[Commits] [svn:einsteintoolkit] Workshop_Spring_2012/genrel/ (Rev. 35)

Wed Apr 4 07:52:24 CDT 2012

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 at framepages {21}{21}}}
 \@writefile{nav}{\headcommand {\slideentry {0}{0}{22}{22/22}{}{0}}}
 \@writefile{nav}{\headcommand {\beamer at framepages {22}{22}}}
-\@writefile{nav}{\headcommand {\beamer at partpages {1}{22}}}
-\@writefile{nav}{\headcommand {\beamer at subsectionpages {1}{22}}}
-\@writefile{nav}{\headcommand {\beamer at sectionpages {1}{22}}}
-\@writefile{nav}{\headcommand {\beamer at documentpages {22}}}
-\@writefile{nav}{\headcommand {\def \inserttotalframenumber {22}}}
+\@writefile{nav}{\headcommand {\slideentry {0}{0}{23}{23/23}{}{0}}}
+\@writefile{nav}{\headcommand {\beamer at framepages {23}{23}}}
+\@writefile{nav}{\headcommand {\beamer at partpages {1}{23}}}
+\@writefile{nav}{\headcommand {\beamer at subsectionpages {1}{23}}}
+\@writefile{nav}{\headcommand {\beamer at sectionpages {1}{23}}}
+\@writefile{nav}{\headcommand {\beamer at 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 at 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 @@
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-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 at framepages {21}{21}}
 \headcommand {\slideentry {0}{0}{22}{22/22}{}{0}}
 \headcommand {\beamer at framepages {22}{22}}
-\headcommand {\beamer at partpages {1}{22}}
-\headcommand {\beamer at subsectionpages {1}{22}}
-\headcommand {\beamer at sectionpages {1}{22}}
-\headcommand {\beamer at documentpages {22}}
-\headcommand {\def \inserttotalframenumber {22}}
+\headcommand {\slideentry {0}{0}{23}{23/23}{}{0}}
+\headcommand {\beamer at framepages {23}{23}}
+\headcommand {\beamer at partpages {1}{23}}
+\headcommand {\beamer at subsectionpages {1}{23}}
+\headcommand {\beamer at sectionpages {1}{23}}
+\headcommand {\beamer at 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:

