# Bully Metric

Six base units are defined in the Bully Metric system. Two variants of the apan are defined as spacetime units. Three variants of the nat are defined as transformation units. And the symbol "e" is used to represent elementary charge (the charge of a single electron).
The Bully Metric system was named in honor of actor Robin Williams' portrayal of US president Teddy Roosevelt. Roosevelt frequently used of the word "bully" and coined the phrase "bully pulpit". As noted in Merriam-Webster's dictionary, bully had a positive connotation through much of history.
Bully spacetime units were originally derived from the orbital periods of various Solar System bodies. In particular, the number of seconds in Earth's sidereal year is 31558150 s = 10330 * 3055 s. Large astronomical objects, such as Sagittarius A*, the Sun, and the Solar System's giant planets, can be thought of as bullies both in the traditional meaning of "beautiful", but also in the modern meaning of being intimidating and threatening. The bullies, in Bully Metric, are Sagittarius A*, the Sun, and giant planets like Jupiter and Saturn.
The Foundations of Bully Metric

### Spacetime Units

The time apan (or timepan) (symbol ta) is by definition exactly 30.55 femtoseconds. The length apan (or lightpan or lengthpan) (symbol la) is by definition the distance light travels in vacuum in 30.55 femtoseconds. The scale of the Apan was selected so that the age and diameter of the visible Universe are approximately thirty orders of magnitude larger than the Apan, whereas the Planck time and Planck length are approximately thirty orders of magnitude smaller than the Apan.
The Bully Metric time unit The Bully Metric length unit Bully Timestamps (GitHub)

### Transformation Units

Table 1: Gravitational Mass :
| Body | mass | mass |
| Sun | 161227199.623(5) | Rn ta |
| Earth | 484.2442275(10) | Rn ta |
| Moon | 5.9587358(11) | Rn ta |
The rapinat (natural unit of rapidity) (symbol Rn) is defined such that an object with a standard gravitational parameter equal to the speed of light in vacuum cubed multiplied by 30.55 femtoseconds, will have a gravitational mass of one rapinat timepan. The dwarf planet Pluto has a gravitational mass of roughly one rapinat timepan. Earth's moon has a gravitational mass of approximately six rapinat timepan. It would take roughly six Pluto sized objects smashed together to form something with the mass of the Earth's moon. The first three digits of the Earth's mass can be approximated using the following: 1 Rn kta / (2 * 1.033) = 484 Rn ta. A few example masses are shown in Table 1.
The infonat (natural unit of entropy) (symbol En) is defined such that for an ideal gas in a given macrostate, the entropy of the gas divided by the natural logarithm of the number of real microstates would be equivalent to one infonat.
Table 2: Quantum Rest Energy :
| Particle | rest energy | rest energy |
| Neutron | 43608632955 | An / ta |
| Proton | 43548604715 | An / ta |
| Electron | 23717311.411 | An / ta |
| Neutrino | < 5.57 | An / ta |
| Graviton | < 3.6 | An / Zta |
The actionat (natural unit of action) (symbol An), and elementary charge (symbol e), are defined such that if a Josephson Junction were exposed to microwave radiation of frequency 2 / 30.55 picoseconds (≈ 65.4664484 gigahertz), then the junction would form equidistant Shapiro steps with separation of 2π actionats per kilo-time-apan electron. Also,the quantum Hall effect will have resistance steps of multiples of 2π actionats per electron squared. A few example rest energies are listed in Table 2.
The Bully Metric rapidity unit

### Normalized Physical Constants

The definitions of the Bully Metric system ensure normalization of the speed of light (c), Newton's gravitational constant (G), the Boltzmann constant (k$_B$), the reduced Planck constant (ħ), and the elementary charge (e):
$c=1.0\,{\frac {la}{ta}}$ (exact)
$G=1.0\,{\frac {{la}^{3}}{Rn\,ta^{3}}}$ (exact)
$k_{B}=1.0\,En$ (exact)
$\hbar =1.0\,An$ (exact)
$elementary\,charge=1.0\,e$ (exact)

