My 2001 paper "Parallel Universes In Math".
Here is my 2001 paper entitled "Parallel Universes In Math". I finished it in June of 2001 and put it on my website(nolonger exists) at that point in time. The website stayed up until the Fall of 2001. The primary concern of the website was to publish webpages I had created, for a course "final" in an HTML college course I had taken. That "final" had a bunch of poems I had written on it and links to the Poetry Webring. I added my parallel universe paper because I didn't want to publish it in the same place as my 1997 paper. By the way, the status on my 2006 paper is less then what I alluded to in my previous post. I think I said I was writing it. A more accurate statement might be, I'm writing it - in my head. I hope you enjoy my 2001 paper.
Parallel Universes In Math.
By Jim Akerlund
This paper will try to show that there are parallel universes in math. In order to do this, we will need to explain to you what we mean by the statement; "there are parallel universes in math". The concept of parallel universes as brought down to us from history basically says; given the same set of circumstances, a different set of events may have happened in a different universe. That previous statement divides the set of events in our universe into two different types; events that are "universe invariant" and events that are "universe specific". We will define universe invariant as; events, ideas, memes, rules, and laws that are the same in all parallel universes. And we will define universe specific as; events, ideas, memes, rules, and laws that are specific to exactly one universe. Our current thinking of math is that all of math is universe invariant. This paper will try to show that some of math is universe specific and therefore there are parallel universes in math.
Summary
A model of our universes expansion will be developed. It will be shown that this model is a derivative of Hubble's model of universal expansion. We then go onto create a curved spacetime diagram from this model. From the curved spacetime diagram we calculate an expansion rate and postulate the existance of parallel universes based on different values of π. We then split math into two seperate groups: "universe invariant" and "universe specific" math. We use the universe specific number of π to show that our number system is also universe specific. This leads to another way to define universe specific and universe invariant. In the discussion, we discuss if the paper itself is universe invariant and finally we look for what parts of math that mankind already has developed that maybe universe invariant. One of the goals of this paper is to give you enough information about parallel universes and math so that hopefully you won't be looking into the same mathematical abyss that we were looking into nine years ago when we first conceived of parallel universes based on different values of π.
The Retroverse
There is more then one way to describe how the universe expands. The Hubble model (Peebles 1993) of universal expansion is what we will describe as an event-to-event model of expansion. Meaning that the model is not referenced to any particular event in the universe. A different way of describing the universes expansion would be event-to-specific-event. Meaning that all expansion in the universe is referenced to one specific event in it's history. For this type of model to work effectively, we would need an event that has no preceeding events before it and the specific event is in the histories of all subsequent events. The only event in the universe that fulfills this requirement is the Big Bang event. This model of the universes expansion is unique, because it requires no observation. We imagine a photon emitted at the Big Bang and arriving at some event in the universe. We then ask the question, how far did the event expand from the Big Bang in the time that it took the photon to reach the event. This can be put down into an equation,
d*q = a. (1)
a is the length of the geodesic for the photon and it is also the age of the universe when the photon reached the event. d is the distance the event expanded from the Big Bang and q is the factor of the difference between a and d.
The Hubble model of the universes expansion is based on a scaling factor. This scaling factor, ignoring local irregularities, is thought to be the same for all of spacetime. Recent research that suggests an accelerating expansion(Hogan, Kirshner, and Suntzeff 1999), calls for this scaling to be in question. We will call the Hubble scaling "flat" scaling. It describes a situation where an object twice as far as another object will be expanding at twice the expansion rate. The accelerating expansion model, suggests a "hyperbolic" scaling. Hyperbolic scaling occurs when an object twice as far away will be expanding at greater then twice the expansion rate. The model described in this paper will display "elliptic" scaling. Elliptic scaling occurs when an object twice as far away will be expanding at less then twice the expansion rate.
Research into Einstein Rings(Wambsganss 2001) may resolve whether the universe has "flat", "hyperbolic", or "elliptic" scaling. Einstein Rings provide two seperate ways too evaluate distance and expansion. The expansion rate of the two objects can be determined by redshift. The distance of the two objects can be determined by the geometery of the gravitational lens. If the distance determined by the lens geometery agrees with the expansion via redshift, then this is evidence for flat scaling. If the distance via lens geometery is less then the expected expansion via redshift, then this is evidence of hyperbolic scaling. If the distance via lens geometery is greater then the expected expansion via redshift, then this is evidence of elliptic scaling
If we set q = 1, we get the strange situation where the event is expanding from the Big Bang at the speed of light. To set q > 1, we get the situation where the geodesic of light(a) is longer then the path of the event(d) from the Big Bang. This leads to a conclusion that some how the geodesic of light is curved. General Relativity (Einstein 1956) would tell us that gravity is causing the curvature of the light. In this case, because the light has traversed the universe to get to the event, then the mass of the universe is causing the curvature of the light. Other examples of the curvature of light due to mass include; light bending around the sun and gravitational lensing due to intervening mass between event and observer(Ciufolini and Wheeler 1995). We believe that this is another manifestation of light bending in our observable universe.
