Black holes of type D revisited: relating their various metric forms
Hryhorii Ovcharenko Jiri Podolsky Marco Astorino






LIFT–7-2.24













gregor_ovcharenko@outlook.com, hryhorii.ovcharenko@matfyz.cuni.cz
Charles University, Faculty of Mathematics and Physics,
Institute of Theoretical Physics,
V Holešovičkách 2, 18000 Prague 8, Czechia


jiri.podolsky@mff.cuni.cz  
Charles University, Faculty of Mathematics and Physics,
Institute of Theoretical Physics,
V Holešovičkách 2, 18000 Prague 8, Czechia


marco.astorino@gmail.com
Laboratorio Italiano di Fisica Teoretica (LIFT), 

Via Archimede 20, I-20129 Milano, Italy



§ ABSTRACT 
We investigate a complete family of spacetimes which represent black holes with rotation, NUT twist, acceleration, electric and magnetic charges. These are exact solutions of the Einstein-Maxwell equations with any cosmological constant, such that the (non-null) electromagnetic field is aligned with both the double-degenerate principal null directions of the Weyl tensor. In particular, we explicitly relate various coordinates and the corresponding physical parameters of such solutions, namely the original Plebański-Demiański (PD) form, the convenient Astorino (A) form which was found recently, and formally improved here (A^+), the Griffiths-Podolský (GP), and Podolský-Vrátný (PV) form of the metric. It is demonstrated that, if properly mapped and physically interpreted, all these representations cover the complete class of type D black holes. Using the new A-parameters, the two main PD quartic metric functions are factorized into the product of quadratic expressions, enabling thus an explicit analysis. Moreover, we clarify the role of the twist parameter ω, related to both the Kerr-like rotation and the NUT parameters a and l, respectively. Special attention is payed to the elusive subclass of accelerating NUT black holes with a=0.




04.20.Jb, 04.40.Nr, ...








Black holes of type D revisited:
 relating their various metric forms

    Marco Astorino
    September 9, 2024
======================================================================














§ INTRODUCTION





The aim of this article is to elucidate mutual relations between various metric representations of a large family of solutions in the Einstein-Maxwell theory (with or without the cosmological constant) which belong to the type D of the Petrov-Penrose classification. In particular, we focus on stationary and axisymmetric solutions (i.e., with a couple of commuting Killing vectors ∂_t and ∂_φ) such that two expanding repeated principal null directions of the Weyl tensor are both aligned with the two principal null directions of the electromagnetic field.



Such a family of spacetimes is relevant because it describes the most renowned black holes in general relativity, starting from the static and spherically symmetric metrics, such as the Schwarzschild line element, the C-metric describing accelerating black holes, the Newman-Unti-Tamburino twisting spacetime, to a general stationary rotating, accelerating, and charged Kerr-Newman solution <cit.>. This class is often identified with the Plebański-Demiański family <cit.> (but see also the earlier work of Debever <cit.>), subsequently investigated in detail in <cit.>.[Actually, this is the subclass of all type D spacetimes for the theory under consideration. Other solutions, including non-expanding cases, were studied in a number of works. See, e.g., the review <cit.>.]


Recently a large class of such type D solutions was systematically investigated by means of the solution generating technique, and a nice new metric form was thus obtained <cit.>. This novel spacetime representation has the advantage to directly contain the limits to all the subcases of type D black holes contained in the general Plebański-Demiański solution, including also the peculiar accelerating solutions with (just) the NUT parameter, which  was previously considered to exist only outside the type D class <cit.>, <cit.>. It came as a surprise because the only accelerating black holes with NUT parameter known before <cit.>, namely the Chng-Mann-Stelea metric <cit.> investigated in <cit.>, were of a general algebraic type I (see <cit.> for the rotating and charged generalization).










This discovery of a novel general form of type D metric, which comprises accelerating black holes with NUT parameter, naturally opens the way to questions about the actual generality of the Plebański-Demiański metric and its different parameterizations, namely:


  * Is the Plebański-Demiański solution the most general black hole spacetime of type D, or is the metric presented in <cit.> its extension?

  * Might the metric of <cit.> be just another equivalent reparametrization of the Plebański-Demiański metric, but more suitable for description of all type D black hole specializations?

  * What is the relation between the new spacetime of <cit.> and the type D metrics known so far in the literature, such as those in <cit.>?

  * Why the accelerating NUT black holes have not been explicitly identified in previous works <cit.>?

  * Which is the more appropriate/convenient parametrization to describe the physical and geometrical properties of the whole class of accelerating Kerr-Newman-NUT black holes?






In our paper we address these open questions. In Section <ref> we start by revisiting the solution of <cit.>, denoted here as A, putting it into a simpler metric form which we will denote A^+. In subsequent Section <ref> the transformation from the Astorino metric to the original Plebański-Demiański coordinates, together with explicit relation of the physical A and A^+ parameters to the PD integration constants, is presented (full details can be found in Appendix <ref>). This leads to a factorized form of the PD metric functions, and their simplification for various special cases. Transformation to the Griffiths-Podolský form of this family of black-hole spacetimes is presented in Section <ref>, and their special cases are discussed in Section <ref>, after elucidating the role of the twist parameter ω and clarifying the physical dimensionality of the parameters. Relation to the Podolský-Vratný metric representation is contained in Section <ref>. Section <ref> summarizes and compares the key special cases in A, PD, GP, and PV metric forms, followed by concluding remarks in Section <ref>.




For convenience of the reader, we summarize our nomenclature and conventions for A, PD, GP, and PV coordinates and parameters in Table <ref>, together with references to original articles. These metrics admit any value of the cosmological constant Λ, but in this paper we only investigate black hole spacetimes with Λ=0.













Mutual relations between all these metric forms are shown in the scheme on Figure <ref>. Particular connections are presented in full detail in Sections which are indicated by the corresponding arrows between them. Namely, the double-arrows show equivalence proven already in previous works, the single-arrows are the equivalences proven in this paper.
















§ THE ASTORINO MOST GENERAL METRIC FORM



In 2024, Astorino <cit.> presented a very convenient explicit metric for the most general type D black hole (with a cosmological constant Λ and doubly-aligned electromagnetic field) in the form[We have relabeled the parameters, namely α to -α, l to -l, and p to g.]

    s^2=-f(r,x) [ t-ω(r,x) φ]^2+1/f(r,x)[
    e^2γ(r,x)(   r^2/Δ_r(r)+ x^2/Δ_x(x))
    + ϱ^2(r,x) φ^2],

where the metric functions are

    f=    [1+α^2(l^2-a^2)x^2]^2 Δ_r
    -[a+2α l r+α^2a r^2]^2 Δ_x/(1-α r x)^2ρ^2, 
    ω=    (a+2l x+a x^2)[1+α^2(l^2-a^2)x^2] Δ_r
    +(r^2+l^2-a^2)(a+2α l r+α^2a r^2) Δ_x/[1+α^2(l^2-a^2)x^2]^2 Δ_r-[a+2α l r+α^2a r^2
    ]^2 Δ_x,
    
    e^2γ=    C_f [1+α^2(l^2-a^2)x^2]^2 Δ_r-(a+2α l  r+α^2 a  r^2)^2 Δ_x(1-α r x)^4,
    ϱ^2=    Δ_r Δ_x(1-α r x)^4,
    Δ_r=    (1-α^2r^2)[(r-m)^2 - (m^2+l^2-a^2-e^2-g^2)] 
        - Λ/3( 3l^2/1+α^2 a^2 r^2
       - 4α a l/1+α^2 a^2 r^3 + r^4 ) ,
    Δ_x=    (1-x^2)[(1-α m  x)^2-α^2x^2(m^2+l^2-a^2-e^2-g^2)] 
        - Λ/3( 3l^2/1+α^2 a^2 x^2
        +4al/1+α^2 a^2 x^3 + a^2+α^2(a^2-l^2)^2/1+α^2 a^2 x^4 ) ,

and ρ^2 in the numerator of (<ref>) is defined as

    ρ^2= (l+a x)^2 + 2α l (a+2l x+a x^2) r + α^2(a^2-l^2)^2 x^2 + [1+α^2(a+l x)^2 ] r^2 .



It was argued in <cit.> that m is the mass parameter, a denotes rotation, l in the NUT parameter, α is acceleration, e and g are electric and magnetic charges,[More precisely, they are directly related to physical conserved charges only in some special cases, not in the general case.] and C_f is an additional normalization constant (which could be related to conicity).[Here, without loss of generality, we have set the auxiliary constant parameters ϵ, κ, ξ, χ which appear in Eq. (2.7) in <cit.> to zero. See also the Wolfram Mathematica Notebook in Supplementary Material.] Here  we naturally assume that m, a, α are positive (or zero), while l, e, g, Λ can take any value.

We should also clarify that these quantities have usual physical dimensions, namely m, a, l, e, g have the dimension of length, α and √(Λ) are inverse length, and C_f is dimensionless. The coordinates t, r have dimension of length while x, φ are dimensionless. Consequently, Δ_x, f are dimensionless, Δ_r, ϱ^2, ρ^2, e^2γ  have the dimension of length squared, and ω(r,x) has the dimension of length.




The electromagnetic field is given by the vector potential A(r,x)=A_t  t + A_φ φ with

    A_t =   √(1+α^2(a^2-l^2)/1+α^2a^2) 1/ρ^2[ [ g (r - α a l +  α^2 a^2 r) + e l  ] r  
        + l (g+α a e) (a-2 α l  r+α^2a r^2) x  
        +(a - α l r)[ ag + α l(el-gr) + α^2a [g(a^2-l^2)-el r] ] x^2]
       + g/l ,
    
     A_φ =    -√(1+α^2(a^2-l^2)/1+α^2a^2) 1/ρ^2 x/l[(1+α^2a^2)[ ae(x-α r) r - gal x + g(r+α a^2x) r] 
        - α l^3 (e-α ag)x  + l^2[α^2a(e-α a g) xr - (g+α a e)] ] - (a+ω_c)  A_t ,

where ω_c is a constant (for the choice ω_c=-a the last term vanishes).
The apparently problematic terms l^-1 are included to remove the divergencies in the limit l → 0.

Now, to simplify the metric (<ref>)–(<ref>), it is useful to define the following functions

    A(x)    := 1 + α^2(l^2-a^2) x^2,
    
        B(x)    := a + 2l x + a x^2,
    
        C(r)    := a + 2α l r + α^2 a  r^2,
    
        D(r)    := (l^2-a^2) + r^2,

and

    Ω(r,x)    := 1-α  r  x .

Interestingly, their specific combination,

    ρ^2 =  AD+BC,

gives exactly the complicated expression introduced in (<ref>). Notice also that (<ref>)–(<ref>) are just quadratic polynomials, and (<ref>) also includes up to the second power, both in x and r.

With these shorthands, we can rewrite the metric functions f, ω and γ as

    f=    A^2Δ_r-C^2Δ_xΩ^2ρ^2,
    ω=     AB Δ_r + CD Δ_xA^2 Δ_r-C^2 Δ_x,
    
        e^2γ=    C_f A^2 Δ_r - C^2 Δ_xΩ^4,
    ϱ^2=    Δ_r Δ_xΩ^4,

and after substituting (<ref>)–(<ref>) into (<ref>) we get the metric

    s^2 =    1Ω^2[-A^2Δ_r-C^2Δ_xρ^2( t-AB Δ_r + CD Δ_xA^2Δ_r-C^2Δ_x φ)^2
        
            +ρ^2Δ_r Δ_xA^2Δ_r-C^2Δ_x φ^2
        + C_f ρ^2 (  r^2Δ_r +  x^2Δ_x )].




Next, we simplify the parts depending on t and φ, namely

    -A^2Δ_r-C^2Δ_xρ^2  t^2
     + 2 AB Δ_r +CD Δ_xρ^2  t  φ
       -(AB Δ_r + CD Δ_x)^2ρ^2(A^2Δ_r-C^2Δ_x) φ^2
    +ρ^2Δ_r Δ_xA^2Δ_r-C^2Δ_x φ^2.

