Gravity

A Rigorous Derivation of the Gravitational Field

By:- H. E. Retic

Published by-

The H. E. Retic Co.
West Caldwell, NJ

Copyright:- 1987
All Rights Reserved

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• The symbol "*" represents a multiplication.
  
  
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If it is found that a hardcopy of the material at this website is desired but printing the text on your computer is too tedious, a softcover book which contains the complete texts ("The Einstein Hoax", with "Gravity" and "Corrections of Residual Errors in Special Relativity" as appendices) is available at AuthorHouse.com under the title "The Einstein Hoax".

"Gravity" - Index of Topics

Part Number	Part Subject
Part 1	Introduction to Discussion of Gravity
Part 2	Laying the Groundwork
Part 3	Evaluating the Gravitational Conversion Factors
Part 4	Comparison with the 'Real World'
Part 5	The Complete Gravitational Field

Part 1 - Introduction to Discussion of Gravity

1.0.1 -In high school, the author asked the question: "Where is the energy stored when a weight is raised?" The answer which was received, that it was stored as potential energy, was frustrating. It seemed, at the time, to be a face saving means of saying that the answer was unknown. The author's interest in gravitation was rekindled in the early 1960's by an article in the 'Scientific American'. In the hope that General Relativity might provide an answer to that old high school question, the author began to study the subject. The effort ranged from reading popular books by reputable workers in the field to laboriously studying undergraduate and postgraduate level texts. As the activity proceeded, the authors initial open acceptance changed to bewilderment, then to disbelief and then to an overwhelming sense of disappointment as internal and external contradictions associated with General Relativity became more and more apparent.

• General Relativity provides an expression for time dilation which is additively rather than multiplicatively commutative. As a result, it cannot yield correct results in strong gravitational fields(e.g.- neutron stars, gravitational collapse) even though its conclusions are verified to the limits of observational accuracy within the weak fields of the Solar System.
  
  
• The time dilation provided by General Relativity implies the existence of equivalent dilations for other basic parameters such as length and force. The possible existence of such dilations is not only ignored, but seems to be denied.
  
  
• General Relativity teaches that space in a gravitational field is curved (non-Euclidian). Older texts teach that the straight line of that curved space is the "null geodesic", which is the path of a ray of light in the gravitational field and represents the least time path between two points. Recent texts recognize than an ideal massless string stretched between points in a gravitational field would follow a path having half the curvature of the null geodesic. In keeping with the accepted definition of a straight line, that path is the straight line of the non-Euclidian geometry of the curved space since it is the shortest distance between two points. This viewpoint apportions the observed curvature of a ray of light in the gravitational field as resulting from equal parts of conventional refraction and spatial curvature. However, it is easily shown that, if space is non-Euclidian in the gravitational field, a closed cycle perpetual motion machine is possible in principle.
  
  
• Some texts teach that, under General Relativity, the force which we experience as gravity results from the curvature of space. Curved space, of itself, would seem to be incapable of producing an observable force either on a moving or stationary object. Where the object is moving within the space, the inertia forces resulting from motion within the curved geometry would occur in a direction orthogonal to that geometry and cannot be observed within it. Other texts have taken a different view and state that General Relativity has shown that the force (energy) of gravity has been shown not to exist! This seems to contradict the most rudimentary experimental data, such as is obtained when one slips on the ice.
  
  
• Under both Newtonian Theory and General Relativity, the gravitational field is capable of creating energy. Indeed, in an early text, the statement is made that "the presence of mass calls into being additional mass" (in the form of gravitational energy), but this does not constitute a violation of the Law of Conservation of Energy because, if that energy were to escape the field, the work required for that escape would reduce the energy which would reach the external universe to a level which was no greater than the original mass energy which entered the field." Some individuals may find such a statement satisfactory, to the author it appears to be a rationalization.

1.0.2 - As a result of these and other considerations, the author decided to facilitate his understanding by attempting to generate an energy balance for the process of lowering a weight from the ceiling to the floor. At first glance, this appears to be simple. At the ceiling, the weight has a mass energy equal to its mass times the square of the velocity of light. At the floor, it has the same mass energy and has released an energy of fall equivalent to its mass energy times the gravitational potential between the ceiling and the floor. However, in providing a time dilation as a function of gravitational potential difference, General Relativity opened a door. Time dilation, which has been experimentally verified, may be equivalent to a change in the size of the units of measurement (Section 3.2 shows that this is actually the case). With this door opened, the possibility must be considered that other quantities are subject to a similar dilation. Thus, performing the desired energy balance requires that the relationship between the size of the unit of measurement at the ceiling and the size of the unit of measurement at the floor be established by a method which does not involve circular reasoning. It is not valid to assume that they are equal.

1.0.3 -The result of this effort was more fruitful than could have been hoped. Not only did the source of gravitational energy become apparent, it was seen to be released in complete compatibility with the Laws of Conservation of Energy and Momentum, to occur in three dimensional Euclidian space, and to be compatible with the second order gravitational effects which are cited as proofs of the validity of General Relativity. At the low levels of gravitational potential observable within the Solar System  (10^-6),  the difference between the conclusions provided herein and those of General Relativity are, and may always remain, undetectable. At much higher levels, such as in the vicinity of a neutron star, the differences are quite pronounced and lead to an unexpected bonus. They demonstrate that, contrary to the predictions of General Relativity, gravitational collapse has an end limit and does not proceed to a singularity. Near this end limit, the conditions which would be observed within a collapsed object bear a startling resemblance to present descriptions of our external universe!

Section 1.1 - A Tabular Comparison with General Relativity

1.1.1 - A tabular comparison of the compatibility of General Relativity and of the results of this discussion with our observed and/or currently accepted conceptual external reality is provided in Table 1.1.1 below:

Item	General Relativity	Discussion
Consistent with Principle of Relativity	Yes	Yes
Consistent with Unnormalized Principle of Equivalence	Yes	No
Consistent with Renormalized Principle of Equivalence	No	Yes
Locally Consistent with Law of Conservation of Energy	No	Yes
Globally Consistent with Law of Conservation of Energy	No	Yes
Locally Consistent with Law of Conservation of Momentum	Yes	Yes
Globally Consistent with Law of Conservation of Momentum	No	Yes
Consistent with Eotvos Experiment within Experimental Limits	Yes	Yes
Consistent with Observed Time Dilation within Experimental Limits	Yes	Yes
Time Dilation Multiplicatively Commutative	No	Yes
Other Dilations Multiplicatively Commutative	Unstated	Yes
Consistent with Orbital Precession Data within Experimental Limits	Yes	Yes
Consistent with Refraction of Starlight within Experimental Limits	Yes	Yes
Reveal Source of Gravitational Force/Energy	No	Yes
Valid for Intense Fields	No	Yes
Singularity at Final State	Yes	No
Gravitational Collapse Self Limiting	No	Yes
Leads Directly to Observed Cosmology	No	Yes

1.1.2 - A comparison of the most significant dilations, hereafter designated as gravitational transformations, as provided by General Relativity and by this discussion, is made in Table 1.1.2 below:

Notes on Table 1.1.2:

1. A force-length-time system of units is employed rather than the conventional  mass-length-time  system. The reasons for this choice are outlined in Section 2.3.
   
   
2. The symbol '$' is employed to represent the gravitational potential. This symbol is defined in Section 3.1 and is identical with the equivalent function employed by General Relativity.
   
   
3. Two values are provided for the transformations for length, force, and space under General Relativity. The ambiguity results from a conflict implicit in the use of Tensor Calculus in its derivation.
   
   
4. Time dilation, refraction of starlight, and orbital precession differ between the two approaches in proportion to  $².  The difference is unobservable in the weak fields of the Solar System but are very pronounced in the region around a neutron star.
   
   
5. Energy differs in proportion to $. The difference is grossly observable in the weak fields in the Solar System and manifests itself as the energy of fall. It allows the discussion to pinpoint the source of gravitational energy.

Transformation	General Relativity	Discussion
Time	1/(1+$)	(1-$)
Length	1 or 1/(1+$)	1/(1-$)
Force	1 or (1+$)	1
Energy (Force times Length)	1	1/(1-$)
Space	(1+$) or (1+$)2	1

Section 1.2 - Concerns with the Derivation of General Relativity

1.2.1 - In examining the derivation and the usage of General Relativity, the author has been left with a feeling of uneasiness by certain questions which present themselves and which, when they have been addressed, seem to have been answered ineffectively.

