Using a Grandfather Pendulum Clock to Measure the World’s Shortest Time Interval, the Planck Time (With Zero Knowledge of G )

Haug has recently introduced a new theory of unified quantum gravity coined “Collision Space-Time”. From this new and deeper understanding of mass, we can also understand how a grandfather pendulum clock can be used to measure the world’s shortest time interval, namely the Planck time, indirectly, without any knowledge of G. Therefore, such a clock can also be used to measure the diameter of an indivisible particle indirectly. Further, such a clock can easily measure the Schwarzschild radius of the gravity object and what we will call “Schwarzschild time”. These facts basically prove that the Newton gravitational constant is not needed to find the Planck length or the Planck time; it is also not needed to find the Schwarzschild radius. Unfortunately, there is significant inertia towards new ideas that could significantly alter our perspective on the fundamentals in the current physics establishment. Howev-er, this situation is not new in the history of science. Still, the idea that the Planck time can be measured totally independently of any knowledge of Newton’s gravitational constant could be very important for moving forward in physics. Interestingly, an old instrument that today is often thought of as pri-mitive instrument can measure the world’s shortest possible time interval. No atomic clock or optical clock is even close to be able to do this.

will briefly summarize the key points in the history of gravity that are relevant to this paper.
• 1673-Christiaan Huygens [1] publishes theory on how to calculate the pendulum periodicity from gravity acceleration: • 1687, 1713, and 1726-Three versions of the Principia. Newton's [2] gravitational formula Newton only states this formula in words. Newton never introduced a gravitational constant per se, nor did he have any use for one. Newton stated that mass is proportional to weight. Further, he showed how to extract the relative mass of astronomical objects easily. Based on the size of the planets, he could find their relative densities. Newton was focusing on relative masses, as also pointed out by Cohen [3]; "That is, since Newton is concerned with relative masses and densities, the test mass can take any unity, so that weight-force may be considered the gravity or gravitational force per unit mass...".
• 1796-The introduction of the kilogram (kg), but the kilogram gets widely accepted first in Europe after the Metre Convention was signed in 1875, which again leads to the production of The International Prototype of the Kilogram. • 1798-Cavendish [4] calculates the density of the Earth relative to known uniform elements such as water, lead, or gold using a torsion balance. This torsion balance was invented by Cavendish's friend John Michell sometime before 1783, but Michell was not able to perform the experiment before he died. However, Cavendish never mentions a gravitational constant.
• 1873-The idea that one needed a constant, now known as Newton's gravitational constant to obtain the gravity force.
2 Mm F f r = was mentioned explicitly by Cornu, and Baille [5], with notation f for the gravity constant.
It was needed because of the practice of defining mass in terms of the kg. The gravitational constant was then used to remove an element from an arbitrary mass that had nothing to do with gravity and include what had been missing in the kg definition of mass, namely the Planck length. This inclusion appears to have occurred subtly, without physicists actually examining or challenging this evolution in the conception of mass, see [6].
• 1894-The gravitational constant was first called G by Boys in August 1884, see [7]. Here he stated the following formula:  [13] produces his theory of general relativity, where he "naturally" is heavily dependent on G (he uses notation K) in his formulas.
• 1916-Einstein [13] suggests that the next big step in the progress of gravity will be to develop a quantum gravity, or in his own words "Because of the intra-atomic movement of electrons, the atom must radiate not only electromagnetic but also gravitational energy, if only in minute amounts. Since, in reality, this cannot be the case in nature, then it appears that the quantum theory must modify not only Maxwell's electrodynamics but also the new theory of gravitation." Einstein • 1918-Eddington [14] suggests that quantum gravity must be linked to the Planck length, or in his own words: "But it is evident that this length (the Planck length) must be the key to some essential structure. It may not be an unattainable hope that someday a clearer knowledge of the process of gravitation may be reached." However, this contention was ridiculed by other prominent physicists like Bridgman [15] (who received the 1946 Nobel Prize in Physics).
