Newton and Einstein’s Gravity in a New Perspective for Planck Masses and Smaller Sized Objects

In a recent paper, Haug [1] has rewritten many of Newton’s and Einstein’s gravitational results, without changing their output, into a quantized Planck form. However, his results only hold down to the scale of Planck mass size objects. Here we derive similar results for any mass less than or equal to a Planck mass. All of the new formulas presented in this paper give the same numerical output as the traditional formulas. However, they have been rewritten in a way that gives a new perspective on the formulas when working with gravity at the level of the subatomic world. To rewrite the well-known formulas in this way could make it easier to understand strength and weakness in Newton and Einstein gravitation formulas at the subatomic scale, potentially opening them up for new interpretations.


Introduction
Based on dimensional analysis, [2] the Newton gravitational constant [3] can be written as a function of the reduced Planck constant, the Planck length, and the speed of light: Alternatively, this way of writing the gravitational constant can be obtained directly by solving the Planck length formula, [4] with respect to G. One could argue that this leads to a circular argument, since the Planck length is obtained from the gravitational constant. This circular problem is discussed and solved by Haug, [2] where he gives new theoretical insight that strongly indicates one can find l p independent of any knowledge of G. See also [5]. Further, based on the recent developments in mathematical atomism [6,7], it is reasonable to think that the Planck length is among the most fundamental constants that could represent the diameter of an indivisible particle. We are not questioning if big G is a universal constant; we are asking if G could be a universal composite constant consisting of even more fundamental constants, and we have reason to think these are c,h, and l p . This paper actually confirms that big G is also valid, and therefore Universal, for any subatomic particles, at least inside the theory o↵ered here.
There is still considerable uncertainty about the exact measurement of the gravitational constant. Experimentally, substantial progress has been made in recent years based on various methods. See, for example, [8,9,10,11,12]. Also, the relationship between physical constants from the microcosmos (subatomic world) and the macrocosmos (cosmos) plays an important role in physics. A continuous e↵ort is going into improving our measurements and understanding these relationships. See, for example, [13].
The Planck form of the gravitational constant enables us to rewrite the Planck length as and the mass of any subatomic particle with mass up or equal to the Planck mass, m p , can be written as ⇤ e-mail espenhaug@mac.com. Thanks to Victoria Terces for helping me edit this manuscript.
Using the gravitational constant in the Planck form, as well as the mass formula relationship above, we can easily rewrite a series of mathematical end results from Newton and Einstein gravitation without changing their output values. The elements that will change are the input parameters and their mathematical form. Seeing well-known formulas in this new perspective seems to give new and additional insight that we believe can be useful.

Newton's Universal Gravitational Force
Newton's gravitational force is given by Using the gravitational constant of the form G p = l 2 p c 3 h and two subatomic particles with the same reduced Compton wavelength, we can rewrite Newton's gravitational force for two subatomic particles as In the case where r =¯ , we get In the special case of two Planck masses, we have¯ = l p and we get the well-known Planck force

Escape Velocity
The traditional Newton escape velocity [14,15] is given by where G is the traditional gravitational constant, m is the mass of the object we are "trying" to escape from, and r is the radius we (for example a particle) are leaving from. Exactly the same escape velocity formula can be derived directly from Einstein's general relativity using the Schwarzschild metric, see [16].
Based on the gravitational constant written in the Planck form and the mass as written on the form given in the introduction, we can derive the escape velocity for a particle with mass equal or less than the Planck mass. We will assume the escape happens at a radius equal to the reduced Compton wavelength of the particle we are escaping from, r =¯ , this gives (12) This is sometimes called the Planck particle; it is the only particle where the Schwarzschild radius is equal to the reduced Compton wavelength.
Another interesting special case for the escape velocity is related to a particle with half the Planck mass in radius of l p ; this gives an escape velocity of c Motz and Epstein suggested in 1979 that there likely exists a very fundamental particle with half the Planck mass, [23]. However, they had no good explanation why such a particle was so much heavier than any known observed particle. Haug [6,7] has recently built a new fundamental theory around an indivisible particle with mass equal to half the Planck mass, where he has speculated on how such a particle could be the fundament of every observable mass. The Haug particle always moves at speed c and is quite di↵erent from what modern physics thinks of as particles. The indivisible particle is half a Planck mass when counterstriking (colliding) with another indivisible particle, but only for a Planck second. When non-counter striking it is mass-less and the origin of energy.