1.2.2 - To be useful in generating a gravitational theory, the Principle of Equivalence must have more significance than the fact that all objects (particles) gravitate equally. That requirement need only mean that things (inertial mass, gravitational mass) proportional to the same thing are proportional to each other, and is, of itself, rather trite. To be significant, it is necessary that the Principle of Equivalence means that the effects of an 'inertial field' are identical to the effects of a gravitational field. This interpretation, as it is employed in the generation of General Relativity, seems to present difficulties which can be described in words by reference to Einstein's elevator model. That model asserts that observations made within a closed room on the Earth's surface are indistinguishable from the observations which would be made in an elevator being accelerated 'upwards' in free space. It also assumes that the room is sufficiently small so that the effects of the Earth's curvature are undetectable.

• The first difficulty occurs from the fact that, with the elevator being accelerated upwards, and external force must be applied to the elevator. This force acts through the distance that the elevator moves under the applied inertial acceleration. Since a force applied to a moving body results in a transfer of energy to or from that body, the elevator model implies that the elevator is receiving a steady input of energy from a source external to the model. In model analysis, any relevant quantity which crosses the boundary of the model must be taken into account as an input or an output. For the treatment to be correct, an accounting must be made of the energy transfer to or from the elevator. The derivation of General Relativity does not take this energy transfer into account, and, by failing to do so, produces internal contradictions which, in turn requires a non-Euclidian geometry for space for their resolution. Such a geometry can be shown to violate the Law of Conservation of Momentum as applied to a closed system (Section 4.2).
  
  
• The second difficulty with the usage of the Principle of Equivalence results from the assertion that the effects of curvature can be reduced to insignificance by considering a sufficiently small region of space. While Cartesian coordinates tend to conceal its nature, polar coordinates clearly show curvature to be a first derivative. Its effects are therefore scale independent and this assertion cannot be true. In a field of a given curvature, the same results is obtained, in proportion to the gravitational potential difference, regardless of the size of the region of space considered.
  
  
• A final difficulty is that if the Principle of Equivalence (i.e.- that a gravitational field is equivalent to an 'inertial field') is validly applied in General Relativity, it should be possible to describe a single inertial reference frame which accounts for the gravitational and inertial accelerations observed on opposite sides of the Earth. Such a reference frame can be described in mathematical terms  (Section 4.9),  but it is rather bizarre.
  
  

1.2.3 - General Relativity employs Tensor Calculus in its derivation. In so doing, it utilizes the mathematical process of integration. When one remembers that, in performing a mathematical integration, it is not valid to assume that the integral of K times  (dX)  is equal to K times the integral of  (dX)  without first determining that K is independent of X, the use of Tensor Calculus becomes highly suspect. By using the Principle of Relativity as a postulate, the derivation of General Relativity admits of the possibility that the size of the units of measurement for various quantities are a function of the gravitational potential difference between reference frames. (Many descriptions of Special Relativity refer to the shrinkage of metersticks and the slowing of clocks. Section 3.2 shows that the gravitational time dilation can only be interpreted as a change in the size of the units of measurement for time.) In order to validly use Tensor Calculus in the gravitational field then, it is necessary to establish the effects, if any, of the field on the units of measurement for the relevant quantities. Once that effect is determined, Tensor Calculus is no longer required except as convenient computational tool. The manner in which Tensor Calculus was employed in the derivation of General Relativity leads to the dual tabulations in Table 1.1.2.

Section 1.3 - The Derivation of a Gravitational Theory

1.3.1 - In introducing the concept of 'invariance' between reference frames, the Principle of Relativity introduced the concept of 'constancy' between reference frames as a corollary. To be rigorous, therefore, it is not adequate to just consider whether relationships or quantities change between reference frames in terms of units of measurements as they exist within each reference frame. It is also necessary to consider whether or not those relationships or quantities change between reference frames after a compensation has been made for any change in the size of the units of measurement which may occur as a result of the change in reference frame. The Principle of Relativity implies four, not two possibilities. They are ' invariant' vs.  'non-invariant'  and, independently, 'constant' vs.  'not-constant'.  Inclusion of both of these concepts may lead to unnecessary complexity where both are not required, but will produce no error. Omission of one of these concepts where it is required will force an incorrect solution to the problem under consideration. The importance of the concept of 'constancy' will become apparent as the discussion proceeds. The force and energy of gravitation are shown (without circular reasoning) to result from the fact that total energy is 'constant' rather than 'invariant' in the gravitational field.

1.3.2 - It is normally considered that the time dilation predicted by General Relativity results from the application of the Principle of Equivalence for its derivation. In this discussion, the necessary derivations will be performed without reliance on that principle. It will be shown, after the derivations have been completed and verified by comparison with external observations, that the results are consistent with the Principle of Equivalence after it has been renormalized to account for the energy input to the accelerated system as described in Section 1.2.

1.3.3 - In the discussion which follows, the relationship between the gravitational time dilation, an equivalent energy dilation, and the gravitational potential are derived based upon the following precepts:

• The Principle of Relativity is valid.
  
  
• The energy of a photon is proportional to its frequency.
  
  
• Energy which is capable of existence 'at rest' gravitates.
  
  
• A perpetual motion machine of the first kind is impossible in principle.

The equation which results provides a family of solutions which include both the time dilation expression of General Relativity and the time dilation expression provided by the present discussion. Adding the requirement that the time dilation be multiplicatively commutative parses the solution into separate expressions for the time and for the energy dilations as a function of the gravitational potential. It is at this point where the conclusions of this discussion diverge from those of General Relativity.

1.3.4 - A complete description of the gravitational field requires a third dilation expression. This is obtained by factoring the dilation expression for energy into expressions for force and length dilations. Symmetry considerations suggest that a ' Law of Conservation of Existence' should hold in addiction to the conservation laws for energy , momentum, and angular momentum. Such a law is therefore postulated and employed to permit the required factoring. With the three dilations determined, the gravitational field is defined completely.

1.3.5 - In Part 4, the predictions which these dilations make of the gravitational field are compared with experimental results. They are shown to be in complete agreement with observation to the limits of accuracy of measurement possible within the Solar System, to be compatible with three dimensional Euclidian Space, and to be consistent with the absolute validity of the Law of Conservation of Energy. The propose 'Law of Conservation of Existence' is therefore considered to be verified. In Part 5, the gravitational dilations which were developed are employed to describe the complete field. Among what is shown is that gravitational collapse does not proceed to a cataclysmic singularity within a 'black hole'. Instead, it is a self limiting process which proceeds to a state which, when observed internally, bears a striking resemblance to our own universe.

Section 1.4 - References

1.4.1 - No references are provided for the material presented since the development of the arguments which follow requires only those relationships which are accepted as general knowledge at the undergraduate level in the physical sciences and in engineering. The treatment of the material which follows is based solely on the requirement that these relationships obey the Principle of Relativity, that all material will be both internally and externally consistent, and that all currently accepted physical laws will be followed. (It shall be impossible in principle, for example, to construct a perpetual motion machine of the first kind.) The relationships which are employed are chosen so as to be verifiable within a given reference frame and may be tested experimentally if in doubt.

Section 1.5 - Author's Notes:-

1.5.1 - The statement that curvature is a first derivative has raised a question among some readers who, interestingly enough, possessed Ph.D. degrees. These individuals cited the handbook expressions for curvature. When curvature is expressed in terms of polar or Cartesian coordinates, second derivatives are present, however, in terms of Cartesian coordinates, second derivatives must appear. This coordinate system is incapable of dealing with curvature without introducing second derivatives even if they are not related to the nature of that curvature. In the polar coordinate system, second derivatives appear as a function of the distance between the instantaneous center of curvature and the origin of the polar coordinate system. If the coordinate system is translated to the instantaneous center of curvature, those second derivatives vanish. Obviously, if translating the coordinate system causes the second derivatives to vanish, they are artifacts introduced by the choice of the coordinate system rather than properties of the curvature itself. This conclusion is verified by the fact that handbooks also provide curvature as the rate of change of direction with respect to distance. Curvature, by its nature, is a first derivative.  (1992) 

1.5.2 - Some question has been raised as to whether the need for multiplicative commutivity has been proven. A brief reflection should demonstrate that multiplicative commutivity is a requirement of the Principle of Relativity and it is the fact that the Lorentz Transformations are multiplicative commutive that allows Special Relativity to work. The product of the time dilations between levels A and B (in terms of the gravitational potential between those levels measured in terms of level A) times the time dilation between levels B and C (in terms of the gravitational potential between those levels measured in terms of level B) must equal the time dilation between levels A and C (in terms of the gravitational potential between those levels measured in terms of level A). General Relativity does not meet this requirement and must therefore be in conflict with one of its postulates, the Principle of Relativity. General Relativity's lack of a length dilation corresponding the Lorentz Transformation for Length of Special Relativity means that General Relativity does not satisfy the Principle of Equivalence either. It would seem that General Relativity is a theory which contradicts its own postulates. (1999 comment)

1.5.3 - The copyrighted text of "Gravity" was sent in 1988 both to individuals and publications identified as having a reputation in the field of gravitation.

