William Schieber (Physical Science, American Public University)
Published in physic.philica.com Abstract A theory of gravity was developed using four spatial dimensions that leads to Newton’s Universal Law of Gravitation but with a Gravitational Constant (G) that is proportional to the radius of the observable universe that increases with time. A timevarying G implies that calculations of the properties of very distant astronomical objects using a constant G need to be corrected. The theory indicates that only a finite value of gravity is possible so that singularities with infinite density and infinite gravity cannot occur in nature. Lastly, the theory offers an explanation for the small sunward acceleration (8.7x1010 ms2) measured for the two Pioneer spacecrafts and predicts that the added acceleration due to an increasing G is equal to the Hubble constant times of speed of light in vacuum (Hc = 7x1010 ms2). Article body
Background The variability of the gravitational constant of the universe (G) is a fascinating topic where many physicists and astronomers support the view in Newton's law of gravity and Einstein's general theory of relativity that G is a constant but a few prominent physicists dating back to the development of general relativity believed that G actually varies with time. A time varying gravitational constant would lead to a modification of the laws of physics with profound implications to astronomy. The force of gravity is much weaker than the other three fundamental forces. This paper asserts that gravity is much weaker because it acts over four spatial dimensions, whereas, the other three forces are confined to our three dimensional universe. Thus, what is experienced as gravity is only a small fraction of the total force that is comparable to the three other fundamental forces. Theories that consider gravity as acting in four spatial dimensions are not new in physics. The KaluzaKlein theory of 1921 attempted to combine gravitation and electromagnetism by extending general relativity to fivedimensional spacetime. In 1935, astrophysicist Edward Arthur Milne suggested the relation G = c^{3 }t/ M_{U} for the gravitational constant, where M_{U }is the mass of the universe and t is the age of the universe (Milne 1935). Milne concluded that G increases over time. Paul Dirac was led to a timevarying gravitational constant from the large number coincidence. The large number coincidence can be summarized as the ratio of the product of the speed of light and the age of the universe divided by the Classical electron radius is about 10^{40} and the ratio of the electrical to gravitational forces between a proton and an electron is also about 10^{40}. Dirac found that in atomic units, G had a value of 10^{40}. He interpreted this as G varies inversely with time. In general relativity, G needs to be constant or conservation of energy is violated. Dirac resolved this problem by introducing a gauge function and proposed the concept of continuous creation of matter in the universe. Theoretical physicist Carl Brans' weak field analysis of Einstein's field equations suggested to him that G ~ R/M, where M and R are the mass and radius of the observable universe, respectively (Brans 1961). In the JordanBransDicke theory of gravity, G becomes a dynamical variable represented by a scalar field. There are many other timevarying theories available in the literature. Some examples include theories that reconcile timevarying G with general relativity (Canuto 1979) and higher dimensional theory and Mtheory (Brax et al. 2006). Recent articles on timevarying G are, for example, Berman (2010) and Roman (2010). Theoretical Development In this paper a theory of gravity is presented based on 4dimensional space or hyperspace. The theory is developed without referring to Einstein's general relativity or Newton's theory of gravitation. The development of the theory is as follows. Consider that our universe lies inside a fourdimensional hyperspace. If we assume for simplicity that our universe is closed and hyperspace is also closed, than the model of our universe embedded in a higher dimensional space could be visualized by neglecting one dimension as a twodimensional disk within a much larger threedimensional sphere. Next, an arbitrary point in hyperspace is chosen as an inertial reference frame. From this point of reference, the claim is made that our universe and every object in our universe is rotating about its center of mass relative to hyperspace. An example of a model of gravity where our universe is rotating relative to higherdimensional space is described by (Ivanenko 2003). The rotation is such that the tangential speed is at its maximum possible value, the speed of light in vacuum. Then every object with finite mass and size creates a centripetal acceleration due to its rotation