# Physics Applications

The Bohr Atomic Model using Bully Metric units

# Planck units and the Bully Metric

Table 3 below was taken from the Wikipedia Planck units article:
Table 3: Modern values for Planck's original choice of quantities :
| Name | Expression | Value (SI units) |
| Planck time | $t_{\text{P}}={\sqrt {\frac {\hbar G}{c^{5}}}}$ | 5.391247(60)×10$^{−44}$ s |
| Planck length | $l_{\text{P}}={\sqrt {\frac {\hbar G}{c^{3}}}}$ | 1.616255(18)×10$^{−35}$ m |
| Planck mass | $m_{\text{P}}={\sqrt {\frac {\hbar c}{G}}}$ | 2.176434(24)×10$^{-8}$ kg |
| Planck temperature | $T_{\text{P}}={\sqrt {\frac {\hbar c^{5}}{Gk_{\text{B}}^{2}}}}$ | 1.416784(16)×10$^{32}$ K |

### Planck to Bully conversion constant

Since c, G, k$_B$, and ħ are all normalized in the Bully system, this ensures that Bully units have a simple relationship with Planck's units. In fact, multiplying each value from Table 3 by 0.566660, results in the corresponding Bully value multiplied by 10$^{-30}$:
Table 4 below uses algebraic substitution to illustrate that there is one unique multiplicative constant that converts between Planck and Bully values. When Planck energy is included in the table (see "Planck energy" row in Table 4), one finds that the Planck to Bully conversion factor for energy is the inverse of the mass, time, and length conversion factor.
Table 4: Planck's units relationship with Bully units :
| Name | Expression |
| Planck time | $t_{\text{P}}={\sqrt {\frac {\hbar G}{c^{5}}}}={\sqrt {\frac {An{\frac {la^{3}}{Rn\,ta^{3}}}}{\frac {la^{5}}{ta^{5}}}}}={\sqrt {\frac {An}{Rn\,la^{2}}}}\,ta$ |
| Planck length | $l_{\text{P}}={\sqrt {\frac {\hbar G}{c^{3}}}}={\sqrt {\frac {An{\frac {la^{3}}{Rn\,ta^{3}}}}{\frac {la^{3}}{ta^{3}}}}}={\sqrt {\frac {An}{Rn\,la^{2}}}}\,la$ |
| Planck mass | $m_{\text{P}}={\sqrt {\frac {\hbar c}{G}}}={\sqrt {\frac {An{\frac {la}{ta}}}{\frac {la^{3}}{Rn\,ta^{3}}}}}={\sqrt {\frac {An}{Rn\,la^{2}}}}\,Rn\,ta$ |
| Planck energy | $m_{\text{P}}c^{2}={\sqrt {\frac {\hbar {c^{5}}}{G}}}={\sqrt {\frac {An{\frac {la^{5}}{ta^{5}}}}{\frac {la^{3}}{Rn\,ta^{3}}}}}={\sqrt {\frac {Rn\,la^{2}}{An}}}\,{\frac {An}{ta}}$ |
| Planck temperature | $T_{\text{P}}\times k_{\text{B}}=m_{\text{P}}c^{2}={\sqrt {\frac {Rn\,la^{2}}{An}}}\,{\frac {An}{ta}}$ |
| ∴ | ${\frac {t_{\text{P}}}{ta}}={\frac {l_{\text{P}}}{la}}={\frac {m_{\text{P}}}{Rn\,ta}}={\frac {\frac {An}{ta}}{m_{\text{P}}c^{2}}}={\sqrt {\frac {An}{Rn\,la^{2}}}}$ |

### The meaning of Planck units

The Planck length and time are understood to represent the smallest meaningful size of each quantity. Looking at small objects through a microscope requires energy. If one were to build a microscope powerful enough to see objects at Planck length or smaller, the microscope would use so much energy that a black hole would form. In fact, the existence of objects on the Planck scale would cause a black hole.
Unlike the Planck length and time, the Planck mass of 2.176434(24)×10$^{-8}$ kg is not a minimum value, but rather, it is a crossover point. The Planck mass represents the boundary between gravitation and quantum mechanics. If an object has a mass much larger than the Planck mass then gravitational effects will become more important. If the mass is much smaller than the Planck mass then quantum mechanical effects will be more important.

### Visible universe and the Bully Metric

The scale of the Apan was selected so that the age and diameter of the visible Universe are approximately thirty orders of magnitude larger than the Apan, whereas the Planck time and Planck length are approximately thirty orders of magnitude smaller than the Apan. The universe is currently understood to be 13.7 billion years old, which is 14.15 × 10$^{30}$ ta in Bully units. The radius of the visible universe is 46.508 billion light years, which is 48.04 × 10$^{30}$ la in Bully units.

# The apan prefix table

SI prefixes have the same meaning and conventions when used with apan variants as they have when used with standard SI units. See Table 5 below for the list of SI prefixes used with apan variants. Also shown in the table are the smallest (Planck scale) and largest (Visible Universe) values for each unit.
Table 5: The apan prefix table :
| Prefix | Prefix | Prefix | Spacetime Symbols | Spacetime Symbols | Spacetime Symbols |
| Name | Symbol | Base 10 | Time | Length | Charge |
| Maximum Value (Observable Universe) | Maximum Value (Observable Universe) | Maximum Value (Observable Universe) | $14.15\,Qta$ | $48.04\,Qla$ | — |
| quetta | Q | 10$^{30}$ | Qta | Qla | Qe |
| ronna | R | 10$^{27}$ | Rta | Rla | Re |
| yotta | Y | 10$^{24}$ | Yta | Yla | Ye |
| zetta | Z | 10$^{21}$ | Zta | Zla | Ze |
| exa | E | 10$^{18}$ | Eta | Ela | Ee |
| peta | P | 10$^{15}$ | Pta | Pla | Pe |
| tera | T | 10$^{12}$ | Tta | Tla | Te |
| giga | G | 10$^9$ | Gta | Gla | Ge |
| mega | M | 10$^6$ | Mta | Mla | Me |
| kilo | k | 10$^3$ | kta | kla | ke |
| — | — | 10$^0$ | ta | la | e |
| milli | m | 10$^{−3}$ | mta | mla | me |
| micro | μ | 10$^{−6}$ | μta | μla | μe |
| nano | n | 10$^{−9}$ | nta | nla | ne |
| pico | p | 10$^{−12}$ | pta | pla | pe |
| femto | f | 10$^{−15}$ | fta | fla | fe |
| atto | a | 10$^{−18}$ | ata | ala | ae |
| zepto | z | 10$^{−21}$ | zta | zla | ze |
| yocto | y | 10$^{−24}$ | yta | yla | ye |
| ronto | r | 10$^{−27}$ | rta | rla | re |
| quecto | q | 10$^{−30}$ | qta | qla | qe |
| Minimum value (Planck Scale) | Minimum value (Planck Scale) | Minimum value (Planck Scale) | ${\frac {qta}{0.566660}}$ | ${\frac {qla}{0.566660}}$ | — |