Part 2 - Laying the Groundwork 

2.5.4 - It should be noted that the use of factor X, Y, and Z as conversion factors for 'dimensional entities' between reference frames is not new to the Science of Physics. The Lorentz Transformations provided by Special Relativity are the equivalent transformations for reference frames having a relative velocity but based upon the 'dimensional entities' of mass, length, and time.

• The quantities of velocity, acceleration, and mass must be treated carefully when considering reference frames having relative velocity due to the effects introduced by the finite velocity of light as evidenced by the correction factor,  (1+V₁*V₂/C²)  in the denomenator of Special Relativity's expression for the addition of velocities. The transformations for velocity and for mass are only rigorously valid when the product of the velocities,  V₁  and  V₂ , is sufficiently small compared to the square of the velocity of light. The expression for acceleration is valid for all levels of acceleration provided that the duration of that acceleration is sufficiently small so as to result in a small velocity change. This subject is discussed completely in the text entitled "Corrections Special Relativity" accessible on this website.

Table 2.5.1 - Conversion Factors for Physical Quantities

Quantity	Conversion Factor
Length	l = (X)*L
Time	t = (Y)*T
Force	f = (Z)*F
Charge	q = (X)*Q
Energy	e = (X*Z)*E
Angular Momentum	j = (X*Y*Z)*J
Velocity*	v = (X/Y)*V
Acceleration*	a = (X/Y2)*A
Mass*	m = (Y2*Z/X)*M
Gravitational Constant	g = (X4/[Z*Y4])*G
Dielectric Constant of Space	e' = (Z)*E'
Permeability of Space	u' = (Y2*Z/X2)*U'
Existence	b = (X*Y)*B
Ergo-Gravitational Constant	d = (1/Z)*D

2.7.1 - In order to draw meaningful conclusions as to the nature and behavior of the gravitational field, it is necessary to be able to evaluate the effects of changes in elevation (reference frame) within the field upon the various physical quantities. Since, in apparent agreement with the results of observation, quantities and relationships accepted as laws and constants of the Science of Physics are invariant between reference frames, the results of measurements of these quantities and relationships will be identical at all points within the field. Any changes which actually occur as a result of a change in elevation must therefore be concealed by suitable changes in the units of measurement between these elevations. (The laws and constants of the Science of Physics are invariant between reference frames, in accordance with the Principle of Relativity.)

2.7.2 - Changes in physical quantities or constants which occur as a result of changes in elevation will be revealed only when compensation has been made for the effects of any changes in the magnitudes of the units of measurement which result from the change in elevation. such a compensation permits the results of measurement to be recorded in terms of units of measurement which are unaffected by the change in elevation. The required compensation of the results of experiment may be accomplished between any two elevations providing the factors X, Y, and Z can be evaluated unambiguously between these elevations. It is toward the evalutation of these factors that Part 3 of this discussion id directed. The upper elevation will be designated as the upper reference frame and the lower elevation will be designated as the lower reference frame.

2.7.3 - Much of the discussion which follows is based upon 'ideal thought experiments' which are assumed to employ ideal error free instruments with the requirement that their results must be consistent with the laws and constants already accepted by the Science of Physics. The justification for the use of these laws and constants is that, if their validity is questioned, they may be verified by observation using local units of measurement. These 'thought experiments' will be considered to take place under the following conditions:

• The determinations are made on an ideal, non-rotating planet located sufficiently remote from all other gravitating objects so that their effects may be ignored.
  
  
• The determinations are made between two elevations, a fixed distance apart, which are separated by an unchanging gravitational potential which is sufficiently small as compared to unity so that the gravitational field may be considered to be ideally linear.
  

2.7.4 - In addition to the postulates regarding invariance, constancy, and dimensional analysis provided in the earlier Sections of this Part, the following statements are considered to be true, subject, of course, to reverification by physical observation in a single reference frame with local units of measurement:

1. The Law of Conservation of Energy (including the energy represented by 'rest mass') is valid for closed systems. If this were not true, it would be possible in principle for a closed system to continuously export energy without depleting its internal resources.
   
   
2. The Law of Conservation of Momentum is also valid for closed systems. If this were not true, it would be possible, in principle, to construct a machine which would violate the Law of Conservation of Energy in contradiction to the above.
   
   
3. The energy of the photon, as measured with local units of measurement, is equal to Planck's Constant times its frequency.
   
   
4. The inertial mass of the photon, as measured with local units of measurement, is equal to its energy divided by the square of the velocity of light. This inertial mass is evidenced by the radiation pressure observed when light is reflected or absorbed.
   
   
5. The energy stored in material objects is invariant between reference frames. For example, one might consider, as a thought experiment, that the energy was stored in material particles by breaking up a helium atom into four hydrogen atoms and was recovered by recombining the hydrogen atoms into a helium atom. the energy represented by the mass difference between the hydrogen atoms and the helium atom would then be considered to be 'stored energy'.
   
   
6. The gravitational transformations, such as represented by Table 2.5.1, must be multiplicatively commutative. If this were not true, one would obtain the absurdity of a different result occurring between the first floor and the third floor of a building depending upon whether the elevator happened to stop at the second floor. (It should also be noted that, unless the gravitational transformations meet this requirement, the resultant conclusions will not be consistent with the Principle of Relativity [1999 comment])
   
   
7. 'Existence' is conserved between reference frames. This postulate is employed to allow a prediction of the results of astronomical observation to be made based upon basic principles. If one wished, the procedure could be inverted, and the results of astronomical observation could be used to verify the 'Law of Conservation of Existence'. (It should also be noted that, under Special Relativity, 'existence' is conserved in a direction parallel to the relative velocity vector. If 'existence' is not conserved in the gravitational field, then the Principle of Equivalence would not be valid [1999 comment]).
   
   
8. The gravitational field is isotropic. This assumption provides a considerable simplification of the discussion since it eliminates the need to consider horizontal and vertical transformations separately. In order to provide rigor, Section 2.8 which follows discusses the isotropicity of space in the gravitational field and provides a description of the observable effect which would occur if space were not isotropic. An experimental verification may be made if desired, otherwise Section 2.8 may be omitted.
   
   
9. The derivations will be made without consideration of the 'Principle of Equivalence' which forms the basis of General Relativity. After it is shown that the conclusions which are derived are in complete agreement with the results of observation, the conflict between the results which are obtained herein and those of General Relativity will be shown to result from the failure of General Relativity to perform a required renormalization in applying that principle.
   
   
10. Where it is desired to duplicate a thought experiment as a real experiment performed on the Earth, compensations can be made for the effects of the Earth's motion through space and for the effects of other astronomical bodies through the laws of celestial mechanics and through the use of the Special Theory of Relativity.
    

Note:- A sufficiently large number of particles are assumed so as to permit quantum uncertainties to be ignored.

Section 2.8 - The Isotropic Nature of the Gravitational Field

2.8.1 - It is necessary to assume that space is isotropic since, in making physical measurements, the orientation of the measuring apparatus in space produces no detectable effect on the result of measurement. While this insures that space is isotropic in terms of 'invariant' units of measurement, the possibility exists that space may not be isotropic in terms of 'constant' units of measurement in the presence of the gravitational field. Changes in the units of measurement between orientations may occur which serve to conceal a lack of isotropicity. It is the purpose of this Section to examine the isotropicity of space in the gravitational field in terms of units of measurement which are corrected for the effects of changes in orientation. This examination must be performed n order to permit the factors X, Y, and Z to be evaluated in the Sections which follow. Should observation show that space is not isotropic in the gravitational field, this evaluation is probably still possible, but will be much more complicated.

