The Natural Theory of Space Quantum t=cB
Table of Contents in reverse order:
XII. Appendix M: General Spatial Intersections
XI. Appendix L: Mass and Dimensional Geometry
X. Appendix K: Spatial Curvature
IX. Appendix H: Definition of the EventHorizon for Black Holes
VIII. Appendix G: Boundary Conditions Regarding Black Holes and Range of Density
VII. Appendix E: Energy Relationship Regarding Quantum Mechanics and Black Holes
VI. Appendix D: Definition of Planck’s Constant
V. Appendix C: General Calculations
IV. Appendix B: Alternate Calculations
III. Appendix A: Time vs. Space Calculations
II. Manuscript and Main Text
I. Abstract
I. Manuscript Abstract:
A variable transformation for time t is supported by wave mechanics and relativity theory and shows that time and space can be related and connected by the concept of physical events per unit space. The transformation confirms our daily macroscopic experience as unchanged from classical physics while still suggests new physics regarding small energies and large spaces. Perceived time can be altered relative to Earthbound clocks in regions of lower or higher gravitational force. A series of calculationverifications proves the theory, derives Planck’s constant and defines quantum mechanics. Black holes and their massradius relationship are defined. The Schwarzschild radius is defined. Minimum and maximum energies are defined.
II. Manuscript:
A Mathematical Transformation of Variables Defining Space – Time and the Constant h Marc E. King Silicon Valley, California January, 2012
Abstract Introduction We consider a new unit system using the transformation t = cB with c = speed of light, b (meter) = 1 (event) / B (events meter^1) The transformation t = cB implies the units t (sec) = c (met/sec) x B (sec^2 met^1) Then B events per meter = B sec^2 per meter, Derivation dψ/dt = +/ 2πi/h x Eψ as a partial derivative Per unit mass, then Further defining bminimum as the minimum Δx and using the positive root in this analysis, then Subject to the further justification below, we assert: E = FsubB x b E / m = 9.8b / (c/b)^2 and E / m = 9.8 / 9 (10^16) b^3 Jkg^1, Justification for Spatial Dimension b The neurological bound approximates the largest frame or spatial size that could be perceived as the continuity of time and accommodates the calculated boundary dimension Δx = b ~ 10^17 meters. b(min) = 1.027E17 meters was derived assuming t = cB so that c is assumed to be the maximum achievable velocity^{iii} and as such defines a maximum sequential rate and a minimum allowable b. And for the planet surface EsubB, we verify our unit of measure calculations: Independently, we can recalculate the value of b using FsubG on the planet surface:
b = EsubB / m x (a)^1 = 1.089E16 / 9.8 = 1.111E17 meters.
This should be the universal value of b and is independent of FsubG since the accelerations “g” cancel for any spatial position.
This value is larger than the allowed minimum calculated b(min) = 1.027E17 by the difference 8.4E19 meters and we find the calculated surface value to be approximately 8% larger than the minimum allowed value b(min) using the Earth gravitational acceleration a = g = 9.8 ms^2 and using no transformations in this calculation. We do not pursue further calculations in the present scope. (See Appendices for calculations.)
Energy Change as a Function of FsubG = FsubB
EsubB = F x b. With different EsubB, clocks should appear to run at different rates in regions of higher or lower FsubG relative to the planet surface. A fictitious force, like the Coriolis force or the weightlessness of orbit, should not affect the real force FsubG = FsubB. 3 x 10^8 physical events in one second of time t. The surface barrier energy E subB per unit mass = 680 eV / kg has been defined. Perceptions of Earthtime and clocks are expected to experience different rates in regions of lower or higher gravitational force relative to the planet surface.

III. Appendix A
The difference in value between b(min) derived from the Schrödinger equation and b calculated empirically from Earthsurface FsubG is 8.4E19 meters and proves to be the mean factor 1.079 or 7.9%.
The concept of continuous time t leads to exponential growthdecay:
a = asub0 x e^ (rate x time) where e = lim (nà infinity) (1 + 1/n)^n = 2.718.
Applying the expression for time itself, then
T(new) / T(old) = e^(r x t) = e^(0) = 1.
