The approach I have take, wrt time dilation as the cause of the gravitational effect (from one of my publications) is as follows (it not complete yet and needs more work)
(Note the sub and super scripting has been distorted)
3.1 Introduction
The laws of physics are invariant with respect to frame of reference because the cumulative probability density of a particle cloud is unity for any frame of reference. Time dilation causes gravity and not the other way around. With a stationary particle, I demonstrate how the particle’s probability cloud distorts under time dilation. This distortion of the probability cloud causes the centre of mass to shift in the direction of time contraction. This effect in the presence of a continuous, non-linear time dilation well is called gravity. I have shown that the error between gravitational escape velocity and equivalent Lorentz/time dilation velocity is less than ± two parts per million. I have not addressed the relativistic nature of this model in this paper. I hope to spark new theoretical approaches to gravity that is not centred about gravitons, Higgs particle or superstrings to explain my experimental results.
3.2 The Uncertainty Principle
Quantum mechanics (Bethe [2]) states that for a free particle, (e.g. free electron) “a wave function (wave packet) may be constructed that puts the main probability near a position, xo, and near momentum po”. The uncertainty principle, dictates that position and momentum cannot be simultaneously determined accurately, their uncertainties are related by
∆x∆p ³ ½ ħ (3.1)
Since the mass of a free electron is known, the uncertainty principle dictates uncertainty of both position and velocity simultaneously. Let’s take this velocity/position concept a little further.
3.3 The Axioms
Axiom 1: Principle of the Particle Probability Cloud. This axiom states that, a particle can be represented by a probability cloud. The probability of detecting a particle at any point in its probability cloud is some value between zero and one. That is, there is a possibility of passing right through a particle without detecting it. However, the cumulative probability of detecting the particle at any point and within any duration, in its probability cloud must be one.
Axiom 2: Principle of Probability Density Invariance. This principle states that every mass particle, at rest, is a probability cloud with a probability density, ρ, that is invariant in velocity-space (sx/tx, sy/ty, sz/tz) corresponding to x, y, and z axes in spacetime and allowing for different amounts of time dilation, tx, ty, and tz, along each axis, x, y, and z respectively.
Axiom 3: Probability Volume. The volume occupied by a particle in velocity-space, is the volume of its probability function or probability cloud. Therefore, any frame of reference has to observe the cumulative probability of one of detecting the particle in its probability cloud.
Axiom 4: Principle of Mass Density Invariance. This principle states that the mass of a particle is equivalent to its volume occupied by the particle’s cumulative probability in velocity-space.
3.4 The Principle of Probability Density Invariance
The probability of detecting the particle in the region of space sx, along the x-axis, within a time duration dx, is given by, P(sx/dx). Similarly, P(sy/dy) and P(sz/dz) are the probabilities associated with the y and z-axes respectively.
The cumulative probability of finding this particle is one,
Cum P(sx/dx, sy/dy, sz/dz) = 1 (3.2)
Since the probability distribution is identical along any axis, the cumulative probability is formed by the rotation about y and z-axes, which must be one. Therefore, the volume formed by the probability function is,
(4/3) π P(sx/dx)^3 = 1 (3.3)
Or,
P(sx/dx), ρ = [1/(4/3) π]^–3 = 0.024189, (3.4)
a constant
That is, the probability density in velocity-space is independent of the nature of the particle’s probability function and is invariant, when the particle is at rest.
3.5 The Time Dilation Effect
Since “the laws of nature are the same in all frames moving with constant velocity with respect to one another” (Shapiro [1]), one can substitute an external observer with a stationary observer who is internal to the particle’s probability cloud. In effect we have shifted from observing the particle as a probability cloud to observing the probability cloud itself.
Let’s say that time is normal on the left half of the particle probability cloud and dilated on its right half. Fig. 3.1 depicts the distortion introduced by time contraction, for a stationary particle, along the x-axis, on the right half of the particle probability cloud. Contracted time allows the right half of the particle probability cloud to spread further out in space than the left half.
That is, even though the probability of detecting the particle on the left half is P(sxo/dxo, syo/dyo, szo/dzo) / 2, the probability of detecting the particle on the right half is now P(sxd/dxd, syd/dyd, szd/dzd) / 2, where the subscript ‘o’ represents undilated time and ‘d’ represents contracted time.
