's World of Ideas: Math, Science and technicalitiesSpace & Time
•Working definition of space & time
•What really is Time Travel?
•Multidimensional Time
•Bending of time: why faster than light travel would compromise causality
•P Theory/F Theory (Patrix/Father): The Third Continuum
•Inertial Frame of Reference for the Speed of Light
•Noncommutative Space
•Representations of space and time in two scifi stories
•Forgotten 4D object: the Duo-cone
Various topics:
Alternate Hexadecimal codes
Definitions of "Fortune/Luck" and "Money"
Division between man's soul and spirit: the 12 primary emotions
Metric Time
When did the new century/millenium begin?
Color mixing and perception
Math Trivia
•"Great Zero" (Lowest common multiple of all numbers from 1 to 100)
•Alternate negative number system
•Why is the "powers of ten" line asymmetrical?
•Grownup name of "googol"
•How many sides does a circle have?
A working definition of space and time
SPACE:
If I take a ruler and show one foot, or one inch, that is based on this wood, plastic or metal ruler, with markings etched into it. I can point to objects in the room several feet away, or a building a mile away in one direction, a body of water a mile away in another direction, or some other land feature in another direction. All of this is planted on raised levels of earth exposed above the sea level which remain the same shape. And these lands are all positioned on the globe of earth. They are basically stable and do not move, so this is how space (places) are reckoned on earth. It is all MATTER.
Now beyond earth it is not as definite. The entire surface of the earth is "moving"; rotating around the axis. Then, the earth itself is revolving around the sun, and the sun revolves around the center of the galaxy, and even the galaxies are moving in relation to one another. So as one book says you cannot drop anchor in the universe and determine a universal "location" here. So location and distance can be defined by local matter. Space is marked by matter in a given arrangement. So the definition of space we arrive at is the medium in which the relative positions of matter are measured. (As another book says, if you had a universe with no matter in it at all, distance and location would have no meaning or be undeterminable).
TIME:
I know one second or one minute has passed by looking at a clock; once again, matter. Naturally, the point on the surface of the earth I am standing on will turn till it faces the sun, and then after a period of time will turn away. This is day and night. In a few hundred of these rotations, the angle of sunlight will change, making it hotter or colder. This is the seasons making up a year. We are born, and our bodies change as they get older. On a given day, I get up, and do this thing now, and then I go and do something else afterwards, until the day is filled up. Meanwhile, all kinds of things are happening in the outside world, and in the universe. Certain people are alive now who weren't in the past, or won't be in the future. What first got me thinking about this, is seeing how the subways and many buildings were in one shape 25-30 years ago, but have been renovated now. When I think of a given year, I think about what everything looked like, and what the culture (music, TV, what was in style, etc) was like, as well as what school or job I was in, and where I lived. Stars have become novas, and the planets line up in different ways. We have sent spacecraft to them, and beyond. All of this is also MATTER. Even our perception of these things comes from our brains; also matter. So time is marked by circumstances (states of matter), or events (changes in matter or energy). So time can be defined as an imaginary medium in which the state of or changes in matter is measured. (It can be considered imaginary because you can only be in one point at that given time; you are not really "moving through" anything). This too is basically local, because at different speeds, and in or out of gravity fields it has been shown that the changes to matter (natural processes or even our perception) occur at different rates. So for this reason, time appears to move at different speeds, or even stop completely in certain extreme cases (light speed, black hole event horizon).
TIME TRAVEL?
Because time has been observed slowing down or speeding up, and we seem to move through it like we do through space, many feel why can't we turn around and go back in it? But now with our definition of time as a[n imaginary] medium in which change of matter is measured, going back in time means reversing all changes to every particle of matter around us. For something that was burned up, each carbonized particle must be un-carbonized and come back together as it was at the given time you want to return to. Someone who died must come back to life, his body un-decay and get younger. Someone born must grow down and return to an unfertilized egg and separate sperm cell. Everything else in the universe must simultaneously revert to prior states. Perhaps you could localize it and only make everything in one room revert, but as soon as you stepped out of that room, it would be the present again. What point in time would it really be? (This would be analogous to twisting or bending time).
And it seems then, that one person could not even go back in time by himself (if on a universal scale). Everything would go backwards and everyone with it. Their consciousnesses too (since that is a material process)! So perhaps, no one could even perceive a reversal of time. Unless the one traveling exempted himself from the reversing process; remaining in the present of his own consciousness. So actually, that person is not traveling in time, but actually sending everyone else; the entire universe around him, back in time. (What would happen to stuff he changed? It would still probably un-change, but it would appear to be a ghost where he was if he doesn't un-enact everything he did. If he could then restart everything forward again, perhaps he could alter the past. But what would happen to everyone else's consciousness? I'm not exactly sure.)
But it does not seem possible to for someone to reverse the entire universe around him, so it does not seem that going back in time is possible. What are you going "back" to? Just a particular set of simultaneous states of matter that have since been changed. It's not even really "back"; as if it were a location in space that still exists and thus could be returned to. Existence is only in one point of time. No-one knows how to recreate this simultaneous state of all matter, since matter constantly changes and no two states of it are ever exactly alike (just like no two pieces of matter are ever just alike). And then you have the fact that even "simultaneity" is reckoned differently to different observers at different speeds, as General Relativity teaches. So again, what would even determine a "simultaneous" point of the "past"?
Scientists have theorized possible ways to return to the past, such as wormholes, and something about a rotating mass of dust spanning the universe (?); but these are all conjectural and un-provable.
MULTIDIMENSIONAL TIME?