Using the definition (<ref>) the last two terms combine to

    D^2Δ_x - B^2Δ_rρ^2 φ^2,

so that the whole complicated expression (<ref>) “miraculously” simplifies to

    -Δ_rρ^2(A  t-B φ)^2+Δ_xρ^2(C  t+D φ)^2.


This allows us to finally write the Astorino complete metric in a very compact and explicit form as

    s^2=1Ω^2[-Δ_rρ^2(A  t - B φ)^2
        + Δ_xρ^2(C  t + D φ)^2
        + C_f ρ^2 (  r^2Δ_r +  x^2Δ_x )]
        .

It may naturally be given a nickname “A^+ metric”.
Recall that ρ^2 is the polynomial expression (<ref>), the quadratic functions A, B, C, D, Ω have the form (<ref>)–(<ref>), and the two quartics Δ_r, Δ_x  are given by (<ref>), (<ref>).

The Astorino (A) solution (<ref>) rewritten in the new compact (A^+) form of the metric  (<ref>) resembles the Griffiths-Podolský (GP) representation <cit.> of the family of type D black holes in the Plebański-Demiański (PD) class of electrovacuum solutions with Λ. An exact relation between these metric forms, and the physical parameters of the black holes, will be derived and investigated later in Sections <ref>–<ref>.



However, first we will present an explicit and direct transformation from the Astorino metric (<ref>), that is (<ref>), to the original Plebański-Demiański form of the metric. This will demonstrate, that both the Astorino class of solutions and the Plebański-Demiański class of solutions are equivalent, and that represent all solutions of a given type, including the elusive accelerating (purely) NUT black holes of algebraic type D.









§ TRANSFORMATION TO THE PLEBAŃSKI-DEMIAŃSKI METRIC FORM



A general Plebański-Demiański metric representing all solutions of Einstein-Maxwell-Λ equations of algebraic type D (with double aligned, non-null electromagnetic field) — which includes black holes of this type — is originally given by Eq. (3.30) in the seminal paper <cit.>.
It is also repeated (with a slight modification of the symbols used) in Chapter 16 of <cit.> as Eqs. (16.1) and (16.2), namely







































  
    [ s^2=1/(1-p' r')^2[
     -Q' (τ'-p'^ 2 σ')^2/r'^ 2+p'^ 2
     +P' (τ'+r'^ 2 σ')^2/r'^ 2+p'^ 2;                              12pc  +r'^ 2+p'^ 2/P'  p'^ 2
     +r'^ 2+p'^ 2/Q'  r'^ 2] . ]

This contains two quartic functions
  
    [ P'(p') = k' +2 n'p' - ϵ'p'^ 2 +2 m'p'^ 3-(k'+e'^2+g'^2+Λ/3) p'^ 4 ,;   Q'(r') =(k'+e'^2+g'^2) -2m'r' +ϵ'r'^ 2 -2n'r'^ 3-(k'+Λ/3) r'^ 4 , ]

with 7 arbitrary real parameters Λ, e', g', m', n', ϵ', k' (the parameter γ of <cit.> is obtained by putting k'=γ-g'^2-Λ/6). Here Λ is the cosmological constant, while e' and g' represent electric and magnetic charges, as the vector potential reads

    A = - e' +  i g'/r' +  i p' ( τ'  -   i p' r' σ'  ).



In <cit.> a convenient rescaling of the original PD metric (<ref>), (<ref>) was performed, namely
p' ↦√(αω) p', r' ↦√(α/ω) r', σ' ↦√(ω/α^3) σ', τ' ↦ =√(ω/α) τ',
with the relabelling of constants
m'+ n' ↦ (α'/ω)^3/2(m'+  n'),
e'+ g' ↦ (α'/ω)(e'+  g'),
ϵ' ↦ (α'/ω) ϵ',
k' ↦α'^2k.
This introduced two important kinematic parameters α and ω, later interpreted as the acceleration and the twist of the black hole, respectively. Such a rescaled metric, we will denote as PD_αω, reads
  
    -3pc s^2=1/(1-α'  p' r')^2[
     - Q/r'^ 2+ω^2p'^ 2 (τ'-ω p'^ 2σ')^2
     + P/r'^ 2+ω^2p'^ 2 (ωτ'+r'^ 2σ')^2 
          12pc
      +r'^ 2+ω^2p'^ 2/ Q  r'^ 2
      +r'^ 2+ω^2p'^ 2/ P  p'^ 2],

  where the key functions are
 
    [ P(p') = k' +2ω^-1n'p' -ϵ'p'^ 2 +2α' m'p'^ 3
    -[α'^ 2(ω^2 k'+e'^ 2+g'^ 2)+ω^2Λ/3] p'^ 4 ,;    Q(r') = (ω^2 k'+e'^ 2+g'^ 2) -2m'r' +ϵ' r'^ 2 -2α'ω^-1n'r'^ 3
    -(α'^ 2k'+Λ/3) r'^ 4 , ]

see Eqs. (16.5) and (16.6) in <cit.>, with the vector potential

    A = - e' +  i g' /r' +  i ω p' ( τ'  -   i p'r' σ'  ).




Here we considered the Plebański-Demiański metric, denoted as PD_α,

    s^2=1(1-α' r' x')^ 2[
           - Q'r'^ 2+x'^ 2 (τ'  -  x'^ 2 ϕ' )^2
         + P'r'^ 2+x'^ 2 (τ'  +  r'^ 2 ϕ' )^2 
       + C^2 (r'^ 2+x'^ 2) ( r'^ 2Q' + x'^ 2P' )],

with the metric functions
 
    [      P'(x') = k'+2n' x' - ϵ'x'^ 2 +2α'm' x'^ 3
    - [α'^ 2 (k'+e'^2 + g'^2)+Λ'/3] x'^ 4 ,; Q'(r') = (k'+e'^2 + g'^2)-2m' r' + ϵ'r'^ 2 - 2α' n'  r'^ 3 - (α'^ 2 k' + Λ'/3) r'^ 4 . ]

Changing x' to p', ϕ' to σ, and unprimed, these are exactly the expressions (<ref>) for the twist parameter ω=1. They are equivalent to the metric functions (16.6) in <cit.>.



From now on, we will only consider the case Λ'=0. Generalization to any value of the cosmological constant will be presented in our subsequent paper elsewhere.




A direct transformation of coordinates between the original Astorino metric (<ref>)–(<ref>), equivalent to the A^+ metric (<ref>), and the Plebański-Demiański metric (<ref>), (<ref>) is

    t    = aα(a^2-l^2) √(K-1√(I^2 ∓ J^2)) [[K-α^2(a^2-l^2)] τ' + K-1/α^2(a^2-l^2) ϕ' ],
    φ   = 1α(a^2-l^2) √(K-1√(I^2 ∓ J^2)) [ϕ'-α^2(a^2-l^2) τ' ],
    
    x            = aK x'-l/aK-lα^2(a^2-l^2) x' , 
    α r    = aK(a^2-l^2) α r'-al(K-1)/(a^2K-l^2) - l(a^2-l^2) α r' ,

in which the convenient (auxiliary) dimensionless constants I, J, and K are defined as

    I :=     1+α^2(a^2-l^2)  , 
    
        J :=     2 α la √(|a^2-l^2|) ,
    
        2K :=     I + √(I^2 ∓ J^2) .

Here the upper sign is used when a^2>l^2, while the lower sign is used in the complementary case a^2<l^2 (and a=0 in particular). Therefore,

    a^2(I^2 ∓ J^2) = a^2 [1+α^2(a^2-l^2)]^2 + 4α^2 l^2(l^2-a^2) .

and

    2aK = a [1+α^2(a^2-l^2)] + √( a^2[1+α^2(a^2-l^2)]^2 + 4α^2 l^2(l^2-a^2)) ,

which are useful explicit relations to be employed later.




Actually, the transformation has a simple structure. For t and φ it is just a linear combination, and for x and r it is a fraction of linear expressions of the respective coordinates (a real version of the Möbius transformation). For l=0 this reduces to x=x', r=a r'.


Let us observe that the coordinates τ', ϕ' and the constant C in the metric (<ref>) have the physical dimension of length, while r', x' are dimensionless. Also the PD coefficients α', k', n', ϵ', m', e', g', and thus both the metric functions P', Q', are dimensionless.[There are other possibilities.
By a rescaling of the coordinates τ' ↦ C τ' and ϕ' ↦ C ϕ', the constant C^2 can be removed from the last two terms in (<ref>) because it then becomes an overall constant conformal factor determining the specific physical scale of the metric in the square brackets with dimensionless coordinates τ', ϕ', r', x'. Alternatively, by performing r' ↦ r'/C and x' ↦ x'/C all the new coordinates τ', ϕ', r', x' have the dimension of length (after the PD parameters are properly rescaled).]





As explained in full detail in Appendix, to exactly identify the metrics A and PD_α, that is (<ref>) and (<ref>), it is also necessary to perform a constant dimensionless scaling such that the conformal factor in (<ref>) is

    Ω'^ 2 = S^2 ( 1 -α' r' x' )^2,
        where    
    S^2   = a^2/α^4 |a^2-l^2|^3 K-1/√(I^2 ∓ J^2),

see (<ref>) and (<ref>). This specific rescaling is already included in the transformation of t and φ given by (<ref>) and (<ref>).




Even more importantly, the transformation uniquely relates the PD acceleration parameter α' in  Ω'^ 2 and the  more convenient acceleration parameter α as

    α'    = α a [ K - α^2(a^2-l^2)],

see (<ref>) which follows from (<ref>). Written explicitly it terms of the new Astorino parameters, it reads

    α'    = α  12[ a - α^2 a (a^2-l^2)
    + √(a^2+α^4 a^2(a^2-l^2)^2+2α^2(a^2-l^2)(a^2-2l^2)) ].

By setting α=0 or l=0 we get α'=α a, while by setting a=0 we get α'=α^2 l^2.



Relations between the other six parameters in the Plebański-Demiański metric functions  (<ref>)
are more involved, namely

    k'     = 1/I^2 ∓ J^2  K-1/α^2(a^2-l^2)  I  L , 
    
    
    n'     = -1/I^2 ∓ J^2 [ K-1/α^2(a^2-l^2)  I  M  - [1-α^2(a^2-l^2)] L l/a ], 
    
    
    
    
    ϵ'     = 1/I^2 ∓ J^2 [1-α^2(a^2-l^2)]
          ( I  L  + 4 M l/a)
          - (e^2+g^2)K-1/(a^2-l^2) I/√(I^2 ∓ J^2) ,
    
    
    
    α' m'    = 1/I^2 ∓ J^2 [  (K-1) I  M
           
       + [1-α^2(a^2-l^2)]( I  M
           + α^2 [(a^2-l^2) L+ (e^2+g^2)√(I^2 ∓ J^2) ]l/a)], 
    
    
    
    
    
    e'^2 + g'^2    = (e^2 + g^2) K-1/α^2(a^2-l^2)^2 I/√(I^2 ∓ J^2) ,

where we introduced specific dimensionless combinations of the physical parameters as

    L :=     I+2α m l/a + α^2(e^2+g^2)1/K l^2/a^2  , 
    
        M :=     α m I+α^2(2a^2-2l^2+e^2+g^2)l/a .

Notice that in (<ref>) we can alternatively employ the identity

    1/K l^2/a^2 = I-√(I^2 ∓ J^2)/2α^2(a^2-l^2) .


A systematic, step-by-step derivation of this transformation from the Astorino form to the Plebański-Demiański form of the metric is contained in Appendix <ref>.



To complete the discussion of the mutual relation between the original Plebański-Demiański form (<ref>) of the metric PD_α (with Λ=0) and the new Astorino representation of this entire family of black holes, we will now substitute the parameters k', n', ϵ', m', e', g' and α', as given by (<ref>) and (<ref>), into the metric functions (<ref>), resulting in

    P'=1I^2 ∓ J^2   [ I(K-1)α^2(a^2-l^2)+2 [1-α^2(a^2-l^2)]la x'-I [K-α^2(a^2-l^2)] x'^ 2]
    ×   [ L -2M x' + α^2 [(a^2-l^2)L+(e^2+g^2)√(I^2 ∓ J^2) ] x'^ 2], 
    
     Q'=1I^2 ∓ J^2   [I-2α l [1-α^2(a^2-l^2)] r'-α^2(a^2-l^2)I r'^ 2]
    ×   [ K-1α^2(a^2-l^2)^2 [(a^2-l^2)L + (e^2+g^2)√(I^2 ∓ J^2) ] 
        -2α aM r'+L [K-α^2(a^2-l^2)] r'^ 2].



