CanudaX_ExactID

Cheng-Hsin Cheng

April 15, 2026

Abstract

CanudaX_ExactID provides exact initial data of vacuum spacetimes for use in ADMBaseX. Currently, CanudaX_ExactID can set up initial data for a single Kerr BH, Teukolsky waves, and Apples with Apples tests.

1 Introduction

This thorn provides exact initial data for vacuum spacetimes, which currently includes the following cases.

1.1 Kerr BH in quasi-isotropic coordinates

Based on the KerrQuasiIsotropic thorn from Lean. The original KerrQuasiIsotropic thorn provides analytic initial data for a rotating black hole, spinning around the z-axis, in quasi-isotropic coordinates where the radial coordinate is defined by \begin {equation} r_{\rm BL} = r \left (1 + \frac {r_{+}}{4r} \right )^2. \end {equation} We employ the construction of [1], with our specific construction outlined in Eqs. (46)–(48) of [2].

1.2 Teukolsky waves

Based on EinsteinInitialData/IDLinearWaves. Teukolsky waves are quadrupolar waves in linearized GR in transverse-traceless gauge [3]. Please see the original documentation for IDLinearWaves for more detail.

1.3 Apples with apples tests

Provides initial data for the gauge wave, shifted gauge wave, linearized wave tests. For details, see the references [4] [5] [6].

2 Applications

References

[1]   Y. T. Liu, Z. B. Etienne and S. L. Shapiro, Phys. Rev. D 80, 121503 (2009) doi:10.1103/PhysRevD.80.121503 [arXiv:1001.4077 [gr-qc]].

[2]   H. Okawa, H. Witek and V. Cardoso, “Black holes and fundamental fields in Numerical Relativity: initial data construction and evolution of bound states,” Phys. Rev. D 89, no.10, 104032 (2014) doi:10.1103/PhysRevD.89.104032 [arXiv:1401.1548 [gr-qc]].

[3]   S. A. Teukolsky, “LINEARIZED QUADRUPOLE WAVES IN GENERAL RELATIVITY AND THE MOTION OF TEST PARTICLES,” Phys. Rev. D 26, 745-750 (1982) doi:10.1103/PhysRevD.26.745

[4]   M. Alcubierre, G. Allen, C. Bona, D. Fiske, T. Goodale, F. S. Guzman, I. Hawke, S. H. Hawley, S. Husa and M. Koppitz, et al. “Toward standard testbeds for numerical relativity,” Class. Quant. Grav. 21, no.2, 589-613 (2004) doi:10.1088/0264-9381/21/2/019 [arXiv:gr-qc/0305023 [gr-qc]].

[5]   M. C. Babiuc, S. Husa, D. Alic, I. Hinder, C. Lechner, E. Schnetter, B. Szilagyi, Y. Zlochower, N. Dorband and D. Pollney, et al. “Implementation of standard testbeds for numerical relativity,” Class. Quant. Grav. 25, 125012 (2008) doi:10.1088/0264-9381/25/12/125012 [arXiv:0709.3559 [gr-qc]].

[6]   D. Daverio, Y. Dirian and E. Mitsou, “Apples with Apples comparison of 3+1 conformal numerical relativity schemes,” [arXiv:1810.12346 [gr-qc]].

3 Parameters

eps_r
 Scope: private REAL
Description: Floor value for radial distance
Range Default: 1.0d-12
0:*
any small positive value

kqi_m_plus
 Scope: private REAL
Description: Mass of the Kerr BH
Range Default: 1.0
0.0:*
any positive value

kqi_pos_plus
 Scope: private REAL
Description: Position of the Kerr BH
Range Default: 0.0
*:*
any value possible; for zero spin set pos_plus[0]=0.0001

kqi_spin_plus
 Scope: private REAL
Description: Dimensionless spin of the Kerr BH with z being the rotation axis
Range Default: 0.0
0:1
any value between 0 and 1

schedule_in_admbasex_initialdata
 Scope: private BOOLEAN
Description: Schedule in (instead of after) ADMBaseX_InitialData
Default: yes

teukolsky_mvalue
 Scope: private INT
Description: m value for Teukolsky waves: integer from -2 to 2
Range Default: (none)
-2:2
implemented : m = -2, -1, 0, 1, 2

teukolsky_packet
 Scope: private KEYWORD
Description: Wave packet shape for Teukolsky waves
Range Default: eppley
eppley
Eppley packet: x exp(-xˆ 2 )
evans
Evans packet: wˆ4 (1-xˆ2 /wˆ2 )ˆ6
square
Square packet: (1-xˆ2 /wˆ2 )ˆ2

teukolsky_parity
 Scope: private KEYWORD
Description: Parity for Teukolsky waves: even or odd
Range Default: even
even
even parity
odd
odd parity

