My 1997 paper "Foundations Of Parallel Universe Math"
I have written several papers about Parallel Universes; one in 1997, one 2001, and one I am currently in the process of writing for 2006. The paper I wrote in 2001, I posted to my own webpage which now onlonger exists. I'm guessing also that nobody ever saw it. The paper I wrote in 1997 I posted to the newsgroup Alt.Sci.Physics.New-Theories several times in the July/August time frame of 1997. I do know that some people actually read it there, because two people sent me emails concerning it. The emails were questions concerning early ideas in the paper and not part of the meat and potatos of the paper. This year (2006) I ran across a conversation in a different NG that seems to be related to one of the meat and potatos ideas of the paper, but I have no idea as to whether the conversation was based on my 1997 paper. Anyway I tell you all of this in preperation of the 1997 paper itself. Here it is for your enjoyment.
Foundations Of Parallel Universe Math
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 occured in the universe. The Big Bang, the event(it doesn't matter when or what, as long as it occured 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 universe 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 universe has expanded, and we set these equal to each other, then we will get
D = A. (1)
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. (2)
In Eq. (2), 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 < or = 0, I do not know what to make of it. For Z < or = 0, these numbers would describe expansion rates faster then infinite.
I stumbled across these equations because I came to the conclusion that the universe on the whole was curved.
Here is what I did. Draw two dots. Label one dot "Us" (the event), and the other dot "BB" (the Big Bang). Draw a half circle connecting the two. The half circle represents the geodesic of light traveling from the Big Bang to the event. I started out with the half circle, because it was the easiest curve I could think of. I was expecting to graduate to other curves once I understood what was happening with the simplist. Draw a straight line connecting the two dots. The straight line represents the distance the universe has expanded when the event occured, 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 "BB" to "Us", so that it meets "BB" 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, this can also be extended to a sphere, 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 "Us" event. An event that happens inside the curve from "Us" to "BB", is an event that occured in the past for "Us". An event that happens along the curve between "Us" and "BB", is a simultaneous event as the "Us" event. An event that happens outside the curve, will be in "Us"'s future. Because this is drawn on paper, there is an area on the paper where if events happen there, they can not be in "Us"'s future, past, or present (It is the area defined as: everything on the opposite side of "BB" from "Us", including a line passing through "BB" that is perpendicular to the line from "Us" to "BB".). 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 "Us" 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 "Us" 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. (2) 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 precieved 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 diagramed in this fashion. Robert Osserman in his book "Poetry of the Universe: a mathematical exploration of the cosmos". (1995). Anchor Books., pages 114-120, 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 prespective, 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 (Pi) = (circumference). Our diagram is half of a circle, so the equation becomes: (diameter) x (Pi)/2 = (circumference)/2. In the diagram, (circumference)/2, is distance between "Us" and "BB" as measured along the curve, and diameter, is the distance between "Us" and "BB" as measured along the straight line. Substituting A and D for circumference/2 and diameter, we get this equation
D*(Pi)/2 = A. (3)
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. (4)
Substituting Eq. (3) into Eq. (4), we get this equation
D/D*Pi/2 = R. (5)
Eq. (5) reduces to this equation
2/Pi = R. (6)
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. (6) becomes
2*C/Pi = R (6.1)
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("The Expansion Rate And Size Of The Universe". Wendy L. Freedman. (Nov. 92). Sci. Am.). In other words, galaxies at different distances have different recession rates. This seems to be at odds with the retroverse model. 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" and "BB" again and this time a third set of events, "X". It is observed from "Us" that an event "X1" is moving away form "Us" some rate Q1. "X2" is observed to be further away then "X1", and it's rate of moving away from "Us" 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.
The retroverse model, gives an expansion rate from the Big Bang that is time invariant. Is the upper limit, from the Hubble model, time invariant? Meaning, is the upper limit, the same value at one minute after the Big Bang as a trillion years after the Big Bang? The Hubble model is not designed to answer that question. We shall now see that the Hubble model is very time specific. One of the things the Hubble model comes up with, is the Hubble constant. It is a measure of the recession velocity of a galaxy divided by it's distance. It is measured in kilometers per second per megaparsec. The value of the Hubble constant is not part of the scope of this paper, so we will set the value to Q per second per megaparsec. We continue with this question; for an observer, when she was exactly one megaparsec from the Big Bang, what was the value of the Hubble constant then? If the observer says Q per second per megaparsec, then the universe is speeding up it's expansion rate over time. This is not what we expect of the expanding universe. The other solution is, the Hubble constant is not constant, and is dependent upon when it is measured, or time dependent.
