_{1}

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.

We suggest that Newton’s gravitational constant [

where

As shown by Haug [

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’s gravitational force is given by

Using the gravitational constant of the form

In the special case where

It seems from this that gravity potentially could be related to hits per second, even if the output naturally is the same as from the standard formula. For large masses the form will be

where

The traditional escape velocity [

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 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 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

as the escape velocity for a particle with Planck mass 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

The Earth’s mass in terms of the numbers of Planck masses must be

which is equal to 40,269 km/h, the well-known escape velocity from the Earth’s 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.

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

The gravitational acceleration field in modern physics is given by

This can be rewritten in quantized form as

The standard gravitational parameter is given by

This can be rewritten in quantized form as

The Newtonian “mechanics version” of Kepler’s third law of motion for a circular orbit is given by

where

where

where N is now the number of Planck masses in the Sun.

Einstein’s gravitational time dilation [

where

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

Planck mass must be

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 to say, for every year 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 orbit’s gravitational time dilation

The time dilation for a clock at circular orbit^{1} is given by

where

The Schwarzschild radius [

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 meters

The Schwarzschild mass in terms of meters is given by

This can be rewritten as

The angle of deflection in Einstein’s General Relativity theory is given by

This can be rewritten as

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

If we multiply this by

See for example [

Einstein’s gravitational redshift is given by

where

Further, in the Newtonian limit when

And finally we get to Einstein's field equation. It is given by

Units | Newton and Einstein form | Quantized-form |
---|---|---|

Gravitational constant | ||

Newton’s gravitational force | ||

Newton’s gravitational force | ||

Kepler’s third law | ||

Newton’s escape velocity from any mass | ||

Orbital velocity for any mass | ||

Gravitational parameter | ||

Gravitational acceleration field | ||

Gravitational time dilation | ||

Orbital time dilation | ||

Schwarzschild radius | ||

Newton bending of light | ||

Einstein bending of light | ||

Black holes | Possible | Depends on quantum interpretation |

Gravitational red-shift | ||

Gravitational red-shift approx |

I am far from an expert on Einstein’s field equation, but based on the Planck gravitational constant given in this paper, we can rewrite it as

Bear in mind

The potential interpretation and usefulness of this rewritten version of Einstein’s field equation we leave to other experts for consideration. An interesting question is naturally whether or not it is consistent with some of the derivations given above in this form.

By making the gravitational constant be 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.

Thanks to Victoria Terces for helping me edit this manuscript. Also thanks to an anonymous referee for useful comments.

Espen Gaarder Haug, (2016) Planck Quantization of Newton and Einstein Gravitation. International Journal of Astronomy and Astrophysics,06,206-217. doi: 10.4236/ijaa.2016.62017

Derivation of the escape velocity from Planck scale

where

This is a quantized escape velocity. Since

where N is the number of Planck masses in the mass we are trying to escape from.