Why is Gravity so Weak?

David W. Talmage and Richard J. Sanderson

    Gravity possesses several unique features that distinguish it from the other three fundamental forces.  Most prominent among these is its relative weakness, which at the level of fundamental particles amounts to approximately 40 powers of ten.  In addition to this weakness, gravity also differs from the other forces by affecting all forms of matter and energy and by being effective without being absorbed.  Despite these differences, most theoretical attempts to explain gravity are based on the assumption that the underlying mechanism of the gravitational force is the same as that of the other forces.  The present model is an attempt to explain the unique properties of gravity as well as the exact equivalence of  gravitational and inertial masses.  By borrowing well established concepts from quantum mechanics and general relativity and the universe's scalar field from Brans-Dicke we have proposed a causal model of gravity in which the primary effect of the gravitational field is on the velocity of light.  The other effects of gravity and inertia are considered secondary to changes in light velocity....By using this model and Fermat's principle that light follows the least time path, we have been able to calculate the observed displacement of a star at the rim of the Sun and the time delay of light passing that way...

     The weakness of gravity is forecefully demonstrated by the ease with which a small dime store magnet can lift an iron nail agianst the Earths's mass of 6 x 10³⁴ .  And at the level of the fundamental particles, the force of gravity is approximately 40 powers of ten weaker than the electrostatic force⁽¹⁾.  This unique weakness of gravity has been the major stumbling block to efforts to unify it with the other three forces of nature under the aegis of quantum mechanics (QM).  In describing the difficulties inherent in any quantum theory of gravity, Isham⁽²⁾  quoted a common maxim of workers in this area: "What God has put asunder, let no man join together."

     Before the advent of QM, Einstein in 1915 proposed a theory of gravity,⁽³⁾  called general relativity (GR), that is based on the assumption that matter and energy produce a curvature of the surrounding space-time metric.  the theory is constructed with tensor equations that can be transformed from one set of coordinates to another according to mandates of Einstein's earlier theory, special relativity (SR)⁽⁴⁾ .  GR is based on the observation, dating from Galileo, that the ratio of the inertial to the gravitational masses is the same for all materials.⁽⁵⁾   For this reason, all objects, regardless of size and composition, follow the same path or orbit as they fall in a gravitational field.  The concept that objects follow a geodesic, or path, in curved space-time explains why gravity, unlike the other three forces,  affects all forms of matter and energy.

            GR has been successful in predicting the path and velocity of light as it passes by a massive object such as the Sun, the frequency of atomic clocks, and the precession of the perihelion of Mercury’s orbit.  According to GR the rate at which clocks run is affected by the gravitational field, so that

            dT/T = adr/c²                                                                                                                     (1)

where dT/T is the fractional change in clock rate, a is the gravitational acceleration, dr is the difference in altitude between two clocks, and c is the velocity of light.  This prediction has been amply confirmed by numerous investigators using cesium⁽⁶⁾ and hydrogen maser⁽⁷⁾ clocks and the Mossbauer effect. ⁽⁸⁾  Since the equation for the change in clock rate is identical to the fractional change in kinetic energy (over total energy) of an object falling in the gravitational field (E_k/E = adr/c²), the change in energy could be reasonably attributed to the change in clock rate.  This would be consistent with the de Broglie concept that matter particles, like photons, possess waves⁽⁹⁾ and with its corollary that the rest energy of matter particles, like the energy of photons,  is directly proportional to the frequency of their particular waves.  However, according to GR, the rest mass of fundamental particles remains constant, and the kinetic energy is obtained from the gravitational field, the strength of which is nevertheless not reduced.

            The above discussion illustrates the difficulty in bridging the concepts of QM and GR, the two most successful theories of this century.  Although Einstein played an important role in the early development of quantum theory with his discovery of the quantal nature of light, he could never accept the indeterminacy of quantum mechanics.⁽¹⁰⁾  And attempts to quantize gravity have run into insuperable mathematical problems associated with the appearance of infinite terms in the equations.⁽¹¹⁾

            One assumption that is made by both quantum theorists and relativists is that the gravitational constant, G, is a universal constant   This assumption is necessary for a quantum theory of gravity in order to unify gravity with the other three forces, in which the constant of the force is independent of the charge.⁽¹¹⁾  This is apparently required if gravity is to be explained by a gauge boson, called the graviton, in the same way that electrostatic forces are explained by the exchange of photons.⁽¹²⁾

            Brans and Dicke⁽¹³⁾ (BD) have presented arguments that Mach’s principle requires that G be related to the mass and size of the universe according to the relationship

            G ~ c²R_u/M_u                                                                                                                                                      (2)

where M_u  is the total mass of the visible universe and R_u its size.  Before that, Sciama(¹⁴⁾ had proposed a theory of inertia based on similar ideas.  In addition to the fact that Eq (2) can be shown to be approximately true, it is also relevant that distant matter plays an obvious role in determining the direction of inertial force.  The equivalence of  gravity and inertial forces, as assumed by GR and confirmed by observation, makes it reasonable to expect that distant matter would also affect gravitational forces.

