1 Introduction

This thorn sets up the ADM variables for any of a number of different spacetimes/coordinates (we call the combination of a spacetime and a coordinate system a model), as specified by the Exact::exact_model parameter.

By default, this thorn sets up the ADM variables on an initial slice only. However, setting
ADMBase::evolution_method = "exact" makes this thorn set up the ADM variables at CCTK_PRESTEP every time step of an evolution, so you get an exact spacetime, not just a single slice.

There is an option to Lorentz-boost any vacuum model (more precisely any model which doesn’t set the stress-energy tensor; see table 1 and section 2 for details) in any direction.

There is also a more general option to set up the ADM variables on an arbitrary slice through the spacetime, using arbitrary coordinates on the slice. Given a lapse and shift computed by some other thorn(s), the slice can be evolved through the exact solution, in order to check on an evolution code, or in order to test gauge conditions without the need for an evolution code. This option is documented in doc/slice_evolver.tex.

This thorn is mainly written in a mixture of Fortran 77 and Fortran 90; a few routines are written in C. At present Fortran 90 is used only for the “arbitrary slice” option (described in the previous paragraph). If this option isn’t needed, then the Fortran 90 code can all be #ifdef-ed out, allowing this thorn to be compiled on a system having only Fortran 77 and C compilers (i.e. no Fortran 90 compiler). This can be done by changing a single line in src/include/Exact.inc; see the comments there for details.

1.1 Models Supported

Table 1 shows the models supported by thorn Exact.¹ As a general policy, this thorn includes only cases where the full 4-metric \(g_{ab}\) (and its inverse, although we could 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.

1.2 Naming Conventions

This thorn includes many different spacetimes and coordinate systems, so we use the following naming conventions to help keep the different models and parameters clear:

• If we have multiple coordinate systems for a given spacetime, the models are named with the pattern "spacetime/coordinates". For example, the model "Schwarzschild/EF" is Schwarzschild spacetime in Eddington-Finkelstein coordinates.²

• If we have spacetimes which are identical or very similar, except that one has a cosmological constant and the other doesn’t, we name the with-cosmological-constant one by appending +Lambda to the without-cosmological-constant spacetime name. For example, the cosmological-constant variant of anti-de Sitter spacetime is the model "anti-de Sitter+Lambda".

• All the parameters for individual models have names which begin with the model name (with any slash (/) or hyphen (-) characters converted to underscores (_)), followed by a double underscore (__).

• The description comment for each parameter in the param.ccl file begins with the model name follwed by a colon (:). For example,

       REAL Schwarzschild_EF__mass "Schwarzschild/EF: BH mass"
       {
       *:* :: "any real number"
       } 1.0
  

1.3 The Cosmological Constant and the Stress-Energy Tensor

A number of these models have a cosmological constant. To use these with the Cactus code (which generally is written for the case of no cosmological constant), we use a simple trick: we transfer the term with the cosmological constant to the right hand side of the Einstein equations, introducing fictitious “matter” terms in the stress-energy tensor.

This thorn uses the standard Cactus “CalcTmunu” interface for introducing terms into the stress-energy tensor. See Ian Hawke’s documentation for the ADMCoupling thorn for details.

1.4 Accuracy

This thorn has been found to set up its initial data less accurately than you might think. In particular, …

Denis Pollney has found that if this thorn is used to set up a Schwarzschild or Kerr solution (Schwarzschild/EF or Kerr/Kerr-Schild model), there are low-level (\(\sim 10^{-12}\)) asymmetries between the field variables in the supposedly-symmetric octants.

Jonathan Thornburg has found that for at least the Schwarzschild/EF and Kerr/Kerr-Schild models (and quite likely for all models), this thorn’s values of the extrinsic curvature \(K_{ij}\) have random point-to-point errors of about \(\sim {\textstyle \frac {1}{2}} \times 10^{-10}\) (in regions where the metric is \(O(1)\).

We suspect that these problems may be due to this thorn’s internally first setting up the 4-metric, then computing the Bona-Masso variables by numerically finite differencing the 4-metric³ , then computing the ADM variables from this.

1.5 Time Levels

In the context of Cactus grid functions with (potentially) multiple time levels, this thorn only sets the initial data on the current time level!. This is a bug. :(

If you’re doing a time evolution using the standard Cactus MoL thorn,

• Variables that you’re going to evolve (for example the ADM 3-metric if you’re evolving it directly) only need initial data to be set on the current time level.

• Other variables that are used by the evolution system, but aren’t themselves evolved (for example the ADM 3-metric if you’re evolving some other fields on a fixed background) must have initial data set on all time levels. MoL calls these “save-and-restore” variables.

MoL offers a useful option to help work around problems of this sort (only the current time level is set, when you need all time levels): If you set

MoL::initial_data_is_crap = true

then MoL will copy the current time level to all other time levels, for all grid functions that are registered as “evolved”, “save-and-restore”, and/or “constrained” variables. This happens in the POSTINITIAL schedule bin.

