REFLECTIONS ON GRAVITY
Part 2
Martin Gottschall  PhD  © 2004

DERIVATIONS

In PART 1 we found gravity to be a force after all, and found it necessary to posit the proposition that in all matter, antimatter and energy, including photons, energy and mass (or inertia) are associated exclusively and only with electric and magnetic fields. We also found that the energy released when any object is lowered in a gravity field was derived from the rest or field energy of the objects being lowered. Conversely, when an object is raised in such a field, the energy absorbed becomes part of the rest or field energy of the object being lifted. We will now consider the derivation of the gravity equations.

In deriving the equations of gravity, we will consider two kinds of motion in a gravity field, the free fall of the object under consideration or its steady raising or lowering by some unspecified agency. In free fall, all rest energy absorbed as the object rises is supplied by its kinetic energy and all rest energy released as the object descends is imparted to the object as kinetic energy. In free fall the total energy of the object stays the same, but in steady (low speed) motion the total energy of the object is just its rest or self energy, the kinetic energy being taken as negligible and constant. In this case the raising or lowering agency supplies any increase in rest energy and absorbs any such decrease.

THE SPEED OF LIGHT

To discover the equation for the speed of light in a gravity field, we will consider a photon rising (or falling) vertically in it, as in FIGURE 4(a). Here a photon of mass "m" rises at the speed of light "c" in a gravity field with local value "g" Newtons/kg, through a distance "dr" in time "dt". As said, its total energy "m.c.c" stays constant since it is in free fall, hence:

        d(m.c.c)  =  0                 (10)

The symbol "d" outside the brackets of EQUATION (10) designates "the change in" or the "differential of" the quantity in brackets. This change is zero and the equation states just that. The "d" in "dr" and "dt" has the same meaning. The photon experiences the pull of gravity, and since it is rising, its momentum "m.c" decreases as given in:

        d(m.c)  =  m.g.dt              (11)

Here a minus sign signifies a decrease in momentum, and upward momentum is taken as positive. Note that "g" which acts downwards is a negative number, making the change in momentum negative (the momentum is decreasing).

These equations can be solved by the methods of differential calculus, and yield:

        c.dc  =  -g.dr                 (12)

This equation tells us that the local speed of light "c" when multiplied by the change in that speed "dc" yields a number equal (=) to the number obtained when multiplying the negative of "g" by the change in vertical position "dr". In accordance with well established practice, we define the gravity potential by the symbol "P" and a change of potential as "dP". "P" measures how much energy it takes to lift a unit mass from wherever it is, out of the gravity field to infinity, and we treat it as a positive number, thus:

        dP  =  g.dr                    (13)

tells us that as the photon rises its potential becomes numerically smaller (dP is negative) and hence:

        c.dc  =  -dP                   (14)

This equation can be "integrated" which means that instead of talking about the changes of "c" and "P", we find the magnitudes themselves, by systematically adding up all of these changes. When we do this, and define the speed of light at infinity (when "P" is zero) as "C" we get:

        c.c  =  C.C  -  2.P            (15)

This equation tells us that the speed of light "c" in a gravity field is determined entirely and only by the value of "P". This is also the equation which was published by Weil and others, as described in PART 1. EQUATION (15) and others which we will derive presently, can all be cast in the form of circle equations, and when we do, they teach us things we might otherwise not suspect. This will be deferred till later. Note that if we had considered a photon falling, instead of rising in a gravity field, we would also have obtained EQUATION (15).

To make the speed of light zero in EQUATION (15), "P" must have the value "C.C/2". We will need to explore the meaning of the speed of light being zero later in the context of "black holes". For the present we will note only that EQUATION (2) in PART 1 tells us that when "c" is zero, "e.u" has to be infinite. At this point space is an infinite dielectric making it a perfect conductor of AC as well as being infinitely permeable to a magnetic field. If this condition could be produced or even approached artificially in some convenient way, we would have "superconductors" and "supermagnets" that we can at present only dream about.

