It remains a longstanding problem unsettled for almost two decades in the general relativity community ever since Griffiths and Podolsk demonstrated in their previous paper J. B. Griffiths and J. Podolsk Classical Quantum Gravity 22 3467 2005CQGRDG0264938110.1088026493812217008 that the typeD NUT inlineformulammlmath displayinlinemmlmiCmmlmimmlmathinlineformula metric seems to be absent from the most general family of the typeD PlebaskiDemiaski PD solution. However Astorino in his recent article M. Astorino Phys. Rev. D 109 084038 2024PRVDAQ2470001010.1103PhysRevD.109.084038 presented a different form of rotating and accelerating black holes and showed that all known fourdimensional typeD accelerating black holes without the NUT charge can be recovered via various different limits in a definitive fashion. In particular he provided for the first time the correct expressions for the typeD static accelerating black holes with a nonzero NUT charge which was previously impossible using the traditional parametrization of the familiar PD solution. Nevertheless it still remains elusive how these two different forms of the fourdimensional rotating and accelerating solutions are related. In this paper we aim to fill this gap by finding the obvious coordinate transformations and parameter identifications between the vacuum metrics after two different parametrizations of the generated solution via the inverse scattering method from the seed metricthe Rindler vacuum background. We then resolve this missing puzzle by providing another Mbius transformation and linear combinations of the Killing coordinates which clearly cast the typeD NUT inlineformulammlmath displayinlinemmlmiCmmlmimmlmathinlineformula metric into the familiar form of the PD solution. Additionally we propose an alternative new routine for the normalization of the obtained metric derived via the inverse scattering method from the vacuum seed solution which could be potentially useful for the construction of higherdimensional solutions using the trivial vacuum background as the seed metric.
['General relativity', 'alternative theories of gravity', 'General Relativity and Quantum Cosmology', 'High Energy Physics - Theory']
Is the typeD NUT inlineformulammlmath displayinlinemmlmiCmmlmimmlmathinlineformula metric really missing from the most general PlebaskiDemiaski solution
{"Is the type-D NUT C-metric really 'missing' from the most general Pleba'nski-Demia'nski solution?": "Shuang-Qing Wu ∗ and Di Wu †  \nSchool of Physics and Astronomy, China West Normal University, Nanchong, Sichuan 637002, People's Republic of China (Dated: Received 27 August 2024; Accepted 21 October 2024)  \nIt remains a long-standing problem, unsettled for almost two decades in the general relativity community, ever since Griffiths and Podolsk'y demonstrated in their previous paper [J.B. Griffiths and J. Podolsk'y, Classical Quantum Gravity 22 , 3467 (2005)] that the type-D NUT C-metric seems to be absent from the most general family of the type-D Pleba'nski-Demia'nski (P-D) solution. However, Astorino, in his recent article [M. Astorino, Phys. Rev. D 109 , 084038 (2024)] presented a different form of rotating and accelerating black holes and showed that all known four-dimensional type-D accelerating black holes (without the NUT charge) can be recovered via various different limits in a definitive fashion. In particular, he provided, for the first time, the correct expressions for the type-D static accelerating black holes with a nonzero NUT charge, which was previously impossible using the traditional parametrization of the familiar P-D solution. Nevertheless, it still remains elusive how these two different forms of the four-dimensional rotating and accelerating solutions are related. In this paper, we aim to fill this gap by finding the obvious coordinate transformations and parameter identifications between the vacuum metrics after two different parametrizations of the generated solution via the inverse scattering method from the seed metric - the Rindler vacuum background. We then resolve this 'missing' puzzle by providing another Mobius transformation and linear combinations of the Killing coordinates, which clearly cast the type-D NUT C-metric into the familiar form of the P-D solution. Additionally, we propose an alternative new routine for the normalization of the obtained metric derived via the inverse scattering method from the vacuum seed solution, which could be potentially useful for the construction of higher-dimensional solutions using the trivial vacuum background as the seed metric.", 'I. INTRODUCTION': "The most general type-D solution in general relativity has been well known for more than half a century [1-3], and is often referred to as the Pleba'nski-Demia'nski (P-D) metric. With the advent of the conveniently factorized form [4] of two quartic structure functions, this solution has subsequently been extensively studied during the past twenty years, see the book [5] for a comprehensive review. However, as demonstrated in Ref. [6], the type-D NUT C-metric is apparently accelerating and can be transformed into the usual Taub-NUT solution. Since then, 'such a solution has to date neither been identified nor proved not to exist.' This raised a 'missing' puzzle for almost two decades, as the type-D NUT C-metric appears to be absent from the most general family of the type-D P-D solution.  \nRecently, Astorino [7] presented a different form of the rotating and accelerating black holes that belong to the type-D family (although no obvious proof of this issue is provided in that paper) and showed that all known fourdimensional type-D accelerating black holes without the NUT charge can be obtained via various different limits in a definitive fashion. Moreover, he presented, for the first time, the correct expressions for the type-D NUT C-metric, which are not available in the traditional parametrization for the familiar P-D solution. Lately, this solution has also been generalized in [8] to include the electromagnetic charges and a nonzero cosmological constant.  \nAs far as the NUT C-metric is concerned, apart from the above type-D one, there is also another kind of accelerating Taub-NUT solution [9], which belongs to a more general type I [10] that has attracted some recent interest [11-15]. Although the initial form of this solution is complicated and has been obtained via a hybrid solution-generating method from a specific seed, in fact it can be simply generated [11-13] via the Ehlers transformation from the C-metric. The Ehlers transformation not only endows the black hole horizon with the NUT charge, but also allocates a port of the NUT charge to the accelerating horizon. From the perspective of the double Kerr-Schild formalism, the type-D NUT C-metric can be viewed as a perturbation on the pure Rindler background. It seems that one could regard the NUTcharged Rindler metric as the background spacetime of the type-I NUT C-metric. However, it is unclear whether the type-I C-metric admits a similar double Kerr-Schild formulation. Based on the same reasoning, when the Harrison transformation is applied to the seed C-metric, one gets a