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 = r2 + a2 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

α = 1 1 + 2Mr ρ2 (1) βr = α22Mr ρ2 (2) β𝜃 = βϕ = γ r𝜃 = γ𝜃ϕ = 0 (3) γrr = 1 + 2Mr ρ2 (4) γrϕ = aγrr sin 2(𝜃) (5) γ𝜃𝜃 = ρ2 (6) γϕϕ = r2 + a2 + 2Mr ρ2 a2 sin 2(𝜃)sin 2(𝜃). (7)

Next, we define a few useful quantities,

A = a2 cos (2𝜃) + a2 + 2r2 (8) B = A + 4Mr (9) D = 2Mr a2 cos 2(𝜃) + r2 + 1. (10)

Then the extrinsic curvature Kij = (iβj + jβi)(2α) (see, e.g., Eq. 13 in Ref. [1]) with tγij = 0, may be written in spherical polar coordinates as

Krr = D(A + 2Mr) A2B 4M a2 cos (2𝜃) + a2 2r2 (11) Kr𝜃 = D AB 8a2Mrsin (𝜃)cos (𝜃) (12) Krϕ = D A2 2aMsin 2(𝜃) a2 cos (2𝜃) + a2 2r2 (13) K𝜃𝜃 = D B 4Mr2 (14) K𝜃ϕ = D AB 8a3Mrsin 3(𝜃)cos (𝜃) (15) Kϕϕ = D A2B 2Mrsin 2(𝜃) a4(r M)cos (4𝜃) + a4(M + 3r)+ (16) 4a2r2(2r M) + 4a2rcos (2𝜃) a2 + r(M + 2r) + 8r5 . (17)

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 = r0 > 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 < r0 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 + r0, where r0 > 0 is the (constant) radial shift.

As an example, to compute Kxy at some arbitrary point (x,y,z), we first convert the coordinate (x,y,z) into shifted spherical-polar coordinates via (r = 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 Kij (provided above) in standard spherical polar coordinates at (r = r + r0,𝜃,ϕ). Defining xsph,shi as the ith shifted spherical polar coordinate and xsphi as the ith (unshifted) spherical polar coordinate, Kxy is computed via the standard coordinate transformations:

Kxy = dxsph,shk dx dxsph,shl dy dxsphi dxsph,shk dxsphi dxsph,shlKij. (18)

However, we have dxsphi = dxsph,shi, since the radial shift r0 is a constant and the angles are unaffected by the radial shift. This implies that dxsph,shkdx = dxsphkdx and dxsphidxsph,shk = δki.

So after computing any spacetime quantity at a point (r = r + r0,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_shiftr0,
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   compact0
SKSgrr   descriptionShifted Kerr-Schild 3 metric and shift
SKSgrth   dimensions3
SKSgrph   distributionDEFAULT
SKSgthth   group typeGF
SKSgthph   tagsprolongation=”none”
SKSgphph   timelevels1
SKSbetar  variable typeREAL




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