1.6 Further Sources of Information

This documentation is at best a secondary source of information about this thorn – the primary sources are the param.ccl file and the source code itself. In particular, much of this documentation was developed by reverse-engineering from these primary sources, so it’s quite possible (indeed even likely!) that there are errors or omissions here. Caveat Lector!

2 Lorentz-Boosting a Spacetime

For any of the models which don’t set the stress-energy tensor (i.e. which are vacuum and have no cosmological constant; see section 1.3 and table 1 for details),⁴ you can optionally Lorentz-boost the model by a specified 3-velocity \(v^i\). The parameters for this are boost_vx, boost_vy, and boost_vz.

We define the Cactus spacetime coordinates to be \((t,x^i)\), while the model is at rest in coordinates \((T,X^i)\). The model’s “origin” \(X^i = 0\) is located at the Cactus coordinates \(x^i = v^i t\).

The boost Lorentz transformation is defined by \begin {equation} \renewcommand {\arraystretch }{1.333} \begin {array}{lcl} T & = & \gamma (t - \eta _{ij} v^i x^j) \\ X^i_\parallel & = & \gamma (x^i_\parallel - v^i t) \\ X^i_\perp & = & x^i_\perp \end {array} \end {equation} and the inverse transformation by \begin {equation} \renewcommand {\arraystretch }{1.333} \begin {array}{lcl} t & = & \gamma (T + \eta _{ij} v^i X^j) \\ x^i_\parallel & = & \gamma (X^i_\parallel + v^i T) \\ x^i_\perp & = & X^i_\perp \end {array} \end {equation} where \(\gamma \equiv (1 - v^2)^{-1/2}\) is the usual Lorentz factor, \(\eta _{ij}\) is the flat metric, and \(\parallel \) and \(\perp \) refer to the (flat-space) components of \(x^i\) parallel and perpendicular to \(v^i\), respectively.

In more detail, define the unit vector \(n^i = v^i / \sqrt {\eta _{jk} v^j v^k}\) and the (flat-space) projection operators \begin {equation} \renewcommand {\arraystretch }{1.333} \begin {array}{lclcl} \parallel ^i{}\!_j & = & \eta _{jk} n^i n^k \\ & \equiv & n^i n_j \qquad \hbox {(using $\eta _{ij}$ to raise/lower indices)} \\ \perp ^i{}\!_j & = & \delta ^i{}_j - \parallel ^i{}\!_j \end {array} \end {equation} Then the Lorentz transformations are \begin {equation} \renewcommand {\arraystretch }{1.333} \begin {array}{lcl} T & = & \gamma (t - \eta _{ij} v^i x^j) \\ X^i & = & \gamma (\parallel ^i{}\!_j x^j - v^i t) + \perp ^i{}\!_j x^j \end {array} \end {equation} and \begin {equation} \renewcommand {\arraystretch }{1.333} \begin {array}{lcl} t & = & \gamma (T + \eta _{ij} v^i X^j) \\ x & = & \gamma (\parallel ^i{}\!_j X^j + v^i T) + \perp ^i{}\!_j X^j \end {array} \end {equation} so their coordinate partial derivatives for transforming \(g_{ab}\) and \(g^{ab}\) are \begin {equation} \renewcommand {\arraystretch }{2.5} \begin {array}{lcl@{\qquad \qquad \qquad }lcl} \dfrac {\partial T}{\partial t} & = & \gamma & \dfrac {\partial T}{\partial x^j} & = & - \gamma v^j \\ \dfrac {\partial X^i}{\partial t} & = & - \gamma v^i & \dfrac {\partial X^i}{\partial x^j} & = & \gamma \parallel ^i{}\!_j + \perp ^i{}\!_j \end {array} \end {equation} and \begin {equation} \renewcommand {\arraystretch }{2.5} \begin {array}{lcl@{\qquad \qquad \qquad }lcl} \dfrac {\partial t}{\partial T} & = & \gamma & \dfrac {\partial t}{\partial X^j} & = & \gamma v^j \\ \dfrac {\partial x^i}{\partial T} & = & \gamma v^i & \dfrac {\partial x^i}{\partial X^j} & = & \gamma \parallel ^i{}\!_j + \perp ^i{}\!_j \end {array} \end {equation}

3 Minkowski Spacetime

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

3.1 Minkowski Spacetime

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

3.2 Minkowski Spacetime in Non-Trivial Spatial coordinates

Exact::exact_model = "Minkowski/funny" 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 {w}} \big (1 - a \Gaussian (r_{\text {w}})\big ) \end {equation} where \(\Gaussian (r) = \exp (-\half r^2/\sigma ^2)\) is a Gaussian centered at \(r=0\).

