## ShiftedKerrSchild

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 $\rho ={r}^{2}+{a}^{2}{\mathrm{cos}}^{2}\left(𝜃\right)$, $M$ is the black hole mass, and $a$ is the black hole spin parameter, the Kerr-Schild lapse, shift, and 3-metric are given by

$\begin{array}{rcll}\alpha & =& \frac{1}{\sqrt{1+\frac{2Mr}{{\rho }^{2}}}}& \text{(1)}\text{}\text{}\\ {\beta }^{r}& =& {\alpha }^{2}\frac{2Mr}{{\rho }^{2}}& \text{(2)}\text{}\text{}\\ {\beta }^{𝜃}& =& {\beta }^{\varphi }={\gamma }_{r𝜃}={\gamma }_{𝜃\varphi }=0& \text{(3)}\text{}\text{}\\ {\gamma }_{rr}& =& 1+\frac{2Mr}{{\rho }^{2}}& \text{(4)}\text{}\text{}\\ {\gamma }_{r\varphi }& =& -a{\gamma }_{rr}{\mathrm{sin}}^{2}\left(𝜃\right)& \text{(5)}\text{}\text{}\\ {\gamma }_{𝜃𝜃}& =& {\rho }^{2}& \text{(6)}\text{}\text{}\\ {\gamma }_{\varphi \varphi }& =& \left({r}^{2}+{a}^{2}+\frac{2Mr}{{\rho }^{2}}{a}^{2}{\mathrm{sin}}^{2}\left(𝜃\right)\right){\mathrm{sin}}^{2}\left(𝜃\right).& \text{(7)}\text{}\text{}\end{array}$

Next, we define a few useful quantities,

$\begin{array}{rcll}A& =& \left({a}^{2}\mathrm{cos}\left(2𝜃\right)+{a}^{2}+2{r}^{2}\right)& \text{(8)}\text{}\text{}\\ B& =& A+4Mr& \text{(9)}\text{}\text{}\\ D& =& \sqrt{\frac{2Mr}{{a}^{2}{\mathrm{cos}}^{2}\left(𝜃\right)+{r}^{2}}+1}.& \text{(10)}\text{}\text{}\end{array}$

Then the extrinsic curvature ${K}_{ij}=\left({\nabla }_{i}{\beta }_{j}+{\nabla }_{j}{\beta }_{i}\right)∕\left(2\alpha \right)$ (see, e.g., Eq. 13 in Ref. [1]) with ${\partial }_{t}{\gamma }_{ij}=0$, may be written in spherical polar coordinates as

$\begin{array}{rcll}{K}_{rr}& =& \frac{D\left(A+2Mr\right)}{{A}^{2}B}\left[4M\left({a}^{2}\mathrm{cos}\left(2𝜃\right)+{a}^{2}-2{r}^{2}\right)\right]& \text{(11)}\text{}\text{}\\ {K}_{r𝜃}& =& \frac{D}{AB}\left[8{a}^{2}Mr\mathrm{sin}\left(𝜃\right)\mathrm{cos}\left(𝜃\right)\right]& \text{(12)}\text{}\text{}\\ {K}_{r\varphi }& =& \frac{D}{{A}^{2}}\left[-2aM{\mathrm{sin}}^{2}\left(𝜃\right)\left({a}^{2}\mathrm{cos}\left(2𝜃\right)+{a}^{2}-2{r}^{2}\right)\right]& \text{(13)}\text{}\text{}\\ {K}_{𝜃𝜃}& =& \frac{D}{B}\left[4M{r}^{2}\right]& \text{(14)}\text{}\text{}\\ {K}_{𝜃\varphi }& =& \frac{D}{AB}\left[-8{a}^{3}Mr{\mathrm{sin}}^{3}\left(𝜃\right)\mathrm{cos}\left(𝜃\right)\right]& \text{(15)}\text{}\text{}\\ {K}_{\varphi \varphi }& =& \frac{D}{{A}^{2}B}\left[2Mr{\mathrm{sin}}^{2}\left(𝜃\right)\left({a}^{4}\left(r-M\right)\mathrm{cos}\left(4𝜃\right)+{a}^{4}\left(M+3r\right)+& \text{(16)}\text{}\text{}\\ & & \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}4{a}^{2}{r}^{2}\left(2r-M\right)+4{a}^{2}r\mathrm{cos}\left(2𝜃\right)\left({a}^{2}+r\left(M+2r\right)\right)+8{r}^{5}\right)\right].& \text{(17)}\text{}\text{}\end{array}$

All GiRaFFE curved-spacetime code validation tests adopt shifted Kerr-Schild Cartesian coordinates $\left({x}^{\prime },{y}^{\prime },{z}^{\prime }\right)$, which map $\left(0,0,0\right)$ 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 ${r}^{\prime }$ relates to the standard spherical polar radial coordinate $r$ via $r={r}^{\prime }+{r}_{0}$, where ${r}_{0}>0$ is the (constant) radial shift.

As an example, to compute ${K}_{{x}^{\prime }{y}^{\prime }}$ at some arbitrary point $\left({x}^{\prime },{y}^{\prime },{z}^{\prime }\right)$, we first convert the coordinate $\left({x}^{\prime },{y}^{\prime },{z}^{\prime }\right)$ into shifted spherical-polar coordinates via $\left({r}^{\prime }=\sqrt{{x{}^{\prime }}^{2}+{y{}^{\prime }}^{2}+{z{}^{\prime }}^{2}},{𝜃}^{\prime },{\varphi }^{\prime }\right)=\left({r}^{\prime },𝜃,\varphi \right)$, as a purely radial shift like this preserves the original angles. Next, we evaluate the components of the Kerr-Schild extrinsic curvature ${K}_{ij}$ (provided above) in standard spherical polar coordinates at $\left(r={r}^{\prime }+{r}_{0},𝜃,\varphi \right)$. Defining ${x}_{sph,sh}^{i}$ as the $i$th shifted spherical polar coordinate and ${x}_{sph}^{i}$ as the $i$th (unshifted) spherical polar coordinate, ${K}_{{x}^{\prime }{y}^{\prime }}$ is computed via the standard coordinate transformations:

 ${K}_{{x}^{\prime }{y}^{\prime }}=\frac{d{x}_{sph,sh}^{k}}{d{x}^{\prime }}\frac{d{x}_{sph,sh}^{l}}{d{y}^{\prime }}\frac{d{x}_{sph}^{i}}{d{x}_{sph,sh}^{k}}\frac{d{x}_{sph}^{i}}{d{x}_{sph,sh}^{l}}{K}_{ij}.$ (18)

However, we have $d{x}_{sph}^{i}=d{x}_{sph,sh}^{i}$, since the radial shift ${r}_{0}$ is a constant and the angles are unaffected by the radial shift. This implies that $d{x}_{sph,sh}^{k}∕d{x}^{\prime }=d{x}_{sph}^{k}∕d{x}^{\prime }$ and $d{x}_{sph}^{i}∕d{x}_{sph,sh}^{k}={\delta }_{k}^{i}$.

So after computing any spacetime quantity at a point $\left(r={r}^{\prime }+{r}_{0},𝜃,\varphi \right)$, 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



### 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:

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