A Mathematical Transformation of Variables Regarding Space - Time
Silicon Valley, California
January, 2012
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 verification of calculations is shown.
Introduction
In this model, it is shown that continuous time t and a contiguous view of spatial frames are mathematically the same in the macroscopic sense. A suggested transformation of variables presents interesting differences in concept for small and large energies and spaces.
Concept
We postulate that continuous time as experienced can also be represented, with the same physical result, as a directional spatial sequence or frames of events.
We consider a new unit system using the transformation t = cB with c = speed of light,
where one spatial frame (size b) is related to one physical event B by
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,
and one physical event = one square second = sec^2.
Derivation
From wave mechanics, we have the Schrödinger equation
dψ/dt = +/- 2πi/h x Eψ as a partial derivative
and the related approximation
ΔxΔk ≥ O(1).
This defines the uncertainty in measurements
Δx Δp ≥ h/2π.
Implying Δp = m Δx / Δt and using a transformation for Δt, then
Δp = m Δx / Δ(cB)
This leads to
m (Δx)^2 ≥ (h/2π J-s) (ΔcB) = (h/2π J) (cB) (ΔcB)
Per unit mass, then
(Δx)^2 ≥ (h/2π)(ΔcB)(cB) from transformation.
For a single B (events-meter^-1) the corresponding Δx = b meters and Δ(cB) = 1/(cB)
Then b ≥ (h-bar)^1/2.
Further defining b-minimum as the minimum Δx and using the positive root in this analysis, then
b(min) =1.027E-17 meters.
Subject to the further justification below, we assert:
E = F-sub-B x b
Where F-sub-B = F-sub-G = the gravitational force at the spatial location of event B.
And on the planet surface, F-sub-B = ma = m x 9.8 meters-sec^-2.
Then E / m = a x b = 9.8b meters / (cB)^2 or
E / m = 9.8b / (c/b)^2 and
E / m = 9.8 / 9 (10^-16) b^3 J-kg^-1,
Or we can write the expression:
E / m / b^3 = 1.089E-16 J kg^-1 or
E-sub-B / m = 680 eV / kg for one cubic spatial boundary.
Justification for Spatial Dimension b
Using uncertainty and similar to the derivation above, one estimation using neurological sensory communication as an upper bound on Δx = b (in one dimension) for spatial boundary (frame) size is suggested by:
(Spatial Frame Width)^2 ≤ (h – bar ) x (c) x (Time Required for Sensory Continuity)
Using orders of magnitude 10^-34 J-sec (and adjusting for units) from wave mechanics and estimating the time required for sensory communication in the range 10^-3 sec – 10^-6 sec from synapse switching (potential change) rate, we would then estimate the magnitude:
Δx = b ~ 10^-14 to 10^-16 meters (for example as an upper spatial bound) in order to perceive continuity from actual contiguity of frames.
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 velocity and as such defines a maximum sequential rate and a minimum allowable b.
Justification for Spatial Barrier Energy
Using a one dimensional example,
E = F x distance.
We are using the transformation t = cB where B = 1 / b and b has the spatial dimension of meters.
The energy associated with the distance b is a function of a force F acting upon a mass m at a particular set of spatial coordinates.
It follows, the innate force acting on the mass m in space is the gravitational force.
There are no external forces to be considered for the mass m for our purpose regarding the transformation associating time and space.
Verification of Calculations
E-sub-B is then a function of gravitational force.
Using the calculations above and for a single event B = 1,
we can also write, for the planet surface as an example:
E / m = a x b = b x 9.8 (c / b )^-2 = 9.8 / c^2 / b.
Then 1.089E-16 = 9.8 / c^2 / b.
And for the planet surface E-sub-B, we verify our unit of measure calculations:
b meters = (1 event / B events meter^-1) = 1.000 as a confirmation of the energy calculation 680 eV / kg.
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 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 Appendix A.)
Energy Change as a Function of F-sub-G = F-sub-B
Assuming mass m and boundary b are unchanged, then E-sub-B changes as a function of F-sub-G = F-sub-B the gravitational force at the location of physical event B.
This follows directly from
E-sub-B = F x b.
A smaller gravitational force leads to a smaller E-sub-B relative to the planet surface.
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.
Conclusions
Continuous time can be represented by a contiguous spatial sequence of frames, or boundaries Δx = b, while conforming to existing physics in the macroscopic sense and with our sensory perceptions.
The transformation t = cB leads to the spatial frame dimension b(min) = 1.027E-17 meters and corresponds to
3 x 10^8 physical events in one second of time t.
For this model,
The surface barrier energy E sub-B per unit mass = 680 eV / kg has been defined.
E-sub-B is suggested to be a function of gravitational force and so a function of spatial location.
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.
References