One of the unique things about q is that each value of q determines a seperate mass of the
universe. Because matter can neither be created nor destroyed then the universe has exactly one
mass that is the same from it's creation. In other words, once the value of q is established at the Big Bang it can never change for all events in the same universe.
Of all the possible curvatures for the values q > 1, q = π/2 defines the simplest curve. It also implies a universe that spatially homogeneous and isotropic on the very large scale. When q = π/2, we can
then draw a curved spacetime diagram.
Diagram #1, above simulates two photons traveling from the Big Bang to the event, each from
opposite directions. Distances measured in spacetime are measured along geodesics of light, since d isn't measured along a geodesic of light then it is an imaginary distance (not the mathematical term, imaginary). The diagram shows that the event is at the antipodal from the Big Bang in spacetime. All events are at the antipodal of spacetime from the Big Bang
This diagram is very similiar to a diagram presented by (Osserman 1995) in his book. He calls the
diagram the "retroverse". Mr. Osserman doesn't apply variables to his diagram. This paper will take
the retroverse far beyond what is described in his book.
Also from the above diagram we can eliminate other values of q > 1, because when drawn(Diagrams #2 and #3) they will reveal a universe that isn't spatially homogeneous and isotropic on the very large scale, or they don't satisfy the cosmological principle(Peebles 1993).
The curved spacetime diagram also shows a time relation to all events in the universe relative to the event illustrated. Here is a diagram showing the time relations.
Diagram #4 shows that everything outside the circle is composed of both future events and
"Unknowable events" and everything inside the circle is composed of past events. The area on the
diagram called "Unknowable Events" is a manifestation of the two dimensional diagram that is drawn. We don't know what this would look like in the four dimensional spacetime we live in, nor do we know if the "Unknowable Events" area actually exists in our four dimensional universe.
Parallel Universes
For any value of q we can determine the expansion rate of that event from the Big Bang. Here is
equation 1 again,
d*q = a. (1)
To get a rate of expansion r of the event from the Big Bang, we need to divide the distance the
event expanded d, by the age of the universe a, and we get this equation,
d/a = r. (2)
Substituting equation 1 into equation 2 we get
1/q = r. (3)
Substituting π/2 for q we get,
2/π = r = 0.6366197724... lightyears. (3.1)
When the speed of light is not unity, equation 3.1 becomes,
2c/π = r, (3.2)
where c is the speed of light.
If every event in the universe is expanding at exactly 0.6366197724... lightyears from the Big Bang,
then what about the other expansion rates from the Big Bang? We are faced with two seperate
solutions here. The first solution is that the Big Bang expanded with exactly one expansion rate and
that all other expansion rates are not possible. Or the Big Bang expanded with many expansion
rates and each expansion rate defines it's own separate and complete universe.
The "exactly one expansion rate" solution, suggests an Anthropic principle(Barrow and Tipler 1983) is at work in the Big Bang. This paper will not explore that solution. This paper will explore the seperate expansion rates that define seperate universes, or a parallel universes solution. This solution is based off of an idea originally proposed by Hugh Everett III(Everett 1957) and followed up by(DeWitt and Graham 1973).
Here is equation 3.1 again,
2/π = r. (3.1)
If we change the expansion rate r, then some variable on the left hand side of the equation will also
have to change. That varaible is π. This immediately implies that each seperate universe has it's own different value of π. Universes where π > 3.14159... will be expanding at r < 0.6366197724... lightyears, and universes where π < 3.14159... will be expanding at r > 0.6366197724... lightyears. In equation 3.1, the speed of light is unity, it will also be unity in the parallel universes, so it can not be a variable that determines the existance of parallel universes.
"Universe Invariant" and "Universe Specific" Math
Now we need to ask the question; what do we mean π changes in the parallel universes? π is the
ratio between a circles diameter and it's circumference in Euclidian spacetime. In a parallel universe, the ratio between a circles diameter and it's circumference will be different in that universes Euclidian spacetime.
From the above paragraph we can show you two concepts of parallel universes. These two concepts are that some ideas are specific to each individual universe or "universe specific" and some ideas are applicable to all universes or "universe invariant". We will define universe invariant as; events, ideas, memes, rules, and laws that are the same in all parallel universes. We will define universe specific as; events, ideas, memes, rules, and laws that are specific to one universe. The value of π is universe specific as shown above. Circle, circumference, diameter, ratio, and Euclidian spacetime are universe invariant. If Euclidian spacetime were universe specific then that would lead to the absurd proposition that somehow our universe exhibits Euclidian spacetime and the parallel universes don't or are Non-euclidian spacetimes. There is a "Copernican" view in physics that says that the section of the universe we reside in isn't special from the other sections in the universe. We now extend that "Copernican" view to the relationship between parallel universes, in that, no specific universe is more special then any other universe. This translates to, if this universe exhibits Euclidian spactime, then the parallel universes will also exhibit Euclidian spactimes, or Euclidian Spacetime is universe invariant. Euclidian spacetime becomes a feature to the "flatness" of the spacetime you are measuring, irrespective of the universe you are measuring it in. This same type of arguement can also be applied to circle, circumference, diameter, and ratio.
Circle, circumference, diameter, ratio, π, and Euclidian geometery are all concepts of math, but in the previous paragraph we seperated them into universe specific and universe invariant concepts. That means that somehow math is divided into two seperate groups; math that is universe specific and math that is universe invariant.
That previous paragraph is contrary to our current thinking in math. Our current thinking in math is that all of math is the same in all parallel universes, or that all of math is universe invariant. The only math that discusses parallel universes is math used by Quantum Physics to discuss possible interpretations of quantum theory(Everett 1957) and(DeWitt and Graham 1973). The math used in quantum theory was formulated by John von Neuman(von Neuman 1955). This math was formulated without consideration that there are might be parallel universes and that math may change in these parallel universes.
As we write this, we don't know how deep and extensive the splitting of math into universe invariant and universe specific groups goes, but using the universe invariant concept of ratio we can show you another universe specific concept in math. Suppose we are in a universe where the value of π is not equal to 3.14159... and we decide to use equation 4 to calculate the value of π. We will run into a paradox.
π/4 = 1 - 1/3 + 1/5 - 1/7 + 1/9.... (4)
We can not have two different values of π in the same universe. One value based on the spacetime we are in and another value based on a calculation we did on paper, or computer, or in our heads. Equation 4 is known as Liebniz' equation(Blatner 1997).
Equation 4 shows the very special relationship π has with the number system. We know from above that π is universe specific. In order for equation 4 to calculate a value of π not equal to 3.14159..., then some part of the equation needs to change in some way. We have come up with three possible solutions to this problem.
1. The whole equation is universe specific.
2. The numbers only are universe specific and the equation form is universe invariant.
3. Some, unknown aspect of the equation is universe specific and another, unknown aspect of the equation is universe invariant
Of the three possiblities, #2 the situation where the numbers are universe specific and the equation form is universe invariant, is the only solution where ratio remains a universe invariant concept. Solution #2 also allows for this hypothetical situation to occur. Where a parallel universe equaliant of Gottfried Liebniz formulates an equation to calculate π in his universe and his equation looks like this,
<π>/<4> = <1> - <1>/<3> + <1>/<5> - <1>/<7>... (5)
The symbols inside the < > are this universes intrepretation of the parallel universes Gottfried Liebniz' symbols. The < > symbols do you a disservice. They help us in explaining the thinking, but when actually applied to mathematical equations they will preduce lots of confusion. This universes number system is specific to this universe. Three apples in this universe is in no way equivalent to three apples in another universe. Sure the concept of "three" does exist in the parallel universes, but "three" is universe specific. The best that we have been able to do is to turn Liebniz' equation into a kind of parallel universe number system translator. Here is that equation,
u*π/4 = u/1 - u/3 + u/5 - u/7 + u/9.... (6)
u is the value of 1, relative to this universe, of the universe you are looking at. When u = 1 that defines this universe for equation 6. u*π is the value of π in the parallel universe relative to this universe.
The above demonstration of π showing that the number system is universe specific leads us to another definition of universe invariant and universe specific. A universe invariant concept will not change when π changes. A universe specific concept will change when π changes. This can be generalized too; given an unknown concept(unknown as to whether it is universe invariant or universe specific) and a known universe specific concept. The unknown concept will be universe invariant if when the two concepts are brought together and the universe specific concept is changed, the unknown will not change. If the unknown is universe specific, then it will change under the above conditions. We are not quite sure what we mean when we say "two concepts are brought together".
