Rotation and Time, Part 1

This is a 3-part series. I'll publish the next two parts weekly on the following Wednesdays.

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 Euclidean. The second is the more complex consideration. As complex as Euclidean rotation can be in higher dimensions, it’s pretty well understood mathematically.

Euclidean geometry describes objects (and universes) with only spacial dimensions. Our universe, as described by Einstein’s General Theory of Relativity, has four dimensions of spacetime expressed by Minkowski geometry. Within this geometry, space and time are blended. 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.

What I’m considering is the possibility that the universe as a whole has a rotation which creates our one-directional time. 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 Euclidean. Minkowski 4d spacetime has further complications. I’m going to discuss what I know about 4d Euclidean 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 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 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 is one of the further complications of the geometry of our actual universe over a Euclidean universe. As mentioned, I’m sticking with a Euclidean analysis for now.

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 a 4th 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. 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

I’m not sure how useful this animation is, because the object appears to be constantly deforming, and its points don’t look like they are all maintaining constant distances from the center point, even though that’s what would actually be happening.

There are two initial problems with this model for rotation of our 4d spacetime universe. First, since spacetime is not Euclidean, the extent to which a model like this is relevant is uncertain. Second, 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, it’s not clear how or where this rotational time exists.

This may require 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 three spacial dimensions and one time dimension.

There is a 5d version of General Relativity called the Kaluza-Klein Theory which is now regarded mostly as a precursor of the string theories which require 10 (or 11) dimensions. None of these theories are contradicted by what I’m suggesting, but none include it either. Their way of adding higher dimensions is different. 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. The natural conclusion from current physics is, of course, that I’m just wrong, but bear with me through the three short parts of this series and see what you think.

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

- 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.
- 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