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 Pi/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 exploded at exactly one expansion rate, and all other expansion rates are not possible. The second solution is, the Big Bang exploded with many expansion rates, and each seperate expansion rate defines it's own seperate and complete universe.
If the Big Bang created one expansion rate, then there has to be a physical reason why this occured, or an explanation has to be available why the other expansion rates are not possible. An Anthropic principle will not suffice here("The Anthropic Cosmological Principle", Barrow, J.D. and Tipler, F.J. (1986). Oxford University Press.). For these reasons, this paper will not explore the exacly one expansion rate solution. This paper will explore the seperate expansion rates that define seperate universes, or parallel universes solution. This solution is based off of an idea originally proposed by Hugh Everett III, in his paper ("Relative State" Formulation Of Quantum Mechanics In Quantum Theory And Measurement (1957) Rev. Of Mod. Phys. 29, 454-462).
The equation that gave us the expansion rate is,
2/Pi = R. (6)
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 Pi. This immediately implies that each seperate universe has it's own different value of Pi. The values of Pi approaching infinite as the expansion rate approaches 0, from Eq. (6), and Pi approaching 0 as the expansion rate approaches infinite.
A few things about Pi before we go on. Pi 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 Pi. Pi's expansion is infinite in length, and the numbers do not repeat. Pi is a transendental number. Pi can be calculated both mathematically and physically. A math equation to calculate the value of Pi is this
Pi/4 = 1 - 1/3 + 1/5 - 1/7 + ... (7)
This equation is called Liebniz' series. There are many other equations to calculate Pi. Pi can be calculated by droping sticks on Parallel lines, first observed in the eighteenth century by Count Buffon("One Two Three...Infinity" by George Gamov, Bantom books 1965). In 1995, it was discovered that any number in the expansion of Pi can be extracted without knowledge of the perceeding numbers in the expansion (On The Rapid Computation Of Various Polylogrithmic Constants by Bailey, Bowein, and Plouffe, http://www.mathsoft.com/asolve/plouffe/plouffe.html).
Different values of Pi are not new to math. In both the Geometeries of Riemann(eliptic) and Lobachevsky(hyperbolic), different values of Pi can be derived. But there is a catch, in both of the Geometeries, the smaller the space that is measured, the closer to "Pi" the different values of Pi get. This is the equation for the circumference of a circle in Hyperbolic Geometery
C = 2*Pi*sinh*r. (8)
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*Pi.
This is the equation for the circumference of a circle in Double Eliptic Geometery, sin(a/r) is a measure of the eliptic surface,
C = 2*Pi*r*sin(a/r). (9)
Once again, as r approaches 0, the difference between C and r gets closer to 2*Pi. Equations (8) and (9), were taken from ("An Introduction To Non-Euclidian Geometery". David Gans. (1973). Academic Press.).
We will use Pi-bar to set the values of Pi in different universes. This will eliminate the confusion between Pi in this universe and Pi in another universe. When talking about a specific universe, this notation will show up, Pi-bar = 10, this would indicate that wherever the variable Pi-bar shows up, the value of Pi in that universe is 10. We can also set Pi-bar = Pi, 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*Pi-bar. (10)
As d approaches 0, the difference between d and C does not change, so the changes in Pi, in a parallel universe, is not a change to Non-Euclidian Geometery. To the best of my knowledge, no other math we use to describe spacetime, suggests different values of Pi. I am 95% sure of this about math that was produced before the 20th century, and I am 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 seperate expansion rate is it's own universe with it's own different value of Pi. Humankind has created a math that has different Pi'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 Pi to describe spacetime.
