IDBrillData

Carsten Gundlach, Gabrielle Allen

Date

Abstract

This thorn creates time symmetric initial data for Brill wave spacetimes. It can create both axisymmetric data (in a 3D cartesian grid), as well as data with an angular dependency.

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

d s^{2} = Ψ^{4} [e^{2 q} (d ρ^{2} + d z^{2}) + ρ^{2} d ϕ^{2}] = Ψ^{4} {\hat{d s}}^{2},

(1)

where $q$ is a free function subject to certain regularity and fall-oﬀ conditions, $ρ = \sqrt{x^{2} + y^{2}}$ and $Ψ$ is a conformal factor to be solved for.

Thorn IDBrillData provides three choices for the $q$ function: an exponential form, (IDBrillData::q_function = "exp")

q = a \frac{ρ^{2 + b}}{r^{2}} e^{- {(\frac{z}{σ_{z}})}^{2}} e^{- {(ρ - ρ_{0})}^{2}} [1 + d \frac{ρ^{m}}{1 + e ρ^{m}} {cos}^{2} (n ϕ + ϕ_{0})]

(2)

a generalized form of the $q$ function ﬁrst written down by Eppley (IDBrillData::q_function = "eppley")

q = a {(\frac{ρ}{σ_{ρ}})}^{b} \frac{1}{1 + {[(r^{2} - r_{0}^{2}) ∕ σ_{r}^{2}]}^{c ∕ 2}} [1 + d \frac{ρ^{m}}{1 + e ρ^{m}} {cos}^{2} (n ϕ + ϕ_{0})]

(3)

and the (default) Gundlach $q$ function which includes the Holz form (IDBrillData::q_function = "gundlach")

q = a {(\frac{ρ}{σ_{ρ}})}^{b} e^{- {[(r^{2} - r_{0}^{2}) ∕ σ_{r}^{2}]}^{c ∕ 2}} [1 + d \frac{ρ^{m}}{1 + e ρ^{m}} {cos}^{2} (n ϕ + ϕ_{0})]

(4)

Substituting the metric into the Hamiltonian constraint gives an elliptic equation for the conformal factor $Ψ$ which is then numerically solved for a given function $q$ :

\hat{\nabla} Ψ - \frac{Ψ}{8} \hat{R} = 0

(5)

where the conformal Ricci scalar is found to be

\begin{array}{rcl} \hat{R} = - 2 (e^{- 2 q} (\partial_{z}^{2} q + \partial_{ρ}^{2} q) + \frac{1}{ρ^{2}} (3 {(\partial_{ϕ} q)}^{2} + 2 \partial_{ϕ} q)) & (6) \end{array}

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 $Ψ$ with just the ﬂat space Laplacian,

\nabla_{f l a t} Ψ + \frac{Ψ}{4} (\partial_{z}^{2} q + \partial_{ρ}^{2} q) = 0

(7)

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"
$(a, b, σ_{z}, ρ_{0}) =$ (exp_a, exp_b, exp_sigmaz,exp_rho0)
Eppley $q$ : IDBrillData::q_function = "eppley"
$(a, b, σ_{ρ}, r_{0}, σ_{r}, c) =$ (eppley_a, eppley_b, eppley_sigmarho, eppley_r0, eppley_sigmar, eppley_c)
Gundlach $q$ : IDBrillData::q_function = "gundlach"
$(a, b, σ_{ρ}, r_{0}, σ_{r}, c) =$ (gundlach_a, gundlach_b, gundlach_sigmarho, gundlach_r0, gundlach_sigmar, gundlach_c)
Non-axisymmetric part for each choice of $q$
$(d, m, e, n, ϕ 0) =$ (brill3d_d, brill3d_m, brill3d_e, brill3d_n, brill3d_phi0)

Note that the default $q$ expression is

q = g u n d l a c h_a ρ^{2} e^{- r^{2}}

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.