If they were particle trajectories then we could have some grounds of believing in gravitational waves. But there is no peripherial mass to be found. Whittaker [

The fundamental distinction between electromagnetic and the putative gravitational waves is that the latter are not known to refract. In the presence of media with different indices of refraction, the characteristic hypersurface

Even though Eddington [

The fundamental property of light is its frequency. The wavelength of light keeps changing as light passes through different media having different indices of refraction which depend on the spatial coordinates. This guarantees that the ticks on a clock do not get lost as we pass from one frame to another. Even the eikonal approximation makes a demand on the wavelength irrespective of the frequency. Its validity requires that the wavelength change very slowly in com- parison to the characteristic dimension of the system. In contrast, there is no condition on the frequency that the wave can assume. The wavelength becomes proportional to the velocity of the electromagnetic wave, and, hence, inversely proportional to the index of refraction.

The relation between wavelength and the index of refraction is given by Snell’s law:

The index of refraction for the Schwarzschild metric is usually identified as:

which is a function of the radial distance, r, and differs from (2) by the famous factor of 2. However, we can consider the non-static case in which r remains constant, and the mass is a function of time [

invariant. Introducing

for the frequency.

Again consider the Schwarzschild metric (21), but this time consider the system stationary,

How do we know that (68) is not

on account of (11) for the escape velocity?

If an atom is emitting at a “proper” frequency

If more pulses are transmitted than received then some pulses are lost [

If we introduce (11) into (70) it will look like a second-order Doppler effect,

with the escape velocity, V, replacing the velocity in Kepler’s third law. What about the first-order Doppler effect? It is just a quirk that the time component of the Schwarzschild metric, (68), can be made to look like the Lorentz invariant definition of local time.

Another point is that (71) is not obtained from (70) for a system at rest. The spatial part of the metric is involved since it defines the velocity, but it is clear that it is not the Schwarzschild metric that is involved.

One could rightly argue that the calculation of the total transit time, (37) has no meaning since

Buy why should we contemplate an average of the forward and reverse Doppler shifts on going from coordinate time to proper time?

In other words, what is the relation between the indefinite spacetime metric and (72)? It is usually assumed that

represents the time component of the metric under stationary conditions,

If we consider, along with Faraday, two charges of opposite sign in relative motion, lines of force will emerge from the positive charge and terminate at the negative charge. If the negative charge is removed the lines of force on the positive charge in motion must terminate somewhere. Since the positive charge is confined by the walls of the room, the lines of force must terminate there. So without reference to the walls of the room there can be no Lorentz contraction.

We do not know what changes in frequency have to do with a stationary gravitational field, but we do know what Snell’s law is. If the refractive index can be factored into the product of spatial and temporial terms, then the time component will be equal on both sides of the interface so that it will have no effect on the change in wavelength as the rays cross the interface.

When a light wave travels from a low to a high index of refaction, the wavelength decreases so as to satisfy Snell’s law:

As the wave begins to coil up in the medium with the higher index of refraction, both

Can we imagine a situation in which the index of refraction is the same everywhere at any instant in time and yet is a decreasing function of time? If so, the leading and trailing edges of the wave will not encounter any change in the index of refraction so the wavelength remains the same but the number of pulses will change because the index of refraction is a decreasing function of time. We should thus expect a law of the form (67) because the

Consequently, (70) makes absolutely no sense: A static gravitational field has absolutely no effect upon the frequency. In other words, defining a velocity in (72) as

If it is assumed that c is a constant,

General relativity makes no qualms about being able to measure coordinate times and distances. The “metric” frequencies and wavelengths are obtained from the “locally” measured time intervals

According to Einstein, “nothing compels us to assume that clocks in different gravitational potentials must be regarded as going at the same rate”, but, by the same token, nothing compels us not to do so. Since we know from the work of Ives and Stilwell that velocity affects the ticking of a clock [

There is no logical reason, nor any physical explanation, why it should be in any way caused by the acceleration that produces the motion? There is therefore no logical reason why it should be caused by a gravitational potential, which is assumed to be equivalent to acceleration.

Evidence supporting that the rate of a clock is impervious to accelerations comes from storage ring experiments on muons up to accelerations of the order of 10^{19} g. It is astonishing that Einstein, in one breath, should argue that accelerations should have no effect on the rate of a clock, while, in the next breath, he should argue that a static gravitational potential should affect the rate of ticking of a clock. What about his cherished equivalence principle?