For time t itself, the rate r = 0 and there is no change in continuous time t so that one “second” of “time t” does not change. Time t is absolute.
Differential equations for centuries, e.g. the Schrödinger equation, assume time t is a real and continuous variable.
In the transformation t = cB, we need to treat continuity of time as a slightcontiguity of space.
In that case, we find 3 x 10^8 met/sec to be a large enough frequency (number n) to continue using the calculated value of e = 2.718 = lim as nàinfinity of (1+1/n)^n.
But in a directional spatial sequential model, then space itself advances or grows per some rate different from r = 0.
As a one dimensional chalk line curves in a two dimensional blackboard, and as a twodimensional earthsurface curves in threedimensional space, then a change in 3dimensional space needs to take place in a mathematical dimension higher than 3.
Postulating the higher number of dimension (vertices) to be 5 as in the Fibonacci^{iv} infinite sequence, and considering physical events B = 1 / b = sec^2, then we calculate the following for one second of time t:
(VsubS / Vsub0)^1/5 = e^(r_{V} x t)^1/5 = e^(.618^(53))^1/5 = 1.079
or a 7.9% decrease in physical events B (increase in onedimensional size b) from the continuoustime model used to calculate b(min).
IV. Appendix B
Another Calculation for difference bempirical  b(min) = 7.9%:
In two dimensional physics,
F = ma = m x metsec^2 becomes Fsub2 = m x asub2 = m x met –sec^3/2
and one physical event B would no longer have units of sec^2; instead, sec^3/2.
The uncertainty principle then has a transformed hbar, and now
b(min) = (hbar)^1/2 x c^1/2 = 1.779E13 meters.
Similarly, EsubB / m = 9.8 / c^3/2 Jkg^1 per square boundary = 1.886E12 Jkg^1 per boundary and
b = EsubB / m / 9.8 = 1.924E13 meters.
Then we again have the mean factor
= 1.079 or 7.9%
between b(min) and b (from FsubG) in twodimensional space exactly the same as in threedimensional space.
V. Appendix C
A General Fibonacci Calculation:
The Fibonacci infinite sequence was referenced in Appendix A,
F(n) = F(n1) + F(n2) with seed values F(0) = 0 and F(1) = 1.
Ratios converge, and
lim(nà infinity) F(n+1) / F(n) = φ = (1 + 5^1/2) / 2 = .618 … and
lim(nà infinity) F(n2) / F(n) = γ = .382 … and so on.
Writing an example expression for spatial dimension ≥ 3 per Appendix A
∫∫∫∫∫∫∫∫dV = ∫∫∫∫∫dV(0) x exp(r_{V} x t)
where r_{V} = rsubV = φ^(D(n+1) – D(n))
then
dx /dx(0) = exp(φ ^ (D(n+1) – D(n))^1/(n+1).
Except we are now doing math in another dimension, and while e = 2.718 in three dimensions, the base of natural logarithms should change in higher or lower dimensional space.
For example, in the case of 8 dimensions: e à e^(1/γ)^5/2.
We quickly find dx /dx(0) = 1.08.
A different example, for the case of spatial dimension < 3:
The base e must change as a function of the power of B, i.e. in three dimensional space B ~ sec^2 while in twodimensional space B ~ sec^3/2.
The difference in power of physical events B
2 – 3/2 = 1/2 and the twodimensional e = 2.718^1/2.
Then we quickly find dx /dx(0) = 1.08 similar to the previous mean calculations for the difference between b(min) and bempirical.
The (Fibonacci) calculations hold true for any spatial dimension n moving through n+1 with a dimensional adjustment for e.
VI. Appendix D
Planck’s Constant
Planck’s constant^{v} h = 6.626068E34 met^2 kg sec^1:
From the Schrödinger equation,
h = 13.6eV / (1^2) / ν = (13.6eV / (1^2) / c) x 91.2nm
Then h = (EsubB / c) x (91.2E9 / 50)
Or h = (EsubB / c) x 1.82E9
And h = (EsubB / c) x (b x c / 1.82)
So
h = EsubB x b / 1.82 or
h = e^(3/5) x b x EsubB = b E_{B} / e^{3/5}
where
b = 1.111E17 meters
and
E_{B} = EsubB = Earth surface barrier energy = 680eV/kg = 1.089E16 J/kg
And the calculated h = 2.718^3/5 x 1.111E17 x 1.089E16 = 6.6E34 per event.
