User: jfaber Date: 2012/04/04 07:52 AM
Modified: /genrel/ genrel.aux, genrel.log, genrel.nav, genrel.pdf, genrel.tex
Log: update for today's talk
File Changes:
Directory: /genrel/ ===================
File [modified]: genrel.aux Delta lines: +7 -5 =================================================================== --- genrel/genrel.aux 2012-04-04 12:15:48 UTC (rev 34) +++ genrel/genrel.aux 2012-04-04 12:52:23 UTC (rev 35) @@ -57,8 +57,10 @@ @writefile{nav}{\headcommand {\beamer@framepages {21}{21}}} @writefile{nav}{\headcommand {\slideentry {0}{0}{22}{22/22}{}{0}}} @writefile{nav}{\headcommand {\beamer@framepages {22}{22}}} -@writefile{nav}{\headcommand {\beamer@partpages {1}{22}}} -@writefile{nav}{\headcommand {\beamer@subsectionpages {1}{22}}} -@writefile{nav}{\headcommand {\beamer@sectionpages {1}{22}}} -@writefile{nav}{\headcommand {\beamer@documentpages {22}}} -@writefile{nav}{\headcommand {\def \inserttotalframenumber {22}}} +@writefile{nav}{\headcommand {\slideentry {0}{0}{23}{23/23}{}{0}}} +@writefile{nav}{\headcommand {\beamer@framepages {23}{23}}} +@writefile{nav}{\headcommand {\beamer@partpages {1}{23}}} +@writefile{nav}{\headcommand {\beamer@subsectionpages {1}{23}}} +@writefile{nav}{\headcommand {\beamer@sectionpages {1}{23}}} +@writefile{nav}{\headcommand {\beamer@documentpages {23}}} +@writefile{nav}{\headcommand {\def \inserttotalframenumber {23}}}
File [modified]: genrel.log Delta lines: +15 -13 =================================================================== --- genrel/genrel.log 2012-04-04 12:15:48 UTC (rev 34) +++ genrel/genrel.log 2012-04-04 12:52:23 UTC (rev 35) @@ -1,4 +1,4 @@ -This is pdfTeX, Version 3.1415926-1.40.10 (TeX Live 2009) (format=pdflatex 2009.11.7) 3 APR 2012 21:34 +This is pdfTeX, Version 3.1415926-1.40.10 (TeX Live 2009) (format=pdflatex 2009.11.7) 4 APR 2012 08:50 entering extended mode \write18 enabled. %&-line parsing enabled. @@ -1231,18 +1231,20 @@ File: lapseshift.png Graphic file (type png) <use lapseshift.png> [20
- <./lapseshift.png (PNG copy)>] + <./lapseshift.png (PNG copy)>] [21
-LaTeX Warning: Command \d invalid in math mode on input line 261. +]
+LaTeX Warning: Command \d invalid in math mode on input line 282.
-LaTeX Warning: Command \d invalid in math mode on input line 261.
-[21 +LaTeX Warning: Command \d invalid in math mode on input line 282.
-] <gwpol.png, id=594, 803.0pt x 401.5pt> +[22 + +] <gwpol.png, id=620, 803.0pt x 401.5pt> File: gwpol.png Graphic file (type png) - <use gwpol.png> [22 + <use gwpol.png> [23
<./gwpol.png (PNG copy)>] \tf@nav=\write6 @@ -1261,10 +1263,10 @@
) Here is how much of TeX's memory you used: - 15010 strings out of 493848 - 276828 string characters out of 3149207 + 15014 strings out of 493848 + 276863 string characters out of 3149207 358008 words of memory out of 3000000 - 17806 multiletter control sequences out of 15000+200000 + 17808 multiletter control sequences out of 15000+200000 11753 words of font info for 43 fonts, out of 3000000 for 9000 718 hyphenation exceptions out of 8191 47i,16n,57p,830b,534s stack positions out of 5000i,500n,10000p,200000b,50000s @@ -1282,9 +1284,9 @@ al/texlive/2009/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy6.pfb></usr/local /texlive/2009/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy8.pfb></usr/local/t exlive/2009/texmf-dist/fonts/type1/public/amsfonts/symbols/msam10.pfb> -Output written on genrel.pdf (22 pages, 2415482 bytes). +Output written on genrel.pdf (23 pages, 2422055 bytes). PDF statistics: - 677 PDF objects out of 1000 (max. 8388607) - 45 named destinations out of 1000 (max. 500000) + 703 PDF objects out of 1000 (max. 8388607) + 47 named destinations out of 1000 (max. 500000) 102 words of extra memory for PDF output out of 10000 (max. 10000000)