Ultimately, all concepts of math are concepts that are invented by mankind to deal with problems that he has encountered. Math could be classified as the set of solutions and methods, mankind has invented to solve a certain class of problems(Davis and Hersh 1981), (Skemp 1971). Numbers, are a solution mankind has come up with to solve a problem of counting(Barrow 1992). So, when we say that numbers are universe specific; we are saying that should mankind had been in a parallel universe and once again created the "number" solution to his counting problem and we were to compare that parallel universe solution to the same solution in this universe, then equation 6 would show how they are different. We don't know if there is a universe invariant "number" solution. To reiterate our definition of math again and this time include the concepts of universe invaraint and universe specific, we can say: Math is a set of solutions and methods, mankind has come up with to solve a certain class of problems, some of those solutions and methods are universe invariant and some of those solutions and methods are universe specific.
Discussion
The idea where each different value of π from infinity to zero is stacked like the layers of an onion as a result of the Big Bang is one of the most compelling ideas we have ever run across. The concepts of universe invariant and universe specific are byproducts of the different values of π. Throughout this paper we have struggled with defining, describing, and manipulating the concepts of universe invariant and universe specific. One of the key aspects of universe specific is that universe specific concepts can not be used to describe parallel universes. You can only use universe invariant concepts to describe parallel universes. That begs the question then; is this paper itself universe invariant? The answer to that question we do not have, but if it is it may explain why the math is so simple in it. You can not use universe specific math to define parallel universes, it has to be universe invariant. We believe the ideas in this paper are also simple for the same reason. We don't think this means that all universe invariant concepts are simple. We don't know. The Friedman cosmological models(Ciufolini and Wheeler 1995) all appear to be universe specific and therefore can not define parallel universes. One other thing about universe invariant and universe specific. In our definitions of these two concepts, we said they applied to one universe or all universes. We don't know if there are concepts that may apply to more the one universe but less then all. In any case we have called these types of concepts semi-universe invariant.
One of the questions that was continually raised while working on this model is, has mankind formulated any math where unbeknownst to the mathematician, he has created a universe invariant math. From the conditions outlined above, we would be looking for any maths that don't use; a specific value of π, number, or quantity. The best example of universe invariant math we have run across so far, is a book written in 1854 by George Boole. Here is a quote from "The Laws Of Thought" by George Boole, page 12(Boole 1854) "It is not the essence of mathematics to be conversant with the ideas of number and quantity." We have found threads of universe invariant math in many of the fields of math; Ring theory(Rowen 1991), all sorts of Algebras(Birkhoff and Mac Lane 1977), Euclidian and Non-euclidian geometries(Gans 1973), and even Number theory(Ore 1948) has universe invariant threads running through it. We would need to pull all these universe invariant concepts of math together, before we could go on to create a universe invariant physics. With a universe invariant physics in our hands we could then start building the methods that will take us to all these parallel universes that are described in this paper. We have dubbed the act of crossing into a parallel universe, pi-crossing.
References
Barrow, John, D. and Tipler, F.J., "The Anthropic Cosmological Principle". (1986). Oxford University Press.
Barrow, John, D., "Pi in the Sky: Counting, Thinking, and Being". (1992) Oxford University Press.
Birkhoff, Garrett, and Mac Lane, Saunders., "A Survey of Modern Algebra". (1977). Macmillian.
Blatner, David., "The Joy Of p". (1997). Walker Publishing.
Boole, George., "An Investigation Of The Laws Of Thought On Which Are Founded The Mathematical Theories Of Logic And Probabilities". (1958). Dover. Originally published (1854). Macmillan.
Ciufolini, Ignazio, and Wheeler, John, A., "Gravitation and Inertia". (1995). Princeton University Press.
Davis, Philip, J. and Hersh, Reuben., "The Mathematical Experience". (1981). Birkhäuser.
DeWitt, Bryce, S. and Graham, Neill., "The Many-Worlds Interpretation Of Quantum Mechanics". (1973). Princeton University Press.
Einstein, Albert., "The Meaning Of Relativity", 5th ed. (1956). Princeton University Press.
Everett, Hugh, III., "Relative State, Formulation Of Quantum Mechanics". Rev. Mod. Phys. 29, 454, (1957)
Gans, David., "An Introduction To Non-euclidian Geometery". (1973). Academic Press.
Hogan, Craig, J., Kirshner, Robert, P., and Suntzeff, Nicholas, B., "Surveying Space-time with Supernova". Sci. Am. January (1999).
Ore, Oystein., "Number Theory and It's History". (1948). McGraw-Hill.
Osserman, Robert., "Poetry Of The Universe: A Mathematical Exploration Of The Cosmos". (1995). Anchor books.
Peebles, P. J. E., "Principles Of Physical Cosmology". (1993). Princeton University Press.
Rowen, Louis, H., "Ring Theory". (1991). Academic Press.
Skemp, Richard, R., "The Psychology of Learning Mathematics". (1971). Penguin Books.
von Neuman, John., "Mathematical foundations of Quantum Mechanics". (1955). Princeton University Press.
