The Projection Geometry of a Theoretical Net-Zero Gravitational Field

~ charles

There are some equations I’m working on.[1]  They are intended to describe a projection from the outside surface of a sphere to the inside surface of a concentric spherical shell. This paradigm corresponds to a hypothetical gravitational field that has a net zero gravitational force in a universe with a single black hole.[2]

In two dimensions, the equations are scaled and rotated logarithmic spirals, which preserve the angles between its tangent and the intersecting circle — the polar slope angle is constant, which is consistent with flat planar projections. The equations in three dimensions will be published in a later post, perhaps.

There exists at least one r1(φ1) and a scaled inverse r2(φ1 + φ2), where every product of r1(φ) with r2 (φ) is constant. See the proof at the bottom of the page. 

See equations (1-5) below, which include:
  (1) logarithmic spiral in polar form (red),  
  (2) the inverse of equation 1 (black),  
  (3) equation 2 scaled and rotated (blue), and  
  (4) domain, scaling and rotational factors.

\begin{flushleft}
\begin{small}
{r_a = 1, \quad r_b = 2, \quad r_c = 4} \quad : \quad {r_b^2 = r_a \cdot r_c}
\end{flushleft}

(1)   \begin{equation*} $ ${r(\varphi) = b \cdot e^{\frac{k}{\pi} (\varphi)}}$ \end{equation*}

(2)   \begin{equation*} $ $r(\varphi) = \frac{1}{b} \cdot e^{-\frac{k}{\pi}(\varphi)$ \end{equation*}

(3)   \begin{equation*} $ $r(\varphi) = b \cdot e^k \cdot e^{-\frac{k}{\pi} (\varphi+\pi)}= b \cdot e^{-\frac{k}{\pi} (\varphi)}}$ \end{equation*}

(4)   \begin{equation*} $ $-\frac{\pi \cdot \ln{b}}{k} \leq \varphi \leq \frac{\pi \cdot \ln{b}}{k} $ \end{equation*}

(5)   \begin{equation*} $ $k = P^{-1} \ln{b} $ \end{equation*}

The variable P is the subtended projection angle in units of turns, 1=π (or 1=1/2 turn), and the variable b is the radius of the neutral surface, where all projection lines meet.

Below is an example drawn to represent the projection in three dimensions (not to scale) in a universe containing only a single black hole — a very small universe with a very large black hole!
[Click the image to enlarge.]

[1] For a more dynamic view of these equations, visit desmos.com.
[2] Here is where I am on all this speculation. Click the link to see a recently reported-measurable experiment inside (see page 18). This link has a graphic on page 24 that might be helpful.

Projections onto Projections

I imagine various projections onto projections forming a closed loop across I-dimensional space and/or curved spaces given the letter I, which is a pun for both Integer and Imaginary. Below are some examples.

Matter v. Fields

When I imagine these projections, projections in the holographic universe, there is no distinction between matter and fields. For example, matter and anti-matter are fields and anti-fields.

The well-known attributes of matter, however, are cancelled out: The wave functions are projected anti-wave functions and vice versa, whatever that entails. And these two are topographically connected across some geometry. And this is basically the universe summed across all particles and collection of particles (wave functions) that began under the most entropic and random circumstances imaginable, it is theorized — from primordial space-time fluctuations of quantum fields.

Somehow, conditions from there “gravitationally” coalesced into the clumps we observe today, particles, solar systems, galaxies, etc. Most likely, we will continue to observe this in the future — the continued “gravitational” contraction of matter to form some black holes, trigger some explosions, vaporize some disappearances, and all the rest — alongside the further expansion of the universe.

[Editor’s Note: The words “gravitationally” and “gravitational” are inside quotation marks because the two words are semantic representations of something we do not yet fully understand and the object of study.]

 

PROOF:\begin{flushleft*}

    \begin{align*} b \cdot e^k \cdot e^{-\frac{k}{\pi} (\varphi+\pi)} \quad \times \quad {b \cdot e^{\frac{k}{\pi} (\varphi)}} \quad &= \quad b^2 \\ e^k \cdot e^{-\frac{k}{\pi} (\varphi+\pi)} \quad \times \quad {e^{\frac{k}{\pi} (\varphi)}}\quad &= \quad 1 \\ e^k \cdot e^{-\frac{k}{\pi}(\varphi)-\frac{k}{\pi}(\pi)} \quad \times \quad {e^{\frac{k}{\pi} (\varphi)}}\quad &= \\ e^k \cdot e^{-\frac{k}{\pi} (\varphi)} \cdot e^{-k} \quad \times \quad {e^{\frac{k}{\pi} (\varphi)}}\quad &= \\ e^0 \cdot e^0 \quad &= \\ \end{align*}

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