It has been argued that “it does not matter if we put the refractive index n in the space part or its inverse

The contradiction between frequency shifts in Doppler’s law and wavelength variations in Snell’s law could not be made any more apparent than to consider the Schwarzschild metric in the equatorial plane. The Lagrangian describing this geometry is given by [

The dot denotes differentiation with respect to an affine parameter

The variational principle,

yields the standard equations of motion, and determines

The Euler-Lagrange equations are, thus, given by [

And the equation for the ray trajectory,

describes closed Keplerian orbits that are conic sections when

The constant

Any static spacetime metric, like Schwarzschild’s metric (21), gives the projections

which is a positive-definite line element on a 2-dimensional surface

Since the propagation time does not depend upon the independent variable r,

where the prime denotes differentiation with respect to r, the Euler-Lagrange equation reduces to the first integral

Multiplying numerator and denominator by

which is none other than Snell’s law,

Since

where

Solving (82) for the inverse of

which is analogous to―but not the same as―the equation of the ray trajectory, (79).

Whereas (66) is a bona fide index of refraction, obeying

The space dependency of

This proves conclusively that there are no gravitational frequency shifts in the absence of radiation. Said differently, clocks do not slow down in a static gravitational field, because some form of energy transfer would be necessary to allow the frequency to increase when the body is approaching, or decrease when it is receding. In other words, the change in the frequency in the Doppler effect is sustained by the inertial motion of the body.

Whereas the time component of the full metric, (68), would evoke an analogy with a second-order Doppler shift [

gives Snell’s law. The relevance of Snell’s law, as opposed to Doppler’s law, rests on the fact that the Schwarzschild metric is static, possessing a pair of Killing vectors. Moreover,

In the presence of radiation things are different since the mass in

To predict the frequency shift, we need to determine the ratio of two time intervals: the time interval between two successive wave-crests at the emitter, and those at the receiver. Since we have no way of determining the successive light pulses emitted by the source, we assume that they are given by the same (constant) frequency as that found in the lab,

When the source is infinitely far away,

Taking the time derivative of (85) we get:

Expression (86) says that the rate at which the frequency increases is directly proportional to that at which the mass increases, rather than the expected result that it should be the rate of decrease of frequency.

Consider a charged particle which travels a distance

Remarkably, this, again, is Snell’s law. For a small angle,

If the charge accelerates for a small time

is multiplied by the otherwise static gravitational field,

where

Over a century ago, Barnett [

Larmor’s theorem in electrodynamics is expressed in terms of mechanical quantities: the radiation field corresponds to acceleration, and the magnetic field to the velocity of the moving charge. In the gravitational analogue, the magnetic field would correspond to the angular velocity of the rotation of the mass.

This is certainly non-Newtonian on two counts: In Newtonian gravitation the force acts only in a radial direction, and the speed of propagation of the force is infinite so that there can be no aberration, (87). Consequently, there is no analogue of a magnetic field, and no propagation of gravitational waves at the speed of light.

This was appreciated long ago by Heaviside [

in analogy with the Lorentz force, where

Thus, the gravitational Poynting’s vector is:

where

According to some authors [

Repeating Einstein’s calculation of the energy loss through gravitational radiation, Eddington [

for the linearized metric representing small deviations when a gravitational wave passes.

For LL and LT plane waves, the Riemann tensor,

Although Einstein’s condition of emptiness, (90), has been used to derive gravitational plane waves, the material energy tensor is not zero! For Einstein considers a spinning rod of length 2a, and density per unit length,

In this “region”, the Ricci tensor,

does not vanish. How far from this region do we have to be for (90) to apply? Einstein does not answer the question, but assumes that the metric coefficients are now functions of time only, and only those terms in the second derivatives of

where only the yy and zz stresses act on the rod.

There is nothing abnormal about the stresses

containing only first, and not second, derivatives of the small, perturbed metric,

In the case of electromagnetic waves, the right-hand side of Einstein’s equations (cf., (94) below), partake in the derivation of Poynting’s vector, but, gravitation is not included in the matter-energy stresses,

The pseudo-tensor (93) makes Poynting’s vector a pseudo-vector. And although the flow of energy is mainly along the axis of rotation of the rod, “eight times more intense than in the direction at right angles to the rod” [

As Einstein [

Gravitational waves need a medium to propagate in just like sound waves for they are the “ripples of spacetime”. However, whereas sound waves satisfy the wave equation “unconditionally”, gravitational waves also do so, “but not unconditionally”. The condition is that “sources shall be such as can occur in practical problems, but it none the less limits the number of admissible solutions” [

In contrast to TT plane gravitational waves, electromagnetic waves have a source term in Einstein’s equations [

written in terms of the contracted Riemann tensor (90) instead of the Ricci tensor, (91), where

is the electromagnetic stress tensor which is a bilinear product of the covariant curl of the covariant vector potential,

For (95) there is true Poynting vector, and energy flow is deducible together with the localization of energy. The waves carry away energy―“not a small quantity of second-order but a quantity of first-order” [

Yet, whereas plane gravitational waves need a medium to propagate in, electromagnetic waves can propagate in vacuum. However, all that was done was to add an energy-stress tensor to the right-hand side of (90). The fact that the linearized Einstein equations are used would mean that they are formerly equivalent to Maxwell’s equations so there would be an equivalent to a magnetic field. The effect of the tensorial character of the pseudo tensor is felt beginning at second-order in the perturbed metric. Hence, the covariant character of linearized equations is with respect to Lorentz transformations only [

According to Weyl [

means that there is no matter present and no physical fields except the gravitational field.The gravitational field does not disturb the emptyness [sic]. Other fields do.