We have thus arrived at a very nice result: using the new parameters introduced in <cit.>, the quartic PD polynomials <cit.> are factorized into the product of quadratic expressions. Such a  factorized form is crucial for the geometrical and physical interpretation because the roots of Q' and P', which can now by easily found, represent the horizons and poles (axes of symmetry), respectively, of the the black-hole spacetimes. This was previously achieved in the GP and PV forms of the metric <cit.>, but the new Astorino parametrization has now enabled us to factorize the metric functions P' and Q' directly in the original PD metric.





There is a simplification in special cases when some of the physical parameters vanish:





∙ Special case  l=0: no NUT

For black holes without the NUT twist (when l=0),

    I = 1+α^2a^2  ,     
        J = 0  ,    
        K = I  ,    
        L = I ,    
        M = α m I ,

so that S^-2 = α^2a^2(1+α^2a^2) and the Plebański-Demiański parameters (<ref>) simplify to

    α'    = α a  , 
    
    k'     = 1  , 
    
    n'     = -α m  ,
    ϵ'     = 1 - α^2(a^2+e^2+g^2) , 
    α' m'    = α m  ,
    
    (e'^2 + g'^2)    = a^-2 (e^2 + g^2) .

The two key metric functions (<ref>), (<ref>) reduce to

    P'    =       ( 1 - x'^ 2)( 1 -2α m x' + α^2 (a^2+e^2+g^2) x'^ 2), 
    
     Q'    =a^-2( 1 -α^2a^2 r'^ 2) ( a^2+e^2+g^2 - 2 m a r' + a^2 r'^ 2).

After restoring the correct physical dimensionality of the PD parameters and coordinates (as explained in Subsections <ref> and <ref>, in particular by applying the analogue of the relations (<ref>) for the choice γ=a) we get
Q' = ( 1 -α^2 r^2 ) ( a^2+e^2+g^2 - 2 m r + r^2 ). These results fully agree with Eq. (16.24) in Section 16.3.2 of the monograph <cit.>. Moreover, due to the nice factorization, the roots of P' and Q' — which identify the axes and horizons, respectively — can easily be determined.







∙ Special case α=0: no acceleration


For non-accelerating black holes, that is in the limit α→0, we get

    I = 1 ,     
        J = 0  ,    K-1/α^2(a^2-l^2) = a^2-l^2/a^2 ,    
        L = 1 ,    
        M = α m ,

where the expression for K can be calculated using (<ref>), so that K → 1, S^-2→α^2|a^2-l^2|, and α' →α  a.
The PD parameters thus become

    α'    = 0  , 
    
    k'     = 1 - l^2/a^2 , 
    
    n'     = l/a ,
    ϵ'     = 1   , 
    
    m'    = m/a  ,
    e'^2 + g'^2    = a^-2 (e^2 + g^2) .

Note that setting l=0 in (<ref>) agrees with the previous l=0 case if α=0 is set in (<ref>).
The metric functions (<ref>), (<ref>) simplify to

    P'    = 1 - ( x' - la )^2, 
    
     Q'    = a^-2( a^2-l^2+e^2+g^2 - 2 m a r' + a^2 r'^ 2).

It agrees with Eq. (16.23) in Sec. 16.3.1 of <cit.> after performing a shift x'-l/a ↦ x' in P', and restoring the correct dimensionality in Q' by rescaling it a^2 and redefining r' to r=a r'.







∙ Special case a=0: no rotation

For black holes without the Kerr-like rotation, by setting a = 0 in (<ref>), (<ref>), (<ref>) and (<ref>), we get

    I    =  1-α^2l^2  ,     
        a^2(I^2 ∓ J^2) = 4α^2l^4  ,    
        aK = α l^2    ,  
    
        aL   = α (2ml + e^2+g^2) ,    
        aM = α^2l (-2l^2 +e^2+g^2) .

Consequently, S^-2→ 2α^4l^6/a^2 and

    α'    = α^2  l^2  , 
    
    k'     = α^2 l^2-1/4α^2 l^4 (2m l +e^2+g^2   ) , 
    
    n'     = (m-l) l+(e^2+g^2)/2 α  l^3 + α/2(m+l) ,
    ϵ'     = -2 (1+α^2l^2) + 1/2(e^2+g^2)( α^2 + 3/l^2)  , 
    α' m'    = α/2 [ -(m+l) + α^2l^2(l-m)
        +e^2+g^2/l ] ,
    e'^2 + g'^2    = 1-α^2l^2/2α^2 l^4 (e^2 + g^2) .

The two key metric functions (<ref>), (<ref>) thus take an explicit factorized
form

    P'=14α^2l^4   [ (1-α^2l^2) - 2α l (1+α^2l^2) x' + α^2l^2 (1-α^2l^2) x'^ 2]
    ×   [ - (2ml + e^2+g^2)
        + 2 α l (-2l^2 +e^2+g^2) x'
        - α^2l^2(-2ml +e^2+g^2) x'^ 2], 
    
     Q'=14α^2l^4   [(1-α^2l^2)-2α l (1+α^2l^2) r' + α^2l^2(1-α^2l^2) r'^ 2]
    ×   [ (-2ml+e^2+g^2) -2 α l (-2l^2 +e^2+g^2) r'
          + α^2l^2(2ml + e^2+g^2) r'^ 2].

This particular choice of the PD coefficients, expressed in terms of the genuine Astorino physical parameters, identifies the elusive family of accelerating NUTty black holes without the Kerr-like rotation within the Plebański-Demiański family of metrics. This will be investigated in more detail in Section <ref>. Moreover, we will present these accelerating (possibly charged) purely NUT black holes in the Griffiths-Podolský form, see the metric (<ref>)–(<ref>), which has not been known untill now.







§ TRANSFORMATION TO THE GRIFFITHS-PODOLSKÝ METRIC FORM





In previous section we have proven that, assuming Λ=0, the complete Astorino (A) class of solutions (<ref>) can be equivalently written in the Plebański-Demiański (PD_α) form of the metric (<ref>),

    s^2=1(1-α' r' x')^ 2[
           - Q'r'^ 2+x'^ 2 (τ'  -  x'^ 2 ϕ' )^2
         + P'r'^ 2+x'^ 2 (τ'  +  r'^ 2 ϕ' )^2 
       + C^2(r'^ 2+x'^ 2) ( r'^ 2Q' + x'^ 2P' )],

where the metric functions P'(x') and Q'(r'), explicitly expressed in terms of the very convenient Astorino parameters and factorized, are generally given by (<ref>) and (<ref>), respectively.











Our aim in this section is to relate the PD_α form (<ref>) of the type D black holes to the Griffiths-Podolský (GP) form of these solutions, summarized in <cit.>. This will elucidate the direct relation of the GP metric to the A metric. In particular, it will clarify the relation between the Astorino initial parameters α, a, l, m, e, g and the physical parameters α̃, ã, l̃, m̃, ẽ, g̃ employed in the GP form of the metric previously <cit.>.



We start by performing a simple rescaling of coordinates, bringing the PD_α metric to the PD_αω metric (<ref>) which involves an additional twist parameter ω, namely[Note that the acceleration parameter α' is already included in the metric PD_α so we do not have to include it in the coordinate transformation as was done <cit.> and repeated at the beginning of section <ref>.]

    x' ↦√(ω) x',    
        r' ↦r'√(ω),    τ' ↦√(ω) τ',    ϕ' ↦√(ω) ϕ'.

These rescalings bring the metric (<ref>) to the PD_αω form
  
    -3pc s^2=1/(1-α'  r' x')^2[
     - Q/r'^ 2+ω^2x'^ 2(τ'-ω x'^ 2ϕ')^2
     +r'^ 2+ω^2x'^ 2/ Q  r'^ 2
          6pc
      + P/r'^ 2+ω^2x'^ 2(ωτ'+r'^ 2ϕ')^2
      +r'^ 2+ω^2x'^ 2/ P  x'^ 2],

where
𝒫(x') := P'(√(ω) x') and
𝒬(r') := ω^2 Q'(r'√(ω)).



Following <cit.>, the next step is to perform a coordinate transformation

    x'=l̃ω+ãω x̃,    τ'=t-(ã+l̃)^2ã φ,    ϕ'=-ωã φ,    
        r'=r̃,

where ã represents the Kerr-like rotational parameter, while l̃ represent the NUT-like parameter. After these linear transformations, the metric becomes

    s^2=1Ω̃^2[   -𝒬̃ ρ̃^ 2[ t-(ã(1-x̃^2)+2l̃(1-x̃))φ]^2
         + ρ̃^ 2𝒬̃ r̃^2
       + ρ̃^ 2𝒫̃ x̃^2
         +𝒫̃ ρ̃^ 2[ã  t-(r̃^2+(ã+l̃)^2) φ]^2],

in which

    Ω̃ :=    1-α'ω r̃ (l̃+ã x̃), 
    ρ̃^ 2 :=   r̃^ 2+(l̃+ã x̃)^2, 
    𝒫̃(x̃) :=   ω^2ã^2 P'(l̃+ã x̃√(ω)),
    
    𝒬̃(r̃) :=   ω^2  Q' (r̃√(ω)) ,

where the functions P'(x'), Q'(r') are given by (<ref>), (<ref>). This is the general Griffiths-Podolský form of the metric, as summarized in Eq. (16.12) in <cit.>.



By inspecting the conformal factor Ω̃ we observe that the GP acceleration parameter α̃ is equal to the PD acceleration parameter, α̃=α'. Using (<ref>), we can thus directly relate the GP acceleration to the  A parameters as

    α̃ = α a [ K - α^2(a^2-l^2)].

Expressed explicitly in terms of the new Astorino parameters, this is actually quite an involved expression (<ref>), namely

    α̃ = α  12[ a - α^2 a (a^2-l^2)
    + √(a^2+α^4 a^2(a^2-l^2)^2+2α^2(a^2-l^2)(a^2-2l^2)) ].

Clearly, α=0 implies α̃=0, which is expected. However, by setting l=0 we get α̃=α a. It means that the GP acceleration parameter α̃ also vanishes for l=0=a. Similarly, for a=0 we get α̃=α^2 l^2, so that
α̃ also vanishes for a=0=l. This degeneracy is an unfortunate feature of the original GP representation of the whole class of type D black holes, preventing to identify the genuine subclass of accelerating NUT black holes without the Kerr-like rotation — which exists, and is nicely contained in the Astorino metric (<ref>)–(<ref>).


Let us now concentrate on the GP “rotational” parameters ã and l̃. In the transformation (<ref>) these are arbitrary constants. However, they are naturally constrained by the requirement that in the final form of the GP metric the spherical-like coordinate θ should be introduced instead of x̃ in (<ref>) via the relation x̃=cosθ. To this end, the function 𝒫̃(x̃) must be written in the specific factorized form

    𝒫̃=(1-x̃^2)(1-a_3 x̃-a_4 x̃^2),

and this can be achieved by a unique values of the parameters ã and l̃. Indeed, for the choice[In view of the definition (<ref>), the upper sign applies for a^2>l^2, while the lower sign applies for a^2<l^2.]

    ã   = √(ω)K-α^2(a^2-l^2) √(I^2 ∓ J^2)/I ,
    l̃   = √(ω)K-α^2(a^2-l^2) 
                  F la ,

where F denotes the fraction

    F := 1-α^2(a^2-l^2)1+α^2(a^2-l^2),

the first bracket in P' given by (<ref>), expressed in the new coordinate x̃ such that x'=(l̃+ã x̃)/√(ω), becomes

    I(K-1)α^2(a^2-l^2)+2 [1-α^2(a^2-l^2)]   la x'-I [K-α^2(a^2-l^2)] x'^ 2
        = I^2 ∓ J^2I [K-α^2(a^2-l^2)] (1-x̃^2) .

Notice also that this natural fixing of ã and l̃ can be rewritten using the relation (<ref>) as

    α̃ ã   = α a √(ω) √(I^2 ∓ J^2)I , 
    α̃ l̃   = α l √(ω)  F .




Evaluating also the second bracket in P', we obtain the metric function 𝒫̃ in   (<ref>)

    𝒫̃(x̃)=    (1-x̃^2) ω^2ã^2 1I [K-α^2(a^2-l^2)] 
       ×[ L  - 2Ml̃√(ω)
           +α^2[(a^2-l^2)L+(e^2+g^2) √(I^2 ∓ J^2) ] l̃^2ω
           
           -(2Mã√(ω)-2α^2[(a^2-l^2)L+(e^2+g^2) √(I^2 ∓ J^2) ]
           l̃ãω) x̃
           
           +α^2 ã^2ω[(a^2-l^2)L+(e^2+g^2) √(I^2 ∓ J^2) ] x̃^2 ],

which is indeed of the required factorized form (<ref>) —  up to an overall rescaling which can always be achieved.

Indeed, so far ω has been a free “twist” parameter introduced by (<ref>). To describe a black hole with the horizon topology of a sphere, the function 𝒫̃(x̃) has to satisfy the condition 𝒫̃(x̃=0)=1 which directly follows from (<ref>) (see also <cit.>). For (<ref>), using (<ref>), (<ref>) and (<ref>), we thus derive a special value of ω, namely[Let us note, however, that in  Sec. <ref> we will demonstrate that an arbitrary value of ω in the metric can be restored by a rescaling of the acceleration parameter.]

    ω_0 =
        α aα̃ I^2 ∓ J^2I( L - 2Mα lα̃ F
        +α^4l^2α̃^2[(a^2-l^2)L+(e^2+g^2) √(I^2 ∓ J^2)] F^2
        )^-1.

This brings 𝒫̃ exactly to the desired form (<ref>), that is

    𝒫̃(x̃)=(1-x̃^2) P̃(x̃),
        where    P̃(x̃) :=  1-a_3 x̃-a_4 x̃^2,

with the coefficients a_3 and a_4 given by

    a_3=   2√(I^2 ∓ J^2) ω_0
        ( M - α^3 l α̃[(a^2-l^2)L+(e^2+g^2) √(I^2 ∓ J^2) ] F   ),
    
        a_4=   -α^3 aα̃  I ω_0[(a^2-l^2)L+(e^2+g^2) √(I^2 ∓ J^2) ].




The last metric function 𝒬̃(r̃) easily follows from the relations (<ref>) and (<ref>),

    𝒬̃(r̃)=
        ω_0^2(k'+e'^ 2+g'^ 2)-2 ω_0^3/2m' r̃+ω_0ϵ' r̃^2
        -2√(ω_0) α'n' r̃^3-α'^ 2k' r̃^4 ,

recalling that α̃=α', and introducing the rescaled parameters

    k̃:=k',  m̃:=ω_0^3/2 m',  ñ:=ω_0^3/2 n' ,  ϵ̃:=ω_0 ϵ',  ẽ:=ω_0 e',  g̃:=ω_0 g',

where k', n', ϵ', m', e', g' are given by (<ref>).




To obtain the Griffiths-Podolský form of the metric it now suffices to introduce the angular coordinate
θ in (<ref>) via the simple relation x̃=cosθ, resulting in

    s^2=1Ω̃^2[   -Q̃ ρ̃^ 2[ t-(ãsin^2θ+2l̃(1-cosθ))φ]^2
         + ρ̃^ 2Q̃ r̃^2
       + ρ̃^ 2P̃ θ^2
         +P̃ ρ̃^ 2sin^2θ [ ã  t-(r̃^2+(ã+l̃)^2) φ]^2],

where

    Ω̃   = 1-α̃ω_0 r̃ (l̃+ãcosθ), 
    ρ̃^ 2   = r̃^ 2+(l̃+ã cosθ)^2, 
    P̃(θ)       = 1-a_3cosθ-a_4cos^2θ , 
    Q̃(r̃)    = (ω_0^2 k̃+ẽ^ 2+g̃^ 2)
            -2m̃ r̃+ϵ̃ r̃^ 2
            -2α̃ñω_0 r̃^ 3
            -α̃^2k̃ r̃^ 4 .

It is exactly the GP metric given by Eqs. (6.18), (6.19) in the monograph <cit.>.


Moreover, a straightforward (but somewhat lengthy) calculation proves that the above parameters satisfy the following set of relations

    a_3    = 2α̃ãω_0m̃
          -4α̃^2ãl̃ω_0^2(ω_0^2k̃+ẽ^ 2+g̃^ 2),
          
    
      a_4    =
          -α̃^2ã^2ω_0^2(ω_0^2k̃+ẽ^ 2+g̃^ 2),
          
    ϵ̃   = ω_0^2 k̃ã^2
          -l̃^2+4α̃l̃ω_0m̃
          -(ã^2+3l̃^2)[ α̃^2ω_0^2(ω_0^2k̃+ẽ^ 2+g̃^ 2)],
          
    ñ   = ω_0^2 k̃l̃ã^2-l̃^2
          -α̃ã^2-l̃^2ω_0m̃
          +(ã^2-l̃^2)l̃[ α̃^2ω_0^2(ω_0^2k̃+ẽ^ 2+g̃^ 2)],
          
    (ω_0^2ã^2-l̃^2+3α̃^2 l̃^ 2)k̃   =
          1+2α̃l̃ω_0m̃
          -3α̃^2l̃^ 2ω_0^2(ẽ^ 2+g̃^ 2),

which are exactly the expressions in Eqs. (16.20) and (16.15)–(16.17) in <cit.>. 


This finishes the construction of the Griffiths-Podolský form of the general metric of black holes of algebraic type D. Moreover, it explicitly demonstrates the full equivalence of the GP form with the PD and A forms of this large class of spacetimes.

However, it should be emphasized that there is a subtle but very important difference: the original Griffiths-Podolský physical parameters α̃, ã, l̃ (representing acceleration, Kerr-like rotation, NUT twist) and m̃, ẽ, g̃ (representing mass, electric charge, magnetic charge) in the metric (<ref>) are not the new Astorino genuine parameters α, a, l and m, e, g, which properly separate the corresponding subclasses when they are set to zero.

In fact, the old GP parameters are now explicitly expressed in terms of the new A parameters via the complicated relations (<ref>), (<ref>), (<ref>) and (<ref>) with (<ref>). Moreover, the additional twist parameter ω has a very special value ω_0 given by (<ref>). Due to their highly involved structure, these could not be guessed in previous investigation of this family of spacetimes.

To elucidate the relation between the GP and A (that is also A^+) physical parameters in more detail, it seems to be instructive to consider the special cases α=0, l=0, a=0. It will clearly demonstrate in which situations the two sets of parameters agree, and what are their specific differences.








§ SPECIAL CASES



To analyze various special subcases of black holes, it is first necessary to clarify the freedom in the choice of the twist parameter ω, and also to consider the physical dimensionality of the parameters α̃, l̃ and ã.






 §.§ An issue of the twist parameter ω



In previous section, an explicit transformation from the Astorino form of the metric to the Griffiths-Podolský one was presented. It involves a very special, unique choice of the twist parameter ω=ω_0 given by (<ref>). This may be seen as a contradiction to statements in the published  works, such as <cit.>, where it was argued that ω is a free parameter (with a general restriction that it is related to the twist of the congruence generated by PNDs, and thus to both the Kerr-like rotational parameter ã and the NUT-like parameter l̃.

However, it can be demonstrated that there is no such contradiction because arbitrary (nonzero) value of ω can be restored from (nonzero) ω_0. This is achieved by a simple rescaling of the acceleration parameter,

    α̃  ↦  α̃ ω_0ω ,

while keeping all other physical parameters (that is ã, l̃, m̃, ẽ, g̃) the same. After this substitution, the set of relations (<ref>)–(<ref>) become

    a_3    = 2α̃ãωm̃
          -4α̃^2ãl̃ω^2(ω_0^2k̃+ẽ^ 2+g̃^ 2),
          
    
      a_4    =
          -α̃^2ã^2ω^2(ω_0^2k̃+ẽ^ 2+g̃^ 2),
          
    ϵ̃   = ω_0^2 k̃ã^2
          -l̃^2+4α̃l̃ωm̃
          -(ã^2+3l̃^2)[ α̃^2ω^2(ω_0^2k̃+ẽ^ 2+g̃^ 2)],
          
    ñ   = ω_0^2 k̃l̃ã^2-l̃^2
          -α̃ã^2-l̃^2ωm̃
          +(ã^2-l̃^2)l̃[ α̃^2ω^2(ω_0^2k̃+ẽ^ 2+g̃^ 2)],
          
    (ω^2ã^2-l̃^2+3α̃^2 l̃^ 2) ω_0^2/ω^2 k̃   =
          1+2α̃l̃ωm̃
          -3α̃^2l̃^ 2ω^2(ẽ^ 2+g̃^ 2).

The last equation suggest a rescaling

    k̃  ↦  ω^2ω_0^2 k̃ ,

which replaces the special value of ω_0 in all relations (<ref>)–(<ref>) by an arbitrary value ω (because ω_0^2k̃ is replaced by ω^2k̃).


Concerning the metric functions given by (<ref>)–(<ref>), ρ̃^ 2 and P̃ remain the same, while Ω̃ and Q̃ change to

    Ω̃   = 1-α̃ω r̃ (l̃+ãcosθ), 
    Q̃(r̃)    = (ω^2k̃+ẽ^ 2+g̃^ 2)
            -2m̃ r̃+ϵ̃ r̃^ 2
            -2α̃ñω r̃^ 3
            -α̃^2k̃ r̃^ 4 .

The metric (<ref>) thus takes exactly the form of Eq. (16.18)–(16.20) in <cit.>, that is the original Griffiths-Podolský metric with a general (not fixed) value of ω.


To conclude, the simple rescaling (<ref>) of the acceleration parameter α̃, accompanied by the rescaling (<ref>) of the parameter k̃, restores arbitrariness of ω in the GP metric. In other words, the original GP metrics with different values of ω are equivalent (unless ω=0).







 §.§ Restoring the physical dimensionality of the black-hole parameters




Recall that it is convenient and natural to consider that all the coordinates and genuine physical parameters in the Astorino metric (<ref>)–(<ref>) have the usual physical dimension — as in the Boyer-Lindquist-type coordinates for the Kerr-Newman black holes (and thus their generalization in the Griffiths-Podolský metric form). In particular, the physical parameters m, a, l, e, g have the dimension of length (while α has the dimension 1/length). Also the coordinates r and t have the dimension of length.


On the other hand, the GP parameters m̃, α̃, ã, l̃, ẽ, g̃ in the metric (<ref>) are dimensionless. The reason is that they have been obtained  from the dimensionless PD coefficients k', n', ϵ', m', e', g' — given by (<ref>) — using the relations (<ref>) and  α̃=α'.




Their proper physical dimensionality can be restored by introducing a parameter γ with the dimension of length, namely by rescaling the GP parameters in such a way that

    r̃↦γ r̃,     m̃↦γ m̃,     ã↦γ ã,     l̃↦γ l̃,     ω↦γ ω, 
       ẽ↦γ ẽ,     g̃↦γ g̃,     α̃↦α̃γ,     Q̃↦γ^2  Q̃ .

Let us note that after this rescaling the conformal factor Ω̃ given by  (<ref>) remains the same, while (<ref>) changes to

    Ω̃ = 1-α̃/γω_0 r̃ (l̃+ãcosθ)

if we keep ω_0, fixed by (<ref>), dimensionless.


In the most general case, it is not a priori clear how to choose the unique value of γ. However, it can be easily identified in the particular cases of black holes, to recover the standard forms of these well-known solution.






 §.§ The special case l=0: no NUT



Using (<ref>), from (<ref>) we get ω_0 =
    α a / α̃. Recalling (<ref>), which gives the relation

    α̃   = α a,

we immediately obtain a nice result ω_0 = 1. The expressions (<ref>), (<ref>) then reduce to

    ã = √(ω_0) = 1 ,     l̃ = 0 ,

and the coefficients a_3 and a_4 are simply a_3 = 2α m, a_4 = -α^2 (a^2+e^2+g^2),
so that

    P̃ = 1 - 2α m  cosθ + α^2 (a^2+e^2+g^2) cos^2θ .

The function Q̃ is given by (<ref>) with the coefficients determined by (<ref>) and (<ref>),

    Q̃ = 1/a^2((a^2+e^2+g^2) - 2m a r̃ +a^2r̃^ 2)
                               (1 - α^2a^2 r̃^ 2).

Together with

    Ω̃      = 1-α a r̃cosθ,     ρ̃^ 2  = r̃^ 2+cos^2θ,

it gives an explicit form of the metric (<ref>) in terms of the Astorino physical parameters.


Restoring now the correct dimensionality of the PD parameters by using (<ref>) with the simple choice

    γ = a,

we obtain

    ã=a,     l̃=0,     α̃=α

The metric functions take the form

    Ω̃   = 1-α̃ r̃cosθ, 
    ρ̃^ 2   = r̃^ 2+ã^2 cos^2θ, 
    P̃   = 1 - 2α̃m̃ cosθ + α̃^ 2 (ã^ 2+ẽ^ 2+g̃^ 2) cos^2θ ,
    Q̃   =  ((ã^ 2+ẽ^ 2+g̃^ 2) - 2m̃ r̃ + r̃^ 2) (1 - α̃^ 2r̃^ 2),

which fully agrees with previous GP and PV forms of this class of accelerating Kerr-Newman black holes without the NUT parameter, as presented in Eq. (35)–(39) of <cit.> and also in Sec. 16.3.2 of <cit.>. In this standard form of the metric, any remaining physical parameters can be set to zero, in any order.


For m̃^ 2>ã^ 2+ẽ^ 2+g̃^ 2, both the key metric functions can be written in a factorized form

    P̃   = (1 - α̃ r̃_+ cosθ)(1 - α̃ r̃_- cosθ)   ,
    Q̃   = ( r̃ - r̃_+ )( r̃ - r̃_- )
                                 (1 - α̃ r̃)(1 + α̃ r̃)  ,

where r̃_± = m̃±√(m̃^ 2-ã^ 2-ẽ^ 2-g̃^ 2). The roots of Q̃ define the position of the horizons. As explained in detail in <cit.>, there are two black-hole horizons and two acceleration horizons. Extremal black holes with a degenerate horizon occur when m̃^ 2=ã^ 2+ẽ^ 2+g̃^ 2, while for α̃=0 the acceleration horizons disappear.









 §.§ The special case α=0: no acceleration



In this case

    I = 1 ,     
        J = 0  ,    
        K = 1  ,    
        L = 1 ,    
        M = α m ,

see (<ref>). From (<ref>) we get the limit α̃→α a, so that using (<ref>) we obtain

    α̃=0 ,    ω_0 = 1 .

Consequently, the relations (<ref>), (<ref>) give

    ã = 1 ,     l̃ = l/a ,

and the coefficients a_3 and a_4 are simply a_3 = 0 = a_4. Therefore, employing
(<ref>)–(<ref>) with (<ref>) and (<ref>), the metric functions in the GP metric (<ref>) become

    Ω̃    = 1 ,  
    ρ̃^ 2   = r̃^ 2+1/a^2(l +a cosθ)^2 , 
    P̃   = 1  ,
    Q̃   = 1/a^2((a^2-l^2+e^2+g^2) - 2m a r̃ +a^2r̃^ 2).

This gives an explicit form of the metric in terms of the Astorino physical parameters. Restoring the correct dimensionality of the coordinates and parameters by the choice

    γ = a  ,

we finally obtain

    ρ̃^ 2   = r̃^ 2+(l +a cosθ)^2  , 
    Q̃   = (a^2-l^2+e^2+g^2)-2m r̃+r̃^ 2 ,

which fully agrees with Eq. (16.23) in Sec. 16.3.1 of <cit.>.









 §.§ The special case a=0: no Kerr-like rotation



In such a case the auxiliary parameters are

    I    = 1-α^2l^2,     
        a|J| = 2α l^2,     
        aK = α l^2,     
    
        aL   = α (2m l+e^2+g^2),     
        aM = α^2 l  (-2l^2+e^2+g^2),

so that using (<ref>), (<ref>), (<ref>), (<ref>) we obtain

    α̃   = α^2l^2, 
    ã   = 2√(ω_0) 11-α^2l^2,    l̃= √(ω_0)α l 1+α^2l^21-α^2l^2, 
    ω_0   = 1-α^2l^2/ 1 - 2α^2 ml + α^4l^2(e^2+g^2-l^2) .

Consequently,

    P̃    = 1-a_3cosθ-a_4cos^2θ,   
    Q̃    = ω_0^24α^2l^4[(1-α^2l^2) - 2α l (1+α^2l^2)r̃√(ω_0)
          +α^2l^2 (1-α^2l^2)r̃^ 2ω_0 ]  
       ×[e^2+g^2-2ml - 2α l (e^2+g^2-2l^2)r̃√(ω_0)
                +α^2l^2(e^2+g^2+2ml)r̃^ 2ω_0 ],

where

    a_3     = 2α  m (1+α^2l^2) - l - α^2 l  (e^2+g^2-l^2)/1 - 2α^2 ml + α^4l^2(e^2+g^2-l^2),
    
        a_4     = α^22ml-e^2-g^2 1 - 2α^2 ml + α^4l^2(e^2+g^2-l^2).

Here we have used the definition (<ref>) together with (<ref>) (<ref>), that is

    𝒫̃(x̃) :=
       ω_0^2ã^2 
       P'(√(ω_0) x'=l̃+ã x̃√(ω_0)),     𝒬̃(r̃) :=
       ω_0^2 
       Q' (r'=r̃√(ω_0)),

in which the explicit transformation (<ref>),

    x' = l̃+ã x̃ω_0
           = (1+α^2l^2) + 2α l x̃/α l (1-α^2l^2) √(ω_0),

was inserted. In fact, the resulting metric functions P̃, Q̃ are fully consistent with the expressions (<ref>),  (<ref>). In particular, the first quadratic factor in P' gives the term (1-x̃^ 2), while the second term leads to P̃(x̃) = 1-a_3 x̃-a_4 x̃^2. It is then natural to introduce x̃=cosθ.




Correct dimensionality of the physical parameters can be restored by applying (<ref>) with

    γ = α  l^2,

so that

    α̃   = α,    ã= 2√(ω_0) α  l^21-α^2l^2,    l̃= l √(ω_0) 1+α^2l^21-α^2l^2.

Recall that the mass and charge parameters m,e,g in (<ref>)–(<ref>) already have the proper physical dimension (of length) but it is also necessary to rescale r̃↦γ r̃, Q̃↦γ^2  Q̃. Thus,

    Q̃(r̃)      = 14 [(1-α^2l^2) ω_0 - 2(1+α^2l^2)√(ω_0) r̃l
          +(1-α^2l^2) r̃^ 2l^2  ]  
       ×[(e^2+g^2-2ml) ω_0 - 2(e^2+g^2-2l^2)√(ω_0) r̃l
          +(e^2+g^2+2ml) r̃^ 2l^2  ],

while the function P̃ remains the same as in (<ref>). Also, here we keep the same dimensionless parameter ω_0 given by (<ref>).



We can thus write the metric for accelerating charged NUT black hole without the Kerr-like rotation in the Griffiths-Podolský metric representation (<ref>) as

    s^2=1Ω̃^2[   -Q̃ ρ̃^ 2[  t - ( (ã+l̃) + (l̃+ã cosθ))(1-cosθ) φ ]^2
         + ρ̃^ 2Q̃ r̃^2
       + ρ̃^ 2P̃ θ^2
         +P̃ ρ̃^ 2sin^2θ [ ã  t-(r̃^2+(ã+l̃)^2) φ]^2],

where from (<ref>) it follows that

    ã+l̃   = l √(ω_0)  1+α l1-α l, 
    l̃+ã cosθ   = l √(ω_0) [ 1+α l1-α l - 2α l1-α^2l^2(1-cosθ) ].

Applying these relations we get an explicit metric

    s^2=1Ω̃^2{   -Q̃ ρ̃^ 2[  t -
        2l √(ω_0) [ 1+α l1-α l - α l1-α^2l^2(1-cosθ) ]
        (1-cosθ) φ ]^2
         + ρ̃^ 2Q̃ r̃^2
       + ρ̃^ 2P̃ θ^2
         +P̃ ρ̃^ 2sin^2θ [ 2√(ω_0) α  l^21-α^2l^2  t
        -(r̃^2+l^2ω_0 (1+α l)^2(1-α l)^2) φ]^2},

in which (<ref>) and (<ref>) takes the form

    Ω̃   = 1-[ 1+α l1-α l - 2α l1-α^2l^2(1-cosθ) ]
       r̃l √(ω_0), 
    ρ̃^ 2   = r̃^ 2+l^2ω_0 [ 1+α l1-α l - 2α l1-α^2l^2(1-cosθ) ]^2,

while the functions P̃ an  Q̃ are given by (<ref>) and (<ref>), respectively.



It seems convenient now to perform a rescaling of the coordinates and the metric functions

    r̅   = r̃/√(ω_0),     t̅   = √(ω_0) t,     φ̅ = ω_0 φ,
       
    P̅   = P̃/ω_0,     Q̅   = Q̃/ω_0^2,     ρ̅^ 2 = ρ̃^ 2/ω_0,

which brings the metric to the form

    s^2=1Ω̅^2{   -Q̅ ρ̅^ 2[ t̅ - 2l  (1+α^2l^2)(1-cosθ) + α l sin^2θ/1-α^2l^2 φ̅ ]^2
         + ρ̅^ 2Q̅ r̅^2
       + ρ̅^ 2P̅ θ^2
         +P̅ ρ̅^ 2sin^2θ [ 2 α  l^21-α^2l^2 t̅
        -(r̅^2+l^2 (1+α l)^2(1-α l)^2)φ̅ ]^2},













where

    Ω̅   = 1-
    
      1+α^2l^2+2α lcosθ1-α^2l^2 r̅l, 
    ρ̅^ 2   = r̅^ 2+l^2
    
       [ 1+α^2l^2+2α lcosθ1-α^2l^2 ]^2 , 
    P̅   = 1/1-α^2l^2[ 1 - 2α^2 ml + α^4l^2(e^2+g^2-l^2)
        + 2α(  l - m (1+α^2l^2) + α^2 l  (e^2+g^2-l^2))cosθ
        + α^2 ( e^2+g^2-2ml)cos^2θ],
    Q̅   = 1/4l^4[ (r̅-l)^2 - α^2 l^2(r̅+l)^2]
           [ 2ml(r̅^2-l^2) + 4l^3r̅ +(e^2+g^2)(r̅-l)^2 ].

This is an explicit GP metric form (<ref>) of the class of accelerating charged NUT black holes (of type D) without the Kerr-like rotation. It has not been identified in previous works <cit.> due to the fact that the Kerr-like parameter ã was not properly chosen to cover this special subcase. Its convenient choice is (<ref>), coupled to the NUT parameter l and acceleration α, whereas previously it was incorrectly assumed that ã=0 in such a situation. This lead to a degenerate parametrization of this sector of the complete family of type D black hole spacetimes.

In this metric we can independently set α=0 and l=0, expecting to obtain the (charged) NUT solution without acceleration and the (charged) C-metric without the NUT parameter, respectively. Let us investigate these two special subcases in detail.