The parameters are the perturbation amplitude \(a = \verb |Exact::Minkowski_funny__amplitude|\) and the perturbation width \(\sigma = \verb |Exact::Minkowski_funny__sigma|\).

3.3 Minkowski Spacetime in Non-Trivial Slices with Shift

Exact::exact_model = "Minkowski/shift" 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 {w}} = t - a \Gaussian (r) \end {equation} \(\Gaussian (r) = \exp (-\half r^2/\sigma ^2)\) is a Gaussian centered at \(r=0\). Note this gives a time-indpendent 4-metric.

The parameters are the perturbation amplitude \(a = \verb |Exact::Minkowski_shift__amplitude|\) and the perturbation width \(\sigma = \verb |Exact::Minkowski_shift__sigma|\).

3.4 Minkowski Spacetime in gauge-wave coordinates

Exact::exact_model = "Minkowski/gauge wave" specifies Minkowski spacetime with the “gauge-wave” coordinates suggested by Carlos Bona: The line element is \begin {equation} ds^2=-H dt^2 +Hdx^2+dy^2+dz^2, \end {equation} where \(H=H(x-t)\), for instance \(H=1-A\sin \left ((x-t)/\Lambda \right )\). This is a flat spacetime, but the slice is a planar wave travelling along the x axis.

This thorn implements several possible choices for the \(H\) function, controlled by the Minkowski_gauge_wave__what_fn parameter:

The parameters are

• \(A = \verb |Minkowski_gauge_wave__amplitude|\), the amplitude

• \(\omega = \verb |Minkowski_gauge_wave__omega|\), the angular frequency

• \(\lambda = \verb |Minkowski_gauge_wave__lambda|\), the wavelength

• \(\delta = \verb |Minkowski_gauge_wave__phase|\), the phase shift

A plane wave has \(\omega = \pm \lambda \) for a wave that travels in the \(x\) direction, and \(\omega = \pm \lambda \sqrt {2}\) for a diagonal wave.

If the Boolean parameter Minkowski_gauge_wave__diagonal is true, then we make the gauge wave travel diagonally across the grid by the coordinate transformation

For code testing, the idea is to test evolving this with periodic boundary conditions, to see whether the code is able to cope with that. The tricky part is to make the wave fit the grid exactly (otherwise the periodic boundary wouldn’t make sence), especially in the diagonal case.

3.5 Minkowski Spacetime in shifted gauge-wave coordinates

Exact::exact_model = "Minkowski/shifted gauge wave" specifies Minkowski spacetime with the “shifted gauge-wave” coordinates suggested by Jeff Winicour: The line element is \begin {equation} ds^2 = (H-1)\, dt^2 + (H+1)\, dx^2 - 2H\, dt\, dx + dy^2 + dz^2 \end {equation} where \(H=H(x-t)\), for instance \(H=A\sin \left ((x-t)/\Lambda \right )\). This is a flat spacetime, but the slice is a planar wave travelling along the x axis.

This thorn implements one choice for the \(H\) function, controlled by the Minkowski_gauge_wave__what_fn parameter:

The parameters are

• \(A = \verb |Minkowski_gauge_wave__amplitude|\), the amplitude

• \(\omega = \verb |Minkowski_gauge_wave__omega|\), the angular frequency

• \(\lambda = \verb |Minkowski_gauge_wave__lambda|\), the wavelength

• \(\delta = \verb |Minkowski_gauge_wave__phase|\), the phase shift

A plane wave has \(\omega = \pm \lambda \) for a wave that travels in the \(x\) direction, and \(\omega = \pm \lambda \sqrt {2}\) for a diagonal wave.

If the Boolean parameter Minkowski_gauge_wave__diagonal is true, then we make the gauge wave travel diagonally across the grid by the coordinate transformation

For code testing, the idea is to test evolving this with periodic boundary conditions, to see whether the code is able to cope with that. The tricky part is to make the wave fit the grid exactly (otherwise the periodic boundary wouldn’t make sence), especially in the diagonal case.

3.6 Minkowski Spacetime with \(\sin \) term in conformal factor

Exact::exact_model = "Minkowski/conf wave" specifies Minkowski spacetime with a \(\sin \) term in the Cactus static conformal factor. You have three parameters:

• Minkowski_conf_wave__amplitude (\(a\))

• Minkowski_conf_wave__wavelength (\(l\))

• Minkowski_conf_wave__direction (\(d\))

These control \(\Psi \) in the following form: \begin {equation} \Psi =a\sin \left (\frac {2\pi }{l}D\right )+1 \end {equation} Here \(D\) is x, y or z according to d of 0, 1 or 2.

