Rotation and Time, Part 3 

           Before I conclude, note that what I am attempting to describe is not our universe but a model for thinking about our universe. This is the nature of all understanding. The value of an insight is in how it explains and predicts observable events. Determining what the universe actually is lies beyond the scope of science and, most likely, reason itself.           

            I’ve described two ways that a universe with three spacetime dimensions could expand from an origin point in two spacial dimensions: a rotating 1d line propagating out in both directions like a string and a rotating 1d line propagating as a closed circle expanding in diameter. 

            There is a third way a 2d plus time universe could expand from an origin. That is as a rotating 2d disk expanding from the center like the circle but with its interior as part of its universe. Creatures living on this disk would be fully two-dimensional and could move everywhere on the disk within its expanding circular edge, including crossing the origin, though as with the straight line, it would be harder to move toward the origin and easier away from it. Like the other two 2d (spacial) universes, this disk universe rotation would create the same distinction in the directions of time perpendicular to it. No more dimensions are needed. It’s a 2d plus 1 universe beginning as three spacetime dimensions. 

            It seems to me the 2d disk is a less realistic model. It would take more energy to form and expand than would the 2d circle or series of concentric circles, and like the rotating stringlike line, it would create a complicated physical experience for its creatures. Nevertheless, this step gives me a 2d surface with 2d creatures, so I’m making progress. 

            The next step is to add another dimension and have a 2d surface universe expand as the surface of a 3d sphere from the Big Bang origin point. This is analogous to the line universe expanding as a circle. Every point on the surface of the sphere would be roughly (there are always instabilities) the same distance from the origin, which would not be part of the surface universe, and, like the circular line, the spherical surface would be unbounded though finite. 

            The 2d creatures on this surface could now move in any combination of forward-backward and left-right directions forever without reaching an edge. The creatures and their world would necessarily have a non-zero magnitude in the 3rd dimension. The 2d surface of the sphere would have a tiny thickness. Again, this thickness, like the 3d curvature of the sphere, would be imperceptible to the 2d creatures living in this universe. 

            But what about rotation to define a time dimension? 

            That’s a problem. This spherical surface universe could rotate, but for a 3d object there is no rotational mode possible that can uniquely define a 4th dimension, which is needed by my model for a time dimension that differentiates between past and future. Any plane in which a sphere can rotate defines a perpendicular direction in a 3rd dimension which already exists in this universe. Remember, when a 2d disk rotates, the perpendicular 3rd dimension doesn’t exist in the disk. It’s a new dimension. Not so when rotating with three dimensions. We are obviously using a time dimension for expansion and rotation, but we have no place to put it in the model I’m developing. 

             However, a 4d object does have such a rotational mode. I need another dimension. 

             It doesn’t help to just fill in the sphere and add the experience of a 3rd spacial dimension for the creatures. This would be a 3d (spacial) universe as we experience our own. Including time, it would have three plus one dimensions, a total of four spacetime dimensions, and fully three-dimensional creatures living in it. A 3d (spacial) universe can expand in two basic ways. It can expand as a solid ball or cloud, or it can expand like a donut (called a torus in math) growing in diameter. Neither of these models solve my problem because in either case, like the 2d spherical surface of a 3d sphere, I still have no rotational mode that defines a new dimension for time. 3d rotations always happen in a plane around an axis line in an existing 3rd dimension. By my rotational premise there is still no way to identify the time dimension. 

             I’m stuck again. I need a 4d (spacial) time. I need five spacetime dimensions. 

            I am now past the point where we can easily visualize what is happening. What I propose is not far from what is currently understood about our universe. The critical difference is the use of an extra spacial dimension that doesn’t change our physics. Again, there is a distinction between Euclidean geometry and Minkowski geometry. Hopefully, a future analysis of this difference in the context of rotation will provide a connection to our current understanding of reality, which I am about to leave. 