For α=0 we get

    s^2=
            -Q̅Ω̅^2 ρ̅^ 2[ t̅ -  2l(1-cosθ)φ̅ ]^2
         + ρ̅^ 2Ω̅^2Q̅ r̅^2
         +r̅^2+l^2/Ω̅^2( θ^2  + sin^2θ φ̅^2 ),

with P̅ = 1 and

    ρ̅^ 2   = r̅^ 2+l^2, 
    Ω̅   = 1l (l-r̅), 
    Q̅   = 1/4l^4 (l-r̅)^2
           [ 2ml(r̅^2-l^2) + 4l^3r̅ +(e^2+g^2)(r̅-l)^2 ].

Performing now the following transformation and rescaling of the physical parameters,

    r̅/l   = R-L/R+L,    t̅ = √(2) T,        
    
      l    =√(2) L,    
      m=√(2) M,    
      e=√(2) E,    
      g=√(2) G,

we obtain

    s^2=
        -F  [  T - 2L(1-cosθ)φ̅ ]^2
         + R^2F + (R^2+L^2)( θ^2  + sin^2θ φ̅^2),

where

    F = R^2-2MR-L^2+E^2+G^2/R^2+L^2.

This is the standard form of the NUT solution, see Eqs. (12.3), (12.2) and (12.19) in <cit.>. For vanishing NUT parameter (L=0), it reduces to the Reissner-Nordström solution.

It should also be remarked that the inverse transformation to (<ref>) reads

    R/L = l+r̅/l-r̅,

so that R=∞⇔r̅=l ⇔Ω̅=0. The conformal infinity is thus approached as R→∞ in the standard coordinates of (<ref>), while it is located at r̅=l in the new (unfamiliar) metric representation (<ref>) when α=0. This may be another “technical” reason why it was previously difficult to identify the genuine accelerating purely NUT black holes in the whole type D class.






Finally, let us investigate the complementary special subcase l=0, which should lead to the (charged) C-metric without the NUT parameter. To do so, we consider the metric (<ref>)–(<ref>) and perform the change of coordinates

    r̅  = l  R/R+2l ,     φ̅ = φ + α t̅ .

It is then possible in the transformed line element to take the limit l → 0 to zero NUT parameter, obtaining

    2  s^2 = 1(1-α Rcosθ)^2[ - Q̅ t̅^ 2 +  R^2Q̅
         +R^2P̅ θ^2
         +P̅ R^2sin^2θ φ^2 ],

where

    P̅   = 1 - 2α m cosθ  + α^2 ( e^2+g^2)cos^2θ ,  
    Q̅   = ( 1 - α^2R^2)(1 - 2m/R + e^2+g^2/R^2)  .

This is precisely the standard form of the C-metric solution, as given by Eqs. (14.6), (14.41) in <cit.> (up to a trivial overall conformal rescaling of the line element s^2 by the constant factor 2). Recall that the roots of Q identify horizons. In general, there are two acceleration horizons located at R_ a± = ± 1/α and two black-hole horizons at R_ b± = m̃±√(m̃^ 2-ẽ^ 2-g̃^ 2) <cit.>. For vanishing acceleration (α=0), the metric (<ref>) reduces to the usual form of the Reissner-Nordström solution.












§ TRANSFORMATION TO THE PODOLSKÝ-VRÁTNÝ METRIC FORM




Finally, we will elucidate the relation between the Astorino (A) metric (<ref>) of all type D black holes and the metric presented by Podolský and Vrátný (PV) in 2021 and 2023 <cit.>. The PV metric is an improvement of the GP metric in the sense that the key metric functions are considerably simpler, fully explicit and factorized. This turned out to be convenient for geometrical and physical interpretation of these spacetimes (identification and description of horizons and singularities, finding the global structure, ergoregions, identification of cosmic string, including regions with closed timelike curves if these strings are rotating, etc.).


Actually, the new PV metric has the same general form as the GP metric (<ref>), that is the metric given by Eqs. (6.18) in <cit.>, but the metric functions P̃, Q̃ are much simpler  (this was achieved by introducing a new set of the mass and charge parameters, rescaling the metric by a uniquely chosen constant conformal factor, and making a suitable choice of the twist parameter ω). Therefore, we can employ the same initial steps as in section <ref>, starting form the PD_α form of the metric (<ref>) — that is equivalent to the A form — arriving at  (<ref>)–(<ref>). Changing tildes to hats in all the parameters and coordinates (except t and φ which remain the same), we thus obtain

    s^2=1Ω̂^2[   -𝒬̂ ρ̂^ 2[ t-(â(1-x̂^2)+2l̂(1-x̂))φ]^2
         + ρ̂^ 2𝒬̂ r̂^2
       + ρ̂^ 2𝒫̂ x̂^2
         +𝒫̂ ρ̂^ 2[â  t-(r̂^2+(â+l̂)^2) φ]^2],

with

    Ω̂ :=    1-α'ω r̂ (l̂+â x̂),
    ρ̂^ 2 :=   r̂^ 2+(l̂+â x̂)^2, 
    𝒫̂(x̂) :=   ω^2â^2 P'(l̂+â x̂√(ω)),
    𝒬̂(r̂) :=   ω^2  Q' (r̂√(ω)) ,

where the functions P'(x'), Q'(r') are given by (<ref>), (<ref>). Again, this is the general Griffiths-Podolský metric, see Eq. (16.12) in <cit.>.




The conformal factor Ω̂ already has the PV form, so that we can directly identify the Podolský-Vrátný (dimensionless) acceleration parameter as

    α̂ = α̃ = α' = α a [ K - α^2(a^2-l^2)].

The PV, GP, and PD acceleration parameters are thus the same, and expressed explicitly in terms on the Astorino parameters,

    α̂ = α  12[ a - α^2 a (a^2-l^2)
    + √(a^2+α^4 a^2(a^2-l^2)^2+2α^2(a^2-l^2)(a^2-2l^2)) ].

If α=0 then α̂=0. As in the GP case, for l=0 we get α̂=α a, while for a=0 we get α̂=α^2 l^2.

Concerning the PV rotational parameters â and l̂, as in the previous GP case they are fixed by the condition that the metric function 𝒫̂(x̃) is factorized to 𝒫̂∝ (1-x̂^2), this leads to the same expressions as in (<ref>) and (<ref>), that is

    â = ã   = √(ω)K-α^2(a^2-l^2) √(I^2 ∓ J^2)/I,
    l̂ = l̃   = √(ω)K-α^2(a^2-l^2) 
                  F la .

Recall that the upper (minus) sign in (<ref>) applies when a^2>l^2, while the lower (plus) sign is used when a^2<l^2. With these parameters, the metric function 𝒫̂ in (<ref>) takes the form analogous to (<ref>),

    𝒫̂(x̂)=    (1-x̂^2) ω^2â^2 1I [K-α^2(a^2-l^2)] 
       ×[ L  - 2Ml̂√(ω)
           +α^2[(a^2-l^2)L+(e^2+g^2) √(I^2 ∓ J^2) ] l̂^2ω
           
           -(2Mâ√(ω)+2α^2[(a^2-l^2)L+(e^2+g^2) √(I^2 ∓ J^2) ]
           l̂âω) x̂
           
           +α^2 â^2ω[(a^2-l^2)L+(e^2+g^2) √(I^2 ∓ J^2) ] x̂^2 ].


So far, the steps were the same as those deriving the GP form of the metric in previous section. However now, instead of fixing the twist parameter  ω=ω_0 by (<ref>), we choose a different value ω=ω_1, where

    ω_1 = α  aα̂I^2 ∓ J^2I L≡I^2 ∓ J^2I [ K - α^2(a^2-l^2)] L  .

The function 𝒫̂ then becomes

    𝒫̂=   (1-x̂^2)[1-2MLl̂+â x̂√(ω_1)
        +α^2[a^2-l^2+(e^2+g^2) √(I^2 ∓ J^2)L ](l̂+â x̂√(ω_1))^2].


Now we introduce new mass and charge parameters as

    m̂    := 1α̂ ML √(ω_1) ,
    ê^2+ĝ^2    :=α^2α̂^2 (e^2+g^2) √(I^2 ∓ J^2)L ω_1 ,

and employ an important identity

    α̂^2(â^2-l̂^ 2)=α^2(a^2-l^2) ω_1 ,

so that 𝒫̂ becomes


    𝒫̂=(1-x̂^2)[1-2 α̂ m̂ l̂+â x̂ω_1
        +α̂^2(â^2-l̂^2+ê^2+ĝ^2)(l̂+âx̂ω_1)^2].

Defining

    r̂_± :=m̂±√(m̂^2+l̂^2-â^2-ê^2-ĝ^2),

the metric function takes a compact and fully factorized form

    𝒫̂(x̂)=(1-x̂^2)(1-α̂ω_1 r̂_+(l̂+â x̂))
             (1-α̂ω_1 r̂_-(l̂+â x̂)).

Notice that such a factorization corresponds to the factorization of the Astorino metric function (<ref>),

    Δ_x = (1-x^2)( 1 - α  r_+  x ) ( 1 - α  r_-  x),

where r_± := m ±√(m^2+l^2-a^2-e^2-g^2).





Similarly we  analyze the metric function 𝒬̂ in (<ref>). From the definition (<ref>) with (<ref>), we obtain the expression

    𝒬̂(r̂)=ω_1^2I^2 ∓ J^2   [I-2α l[1-α^2(a^2-l^2)] r̂√(ω_1)
        -α^2(a^2-l^2)I r̂^ 2ω_1]
    ×   [ K-1α^2(a^2-l^2)^2 [(a^2-l^2)L + (e^2+g^2)√(I^2 ∓ J^2) ] 
        -2α aM r̂√(ω_1)+L [K-α^2(a^2-l^2)] r̂^ 2ω_1].

Using the relation α̂ l̂ = α̃ l̃ = α l √(ω_1)  F which follows form  (<ref>), and the identity (<ref>), the first (square) bracket can be rewritten in the factorized form

    I (1-2 α̂ l̂ r̂ω_1-α̂^2(â^2-l̂^ 2)r̂^2ω_1^2)
    =I (1-α̂(â+l̂ )r̂ω_1)
     (1+α̂(â-l̂ )r̂ω_1).

The second (square) bracket, applying the relations

    K-1α^2(a^2-l^2)^2 = αα̂ a ,    
        K-α^2(a^2-l^2) = α̂α a ,

see (<ref>), and the definitions of the “hatted” charges and masses (<ref>) and (<ref>), becomes

    α̂/α a L [α^2(a^2-l^2)α̂^2 + (ê^2+ĝ^2)1ω_1
        - 2  m̂ r̂ω_1 + r̂^2ω_1].

Using the identity (<ref>) we then obtain a factorized expression

    α̂α a Lω_1 [r̂^2-2 m̂ r̂+(â^2-l̂^2+ê^2+ĝ^2)]
      = α̂α a Lω_1 (r̂-r̂_+)(r̂-r̂_-).

Therefore, the metric function  (<ref>) takes the form

    𝒬̂= ω_1 α̂α a I LI^2 ∓ J^2 (r̂-r̂_+)(r̂-r̂_-)
     (1-α̂(â+l̂ )r̂ω_1)
     (1+α̂(â-l̂ )r̂ω_1),

which for the convenient choice (<ref>) of ω_1 simplifies to a nice factorized expression

    𝒬̂(r̂)=
        (r̂-r̂_+)(r̂-r̂_-)
        (1-α̂(â+l̂ )r̂ω_1)    (1+α̂(â-l̂ )r̂ω_1).

This directly corresponds to the factorization of the Astorino metric function (<ref>),

    Δ_r = (r-r_+)(r-r_-)(1-α r)(1+α r) .



Finally, using (<ref>) the conformal factor (<ref>) reads
Ω̂ = 1-α̂ω_1 r̂ (l̂+â x̂). As argued in section <ref>, the special (nonzero) twist parameter ω_1 can by put to any (nonzero) value ω by the simple rescaling (<ref>) of the acceleration parameter (provided it is nonzero),

    α̂  ↦  α̂ ω_1ω .