Alas, the “arbitrary slice evolver” option (documented in doc/slice_evolver.tex) doesn’t work with this model. There’s no warning, you’ll just silently get wrong results. Sigh…

4 Black Hole Spacetimes

This thorn can set up Schwarzschild and Kerr spacetimes in several different types of coordinates, and also a couple of multiple-black-hole spacetimes:

4.1 Schwarzschild Spacetime in Eddington-Finkelstein coordinates

Exact::exact_model = "Schwarzschild/EF" specifies Schwarzschild spacetime in (ingoing) Eddington-Finkelstein coordinates, as described in MTW box 31.2 and figure 32.1. The only physics parameter is the black hole mass \(m = \verb |Schwarzschild_EF__mass|\).

There is also a numerical parameter Schwarzschild_EF__epsilon 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

(Various other \(3+1\) variables for Schwarzschild spacetime in these coordinates are tabulated in appendix 2 of Jonathan Thornburg’s Ph.D thesis, 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}

4.2 Schwarzschild spacetime in Painlevé-Gullstrand coordinates

Exact::exact_model = "Schwarzschild/PG" specifies Schwarzschild spacetime in Painlevé-Gullstrand coordinates, as described by Martel and Poisson, gr-qc/0001069. These coordinates have the interesting property that the spatial metric is flat. The only physics parameter is the black hole mass \(m = \verb |Schwarzschild_PG__mass|\).

There is also a numerical parameter Schwarzschild_PG__epsilon 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}

4.3 Schwarzschild spacetime in Brill–Lindquist coordinates

Exact::exact_model = "Schwarzschild/BL" specifies Schwarzschild spacetime in Brill–Lindquist coordinates. These coordinates have the interesting property that the spatial metric is conformally flat and time-symmetric for the initial data. The only physics parameter is the black hole mass \(m = \verb |Schwarzschild_BL__mass|\).

There is also a numerical parameter Schwarzschild_BL__epsilon 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 given by

with the conformal factor

where \(r\) is the coordinate radius.

4.4 Schwarzschild Spacetime in Novikov coordinates

Exact::exact_model = "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). The only physics parameter is the black hole mass \(m = \verb |Schwarzschild_Novikov__mass|\).

There is also a numerical parameter Schwarzschild_Novikov__epsilon 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.

4.5 Kerr Spacetime in Boyer-Lindquist coordinates

Exact::exact_model = "Kerr/Boyer-Lindquist" specifies Kerr spacetime in (cartesian) Boyer-Lindquist coordinates, as described in MTW box 33.2 (the spin axis is the \(z\) axis). The physics parameters are the black hole mass \(m = \verb |Kerr_BoyerLindquist__mass|\), and the dimensionless spin parameter \(a = J/m^2 = \verb |Kerr_BoyerLindquist__spin|\).

Mitica Vulcanov says: this metric still need some work in order to run properly. Major problems: the convergence and calibration of the units for the parameters and variables.

4.6 Kerr-Schild form of Boosted Rotating Black Hole

Exact::exact_model = "Kerr/Kerr-Schild" specifies Kerr spacetime in Kerr-Schild coordinates, as described in MTW exercise 33.8 (the spin axis is the \(z\) axis), Lorentz boosted in the \(z\) direction so the black hole is centered at the position \(z = vt\). The physics parameters are the black hole mass \(m = \verb |Kerr_KerrSchild__mass|\), the dimensionless spin parameter \(a = J/m^2 = \verb |Kerr_KerrSchild__spin|\), and the boost velocity \(v = \verb |Kerr_KerrSchild__boost_v|\).

There is also a numerical parameter Kerr_KerrSchild__epsilon 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 \((x_K, y_K, z_K)\), but define new spatial coordinates \(x \equiv x_{KS}\) and \(y \equiv y_{KS}\) by \begin {equation} x_{KS} + iy_{KS} = (r + ia) e^{i\phi } \sin \theta \end {equation} so that

In Kerr-Schild 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.

4.7 Schwarzschild-Lemaitre spacetime (Schwarzschild black hole with cosmological constant

Exact::exact_model = "Schwarzschild/Lemaitre"is a metric proposed by Lemaitre in 1932 as a version of the Schwarzschild solution in a universe with cosmological constant. For a history of this metric and a good review see the chapter of Jean Eisenstaedt, “Lemaitre and the Schwarzschild solution” in the book “The attraction of Gravitation: New Studies in the History of General Relativity”, by J. Earman, et.al. Birkäuser, 1993. The line element is \begin {equation} ds^2 = - \left ( 1-\frac {2m}{r} - \frac {\Lambda }{3}r^2\right ) \, dt^2 + \left ( 1-\frac {2m}{r} -\frac {\Lambda }{3}r^2 \right )^{-1} \, dr^2 + r^2 \, d\theta ^2 + r^2 \sin (\theta )^2 \, d\phi ^2 \end {equation} Notice that for \(\Lambda = 0\) this reduces to Schwarzschild spacetime in the usual Schwarzschild coordinates.

The physics parameters are the black hole mass \(m = \verb |Schwarzschild_Lemaitre__mass|\), and the cosmological constant \(\Lambda = \verb |Schwarzschild_Lemaitre__Lambda|\).

