ABSTRACT (original article): The gravitational perturbations of a rotating Kerr black hole are notoriously complicated, even at the linear level. In 1973, Teukolsky showed that their physical degrees of freedom are encoded in two gauge-invariant Weyl curvature scalars that obey a separable wave equation. Determining these scalars is sufficient for many purposes, such as the computation of energy fluxes. However, some applications—such as second-order perturbation theory—require the reconstruction of metric perturbations. In principle, this problem was solved long ago, but in practice, the solution has never been worked out explicitly. Here, we do so by writing down the metric perturbation (in either ingoing or outgoing radiation gauge) that corresponds to a given mode of either Weyl scalar. Our formulas make no reference to the Hertz potential (an intermediate quantity that plays no fundamental role) and involve only the radial and angular Kerr modes, but not their derivatives, which can be altogether eliminated using the Teukolsky–Starobinsky identities. We expect these analytic results to prove useful in numerical studies and for extending black hole perturbation theory beyond the linear regime. CITATION (original article): Roman Berens, Trevor Gravely, Alexandru Lupsasca (2024), Gravitational waves on Kerr black holes: I. Reconstruction of linearized metric perturbations, Class. Quantum Grav. 41 195004. https://doi.org/10.1088/1361-6382/ad6c9c
This notebook accompanies the paper “Gravitational Waves on Kerr Black Holes I: Reconstruction of Linearized Metric Perturbations” https://arxiv.org/abs/2403.20311. GitHub: https://github.com/Metric-Reconstruction/metric-reconstruction/
The first section, “Metric Perturbations”, computes the metric perturbations in ingoing and outgoing radiation gauge in Boyer-Lindquist, ingoing, and outgoing coordinates in terms of the radial and angular mode functions (r) and (θ). It then exports those expressions for later use.
IRG/ORG
h
μν
(±2)
R
ωℓm
(±2)
S
ωℓm
The second section, “Checking the Einstein Field Equations”, computes the linearized Einstein equation and checks that the metric perturbations defined in Notebook 1 are solutions.
The third section, “Examples”, imports the results of the first section and shows how to plot a tetrad or coordinate component of the metric perturbation.
The second and third sections use the SpinWeightedSpheroidalHarmonics package from the BHPToolkit, which can be installed from here: https://bhptoolkit.org/SpinWeightedSpheroidalHarmonics/.
1. Metric Perturbations
1. Metric Perturbations
1. Setup and Definitions
1. Setup and Definitions
2. Background Metric, Tetrad, and Projection to Coordinates
2. Background Metric, Tetrad, and Projection to Coordinates
Here, the background metric and the tetrad vectors are defined in Boyer-Lindquist , ingoing , and outgoing coordinates. Hereafter, any object explicitly defined in one of these coordinate systems will have a variable name ending in BL, In, or Out, respectively. Tetrad vectors with upper and lower indices are denoted by up and down, respectively. In each coordinate system, the various inner products are checked to ensure agreement with Eq. (1.5). The outer products of the tetrad vectors (with lowered indices) are also defined, and the decomposition of the metric as a sum of outer products is checked to ensure agreement with Eq. (1.6).The objects in the three coordinate systems are defined as the 4 x 4 x 4 x 4 arrays CoordProj, where the first pair of indices corresponds to the tetrad indices a and b, and the second pair corresponds to the coordinate indices μ and ν. These objects let us express the coordinate components of the metric perturbation in terms of the tetrad components in accordance with , where we define -, -, , .At the end of each subsection for each coordinate system, the objects and are defined in terms of the tetrad components as 4 x 4 matrices hIRG or hORG (with the appropriate variable suffix indicating the coordinate system). These expressions agree with Eqs. (2.47) and (2.54) in the paper for Boyer-Lindquist coordinates, Eqs. (B.4) and (B.5) in ingoing coordinates, and Eqs. (B.10) and (B.11) in outgoing coordinates.