2.8.2 - Conditions of symmetry dictate that there will be no difference in the characteristics of space in the horizontal directions. A difference may exist, however, between the horizontal and vertical orientations. Further, since length and force are vector quantities while time is a scalar quantity, orientation in the gravitational field is significant only in its effect on the units of measurement of length and force. Evaluation of the isotropicity of space will be based upon determining the change in the units of measurement for length and force between horizontal and vertical orientations, as denoted by the conversion factors for isotropicity,  X_I  and  Z_I,  as defined below:

Eq. 2.8.1      X_I = l_I/L_I

Eq. 2.8.2      Z_I = f_I/F_I

Where:

• 'l_I' is a length measured with horizontal units of length.
  
  
• 'L_I' is the same length measured with vertical units of length.
  
  
• 'f_I' is a force measured with horizontal units of force.
  
  
• 'F_I' is the same force measured with vertical units of force.
  
  

Additional subscripts, 'V' and 'H' will be employed to denoted measurements of length and force made in the vertical and horizontal directions respectively.

2.8.3 - Consider a simple bell crank having both arms of equal length, L', as measured in a horizontal plane. This bell crank is mounted with one arm horizontal and one arm vertical. A horizontal force,  f_IH,  is applied to the vertical arm and is balanced by a vertical force,  F_IV  applied to the horizontal arm such that the bell crank does not rotate on its ideal frictionless pivot Figure 2.8.1.

2.8.4 - The application of the vertical force,  F_IV,  to the horizontal arm produces a clockwise torque on the bell crank equal to  F_IV*l_IH,  while the application of the horizontal force,  f_IH,  to the vertical arm produces a counterclockwise torque on the bell crank equal to  f_IH*L_IV.  The net torque on the bell crank, as measured with uncorrected units of measurement, is the difference between these torques. Since the bell crank is observed not to rotate, the net torque must be zero and we may write:

Eq. 2.8.3      F_IV*l_IH = f_IH*L_IV

In terms of 'constant' units of measurement, the net torque applied to the bell crank must also equal zero, therefore:

Eq. 2.8.4      F_IV*L_IH = F_IH*L_IV

Converting the horizontal units of measurement in Equation 2.8.3 to vertical units of measurement by the use of the conversion factors  X_I  and  Z_I  from Equations 2.8.1 and 2.8.2 provides:

Eq. 2.8.5      X_I*F_IV*L_IH = Z_I*F_IH*L_IV

Combining Equations 2.8.4 and 2.8.5 provides:

Eq. 2.8.6      X_I = Z_I

2.8.5 - Consider next the case of an ideal coil spring which is oriented with its axis in a horizontal plane, compressed, and tied. Since the ideal spring is presumed to obey Hooke's Law, the energy stored in the spring,  e_S,  is provided by:

Eq. 2.8.7      e_S = 0.5*f_IH*l_IH

Where:

• 'f_IH' is the force in the spring in the horizontal direction as measured with horizontal units of measurement.
  
  
• 'l_IH' is the change in length of the spring in the horizontal direction as measured with horizontal units of measurement.
  
  

The spring is then rotated so as to have its axis vertical and the tie is released. The spring then returns an amount of energy equal to  E_S  as provided by:

Eq. 2.8.8      E_s = 0.5*F_IV*L_IV

Where:

• 'F_IV' is the force in the spring in the vertical direction as measured with vertical units of measurement.
  
  
• 'L_IV' is the change in length of the spring in the vertical direction as measured with vertical units of measurement.
  

2.8.6 - Providing that gravity gradient effects are compensated by knowledge of their magnitude, by insuring that the moments of inertia of the spring are equal in all axes, and/or by performing the experiment in a field of infinite radius, any net torque which is observed which tends to align the spring either vertically or horizontally must produce a change in the energy stored in the spring. The existence of such a torque is subject to experimental verification and, to the author's knowledge, has never been reported. We may write, therefore:

Eq. 2.8.9      E_s = e_S

Combining Equations 2.8.1, 2.8.2, 2.8.7, 2.8.8, and 2.8.9 provides:

Eq. 2.8.10      X_I*Z_I = 1

And combining Equations 2.8.6 and 2.8.10 provides:

Eq. 2.8.11      X_I = 1

Eq. 2.8.12      Z_I = 1

Since the factors X_I and Z_I are equal to unity, they may be ignored.

2.8.7 - In the absence of an observed tendency for an object in which energy is stored anisotropically to align itself either vertically or horizontally in the gravitational field after the gravity gradient effects on anisotropic moments of inertia have been compensated, space must be isotropic in the gravitational field both in terms of 'invariant' and of 'constant' units of measurement if the Law of Conservation of Energy is to be satisfied. Since no such tendency has been reported, its seems permissible to proceed with the determination of the factors X, Y, and Z in the following Sections without regard to orientation in the gravitational field.

Part 3 - Evaluating the Gravitational Conversion Factors 

Sections	Section Titles
3.0	Introduction
3.1	The Gravitational Potential, $
3.2	The Relationship Between Y, X*Z, and $
3.3	The Evaluation of the X*Z Product and of Y in Terms of $
3.4	The Evaluation of X and Z in Terms of $
3.5	The Evaluated Gravitational Conversion Factors

Section 3.0 - Introduction

3.0.1 - In this Part, the definition of the gravitational potential, $, is provided and the gravitational conversion factors for time (Y), and for energy (X*Z) are determined using 'ideal thought experiments' and the requirement that the gravitational transformation for time be multiplicatively commutative. A postulated 'Law of Conservation of Existence' is then employed to separate the conversion factor for energy into the component factors for force (Z) and for length (X). (As shall be seen in Part 4, the use of the postulated 'Law of Conservation of Existence' yields results which are in agreement with observations and which agree with the results which are obtained when the Principle of Equivalence is correctly applied.) The gravitational conversion factors listed in Table 2.5.1 are evaluated and are tabulated in Table 3.5.1.

Section 3.1 - The Gravitational Potential

3.1.1 - The behavior of the gravitational field between elevations is best described in terms of the gravitational potential, denoted by the symbol $, existing between those elevations. It is necessary, therefore, to define the meaning of the gravitational potential as used in this discussion.

3.1.2 - The gravitational potential, $, between elevations (reference frames) is a dimensionless quantity which is defined as the energy, as measured with upper elevation units of measurement, released by a material test object in falling from the upper reference frame to the lower reference frame divided by the upper reference frame energy equivalent of the mass of that object  (E=M*C²).  The upper reference frame is chosen as a basis for the definition to facilitate the treatment of the gravitational field as a whole. The choice permits the conditions which exist at an infinite distance to be employed where convenient to the discussion.

3.1.3 - The definition of gravitational potential allows its value between elevations to be determined in terms of the gravitational parameters which exist at the upper elevation and the distance between the upper and lower elevations. For small values of gravitational potential, second order effects may be ignored, and in terms of upper elevation units of measurement, Newton's Law of Gravitation provides:

Eq. 3.1.1      F_T=G*M*M_T/R²

Where:

• 'G' is the gravitational constant.
  
  
• 'M' is the mass of the central attracting object.
  
  
• 'M_T' is the mass of the test object.
  
  
• 'R' is the distance between the centers of the test object and the central attracting object.
  
  
• 'F_T' is the gravitational force on the test object.
  
  

3.1.4 - Multiplying both sides of Equation 3.1.1 by the vertical distance, L, as measured with upper elevation units of measurement, and specifying the mass of the test object in terms of energy equivalence provides:

Eq. 3.1.2      F_T*L/E_T = G*M*L/(R²*C²)

But, F_T*L is the energy of fall of the test object, then:

Eq. 3.1.3      E_F/E_T=G*M*L/(R²*C²)

And since, by definition, E_F/E_T is equal to $ then:

Eq. 3.1.4      $=G*M*L/(R²*C²)

3.1.3 - It should be noted that the preceding equations are approximations which are valid only where the gravitational potential is sufficiently low to permit higher order effects to be ignored. However, since the definition of $ is based upon the upper reference frame units of measurement for both the energy of fall and for the rest mass equivalent energy of the mass which is falling, $ is exact for all fields. (It should be noted that the factor '$' is identical to the factor incorporated by General Relativity in its expression for time dilation).

Section 3.2 - The Relationship Between Y, X*Z, and $

3.2.1 - In this Section, the definition of the gravitational potential, the law of the Science of Physics which provides the energy of the photon in terms of its locally measured frequency, and the Law of Conservation of Energy as applied to a closed system will be employed to establish the relationship between the factor Y, the product of the factors X and Z and the gravitational potential, $. An ideal thought experiment will be described which employs a sufficient number of photons to reduce the uncertainty resulting from quantum effects to insignificance compared to the gravitational potential involved. The diagram for the experiment is provided in Figure 3.2.1.

3.2.2 - At the upper elevation, a spring is compressed and tied, thereby storing an amount of energy  E_U  at the upper elevation. Lowering the spring to the lower elevation releases an amount of gravitational energy due to the reduction of elevation of the energy of compression. The amount of energy release in this manner is equal to  E_U*$ since the energy equivalent of the mass of the stored energy is identically equal to that energy. As measured with the units of measurement of the upper elevation, the energy released by the fall,  E_F,  is provided by:

Eq. 3.2.1      E_F=E_U*$

Converting  E_F  to the units of measurement of the lower elevation by the use of the conversion factors from Table 2.5.1 provides:

Eq. 3.2.2      e_F=X*Z*E_F

And, by combining with the previous expression:

Eq. 3.2.3      e_F=X*Z*$*E_U

3.2.3 - At the lower elevation, the spring is untied and its stored energy is released, the release in the compressional energy of the spring provides an amount of energy at the lower elevation, as measured with lower elevation units of measurement, of  e_L.  Since the energy of compression if the ideal spring obeys the laws of the Science of Physics, the energy released at the lower elevation will be numerically equal to the energy installed in the spring at the upper elevation,  E_U.  (In place of the energy stored in the spring, the energy could be stored, in principle, by separating a helium atom into four hydrogen atoms. It would then be released at the lower elevation by re-combining the hydrogen atoms into a helium atom. The stored energy would then be equal to the energy equivalent of the mass difference between an atom of helium and the four hydrogen atoms. This mass [energy] difference is a constant of the Science of Physics and is therefore invariant between reference frames. Quantum effects may be ignored in this example if a sufficiently large number of atoms are considered.) Then:

Eq. 3.2.4      e_L=E_U

The total energy received at the lower elevation as a result of the lowering of the compressional energy of the spring, as measured with local units of measurement, is equal to the sum of these energies:

Eq. 3.2.5      e_T=e_L+e_F

And, by combining Equations 3.2.3, 3.2.4, and 3.2.5:

Eq. 3.2.6      e_T=(1+X*Z*$)*E_U

3.2.4 - The released spring is then returned to the upper elevation, the energy of fall of the spring itself is exactly equal to the energy required to raise the relaxed spring to the upper elevation. The spring acts as a 'passive working fluid' and it may be ignored.

3.2.5 - The net energy received at the lower elevation,  e_T,  is converted to a quantity of photons numerically equal to K, and of locally measured frequency  w_L.  It follows from the law of the Science of Physics which provides the energy of a photon in terms of its frequency:

Eq. 3.2.7      e_T=K*H'*w_L

Where:

• 'H' is the numerical value of Planck's Constant as measured with local units of measurement. As a constant of the Science of Physics, Planck's Constant is invariant between reference frames.

Combining Equations 3.2.6 and 3.2.7 provides:

Eq. 3.2.8      K*H'*w_L=(1+X*Z*$)*E_U

3.2.6 - At the upper elevation, the frequency of the photons, as measured with local units of measurement, is  W_U.  (A sufficiently large number of photons are assumed so as to allow quantum effects to be ignored.) The energy of these photons, if the Law of Conservation of Energy is be valid over the closed cycle, must be equal to the energy originally installed in the spring:

Eq. 3.2.9      E_U=K*H'*W_U

Combining Equations 3.2.8 and 3.2.9 provides:

Eq. 3.2.10      w_L=(1 +X*Z*$)*W_U

3.2.7 - The symbol  w_L  represents the frequency of the photons as measured at the lower elevation with local units of measurement while the symbol  W_U  represents the frequency of the same photons as measured at the upper elevation with local units of measurement units of measurement. The value of  w_L  may differ from the value of  W_L  as a result of either or both of two effects. The change in elevation may cause a change in the frequency of the photons as measured with constant units of time and/or the units of time which are used to measure frequency may change between elevations. To dispose of the first possibility, let us consider another (physically realizable) thought experiment.

3.2.8 - A radio wave may be considered to be a large group of photons of identical frequency and phase traveling together as a group. The effect on frequency experienced by a radio wave as a result of a change in elevation is therefore identical to the effect of such a change in elevation on the frequency of the photons which compose that wave. Unlike individual photons, however, the radio wave may be observed continuously enroute. Each cycle of that wave may in principle and in practice be observed as it travels from transmitter to receiver.

3.2.9 - Consider a radio transmitter at the bottom of a vertical shaft. Connected to the transmitter is a counter which counts the number of cycles of the radio wave which have been transmitted. At the top of the shaft is a receiver to which is connected another counter which counts the number of cycles which have been received. Initially, both counters are set to zero. The transmitter is then turned on and its counter counts the number of cycles which have been transmitted. When the radio wave reaches the receiver, the upper counter begins to count the number of cycles which have been received. The transmitter is operated for an extended period of time and is then turned off, stopping its counter. When the last cycle of the transmitted wave is received, the receiver counter also stops, as shown in Figure 3.2.2.

3.2.10 - The maximum simultaneity error possible in this experiment is twice the time for the radio wave to travel between the transmitter and the receiver, and, as a result, the duration of the experiment may be made identical for both elevations, to any desired accuracy, by continuing the experiment for a sufficient period of time. (Since the change in the velocity vector of the shaft in space between the beginning and the end of the experiment is known from astronomical data, simultaneity effects can also be calculated and compensated using Special Relativity.) The frequency of the radio wave at each elevation, as measured with 'constant' units of time, will then be proportional to the counts on each counter. The change in frequency of the photons which make up the radio wave, in terms of 'constant' units of time, is then provided by the ratio of the counter readings.

3.2.11 - In order for the outputs of the counter to differ, cycles of the radio wave would have to be created or destroyed enroute. Such an occurrence would involve the appearance or disappearance of discrete observable entities (cycles) without any means to cause such a creation or destruction, rather than just a change in the size of the units of measurement. As a result, it must be concluded that the counts on the counter will be identical since any other conclusion is absurd. We may conclude, therefor, that any observed change in the frequency of the photon between elevations results in a change in the size of the units of measurement for time. Since frequency is defined as cycles (dimensionless) per unit time, Equation 3.2.10 may be written as:

Eq. 3.2.11      t_L=T_U/(1+X*Z*$)

And from the definition of the factor Y:

Eq. 3.2.12      Y=1/(1+X*Z*$)

Note:- The conclusions to this point are identical to those of General Relativity. General Relativity asserts, through the manner in which it uses the Principle of Equivalence, that no transformation akin to the transformation for time (time dilation) exists for energy. This means that the X*Z product would be unity in Equation 3.2.12. Substituting unity for the X*Z product and substituting the value of $ provided by Equation 3.1.4 into Equation 3.2.12 yields the expression for time dilation provided by General Relativity. The divergence between the approach taken in this discussion and that of General Relativity begins in the next Section when the requirement that the expression for time dilation be multiplicatively commutative is added.

Section 3.3 - The Evaluation of the X*Z Product and of Y in Terms of $

3.3.1 - It is a corollary of the Principle of Relativity that the gravitational transformations represented by the factors X, Y, and Z must be multiplicatively commutative. In other words, the results of observations at elevation  #1  made with the units of measurement of elevation #1 , converted to the results of obtained when using the measurements of elevation  #2  and then converted to the results obtained when using the units of measurement of elevation  #3  must be the same as the results which are obtained when the results which are obtained when the results of these observations are converted directly to the results obtained when using the units of measurement of elevation  #3 . (Colloquially, when going form the first floor to the third floor, it must not make a difference if the elevator stops at the second floor.) It is the fact that the Lorentz Transformations associated with Special Relativity are multiplicatively commutative in combination which result from the finite velocity of light which prevents the measurement of an absolute velocity through space. The requirement that the time dilation be multiplicatively commutative is not met by General Relativity and form this point on, General Relativity and this discussion diverge. See Figure 3.3.1.

