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 Event-Horizon 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. 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 Earth-bound clocks in regions of lower or higher gravitational force. A series of calculation-verifications proves the theory, derives Planck’s constant and defines quantum mechanics. Black holes and their mass-radius relationship are defined. The Schwarzschild radius is defined. Minimum and maximum energies are defined.
A Mathematical Transformation of Variables Defining Space – Time and the Constant h
Marc E. King
Silicon Valley, California
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,
dψ/dt = +/- 2πi/h x Eψ as a partial derivative
Per unit mass, then
Further defining b-minimum as the minimum Δx and using the positive root in this analysis, then
Subject to the further justification below, we assert:
E = F-sub-B x b
E / m = 9.8b / (c/b)^2 and
E / m = 9.8 / 9 (10^-16) b^3 J-kg^-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) = 1.027E-17 meters was derived assuming t = cB so that c is assumed to be the maximum achievable velocityiii and as such defines a maximum sequential rate and a minimum allowable b.
And for the planet surface E-sub-B, we verify our unit of measure calculations:
Independently, we can re-calculate the value of b using F-sub-G on the planet surface:
b = E-sub-B / m x (a)^-1 = 1.089E-16 / 9.8 = 1.111E-17 meters.
This should be the universal value of b and is independent of F-sub-G since the accelerations “g” cancel for any spatial position.
This value is larger than the allowed minimum calculated b(min) = 1.027E-17 by the difference 8.4E-19 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 m-s^-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 F-sub-G = F-sub-B
E-sub-B = F x b.
With different E-sub-B, clocks should appear to run at different rates in regions of higher or lower F-sub-G relative to the planet surface.
A fictitious force, like the Coriolis force or the weightlessness of orbit, should not affect the real force F-sub-G = F-sub-B.
3 x 10^8 physical events in one second of time t.
The surface barrier energy E sub-B per unit mass = 680 eV / kg has been defined.
Perceptions of Earth-time 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 Earth-surface F-sub-G is 8.4E-19 meters and proves to be the mean factor 1.079 or 7.9%.
The concept of continuous time t leads to exponential growth-decay:
a = a-sub-0 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 slight-contiguity 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 two-dimensional earth-surface curves in three-dimensional space, then a change in 3-dimensional 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 Fibonacciiv infinite sequence, and considering physical events B = 1 / b = sec^2, then we calculate the following for one second of time t:
(V-sub-S / V-sub-0)^1/5 = e^(rV x t)^1/5 = e^(.618^(5-3))^1/5 = 1.079
or a 7.9% decrease in physical events B (increase in one-dimensional size b) from the continuous-time model used to calculate b(min).
IV. Appendix B
Another Calculation for difference b-empirical - b(min) = 7.9%:
In two dimensional physics,
F = ma = m x met-sec^-2 becomes F-sub-2 = m x a-sub-2 = 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 h-bar, and now
b(min) = (h-bar)^1/2 x c^1/2 = 1.779E-13 meters.
Similarly, E-sub-B / m = 9.8 / c^3/2 J-kg^-1 per square boundary = 1.886E-12 J-kg^-1 per boundary and
b = E-sub-B / m / 9.8 = 1.924E-13 meters.
Then we again have the mean factor
= 1.079 or 7.9%
between b(min) and b (from F-sub-G) in two-dimensional space exactly the same as in three-dimensional space.
V. Appendix C
A General Fibonacci Calculation:
The Fibonacci infinite sequence was referenced in Appendix A,
F(n) = F(n-1) + F(n-2) 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(n-2) / F(n) = γ = .382 … and so on.
Writing an example expression for spatial dimension ≥ 3 per Appendix A
∫∫∫∫∫∫∫∫dV = ∫∫∫∫∫dV(0) x exp(rV x t)
where rV = r-sub-V = φ^(D(n+1) – D(n))
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 two-dimensional space B ~ sec^3/2.
The difference in power of physical events B
2 – 3/2 = 1/2 and the two-dimensional 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 b-empirical.