File [modified]: genrel.nav Delta lines: +7 -5 =================================================================== --- genrel/genrel.nav 2012-04-04 12:15:48 UTC (rev 34) +++ genrel/genrel.nav 2012-04-04 12:52:23 UTC (rev 35) @@ -43,8 +43,10 @@ \headcommand {\beamer@framepages {21}{21}} \headcommand {\slideentry {0}{0}{22}{22/22}{}{0}} \headcommand {\beamer@framepages {22}{22}} -\headcommand {\beamer@partpages {1}{22}} -\headcommand {\beamer@subsectionpages {1}{22}} -\headcommand {\beamer@sectionpages {1}{22}} -\headcommand {\beamer@documentpages {22}} -\headcommand {\def \inserttotalframenumber {22}} +\headcommand {\slideentry {0}{0}{23}{23/23}{}{0}} +\headcommand {\beamer@framepages {23}{23}} +\headcommand {\beamer@partpages {1}{23}} +\headcommand {\beamer@subsectionpages {1}{23}} +\headcommand {\beamer@sectionpages {1}{23}} +\headcommand {\beamer@documentpages {23}} +\headcommand {\def \inserttotalframenumber {23}}
File [modified]: genrel.pdf Delta lines: +0 -0 =================================================================== (Binary files differ)
File [modified]: genrel.tex Delta lines: +22 -1 =================================================================== --- genrel/genrel.tex 2012-04-04 12:15:48 UTC (rev 34) +++ genrel/genrel.tex 2012-04-04 12:52:23 UTC (rev 35) @@ -222,7 +222,10 @@ \end{equation} where $\rho_0$ is the rest-mass density, $\rho = \rho_0[1+\epsilon]$ is the mass-energy density, $\epsilon$ the internal energy density of a fluid, $p$ the pressure, $u^\mu$ the 4-velocity. We are assuming an ideal fluid, measured in its own rest frame, for the matrix expression.
- The stress energy tensor encodes not only the density of mass and momentum, but also their flows (the pressure term). + The stress energy tensor encodes not only the density of mass and momentum, but also their flows (the pressure term). The evolution set is: + \begin{equation} + \nabla_\mu (\rho u^\mu) = 0;~~\nabla_\mu T^{\mu\nu} = 0 + \end{equation} }
\frame{\frametitle{Einstein's equations} @@ -232,6 +235,11 @@ R_{\mu\nu}-\frac{1}{2}g_{\mu\nu} R = 8\pi G T_{\mu\nu} \end{equation} Put simply, the left hand side involves second derivatives, both in time (accelerations) and space (just like Gauss's law for gravitation). The right hand-side incorporates the gravitational contributions of mass, and momentum, and stresses in general. Because we have a second-order system, we need both the metric and its first derivatives to be given as initial data to evolve the system. + +The left hand side is automatically divergence free, and this fact played a critical role in deriving it in the first place: +\begin{equation} +\nabla_{\mu}(R_{\mu\nu}-\frac{1}{2}g_{\mu\nu} R ) = 0 +\end{equation} }
\frame{\frametitle{3+1 formalism} @@ -245,6 +253,19 @@ \includegraphics[width=1.5in]{lapseshift} }
+\frame{\frametitle{3+1 formalism} + +Much of numerical relativity comes from determining how to do the following steps: +\begin{itemize} +\item The 3+1 formalism is not numerically stable, but by evolving additional variables whose behavior we should know, it can stabilize the system: +\begin{equation} +\partial_t \gamma_{ij} = ... +\end{equation} +\item We need an evolution for the time derivative of the (spatial) metric. This is captured by the {\em extrinsic curvature} $K_{ij}$, which is closely related to the metric derivatives. Think of it as how the metric evolves from slice to slice. +\item We need gauge evolution equations for $\alpha$ and $\beta^i$. We can choose these freely, but our choice affects the resulting evolution set. This is the choice of coordinates for the spacetime. +\end{itemize} +} + \frame{\frametitle{Black holes}
The most familiar black hole solution, for an uncharged BH without any angular momentum, is the Schwarzschild solution: