The Unified Theoretical Definition of Newton's Gravitational Constant G (Ideas for Research #7)

Most physicists assume that Einstein's theory of gravity (general relativity) made Newton's gravitational theory an outdated theory. In contrast, this article reveals the Einstein-Newton connections that enabled me to discover the theoretical definition of Newton's G. I also show how Einstein’s use of Newton’s G in general relativity defines the Conservation Law for Force.

Overview of the Discovery

Two key pieces of information were buried in the dustbins of historical physics. First, Einstein noted that Newton’s value for the deflection of a photon by the Sun is half the value he calculated for general relativity. Secondly, Newton instructed that his value be doubled because it calculates the force for only one object in mutual attraction. Obviously, if Newton’s equation calculates one-half of Einstein's and Newton's result is to be doubled, then Newton’s and Einstein’s final values are identical.

Also, Einstein used Newton’s G in general relativity to define a background field of constant force. He ‘eliminated’ gravity as a force by defining its unit force as the constant value of the background field, the universal substrate. The right-hand side of the general-relativity equation is read in SI units as the value of the Energy density tensor per Force. This has been ignored because the values in Einstein's 1/Force coefficient are typically set to "1" in the Natural units.

In the unified Interval Dynamics Theory (IDT), Einstein's universal substrate is the Force/Current A field in Maxwell’s "Theory of Light" (Treatise, Vol. 2, Chap. XX). Maxwell's A field is also the fundamental field in Schrödinger’s quantum-wave equation and Feynman’s Quantum Electrodynamics (QED) (Lectures, Vol. 2, Sect. 15-5).

The unification of electromagnetism, quantum theory, and general relativity into IDT is discussed in my forthcoming book (early 2025). The conceptual and mathematical models can enable physicists to break through the cognitive barriers blocking their progress in physics.

This article describes IDT's theoretical definition of Newton’s G as the ratio of two lightspeeds. The theoretical value differs from the NIST standard value by only 1.7x10^-14. This is down in the uncertain digits, which is amazing given that NIST's standard is an average of numerous experiments. Furthermore, the IDT value is within the error bars of the only value measured using light rays.

The Development of “Newton’s G”

Newton's discussion of gravitation in the Principia in 1687 does not include the gravitational constant. Instead, he expressed gravitational attraction through proportional relationships, demonstrating that the force was proportional to the product of masses and inversely proportional to the square of the distance.

Henry Cavendish made the first experimental measurement of the gravitational force in 1798. However, he measured the Earth's density rather than establishing the gravitational constant. His paper is often cited because a value of Newton’s G can be calculated from his work.

The first explicit mention of a gravitational constant (“f”) came from French physicists A. Cornu and J. B. Baille in 1873. The symbol 'G' was first formally introduced in J. H. Poynting's 1893 Adams Prize essay, "The Mean Density of the Earth." Poynting referred to G as "the constant of gravitation" and "the constant of attraction." In 1894, C. V. Boys published a paper "On the Newtonian Constant of Gravitation."

In 1916, Einstein used Newton's G in general-relativity theory to define the constant-force background field of the Universe. On the right-hand side of Einstein’s summary equation, the coefficient of the stress-momentum (energy) tensor is the scalar value 8πG/c^4, which is 1/Force.

Consequently, rearranging Einstein’s equation defines his inverted force constant as the ratio of (a) the effects of gravity to (b) the sources of gravity. This constant-force value ensures the curvature of spacetime changes as the energy density changes at any point. It also establishes Einstein’s general relativity as IDT's Law of Conservation of Force.

The value c^4/8πG is the Force/Current of the magnetic vector potential, which is Maxwell’s A-field universal substrate. It is the fundamental force Faraday believed manifested as other types of force.

Einstein’s and Newton’s Identical Values

Gravity is unique because it is the only fundamental interaction that quantum theory has not explained. Also, Newton's G is the most difficult physical constant to measure, has no theoretical value, and keeps increasing in uncertainty as technology advances. Its certainty is known for only three significant digits (6.67x10^-11).

When Einstein calculated the Sun’s deflection of a photon’s path, he divided his results into two parts. He wrote, “It may be added that, according to the [general-relativity] theory, half of this deflection is produced by the Newtonian field of attraction of the sun, and the other half by the geometrical modification ("curvature") of space caused by the sun.” (Vol. 6, p. 385 [Doc. 42])

Newton’s focus was on the force of attraction for one object, not mutual attraction. For example, “Gravitation towards the sun is made up out of the gravitations towards the several particles of which the body of the sun is composed.” This is common knowledge because it is explicitly diagrammed in Wikipedia.

However, Poynting (p. 2) wrote, “we conclude that each planet reacts to the sun with a force equal to and opposite to that exerted by the sun on the planet.” Consequently, the force is to be doubled for two-body mutual attraction. A test body of light or matter pulls on the energy of the Sun, and the Sun pulls on the energy of the test body.

Three times in his Principia (pp. 221-225), Newton states that the value calculated from his equation is to be doubled (bold emphasis added):

• “therefore since in all attractions (by Law III) the attracted and attracting point are both equally acted on, the force will be doubled by their mutual attractions, the proportions remaining.” (in Proposition LXXV, Theorem XXXV)

• “The proportions take place also when the attraction arises from the attractive virtue of both spheres mutually exerted upon each other. For the attraction is only doubled by the conjunction of the forces, the proportions remaining as before.” (in Proposition LXXVI, Theorem XXXVI)

• “Let the spheres attract each other mutually, and the force will be doubled, but the proportion will remain.” (in Proposition LXXVII, Theorem XXXVII)

Although photons have no observable mass, the equal gravitational forces in both directions still exist between the photon’s energy and the Sun. This is why, as Einstein noted, Newton's value is precisely half of Einstein's general-relativity value.

Universal and Local Observers

Several blocks to progress in physics are caused by failing to identify two types of observers: (1) the Universal Observer [UO] for fields, and the Local Observer [LO] for particles. Instead of distinguishing between their two types of coordinate systems, physics has been stuck in the "wave-particle duality" paradigm.

Classical physicists use an LO’s local coordinate system to measure particles that move slower than light. In contrast, Schrödinger modeled electrons as moving in a quantum "zitterbewegung" motion at the speed of light. As Dirac described in his Nobel Lecture, “As a result of this oscillatory motion, the velocity of the electron at any time equals the velocity of light.”

This classical Maxwellian A field is the fundamental field in both Schrödinger’s and Feynman’s quantum theories. When discussing this, Feynman thought it “interesting that something like this can be around for thirty years but, because of certain prejudices of what is and is not significant, continues to be ignored.”

IDT's definition of the Universal Observer (UO) reveals the field nature of our Universe as Einstein's general relativity describes it (but without Minkowski's imaginary time value, p. 51). Eliminating "ict" and using "ct" allows modeling the UO in the same finite volumes as human sensory perception. There is no need to model curved surfaces.

Thus, Newton's G is a conversion constant that relates the UO's gravitational force field to the LO's product of two mass linear densities. The dimensions of G are L^3/MT^2 of Kepler’s third law for planets per unit mass. The dimensions simply convert the Local dimensions of particle-mass densities (M^2/L^2) to the Universal field dimensions of gravitational force (ML/T^2).

The Theoretical Value of G

Maxwell’s original theory with 20 equations and seven fields was far ahead of its time. Maxwell mapped two coordinate systems into the physical dimensions of Mass M, Length L, and Time T.

• ElectroStatic Units (ESU) system: Based on the speed of light as "c", it is equivalent to IDT's Local Observer (LO).

• ElectroMagnetic Units (EMU) system: Based on the speed of light as dimensionless "1", it is equivalent to the Universal Observer (UO) with one change.

The one change in IDT is the UO's speed of transmission is "2" rather than "1". Both Maxwell and Schrödinger modeled the unit spinners in the A-fields substrate as pairs. Maxwell modeled solution pairs that form light waves, and Schrödinger modeled non-solution pairs that form matter.

The dimensionless "2" is a coordinate-independent absolute speed of light. It is the Universal-to-Local scaling constant for the dimension of length. It is cubed for volumes, such as those modeled by IDT when replicating Einstein's values for the orbital precessions of planets.

