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\begin{document}
\title{Exact}
\author{Code by many different people,	\\
	this documentation by Jonathan Thornburg}
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\date{$ $Id$ $}
\maketitle

\abstract{
This thorn sets up the $3+1$ ADM field variables for any of a number
of exact spacetimes/coordinates, and even some non-Einstein
spcetimes/coordinates.  It's easy to add more spacetimes/coordinates:
all you have to supply is the 4-metric $g_{ab}$ and the inverse 4-metric
$g^{ab}$, this thorn automagically calculates all the ADM variables
from these.
	 }  

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Introduction}

This thorn sets up the ADM field variables for any of the following
spacetimes/coordinates, as specified by the \verb|exact::exactmodel|
parameter:
\begin{description}
\item[Minkowski spacetime]\mbox{}\\[-\baselineskip]
	\begin{description}
	\item[{\tt "Minkowski"}]
		Minkowski spacetime
	\item[{\tt "flatfunny"}]
		Minkowski spacetime in non-trivial spatial coordinates
	\item[{\tt "flatshift"}]
		Minkowski spacetime in non-trivial slices with shift
	\end{description}
\item[Black hole spacetimes]\mbox{}\\[-\baselineskip]
	\begin{description}
	\item[{\tt "flatSchwarz"}]
		Schwarzschild spacetime with flat spatial metric
	\item[{\tt "Novikov"}]
		Schwarzschild spacetime in Novikov coordinates
	\item[{\tt "Finkelstein"}]
		Schwarzschild spacetime in Eddington-Finkelstein coordinates
	\item[{\tt "Kerr"}]
		Kerr spacetime in cartesian coordinates
	\item[{\tt "KerrSchild"}]
		Kerr spacetime in Kerr-Schild coordinates
	\item[{\tt "fakebinary"}]
		Thorne's ``fake binary'' approximate spacetime
	\item[{\tt "multiBH"}]
		Majumdar-Papapetrou or Kastor-Traschen
		maximally-charged multi-BH solutions
	\end{description}
\item[Cosmological spacetimes]\mbox{}\\[-\baselineskip]
	\begin{description}
	\item[{\tt "Rob-Wal"}]
		Pure-radiation Robertson-Walker cosmology
	\item[{\tt "DeSitter"}]
		Einstein-De~Sitter spacetime
	\item[{\tt "Godel"}]
		G\"{o}del spacetime%%%
\footnote{%%%
	 Note that the parameter is {\em not\/} "Goedel"!
	 }%%%
	\item[{\tt "BianchiI"}]
		Approximate Bianchi type~I spacetime
	\item[{\tt "Kasner"}]
		Kasner-like spacetime
	\item[{\tt "Milne"}]
		Milne spacetime for pre-big-bang cosmology
	\end{description}
\item[Miscellaneous spacetimes]\mbox{}\\[-\baselineskip]
	\begin{description}
	\item[{\tt "boostrot"}]
		Boost-rotation symmetric spacetime
	\item[{\tt "starSchwarz"}]
		Schwarzschild (constant density) star
	\item[{\tt "bowl"}]
		Non-Einstein Bowl (``bag of gold'') spacetime
	\end{description}
\end{description}

As a general policy, this thorn includes only cases where the full
4-metric $g_{ab}$ (and its inverse, although we could probably dispense
with that if needed) is known throughout the spacetime.  Cases where
this is only known on one specific slice, should live in separate
initial data thorns.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Minkowski Spacetime}

This thorn can set up Minkowski spacetime using several different
types of coordinates:

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection{Minkowski Spacetime}

\verb|Exact::exactmodel = "Minkowski"| specifies Minkowski spacetime
in the usual Minkowski coordinates:
\begin{equation}
g_{ab} = \text{diag}
	 \left[
	 \begin{array}{cccc}
	 -1	& 1	& 1	& 1	%%%\\
	 \end{array}
	 \right]
\end{equation}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection{Minkowski Spacetime in Non-Trivial Spatial Coordinates}