Since, time contraction occurs only along the x-axis, for any coordinate in the y-z plane in the left half, there is an equivalent coordinate in the right half, such that, syd = syo, szd = szo, dyd = dyo, dzd = dzo along the y- and z-axes. Therefore, the probability function for the right side reduces to, P(sxd/dxd, syo/dyo, szo/dzo) / 2.
The cumulative probability of observing the stationary particle must be one. Therefore,
0.5 Cum P(sxo/dxo, syo/dyo, szo/dzo) +
0.5 Cum P(sxd/dxd, syo/dyo, szo/dzo) = 1 (3.5)
That is, the probability cloud is symmetrical about the x-axis, and given that the left half is a hemisphere, the right half will be an ellipsoid such that,
(0.5) (4/3) π P(sxo/dxo)^3 + (0.5) (4/3) π P(sxo/dxo)^2 P(sxd/dxd) = 1.0 (3.6)
substituting for (3.3)
P(sxd/dxd) / P(sxo/dxo) = 1.0 (3.7)
or
P(sxo/dxo) = P(sxd/dxd) = [1/(4/3) π]^–3 (3.8)
The probability of detecting a particle within its particle cloud, within a duration, dxd or dxo, is independent of the time contraction distortions, and thus gravitational distortions, it experiences. Therefore,
sxo/dxo = sxd/dxd or sxd = sxo (dxd/dxo) (3.9)
Let us call equation (3.9) the Probability Invariance Transformation (PIT) equation for a stationary particle in velocity-space. This PIT equation can also be interpreted as equivalent to the stretching by tidal gravity [Thorne (1)], as time contraction causes the stretching of a particle.
Thus, sxd > sxo when dxd > dxo (3.10)
The probability cloud has extended itself to compensate for the time contraction with respect to its own frame of reference, given an invariant probability density in velocity-space.
The centre of mass of the left hemisphere and right ellipsoid are (3/8) sxo , (3/8) sxd respectively. If, at the very least, both sides have the same mass, the centre of mass of the particle has shifted (3/8)(sxd - sxo) to the right. The new centre of mass, SCM, is,
SCM = (3/8) sxo (dxd/dxo - 1) (3.11)
Therefore, the centre of mass of the particle probability cloud has shifted further to the right, in keeping with the direction of time contraction; this shifting is linearly dependent on time dilation/contraction. Note, however, that by the Principle of Mass Density Invariance, the mass of the right side should be greater than the mass of the left side, therefore, equation (3.11) depicts a “best” case or lower bound or minimum shifting of the centre of mass. The gravitational effect can be summarized as follows,
1. Time dilation/contraction distorts the shape of a particle’s probability cloud in the direction of increasing time contraction.
2. This distortion of the particle’s probability cloud results in the shifting of the centre of mass of the particle in the direction of increasing time contraction.
3. The net effect is that the centre of mass of the particle moves in the direction of increasing time contraction.
4. This effect in spacetime is called a gravitational field.
In a gravitational field, time dilation on the right hand side is replaced with dxo .tR, and on the left hand side with, dxo .tL, where dxo is the duration of the probability cloud in the centre of the particle. tL and tR represent the time dilation from a point at an infinite distance from the source of gravity. (tL ≠ tR for non-linear time dilation) such that,
(0.5) (4/3) π P(sxo/dxo)2 P(sxL/(dxo .tL)) +
(0.5) (4/3) π P(sxo/dxo)2 P(sxR/(dxo .tR)) = 1.0 (3.12)
substituting (3.3),
P(sxL/(dxo .tL)) + P(sxR/(dxo .tR)) = 2 P(sxo/dxo) (3.13)
That is, the probability gained on one side must be compensated for by the same amount, as a probability loss on the other side of the stationary particle. The new right shifted, centre of mass of the stationary particle in a gravitational field is,
SCM = (3/8) (sxR - sxL) (3.14)
Using (3.9),
SCM = (3/8) sxo (tR - tL) (3.15)
For the short distance of the particle size, the change in time dilation, tL - tR = δt, and distance moved by the particle, δs = SCM, such that,
δs = (3/8) sxo . δt (3.16)
that is, distanced moved by the particle is a function of the change in time dilation at that point. Note that the change in time dilation, δt, is not the same as the duration taken to move. To put it another way, when time dilation is constant with respect to a particle’s frame of reference, the particle is stationary with respect to its own fame of reference. When time dilation is non-linear, the particle is displaced and therefore experiences motion with respect to its own frame of reference.