More precisely from our 3decimal calculations and per appendices A through C,
h à h(1Δh) where Δh = 0.08^5/2 and h = 6.63E34 per event or we can write
h = b E_{B} κ = b E_{B} (1 – Δh) / e^{3/5} = b E_{B} ( γ' / ρ').
Units for transformed h:
h ~ met J Kg1 b3 ~ met2 J Kg1 ~ met2 met2 sec2 Kg Kg1 ~ (one B)^1 = event1.
VII. Appendix E
The Nature of Quantum Mechanics
We postulate that any allowed energy quanta has a wavelength λ = nb where n is an integer and b = 1.111… E17 meters per the main text.
For example, the 13.6eV H ground state transition is λ = 91.2nm
And n = λ / b = 820882088.
Similarly, the H state 1 to state 2 transition is λ = 121.6nm
And n = 109450945.
An H state 3 to state 1 transition is λ = 486.1nm and n = 43753375.
Then 0 < one energyevent ≤ c / b ( = 2.700E25 Jevent = c^{3} Jevent)
and quantum energy = hc/λ is defined as an integral operation of 1/b.
Then the base of all quantum mechanics is 1/b = B where t = cB
and h = h(a_{G}) and becomes a function of F_{B} = F_{G}.
It seems better to write the equivalent expressions:
E = hν = hc/λ leading to
E (J) = (b E_{B} κ) event^1 x c / (nb) energyevent and
E / E_{B} = (κ / n) c or
n E / E_{B} = κ c
where 1 ≤ n ≤ κc
and this more clearly defines the universal nature of quantum mechanics.
Units for the ratio E / E_{B} = kg for one spatial boundary and the freeenergystate becomes E_{B} (per unit mass) instead of “zero.”
VIII. Appendix G
Density of Matter and Black Holes
As the intersection of one dimensional space (a line) with two dimensional space (a surface) is a single point with zero dimension, the intersection of two and three dimensional space is a line with one dimension, and the intersection of three and five dimensional space should be a surface with two dimensions, then the intersection between five and eight dimensional space should be threedimensional (observable in 3dimensions) and is suggested by the spherical volume of a black hole.
From Appendix E, the maximum allowed energyevent is 2.700E+25 (using a J system.)
Then EsubB at a black hole surface should be bounded by the maximum allowed c^3 J kg^1.
For the hole surface:
E_{Bmax} = Gm_{H}r_{H} / r_{H}^2 or EsubBmax = G x m / r for the hole, and
c^3 = G x m / r relating to the hole, or we can write
m_{H }/ r_{H} ≤ c^{3 }/ G
or
m_{H} ≤ r_{H} c^3 / G
where G = 6.673E11 met^3 kg^1 sec^2,
rsubH has units meters, and
c (3E+8 numerical) has units J^1/3.
Then c^3 / G ≤ 2.700E+25 / 6.673E11, and
m_{H }/_{ }r_{H} ≤ 4.046E+35 kg met^1
for any black hole.
If we let EsubB = Δλ EsubBmax = Δλ c^3 where 0 < Δλ ≤ 1, and
Δλ = nb where 1 ≤ n ≤ 1/b = B, then
m_{H} = r_{H} Δλ c^3 / G or
m_{H} = K_{G} r_{H}_{ }where K_{G} = Δλ c^3 / G.
IX. Appendix H
The Spatial Nature of Black Holes
Per Appendix G,
m_{H} / r_{H} = K_{G} or msubH = KsubG x rsubH
where K_{G} = Δλ c^3 / G.
The following surface density boundary conditions should apply for any black hole:
Then boundary conditions require:
4πr_{H}^2 = C_{R38 }x r_{H}^5 and 4/3 πr_{H}^3 = C_{R58 }x r_{H}^5
where C_{R nm} = CsubR for dimension n curving through dimension m
and rsubH = r_{H} = C_{R58} / C_{R38 }from boundary conditions.
