11-03-2026
The "Algebra of Nature"
Those who imagine a future era in which advances in computer science will
allow the study of physical phenomena involving very large amounts of particles
even by means of simulation techniques, including their gravitational effects,
might find interesting a theory oriented toward that, which aids to define
objects and methods underlying elementary particles: the theory of bipolarity.
That can be described and developed with any mathematical formalism deemed
appropriate.
Originally, an elementary algebra was adopted (in anticipation of the
computational needs typical of simulations, to which the theory is explicitly
oriented). The symbols involved in this algebra are related to nature, in the
sense that they represent physical quantities and their units of measurement...
| Symbol | Meaning | Dimensional factor |
|---|
| A | area | m2 |
| B | magnitude of magnetic field vector |
|
| C | coulomb | 1 |
| Ck | capacitance |
|
| F | magnitude of force vector |
|
| G |
Newtonian constant of gravitation
|
|
| I | electric current |
|
| J | joule | 1 |
| LB | inductance |
|
| M | mass |
|
| P | power |
|
| U | energy | J |
| V | voltage |
|
| Vg |
modulus of the conventional gravitational potential |
|
| a | magnitude of acceleration vector |
|
| c |
speed of light in vacuum
|
|
| d | distance | m |
| f | frequency |
|
| g | magnitude of gravitational field vector |
|
| h |
Planck constant
| s · J |
| k | magnitude of electric field vector |
|
| m | meter | 1 |
| n | positive integer | 1 |
| p | momentum |
|
| q | electric charge | C |
| r | radius | m |
| s | second | 1 |
| t | time | s |
| u | energy density |
|
| v | magnitude of velocity vector |
|
| β | normalized velocity | 1 |
| ε0 |
vacuum electric permittivity
|
|
Some algebraic expressions have a known meaning...
| Expression | Meaning |
|---|
|
| ampere |
|
| farad |
|
| henry |
|
| newton |
| √
h · c · ε0 · 2
|
Planck electric charge
(qP)
|
|
|
Planck energy
(EP)
|
|
|
Planck time
(tP)
|
|
| pressure |
|
| tesla |
|
|
vacuum magnetic permeability
(μ0)
|
|
| volt |
|
| volume of a sphere |
|
| watt |
A few examples of the "algebra of nature" follow, from the theory of
bipolarity (without the context presentation)...
Localized electromagnetic resonances
are foundamental in the theory. A notable frequency of resonance is taken as a
"tick rate reference" for time measurement...
The theory suggests
| ns = 2952099196999999999999999999999999954121816 |
and
| nm = 9847142975824962214359642096133052 |
In other terms, the second is defined as the time interval corresponding
to "ns whole local ticks"...
| s | = |
2952099196999999999999999999999999954121816 |
| f0 |
The meter is defined as "the distance covered locally by a light signal"
during a time interval corresponding to "nm whole local ticks"...
| m | = |
c · 9847142975824962214359642096133052 |
| f0 |
That is
giving exactly
| c | = |
ns · m |
= |
2952099196999999999999999999999999954121816 · m
| = |
299792458 · m |
| nm · s |
9847142975824962214359642096133052 · s | s |
while
| f0 | = |
ns | = |
2952099196999999999999999999999999954121816
|
| s | s |
Finally note that
| f0 | ≈ |
2.9521E42 | ; |
√
c5
| ≈ |
2.9521E42 | ; |
1 | ≈ |
2.9521E42 |
| s |
√
G · h · π · 2
| s | tP · π · 2 |
s |
The objects and methods underlying elementary particles, provided by the theory
of bipolarity to simulate virtually all physical phenomena, are of
electromagnetic nature. Thus gravity, for example, emerges from "a chaotic complex of
electromagnetic interactions" whose statistics end up justifying relatively
simple macroscopic effects.
One of these effects is the "gravitational time dilation" in accordance with
General Relativity...
The virtual observer ("out
of context observer" of the simulation) is at rest with respect to the
center of gravity of the simulation context and at infinite distance from it;
he/she measures time by means of a "tick rate reference", that is the frequency
of a vibe at rest in his/her
locality...
| f0 | = |
ns | = |
2952099196999999999999999999999999954121816
|
| s | s |
If another vibe identical to the reference one is at rest and located at
a finite distance from the center of gravity then the virtual observer can note
that its frequency of resonance, f, is less than f0; that is, time
passes more slowly where conventional gravitational potential is more negative.
This conventional potential does not have physical meaning for positive
values... Anyway, in the theory of bipolarity it is adopted the symbol
Vg to represent the modulus of the conventional gravitational
potential.
For example, if in the simulation context the gravitational field is spherically
symmetric then at distance d from the center of gravity the modulus of the
conventional gravitational potential is
where M is a homogeneous mass of a sphere of radius smaller than d.