teukolsky_wavecenter
 Scope: private REAL
Description: Center of propagation for Teukolsky waves
Range Default: 0.0
:

teukolsky_wavelength
 Scope: private REAL
Description: Wave length of Teukolsky waves
Range Default: 2.0
0:
positive wavelength

teukolsky_wavepulse
 Scope: private REAL
Description: Pulse length/duration for purely ingoing Teukolsky waves
Range Default: 1.0
0:
positive pulse

teukolsky_wavesgoing
 Scope: private KEYWORD
Description: Wave propagation direction for Teukolsky waves
Range Default: both
in
Ingoing wave
out
Outgoing wave
both
In and outgoing wave

wave_amplitude
 Scope: private REAL
Description: Amplitude of gauge wave or linear wave
Range Default: 1.0
0.0:*
any positive value

wave_period
 Scope: private REAL
Description: Period of the gauge wave or linear wave
Range Default: 1.0
0.0:*
any positive value

initial_data
 Scope: shared from ADMBASEX KEYWORD
Extends ranges:
KQI_ana
”KerrQuasiIsotropic: Analytic initial data for single Kerr BH in quasi-isotropic coordinates”
GaugeWave
Gauge wave
ShiftedGaugeWave
Shifted gauge wave
LinearizedWave
Linearized wave
TeukolskyWave
Teukolsky wave

initial_lapse
 Scope: shared from ADMBASEX KEYWORD
Extends ranges:
KQI_ana
”KerrQuasiIsotropic: analytically extend lapse through origin”
KQI_prec
”KerrQuasiIsotropic: use precollapsed lapse”
KQI_abs
”KerrQuasiIsotropic: use absolute value of lapse (discouraged; better to use KQI_ana to reproduce QNMs)”
GaugeWave
Gauge wave
ShiftedGaugeWave
Shifted gauge wave
LinearizedWave
Linearized wave

initial_shift
 Scope: shared from ADMBASEX KEYWORD
Extends ranges:
KQI_ana
”KerrQuasiIsotropic: anayltic shift”
GaugeWave
Gauge wave
ShiftedGaugeWave
Shifted gauge wave
LinearizedWave
Linearized wave

4 Interfaces

General

Implements:

canudax_exactid

Inherits:

admbasex

Uses header:

loop_device.hxx

5 Schedule

This section lists all the variables which are assigned storage by thorn CanudaX/CanudaX_ExactID. 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

NONE

Scheduled Functions

ADMBaseX_InitialData (conditional)

  canudax_exactidgroup

  canudax_exactid group for setting up initial data

 

  Type: group

CCTK_INITIAL (conditional)

  canudax_exactidgroup

  canudax_exactid group for setting up initial data

 

  After: admbasex_initialdata
  Before: admbasex_postinitial
  Type: group

CanudaX_ExactIDGroup (conditional)

  canudax_exactid_kqi_analytic

  create analytic initial data for a kerr bh

 

  Language: c
  Options: local
  Sync: admbasex::metric
    admbasex::curv
    admbasex::lapse
    admbasex::shift
  Type: function
  Writes: admbasex::metric(everywhere)
    admbasex::curv(everywhere)
    admbasex::lapse(everywhere)
    admbasex::shift(everywhere)

CanudaX_ExactIDGroup (conditional)

  canudax_exactid_gaugewave

  create gauge wave initial data

 

  Language: c
  Options: local
  Sync: admbasex::metric
    admbasex::curv
    admbasex::lapse
    admbasex::shift
  Type: function
  Writes: admbasex::metric(everywhere)
    admbasex::curv(everywhere)
    admbasex::lapse(everywhere)
    admbasex::shift(everywhere)

CanudaX_ExactIDGroup (conditional)

  canudax_exactid_shiftedgaugewave

  create shifted gauge wave initial data

 

  Language: c
  Options: local
  Sync: admbasex::metric
    admbasex::curv
    admbasex::lapse
    admbasex::shift
  Type: function
  Writes: admbasex::metric(everywhere)
    admbasex::curv(everywhere)
    admbasex::lapse(everywhere)
    admbasex::shift(everywhere)

CanudaX_ExactIDGroup (conditional)

  canudax_exactid_linearizedwave

  create linearized wave initial data

 

  Language: c
  Options: local
  Sync: admbasex::metric
    admbasex::curv
    admbasex::lapse
    admbasex::shift
  Type: function
  Writes: admbasex::metric(everywhere)
    admbasex::curv(everywhere)
    admbasex::lapse(everywhere)
    admbasex::shift(everywhere)

CanudaX_ExactIDGroup (conditional)

  canudax_exactid_teukwave

  create teukolsky wave initial data

 

  Language: c
  Options: local
  Sync: admbasex::metric
    admbasex::curv
  Type: function
  Writes: admbasex::metric(everywhere)
    admbasex::curv(everywhere)