# The Mass/Momentum/Energy prefix table

Mass, Momentum, and Energy are compound units in the Bully system. Table 6 below lists SI prefixes used with the rapinat for gravitational masses, and with the actionat for quantum mechanical masses. Also shown in the table is the Planck scale cross-over value where gravitational and quantum effects meet.
Table 6: The Mass/Momentum/Energy prefix table :
| Prefix | Prefix | Prefix | Bully Metric Symbols | Bully Metric Symbols | Bully Metric Symbols |
| Name | Symbol | Base 10 | Mass | Momentum | Energy |
| quetta | Q | 10$^{30}$ | Rn Qta | Rn Qla | Rn c Qla |
| Observable Universe Mass = 480 Rn Rta ||||||
| ronna | R | 10$^{27}$ | Rn Rta | Rn Rla | Rn c Rla |
| yotta | Y | 10$^{24}$ | Rn Yta | Rn Yla | Rn c Yla |
| zetta | Z | 10$^{21}$ | Rn Zta | Rn Zla | Rn c Zla |
| exa | E | 10$^{18}$ | Rn Eta | Rn Ela | Rn c Ela |
| peta | P | 10$^{15}$ | Rn Pta | Rn Pla | Rn c Pla |
| tera | T | 10$^{12}$ | Rn Tta | Rn Tla | Rn c Tla |
| giga | G | 10$^9$ | Rn Gta | Rn Gla | Rn c Gla |
| mega | M | 10$^6$ | Rn Mta | Rn Mla | Rn c Mla |
| kilo | k | 10$^3$ | Rn kta | Rn kla | Rn c kla |
| Earth Mass = 484 Rn ta ||||||
| — | | 10$^0$ | Rn ta | Rn la | Rn c la |
| milli | m | 10$^{−3}$ | Rn mta | Rn mla | Rn c mla |
| micro | μ | 10$^{−6}$ | Rn μta | Rn μla | Rn c μla |
| nano | n | 10$^{−9}$ | Rn nta | Rn nla | Rn c nla |
| pico | p | 10$^{−12}$ | Rn pta | Rn pla | Rn c pla |
| femto | f | 10$^{−15}$ | Rn fta | Rn fla | Rn c fla |
| atto | a | 10$^{−18}$ | Rn ata | Rn ala | Rn c ala |
| zepto | z | 10$^{−21}$ | Rn zta | Rn zla | Rn c zla |
| yocto | y | 10$^{−24}$ | Rn yta | Rn yla | Rn c yla |
| ronto | r | 10$^{−27}$ | Rn rta | Rn rla | Rn c rla |
| quecto | q | 10$^{−30}$ | Rn qta | Rn qla | Rn c qla |
| Crossover value (Planck Scale) (21.765 micro-grams) | Crossover value (Planck Scale) (21.765 micro-grams) | Crossover value (Planck Scale) (21.765 micro-grams) | ${\frac {Rn\,qta}{0.566660}}$ | ${\frac {Rn\,qla}{0.566660}}$ | ${\frac {Rn\,c\,qla}{0.566660}}$ |
| Crossover value (Planck Scale) (21.765 micro-grams) | Crossover value (Planck Scale) (21.765 micro-grams) | Crossover value (Planck Scale) (21.765 micro-grams) | ${\frac {0.566660\,An}{c\,qla}}$ | ${\frac {0.566660\,An}{qla}}$ | ${\frac {0.566660\,An}{qta}}$ |
| quecto | q | 10$^{−30}$ | An / c qla | An / qla | An / qta |
| ronto | r | 10$^{−27}$ | An / c rla | An / rla | An / rta |
| yocto | y | 10$^{−24}$ | An / c yla | An / yla | An / yta |
| zepto | z | 10$^{−21}$ | An / c zla | An / zla | An / zta |
| atto | a | 10$^{−18}$ | An / c ala | An / ala | An / ata |
| femto | f | 10$^{−15}$ | An / c fla | An / fla | An / fta |
| pico | p | 10$^{−12}$ | An / c pla | An / pla | An / pta |
| nano | n | 10$^{−9}$ | An / c nla | An / nla | An / nta |
| micro | μ | 10$^{−6}$ | An / c μla | An / μla | An / μta |
| milli | m | 10$^{−3}$ | An / c mla | An / mla | An / mta |
| 1.00 electronvolt = 46.414 An / ta ||||||
| — | | 10$^0$ | An / c la | An / la | An / ta |
| kilo | k | 10$^3$ | An / c kla | An / kla | An / kta |
| mega | M | 10$^6$ | An / c Mla | An / Mla | An / Mta |
| giga | G | 10$^9$ | An / c Gla | An / Gla | An / Gta |
| tera | T | 10$^{12}$ | An / c Tla | An / Tla | An / Tta |
| peta | P | 10$^{15}$ | An / c Pla | An / Pla | An / Pta |
| exa | E | 10$^{18}$ | An / c Ela | An / Ela | An / Eta |
| zetta | Z | 10$^{21}$ | An / c Zla | An / Zla | An / Zta |
| yotta | Y | 10$^{24}$ | An / c Yla | An / Yla | An / Yta |
| ronna | R | 10$^{27}$ | An / c Rla | An / Rla | An / Rta |
| quetta | Q | 10$^{30}$ | An / c Qla | An / Qla | An / Qta |

# Traditional Units

Bully variations of traditional units of measure may be accepted for use within the Bully system, provided the traditional unit is not uniquely defined, or used in contexts that will cause confusion with a competing Bully definition, and provided the Bully definition is a simple integer multiple of Bully base units.
The following traditional units are accepted for use within the Buly system:
* 1 Bully Mile = 200 megapan (0.9891 nautical miles)
* 1 Bully Fathom = 200 kilopan (72.115 inches)
* 1 Bully Cubit = 50 kilopan (18.029 inches)
* 1 Bully Span = 25 kilopan (9.014 inches)
* 1 Cubit$^3$ = 25 Bully Gallons (101.47 US quarts)
* 1 Bully Gallon = 5,000 kilopan$^3$ (4.059 US quarts)
* 1 Bully Spoon = 20 kilopan$^3$ (1.039 US tablespoons)
* 1 Bully Dash = 1 kilopan$^3$ (0.7682 milliliter)
* 1 Bully Stone = 500 Rn yta (13.59477 pounds)

# References

* ↑ (Merriam-Webster. (n.d.). Bully. In Merriam-Webster.com dictionary. Retrieved May 16, 2024, from https://www.merriam-webster.com/dictionary/bully)