Like virtually all science fiction writers whose stories involve more than one planet, I was faced with the reality that travel only below light speed did not work. How one overcomes that problem did not seem to me to be trivial. Writers used wormholes, tunnels and spaceships that could just crash through the barrier. I used parallel universe with different values c. It all made me wonder about the place of time in my universe and what kind of universe was out there. Two things struck me; if the universe was expanding and if we could only see into the past, then what we saw from another planet happened at a time when our universe was less expanded. I developed a simple theory that I credited to a primitive girl genius+ with only a knowledge that there was a universe with bodies like suns and planets in it, and that light had a maximum speed.
A SPIRAL UNIVERSE:
TIME AS A GEOMETRIC DIMENSION
Bruce E. Dunn
I devised a one-dimensional universe. It is an Archimedes spiral drawn within a circle of radius 4.08 BY (I shall use BY instead of billions of years and BLY for billions of light years from here on). The spiral was then 13.8 BY long (3.38 times 4.08 which will be explained later), which represents the age of the universe. In fact, there were two spirals one in each direction.
Recent science suggests another way of looking at the universe. Suppose the radius is the 13.8 BY. In that case the spiral will be a 46.68 BLY long, approximately the length of the so-called observable universe (estimated at about 46.6 BLY). In the latter case the universe would be 598.24 square BY in size.
Figure 1
I determined that every body (or just spot) in the universe was not a point but rather a vector in time. This does not imply that every object vector can be traced to the origin or exists now. Also notice that I have made it so that some vectors cross the object vector twice generating two sightings of the “same” objects at different times. Finally, there is the “NOW” universe. It is 86.708 BLY long. It is the circumference of the circle. It cannot be observed by us. Each of us exists at one point on the “NOW” universe.
Figure 2
Thus, I have identified four universes. First is the vector representing the age of the universe (R) which we can see only through our own explorations of history; I have defined an observable universe, 2×46.68 BLY long; the ”NOW” universe (X2+ Y2=R2), 86.708 unit universe containing all objects as they presumably are now), but only visible at any one time to a 3-dimensional being; and the TOTAL universe being all the locations at all times (X2+ Y2 ≤R2), also only visible to a 3-dimensional being. How I came up with an Archimedes spiral was straightforward. Light from a planet one 10th of the way around round the circle must have been produced when the radius was1/10th less than at present. I drew a circle of radius one (circumference 2p) and drew a radius line every 36 degrees or1/10th of the way around the circle (That is an angle of .2p. I drew nine more circles inside all 1/10th of the radius apart. I connected straight lines from one circle to the next as shown below (A spiral of Theodoros looked possible, but it doesn’t go to zero). The result most resembled an Archimedes spiral (r=bq). q is an angle in our case between 0 and 2p. b is a constant. I then made the angles between the vectors smaller and smaller (little delta angles). The straight lines continued to converge to the centre in the same manner as the Archimedes spiral. So, I fitted a spiral to the lines. The result is also shown below in the figure and table.
Figure 3
Table 1
| Proportion of totaldiametre | Circle diametres for 13.8radius | Circlesegment lengths for 13.8 radius | Length line Segments for 13.8 radius | Length spiralSegments for 13.8 radius | Length Spiral for 13.8 radius |
| 1 | 27.565 | 8.660 | 8.102 | 8.352 | 46.685 |
| .9 | 24.814 | 7.794 | 7.247 | 7.400 | 38.333 |
| .8 | 22.560 | 6.928 | 6.396 | 6.651 | 30.836 |
| .7 | 19.253 | 6.062 | 5,541 | 5.802 | 24.196 |
| .6 | 16.543 | 5.193 | 4.689 | 4.966 | 18.383 |
| .5 | 13.786 | 4.330 | 3.787 | 4.141 | 13.341 |
| .4 | 11.028 | 3.464 | 3.081 | 3.309 | 9.274 |
| .3 | 8.271 | 2.598 | 2.378 | 2.568 | 5.948 |
| .2 | 5.514 | 1.732 | 1.808 | 1.905 | 3.368 |
| .1 | 2.757 | 0.866 | 1.384 | 1.462 | 1.462 |
Of course, the problem was that I had not been working in three dimensions, not even two. However, that was fixable. So, now to deal with 2- and 3-dimension universes in 3- and 4-dimensionl space. It is easy to visualize our spiral becoming a two-dimensional spiral just by rotating the circle in the first graphic through a third dimension. Rotating the three-dimensional spiral through a fourth dimension is more difficult, for me impossible, to visualize. That it is true is supported by an interesting fact. If you multiply the volume of a sphere of size n (in this discussion, consider an orange a sphere of size 3 and its cover (skin) size 2) by the volume of a cover of size 1 (covers a two-dimensional sphere – a circle covering a disk), you will obtain a size n+1 cover of a size n+2 ball. That is true no matter the value of n: 2, 4 or a billion. Of course, a circle represents a 1-dimensional rotation.
There are equations that I think work for the two-dimensional and three dimensional case.
________
r=\/ a2+b2+c2 or
_________
rq=b\/ x2+y2+z2
Recalling that cos²θ+ sin²θ=1, the following relations hold for the two-dimensional spiral.
x=q cosq
y=q sinq cosf
z=q sinq sinf
rq=bq = time distance
Another way to visualize this is to set the radius equal one and write:
1-z2 = x2+y2 and let z go from -1 to +1 continually stacking the circles.
This can be done for the spiral as well.
The figure shows the circles at a distance apart but in composing a sphere they would be contiguous.
Figure 4
For the 3-dimensional spiral, the following relations hold:
_________________________
rq=b\/x2+y2+z2+w2
x=q cosq
y=q sinq cosf
z=q sinq sinf cosx
w=q sinq sinf sin x
rq=bq = time distance
Using a similar procedure so that
1- w2 =x2+y2+z2 and now stacking the spheres in four dimensions as w goes from -1 to +1. This also can be done for the spiral.
In the two-dimensional case, the position of a planet in space and time is defined by r, (time distance), θ, and φ. In the three-dimensional case the position is defined by r, θ, φ, and ξ. In all three cases, however, the universe is a spiral. We look down the spiral towards the beginning. To see the whole universe including time, we must be five dimensional. The length of the spiral is given by:
_________ __________
L = (b/2)*[q*Ö1+q2)*Ln(qÖ1+q2}].
q approaches 6.283 (2p). The constant b is .159 (1/2p) for a radius of one and 2.196 (billion) for a radius of 13 (billion). It is also true that the length of a spiral is always 3.383 times the radius of a full circle.
Now a very interesting point is that we (as well as anybody on any planet) are always at q=2*p looking down the spiral to smaller values of q. For us
r=bq = b*6.2832, Since r=13.8 and b=2.196, again
L= 46.683.
This equation would apply for any planet in the NOW universe. or for any other planet in the TOTAL Universe. Of course, in neither case will some one elsewhere in the NOW universe or at a b different from 2.219 see the same as us because from their perspective they will be at place where, although q=2*p to them, their spiral will travel through different parts of the Total Universe, and when b is less than 2.219. (See above.) their will be a different length L(also see above). In addition the star we see at say b=1.78 will of course have an observable universe. It will, like us, perceive itself as at q=2*p. and the universe as about 11.28 billion years old. Those in our NOW Universe will have the same b, but will be at a different place on the cover and so will observe a different universe, but it should be about the same age as ours assuming even expansion. In fact if you want to be picky everyone has their own universe.
I guess the meta-physical question is to ask if the univere is really expanding or whether all values of b exist all at once, that is 1.78, 2.196 and perhaps 3.00. In fact our NOW universe is defined by having b=2.219. From the perspective of a 5-dimensional being all values of b may be equally valid to define someone’s NOW universe. It is interesting to speculate in the static case what this 5-dimensional being would see as compared to us. We see process in time, while it sees a state. Perhaps an analogy would be a disc that has a dark-to-light gradient from the centre outwards. Our 3-dimensional observer would see it as a gradient; we 1-dimensional people would see it as a process in time. As the eons passed our 1-dimensional people would see their universe as getting lighter. More of this later.
be a different length L (also see above). In addition the star we see at say b=1.78 will of course have an observable universe. It will, like us, perceive itself as at q=2*p. and the universe as about 11.28 billion years old. Those in our NOW Universe will have the same b, but will be at a different place on the cover and so will observe a different universe, but it should be about the same age as ours assuming even expansion. In fact if you want to be picky everyone has their own universe.
