Abstract
In this paper, via employing the uniformly modified form of the generalized off-shell Helmholtz free energy, we investigate the topological numbers for the four-dimensional neutral Lorentzian Taub–NUT, Taub–NUT–AdS and Kerr–NUT spacetimes, and find that these solutions can also be classified into one of three types of those well-known black hole solutions, which implies that these spacetimes should be viewed as generic black holes from the viewpoint of the thermodynamic topological approach.
Similar content being viewed by others
Avoid common mistakes on your manuscript.
1 Introduction
In the great family of four-dimensional exact solutions in General Relativity, the Lorentzian NUT-charged spacetimes belong to a very important class of asymptotically locally flat solutions to the Einstein field equations [1, 2]. Since its birth, the prevailing view has commonly excluded the Lorentzian NUT-charged spacetimes from the big family of black holes, due to the Misner’s identification of the NUT charge parameter as unphysical [3]. However, in recent years, there has been a surge in the studies that exploit the Lorentzian NUT-charged spacetimes as black holes to explore their physical properties, including thermodynamics [4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25], geodesic motion [26], Kerr/CFT correspondence [27, 28], black hole shadow [29], weak cosmic censorship conjecture [30, 31], holographic complexity [32], and so on [33,34,35,36,37,38,39]. Thus, a question arises naturally as to whether the Lorentzian NUT-charged spacetimes are generic black holes.
On the other hand, the Lorentzian NUT-charged spacetimes have many peculiar properties that are mainly due to the presence of the wire/line singularities at the polar axes (\(\theta = 0\) and \(\theta = \pi \)), which are often called as the Misner strings. For instance: (I) The NUT charge has many different meanings and interpretations [40,41,42,43,44], and there are many different explanations of the physical origin of the NUT-charged spacetimes [45, 46]. Up to date, there is no unified explanation for these facts; (II) The Taub–NUT de Sitter spacetimes not only provide counterexamples to the maximal mass conjecture but also violate the entropic N-bound [47,48,49]; (III) The thermodynamic volume of the Euclidean Taub–NUT–AdS spacetime can be negative [50], thus violating the reverse isoperimetric inequality [51]. Therefore, it is very interesting to explore the physics behind these peculiar properties of the Lorentzian NUT-charged spacetimes.
Very recently, by treating the black hole solutions as topological defects and using the generalized off-shell free energy, Wei et al. [52] constructed the topological numbers which are independent of the intrinsic parameters of black holes and divided some black hole solutions into three categories according to their different topological numbers. Because of its simplicity and easy maneuverability of the procedure, the thermodynamic topological approach proposed in Ref. [52] soon attracted a great deal of attention and was then successfully applied to calculate the topological numbers of other black hole solutions [53,54,55,56,57,58,59,60,61]. It is natural to investigate the topological numbers of the NUT-charged spacetimes so as to decree whether they are black holes or not. In this paper, we shall investigate the topological numbers for the four-dimensional neutral Lorentzian Taub–NUT, Taub–NUT–AdS and Kerr–NUT spacetimes by first employing the uniformly modified form of the generalized off-shell Helmholtz free energy, and find that from the thermodynamic topological standpoint, these spacetimes should be viewed as generic black holes also.
The organization of this paper is outlined as follows. In Sect. 2, we give a brief review of the thermodynamic topological approach proposed in Ref. [52]. In Sect. 3, we first recall the consistent formulation of thermodynamic properties of the four-dimensional static Lorentzian Taub–NUT spacetime and then investigate its topological number. In Sect. 4, we turn to discuss the case of the rotating Lorentzian Kerr–NUT spacetime. In Sect. 5, we then extend to discuss the more general static Lorentzian Taub–NUT–AdS\(_4\) spacetime. Finally, our conclusion and outlook are given in Sect. 6.
2 A brief review of thermodynamic topological approach
Following the thermodynamic topological approach proposed in Ref. [52], one can first introduce the generalized off-shell Helmholtz free energy
for a black hole thermodynamical system with the mass M and the entropy S, where \(\tau \) is an extra variable that can be regarded as the inverse temperature of the cavity surrounding the black hole. Only when \(\tau = T^{-1}\), the generalized Helmholtz free energy (2.1) is on-shell and reduces to the black hole’s ordinary Helmholtz free energy \(F = M -TS\) [62,63,64].
In Ref. [52], a key vector \(\phi \) is defined as
where the two parameters obey \(0< r_h < +\infty \), \(0 \le \Theta \le \pi \), respectively. The component \(\phi ^\Theta \) is divergent at \(\Theta = 0\) and \(\Theta = \pi \), implying that the direction of the vector is outward here.
