1 Purpose

The purpose of this thorn is to create (time symmetric) initial data for a Brill wave spacetime. It does so by starting from a three–metric of the form originally considered by Brill \begin {equation} ds^2 = \Psi ^4 \left [ e^{2q} \left ( d\rho ^2 + dz^2 \right ) + \rho ^2 d\phi ^2 \right ] =\Psi ^4 \hat {ds}^{2}, \label {eqn:brillmetric} \end {equation} where \(q\) is a free function subject to certain regularity and fall-off conditions, \(\rho =\sqrt {x^2+y^2}\) and \(\Psi \) is a conformal factor to be solved for.

Thorn IDBrillData provides three choices for the \(q\) function: an exponential form, (IDBrillData::q_function = "exp") \begin {equation} q = a \; \frac {\rho ^{2+b}}{r^2} e^{-\left ( \frac {z}{\sigma _z} \right )^2} e^{-(\rho - \rho _0)^2} \left [ 1 + d \frac {\rho ^m}{1 + e \rho ^m} \cos ^2 \left ( n \phi + \phi _0 \right ) \right ] \end {equation} a generalized form of the \(q\) function first written down by Eppley (IDBrillData::q_function = "eppley") \begin {equation} q = a \left ( \frac {\rho }{\sigma _\rho } \right )^b \frac {1}{1 + \left [ \left ( r^2 - r_0^2 \right ) / \sigma _r^2 \right ]^{c/2}}\left [ 1 + d \frac {\rho ^m}{1 + e \rho ^m} \cos ^2 \left ( n \phi + \phi _0 \right ) \right ] \end {equation} and the (default) Gundlach \(q\) function which includes the Holz form (IDBrillData::q_function = "gundlach") \begin {equation} q = a \left ( \frac {\rho }{\sigma _\rho } \right )^b e^{-\left [ \left ( r^2 - r_0^2 \right ) / \sigma _r^2 \right ]^{c/2}} \left [ 1 + d \frac {\rho ^m}{1 + e \rho ^m} \cos ^2 \left ( n \phi + \phi _0 \right ) \right ] \end {equation}

Substituting the metric into the Hamiltonian constraint gives an elliptic equation for the conformal factor \(\Psi \) which is then numerically solved for a given function \(q\): \begin {equation} \hat {\nabla } \Psi - \frac {\Psi }{8} \hat {R} = 0 \end {equation} where the conformal Ricci scalar is found to be

Assuming the initial data to be time symmetric means that the momentum constraints are trivially satisfied.

In the case of axisymmetry (that is \(d=0\) in the above expressions for \(q\)), the Hamiltonian constraint can be written as an elliptic equation for \(\Psi \) with just the flat space Laplacian, \begin {equation} \nabla _{flat} \Psi + \frac {\Psi }{4} (\partial _z^2 q + \partial _\rho ^2 q) = 0 \end {equation} If the initial data is chosen to be ADMBase::initial_data = "brilldata2D" then this elliptic equation is solved rather than the equation above.

2 Generating Initial Data with IDBrillData

Brill initial data is activated by choosing the CactusEinstein/ADMBase parameter initial_data to be brilldata, or for the case of axisymmetry brilldata2D can also be used.

The parameter IDBrillData::q_function chooses the form of the \(q\) function to be used, defaulting to the Gundlach expression.

Additional IDBrillData parameters for each form of \(q\) fix the remaining freedom:

• Exponential \(q\): IDBrillData::q_function = "exp"

  \((a, b,\sigma _z,\rho _0)=\) (exp_a, exp_b, exp_sigmaz,exp_rho0)

• Eppley \(q\): IDBrillData::q_function = "eppley"

  \((a, b,\sigma _\rho , r_0,\sigma _r,c)=\) (eppley_a, eppley_b, eppley_sigmarho, eppley_r0, eppley_sigmar, eppley_c)

• Gundlach \(q\): IDBrillData::q_function = "gundlach"

  \((a, b,\sigma _\rho , r_0,\sigma _r,c)=\) (gundlach_a, gundlach_b, gundlach_sigmarho, gundlach_r0, gundlach_sigmar, gundlach_c)

• Non-axisymmetric part for each choice of \(q\)

  \((d, m, e, n, \phi 0)=\) (brill3d_d, brill3d_m, brill3d_e, brill3d_n, brill3d_phi0)

Note that the default \(q\) expression is

IDBrillData can use the elliptic solvers (type LinMetric) provided by CactusEinstein/EllSOR,
AEIThorns/BAM_Elliptic, or CactusElliptic/EllPETSc to solve the equation resulting from the Hamiltonian constraint. In all cases the parameter thresh sets the threshold for the elliptic solve. The choice of elliptic solver is made through the parameter brill_solver:

• sor: Understands the Robin boundary condition, additional parameters control the maximum number of iterations (sor_maxit).