\verb|Exact::exactmodel = "flatfunny"| specifies Minkowski spacetime
with the usual Minkowski time slicing, but using the nontrivial spatial
coordinates defined as follows:  First take the flat metric in polar
spherical coordinates, then define a new radial coordinate by
\begin{equation}
r = r_\text{new} (1 - a {\sf G}(r_\text{new}))
\end{equation}
where $a = \verb|Exact::flatfunny_a|$ is a specified parameter with
$a \in [0,1)$, and ${\sf G}(r) = \exp(-\frac{1}{2} r^2/\sigma^2)$
is a Gaussian centered at $r=0$ with amplitude~1 and width
$\sigma = \verb|Exact::flatfunny_s|$.  Finally, transform back to
Cartesian coordinates.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection{Minkowski Spacetime in Non-Trivial Slices with Shift}

\verb|Exact::exactmodel = "flatshift"| specifies Minkowski spacetime
with the nontrivial time slicing and spatial coordinates defined as
follows:  First take the flat 4-metric in polar spherical coordinates,
then define a new time coordinate by
\begin{equation}
t_\text{new} = t - a {\sf G}(r)
\end{equation}
where $a = \verb|Exact::flatshift_a|$ is a specified parameter with
$a \in (-1,1)$, and ${\sf G}(r) = \exp(-\frac{1}{2} r^2/\sigma^2)$
is a Gaussian centered at $r=0$ with amplitude~1 and width
$\sigma = \verb|Exact::flatshift_s|$.  Finally, transform back to
Cartesian (4-)coordinates.  (N.b.\ this gives a time-independent
4-metric.)

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Black Hole Spacetimes}

This thorn can set up Schwarzschild, Kerr, and even some other types
of black hole or pseudo-black-hole spacetimes, using several different
types of coordinates:

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection{Schwarzschild Spacetime with Flat Spatial Metric}

\verb|Exact::exactmodel = "flatSchwarz"| specifies Schwarzschild spacetime
in Painleve-Gustrand coordinates (these have a {\em flat\/} 3-metric),
transformed to the usual Cactus $(t,x,y,z)$ Cartesian-topology
coordinates.  The only physics parameter is
\begin{equation}
m = \text{\tt KerrSchild\_m}
\end{equation}
(note the name!).
There is also a numerical parameter \verb|KerrSchild_eps| (again note
the name!) which is used to avoid division by zero if a grid point falls
exactly at the origin; the default setting should be ok for most purposes.

In the usual Cactus $(t,x,y,z)$ Cartesian-topology coordinates, the
4-metric is
\begin{equation}
g_{ab} = \left[
	 \begin{array}{cccc}
	 -1 + \frac{2m}{r}
		& \sqrt{\frac{2m}{r}} \frac{x}{r}
			& \sqrt{\frac{2m}{r}} \frac{y}{r}
				& \sqrt{\frac{2m}{r}} \frac{z}{r}
									\\
	 \sqrt{\frac{2m}{r}} \frac{x}{r}
		& 1	& 0	& 0					\\
	 \sqrt{\frac{2m}{r}} \frac{y}{r}
		& 0	& 1	& 0					\\
	 \sqrt{\frac{2m}{r}} \frac{z}{r}
		& 0	& 0	& 1					%%%\\
	 \end{array}
	 \right]
\end{equation}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection{Schwarzschild Spacetime in Novikov Coordinates}

\verb|Exact::exactmodel = "Novikov"| specifies the unit-mass Schwarzschild
spacetime in Novikov coordinates, as described in gr-qc/9608050
(see also MTW section~31.4 and figure~31.2), transformed to the
usual Cactus $(t,x,y,z)$ Cartesian-topology coordinates.  There are
no parameters.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection{Schwarzschild Spacetime in Eddington-Finkelstein Coordinates}

\verb|Exact::exactmodel = "Finkelstein"| specifies Schwarzschild spacetime
in (ingoing) Eddington-Finkelstein coordinates $(t,r,\theta,\phi)$, as
described in MTW box~31.2 and figure~32.1 (that is, $r$ is the usual
areal radial coordinate, and $t+r$ is an ingoing null coordinate),
but transformed to the usual Cactus $(t,x,y,z)$ Cartesian-topology
coordinates.  The only physics parameter is
\begin{equation}
m = \text{\tt KerrSchild\_m}
\end{equation}
(note the name!).
There is also a numerical parameter \verb|KerrSchild_eps| (again note
the name!) which is used to avoid division by zero if a grid point falls
exactly at the origin; the default setting should be ok for most purposes.