REST MASS

Consider now an object of mass "q" which is being lowered at a small, steady speed in a gravity field "g", through a height "dr", FIGURE 4(b). In this case "q" is the rest mass which is doing work on the agency lowering it, by imparting energy to it, and the change in energy "d(q.c.c)" is no longer zero:

        d(q.c.c)  =  -m.g.dr  =  -m.dP   (16)

In this case both "g" and "dr" are negative numbers and multiply to a positive number (dP), so we retain the minus sign to make the change in energy of the object a decrease (negative). We can now solve EQUATIONS (14) and (16) together and obtain:

        q.c  =  Constant  =  Q.C       (17)

We find that the product of the rest mass "q" and local light speed "c" remains the same at all potentials. If we designate the rest mass of the object when at infinity as "Q" and the speed of light there as "C" then we can calculate the value of that constant for any object. Since "c" decreases as we descend in a gravity field, "q" must increase if the product "q.c" is to remain constant. Even though the energy "q.c.c" of the object decreases as it descends, its mass actually increases, becoming infinite when "c" is zero.

EQUATION (17) allows us to calculate the rest mass "q" of an object:

        q  =  Q.C/c                    (18)

And we can obtain "C/c" from EQUATION (15) as:

        C/c  =  1/Sqr(1 - 2.P/(C.C))   (19)
and
        c/C  =  Sqr(1 - 2.P/(C.C))     (20)

in which the operation "Sqr" takes the square root of the quantity in brackets following on.

REST ENERGY

The rest or self energy "re" of an object can be calculated from equations which we already have. We know that:

        re  =  q.c.c                   (21)

and from EQUATION (17) we can replace "q.c" with "Q.C" and multiply by C/C, which is just "1" and so always allowed, so that;

        re  =  Q.C.C.(c/C)             (22)

Which tells us that the rest energy in a gravity field is the rest energy at infinity multiplied by the factor "c/C" which we can obtain from EQUATION (20). This equation tells us how the rest energy of an object changes as it is raised or lowered in a gravity field.

TIME

Since the speed of light slows in a gravity field it will not surprise us that  all activity in general slows at the same rate as the slowing of the speed of light, as we shall see presently.

SIZE

However, the size of objects does not change. If we were to take the assembladge of FIGURE 1(b) in PART 1 and lower it in a gravity field, we would find that the field energy would decrease in the correct way only if the volume of the field stayed the same. If we were to lower a spinning object, its spin energy would decrease in accordance with EQUATION (22) but its angular momentum would remain the same. These conditions also require the radius of gyration of the object to stay the same. General Relativity finds that radial distances and tangential distances are not the same in a gravity field, but we do not find this, although we do find that the effect of gravity on a radial photon is different from its effect on a tangential photon, as indicated in PART 1.
 
Let a mass "q" having a radius of gyration "k" spin at an angular speed "w" in a gravity field. Its spin energy "se" is:

        se  =  q.k.k.w.w/2             (23)

and its angular momentum "am" is:

        am  =  q.k.k.w                 (24)

At infinity, when "P" is zero, "q" is "Q", "k" is "K", "w" is "W", "se" is "SE" and "am" is "AM". Since angular momentum remains the same:

        q.k.k.w  =  Q.K.K.W            (25)

and from (22) the ratio of rest energy in the gravity field to rest energy at infinity is "c/C". The spin energy is a part of the rest energy of the objects, so that:

        (q.k.k.w.w)/(Q.K.K.W.W)  =  c/C   (26)

EQUATIONS (25) and (26) will solve, giving:

        w/W  =  c/C                    (27)

And EQUATIONS (18), (25) and (27) will solve to yield:

        k/K  =  1                      (28)

telling us that the rate of spin of the object slows in step with the slowing of the speed of light, and that the radius of gyration stays the same. It is said to be "invariant" with respect to the gravity potential.

PHOTONIC ENERGY

We have already stated that for any object in free fall in a gravity field, the total energy remains the same. This is particularly significant in the case of photons. Whether rising or falling, photons retain a constant energy, even though their mass, wavelength and frequency changes. Clearly, photonic energy is also invariant with respect to the gravity potential. When matter objects rise or fall in a gravity field, there is an exchange of kinetic energy and self energy, and the rest energy of any object is not invariant in a gravity field. For a photon, however, kinetic energy and rest energy can not be distiguished one from the other.