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 constantv h = 6.626068E-34 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 = (E-sub-B / c) x (91.2E-9 / 50)
Or h = (E-sub-B / c) x 1.82E-9
And h = (E-sub-B / c) x (b x c / 1.82)
h = E-sub-B x b / 1.82 or
h = e^(-3/5) x b x E-sub-B = b EB / e3/5
b = 1.111E-17 meters
EB = E-sub-B = Earth surface barrier energy = 680eV/kg = 1.089E-16 J/kg
And the calculated h = 2.718^-3/5 x 1.111E-17 x 1.089E-16 = 6.6E-34 per event.
More precisely from our 3-decimal calculations and per appendices A through C,
h à h(1-Δh) where Δh = 0.08^5/2 and h = 6.63E-34 per event or we can write
h = b EB κ = b EB (1 – Δh) / e3/5 = b EB ( γ' / ρ').
Units for transformed h:
h ~ met J Kg-1 b-3 ~ met-2 J Kg-1 ~ met-2 met2 sec-2 Kg Kg-1 ~ (one B)^-1 = event-1.
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… E-17 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 energy-event ≤ c / b ( = 2.700E25 J-event = c3 J-event)
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(aG) and becomes a function of FB = FG.
It seems better to write the equivalent expressions:
E = hν = hc/λ leading to
E (J) = (b EB κ) event^-1 x c / (nb) energy-event and
E / EB = (κ / n) c or
n E / EB = κ c
where 1 ≤ n ≤ κc
and this more clearly defines the universal nature of quantum mechanics.
Units for the ratio E / EB = kg for one spatial boundary and the free-energy-state becomes EB (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 three-dimensional (observable in 3-dimensions) and is suggested by the spherical volume of a black hole.
From Appendix E, the maximum allowed energy-event is 2.700E+25 (using a J system.)
Then E-sub-B at a black hole surface should be bounded by the maximum allowed c^3 J kg^-1.
For the hole surface:
EBmax = GmHrH / rH^2 or E-sub-Bmax = G x m / r for the hole, and
c^3 = G x m / r relating to the hole, or we can write
mH / rH ≤ c3 / G
mH ≤ rH c^3 / G
where G = 6.673E-11 met^3 kg^-1 sec^-2,
r-sub-H has units meters, and
c (3E+8 numerical) has units J^1/3.
Then c^3 / G ≤ 2.700E+25 / 6.673E-11, and
mH / rH ≤ 4.046E+35 kg met^-1
for any black hole.
If we let E-sub-B = Δλ E-sub-Bmax = Δλ c^3 where 0 < Δλ ≤ 1, and
Δλ = nb where 1 ≤ n ≤ 1/b = B, then
mH = rH Δλ c^3 / G or
mH = KG rH where KG = Δλ c^3 / G.
IX. Appendix H
The Spatial Nature of Black Holes
Per Appendix G,
mH / rH = KG or m-sub-H = K-sub-G x r-sub-H
where KG = Δλ c^3 / G.
The following surface density boundary conditions should apply for any black hole:
Then boundary conditions require:
4πrH^2 = CR3-8 x rH^5 and 4/3 πrH^3 = CR5-8 x rH^5
where CR n-m = C-sub-R for dimension n curving through dimension m
and r-sub-H = rH = CR5-8 / CR3-8 from boundary conditions.
Then Δλ = C-sub-R3-8 / C-sub-R5-8 or
Δλ = CR3 / CR5
where C-sub-R3 and C-sub-R5 represent the curvature rates of 3 and 5 dimensional space respectively through 8-dimensional space, where the ratio mH / rH is proportional to Δλ, and where we assume C-sub-R3 ≤ C-sub-R5.
Then there is only an effective zero-density “black hole” for CR3 = 0 while the highest density black hole occurs where CR3 = CR5.
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 3-dimensions. 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 C-sub-R5 is closed (curvature 1) in 8-dimensions, then C-sub-R3 has the possible range 0 à 1 in 8-dimensions 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 (3-dimensional) 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 mH / rH” 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:
CR3min = 1 / cB where B = 1 and
Δλmin = 1 meter / c meters = 3.333E-9.
c2 / G ≤ mH / rH ≤ c3 / G or
c^2 ≤ mH / rH ≤ c^3 kg met^-1
or we can write the expression in the Schwarzschildvi form,
rH = k(λ) mH G / c^2 meters
where k(λ) = 1 / cΔλ.
