From: hoyt@isus.UUCP (Hoyt A. Stearns jr.)
Subject: Calculation of G
Followup-To: sci.physics
Keywords: Reciprocal System, Derivation of gravitational constant
Organization: International Society of Unified Science
Date: Sun, 22 Nov 1992 21:37:16 GMT

In article <9NOV199212382504@csa2.lbl.gov> sichase@csa2.lbl.gov (SCOTT I CHASE) writes:
>>G can be calculated from basic Reciprocal System principles to
>> the minimum accuracy of the three numbers used as a starting point:
>>	c, Avogadro's number, and the Rydberg frequency.
>> 
>Would you please demonstrate a derivation of G from this data?
>
Here is a derivation:

In the Reciprocal System (RS), there is a consistent set of natural units.
All physical constants are derived from these.
(They usually have a value of 1 in RS.)

A complication is that in non-RS physics, mass is treated as an
independent entity, whereas in RS, it is in units of time^3/length^3.

All units are in terms of space and time only. All physical units can
be derived consistently in these terms, since everything is made up
of combinations of actual units of space and time only,
in various combinations in 3 dimensions; i.e. the fundamental particle is
an actual discreet unit of space-time).

First we'll derive some of the Reciprocal System's natural units <nu>.

Unit speed is c = s/t, s=space(length), t=time.
Acceleration is s/t^2
Force is in units of t/s^2.
Unit mass in RS is t^3/s^3.

To convert the unit natural values to cgs, we have to equate these to some
physical phenomena known in cgs:

Unit speed is obviously c.
Unit time is 1/2 cycle of R (because a full cycle includes a unit of time
alternating in opposite directions to make a full cycle).

For R in Hz (sec^-1), unit-time t is:
1/(2R) (sec/unit-time<nu>).

Unit length is therefore:
#	c/2R (unit-length<cs>/unit-length<nu>).

Unit acceleration:
#	2Rc (unit-acceleration<cs>/unit-acceleration<nu>)

Because mass in RS is defined in terms of space and time, and in other
systems it is an independent unit, we derive hybrid conversion factors:

Define M the mass in amu of a natural unit of mass.
M = 1 + 1/(128*(1+2/9)) = 1.00639205 amu.

The term (128*(1+2/9)) = 156.4444  is called the inter-regional ratio in RS,
and is derived from the degrees of freedom the motions constituting
an atom have.

<cs> means unit expressed in  cm and seconds.
<nu> means natural units.
<gram> means unit mass expressed in grams.

1 mole of natural mass units = M  grams<cgs>
#	M/N (unit-mass<gram>/unit-mass<nu>).

1 natural mass unit in space-time units (sec^3/cm^3) is:
#	1/c^3 (unit-mass<cs>/unit-mass<nu>) 

The ratio of unit mass expressed in sec^3/cm^3
to unit mass expressed in grams:

#	N/Mc^3 (unit-mass<cs>/unit-mass<gram>)


Using Force=ma, we must use the hybrid mass conversion factor to get the
ratio Force<nu>/dyne:

dyne = unit-mass<cgs> unit-acceleration<cgs>
F<nu> = unit-mass<nu> unit-acceleration<nu> =  t/s^2 = (1/2R)/(c/2R)^2 = 2R/c^2.

#	2R/c^2 (unit-force<cs>/unit-force<nu>)

dynes = grams<cgs> cm / sec^2 

F<nu> = unit-mass<nu> unit-length<nu> / unit-time<nu>^2 =
			 t/s^2 = (1/2R)/(c/2R)^2 = 2R/c^2.

dynes = F<nu> M/N(grams<cgs>/unit-mass<nu>) c/2R (cm/unit-length<nu>) /
				(1/(2R))^2 (sec/unit-time<nu>)^2.

unit force<cgs> is:
#	2RcM/N (dynes/unit-force<nu>)

-----

Now that the conversion factors are defined, we'll work with s, and t
which are unit-length<nu> and unit-time<nu>.

(unit-force<nu>) = (unit-mass<nu>) * (unit-acceleration<nu>)
t/s^2			=	t^3/s^3				s/t^2

The force of one unit-mass<nu> toward another at unit-length<nu> is
3 * unit-force<nu>. The factor 3 is due to gravity being effective in
three scalar dimensions.

Converting s and t from unity natural units to cm and seconds, we get

(c^2/2R) sec/cm^2 == c^3 unit-mass<cs> 	*	1/2Rc cm/sec^2	

The force of m units of mass toward n units of mass at unit-length is
just mn times the force for one mass unit (unitless m,n).

The force of two masses reduces as the square of the number d of unit-lengths
separating them (unitless d).

The general force equations are:

F<nu> = G<nu> mn/d^2 (unit-force<nu>); where G<nu> = unitless 3,
and m,n,d are multipliers of natural units.

Converting natural units to cm and seconds:

F(mass<cs> cm/sec^2)  = 3 mn/d^2  unit-mass<cs>  cm<nu>/(sec<nu>)^2

Converting mass in space-time units to grams:

F(gram cm/sec^2) = F<dyne> =
	3 mn/d^2  N/Mc^3 unit-mass<gram>  cm<nu>/(sec<nu>)^2

So G<cgs> = 3N/Mc^3 = 6.66559478 E-8 dyne-cm^2/g^2

The gravitational constant is 3 times the
ratio of natural unit mass in space-time units to unit mass expressed in grams.

(Rydberg dropped out, so I probably shouldn't have mentioned it in the
original post).
---
The above derivation can be extended to electric and magnetic phenomena,
as they are the 1 and 2 dimensional equivalents of gravity.:

Electric charge in RS is t/s
Magnetic pole strength is t^2/s^2

Coulomb's law of electrostatic attraction is:

F = k Q1 Q2 / d^2

In natural units t/s^2 = k (t/s)(t/s)/s^2
so k is s^2/t = 136670.11 cm^2/sec.

Coulomb's law of magnetostatics is

F = km M1 M2 / d^2, where M is t^2/s^2

so km is s^4/t^3 = 1.228333 E26
__
References:

Permittivity, Permeability, and the speed of Light in the Reciprocal System,
	-Ronald W. Satz, Reciprocity Autumn, 1988

Basic Properties of Matter -Dewey B. Larson, ISUS, Inc. 1988.


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