and a reactionary centrifugal force. The centrifugal force points into the fourdimensional hyperspace. All materials deform their shape if a sufficient force is applied. Space itself will deform or warp into hyperspace because of this centrifugal force. Then the warping of space is what is experienced as gravity. It is the projection of the centrifugal force onto our universe. The centrifugal force for a mass M, radius R and tangential speed v is defined as F = Mv^{2}/R (1) The tangential speed is the speed of light in a vacuum, c, then we obtain F = Mc^{2}/R (2) From the geometry, the force of gravity F_{g} is the projection of the centrifugal given by F_{g}= F sinα = Mg (3) Substituting equation (1) into (3) and solving for g, the acceleration of gravity becomes g = F sinα /M = (c^{2}/R)sinα (4) The angle α is the deflection of space into the 4^{th} spatial dimension. Thus the acceleration of gravity produced by an object is equal to the sine of the deflection angle α of space into hyperspace times the square of the speed of light in vacuum c divided by the distance R from the center of mass of the object. The next step is to estimate how space is deformed by an object's centrifugal force. If the deformation of space is treated as elastic, then the force is proportional to a change in displacement into the 4^{th} dimension. To estimate how space is deformed the following assumption is introduced. The warping of space caused by the centrifugal force of an object rotating relative to hyperspace is resisted by the total centrifugal force of the rotating universe. This assumption is a restatement of a conjecture was first proposed by Ernst Mach in the statement "local physical laws are determined by the largescale structure of the universe." The centrifugal force of the entire universe rotating relative to the hyperspace is given by F_{U} = M_{U}c^{2}/R_{U} (5) Where, M_{U} and R_{U} are the mass and radius of the known universe, respectively. Next, if equations (2) and (5) are substituted into (3), then the deflection of space by an object's centrifugal force F is related to the total centrifugal force of the universe by sinα = F/F_{U} = (Mc^{2}/R) / (M_{U}c^{2}/R_{U}) = (M/M_{U})(R_{U}/R) (6) Substituting this deflection relationship (6) into (4), the acceleration caused by a mass M becomes g = (c^{2}/R)(MR_{U}/M_{U}R) = (R_{U}c^{2}/M_{U})M/R^{2 }(7) The model leads to an inverse square law for the acceleration of gravity with a constant of proportionality k = (R_{U}c^{2}/M_{U}) (8) The constant of proportionality can be estimated from the known values of the mass and radius of the observable universe. Unfortunately, the mass of the observable universe is not well known. The mass of the universe was tabulated for various researchers (McPherson 2006) and it was found that it varies by 7 orders of magnitude between 10^{53} and 10^{60} kg. In this paper, the value of G was calculated using values of R_{U} and M_{U} found on the website http://en.wikipedia.org/wiki/Observable_universe. The total mass of stars in the visible universe is estimated as 3x10^{52 }kg^{ }and this mass is only 5% of the total mass of the universe. Then the total mass is estimated as 6.0x10^{53} kg. The radius of the observable universe is given as 46.5 billion light years, or 4.4x10^{26} m. This radius is the comoving distance or the distance now to the edge of the observable universe. Substituting the values of R_{U} and M_{U }into (8) gives k = (R_{U}c^{2}/M_{U}) = (4.4x10^{26} m)(3x10^{8} ms^{1})^{2}/(6x10^{53} kg) = 6.60x10^{11} Nm^{2}kg^{2} The proportionality constant is in exceptionally good agreement with the known value of the gravitational constant, G = 6.673x10^{11} Nm^{2}kg^{2}, considering the uncertainty in M_{U} and R_{U}. Thus it was found that the constant of proportionality k is the same as G, the universal constant of gravitation. Therefore, from this theoretical analysis G = R_{U}c^{2}/M_{U} (9) Where G = Gravitational Constant , R_{U} = Radius of the known universe, M_{U }= Mass of the known universe, c = Speed of light in vacuum Substituting (9) into (7) gives g = GM/R^{2} (10) Equation (10) shows that the theory presented has led to a derivation of Newton's law of gravitation. But it also shows in (9) that the gravitational constant G is proportional to the radius of the known universe and inversely proportional to the mass of the known universe. The theoretical model leads to two assertions: 1. There is an inertial reference frame called hyperspace in which all objects in our universe and the universe as a whole are rotating with a tangential speed equal to the speed of light in vacuum. This rotation creates a centrifugal force that is perpendicular to our universe. 