Rotation and Time, Part 1       

           This is a 3-part series with an Afterthought and a follow-on, making it actually a 5-part series.

           As I wrote in Pencils 2019 05-Rotation, rotation of the universe seems inevitable to me. Not rotating is just a unique case of rotation with angular momentum of zero. With zero rotation, the center of rotation is undefined. That means there are infinite possible centers around which rotation could happen and infinite opportunities for instabilities. Nature abhors instabilities in a dynamic system and tends to resolve them with motion. 

           Analyzing possible rotational modes is complicated for our universe for two reasons. First, because the universe has more than three dimensions, and second, because its geometry is not ordinary space but Minkowski spacetime. The second is the more complex consideration. I’ll start with ordinary space of undetermined dimensions and attempt to introduce the characteristics of spacetime as we perceive it. Anyway, as complex as rotation can be in higher dimensions, it’s pretty well understood mathematically.

           Ordinary geometry describes objects (and universes) with only spacial dimensions. Our universe, as described by Einstein’s Theories of Relativity, has four dimensions of spacetime expressed by Minkowski geometry. Within this geometry, space and time are blended mathematically. Some directions are more spacelike, and some are more timelike, but the limiting cases of pure space and pure time don’t seem to have much meaning. Why they are separated in our perception is a mystery to solve.

           What I’m considering is the possibility that the universe as a whole has a rotation which creates our one-directional time out of one of the spacetime dimensions. This rotation creates a preferred dimension, our time, in Minkowski spacetime and the direction of the rotation creates a distinction between the two directions along that dimension, our past and future.

           Intuition fails in higher dimensions, even when they’re ordinary space. I’m going to discuss what I know about 4d spacial rotations and see where it leads with the hope that it will translate into our actual universe.

           Rotation of a rigid object means that all points of the object maintain their original distance from some center of rotation as they move. A flat disk in 2d space can rotate around a point. Commonly we visualize this as rotation around its center, like a wheel turning. A solid 3d object like a sphere in 3d space can rotate around a line axis, usually thought of as passing through the center of the sphere and creating poles at two opposite points. Every point of a rotating sphere remains at its original distance from the axis of rotation defined by these poles, which doesn’t move. This is approximately what our Earth does as it rotates.

           Of course, the universe is not rigid. Rotation creates angular momentum which results in linear momentums in points of mass which appear as a centrifugal force pushing outwards from the center of rotation. A rotating 3d non-rigid body, like a space with lumps of matter, will tend to expand limited by the opposing gravitational forces from its mass. There will still be a defined stationary center axis of rotation even though internal distances are changing.

           General Relativity tells us gravity isn’t really a force but an effect of the bending of space by mass. This bending is another complication of the geometry of our actual universe over a Euclidean universe, however on large scales the universe is very flat, not perceptibly curved, so I’m sticking with a Euclidean analysis.

           In 4d Euclidean space, rotation cannot happen around an axis (as in three dimensions) but in a basic rotation around a plane, or a double rotation around a point. This is counter-intuitive for our 3d-plus-time minds. One way to try to see a double rotation is as an object rotating as usual around a polar axis while at the same time rotating in another dimension around another polar axis. These two polar axes intersect at the center of the object, and all points of the object maintain a constant distance from this center point as rotation occurs around it.

           The object is a 4d object, not our common 3d object, so each rotation is around a plane, and the two planes intersect at a point. When this type of rotation is animated in an attempt to illustrate what it would look like to us, it seems the rotating 3d object is also turning inside out. 

           Here’s an example with a 4d cube. Click on the link and scroll down the page about 3/4 of the way to the heading "Visualizing 4D Rotations".

           http://eusebeia.dyndns.org/4d/vis/10-rot-1

           In this animation the object appears to be constantly deforming, and its points don’t look like they are all maintaining constant distances from the center, even though that’s what would actually be happening. We can’t see that because we only see its projection into our 3d world.

           The initial problem with this model for rotation of our 4d spacetime universe is that rotation is a motion, and motion requires time. If I am postulating that rotation of our 4d Minkowski spacetime universe crystalizes our three spacial dimensions and one time dimension out of 4 equivalent spacetime dimensions, I have to explain how and where this rotational time exists. 

           This requires the existence of a 5th dimension, which would mean we have at least a five-dimensional universe in which a 4d sub-universe (ours) is rotating in a way that creates the three spacial dimensions and one time dimension that we know plus one extra spacial dimension that we cannot perceive. I’ll explain how I get to this in part 3.

           There is a 5d version of Relativity called the Kaluza-Klein Theory which is now regarded mostly as a precursor of string theories which require 10 (or 11) dimensions. None of these theories are contradicted by what I’m suggesting, and they offer something that will be helpful later on. Their way of adding higher dimensions is that the extra dimensions are “rolled up” so small that we are unable to detect the deviations from 3d plus time physics that they could cause. Bear with me through the three short parts of this series and see if you think I’m on to something.

           I’ll end this part with a statement of my two premises:

1. When a body has a preferred dimension and a preferred direction in that dimension, such as time in our universe, the simplest explanation is a rotation. 
2. Non-rotation of an unbound body, including a universe, seems so unlikely as to be essentially an impossible state.

           This is where I begin and end. 

           Next, I’ll go through some lower dimensional models and work my way up to our universe and my general idea.

Hugh Moffatt
Watertown, Massachusetts
November 9, 2020

           *Eusebeia. (2020, July 19). 4D Visualization. 4D Euclidean space. http://eusebeia.dyndns.org/4d/vis/10-rot-1.