ShiftedKerrSchild

Zacharia E. Etienne <zachetie *at* gmail *dot* com>
Roland Haas <rhaas@illinois.edu>

November 04 2019

Abstract

To set up Kerr-Schild initial data, with a shifted radial coordinate.

1 Introduction

A complete description of a black hole spacetime in Kerr-Schild spherical polar coordinates that includes an explicit analytic form of the extrinsic curvature for arbitrary spin parameters does not exist in the literature, so we first include it here for completeness. Then we present our strategy for transforming spacetime quantities into shifted Kerr-Schild Cartesian coordinates.

2 Physical System

In unshifted spherical polar coordinates, where $ρ = r^{2} + a^{2} \cos^{2} (𝜃)$ , $M$ is the black hole mass, and $a$ is the black hole spin parameter, the Kerr-Schild lapse, shift, and 3-metric are given by

\begin{array}{rcl} α & = & \frac{1}{\sqrt{1 + \frac{2 M r}{ρ^{2}}}} & (1) \\ β^{r} & = & α^{2} \frac{2 M r}{ρ^{2}} & (2) \\ β^{𝜃} & = & β^{ϕ} = γ_{r 𝜃} = γ_{𝜃 ϕ} = 0 & (3) \\ γ_{r r} & = & 1 + \frac{2 M r}{ρ^{2}} & (4) \\ γ_{r ϕ} & = & - a γ_{r r} \sin^{2} (𝜃) & (5) \\ γ_{𝜃 𝜃} & = & ρ^{2} & (6) \\ γ_{ϕ ϕ} & = & (r^{2} + a^{2} + \frac{2 M r}{ρ^{2}} a^{2} \sin^{2} (𝜃)) \sin^{2} (𝜃) . & (7) \end{array}

Next, we define a few useful quantities,

\begin{array}{rcl} A & = & (a^{2} \cos (2 𝜃) + a^{2} + 2 r^{2}) & (8) \\ B & = & A + 4 M r & (9) \\ D & = & \sqrt{\frac{2 M r}{a^{2} \cos^{2} (𝜃) + r^{2}} + 1} . & (10) \end{array}

Then the extrinsic curvature $K_{i j} = (\nabla_{i} β_{j} + \nabla_{j} β_{i}) ∕ (2 α)$ (see, e.g., Eq. 13 in Ref. [1]) with $\partial_{t} γ_{i j} = 0$ , may be written in spherical polar coordinates as

\begin{array}{rcl} K_{r r} & = & \frac{D (A + 2 M r)}{A^{2} B} [4 M (a^{2} \cos (2 𝜃) + a^{2} - 2 r^{2})] & (11) \\ K_{r 𝜃} & = & \frac{D}{A B} [8 a^{2} M r \sin (𝜃) \cos (𝜃)] & (12) \\ K_{r ϕ} & = & \frac{D}{A^{2}} [- 2 a M \sin^{2} (𝜃) (a^{2} \cos (2 𝜃) + a^{2} - 2 r^{2})] & (13) \\ K_{𝜃 𝜃} & = & \frac{D}{B} [4 M r^{2}] & (14) \\ K_{𝜃 ϕ} & = & \frac{D}{A B} [- 8 a^{3} M r \sin^{3} (𝜃) \cos (𝜃)] & (15) \\ K_{ϕ ϕ} & = & \frac{D}{A^{2} B} [2 M r \sin^{2} (𝜃) (a^{4} (r - M) \cos (4 𝜃) + a^{4} (M + 3 r) + & (16) \\ 4 a^{2} r^{2} (2 r - M) + 4 a^{2} r \cos (2 𝜃) (a^{2} + r (M + 2 r)) + 8 r^{5})] . & (17) \end{array}

All GiRaFFE curved-spacetime code validation tests adopt shifted Kerr-Schild Cartesian coordinates $(x^{'}, y^{'}, z^{'})$ , which map $(0, 0, 0)$ to the finite radius $r = r_{0} > 0$ in standard (unshifted) Kerr-Schild spherical polar coordinates. So, in many ways, this is similar to a trumpet spacetime. Though this radial shift acts to shrink the black hole’s coordinate size, it also renders the very strongly-curved spacetime fields at $r < r_{0}$ to vanish deep inside the horizon, which can contribute to numerical stability when evolving hydrodynamic, MHD, and FFE fields inside the horizon.

The shifted radial coordinate $r^{'}$ relates to the standard spherical polar radial coordinate $r$ via $r = r^{'} + r_{0}$ , where $r_{0} > 0$ is the (constant) radial shift.

As an example, to compute $K_{x^{'} y^{'}}$ at some arbitrary point $(x^{'}, y^{'}, z^{'})$ , we first convert the coordinate $(x^{'}, y^{'}, z^{'})$ into shifted spherical-polar coordinates via $(r^{'} = \sqrt{{x^{'}}^{2} + {y^{'}}^{2} + {z^{'}}^{2}}, 𝜃^{'}, ϕ^{'}) = (r^{'}, 𝜃, ϕ)$ , as a purely radial shift like this preserves the original angles. Next, we evaluate the components of the Kerr-Schild extrinsic curvature $K_{i j}$ (provided above) in standard spherical polar coordinates at $(r = r^{'} + r_{0}, 𝜃, ϕ)$ . Defining $x_{s p h, s h}^{i}$ as the $i$ th shifted spherical polar coordinate and $x_{s p h}^{i}$ as the $i$ th (unshifted) spherical polar coordinate, $K_{x^{'} y^{'}}$ is computed via the standard coordinate transformations:

K_{x^{'} y^{'}} = \frac{d x_{s p h, s h}^{k}}{d x^{'}} \frac{d x_{s p h, s h}^{l}}{d y^{'}} \frac{d x_{s p h}^{i}}{d x_{s p h, s h}^{k}} \frac{d x_{s p h}^{i}}{d x_{s p h, s h}^{l}} K_{i j} .

