# Different Mass Definitions and Their Pluses and Minuses Related to Gravity

## Abstract

**:**

## 1. Mass and Gravity: A Short Historical Perspective

“All these things being consider’d it seems probable to me, that God in the Beginning form’d Matter in solid, massy, hard, impenetrable, movable Particles, of such Sizes and Figures, and in such Proportion to Space, as most conduce to the End for which he form’d them; and that these primitive Particles being Solids, are incomparably harder than any porous Bodies compounded of them; even so very hard, as never to wear or break in pieces; no ordinary Power being able to divide what God himself made one in the first Creation. While the Particles continue entire, they may compose bodies of one and the same Nature and Texture in all Ages; But should they wear away, or break in pieces, the Nature of Things depending on them, would be changed. Those minute rondures, swimming in space, from the stuff of the world: the solid, coloured table I write on, no, less than the thin invisible air I breathe, is constructed out of small colourless corpuscles; the world at close quarters looks like the night sky–a few dots of stuff, scattered sporadically through and empty vastness. Such is modern corpuscularianism.”

“The extension, hardness, impenetrability, mobility, and vis inertiae of the whole, result from the extension, hardness, impenetrability, mobility, and vires inertiae of the parts; and thence we conclude the least particles of all bodies to be also all extended and hard and impenetrable, and moveable, and endowed with their proper vires inertia. And this is the foundation of all philosophy. Moreover, that the divided but contiguous particles of bodies may be separated from.”

“The quantity of matter is the measure of the same, arising from its density and bulk conjunctly. It is this quantity that I mean hereafter everywhere under the name of body or mass.”

## 2. Different Mass Definitions in Gravity; Which One Is Preferable?

#### 2.1. Kilogram Mass

^{3}water at the temperature of its maximum density, about 4 Celsius. Later, a kilogram prototype stood in Paris. However, the kilogram was hardly used yet in the rest of Europe. The kilogram mass definition of mass was first accepted in scientific circles across Europe after 20 May 1875 when the meter convention was signed in Paris by 17 nations. The international prototype of the kilogram was the kilogram standard from 1889 to 2019.

#### 2.2. Deeper Understanding of Kilogram Mass

#### 2.3. Collision-Time Mass

#### 2.4. Newton Gravitational Mass

#### 2.5. Time-Speed Mass

## 3. Mass Definition Comparison

## 4. Kilogram Mass Versus Gravitational Mass

## 5. Weak Equivalence Principle

## 6. Einstein’s Field Equation

## 7. Gravity Predictions

## 8. Conclusions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Table 1.**The table shows different mass definitions, their dimensions or units, and our comments on the mass intuition, completeness in relative to model gravity, as well as how easy the mass is to measure for macroscopic object.

Mass or Energy Label | Mass | Dimensions or Energy | Intuition | Completness | Measure for Gravity |
---|---|---|---|---|---|

Collision-time | ${m}_{t}=\frac{{l}_{p}}{c}\frac{{l}_{p}}{\overline{\lambda}}$ | $\left[T\right]$ | High | High | Easy |

Collision-length | ${\mathit{E}}_{g}={l}_{p}\frac{{l}_{p}}{\overline{\lambda}}$ | $\left[L\right]$ | High | High | Easy |

Kilogram modern physics | m | $kg$$\left[M\right]$ | Low | Low | Medium |

Kilogram deep understanding | $m=\frac{\hslash}{\overline{\lambda}}\frac{1}{c}$ | $kg$$\left[M\right]$ | Medium | Low | Medium |

Joule (deep understanding) | $E=h\frac{c}{\lambda}=\hslash \frac{c}{\overline{\lambda}}$ | $kg\xb7{m}^{2}\xb7{s}^{-2}$ | Medium | Low | Medium |

$[M\xb7{L}^{2}\xb7{T}^{-2}]$ | |||||

“Newton” alternative-1 | ${m}_{n}={c}^{2}{l}_{p}\frac{{l}_{p}}{\overline{\lambda}}$ | $[{L}^{3}\xb7{T}^{-2}]$ | Medium | High | Easy |

“Newton” alternative-2 | ${m}_{n}=\frac{{l}_{p}}{c}\frac{{l}_{p}}{\overline{\lambda}}$ & $c=1$ | $\left[T\right]$ | High | High | Easy |

Time-speed | ${m}_{s}=\frac{{l}_{p}}{\overline{\lambda}}\frac{1}{c}$ | $[T\xb7{L}^{-1}]$ | Medium | Medium | Medium |

Frequency per Planck time | ${m}_{f}=\frac{{l}_{p}}{\overline{\lambda}}$ | High | Medium | Medium | |

Planck length times Frequency | ${m}_{e}={l}_{p}\frac{c}{\overline{\lambda}}$ | $[L\xb7{T}^{-1}]$ | Medium | Medium | Medium |

Frequency per second | ${m}_{y}=\frac{c}{\overline{\lambda}}$ | $\left[{T}^{-1}\right]$ | High | Low | Medium |

${c}^{2}$ times Planck time frequency | ${m}_{x}={c}^{2}\frac{{l}_{p}}{\overline{\lambda}}$ | $[{L}^{2}\xb7{T}^{-2}]$ | Medium | Medium | Medium |

Joule per Planck time | ${m}_{z}=\hslash \frac{{l}_{p}}{\overline{\lambda}}$ | $kg\xb7{m}^{2}\xb7{s}^{-1}$ | Medium | Low-medium | Medium |

Kilogram per Planck time | ${m}_{k}=\frac{\hslash}{\overline{\lambda}}\frac{1}{c}\frac{{l}_{p}}{c}$ | $kg\xb7s$ | Low | Low-medium | Medium |

**Table 2.**The table shows different mass definitions and gravity constants needed for each mass definition. All these mass definitions, when multiplied with their corresponding gravity constant, give the same results both in output units and value. The M subscript on the reduced Compton wavelength is just to make it clear that this is the reduced Compton wavelength of the large mass in the gravity force formula.

Mass or Energy Label | Mass or Energy | Gravity Constant | Gravity Contant Times Mass |
---|---|---|---|

Kilogram | M | G | $GM={c}^{2}{l}_{p}\frac{{l}_{p}}{{\overline{\lambda}}_{M}}$ |

Kilogram (deep) | $M=\frac{\hslash}{{\overline{\lambda}}_{M}}\frac{1}{c}$ | $G=\frac{{l}_{p}^{2}{c}^{3}}{\hslash}$ | $\frac{{l}_{p}{c}^{3}}{\hslash}M={c}^{2}{l}_{p}\frac{{l}_{p}}{{\overline{\lambda}}_{M}}$ |

Collision-time | ${M}_{t}=\frac{{l}_{p}}{c}\frac{{l}_{p}}{{\overline{\lambda}}_{M}}$ | ${c}^{3}$ | ${c}^{3}{M}_{g}={c}^{2}{l}_{p}\frac{{l}_{p}}{{\overline{\lambda}}_{M}}$ |

“Newton” alternative-1 | ${M}_{n}={c}^{2}{l}_{p}\frac{{l}_{p}}{{\overline{\lambda}}_{M}}$ | 1 | ${M}_{N}={c}^{2}{l}_{p}\frac{{l}_{p}}{{\overline{\lambda}}_{M}}$ |

“Newton” alternative-2 | ${M}_{n}=\frac{{l}_{p}}{c}\frac{{l}_{p}}{{\overline{\lambda}}_{M}}$ | ${c}^{3}=1$ | ${c}^{3}{M}_{n}={c}^{2}{l}_{p}\frac{{l}_{p}}{{\overline{\lambda}}_{M}}$ |