In the usual polar spherical $(t,r,\theta,\phi)$ coordinates, the 4-metric
and ADM variables are
\begin{align}
g_{ab} & = \left[
	   \begin{array}{cccc}
	   - \left( 1 - \frac{2m}{r} \right)
			& \frac{2m}{r}	& 0	& 0	\\
	   \frac{2m}{r}	& 1 + \frac{2m}{r}
					& 0	& 0	\\
	   0		& 0		& r^2	& 0	\\
	   0		& 0		& 0	& r^2 \sin^2 \theta
							%%%\\
	   \end{array}
	   \right]
									\\
g_{ij} & = \text{diag}
	   \left[
	   \begin{array}{ccc}
	   1 + \frac{2m}{r}	& r^2	& r^2 \sin^2 \theta	%%%\\
	   \end{array}
	   \right]
									\\
K_{ij} & = \text{diag}
	   \left[
	   \begin{array}{ccc}
	   - \frac{2m^2}{r^2} \frac {1 + \frac{m}{r}} {\sqrt{1 + \frac{2m}{r}}}
				& \frac{2m^2}{\sqrt{1 + \frac{2m}{r}}}
					& \frac{2m^2}{\sqrt{1 + \frac{2m}{r}}}
					  \sin^2 \theta		%%%\\
	   \end{array}
	   \right]
									\\
\alpha & = \frac{1}{\sqrt{1 + \frac{2m}{r}}}
									\\
\beta^i& = \left[
	   \begin{array}{ccc}
	   \frac{2m}{r} \frac{1}{\sqrt{1 + \frac{2m}{r}}}
				& 0	& 0			%%%\\
	   \end{array}
	   \right]
									%%%\\
\end{align}
(Various other $3+1$ variables for Schwarzschild spacetime in these
coordinates are tabulated in appendix~2 of Jonathan Thornburg's Ph.D
thesis, \verb|http://www.aei.mpg.de/~jthorn/phd/html/phd.html|.)

In the Cactus $(t,x,y,z)$ Cartesian-topology coordinates the 4-metric is
\begin{equation}
g_{ab} = \left[
	 \begin{array}{cccc}
	 - \left( 1 - \frac{2m}{r} \right)
			& \frac{2m}{r} \frac{x}{r}
				& \frac{2m}{r} \frac{y}{r}
					& \frac{2m}{r} \frac{z}{r}	\\
	 \frac{2m}{r} \frac{x}{r}
			& 1 + \frac{2m}{r} \frac{x^2}{r^2}
				& \frac{2m}{r} \frac{xy}{r^2}
					& \frac{2m}{r} \frac{xz}{r^2}	\\
	 \frac{2m}{r} \frac{y}{r}
			& \frac{2m}{r} \frac{xy}{r^2}
				& 1 + \frac{2m}{r} \frac{y^2}{r^2}
					& \frac{2m}{r} \frac{yz}{r^2}	\\
	 \frac{2m}{r} \frac{z}{r}
			& \frac{2m}{r} \frac{xz}{r^2}
				& \frac{2m}{r} \frac{yz}{r^2}
					& 1 + \frac{2m}{r} \frac{z^2}{r^2}
									%%%\\
	 \end{array}
	 \right]
\end{equation}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection{Kerr Spacetime in Cartesian Coordinates}

\verb|Exact::exactmodel = "Kerr"| specifies Kerr spacetime in
FIXME coordinates.

