In 2009, I gritted my teeth and finished "Foundations Of Parallel Universe Math" to all the logical conclusions I knew it lead to but were afraid to actually put down in words. My main problem with the conclusions, is that you can find an age of the universe from it, but this age is outrageously large. The age of the universe is now in it. Hopefully all the covering explanations I give for that large age will make you wonder if I am right. To tell you the truth; I still don't believe it, and I came up with the explanations. This new version now has diagrams that the previous version didn't have because of where I posted it.
Foundations Of Parallel Universe Math (Update 2009)
by Jim Akerlund
Abstract
A simple mathematical relationship is developed between an event and the Big Bang. This relationship is then compared with known rules of universe expansion rates to extrapolate parallel universes. A curvature of spacetime based on the events relationship to the Big Bang is shown, and an explanation of the "matter in the universe seems older then the universe it resides in" is presented, also the curvature of spacetime that is derived suggests an "arrow of time". The curvature of spacetime is then shown to require an "advanced" light particle to complete the transaction. Finally, a connection is developed between math in this universe, and math in the Parallel universes.
The Geodesic Of Light
What, you may ask, is a simple relationship between an event and the Big Bang? It is the thought that there are only three things that occurred in the universe. The Big Bang, the event(it doesn't matter when or what, as long as it occurred after the Big Bang), and one photon of light traveling between the two(from the Big Bang to the event). We now ask the question, how long is the length of the path that the photon traveled(called a Geodesic)? The answer is, the age of the universe at the time of the event. But if we also ask, how far has the event expanded, when the length of the geodesic is questioned. We get an answer that will suggest a shape to the universe. If we let A be the length of the geodesic, and D be the distance the event has expanded, and we set these equal to each other, then we will get
D = A.
This equation shows a very flat universe, with the expansion rate being equal to the speed of light. Our universe is not like this, so that means that the light that traveled from the Big Bang to an event had to get curved in some way. This curvature could be represented in this equation
DZ = A. (Eq. 1)
In Eq. (1), Z is some number greater then 1. In the case where 1>Z>0, this would describe a universe where the expansion rate exceeds the speed of light. For the case where Z ≤ 0, we do not know what to make of it. For Z ≤ 0, these numbers would describe expansion rates faster then infinite.
Here is what we did. Draw two dots. Label one dot the Event, and the other dot the Big Bang. Draw a half circle connecting the two dots. Diagram # 1 is an example. The half circle represents the geodesic of light traveling from the Big Bang to the Event. We started out with the half circle, because it was the easiest curve we could think of. We were expecting to graduate to other curves once we understood what was happening with the simplest. Draw a straight line connecting the two dots. The straight line represents the distance the universe has expanded when the event occurred, it is an imaginary distance(not to be confused with imaginary numbers), and is not something that can be measured physically.
If we were to extend the curve from Big Bang to the Event, so that it meets Big Bang again, then we would have a circle. The completed circle shows a special relationship between the Event and the Big Bang, the Big Bang is at the antipodal from the Event. An antipodal is the point opposite another point 
on a circle, sphere, or hypersphere, an example on the earth is the antipodal of the North Pole, is the South Pole. We could also draw other events, all at different distances and different angles from the Big Bang, with there corresponding circles. All of these events would also be at their own antipodal from the Big Bang
The areas of this drawing can be labeled, and show a time relationship between other events in the universe and the Event, as shown in diagram # 2. An event that happens inside the curve from the Event to Big Bang, is an event that occurred in the past for the Event. An event that happens along the curve between the Event and Big Bang, is a simultaneous event as the Event according to Einstein's definition of simultaneous[14]. An event that happens outside the curve, will be in the Events future. Because this is drawn on paper, there is an area on the paper where if events happen there, they can not be in the Event's future, past, or present (It is the area defined as; everything on the opposite side of Big Bang from the Event, including a line passing through the Big Bang that is perpendicular to the line from the Event to Big Bang.). The author does not know if this type of area also exists in reality.