2. The warping of space due to mass is proportional to the centrifugal force of an object and inversely proportional to the centrifugal force of the universe as a whole rotating relative to the hyperspace. Discussion Gravity is Getting Stronger with Time The observable universe is known to be expanding so R_{U} is increasing with time. The consensus in the scientific community is that the total mass of the universe has not changed significantly since shortly after the Big Bang and the speed of light has not changed from its present value. Therefore, the derived expression in equation (9) shows that G is proportional to R_{U} and thus the gravitational constant is increasing with time. In the Appendix an equation for the comoving distance was derived and found to be R_{U} = ct(2+ln4) ≈ 3.386ct (11) Substituting (11) into (9) then gives G = (2+ln4)c^{3 }t/M_{U} (12) The result in Equation (12) is the same as that found by Edward Milne in 1935, except for the constant (2+ln4). The Gravitational "Constant" G is increasing as the age of the universe "t" increases over time. An important consequence of equation (12) is that the force of gravity was weaker in the past and is becoming stronger as the universe ages. An implication to this discovery is that the size and mass of distant astronomical objects determined by Newton's form of Kepler's Third Law need to be corrected using equation (12) because the Gravitational Constant is actually a variable that was considerably different early in the age of the universe. For example, a distant galaxy that is 10 billion lightyears from Earth, we observe it the way it looked 10 billion years ago. From equation (12), the Gravitational Constant and thus the force of gravity for that galaxy will be only 27% of the present value. Singularities are Impossible In this theory, the warping of space by an object with mass M and radius R is given by equation (6) as sinα = (M/M_{U})(R_{U}/R) (6) But sinα ≤ 1, therefore (M/M_{U})(R_{U}/R) ≤ 1 (13) Or rearranging gives R ≥ MR_{U}/M_{U }(14) Substituting (9) into (14) gives R ≥ GM/c^{2} (15) Thus, the theory leading to equation (15) shows that an object of mass M must have a finite minimum radius given as R_{min} = GM/c^{2} (16) From General Relativity (see Carroll 1996), the Schwarzschild Radius is given as R_{Sch} = 2GM/c^{2} (17) Then in terms of the Schwarzschild Radius, an object's minimum radius is R_{min} = (1/2) R_{Sch} (18) Therefore, an object with finite mass cannot exist as a singularity with zero radii. This would imply infinite density and infinite gravitational force. However there is only a finite gravitational force available and that force is given by equation (5) as F_{U} = M_{U}c^{2}/R_{U} (5) Substituting (9) into (5) gives the maximum gravitational force that an object can possess. F_{U} = c^{4}/G = (3.00x10^{8} m/s)^{4}/(6.673x10^{11} Nm^{2}kg^{2}) = 1.2x10^{44} N (19) Equation (6) shows why the force of gravity appears so weak. The deflection or warping of space due to mass is resisted by the total mass and size of the universe. The theory leads to the conclusion that a black hole cannot have its mass confined to a singularity, rather its mass must be distributed over a radius that is at least onehalf the Schwarzschild Radius. Explanation for the Pioneer Anomaly The Pioneer anomaly is the name given to the observed difference in velocities and trajectories of spacecraft in the outer solar system from what is expected from the current theory of gravity. A small, additional sunward acceleration of (8.74 ± 1.33)×10^{10} m/s^{2} has been observed for both Pioneer 10 and 11 and there is presently no accepted explanation for this anomaly. Both spacecrafts are closer to earth than should be expected when all the known forces acting on the spacecrafts are taken into consideration. It is also noted that the unexplained acceleration magnitude is close to the value of the product of the Hubble constant times the speed of light in a vacuum but the significance of this is unknown (Anderson 2009). The explanation could simply be measurement error, gas leakage, effects of uneven heating or other possibilities. However, the unexplained acceleration can also be explained by a small variation in the gravitational constant with time. Deriving the acceleration observed in the Pioneer anomaly The first step in the derivation is to differentiate (9) with respect to time, giving dG/dt = (c^{2}/M_{U}) dR_{U}/dt (20) From Hubble's Law: dR_{U}/dt = HR_{U} (21) Substituting (21) into (20) gives: dG = (c^{2}/M_{U})(HR_{U})dt = (H R_{U}c^{2}/M_{U}) dt (22) The change in acceleration from equation (10) is: dg = dG(M_{U}/R_{U}^{2}) (23) Substituting (22) into (23) gives dg = (H R_{U}c^{2}/M_{U})dt(M_{U}/R_{U}^{2}) (24) The radius R_{U }used in Hubble's Law is the proper distance, not the comoving distance. Then using R_{U} = cdt gives: dg = Hc^{2}dt / R_{U} = Hc^{2}dt/cdt = Hc (25) The theory predicts an added sunward