The Curvature of Spacetime
The equation for the curvature of a circle is
k = 1/r. (11)
Where k is the curvature of the circle, and r is the circles radius. Knowing that the retroverse is a half circle, we can apply Eq. (11) to this model, and we get this
k = 2/D. (12)
This is an equation for the curvature of the geodesic when we assume a non-convoluted shape to the geodesic. The geodesic is a part of spacetime, so we will say the curvature of spacetime. We can also set k equal to other variables from Eq. (3)
k = 2/D = Pi/A. (13)
This shows an important thing about the curvature of spactime, remembering that A is the age of the universe, the curvature of spacetime is getting smaller as the universe gets older, and an event determines the curvature of spacetime. Eq. (13) 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 to get larger, so that the geodesics 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 occured, 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 noticable. It is the, "matter in the universe seems to be older then the universe it resides in" problem ("Hubble Space Telescope measures precise distance to the most remote Galaxy yet". Press release No. STScI-PR94-49. (10/26/94). http://oposite.stsci.edu/pubinfo/press-release/94-49.txt).
Here are three equations to determine the actual age of the event, versus the preceived age of the event. A is the age of the universe for the observer. B is the preceived age of the universe, relative to the Big Bang, of the observed event. F is the actual age of the event, relative to the Big Bang. D = 2*A/Pi. This is the equation when B is older then A/2, but younger then A,
F = Pi((90(2B-1)sin*D+D)^2+(90(2B-1)cos*D)^2)^-1/2
_ _ _ _ _ _
2 A 2 2 A 2. (14)
This is the equation, when B is exactly A/2
F = Pi/2*((D/2)^2+(D/2)^2)^-1/2. (15)
This is the equation, when B is younger then A/2
F = Pi/2((D-90(2B)cos(D))^2+(90(2B)sin(D))^2)^-1/2
_ _ _ _ _
2 A 2 A 2. (16)
Equations (14, 15, & 16) are corrected in a post on this blog called Corrections to Equations 14, 15, and 16 for My 1997 paper "Foundations Of Parallel Universe Math".
This time illusion is a feature of curved spacetime. The difference between observed time of the event, and actual time of the event is a measure of the curvature of spacetime. The time illusion discrepancey is governed by the value of Z from Eq. (2). When Z = 1, there will be no time illusion. The three equations (14), (15), and (16) are for Z = Pi/2 only.
The author suspects that there are shorter equations for (14), (15), and (16), but he is unable to derive them.
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 ("Faster Than Light: Superluminal Loopholes In Physics". Nick Herbert. (1988). Plume.). 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 = Pi/A (Eq. (13)). 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 = Pi/A where the value of A is based on the when the receiving event occured, relative to the Big Bang. Relative to the emitter, the emitted light has the curvature of
k = Pi/(A + t), (17)
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 = Pi/(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 transferance 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 = -Pi/A (18)
relative to the receiver.
This transaction is very similiar to an advanced light and retarded light transaction that was originally proposed in the paper ("Interaction with the Absorber as the Mechanism of Radiation." Wheeler, J.A. and Feynman, R.P. (1945) Reviews Of Modern Physics 17, 157). 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 ("Advanced Effects In Particle Physics." Csonka, Paul L. (1969) Physical Review 180, 1266), and ("The Transactional Interpretation Of Quantum Mechanics." Cramer, John G. (1986) Reviews Of Modern Physics 58, 647). Mr. Cramer's paper can also be found at http://mist.npl.washington.edu/npl/int_rep/tiqm/ti_toc.html.
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. (2). 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 Pi/2) that allow this closed "circuit" to be completed geometerically, for all events since the Big Bang. It is also believed that this can be proved mathmatically, but the author does not know how to go about it. If this can be proved, then the Z = Pi/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 seperate universe has it's own seperate value of Pi. In this universe, the number system and Pi are intimately connected, and that is most vividly shown in Eq. (7).
Pi/4 = 1 - 1/3 + 1/5 - 1/7 + ... (7)
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. (7) in the Parallel universe will produce the given value of Pi in that universe. This equation extends the Liebniz' series to Parallel universes
U*Pi/4 = U/1 - U/3 + U/5 - ... (19)
The U*Pi is the value of Pi 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 Eq. (10), 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 Pi. The Axioms of Euclid do not give a specific value of Pi, and are not dependent on what the value of Pi is. That suggests that the Axioms of Euclid are valid in the Parallel Universes along with the field of Non-Euclidian Geometery 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 gallions 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 know where Einstein went wrong.