g
μν
{l,n,m,}
m
(t,r,θ,ϕ)
(v,r,θ,ψ)
(u,r,θ,ψ)
a
e
μ
b
e
ν
h
μν
a
e
μ
b
e
ν
h
ab
1
e
μ
ϵ
g
n
μ
2
e
μ
ϵ
g
l
μ
3
e
μ
ϵ
g
m
μ
4
e
μ
ϵ
g
m
μ
IRG
h
μν
ORG
h
μν
h
ab
2a. Boyer-Lindquist Coordinates
2a. Boyer-Lindquist Coordinates
Background Metric and Tetrad Vectors
gBL=-ϵg*1-,0,0,,0,-,0,0,{0,0,-Σ,0},,0,0,--Δ***;lupBL=+,1,0,;nupBL=*{+,-Δ,0,a};mupBL=**a*Sin[θ],0,1,;mbarupBL=*-*a*Sin[θ],0,1,-;ldownBL=gBL.lupBL//Simplify;ndownBL=gBL.nupBL//Simplify;mdownBL=gBL.mupBL//Simplify;mbardownBL=gBL.mbarupBL//Simplify;
2M*r
Σ
a**(2M*r)
2
Sin[θ]
Σ
Σ
Δ
a**(2M*r)
2
Sin[θ]
Σ
2
(+)
2
r
2
a
2
a
2
Sin[θ]
Σ
2
Sin[θ]
2
r
2
a
Δ
a
Δ
1
2*Σ
2
r
2
a
1
Sqrt[2]*ζbar
Sin[θ]
1
Sqrt[2]*ζ
Sin[θ]
Checking Inner Products
In[]:=
{lupBL.gBL.lupBL,nupBL.gBL.nupBL,mupBL.gBL.mupBL,mbarupBL.gBL.mbarupBL}//Simplify{lupBL.gBL.mupBL,lupBL.gBL.mbarupBL}//Simplify{lupBL.gBL.nupBL,mupBL.gBL.mbarupBL}//Simplify
Out[]=
{0,0,0,0}
Out[]=
{0,0}
Out[]=
{-ϵg,ϵg}
Outer Products
In[]:=
lldownBL=Simplify[Outer[Times,ldownBL,ldownBL]]/.->1;lndownBL=Simplify[Outer[Times,ldownBL,ndownBL]]/.->1;lmdownBL=Simplify[Outer[Times,ldownBL,mdownBL]]/.->1;lmbardownBL=Simplify[Outer[Times,ldownBL,mbardownBL]]/.->1;nldownBL=Simplify[Outer[Times,ndownBL,ldownBL]]/.->1;nndownBL=Simplify[Outer[Times,ndownBL,ndownBL]]/.->1;nmdownBL=Simplify[Outer[Times,ndownBL,mdownBL]]/.->1;nmbardownBL=Simplify[Outer[Times,ndownBL,mbardownBL]]/.->1;mldownBL=Simplify[Outer[Times,mdownBL,ldownBL]]/.->1;mndownBL=Simplify[Outer[Times,mdownBL,ndownBL]]/.->1;mmdownBL=Simplify[Outer[Times,mdownBL,mdownBL]]/.->1;mmbardownBL=Simplify[Outer[Times,mdownBL,mbardownBL]]/.->1;mbarldownBL=Simplify[Outer[Times,mbardownBL,ldownBL]]/.->1;mbarndownBL=Simplify[Outer[Times,mbardownBL,ndownBL]]/.->1;mbarmdownBL=Simplify[Outer[Times,mbardownBL,mdownBL]]/.->1;mbarmbardownBL=Simplify[Outer[Times,mbardownBL,mbardownBL]]/.->1;
2
ϵg
2
ϵg
2
ϵg
2
ϵg
2
ϵg
2
ϵg
2
ϵg
2
ϵg
2
ϵg
2
ϵg
2
ϵg
2
ϵg
2
ϵg
2
ϵg
2
ϵg
2
ϵg
Checking Decomposition of Background Metric
In[]:=
gBL-ϵg*(-(lndownBL+nldownBL)+(mmbardownBL+mbarmdownBL))//Simplify
Out[]=
{{0,0,0,0},{0,0,0,0},{0,0,0,0},{0,0,0,0}}
Projection from Tetrad Components to Coordinate Components
In[]:=
CoordProjBL={{nndownBL,nldownBL,-nmbardownBL,-nmdownBL},{lndownBL,lldownBL,-lmbardownBL,-lmdownBL},{-mbarndownBL,-mbarldownBL,mbarmbardownBL,mbarmdownBL},{-mndownBL,-mldownBL,mmbardownBL,mmdownBL}};
IRG and ORG Coordinate Components of Metric Perturbation in terms of Tetrad Components. (Click on the output to view the matrices in full size)
In[]:=
Clear[hll,hln,hlm,hlmbar,hnn,hnm,hnmbar,hmm,hmmbar,hmbarmbar]
In[]:=
hIRGBL[t_,r_,θ_,ϕ_]:=Collect[Sum[CoordProjBL[[a1]][[b1]]*htetrad[t,r,θ,ϕ][[a1]][[b1]],{a1,1,4},{b1,1,4}]/.IRGtetrad,Flatten[htetrad[t,r,θ,ϕ]],Simplify]MatrixForm[hIRGBL[t,r,θ,ϕ]]hORGBL[t_,r_,θ_,ϕ_]:=Collect[Sum[CoordProjBL[[a1]][[b1]]*htetrad[t,r,θ,ϕ][[a1]][[b1]],{a1,1,4},{b1,1,4}]/.ORGtetrad,Flatten[htetrad[t,r,θ,ϕ]],Simplify]MatrixForm[hORGBL[t,r,θ,ϕ]]
2b. Ingoing Coordinates
2b. Ingoing Coordinates
Background Metric and Tetrad Vectors
gIn=ϵg*-1-,1,0,-a**,{1,0,0,-a*},{0,0,Σ,0},-a**,-a*,0,-Δ***;lupIn=,1,0,;nupIn=0,-,0,0;mupIn=*aSin[θ],0,1,;mbarupIn=-*aSin[θ],0,1,-;ldownIn=gIn.lupIn;ndownIn=gIn.nupIn;mdownIn=gIn.mupIn;mbardownIn=gIn.mbarupIn;