FIXME: more detail needed from Mitica

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection{Kerr-Schild form of Boosted Rotating Black Hole}

\verb|Exact::exactmodel = "KerrSchild"| specifies Kerr spacetime in
Kerr-Schild coordinates, as described in MTW exercise~33.8, Lorentz
boosted in the $z$ direction so the black hole is centered at the
position $z = vt$, and transformed to the usual Cactus $(t,x,y,z)$
Cartesian-topology coordinates.  The physics parameters are
\begin{align}
a & = \text{\tt KerrSchild\_a}						\\
m & = \text{\tt KerrSchild\_m}						\\
v & = \text{\tt KerrSchild\_boostv}					%%%\\
\end{align}
There is also a numerical parameter \verb|KerrSchild_eps| which is
used to avoid division by zero if a grid point falls exactly at the
black hole center; the default setting should be ok for most purposes.

Kerr-Schild coordinates use the same time slicing (n.b.\ non-maximal!)
and $z$~spatial coordinate as Kerr coordinates, but define new spatial
coordinates $x$ and~$y$ by
\begin{equation}
x + iy = (r + ia) e^{i\phi} \sin\theta
\end{equation}
so that
\begin{align}
x	& = x_\text{Kerr} - a \sin\theta \sin\phi			\\
y	& = y_\text{Kerr} + a \sin\theta \cos\phi			%%%\\
\end{align}

In these coordinates the 4-metric can be written
\begin{equation}
g_{ab} = \eta_{ab} + 2 H k_a k_b
\end{equation}
where
\begin{equation}
H = \frac{mr}{r^2 + a^2z^2/r^2}
\end{equation}
and where
\begin{equation}
k^a = - \frac{r(x\,dx + y\,dy) - a(x\,dy - y\,dx)}{r^2 + a^2}
      - \frac{z\,dz}{r}
      - dt
\end{equation}
is a null vector.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection{Thorne's ``Fake Binary'' Approximate Spacetime}

\verb|Exact::exactmodel = "fakebinary"| specifies Thorne's ``fake binary''
approximate spacetime, as described in gr-qc/9808024.  This is not an
exact solution of the Einstein equations, but has qualitative features
designed to mimic those of an inspiralling binary black hole spacetime.
The physics parameters are
\begin{align}
m	& = \text{\tt fakebinary\_m}					\\
a_0	& = \text{\tt fakebinary\_a0}					\\
\Omega_0& = \text{\tt fakebinary\_Omega0}				%%%\\
\end{align}
There are also algorithm parameters \verb|fakebinary_atype|,
\verb|fakebinary_retarded|, and \verb|fakebinary_bround|.
There is also a numerical parameter \verb|fakebinary_eps| which is
used to avoid division by zero if a grid point falls exactly at either
black hole's center; the default setting should be ok for most purposes.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection{Majumdar-Papapetrou or Kastor-Traschen
	    Maximally-Charged multi-BH Solutions}

\verb|Exact::exactmodel = "multiBH"| specifies the Majumdar-Papapetrou
or Kastor-Traschen solution.  The file \verb|KTsol.tex| in the
documentation directory of this thorn gives more details/references
about these solutions.

The Majumdar-Papapetrou solution is a multi-black-hole static solution
to Einstein's equation, containing $N$ maximally charged ($Q=M$) black
holes.  The balance between gravitational attraction and electrostatic
repulsion among the black holes causes each to maintain its position
relative to the others eternally.  (The Majumdar-Papapetrou solution
somewhat resembles Brill-Lindquist initial data, but with the black
holes being charged to make the solution static.)  The line element is
\begin{equation}
ds^2=-\frac{1}{\Omega^2} dt^2+ \Omega^2(dx^2+dy^2+dz^2)
\end{equation}
where
\begin{align}
\Omega	&= 1+\sum_{i=1}^N \frac{M_i}{r_i}				\\
r_i	&= \sqrt{(x-x_i)^2+(y-y_i)^2+(z-z_i)^2}				%%%\\
\end{align}
where $M_i$ and $(x_i, y_i, z_i) \in \Re^3$ are the masses and
locations of the individual black holes.

