ORIGINAL THESIS: Swearngin, J. (2022). From Linked and Knotted Plasma, Electromagnetic, and Gravitational Radiation to Shockwaves on Rotating Black Holes. UC Santa Barbara. ProQuest ID: Swearngin_ucsb_0035D_15721. Merritt ID: ark:/13030/m5jm9mr8. Retrieved from https://escholarship.org/uc/item/2rc3n8hk
The most general asymptotically flat black hole in 3+1 dimensions has three physical attributes: mass, angular momentum, and electric charge. It is called the Kerr-Newman solution. In this notebook we will review its properties in the spin coefficient formalism, as introduced in E. Newman and R. Penrose, “An approach to gravitational radiation by a method of spin coefficients,” Journal of Mathematical Physics 3, 566 (1962) pp. 566-578 and further refined in R. Geroch, A. Held, and R. Penrose, “A space-time calculus based on pairs of null directions,” Journal of Mathematical Physics 14, 874 (1973) pp. 874-881.
​
In addition to these original papers, our references for this notebook will be the following textbooks on general relativity: S. Chandrasekhar, The Mathematical Theory of Black Holes, Oxford University Press (1983), R. Penrose and W. Rindler, Spinors and space-time, Cambridge University Press (1987), and A. Zee, Einstein Gravity in a Nutshell, Princeton University Press (2013).

Initialization

The global assumptions will be contained in the following list:
In[]:=
variables={t,r,θ,φ,M,a,Q,rPlus,rMinus,α,dt,dr,dθ,dφ,ψ[θ]};​​assumptions=Flatten@{Thread[#∈Table[Reals,Length@#]],0<2M<r,0<a<M,0<Q<M,M<rPlus<2M,0<rMinus<M}&@variables;​​$Assumptions=assumptions
Out[]=
{t∈,r∈,θ∈,φ∈,M∈,a∈,Q∈,rPlus∈,rMinus∈,α∈,dt∈,dr∈,dθ∈,dφ∈,ψ[θ]∈,0<2M<r,0<a<M,0<Q<M,M<rPlus<2M,0<rMinus<M}

Coordinates

We will define two sets of coordinates, which we will call the “asymptotic” and “horizon” coordinates, which will be denoted by x and  (“scx”) respectively:
In[]:=
x={t,r,θ,φ};={U,V,θ,χ};
The dimensionality of spacetime will be set by these definitions:
In[]:=
dim=Length@x
Out[]=
4
We will also define what Zee calls the “ur-vectors,” which serve as a coordinate basis in which to expand differential forms:
In[]:=
dx={dt,dr,dθ,dφ};d={dU,dV,dθ,dχ};
It is also useful to define a coordinate basis in which to expand vectors:
In[]:=
BasisVectors={"
∂
t
","
∂
r
","
∂
θ
","
∂
φ
"};ℬℯℴ={"
∂
U
","
∂
V
","
∂
θ
","
∂
χ
"};
Note that since the symbol ∂ is reserved for the partial derivative operator (rightly so), we decided to express the basis vectors as a list of strings instead of a list of symbols.
The signature of the metric will be set by the quantity
ζ =
+1
for(-+++)(ourpreferredsignature)
-1
for(+---)(thesignatureofPenroseandRindlerandofChandrasekhar)
.
Choose your own adventure:
In[]:=
ζ=+1;
Finally, we will find it convenient to define a list of partial derivative operators.

Derivatives in asymptotic coordinates

In[]:=
AsymptoticFirstDerivatives[function_]:={
∂
t
#,
∂
r
#,
∂
θ
#,
∂
φ
#}&@function
In[]:=
AsymptoticSecondDerivatives[function_]:=
∂
t,t
#,
∂
r,r
#,
∂
θ,θ
#,
∂
φ,φ
#,
∂
t,r
#,
∂
r,θ
#,
∂
θ,φ
#,
∂
φ,t
#,
∂
t,θ
#,
∂
r,φ
#&@function

Classical fields

It is often convenient to test out various operators on an explicit function, either just as a preliminary test or as an intermediate step towards understanding the final output of a calculation. In this notebook, we will specify two classical fields at the outset: a function of all of the asymptotic coordinates, and a function only of the angular coordinates on the horizon:
In[]:=
Shift=S[t,r,θ,φ];HorizonField=f[θ,χ];​​ShiftGHP={Shift,{-1,-1}};HorizonFieldGHP={HorizonField,{-1,-1}};
We choose the names “Shift” and “HorizonField” just to anticipate the terminology of a later notebook and provide a uniformity of notation. The GHP notation will be explained later in this notebook.