The easiest way to conceive of two-dimensional time is to picture a bunch of people living and moving about in an area filled with statues. But these statues are really the inhabitants of a perpendicular (2nd) time dimension. Our whole time line lies only in an instant of theirs, so we see them frozen in an instant of time; and their whole time line lies in an instant of ours. So we are frozen in time to them. Now how could we appear frozen to each other? Well, if you could use a stopwatch (like in Twilight Zone story "A Kind of Stopwatch", or the made for syndicated TV movie The Girl, the Gold Watch and Everything, and it happened automatically without a stopwatch in Stephen King's "the Langoliers"), and leave our time, and enter theirs, then everything in this world would stop, and the frozen statues of that world would start up, moving around. If you stopped the watch, then that world would stop, and this one would start again, all the objects in this time dimension picking up exactly where you left off. But now all the people in the 2nd time dimension have changed position from where you left off. So you're really not back in our time dimension, but in a parallel time dimension to ours, in a different position in the 2nd time dimension. From this dimension, you would disappear; going into hypertime is pretty much like disappearing into hyperspace. You could travel through both dimensions at the same time, in which you would see both times running in slow motion. With the right trajectory, you could reach the same point described above by stopping one time, living completely in another, and entering the parallel-to-the-first time. If you change your path to a smaller angle to one of the time lines, that one speeds up, and the other one slows down.
Now all of this could be taking place amidst other statues which are inhabitants of a third time dimension, and you could carry it to any number of dimensions, getting more complex as you increase the number, just like with space dimensions.
Where this might fail, is that the most popular definition of time, is in terms of the causality of events. Events only have one cause. Two dimensional time would be defined with events having two causes. And 3D time would have events with 3 causes. Multidimensional events are not clearly defined in the scenario I've given. Instead, it is more like our time, where if you approach the speed of light relative to earth, then the other frame of reference's proper time will stop. One Scientific American article on the subject a few years ago ("Parallel Universes", Max Tegmark, 5-2003, p.45 http://www.scribd.com/doc/17662852/Parallel-Universes), had a table with several space and time dimension number combinations, and only 3D space/1D time could sustain reality as we know it. Less dimensions of space, "complex structures cannot exist". 0-D space and/or time, a red backgrounded "events are completely unpredictable". More dimensions of space "atoms are unstable". 2 or more dimensions of space with more dimensions of time together, had a green-backgrounded "events are completely unpredictable". I wonder if the different color meant it was not as bad as the 0-D scenarios (which obviously could not have any events at all). The flipside of our universe: 3D time/1D space, had "fields are unstable". I don’t know what that means, but it is funny that that does not seem to be as restricted as the other combinations. So complex structures can hypothetically exist in that 1D space, unlike ones with less time dimensions? Likewise, 1D space with 4 or 5 dimensions of time also had "atoms are unstable", like the reverse.
I wonder what would those universes be like. Why would three or more dimensions of time only be viable with one dimension of space, and not more? Who knows. And who says such "universes" would even have the same laws ("the standard Model"), which governs things such as atomic stability? With different laws; atoms might be stable with more dimensions! (And Michio Kaku and others have theorized, apart of string theory, on other 3D space/1D time universes like this one that might have other than Standard Model laws, including a cataclysmic condition where the laws in this universe could change from the standard model, and matter would all break down and reform!)
Here, btw, is a great way of explaining the fourth dimension, from scratch, for beginners: Fourth Dimension: Tetraspace
Another good explanation of the fourth dimension: "A Study of Dimensions" by Bill Price. And a video building up to higher dimensions in a similar fashion is at the bottom.
Explanation of "bending" of time
Micho Kaku's Hyperspace: A scientific Odyssey of Parallel Universes, Time Warps and the 10th Dimension (NY, Oxford Press, 1994) renewed interest in alot of this stuff for me. (I first got into it through Carl Sagan's Cosmos series, which among other things, explained the fourth dimension, and it's hypercube, the "tesseract", by comparing it to how the second dimension relates to our space. This is instrumental in trying to understand higher dimensions).
One curiosity Kaku mentioned, to try to explain why time supposedly slows down when you approach the speed of light, is an imaginary scenario where the speed of light was only 30mph; the speed a subway train enters the station. (p.84; note, p.341). It would appear compressed down to maybe about 1 foot long, yet to the people on the train, it would be the station that appears compressed to one foot long. The foot would expand to the full 600ft as the train slowed down. This seems paradoxical, as how can both be shorter than the other? But he points out that it takes time to do the measurement. So if a person on the platform takes out a foot long ruler, it will measure the entire train. Yet if he drops the ruler onto the platform so that both ends hit the platform simultaneously as the train passes by, the people on the train will not see the ends hit the ground simultaneously. One end of the (compressed) ruler hits the floor first as the front of the train goes by, and the other end hits as the station is completely past. So this is how the paradox of a one foot ruler measuring a 600 ft train is resolved. But now it rasises another problem.
If the ruler is dropped so that both ends touch the ground at the same time, then it is always parallel to the ground. So how do the observers on the train see one side touch the ground first —i.e. it is apparently not parallel with the ground? Well, the compressed ruler is seen in the compressed station standing up on one end, so it measures 1/600 ft horizontally, but one foot vertically. The 1 inch mark is one inch off the ground; the 2 inch mark is 2 inches off the ground; the other end is one foot off the ground. How does this happen when the entire ruler was the same distances off the ground to the other observer? Well, think of it this way: was the one inch marker ever one inch off the ground? Yes; it had to pass that height when the entire ruler passed that point right before it hit the ground. Was the 2 inch mark ever 2 inches off the ground? Yes; right before it was one inch; when the entire ruler (including the 2 inch mark) passed a two inch height off the ground. Was the other end of the ruler ever one foot off the ground? Yes, even more time before that, when the entire ruler passed that height of one foot off the ground.
