Rotating black holes: corrections to the Kerr metric package​
​by Pablo A. Cano
The Kerr metric, representing the geometry of a rotating black hole, is the most important solution of General Relativity. However, it is in general not a solution of modified theories of gravity and obtaining the corresponding rotating black hole metrics is a challenging problem. The package CorrectionsKerr.mx allows one to obtain the leading perturbative corrections to the Kerr metric due to the presence of higher-derivative terms in the gravitational action, and to study some properties of these modified black holes. The solution and its properties are obtained analytically as a power series in the spin parameter χ=a/M.The only external input needed is the effective energy-momentum tensor of the corrections, such that
G
μν
=
T
μν
, evaluated on the Kerr metric. It should have the indices down and expressed in Boyer-Lindquist coordinates {t,r,x=Cos[θ],ϕ} and in terms of M and χ. The energy-momentum tensor of the most general cubic- and quartic-curvature Lagrangians are included in the package.
Please download the package file attached below and make sure it is in the same folder with this notebook.
SetDirectory[NotebookDirectory[]];
We load the package CorrectionsKerr.mx which contains a number of subroutines
In[]:=
<<CorrectionsKerr.mx
The package includes the following objects (plus auxiliary ones), all of which are documented: SolveH, Tcubic1, Tcubic2, Tquartic1, Tquartic2, Tquartic3, Omega, AreaH, SurfaceGravity, PhotonRing, ExpandLaplacian, Potential, FindW
In[]:=
?SolveH
Out[]=
We include the tensors Tcubic1, Tcubic2, Tquartic1, Tquartic2, Tquartic3 associated to cubic and quartic interactions
In[]:=
?Tquartic1
Out[]=
In[]:=
?Tquartic2
Out[]=
Example: parity-even quartic corrections.
In[]:=
T=ξ1Tquartic1+ξ2Tquartic2;
It is convenient to define the array “parameters”, containing the couplings on which the energy-momentum tensor depends. This is used by several of the functions in the package and helps them to simplify the computations.
In[]:=
parameters={ξ1,ξ2};
We solve for the corrected Kerr metric expanded to order χ^chimax
In[]:=
chimax=8;​​AbsoluteTiming[SolveH[T,chimax,2]]
Out[]=
{52.6423,Null}
We check the equations are satisfied
In[]:=
Collect[Series[Eq,{χ,0,chimax}],parameters,Simplify]
Out[]=
{{
9
O[χ]
,0,0,
9
O[χ]
},{0,
9
O[χ]
,
9
O[χ]
,0},{0,
9
O[χ]
,
9
O[χ]
,0},{
9
O[χ]
,0,0,
9
O[χ]
}}
We can compute a few properties of the solution
◼
  • Angular velocity of the horizon expanded at order χ^chimax
    • In[]:=
    Omega[chimax]
    Out[]=
    1
    176
    (44+35ξ1-108ξ2)χ+
    (2860-35623ξ1-7452ξ2)
    3
    χ
    45760
    +
    1
    32
    -
    3(13924ξ1+8061ξ2)
    45760
    5
    χ
    +
    (48620-2630965ξ1-2020784ξ2)
    7
    χ
    2489344
    ◼
  • Area of the horizon
    • In[]:=
    AreaH[chimax]
    Out[]=
    16π-20πξ1+-4π+
    53πξ1
    11
    -
    288πξ2
    11
    2
    χ
    +-π-
    16047πξ1
    2860
    -
    2025πξ2
    143
    4
    χ
    +-
    π
    2
    -
    4907πξ1
    715
    -
    28923πξ2
    2860
    6
    χ
    +-
    5π
    16
    -
    2373531πξ1
    388960
    -
    370533πξ2
    48620
    8
    χ
    ◼
  • Surface gravity of the horizon
    • In[]:=
    SurfaceGravity[chimax]
    Out[]=
    2+ξ1
    8
    +
    1
    176
    (-11-24ξ1+108ξ2)
    2
    χ
    +
    (-715+12083ξ1+8937ξ2)
    4
    χ
    22880
    +
    (-325+13303ξ1+9312ξ2)
    6
    χ
    16640
    +
    (-340340+24673533ξ1+19055492ξ2)
    8
    χ
    24893440
    ◼
  • Several properties of the equatorial photon rings
    • In[]:=
    ?PhotonRing
    Out[]=
    Symbol
    PhotonRing[chimax,(optional)sign] returns an array with the frequency, Lyapunov exponent and coordinate radius of equatorial photon rings expanded at order χ^chimax. (Optional) sign: +1 for counterclockwise orbits and -1 for clockwise orbits (default=1).
    This can take some time, so best try first with a low “chimax”
    In[]:=
    PhotonRing[2,1]
    Out[]=
    
