# Progress in the Composite View of the Newton Gravitational Constant and Its Link to the Planck Scale

## Abstract

**:**

_{p}and c to predict observable gravitational phenomena. While there have been several review papers on the Newton gravitational constant, for example, about how to measure it, we have not found a single review paper on the composite view of the gravitational constant. This paper will review the history of, as well as recent progress in, the composite view of the gravitational constant. This should hopefully be a useful supplement in the ongoing research for understanding and discussion of Newton’s gravitational constant.

## 1. Short History on the Newton Gravitational Constant and the Planck Units

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?

## 2. History of the Composite View of G and the Circular Problem

“Is Newton’s gravitational constant G a, fundamental parameter or is it calculable in terms of other fundamental parameters? In this paper I would like to argue the latter view and to present a calculation of G, unfortunately not in the real world, but in a toy world, just to demonstrate that G is indeed calculable.”

“The actual distribution of energy throughout space-time causes the tetrads to assume vacuum expected values of the order of the Planck mass, ${m}_{p}$. Thus the gravitational constant, $G=\frac{\hslash c}{{m}_{p}^{2}}$, may be viewed not as a fundamental constant, but as a mass scale that is dynamically determined by the large-scale structure of the Universe.”

“Dimensional analysis let us write $G=hc/{m}_{pl}^{2}$, where ${m}_{pl}$ is the Planck mass $21.77\times {10}^{-9}kg$, but this is of no help of determining G since there are no independent determination of ${m}_{pl}$.” (Page 74. Note that we will use notation ${m}_{p}$ for the Planck mass, while several papers also use notation ${m}_{pl}$)

“In the above gravitational derivation, the correct value for the gravitational constant G can only be obtained when it is assumed that the gravitational interaction occurs between whole multiples of the Planck mass, but this last part of the derivation involves some circular reasoning, since the Planck mass is defined using the value for G.”

## 3. The Breakthrough in the Circular Problem

## 4. Putting the Pieces Together

**Table 3.**The table shows the standard gravitational prediction formulas re-written when we assume $G=\frac{\hslash c}{{m}_{p}^{2}}$. We can see that the end results are likely even less intuitive than the existing results, and that we basically only have swapped one constant for a new one (G for ${m}_{p}$).

Gravity with $G=\frac{\hslash c}{{m}_{p}^{2}}$: | |

Mass | M and m (kg) |

Gravity force | $F=G\frac{Mm}{{R}^{2}}=\frac{\hslash c}{{m}_{p}^{2}}\frac{Mm}{{R}^{2}}(\mathrm{kg}\xb7\mathrm{m}\xb7{\mathrm{s}}^{-2})$ |

Gravity acceleration | $g=\frac{GM}{{R}^{2}}=\frac{\hslash cM}{{m}_{p}^{2}{R}^{2}}$ |

Orbital velocity | ${v}_{o}=\sqrt{\frac{GM}{R}}=\frac{1}{{m}_{p}}\sqrt{\frac{\hslash cM}{R}}$ |

Orbital time | $T=\frac{2\pi R}{\sqrt{\frac{GM}{R}}}=\frac{2\pi R{m}_{p}}{\sqrt{\frac{\hslash cM}{R}}}$ |

Periodicity pendulum3 (clock) | $T=2\pi \sqrt{\frac{L}{g}}=2\pi R\sqrt{\frac{L}{GM}}=2\pi R{m}_{p}\sqrt{\frac{L}{\hslash cM}}$ |

Frequency Newton spring | $f=\frac{1}{2\pi}\sqrt{\frac{k}{m}}=\frac{1}{2\pi R{m}_{p}}\sqrt{\frac{\hslash cM}{x}}$ |

Velocity ball Newton cradle4 | ${v}_{out}=\sqrt{2\frac{GM}{{R}^{2}}H}=\frac{1}{R{m}_{p}}\sqrt{2\hslash cMH}$ |

Predictions from GR: | |

Advance of perihelion | $\sigma =\frac{6\pi GM}{a(1-{e}^{2}){c}^{2}}=\frac{6\pi \hslash cM}{a(1-{e}^{2}){c}^{2}{m}_{p}^{2}}$ |