So do you see what is happening? The observers on the train are witnessing the actual bending of time. They are seeing each point of the ruler at a different time, when it was at different heights from the ground! (Going further into the past the further towards the rear you go). Likewise, the observers in the station see the rear of the train in the present; what we can call event B; but the front of the train apparently one foot ahead is actually being seen at event A, in the past; because the front is really 600 feet ahead, having already passed the one foot point. So the train is compressed because each point on it is being seen at different points in its own proper time, going further into the past, the further to the front you go.
My reason for discussing this, is not only to draw further on Kaku's illustration; but also to illustrate the supposed problem with faster than light travel, as also discussed by Kaku, Rudy Rucker's similar book The Fourth Dimension, and especially William Kaufmann's The Cosmic Frontiers of General Relativity. If time supposedly stops at the speed of light, then you may think it would go backwards, faster than the speed of light. Some have created stories, and even a poem based on this. But it is not necessarily true. If you headed towards Alpha Centauri at four times the speed of light, it would still take a whole year on earth's "coordinate time" (the time of observers who maintain the same relative velocity from the start of the observation, or in their own frame of reference, remain "at rest") to get there. You then turn around and return at the same speed, and your total journey would still be two years.
Time would not become negative, because the equations governing this phenomenon include square roots; so when one factor falls below 0, you end up with not negative numbers, but rather square roots of negative numbers, which are known as imaginary numbers. That is a fitting description of FTL mass, length and time when compared between proper and coordinate time! Something moving AT the speed of light supposedly would be like infinite speed. You could go anywhere in the universe instantly in your proper time, though it would be billions of years in coordinate time. What we would normally consider infinite speed, like the "Kind of Stopwatch" example, would be well into the FTL range (of course), and cause problems of causality, even though you would THINK you were still not going into the past, as shall be explained. It seems like some sort of disconnect between proper and coordinate time. A very finite speed of 670,616,629 miles per hour becomes like infinite speed in several respects. You would think light would move at infinite speed. It is not even light itself that determines this speed. It's that light has no mass, and any particle having no mass will normally move at the fastest possible velocity, and this for some reason is not instant to the rest of the universe, though it is to the one traveling at that speed. I have no idea what the FTL traveler's "imaginary [proper] time" would be like, as we watched him make the two year round trip in our coordinate time. [I've since surmised what this would be like, below!]
The problem with the 4c trip occurs if on the way out, you "catch up to" another astronaut, heading that same direction just under the speed of light. Instead of floating around Alpha Centauri, you beam aboard this other ship, and ride with him for a couple of seconds, and then try to return to earth at the same FTL speed, from his frame of reference!
With the "bent time subway station" illustration; I can now easily show what happens. Let's say people on the train and station try to fire a tachyon (a hypothetical particle that can only move faster than light) to each other. This was conceived to get around the issue that we could never accelerate to the speed of light, let alone beyond it, because of the time contraction (as well as mass and required energy increasing as well). So they might have some sort of receptors that emit and register its reception with an indicator light or something, since they could not actually "hold", "throw" or "catch" such a particle at velocities less than light. If the person in the station at event A passes the tachyon to someone at the front of the train as it passes by him, and then that person passes it to someone at the rear of the train (at the same speed. Perhaps infinite speed, even, or it can be any finite multiple of the speed of light), and then that person tries to fire it at back to the station, what happens?
When event A is seen by the front of the train, event B hasn't happened yet!!. He sees the back end of the ruler still off the ground, like before the train passed it. And this makes sense, because to the observer at the front of the train, the back of the train is not 1 foot, or 1/600 ft back (the distance of the back end of the ruler from the front end) — it's 600 ft back! — not even in the station yet! From the viewpoint of the simultaneous event A-B in the station, it is in the past. The tachyon, leaving the back of the train, which is not in the station yet, will get to the station before the back of the train, and events A and B! It jumped into the past and turned up before it was originally even omitted!
Also, when the back of the train reaches event B, from that frame of reference, the front of the train is not 1 foot or 1/600 ft ahead, it is 600 feet ahead —already past the station. So from one viewpoint, when the tachyon is first passed from the station to the train at event A, it is jumping into the past (because the station sees event B simultaneously with event A, yet from event B, the front of the train was really past the station already, and thus only in the station in the past). From the other viewpoint, it moves into the past going from the front of the train to the back.
So basically, scientists conclude that if tachyons exist, they do not interact with the familiar "tardyon" matter of the visible universe. So a receptor would never even be able to register receipt of a tachyon. They all share the same space, but have the space and time coordinates sort of reversed from each other. I wonder if there might be some other "cosmic censorship" principle that might prevent that scenario. Like spacetime itself "remembers" where/when you or the tachyon originated, and your "world line" (projection of locations in space through time generated as a path through "space-time") simply would never be allowed to cross itself, (as would happen if you went back in time, returned to your starting place, and then resumed normal time), and you would simply appear at your starting point at the next allowable time after you left.
P Theory/F Theory (Patrix/Father): The Third Continuum
Shorter version with illustrations
Some theorists have speculated on alternate realities. String theorist Brian Greene even says these are implied by quantum mechanics! Particularly when the subject of reverse time travel comes up. If you go back into the past, and begin altering reality, causality is violated, and the universe would make no sense. So they speculate that if you do go back into the past, you would in fact enter a parallel spacetime universe in which events are different. Each instant in time, when an event occurs, such as when we make one decision or another, becomes like a "fork" where multiple realities split off, and only one becomes realized in our experience; the one "chosen" over the others.
So there is a notion that the other, unchosen realities ("counterfactuals") might exist in some way, yet we simply don't have access to them. If you accept that the universe and all its matter, events and space and time itself consists of vibrating loops of string, then it doesn't seem farfetched to believe that all of these other states of them (collective vibrational patterns) exist, yet only one path through them has been chosen. I have seen two articles, using both space and time to access these counterfactuals. One, in Scientific American, proposes an infinite space, and the further out you go, you run out of possible configurations of matter and energy. So all matter can do is to start repeating itself. Eventually you will run across exact and near copies of everything we see around us, including ourselves.