• bam: BAM_Elliptic does not properly implement the elliptic infrastructure of EllBase, and the BAM_Elliptic parameter to use the Robin boundary condition must be set independently of
  IDBrillWave::brill_bound.

3 Notes

Thorn IDBrillData understands both the “physical” and “static conformal” metric_type. In the case of a conformal metric being chosen, the conformal factor is set to \(\Psi \). Currently the derivatives of the conformal factor are not calculated, so that only staticconformal::conformal_storage = "factor" is supported.

4 References

4.1 Specification of Brill Waves

1. Dieter Brill, Ann. Phys., 7, 466, 1959.

2. Ken Eppley, Sources of Gravitational Radiation, edited by L. Smarr (Cambridge University Press, Cambridge, England, 1979), p. 275.

4.2 Numerical Evolutions of Brill Waves

1. Gravitational Collapse of Gravitational Waves in 3D Numerical Relativity, Miguel Alcubierre, Gabrielle Allen, Bernd Bruegmann, Gerd Lanfermann, Edward Seidel, Wai-Mo Suen, Malcolm Tobias, Phys. Rev. D61, 041501, 2000.

5 Parameters

brill3d_d	Scope: private	REAL
Description: 3D Brill wave: d rho cos(n (phi + phi0))
Range		Default: 0.0
:	Anything

brill3d_e	Scope: private	REAL
Description: 3D Brill wave: d rho cos(n (phi + phi0))
Range		Default: 1.0
:	Anything

brill3d_m	Scope: private	REAL
Description: 3D Brill wave: d rho cos(n (phi + phi0))
Range		Default: 2.0
:	Anything

brill3d_n	Scope: private	REAL
Description: 3D Brill wave: d rho cos(n (phi + phi0))
Range		Default: 2.0
:	Anything

brill3d_phi0	Scope: private	REAL
Description: 3D Brill wave: d rho cos(n (phi + phi0))
Range		Default: 0.0
:	Anything

eppley_b	Scope: private	INT
Description: Eppley Brill wave: used in exponent in rho: rho
Range		Default: 2
:	Anything

eppley_c	Scope: private	INT
Description: Eppley Brill wave: (r - r0)c/2)
Range		Default: 2
:	Anything

exp_b	Scope: private	INT
Description: Exp Brill wave: used in exponent in rho: rho2+b)
Range		Default: 2
:	Anything

gundlach_b	Scope: private	INT
Description: Gundlach Brill wave: used in exponent in rho: rho
Range		Default: 2
:	Anything

gundlach_c	Scope: private	INT
Description: Gundlach Brill wave: (r - r0)c/2)
Range		Default: 2
:	Anything

q_function	Scope: private	KEYWORD
Description: Form of function q [0,1,2]
Range		Default: gundlach
exp	contains e-z} factor
eppley	contains 1/(1+r) factor
gundlach	contains e-r} factor

6 Interfaces

General

Implements:

idbrilldata

Inherits:

grid

admbase

staticconformal

ellbase

Grid Variables

6.0.1 PRIVATE GROUPS

7 Schedule

This section lists all the variables which are assigned storage by thorn EinsteinInitialData/IDBrillData. Storage can either last for the duration of the run (Always means that if this thorn is activated storage will be assigned, Conditional means that if this thorn is activated storage will be assigned for the duration of the run if some condition is met), or can be turned on for the duration of a schedule function.

Storage

Scheduled Functions

CCTK_PARAMCHECK (conditional)

  idbrilldata_paramchecker

  check that the metric_type is recognised

CCTK_WRAGH (conditional)

  brilldata_initsymbound

  set up symmetries for brillpsi

ADMBase_InitialData (conditional)

  brilldata

  construct brill wave initial data