The Projection Geometry of a Theoretical Net-Zero Gravitational Field

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.

There exists at least one r11) and a scaled inverse r21 + 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.

{r_a = 1,  \quad r_b = 2, \quad r_c = 4} \quad : \quad {r_b^2 = r_a \cdot r_c}

(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
[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.

Happy Pi Day, everyone!


    \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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