            In order to combine Mach’s principle with GR, Brans and Dicke introduced a scalar field equivalent to the sum of all the m/r derived from all the matter and energy of the visible universe.  The scalar field was coupled to the tensor equations of GR through a constant, w, that Brans and Dicke proposed would be near unity.  However, observations of the time delay produced by the sun in laser ranging experiments to Venus and Mars indicate that w cannot be less than 500.^{(15, 16)}

             The present essay is an exploration of the possibility of bridging the gap between QM, GR and BD through the construction of causal hypotheses of gravity that borrow from all three theories, but are based on the notion that the unique properties of gravity that distinguish it from the other fundamental forces suggest a unique mechanism.  We will show that the approach has promise by calculating the displacement of a star on the rim of the Sun, the time delay of light passing that way, and the precession of Mercury’s orbit.  However, this is not a mathematical theory of gravity, although we have used mathematics and computers to test the hypotheses.

            Traditionally, physicists have been  reluctant to pose causal questions such as the title of this essay, preferring instead to follow the example of Newton, who said, “I make no hypotheses.”⁽¹⁷⁾  This tradition has been supported by the success of Newton and others in describing nature with mathematical equations and by the logical positivism of Comte and Mach who held that the only meaningful reality was that obtained directly from the senses.^{(18, 19)}

                But causal questions have been useful in all branches of science when a mathematical description of observed events was impossible.  Newton,  himself, began his study of gravity by asking if the forces that pulled the apple to the ground and kept the Moon in its orbit were based on the same mechanism.  And in the summary of his philosophy he said, “We are to admit no more causes of natural things than such as are sufficient to explain their appearances.  Therefore, to the same natural effects we must, as far as possible, assign the same natural causes.”⁽²⁰⁾

1.      A CAUSAL MODEL OF GRAVITY

The model that we are proposing is based on the following five hypotheses regarding the velocity of light:

Hypothesis 1: The primary effect of the gravitational field is on the velocity of light (n), which can be determined by the time delay of light passing through that field.⁽²¹⁾

Hypothesis 2: The velocity of light at any point is inversely proportional to the total Newtonian potential (P), which is the integral of all the m/r for all the matter and energy in the visible universe.  Thus

dn/n = -dP/P                                                                                                               (3)

Hypothesis 3: The fractional change in clock rate (dT/T) produced by a change in P (e.g. from the m/r of the Sun) is one-half the fractional change in n (dn/n).  Thus

2dT/T = dn/n                                                                                                               (4)

Hypothesis 4:  We adopt Fermat’s principle formulated in 1661 that light travels by the least time path.  More recently this principle has been applied to light traveling in a gravitational field^{(22, 23)} and shown to be a prediction of  quantum electrodynamics.⁽²⁴⁾

Hypothesis 5: The rest mass of matter particles, like the energy of photons, is proportional to their particular wave frequency. Thus Eq. (1), which is a prediction of GR, is a statement that energy is conserved in a gravitational fall and that the kinetic energy is derived from a loss of rest mass.

                As indicated in Hypothesis 3 (and also predicted by GR), the fractional change in clock rate (T) is one-half the fractional change in n. For this reason, in order to make the measured value of n constant, there must be a fractional contraction of length of measuring sticks equal to the fractional change in clock rate.  Thus  when a matter particle encounters a reduction in n due  to an increase of P, it divides the effect symmetrically between a reduction in frequency and a reduction in length.  The result is that a measurement of the velocity of light within one gravitational field (e.g., on the surface of the Moon) gives the same result as in any other field (e.g., on the surface of the Earth) because the changes in clock rate and measuring sticks exactly balance the change in velocity of light.

