W1 · Relativistic π of Sgr A*

Light takes 70 minutes to escape the outermost ring of this chart, then 27,000 years to reach the Earth. Everything strange about π near Sgr A* lives in a region the size of our planetary system; everything else is the long quiet flight home.

$$\pi (r)=\frac{C(r)}{D(r)}=\frac{2{\pi }_{\infty }r}{2\underset{r_s}{\overset{r}{\int }}\frac{d{r}^{\prime }}{\sqrt{1-\frac{r_s}{r^{\prime }}}}}=\frac{{\pi }_{\infty }}{[1-\frac{r_s}{r}+\frac{r_s}{2r}·\ln (\frac{r}{r_s})+\frac{3r_s}{8r}+O(\frac{1}{r^2})]}$$

$r$ is the coordinate of a ring; $C$ and $D$ are ruler measures; $\pi (r)$ is the ratio of measures at ring $r$.

The equatorial Schwarzschild slice: $r_s=\frac{2GM}{c^2}$.

The proper line element: $${ds}^2={(1-\frac{r_s}{r})}^{-1}{dr}^2+r^2{d\phi }^2$$

$${\pi }_{\infty }=3.14159\dots$$