The fictitious “matter” stress-energy tensor representing \(\Lambda \) is \begin {equation} T_{ij}= - \frac {\Lambda }{8 \pi } g_{ij} = \left ( \begin {array}{cccc} -\frac {1}{24}\frac {\Lambda A}{r\pi } & 0 & 0 & 0\\ 0 & \frac {3}{8}\frac {r\Lambda }{8\pi A }& 0 & 0\\ 0 & 0 &-\frac {1}{8}\frac {\Lambda r^2}{\pi }& 0\\ 0 & 0 & 0 & -\frac {1}{8}\frac {\Lambda r^2 \sin (\theta )^2}{\pi } \end {array} \right ) \end {equation} where \(A = (-3r +6m+\Lambda r^3)\).

Alas, this metric doesn’t seem to give proper finite difference convergence for \(\Lambda \ne 0\). It works fine for \(\Lambda = 0\).

4.8 Majumdar-Papapetrou or Kastor-Traschen Maximally-Charged (extreme Reissner-Nordstrom) multi-BH Solutions

Exact::exact_model = "multi-BH" specifies the Majumdar-Papapetrou or Kastor-Traschen solution. The file 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 (\(Q=M\), i.e. extreme Reissner-Nordstrom) 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, so the spacetime is static. (The Majumdar-Papapetrou solution somewhat resembles Brill-Lindquist initial data, but with the black holes being charged.) 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

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

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 the number of black holes \(N = \verb |multi_BH__nBH|\) and the Hubble constant \(H = \verb |multi_BH__Hubble|\), and then for each black hole \(i = 1, \dots , N\), the mass \(m_i = \verb |multi_BH__mass|\,i\) and the \(x\), \(y\), and \(z\) positions \(x_i = \verb |multi_BH__x|\,i\), \(y_i = \verb |multi_BH__y|\,i\), and \(z_i = \verb |multi_BH__z|\,i\) respectively.

Note that this thorn does not set \(T_{\mu \nu }\). FIXME: does treating this metric as vacuum still give a solution to the Einstein equations?

4.9 Alvi post-Newtonian 2BH spacetime (not fully implemented yet)

Exact::exact_model = "Alvi" specifies the Alvi post-Newtonian binary black hole metric, as described in gr-qc/9912113. This uses different approxamintion methods to describe different regions of a binary black hole system: Near the holes, one uses a distorted Schwarzschild black hole metric, while in the region around them (divided into a near zone and a wave zone) one uses a 1st order post-Newtonian approximation. There are discontinuities at the boundaries between the zones.

This model has physics parameters giving the masses of the two black holes, \(m_1 = \verb |Alvi__mass1|\) and \(m_2 = \verb |Alvi__mass2|\), and their spatial separation \(b = \verb |Alvi__separation|\).

Unfortunately, this metric isn’t fully implemented yet. See Nina Jansen for details.

4.10 Thorne’s “Fake Binary” Approximate Spacetime

Exact::exact_model = "fakebinary" specifies Thorne’s “fake binary” approximate binary-black-hole 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:

• \(m = \verb |Thorne_fakebinary__mass|\), the mass
  (FIXME: is this the mass of the whole spacetime, or of an individual BH?)

• \(a_0 = \verb |Thorne_fakebinary__separation|\), the initial binary separation

• \(\Omega _0 = \verb |Thorne_fakebinary__Omega0|\), the initial angular frequency of the binary orbit

• Thorne_fakebinary__retarded, a Boolean parameter which controls whether or not to use a retarded time coordinate

• Thorne_fakebinary__atype, a keyword parameter to select a constant ("constant") or quadrupole ("quadrupole") solution

• Thorne_fakebinary__smoothing, a smoothing length for the Newtonian potential

There is also a numerical parameter Thorne_fakebinary__epsilon 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.

5 Cosmological Spacetimes

The code for most of these models was written by Mitica Vulcanov <vulcan@aei.mpg.de>.

5.1 Lemaitre-type spacetime

Exact::exact_model="Lemaitre" specifies a Lemairre spacetime, version of the Friedmann-Robertson-Walker model with flat space (i.e. \(k=0\)), possibly a cosmological constant, \(\Lambda \), and a linear dependence between the energy density \(\epsilon \) and the pressure, \(p\), namely \(p=\kappa \epsilon \). Thus the metric is the Robertson-Walker metric with \(k =0\) and (see gr-qc/0110030, astro-ph/9910093 and references cited here), \begin {equation} R(t) = R_0 \left [ \cosh \left (\frac {\sqrt {3\Lambda }}{2}(\kappa +1) t \right ) + \sqrt {1+\frac {8\pi G\,\epsilon _{0}}{\Lambda }} \sinh \left ( \frac {\sqrt {3\Lambda }}{2}(\kappa +1) t \right ) \right ]^{2/3(\kappa +1)} \end {equation} where \(R_0\) is the scale factor of the universe (“radius”) at \(t=0\); the density of energy reads \begin {equation} \relax \expandafter \ifx \csname cur:th\endcsname \relax \expandafter \:label \else \expandafter \l:bel \fi {dens} \epsilon (t)=\epsilon _0\,a(t)^{-3(\kappa +1)}\,. \end {equation} The stress-enegy tensor is one of a perfect fluid, \begin {equation} T_{\mu }^{\nu }=(\epsilon +p)u^{\nu }u_{\mu }-p \delta _{\mu }^{\nu }\,, \end {equation} which depends on the covariant four-velocity \(u^{\mu }=dx^{\mu }/ds\) (remember \(p=\kappa \epsilon \)).