Parallelization

It is convenient to introduce new Simplify and FullSimplify commands that explicitly include the global assumptions, so that those assumptions will be passed to subkernels in parallelized computations:
In[]:=
BetterSimplify[expression_]:=Simplify[expression,Assumptionsassumptions]​​BetterFullSimplify[expression_]:=FullSimplify[expression,Assumptionsassumptions]
It is also convenient to do this in case we want to consistently use a non-standard configuration for Simplify, such as a different TimeConstraint from the default value of 300 seconds.

Series expansions

We will focus on the properties of the outer horizon. Sometimes the limit is not completely straightforward, so we will find ourselves using the following Series function fairly often:
In[]:=
HorizonSeries[function_]:=Normal@Series[function,{r,rPlus,0}]
It is also convenient to consider a series expansion in angular momentum:
In[]:=
StaticLimit[function_]:=Series[function,{a,0,1}]

Formatting output

The principal advantage of using Mathematica as opposed to any other computing language is that we can perform analytical calculations much in the same way as we do by hand. But in addition to this, another enormous benefit is that the output of a calculation can also be expressed in a manner similar to the result of a pencil-and-paper calculation.
With this in mind, we will define the following functions that convert ordinary symbols to human readable string representations.

Horizons

In[]:=
HorizonsToStrings={rPlus"
r
+
",rMinus"
r
-
"};

Directional derivatives

Newman-Penrose derivatives at the horizon:
In[]:=
FirstDerivativesNPStrings={"δ","δ'"};SecondDerivativesNPStrings={"δδ","δδ'","δ'δ'"};
Geroch-Held-Penrose derivatives at the horizon:
In[]:=
FirstDerivativesGHPStrings={"ð","ð'"};SecondDerivativesGHPStrings={"ðð","ðð'","ð'ð'"};

Classical fields

In[]:=
HorizonFieldString=ToString@Head@HorizonField
Out[]=
f
In[]:=
HorizonFieldNPFirstDerivativesString=Table[FirstDerivativesNPStrings[[]]<>#,{,1,Length@FirstDerivativesNPStrings}]&@HorizonFieldString
Out[]=
{δf,δ'f}
In[]:=
HorizonFieldNPSecondDerivativesString=Table[SecondDerivativesNPStrings[[]]<>#,{,1,Length@SecondDerivativesNPStrings}]&@HorizonFieldString
Out[]=
{δδf,δδ'f,δ'δ'f}
In[]:=
HorizonFieldGHPFirstDerivativesString=Table[FirstDerivativesGHPStrings[[]]<>#,{,1,Length@FirstDerivativesGHPStrings}]&@HorizonFieldString
Out[]=
{ðf,ð'f}
In[]:=
HorizonFieldGHPSecondDerivativesString=Table[SecondDerivativesGHPStrings[[]]<>#,{,1,Length@SecondDerivativesGHPStrings}]&@HorizonFieldString
Out[]=
{ððf,ðð'f,ð'ð'f}

Directory

Set the directory for the notebook using the following command:
(*SetDirectory["Add directory here"]*)

Null tetrad

The method of spin coefficients begins with the definition of a null tetrad. Here there is a choice as to whether we first specify the vectors that define the inverse metric, or the forms that define the metric. Our formulation of the problem focuses on the metric itself, so we will define the forms and derive the vectors by matrix inversion.
It will be useful to define the “horizon function”
In[]:=
Δ=(r-rMinus)(r-rPlus);
The actual algebraic expressions for the horizons (defined by Δ = 0) will be encoded by the following substitution rules:
In[]:=
rMinusRule=rMinusM-
1/2
(
2
M
-
2
a
-
2
Q
)
;rPlusRule=rPlusM+
1/2
(
2
M
-
2
a
-
2
Q
)
;
These are the horizons for the Kerr-Newman black hole, which has mass M, charge Q, and angular momentum J = Ma. To recover the Kerr solution, keep a ≠ 0 and set Q = 0; to recover the Reissner-Nordström solution, keep Q ≠ 0 and set a = 0. To recover the Schwarzschild solution, set a = Q = 0, in which case the inner horizon
r
-
goes to zero, and the outer horizon
r
+
becomes the Schwarzschild radius,
r
S
=
2M.

Input the forms

We will specify a collection of null 1-forms
l
μ
,
l'
μ
,
m
μ
,
m'
μ
for which the metric tensor
g
μν
≡ζ(-
l
μ
l'
ν
-
l'
μ
l
ν
+
m
μ
m'
ν
+
m'
μ
m
ν
)
describes the Kerr-Newman spacetime. Since one purpose of this notebook is to verify standard known results (see, for example, the explanation on p. 579 of Chandrasekhar), we will begin with the asymptotic coordinates (t,r,θ,φ).