Wambsganss, Joachim., "Gravity's Kaleidoscope". Sci. Am. November. (2001).
The ideas in this paper that were originated by Jim Akerlund were originally published in 1997 in the Newsgroup Alt.sci.physics.new-theories in the July/August time frame under the title "Foundations of Parallel Universe Math".
Parallel Universes In Math.
By Jim Akerlund
This paper will try to show that there are parallel universes in math. In order to do this, we will need to explain to you what we mean by the statement; "there are parallel universes in math". The concept of parallel universes as brought down to us from history basically says; given the same set of circumstances, a different set of events may have happened in a different universe. That previous statement divides the set of events in our universe into two different types; events that are "universe invariant" and events that are "universe specific". We will define universe invariant as; events, ideas, memes, rules, and laws that are the same in all parallel universes. And we will define universe specific as; events, ideas, memes, rules, and laws that are specific to exactly one universe. Our current thinking of math is that all of math is universe invariant. This paper will try to show that some of math is universe specific and therefore there are parallel universes in math.
Summary
A model of our universes expansion will be developed. It will be shown that this model is a derivative of Hubble's model of universal expansion. We then go onto create a curved spacetime diagram from this model. From the curved spacetime diagram we calculate an expansion rate and postulate the existance of parallel universes based on different values of π. We then split math into two seperate groups: "universe invariant" and "universe specific" math. We use the universe specific number of π to show that our number system is also universe specific. This leads to another way to define universe specific and universe invariant. In the discussion, we discuss if the paper itself is universe invariant and finally we look for what parts of math that mankind already has developed that maybe universe invariant. One of the goals of this paper is to give you enough information about parallel universes and math so that hopefully you won't be looking into the same mathematical abyss that we were looking into nine years ago when we first conceived of parallel universes based on different values of π.
The Retroverse
There is more then one way to describe how the universe expands. The Hubble model (Peebles 1993) of universal expansion is what we will describe as an event-to-event model of expansion. Meaning that the model is not referenced to any particular event in the universe. A different way of describing the universes expansion would be event-to-specific-event. Meaning that all expansion in the universe is referenced to one specific event in it's history. For this type of model to work effectively, we would need an event that has no preceeding events before it and the specific event is in the histories of all subsequent events. The only event in the universe that fulfills this requirement is the Big Bang event. This model of the universes expansion is unique, because it requires no observation. We imagine a photon emitted at the Big Bang and arriving at some event in the universe. We then ask the question, how far did the event expand from the Big Bang in the time that it took the photon to reach the event. This can be put down into an equation,
d*q = a. (1)
a is the length of the geodesic for the photon and it is also the age of the universe when the photon reached the event. d is the distance the event expanded from the Big Bang and q is the factor of the difference between a and d.
The Hubble model of the universes expansion is based on a scaling factor. This scaling factor, ignoring local irregularities, is thought to be the same for all of spacetime. Recent research that suggests an accelerating expansion(Hogan, Kirshner, and Suntzeff 1999), calls for this scaling to be in question. We will call the Hubble scaling "flat" scaling. It describes a situation where an object twice as far as another object will be expanding at twice the expansion rate. The accelerating expansion model, suggests a "hyperbolic" scaling. Hyperbolic scaling occurs when an object twice as far away will be expanding at greater then twice the expansion rate. The model described in this paper will display "elliptic" scaling. Elliptic scaling occurs when an object twice as far away will be expanding at less then twice the expansion rate.
Research into Einstein Rings(Wambsganss 2001) may resolve whether the universe has "flat", "hyperbolic", or "elliptic" scaling. Einstein Rings provide two seperate ways too evaluate distance and expansion. The expansion rate of the two objects can be determined by redshift. The distance of the two objects can be determined by the geometery of the gravitational lens. If the distance determined by the lens geometery agrees with the expansion via redshift, then this is evidence for flat scaling. If the distance via lens geometery is less then the expected expansion via redshift, then this is evidence of hyperbolic scaling. If the distance via lens geometery is greater then the expected expansion via redshift, then this is evidence of elliptic scaling
If we set q = 1, we get the strange situation where the event is expanding from the Big Bang at the speed of light. To set q > 1, we get the situation where the geodesic of light(a) is longer then the path of the event(d) from the Big Bang. This leads to a conclusion that some how the geodesic of light is curved. General Relativity (Einstein 1956) would tell us that gravity is causing the curvature of the light. In this case, because the light has traversed the universe to get to the event, then the mass of the universe is causing the curvature of the light. Other examples of the curvature of light due to mass include; light bending around the sun and gravitational lensing due to intervening mass between event and observer(Ciufolini and Wheeler 1995). We believe that this is another manifestation of light bending in our observable universe.
