Planck Quantization of Newton and Einstein Gravitation

In this paper we rewrite the gravitational constant based on its relationship with the Planck length and based on this, we rewrite the Planck mass in a slightly different form (that gives exactly the same value). In this way we are able to quantize a series of end results in Newton and Einstein’s gravitation theories. The formulas will still give exactly the same values as before, but everything related to gravity will then come in quanta. This also gives some new insight; for example, the gravitational deflection of light can be written as only a function of the radius and the Planck length. Numerically this only has implications at the quantum scale; for macro objects the discrete steps are so tiny that they are close to impossible to notice. Hopefully this can give additional insight into how well or not so well (ad hoc) quantized Newton and Einstein’s gravitation is potentially linked with the quantum world.


Foundation
We suggest that the gravitational constant should be written as a function of Planck's reduced constant where h is the reduced Planck's constant and c is the well tested round-trip speed of light.We could call this Planck's form of the gravitational constant.The parameter @ is an unknown constant that is calibrated so that Gp matches our best estimate (measurement) for the gravitational constant.
As shown by Haug (2016), the Planck form of the gravitational constant enables us to rewrite the Planck length as and the Planck mass as Using the gravitational constant in the Planck form, as well as the rewritten Planck units, we are easily able to modify a series of end results from Newton and Einstein's gravitational theories to contain quantization as well.

Newton Universal Gravitational Force
The Newton gravitational force is given by ⇤ e-mail espenhaug@mac.com.Thanks to Victoria Terces for helping me edit this manuscript.In version 5 a mathematical typo in the gravitational acceleration and Newtons version of Kellers third law was fixed.If you find this paper of interest you will possibly also find my recent paper in the Relativity and Cosmology section "The Collapse of the Schwarzschild Radius: The End of Black Holes" of interest.
Using the gravitational constant of the form Gp = @ 2 c 3 h and the Planck mass of mp = h @ 1 c we can rewrite the Newton gravitational force for two Planck masses as In the special case where r = @ we get It seems from this that gravity could be interpreted as hits per second.For large masses the form will be where N1 is the number of Planck masses in object one and N2 is the number of Planck masses in object two.In the case when the two masses are of equal size we have 3 Escape Velocity at the Quantum Scale The traditional escape velocity is given by where G is the traditional gravitational constant and M is the mass of the object we are "trying" to escape from, and r is the radius of that object.In other words, we stand at the surface of the object, for example a hydrogen atom or a planet.Based on the gravitational constant written in the Planck form we can find the escape velocity at Planck scale; see also the Appendix for a derivation from "scratch".It must be where N is the number of Planck masses in the planet or mass in question.
A particularly interesting case is when we only have one Planck mass N = 1 and r = 2@ (this is actually the Schwarzschild radius of a Planck mass object).This gives us as the escape velocity for a particle with Planck mass with radius 2@ is c.Next we will see if the formula above can also be used to calculate the escape velocity of Earth.The Earth's mass is 5.972 ⇥ 10 24 kg.We must convert this to the number of Planck masses.The Planck mass is The Earth's mass in terms of the numbers of Planck masses must be 5.972⇥10 24 2.17651⇥10 8 ⇡ 2.74388 ⇥ 10 32 .Further the radius of the Earth is r ⇡ 6 371 000 meters.We can now just plug this into the Planck scale escape velocity: ve,p = 299 792 458 ⇥ r 2 ⇥ 2.74388 ⇥ 10 32 ⇥ 1.61622837 ⇥ 10 35 6 371 000 ⇡ 11 185.7 meters/second which is equal to 40,269 km/h, the well-known escape velocity from the Earths gravitational field.We think our new way of looking at gravity could have consequences for the understanding of gravity.Gravitation must come in discrete steps and the escape velocity must also come in discrete steps for a given radius; this is because the amount of matter likely comes in discrete steps.

Orbital Speed
The orbital speed 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

Gravitational Acceleration
The gravitational acceleration field in modern physics is given by This can be rewritten in quantized form as 6 Gravitational Parameter The standard gravitational parameter is given by This can be rewritten in quantized form as 7 Kepler's Third Law of Motion The Newton "mechanics version" of Kepler's third law of motion for a circular orbit is given by Where Ms is the mass of the Sun, m the mass of the planet, P is the period, and a is the semi-major axis.This can be re-written as where N1 is the number of Planck masses in the mass of the Sun Ms and N2 is the number of Planck mass of the planet m.In the case the planets mass is much smaller than the Suns mass, we can use the following approximation where N is now the number of Planck masses in the Sun.