X. Appendix K
The Various Sizes of Black Holes and Curvatures of Three-Dimensional Space
Appendix H defines the mass-radius relationship as observed in three-dimensional space:
mH / rH = KG(λ)
where λ = Δλ = CR3 / CR5
and represents the ratio of curvatures from 3-dimensional space and 5-dimensional space through 8-dimensional space respectively.
We assume, for the three dimensional intersections, that CR5 = 1.
The minimum CR3 = 1 / c and the maximum CR3 = 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 CR5 = 1 and CR3 = CR3(min) = 1 / c.
The next “largest” (more dense) intersection should occur for CR3 = 2 / c and so on.
The most dense intersection occurs where CR3 = 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
CR1-3 = 0 while the circumference (length πd) closes upon itself (runs into the back of itself) and has the curvature CR1-3 = 1.
The curvature CR2-3 is closed in 3-dimensions visualized as a spherical (or elliptical, not reviewed in this scope) surface area that has closed itself around a center-of-mass cM.
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 (CR1-3 = 0) or the line of a planetary satellite CR1-3 = 1) both curve (or bend) around mass in three dimensions to the two extreme degrees of curvature.
Then the ratio mH / rH = KG should represent a curvature of three-dimensional space through eight-dimensional space.
XI. Appendix L
Definition of Mass and Geometry for Black Holes
From the equation EB = aG J kg^-1 bn^-n x bn ( = 1.089E-16 on Earth surface,)
and from the definition of physical events in dimension n = B where B ~ sec^2 in 3-dimensions and sec^(D-1) per appendix C, then bn has the following value:
3-dimensions: b3 = 1.111E-17 meters (per the main text)
5-dimensions: b5 = b3 / c = 3.703E-26 meters
8-dimensions: b8 = b5 / c^2 = 4.114E-43 meters and so on.
Per Appendices G, H and K, the black hole geometry is a function of mass and becomes a series of symmetrically-closed concentric surfaces having internal densities:
8-dimensional volume in 3-dimensions = 4 / 3 π r8^3
5-dimensional volume ( = ∫ (r5-8-to-r5) 4πr^2 ) = 4 / 3 π r5^3 (including the volume of 8)
3-dimensional volume = 4 / 3 π rH^3 where rH = r5 (including volumes of 5 and 8)
the boundary condition for the most-dense black hole is then:
mH / (4πrH^2) = mH / (4/3 πrH^3)
where rH = mH / KG, then
mH(max) = 3KG kg and the corresponding
rH = 3 km.
The black hole mass mH for the general case rH = r5:
mH(λ) = ∫ (from r5 – r8 -to- r5) 4πr^2 dr = 3KG(λ)
λ = r8 / r5,
rH = r5 and
r8 is the 3-dimensional-radius of zero-mass 8-dimensional space (b = b8) at the center of the hole.
XII. Appendix M
A General Case of Intersections
Per Appendix L, the adjacent intersections to three-dimensional space D = 3 are with dimensions D = 2 and D = 5, and the intersection between three and five dimensional space is a two-dimensional 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 transmission-ability proportional to r^-2:
TS = G' FG(r)
or more generally,
TS = ∑ G' FG(r) for masses M at distances r.
In the absence of continuous time t, then the 3-dimensional spatial sequence progresses through 5-dimensional space. In order to access a prior 3-dimensional spatial frame, an interaction is required through a two-dimensional intersection.
As an example, the reader can think of any important memory. Subsequently, a photograph-like image immediately appears to the reader’s mind. In the absence of continuous time, the three-dimensional spatial frame would need to be accessed in order to review the memory.
The access required is through a two-dimensional intersection and through five-dimensional space in order to obtain information contained in a prior three-dimensional 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 self-contained. Electro-magnetic communication can take place as a result of hydro-carbon molecular energy state transitions and with the surface(s) having the combined intensity ∑ TS as above. The communication rate (not the perceptibility rate) should be the speed of light c.
∑ TS is better defined as follows for an altitude A above the surface of mass M:
TS = (rCM à rS) ∫ G'(ε,r) FG(r) dr + (rS à rA) ∫ G'(ε,r) FG(r) dr
+ (rA à infinity) ∫ G'(ε,r) FG(r) dr.
[i] Erwin Schrödinger, Quantisierung als Eigenwert Problem, 1926.
[ii] Werner Heisenberg, Über die Grundprinzipien der 'Quantenmechanik, 1927.
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.