If we were to rotate the circle through the third dimension, then we would get a sphere, with the Event at the antipodal from the Big Bang. If we were to then rotate the sphere through the fourth dimension, then we would get a hypersphere, with, once again, the Event at the antipodal from the Big Bang. And that is the complete picture of the relationship between an Event and the Big Bang. We are rotating the geodesic through these dimensions to show that the geodesic can be rotated. The results from Eq. (1) will not change when rotated into other dimensions. The shape of the geodesic doesn't have to be a smooth curve either. It can be any convoluted shape, just as long as the difference between A and D remain the same. But, we are then faced with the question of why the convoluted shape when a smooth curve satisfies the same equation? As a general rule, if the universe can be perceived to be doing something a complex way, or a simple way, the universe will choose the simple way. But this in no way eliminates the convoluted shape.
This is not the first time the universe has been diagrammed in this fashion. Robert Osserman in his book "Poetry of the Universe: a mathematical exploration of the cosmos"[1] describes essentially the same thing. He even gives a name to the curved universe he describes, he calls it the "retroverse", we will also use the same name. Mr. Osserman arrives at the retroverse from a different perspective, and does not derive an equation from his model, nor does he label the parts of the model other then the event and the Big Bang. This model of the retroverse will be different then what is explained in Mr. Osserman's book.
The Hubble model of Universe expansion versus the retroverse model.
The equation for the circumference of a circle is: (diameter) x (π) = (circumference). Our diagram is half of a circle, so the equation becomes: (diameter) x (π)/2 = (circumference)/2. In the diagram, (circumference)/2, is distance between the Event and Big Bang as measured along the curve A, and diameter is the distance between the Event and Big Bang as measured along the straight line D. Substituting A and D for circumference/2 and diameter, we get this equation
D*(π)/2 = A. (Eq. 2)
Since this model uses light as the measuring unit, we can set the speed of light to equal 1. That is the same as saying, the speed of light is unity. So if light were to travel for one light year, we would get two pieces of information from that; the distance it has traveled, and the time it traveled in. We get the same type of information when light has traveled one light second. When that is applied to this model, "A" becomes two different values at the same time; a distance, and a time. With that in mind, we can proceed.
The equation for the rate of expansion of an event from the Big Bang, in this model is: (distance the universe has expanded at the time of the event) / (age of the universe at the time of the event) = (rate of expansion). In the diagram, the age of the universe, is A, and the distance the universe has expanded, is D. We will set R to be the, rate of expansion, and we get this equation
D/A = R. (Eq. 3)
Substituting the left hand side of Eq. (2) into Eq. (3), we get this equation
D/D*π/2 = R. (Eq. 4)
Eq. (4) reduces to this equation
2/π = R. (Eq. 5)
According to this model, every event that has and will occur in this universe, since the Big Bang, is expanding at .6366197724... lightyears from the Big Bang. When the speed of light is not unity, Eq. (5) becomes
2*C/π = R,
where C is the speed of light.
Edwin P. Hubble showed by observation that the velocity of recession is proportional to the distance of a galaxy [2]. In other words, galaxies at different distances have different recession rates from us. This seems to be at odds with the retroverse model above. Well, actually they are not at odds with each other. The Hubble model determines "event to event" expansion rates. The Retroverse model determines "event to Big Bang" expansion rates. If we were to turn the Hubble model into an "event to Big Bang" type of model, the model would produce the same type of results as the retroverse model. Mainly, all events are expanding at one rate from the Big Bang.
Here is the Hubble model of universe expansion when the Big Bang is the other event to be measured. We shall use “Us”, X (for a series of events) and Q for expansion rates. It is observed from “Us” that an event X1 is moving away from “Us” at some rate Q1. X2 is observed to be further away then X1, and it's rate of moving away from "Us" that is proportional to the distance. There is some X at the Big Bang where it's moving away from "Us" is the absolute limit. Meaning, there is no event further back in time then the Big Bang, so nothing can expand faster then an event at the Big Bang. This puts an upper limit on the expansion rate one event can expand from another event. What that upper limit is, the Hubble model does not say.