The Kastor-Traschen solution is a cosmological generalization of the
Majumdar-Papapetrou solution, where there is a cosmological constant
and the black holes participate in an overall De~Sitter expansion or
contraction.  For $\Lambda = 0$ the Kastor-Traschen solution reduces
to the Majumdar-Papapetrou solution.

The Kastor-Traschen line element is
\begin{equation}
ds^2=-\frac{1}{\Omega^2} dt^2+a(t)^2 \Omega^2(dx^2+dy^2+dz^2)
\end{equation}
where
\begin{align}
\Omega	&= 1+\sum_{i=1}^N {\frac{M_i}{a r_i}}				\\
a	&= e^{Ht}							\\
H	&= \pm \sqrt{\frac{\Lambda}{3}}					\\
r_i	&= \sqrt{(x-x_i)^2 + (y-y_i)^2 + (z-z_i)^2}			%%%\\
\end{align}
This solution represents ``incoming'' (``outgoing'') charged BHs
if $H < 0$ ($H > 0$).  We interpret $M_i$ as the mass of the
$i{\rm th}$ black hole, although we have neither an asymptotically
flat region nor event horizons available to convert this naive
interpretation into a rigorous one.

This thorn supports up to 4~black holes; the physics parameters are
\begin{align}
N	& = \text{\tt KT\_nBH}						\\
H	& = \text{\tt KT\_Hubble}					%%%\\
\end{align}
and
\begin{align}
M_i	& = \text{\tt m\_bh$i$}						\\
x_i	& = \text{\tt co\_bh$i$x}					\\
y_i	& = \text{\tt co\_bh$i$y}					\\
z_i	& = \text{\tt co\_bh$i$z}					%%%\\
\end{align}
for each $i = 1$, $2$, $3$, $4$.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Cosmological Spacetimes}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection{Pure-Radiation Robertson-Walker Cosmology}

\verb|Exact::exactmodel = "Rob-Wal"| specifies a pure-radiation
Robertson-Walker spacetime ($p = \frac{1}{3} \rho$, $k=0$), as described in
Hawking and Ellis section~5.3 and MTW section~27.11 (see also gr-qc/0110031),
transformed to the usual Cactus $(t,x,y,z)$ Cartesian-topology coordinates.
The physics parameters are
\begin{align}
a &= \text{\tt Desitt\_a}						\\
b &= \text{\tt Desitt\_b}						%%%\\
\end{align}
(note the names!).

The general Robertson-Walker line element in $(t,r,\theta,\phi)$ coordinates
is
\begin{equation}
ds^2 = -dt^2 + R(t)^2 \left[ \frac{dr^2}{1 - kr^2} + r^2 \, d\Omega^2 \right]
\end{equation}

For the special case here, $R(t) = \sqrt{at}$, so
\begin{equation}
ds^2 = -dt^2 + a t \left[ dr^2 + r^2 \, d\Omega^2 \right]
\end{equation}

This thorn also sets up the stress-energy tensor for this spacetime.

{\bf This solution doesn't work properly yet.  See Mitica Vulcanov for
further information.}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection{De~Sitter Spacetime}

\verb|Exact::exactmodel = "DeSitter"| specifies an Einstein-De~Sitter
spacetime (a zero-pressure spatially-flat Robertson-Walker spacetime),
as described in Hawking and Ellis section~5.3 and MTW section~27.11
(see also gr-qc/0110031), transformed to the usual Cactus $(t,x,y,z)$
Cartesian-topology coordinates.  The physics parameters are
\begin{align}
a &= \text{\tt Desitt\_a}						\\
b &= \text{\tt Desitt\_b}						%%%\\
\end{align}
(note the name!).

The Einstein-De~Sitter spacetime is the special case
$R(t) = \sqrt{a}\,t^{2/3}$, $k = 0$ of the more general Robertson-Walker
spacetime, so the line element in $(t,r,\theta,\phi)$ coordinates is
\begin{equation}
ds^2 = -dt^2 + a t^{4/3} \left[ dr^2 + r^2 \, d\Omega^2 \right]
\end{equation}
The only non-vanishing component of the stress-energy tensor is
\begin{equation}
T_{tt} = \frac{1}{6 \pi t^2}
\end{equation}
This is properly set up by this thorn.