The other one (http://www.exitmundi.nl/eternity.htm) does the same thing using infinite time. After the entire universe burns itself out and cools into nothingness, the quantum uncertainty principle proposes matter will be randomly popping in and out of existence. Given enough time, more complex objects will appear, including eventually, copies of everything we see around us, including ourselves and a new big bang.
One problem I have with infinite space or time, is that matter and events become zero (infinitessimal) in the overall scale of the universe.
But to me, parallel realities created from being "unchosen" in the here and now (and not cast off into infinite distance or future simply because the universe has run out of possible combinations) creates a new continuum or "vector space", in addition to space and time. Space is the medium in which we measure the relative locations of matter and events. We use it to get from one location (marked by matter or an event) to another (see above). Time is the medium in which we measure the chain of causality or simultaneity between events. We use it as we live and experience one event after another.
So think, what medium would be the one "travelled" by jumping straight from the here and now in our universe, to a parallel universe where we wore red instead of blue in the here and now? To simply this, which medium connects these two corresponding points marked by these alternate events?
You might think time, but time is marked by a causal chain. One event causes the other. Yet an event we are experiencing now did not cause an alternate event in a parallel universe. They are results of a different choice (event) at a point further back in time where the two realities merge. The causal chain lies in the forward time dimension itself, not in the perpedicular dimension in which you jump from one to another.
It is also not space. When we think of "parallel universes", we are usually thinking of space, where one space is embedded in another space with more dimensions, containing other lower dimensional spaces ("branes") parallel to the first one. Still, the higher dimension space would still have the first dimension, of length, so you could measure the distance between parallel branes. If you had access to that hyperspace, you coul d move freely back and forth between them. Yet, alternative realities have no logical spatial 'distance" between them. And no necessary way to move back and forth between them.
You might think that alternate realities would be the temporal counterpart to embedding in higher spaces. But again, in higher dimensions, the medium in which the lower spaces are located relative to each other is still space. The medium between counterfactuals is not time, because, once again, the relationship between parallel corresponding events is not causal.
So this is an all new medium. To give it a familiar monosyllabic name like "space" and "time", I would call it "chance".
I always like things like this in threes. So we have space, time and chance. It sort of parallels the Christian concept of the Trinity. And creationist Henry Morris and others have even linked the concept. While the Trinity is often thought of as three equal beings sitting side by side, Morris framed them in terms of a reference, a visual form and an experential form. So the Father is what God is, or who we reference when speaking of God. The Son is God made flesh, visible and tangible in the world (the Father cannot be contained in space and time). The Holy Spirit is how God is experienced (in the heart).
Morris ultimately still holds the "traditional" symmetrical view of three "equals" side by side. But when I researched all of this, I found that the pre-Nicene church fathers actually held a non-symmetrical view in which the Father was the Godhead from eternity, and the Son and Spirit were manifested from Him in time (i.e. at the birth of Christ). Forms of this were later revived by the likes of Marcellus and Servetus, but the church by then condemned them in favor of the symmetrical view, which has become the official dogma ever since. (More on this at Triune.html)
Morris had linked this tri-unity to the universe, which he said was referenced to space, seen in matter and experienced through time. (and space had its three dimensions; and time had past, present and future, etc).
I did not like making space the "Father-like" element and matter the "Son-like" element. I had already started coming to see space and time as the visible/experential counterparts, and expected the third continuum to be another kind of continuum like space and time, unlike matter. Matter is what occupies space, and that may appear to fit the "visible manifestation" role, but it is not the same sort of thing; i.e. not a vector space. (Though mass is often the third measurement next to distance and duration in equations). You can imagine a universe without matter. Measurement of distance and events then becomes irrelevant, but it is still hypothetically possible. But a universe with no space or time is a whole different kind of existence.
So if we look at the universe as the entire set of possibilities, then this new "chance" continuum is what it is "referenced" to, and space and time are manifestations of it, with space as what it is seen in and time what it is experienced in. This makes sense, as space tells you "where" and time tells you "when", but chance tells you "WHAT", in the first place!
Just as you can get "close" to a point in space (with gradual changes as you pass one material object after another. Think of the transition from country to city as you get closer and the density of people and buildings gradually increases), and closer to a point in time (as one event leads to another, and a new "present" takes shape. Think of any transformation in time. A flower or other living thing growing, etc), you can also hypothetically/theoretically move closer to a point in chance by changing things to alternate states.
Like if I chart my position using the four dimensions of spacetime. At such and such time, I am at a particular longitude, latitude and altitude. In an alternate reality, I may have moved to a point five feet away in latitude. Or maybe ten feet away. That would indicate a further "distance" in chance, since five feet away is "closer" to the starting reality than ten feet away. Of course this will affect the choices I would have had to have made in space and time, in order to get to that point. Just like space and time determine where you can go in each other. So this is interchangeable, just like space and time, and can also be measured. While space has three dimensions, and time has only one dimension, the number of dimensions in chance seems to be unlimited.
I've recently been thinking more on this, and trying to come up with even more fundamental definitions.
I would say that with strings as the fundamental fabric of space-time, any given point in time is a collection of string-vibrations (i.e. all the strings making up space). Some strings are just empty space-time. Some are gravitational fields, some are matter, and some are energy. They are all arranged a particular way at any given time, measured by relative location in the medium called "space".
The entire "matrix" is a set of all possible arrangements of string patterns, yet only certain arrangements are "actualized". "Events" are changes in matter and energy (either changing form, or changing location in relative to other events). Space, time and chance are simply the continuums in which events are located relative to one another. This can be space, time or chance (which would be counterfactual events, or "alternate reality", which are simply the possible events not actualized).