Then Δλ = CsubR38 / CsubR58 or
Δλ = C_{R3} / C_{R5}
where CsubR3 and CsubR5 represent the curvature rates of 3 and 5 dimensional space respectively through 8dimensional space, where the ratio m_{H} / r_{H} is proportional to Δλ, and where we assume CsubR3 ≤ CsubR5.
Then there is only an effective zerodensity “black hole” for C_{R3} = 0 while the highest density black hole occurs where C_{R3} = C_{R5}.
Then Δλ represents the ratio of curvatures of 3 and 5 dimensional space through 8 dimensions for the spatial intersections known as black holes.
The higher the mass density in a spatial location, the more the effective radius of curvature should change. With dense enough matter, then curvatures among dimensions become more closely equivalent as density becomes large.
To visualize in two dimensions, πr^2 and 4πr^2 are both two dimensional surfaces that curve in 3dimensions. The curvature (lack of) for a flat circle is 0 while the curvature for the closed spherical surface is 1.
Appendix H (cont.)
If we assume CsubR5 is closed (curvature 1) in 8dimensions, then CsubR3 has the possible range 0 à 1 in 8dimensions where 0 represents no intersection at all and 1 represents a closure of the five and eight dimensional intersection.
To see/observe the intersection (black hole) it must be at least a 5 and 8 dimensional intersection (3dimensional) or constitute the five dimensional surface intersection represented by the integral ∫4πr^2dr throughout r for the continual surfaces.
Then the “smallest” black hole is the “least dense m_{H} / r_{H}” black hole having Δλ ~ 0 but still large enough to represent an intersection of 5 and 8 dimensional space.
Then the boundary condition is a single event:
C_{R3}min = 1 / cB where B = 1 and
Δλmin = 1 meter / c meters = 3.333E9.
Then
c^{2 }/ G ≤ m_{H} / r_{H} ≤ c^{3} / G or
c^2 ≤ m_{H} / r_{H} ≤ c^3 kg met^1
or we can write the expression in the Schwarzschild^{vi} form,
r_{H} = k(λ) m_{H} G / c^2 meters
where k(λ) = 1 / cΔλ.
X. Appendix K
The Various Sizes of Black Holes and Curvatures of ThreeDimensional Space
Appendix H defines the massradius relationship as observed in threedimensional space:
m_{H} / r_{H} = K_{G}(λ)
where λ = Δλ = C_{R3} / C_{R5}
and represents the ratio of curvatures from 3dimensional space and 5dimensional space through 8dimensional space respectively.
We assume, for the three dimensional intersections, that C_{R5} = 1.
The minimum C_{R3} = 1 / c and the maximum C_{R3} = c / c = 1.
Allowed quantum are then n / c for n = 1 to c.
The minimum (least dense) intersection is an intersection among 3, 5 and 8 dimensional space where C_{R5} = 1 and C_{R3} = C_{R3}(min) = 1 / c.
The next “largest” (more dense) intersection should occur for C_{R3} = 2 / c and so on.
The most dense intersection occurs where C_{R3} = c / c = 1 and represents a closed third dimension in both eight dimensional and five dimensional space.
To visualize curvatures, the diameter of a circle = d is a straight line with curvature
C_{R13} = 0 while the circumference (length πd) closes upon itself (runs into the back of itself) and has the curvature C_{R13} = 1.
The curvature C_{R23 }is closed in 3dimensions visualized as a spherical (or elliptical, not reviewed in this scope) surface area that has closed itself around a centerofmass c_{M}.
The two dimensional surface does not alter or “grow” in three dimensions, but the one dimensional line, e.g. the straight path of a distant comet or ray of light (C_{R13} = 0) or the line of a planetary satellite C_{R13} = 1) both curve (or bend) around mass in three dimensions to the two extreme degrees of curvature.
Then the ratio m_{H} / r_{H} = K_{G} should represent a curvature of threedimensional space through eightdimensional space.