Gravitational redshift | $z=\frac{\sqrt{1-\frac{2GM}{{R}_{1}{c}^{2}}}}{\sqrt{1-\frac{2GM}{{R}_{2}{c}^{2}}}}-1=\frac{\sqrt{1-\frac{2\hslash M}{{R}_{1}c{m}_{p}^{2}}}}{\sqrt{1-\frac{2\hslash M}{{R}_{2}c{m}_{p}^{2}}}}-1$ |

Time dilation | ${T}_{R}={T}_{f}\sqrt{1-{\sqrt{\frac{2GM}{R}}}^{2}/{c}^{2}}=$ |

Deflection | $\delta =\frac{4GM}{{c}^{2}R}=\frac{4\hslash M}{cR{m}_{p}^{2}}$ |

Microlensing | ${\theta}_{E}=\sqrt{\frac{4GM}{{c}^{2}}\frac{({d}_{S}-{d}_{L})}{{d}_{S}{d}_{L}}}=\sqrt{\frac{4\hslash M}{c{m}_{p}^{2}}\frac{({d}_{S}-{d}_{L})}{{d}_{S}{d}_{L}}}$ |

## 5. Is the Inertial Mass Really Identical to the Gravitational Mass?

## 6. The Gravity Constant Calculated from Cosmological Entities

## 7. The Composite View of G with Respect to Alternative Gravitational Theories

## 8. Conclusions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Notes

1 | It is impossible for anyone today to know the full literature on physics, so there could be other authors already publishing these formulas; however, we have made a very serious attempt to search and find anyone who might have published these results first. |

2 | |

3 | The formula is a very good approximation when the angle of the pendulum is small, as it is in most pendulum clocks. It is not accurate for large angles, but is again exact for an angle of 360; that is to say, for full circle, see [71]. |

4 | Where H is the height of the ball drop. |

5 | The formula is a very good approximation when the angle of the pendulum is small, as it is in most pendulum clocks. It is not accurate for large angles, but is again exact for an angle of 360; that is to say, for a full circle; see [71]. |

6 | Where H is the height of the ball drop. |

7 | The formula is a very good approximation when the angle of the pendulum is small, as it is in most pendulum clocks. It is not accurate for large angles, but is again exact for an angle of 360; that is to say, for a full circle, see [71]. |

8 | Where H is the height of the ball drop. |

9 | That not should be confused with gravitational potential energy. |

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From | Gravity Constant Formula | Likely First Described1 |
---|---|---|

Planck mass ${m}_{p}=\sqrt{\frac{\hslash c}{G}}$ | $G=\frac{\hslash c}{{m}_{p}^{2}}$ | Cahill [25] 1984 and Cohen2 [27] 1987 |

Planck time ${t}_{p}=\sqrt{\frac{G\hslash}{{c}^{5}}}$ | $G=\frac{{t}_{p}^{2}{c}^{5}}{\hslash}$ | Nastasenko [61] 2013 |

Planck length ${l}_{p}=\sqrt{\frac{G\hslash}{{c}^{3}}}$ | $G=\frac{{l}_{p}^{2}{c}^{3}}{\hslash}$ | Haug [21] 2016 |

Planck energy ${E}_{p}=\sqrt{\frac{\hslash {c}^{5}}{G}}$ | $G=\frac{\hslash {c}^{5}}{{E}_{p}^{2}}$ | this paper Haug [64] 2020 |

Planck temperature ${T}_{p}=\sqrt{\frac{\hslash {c}^{5}}{G{k}_{b}^{2}}}$ | $G=\frac{\hslash {c}^{5}}{{T}_{p}^{2}{k}_{b}}$ | this paper |

Planck mass ${a}_{g}=\frac{{m}^{2}}{{m}_{p}^{2}}$ | $G=\frac{{a}_{g}\hslash c}{{u}^{2}}=\frac{\hslash c}{{m}_{p}^{2}}$ | Clark [33] 2003 |

