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

ds2 = Ψ4 e2q dρ2 + dz2 + ρ2dϕ2 = Ψ4dŝ2, (1)

where q is a free function subject to certain regularity and fall-off conditions, ρ = x2 + y2 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 = aρ2+b r2 e z σz2 e(ρρ0)2 1 + d ρm 1 + eρm cos2 nϕ + ϕ 0 (2)

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

q = a ρ σρ b 1 1 + r2 r02 σr2 c2 1 + d ρm 1 + eρm cos2 nϕ + ϕ 0 (3)

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

q = a ρ σρ ber2r 02 σ r2 c2 1 + d ρm 1 + eρm cos2 nϕ + 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:

̂Ψ Ψ 8 R̂ = 0 (5)

where the conformal Ricci scalar is found to be

R̂ = 2 e2q( z2q + ρ2q) + 1 ρ2(3(ϕq)2 + 2 ϕq) (6)

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,

flatΨ + Ψ 4 (z2q + ρ2q) = 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:

Note that the default q expression is

q = gundlach_aρ2er2

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: