Bully Metric

Six base units are included in the Bully Metric system. Two variants of the apan are defined as space-time 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 time-pan (or time apan) (symbol ta) is by definition exactly 30.55 picoseconds. The light-pan (or light apan or length apan) (symbol la) is by definition the distance light travels in vacuum in 30.55 picoseconds.

The info-nat (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 info-nat.

The rapid-nat (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 picoseconds, will have a gravitational mass of one rapid-nat time-pan.

The action-nat (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 action-nats per time-pan electron. Also,the quantum Hall effect will have resistance steps of multiples of 2π action-nats per electron squared.

ta = 30.55 picoseconds (exact)

la = c × 30.55 picoseconds (exact)
   = 9.1586595919 millimeters (exact)

En = 1.380649 x 10-23 joule / kelvin (exact)

Rn = (c3 / G) (exact)
   ≈ 4.0370 × 1035 kilogram / second (approximate)

An = 4 / (2π × KJ2 × RJ)  (exact)
   = 1.05457182 × 10-34 joule second (approximate)

e = 2 / (KJ × RJ) (exact)
  = 1.60217663 × 10-19 coulombs (approximate)

The above definitions 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)

Planck units and the Bully Metric

Table 1 below was taken from the Wikipedia Planck units article:

Table 1: 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)×1032 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 1 by 566.660, results in the corresponding Bully value multiplied by 10^-30:

566.660 × tP = 1.00001(11) × 10-30 ta
566.660 × lP = 1.00001(11) × 10-30 la
566.660 × mP = 1.00001(11) × 10-30 Rn ta

Table 2 below uses algebraic substitution to illustrate that there is a single multiplicative constant that converts between Planck and Bully values. When Planck energy is included in the table (see "Planck energy" row in Table 2), one finds that the Planck to Bully conversion factor for energy is the inverse of the mass, time, and length conversion factor.

Table 2: 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 units are understood to represent the smallest meaningful size of each quantity. For example, the Planck length is the smallest meaningful length because 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, it is generally understood that the existence of objects on the Planck scale would cause a black hole.

The Planck mass of 2.176434(24)×10^-8 kg is not a minimum value. In the case of mass, the Planck value is a crossover point. The Planck mass represents the boundary between gravitation and quantum mechanics. If an object has a mass larger than the Planck mass then gravitational effects will become more important. If the mass is smaller than the Planck mass then quantum mechanical effects will be more important.

Visible universe and the Bully Metric

Since Planck units represent the smallest meaningful length and time values, it seems appropriate to next consider the largest meaningful length and time value, and situate these within the Bully system. The universe is currently understood to be 13.7 billion years old, which is 14.15 × 10²⁷ ta in Bully units. The radius of the visible universe is 46.508 billion light years, which is 48.04 × 10²⁷ 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 3 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 3: The apan prefix table

Prefix			Spacetime Symbols
Name	Symbol	Base 10	Time	Length	Charge
quetta	Q	1030	Qta	Qla	Qe
Visible Universe			$14.15\,Rta$	$48.04\,Rla$	—
ronna	R	1027	Rta	Rla	Re
yotta	Y	1024	Yta	Yla	Ye
zetta	Z	1021	Zta	Zla	Ze
exa	E	1018	Eta	Ela	Ee
peta	P	1015	Pta	Pla	Pe
tera	T	1012	Tta	Tla	Te
giga	G	109	Gta	Gla	Ge
mega	M	106	Mta	Mla	Me
kilo	k	103	kta	kla	ke
—	—	100	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)			${\frac {qta}{566.66}}$	${\frac {qla}{566.66}}$	—

The Mass/Momentum/Energy prefix table

Mass, Momentum, and Energy are compound units in the Bully system. Table 4 below lists SI prefixes used with the rapid-nat for gravitational masses, and with the action-nat for quantum mechanical masses. Also shown in the table is the Planck scale cross-over value where gravitational and quantum effects meet.

Table 4: The Mass/Momentum/Energy prefix table

Prefix			Bully Metric Symbols
Name	Symbol	Base 10	Mass	Momentum	Energy
quetta	Q	1030	Rn Qta	Rn Qla	Rn c Qla
ronna	R	1027	Rn Rta	Rn Rla	Rn c Rla
Visible Universe Mass = 480 Rn Yta
yotta	Y	1024	Rn Yta	Rn Yla	Rn c Yla
zetta	Z	1021	Rn Zta	Rn Zla	Rn c Zla
exa	E	1018	Rn Eta	Rn Ela	Rn c Ela
peta	P	1015	Rn Pta	Rn Pla	Rn c Pla
tera	T	1012	Rn Tta	Rn Tla	Rn c Tla
giga	G	109	Rn Gta	Rn Gla	Rn c Gla
mega	M	106	Rn Mta	Rn Mla	Rn c Mla
kilo	k	103	Rn kta	Rn kla	Rn c kla
—		100	Rn ta	Rn la	Rn c la
Earth Mass = 484 Rn mta
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)			${\frac {Rn\,qta}{566.66}}$	${\frac {Rn\,qla}{566.66}}$	${\frac {Rn\,c\,qla}{566.66}}$
			${\frac {566.66\,An}{c\,qla}}$	${\frac {566.66\,An}{qla}}$	${\frac {566.66\,An}{qta}}$
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
1.00 electronvolt = 46.414 An / mta
milli	m	10−3	An / c mla	An / mla	An / mta
—		100	An / c la	An / la	An / ta
kilo	k	103	An / c kla	An / kla	An / kta
mega	M	106	An / c Mla	An / Mla	An / Mta
giga	G	109	An / c Gla	An / Gla	An / Gta
tera	T	1012	An / c Tla	An / Tla	An / Tta
peta	P	1015	An / c Pla	An / Pla	An / Pta
exa	E	1018	An / c Ela	An / Ela	An / Eta
zetta	Z	1021	An / c Zla	An / Zla	An / Zta
yotta	Y	1024	An / c Yla	An / Yla	An / Yta
ronna	R	1027	An / c Rla	An / Rla	An / Rta
quetta	Q	1030	An / c Qla	An / Qla	An / Qta

p

Defining Constants

Symbol	SI value	Bully
$c\,$	$299792458\$	$1$
$h\,$	${\frac {4\times 10^{-18}}{(25812.807)(483597.9)^{2}}}\$	$2\pi \,$
$\hbar ={\frac {h}{2\pi }}$	${\frac {2\times 10^{-18}}{\pi (25812.807)(483597.9)^{2}}}\$	$1\,$
$e\,$	${\frac {2\times 10^{-9}}{(25812.807)(483597.9)}}\$	$1\,$
$K_{J}={\frac {2e}{h}}\,$	$483597.9\times 10^{9}\,$	${\frac {1}{\pi }}\,$
$R_{K}={\frac {h}{e^{2}}}\,$	$25812.807\,$	$2\pi \,$
$Z_{0}=2\alpha R_{K}\,$	$2\alpha (25812.807)\,$	$4\pi \alpha \,$
$\varepsilon _{0}={\frac {1}{Z_{0}c}}\,$	${\frac {1}{2\alpha (25812.807)(299792458)}}\$	${\frac {1}{4\pi \alpha }}\,$
$\mu _{0}={\frac {Z_{0}}{c}}\,$	${\frac {2\alpha (25812.807)}{299792458}}\$	$4\pi \alpha \,$
$G\,$	$-\,$	$1\,$

The Bully system includes units of transformation which are defined by analogy with units of information. These include the nat (n), bit (b), trit(t), and dit or digit (d). For each type of transformation unit, one may convert from nats to bits, trits, or dits, by multiplication with the natural logarithm as shown below:

b = n × loge(2)
t = n × loge(3)
d = n × loge(10)
where loge is the natural logarithm.