I guess the meta-physical question is to ask if the univere is really expanding or whether all values of b exist all at once, that is 1.78, 2.196 and perhaps 3.00. In fact our NOW universe is defined by having b=2.219. From the perspective of a 5-dimensional being all values of b may be equally valid to define someone’s NOW universe. It is interesting to speculate in the static case what this 5-dimensional being would see as compared to us. We see process in time, while it sees a state. Perhaps an analogy would be a disc that has a dark-to-light gradient from the centre outwards. Our 3-dimensional observer would see it as a gradient; we 1-dimensional people would see it as a process in time. As the eons passed our 1-dimensional people would see their universe as getting lighter. More of this later.
Figure 5
One very important thing to recognize is that the vectors are much further apart than they were billions of years ago. They are spreading. Let me illustrate this and a few implications by following three vectors. Vector S (VS) is the one our sun is on. I’ll set its angle at 0 or 2p. I will pick another that is 360 (8.66 BLY) away around the NOW universe (Call it VA) It’s seen when it was 12.42 BY old. At that time, it was only 7.94 BLY away from VS around the NOW universe. If we went the other way around the NOW universe, the distance would be 78.05 BLY. The third vector is 180o around the NOW universe from VS. Call it VB. In our NOW universe it is 43.3 BLY no matter which direction, but when the universe was 12.42 BY old, it was only 38.97 BLY. To us in VS, it is about 28 BLY in the observable universe.
Let us go back to when the universe was 100 million years (not light years) old. These vectors may be said to have had stars. Our VS was not one to have them, but stars were perhaps being created. At that time, the distance between VS and VA was 628,318 miles, less than the distance from the present earth to the moon and back. It would take light 3.28 seconds to travel from VA to VS as opposed to 8.352 billion years now. As important is the fact that in our universe VA is 1.38 BY behind VS, whereas in year 100 million it was only a little over three seconds behind. Our own sun is around 7 minutes behind us. Looking back the other way around the circle, VB was much further behind but still only about 66 seconds behind as opposed to 45.223 billion years behind now assuming we could even see it. VB now is 38.341 billion years younger no matter which way we look. After100 million years it would have been about 50 seconds behind VS. Let us assume we started looking at either one from VS at 100 million years and kept continuously looking to the present. Time would have seemed to move more slowly on VA so that from our present position on NOW, it would appear 8.352 billion years behind us.
Pushing this further, let us assume that 180 further around NOW, there is another vector (VC) with an observer who started watching VA at the same time as the observer on VS. When the observer started watching the points VA would have appeared to be only about 1.5 seconds behind VC. At present, VA would appear 4.176 BY younger than VC to the observer on VC. The interesting part is that it would appear 12.102 BY younger to us. It doesn’t add up. That is because any message we or any other point on a vector receives now is always from an earlier stage in the history of the sender. The Arecibo message was sent about 48 years ago. It would not have reached a planet 50 light years away. When it does in two years, it will convey us as we were in 1974. A planet 300 light years away if it could somehow detect us, it might conclude we were preindustrial. If we should receive signal from a planet 5000 light years away, who could guess what their level of development would be now. If they were looking at us, they would see the beginning of Egyptian hieroglyphs. In fact, no adult over twenty could expect to send a signal to say Trappist 1 (39 LY away with7 planets) and expect an answer in their lifetime. Even a message sent in 1974 could not have been returned yet.
Here is an interesting aside. Many scientists speak of a ten dimensional universe. They curl up the extra dimensons because we can’t see them. Of course we can’t. We’d have to be at least 11-dimensional beings. We can’t even truly see three dimensions. We use off-set two-dimensional pictures. Instead, suppose our four dimensions (including time) were the result of something like below. There are 10-dimensions (a,b,c,……j). A linear transformation converts them into four linear orthogonal combinations (W, X, Y, Z). using four orthogonal vectors. Cjr second matrix is a matrix of constants that renders the radius of the sphere to equal 13.8 (b=2.219). The mi s are the ten coordinates of a point in the original system. At the end of the transformation our point in the 2.219 NOW universe is defined by W,X,Y,Z.
V C D=V*C
he coordinates of a 4-dimensional point.