Planck frequency ${f}_{p}=\sqrt{\frac{{c}^{5}}{G\hslash}}$ | $G=\frac{{c}^{5}}{{f}_{p}^{2}\hslash}$ | Nastasenko [61] 2013 |

Planck acceleration ${a}_{p}=\sqrt{\frac{{c}^{7}}{G\hslash}}$ | $G=\frac{{c}^{7}}{{a}_{p}^{2}\hslash}$ | this paper |

Planck density ${\rho}_{p}=\frac{{c}^{5}}{\hslash {G}^{2}}$ | $G=\sqrt{\frac{{c}^{5}}{{\rho}_{p}\hslash}}$ | this paper |

Planck momentum ${p}_{p}=\sqrt{\frac{\hslash {c}^{3}}{G}}$ | $G=\frac{\hslash {c}^{3}}{{p}_{p}^{2}}$ | this paper |

Planck force ${F}_{p}=\frac{{E}_{p}}{{l}_{p}}$ | $G=\frac{{c}^{4}}{{F}_{p}}$ | this paper |

Planck length, time and mass | $G=\frac{{l}_{p}^{3}}{{m}_{p}{t}_{p}^{2}}$ | Zwiebach [34] 2004 and Nastasenko [35] 2004 |

Planck length and Planck time | $G=\frac{{l}_{p}{c}^{2}}{{m}_{p}}$ | Zivlak [41] 2013 |

Planck mass and Planck time | $G=\frac{{t}_{p}{c}^{3}}{{m}_{p}}$ | Eldred [42] 2019 |

Planck length, time and Planck energy | $G=\frac{{l}_{p}^{3}{c}^{2}}{{E}_{p}{t}_{p}^{2}}$ | this paper |

Planck time and Planck length | $G=\frac{{t}_{p}{l}_{p}{c}^{4}}{\hslash}$ | this paper |

Planck frequency Planck mass | $G=\frac{{c}^{3}}{{f}_{p}{m}_{p}}$ | this paper |

Planck acceleration and mass | $G=\frac{{c}^{4}}{{a}_{p}{m}_{p}}$ | this paper |

Planck charge and Planck length | $G=\frac{{l}_{p}^{2}{c}^{2}{10}^{7}}{{q}_{p}^{2}}$ | this paper |

Planck charge and Planck mass | $G=\frac{{10}^{7}}{{m}_{p}^{2}{q}_{p}^{2}}$ | this paper |

Planck charge and Planck time | $G=\frac{{t}_{p}^{2}{c}^{4}{10}^{7}}{{q}_{p}^{2}}$ | this paper |

From | Gravity Constant Formula | Likely First Described |
---|---|---|

when $\hslash =c=1$ | $G=1/{m}_{p}^{2}$ | Kiritsis 1997 [43] and Cerdeno and Munoz 1998 [44] |

when $\hslash =c=1$ | $G={l}_{p}^{2}$ | Schwarzschild 2000 [52] |

when $\hslash =c=1$ | $G={t}_{p}^{2}$ | this paper |

when $\hslash =c=1$ | $G=1/{a}_{p}^{2}$ | this paper |

when $\hslash =c=1$ | $G=1/{E}_{p}^{2}$ | this paper |

when $\hslash =c=1$ | $G=1/{p}_{p}^{2}$ | this paper |

when $c=1$ | $G={l}_{p}/{m}_{p}$ | Casadio 2009 [54] |

when $c=1$ | $G={t}_{p}/{m}_{p}$ | this paper |

when $c=1$ | $G={l}_{p}/{E}_{p}$ | this paper |

when $c=1$ | $G={t}_{p}/{E}_{p}$ | this paper |

when $c=1$ | $G={l}_{p}/{a}_{p}$ | this paper |

when $c=1$ | $G={t}_{p}/{a}_{p}$ | this paper |

**Table 4.**The table shows that any observable gravity phenomena contains $GM$ and not $GMm$ and, further, that when assuming G is a composite, then we end up being able to predict all observable gravity phenomena only from ${l}_{p}$ and c.