This is what can be concluded with both the Hubble model of universe expansion, and the retroverse model of universe expansion. The Hubble model does not contradict the retroverse model. The retroverse model reveals some limitations of the Hubble model. The Hubble model can not confirm nor deny that the value of Z from Eq. (2) is equal to π/2. The Hubble model and the retroverse model are two different perspectives of an expanding universe from a single Big Bang. The Hubble model is based on "event to event" expansion rates, and the retroverse model is based on "event to Big Bang" expansion rates.
The Wow Section
If every event that has and will occur in this universe is expanding at .6366197724... lightyears, then what about the other expansion rates? We are faced with two solutions here. The first one, is that the Big Bang produced exactly one expansion rate, and all other expansion rates are not possible. The second solution is, the Big Bang produced many expansion rates, and each separate expansion rate defines it's own separate and complete universe.
If the Big Bang created one expansion rate, then there has to be a physical reason why this occurred, or an explanation has to be available why the other expansion rates are not possible. We do not believe an Anthropic principle will suffice here [3]. For these reasons, this paper will not explore the exactly one expansion rate solution. This paper will explore the separate expansion rates that define separate universes, or parallel universes solution. This solution is based off of an idea originally proposed by Hugh Everett III [4].
The equation that gave us the expansion rate is,
2/π = R. (Eq. 5)
If we change the expansion rate R, then some variable on the left hand side of the equation will also have to change. the only variable available to us is π. This immediately implies that each separate universe has it's own different value of π. The values of π approaching infinite as the expansion rate approaches 0, from Eq. (5), and π approaching 0 as the expansion rate approaches infinite.
A few things about π before we go on. π is the ratio between a circles circumference and it's diameter. If we take the circumference of a circle and divide it by the diameter, we will get π. π's decimal expansion is infinite in length, and the numbers do not repeat. π can be calculated both mathematically and physically. One math equation to calculate the value of π is this
π/4 = 1 - 1/3 + 1/5 - 1/7 + ... (Eq. 6)
This equation is called Gregory-leibnitz' series. There are many other equations to calculate π. π can be calculated by dropping sticks on Parallel lines, first observed in the eighteenth century by Count Buffon [5]. In 1995, it was discovered that any binary number in the binary expansion of π can be extracted without knowledge of the preceding numbers in the expansion [6].
Different values of π are not new to math. In both the Geometries of Riemann(elliptic) and Lobachevsky(hyperbolic), different values of pπ can be derived. But there is a catch, in both of the Geometries, the smaller the space that is measured, the closer to “π” the different values of π get. This is the equation for the circumference of a circle in Hyperbolic Geometry
C = 2*π*sinh*r. (Eq. 7)
C is the circumference of the circle, r is the radius of the circle, and sinh is measure of the hyperbolic surface. As r approaches 0, the difference between C and r gets closer to 2*π .
This is the equation for the circumference of a circle in Double Elliptic Geometry, sin(a/r) is a measure of the elliptic surface,
C = 2*π*r*sin(a/r). (Eq. 8)
Once again, as r approaches 0, the difference between C and r gets closer to 2*π . Equations (7) and (8), can be found in the book, "An Introduction To Non-Euclidean Geometry" [7].
We will use π-bar to set the values of π in different universes. This will eliminate the confusion between π in this universe and π in another universe. When talking about a specific universe, this notation will show up, π-bar = 10, this would indicate that wherever the variable π-bar shows up, the value of π in that universe is 10. We can also set π-bar = π, and that would describe situations in this universe.
For reasons that will be explained later, we can not use 2*r in the circle equation for parallel universes, we will use d for the diameter, instead. This is the equation for the circumference of a circle in a parallel universe
C = d*π-bar.
As d approaches 0, the difference between d and C does not change, so the changes in π, in a parallel universe, is not a change to Non-Euclidean Geometry. To the best of my knowledge, no other math we use to describe spacetime, suggests different values of π. We are 95% sure of this about math that was produced before the 20th century, and 51% sure of this, for the 20th century itself.