Orbital Velocity
The orbital velocity is given by We can rewrite this in the form of the Planck gravitational constant and the Planck mass as This can also be written as In the special case where¯ = l p , then the orbital velocity is equal to c. The reduced Compton wavelength of a Planck mass is l p . This also means the orbital velocity of a particle is equal to the mass of the particle in question divided by the Planck mass and then multiplied by the speed of light

Gravitational Acceleration Parameter
The gravitational acceleration field in modern physics is given by This can be rewritten in quantized form for particles with a mass less than or equal to a Planck mass as In the special case of a Planck mass,¯ = l p we get the well-known Planck mass acceleration a p = l 2 p l 3 p c = c 2 lp ; see [24] and [25]. If we have this maximum acceleration for one Planck second, we get to the speed of light: This should be interpreted to mean that the Planck acceleration can only last one Planck second, as nothing can move faster than the speed of light. This likely also means that a Planck acceleration field can only last for one Planck second, but more research is needed here.
We can also find big G from the gravitational acceleration parameter of any fundamental particle For example, we can find find big G from the gravitational acceleration field and the mass of a Planck mass particle Or we can find big G from the gravitational acceleration field and the mass of an electron That is to say that G is always the same no matter the particle; this even holds true for a half-Planck mass particle: We can conclude that big G is independent of the reduced Compton wavelength of the particle in question. With respect to this concept, we can call big G a Universal constant at the subatomic level as well, since it must be the same for any subatomic particle, at least inside our theoretical model, which is based on Newton.

Gravitational Parameter
The standard gravitational parameter is given by This can be rewritten in quantized form as where N is the number of particles with mass m =h 1 c that make up the object in question. This can further be rewritten as where v max = q 1 l 2 p 2 is the maximum velocity for any particle with mass equal to or less than the Planck mass. This maximum velocity has recently been introduced by [2,5,7], and it seems to play a central role in understanding the Planck mass and its relation to other particles. See the suggested papers for an in-depth discussion on this topic.

Gravitational Time Dilation at the Subatomic Level
Einstein's gravitational time dilation [26] is given by where v e is the traditional escape velocity. This means that the gravitational time-dilation could have been derived from the Newton escape velocity, but to my knowledge Einstein is the first one to mention gravitational time dilation. We can rewrite this in the form of the quantized escape velocity (derived above).
Notice that for a reduced Compton wavelength equal to the Planck length¯ = l p , the formula above is invalid, or more precisely the time-dilation becomes imaginary: However, since v e for a Planck mass is > c, it is more precise to say that Einstein's time dilation formula breaks down for a Planck mass. The solution to the problem lies in Haug's maximum velocity formula. The maximum velocity for a Planck mass as observed from any reference frame (no matter how strange it first seems) is zero. This gives a time dilation of zero for a Planck mass. Bear in mind that the Planck mass only lasts for a Planck second. Again see [2,5,7] for an in-depth discussion on the maximum velocity. The Planck mass is the turning point of a "photon; when two photons collide, the collision only lasts for an instant, but will last for one Planck second,as observed with clocks, due to a minimum synchronization error. See [6].
Interestingly, for a half-Planck mass when at the Schwarzschild radius we get In other words, only for half-Planck mass particles does time stand still. Haug has derived a new theory based on an indivisible particle with potential mass equal to half the Planck mass, and half the Planck rest-mass when colliding. Under atomism time is simply counter-strikes between indivisible particles. For a single indivisible particle when not counter-striking, time also stands still; this is due to the concept that time under atomism is counter-strikes (clock ticks). Based on atomism as well as the Newton corpuscular theory, light consists of indivisible particles traveling one after another. When they travel after each other they cannot counter-strike as they all travel with the same speed, and time stands still as "observed" from these particles.