A 3-dimensional cover of a 4-sphere (ball) defined by all points fitting
r42 = W2 + X2 + Y2 + Z2 = 13.82 , is our NOW universe. Note
r42 < W2 + X2 + Y2 + Z2< 13.82 , is our history.
r42 > W2 + X2 + Y2 + Z2> 13.82 , is our future.
Note that each of new coordinates is a linear combination of all ten-dimensions. This new sphere can have a spiral defined upon it. One implication is that the influence of each of the ten dimensions might be different in different parts of our 3-d NOW universe as well as our observable universe. Would this mean that the laws of the universe might differ in different places? Also it is evident that by choosing different vectors V there can be many universes. It also important to note that the 13.8 is continuous if minisculy growing.
This leads one to ask whether there is a greater universe, a ?- dimensional (10 has appeal) infinite domain in which our 4-dimensional universe is imbedded as above or are we part of a 4-dimensiona space that is all that can be meaningfully talked about? Is our NOW layer really the cover of the sphere or are we just at a point 13.8 BLY from the centre of an infinite sphere? There could be many universes of various dimensions, but we are interested in just one. Outside the 3-dimensional cover of the the 4-dimensional space may be raw space if the greater universe is truly many dimensional. It is presumably rather chaotic except that the closer events are to each other the more highly correlated they may be. Inside the cover is history. The cover is the NOW universe. It is 3- dimensional. It grows cubically as the 4-dimensional ball expands
(4-dimensions: (1/2)p2R4 =4-d size of ball; 2p2R3=3-d size of cover)
with the growth of the radius. The cover is so transitory, it can hardly be thought of as existing. It is important to note that it is only an infinitely thin skin with respect to the fourth dimension. In fact the 3-dimensional cover of a 4-dimensional ball has about 4.71 (1.5p) times greater volume than a 3-dimensional ball having the same radius. Because of its proximity to the raw outer universe, it must be highly correlated with the close events. Once it passes the events in the raw universe, these events collapse and become history basicly immutable, much as described in the Copenhagen model. A problen is that while the cover bounds our universe from the fourth dimension, it does not do so against higher dimensions. Perhaps there are no higher dimensions or perhaps once collapsed into history it can no longer be changed by them or perhaps even recognized.
Here is an interesting (at least to me) thought regarding a 5-dimensional super universe (It does not eliminate a many-more dimensioned universe. We could be visualizing a 5-dimensional ball in an n-dimensonal universe.). Let us start by writing the equation for the cover of a 4-dimensional ball. First, we must define a ball as opposed to a cover. A baseball is a 3-dimensonal ball with a two-dimensional cover. A 4-dimensional ball’s equation is written as:
X2+Y2+Z2 +W2≤R2
The corresponding 3-dimensional cover would be written as:
X2+Y2+Z2+W2 =R2
Suppose we now write the equation as:
X2+Y2+Z2 = R2- W2
This is a 2-dimensional cover for a 3-dimensonal ball. It is a different cover for every value of W while holding R constant, an infinite number of covers then. However, they are all in the space of the 4-dimensional ball. The 4-dimensional ball itself is made up of an infinite number of 3-dimensional balls:
X2+Y2+Z2 ≤R2- W2
Note that as W goes from zero to R the size of the ball decreases. Incidentally W is arbitrary, any dimension would be the same.
To be able to visualize this better consider the 3-dimenional ball:
X2+Y2+Z2 ≤R2
Its cover is:
X2+Y2+Z2 = R2.
Continuing as before:
X2+Y2 = R2- Z2
Is an infinitude of circles, the diameter depending on Z. Also:
X2+Y2 ≤R2- Z2
It is a disc or a stack of discs as the values of Z changes.
An interesting sidelight is that if we start with:
X2+Y2+Z2+V2+W2 =R2,
It is a 4-dimensional cover of a 5-dimensional ball. Now we could write:
X2+Y2+Z2+V2 =R2-W2
That would be infinitude of 3-dimensional covers like or postulated universes. So, if a super universe were 5- dimensions there could be in infinite number of 3-dimensional universes. That would probably make the many-universe proponents happy except this is mathematics and may have nothing to do with reality.