A topological current can be defined using the Duan’s \(\phi \)-mapping topological current theory [65,66,67,68] as follows:
where \(\partial _{\nu }= \partial /\partial x^{\nu }\) and \(x^{\nu }=(\tau ,~r_h,~\Theta )\). The unit vector n reads as \(n = (n^r, n^\Theta )\), where \(n^r = \phi ^{r_h}/\Vert \phi \Vert \) and \(n^\Theta = \phi ^{\Theta }/\Vert \phi \Vert \). Since it is easy to demonstrate that the aforementioned current (2.3) is conserved, and one can quickly arrive at \(\partial _{\mu }j^{\mu } = 0\) and then demonstrate that the topological current is a \(\delta \)-function of the field configuration [67,68,69]
where the three dimensional Jacobian \(J^{\mu }(\phi /x)\) satisfies: \(\epsilon ^{ab}J^{\mu }(\phi /x) = \epsilon ^{\mu \nu \rho }\partial _{\nu }\phi ^a\partial _{\rho }\phi ^b\). It is simple to show that \(j^\mu \) equals to zero only when \(\phi ^a(x_i) = 0\), and one can deduce the topological number W as follows:
where \(\beta _i\) is the positive Hopf index counting the number of the loops of the vector \(\phi ^a\) in the \(\phi \)-space when \(x^{\mu }\) are around the zero point \(z_i\), while \(\eta _{i}= {\textrm{sign}} (J^{0}({\phi }/{x})_{z_i})=\pm 1\) is the Brouwer degree, and \(w_{i}\) is the winding number for the i-th zero point of \(\phi \) that is contained in the domain \(\Sigma \).
3 Four-dimensional Lorentzian Taub–NUT spacetime
As the simplest case, we will investigate the four-dimensional Lorentzian Taub–NUT spacetime solution [2], and adopt the following line element in which the Misner strings [3] are symmetrically distributed along the polar axis:
where \(f(r) = r^2 -2mr -n^2\), in which m and n are the mass and NUT charge parameters, respectively. The event horizon radius \(r_h\) is the largest root of the equation \(f(r_h) = 0\), namely, \(r_h = m + \sqrt{m^2 +n^2}\).
3.1 Consistent thermodynamics
First, let’s briefly recapitulate the \((\psi -{\mathscr {N}})\)-pair formalism [4, 5] of the consistent thermodynamic of the four-dimensional Lorentzian Taub–NUT spacetime.
The Bekenstein–Hawking entropy is taken as one quarter of the area of the event horizon
and the Hawking temperature is thought of as being proportional to the surface gravity \(\kappa \) on the event horizon
in which a prime denotes the partial derivative with respective to its variable.
In the \((\psi -{\mathscr {N}})\)-pair formalism, one intentionally divides the Komar integral into three patches: the spatial infinity, the horizon, and two Misner string tubes, and defines a non-globally conserved Misner charge. For the global conserved charge, the Komar mass related to the timelike Killing vector \(\partial _t\) is: \(M = m\), which is computed via the Komar integral at infinity.
There are also other Killing horizons associated with the Misner strings in the Lorentzian Taub–NUT spacetime, namely, the north/south pole axis is a Killing horizon related to the Killing vector [5]: \(k = \partial _t +\partial _\varphi /(2n)\), whose corresponding surface gravity can be calculated as
and the associated Misner potential is
It is a simple matter to check that the above thermodynamic quantities simultaneously fulfil both the differential and integral mass formulae:
with the gravitational Misner charge
being conjugate to the Misner potential \(\psi \).
The expression of the Helmholtz free energy is then given as [4, 5, 15]
3.2 Topological number
Next, we will obtain the topological number of the four-dimensional Lorentzian Taub–NUT spacetime. The expression for the Helmholtz free energy of the Lorentzian Taub–NUT spacetime can be recovered via a Wick-rotated back procedure from the Euclidean action of the Euclidean Taub–NUT spacetime:
where h is the determinant of the induced metric \(h_{ij}\), K is the trace of the extrinsic curvature tensor defined on the boundary with this metric, and \(K_0\) is the subtracted one of the massless Taub–NUT solution as the reference background.
The free energy calculated by the action integral is: \(I/\beta = m/2 = F\), where \(\beta = 4\pi {}r_h\) is the interval of the time coordinate. Replacing T with \(1/\tau \) in Eq. (3.9) and using \(m = (r_h^2 -n^2)/(2r_h)\), the generalized off-shell Helmholtz free energy is modified as
Utilizing the definition of Eq. (2.2), the components of the vector \(\phi \) can be easily computed as follows:
By solving the equation: \(\phi ^{r_h} = 0\), one can obtain a curve on the \(r_h-\tau \) plane, which is just a straight line for the four-dimensional Taub–NUT spacetime:
For the Lorentzian Taub–NUT spacetime, we plot, respectively, the zero points of the component \(\phi ^{r_h}\) in Fig. 1, and the unit vector field n on a portion of the \(\Theta -r_h\) plane in Fig. 2 with \(\tau = 4\pi r_0\) in which \(r_0\) is an arbitrary length scale set by the size of a cavity enclosing the Taub–NUT spacetime. From Fig. 1, one can observe that the behavior of the Taub–NUT spacetime resembles that of the Schwarzschild black hole, and this indicates that the NUT charge parameter seems to have no effect on the thermodynamic topological classification for neutral static asymptotically locally flat spacetime. Therefore, it would be interesting to explore the connection between the geometric topology and the thermodynamic topology, for instance, to investigate the topological number of the ultraspinning black holes [70,71,72,73,74,75,76,77] and their usual counterparts.