• Recent time-Many researchers [16]- [21], particularly those working on quantum gravity, are more or less convinced that the Planck scale is the smallest possible scale in terms of space and time, and that it is essential for unifying gravity with quantum mechanics. Others have been questioning this view because it seems like we only can find the Planck units through dimensional analysis based on the knowledge of G,  , and c. See for example [22].
If we not can detect the Planck scale and only calculate the Planck units from dimensional analysis, why not simply abandon the hypothesis of their existence? Our findings in this paper show that the Planck scale is real and indirectly easily detectable from gravity phenomena.
• 2019/2021-For more than 100 years, physicists have tried to unify theories of gravity and quantum mechanics with minimal success and little progress.
The recently published theory of collision space-time [6] [23] [24] offers a fresh, new perspective that seems to unify a modified relativity theory with gravity and quantum mechanics in a fully consistent way. This process should not be taken for granted, and further careful study of this theory is warranted. In the end, discussions and investigations by many researchers over time will be the best way to evaluate its merits. The Planck length and The importance of further understanding the Planck time is expressed by, for example, Ball, for example, [25] in 1999 where he states: "A physics to match the Planck timescale is the biggest challenge to physicists in the coming century." We will show that the Planck time can be measured from a pendulum clock (grandfather clock) with no knowledge of G.
It is important to understand that Newton never invented a gravitational constant, nor did he use one himself. Still, from his theories and insights, he was able to calculate the relative masses of heavenly objects, such as the Earth, the Sun, Jupiter, and Saturn; see the Principia and [3], for example. He was also able to calculate their approximate relative densities. Newton claimed and showed that the weight was proportional to mass. Mass for Newton was a quantity of matter, and he thought matter consisted of indivisible particles. Weight was not linked to kg in those days because the kilogram definition of mass was first was put forward in 1796, long after Newton's death.
At a deeper level, The kg is a collision frequency ratio, as shown by Haug [6] [26], and the rest mass in kg for any mass (from the subatomic particle to super massive galaxies, and even the mass of the observable universe) can be described where  is the Planck constant, λ is the reduced Compton wavelength [27] of the mass in question, and c is the speed of light. This calculation is simply the Compton wavelength formula solved for m. We do not need to know m to find the Compton wavelength first. Some people will likely question why we are not using the de Broglie [28] [29] matter wavelength, as it is normally the wavelength linked to matter. We think one of the greatest mistakes in physics has been to build quantum and matter theory around the de Broglie wavelength rather than the Compton wavelength. For example, the de Broglie wavelength, h mv λ γ = , is not mathematically defined for rest-mass particles because we cannot divide by zero ( 0 v = ), or alternatively, we [30] can claim it converges to infinity as the particle almost comes to rest-something that is absurd. However, the de Broglie wavelength has been accepted as the matter wavelength without hardly anyone questioning it, partly because it has been in use in theoretical physics for almost 100 years. It was assumed in 1927 by the Davisson-Germer [31] experiment for electrons and the experiment by Thomson and Reid [32] that the de Broglie hypothesis was confirmed. This situation is not the case in our view.
These experiments only confirm that matter also had wave-like properties, but not that this wavelength was equal to the de Broglie wavelength, one could just as well have claimed it confirmed that matter is linked to the Compton wavelength.
One obtains (mostly) the correct predictions when building a theory using the de Broglie wavelength, but at an unnecessarily complex cost. In our view, the de

E. G. Haug Journal of Applied Mathematics and Physics
Broglie wavelength is a mathematical derivative of the true matter wavelength, which we claim is the Compton wavelength; this hypothesis is discussed in more detail in [6] [23]. The relationship between the Compton wavelength and the de Broglie wavelength is that the de

