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You will find here the necessary mathematical arsenal currently available for estimating some effects we should expect. We are more interested into time dilation effects than length contractions because it is a measurable and thus tangible.
The solutions presented in this chapter do not use the complete General Relativity modeling because of its unnecessary complexity. What is being used is the simplified Special Relativity formulation that is basically the roots of its success or and there fore should be necessary to present the problem and the approach that should be taken to resolve it.
Quick presentations are made and it is assumed knowledge of their origin is known.
1.1. Lorentz Time Transformations
The accepted and most precise equation explaining time dilation is given by a derivation of the famous Lorentz transformations:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)
v is the relative velocity between the observer and the moving clock c is the speed of light
This equation can be represented with the following graph:
[FIGURE 1 OMITTED]
1.2. Hubble's Law
Additionally according to Hubble's law the galaxies are traveling away from each other at a constant acceleration given by:
v = H0 D (2)
The observational data resulting to this linear regression constitutes the speed at which galaxies are traveling away from each other relative to the distance from the observer.
1.3. Twin Paradox
Special Relativity is a subjective approach to demonstrate effects caused by high-speed velocities and instantly leads to paradoxes such as the famous twin paradox. Taking time dilation into account when comparing accelerated frames altogether is fundamental.
By taking for example a variation of the twin paradox and consider the simplified scenario of a spaceship traveling nearly the speed of light in relation to stationary clocks:
[FIGURE 3 OMITTED]
According to Special Relativity if clock 1 and 2 are perfectly synchronized and their adjacent camera takes a picture of the spaceship at t = 0 and t = 1 then they will both return the picture of a spaceship having clocks indicating a time almost identical.
Where Special Relativity fails is to represent the perception of our spacecraft. Pursuant to Special Relativity:
"The laws by which the states of physical systems undergo change are not affected, whether these changes of state be referred to the one or the other of two systems in uniform translatory motion relative to each other."
Well because of the time dilation effects we can see the lack of change in time on the clock number 3. This means the observer driving the ship must see the external clock 1 and clock 2 running very fast in relation to his own. Thus the time outside his reference frame will be viewed as being nearly infinitely much quicker.
1.4. Length Contraction Paradox
Having a paradox in the heart of a theory ultimately disproves it. Notwithstanding infinite masses is also predicted by SR, length contraction is enough showing inconsistency.
According to SR length of a body will contract in the direction of its velocity when its amplitude approaches the speed of light as we can see from the formula:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)
If we study further in details the behavior of this effect by having two cannons adjacent to each other and an observer taking in picture the length of the objects thrown by the cannons, then we will get very different results out of distinct scenarios.
In essence a projectile when traveling at high velocity as predicted by SR will have a velocity of c- m/s in our example:
[FIGURE 5 OMITTED]
1.4.1. Thought Experiment #1
We then decide adding a second cannon next to the first one and exactly 1 meter in front of it. A clerk behind the cannons now fires the cannonballs at the exact same moment and the photographer standing on the ground should measure the distance of the two bullets to be 1 meter:
[FIGURE 6 OMITTED]
By now tying the bullets with a chain, the photographer won't see any difference in the distance of the two bodies in motion:
[FIGURE 7 OMITTED]
1.4.2. Thought Experiment #2
The leading cannon in this scenario is moved back and placed side by side to the other one. 3.33×10-9 second after the first cannon fires, the second one is then ejected making it 1 meter away from the leading one:
[FIGURE 8 OMITTED]
Up to now everything is consistent with SR but here comes the tricky part. What would exactly happen if the two bullets were tied together with a rope before they are ejected from their respective cannon as shown below?
This is controversial because now the 2 bullets become 1 object only. According to SR this basically means the 2 bullets and the rope must all contract altogether.
It makes the two last events incongruous by defining the behavior of the objects already in motion with any precondition.
Finite Theory (FT) defines a new representation of the actual formulas derived from General Relativity (GR). Where it differs from it is how time is defined and will help understand the implications previously stated.