In Fig. 2, the zero point is located at \(r_h = r_0\) and \(\Theta = \pi /2\). one can determine the winding number w for an arbitrary loop since it is independent of the loops that surround the zero point; for instance, through a look at the blue contour C in Fig. 2. If the calculation is made for the forward rotation in the anticlockwise direction, then the winding number is \(w = -1\), which coincides with those of the Schwarzschild black hole [52] and the \(d \ge 6\) singly rotating Kerr black hole [58]. In this sense, the Lorentzian Taub–NUT spacetime behaves like a genuine black hole. Since the winding number is related to local thermodynamic stability, with positive and negative values corresponding to stable and unstable black hole solutions [52] respectively, one can naturally conclude that the Taub–NUT spacetime is an unstable black hole in an arbitrary given temperature just like the Schwarzschild black hole. Turning to the topological global properties, we have the topological number \(W = -1\) for the Taub–NUT spacetime from Fig. 2, which is also the same as the results of the Schwarzschild black hole and the \(d \ge 6\) singly rotating Kerr black hole. Therefore, from the point of view of topological numbers, the Lorentzian Taub–NUT spacetime should be accepted into the black hole family. In addition, it can be concluded that although the Taub–NUT spacetime and Schwarzschild black hole are obviously different in geometric topology aspect, they are the same class from the perspective of the thermodynamic topology.
4 Lorentzian Kerr–NUT spacetime
In this section, we will extend the above discussion to the case of a rotating NUT-charged spacetime by considering the four-dimensional Kerr–NUT spacetime [78,79,80,81], whose line element with the Misner strings symmetrically distributed along the rotation axis is
where
in which m and a are the mass and rotation parameters, respectively. The horizon is determined by: \(\Delta (r_h) = 0\), which gives \(r_h = m \pm \sqrt{m^2 +n^2 -a^2}\).
4.1 Consistent thermodynamics
Now we briefly recall the \((\psi -{\mathscr {N}})\)-pair formalism [7] of the consistent thermodynamics of the four-dimensional Lorentzian Kerr–NUT spacetime. The Bekenstein–Hawking entropy is taken as one quarter of the event horizon area:
while the Hawking temperature is proportional to the surface gravity \(\kappa \) on the event horizon
The angular velocity at the event horizon and the Misner potential are given by
As for the global conserved charge, the Komar mass \(M = m\) at infinity is related to the timelike Killing vector \(\partial _t\).
It is easy to check that the above thermodynamic quantities satisfy the differential first law and integral Bekenstein–Smarr mass formula simultaneously,
with the gravitational Misner charge and the angular momentum
being conjugate to the Misner potential \(\psi \) and the angular velocity, respectively. Note that both of them do not have a global character, due to having a location-dependent factor \(1/r_h\).
With the help of the above expressions and using \(m = (r_h^2 -n^2 +a^2)/(2r_h)\), the Gibbs free energy of the Kerr–NUT spacetime reads
which is identical to the one calculated via the action integral [7], just like the non-rotating case.
4.2 Topological number
In order to calculate the thermodynamical topological number of the Kerr–NUT spacetime, we then consider the Helmholtz free energy which is given by
It coincides with the result of Eq. (4.19) given in Ref. [8] in the case of the parameters \(e = g = 0\) are turned off.
It is straightforward to modify the generalized Helmholtz free energy as
Then, the components of the vector \(\phi \) are given by
Therefore, using the thermodynamic topological approach and solving the equation: \(\phi ^{r_h} = 0\), we get
as the zero point of the vector field \(\phi ^{r_h}\).
Taking \(a = r_0\) for the Kerr–NUT spacetime, we plot the zero points of the component \(\phi ^{r_h}\) in Fig. 3, and the unit vector field n on a portion of the \(\Theta -r_h\) plane in Fig. 4 with \(\tau /r_0 = 40\), respectively. In Fig. 3, one generation point can be found at \(\tau /r_0 = \tau _{c}/r_0 = 32.65\). In Fig. 4, the zero points are located at \((r_h/r_0, \Theta ) = (1.30,\pi /2)\), and \((2.77,\pi /2)\), respectively. Based upon the local property of the zero points, we can obtain the topological number: \(W = -1 +1 = 0\) for the Kerr–NUT spacetime, which is same as that of the Kerr black hole [58]. Thus, the big family of the black holes should contain the four-dimensional Kerr–NUT spacetime. In addition, it also can be concluded that although the Kerr–NUT spacetime and Kerr black hole are obviously different in geometric topology aspect, they seem to be the same kind from the perspective of the thermodynamic topology, just like the Taub–NUT spacetime and Schwarzschild black hole as discussed in Sect. 3 and Ref. [52], respectively.
5 Lorentzian Taub–NUT–AdS\(_4\) spacetime
In this section, we turn to explore the Lorentzian Taub–NUT spacetime with an negative cosmological constant, namely, the Lorentzian Taub–NUT–AdS\(_4\) spacetime, whose metric is still given by Eq. (3.1), but now \(f(r) = r^2 -2mr -n^2 +(r^4 +6n^2r^2 -3n^4)/l^2\), in which the AdS radius l appears in the thermodynamic pressure \(P = 3/(8\pi {}l^2)\) of the 4-dimensional AdS black hole [51, 82].
5.1 Consistent thermodynamics
We now recollect the thermodynamical properties of the four-dimensional Taub–NUT–AdS spacetime. The Bekenstein–Hawking entropy is taken as one quarter of the event horizon area:
and the Hawking temperature is assumed to be proportional to the surface gravity \(\kappa \) on the event horizon
where \(r_h\) is the location of the event horizon.
The conformal mass is: \(M = m\), and the Misner potential is
It is easy to verify that the above thermodynamic quantities satisfy the differential first law and integral Bekenstein–Smarr mass formula simultaneously [4, 5],
with the gravitational Misner charge and the thermodynamic volume
being conjugate to the Misner potential \(\psi \) and the pressure \(P = 3/(8\pi l^2)\), respectively.
