Relativistic Symmetry

Here we examine the symmetries that exist between the specialized and generalized theories of relativity, how they affect our experience on Earth, and how they relate to the model of relativistic spatial dilation.

One mystery that has bothered me more than most was the manner in which the time dilation effects due to special relativity and general relativity are almost perfectly equivalent here on Earth. The centripetal force of the Earth's rotation is believed to be responsible for the Earth being not quite spherical, with a bulge forming near the equator. As the Earth spins, the excess time dilation at the equator due to the additional velocity created by the Earth's rotation, is almost perfectly offset by the excess gravity at Earth's poles which is a consequence of a slightly shorter radius.

Surely there must be some physics at play here that binds the rotation of the Earth to it's radius and mass density, that's really not much in question. Perhaps more profoundly, surely these to sources of time dilation must be the same.

We can start by analyzing this symmetry from the perspective of a single, simple equation:

This is to say that is almost precisely our rotational velocity at the equator.

Looking at this in the context of the model of relativistic spatial dilation, recall that can be made to apply as a scalar of by taking the average of it's integral. This gives:

If we are not concerned with the new radius, but only the change in radius, the disappears which gives us a scalar of , such that .

On the other side of this equation, we have the rotational velocity at the equator. If instead of a set of equations that apply uniquely to the Earth's equator, consider the following:

This is to say that the sine of the integral from the pole to the equator is . This in turn allows that

Which when put together, gives

Interpretation

This result can be viewed from several perspectives. First, acts like a scalar of distance as it's applied only to the shift in spatial density. When applied to however, this might be viewed as a scalar of ¹, as would be inversely proportional to spatial density and scale proportionally to . This dilation of is required to maintain the symmetries that have carried existing models of relativity this far, and is foundational to the concept of relativity itself.

The left side of this equation is more physically obvious, but interesting none the less as we integrate not just over the rotational velocity, but of all of the rotational velocities on the Earth's surface.

Lastly, this relationship is time independent. While is obviously time dependent, represents a time dependent scalar of in the form of . This allows the time dependence to offset, making this relationship completely time independent.

In relation to

This section relates closely to, but extends upon the principles covered in the article available here.

Recall that was the variable I assigned to the velocity found through the proposed relationship between and our velocity to the cosmic rest reference frame.² can be found by the following:

If we take a closer look at how applies to this relationship, we find that

While this relationship between and indicates that there may be something to this relationship between spatial dilation, and the magnitude of velocity, there is clearly a dependency upon the observer's trajectory through space.

If this relationship exists, it must be dependent upon a body's motion in respect to the local spatial dilation. Consider the fact that is perpendicular to the direction of spatial dilation, . on the other hand, is parallel to the direction of spatial dilation, .³

Relating this symmetry to the relationship between gives the following, where is the angle between and .

in turn yields

This relationship appears to lend credence to the principle that allows to indicate a pure magnitude in some situations, as described in the Center of Mass Shift post, where instaneous velocities can represent a magnitude relative to other velocities, rather than a change of position with respect to time.

Footnotes

1. The principle that being equivalent in each reference frame is not necessarily the same as being equivalent between reference frames plays a significant role in the model of relativity being proposed. See Predictions and the theory of gravitational time for more. ↩

2. If you have yet to read the post that summarizes the model that inspired the creation of ULLD you can find that here. ↩

3. The spatial dilation would actually radiate from that body, assuming this relationship takes place in a vacuum in empty space. However, the portion of that radiating spatial dilation would likely have no effect on the body in motion. ↩