% Bernd Berg Nov 9 2018.
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\centerline{\bf Special and General Relativity (PHZ 4601/5606) Fall 2018 
Solutions} \medskip

\centerline{\bf Set 12} 

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\item {\bf 30. Negative energy density due to the cosmological constant.}
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\noindent The modified Poisson [Rindler equation (14.20), p.304]
can be re-written as
\begin{eqnarray} \nonumber
  \sum_i \Phi_{ii} = 4\pi\,G\,\left(\rho_o
                   - \frac{\Lambda\,c^2}{4\pi\,G}\right)\,.
\end{eqnarray} 
The absolut volue of the energy density due to the cosmological 
constant is therefore
\begin{eqnarray} \nonumber
  \rho_L = \frac{\Lambda\,c^2}{4\pi\,G}\,.
\end{eqnarray} 
Using the numbers given in the problem we obtain
\begin{eqnarray} \nonumber
  \rho_o = 2.8\times 10^{-31}\,[g/cm^3]~~~{\rm and}~~~
  \rho_L = 1.2\times 10^{-29}\,[g/cm^3]\,.
\end{eqnarray} 
In percent the relative values are then given by
\begin{eqnarray} \nonumber
  p_o = 100\,\frac{\rho_o}{\rho_o+\rho_L}=2.3\%~~~{\rm and}~~~
  p_L = 100\,\frac{\rho_o}{\rho_o+\rho_L}=97.7\%\,.
\end{eqnarray} 
Our present consideration does not include dark matter.
Besides, there is something wrong with the input numbers.

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