acceleration of Hc = 7x10^{10} ms^{2} and the observed sunward acceleration measured for the two Pioneer spacecrafts is 8.7x10^{10} ms^{2}. Using the timevarying equation for G, it was found that the added acceleration caused by the change in gravitational constant to be the Hubble constant times the speed of light. The acceleration closely predicts the unexplained acceleration observed by the two Pioneer spacecrafts. The proposed theory suggests that the expansion of space is causing the gravitational constant of the universe to slowly increase, which would over time affect the motion of all objects. Conclusion The objective of this paper was to develop a theory of gravitation with a timevarying Gravitational Constant G. It was found that G is proportional to the radius of the known universe and inversely proportional to the mass of the known universe. The theory offers an explanation for the present value of the gravitational constant and makes predictions of past and future values of G. A timevarying G has important implications to physics and astronomy. The size of distant objects such as QSOs and galaxies that are determined using G in Newton's form of Kepler's third law will need to be revised if this theory is found to be correct. It is generally believed that nearearth objects and objects within our galaxy are not affected by the expansion of the universe. However a timevarying G offers a viable explanation for the unexplained sunward acceleration observed by the Pioneer spacecraft. Important outcomes of this timevarying G theory include: 1. A new theory of gravity was developed independent of Einstein's and Newton's theory of gravity that found that the Gravitational Constant is increasing with time. 2. An explanation why the force of gravity is relatively weak compared to the three other fundamental forces. 3. An explanation why black holes cannot have singularities. 4. The small added sunward acceleration observed in the Pioneer and other spacecrafts can be explained in terms of a timevarying Gravitational Constant. 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Astrophysics and Space Science, 2010, vol. 325, n^{o}2, pp. 259275 [17 page(s) (article)] Ivanenko, Sergey, "Big Spin Model of Gravity." March 2003 http://en.wikipedia.org/wiki/Mach's_principle http://en.wikipedia.org/wiki/Observable_universe Edward Wright, 2009  http://www.astro.ucla.edu/~wright/cosmology_faq.html#DN http://hypertextbook.com/facts/2006/KristineMcPherson.shtml http://hypertextbook.com/facts/2000/ChristinaCheng.shtml Carroll, Bradley W. and Ostlie, Dale A., 1996. An Introduction to Modern Astrophysics, 1^{st} Ed., AddisonWesley Publishing John Anderson, "Is there something we don't know about gravity?", Astronomy Magazine, vol 37, # 3, March 2009, pp. 2227. Appendix: Derivation of the Comoving Radius Equation The objective in this section is to derive an equation for the radius of observable universe. If the universe is at the critical density, the comoving distance or distance now to the edge of the observable universe is given by the equation R_{U} = 3ct (Wright 2009). To derive the comoving radius, let t be the look back time or time it takes light to travel from the object to Earth and c is the speed of light in vacuum. The R_{0} is the distance to the object now if the universe was not expanding, the term R^{'}t is due to the expansion at constant acceleration, and the additional terms are due to the acceleration and the time variation in the acceleration. The acceleration term is the Hubble variable H times the speed of light c. It is treated that the acceleration, rate of change of acceleration, and higher order terms are varying with time ad infinitum. Then we have the comoving radius equation R(t) given by the Taylor Series Expansion: R(t) = R_{0} + R_{0} + R^{'}t + R^{''}t^{2}/2! + R^{'''}t^{3}/3! + R^{IV}t^{4}/4! + R^{V}t^{5}/5! + … + R^{n}t^{n}/n! R_{0} = ct R'^{ }= c R^{''} = Hc = ct^{1} R^{'''} = ct^{2} R^{IV} = 2ct^{3} R^{V} = 6ct^{4} = 3!ct^{4} Then for n = 2, ∞, we R^{n} = (1)^{n}(n2)!ct^{(n1)} Substituting gives R(t) = 2ct + ct + ct/2  ct/(2*3) + ct/(3*4)  ct/(4*5) + … R(t) = ct{2+[1 + ∑^{∞}_{1}(1)^{n+1}/n(n+1)]} R(t) = ct(2 + ln4) R(t) = 13.7 bly(2+ln4) = 46.4 bly Thus an object with a lookback t is now at a distance of R(t) = ct(2 + ln4). Conversely, if an object is at the edge of the observable universe with a look back time t = 13.7 billion years, the comoving radius is 46.4 billion lightyears. Information about this Article This Article has not yet been peerreviewed This Article was published on 15th April, 2011 at 04:16:40 and has been viewed 2777 times. This work is licensed under a Creative Commons Attribution 2.5 License. 
The full citation for this Article is: Schieber, W. (2011). Time Variation in the Gravitational Constant. PHILICA.COM Article number 235. 