The way to understand this, is to imagine the event of the beginning of the universe (supposedly expanded out from a single point). Now, the string pattern begins changing, and one pattern leads to another in a causative chain. First, matter is compact; then it has moved further apart. There are three coordinates in which it moves, and a different arrangement of strings can be found.
So there are actually four paths connecting to different string arrangements. One is causative, and you can only experience events in a chain once. The other three consist of all the displaced chains of causation, where matter and energy has taken other shapes in other events, yet these chains can be accessed from one another. I say chains and not events, because you cannot travel between simultaneous events, because that is faster than light, and even simultaneity is relative! So instead, you can access another event that stems from the event you observed from a distance. Hence, a different "chain". So each piece of matter you look at in space is apart of another chain of events (its creation, changes made to it, things done with it, etc), displaced by whatever "distance" you measure, from the chain of events you are apart of.
Each collective pattern arrangement is actualized when all the chains of events leads to that particular pattern. (You scratch your head, as the object you are looking at reflects a flash of light from outside, and everything else occurring at that time).
So the continuum along which the chains progress becomes causative, and called "time", and the continuum (with its three coordinates) in which all the different chains are simultaneously displaced is called "space", and we have freedom to move in any direction. Chance would be the hypothetical continuum connecting to the un-actualized arrangements.
Like if I wear a red shirt, that is one pattern of strings actualized at the same time as every other event occurring in the universe. But a parallel timeline where I wore a blue shirt instead could be thought of as possible, and even "existing" in some hypothetical way, but it is simply not actualized. (Putting on a blue shirt after the red one doesn't count as actualizing, because then in that time frame, everything else in the universe has progressed (changed) from what it was at the time of the initial event, so it would be a yet different arrangement).
You can see this stuff in theoretical physics' books, but I have never seen any of them speak of this continuum in which parallel timelines exist, as a third continuum. It's not time, as there is no causal chain between parallel [counterfactual] events. It's not space (e.g. hyperspace), as space only contains actualized (tangible, visible) events that can be measured in distance relative to each other (even if not visible from lower spaces).
Time and space are frequently said to be interchangeable, and I would say chance is interchangeable with them. All they are is three different methods of arriving to different potential universal collections of states of matter and energy. One of them you have free movement in, another you are involuntarily "dragged" along, and the other you have no known access to at all.
It's said that if you move at the speed of light, then that direction of space then becomes like time. (I'm not sure if time then becomes like space, since when you move at the speed of light in any one direction, you are still in the others. Within a black hole, space and time are said to switch like that as well, with some sort of freer movement in time, but then you don't have much space or time left, as you are pulled into the singularity and torn apart by the tidal forces).
So while we often hypothesize about "time machines", which take us to points in time outside our chain of actualization, and we can also call any faster-than-light ship a "space machine", since it travels in a "space-like trip" which is also outside of our range of actualization (such as being two places at the same time).
We can thus also imagine a "chance machine", which can take us to these alternate realities. Rudy Rucker points out that "any FTL machine can be adapted to become a time machine" (The Fourth Dimension, p. 171), because motion through space affects time as discussed above. So if we did end up in the past through it, it would also have become a chance machine! We have now actualized an alternate set of events, appearing in a particular time and space we did not appear in before the trip!
On the flipside, he shows how a time machine becomes like hyperspace travel, where if you are blocked by something, you can reach to a time where the barrier is not there, then return to your time, and have thus passed "around" the object just like if you had moved through the fourth [space] dimension. (p.215-6)
The same thing could be done with a counterfactual in the "chance" realm, if you could move back and forth between the two realities. Jump to an alternate reality where that wall wasn't built, move, and then jump back to our reality. You are now on the other side.
So all across the board, we see three different vector spaces that are interchangeable. The difference between them lies quarely in the means of actualization of events.
I've also been looking into the notion of a "chancelike wormhole". A spacelike wormhole is the typical one you hear about. Looked at relativistically, it would only be "open" for a fraction of a second, requiring faster than light "spacelike" travel to get through. The singularity is spacelike, cutting off the future, as the gravity basically turned that direction of space into time. This is if the hole has mass only (Schwarzschild solution).
If you add one other factor; either spin (Kerr) or electric charge (Reissner-Nordstrøm), then a second event horizon forms, swapping space and time again, and thus realigning the interior of the hole so that the singularity now becomes a timelike barrier you can "pass by" just by sitting still in the center of the hole, as this universe and its spacelike twin pass away (replaced by the split singularity pair), and new ones are born in their place. So it's basically a "timelike worhmole".
There is also a hole with both charge and spin (Kerr-Newman). Since each factor supposedly generates another event horizon, we would expect this one to have three. But the only descriptions of it I can find show it as identical to the plain Kerr hole, with spin only (see http://lpsc.in2p3.fr/ams/aurelien/aurelien/CCDecMULTIV.pdf)
Perhaps, it would have three horizons, and the third one would now turn space and/or time into chance. (Because theorists didn't recognize a third continuum, they just figured there would be no third horizon).
In this idea, chance is basically the primary continuum of the universe; in which are defined the connections between all possible events or collective states of matter and energy. Space and time are generated from it depending on the relationshipe of events that have been actualized.
Space is the continuum of the connections between actualized events that exist independently of each other.
Time is the continuum of the connections between actualized events that generate (cause, lead to) one another.
To further elaborate on something said before; an alternate causal chain springing from an event is not time because it is not actualized, and we are defining the continuum of "time" as causal connection between actualized events. So such a chain would simply be a veering off into this new "chance" continuum. Yet if we could actualize that alternate chain of events (at the same time as the first chain), then it would sweep out a causal plane rather than line, and thus generate a second dimension of time. So in this, a chance dimension has transformed into time.