FT postulates time dilation is directly proportional to its energy, where the former will be later shown to be sufficient in describing universal phenomena:
1. The kinetic energy of body relative to its maxima induces dilation of time 2. A gravitational time dilation is the direct cause of the superposed gravitational potentials
Indeed in contrast to GR where space is ultimately variable to keep the speed of light constant, FT considers time to be variable and therefore the space can be represented with the standard Cartesian coordinate system. No effective result deriving from GR is in violation.
We herein will very commonly use the term "gravity density" which can be directly translated to: "space density" in GR; "graviton flux" in quantum mechanics; or simply "gravitational potential" in classical mechanics.
2.1. Black Hole Radius
A black hole is a region in space where all matter and energies, including light, cannot escape from its gravitational force. The Schwarzschild radius defines the event-horizon where the gravitational pull exceeds the escape velocity of the speed of light. This is given by:
rs = 2GM/c2 (4)
Given that Schwarzschild radius derives from GR formulation, FT will need its own definition. Satisfyingly, this event horizon can easily be found with the amount of kinetic energy needed to overtake the gravitational potential energy:
1/2 mv2 = GMm/rb (5)
By solving the equation with the maximum escape velocity a photon can have, where the mass is of non-importance we get:
rb = 2GM/c2 (6)
Despite the fact the resulting equation is exactly the same as the Schwarzschild radius, we will use a different notation given that its origin differs.
2.2. Composition of a Black Hole
Given that nothing can cross the event horizon because the mass basically freeze in time, halts and thus gradually cumulate layer by layer. At an infinitesimal level we can calculate the smallest event horizon with the mass of a proton:
rb = 2GM/c2 (7)
M = 1.674×10-25 kg (mass of a proton)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)
The radius of a normal proton is:
rb = 2.483×10-54 m (9)
This means the compression of protons inside a black hole will combine them. When the black hole runs out of gravitons to hold it together its explosion will therefore be much more violent than a simple atomic reaction given this subatomic instability.
Black holes therefore consist of very unstable subatomic particles. Black holes also have a lifetime.
2.3. Time Dilation Reengineered
We can represent time dilation using simpler techniques by interpolating dilation. Indeed if we rationalize the kinetic energy gained by the object in motion according to the maximum one it can experience at the speed of light then:
pv = 1/2 mv2 / 1/2 mc2 (10)
Since the time dilation percentage is the exact opposite of the speed ratio then:
pt = 1 - pv (11)
We consequently define general time dilation in direct relation to the proportion as follows:
to = tf / [1 - v2/c2] (12)
We see clearly the differences upon comparing the derived graph:
[FIGURE 10 OMITTED]
2.4. Gravitational Time Contraction
Since in the candidate theory the acceleration is defined by gravitons pulling the body in the opposite direction of their velocity, the net effect of the gravitational acceleration already defines the flux. Unlike kinetic time dilation this is not an incident event but the residuum of the modus operandi by the acceleration vector magnitude.
In contrast to kinetic time dilation, gravitational time contraction will be used interdependently with the non-trivial ambient gravity field of the observer, or fractionalized.
2.4.1. Inverse Square Law—Method 1
Given that FT gravitational time dilation and the Newtonian gravity force are similar, the standard model of gravity inside a sphere cannot be directly linked with FT because factors applied in one direction will not cancel their equivalent in the opposite direction. This means no simplification can be made and all infinitesimal elements of the mass will affect the net amplitude at one particular location.
First, we can represent the respective factor with a triple integral in the following way, using spherical coordinates:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (13)
The square of the distance between the observer located at (x2, y2, z2) and the infinitesimal element being integrated is equivalent to:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (14)
By mapping the location of the observer from the center of the sphere on to an arbitrary axis with the same radius, we can simplify our denominator. In this case (0, 0, d2) will be used to map a radius into Cartesian coordinates:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (15)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (16)
r1 is the spherical mass radius
d2 is the distance of the observer from the center
For example, the respective factor of an observer at position x inside a sphere of radius 20 m will be proportional to:
[FIGURE 11 OMITTED]
2.4.2. Inverse Square Law—Method 2
Different means of calculating the inner gravitational time dilation factor with no relation with the aforementioned procedure can also be used. It consists of calculating the intersection between a growing sphere held within the spherical body in question.