It explicitly restores a freedom to choose ω in Ω̂, and also in
the metric functions (<ref>) and (<ref>). The metric  (<ref>) thus becomes

    s^2=1Ω̂^2[   -𝒬̂ ρ̂^ 2[ t-(â(1-x̂^2)+2l̂(1-x̂))φ]^2
         + ρ̂^ 2𝒬̂ r̂^2
       + ρ̂^ 2𝒫̂ x̂^2
         +𝒫̂ ρ̂^ 2[â  t-(r̂^2+(â+l̂)^2) φ]^2],

with

    Ω̂   = 1-α̂ω r̂ (l̂+â x̂),
    ρ̂^ 2   = r̂^ 2+(l̂+â x̂)^2, 
    𝒫̂(x̂)    =(1-x̂^2)(1-α̂ω r̂_+(l̂+â x̂))
             (1-α̂ω r̂_-(l̂+â x̂)),
    𝒬̂(r̂)    =   (r̂-r̂_+)(r̂-r̂_-)
        (1-α̂(â+l̂ )r̂ω)    (1+α̂(â-l̂ )r̂ω) .




It  now only remains to introduce the angular coordinate via the relation x̂=cosθ, and  choose the twist parameter to be

    ω :=â^2+l̂^ 2â .

This finally leads to

    s^2=1Ω̂^2[   -Q̂ ρ̂^ 2[ t-(âsin^2θ+2l̂(1-cosθ))φ]^2
         + ρ̂^ 2Q̂ r̂^2
       + ρ̂^ 2P̂ θ^2
         +P̂ ρ̂^ 2sin^2θ [ â  t-(r̂^2+(â+l̂)^2) φ]^2],

where

    Ω̂   = 1-α̂ ââ^2+l̂^ 2 r̂ (l̂+âcosθ) , 
    ρ̂^ 2   = r̂^ 2+(l̂+â cosθ)^2, 
    P̂(θ)    =(1-α̂ ââ^2+l̂^ 2 r̂_+(l̂+âcosθ))
                      (1-α̂ ââ^2+l̂^ 2 r̂_-(l̂+âcosθ)),
    Q̂(r̂)    =   (r̂-r̂_+)(r̂-r̂_-)
        (1-α̂ â â+l̂â^2+l̂^ 2 r̂)    (1+α̂ â â-l̂â^2+l̂^ 2 r̂).

This is exactly the Podolský-Vrátný form of the metric, as expressed by  Eqs. (1)–(5) in <cit.>.



Let us emphasize that the choice (<ref>), although convenient in many situations, prevents us to identify accelerating “purely NUT” black holes because â=0 completely removes the acceleration parameter α̂ from the metric (<ref>)–(<ref>). However, because ω is an additional free parameter, the PV metrics (<ref>) with different values of the twist ω are equivalent by the rescaling (<ref>) of the acceleration parameter (unless ω=0). This keeps a possibility to explicitly identify the accelerating purely NUT black holes (without the Kerr-like rotation, charged, of type D) even within the PV class of metrics for a better choice of ω than (<ref>), such that the degeneracy â=0 ⇒α̂=0 is removed. This possibility was explicitly demonstrated in the GP form of the metric, see Eqs. (<ref>)–(<ref>).














§ SUMMARY OF THE SPECIAL CASES



Let us summarize special cases of the Astorino (A, or A^+) metric in which the physical parameters are set to zero, and the corresponding Plebański-Demiański (PD), Griffiths-Podolský (GP) and Podolský-Vratný (PV)  parameters.






 §.§ The limit l → 0

As derived in (<ref>), the dimensionless PD acceleration parameter is α' = α  a. After restoring the correct physical dimensionality of the GP and PV parameters in Subsection <ref>, the result is simple, namely

    α̃    = α̂ = α , 
    ã    =â=a ,  
    l̃    =l̂=l=0 .

The acceleration, Kerr-like and NUT parameters in A, A^+, GP, and PV metric forms are thus identical.






 §.§ The limit α→ 0


From (<ref>) we obtain

    α' →α  a,    so that    α' = α̃ = α̂→ 0.

The Kerr-like and NUT parameters (rescaled to a correct physical dimension) are

    ã   =â=a ,
    l̃   =l̂=l ,

respectively. This gives us the Kerr-Newman-NUT solution in A, A^+, GP, and PV metrics.







 §.§ The limit a → 0


From (<ref>)–(<ref>) it follows that

    ãl̃ = √(I^2 ∓ J^2)/1-α^2(a^2-l^2) a/l  →  2 α l1+α^2 l^2.

In the limit a → 0 the ratio ã/l̃ thus remains finite, and nonzero (unless α l=0). It means that taking the limit in which Astorino's Kerr-like parameter a tends to zero does not bring either of the PV parameters ã or l̃ to zero.

Thus we conclude that the accelerating purely NUT black hole solution with a=0, identified by Astorino, corresponds (after a suitable  coordinate transformations presented in previous sections) to “accelerating Kerr-NUT solution” with a non-zero Kerr-like GP parameter ã0 (and l̃0), and PV parameter â0 (and l̂0).


This also explains why in the A (and A^+) metric form the acceleration  parameter α is not redundant in the case a=0, as opposed in the GP (and PV) form. Similar considerations show that the GP, PV, and PD acceleration parameters are non-zero in the case a=0. Indeed, from (<ref>), (<ref>) we obtain that the dimensionless acceleration parameters are given by

    α̃ = α̂ = α' = α^2 l^2 ,

which does not vanish when α l  0.








§ CONCLUSIONS





We thoroughly studied a large class of spacetimes representing black holes with mass m, rotation a, NUT parameter l, acceleration α, and electric and magnetic charges e and g (generating electromagnetic field aligned with both principal null directions of the type D Weyl tensor).

In particular, we found relations between coordinates and physical parameters of various metric forms of such exact solutions to the Einstein-Maxwell system, namely those of Plebański-Demiański (PD), Astorino (A, improved here to a more compact metric A^+), Griffiths-Podolský (GP), and Podolský-Vrátný (PV). The references to original articles, nomenclature and conventions are summarized in Table <ref>.


Main conclusions resulting from our investigation are:




  * If properly mapped and physically interpreted, all these representations cover the complete class of such type D black holes.


  * The physical parameters of the A metric representation (and thus also A^+) are a very good choice for describing the type D black holes. Moreover, they can be set to zero in any order, leading to expected special cases (and further subcases), without any unpleasant coordinate degeneracies or divergencies.


  * Explicit coordinate transformations and relations between the parameters of A, A^+, PD, GP and PV metric forms (and their variants which also include the twist parameter ω) were found and discussed. These mutual relations are shown in the scheme on Figure <ref> by arrows, with the references to the corresponding Sections of our paper.


  * We clarified the role of the twist parameter ω, related to both the Kerr-like rotation parameter a and to the NUT parameter l.


  * Expressed in terms of the new convenient physical parameters m, a, l, α, e, g, the key A^+, PD, GP and PV metric functions are explicitly factorized into the product of quadratic expressions. This is very helpful for the physical interpretation, namely the identification of horizons and axes.


  * Special attention was payed to the main subclasses of type D black holes, namely those with no NUT (l=0), no acceleration (α=0) and — until recently elusive — black holes with no Kerr like-rotation a=0.


  * We proved that in the subclasses l=0 and α=0, the rotation, NUT, and acceleration parameters are the same in  A, GP, and PV metric forms, that is a=ã=â, l=l̃=l̂, and α=α̃= α̂.


  * On the other hand, in the non-rotating subclass a=0, the Kerr-like parameters ã=â and the NUT parameters l̃=l̂ were not properly identified in previous GP and PV metric forms <cit.>. The correct general relations are given by (<ref>)–(<ref>), which in the case a=0 reduce to (<ref>).


  * The accelerating (charged) purely NUT black holes of type D (a=0, but α l nonzero) thus have ã=â0 in the GP and PV forms. With this correction, we were able to derive the GP (equivalent to PV) metric form of such black holes (<ref>)–(<ref>).


  * In this metric it is possible to independently set α=0 and l=0, obtaining thus the (charged) NUT solution without acceleration and the (charged) C-metric without the NUT parameter, respectively, in their usual form.












§ ACKNOWLEDGMENTS


This work has been supported by the Czech Science Foundation Grant No. GAČR 23-05914S.











10

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§ DERIVATION OF THE TRANSFORMATION TO THE PLEBAŃSKI-DEMIAŃSKI FORM OF THE METRIC, AND IDENTIFICATION OF THE PHYSICAL PARAMETERS




The  A^+ metric (<ref>) has a general form which is very similar to the Griffiths–Podolský representation <cit.> (see also Eq. (16.18) in <cit.>, and  <cit.>)
of the family of type D black holes in the Plebański-Demiański class of electrovacuum solutions with Λ.
Nevertheless, a closer look at both the metrics reveals a crucial difference. In the Griffiths–Podolský representation, the functions A(x) and C(r) are  constants, namely

    A ≡ A_c = 1,
               
        C ≡ C_c = a

(there is a nice agreement with (<ref>), (<ref>) for α=0).
However, we can remedy this problem by a suitable transformation of the coordinates x ↦ X and r ↦ R given by

    X ≡ A_c  xA(x),
                 R ≡ C_c  rC(r).

We also introduce new metric functions, expressed in these coordinates, as

    Δ_X(X) ≡A_c^2A^2 Δ_x,
                Δ_R(R) ≡C_c^2C^2 Δ_r,

and analogously

    Ω̃^2(R,X) ≡A_c C_cA C Ω^2,
                ρ̃^ 2(R,X)   ≡A_c C_cA C ρ^2,

so that

    ρ̃^ 2Ω̃^2 = ρ^2/Ω^2.

Then the A^+ metric (<ref>) takes the form

    s^2=1Ω̃^2[
        - Δ_Rρ̃^ 2(A_c  t - ℬ φ)^2
        + Δ_Xρ̃^ 2(C_c  t + 𝒟 φ)^2
        + C_f ρ̃^ 2(  R^2Δ_R +  X^2Δ_X )]
        ,

where A_c, C_c are constants, as desired, and the new metric functions are

    ℬ(X) ≡ A_c BA,
                𝒟(R) ≡ C_c DC.


It is now possible to find an explicit relation between the Astorino representation of all type D black holes, written in the form (<ref>), and the Plebański-Demiański class of the same solutions, written in the Griffiths–Podolský representation. In particular, this will elucidate the problem of purely NUT accelerating black holes of type D without the Kerr-like rotation (i.e., black holes with m, l, α 0 and a=0, admitting also the charges e, p and a cosmological constant Λ). These are clearly contained in the Astorino class of solutions (<ref>) but so far have not been identified in the Plebański-Demiański family.

The procedure starts with an integration of (<ref>),

    X=A_c∫ xA(x),
               
        R=C_c∫ rC(r),

of the specific quadratic functions A(x) and C(r) given by (<ref>) and (<ref>), respectively. Fortunately, it can be done explicitly, yielding for a^2>l^2 quite a simple transformation

    x =    1α√(a^2-l^2) tanh[α√(a^2-l^2)A_c (X + X_0) ], 
    
     r =    √(a^2-l^2)α a tan[α√(a^2-l^2)C_c  (R + R_0) ] - lα a,

where X_0, R_0 are free constants of integration. Notice that for very small values of X and R we obtain just linear relations
x ≈1A_c (X + X_0) and
r + lα a≈a^2-l^2C_c a (R + R_0).


Here we assume a^2>l^2, i.e., that the Kerr-like rotation parameter a is greater than the NUT parameter l, and a>0. The complementary case a^2<l^2 can be treated similarly by changing the trigonometric functions to hyperbolic ones, and vice versa. It leads to the same expressions, but with the term  √(a^2-l^2) generalized to √(|a^2-l^2|), and 2K := I + √(I^2+J^2) instead of the definition 2K := I + √(I^2-J^2) employed in (<ref>). It also covers various special cases when (some of) parameters are zero.


Next we have to evaluate the functions  A(X), B(X), C(R), D(R) given by (<ref>)–(<ref>). Then we obtain ℬ(X), 𝒟(R) by using (<ref>), and finally
Δ_X(X), Δ_R(R), Ω̃^2(R,X), ρ̃^ 2(R,X)  by using (<ref>), (<ref>).