Time-speed | ${M}_{s}=\frac{{l}_{p}}{{\overline{\lambda}}_{M}}\frac{1}{c}$ | ${l}_{p}{c}^{3}$ | $\frac{{l}_{p}{c}^{2}}{\hslash}{M}_{s}={c}^{2}{l}_{p}\frac{{l}_{p}}{{\overline{\lambda}}_{M}}$ |

Frequency per Planck time | ${M}_{f}=\frac{{l}_{p}}{{\overline{\lambda}}_{M}}$ | ${l}_{p}{c}^{2}$ | ${l}_{p}{c}^{2}{M}_{f}={c}^{2}{l}_{p}\frac{{l}_{p}}{{\overline{\lambda}}_{M}}$ |

Planck length times Frequency | ${M}_{e}={l}_{p}\frac{c}{{\overline{\lambda}}_{M}}$ | ${l}_{p}c$ | ${l}_{p}c{M}_{e}={c}^{2}{l}_{p}\frac{{l}_{p}}{{\overline{\lambda}}_{M}}$ |

Frequency per second | ${M}_{y}=\frac{c}{{\overline{\lambda}}_{M}}$ | ${l}_{p}^{2}c$ | ${l}_{p}c{M}_{y}={c}^{2}{l}_{p}\frac{{l}_{p}}{{\overline{\lambda}}_{M}}$ |

${c}^{2}$ times Planck time frequency | ${M}_{x}={c}^{2}\frac{{l}_{p}}{{\overline{\lambda}}_{M}}$ | ${l}_{p}$ | ${l}_{p}{M}_{x}={c}^{2}{l}_{p}\frac{{l}_{p}}{{\overline{\lambda}}_{M}}$ |

Joule per Planck time | ${M}_{z}=\hslash \frac{{l}_{p}}{{\overline{\lambda}}_{M}}$ | $\frac{{l}_{p}{c}^{2}}{\hslash}$ | $\frac{{l}_{p}{c}^{2}}{\hslash}{M}_{z}={c}^{2}{l}_{p}\frac{{l}_{p}}{{\overline{\lambda}}_{M}}$ |

Kilogram per Planck time | ${M}_{k}=\frac{\hslash}{{\overline{\lambda}}_{M}}\frac{1}{c}\frac{{l}_{p}}{c}$ | $\frac{{l}_{p}{c}^{4}}{\hslash}$ | $\frac{{l}_{p}{c}^{4}}{\hslash}{M}_{k}={c}^{2}{l}_{p}\frac{{l}_{p}}{{\overline{\lambda}}_{M}}$ |

**Table 3.**The table shows different mass definitions and the corresponding gravity constant and gravity force formula. All these formulae, when divided by the small mass, give the same output units and predictions, except some of the formulae give higher uncertainty in the output. Some of the mass definitions are much more intuitive and so is the gravity constant for these.

Mass or Energy Label | Mass or Energy | Gravity Constant | Gravity Force | Dimensions | Accuracy | Intuition |
---|---|---|---|---|---|---|

Kilogram | M | G | $F=G\frac{Mm}{{R}^{2}}$ | $kg\xb7m\xb7{s}^{-2}$ | Less | Low |

Kilogram (deep) | $M=\frac{\hslash}{{\overline{\lambda}}_{M}}\frac{1}{c}$ | $G=\frac{{l}_{p}^{2}{c}^{3}}{\hslash}$ | $F=\frac{{l}_{p}^{2}{c}^{3}}{\hslash}\frac{Mm}{{R}^{2}}$ | $kg\xb7m\xb7{s}^{-2}$ | Top | Less |

Joule (deep) | $E=h\frac{c}{{\lambda}_{M}}=\hslash \frac{c}{{\overline{\lambda}}_{M}}$ | ${G}_{E}=\frac{{l}_{p}^{2}}{\hslash c}$ | $F=\frac{{l}_{p}^{2}}{\hslash c}\frac{E\mathit{E}}{{R}^{2}}$ | $kg\xb7m\xb7{s}^{-2}$ | Less | Less |