Undoubtedly what Dirac is referring to is what is “left-over” of the Riemann tensor when its trace vanishes on account of Einstein’s condition of emptiness, (90). The remainder, or trace-free part of the Riemann tensor, known as the Weyl tensor, can even be non-zero in the case of no sources. Such would be the case for gravitational waves propagating in spacetime.

The proponents of such an interpretation draw the analogy with electrody- namics waves propagating in a vacuum. However, we know what produced such waves―charges and currents―even if they are not physically present. Yet, we do not know what caused the non-vanishing of the Weyl tensor so the analogy is less than perfect, to say the least. If gravitational waves are the “ripples of spacetime”, or small variations in the curvature of spacetime, what created them? The role of matter can’t be subtracted that easily. And, as we have seen, it isn’t: the TT gravitational waves are derived from Einstein’s condition of emptiness, (90), yet when it comes to calculating energy losses, energy and matter suddenly appear!

It is obvious that if the metric coefficients, ^{3} will vanish. What causes the time-independent coefficients to vary is precisely mass, or mass density, which defines the non- Euclidean disc. In that case both Ricci and Riemann will not vanish.

Loinger [

Without an analogue of a magnetic field in the linearized equations there is no energy flow, and, consequently, no gravitational waves. Moreover, the aberration of gravitational waves should be easier to observe than the gravitational waves themselves, since it is basically an optical property due to their finite speed of propagation. There is no need to fabricate an explanation of why the Riemann curvature tensor should be different from zero when its comtraction vanishes [

Their distinction with electromagnetic waves are non-existent since the linearized Einstein equations are equivalent to Maxwell’s vector theory of elec- tromagnetic propagation except for the additional condition that Einstein’s equations be satisfied. The existence, or lack thereof, of a medium in which to propagate is not dealt with, and the conclusion that “light cannot be propagated without a change in wave-form” [

The metrics of general relativity are non-Euclidean and hence incompatible with the classical laws of physics which live on Euclidean spaces. Newton’s law is universal whereas those derived from non-Euclidean metrics are limited in space because of the finiteness of the hyperbolic disc. Never has so much effort been consumed in trying to make them compatible, at least in limited domains. However, in no domain can gravity become repulsive nor centrifugal forces be attractive.

The metric itself is incompatible with the nature of the phenomenon being described. The two-time metric should be interpreted as a two-space metric, and phenomena arising from a static gravitational potential should be analyzed as optical phenomena in the realm of geometrical optics. It has been shown that this is the only consistent interpretation in which light rays pass through an inhomogeneous medium created by a static gravitational potential. Time does not enter into the considerations, but the in homogenity of the medium does.

In particular, we have shown that what has been interpreted as a gravita- tional frequency shift, at a constant speed of light, is a shift in the wavelength of light, at a variable speed of light. A discontinuity in the wave surfaces is due to a change in the index of refraction: it is Snell’s―and not Doppler’s―law that applies.

As Weyl [

The geodesic motion that determines the law of wavefronts concerns light rays and not test particles. Test particles are not sources of the gravitational field, and do not exist because no matter how small they would be they would have an associated gravitational field. The assumed smallness is not an argument to exclude such a field. It is inconceivable that such small effects, like the deflection of light by a massive body is being sought, and, yet, the gravitational field of a test particle is completely neglected.

The fact that Einstein’s equations are amenable to a geometric optics inter- pretation does not allow one to replace the propagation of an electromagnetic disturbance by a putative gravitational one. There is a well-defined energy tensor of the electromagnetic field whereas there is no corresponding tensor of the gravitational field, as Einstein realized and Levi-Civita [

To first-order, there is no distinction between plane gravitational waves and electromagnetic waves. In general relativity there is no distinction between the presence or absence of a propagating medium. The distinction between the vanishing of the full Riemann tensor and its contraction is a red-herring since the material stresses for gravitational waves exist, and, consequently, both are non-zero, even though gravitational waves have been derived under condition (90). A finite propagation speed and a finite flow of energy of gravitational waves means that they behave exactly like electromagnetic waves.

Only geodesic motion has been discussed for which no radiation can occur. The finite propagation of gravitational waves would mean that they manifest the same optical phenomena as electromagnetic waves, which they don’t. Numerical relativity does not reflect the limitations imposed by general relativity nor is it correct in its prediction of gravitational waves. The interference fringes that LIGO [

Lavenda, B. (2017) The Optical Properties of Gravity. Journal of Modern Physics, 8, 803-838. https://doi.org/10.4236/jmp.2017.85051