Foundations Of Parallel Universe Math
Jim Akerlund
This is my second posting of this. I am looking for comments, or
suggestions. I would appreciate any responces sent by email.
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)
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
discrepencey 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.
Jim Akerlund