Mass | $M=\frac{\hslash}{{\overline{\lambda}}_{M}}\frac{1}{c}$ (kg) |

Non observable (contains $GMm$) | |

Gravitational constant | $G,\left(G=\frac{{l}_{p}^{2}{c}^{3}}{\hslash}\right)$ |

Gravity force | $F=G\frac{Mm}{{R}^{2}}(\mathrm{kg}\xb7\mathrm{m}\xb7{\mathrm{s}}^{-2})$ |

Observable predictions: (contains only $GM$) | |

Gravity acceleration | $g=\frac{GM}{{R}^{2}}=\frac{{c}^{2}}{{R}^{2}}\frac{{l}_{p}^{2}}{{\overline{\lambda}}_{M}}$ |

Orbital velocity | ${v}_{o}=\sqrt{\frac{GM}{R}}=c{l}_{p}\sqrt{\frac{1}{R{\overline{\lambda}}_{M}}}$ |

Orbital time | $T=\frac{2\pi R}{\sqrt{\frac{GM}{R}}}=\frac{2\pi \sqrt{{\overline{\lambda}}_{M}{R}^{3}}}{c{l}_{p}}$ |

Periodicity pendulum5 (clock) | $T=2\pi \sqrt{\frac{L}{g}}=2\pi R\sqrt{\frac{L}{GM}}=\frac{2\pi R}{c{l}_{p}}\sqrt{L{\overline{\lambda}}_{M}}$ |

Frequency Newton spring | $f=\frac{1}{2\pi}\sqrt{\frac{k}{m}}=\frac{1}{2\pi R}\sqrt{\frac{GM}{x}}=\frac{c{l}_{p}}{2\pi R}\sqrt{\frac{1}{{\overline{\lambda}}_{M}x}}$ |

Velocity ball Newton cradle6 | ${v}_{out}=\sqrt{2\frac{GM}{{R}^{2}}H}=\frac{c{l}_{p}}{R}\sqrt{\frac{2H}{{\overline{\lambda}}_{M}}}$ |

Observable predictions (from GR): (contain only $GM$) | |

Advance of perihelion | $\sigma =\frac{6\pi GM}{a(1-{e}^{2}){c}^{2}}=\frac{6\pi}{a(1-{e}^{2})}\frac{{l}_{p}^{2}}{{\overline{\lambda}}_{M}}$ |

Gravitational redshift | $z=\frac{\sqrt{1-\frac{2GM}{{R}_{1}{c}^{2}}}}{\sqrt{1-\frac{2GM}{{R}_{2}{c}^{2}}}}-1=\frac{\sqrt{1-\frac{2{l}_{p}^{2}}{{R}_{1}{\overline{\lambda}}_{M}}}}{\sqrt{1-\frac{2{l}_{p}^{2}}{{R}_{2}{\overline{\lambda}}_{M}}}}-1$ |

Time dilation | ${T}_{R}={T}_{f}\sqrt{1-{\sqrt{\frac{2GM}{R}}}^{2}/{c}^{2}}={T}_{f}\sqrt{1-\frac{2{l}_{p}^{2}}{R{\overline{\lambda}}_{M}}}$ |

Deflection | $\delta =\frac{4GM}{{c}^{2}R}=\frac{4}{R}\frac{{l}_{p}^{2}}{{\overline{\lambda}}_{M}}$ |

Microlensing | ${\theta}_{E}=\sqrt{\frac{4GM}{{c}^{2}}\frac{({d}_{S}-{d}_{L})}{{d}_{S}{d}_{L}}}=2{l}_{p}\sqrt{\frac{{d}_{S}-{d}_{L}}{{\overline{\lambda}}_{M}\left({d}_{S}{d}_{L}\right)}}$ |

**Table 5.**The table shows that we can write the gravitational constant as ${c}^{3}$ when using, in our view, a more complete mass definition, $\overline{m}=\frac{{l}_{p}}{c}\frac{{l}_{p}}{\overline{\lambda}}$. That is, mass is related to time, or what Haug has called collision-time. Different mass sizes then only differ in different Compton wavelengths. Writing the gravitational force formula this way yields the same predictions as standard Newton gravity except we only rely on two constants, ${l}_{p}$ and c, to describe mass and any observable gravity phenomena. In addition, in general relativity predictions, we can replace the mass with this mass definition if we replace G with ${c}^{3}$. The reason we can do this is that ${c}^{3}\overline{M}=GM$. This is clear when we understand that G is a composite constant and, in addition, understand that the kilogram mass can be written by simply solving the Compton wavelength formula with respect to m.