            In my model, our universe is the 3d (spacial) surface of a 4d (spacial) hypersphere expanding in time in four spacial dimensions from a central origin point. We are 3d (spacial) creatures living on/in this surface with no perception of the 4th (spacial) dimension which contains the origin at the Big Bang. Though we cannot visualize this, this geometry is well understood mathematically and is analogous to the 1d (spacial) creatures living on the 1d (spacial) line curved in a 2d (spacial) dimension into a circle universe. Just like those creatures, our universe is finite (curved into the surface of a hypersphere in our case, a circle in theirs) and unbounded. Like them, we can travel in any direction in our universe and never come to an edge. 

            This is the same as the current understanding of our universe, except we now think of time as the 4th dimension, the expansion and curvature is described by Minkowski geometry, and there is no rotation. 

            Please take a moment to consider a 3d surface. It’s not at all easy to visualize. It is fully three dimensional, and in our experience would be no different from the 3d world we recognize. What makes it a surface is a curvature in a 4th spacial dimension that we can’t perceive. Living on/in it we would be like the 2d (spacial) creatures on the surface of a 3d (spacial) sphere. The only possible direct experience of this geometry would be if either they in their 2d world or we in our 3d world traveled far enough in a “straight” line. We would each eventually return to where we started, arriving from the opposite direction. They would go around their sphere and we would go around our hypersphere, each traveling along an unperceived curve in an extra spacial dimension. If the distances are great enough, that will be effectively impossible. 

            The advantage of a world in four spacial dimensions is that there is a 4d rotational mode (a double rotation as described in part 1) that can define an extra dimension of time, in this case a 5th dimension. This rotational mode, which is two independent rotations in two different planes, sums as a rotation around a single point.* This point is the intersection of a 5th dimensional line with our four-dimensional universe. This 5th dimension, orthogonal (perpendicular) to our four spacial dimensions, operates as our time dimension with the rotation defining its orientation and our past and future directions along it, just like the 3rd dimension does for the rotating 2d (spacial) circle universe. The rotations would appear differently in each direction of travel along the axis line. 

            A double rotation in four dimensions has more possible modes than the rotation of a circle in two dimensions, but each 4d rotational mode would create an observable physical difference between the two possible directions of travel along the axis of the 5th dimension. Any one of the modes could define a past and a future. 

            NOTE – No matter how many dimensions there are, a rotation always happens in a plane around a center point on an axis line which may or may not be a new dimension depending on the number of dimensions originally available. Complications arise because in higher dimensions multiple planes of rotation may be involved, each with its own center of rotation. Each existing rotation still has only two choices of direction, clockwise or counterclockwise, and which direction is observed will always depend on the observer’s direction of travel along the axis of rotation. There will always be an observable difference when traveling along that axis even if there are multiple rotations involved. My contention is that we perceive this difference as past and future as we travel along a timelike axis. 

             My model supposes a universe originating with five spacetime dimensions. Our universe expands from the Big Bang as the three-dimensional surface of a four-dimensional hypersphere that rotates with a double rotation that defines the 5th dimension as the dimension of time that we perceive and travel along in only one direction. To be clear, the time dimension is integrated with all the other dimensions and is orthogonal to each. It isn’t “separate” in any other way, just uniquely defined by rotation. Also, numbering the dimensions makes it seem like there is an order to them, but there isn’t. It’s the process of developing my thinking about this model stage by stage that can give the impression of a creation hierarchy. 

            From our point of view, time is the 4th dimension of our universe and the extra spacial dimension is unobserved. Living on an expanding 3d surface with 3d plus time physics, we are unaware of a 4th spacial dimension. It has no impact on our physics, including the observed inverse-square intensity to distance relationship of propagating fields. 

             Why is this? An electromagnetic wave doesn’t need a medium to propagate and should move freely in all available spacial dimensions. If it could move in four spacial dimensions the intensity would diminish with an inverse-cube relationship to distance. 