One of the unique things about q is that each value of q determines a seperate mass of the
universe. Because matter can neither be created nor destroyed then the universe has exactly one
mass that is the same from it's creation. In other words, once the value of q is established at the Big Bang it can never change for all events in the same universe.
Of all the possible curvatures for the values q > 1, q = π/2 defines the simplest curve. It also implies a universe that spatially homogeneous and isotropic on the very large scale. When q = π/2, we can
then draw a curved spacetime diagram.
Diagram #1, above simulates two photons traveling from the Big Bang to the event, each from
opposite directions. Distances measured in spacetime are measured along geodesics of light, since d isn't measured along a geodesic of light then it is an imaginary distance (not the mathematical term, imaginary). The diagram shows that the event is at the antipodal from the Big Bang in spacetime. All events are at the antipodal of spacetime from the Big Bang
This diagram is very similiar to a diagram presented by (Osserman 1995) in his book. He calls the
diagram the "retroverse". Mr. Osserman doesn't apply variables to his diagram. This paper will take
the retroverse far beyond what is described in his book.
Also from the above diagram we can eliminate other values of q > 1, because when drawn(Diagrams #2 and #3) they will reveal a universe that isn't spatially homogeneous and isotropic on the very large scale, or they don't satisfy the cosmological principle(Peebles 1993).
The curved spacetime diagram also shows a time relation to all events in the universe relative to the event illustrated. Here is a diagram showing the time relations.
Diagram #4 shows that everything outside the circle is composed of both future events and
"Unknowable events" and everything inside the circle is composed of past events. The area on the
diagram called "Unknowable Events" is a manifestation of the two dimensional diagram that is drawn. We don't know what this would look like in the four dimensional spacetime we live in, nor do we know if the "Unknowable Events" area actually exists in our four dimensional universe.
Parallel Universes
For any value of q we can determine the expansion rate of that event from the Big Bang. Here is
equation 1 again,
d*q = a. (1)
To get a rate of expansion r of the event from the Big Bang, we need to divide the distance the
event expanded d, by the age of the universe a, and we get this equation,
d/a = r. (2)
Substituting equation 1 into equation 2 we get
1/q = r. (3)
Substituting π/2 for q we get,
2/π = r = 0.6366197724... lightyears. (3.1)
When the speed of light is not unity, equation 3.1 becomes,
2c/π = r, (3.2)
where c is the speed of light.
If every event in the universe is expanding at exactly 0.6366197724... lightyears from the Big Bang,
then what about the other expansion rates from the Big Bang? We are faced with two seperate
solutions here. The first solution is that the Big Bang expanded with exactly one expansion rate and
that all other expansion rates are not possible. Or the Big Bang expanded with many expansion
rates and each expansion rate defines it's own separate and complete universe.
The "exactly one expansion rate" solution, suggests an Anthropic principle(Barrow and Tipler 1983) is at work in the Big Bang. This paper will not explore that solution. This paper will explore the seperate expansion rates that define seperate universes, or a parallel universes solution. This solution is based off of an idea originally proposed by Hugh Everett III(Everett 1957) and followed up by(DeWitt and Graham 1973).
Here is equation 3.1 again,
2/π = r. (3.1)
If we change the expansion rate r, then some variable on the left hand side of the equation will also
have to change. That varaible is π. This immediately implies that each seperate universe has it's own different value of π. Universes where π > 3.14159... will be expanding at r < 0.6366197724... lightyears, and universes where π < 3.14159... will be expanding at r > 0.6366197724... lightyears. In equation 3.1, the speed of light is unity, it will also be unity in the parallel universes, so it can not be a variable that determines the existance of parallel universes.
"Universe Invariant" and "Universe Specific" Math
Now we need to ask the question; what do we mean π changes in the parallel universes? π is the
ratio between a circles diameter and it's circumference in Euclidian spacetime. In a parallel universe, the ratio between a circles diameter and it's circumference will be different in that universes Euclidian spacetime.
From the above paragraph we can show you two concepts of parallel universes. These two concepts are that some ideas are specific to each individual universe or "universe specific" and some ideas are applicable to all universes or "universe invariant". We will define universe invariant as; events, ideas, memes, rules, and laws that are the same in all parallel universes. We will define universe specific as; events, ideas, memes, rules, and laws that are specific to one universe. The value of π is universe specific as shown above. Circle, circumference, diameter, ratio, and Euclidian spacetime are universe invariant. If Euclidian spacetime were universe specific then that would lead to the absurd proposition that somehow our universe exhibits Euclidian spacetime and the parallel universes don't or are Non-euclidian spacetimes. There is a "Copernican" view in physics that says that the section of the universe we reside in isn't special from the other sections in the universe. We now extend that "Copernican" view to the relationship between parallel universes, in that, no specific universe is more special then any other universe. This translates to, if this universe exhibits Euclidian spactime, then the parallel universes will also exhibit Euclidian spactimes, or Euclidian Spacetime is universe invariant. Euclidian spacetime becomes a feature to the "flatness" of the spacetime you are measuring, irrespective of the universe you are measuring it in. This same type of arguement can also be applied to circle, circumference, diameter, and ratio.