            Since the value of P cannot be determined directly, we must determine the fractional change in n from (4) and (1).  Thus the acceleration on and beyond the surface of the Sun is

            a = GM/r²                                                                                                                   (5)

where M is the mass of the Sun and r the distance to its center.  Combining (5) with (1) and integrating, we get

            dT/T = -GM/rc²                                                                                                           (6)

and from (4) and (6)

            n/c = 1 – 2GM/rc²                                                                                                                                          (7)

where n is the local velocity of light, and c is the velocity of light in space outside the Sun’s gravitational field

2.      CALCULATION OF THE DISPLACEMENT (q)  OF LIGHT PASSING BY THE RIM OF THE SUN AND TIME DELAY (d)

The time required for light to travel from the rim of the Sun to the Earth along a path S is given by T_m = dS/n.

In polar coordinates with the Sun’s center as the origin and F = 0 at the tangent,

dS = [r² + (dr/dF)²]^1/2dF                                                                                             (8)

From (7)

            T_m = 1/cò(1 – 2GM/rc²)^-1[r² + dr/dF)²]^1/2

And

            T_m =  1/cò(1 + 2GM/rc²)[r² + (dr/dF)²]^1/2                                                                     (9)

since 2GM/rc² is small.

            The path for which T_m  is a minimum must satisfy Euler’s equation,^{(25, 26)} which in this particular case is

            (dr/dF)² = r⁴/B²(1 + 4GM/rc²) – r²                                                                               (10)

where B is a constant of length dimension.  This an equation for a hyperbola  in polar coordinates, which in the essentially straight line portion (beyond 3r) makes an angle to the tangent of 0.87”.¹  This leads to a total displacement of starlight passing the rim of the sun (q) of twice 0.87”, or 1.74”.   At r  =  r_s  =  1, F  =  0 and dr/dF  =  0.

            B = (r_s + 4GM/c²)^1/2  = (1 + 2GM/c²)                                                                           (11)

            Substitution of (10) and (11) into (8) and (9) yields

            S = ò rdr/X  +  ½ ò Adr/X                                                                                             (12)

            T_m  =  1/cò rdr/X  +  1/cò Adr/X                                                                                    (13)

Where A = 4GM/c² and X = (r² +Ar – B²)^1/2.

            Integration of (12) and (13) (which are standard forms) gives the time delay

            d = T_m – S/c                                                                                                                (14)

If the velocity of light in outer space (c = 0.4305 r/s) is used in (14), the time delay from r =1 to r = 215 is calculated to be 60 ms.  If the velocity of light at the Earth’s orbit, c(1 – 2GM/rc²), is used, the time delay is 50ms

            We have also used a straight line approximation in Cartesian coordinates to calculate the least time path and time delay.  This has the advantage of ease of determining the effect of of small changes in q/2.  Using this method we obtained a minimum time when q/2 was 0.873” and a time delay using c of 60.8 ms.  The time delay using c(1 – 2GM/rc²)  is 50.2 ms.

            It has not been possible to make an exact comparison between these results and the observations reported by Shapiro  et al.⁽¹⁵⁾  According to that reference, the delay for the round-trip from the Earth to Venus near its superior conjunction was slightly less than 200 ms, which corresponds approximately to the 50 ms for one-quarter that route calculated here.

3.      THE RELATIONSHIP BETWEEN THE GRAVITATIONAL CONSTANT G AND THE TOTAL NEWTONIAN POTENTIAL P

From (4) and (6)

dP/P  =  2GM/rc²                                                                                                         (15)

Where M and r are the mass and radius of the Sun.  Since dP  = M/r

            P  =  c²/2G  =  6.7 x 10²⁶ kg/m                                                                                     (16)

            Since P  =  3M_u/2R_u where M_u and R_u are the mass and radius of the universe,

            G  = R_uc²/3M_u                                                                                                              (17)

which differs from Eq. (2) by a factor of 3.

            Since according to our hypothesis 2, c varies inversely with P,  Eqs. (16 and (17) imply that G is proportional to c³ and inversely proportional to P³.

4.      CALCULATION OF THE PRECESSION OF MERCURY’S ORBIT FROM A COMPUTER MODEL

GR  predicts that the perihelion of a planetary orbit will precess in the direction of motion by an angle equal to 6pMG/lc² rad/rotation, where l = a(1 – e²), a is the semimajor axis of the ellipse, and e is its eccentricity.⁽²⁷⁾

            We have  used a numerical solution of the equations of motion of a body orbiting under a central force.  When the acceleration is modified to correct for the value of G [a = a₀(1 – 2MG/rc²)²], the axis of the normal elliptical orbit coreesponding to a constant G precesses in the direction opposite to the motion of the orbiting body.  The amount of the negative precession is 6pMG/lc² over the straight portion of curve obtained by varying  a₀ and e. The precession attributable to SR and GR may be obtained by applying a correction to velocity of [(1 – v²/c²)²(1  -  2GM/rc²)], yielding a value of 6pMG/lc²

5.      DISCUSSION

The present essay is an attempt to explain the unique properties of gravity that distinguish it from the other fundamental forces by postulating unique mechanisms.  We have called these causal hypotheses to indicate that they are not mathematical equations, although we have used mathematics to test them.