If we start with an old neighborhood, we have our coordinates of how far east or west (longitude), north or south (lattitude), and of course that it is on the ground (up or down; altitude), and then choose a particular time, then we have an event with at least five coordinates; the first three telling you where, a fourth telling you when, and a fifth (and more) telling you WHAT: a particular state of matter. After this initial event, the neighborhood can either have its buildings stay the same, or be renovated, or it can be torn down for an all new development. Either way, we will arrive at one of two different possible events, separated from the starting event by a causal chain, which will replace the starting event with the new event, eventually. Only one of the events will be actualized. The other will then lie outside of the four dimensional spacetime continuum, in the "chance" dimension, and we will say it doesn't exist.
Now, if we had a dimension in which we could actually lay out the entire causal chain (world lines springing from both events) along it, both events could be seen at the same time. Both would be actualized, and the old neighborhood and new neighborhood would exist side by side in the same spacetime continuum, yet occupying the same lattitude, longitude, altitude and time. You could now freely go back and forth between the old neighborhood and the new one. New causal chains could then spring forth from both that do not replace one with the other. (Like the old buildings being renovated). The displacement between them would no longer be causal. So then, a time dimension has become a fourth space dimension, and a separate time dimension continues to march on.
Here in this video, we see another good step by step buildup of dimensions like the ones linked above. He's building up to ten dimensions, and using time as the fourth, and different kinds of dimensions besides space and time after that. Dimensions 5 and 6 would be the realm I am calling "chance".
In part 2 of the video, the 7th dimension connects all the infinite possible "chance" paths of our universe (from big bang to all the possible endings) with all the possible chance paths of entirely separate universes or "infinities". After this, it seems to get a bit redundant, as the 8th dimension is just described as "another line branching off the 7th dimension line to connect to yet another infinity", and the 9th dimension is "all the possible branches for all the possible timelines of all the possible universes". The tenth dimension treats all of this as a single point, but then there is nowhere left to go. The author ties this to the ten dimensions of string theory, yet acknowledging that this imagining is "not the accepted explanation for string theory", in which the other six dimensions are strictly spatial, and collapsed to the Planck length.
In the Powerpuff Girls episode, "Get Back Jojo", the professor explains time as the fourth dimension (showing an orthogonal diagram of a tesseract and a mouse in a maze as the lower dimensional analogy), and the dimension used to jump out of our timeline and then move parallel to it back in time and alter the present/future as the fifth. This would basically be another kind of "chance" dimension.
If the past is altered, that would then be what branches off a new counterfactual, separated from us by a new dimension or "chance" continuum. (Though no other counterfactual branes are represented in this dimension. It only exists to get out of the 4D confines of our timeline, so it would basically be a tiny dimension, like the ones discussed in string theory).
As this video explains, going back in time is "the long way" to reaching a counterfactual, analogous to traveling around the universe from one point to another, instead of just using the wormhole more directly connecting the two points.
However, in that story, since it is seen as having potentially damaging effects on the present, this would represent the other version of time travel theory, where there are no parallel counterfactual universes, and tampering with the past damages causality in this timeline. The "Back To the Future" series is like a combination of this, where changing the past causes things to instantly change in our timeline (from the point right after the trip is made). So there are no parallel counterfactual universes, however, counterfactual events and aftermaths automatically appear in our universe after a reverse time jump in which something was changed. In the PPG story, it was not revealed whether the negative effects would be instant changes in reality, or just a total breakdown of reality.
However, the trip into the past does lead to what is known as a "causal loop" (a self-causing event chain), where the PPG's enemy ends up causing the creation of the girls, rather than erasing them as he intended! The professor had created these girls pattered after the ones he remembered rescuing him in the past, which were actually the created girls going into the past to rescue him from the villain who was trying to destroy him!
Inertial Frame of Reference for the Speed of Light
While many discussions of relativity will discuss the increasing disparity between inertial frames of reference as one approaches the speed of light, one thing I've never seen anywhere, is what it is like AT the speed of light.
Since you cannot reach the speed of light through acceleration, we will have to imagine a hypothetical and basically supernatural situation where people could be "beamed" whole at the speed of light. (Sort of like we imagine that matter could move faster than light; but only if it always moves faster than light. i.e "tachyons").
Extending it to the hypothetical limit of what happens as you approach the speed of light, the entire universe, in the direction you are traveling, flattens down to ZERO. People often describing it as extending in that dimension to be "everywhere at once". That is true only in a limited sense. It's not you who are extending, it is that dimension of space that shrinks. You're still in one point at a time, but the points have all compacted together.So the traveller basically sees himself passing through a "Flatland" having only height and width, and no depth. Its time is also frozen in one point (The point in which he was "emitted").
Really, because all three of his proper space dimensions are unchanged to him, and volume is h, w and d multiplied together, then the universe will be h × w × 0, which reduces the entire volume of the universe to zero. The universe basically doesn't exist from that frame of reference!
Even though a photon can be deflected, changing direction, in our coordinate time; in its proper time, it has simply passed through an infinitely thin membrane. While photons do exist in our coordinate time, their "proper" frame of reference does NOT exist. It is flattened down to zero in the direction it is travelling.
Since the "forward" direction being travelled is still the same in proper time, yet the coordinate universe has been flattened down to zero in that same direction, then what in heavens is this "forward" dimension, (as well as "backward")? That is really the central question here. The answer is a bit shocking, but it should figure!
It comprises a new space dimension perpendicular to all three of the coordinate universe's dimensions! So one dimension of space has flattened down, and been replaced by another!
