Isaac Newton
The Gravitational Constant
Abstract
The presented analysis is in direct opposition to the current opinion that the Gravitational constant itself is simply a conversion factor between systems
but rather it has been shown that it is an essential fundamental value representative of a much deeper fundamental principle from the quantum
scale upto stars and galaxies.
Introduction
Isaac Newton was a brilliant English mathematician, physicist, and astronomer, born on December 25, 1642 (January 4, 1643, in the Gregorian calendar).
He is best known for his laws of motion and universal gravitation, which laid the foundation for classical mechanics. His book, "Philosophiæ Naturalis Principia Mathematica" (Mathematical Principles of Natural Philosophy), published in 1687, is considered one of the most important works in the history of science.
Newton also made significant contributions to optics, including his work on the nature of light and color, and he shares credit with Gottfried Wilhelm Leibniz
for developing calculus,
Newton's discovery of the gravitational constant (𝐺) is perhaps one of his greatest contributions, fundamentally crucial in understanding the gravitational
interactions between celestial bodies and significantly impacting astrophysics and cosmology.
$$ F = \frac{G M_1 M_2}{r^2}$$
The equation states that the gravitational force \(F\) between two objects is directly proportional to the product of their masses \(𝑀_1\) and \(M_2\)
and is inversely proportional to the square of the distance \(r_2\) between them. In simpler terms, the force increases with larger masses and decreases
with greater distances. The Gravitational constant appears in his his most famous equation despite this, its precise measurement continues to be a
challenge in experimental physics. This ongoing difficulty is puzzling, given that Newton's method for determining its value, using Kepler's law, is well
understood and the constant's dimensions are well known. The gravitational constant is one of the least accurately measured constants in physics, with an accuracy limited to three decimal places,
as determined through basic dimensional analysis.
$$G = [M^{-1}][L^{3}][T^{-2}]$$
Physicists generally agree that the Gravitational constant cannot be derived mathematically and must be obtained through direct measurement.
Its small value does not influence the theoretical ability to actually calculate it, given that the dimensions and method for its calculation
are well known.
The main question becomes which fundamental values should be used to establish its value, the most obvious choice
appearing to be the Planck constants. In this context, Kepler's law is evident, as a length cubed divided by a time squared appears in the very
dimensions of the Gravitational constant.
Utilizing this in the following equation the only additional requirement is the inclusion of a suitable mass.
$$G = \frac{{l_p}^3}{m_p {t_p}^2}=6.674\ 787\ 6501 × 10^{-11}\ \text{m}^3\text{kg}^{-1}\text{s}^{-2}$$
Where:
\(G = \text{Newtons Gravitational constant}\)
\(l_p = \text{Planck length}\)
\(m_p = \text{Planck mass}\)
\(t_p = \text{Planck time}\)
It's astonishing that the straightforward application mentioned has gone unnoticed for centuries.
Although the Planck constants were only introduced in the early 20th century, there appears to have been a general disinterest in fundamental physics
within the scientific community, with a greater focus being placed upon more abstract and esoteric topics.
The value proposed above, being no less than seven orders of magnitude greater than the currently measured value, will undoubtedly face scrutiny.
The primary objection is that Planck units necessitate the use of the measured Gravitational constant. However, it is noteworthy that the values of
Planck units correlate when calculated from first principles using unrelated values, as previously discussed.
Additionally, using only the basic properties of the ground state electron in the Hydrogen atom, the result aligns precisely with the calculated value.
$$G = \frac{a_o{c}^3}{v_e m_e {q_n}^3}=6.674\ 787\ 6501 × 10^{-11}\ \text{m}^3\text{kg}^{-1}\text{s}^{-2}$$
Where:
\(G = \text{Newtons Gravitational constant}\)
\(a_o = \text{Bohr radius}\)
\(c = \text{Speed of light}\)
\(v_e = \text{Electron Velocity}\)
\(m_e = \text{Electron mass}\)
\(q_n = \text{Count of Quantum elements (quanta)}\)
Conclusion
Calculating the Gravitational constant using subatomic quantum scale elements introduces one of the most profound questions in physics: is gravity quantized?
Previous calculations suggest that this may be the case, as the value of the Gravitational constant at the subatomic scale depends on Planck units,
particularly the Planck length. However, this alone is not enough to confirm the hypothesis.
It is also evident that the
Planck length can be derived from first principles, thereby negating any argument that its exact
value is implicitly included.