Circle, circumference, diameter, ratio, π, and Euclidian geometery are all concepts of math, but in the previous paragraph we seperated them into universe specific and universe invariant concepts. That means that somehow math is divided into two seperate groups; math that is universe specific and math that is universe invariant.
That previous paragraph is contrary to our current thinking in math. Our current thinking in math is that all of math is the same in all parallel universes, or that all of math is universe invariant. The only math that discusses parallel universes is math used by Quantum Physics to discuss possible interpretations of quantum theory(Everett 1957) and(DeWitt and Graham 1973). The math used in quantum theory was formulated by John von Neuman(von Neuman 1955). This math was formulated without consideration that there are might be parallel universes and that math may change in these parallel universes.
As we write this, we don't know how deep and extensive the splitting of math into universe invariant and universe specific groups goes, but using the universe invariant concept of ratio we can show you another universe specific concept in math. Suppose we are in a universe where the value of π is not equal to 3.14159... and we decide to use equation 4 to calculate the value of π. We will run into a paradox.
π/4 = 1 - 1/3 + 1/5 - 1/7 + 1/9.... (4)
We can not have two different values of π in the same universe. One value based on the spacetime we are in and another value based on a calculation we did on paper, or computer, or in our heads. Equation 4 is known as Liebniz' equation(Blatner 1997).
Equation 4 shows the very special relationship π has with the number system. We know from above that π is universe specific. In order for equation 4 to calculate a value of π not equal to 3.14159..., then some part of the equation needs to change in some way. We have come up with three possible solutions to this problem.
1. The whole equation is universe specific.
2. The numbers only are universe specific and the equation form is universe invariant.
3. Some, unknown aspect of the equation is universe specific and another, unknown aspect of the equation is universe invariant
Of the three possiblities, #2 the situation where the numbers are universe specific and the equation form is universe invariant, is the only solution where ratio remains a universe invariant concept. Solution #2 also allows for this hypothetical situation to occur. Where a parallel universe equaliant of Gottfried Liebniz formulates an equation to calculate π in his universe and his equation looks like this,
<π>/<4> = <1> - <1>/<3> + <1>/<5> - <1>/<7>... (5)
The symbols inside the < > are this universes intrepretation of the parallel universes Gottfried Liebniz' symbols. The < > symbols do you a disservice. They help us in explaining the thinking, but when actually applied to mathematical equations they will preduce lots of confusion. This universes number system is specific to this universe. Three apples in this universe is in no way equivalent to three apples in another universe. Sure the concept of "three" does exist in the parallel universes, but "three" is universe specific. The best that we have been able to do is to turn Liebniz' equation into a kind of parallel universe number system translator. Here is that equation,
u*π/4 = u/1 - u/3 + u/5 - u/7 + u/9.... (6)
u is the value of 1, relative to this universe, of the universe you are looking at. When u = 1 that defines this universe for equation 6. u*π is the value of π in the parallel universe relative to this universe.
The above demonstration of π showing that the number system is universe specific leads us to another definition of universe invariant and universe specific. A universe invariant concept will not change when π changes. A universe specific concept will change when π changes. This can be generalized too; given an unknown concept(unknown as to whether it is universe invariant or universe specific) and a known universe specific concept. The unknown concept will be universe invariant if when the two concepts are brought together and the universe specific concept is changed, the unknown will not change. If the unknown is universe specific, then it will change under the above conditions. We are not quite sure what we mean when we say "two concepts are brought together".
Ultimately, all concepts of math are concepts that are invented by mankind to deal with problems that he has encountered. Math could be classified as the set of solutions and methods, mankind has invented to solve a certain class of problems(Davis and Hersh 1981), (Skemp 1971). Numbers, are a solution mankind has come up with to solve a problem of counting(Barrow 1992). So, when we say that numbers are universe specific; we are saying that should mankind had been in a parallel universe and once again created the "number" solution to his counting problem and we were to compare that parallel universe solution to the same solution in this universe, then equation 6 would show how they are different. We don't know if there is a universe invariant "number" solution. To reiterate our definition of math again and this time include the concepts of universe invaraint and universe specific, we can say: Math is a set of solutions and methods, mankind has come up with to solve a certain class of problems, some of those solutions and methods are universe invariant and some of those solutions and methods are universe specific.