The most unique of the causal hypotheses is that the primary effect of the gravitational field is to determine the velocity of light.  As an addendum to that hypothesis (2), but not absolutely essential to it, we have proposed that the velocity of light is inversely proportional to the total Newtonian potential (P), or the sum of all the m/r of the entire visible universe.  The reason for proposing the addendum  is that it explains the remarkable weakness of gravity and requires a universe of a size and mass within current estimates.⁽²⁸⁾  In addition it is clear from observations of the time delay of light passing the rim of the Sun (and a prediction of GR) that the velocity of light is an inverse function of P.  A straight inverse proportionality  is a much simpler function than that required by GR, which assumes that the velocity of light will vary inversely with P in such a way that G is kept constant.

The negative precession that results from the variable G that is a consequence  of  our second hypothesis requires a total accounting of all the contributions to precession.  Seventeen years before Einstein formulated the equation of precession from GR, Gerber⁽²⁹⁾ proposed that the precession of Mercury’s orbit could be explained by the concept of delayed gravitational potential, which is similar to a established phenomenon in electromagnetism and is based on the assumption that gravity travels at the speed of light.  The effect of the delay is a reduced acceleration that is dependent on distance and produces a precession equal to 6pMG/lc².  This result has been rederived recently by Beckmann.⁽³⁰⁾  It appears to us that the contribution from delayed potential is completely independent of the effects of GR and SR and thus should be additive to those effects.  The former are based on a change of acceleration, whereas the latter are based on a change of velocity.

The opposite effects of our second hypothesis and that caused by delayed potential cancel and result in an acceleration toward the Sun and result in an acceleration toward the Sun that is inversely proportional to the square of the distance.  Thus there is no evidence that either hypothesis is correct, but the two must be accepted or rejected together.  If, however, they are both correct, this should not affect the value of G that is produced by G that is produced by a change in P and measured locally.  Thus it is conceivable that a measurement of G that is independent of the solar acceleration could show a variation, for example, a comparison of G made on the Earth at it aphelion and perihelion.  However the expected difference between these measurements is several orders of magnitude less than the accuracy of current measurements.

A precession of 6pMG/lc² was obtained with our computation method using a correction to velocity equal to (1 – v²/c²)²(1 – 2GM/rc²).  Each (1 – v²/c²)^1/2  in the correction produces a precession of p and the correction of (1 – 2GM/rc²)  produces a precession of 2p.  The justification of using the fourth power of (1 – v²/c²)^1/2  is that the increase in mass absorbs kinetic energy equivalent to two powers, and the reductions in clock rate and distance metric account for one each.  The correction to velocity of (1 – 2GM/rc²) assumes that matter particles slow down as they approach the Sun to the same proportion that light does.

Although tha variation in the value of G has been proposed many times before,^{(31, 32)} laser ranging experiments that measure the distance to the Moon indicate that at the present time any variation in the value of G is probably less than a few parts per 10¹¹ per year.^{(33, 34)}  And the fact that the temperature of the Earth has been compatible with life for more than three billion years sets a limit on the amount of change in G that could have occurred during that time.⁽³⁵⁾

The relative constancy of G during the current era may be compatible with our model if P is also constant.  It is possible that P is constant because the universe is not expanding as rapidly as before and with the present dominance of matter over free energy the edge of the visible (and gravitational) universe is expanding  more rapidly than matter at its edge.  Thus new matter being brought into the visible universe may add to the value of P and cancel the reduction of P caused by the reduced density.

It might seem unimportant whether G is constant or not, as long as current measurements cannot detect a change.  But with ever more precise instruments applied over longer stretches of time it is certainly conceivable that a change in G will be detected.  And gravity is the most obvious of the fundamental forces of nature and the most mysterious.  A great deal of effort has been expended in trying to develop a quantum theory  of gravity, and most of this effort has been based on the assumption that gravity is gauge force like the others, implying that G is independent of P.  Thus distinguishing between these two concepts will either confirm or change our most fundamental concepts of the universe.  The concept that G is inversely proportional to P³ will also affect calculations of events in the early universe when P was probably higher than it is today.  For example the amount of redshift in light from distant galaxies that can be explained by expansion  and the amount explained by a higher P differs in the two models.