Mass | $M=\frac{\hslash}{{\overline{\lambda}}_{M}}\frac{1}{c}$ (kg) |

Non observable: | |

Gravitational constant | ${c}^{3}$ |

Gravity force | $F={c}^{3}\frac{\overline{M}\overline{m}}{{R}^{2}}(\mathrm{kg}\xb7\mathrm{m}\xb7{\mathrm{s}}^{-2})$ |

Observable predictions: | |

Gravity acceleration | $g=\frac{{c}^{3}\overline{M}}{{R}^{2}}=\frac{{c}^{2}}{{R}^{2}}\frac{{l}_{p}^{2}}{{\overline{\lambda}}_{M}}$ |

Orbital velocity | ${v}_{o}=\sqrt{\frac{{c}^{3}\overline{M}}{R}}=c{l}_{p}\sqrt{\frac{1}{R{\overline{\lambda}}_{M}}}$ |

Orbital time | $T=\frac{2\pi R}{\sqrt{\frac{{c}^{3}\overline{M}}{R}}}=\frac{2\pi \sqrt{{\overline{\lambda}}_{M}{R}^{3}}}{c{l}_{p}}$ |

Periodicity pendulum7 (clock) | $T=2\pi \sqrt{\frac{L}{g}}=2\pi R\sqrt{\frac{L}{{c}^{3}\overline{M}}}=\frac{2\pi R}{c{l}_{p}}\sqrt{L{\overline{\lambda}}_{M}}$ |

Frequency Newton spring | $f=\frac{1}{2\pi}\sqrt{\frac{k}{\overline{M}}}=\frac{1}{2\pi R}\sqrt{\frac{{c}^{3}\overline{M}}{x}}=\frac{c{l}_{p}}{2\pi R}\sqrt{\frac{1}{{\overline{\lambda}}_{M}x}}$ |

Velocity ball Newton cradle8 | ${v}_{out}=\sqrt{2\frac{{c}^{3}\overline{M}}{{R}^{2}}H}=\frac{c{l}_{p}}{R}\sqrt{\frac{2H}{{\overline{\lambda}}_{M}}}$ |

Observable predictions (from GR): | |

Advance of perihelion | $\sigma =\frac{6\pi {c}^{3}\overline{M}}{a(1-{e}^{2}){c}^{2}}=\frac{6\pi}{a(1-{e}^{2})}\frac{{l}_{p}^{2}}{{\overline{\lambda}}_{M}}$ |

Gravitational redshift | $z=\frac{\sqrt{1-\frac{2{c}^{3}\overline{M}}{{R}_{1}{c}^{2}}}}{\sqrt{1-\frac{2{c}^{3}\overline{M}}{{R}_{2}{c}^{2}}}}-1=\frac{\sqrt{1-\frac{2{l}_{p}^{2}}{{R}_{1}{\overline{\lambda}}_{M}}}}{\sqrt{1-\frac{2{l}_{p}^{2}}{{R}_{2}{\overline{\lambda}}_{M}}}}-1$ |

Time dilation | ${T}_{R}={T}_{f}\sqrt{1-{\sqrt{\frac{2{c}^{3}\overline{M}}{R}}}^{2}/{c}^{2}}={T}_{f}\sqrt{1-\frac{2{l}_{p}^{2}}{R{\overline{\lambda}}_{M}}}$ |

Deflection | $\delta =\frac{4{c}^{3}\overline{M}}{{c}^{2}R}=\frac{4}{R}\frac{{l}_{p}^{2}}{{\overline{\lambda}}_{M}}$ |

Microlensing | ${\theta}_{E}=\sqrt{\frac{4{c}^{3}\overline{M}}{{c}^{2}}\frac{({d}_{S}-{d}_{L})}{{d}_{S}{d}_{L}}}=2{l}_{p}\sqrt{\frac{{d}_{S}-{d}_{L}}{{\overline{\lambda}}_{M}\left({d}_{S}{d}_{L}\right)}}$ |

**Table 6.**The table shows various ways we can express the gravity constant from cosmological units, as well as from units related to black holes.