5.2 Topological number
One can find that the Helmholtz free energy reads [4]
which coincides with those computed via the action integral, namely \(F = I/\beta \). In order to get this result, one can calculate the Euclidean action [4] for the Euclidean spacetime
where K and \({\mathscr {R}}(h)\) are the extrinsic curvature and Ricci scalar of the boundary metric, respectively. In order to cancel the divergence, the above action also contains the Gibbons–Hawking boundary term and the necessary AdS boundary counterterms [83,84,85,86,87], apart from the standard Einstein–Hilbert term.
It is now a position to investigate the topological number of the four-dimensional Lorentz Taub–NUT–AdS spacetime. Replacing T with \(1/\tau \) and substituting \(l^2 = 3/(8\pi P)\), thus the modified form of the generalized off-shell Helmholtz free energy is
Then, the components of the vector \(\phi \) are obtained as follows:
from which one can get the zero point of the vector field \(\phi ^{r_h}\) as
Note that Eq. (5.11) consistently reduces to the one obtained in the four-dimensional Schwarzschild–AdS black hole case [61] when the NUT charge parameter n vanishes.
Taking the pressure \(Pr_0^2 = 0.002\) and the NUT charge parameter \(n/r_0 = 1\), in Figs. 5 and 6, we plot the zero points of \(\phi ^{r_h}\) in the \(r_h-\tau \) plane and the unit vector field n on a portion of the \(\Theta -r_h\) plane for the Taub–NUT–AdS\(_4\) spacetime. For the Taub–NUT–AdS\(_4\) spacetime, we observe that the topological number is \(W = 0\), which is the same as that of the Schwarzschild–AdS\(_4\) black hole [55, 61], and this implies that the NUT charge parameter also seems to have no impact on the thermodynamic topological classification for the neutral static asymptotically local AdS spacetime. Furthermore, it indicates that the Taub–NUT–AdS\(_4\) solution can still be categorized as one of the three types of known black hole solutions [52]. As a result, at least according to the thermodynamic topological approach, the Lorentzian Taub–NUT–AdS\(_4\) spacetime should be included into a member of the black hole family.
6 Conclusion and outlook
Our results obtained in this paper are summarized in Table 1. In this work, we have employed the uniformly modified form of the generalized off-shell Helmholtz free energy and investigated the topological numbers for the four-dimensional uncharged Lorentzian Taub–NUT, Taub–NUT–AdS and Kerr–NUT spacetimes. We showed that the Taub–NUT spacetime has: \(W = -1\), which is the same as that of the Schwarzschild black hole [52]. We found that the Kerr–NUT spacetime has: \(W = 0\), which is identical to that of the Kerr black hole [58]. We also demonstrated that the Taub–NUT–AdS\(_4\) spacetime has: \(W = 0\), which coincides with that of the Schwarzschild–AdS\(_4\) black hole [61]. Therefore, one can conclude that the existence of the NUT charge parameter seems has no impact on the topological number of the neutral asymptotically locally flat/AdS spacetimes. It can be inferred that the four-dimensional Taub–NUT, Taub–NUT–AdS and Kerr–NUT spacetimes should be viewed as generic black holes from the viewpoint of the thermodynamic topological approach.
There are two promising topics to be pursued in the future. The first intriguing topic is to explore the phase transitions of the NUT-charged AdS spacetimes, such as the Hawking–Page phase transitions [88] and the P–V criticality [89], and we expect that this will help to unravel the mystery of the NUT-charged spacetimes. The second interesting topic is to extend the present work to the more general charged RN–NUT(–AdS) and rotating charged Kerr–Newman–NUT cases.
Perhaps, one has just touched the tip of an iceberg, much more needs to be explored.
Data Availability
This manuscript has no associated data or the data will not be deposited. [Authors’ comment: There are no external data associated with the manuscript.]
References
A.H. Taub, Empty space-times admitting a three parameter group of motions. Ann. Math. 53, 472 (1951). https://doi.org/10.2307/1969567
E.T. Newman, L. Tamburino, T. Unti, Empty-space generalization of the Schwarzschild metric. J. Math. Phys. (N.Y.) 4, 915 (1963). https://doi.org/10.1063/1.1704018
C.W. Misner, The flatter regions of Newman, Unti, and Tamburino’s generalized Schwarzschild space. J. Math. Phys. (N.Y.) 4, 924 (1963). https://doi.org/10.1063/1.1704019
R.A. Hennigar, D. Kubizňák, R.B. Mann, Thermodynamics of Lorentzian Taub–NUT spacetimes. Phys. Rev. D 100, 064055 (2019). https://doi.org/10.1103/PhysRevD.100.064055. arXiv:1903.08668
A.B. Bordo, F. Gray, R.A. Hennigar, D. Kubizňák, Misner gravitational charges and variable string strengths. Class. Quantum Gravity 36, 194001 (2019). https://doi.org/10.1088/1361-6382/ab3d4d. arXiv:1905.03785
A.B. Ballon, F. Gray, D. Kubizňák, Thermodynamics and phase transitions of NUTty dyons. JHEP 07, 119 (2019). https://doi.org/10.1007/JHEP07(2019)119. arXiv:1904.00030
A.B. Bordo, F. Gray, R.A. Hennigar, D. Kubizňák, The first law for rotating NUTs. Phys. Lett. B 798, 134972 (2019). https://doi.org/10.1016/j.physletb.2019.134972. arXiv:1905.06350
A.B. Ballon, F. Gray, D. Kubizňák, Thermodynamics of rotating NUTty dyons. JHEP 05, 084 (2020). https://doi.org/10.1007/JHEP05(2020)084. arXiv:2003.02268
Z.H. Chen, J. Jiang, General Smarr relation and first law of a NUT dyonic black hole. Phys. Rev. D 100, 104016 (2019). https://doi.org/10.1103/PhysRevD.100.104016. arXiv:1910.10107
E. Frodden, D. Hidalgo, The first law for the Kerr-NUT spacetime. Phys. Lett. B 832, 137264 (2022). https://doi.org/10.1016/j.physletb.2022.137264. arXiv:2109.07715
G. Clément, D. Gal’tsov, On the Smarr formulas for electrovac spacetimes with line singularities. Phys. Lett. B 802, 135270 (2020). https://doi.org/10.1016/j.physletb.2020.135270. arXiv:1908.10617
R. Durka, The first law of black hole thermodynamics for Taub–NUT spacetime. Int. J. Mod. Phys. D 31, 2250021 (2022). https://doi.org/10.1142/S0218271822500213. arXiv:1908.04238
N. Abbasvandi, M. Tavakoli, R.B. Mann, Thermodynamics of dyonic NUT charged black holes with entropy as Noether charge. JHEP 08, 152 (2021). https://doi.org/10.1007/JHEP08(2021)152. arXiv:2107.00182
A.M. Awad, S. Eissa, Topological dyonic Taub-Bolt/NUT-AdS solutions: Thermodynamics and first law. Phys. Rev. D 101, 124011 (2020). https://doi.org/10.1103/PhysRevD.101.124011. arXiv:2007.10489
A.M. Awad, S. Eissa, Lorentzian Tuab–NUT spacetimes: Misner string charges and the first law. Phys. Rev. D 105, 124034 (2022). https://doi.org/10.1103/PhysRevD.105.124034. arXiv:2206.09124
S.-Q. Wu, D. Wu, Thermodynamical hairs of the four-dimensional Taub–Newman–Unti–Tamburino spacetimes. Phys. Rev. D 100, 101501(R) (2019). https://doi.org/10.1103/PhysRevD.100.101501. arXiv:1909.07776
D. Wu, S.-Q. Wu, Consistent mass formulas for the four-dimensional dyonic NUT-charged spacetimes. Phys. Rev. D 105, 124013 (2022). https://doi.org/10.1103/PhysRevD.105.124013. arXiv:2202.09251
D. Wu, S.-Q. Wu, Consistent mass formulae for higher even-dimensional Taub–NUT spacetimes and their AdS counterparts. arXiv:2209.01757
D. Wu, S.-Q. Wu, Revisiting mass formulae of the four-dimensional Reissner–Nordström–NUT–AdS solutions in a different metric form. arXiv:2210.17504
L. Ciambelli, C. Corral, J. Figueroa, G. Giribet, R. Olea, Topological terms and the Misner string entropy. Phys. Rev. D 103, 024052 (2021). https://doi.org/10.1103/PhysRevD.103.024052. arXiv:2011.11044
R.B. Mann, L.A.P. Zayas, M. Park, Complement to thermodynamics of dyonic Taub–NUT–AdS spacetime. JHEP 03, 039 (2021). https://doi.org/10.1007/JHEP03(2021)039. arXiv:2012.13506
M. Godazgar, S. Guisset, Dual charges for AdS spacetimes and the first law of black hole mechanics. Phys. Rev. D 106, 024022 (2022). https://doi.org/10.1103/PhysRevD.106.024022. arXiv:2205.10043
G.G.L. Nashed, Kerr–NUT black hole thermodynamics in f(T) gravity theories. Eur. Phys. J. Plus 130, 124 (2015). https://doi.org/10.1140/epjp/i2015-15124-3
N.H. Rodriguez, M.J. Rodriguez, First law for Kerr Taub–NUT AdS black holes. JHEP 10, 044 (2022). https://doi.org/10.1007/JHEP10(2022)044. arXiv:2112.00780
H.-S. Liu, H. Lü, L. Ma, Thermodynamics of Taub–NUT and Plebanski solutions. JHEP 10, 174 (2022). https://doi.org/10.1007/JHEP10(2022)174. arXiv:2208.05494
H.M. Siahaan, Kerr–Sen–Taub–NUT spacetime and circular geodesics. Eur. Phys. J. C 80, 1000 (2020). https://doi.org/10.1140/epjc/s10052-020-08561-z. arXiv:1905.02622
M.F.A.R. Sakti, A. Suroso, F.P. Zen, Kerr/CFT correspondence on Kerr–Newman–NUT–Quintessence black hole. Eur. Phys. J. Plus 134, 580 (2019). https://doi.org/10.1140/epjp/i2019-12937-x. arXiv:1909.06595
H.M. Siahaan, Magnetized Kerr–Taub–NUT spacetime and Kerr/CFT correspondence. Phys. Lett. B 820, 136568 (2021). https://doi.org/10.1016/j.physletb.2021.136568. arXiv:2102.04345
M. Zhang, J. Jiang, NUT charges and black hole shadows. Phys. Lett. B 816, 136213 (2021). https://doi.org/10.1016/j.physletb.2021.136213. arXiv:2103.11416
S.-J. Yang, J. Chen, J.-J. Wan, S.-W. Wei, Y.-X. Liu, Weak cosmic censorship conjecture for a Kerr–Taub–NUT black hole with a test scalar field and particle. Phys. Rev. D 101, 064048 (2020). https://doi.org/10.1103/PhysRevD.101.064048. arXiv:2001.03106
S. Cremonini, C.R.T. Jones, J.T. Liu, B. McPeak, Y.Z. Tang, NUT charge weak gravity conjecture from dimensional reduction. Phys. Rev. D 103, 106011 (2021). https://doi.org/10.1103/PhysRevD.103.106011. arXiv:2011.06083
J. Jiang, B.L. Deng, X.-W. Li, Holographic complexity of charged Taub–NUT–AdS black holes. Phys. Rev. D 100, 066007 (2019). https://doi.org/10.1103/PhysRevD.100.066007. arXiv:1908.06565
F. Long, S.B. Chen, J.C. Wang, J.L. Jing, Electromagnetic emissions from near-horizon region of an extreme Kerr–Taub–NUT black hole. Eur. Phys. J. C 79, 466 (2019). https://doi.org/10.1140/epjc/s10052-019-6989-8. arXiv:1812.11463
W.B. Feng, S.-J. Yang, Q. Tan, J. Yang, Y.-X. Liu, Overcharging a Reissner–Nordström Taub–NUT regular black hole. Sci. China: Phys. Mech. Astron. 64, 260411 (2021). https://doi.org/10.1007/s11433-020-1659-0. arXiv:2009.12846
S. Mukherjee, S. Chakraborty, N. Dadhich, On some novel features of the Kerr–Newman–NUT spacetime. Eur. Phys. J. C 79, 161 (2019). https://doi.org/10.1140/epjc/s10052-019-6662-2. arXiv:1807.02216
S. Mukherjee, S. Chakraborty, Multipole moments of compact objects with NUT charge: theoretical and observational implications. Phys. Rev. D 102, 124058 (2020). https://doi.org/10.1103/PhysRevD.102.124058. arXiv:2008.06891
G. Kalamakis, R.G. Leigh, A.C. Petkou, Aspects of holography of Taub–NUT–AdS\(_4\) spacetimes. Phys. Rev. D 103, 126012 (2021). https://doi.org/10.1103/PhysRevD.103.126012. arXiv:2009.08022
M. Ghezelbash, H.M. Siahaan, Magnetized Kerr–Newman–Taub–NUT spacetimes. Eur. Phys. J. C 81, 621 (2021). https://doi.org/10.1140/epjc/s10052-021-09430-z. arXiv:2103.04865
H.M. Siahaan, Magnetized Reissner–Nordström–Taub–NUT spacetime and microscopic entropy. Eur. Phys. J. C 81, 838 (2021). https://doi.org/10.1140/epjc/s10052-021-09643-2. arXiv:2101.10302
J.S. Dowker, The nut solution as a gravitational dyon. Gen. Relativ. Gravit. 5, 603 (1974). https://doi.org/10.1007/BF02451402
S. Ramaswamy, A. Sen, Dual-mass in general relativity. J. Math. Phys. (N.Y.) 22, 2612 (1981). https://doi.org/10.1063/1.524839
L. Bombelli, R.K. Koul, G. Kunstatter, J. Lee, R.D. Sorkin, On energy in 5-dimensional gravity and the mass of the Kaluza–Klein monopole. Nucl. Phys. B 289, 735 (1987). https://doi.org/10.1016/0550-3213(87)90404-4
S. Deser, M. Soldate, Gravitational energy in spaces with compactified dimensions. Nucl. Phys. B 311, 739 (1989). https://doi.org/10.1016/0550-3213(89)90175-2
D. Lynden-Bell, M. Nouri-Zonoz, Classical monopoles: Newton, NUT space, gravomagnetic lensing and atomic spectra. Rev. Mod. Phys. 70, 427 (1998). https://doi.org/10.1103/RevModPhys.70.427. arXiv:gr-qc/9612049
W.B. Bonnor, A new interpretation of the NUT metric in general relativity. Proc. Camb. Philos. Soc. 66, 145 (1969). https://doi.org/10.1017/S0305004100044807
A. Sackfield, Physical interpretation of NUT metric. Proc. Camb. Philos. Soc. 70, 89 (1971). https://doi.org/10.1017/S0305004100049707
R. Clarkson, A.M. Ghezelbash, R.B. Mann, Mass, action and entropy of Taub-Bolt–de Sitter spacetimes. Phys. Rev. Lett. 91, 061301 (2003). https://doi.org/10.1103/PhysRevLett.91.061301. arXiv:hep-th/0304097
R. Clarkson, A.M. Ghezelbash, R.B. Mann, Entropic N bound and maximal mass conjectures violation in four-dimensional Taub-Bolt(NUT)-dS space-times. Nucl. Phys. B 674, 329 (2003). https://doi.org/10.1016/j.nuclphysb.2003.09.039. arXiv:hep-th/0307059
R. Clarkson, A.M. Ghezelbash, R.B. Mann, A review of the N-bound and the maximal mass conjectures using NUT-charged dS spacetimes. Int. J. Mod. Phys. A 19, 3987 (2004). https://doi.org/10.1142/S0217751X04019822. arXiv:hep-th/0408058
C.V. Johnson, Thermodynamic Volumes for AdS–Taub–NUT and AdS–Taub–Bolt. Class. Quantum Gravity 31, 235003 (2014). https://doi.org/10.1088/0264-9381/31/23/235003. arXiv:1405.5941
M. Cvetič, G.W. Gibbons, D. Kubizňák, C.N. Pope, Black hole enthalpy and an entropy inequality for the thermodynamic volume. Phys. Rev. D 84, 024037 (2011). https://doi.org/10.1103/PhysRevD.84.024037. arXiv:1012.2888
S.-W. Wei, Y.-X. Liu, R.B. Mann, Black hole solutions as topological thermodynamic defects. Phys. Rev. Lett. 129, 191101 (2022). https://doi.org/10.1103/PhysRevLett.129.191101. arXiv:2208.01932
P.K. Yerra, C. Bhamidipati, Topology of black hole thermodynamics in Gauss–Bonnet gravity. Phys. Rev. D 105, 104053 (2022). https://doi.org/10.1103/PhysRevD.105.104053. arXiv:2202.10288
P.K. Yerra, C. Bhamidipati, Topology of Born–Infeld AdS black holes in 4D novel Einstein–Gauss–Bonnet gravity. Phys. Lett. B 835, 137591 (2022). https://doi.org/10.1016/j.physletb.2022.137591. arXiv:2207.10612
P.K. Yerra, C. Bhamidipati, S. Mukherji, Topology of critical points and Hawking–Page transition. Phys. Rev. D 106, 064059 (2022). https://doi.org/10.1103/PhysRevD.106.064059. arXiv:2208.06388
N.-C. Bai, L. Li, J. Tao, Topology of black hole thermodynamics in Lovelock gravity. Phys. Rev. D 107, 064015 (2023). https://doi.org/10.1103/PhysRevD.107.064015. arXiv:2208.10177
C.H. Liu, J. Wang, The topological natures of the Gauss–Bonnet black hole in AdS space. Phys. Rev. D 107, 064023 (2023). https://doi.org/10.1103/PhysRevD.107.064023. arXiv:2211.05524
D. Wu, Topological classes of rotating black holes. Phys. Rev. D 107, 024024 (2023). https://doi.org/10.1103/PhysRevD.107.024024. arXiv:2211.15151
C.X. Fang, J. Jiang, M. Zhang, Revisiting thermodynamic topologies of black holes. JHEP 01, 102 (2023). https://doi.org/10.1007/JHEP01(2023)102. arXiv:2211.15534
N.-C. Bai, L. Song, J. Tao, Reentrant phase transition in holographic thermodynamics of Born–Infeld AdS black hole. arXiv:2212.04341
D. Wu, S.-Q. Wu, Topological classes of thermodynamics of rotating AdS black holes. Phys. Rev. D 107, 084002 (2023). https://doi.org/10.1103/PhysRevD.107.084002. arXiv:2301.03002
J.W. York, Black-hole thermodynamics and the Euclidean Einstein action. Phys. Rev. D 33, 2092 (1986). https://doi.org/10.1103/PhysRevD.33.2092
S.-J. Yang, R. Zhou, S.W. Wei, Y.-X. Liu, Dynamics and kinetics of phase transition for Kerr AdS black hole on free energy landscape. Phys. Rev. D 105, 084030 (2022). https://doi.org/10.1103/PhysRevD.105.084030. arXiv:2105.00491
R. Li, J. Wang, Generalized free energy landscape of a black hole phase transition. Phys. Rev. D 106, 106015 (2022). https://doi.org/10.1103/PhysRevD.106.106015. arXiv:2206.02623
Y.-S. Duan, M.-L. Ge, \(SU\) (2) gauge theory and electrodynamics of \(N\) moving magnetic monopoles. Sci. Sin. 9, 1072 (1979). https://doi.org/10.1142/9789813237278_0001
Y.-S. Duan, The structure of the topological current. Report No. SLAC-PUB-3301 (1984)
Y.-S. Duan, S. Li, G.-H. Yang, The bifurcation theory of the Gauss–Bonnet–Chern topological current and Morse function. Nucl. Phys. B 514, 705 (1998). https://doi.org/10.1016/S0550-3213(97)00777-3
L.-B. Fu, Y.-S. Duan, H. Zhang, Evolution of the Chern–Simons vortices. Phys. Rev. D 61, 045004 (2000). https://doi.org/10.1103/PhysRevD.61.045004. arXiv:hep-th/0112033
S.-W. Wei, Topological charge and black hole photon spheres. Phys. Rev. D 102, 064039 (2020). https://doi.org/10.1103/PhysRevD.102.064039. arXiv:2006.02112
D. Klemm, Four-dimensional black holes with unusual horizons. Phys. Rev. D 89, 084007 (2014). https://doi.org/10.1103/PhysRevD.89.084007. arXiv:1401.3107
R.A. Hennigar, R.B. Mann, D. Kubizňák, Entropy inequality violations from ultraspinning black holes. Phys. Rev. Lett. 115, 031101 (2015). https://doi.org/10.1103/PhysRevLett.115.031101. arXiv:1411.4309