Let's remember the primary rule of general relativity. In that direction, he [the "c" traveller] will by default see himself at rest, just like all other observers.
In his frame of reference, a beam of light would still move forward at c. Coordinate time observers would "see" an infinitely flat ship (or whatever), frozen in time, moving at c.However, in proper time, any light that moves past him at c, is really moving in a new dimension not shared by coordinate space, and the coordinate dimension the photon is travelling is perpendicular to all the dimensions that can be can measured.
This is interesting, as M-Theory now proposes a superspace consisting of a large fourth spatial dimension in addition to the six compacted ones. But, IIRC, it is only gravitons whose strings are free to move in this dimension; I don't think electromagnetism. (All other strings are believed to be attached to the three-brane we know as space, and thus limited to 3D). Still, since these hypothetical gravitons are also said to move at c, then perhaps that is the reason they have access to a large additional dimension.
That's how a paradox I noted was resolved as I pondered this question for years. I had not before completed the thoughts on this, because of being stumped by what it would mean for c to still hold true in the proper time frame of reference of c, while still being c in our coordinate time. "Where" would the light in the c frame be "going"? This explanation answers that!
(Though receiving feedback to the idea in one place so far, I see people just write this off as "self-contradictory", and won't even consider a new dimension, which eliminates the contradiction. This was what I had done previously, when I reached that contradiction and figured it couldn't be solved. some of this might be from the fact that it is connected to string theory, which some view as still too speculative; or, as they call it, "not even wrong").
Also, suspecting that if one space dimension begins flattening down as you approach c, then I wondered if time would then become a new space dimension (since time and space are said to be "exchanged" at or beyond the speed of light; and like inside the Schwarzschild radius of a black hole), and would you see four space dimensions (h, w, and d and t partially flattened with one shrinking and the other unfurling) as you get closer to c? I couldn't make sense of it.
But then I remembered that you really can't accelerate to the speed of light. So as you are accelerating, you see the coordinate depth dimension shrink, while coordinate time slows down. All they'll do is continue to shrink, forever, down to fractions of spacelike units, but never reaching 0. You're not "getting [any] closer" to the speed of light (as I mistakenly was assuming), so you don't encounter any "inbetween" state with a partially extended extra dimension! You're always in your own proper spacetime, where the three dimensions plus time are always the same as they always were. It's the coordinate spacetime you see collapse, and the new dimension doesn't appear as such until the old space has collapsed completely to zero, and that only occurs AT the speed of light.
And again, it is really nothing more than your familiar old back/forth dimension, which is now no longer part of coordinate space.(If space is a hypersphere, you'll see it flatten down to a disk. If its infinite, it won't shrink at all; you'll only see the matter in it shrink, until you reach then end of matter's distribution. If it has some sort of "edge", beyond which there is no space, then there will be a shrinking distance you'll move in before you just cease to exist.
To be AT the speed of light is a quantum jump, in which you'll find one dimension completely collapsed, and a new one filling its place in the "ahead/behind" directions.
If you move in that dimension backwards, even close to c, you still do not come "closer" to reentering the old dimension; you are still moving in the new dimension. All of coordinate space has now become the unapproachable c to you! Likewise, if you move forward, you're not going faster than light. FTL was based on the coordinate frame of reference; and that no longer exists in proper space-time, remember. Your motion in the new dimension has no bearing on the old one.
Now, if space is a hypersphere, it will be an infinitely thin disk that you're perpetually "stuck" going through as you make an infinite number of laps in that direction, in zero proper time, and circumnavigating at 186,000m/s forever in coordinate time. If it has an edge, you'll instantly cease to exist. If its endless, then the distribution of matter (superclusters of galaxies; observable universe, etc) will be infinitely thin, but as for space itself; you'll have a 0×∞ paradox. (Another reason I don't believe in infinite space).
In any scenario, you'll be at the "lightlike infinity" at the boundaries of the Penrose diagram of the universe.
What about other c frames of reference traveling ahead or behind you? Like if you are "emitted" at one point in time, and then another c traveller is "emitted" one second behind you. If you're traveling less than c, yet accelerating towards c, then in coordinate time, the distance between you approaches 186,000 miles. Yet the proper distance collapses towards zero. You in essence, would occupy the same proper space. However, the proper time approaches infinity! Remember, coordinate time grinds to a halt, so "one second behind you" increases towards forever.
You can see why, by graphing this on the Minkowski diagram. Coordinate space runs along at 0° to the right, while coordinate time is 90° (vertical), and c is 45°. However, proper time collapses towards 45° in both dimensions. As your timelike proper "here" heads outward at a decreasing angle, your spacelike proper "now" moves upward at an increasing angle.
At c, the space and time dimensions have collapsed into the c line at 45°. So another c traveller emitted one second behind you will be a parallel 45° line that never intersects your line.
—at least not in that coordinate space direction being travelled. Since the space between you decreases to nothing at the same time, this might balance itself out. How?
What I'm wondering, is that that new "forward" proper space dimension might be what represents coordinate "time". Thus fulfilling the theory of space and time being swapped at or beyond the speed of light (and just like inside the Schwarzschild radius of a black hole).
So you might see your "follower" displaced (perhaps at 186,000 miles?) in that new "behind" dimension. If you both head backward approaching the speed of light in proper time, the distance would shrink, and proper time would also slow down. (Iimagine if you could make the "quantum jump" in that direction, you hypothetically could reemerge in coordinate time, I imagine at the point you made the quantum jump into the c frame; but not sure of this).
Again, this is a self-contradiction only when you look at it from a less-than-c frame of reference, and the coordinate spacetime dimensions. There would obviously be some sort of total disconnect from any less than c frame. sO, it could be argued that light which must always move at c, would be at rest at the same time.