             If there is a 4th spacial dimension, what we observe as 3d propagation could be a 3d projection of a more fundamental 4d propagation. Or we could be wrong about space in our universe not being a medium, and it is needed for light to propagate even though it doesn’t affect the motion of light in the way a normal medium would. If this is somehow the case, it would indicate there is “emptier” space along the 4th spacial dimension toward the center of expansion, and propagation is not possible in that region. 

             Another way to think of this is to remember that this isn’t an expansion of a 4d hypersphere but an expansion of the 3d surface of a 4d hypersphere. It’s like the analog of the expansion of the closed circle 1d line universe in two dimensions from an origin that is not a part of the circle universe. It isn’t actually space as we know it outside our universe in the extra dimension. There is no momentum, so no space. In order even to say there is nothing there, we have to refine our definition of nothing. Our physics doesn’t apply there. (We also have to refine our definition of “there”.) 

             I’m calling it a spacial dimension because there would be a finite radius of our expanding 3d surface—that we may be able to calculate—which measures the distance our universe has moved radially from its origin in the Big Bang, though the origin point doesn’t exist in our universe, and the distance isn’t measured through the same kind of space we have.   

             Our universe must necessarily have a non-zero magnitude in this dimension in order to exist, but it would be very small, as are the extra dimension(s) in the Kaluza-Klein Theory and the various String Theories. The rest of the radial 4th spacial dimension is ______. 

             Just as with the circle universe, there may be other unconnected concentric 3d surfaces expanding from the same center of origin as ours. This model would be preferred to an expanding hypersphere because it is lower energy. It takes less energy to create and expand a series of 3d surfaces with tiny 4th dimensional magnitudes then it would a full blown 4d hypersphere. 

             Why a 3d surface? Why not the 1d circle universe with even lower energy or maybe some higher dimensional and higher energy surface beyond 3d? (Yes, there are such things.) I don’t know, but just like the energy levels of electrons in an atom, there would be energy leaps between each potential surface as the dimensions get higher in number. If my rotation principle is correct, many of them aren’t allowed because like the three spacial dimension models, they don’t all permits rotations that designate an orthogonal time dimension. Our universe originated by using the available energy in the most efficient way. That’s all I can say. 

             I don’t know enough math to work through all the questions this model raises. My hope is that it is not inconsistent with the current formulation of General Relativity, but I don’t know. 

             As a side note, a surface has more than one side. Which side would we be living on, the "inside" or the "outside" of the expanding hypersphere? There is an existing precedent for this question. It’s the contrast between de Sitter space and anti-de Sitter space. Look it up. 

            I have one main purpose in this series. That is to propose a geometric-dynamic reason for our perceived directionality in time. I come back to my initial premises. Whatever else of value there is (or isn’t) in my discussion, this is the heart of it and what I think is important: 

             When a body has a preferred dimension and a preferred direction in that dimension, such as time has 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. 

             An alternative cosmological model has to clear a few important bars to be useful. It must explain something that other models don’t, and it must include current understandings in a reasonable way. It should also predict something uniquely that can be tested by observations. I can’t really claim that my rotation model satisfies any of those conditions, but from my limited perspective, it might. I hope others will consider it and investigate further. 

             Let me know what you think! 

Hugh Moffatt 
Watertown, Massachusetts 
November 16, 2020 

*Here’s an example of a double rotation in our 3d world. Visualize a cylinder (like a soft drink can) spinning on its axis. This rotation defines a center line of rotation, the axis. Now let the spinning cylinder tumble end over end. This defines another axis of rotation through the sides of the cylinder intersecting the first axis at a point. These two rotations sum as a rotation around that point while each rotation continues independently rotating around its own axis line. The difference with a 4d double rotation is that each rotation is around a center plane and the two center planes intersect at a point, the center of the double rotation. I know. Planes intersect at a point?!! That’s four-dimensional geometry for you.