3.3.2 - Consider an ideal thought experiment between three elevations, the upper (U), the middle (M), and the lower (L), which are spaced vertically in a gravitational field. The elevations are chosen so that both the gravitational potential between the upper and middle elevation, as measured with the units of measurement of the upper elevation, and the gravitational potential between the middle and lower elevation, as measured with the units of measurement of the middle elevation are numerically equal to  $'.  The gravitational potential between the upper and lower elevations, as measured with the units of measurement of the upper elevation is designated as  $_T.  (The gravitational potential is assumed to be sufficiently small compared to unity between elevations so that higher order effects may be ignored.

3.3.3 - Since the locally measured gravitational potential represented by the level changes (upper to middle, middle to lower are both equal to $', the locally measured (upper to middle, middle to lower) changes in the units of measurement fot time and for energy associated with the two level changes are identical and may be denoted as 'Y' and  X'*Z'  respectively. (The relationship between these quantities and the gravitational potential is considered to be a law of the Science of Physics and therefore to be invariant between reference frames in accordance with the Principle of Relativity.)

3.3.4 - Since the units of measurement for energy at the energy at the middle elevation are X'*Z' times as large as the units of measurement at the upper elevation, we may write:

Eq. 3.3.1      $_T=$_1U+$_2U

Where:

• '$_1U' is the gravitational potential between the upper and middle elevation as measured with upper elevation units of measurement.
  
  
• '$_2U' is the gravitational potential between the middle and the lower elevation as measured with upper elevation units of measurement.

But:

Eq. 3.3.2      $_1U=$'

Eq. 3.3.3      $_2U=$'/(X'*Z')

Then:

Eq. 3.3.4      $_T=(1+1/[X'*Z'])*$'

3.3.5 - In order for the conversion factors for time and for energy to be multiplicatively commutative, it is necessary that:

Eq. 3.3.5      Y_T=Y'²

Eq. 3.3.6      X_T*Y_T=(X'*Z')²

From Equation 3.2.12:

Eq. 3.3.7      Y=1/(1+X*Z*$)

Then:

Eq. 3.3.8      Y_T=1/(1+X'*Z'*$)²

And:

Eq. 3.3.9      Y_T=1/(1+X_T*Z_T*$_T)

Combining Equations 3.3.4, 3.3.6, 3.3.8, and 3.3.9 provides:

Eq. 3.3.10      X'*Z'=1/(1-$')

Or, for a single elevation change:

Eq. 3.3.11      X*Z=1/(1-$)

Combining Equations 3.3.7 and 3.3.11 provides:

Eq. 3.3.12      Y=(1-$)

3.3.6 - It will be noted that the requirement for multiplicative commutivity requires an energy dilation which is the inverse of the time dilation associated with the gravitational field. It will be further noted that the time dilation differs from the time dilation of General Relativity by  $²,  an amount which is currently undetectable. More significantly, the energy dilation differs from that provided by General Relativity (i.e.- unity) by $, an amount which is highly detectable, and will be the subject of further discussion in Section 4.7.

Section 3.4 - The Evaluation of X and Z in Terms of $

3.4.1 - In the preceding Sections, the value of Y and the combined factor X*Z have been evaluated in terms of the gravitational potential, $. This determination is basic and requires only that the Principle of Relativity and locally testable laws of the Science of Physics be valid. It is now necessary to separately determine the factors X and Z. With the correct form for the transformations known it would be possible to apply the results of astronomical observation and the proof of the Euclidean nature of space in the gravitational field, which will be provided in Section 4.2, to make the necessary evaluation. It is more satisfying however, to perform the evaluation from basic principles.

3.4.2 - Separation of the factor  X*Z  into its components without relying upon the Principle of Equivalence, requires the application of a separate conservation law. This law must be independent of the Law of Conservation of Energy since that law has already been used to separate the energy and non-energy effects of the gravitational field. Of the commonly accepted macroscopic conservation laws, there remains only the Law of Conservation of Momentum and the Law of Conservation of Angular Momentum. (The Law of Conservation of Mass is not suitable since it is another form of the Law of Conservation of Energy.) The Law of Conservation of Angular Momentum is unsuitable. Its conversion factor is  X*Y*Z  and, since the  X*Z  product is the reciprocal of Y, the Law of Conservation of Angular Momentum is satisfied for all possible values of X and Z. It might be hoped that the Law of Conservation of Momentum would be useful, but it yields indeterminate results since it may be derived from the Law of Conservation of Energy. Since momentum is conserved as a vector quantity and not as a magnitude, it is not required to be constant between reference frames. (The magnitude of the momentum present in a system is not conserved, only its vector sum, and, as a result, it is not a true conservation law. Colloquially, one cannot satisfy hunger by creating a positive apple and a negative apple, discarding the negative apple, and eating the positive apple.)

3.4.3 - In Section 2.3, it was suggested that conditions of symmetry required that an entity having the dimensional content of a length-time product  (L*T)  should also obey a conservation law. To facilitate the discussion, the proposed entity has be designated as 'existence'. It is not a vector quantity and should therefore, like energy, be conserved in the absolute rather than the vector sense. Unlike energy, it is not subject to partition as a result of a change in elevation since one of its components is known. (The conservation of energy in the gravitational field is discussed further in Section 4.7.)

3.4.4 - On might consider, for example, that an unstable atomic particle, such as a neutron, possesses 'existence' which might be defined as the product of its diameter and its half-life. The 'existence' of the neutron would be a constant of the Science of Physics and would therefore be invariant between reference frame. (as previously, it is assumed that a sufficient number of particles are involved to permit quantum effects to be ignored.) If the 'existence' of the neutron is to be conserved between the reference frames which represent the different elevations, then its 'existence' must also be constant between those reference frames. For 'existence' to be both constant and invariant in the gravitational field, the gravitational conversion factor for 'existence',  X*Z,  must be equal to unity. Therefore:

Eq 3.4.1      X*Z=1

Combining Equation 3.3.12 and 3.4.1 provides:

Eq. 3.4.2      X= 1/(1-$)

And combining Equations 3.3.11 and 3.4.2 provides:

Eq.3.4.3      Z=1

3.4.5 - Confidence in the proposed 'Law of Conservation of Existence' is provided by examining the 'Law' as applied to reference frames having a relative velocity. In the direction of the relative velocity vector, the equivalent conversion factors are the Lorentz Transformation for Length and for time associated with the Special Theory of Relativity are:

Eq. 3.4.4      X_V=1/(1-V²/C²)^0.5

Eq. 3.4.5      Y_V=(1-V²/C²)^0.5

Where:

• 'X_V' is the Lorentz Transformation for Length.
  
  
• 'Y_V' is the Lorentz Transformation for Time.
  
  
• 'V' is the relative velocity.
  
  
• 'C' is the velocity of light.
  
  

The Lorentz Transformation for 'existence' (in the direction of relative motion) is the product of the Lorentz Transformations for length and for time and is equal to unity. The proposed "Law of Conservation of Existence is therefore valid for reference frames having relative velocity. If the Principle of Equivalence is to be valid, the 'Law of Conservation of Existence' must also be valid for reference frames which differ in elevation. Final verification will occur when it is shown in Part 4 that the values of X, Y, and Z which result are in agreement with the results of observation.

Note:- Some readers may experience misgivings at this point. It will be noted that the reciprocal identity between the factors X and Y, and the Lorentz Transformations for length and time mean that the conversion factor for velocity, X/Y, cannot be equal to unity unless both X and Y are also equal to unity. (This is certainly not the case.) This means that the velocity of light cannot be both constant and invariant between reference frames having relative velocity. The velocity of light is invariant between these reference frames. Special Relativity then must be then be telling us that it is not constant between moving reference frames. The assumption that the velocity of light is both constant and invariant between elevations in the gravitational field is exactly that, an unproven assumption, which if early writings are to be believed, was made because "we have to keep something constant".

Section 3.5 - The Evaluated Gravitational Conversion Factors for Physical Quantities

Part 4 - Comparison with the 'Real World' 

Sections	Section Titles
4.0	Introduction
4.1	The Observational Verification of Y
4.2	The Euclidian Nature of Space in the Gravitational Field
4.3	The Gravitational Acceleration of the Photon
4.4	The Observational Verification of X by the Bending of Starlight
4.5	The Gravitational Effects on the Velocity Vector and the Precession of Mercury's Orbit
4.6	The Gravitational Acceleration of the Confined Photon and the Equivalence of Inertial and Gravitational Mass
4.7	Conservation of Energy in the Gravitational Field
4.8	Conservation of Momentum in the Gravitational Field
4.9	Gravitation and the Principle of Equivalence
4.10	A Speculation on the Gravitation and Nature of Rest Mass

4.0.1 - To be considered valid, a physical theory must be both internally and externally consistent. It must agree internally with itself and it must be consistent with all external realities. The discussion up top this point meets the first requirement, it is internally consistent. The purpose of this Part is to show that the conclusions presented are consistent with the results of observations made in the external universe, to highlight the points where General Relativity is in conflict with the reality represented by that universe, and to show where errors in General Relativity arise.