The theoretical value of G uses the speed of light in the older CGS system of units for the LO, the denominator of the ratio. Consequently, the value for the LO is 29,979,245,800 centimeters per second. If the LO were using SI units, then the LO's observed speed of light would be 299,792,458 meters per second.

For Newton’s G, the UO/LO ratio is 2/29,979,245,800, which equals 6.6713x10^-11. This differs from the NIST standard by only 1.7x10^-14, an amazingly small difference. NIST's standard value is the mean of 14 experiments using six different types of instruments.

Schlamminger's graph from NIST in Figure 2 identifies the 14 experimental values and shows their error bars. The vertical black line is the NIST average. The black dot I added shows IDT’s theoretical value. Thirteen of the 14 experimental values were measured by mechanical motion.

In contrast, the Rosi et al. 2014 experiment used light (an interferometer) and atoms in free-fall motion to measure G. This is important for two reasons: (1) These two characteristics, light and free-fall motion, are the characteristics IDT uses from general relativity to define the UO's coordinate system, and (2) IDT's theoretical value, which is the ratio of two lightspeeds, is well within the error bars of the experiment.

The slight variations in mechanically-based measurements may (a) reflect the inherent limitations of mechanical motion in capturing field effects, or (b) involve the value of the cosmological constant as the energy-density field of the vacuum.

The close agreement of the theoretical and observed values suggests that IDT's theoretical framework provides a unified understanding of gravitational phenomena. Theoretical and experimental physicists can use this as the foundation for breaking through the conceptual blocks that have been stopping progress in physics.

Conclusion

Newton's and Einstein's approaches to gravity are unified in Interval Dynamics Theory (IDT). This reveals one of the fundamental unities in physical theory that have been obscured by forgetting or misinterpreting the models and equations of the original theorists.

This suggests new approaches to theoretical physics, particularly in areas where quantum and gravitational effects intersect. Future research can explore how IDT's framework could contribute to additional theoretical advances and provide new experimental methods for measuring fundamental constants.

My book Interval Dynamics is to be available on Amazon in January. If this topic interests you, purchase it and leave a review. Thanks in advance!

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To cite this article

Lindeman, M. J. (2024, December 31). “The Unified Theoretical Definition of Newton's Gravitational Constant G (Ideas for Research #7).” [Article]. LinkedIn. https://www.linkedin.com/pulse/unified-theoretical-definition-newtons-gravitational-g-martha-m-j--smjwc/

Previous Articles in the Ideas for Research Series

6. Lindeman, M. J. (2024, December 18). "Beyond Position and Velocity: The Quantum Spinner Foundation of Our Universe (Ideas for Research #6)." [Article]. LinkedIn. https://www.linkedin.com/pulse/beyond-position-velocity-quantum-spinner-foundation-lindeman-ph-d--uvkhc/

5. Lindeman, M. J. (2024, December 14). "Step-by-Step Classical Derivation of Schrödinger’s Equation (Ideas for Research #5)." [Article]. LinkedIn. https://www.linkedin.com/pulse/step-by-step-classical-derivation-schr%25C3%25B6dingers-lindeman-ph-d--g275c/

4. Lindeman, M. J. (2024, December 3). "Maxwell's Forgotten Foundation of Quantum Mechanics (Ideas for Research #4). [Article.] LinkedIn. https://www.linkedin.com/pulse/maxwells-forgotten-foundation-quantum-mechanics-lindeman-ph-d--aja8c/

3. Lindeman, M. J. (2024, November 27). “Imaginary Numbers & the Geometry That Rules Our Universe (Ideas for Research #3).” [Article]. LinkedIn. https://www.linkedin.com/pulse/imaginary-numbers-geometry-rules-our-universe-lindeman-ph-d--lkjsc/

2. Lindeman, M. J. (2024, November 22). “How to Unify General Relativity and Quantum Mechanics (Ideas for Research #2)”. [Article]. LinkedIn. https://www.linkedin.com/pulse/how-unify-general-relativity-quantum-mechanics-lindeman-ph-d--suvwc/

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