The physics parameters are the equation of state parameter \(\kappa = \verb |Lemaitre__kappa|\), the cosmological constant \(\Lambda = \verb |Lemaitre__Lambda|\), the energy density of the universe at time \(t = 0\), \(\epsilon _0 = \verb |Lemaitre__epsilon0|\), and the scale factor (radius) of the universe at time \(t = 0\), \(R_0 = \verb |Lemaitre__R0|\).

5.2 de Sitter spacetime

Exact::exact_model = "de Sitter" 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 for some tests of Cactus with this model). The only physics parameter is the multiplicative scale factor \(a = \verb |de_Sitter__scale|\).

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.

5.3 de Sitter spacetime with cosmological constant

Exact::exact_model="de Sitter+Lambda" specifies an Einstein-de Sitter spacetime with a cosmological constant, with the line element \begin {equation} ds^2 = - dt^2 + e^{2/3\sqrt {3\Lambda }t} \left ( dx^2 + dy^2 + dz^2 \right ) \end {equation} where \(\Lambda \) is the cosmological constant. FIXME: how is \(\Lambda \) determined?

The only physics parameter is the multiplicative scale factor \(a = \verb |de_Sitter_Lambda__scale|\).

The fictitious “matter” stress-energy tensor representing \(\Lambda \) is \begin {equation} T_{ij}= - \frac {\Lambda }{8 \pi } g_{ij} = \left ( \begin {array}{cccc} \frac {1}{8}\frac {\Lambda }{\pi } & 0 & 0 & 0\\ 0 & -\frac {1}{8}\frac {\Lambda e^{2/3 \sqrt {3\Lambda }t}}{\pi }& 0 & 0\\ 0 & 0 &-\frac {1}{8}\frac {\Lambda e^{2/3 \sqrt {3\Lambda }t}}{\pi }& 0\\ 0 & 0 & 0 & -\frac {1}{8}\frac {\Lambda e^{2/3 \sqrt {3\Lambda }t}}{\pi }\end {array}\right ) \, \end {equation}

5.4 anti-de Sitter spacetime with cosmological constant

Exact::exact_model="anti-de Sitter+Lambda" specifies an anti-de Sitter spacetime with a cosmological constant, with the line element \begin {equation} ds^2 = dx^2 + e^{2/3\sqrt {-3\Lambda }t} \left ( -dt^2 + dy^2 + dz^2 \right ) \end {equation} FIXME: how is \(\Lambda \) determined?

The only physics parameter is the multiplicative scale factor \(a = \verb |anti_de_Sitter_Lambda__scale|\).

5.5 Approximate Bianchi type I spacetime

Exact::exact_model = "Bianchi I" specifies an approximation to a Bianchi type I spacetime, setting the spacetime metric components as harmonic functions. Thus this is not a proper solution of Einstein equations. The only physics parameter is the multiplicative scale factor \(a = \verb |Bianchi_I__scale|\).

This solution doesn’t work properly yet. See Mitica Vulcanov for further information.

5.6 Gödel spacetime

Exact::exact_model = "Goedel" specifies a Gödel spacetime, as described in Hawking and Ellis section 5.7. The only physics parameter is the multiplicative scale factor \(a = \verb |Goedel__scale|\).

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

This solution doesn’t work properly yet. See Mitica Vulcanov for further information.

5.7 Bertotti spacetime

Exact::exact_model = "Bertotti" specifies a Bertotti spacetime. This a spacetime metric with cosmological constant (see Gravitation and Geometry by Rindler and Trautman, Bibliopolis, Napoli, 1987, page 309), with the line element \begin {equation} ds^2 = -e^{2\sqrt {-\Lambda }x}dt^2 +dx^2 + e^{2\sqrt {-\Lambda }z}du^2 + dz^2 \end {equation} The only physics parameter is the cosmological constant \(\Lambda = \verb |Bertotti__Lambda|\).