4.0.2 - Part 4 opens with a demonstration that the time dilation factor predicted in this discussion and the same prediction by General Relativity differ in a second order term which is too small to be detected by present day technology within the confines of the Solar System. The observed time dilation is thus consistent with both concepts. It is next shown that space, in the presence of the gravitational field, must be Euclidian if the Laws of Conservation of Energy and of Momentum are to be valid for a closed system. It follows, therefore, that any observed bending of the path of a ray of light or precession of orbits due to the gravitational field must result from conventional refraction rather than from a 'curvature of space'.

4.0.3 - It is then shown that unconfined photons experience a gravitational acceleration equal to twice that experienced by a material particle and that this acceleration is the correct acceleration to produce the observed refraction of light by the gravitational field. It is also shown that the velocity vector of a moving object is refracted by the gravitational field to the same degree as is the path of a ray of light and that the degree of refraction is the amount necessary to cause the observed anomalous precession of Mercury's orbit.

4.0.4 - It is next demonstrated that, while both free electromagnetic and free kinetic energy experience a gravitational acceleration twice that which is experienced by material particles, when such energy is confined by matter, lowering the elevation causes the confined energy to do work on the confining matter equal in magnitude to that of conventional gravitational acceleration but of reversed direction. The net effect is to cause confined electromagnetic and/or kinetic energy, in combination with the matter which confines it, to experience a net gravitational acceleration equal to that experienced by rest mass equivalent energy. As a result, all matter gravitates equally, regardless of the fraction of its total inertial mass that results from electromagnetic or kinetic energy associated with matter.

4.0.5 - The primary deficiency in both Einsteinian and Newtonian gravitational theory is their inability to deal with gravitational energy. Under both of these concepts, the gravitational field is ultimately capable of creating an infinite amount of energy, in flagrant violation of the Law of Conservation of Energy. It is shown that such a creation of energy does not occur, but rather, that energy which is released in falling is provided by a reduction in the energy content of the falling object due to the combination of the Principle of Relativity and the reduction in the size of the unit of measurement for length. Energy is conserved absolutely in the gravitational field, as is the momentum-velocity of light product.

4.0.6 - It is disturbing that the combination of the Principle of Relativity and the Principle of Equivalence, as applied in the derivation of General Relativity, did not yield correct results for the gravitational field. Accordingly, the Principle of Equivalence is examined, and it is shown that the incorrect results are caused by an improper application of the concept. When the Principle of Equivalence is renormalized to take into account the work done on the 'accelerating' reference, it, in combination with the Principle of Relativity, yields results which are identical to those derived independently in this discussion.

Section 4.1 - The Observational Verification of Y

4.1.1 - The factor Y, is, by definition, the 'time dilation' in the gravitational field. General Relativity predicted that that 'time dilation' would occur and provided an expression, as reported in reliable texts, for the 'time dilation' as a function of the field:

Eq. 4.1.1      Y_GR = 1/(1+G*M*L/{[R*C]²)R>
Where:

• 'Y_GR' is the gravitational transformation for time (time dilation) under General Relativity.
  
  
• 'G' is the gravitational constant.
  
  
• 'M' is the mass of the attracting body.
  
  
• 'R' is the distance to the center of the attracting body.
  
  
• 'C' is the velocity of light.
  
  
• 'L' is the difference in elevation.
  
  

4.1.2 - Combining Equations 3.1.4 and 4.1.1 provides the time dilation predicted by General Relativity in terms of the gravitational potential as defined by this discussion, $, between elevations:

Eq. 4.1.2      Y_GR = 1/(1+$)

Subtracting Equation 3.3.12 from Equation 4.1.2 provides, for the difference, (dY), between these predicted time dilations:

Eq. 4.1.3      (dY) = $²/(1+$)

Or, for small values of $:

Eq. 4.1.4      (dY) = $²

4.1.3 - The gravitational potential, $, from an infinite distance to the surface of the Earth is on the order of 10^-9  and to the surface of the Sun is on the order of  10^-6 . As a result, a discrepancy on the order of  $²  is undetectable in our Solar system at the present or foreseeable state of the art. It may be considered, therefore, that the observations which are cited as a verification of the time dilation predicted by General Relativity also verify the value of Y provided by Equation 3.3.12.

Section 4.2 - The Euclidian Nature of Space in the Gravitational Field

4.2.1 - In order to use the results of astronomical observation to verify the values of X and Z provided in Equations 3.4.2 and 3.4.3. it is necessary to determine what portion of any observed 'refraction' of light by the field results from true refraction caused by a reduction of its velocity and what portion results from the curvature of a possible non-Euclidian space resulting from the field, as asserted by General Relativity. True refraction in Euclidian space requires that the spatial acceleration (the second derivative of position with respect to time) of the light be accompanied by an equivalent inertial force acting on the mass equivalent of its energy and which lies within the Euclidian space and is therefore is observable. An apparent refraction which results from the curvature of a non-Euclidian space will also produce an inertia force. This force, however, will be unobservable since it will occur along an axis which is orthogonal to the observable spatial axes. To observers such as ourselves, gravitational refraction of light which is produced by a curved non-Euclidian space will not be accompanied by an inertia force. (It should be noted that the observed existence of radiation pressure verifies that the energy represented by light possesses inertial mass in accordance with  E=M*C².) 

4.2.2 - An illustration of the above in terms which permit visualization may be helpful at this point. Let us consider an ideal, non-rotating perfectly spherical planet having a frictionless surface and located at a sufficiently remote location so as to be free of disturbing influences. The surface of such a planet is gravitationally equipotential, and an ideal frictionless spherical ball will either remain stationary at any point, or if is set in motion, will roll along a great circle path of the planet indefinitely. Infinitesimally inscribed on the planet are two great circles which cross at right angles and represents the axes of a spherical coordinate system. We can consider that the surface of the planet represents a two dimensional non-Euclidian geometry which is embedded in a three dimensional Euclidian geometry. (A non-Euclidian geometry of N dimensions may be considered to be a subset of a Euclidian geometry of  N+1  dimensions.)

4.2.3 - Let us move along one of the two great circles  (#1)  for a fraction of its circumference, perhaps 30 degrees from the point of intersection with the other great circle  (#2) , and start the ball rolling at right angles to great circle  #1  and note that initially it is moving parallel to great circle  #2.  It will be observed that, in terms of the two dimensional non-Euclidian geometry represented by the surface of the plane, the path of the ball undergoes a spatial acceleration (second derivative of position with respect to time) towards great circle #2 and eventually crosses it. This observed spatial acceleration is not accompanied by an equivalent inertial force in the two dimensional non-Euclidian space as evidenced by the fact that if one stops the ball at any point along its path, it will remain stationary. An inertial force does occur as the ball moves along the great circle, but this force is at all points normal to the two dimensional non-Euclidian geometry representing the surface of the planet and is therefore undetectable within the 'space' represented by that geometry. A two dimensional observer would conclude that the ball was following the 'null geodesic' which was the 'straight line' of his non-Euclidian universe.

4.2.4 - In terms of our familiar three dimensional space, if we observe that light is 'refracted' by a gravitational field, that refraction will be accompanied by an inertial force to the degree that it results from conventional refraction. The inertial force will be absent to the degree that the observed 'refraction' results from the curvature of a non-Euclidian three dimensional space. (Remember, light possesses inertial mass as evidenced by its ability to exert radiation pressure.) Should all of the observed 'refraction' be consistent with the observed inertial force, it must be concluded that space, in the presence of the gravitational field, is represented by three dimensional Euclidian geometry.

4.2.5 - Consider an ideal thought experiment within a closed system in which two perfect and lossless retroreflectors are located at the ends of ideal, rigid and massless booms of length L which are attached to opposite sides of an ideal non-rotating planet located far from any disturbing influences, as shown in Figure 4.2.1. A beam of photons is assumed to be traveling endlessly back and forth between the retroreflectors along a path which passes close to the surface of the planet at a distance 'r' from its center. (It is assumed that the planet does not possess an atmosphere.) As the beam of photons passes through the planet's gravitational field, it is refracted through an angle '~' by the field as observed by an external observer. In the diagram, the angle '~' may be positive, negative, or zero. In accordance with the Law of Conservation of Momentum, the momentum of this closed system must remain constant. If this were not true, it would be possible, in principle to attach a drawbar to the system and use it as a space tug. Such an implementation would be capable of providing energy continuously to the external universe without depletion of its internal resources, a clear absurdity.