From | Gravity Formula | Comments |
---|---|---|

From universe mass and Hubble time | $G=\frac{{c}^{3}{T}_{H}}{{M}_{u}}$ | Milne 1936 [85] |

From universe mass and Hubble radius | $G=\frac{{R}_{u}{c}^{2}}{{M}_{u}}$ | Bleksley 1951 [89] |

From universe mass and universe radius | $G=\frac{{R}_{u}{c}^{2}}{6{M}_{u}}$ | Unzicker 2020 [19] |

Hubble constant, Friedmann critical mass | $G=\frac{{c}^{3}}{2{H}_{0}{M}_{c}}$ | |

Hubble radius, Friedmann critical mass | $G=\frac{{R}_{H}{c}^{2}}{2{M}_{c}}$ | |

Hubble constant, Friedmann critical mass | $G=\frac{{T}_{H}{c}^{3}}{2{M}_{c}}$ | |

Hubble time, Friedmann critical mass | $G=\frac{{c}^{3}}{2{H}_{0}{M}_{c}}$ | |

Hubble radius, Hubble time, and Friedmann critical mass | $G=\frac{{R}_{H}^{3}}{2{M}_{c}{T}_{H}^{2}}$ | |

Hubble constant, Haug universe mass, | $G=\frac{{c}^{3}}{{H}_{0}{M}_{u}}$ | |

Hubble radius Hubble time and Haug universe mass | $G=\frac{{R}_{H}{c}^{2}}{{M}_{u}}$ | |

Hubble radius, Haug universe mass, | $G=\frac{{T}_{H}{c}^{3}}{{M}_{u}}$ | |

Hubble time, Hubble time, and Haug universe mass | $G=\frac{{T}_{H}{c}^{3}}{{M}_{u}{T}_{H}^{2}}$ | |

Hubble constant, Friedmann critical mass, | $G=\frac{{c}^{3}}{2{H}_{0}{M}_{c}}$ | |

Hubble time and Haug universe mass | $G=\frac{{T}_{H}{c}^{3}}{{M}_{u}}$ | |

Schwarzschild radius, mass, | $G=\frac{{R}_{s}{c}^{2}}{2M}$ | ${R}_{s}=\frac{2GM}{{c}^{2}}$ |

Schwarzschild time, mass, | $G=\frac{{T}_{s}{c}^{3}}{2M}$ | ${T}_{s}=\frac{{R}_{s}}{c}$ |

Haug escape velocity radius, mass, | $G=\frac{{R}_{h}{c}^{2}}{M}$ | ${R}_{h}=\frac{GM}{{c}^{2}}$ |

Haug radius time, mass, | $G=\frac{{T}_{h}{c}^{3}}{M}$ | ${T}_{h}=\frac{{R}_{h}}{c}$ |

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## Share and Cite

**MDPI and ACS Style**

Haug, E.G.
Progress in the Composite View of the Newton Gravitational Constant and Its Link to the Planck Scale. *Universe* **2022**, *8*, 454.
https://doi.org/10.3390/universe8090454

**AMA Style**

Haug EG.
Progress in the Composite View of the Newton Gravitational Constant and Its Link to the Planck Scale. *Universe*. 2022; 8(9):454.
https://doi.org/10.3390/universe8090454

**Chicago/Turabian Style**

Haug, Espen Gaarder.
2022. "Progress in the Composite View of the Newton Gravitational Constant and Its Link to the Planck Scale" *Universe* 8, no. 9: 454.
https://doi.org/10.3390/universe8090454