Taken two points in the free space of this simulation, A and B, we can define
| VgA |
= |
- | M · G |
| ; |
VgB |
= |
- | M · G |
|
| dA |
dB |
and get
| fA = |
√ |
1 - |
VgA · 2
| · f0 |
; |
fB = |
√ |
1 - |
VgB · 2
| · f0 |
| c2 |
c2 |
So, while 1 second elapses locally at point B, the virtual observer
counts
| f0 · ns | = |
ns · c |
| fB |
√
c2 - VgB · 2
|
own ticks, corresponding to a time interval of
While 1 second elapses locally at point B an observer at rest in point A
counts
| fA · ns | = |
√
c2 - VgA · 2 · ns
|
| fB |
√
c2 - VgB · 2
|
own ticks, corresponding to a time interval of
| √
c2 - VgA · 2 · s
|
| √
c2 - VgB · 2
|
Let's put
| M = |
5.9722E24 · s2 · J
| ; |
dA = 6.3596E6 · m
| ; |
dB = 2.6541E7 · m
|
| m2 |
Thus
| VgA | ≈ |
6.2677E7 · m2 |
; | VgB |
≈ |
1.5018E7 · m2 |
| s2 | s2 |
and
| √
c2 - VgA · 2 · s
| ≈ |
( 1 - 5.3028E-10 ) · s |
| √
c2 - VgB · 2
|
This time interval is smaller than 1 second because, according to
General Relativity, time passes more slowly at point A with respect to point B;
"M curves space-time" more at A than at B because dA is less than
dB. (The space-time at the virtual observer, instead, is always flat
as his/her distance is virtually infinite by definition; no clock can tick
faster than the virtual observer's one.)
"Curved space-time" also affects the velocity at which light signals travel. To
deal with this, in the theory of bipolarity, a limit of velocity magnitude is
defined as
Note that vl is "the speed limit remotely observable" by the
virtual observer; according to General Relativity the limit is always c for an
observer local to a zone with homogeneous Vg.
This is consistent with the circumstance that in the theory of bipolarity second
and meter are both based on the same tick rate reference, so it is not relevant
how much slow is the tick rate... locally Special Relativity applies with the
well known value of c as "the speed limit". Nevertheless, when the observer is
"located at a Vg" different than the one related to the observed
phenomenon he/she must take into account the normalized velocity
where v is the magnitude of a velocity vector as remotely observed by
the virtual observer.
Taking into account the normalized velocity is a must for the observer at point
A if he/she wants to apply Special Relativity to a moving vibe in "the
VgB zone". In this zone the speed limit is
| vlB | = |
√
c2 - VgB · 2
| = |
√ |
1 - |
VgB · 2
| · c |
; | vlB |
≈ |
2.9979245795E8 · m |
| c2 |
s |
Note that
| vlB | = |
fB | = |
√ |
1 - |
VgB · 2
|
| c |
f0 |
c2 |
That is, more generally, Vg affects the speed limit in the
same way it affects the tick rate reference, because speed measurements are
based on the latter... Another consequence of this is that β (a ratio of
speeds) is "Vg invariant".
Instead, β is relative to the frame of reference of the observer. In the
present example β is the normalized velocity of a vibe at point B with
respect to a frame of reference that has the origin at the center of gravity;
the virtual observer and the observer at point A are declared at rest in this
frame of reference, thus both observe the same value of β.
According to Special Relativity there is a "kinetic time dilation" associated
with the normalized velocity of the moving vibe in this example; this is a
concept different from the "gravitational time dilation" but, again, it affects
how the frequency of resonance is observed by a hypothetical observer at rest in
"the VgB zone"...
For the virtual observer it appears
| fβB | = |
√
1 - β2 · fB
| = |
| √
1 - β2 ·
| √ |
1 - |
VgB · 2
| · f0 |
| c2 |
|
And, while in the own frame of reference of the moving vibe 1 second
elapses, an observer at rest in point A counts
| fA · ns
| = |
√
c2 - VgA · 2 · ns
|
| fβB |
√
1 - β2 ·
√
c2 - VgB · 2
|
own ticks, corresponding to a time interval of
| √
c2 - VgA · 2 · s
|
| √
1 - β2 ·
√
c2 - VgB · 2
|
Let's put
Thus
| √
c2 - VgA · 2 · s
| ≈ |
( 1 - 4.4672E-10 ) · s |
| √
1 - β2 ·
√
c2 - VgB · 2
|
According to Wikipedia the operating clock of a satellite for GPS
functions is calibrated prior to launch "making it run slightly slower than the
desired frequency on Earth", "specifically, at 10.22999999543 MHz instead of
10.23 MHz".
Note that
| 10.22999999543 |
≈ | 1 - 4.4673E-10 |
| 10.23 |
This "coincidence" between the practice and the current example is due
to the fact that GPS satellites orbit at distance dB from the center
of the terrestrial gravity, with a normalized velocity β... and that the
Earth's mass is M.
Furthermore, the example does not take into account the contribution to
VgB (and to VgA) from any other mass in the universe but
the results are "realistic" because, for most of the scope of the algebraic
expression, this approximation holds...
|
√
c2 - VgA · 2
| ≈ |
| √ |
1 - |
( VgA - VgB ) · 2
|
| c2 |
|
| √
c2 - VgB · 2
|
In fact, if not extreme, the curvature of space-time does not influence
the ratio fA to fB for the "common part" in both points A
and B... the "difference in curvature" does.