Gravitational Time Dilation at Planck Scale
Einstein's gravitational time dilation is given by where ve is the traditional escape velocity.We can rewrite this in the form of quantized escape velocity (derived above).
Let's see if we can calculate the time dilation at, for example, the surface of the Earth from Planck scale gravitational time dilation.The Earth's mass is 5.972 ⇥ 10 24 kg.And again, the Earth's mass in terms of the Planck mass must be 5.972⇥10 24 2.17651⇥10 8 ⇡ 2.74388 ⇥ 10 32 .Further, the radius of the Earth is r ⇡ 6 371 000 meters.We can now just plug this into the quantized gravitational time dilation That is for every second that goes by in outer space (a clock far away from the massive object), 0.99999999930391500 seconds goes by on the surface of the Earth.That is for every year in in outer space (very far from the Earth), there are about 22 milliseconds left to reach an Earth year.This is naturally the same as we would get with Einstein's formula.Still, the new way of writing the formula gives additional insight.

Circular orbits gravitational time dilation
The time dilation for a clock at circular orbit1 is given by where ve is the traditional escape velocity.We can rewrite this in the form of quantized escape velocity (derived above).
9 The Schwarzschild Radius The Schwarzschild radius of a mass M is given by Rewritten into the quantum realm as described in this article it must be For a clock at the Schwarzschild radius we get a time dilation of At the Schwarzschild radius, time stands still.For a radius shorter than that the gravitational time dilation equation above breaks down.2

Mass in Schwarzschild meter
The Schwarzschild mass in terms of meters is given by meter This can be re-written as 10 Quantized Gravitational Bending of Light The angle of deflection in Einstein's General relativity theory is given by where N is the number of Planck masses making up the mass we are interested in.From the formula above, this means that the deflection of angles comes in quanta.Lets also "control" that our Planck scale deflection rooted in Planck and GR is consistent for large bodies like the Sun, for example.The solar mass is Ms ⇡ 1.988 ⇥ 10 30 kg.The Sun's mass in terms of the number of Planck masses must be 1.988⇥10 30 2.17651⇥10 8 ⇡ 9.134 ⇥ 10 37 .Further, the radius of the Sun is rs ⇡ 696 342 000 meters.We can just plug this into the Planck scale deflection: we get a bending of light of about 1.75 arcseconds or 1.75 3 600 of a degree.This is the same as has been confirmed by experiments and helped make Einstein famous, as Newton gravitation supposedly only predicted half of the bending of light.Newton bending of light is given by See for example Soares (2009) and Momeni (2012) for derivations of bending of light under Newton gravitation.

Gravitational Redshift
The Einstein gravitational redshift is given by lim where Re is the distance between the center of the mass of the gravitating body and the point at which the photon is emitted.This we can rewrite as lim Further in the Newtonian limit when Re is su ciently large compared to the Schwarzschild radius we can approximate the above expression with lim The potential interpretation and usefulness of this rewritten version of Einstein's field equation we leave up to other experts to consider.An interesting question is naturally whether or not it is consistent with some of the derivations given above in this form.

Table Summary
The table below summarizes our rewriting of some gravitational formulas.The output is still the same, but based on this view of gravity, masses, gravitational time dilation, and even escape velocity all come in discrete steps.

Conclusion
By making the gravitational constant a function form of the reduced Planck constant one can easily rewrite many of the end results from Newton and Einstein's gravitation in quantized form.Even if this is seen as an ad hoc method, it could still give new insight into what degree quantized Newton's gravitation and General relativity are consistent with the quantum realm.
Table 1: The table shows some of the standard gravitational relationships given by Newton and Einstein and their expression in quantized form.

Units:
Newton and Einstein form: Quantized-form: Newton's gravitational force Kepler's third law Newton's Escape velocity from any mass Orbital velocity for any mass where N1 is the number of Planck masses in the smaller mass m (for example a rocket) and N2 is the number of Planck masses in the other mass.This we have to set to 0 and solve with respect to v to find the escape velocity: This is a quantized escape velocity.Since N1 cancels out we can simply call N2 for N and write the escape velocity as where N is the number of Planck masses in the mass we are trying to escape from.
Derivation of the escape velocity from Planck scale Einstein's Field EquationAnd finally we get to Einstein's field equation.It is given by