The fictitious “matter” stress-energy tensor representing \(\Lambda \) is \begin {equation} T_{ij}= - \frac {\Lambda }{8 \pi } g_{ij} = \left ( \begin {array}{cccc} \frac {1}{8}\frac {\Lambda e^{2\sqrt {-\Lambda } x}}{\pi } & 0 & 0 & 0\\ 0 & -\frac {1}{8}\frac {\Lambda }{\pi }& 0 & 0\\ 0 & 0 &-\frac {1}{8}\frac {\Lambda e^{2\sqrt {-\Lambda }z}}{\pi }& 0\\ 0 & 0 & 0 & -\frac {1}{8}\frac {\Lambda }{\pi }\end {array}\right ) \, \end {equation}

Mitica Vulcanov says: This metric is not working properly. We suspect that it is not a solution of the vacuum Einstein equations with cosmological constant, thus somebody else can try to calculate properly the above components of the \(T_{ij}\) - ask Mitica D.N. Vulcanov for more details.

5.8 Kasner-like spacetime

Exact::exact_model="Kasner-like" is the so-called “Kasner-like” metric, as described in L. Pimentel, Int. Journ. of Theor. Physics, 32, No. 6, p. 979, (1993) and the references cited here. (See also MTW section 30.2, gr-qc/0110031, and S. Gotlober et al., “Early Evolution of the Universe and Formation [of] Structure”, Akad. Verlag, 1990.) The Kasner-like line element is \begin {equation} ds^2 = -dt^2 + t^{2q} (dx^2 +dy^2) + t^{2 - 4q}dz^2 \end {equation} Here we have a stress-energy tensor which has all off-diagonal components vanishing:

There is one parameter \(q = \verb |Kasner_like__q|\).

This metric forms a one parameter family of solutions of Einstein’s equations with a perfect stiff fluid. The parameter \(q\) is related to the energy density, as is obvious from the last equation. The qualitative features of the expansion depend on \(q\) in the following way: for \(q > 1/2\) the universe expands from a “cigar” singularity; for \(q = 1/2\), the universe expands purely transversally from an initial “barrel” singularity; for \(0 < q < 1/2\) the initial singularity is “point-like” and if \(q \leq 0\) we have a “pancake” singularity. The case \(q=1/3\) corresponds to an isotropic universe with a stiff fluid; the case \(q=0\) is a region of Minkowski spacetime in non-Cartesian coordinates. This family of metrics is “Kasner-like” in the sense that the sum of the exponents is equal to one, but the sum of the squares is not equal to one except in the cases when \(q=0\) or \(q=2/3\), when we have the vacuum case.

5.9 axisymmetric Kasner spacetime

Exact::exact_model="Kasner-axisymmetric" specifies an axisymmetric Kasner spacetime, as described in S. D. Hern, Numerical Relativity and Inhomogeneous Cosmologies, PhD thesis, Cambridge (gr-qc/0004036), and S. D. Hern, J. M. Stewart, Class. Quantum Grav, 15, 1581, (1998). The line element is \begin {equation} ds^2 = -\frac {dt^2}{\sqrt {t}} + \frac {dx^2}{\sqrt {t}} + t dy^2 + t dz^2 \end {equation} This is an exact solution of the vacuum Einstein equations, explicitly homogeneous, and features a cosmological singularity at \(t=0\).

There are no parameters for this model.

5.10 generalized Kasner spacetime

Exact::exact_model="Kasner-generalized" specifies a generalized Kasner spacetime, as described in MTW section 30.2, where the line element is \begin {equation} ds^2 = -dt^2 +t^{2p_1}dx^2 + t^{2p_2}dy^2 + t^{2p_3}dz^2 \end {equation} The Kasner parameters \(p_1\), \(p_2\) and \(p_3\) must satisfy the relations \(p_1+p_2+p_3 = 1\) and \(p_1^2+p_2^2+p_3^2 = 1\). Restricting ourselves only to two parameters, \(p_1\) and \(p_2\), we have the following stress-energy tensor: \begin {equation} T_{ij} = \left ( \begin {array}{cccc} \frac {A}{8\pi t^2} & 0 & 0 & 0\\ 0 & \frac {A t^{2p_1-2}}{8 \pi } & 0 & 0\\ 0 & 0 & \frac {A t^{2p_2-2}}{8\pi } & 0 \\ 0 & 0 & 0 & \frac {A t^{-2p_1-2p_2}}{8 \pi }\end {array} \right ) \end {equation} where \(A = p_1 - p_1^2 +p_2 - p_2^2 - p_1 p_2\) (note the use of the above first condition on the parameters, thus we have \(p_3 = 1-p_1-p_2\)).

The parameters are \(p_1 = \verb |Kasner_generalized__p1|\) and \(p_2 = \verb |Kasner_generalized__p2|\).

Mitica Vulcanov has done several simulations with various Kasner spacetimes, see gr-qc/0110031.

5.11 Gowdy-wave Spacetime

Exact::exact_model = "Gowdy-wave" specifies a Gowdy spacetime, which gives a polarized wave in an expanding universe. See K. New, K. Watt, C. W. Misner, and J. Centrella, “Stable 3-level leapfrog integration in numerical relativity”, PRD 58, 064022.