2M*r
Σ
2
Sin[θ]
(2M*r)
Σ
2
Sin[θ]
2
Sin[θ]
(2M*r)
Σ
2
Sin[θ]
2
(+)
2
r
2
a
2
a
2
Sin[θ]
Σ
2
Sin[θ]
2(+)
2
a
2
r
Δ
2a
Δ
Δ
2*Σ
1
Sqrt[2]*ζbar
Sin[θ]
1
Sqrt[2]*ζ
Sin[θ]
Checking Inner Products
In[]:=
{lupIn.gIn.lupIn,nupIn.gIn.nupIn,mupIn.gIn.mupIn,mbarupIn.gIn.mbarupIn}//Simplify{lupIn.gIn.mupIn,lupIn.gIn.mbarupIn}//Simplify{lupIn.gIn.nupIn,mupIn.gIn.mbarupIn}//Simplify
Out[]=
{0,0,0,0}
Out[]=
{0,0}
Out[]=
{-ϵg,ϵg}
Outer Products
Checking Decomposition of Background Metric
Projection from Tetrad Components to Coordinate Components
IRG and ORG Coordinate Components of Metric Perturbation in terms of Tetrad Components. (Click on the output to view the matrices in full size)
2c. Outgoing Coordinates
2c. Outgoing Coordinates
Background Metric and Tetrad Vectors
Checking Inner Products
Outer Products
Checking Decomposition of Background Metric
Projection from Tetrad Components to Coordinate Components
IRG and ORG Coordinate Components of Metric Perturbation in terms of Tetrad Components. (Click on the output to view the matrices in full size)
3. Splitting Tetrad Components into Plus and Minus Pieces
3. Splitting Tetrad Components into Plus and Minus Pieces
2. Checking the Einstein Field Equations
2. Checking the Einstein Field Equations
This is the second of three notebooks which accompany the paper “Gravitational Waves on Kerr Black Holes I: Reconstruction of Linearized Metric Perturbations” https://arxiv.org/abs/2403.20311. This notebook computes the linearized Einstein equation and checks that the metric perturbations defined in Notebook 1 are solutions. This notebook uses the SpinWeightedSpheroidalHarmonics package from the BHPToolkit, which can be installed from here: https://bhptoolkit.org/SpinWeightedSpheroidalHarmonics/.
3. Radial and Angular Mode Functions
3. Radial and Angular Mode Functions
We first define lists containing the radial and angular functions to be used in later simplifications.
We define the radial and angular Teukolsky equations.
Using the above equations of motion, we define rules to remove second radial and angular derivatives of the mode functions. We then check that these rules yield zero when applied to the equations of motion.
3. Examples
3. Examples
4. Plotting a Tetrad Component of a Metric Perturbation
4. Plotting a Tetrad Component of a Metric Perturbation
First, we import our expressions.
We define a substitution rule that replaces complex valued parameters and their complex conjugates with corresponding real and imaginary parts.
We set specific values of the remaining parameters.
5. Metric Perturbations for Single Modes of Weyl Scalars
5. Metric Perturbations for Single Modes of Weyl Scalars
We define a symbolic conjugation function.
We import the Boyer-Lindquist components of a metric perturbation.
CITE THIS NOTEBOOK
CITE THIS NOTEBOOK
Gravitational waves on Kerr black holes: I. Reconstruction of linearized metric perturbations
by Roman Berens, Trevor Gravely & Alexandru Lupsasca
Wolfram Community, STAFF PICKS, September 20, 2024
https://community.wolfram.com/groups/-/m/t/3278362
by Roman Berens, Trevor Gravely & Alexandru Lupsasca
Wolfram Community, STAFF PICKS, September 20, 2024
https://community.wolfram.com/groups/-/m/t/3278362