\documentclass{article}

\def\ZZ{\hbox{\bf Z}}
\def\RR{\hbox{\bf R}}
\def\CC{\hbox{\bf C}}
\def\QQ{\hbox{\bf Q}}
\def\HH{\hbox{\bf H}}

\def\Ga{\Gamma}
\def\ga{\gamma}
\def\al{\alpha}
\def\De{\Delta}
\def\de{\delta}
\def\La{\Lambda}
\def\la{\lambda}
\def\eps{\varepsilon}
\def\ka{\kappa}
\def\oM{\overline{M}}
\def\oX{\overline{X}}
\def\onabla{\overline{\nabla}}
\def\om{\omega}
\def\Om{\Omega}
\def\sup{\rm sup}
\def\grad{\rm grad}
\def\pa{\partial}
\def\bpa{\bar{\partial}}
\def\spec{\hbox{spec}}
\def\Dom{\hbox{Dom}}
\def\bb{\bar{b}}
\def\bB{\bar{B}}
\def\bM{\overline{M}}
\def\bx{\bar{x}}
\def\bw{\bar{w}}
\def\bxi{\bar{\xi}}
\def\bz{\bar{z}}
\def\dist{{\rm dist}}
\def\bpar{\bar{\partial}}
\def\dimg{\dim_{\Gamma}}
\def\codimg{\hbox{codim}_{\Gamma}}
\def\cO{{\cal O}}
\def\Re{\hbox{Re}\,}
\def\Im{\hbox{Im}\,}
\def\Ker{\hbox{Ker}\,}

\begin{document}

\centerline {\bf Riemannian Geometry and General Relativity}
\centerline{\bf (Differential Geometry-2, MTH 3412)}
\centerline{\bf Home Assignement 4}
\centerline{\bf Professor M.Shubin}
\centerline{\bf Winter 1997}
%\medskip\noindent
%{\bf Textbooks:}
%\smallskip\noindent
%1) {\it Riemannian Geometry}, by Manfredo Perdig\~ao do Carmo.
%Birkh\"auser, Boston, 1993.
%\smallskip\noindent
%2) {\it General Theory of Relativity}, by P.A.M.Dirac. Princeton
%University Press, 1996.

\medskip\noindent
{\bf 1.} Prove that in 4-dimensional case the cyclic relation
$$R_{0123}+R_{0231}+R_{0312}=0\leqno (1)$$
is independent of the antisymmetry and symmetry relations
$$R_{\mu\nu\rho\sigma}=-R_{\nu\mu\rho\sigma},\quad
R_{\mu\nu\rho\sigma}=-R_{\mu\nu\sigma\rho},
\quad R_{\mu\nu\rho\sigma}=R_{\rho\sigma\mu\nu}\;,\leqno (2)$$
i.e. there exists a tensor of type (0,4) in $\RR^4$ such that it satisfies
(2)  but  not (1).

\medskip\noindent
{\bf 2.} Prove that in 2-dimensional case
$$R_{\mu\nu\rho\sigma}={S\over
2}(g_{\mu\sigma}g_{\nu\rho}-g_{\mu\rho}g_{\nu\sigma})\;,$$
where $S=g^{\mu\nu}R_{\mu\nu}$ is the scalar curvature.

\medskip\noindent
{\bf 3.} Show that the curvature tensor of the Schwarzschild metric does
not vanish.

\medskip\noindent
{\bf 4.} Find the mass of a black hole if  its horizon sphere has the size
of an atom i.e.
the radius about $10^{-8}$ cm.

\medskip\noindent
{\bf 5.} Prove that the Schwarzschild solution indeed has a singularity
(which can not be
corrected by changing coordinates).

\medskip\noindent
{\bf 6.} Consider a particle moving in the Schwarzschild gravitation field.
Prove that
its trajectory is a plane curve.

\medskip\noindent
{\bf 7.} Consider a satellite rotating around a black hole of the given
mass $m$ (or given
Schwarzschild radius $2m$) along  a circular orbit of a radius $R>2m$
(outside  the horizon sphere). Find
the period of the rotation by a distant observer clock and also by the clock in
the satellite. What happens with these periods as $R\to 2m$?

\medskip\noindent
{\bf 8.} Let $M$ be a Riemannian manifold. Define canonical coordinates
near  $p\in M$
by transfering arbitrary linear orthonormal coordinates in $T_pM$ near $0$ to
a neighborhood of $p$ in $M$ via the geodesic exponential map
$\exp_p:T_pM\to M$.
Prove that in these coordinates $g_{\mu\nu,\sigma}(0)=0$,
$\Ga^\sigma_{\mu\nu}(0)=0$,
$$\Ga^\sigma_{\mu\nu}(tv)v^\mu v^\nu=0\;,$$
for any $v\in\RR^n$ and sufficiently small $t\in\RR$,
and
$$R_{\mu\nu\rho\sigma}(0)={1\over
2}(g_{\mu\sigma,\nu\rho}(0)-g_{\nu\sigma,\mu\rho}(0)-
g_{\mu\rho,\nu\sigma}(0)+g_{\nu\rho,\mu\sigma}(0))\;.$$

\medskip\noindent
{\bf 9.} Let $G$ be a compact Lie group with a biinvariant Riemannian
metric, $X,Y,Z$
left-invariant vector fields on $G$. Prove that $$\nabla_XY={1\over 2}[X,Y]$$
 and
$$R(X,Y)Z=-{1\over 4}[[X,Y],Z].$$

\medskip\noindent
{\it Hint.} Use the fact that one-parametric subgroups and their left translations are geodesics.

\medskip\noindent
{\bf 10.} Let $G$ be a compact Lie group with a biinvariant Riemannian
metric, $X,Y,Z,W$
left-invariant vector fields on $G$. Prove that $$([X,Y],Z)=(X,[Y,Z])$$ and
$$(R(X,Y)Z,W)=-{1\over4}([X,Y],[Z,W]).$$

\end{document}