4.2.6 - In the above diagram, symmetry requires that the horizontal forces at the retroflectors will balance, and momentum is conserved along that axis. For momentum to be conserved in the vertical direction, it is necessary that the downward component of the reaction forces at the retroreflectior caused by the refraction of the beam of photons be matched by an upward force on the planet exerted by the photons as they pass by. There must, in other words, be a gravitational force exerted on the photons by the planet which is proportional to the angle '~'.

4.2.7 - For the purposes of this discussion, it will be assumed that the gravitational potential from an infinite distance to the surface of the planer is sufficiently small compared to unity that second order effects may be ignored. It is valid, therefore, to eliminate considerations of upper and lower elevation from the remainder of the discussion in this Section.

4.2.8 - Let us consider that N photons per second of energy E_P strike each retroreflector. (It is assumed that N is sufficiently large that quantum effects may be ignored.) As verified by observations of radiation pressure, each photon imparts an impulse to the surface which it strikes that is equal to its mass equivalent energy times the change in its velocity vector. When the photon is absorbed, its change in velocity is equal to the velocity of light. When it is reflected, its change in velocity is equal to twice the velocity of light times the sine of the angle of incidence. Since, for the case of the retroreflectors, the angle of incidence is effectively equal to 90 degrees, the force, F, on the retroreflector, which is equal to the impulse imported per second from the beam of photons, is given by:

Eq. 4.2.1      F = 2*N*E_P/C

Where:

• 'N' is the number of photons striking the retroreflector per second.
  
  
• 'E_P' is the energy of the photons striking the retroreflector.
  
  
• 'C' is the velocity of light.
  
  

The vertical force, F_V', at each retroreflector is therefore:

Eq. 4.2.2      F_V' = 2*N*E_P*sin(~/2)/C

Or, since ~ is a small angle:

Eq. 4.2.3      F_V' = N*E_P*~/C

And for the two retroreflectors:

Eq. 4.2.4      F_V = 2*N*E_P*~/C

But, since the beam consists of both arriving and departing photons, the energy per unit length of the beam of photons,  E_L , is given by:

Eq. 4.2.5      E_L = 2*N*E_P/C

Then:

Eq. 4.2.6      F_V = E_L*~

4.2.9 - Now let us consider the bending of the beam of photons by the gravitational field. If the beam of photons is subject to a spatial acceleration of  a_N  normal to its path, then in an incremental distance, (dL), it will change in direction by an incremental angle (d~). To determine the effect, let us consider a particle traveling horizontally at a velocity V for an incremental time (dT) and subjected to a vertical acceleration  a_N , as shown in Figure 4.2.2. As a result of the acceleration, the particle will move in a circular path of radius R, as shown in Figure 4.2.3. In time (dT), the particle will travel a distance (dL) and its path will curve through an angle (d~)

4.2.10 - The angle through which the beam of photons is deflected as it passes the planet is extremely small and, in determining the normal acceleration experienced by the photon due to the planet's gravitational field, we may ignore the bend, since the effects of the bend are proportional to the cosine of a small angel. Figure 4.2.4 then applies. Where:

• '(dL)' is an incremental length of the photon beam.
  
  
• 'L' is the distance of element (dL) from the point of closest approach to the planet.
  
  
• 'r' is the distance form the center of the planet from the photon beam.
  
  
• 'R' is the distance of the element (dL) from the center of the planet.
  
  

If we define a_N' as the acceleration of the element (dL) at the point of closest approach, then, allowing that the local acceleration is proportional to the local gravitational acceleration as required by the Principle of Relativity, at any point along the beam the normal acceleration,  a_N,  is given by:

Eq. 4.2.11      a_N = a_N'*r³/R³

4.2.11 - Now consider a beam of photons as a gravitating mass having an inertial mass per unit length of  E_L/C².  The incremental force,  (dF_NG),  in the vertical direction of Figure 4.2.4 is:

Eq. 4.2.18      (dF_NG) = K_P*a_GN*E_L*r³*(dL)/(C²*R³)

Where:

• 'a_GN' is the acceleration of gravity at the point of closest approach of the beam to the planet.
  
  
• 'K_P' is the ratio of the gravitational force acting on the photon to the gravitational force acting on the material particle of the same mass equivalent energy. (Note: This does not imply that a photon does or does not possess gravitational mass since  K_P  may be positive, negative, or zero.)
  
  

Combining Equations 4.2.12 and 4.2.18, integrating between the limits of +/-L, and allowing L to be large compared to r provides:

Eq. 4.2.19      F_NG = 2*K_P*a_GN*E _L*r/C²

4.2.12 - If the Law of Conservation of Momentum is to be valid for the closed system represented by Figure 4.2.1, then:

Eq. 4.2.20      F_V = F_NG

Combining Equations 4.2.17, 4.2.19, and 4.2.20 provides:

Eq. 4.2.21      a_N' = K_P*a_GN

It should be noted that, since the quantity to be integrated in Equation 4.2.14 is identical to the quantity to be integrated in the combined Equations 4.2.12 and 4.2.18, the approximation which occurs as a result of ignoring the effect of the changing gravitational potential along the beam cancels. The only approximation remaining in Equation 4.2.21 is the small angle approximation for '~'.

4.2.13 - Consider the significance of the preceding equation, The term  a_N'  represents the normal acceleration, where acceleration is defined as the second derivative of position with respect to time, required to cause the beam of photons to bend in terms of three dimensional Euclidian space. The term  K_Pa_GN  represents the normal acceleration, where acceleration is defined as the applied force divided by the mass equivalence of the energy of the beam, required to produce a force which is equal to the net force applied to the retroreflectors. The equality of the preceding equation means that, to the first order at least, any and all gravitationally induced refraction observed for light results from gravitational forces which occur in our familiar three dimensional space. The is no first order effect produced by a component of force acting along an axis normal to our observable three spatial axes. It is concluded, therefore, that space, in the presence of the gravitational field, must be three dimensional Euclidian. This conclusion is basic and stands alone, it is independent of the values of X, Y, and Z found previously and of the arguments presented previously. The curved space of General Relativity cannot be a valid representation of reality.

Section 4.3 - The Gravitational Acceleration of the Photon

4.3.1 - In this Section, The Law of Conservation of Momentum as applied to a closed system will be employed, this time to evaluate the gravitational acceleration of the photon. Let us again consider a closed system consisting of the ideal planet and a pair of ideal retroreflectors which endlessly reflect a beam of photons. This time, the retroreflectors will be mounted along a vertical axis with one retroreflector at the planet's surface and one retroreflector mounted directly above it. For this thought experiment, it will be assumed that the support for the upper retroreflector is made of an ideal material whose length, in terms of 'constant' units of measurement, is unaffected by the effects of the gravitational field. (This assumption is necessary to prevent the radiation pressure contained between the retroreflectors form acting on the support so as to produce a spurious force into the thought experiment, as will be described in a later Section.)

4.3.2 - In the thought experiment of Figure 4.3.1, a beam of photons is continuously reflected between the upper and lower retroreflectors without loss. (A sufficient number of photons is assumed so that quantum effects may be ignored.) At each reflection, the photons impart momentum to the retroreflectors. In order for the Law of Conservation of Momentum to be satisfied for the closed system, the algebraic sum of the momentum applied to the upper retrorelector and the lower retroreflector, minus the momentum applied to the photon stream by the gravitational field must be equal to zero, as measured with 'constant' units of measurement. Since the momentum imparted is the integral of the force applied and the time for which it is applied, and since we are interested in the momentum imparted per unit time, we need only consider the force.

4.3.3 - Equation 4.2.1 provides, for the force produced by a beam of photons on a retroreflector:

Eq. 4.3.1      F = 2*N*E_P/C

Which we shall rewrite for the upper retroreflector as:

Eq. 4.3.2      F_U = 2*N*E_P/C_U

And, for the lower retroreflector, as:

Eq. 4.3.3      F_L = 2*N*E_P/C_L

Where:

• 'N' is the number o