Components

A collection of 1-forms aligned with the principal null geodesics of the Kerr-Newman solution is:
In[]:=
lDown=ζ-1,+
2
r
+
2
a
2
(Cos@θ)
Δ
,0,a
2
(Sin@θ)
;​​lpDown=ζ
Δ
2(
2
r
+
2
a
2
(Cos@θ)
)
-1,-
2
r
+
2
a
2
(Cos@θ)
Δ
,0,a
2
(Sin@θ)
;​​mDown=ζ
Exp[ψ[θ]]
(r+aCos@θ)Sqrt@2
{-aSin@θ,0,(
2
r
+
2
a
2
(Cos@θ)
),(
2
r
+
2
a
)Sin@θ};​​mpDown=ζ
Exp[-ψ[θ]]
(r-aCos@θ)Sqrt@2
{+aSin@θ,0,(
2
r
+
2
a
2
(Cos@θ)
),-(
2
r
+
2
a
)Sin@θ};
Our conventions with respect to changing the metric signature [by which we mean (+ - - -) ↔ (- + + +), not changing from Lorentzian signature to Euclidean or split signature] is that the 1-forms that define the metric change signs, but the vectors that define the inverse metric do not.
It is convenient to collect the 1-forms into a matrix:
In[]:=
e=Simplify/@Transpose[{lDown,lpDown,mDown,mpDown}];
It is very important to note that this corresponds to the following index structure:
e
μ
=(
l
μ
,
l'
μ
,
m
μ
,
m'
μ
)
where μ labels the base space and  (“sca”) labels the tangent space. This is actually the most convenient matrix to work with when calculating the spin coefficients.

Forms

Strictly speaking, much as we often speak of “vectors” when we really mean “components of vectors,” the above are only the components of the differential forms expanded in the basis of asymptotic coordinates. Because we will eventually consider another coordinate system, it is useful to define the actual 1-forms as follows:
In[]:=
{lForm,lpForm,mForm,mpForm}=Sum[#[[μ]]dx[[μ]],{μ,1,dim}]&/@{lDown,lpDown,mDown,mpDown}
Out[]=
-dt+
dr(
2
r
+
2
a
2
Cos[θ]
)
(r-rMinus)(r-rPlus)
+adφ
2
Sin[θ]
,-
dr
2
-
dt(r-rMinus)(r-rPlus)
2(
2
r
+
2
a
2
Cos[θ]
)
+
adφ(r-rMinus)(r-rPlus)
2
Sin[θ]
2(
2
r
+
2
a
2
Cos[θ]
)
,
dθ
ψ[θ]

(
2
r
+
2
a
2
Cos[θ]
)
2
(r+aCos[θ])
-
adt
ψ[θ]

Sin[θ]
2
(r+aCos[θ])
+
dφ
ψ[θ]

(
2
a
+
2
r
)Sin[θ]
2
(r+aCos[θ])
,
dθ
-ψ[θ]

(
2
r
+
2
a
2
Cos[θ]
)
2
(r-aCos[θ])
+
adt
-ψ[θ]

Sin[θ]
2
(r-aCos[θ])
-
dφ
-ψ[θ]

(
2
a
+
2
r
)Sin[θ]
2
(r-aCos[θ])


Metric

Derive the null vectors

Components

It is this quantity ℯ whose rows are the standard null vectors:
Therefore, the vectors of the NP tetrad in this basis are:

Vectors

Inverse metric

Directional derivatives

The Newman-Penrose directional derivatives are the ordinary covariant derivatives contracted with each of the vectors of the null tetrad:
Since all quantities in the spin coefficient formalism are spacetime scalars, the covariant derivatives can be replaced by ordinary partial derivatives. This allows us to define the NP derivatives as the following operators:

Spin coefficients

We will follow Chandrasekhar’s algorithm for calculating the spin coefficients, and then we will name them according to the priming-covariant notation in Penrose and Rindler.

Chandrasekhar’s algorithm

As a first step, we define the λ-symbols [Eq. (266) on p. 38]:
Note the ordinary letter e versus the script letter ℯ, consistent with our notation in the previous section. For convenience, define the 3-index object
Now define the λ-symbols:
We now define the matrix of spin coefficients

Give the spin coefficients their traditional names

Now let us give these components their traditional names:
To avoid any potential simplification issues later on, we should explicitly define the complex conjugates of the spin coefficients:
Display them in a human-readable format:

Save the results

The GHP formalism

A quantity that does not transform according to this rule is said to be nonweighted. (A quantity that does not transform at all is considered to be weighted, just with weights h = k = 0.) For example, a nonweighted quantity might have a transformation law that involves derivatives of λ(x).