XI. Appendix L
Definition of Mass and Geometry for Black Holes
From the equation E_{B} = a_{G} J kg^1 b_{n}^n x b_{n} ( = 1.089E16 on Earth surface,)
and from the definition of physical events in dimension n = B where B ~ sec^2 in 3dimensions and sec^(D1) per appendix C, then b_{n} has the following value:
3dimensions: b_{3} = 1.111E17 meters (per the main text)
5dimensions: b_{5} = b_{3} / c = 3.703E26 meters
8dimensions: b_{8} = b_{5} / c^2 = 4.114E43 meters and so on.
Per Appendices G, H and K, the black hole geometry is a function of mass and becomes a series of symmetricallyclosed concentric surfaces having internal densities:
8dimensional volume in 3dimensions = 4 / 3 π r_{8}^3
5dimensional volume ( = ∫ (r_{58}tor_{5}) 4πr^2 ) = 4 / 3 π r_{5}^3 (including the volume of 8)
3dimensional volume = 4 / 3 π r_{H}^3 where r_{H} = r_{5} (including volumes of 5 and 8)
and
the boundary condition for the mostdense black hole is then:
m_{H} / (4πr_{H}^2) = m_{H} / (4/3 πr_{H}^3)
where r_{H} = m_{H} / K_{G}, then
m_{H}(max) = 3K_{G }kg and the corresponding
r_{H} = 3 km.
The black hole mass m_{H} for the general case r_{H} = r_{5}:
m_{H}(λ) = ∫ (from r_{5} – r_{8} to r_{5}) 4πr^2 dr = 3K_{G}(λ)
where
λ = r8 / r5,
r_{H} = r_{5} and
r_{8} is the 3dimensionalradius of zeromass 8dimensional space (b = b_{8}) at the center of the hole.
XII. Appendix M
A General Case of Intersections
Per Appendix L, the adjacent intersections to threedimensional space D = 3 are with dimensions D = 2 and D = 5, and the intersection between three and five dimensional space is a twodimensional surface, e.g. a closed surface around a large mass M.
There are infinite concentric closed surfaces around the mass M, so we can define a surface “intensity” or transmissionability proportional to r^2:
T_{S} = G' F_{G}(r)
or more generally,
T_{S} = ∑ G' F_{G}(r) for masses M at distances r.
In the absence of continuous time t, then the 3dimensional spatial sequence progresses through 5dimensional space. In order to access a prior 3dimensional spatial frame, an interaction is required through a twodimensional intersection.
As an example, the reader can think of any important memory. Subsequently, a photographlike image immediately appears to the reader’s mind. In the absence of continuous time, the threedimensional spatial frame would need to be accessed in order to review the memory.
The access required is through a twodimensional intersection and through fivedimensional space in order to obtain information contained in a prior threedimensional spatial frame. The previous spatial frame(s) is still there. It didn’t go anywhere. The frames should be available for continual access.
There can be access/communication at the speed of light c by energy state transitions, including hyperfine transitions, in hydrogen atoms and other molecular state transitions as well as low energy magnetic dipole moment interactions from molecular charges within large hydrocarbon (organic) molecules.
Arguably, not all memories are selfcontained. Electromagnetic communication can take place as a result of hydrocarbon molecular energy state transitions and with the surface(s) having the combined intensity ∑ T_{S }as above. The communication rate (not the perceptibility rate) should be the speed of light c.
∑ T_{S} is better defined as follows for an altitude A above the surface of mass M:
T_{S} = (r_{CM} à r_{S}) ∫ G'(ε,r) F_{G}(r) dr + (r_{S} à r_{A}) ∫ G'(ε,r) F_{G}(r) dr
+ (r_{A} à infinity) ∫ G'(ε,r) F_{G}(r) dr.
References:
[i] Erwin Schrödinger, Quantisierung als Eigenwert Problem, 1926.
[ii] Werner Heisenberg, Über die Grundprinzipien der 'Quantenmechanik, 1927.
^{iii }Albert Einstein, Zur Elektrodynamik bewegter Körper, 1905.^{} ^{ }
^{iv} Leonardo Fibonacci, Liber Abbaci, 1202.
^{v }Max Planck, The Genesis and Present State of Development of the Quantum Theory, 1920.
^{vi} Karl Schwarzschild, Uber das Gravitationsfeld eines Massenpunktes nach der Einsteinschen Theorie, 1916.