(18)

However, we have $d x_{s p h}^{i} = d x_{s p h, s h}^{i}$ , since the radial shift $r_{0}$ is a constant and the angles are unaffected by the radial shift. This implies that $d x_{s p h, s h}^{k} ∕ d x^{'} = d x_{s p h}^{k} ∕ d x^{'}$ and $d x_{s p h}^{i} ∕ d x_{s p h, s h}^{k} = δ_{k}^{i}$ .

So after computing any spacetime quantity at a point $(r = r^{'} + r_{0}, 𝜃, ϕ)$ , we need only apply the standard spherical-to-Cartesian coordinate transformation to evaluate that quantity in shifted Cartesian coordinates.

3 Using This Thorn

ShiftedKerrSchild defines the following parameters for the constants defined in section 2:

parameter name	constant

KerrSchild_radial_shift	$r_{0}$ ,
BH_mass	$M$ ,
BH_spin	$a$ .

3.1 Obtaining This Thorn

This thorn is part of the Einstein Toolkit and included in the WVUThorns arrangement.

3.2 Interaction With Other Thorns

This thorn sets up initial data for use by thorn ADMBase based on ADMBase’s parameters initial_data, initial_lapse and initial_shift.

3.3 Examples

For a complete example, please see the GiRaFFE test GiRaFFE_tests_MagnetoWald.par.

# Shifted KerrSchild
ActiveThorns = "ShiftedKerrSchild"
ShiftedKerrSchild::BH_mass = 1.0
ShiftedKerrSchild::BH_spin = 0.9
ShiftedKerrSchild::KerrSchild_radial_shift = 0.4359

ADMBase::initial_data  = "ShiftedKerrSchild"
ADMBase::initial_lapse = "ShiftedKerrSchild"
ADMBase::initial_shift = "ShiftedKerrSchild"

3.4 Support and Feedback

Please contact the author or the Einstein Toolkit mailing list users@einsteintoolkit.org.

References

[1] G. B. Cook, Living Rev. Relativity, 3(1):5 (2000).

4 Parameters


bh_mass	Scope: restricted	REAL

Description: The mass of the black hole. Let’s keep this at 1!

Range		Default: 1.0
0.0:*	Positive


bh_spin	Scope: restricted	REAL

Description: The z-axis dimensionless spin of the black hole

Range		Default: 0.0
-1.0:1.0	Anything between -1 and +1


kerrschild_radial_shift	Scope: restricted	REAL

Description: Radial shift for Kerr Schild initial data. Actual shift = KerrSchild_radial_shift*BH_mass

Range		Default: 0.0
0.0:*	Positive


initial_data	Scope: shared from ADMBASE	KEYWORD

Extends ranges:

ShiftedKerrSchild	Initial data from ShiftedKerrSchild solution


initial_lapse	Scope: shared from ADMBASE	KEYWORD

Extends ranges:

ShiftedKerrSchild	Initial lapse from ShiftedKerrSchild solution


initial_shift	Scope: shared from ADMBASE	KEYWORD

Extends ranges:

ShiftedKerrSchild	Initial shift from ShiftedKerrSchild solution


metric_type	Scope: shared from ADMBASE	KEYWORD

5 Interfaces

General

Implements:

shiftedkerrschild

Inherits:

admbase

boundary

grid

Grid Variables

5.0.1 PUBLIC GROUPS


Group Names	Variable Names	Details

shiftedkerrschild_3metric_shift		compact	0
	SKSgrr	description	Shifted Kerr-Schild 3 metric and shift
	SKSgrth	dimensions	3
	SKSgrph	distribution	DEFAULT
	SKSgthth	group type	GF
	SKSgthph	tags	prolongation=”none”
	SKSgphph	timelevels	1
	SKSbetar	variable type	REAL

6 Schedule

This section lists all the variables which are assigned storage by thorn WVUThorns/ShiftedKerrSchild. 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

Always:
ShiftedKerrSchild_3metric_shift[1]

Scheduled Functions

CCTK_PARAMCHECK (conditional)

shiftedkerrschild_paramcheck

check parameters for consitency and unsupported values

	Language:	c
	Type:	function

CCTK_INITIAL (conditional)

shiftedkerrschild_initial

schedule shiftedkerrschild initial data group

	After:	admbase_initialgauge
	Before:	hydrobase_initial
	Type:	group

ShiftedKerrSchild_Initial (conditional)

shiftedks_id

set up shifted kerr-schild initial data

	Language:	c
	Type:	function

CCTK_PRESTEP (conditional)

shiftedks_id

set up shifted kerr-schild initial data

	Language:	c
	Type:	function

CCTK_ANALYSIS (conditional)

shiftedks_id

set up shifted kerr-schild initial data

	Language:	c
	Type:	function