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%TCIDATA{<META NAME="Title" CONTENT="Reference/Physics/Riemannian Geometry, the Einstein Tensor">}
%TCIDATA{Created=Mon Aug 19 14:52:24 1996}
%TCIDATA{LastRevised=Thu Feb 13 16:20:53 1997}
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\section{Riemannian Geometry, the Einstein Tensor\label{Riemann geometry}}

The basic principles of general relativity are

\begin{enumerate}
\item  The geodesic postulate: free falling particles move along geodesics
of space-time with the proper time $\tau $ or arc length $s$ as parameter.
For particles with zero rest mass (photons), the use of a free parameter is
required because for them the equation $ds=0$ holds. From $\delta \int ds=0$
the equations of motion can be derived: 
\[
\frac{d^{2}x^{\alpha }}{ds^{2}}+\Gamma _{\beta \gamma }^{\alpha }\frac{%
dx^{\beta }}{ds}\frac{dx^{\gamma }}{ds}=0
\]

\item  The \textit{principle of equivalence}: inertial mass $\equiv $
gravitational mass $\Rightarrow $ gravitation is equivalent with a curved
space-time where particles move along geodesics.

\item  By a proper choice of the coordinate system it is possible to make
the metric locally flat in each point 
\[
x_{i}:g_{\alpha \beta }(x_{i})=\eta _{\alpha \beta }:=\limfunc{diag}%
(-1,1,1,1)
\]
\end{enumerate}

\medskip \label{Riemann tensor}The \textit{Riemann tensor} is defined as 
\[
R_{\nu \alpha \beta }^{\mu }T^{\nu }:=\nabla _{\alpha }\nabla _{\beta
}T^{\mu }-\nabla _{\beta }\nabla _{\alpha }T^{\mu } 
\]
where the covariant derivative is given by 
\[
\nabla _{j}a^{i}=\partial _{j}a^{i}+\Gamma _{jk}^{i}a^{k} 
\]
and 
\[
\nabla _{j}a_{i}=\partial _{j}a_{i}-\Gamma _{ij}^{k}a_{k} 
\]
\label{Christoffel symbols}Here, 
\[
\Gamma _{jk}^{i}=\frac{\partial ^{2}\bar{x}^{l}}{\partial x^{j}\partial x^{k}%
}\frac{\partial x^{i}}{\partial \bar{x}^{l}}=\frac{1}{2}g^{il}(\partial
_{k}g_{lj}+\partial _{j}g_{lk}-\partial _{l}g_{jk}) 
\]
are the \textit{Christoffel symbols}.

\label{Second-order tensor}For a second-order tensor the following holds: 
\begin{eqnarray*}
[\nabla _{\alpha },\nabla _{\beta }]T_{\nu }^{\mu } &=&R_{\sigma \alpha
\beta }^{\mu }T_{\nu }^{\sigma }+R_{\nu \alpha \beta }^{\sigma }T_{\sigma
}^{\mu } \\
\nabla _{k}a_{j}^{i} &=&\partial _{k}a_{j}^{i}-\Gamma
_{kj}^{l}a_{l}^{i}+\Gamma _{kl}^{i}a_{j}^{l} \\
\nabla _{k}a_{ij} &=&\partial _{k}a_{ij}-\Gamma _{ki}^{l}a_{lj}-\Gamma
_{kj}^{l}a_{jl} \\
\nabla _{k}a^{ij} &=&\partial _{k}a^{ij}+\Gamma _{kl}^{i}a^{lj}+\Gamma
_{kl}^{j}a^{il}
\end{eqnarray*}
The following also holds: 
\[
R_{\beta \mu \nu }^{\alpha }=\partial _{\mu }\Gamma _{\beta \nu }^{\alpha
}-\partial _{\nu }\Gamma _{\beta \mu }^{\alpha }+\Gamma _{\sigma \mu
}^{\alpha }\Gamma _{\beta \nu }^{\sigma }-\Gamma _{\sigma \nu }^{\alpha
}\Gamma _{\beta \mu }^{\sigma } 
\]

\label{Ricci tensor}The \textit{Ricci tensor} is a contraction of the
Riemann tensor: 
\[
R_{\alpha \beta }:=R_{\alpha \mu \beta }^{\mu } 
\]
which is symmetric: $R_{\alpha \beta }=R_{\beta \alpha }$.