Quantum theories of gravity  have generally tried to unify gravity with the other three forces by assuming that it involves the exchange of a gauge boson, the graviton.  If the exchange of such a particle with matter is responsible for gravity, this should lead to a constant value of G.  However, in the absence of a satisfactory quantum theory of gravity and a demonstration of the existence of the graviton, we feel that a proposal of an alternative model is justified, for gravity differs from the other three forces  by a demonstrated effect on the velocity of light and by its effectiveness without being absorbed.  Since the effect of gravity on the velocity of light is well established and adequate to explain the gravitational force, the postulation of a gauge boson acting directly on matter particles seems unnecessary.  The possibility that gravitons affect the velocity of light without being absorbed is not incompatible with our model.

We assume that GR predicts the correct displacement of a star on the rim of the Sun and the time delay of light passing that way.  The complex tensor equations used to make these predictions are necessary to transform the laws of physics from one set of coordinates to another according to the mandates of SR.  However, with the evidence for the existence of a preferred frame established by the cosmic microwave radiation, we believe that the requirement for covariance can be removed.  This has eliminated the necessity of coupling the BD scalar field to the GR tensor equations.  Thus by using the scalar field alone, plus the Fermat principle that light takes the least time path, we can achieve the same results.

A major advantage of the model is that it incorporates Mach’s principle by means of the universe’s scalar field.  The equivalence principle is explained by postulating that both gravity and inertia involve a proportional change in the energy of all matter particles based on their interaction with the scalar field.  In both cases there is a change in the velocity of light that is divided evenly between clock rate and space measurements.  In the case of gravity the reduction in clock rate is associated with a proportional decrease in energy,  and in the case of motion there is a proportional increase in energy.  But in both cases the change in energy (and mass) is proportional  and affects all types of matter particles equally.

Another important advantage of the model is that it explains the three unique features of gravity, its relative weakness compared to the other forces, its effect on all forms of matter and energy, and its effect without absorption.  According to the mode, the weakness of gravity is due to the enormous size of the universe.  Since G is inversely proportional to P³, gravity would be much stronger in a small universe.   The effect of gravity on all forms of matter and energy is due to its control of the velocity of light (free energy) and through that all forms of matter.  Its lack of absorption is thought to be due to the fact that the scalar field only controls the velocity of light, but does not attract or repel it.

The calculations made above show that a determination of the least time path of light in flat space produces equivalent results to the calculation of a geodesic in curved space-time.  In a similar fashion the elliptical orbits of matter traveling around a gravitating mass such as the Sun can be considered constant energy paths in which there is a continuous exchange of kinetic (inertial) and rest (gravitational) energy.  Although the elegant mathematics of GR correctly predicts the observations, the more obvious concepts of three-dimensional space and a separate dimension of time along with energy conservation make the introduction of additional factors such as a variable G fairly simple and natural.

It is not our intention to replace or improve on GR.  In fact, the calculations we have undertaken here have mude us  appreciate it elegance.  The variable velocity of light plays the same role in our model as the space-time metric plays in GR.  However, coordinate systems, whether Cartesian, polar, or Riemannian, are an invention of the human mind, and thus the choice of which one best accomplishes a human purpose is clearly up to the individual.

Acknowledgement

The authors acknowledge the contribution of David H. Talmage in writing the Adjustable Precision Arithmetic Package in C that mad the calculations reported here possible.  The authors also acknowledge the reviewers’ numerous suggestions and criticisms and Howard Hayden’s review of an early version of this manuscript.

Received 15 December 1992

Endnotes

1.      The total displacement of light predicted by GR is 1.75”.

2.      The mass contained in a large spherical shell of radius R and thickness dR is dM =4psR²dR and dP = dM/R = 4psRdR.  Integration yields P = 2psR².  Since M = 4/3psR³, P =3M/2R.

References

1.      R. P. Feynman, R. B. Leighton, and M. Sands, The Feynman Lectures on Physics  (Addison-Wesley, Reading, MA, 1968), Vol. 1, p.7.

2.      C Isham, The New Physics, edited by P. Davies (Cambridge University Press, 1989), p. 70

3.      A. Einstein, Ann. Phys. 49, 769 (1916); English translation in H.A. Lorentz et al, The Principle of Relativity Dover, NY, 1952.