In the book "Relativity, The Special And The General Theory" by Albert Einstein, 1961 Crown Trade Paperbacks, 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 Pi on the disc, arriving at a value of Pi larger then 3.1415... If this model is correct, then that is wrong. This model is based one the idea that there is exactly one value of Pi in this universe no matter how relativity may effect a rotating disc. The invariance of Pi in the frame of reference.
This was monkey # e * 10^googool typing paper # Pi * 10^googool.
This paper was posted to Alt.Sci.Physics.New-Theories several times in July/August of 1997.
Foundations Of Parallel Universe Math
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 occured in the universe. The Big Bang, the event(it doesn't matter when or what, as long as it occured 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 universe 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 universe has expanded, and we set these equal to each other, then we will get
D = A. (1)
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. (2)
In Eq. (2), 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 < or = 0, I do not know what to make of it. For Z < or = 0, these numbers would describe expansion rates faster then infinite.
I stumbled across these equations because I came to the conclusion that the universe on the whole was curved.
Here is what I did. Draw two dots. Label one dot "Us" (the event), and the other dot "BB" (the Big Bang). Draw a half circle connecting the two. The half circle represents the geodesic of light traveling from the Big Bang to the event. I started out with the half circle, because it was the easiest curve I could think of. I was expecting to graduate to other curves once I understood what was happening with the simplist. Draw a straight line connecting the two dots. The straight line represents the distance the universe has expanded when the event occured, 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 "BB" to "Us", so that it meets "BB" 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, this can also be extended to a sphere, 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 "Us" event. An event that happens inside the curve from "Us" to "BB", is an event that occured in the past for "Us". An event that happens along the curve between "Us" and "BB", is a simultaneous event as the "Us" event. An event that happens outside the curve, will be in "Us"'s future. Because this is drawn on paper, there is an area on the paper where if events happen there, they can not be in "Us"'s future, past, or present (It is the area defined as: everything on the opposite side of "BB" from "Us", including a line passing through "BB" that is perpendicular to the line from "Us" to "BB".). 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 "Us" 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 "Us" 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. (2) 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 precieved 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 diagramed in this fashion. Robert Osserman in his book "Poetry of the Universe: a mathematical exploration of the cosmos". (1995). Anchor Books., pages 114-120, 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 prespective, 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 (Pi) = (circumference). Our diagram is half of a circle, so the equation becomes: (diameter) x (Pi)/2 = (circumference)/2. In the diagram, (circumference)/2, is distance between "Us" and "BB" as measured along the curve, and diameter, is the distance between "Us" and "BB" as measured along the straight line. Substituting A and D for circumference/2 and diameter, we get this equation
D*(Pi)/2 = A. (3)
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. (4)
Substituting Eq. (3) into Eq. (4), we get this equation
D/D*Pi/2 = R. (5)
Eq. (5) reduces to this equation
2/Pi = R. (6)
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. (6) becomes
2*C/Pi = R (6.1)
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("The Expansion Rate And Size Of The Universe". Wendy L. Freedman. (Nov. 92). Sci. Am.). In other words, galaxies at different distances have different recession rates. This seems to be at odds with the retroverse model. 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" and "BB" again and this time a third set of events, "X". It is observed from "Us" that an event "X1" is moving away form "Us" some rate Q1. "X2" is observed to be further away then "X1", and it's rate of moving away from "Us" 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.
The retroverse model, gives an expansion rate from the Big Bang that is time invariant. Is the upper limit, from the Hubble model, time invariant? Meaning, is the upper limit, the same value at one minute after the Big Bang as a trillion years after the Big Bang? The Hubble model is not designed to answer that question. We shall now see that the Hubble model is very time specific. One of the things the Hubble model comes up with, is the Hubble constant. It is a measure of the recession velocity of a galaxy divided by it's distance. It is measured in kilometers per second per megaparsec. The value of the Hubble constant is not part of the scope of this paper, so we will set the value to Q per second per megaparsec. We continue with this question; for an observer, when she was exactly one megaparsec from the Big Bang, what was the value of the Hubble constant then? If the observer says Q per second per megaparsec, then the universe is speeding up it's expansion rate over time. This is not what we expect of the expanding universe. The other solution is, the Hubble constant is not constant, and is dependent upon when it is measured, or time dependent.
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 Pi/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 exploded at exactly one expansion rate, and all other expansion rates are not possible. The second solution is, the Big Bang exploded with many expansion rates, and each seperate expansion rate defines it's own seperate and complete universe.