\label{Bianchi identities}The \textit{Bianchi identities} are 
\[
\nabla _{\lambda }R_{\alpha \beta \mu \nu }+\nabla _{\nu }R_{\alpha \beta
\lambda \mu }+\nabla _{\mu }R_{\alpha \beta \nu \lambda }=0 
\]

\label{Einstein tensor}The \textit{Einstein tensor} is given by 
\[
G^{\alpha \beta }:=R^{\alpha \beta }-\frac{1}{2}g^{\alpha \beta }R 
\]
\label{Ricci scalar}where $R:=R_{\alpha }^{\alpha }$ is the \textit{Ricci
scalar}, for which the following holds: 
\[
\nabla _{\beta }G_{\alpha \beta }=0 
\]
With the variational principle 
\[
\delta \int (L(g_{\mu \nu })-Rc^{2}/16\pi \kappa )\sqrt{|g|}d^{4}x=0 
\]
\label{Einstein field equations}for variations $g_{\mu \nu }\rightarrow
g_{\mu \nu }+\delta g_{\mu \nu }$ the \textit{Einstein field equations} can
be derived: 
\[
\fbox{$G_{\alpha \beta }=\dfrac{8\pi \kappa }{c^{2}}T_{\alpha \beta }$} 
\]
which can also be written as 
\[
R_{\alpha \beta }=\frac{8\pi \kappa }{c^{2}}(T_{\alpha \beta }-\frac{1}{2}%
g_{\alpha \beta }T_{\mu }^{\mu }) 
\]
For empty space this is equivalent to $R_{\alpha \beta }=0$. The equation $%
R_{\alpha \beta \mu \nu }=0$ has as its only solution a flat space.

The Einstein equations are 10 independent equations, which are of second
order in $g_{\mu \nu }$. From this, the\label%
{Laplace equation from Newtonian gravitation} Laplace equation from
Newtonian gravitation can be derived by stating: $g_{\mu \nu }=\eta _{\mu
\nu }+h_{\mu \nu }$, where $|h|\ll 1$. In the stationary case, this results
in 
\[
\nabla ^{2}h_{00}=8\pi \kappa \varrho /c^{2} 
\]

The most general form of the field equations is: 
\[
R_{\alpha \beta }-\frac{1}{2}g_{\alpha \beta }R+\Lambda g_{\alpha \beta }=%
\frac{8\pi \kappa }{c^{2}}T_{\alpha \beta } 
\]
where $\Lambda $ is the\label{Cosmological constant} \textit{cosmological
constant}. This constant plays a role in inflationary models of the universe.

\subsection{\ Related Reference Topics}

{\small \hyperref{Calculus}{}{}{../Mathematics/cal_contents.tex}}

{\small \hyperref{Differential Equations}{}{}{../Mathematics/ODE_contents.tex%
}}

{\small \hyperref{Table of Integrals}{}{}{../Mathematics/TAB_integrals.tex}}

{\small \hyperref{Table of Derivatives}{}{}{%
../Mathematics/TAB_derivatives.tex}}

{\small \hyperref{Table of Physical Constants}{}{}{PhysicalConstants.tex}}

{\small \hyperref{Table of SI Units}{}{}{SIUnits.tex}}

\subsection{\ Related Scientific Notebook Topics}

{\small \hyperref{Numbers, Functions, and Units}{}{}{../../Help/DM2.tex}}

{\small \hyperref{Calculus}{}{}{../../Help/DM7.tex}}

{\small \hyperref{Vector Calculus}{}{}{../../Help/DM9.tex}}

{\small \hyperref{Differential Equations}{}{}{../../Help/DM10.tex}}

\noindent \hrulefill

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\noindent \hrulefill

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