{\bf This solution doesn't work properly yet.  See Mitica Vulcanov for
further information.}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection{G\"{o}del Spacetime}

\verb|Exact::exactmodel = "Godel"| (sic) specifies a G\"{o}del
spacetime, as described in Hawking and Ellis section~5.7, transformed
to the usual Cactus $(t,x,y,z)$ Cartesian-topology coordinates.  The
only physics parameter is
\begin{equation}
a = \text{\tt Godel\_a}
\end{equation}
(note the name!).

At present this thorn doesn't set up the stress-energy tensor;
you have to do this ``by hand''.

{\bf This solution doesn't work properly yet.  See Mitica Vulcanov for
further information.}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection{Approximate Bianchi type~I Spacetime}

\verb|Exact::exactmodel = "BianchiI"| specifies an approximation to
a Bianchi type~I spacetime.

At present this thorn doesn't set up the stress-energy tensor;
you have to do this ``by hand''.

{\bf This solution doesn't work properly yet.  See Mitica Vulcanov for
further information.}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection{Kasner-like Spacetime}

\verb|Exact::exactmodel = "Kasner"| specifies a Kasner-like spacetime,
as described in gr-qc/0110031, and in more detail in
L. Pimentel,
International Journal of Theoretical Physics {\bf 32}(6) [1993], 979,
and
S. Gotlober, {\it et.~al.\/},
``Early Evolution of the Universe and Formation [of] Structure'',
Akad. Verlag, 1990.
There is one physics parameter,
\begin{equation}
q = \text{\tt Kasner\_q}						%%%\\
\end{equation}

In the usual Cactus $(t,x,y,z)$ Cartesian-topology coordinates, the
4-metric is
\begin{equation}
g_{ab} = \text{diag}
	 \left[
	 \begin{array}{cccc}
	 -1	& t^{2q}	& t^{2q} 	& t^{2-4q}	%%%\\
	 \end{array}
	 \right]
\end{equation}
and the stress-energy tensor is
\begin{equation}
T_{ab} = \text{diag}
	 \left[
	 \begin{array}{cccc}
	 q \displaystyle\frac{2 - 3q}{8\pi t^2}
	     & q \displaystyle\frac{(2 - 3q) t^{2q}}{8\pi t^2}
	         & q \displaystyle\frac{(2 - 3q) t^{2q}}{8\pi t^2}
		     & q \displaystyle\frac{(2 - 3q) t^{2-4q}}{8\pi t^2}
								%%%\\
	 \end{array}
	 \right]
\end{equation}

{\bf At the moment only $T_{00}$ is set properly, the other
components are (incorrectly) 0.  See Mitica Vulcanov for
further information.}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection{Milne Spacetime for Pre-Big-Bang Cosmology}

\verb|Exact::exactmodel = "Milne"| specifies a De~Milne spacetime,
as described by gr-qc/9802001 (see in particular reference~14, which
in turn points to Zeldovich and Novikov volume~2 section~2.4):
\begin{equation}
g_{ab} = \left[
	 \begin{array}{cccc}
	 -1	& 0		& 0		& 0		\\
	 0	& V(1+y^2+z^2)	& -Vxy		& -Vxz		\\
	 0	& -Vxy		& V(1+x^2+z^2)	& -Vyz		\\
	 0	& -Vxz		& -Vyz		& V(1+x^2+y^2)	%%%\\
	 \end{array}
	 \right]
\end{equation}
where
\begin{equation}
V = \frac{t^2}{1 + x^2 + y^2 + z^2}
\end{equation}

{\bf The $g_{ab}$ given here is indeed what the code computes, but
noone seems to know whether this is indeed a Milne (De~Milne?) spacetime.}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Miscellaneous Spacetimes}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection{Boost-Rotation Symmetric Spacetime}

\verb|Exact::exactmodel = "starSchwarz"| specifies a boost-rotation
symmetric spacetime.  Pravda and Pravdov\'{a}, gr-qc/0003067, give a
general review of boost-rotation symmetric spacetimes.