A. Gnecchi, K. Hristov, D. Klemm, C. Toldo, O. Vaughan, Rotating black holes in 4d gauged supergravity. JHEP 01, 127 (2014). https://doi.org/10.1007/JHEP01(2014)127. arXiv:1311.1795
D. Wu, P. Wu, Null hypersurface caustics for high-dimensional superentropic black holes. Phys. Rev. D 103, 104020 (2021). https://doi.org/10.1103/PhysRevD.103.104020. arXiv:2104.10476
D. Wu, P. Wu, H. Yu, S.-Q. Wu, Notes on the thermodynamics of superentropic AdS black holes. Phys. Rev. D 101, 024057 (2020). https://doi.org/10.1103/PhysRevD.101.024057. arXiv:1912.03576
D. Wu, P. Wu, H. Yu, S.-Q. Wu, Are ultraspinning Kerr–Sen–AdS\(_4\) black holes always superentropic? Phys. Rev. D 102, 044007 (2020). https://doi.org/10.1103/PhysRevD.102.044007. arXiv:2007.02224
D. Wu, S.-Q. Wu, P. Wu, H. Yu, Aspects of the dyonic Kerr–Sen–AdS\(_4\) black hole and its ultraspinning version. Phys. Rev. D 103, 044014 (2021). https://doi.org/10.1103/PhysRevD.103.044014. arXiv:2010.13518
D. Wu, S.-Q. Wu, Ultra-spinning Chow’s black holes in six-dimensional gauged supergravity and their thermodynamical properties. JHEP 11, 031 (2021). https://doi.org/10.1007/JHEP11(2021)031. arXiv:2106.14218
M. Demianski, E.T. Newman, A combined Kerr–NUT solution of the Einstein field equation. Bull. Acad. Pol. Sci. Ser. Sci. Math. Astron. Phys. 14, 653 (1966). http://adsabs.harvard.edu/abs/1966BAPSS...14.653N
B. Carter, Hamilton–Jacobi and Schrodinger separable solutions of Einstein’s equations. Commun. Math. Phys. 10, 280 (1968). https://doi.org/10.1007/BF03399503
W. Kinnersley, Type D vacuum metrics. J. Math. Phys. (N.Y.) 10, 1195 (1969). https://doi.org/10.1063/1.1664958
J.G. Miller, Global analysis of the Kerr–Taub–NUT metric. J. Math. Phys. (N.Y.) 14, 486 (1973). https://doi.org/10.1063/1.1666343
D. Kastor, S. Ray, J. Traschen, Enthalpy and the mechanics of AdS black holes. Class. Quantum Gravity 26, 195011 (2009). https://doi.org/10.1088/0264-9381/26/19/195011. arXiv:0904.2765
R. Emparan, C.V. Johnson, R.C. Myers, Surface terms as counterterms in the AdS/CFT correspondence. Phys. Rev. D 60, 104001 (1999). https://doi.org/10.1103/PhysRevD.60.104001. arXiv:hep-th/9903238
A. Chamblin, R. Emparan, C.V. Johnson, R.C. Myers, Holography, thermodynamics and fluctuations of charged AdS black holes. Phys. Rev. D 60, 104026 (1999). https://doi.org/10.1103/PhysRevD.60.104026. arXiv:hep-th/9904197
R.B. Mann, Misner string entropy. Phys. Rev. D 60, 104047 (1999). https://doi.org/10.1103/PhysRevD.60.104047. arXiv:hep-th/9903229
V. Balasubramanian, P. Kraus, A stress tensor for anti-de Sitter gravity. Commun. Math. Phys. 208, 413 (1999). https://doi.org/10.1007/s002200050764. arXiv:hep-th/9902121
S. Haro, S.N. Solodukhin, K. Skenderis, Holographic reconstruction of space-time and renormalization in the AdS/CFT correspondence. Commun. Math. Phys. 217, 595 (2001). https://doi.org/10.1007/s002200100381. arXiv:hep-th/0002230
S.W. Hawking, D.N. Page, Thermodynamics of black holes in anti-de Sitter space. Commun. Math. Phys. 87, 577 (1983). https://doi.org/10.1007/BF01208266
D. Kubizňàk, R.B. Mann, P–V criticality of charged AdS black holes. JHEP 07, 033 (2012). https://doi.org/10.1007/JHEP07(2012)033. arXiv:1205.0559
Acknowledgements
We thank Professor Shuang-Qing Wu for helpful suggestions and detailed discussions. We are also greatly indebted to the anonymous referee for his/her constructive comments to improve the presentation of this work. This work is supported by the National Natural Science Foundation of China (NSFC) under Grant No. 12205243, No. 11675130, by the Sichuan Science and Technology Program under Grant No. 2023NSFSC1347, by the Sichuan Youth Science and Technology Innovation Research Team (21CXTD0038), and by the Doctoral Research Initiation Project of China West Normal University under Grant No. 21E028.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
Funded by SCOAP3. SCOAP3 supports the goals of the International Year of Basic Sciences for Sustainable Development.
About this article
Cite this article
Wu, D. Classifying topology of consistent thermodynamics of the four-dimensional neutral Lorentzian NUT-charged spacetimes. Eur. Phys. J. C 83, 365 (2023). https://doi.org/10.1140/epjc/s10052-023-11561-4
Received:
Accepted:
Published:
DOI: https://doi.org/10.1140/epjc/s10052-023-11561-4