% Nov 20 2018.

\documentclass[12pt,titlepage,oneside]{book}
\usepackage{epsfig}
\setlength{\textwidth}{6.5truein}
\setlength{\textheight}{9truein}
% \voffset=-.75truein
\voffset=.0truein
\hoffset=-.75truein
\widowpenalty=10000
\clubpenalty=10000
\renewcommand{\topfraction}{0.9}
\renewcommand{\bottomfraction}{0.5}
\renewcommand{\floatpagefraction}{0.8}
\renewcommand{\textfraction}{0.1}
% \setcounter{bottomnumber}{2}
% \setcounter{totalnumber}{4}
\renewcommand{\dbltopfraction}{0.9}
\renewcommand{\dblfloatpagefraction}{0.8}

\pretolerance=500
\tolerance=1000
\hbadness=3000
\vbadness=3000
\hyphenpenalty=400

\begin{document}
\def\sqr#1#2{{\vcenter{\hrule height.#2pt
   \hbox{\vrule width.#2pt height#1pt \kern#1pt
      \vrule width.#2pt}
   \hrule height.#2pt}}}
\def\square{{\mathchoice\sqr64\sqr64\sqr{4.2}3\sqr{3.0}3}}
\baselineskip 16pt

\setcounter{page}{1}
\renewcommand{\thepage}{\arabic{page}} % \bf 
\Large

Newton gravitational potential:
\begin{eqnarray}
  V = V_N(r) = \frac{h^2}{r^2} - \frac{2m}{r}\ .
\end{eqnarray}
Schwarzschild gravitational potential:
\begin{eqnarray}
  V = V_{GR}(r) = \frac{h^2}{r^2} - \frac{2m}{r} - \frac{2m\,h^2} {r^3}
    = V_N(r) - \frac{2m\,h^2}{r^3}\ .
\end{eqnarray}
Using gravitational units $G=c=1$ and as mass unit [sm], the 
central (solar) mass $m$:
\begin{eqnarray}
  V_N(r) = \frac{h^2}{r^2} - \frac{2}{r}~~~{\rm and}~~~
  V_{GR}(r) = \frac{h^2}{r^2} - \frac{2}{r} - \frac{2 h^2}{r^3}\ .
\end{eqnarray}
Here $h$ is the specific angular moment and the number 2 carries the
dimsion of a mass. In these units we have $h\approx 1000$ for the orbit 
of the earth about the sun.

Consider a point mass at the center.

Small $h$: Barrier against falling into the black hole disappears.

Large enough, increasing $h$: Barrier againt falling into the black

hole increases.

The maximum and minimum of the $V_{GR}(r)$ potential are at
\begin{eqnarray} \label{rpm}
  r_{\pm} = \frac{h^2}{2} \pm \sqrt{\left(\frac{h^2}{2}\right)^2-3\,h^2}\,,
\end{eqnarray}
where $V_{\min}=V_{GR}(r_+)$ and $V_{\max}=V_{GR}(r_-)$. From Eq.~(\ref{rpm})
it follows that there are no extrema for $h^2\le 12$.
\eject

\centerline{\bf Increasing $h\ge 4$} \medskip

\begin{figure}[ht] \begin{center}
\includegraphics[scale=1.00]{VGR000040}
\end{center} \end{figure}
\centerline{Gravitational Potential.}
\bigskip

\begin{figure}[ht] \begin{center}
\includegraphics[scale=1.00]{VGR000050}
\end{center} \end{figure}
\centerline{Gravitatonal Potential.}
\eject

\begin{figure}[ht] \begin{center}
\includegraphics[scale=1.00]{VGR000100S}
\end{center} \end{figure}
\centerline{Schwarzschild Singularity.}
\bigskip

\begin{figure}[ht] \begin{center}
\includegraphics[scale=1.00]{VGR000100N}
\end{center} \end{figure}
\centerline{Newton Potential.}
\eject

\begin{figure}[ht] \begin{center}
\includegraphics[scale=1.00]{VGR001000S}
\end{center} \end{figure}
\centerline{Schwarzschild Singularity.}
\bigskip

\begin{figure}[ht] \begin{center}
\includegraphics[scale=1.00]{VGR001000N}
\end{center} \end{figure}
\centerline{Newton Potential.}
\eject

\begin{figure}[ht] \begin{center}
\includegraphics[scale=1.00]{VGR010000S}
\end{center} \end{figure}
\centerline{Schwarzschild Singularity.}
\bigskip

\begin{figure}[ht] \begin{center}
\includegraphics[scale=1.00]{VGR010000N}
\end{center} \end{figure}
\centerline{Newton Potential.}
\eject

\centerline{\bf Drecreasing $h\le 4$} \medskip

\begin{figure}[ht] \begin{center}
\includegraphics[scale=1.00]{VGR000040}
\end{center} \end{figure}
\centerline{Gravitational Potential.}
\bigskip

\begin{figure}[ht] \begin{center}
\includegraphics[scale=1.00]{VGR000035}
\end{center} \end{figure}
\centerline{Gravitational Potential.}
\eject

\begin{figure}[ht] \begin{center}
\includegraphics[scale=1.00]{VGR000034}
\end{center} \end{figure}
\centerline{Gravitational Potential.}
\bigskip

\begin{figure}[ht] \begin{center}
\includegraphics[scale=1.00]{VGR000030}
\end{center} \end{figure}
\centerline{Gravitational Potential.}
\eject

\relax
\end{document}

\relax
\end{document}