Discussion
The idea where each different value of π from infinity to zero is stacked like the layers of an onion as a result of the Big Bang is one of the most compelling ideas we have ever run across. The concepts of universe invariant and universe specific are byproducts of the different values of π. Throughout this paper we have struggled with defining, describing, and manipulating the concepts of universe invariant and universe specific. One of the key aspects of universe specific is that universe specific concepts can not be used to describe parallel universes. You can only use universe invariant concepts to describe parallel universes. That begs the question then; is this paper itself universe invariant? The answer to that question we do not have, but if it is it may explain why the math is so simple in it. You can not use universe specific math to define parallel universes, it has to be universe invariant. We believe the ideas in this paper are also simple for the same reason. We don't think this means that all universe invariant concepts are simple. We don't know. The Friedman cosmological models(Ciufolini and Wheeler 1995) all appear to be universe specific and therefore can not define parallel universes. One other thing about universe invariant and universe specific. In our definitions of these two concepts, we said they applied to one universe or all universes. We don't know if there are concepts that may apply to more the one universe but less then all. In any case we have called these types of concepts semi-universe invariant.
One of the questions that was continually raised while working on this model is, has mankind formulated any math where unbeknownst to the mathematician, he has created a universe invariant math. From the conditions outlined above, we would be looking for any maths that don't use; a specific value of π, number, or quantity. The best example of universe invariant math we have run across so far, is a book written in 1854 by George Boole. Here is a quote from "The Laws Of Thought" by George Boole, page 12(Boole 1854) "It is not the essence of mathematics to be conversant with the ideas of number and quantity." We have found threads of universe invariant math in many of the fields of math; Ring theory(Rowen 1991), all sorts of Algebras(Birkhoff and Mac Lane 1977), Euclidian and Non-euclidian geometries(Gans 1973), and even Number theory(Ore 1948) has universe invariant threads running through it. We would need to pull all these universe invariant concepts of math together, before we could go on to create a universe invariant physics. With a universe invariant physics in our hands we could then start building the methods that will take us to all these parallel universes that are described in this paper. We have dubbed the act of crossing into a parallel universe, pi-crossing.
References
Barrow, John, D. and Tipler, F.J., "The Anthropic Cosmological Principle". (1986). Oxford University Press.
Barrow, John, D., "Pi in the Sky: Counting, Thinking, and Being". (1992) Oxford University Press.
Birkhoff, Garrett, and Mac Lane, Saunders., "A Survey of Modern Algebra". (1977). Macmillian.
Blatner, David., "The Joy Of p". (1997). Walker Publishing.
Boole, George., "An Investigation Of The Laws Of Thought On Which Are Founded The Mathematical Theories Of Logic And Probabilities". (1958). Dover. Originally published (1854). Macmillan.
Ciufolini, Ignazio, and Wheeler, John, A., "Gravitation and Inertia". (1995). Princeton University Press.
Davis, Philip, J. and Hersh, Reuben., "The Mathematical Experience". (1981). Birkhäuser.
DeWitt, Bryce, S. and Graham, Neill., "The Many-Worlds Interpretation Of Quantum Mechanics". (1973). Princeton University Press.
Einstein, Albert., "The Meaning Of Relativity", 5th ed. (1956). Princeton University Press.
Everett, Hugh, III., "Relative State, Formulation Of Quantum Mechanics". Rev. Mod. Phys. 29, 454, (1957)
Gans, David., "An Introduction To Non-euclidian Geometery". (1973). Academic Press.
Hogan, Craig, J., Kirshner, Robert, P., and Suntzeff, Nicholas, B., "Surveying Space-time with Supernova". Sci. Am. January (1999).
Ore, Oystein., "Number Theory and It's History". (1948). McGraw-Hill.
Osserman, Robert., "Poetry Of The Universe: A Mathematical Exploration Of The Cosmos". (1995). Anchor books.
Peebles, P. J. E., "Principles Of Physical Cosmology". (1993). Princeton University Press.
Rowen, Louis, H., "Ring Theory". (1991). Academic Press.
Skemp, Richard, R., "The Psychology of Learning Mathematics". (1971). Penguin Books.
von Neuman, John., "Mathematical foundations of Quantum Mechanics". (1955). Princeton University Press.
Wambsganss, Joachim., "Gravity's Kaleidoscope". Sci. Am. November. (2001).
The ideas in this paper that were originated by Jim Akerlund were originally published in 1997 in the Newsgroup Alt.sci.physics.new-theories in the July/August time frame under the title "Foundations of Parallel Universe Math".
posted by picrosser at 10:38 PM
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