Directional derivatives

It is in this sense in which the non-weighted spin coefficient ϵ is a gauge field, and the modified derivative þ is a covariant derivative with respect to the GHP gauge group. The new feature that is not shared by the comparison to covariant differentiation for internal local symmetry groups is that here the derivatives themselves carry charges, as we can see from the above transformation law for þ.
Concretely, we will define the four GHP-covariant modified directional derivatives as:
We will not make any effort to anticipate and correct for possible user errors in formatting the input. Just do it correctly and it will work.

GHP weights

We will compute the curvature scalars using the GHP formalism. To do this, we will first define a “GHP version” of each spin coefficient ω with weights (h,k) as a list of the form
We will hard code (rather than compute) the weights of the spin coefficients [compare with Eq. (4.12.13) in Penrose & Rindler, where, compared to the notation of their Eq. (4.12.18), we prefer to define h = p/2 and k = q/2].
If a spin coefficient ω has weights (h, k), then its primed version ω’ has weights (-h, -k):
Complex conjugates:

Define the addition and multiplication of weighted quantities

We need to define versions of addition and multiplication that know how to properly deal with weighted quantities.
​
In the GHP formalism, only quantities with the same weights can be added together. Moreover, if two quantities do have the same weights, their sum should also retain that weight. So a GHP-covariant addition function must check the weights and then add only the actual quantity itself without changing the weight. We will use the “doublestruck” font (“dsP” etc) to denote the new Plus function:
As for multiplication, any two quantities can be multiplied, and this product must have a weight that is the sum of the weights of the quantities being multiplied. We will denote the new Times function by the doublestruck x:
As a side remark, we note that we will often use the symbols  (“wolf”) and  (“math”) as dummy indices, in order to avoid any risk of unintentionally summing over a variable that is not supposed to be an index.
​
Purely for later convenience, we will also define GHP versions of some numbers:
The syntax for using our GHP functions is a bit cumbersome, but we have constructed it to mimic the FullForm syntax for Plus and Times:
So, for example,

Curvature scalars

We will now compute the curvature scalars, beginning with those of GHP weights (0,0).

Spacelike and timelike curvatures of 2-surfaces

The curvature scalars of weight zero are defined by the commutators of GHP derivatives. We begin by defining the complex generalization of the Gaussian curvature and its timelike analogue.

Complex generalization of Gaussian curvature

Define a test function ξ with GHP weights (h,0):
The complex generalization of the Gaussian curvature, for which we use the symbol  (“scK”), is defined by Eq. (4.14.20) in Spinors and Spacetime:
Note that all partial derivatives of the test function ξ as well as the arbitrary h canceled out of the final expression for . Both of these things are required for the formalism to be consistent. Note also that the final weight of ξ is the same as that of ξ itself, which shows that  is indeed a weight (0,0) operator.
Now that the test function has served its purpose, we can get rid of it:
Save it:

Timelike curvature

Penrose and Rindler mention in Sec. 4.14 of Spinors and Spacetime that “one may also apply the compacted spin-coefficient formalism equally to the study of timelike 2-surfaces.” They go on to say, however, that they “shall not give a detailed discussion of the timelike case, since most useful applications seem to involve only spacelike 2-surfaces.” It turns out that shockwaves on the surface of rotating black holes provide exactly a situation in which the timelike case becomes just as important as the spacelike case. We begin with Penrose and Rindler’s observation in the same paragraph that “it is straightforward to obtain the relevant local formulae for the timelike case from the corresponding spacelike ones, since the Sachs *-operation (cf. (4.12.47)-(4.12.55)) can be applied [...].”
The Sachs transformation is defined as follows [see, for our purposes, Eq. (4.12.49) in Spinors and Spacetime]:
Once again, the derivatives of ξ and the arbitrary h have canceled out. Now that the test function has served its purpose, get rid of it:
Save it:

An intermediate operator

Define its primed version:
This operator is actually self-prime:

Curvature scalars of weight zero

Weyl scalars of nonzero weight

Ricci scalars of nonzero weight

Simplify the curvature scalars

Now that we have defined all of the curvature scalars [5 complex Ψs (10 real numbers), a 3x3 Hermitian matrix Φ (9 real numbers), and Π (1 real number)], we can simplify them.
The curvature scalars for the Kerr-Newman solution are:
Save them:

Petrov classification

With the explicit values of the horizons:

CITE THIS NOTEBOOK

The Kerr-Newman black hole​
by Yoni BenTov and Joe Swearngin​
Wolfram Community, STAFF PICKS, September 26, 2023
​https://community.wolfram.com/groups/-/m/t/3021977