This is one point on which our theory differs from General Relativity, which asserts for example that photons leaving the event horizon of a black hole can not escape to infinity. At the event horizon, the gravity potential is C.C/2 for General Relativity and for our theory. We say that at this potential photons have zero energy left, and since their energy remains constant, it is still zero when the photon has risen to infinity.

It is probably more instructive to consider a situation where the speed of light is small but not zero. In this case photons can still move and so can embark on an escape trajectory (say vertically up). As the photon rises, its wavelength inreases, but its energy remains constant throughout. We can now decrease the speed of light further and move progressively as close as we like to the point "c=0", and observe the trends we obtain. We can do the same with particles.  

We will argue later that for various reasons the gravity potential never attains the value C.C/2, and certainly can not go beyond this value. With General Relativity the potential is assumed to go beyond this value, giving rise to the so-called "naked singularity", and all sorts of other theoretical problems.

Although the rest energy of objects varies in gravity fields, photons and objects in free fall retain the same energy at all values of the gravity potential, giving us a ready-made means for comparing energies from different potentials, and we can therefore say that the total energy is also invariant with respect to gravity potential.

FORCE

Since length is invariant with respect to gravity, and rest energy is not, then force, which is tied to energy and distance by the formula Energy = Force x Distance, must also be not invariant. Thus, a spring streched to a certain degree exerts a diminishing force in a gravity field as it is lowered since the stored elastic energy (a form of rest energy) decreases.

THE ENERGY OF A GRAVITY FIELD

So far we have looked at only one side of the gravity interaction. We have considered photons and objects in free fall or being slowly raised or lowered in the gravity field of a relatively massive object like a planet or sun. However, the interaction affects the larger body too. Not only does the smaller object experience a change in gravity potential as it moves in the field of the larger body, but the larger body undergoes a potential change due to the approach of the gravity field of the smaller body. While the potential change may be very small, a large mass experiences this change, and the total change in rest energy for the larger body turns out to equal the change in rest energy of the smaller one. This is demonstrated readily using Newton's inverse square law of gravity.

FIGURE 5 illustrates Newton's law of gravity. In the space surrounding any mass "m" kg, there is a field of force which attracts a mass of 1 kg with a force of "g(m)" Newtons (this unit of force was named in honour of Isaac Newton), which is calculated with the formula:

        g(m)  =  m.G/(d.d)             (29)

in which "G" is known as the gravitational constsnt (about 6.7E-11 N.m.m/(kg.kg). Note that the attraction increases in direct proportion to the mass "m" and decreases as the inverse square of the distance "d". Consequently, a mass of "M" kg will be attracted with a force "F":

        F  =  M.g(m)  =  M.m.G/(d.d)   (30)

FIGURE 6 shows the two masses "M" and "m", seperated by a distance "d". We could also consider the field of attraction about the mass "M", for which the value of "g(M)" will be:

        g(M)  =  M.G/(d.d)             (31)

The force "F" exerted by the large body upon the small one is "g(M)" Newtons for each kg of the small body making the total force "m.g(M)" Newtons, which is identical to the value for equation (30). The two forces are of equal magnitude but their directions are opposed, so there is no question as to which is which.

In our deliberations to date, the object we considered was vastly less massive than the object which created the field we were studying. The force of attraction was large enough to move the smaller object, but far too small to displace the larger one by a measureable amount in the available time. Suppose that after some time, the smaller object has moved closer to the large object by a distance "x". The kinetic energy gained by the small object is "F.x". The larger object experiences a change in potential of "g(m).x", so that the total change in energy for it is "M.g(m).x" which also equals "F.x". This is the amount by which the rest energy of the larger object is decreased.

Thus, we find that Newton's gravity is such that both objects release equal amounts of rest energy when they approach and gain equal amounts when they move apart. We know that the energy released by the smaller object adds to its kinetic energy, and we postulate that the energy released by the larger object  is fed into or builds up, the slightly intensified gravity field associated with the two objects moving together by the distance "x". (For a time I speculated that the rest energy released by the larger object appeared as heat, and that would have made possible a most interesting "gravity heat engine", and would have had significant geophysical implications).

We have considered a special case here where only one of the objects undergoes an appreciable displacement. In the general case, when the interacting objects are both planets and/or suns, we find that each body releases rest energy which becomes partly kinetic energy and partly gravity field energy. However, in total exactly half the energy is kinetic and the other half is absorbed in the composite gravity field.