Circular orbits gravitational time dilation
The time dilation for a clock at circular orbit 1 is given by where v e is the traditional escape velocity. We can rewrite this in the form of the escape velocity for a subatomic particle, where we set the "radius" equal to the reduced Compton wavelength r =¯ , 1 At orbital radius larger then 3 2 rs

The Schwarzschild Radius
The so-called Schwarzschild radius [17,18,19,20,21] of a mass m is given by Rewritten as a function of the Planck units and the reduced Compton wavelength, it is given by We think this formula only make sense for sizes down to half-Planck masses and the Planck mass, but not for any mass smaller than half the Planck mass. In 1979, Motz and Epstein [23] introduced a hypothetical particle with half the Planck mass. Haug has derived an extensive theory around such a particle [6,7]. For the half Planck mass particle, we have a Schwarzschild radius of This indivisible particle has according to Haug a "rest" mass equal to half the Planck mass and is likely the lowest mass a particle can take while the escape velocity still is c. It is likely meaningless to talk about a Schwarzschild radius for fundamental particles with a mass less than half the Planck mass. An electron, for example, does not have a valid Schwarzschild radius even if formula 34 above could be used to calculate a hypothetical Schwarzschild radius for an electron. The Schwarzschild radius for any "object" with mass less than half the Planck mass would be smaller than l p . This is likely impossible and may be best understood from the recent renewal of atomism. The Schwarzschild radius of an indivisible particle is simply the diameter of the indivisible particle. Further, the Schwarzschild "radius" of a Planck mass is the length of two indivisible particles lying next to each other. Only for the indivisible particle is the Schwarzschild radius truly a radius in the sense that, at the depth of reality, it has to do with a perfectly spherical shaped particle.

Gravitational Light Deflection
In 1884, Soldner derived work based on Newton classical mechanics that predicted the following deflection of light The angle of deflection in Einstein's general relativity theory [26] is twice that of the Soldner formula (1884) The solar eclipse experiment of Dyson, Eddington, and Davidson performed in 1919 confirmed [27] the idea that the deflection of light was very close to that predicted by Einstein's general relativity theory. That is 1.75 arcseconds compared to the 0.875 as predicted by the Soldner 1884 formula. 2 This was one of the main reasons general relativity took o↵ and partly replaced Newton gravitation.
Interestingly, Accioly and Ragusa [28] have shown that in semi-classical general relativity, the bending of light is dependent on v c , and that when v = 0 one gets the Soldner formula for Newtonian bending of light, and when v = c one gets the Einsteinian bending of light result. See also [37]. Sato and Sato [29] have recently pointed out that it looks like the two factors (double of Newton) in the light deflection likely are due to a unknown property of the photon rather than the bending of space. Further, theoretical and experimental research will hopefully give us deeper insight into the deflection of light.
In the special case where we deal with half a Planck mass¯ = 2l p we get The light deflection in relation to a half Planck mass seems to be 180 degrees. The gravitational deflection formula can also be written for any particle as = 4 l 2 Only for a Planck mass we have v max = 0 and therefore = 4, and for a half-Planck mass we get = 1 again. In terms of degrees, this is 4 ⇥ 648000 ⇡3600 ⇡ 229.1 and for the half-Planck mass 114.59 . Whether or not Einstein's light deflection formula truly makes sense at such short distances is an open question.

Gravitational Redshift
Einstein's gravitational redshift is given by where r is the distance between the center of the mass of the gravitating body and the point at which the photon is emitted. For a subatomic particle, when we assume r =¯ , we can rewrite this as where v max is the maximum speed a uniform mass can take. At this maximum speed, a fundamental mass (an electron, for example) will have reached a relativistic mass equal to the Planck mass. This is discussed in detail in the recent papers by Haug [7,5,2]. If correct, this means that many of the lp and l 2 p