We will now sum up what has just been presented. The Big Bang "exploded" with many different rates of expansion, and each separate expansion rate is it's own universe with it's own different value of π. Humankind has created a math that has different π's, but that math is not the same as what is presented here, and as far as this author knows, no other math uses different values of π to describe spacetime.
The Curvature of Spacetime
The equation for the curvature of a circle is
k = 1/r. (Eq. 9)
Where k is the curvature of the circle, and r is the circles radius. Knowing that the retroverse is a half circle, we can substitute D/2 for r in Eq. (9), and we get this
k = 2/D.
This is an equation for the curvature of the geodesic when we assume a non-convoluted shape to the geodesic. All photons received at an event will have this same curvature, so this is also the curvature of spacetime for that event. We can also set k equal to other variables from Eq. (2)
k = 2/D = π/A. (Eq. 10)
This shows an important thing about the curvature of spactime, remembering that A is the age of the universe, the curvature of spacetime k is getting smaller as the universe gets older, and an event determines the curvature of spacetime. Eq. (10) also gives a reason why events are not time reversible, the so called, arrow of time. In order for a series of events to reverse process, the universe would also have to reverse it's expansion to get the curvature of spacetime k to get larger, so that the particles can return to their original paths.
If the event is an observer, then the observer is faced with a time illusion about the universe. The observer will observe an event in the past from her curvature, and will falsely assume that the event in the past is at her same curvature. The observer will then falsely give an age after the Big Bang when the event occurred, when in fact, the event in the past has it's own different curvature, which will determine a different age after the Big Bang. The Hubble Space Telescope seems to be reaching the distances where this effect is most noticeable. It is the, "matter in the universe seems to be older then the universe it resides in" problem [8].
Now all you need to know is the age of the universe for you, A, and the distance to the event, Y. Both of these numbers need to be recorded in the same units. Here are four equations to determine the actual age of the event, versus the perceived age of the event. The perceived age of the event after the Big Bang is A – Y. The actual age of the event is,
A/π = F, (Eq. 11)
90-90(Y/A) = Θ, (Eq. 12)
FsinΘ = B, (Eq. 13)
B*π = Age of universe for event Y. (Eq. 14).
Here is a diagram that gives the reason for the equations 11 thru 14.
This time illusion is a feature of curved spacetime. The difference between the observed time after the Big Bang, of the event, and actual time after the Big Bang, of the event is a measure of the curvature of spacetime. The time illusion discrepancy is governed by the value of Z from Eq. (1). When Z = 1, there will be no time illusion. The four equations 11 thru 14 are for Z = π/2 only.
Maxwell's advanced light and the Wheeler-Feynman absorber theory
When James Clerk Maxwell produced his wave equation for light, it had two solutions; the "retarded solution" for light that travels forward in time, and the "advanced solution" for light that travels backward in time; from the book by Nick Herbert,"Faster Then Light: Superluminal Loopholes In Physics" [9]. The "advanced solution" will be the one we are talking about when we say advanced light.
For an observer(receiver), all light that arrives to her, arrives with the curvature k = π/A (Eq. (10). What about light that is emitted by the observer, what is the curvature of the emitted light? This model is past based, so the only way to determine the curvature of an emitted light, is to look at the receiver of the light, and the receiver always receives her light with the curvature k = π/A where the value of A is based on the when the receiving event occurred, relative to the Big Bang. Relative to the emitter, the emitted light has the curvature of
k = π/(A + t),
where t is the time between the emitter and the receiver. Some how the emitter has to "know" what curvature to emit the light for it to reach the receiver. But that is only for the photons that reach that event, there are other photons emitted by the same emitter, that will have a curvature of k = π/(A + ?), where ? is the time between the emitter and any other future receiver. We are presented with two possible solutions here; a "non-local" model, or Maxwell's advanced light/retarded light. The author does not believe in "non-local" models, suggesting some magical transference of information, so that leaves Maxwell's solution as the only solution that fits, where "advanced light" being emitted by the receiver travels backward in time and "tell" the emitter what curvature to emit at. The curvature for the advanced light is
k = -π/A,
relative to the receiver.