Even this case is somewhat special in that we have been at rest relative to one of the objects, usually the larger one. A more general case has both of the interacting objects moving relative to the observer. Thus, if we are studying the interaction of a number of moving objetcs, we can not be at rest relative to most of them, and will need to use this more general approach. We will not present this derivation, but the interested reader may readily do this using the methods outlined here.

A particularly striking demonstration of the energy associated with gravity fields is afforded by another special (and hence mathematically simple) case. Consider a very thin spherical shell of particles of matter collapsing under its own gravity, FIGURE 7. When we do the calculations, we find that as the shell collapses, the gain in kinetic energy of each particle in the shell is exactly half the change in potential energy which that particle undergoes at the same time. In this case each particle of the collapsing shell contributes equally to the kinetic energy of the shell and to its gravitational field energy.

GRAVITY IS NON-LINEAR

EQUATION (18) tells us that the rest mass of an object increases as its gravity potential is increased (which increases the ratio "C/c"). This means that in the case of FIGURE 7 for example, as the shell collapses, its total mass must increase, and its gravity field exhibits a kind of gravity run-away effect, which is only important at so-called "relativistic" potentials when "P" is no longer small compared to "C.C/2". This "non-linearity" is present in General Relativity and our theory but is absent from Newton's theory of gravity, which considered mass to be invarient with respect to gravity potential. We will be able to explore this non-linearity fully when we have defined how the gravity field is produced, and hence can say something about how it radiates into the space about the gravitating body.

GRAVITATIONAL "RADIATION"

In PART 1 we posited that the inertia and energy of all particles of matter was embodied in an electromagnetic field structure which constitutes its rest energy. In PART 2 we have found that when matter is drawn together by gravitational attraction, half the rest energy released is used to build the resulting gravity field. Since the gravity field occupies the space around the objects involved out to "infinity", and the energy fed into it comes from the highly concentrated self energy of the atoms of the masses involved, it must be radiated into the surrounding space in a manner fundamentally different from what we ordinarily call "electromagnetic radiation", since it remains trapped in the space about the gravitating body indefinitely, and yet is reabsorbed by those atoms again when the masses move apart, decreasing the gravity potential.

In Newton's day many scientists had a hard time accepting the notion that one astronomical body could exert a force on another through millions of miles of "empty" space. The concept was called "action at a distance". Since that day we have learnt that "energy" IS "matter" (E=m.c.c), and our theory requires us to consider how a gravity field (a kind of tenuous matter, like electric and magnetic fields) is built into the space around all bodies. Those scientists who had a problem with "action at a distance" had a valid point. What they could not know then was that a space containing a gravity field was no longer "empty", and contained just what was needed to generate the gravity force.

Note that scientists using General Relativity also speak of "gravity waves" or radiation. This is not at all what we are referring to when we consider the energy flows associated with the increase or decrease of a gravity field when masses come together or move apart. This other radiation is associated with similar but very intense gravitational changes which are thought to emit a stream of energy into the surrounding universe, never to return, much like for example radio waves.

CONCLUSIONS TO PART 2

In PART 1 we explored the nature of the gravity force in such a way that its operation could be visualized. In PART 2 we set out to quantify the properties of the gravity field. We are now able to predict the outcomes of a variety of gravity experiements such as the deflection of light and radio signals passing near a body, how gravity fields delay such signals passing through them, the reduction in rest energy, increse in mass, slowing of clocks etc.

We can also explore "relativistic" gravity - when the potential is so high that the speed of light is noticeably samller than its value at infinity - where the change in mass becomes prominent also, and the gravity interaction becomes "non-linear" and mathematically more complicated.

Most important perhaps for our greater understanding of gravity, we have found that when gravity fields are built up, energy is radiated from the matter involved, into the surrounding space where it remains, and must flow back into that matter again when the gravity field is subsequently reduced.

PART 3 will explore the remarkable fact that the equations of Special Relativity and our gravity equations can all be graphed as circles. When presented in this way, these equations confront us with certain fundamental questions that go beyond energy, matter and gravity, and point towards well defined inferences.

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