The above definitions ensure normalization of Boltzmann's constant and Planck's constant when using Bully units:

kB = 1.0 Tn (exact)
${\textstyle \hbar }$ = 1.0 An (exact)

Planck units and the Bully Metric

The following (table 1) was taken from the Wikipedia Planck units article:

Table 1: Modern values for Planck's original choice of quantities

Name	Expression	Value (SI units)
\frac{Planck time}{time apan}	$t_{\text{P}}={\sqrt {\frac {\hbar G}{c^{5}}}}$	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)×1032 K

Since c and G are normalized in the Bully system, this ensures that Bully units should have a simple relationship with Planck's units. As illustrated below, multiplying each SI value from Table 1 by 566.66, results in the corresponding Bully value multiplied by 10^-30:

566.66 × tP = 566.66 × 5.391247(60)×10−44 s
                       = 3055.004(34)×10−44 s
                       = 30.55004(34)×10−30 ps
                       = 1.00001(11)×10−30 ta

566.66 × lP = 566.66 × 1.616255(18)×10−35 m
                       = 915.867(10)×10−35 m
                       = 9.15867(10)×10−30 mm
                       = 1.00001(11)×10−30 la

566.66 × mP = 566.66 × 2.176434(24)×10-8 kg
                       = 1233.298(14)×10−8 kg
                       = 12.33298(14)×10−30 rg
                       = 1.00001(11)×10−30 ma

SI prefixes will have the same meaning and conventions when used with apan variants as they have when used with standard SI units (see table in subsequent section for a list of SI prefixes). The "quecto" (symbol "q") metric prefix means 10^-30. The relationship between Bully units and Planck's units can be summarized as:

566.66 × tP = 1.00001(11) qta
566.66 × lP = 1.00001(11) qla
566.66 × mP = 1.00001(11) qma

Planck units are understood to represent the smallest meaningful size of each quantity. For example, the Planck length is the smallest possible length because looking at small objects requires energy. If one were to build a microscope powerful enough to see objects of Planck length or smaller, the microscope would use so much energy that a black hole would form. In terms of Bully units, the "quecto" of each unit is 566.66 times larger than the absolute minimum size for that unit.

The Planck mass may seem unexpectedly large for a minimum mass value, but keep in mind that in this case the unit is for gravitational mass. There obviously are well defined and detectable masses that are smaller than the Planck mass (for example the electron and proton masses), but the Planck mass represents the boundary between gravitational mass and quantum mass. If an object has a mass larger than the Planck mass then gravitational effects will dominate. If the mass is smaller than the Planck mass then quantum mechanical effects will dominate.

The 'Bully Metric is an extremely efficient set of time and distance measurement units for representing earth's physical parameters. Bully units can efficiently represent the Earth's sidereal year and tropical year to eight digits; The Bully Metric can also efficiently represent four digit approximations for the Earth's radius (r ≈ 6371), Schwarzschild radius (R), standard gravitational parameter (μ = MG ≈ 3.984e14), and a typical gravitational acceleration on earth's surface (g ≈ 9.813 ).

$${1\,Sidereal\,Year}={1.033\,Eta}={31,558,150\,s}$$

$${1\,Tropical\,Year}=(1.033-0.00004)\,Eta-2\,s=31,556,926\,s$$

$$r_{earth}\approx {\frac {1\,Gla}{\sqrt {2\times 1.033}}}=6371\,km$$

$$R_{earth}\approx {\frac {la}{1.033}}\approx {8.866\,mm}$$

$${\mu }_{earth}\approx {\frac {1}{2\times 1.033}}\,{\frac {la^{3}}{ta^{2}}}\approx {398,400,000,000,000\,{\frac {m^{3}}{s^{2}}}}$$

$$g_{earth}\approx 1{\frac {Ela}{Eta^{2}}}\approx {9.813\,{\frac {m}{s^{2}}}}$$

The Bully Constants

There are a surprising number of physical constants that can be approximated using various algebraic combinations of the following four numbers (click here to learn more):

1.033
30.55
2
0.00004

Sidereal year

The number of seconds in the Earth's sidereal year can be approximated as:

$${1\,Sidereal\,Year}=1.033\times 30.55\,Ms={31,558,150\,s}$$

Tropical year

The number of seconds in the Earth's tropical year can be approximated as:

$${1\,Tropical\,Year}=((1.033-0.00004)\times 30.55\,Ms)-2\,s=31,556,926\,s$$

Great year

The number of tropical years in a Great Year can be approximated as:

$${1\,Great\,Year}\approx ({\frac {1.033}{0.00004}})\,{Tropical\,Years}={25,825\,Tropical\,Years}$$

Earth's radius (r)

An approximate relationship of the speed of light to the Earth's radius (r):

$$r_{earth}\approx {\frac {c\times 30.55\,ms}{\sqrt {2\times 1.033}}}=6371\,km$$

Earth's Schwarzschild radius (R)

An approximate relationship of the speed of light to the Earth's Schwarzschild radius (R):

$$R_{earth}={\frac {2\times GM_{earth}}{c^{2}}}\approx {\frac {c\times 30.55\,ps}{1.033}}\approx {8.866\,mm}$$

Earth's standard gravitational parameter (μ = MG)

An approximate relationship of the speed of light to the Earth's standard gravitational parameter (μ = MG):

$${\mu }_{earth}=GM_{earth}\approx {\frac {c^{3}\times 30.55\,ps}{2\times 1.033}}\approx {398,400,000,000,000\,{\frac {m^{3}}{s^{2}}}}$$

Earth's gravity (g)

A typical gravitational acceleration on earth's surface can be approximated as:

$$g_{earth}={\frac {{\mu }_{earth}}{{r_{earth}}^{2}}}\approx {\frac {c}{30.55Ms}}\approx {9.813\,{\frac {m}{s^{2}}}}$$

Definition of the apan

The above approximations for earth's physical parameters can be further simplified by introducing new measurement units. The apan will be defined with two variants. The time apan (symbol ta) is by definition exactly 30.55 picoseconds. The length apan (or light apan) (symbol la) is by definition the distance light travels in vacuum in exactly 30.55 picoseconds.

SI prefixes will have the same meaning and conventions when used with the apan as the have when used with standard SI units (see table in subsequent section). For example:

One million ta = 1,000,000 ta = 1 mega time apan = 1 Mta = 30.55 μs
One million la = 1,000,000 la = 1 mega light apan = 1 Mla = 9.1586595919 km

Approximations for earth's physical parameters can be written in terms of the apan as follows:

Sidereal year

The number of seconds in the Earth's sidereal year can be approximated as:

$${1\,Sidereal\,Year}={1.033\,Eta}={31,558,150\,s}$$

Tropical year

The number of seconds in the Earth's tropical year can be approximated as:

$${1\,Tropical\,Year}=(1.033-0.00004)\,Eta-2\,s=31,556,926\,s$$

Earth's radius (r)

An apan based approximation of Earth's radius (r):

$$r_{earth}\approx {\frac {1\,Gla}{\sqrt {2\times 1.033}}}=6371\,km$$

Earth's Schwarzschild radius (R)

An apan based approximation of the Earth's Schwarzschild radius (R):

$$R_{earth}\approx {\frac {la}{1.033}}\approx {8.866\,mm}$$

Earth's standard gravitational parameter (μ = MG)

An apan based approximation of the Earth's standard gravitational parameter (μ = MG):

$${\mu }_{earth}\approx {\frac {1\,{\frac {la^{3}}{ta^{2}}}}{2\times 1.033}}\approx {398,400,000,000,000\,{\frac {m^{3}}{s^{2}}}}$$

Earth's gravity (g)

A typical gravitational acceleration on earth's surface can be approximated as:

$$g_{earth}={\frac {{\mu }_{earth}}{{r_{earth}}^{2}}}\approx 1{\frac {Ela}{Eta^{2}}}\approx {9.813\,{\frac {m}{s^{2}}}}$$