A direct substitution into (<ref>), (<ref>) gives

    1/A(X) = cosh^2ξ ,
               1/C(R) = aa^2-l^2 cos^2χ,

where we introduced convenient dimensionless parameters

    ξ≡α √(a^2-l^2)A_c (X + X_0) ≡ξ' + ξ_0  ,
               χ≡α √(a^2-l^2)C_c (R + R_0) ≡χ' + χ_0 .

General expressions for B(X), D(R)  look more complicated, namely

    B(X) =    aα^2(a^2-l^2)tanh^2 ξ + 2lα√(a^2-l^2)tanhξ + a  ,
    
    
    D(R) =     1α^2a^2[(a^2-l^2)tan^2χ - 2l√(a^2-l^2) tanχ + [l^2+α^2 a^2(l^2-a^2) ] ] ,

and thus using (<ref>)

    ℬ(X) =     A_c aα^2(a^2-l^2)[
        sinhξ (I sinhξ + J coshξ) + α^2(a^2-l^2)
        ],
    𝒟(R) =     C_cα^2(a^2-l^2)a[
        sinχ ( (a^2 I - 2 l^2) sinχ - 2l√(a^2-l^2)cosχ)
        +l^2-α^2(a^2-l^2)a^2],

where the constants I, J, and K are defined as

    I     := 1+α^2(a^2-l^2)  , 
    
        J     := 2 l αa √(a^2-l^2) ,
    
        2K    := I + √(I^2-J^2) .


Following the idea employed previously in 
[Debever et al 1984] (see Section 4 in and subcases A2 and A3 therein), these involved functions can be considerably simplified by employing a remaining coordinate freedom encoded in the integration constants X_0, R_0 of (<ref>). By their unique choice

    tanh(2 ξ_0) =     -J/I,
    tan(2 χ_0)=    2l√(a^2-l^2)a^2 I - 2 l^2    ≡aα Ja^2 I - 2 l^2,

(assuming a^2 I > 2 l^2, which is natural because it admits a special case l=0) we achieve

    ℬ(X) =      A_c  (  b_0 + b_1 sinh^2 ξ'  ),
    𝒟(R) =      C_c  (  d_0 + d_1 sin^2 χ'  ),

where





the constant coefficients read

    b_1 = + d_1 =     a/α^2(a^2-l^2) √(I^2-J^2),
    
        b_0 = - d_0 =     a/α^2(a^2-l^2) (K - 1).



Next step is to apply a transformation of all the coordinates

    t=    12[α√(a^2-l^2) (√(I^2-J^2)-I+2) τ'+(√(I^2-J^2)+I-2) ϕ'/α√(a^2-l^2) ],
        
    φ=   1a[α√(a^2-l^2) ϕ'-(α√(a^2-l^2) )^3 τ'],
        
    tanhξ'=    α √(a^2-l^2) x' ,
    tanχ'=    α √(a^2-l^2) r' ,

so that

    ( t - b_0 φ) = ( t + d_0 φ) = α√(a^2-l^2) √(I^2-J^2)  τ'   ,

and

    b_1 sinh^2 ξ'  = a √(I^2-J^2) x'^2/1-α^2(a^2-l^2) x'^2 ,     
    d_1 sin^2 χ'  = a √(I^2-J^2) r'^2/1+α^2(a^2-l^2) r'^2 .

This leads to a great simplification of the key combinations of the functions

    A_c ( t - (  b_0 + b_1 sinh^2 ξ'  ) φ) =      A_c α√(a^2-l^2)√(I^2-J^2)/1-α^2(a^2-l^2) x'^2 ( τ'  -  x'^2 ϕ'  ) ,
    
    C_c ( t + (  d_0 + d_1 sin^2 χ'  ) φ) =      C_c α√(a^2-l^2)√(I^2-J^2)/1+α^2(a^2-l^2) r'^2 ( τ'  +  r'^2 ϕ'  ).


Using (<ref>), (<ref>), (<ref>)  we also obtain ρ̃^ 2=A_c 𝒟+C_c ℬ. The functions 𝒟, ℬ are given by (<ref>), (<ref>), (<ref>), so that (with the help of the relation d_0+b_0=0)

    ρ̃^ 2 = A_c C_c a  √(I^2-J^2) (r'^2+x'^2)/[1+α^2(a^2-l^2) r'^2][1-α^2(a^2-l^2) x'^2] .


If we now conveniently introduce metric functions Ω'(r', x'), P(x'), Q(r') as

    Ω'^ 2≡    a [1+α^2(a^2-l^2) r'^2]
    [1-α^2(a^2-l^2) x'^2]/A_c C_c α^2(a^2-l^2) √(I^2-J^2)  Ω̃^2 , 
    
    P' ≡    A_c^-2  [1-α^2(a^2-l^2) x'^2]^2 Δ_X , 
    
    Q' ≡    C_c^-2  [1+α^2(a^2-l^2) r'^2]^2 Δ_R ,

the metric (<ref>)  takes a nice form

    s^2=1Ω'^ 2[
           - Q'r'^2+x'^2 ( τ'  -  x'^2 ϕ'  )^2
         + P'r'^2+x'^2 ( τ'  +  r'^2 ϕ'  )^2 
       + a^2 C_f/α^2(a^2-l^2) (r'^2+x'^2)
        ( r'^2Q' + x'^2P' )].

Actually, this is a general Plebański-Demiański metric representing all (double aligned, non-null) solutions of Einstein-Maxwell-Λ equations of algebraic type D, including black holes of this type, see Eq. (3.30) in <cit.>, and Chapter 16 of <cit.> in which a special gauge α^2(a^2-l^2)=-1 and a^2 C_f=-1 is considered. The former condition can always be obtained by a rescaling on the angular coordinate ϕ', while the latter is achieved by the choice of the free constant C_f. Also, these operations relate the metrics (<ref>) and (<ref>).

To complete the transformation from the Astorino new metric representation (<ref>)-(<ref>) to the Plebański-Demiański metric representation (<ref>), it remains to prove that the conformal factor (<ref>) has the form Ω' = 1-α' r'x', with a suitable acceleration parameter α', and also that the metric functions P'(x'), Q'(r') are quartic functions of the respective coordinates.

Moreover, this explicit transformation will yield a unique relation between the physical parameters of the Astorino metric and the (purely mathematical) Plebański-Demiański  parameters, and with the physical parameters in the Griffiths-Podolský form of these black-hole metrics. In particular, it will give their fully general form, which will identify the elusive (overlooked) accelerating NUT black holes without the Kerr-like rotation parameter a.


To this end, we have to explicitly evaluate these metric functions. We start with the conformal factor Ω̃^2 given by (<ref>), (<ref>),

    Ω̃^2=    A_c C_c/A C  (1-α  r  x)^2.

In view of (<ref>), (<ref>), (<ref>) we get

    Ω̃^2 =  A_c C_c aa^2-l^2 [ cosχ coshξ
     - 1α√(a^2-l^2) sin (χ - β) sinhξ ]^2,

where we defined a useful auxiliary constant β as

    sinβ≡l/a ,
        so that  cosβ = √(a^2-l^2)/a,
      and  tanβ = l/√(a^2-l^2).

Recalling (<ref>)

 we rewrite (<ref>) as

    Ω̃^2 =  A_c C_c aα^2(a^2-l^2)^2 [    -α√(a^2-l^2) cos(χ' + χ_0) cosh(ξ' + ξ_0) 
          +sin(χ' + (χ_0 - β)) sinh(ξ' + ξ_0)  ]^2,

where

    tanh(2 ξ_0) = -J/I,
               tan(2 χ_0) = aα Ja^2 I - 2 l^2

see  (<ref>), (<ref>). Using standard identities for goniometric and hyperbolic functions, one derives the equivalent expressions (to be employed below), namely

    cosh(2 ξ_0) = I/√(I^2-J^2),
               cos(2 χ_0) = a^2 I - 2 l^2/a^2 √(I^2-J^2),

so that

    cosh^2ξ_0 = I+√(I^2-J^2)/2√(I^2-J^2),
              cos^2χ_0 =    a^2(√(I^2-J^2)+I) - 2l^2/2a^2 √(I^2-J^2),
        
    sinh^2ξ_0 = I-√(I^2-J^2)/2√(I^2-J^2),
              sin^2χ_0 =    a^2(√(I^2-J^2)-I) + 2l^2/2a^2 √(I^2-J^2),

and thus

    tanhξ_0 =     -J/2K, 
    tanχ_0 =    a/α J/2(a^2 K-l^2).

Notice that l=0 implies J=0=β, so that ξ_0 = 0 = χ_0.


Applying these relations, it is possible to prove that

    α√(a^2-l^2) sinχ_0 coshξ_0 =     -cos(χ_0-β) sinhξ_0,
    α√(a^2-l^2) cosχ_0 sinhξ_0 =     +sin(χ_0-β) coshξ_0.

Using these two identities, after employing in (<ref>) usual formulae for the sum in the argument of trigonometric and hyperbolic function , the terms containing cosχ' sinhξ' and sinχ' coshξ' vanish, while the two remaining combine in such a way that

    Ω̃^2 =  A_c C_c aa^2-l^2 [ cosχ_0/coshξ_0 cosχ' coshξ'
    +sinχ_0/sinhξ_0 sinχ' sinhξ'  ]^2.

Performing now the transformation (<ref>), (<ref>) we get

    Ω̃^2 =  A_c C_c a c_0^2 a^2-l^2 ( 1 - α' r' x' )^2/[1+α^2(a^2-l^2) r'^2][1-α^2(a^2-l^2) x'^2],

where

    c_0^2 =  cos^2χ_0/cosh^2ξ_0,     α' = α^2(a^2-l^2) tanχ_0/tanhξ_0 .

The conformal factor (<ref>) thus, using (<ref>) and (<ref>),
takes the form

    Ω'^ 2 = S^2 ( 1 - α' r' x' )^2,

where the dimensionless parameters are

    α'    = α a (a^2-l^2)K/a^2K-l^2, 
    
    S^2        = 1/α^2(a^2-l^2)^2 a^2K-l^2/K √(I^2-J^2)  ≡a^2α^4(a^2-l^2)^3K-1√(I^2-J^2).




For l=0, these relations simplify considerably to  α' =  α a and S^-2=α^2 a^2 (1+α^2a^2). On the contrary, for a=0, it reduces to α' = α^2 l^2.

Using these relations, we can rewrite the metric (<ref>) in the form:

    s^2=1S^2(1-α'r'x')^2[
           - Qr'^2+x'^2 ( τ'  -  x'^2 ϕ'  )^2 
       + Pr'^2+x'^2 ( τ'  +  r'^2 ϕ'  )^2 
       + a^2 C_f/α^2(a^2-l^2) (r'^2+x'^2) ( r'^2Q + x'^2P )].

By rescaling time and angular coordinates

    τ' ↦ S τ' ,    ϕ' ↦ S ϕ' ,

and introducing a new constant

    C :=a^2 C_fα^2(a^2-l^2)S^2 ,

the metric takes exactly the form (<ref>). Thus, to conclude, the required transformation of coordinates from the A to the PD form of the spacetimes is given by

    t    = aα(a^2-l^2) √(K-1√(I^2-J^2)) [[K-α^2(a^2-l^2)] τ' + K-1/α^2(a^2-l^2) ϕ' ],
    φ   = 1α(a^2-l^2)√(K-1√(I^2-J^2)) [ϕ'-α^2(a^2-l^2)^2 τ' ],

together with the transformation of x and r given by (<ref>) and (<ref>).



Finally, the metric functions P' and Q' are explicitly obtained from the Astorino metric functions Δ_X and Δ_R using the relations (<ref>) and (<ref>), respectively. It turns out to be the quartics (<ref>), where the parameters in the Plebański-Demiański metric functions  are given by (<ref>).


Recall that throughout this Appendix we assumed a^2>l^2. The case a^2<l^2 leads to similar expressions, but with √(a^2-l^2) replaced by √(|a^2-l^2|), and I^2-J^2 replaced by I^2+J^2. In particular, it generalizes the definition of the parameters J and K in (<ref>) to (<ref>), and yields a more general transformation (<ref>).