The way light can both be at rest and move at c at the same time is that time completely stops from one such frame of reference to the other frame, so in zero time, all speeds are the same: zero. The very concepts of "motion" and "rest" become indistinguishble (to the other frame, that is). Think about what "frozen in time" would mean. Anything "moving" is now just as frozen as any original "coordinate" standard.
So likewise, what one sees as the c speed moving past himself, can include many different relative speeds frozen in time.
In the c frame of reference, you would see that you were really at rest, and that there was still a c that moved at the same speed, in a new dimension. But to us, all of this would be an infinitely flat completely time-frozen perspective moving past us at c (yet with everything in it appearing to move at zero relative to each other).
So in a c frame (which again, is relative to us), the observer (light itself cannot observe) would see himself at rest, while light still moves at its normal speed, but in a dimension different from ours.
And again, that paper the other person linked to appears to support this to some extent. And also, that theorists do allow the possibility of tachyons existing, but that they likely would occupy, essentially, another kind of spacetime dimension. This would be the same dynamic.
I even found a bit of support from this book: http://books.google.com/books?id=XmsbvP1uUeIC&source=gbs_navlinks_s (A First Course in String Theory, Barton Zwiebach), p.20 discussing "non-Lorentz-inertial coordinates that travel at light speed" or "light cone coordinates".
It has a diagram of two light rays heading away from each other (X-, X+; forming "the cone"), and mentions the ideas of one being a new time or space coordinate relative to the other, and that in either case, it wouldn't be ordinary. This may well correspond to what I'm suggesting! Obviously, any new dimension would not be one of ours, since ours would have collapsed down to zero relative to the light cone coordinate.
This should give us an idea of FTL travel, in which coordinate/proper space and time become "imaginary" relative to each other.Now, the collapsed space and time are not even zero; they are beyond zero. That is, in a multiplicative, not additive sense. It doesn't become simply "negative", which is the additive "beyond zero" realm. (Like take 1 and keep dividing it by finite positive numbers until you reach zero. And then imagine continuing to "past" zero. You can't even reach it in the first place). It is analogous to "greater than infinity". Thus, truly "imaginary".In the FTL realm, the rules are the same. A beam of light will still pass you at c, but it will be a different dimension from the direction being traveled in coordinate space.
I believe that the so-called "tachyons" will thus not interact with coordinate spacetime at all (as has been speculated). Light speed is like the "bridge" between the sub- and super- luminous realms, where time and space are reversed like FTL, but you can still interact somewhat with the subluminous universe.
On the Minkowski diagram, while FTL's "here" (the actual world line) would lie in its expected spacelike (less than 45°) line; its "now", rather than lying in a timelike orientation, would generate a third dimension of the diagram; sticking out in the Z axis as a second space axis.
NON-COMMUTATIVE SPACE (What are Space and Time, Really, and Can We Do Without them?)
http://erictb.wordpress.com/2012/05/01/non-commutative-space
Illustrations of two sci-fi stories involving space or time
http://erictb.wordpress.com/2012/05/01/illustrating-little-girl-lost-and-the-langoliers
Forgotten 4D object (http://eusebeia.dyndns.org/4d/geom.html): The duo-cone
http://erictb.wordpress.com/2012/05/01/forgotten-4d-object-the-duo-cone
Alternate Hex Codes
http://erictb.wordpress.com/2012/05/01/alternate-hex-codes
Definitions of "Fortune/Luck" and "Money"
http://erictb.wordpress.com/2012/05/01/definitions-of-fortuneluck-and-money-and-the-new-century
Division between man's soul and spirit
http://erictb.wordpress.com/2012/05/01/division-between-mans-soul-and-spirit
Metric Time:
http://erictb.wordpress.com/2012/05/01/metric-time
When did the new century/millenium begin?
http://erictb.wordpress.com/2012/05/01/definitions-of-fortuneluck-and-money-and-the-new-century
Monochrome yellow vs. filtered or red+green mixing
http://erictb.wordpress.com/2012/05/01/monochrome-yellow-vs-filtered-or-redgreen-mixing
http://erictb.wordpress.com/2012/05/01/great-zero-and-the-alternative-negative-number-system
Why is the "powers of ten" line asymmetrical?
How many sides does a circle have?
http://erictb.wordpress.com/2012/05/01/how-many-sides-does-a-circle-have
(Have to figure out how to get this table to fit on the blog):
polygon name n (Number of sides) C (angle at center: 360/n) A (interior angle of polygon: (180-c) S (length of sides) if r=1
(2 cos A/2)r (radius) if s=1 (.5/cos A/2) common "shape" name trigon 3 120° 60° 1.732 0.57735 triangle tetragon 4 90° 90° 1.414 0.7071 square pentagon 5 72° 108° 1.17557 0.85065 hexagon 6 60° 120° 1 1 octagon 8 45° 135° 0.765 1.30656 decagon 10 36° 144° 0.618 1.618 hecatontagon 100 3.6° 176.4° 0.06282 15.918 hecatonogdocontagon 180 2° 178° 0.0349048 28.649 triacosiahexacontagon 360 1° 179° 0.017453 57.296 chiliagon 1000 .36° 179.64° .006283 159.155 megagon 1,000,000 .036° 179.999964° .00000628318 159,915.5077 gigagon 1,000,000,000 .000036° 179.999999964° .00000000000000628318 159,155,077.52 teragon 1,000,000,000,000 .000000036° 179.999999999964° .00000000000000000628318 159155077524.438 ["Great Zero"]-agon) 69720375229712
477164533808935
312303556800 .000000000000000000000
0000000000000000051634°179.99999999999
999999999999999
99999999999948365°.00000000000000
00000000000000
000000000000901110963070498650
62522983129408
834xxxxxxxxxxx.xapeirogon ∞ 0° 180° 0 ∞ circle