If the Big Bang created one expansion rate, then there has to be a physical reason why this occured, or an explanation has to be available why the other expansion rates are not possible. An Anthropic principle will not suffice here("The Anthropic Cosmological Principle", Barrow, J.D. and Tipler, F.J. (1986). Oxford University Press.). For these reasons, this paper will not explore the exacly one expansion rate solution. This paper will explore the seperate expansion rates that define seperate universes, or parallel universes solution. This solution is based off of an idea originally proposed by Hugh Everett III, in his paper ("Relative State" Formulation Of Quantum Mechanics In Quantum Theory And Measurement (1957) Rev. Of Mod. Phys. 29, 454-462).
The equation that gave us the expansion rate is,
2/Pi = R. (6)
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 Pi. This immediately implies that each seperate universe has it's own different value of Pi. The values of Pi approaching infinite as the expansion rate approaches 0, from Eq. (6), and Pi approaching 0 as the expansion rate approaches infinite.
A few things about Pi before we go on. Pi 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 Pi. Pi's expansion is infinite in length, and the numbers do not repeat. Pi is a transendental number. Pi can be calculated both mathematically and physically. A math equation to calculate the value of Pi is this
Pi/4 = 1 - 1/3 + 1/5 - 1/7 + ... (7)
This equation is called Liebniz' series. There are many other equations to calculate Pi. Pi can be calculated by droping sticks on Parallel lines, first observed in the eighteenth century by Count Buffon("One Two Three...Infinity" by George Gamov, Bantom books 1965). In 1995, it was discovered that any number in the expansion of Pi can be extracted without knowledge of the perceeding numbers in the expansion (On The Rapid Computation Of Various Polylogrithmic Constants by Bailey, Bowein, and Plouffe, http://www.mathsoft.com/asolve/plouffe/plouffe.html).
Different values of Pi are not new to math. In both the Geometeries of Riemann(eliptic) and Lobachevsky(hyperbolic), different values of Pi can be derived. But there is a catch, in both of the Geometeries, the smaller the space that is measured, the closer to "Pi" the different values of Pi get. This is the equation for the circumference of a circle in Hyperbolic Geometery
C = 2*Pi*sinh*r. (8)
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*Pi.
This is the equation for the circumference of a circle in Double Eliptic Geometery, sin(a/r) is a measure of the eliptic surface,
C = 2*Pi*r*sin(a/r). (9)
Once again, as r approaches 0, the difference between C and r gets closer to 2*Pi. Equations (8) and (9), were taken from ("An Introduction To Non-Euclidian Geometery". David Gans. (1973). Academic Press.).
We will use Pi-bar to set the values of Pi in different universes. This will eliminate the confusion between Pi in this universe and Pi in another universe. When talking about a specific universe, this notation will show up, Pi-bar = 10, this would indicate that wherever the variable Pi-bar shows up, the value of Pi in that universe is 10. We can also set Pi-bar = Pi, 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*Pi-bar. (10)
As d approaches 0, the difference between d and C does not change, so the changes in Pi, in a parallel universe, is not a change to Non-Euclidian Geometery. To the best of my knowledge, no other math we use to describe spacetime, suggests different values of Pi. I am 95% sure of this about math that was produced before the 20th century, and I am 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 seperate expansion rate is it's own universe with it's own different value of Pi. Humankind has created a math that has different Pi'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 Pi to describe spacetime.
The Curvature of Spacetime
The equation for the curvature of a circle is
k = 1/r. (11)
Where k is the curvature of the circle, and r is the circles radius. Knowing that the retroverse is a half circle, we can apply Eq. (11) to this model, and we get this
k = 2/D. (12)
This is an equation for the curvature of the geodesic when we assume a non-convoluted shape to the geodesic. The geodesic is a part of spacetime, so we will say the curvature of spacetime. We can also set k equal to other variables from Eq. (3)
k = 2/D = Pi/A. (13)
This shows an important thing about the curvature of spactime, remembering that A is the age of the universe, the curvature of spacetime is getting smaller as the universe gets older, and an event determines the curvature of spacetime. Eq. (13) 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 to get larger, so that the geodesics 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 occured, 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 noticable. It is the, "matter in the universe seems to be older then the universe it resides in" problem ("Hubble Space Telescope measures precise distance to the most remote Galaxy yet". Press release No. STScI-PR94-49. (10/26/94). http://oposite.stsci.edu/pubinfo/press-release/94-49.txt).
