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final_expressions.tex
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15 lines (12 loc) · 4.81 KB
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\documentclass{article}\usepackage{amsmath}\usepackage[margin=0.5in]{geometry}\begin{document}
\section*{Metric}
\[ ds^2 = -\,dt^2 + \bigl(1 - f(r,t)\bigr)\,dr^2 + r^2\,d\theta^2 + r^2\sin^2\theta\,d\phi^2 \]
\section*{Curvature Invariants}
\[ R = \frac{r^{2} f{\left(r,t \right)} \frac{\partial^{2}}{\partial t^{2}} f{\left(r,t \right)} - \frac{r^{2} \left(\frac{\partial}{\partial t} f{\left(r,t \right)}\right)^{2}}{2} - r^{2} \frac{\partial^{2}}{\partial t^{2}} f{\left(r,t \right)} - 2 r \frac{\partial}{\partial r} f{\left(r,t \right)} + 2 f^{2}{\left(r,t \right)} - 2 f{\left(r,t \right)}}{r^{2} \left(f^{2}{\left(r,t \right)} - 2 f{\left(r,t \right)} + 1\right)} \]
\[ R_{\mu\nu}R^{\mu\nu} = \frac{r^{4} \left(2 \left(f{\left(r,t \right)} - 1\right) \frac{\partial^{2}}{\partial t^{2}} f{\left(r,t \right)} - \left(\frac{\partial}{\partial t} f{\left(r,t \right)}\right)^{2}\right)^{2} + r^{2} \left(r \left(2 \left(f{\left(r,t \right)} - 1\right) \frac{\partial^{2}}{\partial t^{2}} f{\left(r,t \right)} - \left(\frac{\partial}{\partial t} f{\left(r,t \right)}\right)^{2}\right) - 4 \frac{\partial}{\partial r} f{\left(r,t \right)}\right)^{2} + 32 r^{2} \left(f{\left(r,t \right)} - 1\right) \left(\frac{\partial}{\partial t} f{\left(r,t \right)}\right)^{2} + 8 \left(r \frac{\partial}{\partial r} f{\left(r,t \right)} - 2 \left(f{\left(r,t \right)} - 1\right) f{\left(r,t \right)}\right)^{2}}{16 r^{4} \left(f{\left(r,t \right)} - 1\right)^{4}} \]
\section*{Stress--Energy Tensor}
\[ T_{\mu\nu} = \begin{pmatrix}\frac{2 r \left(f{\left(r,t \right)} - 1\right)^{3} \frac{\partial^{2}}{\partial t^{2}} f{\left(r,t \right)} + r \left(f{\left(r,t \right)} - 1\right)^{2} \left(\frac{\partial}{\partial t} f{\left(r,t \right)}\right)^{2} - 2 r \left(f{\left(r,t \right)} - 1\right) \frac{\partial^{2}}{\partial t^{2}} f{\left(r,t \right)} - r \left(\frac{\partial}{\partial t} f{\left(r,t \right)}\right)^{2} + 4 \left(f{\left(r,t \right)} - 1\right)^{3} \left(- 2 f{\left(r,t \right)} - \frac{\partial}{\partial r} f{\left(r,t \right)} + 2\right) - 4 \left(f{\left(r,t \right)} - 1\right)^{2} \frac{\partial}{\partial r} f{\left(r,t \right)}}{64 \pi r \left(f{\left(r,t \right)} - 1\right)^{4}} & - \frac{\frac{\partial}{\partial t} f{\left(r,t \right)}}{16 \pi r \left(f{\left(r,t \right)} - 1\right)} & 0 & 0 \\
- \frac{\frac{\partial}{\partial t} f{\left(r,t \right)}}{16 \pi r \left(f{\left(r,t \right)} - 1\right)} & \frac{- 2 r \left(f{\left(r,t \right)} - 1\right)^{3} \frac{\partial^{2}}{\partial t^{2}} f{\left(r,t \right)} - r \left(f{\left(r,t \right)} - 1\right)^{2} \left(\frac{\partial}{\partial t} f{\left(r,t \right)}\right)^{2} + 2 r \left(f{\left(r,t \right)} - 1\right) \frac{\partial^{2}}{\partial t^{2}} f{\left(r,t \right)} + r \left(\frac{\partial}{\partial t} f{\left(r,t \right)}\right)^{2} - 4 \left(f{\left(r,t \right)} - 1\right)^{3} \left(2 f{\left(r,t \right)} + \frac{\partial}{\partial r} f{\left(r,t \right)} - 2\right) + 4 \left(f{\left(r,t \right)} - 1\right)^{2} \frac{\partial}{\partial r} f{\left(r,t \right)}}{64 \pi r \left(f{\left(r,t \right)} - 1\right)^{3}} & 0 & 0 \\
0 & 0 & \frac{r \left(2 r \left(f{\left(r,t \right)} - 1\right)^{3} \frac{\partial^{2}}{\partial t^{2}} f{\left(r,t \right)} + r \left(f{\left(r,t \right)} - 1\right)^{2} \left(\frac{\partial}{\partial t} f{\left(r,t \right)}\right)^{2} + 2 r \left(f{\left(r,t \right)} - 1\right) \frac{\partial^{2}}{\partial t^{2}} f{\left(r,t \right)} + r \left(\frac{\partial}{\partial t} f{\left(r,t \right)}\right)^{2} - 8 \left(f{\left(r,t \right)} - 1\right)^{4} + 4 \left(f{\left(r,t \right)} - 1\right)^{3} \left(2 f{\left(r,t \right)} + \frac{\partial}{\partial r} f{\left(r,t \right)} - 2\right) - 4 \left(f{\left(r,t \right)} - 1\right)^{3} \frac{\partial}{\partial r} f{\left(r,t \right)} + 4 \left(f{\left(r,t \right)} - 1\right)^{2} \frac{\partial}{\partial r} f{\left(r,t \right)}\right)}{64 \pi \left(f{\left(r,t \right)} - 1\right)^{4}} & 0 \\
0 & 0 & 0 & - \frac{r \left(4 \left(f{\left(r,t \right)} - 1\right)^{4} \left(2 f{\left(r,t \right)} + \frac{\partial}{\partial r} f{\left(r,t \right)} - 2\right) - \left(f{\left(r,t \right)} - 1\right) \left(2 r \left(f{\left(r,t \right)} - 1\right)^{3} \frac{\partial^{2}}{\partial t^{2}} f{\left(r,t \right)} + r \left(f{\left(r,t \right)} - 1\right)^{2} \left(\frac{\partial}{\partial t} f{\left(r,t \right)}\right)^{2} + 2 r \left(f{\left(r,t \right)} - 1\right) \frac{\partial^{2}}{\partial t^{2}} f{\left(r,t \right)} + r \left(\frac{\partial}{\partial t} f{\left(r,t \right)}\right)^{2} + 4 \left(f{\left(r,t \right)} - 1\right)^{3} \left(2 f{\left(r,t \right)} + \frac{\partial}{\partial r} f{\left(r,t \right)} - 2\right) + 4 \left(f{\left(r,t \right)} - 1\right)^{2} \frac{\partial}{\partial r} f{\left(r,t \right)}\right)\right) \sin^{2}{\left(\theta \right)}}{64 \pi \left(f{\left(r,t \right)} - 1\right)^{5}}\end{pmatrix} \]
\end{document}