and now for something completely different...

This is the key identity:

We know that, since space is isotropic, the identity must hold true both locally and globally. This is not an issue in flat Euclidean 3-space. However, once we include time into the model and are confronted with relativity, some issues arise. How does the identity remain true both locally and globally?

The answer appears to be that , and are not only dimensionless quantities as we tend to think they are, but they can also be combined in order to make an actual quantity (or scalar, or magnitude, or a measurable entity, or however you wish to say it.)

Because this is the case, the addition of time into the coordinate system can be accomplished by understanding that when this is done (when time is introduced), the coordinate system must wrap back on itself. This is an abstract solution, but it illustrates the problem with having this new identity hold true in the global and local domain. By this I mean that the equation holds true for both sets of points, those that are adjacent and those that are not.

Once you grasp the significance of this reality it becomes clear that the dark energy doesn't exist in the "model" and that's why we don't see it. Once energy becomes oriented such that it is perpendicular to the "model," it disappears because its amplitude goes to zero while its wavelength goes to infinity. It's a geometric artifact created by the way we choose to make the "model" mathematically or, more specifically, geometrically.

An alternative model uses time, length, and direction as the three dimensions. We have coined the term "synchronous geometry" to differentiate this model from the standard model. In this model things flip symmetrically with the way they are represented in the standard model. In synchronous geometry the length triplet that is used to identify a point or event is replaced with a direction triplet. The vector quantity that is present in the standard model (direction) becomes the scalar quantity in the synchronous model.

Following along, the scalar value (length) in the standard model becomes the vector quantity. In the standard model the direction has no magnitude or scalar quantity or length. In the synchronous geometry, where things are swapped, the length has no direction (think of it as the radius of a sphere) since direction is the scalar quantity in this version of spacetime (or more accurately, Euclidean 3-space).

Because this can be done, it becomes clear (or murky) that there should be an additional effect on the phenomenon that we call frame-dragging, which is more clearly another artifact of the way we are choosing to do the math. Rather than try and explain the math (since I don't have the algebra skills to write the expressions that would be necessary to get the point across), I can suggest another thought experiment.

The rings of Saturn are orbiting the gas giant and are frame-dragging spacetime around with them. If we were to look closely at the interface between the plane in which the frame-dragging is represented mathematically and the adjacent spacetime, we should see one of two things: either the spacetime that the gas giant occupies adjacent to this plane interacts identically with the frame that is being dragged, or, the two sides interact differently due the effect on spacetime caused by the Sun's gravity. In other words, the advancing and retreating sides of the frame, relative the Sun's field, will produce different effects. Doesn't this mean that the spacetime itself is rotating in the frame? I think it does mean that, except that I don't have the math skills to show it one way or the other.

In any event, the dark matter question is another effect caused by the same issue with how we do the math. It isn't a trivial thing to understand this or to explain it to someone. It isimpossibleto explain it to anyone who doesn't understand the identity above, or what it means. My best description of it is to say that directions or angles can be used to construct a coordinate system in a symmetrically identical manner as what is done in the Cartesian system.

Having briefly looked at how we model quanta mathematically, it sure looks like the same question arises there as regards to how we are using the standard model of spacetime. In the synchronous model it is almost trivial to show how energy states transition at the rates at which they occur. There isn't really a step-function, but rather a hyperbolic function that goes to zero. Also, once direction is viewed as a quantity it becomes clear (or murky) that direction has no sign associated with it, which very likely explains spooky action at a distance. We have time and distance which each have signs associated with them and direction which does not have a sign associated with it. These are the three quantities that exist in synchronous geometry.

When distance (ength) and ime are combined in spacetime the result is bounded by and :

When direction (urns) and ime are combined in spacetime the result is bounded by zero and infinity:

Note that the sign is actually associated with length rather than direction. This has an effect when spacetime is curved.

The turn isn’t considered to be a physical quantity yet. Nevertheless, we know that it can be used to quantify other physical phenomena such as fictitious force, magnetomotive force, and of course, frame-dragging.

Recommended reading: The Tau Manifesto

Recommended behavior: If you don't understand the math behind the key identity, don't argue that the mathematical argument presented here is incorrect.