If our universe started as a bubble, there is modified chaos outside these boundries and, what happens at that boundry? Clearly, at least at the classical level, the correlation between R and R+D is essentially one. The interaction between the cover and chaos outside would occur at the micro level. Such an interaction would not carry on to the classical level due a process much like central limit theorem (eg. There are 7×1027 atoms in the body of a 70 kg. man.). However, at the micro level there might be far more interaction at the boundry. I might also suggest that if the higher dimensions do impact our historical universe at the micro level and have little impact on the classical world we see, it is possible that the mass of the partcles outside the NOW universe might have a gravitatinal force facilitating expansion. How the past might effect gravity is a question.
There are several theories regarding the beginning, most centre around the ‘Big Bang’. One is appealing because it solves the anthropic problem. Suppose bubbles appear in the ?-dimensioned maelstrom now outside our 3-dimensional cover which is us. Almost invariably they break up as in turbulent water, but at least once, maybe more often, a 4(or 5)-dimensional bubble appears that has a 3 (or 4)-dimensional cover(s) with properties that allow the bubble to persist. At first the inside of the bubble would be much like the outside, but as it stabilized, it would become its own thing and eventually develop what I call history. As it did so, it would exert more and more influence over the interaction at the boundary, although its dominance may have been less at the micro level. The characteristics that allowed it to stay together would become the constants we see today and realize are necessary for the universe as we know it (if the bubble is 5-dimensional at least one of the many 3-dimensional covers has the right constants for us). Then, the bubble grew as the mass outside pulled it. The end? Perhaps the skin (cover) as it expands will be come too weak and the bubble will burst.
Here are two classical metaphors for probability collapse. I go to a restaurant. After looking at the menu, I decide I will have one of five items. I mull this in my mind until the waiter arrive. At that point the five possible choices collapse to one. The waiter is the measuring devise. Similarly, assume my date and I go to a restaurant and see two items we like and decide they are so good we will each choose a different one and split them. The waiter comes and I choose item A. I instantly know my date chooses B. This does not require faster than light information transmission. Depending on her distance, it will take some time for her to know that she will make that decision, but I will know instantly and I could even order correctly for her before she finds out. When she does find out she will immediately know what I have ordered. Equally as important is the fact that neither choice exists before the measurement.
Now let’s return to the question of what happens to our NOW universe at the meeting point of our NOW universe and whatever if anything is beyond. There seem to be three basic possibilities that are discussed.
1) There is nothing, an absolute nothing, beyond.
In this interpretation, there is no point in talking about what’s outside. Our universe simply expands. There is no boundary at which one must theorize about the nature of the interaction between the universe and an outside. Most quantum theories appear to subscribe to this point of view
2) The 4-dimensional Total universe is infinite.
A 5-dimensional person would see it as a static infinite landscape on which we occupy a small spot at b=2.219. We perceive it like we perceive a movie or a DVD. We see it all happening in sequence, but the story is really already there. Unlike a film, there re no reruns. There still exists the spiral observable universe, however. Our observable universe just doesn’t start at a cover. The problem is that it is a completly deterministic universe. As part of it we can’t even independently speculate about it.
3) On the other side of the cover there is something.
Certainly, the first two are the most generally supported by scientists, but the third one fits well with the idea of the universe beginning as a 4- or 5-dimensional bubble. At least two questions arise. From what kind of mileu did the bubble form, and what is the nature of the transition. The most plausible answer to what is out there is to admit not knowing, but let’s take a guess. My guess is an n-dimensional chaotic (meant technically) field, that collapses at the boundry to a value, but one that is highly influenced by the state of the part of the Now universe it contacts. One can see the boundry as the waiter in the above metaphor. But the boundry is truly infinitly thin in the four-dimensional universe, essentially not there, but of course it is and quite large with respect to a three-dimensional universe, but passes into history immediately. All an observor normally sees is streaming history. Suppose, they take a picture (a measurement) and freeze time at one instant. We do that with film and the pause function on television. If not, events are a stream moving into history. This must be even more true at the micro level since change is the name of the game. At the macro level we miss most of the change. Even when something is history, it only appears in a given state for an instant.
In the last case we must realize that all our research about our universe produce data from:
X2+Y2+Z2+W2< R2 or more precisely rq=bq (q<2p), not X2+Y2+Z2+W2=R2
We cannot see or directly measure within the NOW universe.
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A Link to the Beginning 