Collision-time | ${M}_{t}=\frac{{l}_{p}}{c}\frac{{l}_{p}}{{\overline{\lambda}}_{M}}$ | ${c}^{3}$ | $F={c}^{3}\frac{{M}_{t}{m}_{t}}{{R}^{2}}$ | $L\xb7{T}^{-1}$ | Top | Good |

Collision-length | ${E}_{g}={l}_{p}\frac{{l}_{p}}{{\overline{\lambda}}_{M}}$ | c | $F=c\frac{{E}_{g}{\mathit{E}}_{g}}{{R}^{2}}$ | $L\xb7{T}^{-1}$ | Top | Good |

“Newton” alternative-1 | ${M}_{n}={c}^{2}{l}_{p}\frac{{l}_{p}}{{\overline{\lambda}}_{M}}$ | 1 | $F=\frac{{M}_{n}{M}_{n}}{{R}^{2}}$ | ${L}^{4}\xb7{T}^{-4}$ | Top | Less |

“Newton” alternative-2 | ${M}_{n}=\frac{{l}_{p}}{c}\frac{{l}_{p}}{{\overline{\lambda}}_{M}}$ | ${c}^{3}=1$ | $F={c}^{3}\frac{{M}_{n}{M}_{n}}{{R}^{2}}=\frac{{M}_{n}{M}_{n}}{{R}^{2}}$ | ${L}^{4}\xb7{T}^{-4}$ | Top | Less |

Time-speed | ${M}_{s}=\frac{{l}_{p}}{{\overline{\lambda}}_{M}}\frac{1}{c}$ | ${l}_{p}{c}^{3}$ | $F={l}_{p}{c}^{3}\frac{{M}_{s}{m}_{s}}{{R}^{2}}$ | ${T}^{-1}$ | Less | Less |

Frequency per Planck time | ${M}_{f}=\frac{{l}_{p}}{{\overline{\lambda}}_{M}}$ | ${l}_{p}{c}^{2}$ | $F={l}_{p}{c}^{2}\frac{{M}_{f}{m}_{f}}{{R}^{2}}$ | $L\xb7{T}^{-2}$ | Less | Less |

Planck length times Frequency | ${M}_{e}={l}_{p}\frac{c}{{\overline{\lambda}}_{M}}$ | ${l}_{p}c$ | $F={l}_{p}c\frac{{M}_{e}{m}_{e}}{{R}^{2}}$ | ${L}^{3}\xb7{T}^{-3}$ | Less | Less |

Frequency per second | ${M}_{y}=\frac{c}{{\overline{\lambda}}_{M}}$ | ${l}_{p}^{2}c$ | $F={l}_{p}^{2}c\frac{{M}_{y}{m}_{y}}{{R}^{2}}$ | $L\xb7{T}^{-3}$ | Less | Less |

${c}^{2}$ times Planck time frequency | ${M}_{x}={c}^{2}\frac{{l}_{p}}{{\overline{\lambda}}_{M}}$ | ${l}_{p}$ | $F={l}_{p}\frac{{M}_{x}{m}_{x}}{{R}^{2}}$ | ${L}^{3}\xb7{T}^{-4}$ | Less | Less |

Joule per Planck time | ${M}_{z}=\hslash \frac{{l}_{p}}{{\overline{\lambda}}_{M}}$ | $\frac{{l}_{p}{c}^{2}}{\hslash}$ | $F=\frac{{l}_{p}{c}^{2}}{\hslash}\frac{{M}_{z}{m}_{z}}{{R}^{2}}$ | $kg\xb7m\xb7{s}^{-3}$ | Less | Less |

Kilogram per Planck time | ${M}_{k}=\frac{\hslash}{{\overline{\lambda}}_{M}}\frac{1}{c}\frac{{l}_{p}}{c}$ | $\frac{{l}_{p}{c}^{4}}{\hslash}$ | $F=\frac{{l}_{p}{c}^{4}}{\hslash}\frac{{M}_{k}{m}_{k}}{{R}^{2}}$ | $kg\xb7m\xb7{s}^{-1}$ | Less | Less |