There is only a single parameter, the wave amplitude Gowdy_wave__amplitude.

5.12 Milne Spacetime for Pre-Big-Bang Cosmology

Exact::exact_model = "Milne" specifies a 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}

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

6 Miscellaneous Spacetimes

6.1 Boost Rotation Symmetric Spacetime

Exact::exactmodel = "starSchwarz" specifies a boost-rotation symmetric spacetime, as described in Jiri Bicak and Bernd Schmidt, ”Asymptotically flat radiative space-times with boost-rotation symmetry”, Physical Review D 40, 1827 (1989). Pravda and Pravdová, gr-qc/0003067, give a general review of boost-rotation symmetric spacetimes.

FIXME: the parameters are …

6.2 Schwarzschild (Constant Density) Star

Exact::exact_model = "constant density star" specifies a constant-density “Schwarzschild” star, as described in MTW Box 23.2. The stress-energy tensor is also properly set up.

The parameters are the star’s mass constant_density_star__mass and its radius constant_density_star__radius.

6.3 Non-Einstein Bowl (“Bag of Gold”) Spacetime

Exact::exact_model = "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 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 physics parameters are

• \(a = \verb |bowl__strength|\), the deformation strength

• \(c = \verb |bowl__center|\), the deformation center

• bowl__shape is a keyword parameter to specify the type of function to use to specify the bowl (see below)

• \(\sigma = \verb |bowl__sigma|\), the deformation width (N.b. for bowl__shape = "Gaussian" the function is actually \(\exp \big ( \!-(x-c)^2/\sigma ^2 \big )\), not \(\exp \big ( -\half (x-c)^2/\sigma ^2 \big )\). Thus for this case \(\sigma \) is actually \(\sqrt {2}\) times the standard deviation of the Gaussian.)

• bowl__x_scale, bowl__y_scale, and bowl__z_scale, which set the \(x\), \(y,\) and \(z\) scales of the bowl (i.e. all the computations actually use \(x/\verb |bowl__x_scale|\), \(y/\verb |bowl__y_scale|\), and \(z/\verb |bowl__z_scale|\))

• bowl__evolve is a Boolean parameter which controls whether the bowl should be time-dependent; the remaining parameters are only used if bowl__evolve is true

• \(t_0 = \verb |bowl__t0|\), the center of the Fermi step in time

• \(\sigma _t = \verb |bowl__sigma_t|\), the width of the Fermi step in time

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

In detail, we choose \begin {equation} R(r) = r - A f(r) g(r) \end {equation} Here \(A = a\) if 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 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) = \left \{ \begin {array}{ll} \displaystyle \exp \big ( (r-c)^2/\sigma ^2 \big ) & \hbox {if {\tt bowl\_type = "Gaussian"}} \\[1ex] \displaystyle \frac {1}{1 + \exp (-\sigma (r-c))} & \hbox {if {\tt bowl\_type = "Fermi"}} \end {array} \right . \end {equation} \(g(r) = 1 - \sech 4r\) is a fixup factor to ensure that \(\displaystyle \lim _{r \to 0} R(r) = r\).

The three paramters bowl__x_scale, bowl__y_scale, and bowl__z_scale 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.

7 Acknowledgments

The original code, including the boost-rotation symmetric metric and the slice evolver, was written by Carsten Gundlach and Miguel Alcubierre. Many different people have contributed exact solutions. The Schwarzschild/Lemaitre solution and most (all?) of the cosmological solutions were written by Mitica Vulcanov. The Minkowski/gauge wave model was written by Michael Koppitz. In May-June 2002 Jonathan Thornburg cleaned up a lot of the code, systematized the spacetime/coordinate and parameter names, and wrote most of this documentation (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 model is adapted from the file KTsol.tex in this same directory, by Hisa-aki Shinkai. The Gowdy model was written by Denis Pollney. The ADMBase::evolution_method = "exact" code was written by Peter Diener. The “boost any vacuum solution” code was written by Jonathan Thornburg.

8 Parameters

[1]

Minkowski/shifted gauge wave

[1]

Schwarzschild/Novikov

[1]

Schwarzschild-Lemaitre

[1]

Kerr/Kerr-Schild/spherical

[1]

anti-de Sitter+Lambda

[1]

boost-rotation symmetric

[1]

constant density star

kerr_boyerlindquist__spin	Scope: restricted	REAL
Description: Kerr/Boyer-Lindquist: dimensionless spin parameter a = J/m
Range		Default: 0.6
-1.0:1.0	dimensionless spin parameter a = J/m for Kerr black hole

kerr_kerrschild__spin	Scope: restricted	REAL
Description: Kerr/Kerr-Schild: dimensionless spin parameter a = J/m
Range		Default: 0.6
-1.0:1.0	dimensionless spin parameter a = J/m for Kerr black hole