This transaction is very similar to an advanced light and retarded light transaction that was originally proposed in the paper [10]. The only thing this paper adds to the transaction, is the curvature of the geodesic.
Mr. Herbert, in his book "Faster Then Light: Superluminal Loopholes In Physics", also mentions two other Absorber theories using advanced and retarded light [11] and [12].
The advanced light solution of how a photon "knows" what curvature to follow also might lead to a solution to a question raised from Eq. (1). When this model was first conceived, we drew a half circle, because it was "the easiest curve I could think of",and after we understood what was happening with the easiest, we could advance on to other values of Z. This is a conjecture, but advanced and retarded light geodesics create a closed "circuit", it is believed that there are only two values of Z(Z = 1 or π/2) that allow this closed "circuit" to be completed geometrically, for all events since the Big Bang. It is also believed that this can be proved mathematically, but the author does not know how to go about it. If this can be proved, then the Z = π/2 is the actual shape of the universe we reside in; because, due to the Time Illusion mentioned earlier, we know that we do not live in Z = 1.
Parallel Universe Math
We stated earlier that each separate universe has it's own separate value of π. In this universe, the number system and π are intimately connected, and that is most vividly shown in Eq. (6).
π/4 = 1 - 1/3 + 1/5 - 1/7 + ... (Eq. 6)
This universe does not have a special mathematical distinction over any of the Parallel universes, so the number system in the Parallel universes must change in some way so that an Eq. (6) in the Parallel universe will produce the given value of π in that universe. This equation extends the leibnitz' series to Parallel universes
U*π/4 = U/1 - U/3 + U/5 - ...
The U*π is the value of π in the Parallel universe, and the U is the unit value in the parallel universe relative to this universe. This is the reason why we could not use 2*r in the equation C = d*p-bar, the equation for a circle in a parallel universe; the "value", "quantity" of all numbers is specific to this universe only.
The dividing line that we seem to be looking at when we consider as to weather the math in the Parallel Universe is different from this universe, is if the math is somehow derived or connected to π. The Axioms of Euclid do not give a specific value of π, and are not dependent on what the value of π is. That suggests that the Axioms of Euclid are valid in the Parallel Universes along with the field of Non-Euclidean Geometry with very little alteration required. The author is not sure about other Fields in Math.
I close this Paper with a Poem I wrote.
You're Nuts
by
Jim Akerlund
You're not playing with a full deck.
Your elevator doesn't go all the way to the top.
You're a few pancakes short of a stack.
Your antenna isn't receiving all stations.
You're out of your tree.
Your cart isn't rolling on all wheels.
You're one brain short of a brainstorm.
Your train isn't pulling as much weight anymore.
You don't have a clue.
You're a few pieces short of a puzzle.
You're a few degrees short of a summer day.
If sanity were a holiday, yours would be April first.
You're a few notes short of a song.
You're a few gallons short of an ocean.
It seems to me your toilet doesn't flush anymore.
You're a few birds short of a flock.
You're a few cars short of a traffic jam.
You've come to a gun fight with a knife.
Your legs don't reach all the way to the ground.
But all of this is a matter of opinion,
by a guy who thinks he knows, where Einstein went wrong.
In the book "Relativity, The Special And The General Theory" by Albert Einstein [13], Mr. Einstein describes a rotating disc and how it is effected by relativity. He says that a measuring rod used to measure the the circumference of the rotating disc will be shortened by the rotation of the disc, and this will effect the total value of π on the disc, arriving at a value of π larger then 3.1415... If this model is correct, then that is wrong. This model is based on the idea that there is exactly one value of π in this universe no matter how relativity may effect a rotating disc. The invariance of π in the frame of reference.
This was monkey # e * 10^googool typing paper # π * 10^googool.