**Table 4.**The table demonstrates that when different mass or energy definitions are used in conjunction with their corresponding gravitational constant, they all yield the same gravitational predictions and ultimately converge to the same formula. Although we have only displayed this phenomenon for four of the mass definitions provided in the table above, we have verified that it holds true for all of the definitions listed in Table 2. The variable H in the Newton’s cradle formulae below represents the height of the ball drop. Additionally, the variable x represents the spring displacement.

Prediction | From Macroscopic Surface Level to Deepest Level |
---|---|

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

Orbital velocity | ${v}_{o}=\sqrt{\frac{GM}{R}}=\sqrt{\frac{{c}^{3}{M}_{t}}{R}}=\sqrt{\frac{{c}^{2}{E}_{g}}{R}}=c\sqrt{\frac{{l}_{p}}{R}\frac{{l}_{p}}{{\overline{\lambda}}_{M}}}$ |

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

Velocity ball Newton cradle | ${v}_{out}=\sqrt{2\frac{GM}{{R}^{2}}H}=\sqrt{2\frac{{M}_{n}}{{R}^{2}}H}=\sqrt{2\frac{{c}^{3}{M}_{t}}{{R}^{2}}H}=\sqrt{2\frac{{c}^{2}{E}_{g}}{{R}^{2}}H}=\frac{c}{R}\sqrt{2H{l}_{p}\frac{{l}_{p}}{{\overline{\lambda}}_{M}}}$ |

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

Gravitational red shift | $z=\frac{\sqrt{1-\frac{2GM}{{R}_{1}{c}^{2}}}}{\sqrt{1-\frac{2GM}{{R}_{2}{c}^{2}}}}-1=\frac{\sqrt{1-\frac{2{M}_{n}}{{R}_{1}{c}^{2}}}}{\sqrt{1-\frac{2{M}_{n}}{{R}_{2}{c}^{2}}}}-1=\frac{\sqrt{1-\frac{2c{M}_{t}}{{R}_{1}}}}{\sqrt{1-\frac{2c{M}_{t}}{{R}_{2}}}}-1=\frac{\sqrt{1-\frac{2{E}_{g}}{{R}_{1}}}}{\sqrt{1-\frac{2{E}_{g}}{{R}_{2}}}}-1=\frac{\sqrt{1-\frac{2{l}_{p}}{{R}_{1}}\frac{{l}_{p}}{{\overline{\lambda}}_{M}}}}{\sqrt{1-\frac{2{l}_{p}}{{R}_{2}}\frac{{l}_{p}}{{\overline{\lambda}}_{M}}}}-1$ |

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

Gravitational deflection (GR) | $\theta =\frac{4GM}{{c}^{2}R}=\frac{4{M}_{n}}{{c}^{2}R}=\frac{4c{M}_{g}}{R}=\frac{4{E}_{g}}{R}=4\frac{{l}_{p}}{R}\frac{{l}_{p}}{{\overline{\lambda}}_{M}}$ |

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

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## Share and Cite

**MDPI and ACS Style**

Haug, E.G.
Different Mass Definitions and Their Pluses and Minuses Related to Gravity. *Foundations* **2023**, *3*, 199-219.
https://doi.org/10.3390/foundations3020017

**AMA Style**

Haug EG.
Different Mass Definitions and Their Pluses and Minuses Related to Gravity. *Foundations*. 2023; 3(2):199-219.
https://doi.org/10.3390/foundations3020017

**Chicago/Turabian Style**

Haug, Espen Gaarder.
2023. "Different Mass Definitions and Their Pluses and Minuses Related to Gravity" *Foundations* 3, no. 2: 199-219.
https://doi.org/10.3390/foundations3020017