This is done by first calculating all sphere surfaces fitting inside the largest sphere not in intersection with the spherical body. This represents the following area:
[FIGURE 12 OMITTED]
Now for the second part the spherical cap surface area resulting from the intersection of the two spheres will have to be considered only. This will cover the next section:
[FIGURE 13 OMITTED]
By summing both areas we will have:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (17)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (18)
r1 is the spherical mass radius
d2 is the distance of the observer from the center
For instance, the same factor of an observer at position x inside a spherical body radius of 20 m will be corresponding to:
[FIGURE 14 OMITTED]
This results in exactly the same inner curve as the one found by FIGURE 11. Henceforth this confirms the validity of the equation.
As a means to compare previous formulation with the more common Newtonian gravitational acceleration, we will before all else find the corresponding mass with the volume of a sphere with the respective radius at a given mass density ρ. Hence:
m = 4]πr3 ρ / 3 (19)
The general acceleration formula contains a discrepancy constant we will call B:
a = Bm / r2 (20)
By solving the equation using the lowest factor found with Equation (18), the constant B turns out to be 3/2 for a given mass density ρ. By placing side by side both plots we have:
[FIGURE 15 OMITTED]
Here, the outer and inner acceleration factors can be converted by multiplying the mass squared to lay hold of the FT gravitational time dilation factor.
2.4.4. Inside a Sphere
At this point the equivalence of method 1 and 2 is made and the latter can be preferred given its much greater flexibility for more complex calculations. Thus by putting Equation (18) into FT's context we will have to reduce the degree of the inverse radius down to 1:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (21)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (22)
r1 is the spherical mass radius
d2 is the distance of the clock from the center
Or more generically for a clock at a specific position inside one spherical mass, as seen from an observer positioned in a null environment:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (23)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (24)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (25)
r is the location of the clock
rs is the radius of the spherical mass
m is the mass of the sphere
For example, given a sphere of null density and radius of 30 meters then:
[FIGURE 16 OMITTED]
2.4.5. Outside a Sphere
We can now estimate the amplitude of the gravitational potential by sampling anchored bodies at an infinitesimal position by consequently rationalizing the measurement with the amplitude derived from the location of the observer.
Since an inertial body being subject to a specific gravitational force is responsible for gravitational time dilation and that gravity is a superposable force, we will translate the same conditions of all gravitational potentials into the sum of all surrounding fields of an observed clock and the observer:
to = Φ(r)/Φ(ro) × tf (26)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (27)
r is the location of the observed clock
ri is the location of the center of mass i
ro is the location of the observer (typically 0)
mi is the mass i
to is the observed time of two events from the clock
tf is the coordinate time between two events relative to the clock
By juxtaposing the same spherical mass with an internal and external gravitational time dilation factor we have the following, for a spherical mass of 20 meters in radius:
[FIGURE 17 OMITTED]
2.5. Universe of 1 Galaxy
We will now study the effects of gravitational time contraction over a bullet thrown away from the edge of a stationary galaxy. Furthermore the galaxy will be solitary in our fictional Universe. Let's consider:
[FIGURE 18 OMITTED]
m is the mass of the galaxy (1.1535736×1042 kg)
r is the radius of the galaxy (4.7305×1020 m)
v is the observed speed from the galaxy's edge
x is the variable distance between the two bodies
Excerpted from Finite Theory of the Universe, Dark Matter Disproof and Faster-Than-Light Speed by Phil Bouchard Copyright © 2012 by Phil Bouchard . Excerpted by permission of AuthorHouse. All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.
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