One spatial/temporal dimension

[Prashantkrishnan]

This is a continuation of a small discussion I had with quickfur about a year back in the thread http://hddb.teamikaria.com/forum/viewtopic.php?f=30&t=1808

A new realisation came to me related to this.

Imagine a practical situation of life in Lineland.
There are some flaws in the way it is described in Flatland. The people who lived in Lineland, the line segments, could move back and forth but actually this is impossible in a truly 1D universe. This is because there is no friction. The reasons why there cannot be friction in a 1D universe are as follows:

1. For friction, there needs to be a surface of contact. In 1D, contact is not possible except with points of contact.
2. There need to be irregularities in surfaces for friction and irregularities require at least two dimensions.

Of course, these rules would not hold true in a 1D universe that is embedded in a desk under a glass slab in planespace, because the glass slab and the desk would have irregular surfaces. But considering a completely Euclidean line with no irregularities friction is impossible. Thus no force can be exerted until and unless two bodies collide. And collision can be avoided if all bodies in the line are moving with the same velocity. I think we are in a similar situation in a temporal dimension.

The trick with dealing with temporal dimensions is that all of your actions, past, present, future, are all already determined (as least from a mathemetical POV), so that the present you forms a single slice of a continuum that forms an (n+1)-dimensional "time-sweep" that traces your path from past to future. So there is no "motion" at all; the whole thing is just a static, unchanging manifold of history and future already pre-written (at least from the mathematical sense). So it makes no sense to speak of motion in this context: for anything to move, requires an independent time dimension, but when you're viewing the whole of spacetime as a single object, there is no other time dimension in which you could move. Speed is basically just the slope of your "time-sweep", a purely geometrical feature. So there's no "mysterious force" that prevents you from going anywhere -- you can't "go" anywhere because going requires motion, but when viewed as an object in itself, spacetime doesn't have motion anymore than a straight line on graph paper has "motion".

The only reason we experience time going forward is because that's the direction in which our brain processes input. Viewed from an observer outside of space-time, time has no special direction at all.

So dealing with two temporal dimensions just produces something that is probably incomprehensible to us, because we have no concept of what "time moving forward" means when there's a 2D area in which events can progress. The closest I got to two temporal dimensions is the (very) crude analogy of the evolution of a story's plot (in "internal time") over (external) time. That is, your first draft of the story has the plot going forward in a certain way, so you can say at t1=0, time in your story's universe progresses from t0=0..n in a certain way. But as you revise your story, the new version of the plot now progresses from t0=0..n in a different way. So if you collect all of your story's drafts together, you can lay out the different versions of the plotline across a second time dimension, that is, the external time ("real world time"), which is perpendicular to internal time ("in-story time"). Then you can consider a perpendicular cross-section of these different versions of the storyline, and look at, for example, how a particular character changed from its original conception to its final version, at the same point in the story's time. This then represents the progression of the second time dimension: where time in the story's universe has stopped, but the characters and scenery still change because now you're moving from the original version of this event to newer revisions of the same event in your story. Obviously, the characters in your story are oblivious to this perpendicular motion, since they live only in story-time. But conceivably, one could write a story where a character acquires a 90° rotation from in-story time to revision-time, wherein he travels to a "parallel storyline" where characters, scenes, and objects have changed while story-time stays still, and then he can rotate 90° back into in-story time and then proceed to experience the rest of the story in a new version of the plotline. (But as I said, this is just a very crude analogy of 2D time -- I haven't been able to get any further than this 'cos the analogy breaks down.)

Let it be that there is no motion and that we experience time going forward only due to the direction in which our brain processes input. But I still think that I can relate this to the velocity of a mononian in linespace. To change the rate at which our brain processes input, we need temporal friction. For that we need irregularities in temporal surfaces. For that we need two temporal dimensions. That is beyond our comprehension since we are temporal mononians.

(I have not considered the effects of relativity in the above paragraph)

It is even more difficult to conceptualise a second temporal dimension than a fourth spatial dimension because, for spatial dimensions, we already know three and we can use the analogy of lower dimensions to expand our understanding of higher dimensions, while for temporal dimensions, we know only one and we cannot use analogy in a proper manner. The analogy used by quickfur in the post quoted above is quite logical (Even the dot products of the temporal vectors seem to be 0, indicating perpendicular directions), though, as he has himself said, crude. Still, I cannot think of any better analogy.

PS: Don't ask me what temporal vectors are, I just made up the term and I don't know what it means.
What I have written in that bracket is just an intuition.

[Prashantkrishnan]

This analogy seems similar to comparing a 4D world to a situation in which 3D beings are in front of a television where 3D objects are being projected to 2 dimensions and having the extra dimensions of into the screen and out of the screen. Counting it, it comes to four dimensions

[quickfur]

Lineland, as described in the Flatland novel, is basically just a convenient analogy to bring across the author's point. I wouldn't read too much into its actual physics... because if you really want to do physics in 1D, you'll eventually reach the conclusion that the 1D universe consists of a single line segment that contains all the matter in the universe.

Here's why: we know that in 3D, gravity attenuates according to the inverse square law, that is, the force of gravity is proportional to 1/r^2, where r is the distance between the two objects. Why inverse square, as opposed to, say, inverse exponential, or reciprocal, or something else? The currently accepted interpretation of this, is the flux of force carriers decrease in proportion to the surface area of a sphere of radius r from the origin of the force. Imagine if you will, a heavy object from which gravitons are emanating. Suppose it emanates N units of gravitatons per unit time. At r=0, all N gravitons are concentrated on the object, so the density is maximal. As they propagate outwards, though, they will lie on the surface of an expanding sphere, with area A. As they move away from the source object, the surface area of the sphere increases, but the number of gravitons remains constant. Since the surface area of a sphere increases in proportion to r^2, it follows that the density of gravitons at distance r from the source is proportional to 1/r^2, which is what we observe.

Now, this means that in 2D, force should attenuate according to a reciprocal law, because now the force carriers lie on the circumference of an expanding circle, which increases in proportion to the distance r. So in 2D, gravity should obey an inverse reciprocal law (1/r). But in 1D, gravity doesn't attenuate at all, because the density of gravitons from the source at distance r is constant: the "surface area" of a line segment is the "area" of its two ends, which is constant! So the density of gravitons at any distance from an object does not decrease at all. This means that any two objects in a 1D universe, no matter how far they are, will feel each other's gravitational attraction as though they have no distance between them. The ultimate result is that every particle will attract every other particle, and, assuming a finite number of particles in the 1D universe, all of them will eventually merge with each other to form a single universal line that contains all the matter in the universe. You wouldn't be able to move at all, since you're either inside the line, which means you're buried underneath a whole pile of stuff, or you're on one of its two ends, which means that to have any movement, you have to overcome the weight of the entire rest of the universe.

So yeah, nothing interesting could exist in a 1D universe. Unless you drastically revise the laws of physics, of course.