Using a Pendulum Clock to Find the Planck Time
Haug [6] [34] has shown how to find the Planck length independently of both G and  using a Cavendish apparatus, and without G using a Newton force spring, see [35]. We extend this work to show how a grandfather pendulum clock can be used to find the Planck time (and thereby naturally the Planck length as well) independently of any knowledge of G. Huygens [1] was the first to derive the formula for the period of an ideal mathematical pendulum where L is the length of the pendulum, and g is the gravitational acceleration. The gravitational acceleration at the surface of the Earth is experimentally known to be about 2 9.81 m s g ≈ . From modern Newtonian gravity, including the gravitational constant G, we also know that the gravitational acceleration is given by However, based on our new collision space-time quantum gravity theory, any rest-mass can be viewed as collision-time, which is given by where p l is the diameter of an indivisible particle and λ is the reduced Compton wavelength of the mass in question. Be aware that this formula holds for any mass, even large astronomical bodies like the Earth and the Sun. The Earth does not have a physically reduced Compton wavelength, but we predict that each elementary particle making up the Earth has one. These particles are aggregated in the following formula Based on this fact, the gravitational acceleration of any object must be Thus, we have simply replaced the standard mass measure with our mass measure, and we have replaced the gravitational constant with c 3 , which is the much simpler "gravity" constant in our reformulated quantum gravity theory-the speed of light cubed. In other words, we are getting rid of a constant, namely G. Actually G,  and c is replaced with only p l and c. Now we can rewrite Huygens formula and solve it with respect to the mass, this calculation gives: Next keep in mind that we claim  However, some people may claim that we cannot know the Compton wavelength of the Earth λ without knowing G. This situation is not the case. We can measure the Compton wavelength of an electron from Compton [27] scattering, 1 2 1 cos e λ λ λ θ − = − . That is, we need to shoot photons at an electron and measure the wavelength (frequency) of the photon before, 1 λ , and after, 2 λ , the impact with the electron. In addition, we need to measure the angle between the incoming and outgoing photons (θ ). Alternatively, we could use hydrogen spectrum lines to extract the Compton wavelength of the electron; see the Appendix.
The cyclotron frequency is linearly proportional to the reduced Compton frequency. One can find the reduced Compton frequency ratio between the proton and the electron by conducting a cyclotron experiment. For example, [36] measured it to be about (see also [37])   (12) This number is the Schwarzschild radius from the large lead ball in a Cavendish apparatus, where R is the distance between the large and the small ball in the apparatus. Again G or  is not needed. Now we can count the number of protons in the large lead ball. This number also represents a practical challenge, but it is theoretically and even practically possible. For example, one has recently been able to count the number of atoms in 28 Si crystal spheres, see [38] [39]. Alternatively, we can use the Planck constant. Assume our large lead ball is half a kg; the Compton frequency in half a kg is given by where T s is what we will call Schwarzschild time, R is the radius of the Earth, T is the measured pendulum oscillation time, L is the length of the pendulum, and c We question this view that the Schwarzschild radius is related to black holes. We think half the Schwarzschild radius is the collision length of indivisible particles making up the gravity object. This indivisible particle is incredibly small, but it is the most important element-the essential building block of energy and mass.

Earth Moon as Giant Great Grandfather Clock (Pendulum Clock) Can Be Used to Find the Planck Length
The Huygens pendulum clock formula is, in general, only a good approximation for a small angle. It is an inaccurate approximation when the pendulum angle is large, but it is remarkably exact when the Pendulum angle is 360 degrees, that is, a full circle [40]. We can look at the Moon

Conclusions
We It is also worth mentioning that Newton never invented or used a gravitational constant. We claim that Newton's original theory is, in many ways, superior to today's modified version of his formula. The ancient formula made one wonder what mass truly was, and we now assume we know what mass is. The mass should be relative to the Planck mass that, in reality, likely is a collision between two indivisible particles. To relate the mass to an arbitrary quantity like the kg and, in addition, failing to grasp mass at a deeper level truly has caused much confusion and constrained progress and understanding in physics, or as Jammer [41] has pointed out, "Mass is a mess". As suggested by Newton himself, mass is directly linked to an indivisible particle, which we claim has a diameter equal to the Planck length. This paper is one of a series of papers strongly supporting this view.
In conclusion, we assert that in principle, we cannot measure the smallest possible time with an iPhone, an atomic clock, or even the most advanced optical clock, but rather with an old-fashioned grandfather pendulum clock. The key difference is that the grandfather clock is a gravity clock, and the shortest time interval is directly linked to the building blocks of mass, which are linked to gravity.