Update 2009
In the past the author hoped that you, the readers of this paper, wouldn't connect the dots and draw out the eventual conclusions that this paper points too. The author now believes that that is an ignorance expected of the readers that in fact doesn't exist. So, with that in mind the author is forced to show you the conclusions anyway. The reason why the author didn't want to show you these conclusions, is because the author himself doesn't really believe them. With that in mind, the author now presents what he doesn't want to show you.
The equation k = π/A for the curvature of spacetime is also an energy applied to each photon. But from quantum mechanics all the energy of a photon is found in this equation [15]
E = hν,
where E = energy of the photon, h = Planck's constant, and ν = frequency of the photon. The model presented above says that the energy of π/A also needs to be added to E the energy of a photon, but in the entire history of quantum mechanics there hasn't been a need to change the equation for the energy of a photon is any way. This leads to the conclusion that one of the factors in the energy equation of a photon is in fact a variable that is getting smaller with time π/A. The only variable in the energy equation of a photon that seems to be a candidate is h. This leads to the following equation
E = ν*π/A. (Eq. 15)
This equation immediately implies the Planck's constant is in fact a variable.
From Eq. 15 we can directly determine the age of the universe by
A = π/h, (Eq. 16)
but this equation has several problems that the author has identified, there may be more. The first problem is a dimension problem; h has the dimension of action, A has the dimension of time and π has no dimension. But the above paper suggests that π is actually a fifth dimension. The other four being three of space and one of time. What type of dimension π is is not determined in the above paper, Eq. 16 suggests that π has the dimensions of action*time. A second problem is that Eq. (16) suggests that the value of h at the Big Bang was h = +∞. h having such a large value leads to a very different universe then the one we currently live in. The reason we haven't found changes in h is because h has changed very little in the hundred years we have known its value and all radiation we get from space has our current value of h no matter when it was emitted because of the advanced/retarded transaction outlined above. The third problem is the most difficult to deal with. Roughly it gives the age of the universe to be about 3 * 10^8 times greater then our current estimates for the age of the universe, an age of around 501 quadrillion years old, or 501 * 10^15. It is this huge age discrepancy that causes the author to not believe in this line of reasoning, but he can not get around it. It is a well known phenomon of General Relatitivity that a “clock” in a gravity well will proceed slower then a “clock” not in a gravity well [20], this is called gravitational time dilation. So when we measure the age of the universe, the question needs to be asked, what is the strength of the gravity well that the “clock” is in when the age of the universe is determined. Due to the large length of time that the universe has existed, any gravity well that our measuring “clock” is in, will shorten the length of the age of the universe. This is not saying that our current estimate of 1.3 * 10^11 for the age of the universe is wrong, it is just a “clock” that is sitting in a gravity well. Eq. (16) gives an age of the universe where the “clock” isn't sitting in a gravity well.
There is one aspect of equation (16) that presents a very unusual understanding in our knowledge of the universe. Equation (16) creates the tie in between Relativity and quantum mechanics. The mass of the universe combined with its age creates the quantum mechanical effects we observe in our spacetime. Every time you see h in an equation, the equation is asking what universe you are performing the equation (π) and when in that universe you are performing the equation (A).
On a side note. Two of the reasons for the instigation of the Theory of Inflation [16] (The theory that says the universe went through a period of rapid inflation before it settled down to the expansion we now observe) are explained above. The universe appears to be spatially flat is explained by equation (10) and is getting flatter. The data from COBE [18] and WMAP [19] that indicate a very similar temperature of the universe out to it's horizon can be explained by a large value of h, which would suggest if you will the Big Bang as being one big atom until h got smaller. Instead of the universe being causally connected at the Big Bang it was quantum mechanically connected by a large value of h. Which would evolve over time, as h got smaller, to give the impression that the universe was the same temperature beyond a causally connected horizon. We won't say the the universe was a Bose-Einstein [17] condensate at the Big Bang, because Bose-Einstein condensates only occur at cold temperatures, but with larger values of h the temperature of a Bose-Einstein condensate will occur at higher temperatures.
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