type-I electrically charged C-metric, whose background is the electrified Rindler vacuum. However, a well-known type-D charged static accelerating solution (Reissner-Nordstrom C-metric, or RN C-metric) is already included in the most general P-D family. Like the Ehlers transformation that allocates a port of the NUT charge to the Rindler background, the Harrison transformation also assigns a port of the electric charge to the Rindler background. Thus, in the type-I RN C-metric, the accelerating horizon acquires the electric charge; in other words, its background is a charged Rindler metric. In this sense, the original KinnersleyWalker's interpretation [16] of the uniformly accelerating charged mass as two 'oppositely' accelerated black holes apart from each other seems not to be very accurate, as it cannot distinguish the type-D accelerating black holes from the type-I ones. Rather, one could say that the type-D accelerating solution is a black hole sitting on the pure Rindler background, while the type-I accelerating metric is that on the more general Rindler background with dynamical parameters (the NUT charge, electromagnetic charge) plus other kinetic parameters (such as the rotation parameter and possibly the cosmological constant). It is noted that while the type-D solution admits an extension to include a nonzero cosmological constant, the type-I metric seems unable to permit such a generalization. By the way, we mention that there is also another kind of NUT C-metric presented in Ref. [17], which is dubbed the [4-1]-soliton solution [18]. This solution still remains unexplored to date, but we can anticipate it to be of type I in the Petrov classification.  \nWhat remains opaque in the Astorino's work is how these two different forms of the four-dimensional rotating and accelerating solutions are related, and why the type-D NUT C-metric is 'missing' from the P-D metric. The main purpose of this paper is to resolve this 'missing' puzzle in two different ways. First, after generating the rotating and accelerating Kerr-NUT metric via the inverse scattering method (ISM) [19-21] from the seed Rindler vacuum background, we obtain two distinct forms of the same solution by using two different parametrizations. This clearly shows that both metrics are isometric to each other. Then, we find the obvious Mobius transformation and parameter identifications between them. In the meanwhile, we will illustrate what limit should be taken in the general P-D solution to obtain the type-D NUT C-metric. In particular, we shall present another Mobius transformation and linear combinations of the Killing coordinates that clearly cast the type-D NUT C-metric into the familiar form of the P-D solution, with the quartic structure function being factorized into two quadratic functions in a manner different from that of Griffiths-Podolsk'y.  \nThe main parts of our paper are organized as follows. In Sec. II, we present the ISM construction of the type-D accelerating Kerr-NUT metric as a 2-soliton transformation from the Rindler vacuum background, and two distinct parametrizations to arrive at two different rational expressions of the same solution. As preparation, we take the C-metric as a simple example to compare the double Kerr-Schild formalism with the ISM perspective, and make an attempt to combine these two seemly unrelated schemes. This idea has never been advocated in the past years. Then after a minor retrospect of the ISM algorithm, we devise a new strategy for the normalization of the obtained metric derived via the ISM construction from the vacuum seed solution, which could facilitate the construction of  \nhigher dimensional solutions by using the trivial vacuum background as the seed metric. Our normalization scheme is corroborated in the Appendix V by deriving the five-dimensional Schwarzschild-Tangherlini solution [22] from the flat vacuum background in two different ways. The remaining two sections, III and IV, are entirely new and constitute the main contribution of this work, where the Mobious transformation of the radial and angular coordinates, and linear combinations of the Killing coordinates as well as various parameter identifications between these two metrics are plainly displayed. Our paper ends with a brief summary of this work, followed by the Aappendix with two simple examples to illuminate our new normalization strategy.", 'A. The C-metric and its seed': 'We begin with two different descriptions of the C-metric, whose line element in the spherical-like coordinates ( t, r, x = cos θ, ¯ φ ),  \nd ˜ s 2 = 1 (1 + αrx ) 2 [ -f ( r ) dt 2 + dr 2 f ( r ) + r 2 dx 2 h ( x ) + r 2 h ( x ) d ˜ φ 2 ] , (1)  \ncan be rewritten in terms of the soliton formalism as  \nd ˜ s 2 ≡ αds 2 = -µ 1 µ 3 µ 2 dt 2 + ρ 2 µ 2 µ 1 µ 3 dφ 2 + µ 1 µ 3 R 2 12 R 2 23 (1 -2 mα ) 2 µ 2 R 2 13 R 11 R 22 R 33 ( dρ 2 + dz 2 ) , (2)  \nwhere the Weyl coordinates are defined as  \nρ = √ r 2 f ( r ) h ( x ) (1 + αrx ) 2 , z = ( αr + x )[(1 + mαx ) r -m ] (1 + αrx ) 2 ,  \nwith two structure functions being  \nf ( r ) = ( 1 -α 2 r 2 ) ( 1 -2 m r ) , h ( x ) = ( 1 -x 2 ) (1 + 2 mαx ) . (3)  \nIn the above, we have also made a rescaling ˜ φ = αφ and introduced the following notations:  \nR kl = ρ 2 + µ k µ l , µ k = √ ρ 2 +( z -z k ) 2 -( z -z k ) , (4)  \nwhere the 3-soliton µ k s are algebraically expressed as  \nµ 1 = (1 -αr )( r -2 m )(1 -x ) (1 + αrx ) 2 , µ 2 = (1 -αr ) r (1 -x )(1 + 2 mαx ) (1 + αrx ) 2 , µ 3 = (1 -α 2 r 2 )(1 + 2 mαx ) α (1 + αrx ) 2 , (5)  \nwhich are associated with three rod end points: z 1 = -m,z 2 = m,z 3 = 1 / (2 α ), respectively.  \nThe zero-mass limit µ 2 ↦→ µ 1 is the Rindler vacuum  \nd ¯ s 2 ≡ αds 2 = -µ 3 dt 2 + ρ 2 µ 3 dφ 2 + µ 3 ρ 2 + µ 2 3 ( dρ 2 + dz 2 ) , (6)  \nwhich will be used later as our seed metric. The Rindler metric can be viewed as 1-soliton transformation from the flat Minkowski background with the metric components: g µν = ( -1 , ρ 2 , 1 , 1). However, it is noted that the Rindler vacuum is trivial and is entirely equivalent to the flat Minkowski metric since they are related via some simple coordinate transformations.  \nSo in the soliton formalism, the C-metric can be interpreted either as a 3-soliton transformation to the flat Minkowski background or a 2-soliton transformation to the Rindler background. In this article, we will take the latter viewpoint and derive the accelerating Kerr-NUT solution via the ISM procedure by using the Rindler background (6) as the seed metric. Note that usually the seed solution need not be exactly the background metric and frequently can be taken as a more complicated even singular one for the technical aim. However, a simple background metric is surely more preferable in practice as a seed solution.  \nOn the other hand, the C-metric (1) admits the double Kerr-Schild formulation as follows:  \nd ˜ s 2 = 1 (1 + αrx ) 2 [ -( 1 -α 2 r 2 ) d ¯ t 2 + dr 2 1 -α 2 r 2 + r 2 dx 2 1 -x 2 + r 2 ( 1 -x 2 ) d ¯ φ 2 ] + 2 m ( 1 -α 2 r 2 ) r (1 + αrx ) 2 ( d ¯ t + dr 1 -α 2 r 2 ) 2 + 2 mαx ( 1 -x 2 ) r 2 (1 + αrx ) 2 ( d ¯ φ + I dx 1 -x 2 ) 2 , (7)  \nafter making the coordinate transformations  \nt = ¯ t -∫ 2 m rf ( r ) dr , ˜ φ = ¯ φ + I ∫ 2 mαx h ( x ) dx.  \nThe zero-mass part is just the Rindler vacuum written in the C-metric coordinates ( x, y ) with y = -1 / ( αr ). In view of this, the C-metric can be interpreted as a perturbation around the Rindler background. Incidentally, it should be pointed out that the above Kerr-Schild ansatz is completely applicable to the most general P-D metric [3].  \nHowever, it seems that the following double quasi-Kerr-Schild prescription,  \nd ˜ s 2 = 1 (1 + αrx ) 2 [ -( 1 -α 2 r 2 ) dt 2 + dr 2 f ( r ) + r 2 dx 2 h ( x ) + r 2 ( 1 -x 2 ) d ˜ φ 2 ] + 2 m ( 1 -α 2 r 2 ) r (1 + αrx ) 2 dt 2 + 2 mαx ( 1 -x 2 ) r 2 (1 + αrx ) 2 d ˜ φ 2 , (8)  \nis in more accordance with the above 2-soliton viewpoint by virtue of the two-dimensional metrics are conformal to each other:  \n1 (1 + αrx ) 2 [ dr 2 f ( r ) + r 2 dx 2 h ( x ) ] /similarequal C f ( ρ, z )( dρ 2 + dz 2 ) .', 'B. ISM: A minor preliminary': "In a D -dimensional spacetime with ( D -2) Killing vectors, if its metric, whose components depend on the Weyl coordinates ρ and z only, is written as [23]  \nds 2 = g ab dx a dx b + f ( ρ, z )( dρ 2 + dz 2 ) , (9)  \nand supplemented with the canonical condition,  \ndet g = -ρ 2 , (10)  \nthen the vacuum Einstein equations are equivalent to the following equations for the ( D -2) × ( D -2) matrix g = ( g ab ):  \n∂ ρ U + ∂ z V = 0 , (11)  \nand for the conformal factor f ( ρ, z ), which can be integrated by quadratures:  \n∂ ρ ln f = -1 ρ + 1 4 ρ Tr( U 2 -V 2 ) , ∂ z ln f = 1 2 ρ Tr( UV ) , (12)  \nwhere the following two matrices are introduced  \nU = √ -g ( ∂ ρ g ) g -1 , V = √ -g ( ∂ z g ) g -1 . (13)  \nSystem (11) is completely integrable and can be viewed as the compatibility condition of the Lax pair for the ( D -2) × ( D -2) generating matrix Ψ ( λ, ρ, z ):  \n∂ ρ Ψ + 2 λρ λ 2 + ρ 2 ∂ λ Ψ = ρ U + λ V λ 2 + ρ 2 Ψ , ∂ z Ψ -2 λ 2 λ 2 + ρ 2 ∂ λ Ψ = ρ V -λ U λ 2 + ρ 2 Ψ , (14)  \nwith λ being a complex spectral parameter independent of ρ and z . When λ = 0, one gets g ( ρ, z ) = Ψ (0 , ρ, z ). One can construct new solitonic solutions from a seed metric via the dressing procedure:  \nΨ = χ χ Ψ 0 , χ χ = I + n ∑ k =1 n † ( k ) m ( k ) λ -µ k , (15)  \nin which n † ( k ) denotes the transpose of the BZ vector n ( k ) defined below, µ k represents a soliton with its corresponding antisoliton being ν k = -ρ 2 /µ k , and the constant z k is the turning point (or rod end point) on the z axes.  \nThe expression for the generated matrix of the general n -soliton solutions is  \n˜ g = g 0 -n ∑ k,l =1 l † ( k ) ( Γ -1 ) kl l ( l ) µ k µ l , (16)  \nwith the following notations being introduced  \nΓ kl = m ( k ) g 0 m † ( l ) ρ 2 + µ k µ l , l ( k ) = m ( k ) g 0 , m ( k ) = m ( k ) 0 Ψ -1 0 ( µ k , ρ, z ) , n ( k ) = n ∑ l =1 ( Γ -1 ) kl l ( l ) µ l , (17)  \nwhere the BZ vectors m ( i ) 0 consist of arbitrary constants with one of which can be set to unity for each vector.  \nIn the case of a diagonal seed, the generating matrix Ψ 0 ( µ i , ρ, z ) can be easily obtained from the seed metric g 0 ( ρ, z ) by simply replacing every soliton or antisoliton via µ k → µ k -λ and ρ 2 → ρ 2 -2 λz -λ 2 , or ρ 2 /µ k → ρ 2 /µ k + λ .  \n/negationslash  \nIn general, the generated metric ˜ g is unphysics due to the fact that it does not fulfill the canonical condition (10): det ˜ g = -ρ 2 , so it should be appropriately normalized.  \nAs far as the normalization scheme is concerned, to the best of our knowledge, there exist three different kinds of strategies to date. (I) Uniform normalization [21]: Like the four-dimensional case, all the metric components of the Killing parts in the higher-dimensional case are multiplied by the same normalization factor. The main defect of this method lies in that it yields some fractional singularities on the symmetry axes, so that the final solution has little physics meaning. (II) Partial block uniform normalization [24-27], which stems from early work [28, 29]: This method only applies the above uniform normalization to a part of the block diagonal matrix of the generated metric, so it is only effective in the case when not all rotation parameters are turned on, for instance, it is limited to the singly rotating case in five dimensions, and so on. (III) Pomeransky's remove-re-add tricky [30]: If the seed metric already satisfies the canonical condition, then after removing and re-adding the same number of solitions or antisolitons, the generated metric automatically subjects to the canonical condition also. At the same time, the conformal factor can be simply computed as f = f 0 det Γ / det Γ 0 , where Γ 0 can be obtained from Γ by setting the nontrivial BZ constants to zero. This is a wise strategy that is widely adopted nowadays, for example, the doubly rotating Myers-Perry solution [31] is regenerated via this method from the five-dimensional Schwarzschild-Tangherlini solution [22]. However, there is a fly in the ointment in that it cannot work effectively in the case when the seed solution is a vacuum background metric where there is no soliton that can be removed. In particular, it is claimed [30] that the five-dimensional Schwarzschild-Tangherlini solution [22] cannot be obtained as a 2-soliton solution on the flat Minkowski background.  \nBelow we propose the fourth scheme for the normalization of the obtained metric derived from the vacuum seed solution, which would be potentially useful for the construction of higher dimensional solutions by using the trivial vacuum background as the seed metric. In the Appendix, we will present two simple examples to illuminate this method. First, let us consider a simple ansatz:  \ng = W ˜ gW , W = diag( w 1 , · · · , w D -2 ) , (18)  \nby simply employing a diagonal matrix W with which each of its entries satisfies the Laplace equation:  \n∇ 2 ln w i = ( ∂ 2 ρ + 1 ρ ∂ ρ + ∂ 2 z ) ln w i = 0 . (19)  \nThe solution of each entry in the above equation has the general form: w i = ρ b i µ c i i , in which b i and c i are arbitrary constants. The entries in W are chosen so that -ρ 2 = w 2 1 · · · w 2 D -2 det ˜ g is satisfied. By virtue of the fact that det ˜ g is proportional to det g 0 , actually one only needs to choose different powers in w i .  \nIf the generated metric is still diagonal just as its seed metric, then the above means amounts to g = W 2 ˜ g , and it is easy to compute the conformal factor. On the other hand, if all the diagonal entries are identical w i = w , then the above scheme is equivalent to the original uniform normalization strategy. Suppose that only some of the  \nwhere  \ndiagonal entries are set to be identical, then this amounts to the partial block uniform normalization. Finally, if all w i s are appropriately chosen such as ( -ρ 2 /µ 2 i ) or ( -µ 2 i /ρ 2 ), this scheme is actually equivalent to removing solition or antisoliton at least in the case of the generated metric being static (diagonal).  \nSecond, let us handle the more general rotating case. Since the generated metric in general still needs to make further linear combinations of the Killing coordinates, that is, ˆ g = A † gA , where the rotation matrix A is unimodular and its every entry is a constant, then we can have ˆ g = ˆ W † ˜ g ˆ W , where ˆ W = WA , ˆ W † is the matrix transpose of ˆ W and each of its entry obeys the above Laplace equation (19). Clearly it is troublesome to choose a suitable matrix ˆ W so that the final metric is the expected one. However, one need not worry about this, since the Pomeransky's removere-add tricky is already very effective to generate the rotating solution, so our method just plays a nice complement to the case where the Pomeransky's tricky fails to work.", 'C. ISM construction of the accelerating Kerr-NUT solution': "Although the accelerating Kerr-NUT solution can be derived as a 3-soliton transformation via the ISM procedure from the flat Minkowski background, in accordance with the above double Kerr-Schild perspective, we prefer to rederive it as a 2-soliton transformation by using the Rindler background (6) as the seed metric, whose Killing part is  \ng 0 = diag( -µ 3 , ρ 2 /µ 3 ) = diag( -µ 3 , -ν 3 ) , (20)  \nwith the conformal factor f 0 = µ 3 /R 33 , so that the generating matrix is easily obtained as  \nΨ 0 = diag( λ -µ 3 , λ -ν 3 ) = diag( λ -µ 3 , λ + ρ 2 /µ 3 ) . (21)  \nNow we perform a 2-soliton transformation to this Rindler background: add a soliton µ 1 at z = z 1 with the vector m (1) 0 = ( C 1 , 1) and a soliton µ 2 at z = z 2 with the vector m (2) 0 = (1 , C 2 ), respectively. Note that we take the 2-soliton in the order: µ = ( µ 1 , µ 2 ), so that the generated metric exactly reproduces the same form (2) for the C-metric when the BZ parameters are set to zero: C 1 = C 2 = 0. Then two BZ vectors constitute a 2 × 2 matrix:  \nm = ( C 1 µ 13 µ 3 R 13 1 µ 23 C 2 µ 3 R 23 ) , l = ( -C 1 µ 3 µ 13 ρ 2 R 13 -µ 3 µ 23 C 2 ρ 2 R 23 ) (22)  \nwhere a short notation is included for briefness: µ ij = µ i -µ j . The Γ matrix is also recorded as follows:  \nΓ = ( -C 2 1 µ 3 µ 2 13 R 11 + µ 3 ρ 2 R 2 13 R 11 -C 1 µ 3 µ 13 µ 23 R 12 + C 2 µ 3 ρ 2 R 12 R 13 R 23 -C 1 µ 3 µ 13 µ 23 R 12 + C 2 µ 3 ρ 2 R 12 R 13 R 23 -µ 3 µ 2 23 R 22 + C 2 2 µ 3 ρ 2 R 2 23 R 22 ) . (23)  \nThe final physics metric can be obtained via either the uniform normalization scheme or our prescription proposed in the above  \ng = W ˜ gW = -µ 1 µ 2 ρ 2 ˜ g , (24)  \nin which we have taken W = diag( √ -µ 1 µ 2 /ρ , √ -µ 1 µ 2 /ρ ). The conformal factor is  \nf = -16 k 2 f 0 µ 3 1 µ 3 2 ρ 2 µ 2 12 det Γ = 16 k 2 B H . (25)  \nThen the generated metric represents the accelerating Kerr-NUT solution, whose line element reads  \nds 2 = -µ 3 A µ 1 µ 2 B ( dt + µ 12 C µ 3 A dφ ) 2 + ρ 2 µ 1 µ 2 B µ 3 A dφ 2 +16 k 2 B H ( dρ 2 + dz 2 ) , (26)  \n[ ]  \nH = µ 2 12 µ 2 13 µ 2 23 µ 3 1 µ 3 2 µ 3 3 R 2 12 R 2 13 R 2 23 R 11 R 22 R 33  \nA = C 1 ( C 2 µ 2 µ 23 R 12 -ρµ 12 R 23 ) R 13 -µ 1 µ 13 ( C 2 ρµ 2 µ 12 µ 23 + R 12 R 23 ) × [ C 1 ( C 2 µ 2 µ 23 R 12 + ρµ 12 R 23 ) R 13 + µ 1 µ 13 ( C 2 ρµ 2 µ 12 µ 23 -R 12 R 23 ) ] , B = C 2 1 ( C 2 2 µ 2 23 R 2 12 + µ 2 12 R 2 23 ) R 2 13 -2 C 1 C 2 µ 13 µ 23 R 13 R 23 R 11 R 22 + ( C 2 2 ρ 4 µ 2 12 µ 2 23 + R 2 12 R 2 23 ) µ 2 13 , C = C 1 µ 2 µ 13 ( C 2 2 ρ 2 µ 23 -R 2 23 ) R 11 R 12 R 13 + C 2 µ 1 µ 23 ( C 2 1 R 2 13 -ρ 2 µ 2 13 ) R 12 R 22 R 23 , .  \nWhen C 1 = C 2 = 0, the above generated 2-soliton solution just reduces to the C-metric (2). So according to the perspective in Sec. II A, the generated accelerating Kerr-NUT solution (26) can be interpreted as a Kerr-NUT black hole sitting on the Rindler background.  \nNote that the above accelerating Kerr-NUT solution (26) can also be regenerated via the 'remove-re-add' tricky from the C-metric (2). The process is briefly sketched as follows. We first remove an antisoliton ν 1 at z = z 1 and a soliton µ 2 at z = z 2 both with the trivial vectors (1 , 0), then rescale the metric by a factor µ 1 /µ 2 . This essentially generates the above Rindler seed (20):  \ng 0 = µ 1 µ 2 diag ( µ 2 2 µ 2 1 , 1 ) × diag ( -µ 1 µ 3 µ 2 , ρ 2 µ 2 µ 1 µ 3 ) .  \nNext we re-add the antisoliton ν 1 with the vector m (1) 0 = (1 , -C 1 ) and the soliton µ 2 with the vector m (2) 0 = (1 , C 2 ), and multiply the generated metric by a factor µ 2 /µ 1 so that the final metric is physics because it satisfies the canonical condition: det g = -ρ 2 . The conformal factor can be easily evaluated via f = f 00 det Γ / det Γ ( C 1 = 0 , C 2 = 0), where f 00 is the conformal factor given by Eq. (2).", "D. Astorino's parametrization": "We first transform from the Weyl-Lewis-Papapetrou coordinates ( ρ, z ) to the spherical-like coordinates ( r, x ) via  \nρ = √ ∆( r ) P ( x ) (1 + αrx ) 2 , z = ( x + αr ) [ ( r -m )(1 + mαx ) + σ 2 αx ] (1 + αrx ) 2 ,  \nwhere  \n∆( r ) = [ ( r -m ) 2 -σ 2 ]( 1 -α 2 r 2 ) , P ( x ) = ( 1 -x 2 )[ (1 + mαx ) 2 -σ 2 α 2 x 2 ] , (27)  \nThe locations of three turning points are taken to be  \nin which σ = √ m 2 + n 2 -a 2 .  \nz 1 = -σ, z 2 = σ, z 3 = 1 -α 2 ( m 2 -σ 2 ) 2 α ,  \nso their corresponding solitons are purely algebraic:  \nµ 1 = (1 -αr )( r -m -σ )[1 + ( m -σ ) αx )](1 -x ) (1 + αrx ) 2 , µ 2 = (1 -αr )( r -m + σ )[1 + ( m + σ ) αx )](1 -x ) (1 + αrx ) 2 , (28) µ 3 = ( 1 -α 2 r 2 ) [1 + ( m -σ ) αx )][1 + ( m + σ ) αx )] α (1 + αrx ) 2 ,  \nwhich reduce to Eq. (5) exactly when the static C-metric limit is reached ( σ = m ).  \nIt is very tedious to convert the four functions ( A,B,C,H ) into the expressions in terms of the spherical-like coordinates. However, if we take  \nC 1 = ( m -σ )[1 + ( m + σ ) α ] ( a -n )[1 -( m -σ ) α ] , C 2 = ( m -σ )[1 -( m + σ ) α ] ( a + n )[1 + ( m -σ ) α ] , (29)  \nand  \nk = ( m + σ )[1 + ( m -σ ) α ][1 -( m -σ ) α ] 2 α 3 / 2 √ 1 + α 2 a 2 , (30)  \nthen we are able to first write the two-dimensional metric as  \nf ( dρ 2 + dz 2 ) = Σ ( 1 + α 2 a 2 ) (1 + αrx ) 2 [ dr 2 ∆( r ) + dx 2 P ( x ) ] ,  \nin which  \nΣ = r 2 +( ax -n ) 2 +2 nαr [ a ( 1 + x 2 ) -2 nx ] + α 2 r 2 ( a -nx ) 2 + ( a 2 -n 2 ) 2 α 2 x 2 . (31)  \nFinally, after some cumbersome algebraic simplifications, the Killing parts can be converted into  \ng ab dx a dx b = -F (1 + αrx ) 2 Σ α ( dt -ωα F dϕ ) 2 + Σ∆( r ) P ( x ) α (1 + αrx ) 2 F dϕ 2 ,  \nwhere  \nF = [ 1 -( a 2 -n 2 ) α 2 x 2 ] 2 ∆( r ) -[ a ( 1 + α 2 r 2 ) +2 nαr ] 2 P ( x ) , ω = [ a ( 1 + x 2 ) -2 nx ][ 1 -( a 2 -n 2 ) α 2 x 2 ] ∆( r ) + ( r 2 + n 2 -a 2 )[ a ( 1 + α 2 r 2 ) +2 nαr ] P ( x ) .  \nAfter rescaling the Killing coordinates as t ↦→ √ αt , ϕ ↦→ ϕ/ √ α , the above two parts are collectively written in terms of the Lewis-Papapetrou coordinates as [7]  \nds 2 = -F (1 + αrx ) 2 Σ ( dt -ω F dϕ ) 2 + Σ (1 + αrx ) 2 F { ∆( r ) P ( x ) dϕ 2 + F 1 + α 2 a 2 [ dr 2 ∆( r ) + dx 2 P ( x ) ] } , (32)  \nwhich can also be recast into the Boyer-Lindquist-like form  \nds 2 = 1 (1 + αrx ) 2 { -∆( r ) X 2 + P ( x ) Y 2 Σ + Σ 1 + α 2 a 2 [ dr 2 ∆( r ) + dx 2 P ( x ) ] } , (33)  \nwhere  \nX = [ 1 -( a 2 -n 2 ) α 2 x 2 ] dt + [ 2 nx -a ( 1 + x 2 )] dϕ, Y = [ a ( 1 + α 2 r 2 ) +2 nαr ] dt + ( r 2 + n 2 -a 2 ) dϕ. (34)  \nIn Sec. III (especially Sec. IIIB) of Ref. [7], the above metric (32) was derived through an equivalent limiting procedure ( a 2 → 0 , l 2 → 0 and then w 4 →∞ ) on the double Kerr-NUT solution. It was also claimed there that this spacetime is of type D, and might be included into the P-D family of black holes. Here we have built this metric with two solitons on a Rindler accelerating background. In the subsequent subsections, we will give an explicit verification of the Astorino's insight by showing that the solution (33) is indeed diffeomorphic to the P-D metric.  \nIn the absence of the NUT charge, the above metric simply recovers the accelerating Kerr solution in the familiar form after further shifting the Killing coordinates:  \nt ↦→ t -2 aϕ √ 1 + α 2 a 2 , ϕ ↦→ -α 2 at + ( 1 -α 2 a 2 ) ϕ √ 1 + α 2 a 2 .  \nOn the other hand, the accelerating NUT C-metric takes a simple expression:  \nds 2 = 1 (1 + αrx ) 2 { -∆( r ) Σ [( 1 + n 2 α 2 x 2 ) dt +2 nxdϕ ] 2 + P ( x ) Σ [ 2 nαrdt + ( r 2 + n 2 ) dϕ ] 2 +Σ [ dr 2 ∆( r ) + dx 2 P ( x ) ] } , (35)  \nin which  \n∆( r ) = ( r 2 -2 mr -n 2 )( 1 -α 2 r 2 ) , P ( x ) = ( 1 -x 2 )( 1 + 2 mαx -n 2 α 2 x 2 ) , Σ = ( r 2 + n 2 )( 1 + n 2 α 2 x 2 ) -4 n 2 αrx. (36)  \nWhen the accelerating parameter is set to zero, the above type-D NUT C-metric apparently reduces to the Taub-NUT solution, and also it obviously recovers the usual C-metric when the NUT charge vanishes. There is a natural question remaining, namely, whether it is superficially accelerating just as pointed out in Ref. [6] for the traditional form. Then, if it is so, we still lack a type-D NUT C-metric. However, this case seems unlikely. So, we will assume that the above NUT C-metric is not isometric to the Taub-NUT solution in the remaining context, but clearly an obvious proof is still needed.  \nThe above solution has also been generalized in Ref. [8] to include the electric and magnetic charges as well as a nonzero cosmological constant. However, it suffices enough for our aim to turn off these three parameters. Also, in our opinion, it becomes useful only in the static case (where the rotation parameter vanishes), as we will demonstrate the above rotating and accelerating solution can be related to the traditional form of the general P-D metric.", "E. Griffiths-Podolsk'y's parametrization": 'We next transform from the Weyl-Lewis-Papapetrou coordinates ( ρ, z ) to the spherical-like coordinates (˜ r, ˜ x ) via  \nwhere  \nρ = √ ˜ ∆(˜ r ) ˜ P (˜ x ) [1 + ˜ α ˜ r (˜ n +˜ a ˜ x )] 2 , z = [ ˜ x + ˜ α ˜ r (˜ a + ˜ n ˜ x ) ][ ˜ r -˜ m + ˜ α (˜ n +˜ a ˜ x ) ( ˜ m ˜ r + ˜ n 2 -˜ a 2 )] [ 1 + ˜ α ˜ r (˜ n +˜ a ˜ x ) ] 2 ,  \n˜ ∆(˜ r ) = ( ˜ r 2 -2 ˜ m ˜ r +˜ a 2 -˜ n 2 )[ (1 + ˜ α ˜ n ˜ r ) 2 -˜ α 2 ˜ a 2 ˜ r 2 ] , ˜ P (˜ x ) = ( 1 -˜ x 2 )[ 1 + 2 ˜ m ˜ α (˜ n +˜ a ˜ x ) + ( ˜ a 2 -˜ n 2 ) ˜ α 2 (˜ n +˜ a ˜ x ) 2 ] , (37)  \nThe locations of three rod end points are now taken to be  \nin which ˜ σ = √ ˜ m 2 + ˜ n 2 -˜ a 2 .  \nz 1 = -˜ σ, z 2 = ˜ σ, z 3 = 1 + 2˜ α ˜ m ˜ n -˜ α 2 ( ˜ a 2 -˜ n 2 ) 2 2˜ α ˜ a ,  \nand their corresponding solitons are also purely algebraic:  \nµ 1 = [1 -˜ α (˜ a -˜ n )˜ r ](˜ r -˜ m -˜ σ )[1 + ( ˜ m -˜ σ )˜ α (˜ n +˜ a ˜ x )](1 -˜ x ) [1 + ˜ α ˜ r (˜ n +˜ a ˜ x )] 2 , µ 2 = [1 -˜ α (˜ a -˜ n )˜ r ](˜ r -˜ m + ˜ σ )[1 + ( ˜ m + ˜ σ )˜ α (˜ n +˜ a ˜ x )](1 -˜ x ) [1 + ˜ α ˜ r (˜ n +˜ a ˜ x )] 2 , (38) µ 3 = [1 + ˜ α (˜ a + ˜ n )˜ r ][1 -˜ α (˜ a -˜ n )˜ r ][1 + ( ˜ m -˜ σ )˜ α (˜ n +˜ a ˜ x )][1 + ( ˜ m + σ )˜ α (˜ n +˜ a ˜ x )] ˜ α [1 + ˜ α ˜ r (˜ n +˜ a ˜ x )] 2 .  \nIt is troublesome to convert the four functions ( A,B,C,H ) into the expressions in terms of the spherical-like coordinates. A simple trick to do this relatively easily is to avoid using the square root, such as, exploiting identities: ˜ σ 2 k = ( ˜ m 2 +˜ n 2 -˜ a 2 ) k and ˜ σ 2 k +1 = ( ˜ m 2 +˜ n 2 -˜ a 2 ) k ˜ σ can efficiently facilitate to rationalize the expressions containing the square root. This experience is also very effective to manipulate the phantom soliton whose square-root function could not be reduced to a simple algebraic expression.  \nIn order to let the conformal factor include a familiar multiplier ˜ Σ = ˜ r 2 +(˜ n +˜ a ˜ x ) 2 , we have to take  \nC 1 = (˜ a -˜ n )[1 + ( ˜ m + ˜ σ )˜ α (˜ a + ˜ n )] ( ˜ m + ˜ σ )[1 -( ˜ m -˜ σ )˜ α (˜ a -˜ n )] , C 2 = (˜ a + ˜ n )[1 -( ˜ m + ˜ σ )˜ α (˜ a -˜ n )] ( ˜ m + ˜ σ )[1 + ( ˜ m -˜ σ )˜ α (˜ a + ˜ n )] , (39)  \nand  \nk = ( ˜ m + ˜ σ )[1 + ( ˜ m -˜ σ )˜ α (˜ a + ˜ n )][1 -( ˜ m -˜ σ )˜ α (˜ a -˜ n )] 2(˜ α ˜ a ) 3 / 2 √ 1 + ˜ α 2 ( ˜ a 2 -˜ n 2 ) 2 . (40)  \nFinishing the remaining task of the algebraic process, we find that it is instructive to cast the metric into the Boyer-Lindquist-like form rather than the Lewis-Papapetrou form, which is rather concise and simple enough:  \nd ˜ s 2 = 1 [ 1 + ˜ α ˜ r (˜ n +˜ a ˜ x ) ] 2 { -˜ ∆(˜ r ) ˜ X 2 + ˜ P (˜ x ) ˜ Y 2 ˜ Σ + ˜ Σ [ d ˜ r 2 ˜ ∆(˜ r ) + d ˜ x 2 ˜ P (˜ x ) ] } , (41)  \n˜ X = [ 1 -( ˜ a 2 -˜ n 2 ) ˜ α 2 (˜ n +˜ a ˜ x ) 2 ] dt + ˜ α ˜ a [ 2˜ n ˜ x +˜ a ( 1 + ˜ x 2 )] dϕ √ ˜ α ˜ a √ 1 + ˜ α 2 ( ˜ a 2 -˜ n 2 ) 2 , ˜ Y = ˜ a [ 1 + ( ˜ a 2 -˜ n 2 ) ˜ α 2 ˜ r 2 ] dt -˜ α ˜ a ( ˜ r 2 + ˜ n 2 -˜ a 2 ) dϕ √ ˜ α ˜ a √ 1 + ˜ α 2 ( ˜ a 2 -˜ n 2 ) 2 . (42)  \nwhere ˜ Σ = ˜ r 2 +(˜ n +˜ a ˜ x ) 2 and  \nIn the above frame, the Rindler background or the accelerating horizons is rest relative to the conformal infinity.  \nOne can note that the above metric resembles somewhat the one derived in the last subsection. In particular, they only differ by an overall constant factor in the case without the NUT charge and by further making identifications: α = ˜ α ˜ a , m = ˜ m , a = ˜ a , r = ˜ r , x = ˜ x .  \nThe final step that we need to recast the above solution into the familiar form of the general P-D metric is to make the following linear combinations of the Killing coordinates:  \nt = √ ˜ α ˜ a ( ˜ t -2˜ a ˜ φ ) √ 1 + ˜ α 2 ( ˜ a 2 -˜ n 2 ) 2 , ϕ = ˜ a ˜ α 2 ( ˜ a 2 -˜ n 2 ) ˜ t + [ 1 -˜ α 2 ( ˜ a 4 -˜ n 4 )] ˜ φ √ ˜ α ˜ a √ 1 + ˜ α 2 ( ˜ a 2 -˜ n 2 ) 2 ,  \nwhich yield the familiar expressions:  \n˜ X = d ˜ t + [ 2˜ n ˜ x -˜ a ( 1 -˜ x 2 )] d ˜ φ, ˜ Y = ˜ ad ˜ t -( ˜ r 2 +˜ a 2 + ˜ n 2 ) d ˜ φ.  \nNote that in this new frame, the Rindler horizon is rotating relative to the conformal infinity.  \nTo summarize, we have rederived the accelerating Kerr-NUT solution via the ISM procedure from the Rindler background seed. Then using two different parametrizations, we have recast the solution into two distinct forms in terms of the spherical-like coordinates, namely, the one novelly given by Ref. [7], and another by the traditional one. Since the latter is already known to be of type D, so is the one delivered by Astorino in [7]. However, one still needs to find what coordinate transformations may relate them. We leave this subject to the next section.  \nBy the way, here we would like to mention the related work [18] that dealt with the Euclidean P-D metric. In the Sec. VC of that paper, Chen obtained the Euclidean accelerating Kerr-NUT metric as a 3-soliton solution via the ISM procedure. Then he performed a Mobius transformation on the coordinates ( u, v ) and redefined the Killing coordinates, and further made various parameter identifications so that the 3-soliton solution is brought to the Euclidean P-D solution in the familiar form. This hints that the Mobius transformation may be relevant to the present work.', 'III. M OBIUS TRANSFORMATION AMONGST TWO METRICS': 'In order to find the relation between the above two metrics, a direct routine is to examine the solutions given by Eqs. (33) and (41), respectively, since both of them adopt the same Killing coordinates ( t, ϕ ). Therefore, the remaining most possible coordinate transformation is the Mobius transformation amongst the radial and azimuthal coordinates, respectively, together with further parameter identifications.  \nLet us consider the following Mobius transformation:  \n˜ x = x -ν 1 -ν x , ˜ α ˜ r = αr + ν ˜ a (1 + ναr ) -˜ n ( αr + ν ) , (43)  \nsuch that ˜ x = ± 1 transform to x = ± 1 under the first one, and the second one only involves mapping between the acceleration horizons but not the event horizons among both metrics. The inverse of the second transformation is  \nαr = (˜ a -ν ˜ n )˜ α ˜ r -ν 1 + (˜ n -ν ˜ a )˜ α ˜ r . (44)  \nHere and hereafter we prefer to leave ν to be a freely variable parameter so that the relation between the two mass parameters is left to be determined only in the final step.  \nWe first apply the above transformation to ˜ Σ = ˜ r 2 +(˜ n + ˜ a ˜ x ) 2 and find its relation to Σ. To do so, it is easy to find that if the identity ˜ Σ(1 -ν x ) 2 [˜ a (1 + ναr ) -˜ n ( αr + ν )] 2 = C Σ holds true, then we must have the following conditions:  \nn a = -( 1 + ν 2 ) ˜ n -2 ν ˜ a ( 1 + ν 2 ) ˜ a -2 ν ˜ n , a 2 C = ˜ a (˜ n -ν ˜ a )(˜ a -ν ˜ n ) [( 1 + ν 2 ) ˜ a -2 ν ˜ n ] 2 ( 1 + ν 2 ) ˜ n -2 ν ˜ a , ˜ α 2 = -ν (˜ n -ν ˜ a )(˜ a -ν ˜ n ) ( ˜ a 2 -˜ n 2 ) , α 2 a 2 = -ν (˜ a -ν ˜ n ) [( 1 + ν 2 ) ˜ a -2 ν ˜ n ] 2 ( 1 -ν 2 ) 2 (˜ n -ν ˜ a ) ( ˜ a 2 -˜ n 2 ) . (45)  \nNext, we seek the links between the one-forms ( ˜ X, ˜ Y ) in (42) and their counterparts ( X,Y ) without rescaling the Killing coordinates:  \nX = [ 1 -( a 2 -n 2 ) α 2 x 2 ] dt √ α + [ 2 nx -a ( 1 + x 2 )] √ αdϕ, Y = [ a ( 1 + α 2 r 2 ) +2 nαr ] dt √ α + ( r 2 + n 2 -a 2 ) √ αdϕ.  \nIt is relatively easy to examine √ ˜ α ˜ a √ 1 + ˜ α 2 ( ˜ a 2 -˜ n 2 ) 2 ( ˜ X, ˜ Y ) rather than ( ˜ X, ˜ Y ). We further find  \n√ ˜ α ˜ a √ 1 + ˜ α 2 ( ˜ a 2 -˜ n 2 ) 2 (1 -ν x ) 2 ˜ X = ˜ a √ α ( 1 -ν 2 ) ˜ a -ν ˜ n X, √ ˜ α ˜ a √ 1 + ˜ α 2 ( ˜ a 2 -˜ n 2 ) 2 [˜ a (1 + ναr ) -˜ n ( αr + ν )] 2 ˜ Y = -˜ a 2 α 3 / 2 ( 1 -ν 2 ) ˜ α (˜ a -ν ˜ n ) Y , (46)  \nand must additionally impose  \na = -˜ α (˜ a -ν ˜ n ) [( 1 + ν 2 ) ˜ a -2 ν ˜ n ] α ( 1 -ν 2 ) , (47)  \nthen we can write  \nn = ˜ α (˜ a -ν ˜ n ) [( 1 + ν 2 ) ˜ n -2 ν ˜ a ] α ( 1 -ν 2 ) , C = -α 2 ˜ a (˜ n -ν ˜ a ) 2 ( ˜ a 2 -˜ n 2 )( 1 -ν 2 ) 2 ν [( 1 + ν 2 ) ˜ n -2 ν ˜ a ] . (48)  \nIn practice, if we first find the links between ( ˜ X, ˜ Y ) and ( X,Y ) directly, then all the above relations can be obtained, and it can be checked that the identity Σ(1 -ν x ) 2 [˜ a (1 + ναr ) -˜ n ( αr + ν )] 2 = C Σ is also automatically fulfilled.  \n˜ Note that the relation for α 2 listed above then becomes the identity for ˜ α 2 , so α can now temporarily be viewed as a free parameter too. Alternatively, we can also express this in terms of untilded parameters as follows:  \nα 2 = ν ( a -ν n ) ( n -ν a ) ( a 2 -n 2 ) .  \nOur next step is to find the relation between the mass parameters. Using the following identities,  \n1 -˜ x 2 = 1 -ν 2 (1 -ν x ) 2 ( 1 -x 2 ) , d ˜ x 2 1 -˜ x 2 = 1 -ν 2 (1 -ν x ) 2 dx 2 1 -x 2 ,  \nwe can simply let  \nd ˜ x 2 ˜ P (˜ x ) = ( 1 -ν 2 ) dx 2 P ( x ) , (49)  \nso that after using the above relation for ˜ α 2 , we find that this is true only when we further set  \n˜ m = ν 2˜ α (˜ a -ν ˜ n ) , m = ν ( ν 2 ˜ a -3˜ a +2 ν ˜ n ) 2 α (˜ a -ν ˜ n ) , (50)  \nwhich can also be attained by exploiting the radial part.  \nFinally, with the help of the following useful expressions,  \n1 + ˜ α ˜ r (˜ n +˜ a ˜ x ) = ˜ a ( 1 -ν 2 ) (1 + αrx ) (1 -ν x )[˜ a (1 + ναr ) -˜ n ( αr + ν )] , ˜ Σ = C Σ (1 -ν x ) 2 [˜ a (1 + ναr ) -˜ n ( αr + ν )] 2 , ˜ ∆(˜ r ) = ( 1 -ν 2 ) ˜ a 2 α 2 ˜ α 2 [˜ a (1 + ναr ) -˜ n ( αr + ν )] 4 ∆( r ) , ˜ P (˜ x ) = 1 -ν 2 (1 -ν x ) 4 P ( x ) , d ˜ r 2 ˜ ∆(˜ r ) = ( 1 -ν 2 ) dr 2 ∆( r ) ,  \n(1  \n-  \nν  \n2  \n-  \nν  \n˜  \na  \n)  \nand the relations (46) between ( ˜ X, ˜ Y ) and ( X,Y ), we can show that the two metrics only differ by an overall constant factor:  \nprovided that we set  \nwhich yields an equation to determine α :  \n˜ α α [( 1 + ν 2 ) ˜ n -2 ν ˜ a ] = ( 1 -ν 2 ) 2 ˜ n -ν ˜ a ˜ a -ν ˜ n -ν ˜ a 2 -˜ n 2 [ ( 1 + ν 2 ) ˜ a -2 ν ˜ n ] 2 . (52)  \n˜ α 2 α 2 = ( ν 4 +1 ) ˜ a 2 -ν ( 3 ν 2 +1 ) ˜ a ˜ n + ( 3 ν 2 -1 ) ˜ n 2 ˜ a (˜ a -ν ˜ n ) ( ˜ a 2 -˜ n 2 )( 1 -ν 2 ) . (53)  \nThus we have generally proven that the two metrics given in the last two sections are isometric to each other. It seems likely to let C = ˜ a 2 ( 1 -ν 2 ) / ( 1 + α 2 a 2 ) so that both metrics are completely identical. This condition implies  \nWhen combined with Eq. (52), they yield a cubic equation for ˜ a/ ˜ n :  \nν ( ν 4 -ν 2 +1)˜ a 3 -ν 2 (3 ν 2 -2)˜ a 2 ˜ n + ν (3 ν 2 -4)˜ a ˜ n 2 + ˜ n 3 = 0 . (54)  \nOne can also conduct a similar task as done in the above by using the C-metric-like coordinates ( x, y ) with y = -1 / ( αr ) and (¯ x, ¯ y ) defined via ¯ y = -1 / (˜ α ˜ r ), ¯ x = ˜ n + ˜ a ˜ x , rather than by employing the spherical-like coordinates. In this case, both metrics become more symmetric, and now ˜ a behaves like a scale parameter; for example, ¯ y = ˜ n + ˜ a ( y -ν ) / (1 -ν y ). Since the results remain the same as those given in the above, we will not repeat this work although it might be relatively easy.', 'What limit should be taken in the original P-D metric to get the type-D NUT C-metric?': "Previously, Griffiths and Podolsk'y [6] demonstrated that in the limit of zero angular momentum parameter, the NUT C-metric is apparently accelerating and can be transformed into the Taub-NUT solution; as a result of this, the type-D NUT C-metric seems to be absent from the most general family of the type-D P-D solution, leaving a puzzle for almost two decades. The reason for this is probably that they did not consider the Mobius transformation. On the other hand, it is a simple matter to just set a = 0 in the Astorino's expression [7] to arrive at the type-D C-metric, but this still does not explain how to get the type-D NUT C-metric from the most general P-D solution.  \nTo elucidate the lesson why the type-D NUT C-metric is missing from the P-D metric, let us consider what limit we should take in the original P-D metric to get the expected type-D NUT C-metric. Let us first assume that the transformation parameter ν and all the solution parameters ( ˜ m, ˜ n, ˜ a, ˜ α ) as well as the radial coordinate ˜ r are all finite and nonzero (although ˜ a can be set to zero). To achieve the static NUT C-metric, we must first let the rotation parameter a tend to zero ( a → 0). Then from the first identity in (45), the NUT charge parameter n must also tend to a very tiny constant ( n /similarequal 0) so that the ratio n/a keeps finite. From the fourth one in (45) or (47), we know that the acceleration parameter α must approach the infinity ( α → ∞ ) so that αa is also finite. The second identity in (45) tells us that the overall factor C should tend to the infinity ( C →∞ ) so that Cα can be kept finite. From the inverse Mobius transformation (44), we can recognize that the radial coordinate r must approach zero so that αr remains finite. Finally, from the second equation of Eq. (50), we learn that the mass parameter m should tend to a very tiny constant ( m /similarequal 0). In the a → 0 limit, both ( αa , αn , αm ) and αr remains fixed.  \nTo summarize, in order to get the type-D NUT C-metric, the corresponding limit is the following set:  \na → 0 , n /similarequal 0 , m /similarequal 0 , r → 0 , α →∞ , (55)  \napart from the appropriate rescaling of the whole line element plus the linear combinations of the Killing coordinates ( t , ϕ ). However, the key measure is to consider the above Mobius transformation (43).  \nd ˜ s 2 = C Σ ( 1 -ν 2 ) ˜ a 2 (1 + αrx ) 2 [ dr 2 ∆( r ) + dx 2 P ( x ) ] + ˜ aα 3 ( 1 -ν 2 ) C ˜ α 3 (˜ a -ν ˜ n ) 2 [ 1 + ˜ α 2 ( ˜ a 2 -˜ n 2 ) 2 ] -∆( r ) X 2 + P ( x ) Y 2 (1 + αrx ) 2 Σ = C ˜ a 2 ( 1 -ν 2 ) { Σ (1 + αrx ) 2 [ dr 2 ∆( r ) + dx 2 P ( x ) ] + n (1 -ν 2 )(˜ n -ν ˜ a ) -∆( r ) X 2 + P ( x ) Y 2 (1 + αrx ) 2 Σ } = C ( 1 + α 2 a 2 ) ˜ a 2 ( 1 -ν 2 ) ds 2 , (51)  \n1 +  \nα  \n2  \na  \n2  \n=  \nn  \n)(˜  \nn", 'IV. TRANSFORMING THE TYPE-D NUT C-METRIC INTO THE FAMILIAR P-D FORM': "In this section, we will consider another obvious coordinate transformation that brings the type-D NUT C-metric (35) into the familiar P-D form. To this end, let us introduce the following coordinate transformations:  \nt = 4 nµ ( ˜ t + ˜ φ ) , ϕ = 4 nµα ( ˜ t -˜ φ ) , r = n q +1 q -1 , x = 1 -p nα (1 + p ) , m = n 2 µ , (56)  \nwhich bring the NUT C-metric (35) into the familiar P-D form:  \nds 2 = 8 n 2 µ ( p -q ) 2 { Q ( q ) ( d ˜ t + p 2 d ˜ φ ) 2 + P ( p ) ( q 2 d ˜ t -d ˜ φ ) 2 1 + p 2 q 2 + ( 1 + p 2 q 2 ) [ dq 2 -Q ( q ) + dp 2 P ( p ) ] } , (57)  \nwhere the structure function reads  \nP ( ξ ) = [ n 2 α 2 (1 + ξ ) 2 -(1 -ξ ) 2 ]( 1 + 4 µξ -ξ 2 ) , (58)  \nin which α can be simply set to unity. The only nonzero Kretschmann invariant is  \nR abcd R abcd = 3 α 4 ( p -q ) 6 µ 2 ˜ ν 4 (1 + p 2 q 2 ) 6 [ 2 µ ( 1 -˜ ν 2 ) pq ( p 2 q 2 -3 ) + ( ˜ ν 2 +1 )( 3 p 2 q 2 -1 )] × [ 2 µ ( 1 -˜ ν 2 )( 3 p 2 q 2 -1 ) -( ˜ ν 2 +1 ) pq ( p 2 q 2 -3 )] , (59)  \nwhere ˜ ν = nα . This Kretschmann invariant has a similar factorized structure form as that of the Taub-NUT solution, but it does not coincide with the latter because the former contains a conformal factor ( p -q ) 6 . It demonstrates that the type-D NUT C-metric cannot reduce to the Taub-NUT solution unless the acceleration parameter vanishes ( α = 0).  \nFrom the above expression for the structure function, it is clear that this quartic function is factorized in an entirely distinct manner from the traditional way. This perhaps is the primary reason why the type-D NUT C-mtric is 'lost' from the most general P-D solution.", 'V. CONCLUDING REMARKS': "In this paper, we have shown that the type-D accelerating Kerr-NUT metric presented in Ref. [7] is not a new solution in the sense that it is diffeomorphic to the original P-D solution, but it is still very convenient for further investigations in the nonrotating case where the angular momentum parameter vanishes, namely, the type-D static accelerating solution with a nonzero NUT charge since this solution is unavailable in the previous literature. We have justified this point from two aspects. First, we have applied the ISM procedure to generate the accelerating Kerr-NUT solution from the seed metric - Rindler vacuum background, which can be equivalently obtained as in Ref. [7] via a limiting procedure ( a 2 → 0 , l 2 → 0 and then w 4 → ∞ ) on the double Kerr-NUT solution. Then we adopted two different parametrizations to derive the novel metric given by Astorino in Ref. [7] and the traditional one early delivered by Griffiths and Podolsk'y. This clearly demonstrates that these two metrics are isometric, though displayed in two different forms. Second, we have also provided the obvious coordinate transformations (i.e., Mobius transformation of the radial and angular coordinates and linear combinations of the Killing coordinates) and parameter identifications between these two line elements. Therefore, our work provides an explicit verification of the insight of Ref. [7] that the obtained spacetimes not only are of type D, but also might be included into the P-D family. In particular, we also presented another concrete Mobius transformation and the linear combinations of the Killing coordinates that obviously cast the type-D NUT C-metric into the familiar form of the P-D solution. We anticipate that this work can give a final answer to resolve the 'missing' puzzle of the type-D NUT C-metric.  \nIn the Appendix V, we have given two simple examples to illustrate our new normalization scheme by deriving the five-dimensional Schwarzschild-Tangherlini solution from the flat vacuum via the ISM construction. Our normalization strategy can be viewed as a methodological complement to the Pomeransky 'remove-re-add' tricky.  \nNote added . Recently, an eprint [32] appeared with a partial overlap of the subject for which the Mobius transformation is also used to relate the Astorino's new form to three different forms of the most general P-D solution. Our paper, however, focuses on the proof that the Astorino's novel form can be reproduced from the already-known accelerating Kerr-NUT solution and explains why the type-D NUT C-metric is 'missing' from the rotating and accelerating P-D solution.", 'ACKNOWLEDGEMENTS': 'This work is supported by the National Natural Science Foundation of China (NSFC) under Grants No. 12375053, No. 12205243, and No. 11675130, by the Sichuan Science and Technology Program under Grant No. 2023NSFSC1347, and by the Doctoral Research Initiation Project of China West Normal University under Grant No. 21E028.', 'APPENDIX: ISM CONSTRUCTION OF FIVE-DIMENSIONAL STATIC SOLUTIONS WITH OUR NEW NORMALIZATION SCHEME': 'In this Appendix, we will show that by using our normalization scheme, the five-dimensional SchwarzschildTangherlini solution can be generated from the flat vacuum background via the ISM construction, contrary to the statement made in Ref. [30]. Our derivation strictly follows the spirit of the ISM construction, and is different from what had been done in Ref. [33] where the ansatz of the Killing metric is modified.', 'A. Generating five-dimensional Schwarzschild-Tangherlini solution': 'We first take the five-dimensional flat Minkowski background  \nds 2 = -dt 2 + ρ 2 dφ 2 + dψ 2 + dρ 2 + dz 2 ,  \nas our seed metric:  \ng 0 = diag( -1 , ρ 2 , 1) , (A.1)  \nso the generating matrix is easily written as  \nΨ 0 = diag( -1 , ρ 2 -2 λz -λ 2 , 1) .  \nWe now perform a 2-soliton transformation: add a soliton µ 1 at z = z 1 with the vector m (1) 0 = ( -1 , 0 , 0) and a soliton µ 2 at z = z 2 with the vector m (2) 0 = (0 , -2 z 2 , 0), respectively. Then two BZ vectors constitute a 2 × 3 matrix and Γ is a 2 × 2 diagonal matrix:  \nm = ( 1 0 0 0 1 µ 2 0 ) , l = ( -1 0 0 0 ρ 2 µ 2 0 ) , Γ = ( -1 ρ 2 + µ 2 1 0 0 ρ 2 µ 2 2 ( ρ 2 + µ 2 1 ) ) .  \nThe generated five-dimensional static 2-soliton metric is very simple:  \n/negationslash  \n˜ g = diag ( ρ 2 µ 2 1 , -ρ 4 µ 2 2 , 1 ) , (A.2)  \nIt is suggested to take the normalization diagonal matrix as  \nW = diag ( µ 3 / 2 1 ρ √ µ 2 , µ 2 ρ √ µ 1 , √ -µ 2 ) , (A.3)  \nwith the determinant det ˜ g = -ρ 6 / ( µ 2 1 µ 2 2 ) = -ρ 2 .  \nthen the final physics metric is exactly the expected one:  \ng = -W ˜ gW = -W 2 ˜ g = diag ( -µ 1 µ 2 , ρ 2 µ 1 , µ 2 ) , (A.4)  \nf = µ 2 ( ρ 2 + µ 1 µ 2 ) ( ρ 2 + µ 2 1 )( ρ 2 + µ 2 2 ) . (A.5)  \nwith the conformal factor being recorded as  \nThis trivial example illuminates that the five-dimensional Schwarzschild-Tangherlini solution is feasibly generated via the ISM procedure with our new normalization scheme from the flat Minkowski vacuum seed.', 'B. Generating five-dimensional static Emparan-Reall black ring': "Next, we want to rederive the static Emparan-Reall black ring [34] solution from the five-dimensional flat Minkowski background written in another form:  \nds 2 = -dt 2 + ρ 2 µ 3 dφ 2 + µ 3 dψ 2 + µ 3 ρ 2 + µ 2 3 ( dρ 2 + dz 2 ) . (B.1)  \nClearly, its t = const slice is the Euclidean Rindler metric.  \nFrom the seed metric  \ng 0 = diag ( -1 , ρ 2 µ 3 , µ 3 ) , (B.2)  \nwe rewrite ρ 2 /µ 3 = -ν 3 and make the replacements, µ 3 → µ 3 -λ , ν 3 → ν 3 -λ , then easily obtain the generating matrix:  \nΨ 0 = diag ( -1 , ρ 2 µ 3 + λ,µ 3 -λ ) .  \nWe now perform a 2-soliton transformation: add a soliton µ 1 at z = z 1 with the vector m (1) 0 = (1 , 0 , 0) and a soliton µ 2 at z = z 2 with the vector m (2) 0 = (0 , 1 , 0), respectively. The order for the 3-soliton is aligned as z 1 < z 2 < z 3 . Then two BZ vectors constitute a 2 × 3 matrix and Γ is also a diagonal 2 × 2 matrix:  \nm = ( -1 0 0 0 µ 3 ρ 2 + µ 2 µ 3 0 ) , l = ( 1 0 0 0 ρ 2 ρ 2 + µ 2 µ 3 0 ) , Γ = ( -1 ρ 2 + µ 2 1 0 0 ρ 2 µ 3 ( ρ 2 + µ 2 2 )( ρ 2 + µ 2 µ 3 ) 2 ) .  \nThe generated five-dimensional 2-soliton metric is still diagonal:  \n/negationslash  \n˜ g = diag ( ρ 2 µ 2 1 , -ρ 4 µ 2 2 µ 3 , µ 3 ) , (B.3)  \nwith the determinant det ˜ g = -ρ 6 / ( µ 2 1 µ 2 2 ) = -ρ 2 . Now we suggest to take the normalization matrix as  \nW = diag ( µ 3 / 2 1 ρ √ µ 2 , µ 3 / 2 2 ρ √ µ 1 , √ -1 ) , (B.4)  \nthen the final physics metric is exactly that of the static Emparan-Reall black ring:  \ng = -W ˜ gW = -W 2 ˜ g = diag ( -µ 1 µ 2 , ρ 2 µ 2 µ 1 µ 3 , µ 3 ) , (B.5)  \nwhose conformal factor can be easily computed as  \nf = k µ 3 ( ρ 2 + µ 1 µ 2 ) 2 ( ρ 2 + µ 2 µ 3 ) (1 -ν ) ( ρ 2 + µ 1 µ 3 )( ρ 2 + µ 2 1 )( ρ 2 + µ 2 2 )( ρ 2 + µ 2 3 ) . (B.6)  \nIn the coalescing limit µ 3 ↦→ µ 2 with k = 1 -ν , the solution reduces to the five-dimensional SchwarzschildTangherlini solution. On the other hand, in the merging limit µ 2 ↦→ µ 1 with k = 1 -ν , the solution coalesces to the above five-dimensional Minkowski background metric.  \n- [4] K. Hong and E. Teo, A new form of the rotating C-metric, Classical Quantum Gravity 22 , 109 (2005).\n- [5] J.B. Griffiths and J. Podolsk'y, Exact Space-Times in Einstein's General Relativity (Cambridge University Press, Cambridge, England, 2009), 10.1017/CBO9780511635397.\n- [6] J.B. Griffiths and J. Podolsk'y, Accelerating and rotating black holes, Classical Quantum Gravity 22 , 3467 (2005).\n- [7] M. Astorino, Equivalence principle and generalized accelerating black holes from binary systems, Phys. Rev. D 109 , 084038 (2024).\n- [8] M. Astorino, Most general type-D black hole and accelerating Reissner-Nordstrom-NUT-(A)dS, to appear in Phys. Rev. D, arxiv:2404.06551 [gr-qc].\n- [9] B. Chng, R.B. Mann, and C. Stelea, Accelerating Taub-NUT and Eguchi-Hanson solitons in four dimensions, Phys. Rev. D 74 , 084031 (2006).\n- [10] J. Podolsk'y and A. Vratny, Accelerating NUT black holes, Phys. Rev. D 102 , 084024 (2020).\n- [11] G. Boldi, Ehlers transformation and accelerating spacetimes with a gravomagnetic monopole, M.Sc. thesis, Universit'a degli Studi di Milano, Italia, 2022.\n- [12] J. Barrientos and A. Cisterna, Ehlers transformations as a tool for constructing accelerating NUT black holes, Phys. Rev. D 108 , 024059 (2023).\n- [13] M. Astorino, Accelerating and charged type I black holes Phys. Rev. D 108 , 124025 (2023).\n- [14] M. Astorino and G. Boldi, Pleba'nski-Demia'nski goes NUTs (to remove the Misner string), J. High Energy Phys. 08 (2023) 085.\n- [15] J. Barrientos, A. Cisterna, and K. Pallikaris, Pleba'nski-Demia'nski 'a la Ehlers-Harrison: Exact rotating and accelerating type I black holes, Gen. Relativ. Gravit. 56 , 111 (2024).\n- [16] W. Kinnersley and M. Walker, Uniformly accelerating charged mass in general relativity, Phys. Rev. D 2 , 1359 (1970).\n- [17] Y. Chen and E. Teo, Five-parameter class of solutions to the vacuum Einstein equations Phys. Rev. D 91 , 124005 (2015).\n- [18] Y. Chen, Gravitational multisoliton solutions on flat space, Phys. Rev. D 93 , 044021 (2016).\n- [19] V.A. Belinski and V.E. Zakharov, Integration of the Einstein equations by means of the inverse scattering problem technique and construction of exact soliton solutions, Sov. Phys. JETP 48 , 985 (1978).\n- [20] V.A. Belinski and V.E. Zakharov, Stationary gravitational solitons with axial symmetry, Sov. Phys. JETP 50 , 1 (1979).\n- [21] V. Belinski and E. Verdaguer, Gravitational Solitons (Cambridge University Press, Cambridge, England, 2005), 10.1017/CBO9780511535253.\n- [22] F.R. Tangherlini, Schwarzschild field in N dimensions and the dimensionality of space problem, Nuovo Cimento 27 , 636 (1963).\n- [23] T. Harmark, Stationary and axisymmetric solutions of higher-dimensional general relativity, Phys. Rev. D 70 , 124002 (2004).\n- [24] S. Tomizawa, Y. Morisawa, and Y. Yasui, Vacuum solutions of five dimensional Einstein equations generated by inverse scattering method, Phys. Rev. D 73 , 064009 (2006).\n- [25] S. Tomizawa and M. Nozawa, Vacuum solutions of five dimensional Einstein equations generated by inverse scattering method. II. Production of the black ring solution, Phys. Rev. D 73 , 124034 (2006).\n- [26] S. Tomizawa, H. Iguchi, and T. Mishima, Relationship between solitonic solutions of five-dimensional Einstein equations, Phys. Rev. D 74 , 104004 (2006).\n- [27] T. Azuma and T. Koikawa, An infinite number of stationary soliton solutions to the five-dimensional vacuum Einstein equation, Prog. Theor. Phys. 116 , 319 (2006).\n- [28] W. Bruckman, Stationary axially symmetric exterior solutions in the five-dimensional representation of the Brans-DickeJordan theory of gravitation, Phys. Rev. D 34 , 2990 (1986).\n- [29] T. Azuma, M. Endo, and T. Koikawa, Axially symmetric solution of the Einstein equation with solitons in extra dimensions, Phys. Lett. A 136 , 269 (1989).\n- [30] A. Pomeransky, Complete integrability of higher-dimensional Einstein equations with additional symmetry, and rotating black holes, Phys. Rev. D 73 , 044004 (2006).\n- [31] R.C. Myers and M.J. Perry, Black holes in higher dimensional space-times, Ann. Phys. (N.Y.) 172 , 304 (1986).\n- [32] H. Ovcharenko, J. Podolsk'y, and M. Astorino, Black holes of type D revisited: Relating their various metric forms, arXiv:2409.02308 [gr-qc].\n- [33] T. Koikawa, Infinite number of soliton solutions to 5-dimensional vacuum Einstein equation, Prog. Theor. Phys. 114 , 793 (2005).\n- [34] R. Emparan and H.S. Reall, A rotating black ring in five dimensions, Phys. Rev. Lett. 88 , 101101 (2002)."}