    1
    3
    3
    +
    832ξ1
    59049
    3
    +
    2
    27
    +
    128(1249ξ1-324ξ2)
    5845851
    χ+
    11
    162
    3
    +
    -5606779
    3
    ξ1+4236354
    3
    ξ2
    223205220
    2
    χ
    ,
    1
    3
    3
    -
    7232ξ1
    59049
    3
    -
    25613
    3
    ξ1-8
    3
    ξ2χ
    19683
    3
    +-
    2
    81
    3
    +
    -77953591
    3
    ξ1+170805546
    3
    ξ2
    1227628710
    2
    χ
    ,3+
    20679808ξ1
    26582985
    +-
    2
    3
    -
    5121408463
    3
    ξ1-328185
    3
    ξ2
    717740595
    χ+
    2(-159099165225+2091379285051ξ1-920921539266ξ2)
    2
    χ
    1431892487025
    

    Effective potentials for scalar perturbations

    In order to compute the effective potentials for scalar perturbations, we first have to perform an expansion of the scalar Laplacian by using this command. The functions
    H
    i
    (r,x)
    should have been computed previously.
    In[]:=
    ?ExpandLaplacian
    Out[]=
    In[]:=
    ​​chimax=8;​​ExpandLaplacian[chimax]
    Then we use the following function to obtain the potentials
    In[]:=
    ?Potential
    Out[]=
    Symbol
    Potential[l,m,chimax,cmax] computes the effective potential for scalar perturbations SubscriptBox[\(V\), \(l, m\)]​ expanded at order χ^chimax and c^cmax, where c=χω. Returns the potential stored in the variable 'V' and a rule 'ksol={​{k->value}​}' that is used in the definition of the tortoise coordinate
    In[]:=
    l=2;​​m=2;​​cmax=8;​​AbsoluteTiming[Potential[l,m,chimax,cmax]]
    Out[]=
    {579.616,Null}
    The result of the evaluation is the potential “V” (long expression) and the rule “ksol”
    In[]:=
    V
    In[]:=
    ksol
    Out[]=
    k
    1
    64179356841600
    (-43254404874240ξ1-69651334663200ξ1
    2
    χ
    +38642732510400ξ2
    2
    χ
    +35132490565680ξ1
    4
    χ
    -4489515265728ξ2
    4
    χ
    +97577760435260ξ1
    6
    χ
    +27375495642600ξ2
    6
    χ
    +157681012855135ξ1
    8
    χ
    +80341821380590ξ2
    8
    χ
    )

    Quasinormal mode frequencies

    We also provide a method to compute the fundamental quasinormal mode frequencies of the scalar effective potentials defined above.
    In[]:=
    ?FindW
    Out[]=
    We must first define the potential as a function of the radius and the frequency and assign the rule “ksol” to a variable
    In[]:=
    Clear[χ,μ,ξ1,ξ2,ξ3]​​Vlm[p_,u_]:=V/.{rp,ωu,cχu};​​k0=k/.ksol[[1]];
    Before we execute FindW we have to specify the numeric values for all the variables.
    In[]:=
    M=1;​​χ=0.4;​​wini=0.1+0.2l-0.1I;(*Agoodinitialguessforthefrequencyω_lm*)​​
    First we compute the QNM frequency for the Kerr black hole
    In[]:=
    ξ1=0;​​ξ2=0;​​AbsoluteTiming[w0=ww/.FindW[Vlm,k0,wini]]​​Clear[ξ1,ξ2]
    The dependence of ω on ξ1, ξ2 is linear when these are small, so we only need to evaluate the QNMs for two different values of (ξ1, ξ2)
    Linear dependence can be checked by considering additional (small) values of (ξ1, ξ2)