4.      A. Einstein, Ann. Phys.17, 891 (1905).

5.      R. H. Dicke, Gravitation and Relativity, edited by H.Y. Chiu and W. F. Hoffman (W. A. Benjamin, NY, 1964), p. 1.

6.      J. C. Hafele and R. E. Keating, Science 177, 168 (1972).

7.      R. F. C. Vessot, Contemp. Phys. 25, 355 (1984).

8.      R. V. Pound and G. A. Rebka, Jur., Phys. Rev. Let. 4, 337 (1960),

9.      L. de Broglie, C. R.. Acad Sci. Paris 177, 507 (1923).

10.   A. Einstein, B. Podolsky, and N. Rosen. Phys Rev. 47, 777 (1935).

11.   Ref. 2, p.3.

12.   Ibid, p. 83.

13.  C. Brans and R. H. Dicke. Phys. Rev. 124, 925 (1961).

14.  D. W. Sciama, Mon. Not. R. Astron. Soc. 113,34 (1953).

15.  I. I.  Shapiro, M. E. Ash, R. P. Ingalls, W. B. Smith, D. B. Campbell, R. B. Dyce, R. F. Jurgens, and G. H. Pettengill, Phys. Rev. Let. 26, 1132 (1971).

16.  R. D. Reasenberg, I. I> Shapiro, P. E. MacNeil, R. B. Goldstein, J. C. Breidenthal, J. P. Brenkle, D. L. Cain, T. M. Kaufman, T. A. Korarek, and A. I. Zygielbaum, Astrophys. J. 234, L219 (1979).

17.  I. Newton, Mathematical Principles of Natural Philosophy; translated by A. Motte and revised by F. Cajori, in Great Books of the Western World, edited by R. M. Hutchins (Encyclopedia Gritannica, Chicago, 1952), Vol. 34, p. 371.

18.  S. Hawking, Black Holes and Baby Universes and Other Essays (Bantam, NY, 1993) p.47.

19.  F. Rohrlich, From Paradox to Reality (Cambridge University Press, 1987), p. 20.

20.  Ref. 17, p. 270.

21.  I. I. Shapiro, Phys. Rev. Let. 13, 789 (1964.

22.  L. D. Landau and E. M. Lifshitz, The Classical Theory of Fields, 4^th rev. English ed.; translated from Russian by M. Hamermesh (Pergamon, Oxford, 1975), p. 253

23.  T. Frankel, Gravitational Curvature. An Introduction to Einstein’s Theory (W. H. Freeman, San Francisco, 1979), p. 90.

24.  R. P. Feynman, QED, The Strange Theory of Light and Matter (Princeton University Press, Princeton, 1985), pp52, 63.

25.  I.M. Gelfand and S. V. Fomin, Calculus of Variations, rev. English ed.; translated from Russian and edited by R. A. Silverman (Prentice-Hall, Englewood Cliffs, NJ, 1963).

26.  U. Barechtken-Manderscheid, Introduction to the Calculus of Variations; translated from German by P. G. Engstrom (Chapman & Hall, London, 1991).

27.  I. R. Kenyon, General Relativity (Oxford University Press, Oxford, 1990), p. 92.

28.  A. Lightman and R. Brawer, Origins: The Lives and Worlds of Modern Cosmologists (Harvard University Press, Cambridge, 1990), p. 4.

29.  P. Gerber, Z. Math. Phys. 43, 93 (1898)

30.  P. Beckmann, Einstein Plus Two (The Golem Press, Boulder, CO, 1987), p. 170.

31.  E. A. Milne, Relativity Gravitation and World Structure (Clarendon Press, Oxford, 1935), p. 292.

32.  P. A. M. Dirac, Nature 139, 323 (1937); idem in Cosmology, Fusion and Other Matters; George Gamow Memorial Volume, edited by F. Reines (Colorado Associated University, Boulder, CO, 1972), p. 56.

33.  R.W. Heilings, P. J. Adams, J. S. Anderson, M. S. Keese, E. L. Lau, E. M. Standish, V. M. Canuto and I. Goldman, Int. J. Theor. Phys. 28, 35 (1989).

34.  E. Teller, in Cosmology, Fusion and Other Matters; George Gamow Memorial Volume, Ref. 32, p. 60.

35.  R. H. Dicke, Gravitation and the Universe (The American Philosophical Society, Philadelphia, 1970), p.26