FIXME: get more info from Miguel and Carsten

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection{Schwarzschild (Constant Density) Star}

\verb|Exact::exactmodel = "starSchwarz"| specifies a constant-density
``Schwarzschild'' star, as described in MTW~Box 23.2.  The stress-energy
tensor is also properly set up.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection{Non-Einstein Bowl (``Bag of Gold'') Spacetime}

\verb|Exact::exactmodel = "bowl"| specifies a ``bag of Gold'' metric,
as described in gr-qc/9809004.  This is useful for testing purposes,
but isn't a solution of the Einstein equations.  The physics parameters
are
\begin{align}
a	& = \text{\tt bowl\_a}						\\
\sigma	& = \text{\tt bowl\_s}						\\
c	& = \text{\tt bowl\_c}						\\
\delta x& = \text{\tt bowl\_dx}						\\
\delta y& = \text{\tt bowl\_dy}						\\
\delta z& = \text{\tt bowl\_dz}						\\
t_0	& = \text{\tt bowl\_t0}						\\
\sigma_t& = \text{\tt bowl\_st}						%%%\\
\end{align}
There are also algorithm parameters \verb|bowl_type| and \verb|bowl_evolve|.

The line element in $(t,r,\theta,\phi)$ coordinates is
\begin{equation}
ds^2 = -dt^2 + dr^2 + R^2(r) \, d\Omega^2
\end{equation}

We choose $R(r)$ such that $\displaystyle \lim_{r \ll 1} R(r) = r$
and $\displaystyle \lim_{r \gg 1} R(r) = r$, so we have a flat 3-metric
(and hence 4-metric too) for very small~$r$ and for very large~$r$.
For intermediate values of~$r$, we take $0 < R(r) < r$; this deficit
in areal radius produces the ``bag of gold'' geometry.

The size of the deviation from a flat geometry is controled by the
parameter $a = \verb|bowl_a|$.  If $a = 0$, we are in flat spacetime.
The width of the curved region is controled by $\sigma = \verb|bowl_s|$,
and the place where the curvature becomes significant (the center of
the deformation) is controled by $c = \verb|bowl_c|$.

In detail, we choose
\begin{equation}
R(r) = r - A f(r) g(r)
\end{equation}
Here $A = a$ if \verb|bowl_evolve = "false"|, but is multiplied by
a Fermi factor
\begin{equation}
A = \frac{a}{1 + \exp(-\sigma_t(t-t_0))}
\end{equation}
if \verb|bowl_evolve = "true"|.  For this latter case we have
flat spacetime far in the past, and a static bowl far in the future.
$f(r)$ is either a Gaussian or a Fermi function,
\begin{equation}
f(r) = \begin{cases}
       \exp \left( -\frac{1}{2} (r-c)^2/\sigma^2 \right)
			& \text{if {\tt bowl\_type = "Gauss"}}		\\[1ex]
       \displaystyle
       \frac{1}{1 + \exp(-\sigma(r-c))}
			& \text{if {\tt bowl\_type = "Fermi"}}		%%%\\
       \end{cases}
\end{equation}
$g(r) = 1 - \text{sech}(4r)$ is a fixup factor to ensure that
$\displaystyle \lim_{r \to 0} R(r) = r$.

The three extra paramters $(\delta x, \delta y, \delta z)$
scale the $(x,y,z)$ axes respectively.  Their default values are all~1.
These parameters are useful to hide the spherical symmetry of the metric.

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\section{Acknowledgments}

Many different people have contributed code to this thorn.
Jonathan Thornburg wrote most of this documentation in May 2002 based
on the comments in the code, some reverse-engineering, and querying
various people about how the code works.  The description of the
Kastor-Traschen maximally charged multi-BH solutions is adapted from
the file \verb|KTsol.tex| in this same directory, by Hisa-aki Shinkai.

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%
% Automatically created from the ccl files 
% Do not worry for now.
%
\include{interface}
\include{param}
\include{schedule}

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\end{document}