Rotation and Time, Part 4 – Afterthoughts 

            I want to comment on a few loose ends I left. Speculating about these in the body of the first three parts would have interrupted the flow too much, and these comments are much less developed, so here goes. 

  1. Rotational velocities – In my model, our universe is the 3d surface of a 4d hypersphere expanding from its center and rotating everywhere around that center point. Since that is impossible to visualize, think of the 1d line universe, a circle, expanding from and rotating around its center in a 2d plane universe plus time. As the rotating circle expands, the angular momentum would remain constant, so the angular velocity of a point on the circle would decrease, and the scalar magnitude of its linear velocity tangent to the circle would remain constant. The linear velocities on opposite sides of the circle everywhere cancel leaving a total linear momentum for the universe of zero. This is also true of our 3d surface rotating in a fourth dimension. There would be no appreciable relativistic effects of the linear velocity of masses in our universe from its rotational motion. There is no centripetal force from the center of rotation impeding the expansion, so no accelerations from the rotation. 
  2. Angular momentum – There is a nonzero, constant angular momentum from this extra dimensional rotation that would appear as mass-energy somewhere in our universe. It’s tempting to associate it with dark matter, but without calculations to back that up, it remains only a convenient guess. 
  3. General Relativity – Our spacetime is essentially flat. Since the curvature I propose is in an extrinsic dimension which exists for us as the tiny thickness of an expanding 3d surface, flatness is allowed, possibly even expected. The extrinsic curvature in a fourth spacial dimension wouldn’t necessarily affect the intrinsic localized spacetime curvature which we observe as gravity. As our 3d surface moves out further from the 4d origin, our universe would appear to expand as it does, every point moving away from every other point with masses weakly constrained by gravity. It seems to me that the rate of this expansion would increase with distance like the action of the cosmological constant. 
  4. Special Relativity – Though General Relativity is more complex, Special Relativity is more disturbing. It operates in Minkowski spacetime, a blending of space and time. The following uses a common convention of considering only one space direction, x, in the direction of the motion being considered, and the time direction, t. Velocities less than the speed of light exist in timelike spacetime. Velocities at the speed of light are in lightlike spacetime. Velocities “greater” than the speed of light (with negative time) would be in spacelike spacetime. The Lorentz transformation invariant t’^2-x’^2=t^2-x^2 (expressing velocity in units of c, the speed of light, so that c=c^2=1) measures proper time when timelike (t^2>x^2) and proper distance when spacelike (x^2>t^2). Since t and x are both squared, they can each be positive or negative without changing the value of the invariant relationship. This implies four possible quadrants of which we experience only one, the one in which t and x are both positive and spacetime is timelike. That the quadrants are impassably separated by singularities is expected because any equation of the form of a difference of squares graphs as a hyperbola which goes to infinity at each of four asymptotes which match the lightlike lines offset 45 degrees from the major axes. (Here is where ignoring the other space dimensions oversimplifies a lot, but as a first pass speculation, I’m disregarding that.) What does this have to do with higher dimensional rotation? As its relative velocity increases toward the speed of light, a moving reference frame appears to squeeze the t’ and x’ axes together, like a pair of scissors closing, toward the 45-degree angle of lightlike spacetime where t’^2-x’^2=0. I can’t find a gif to illustrate it, but this is exactly how the 2D projection of a rotating 3D coordinate system appears from a skew angle. Maybe there is a corresponding projection of a 4D rotation onto a 3D surface. Could motion be a twisting of 4D spacetime in a fifth dimension and mass the resistance (gyroscopic?) of 4D spacetime to being twisted? 
  5. Inflation – In current cosmology there is an idea that in the very early universe (way before it was one second old) the universe expanded very quickly, faster than the speed of light. This model is somewhat successful at explaining important parts of the large-scale structure of our current universe. There are three pertinent possibilities for this idea in regard to my model: 
       a) My model accommodates the idea. 
       b) My model contradicts the idea and offers alternate explanations. 
       c) My model contradicts the idea and offers nothing to replace it. 
    I have no idea which is the case. 

Hugh Moffatt 
Watertown, Massachusetts 
February 3, 2021

(P.S. Though written in 2021, this belongs with this series so I slipped it in.)