Here are three equations to determine the actual age of the event, versus the preceived age of the event. A is the age of the universe for the observer. B is the preceived age of the universe, relative to the Big Bang, of the observed event. F is the actual age of the event, relative to the Big Bang. D = 2*A/Pi. This is the equation when B is older then A/2, but younger then A,
F = Pi((90(2B-1)sin*D+D)^2+(90(2B-1)cos*D)^2)^-1/2
_ _ _ _ _ _
2 A 2 2 A 2. (14)
This is the equation, when B is exactly A/2
F = Pi/2*((D/2)^2+(D/2)^2)^-1/2. (15)
This is the equation, when B is younger then A/2
F = Pi/2((D-90(2B)cos(D))^2+(90(2B)sin(D))^2)^-1/2
_ _ _ _ _
2 A 2 A 2. (16)
Equations (14, 15, & 16) are corrected in a post on this blog called Corrections to Equations 14, 15, and 16 for My 1997 paper "Foundations Of Parallel Universe Math".
This time illusion is a feature of curved spacetime. The difference between observed time of the event, and actual time of the event is a measure of the curvature of spacetime. The time illusion discrepancey is governed by the value of Z from Eq. (2). When Z = 1, there will be no time illusion. The three equations (14), (15), and (16) are for Z = Pi/2 only.
The author suspects that there are shorter equations for (14), (15), and (16), but he is unable to derive them.
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 ("Faster Than Light: Superluminal Loopholes In Physics". Nick Herbert. (1988). Plume.). 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 = Pi/A (Eq. (13)). 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 = Pi/A where the value of A is based on the when the receiving event occured, relative to the Big Bang. Relative to the emitter, the emitted light has the curvature of
k = Pi/(A + t), (17)
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 = Pi/(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 transferance 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 = -Pi/A (18)
relative to the receiver.
This transaction is very similiar to an advanced light and retarded light transaction that was originally proposed in the paper ("Interaction with the Absorber as the Mechanism of Radiation." Wheeler, J.A. and Feynman, R.P. (1945) Reviews Of Modern Physics 17, 157). 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 ("Advanced Effects In Particle Physics." Csonka, Paul L. (1969) Physical Review 180, 1266), and ("The Transactional Interpretation Of Quantum Mechanics." Cramer, John G. (1986) Reviews Of Modern Physics 58, 647). Mr. Cramer's paper can also be found at http://mist.npl.washington.edu/npl/int_rep/tiqm/ti_toc.html.
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. (2). 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 Pi/2) that allow this closed "circuit" to be completed geometerically, for all events since the Big Bang. It is also believed that this can be proved mathmatically, but the author does not know how to go about it. If this can be proved, then the Z = Pi/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 seperate universe has it's own seperate value of Pi. In this universe, the number system and Pi are intimately connected, and that is most vividly shown in Eq. (7).
Pi/4 = 1 - 1/3 + 1/5 - 1/7 + ... (7)
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. (7) in the Parallel universe will produce the given value of Pi in that universe. This equation extends the Liebniz' series to Parallel universes
U*Pi/4 = U/1 - U/3 + U/5 - ... (19)
The U*Pi is the value of Pi 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 Eq. (10), 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 Pi. The Axioms of Euclid do not give a specific value of Pi, and are not dependent on what the value of Pi is. That suggests that the Axioms of Euclid are valid in the Parallel Universes along with the field of Non-Euclidian Geometery 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 gallions 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 know where Einstein went wrong.
In the book "Relativity, The Special And The General Theory" by Albert Einstein, 1961 Crown Trade Paperbacks, 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 Pi on the disc, arriving at a value of Pi larger then 3.1415... If this model is correct, then that is wrong. This model is based one the idea that there is exactly one value of Pi in this universe no matter how relativity may effect a rotating disc. The invariance of Pi in the frame of reference.
This was monkey # e * 10^googool typing paper # Pi * 10^googool.
This paper was posted to Alt.Sci.Physics.New-Theories several times in July/August of 1997.
posted by picrosser at 10:00 PM
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