NEW HORIZONS

In 4d Domain Connector I outlined an idea for 4-dimensional fundamental domains. A consequence of this type of 4D tiling is the observed expansion of space. Although I dubbed this tiling O'Neill space, in reality it is just the fabric of space-time understood from a different point of view. There are, however, a few direct consequences that derive from stating that space is expanding from 4th dimensional space. First off, it means that the rate of expansion is linked, in some way, to our perception and knowledge of what constitutes time itself. Second, it shows that time and space are orthogonal entities, existing at right angles to one another. Other speculations that I will address in this and later posts are;

• There are no straight line accelerations in space-time, as viewed from outside the 4d reference frame
• The expansion of space-time becomes negligible, or non-existent, at the speed of light
• Time is not a spatial dimension with vector points other than those of 4d space
• Rest mass is effectively a black hole singularity for 4d space

Imagine you are sitting on this train. When you look out the window you notice that objects in the foreground rush passed you at a seemingly greater speed than objects that are situated on the horizon. If we imagine that the motion of the train is equivalent to time, then we can say that the relative motion of the objects in the foreground constitute events passing us by, while the objects that appear frozen in the distance correspond to a 4d spatial vector point that is equivalent to the speed of light. These spacial vector points exist at right angles to the motion of the train, and are therefore 4 dimensional.

Now imagine that the landscape is tiled with a grid pattern, like the one above. This grid pattern is fixed relative to the motion of the train, but it is expanding out towards you from a point dubbed infinity on the horizon. This is the expansion of space that is happening orthogonally to the passage of time. It is driving the circular motion of time, just as water coming out of a right angled sprinkler nozzle drives the sprinkler around. Accelerating in space is equivalent to zooming out into the horizon where the trees and shrubs appear mostly static, thus achieving the relativistic effect of time dilation. If we imagine that our straight line track is actually the circumference of an infinitely large circle then we can legitimately bend our planar grid pattern into a circle (see below). The advantage of having a circular space-time diagram is so we can include the expansion of space from a central 4d point and still have all the characteristics of an ordinary Minkowskian diagram.

In Minkowskian Spacetime diagrams, time is shown on the vertical axis and space is shown on the horizontal axis. The radial space-time diagram has the time 'axis' for the entire 3d spatial realm around the circumference of the circle, while the spatial axis of the 4d realm goes off towards the centre. The whole diagram is rotating with time. Just like an ordinary wheel or merry-go-round the circumference of the circle is traveling much faster than the centre; indeed at the very centre we can say that no rotational motion (i.e. time) is taking place whatsoever. This means that the centre corresponds to the speed of light c and the horizon as viewed from our train.

If it were possible to build a spacecraft capable of traveling at the speed of light, it would depart from a point in spacetime at the outer periphery of the circle (green line) and travel inwards towards the centre (purple line). It would be traveling in a straight line away from this point, but the departure point is also moving around the circumference as time moves on. An outside observer, like ourselves, would see a curved trajectory; by virtue of what is known as the Coreolis effect. If the outsiders perspective exists, it means that there are no straight line accelerations extant anywhere in the Universe. Not even light could be thought of as traveling in straight lines; it would spiral towards different datums depending upon where the original parameters are set. The parametres are entirely movable so this paints the whole Universe as some kind of giant fractal whirligig.

Each time this radial space-time chart rotates it would wipe out or occlude data from an earlier time. The most obvious solution to this is some kind of 5d Riemann surface similar to Log z that extends indefinitely in the vertical axis. The central 'spine' of this structure denotes the speed of light.
By virtue of the fact that Riemann surfaces include both real and imaginary coordinates they give a much better picture of Einstein's equations beyond the speed of light. This is because superluminal  equations require solutions to the square root of a negative number, something which leads directly to imaginary numbers.

It has been speculated that at faster than light speeds time must begin to flow in reverse. However, from my understanding time is space expanding from the central spine at coordinate zero. Therefore, when you cross this impassable boundary point, the expansion of space would be coming from behind your trajectory; something which I don't believe would alter your perception of time or space very much, if indeed at all. Another example of a Riemann surface that could be useful is this one of the multi-valued cube root, or one such like it;

This image returns us to the concept that there exists two separate Universes side by side one another that interact to drive each other one, like an electromagnetic wave. See last post; Tumbling Toy Universe Theory for more on this.