Misner Gravitational Charges and Variable String Strengths

TL;DR

This paper provides a geometric interpretation of Misner charges in Taub-NUT spacetimes, linking them to surface gravity and Komar integrals, and extends the thermodynamic first law to include variable string strengths.

Contribution

It introduces a geometrical framework for Misner charges using surface gravity and Komar integrals, and generalizes the thermodynamic first law for Taub-NUT solutions.

Findings

Misner potential identified with surface gravity of the string

Misner charge obtained via Komar-type integration

First law extended to include variable string strengths

Abstract

As shown recently (arXiv:1903.08668), consistent thermodynamics of the Lorentzian Taub-NUT solutions with Misner strings present can be formulated provided a new pair of conjugate quantities (related to the NUT parameter) $\psi-N$ is introduced. In (arXiv:1903.08668) this pair was calculated from the Euclidean action but no geometrical interpretation for the new quantities was provided. In this paper we propose that the potential $\psi$ should be identified with the surface gravity of the Misner string and the conjugate Misner charge $N$ can be obtained by a Komar-type integration over the tubes surrounding the string singularities. We show that similar tube contributions also modify the Komar formula for the thermodynamic volume. To render the integrals finite we employ the method of Killing co-potentials. By construction the new charges then satisfy the Smarr relation. Equipped with these geometrical notions, we generalize the first law for the (possibly charged) Taub-NUT spacetimes to account for asymmetric distributions of Misner strings and their potential variable strengths.

Figures

Equations

$$\delta M=T\delta S+\psi \delta N+V\delta P.$$

$$P=-\frac{\Lambda}{8\pi}\,,\quad V=\Bigl{(}\frac{\partial M}{\partial P}\Bigr{)}_{S,N}\,.$$

$$M=2(TS+{\psi }_{+}{N}_{+}+{\psi }_{-}{N}_{-}-PV),$$

$$\delta M=T\delta S+{\psi }_{+}\delta N_{+}+{\psi }_{-}\delta N_{-}+V\delta P.$$

$$d{s}^2$$

$$f$$

$$\Lambda =-\frac{3}{l^2}.$$

$$t\to t+2s\varphi ,$$

$$d{s}^2$$

$$k={\partial }_t.$$

$$T=\frac{f^{\prime}}{4\pi}=\frac{1}{4\pi r_{+}}\Bigl{(}1+\frac{3(n^{2}+r_{+}^{2})}{l^{2}}\Bigr{)}\,,$$

$$\text{mbox}Area=\pi (r_{+}^2+n^2).$$

$$k_{\pm }={\partial }_t-\frac{1}{2(s\pm n)}{\partial }_{\varphi }.$$

$${\kappa }^2=\frac{1}{4}\frac{{\nabla }^{\mu }L{\nabla }_{\mu }L}{L},L=-k^2,$$

$${\kappa }_{\pm }=\frac{1}{2(n\pm s)}.$$

$${\psi }_{\pm }=\frac{{\kappa }_{\pm }}{4\pi }=\frac{1}{8\pi (n\pm s)}.$$

$$\psi =\frac{1}{8\pi n}.$$

$$S=\frac{\text{mbox}Area}{4}=\pi (r_{+}^2+n^2).$$

$$M=m.$$

$$I$$

$$\mathcal{G}=\mathcal{G}(T,\psi ,P)=M-TS-\psi N,$$

$$\mathcal{G}=\frac{I}{\beta }=\frac{m}{2}-\frac{r_{+}(r_{+}^2+3n^2)}{2l^2}.$$

$$\psi N=-\frac{n^{2}}{2r_{+}}\Bigl{(}1+\frac{3(n^{2}-r_{+}^{2})}{l^{2}}\Bigr{)}$$

$$\psi=\frac{1}{8\pi n}\,,\quad N=-\frac{4\pi n^{3}}{r_{+}}\Bigl{(}1+\frac{3(n^{2}-r_{+}^{2})}{l^{2}}\Bigr{)}\,,$$

$$V=\frac{4\pi r_{+}^{3}}{3}\Bigl{(}1+\frac{3n^{2}}{r_{+}^{2}}\Bigr{)}\,.$$

$${\nabla }_a{k}^a=0,{\nabla }_a{\nabla }^a{k}^b=-R^b{}_a{k}^a.$$

$$k^a={\nabla }_c{\omega }^{ca}.$$

$$\omega \to \omega +\nu$$

$$R_{ab}=\frac{2}{d-2}\Lambda g_{ab},$$

$${\nabla }_a{\nabla }^a{k}^b=-\frac{2}{d-2}\Lambda k^b=-\frac{2}{d-2}\Lambda {\nabla }_a{\omega }^{ab},$$

$$d\ast dk+\frac{4}{d-2}\Lambda d\ast \omega =0.$$

Full text

I Introduction

The Lorentzian Taub–NUT spacetime taub1951empty ; newman1963empty is full of surprises. Besides the Schwarzschild-like horizon, it features the unprecedented rotating string like singularities (Misner strings) on the axis and the associated regions with closed timelike curves (CTCs) in their vicinity.

Over the years, two proposals for the physical interpretation of the Taub-NUT spacetime have emerged. Misner misner1963flatter , studying the group of motions generated by the Killing vectors on an $r=const.$ hypersurface, argued that these sections of the spacetime are topologically three-spheres and that the time coordinate should be periodically identified to ensure geometric regularity. This, Misner found, was sufficient to remove the string singularities along the axis, but comes at the price of introducing CTCs everywhere in the spacetime. An alternate interpretation was offered by Bonnor bonnor1969new , who argued that the Misner string singularities—and associated CTC regions—should be replaced by a corresponding matter source, some form of rotating topological defect. This provides a physical interpretation for the ‘lack of regularity’ in the unidentified spacetime, somewhat analogous to spacetimes containing cosmic strings Israel:1976vc .

However, it has been recently understood that the Taub-NUT spacetime with Misner strings present is less pathological than previously thought, and their complete removal may be unnecessary Clement:2015cxa ; Clement:2015aka ; Clement:2016mll . The Misner strings are transparent for geodesics, rendering the spacetime geodesically complete, and despite the occurrence of CTC regions, the spacetime is (at least for sufficiently small string strengths) free of causal pathologies for geodesic observers miller1971taub ; Clement:2015cxa ; Clement:2015aka ; Clement:2016mll . In this paper we thus consider the whole spacetime with Misner strings present as ‘physically relevant’ and study its thermodynamic properties.

The thermodynamics of the Lorentzian Taub–NUT solutions with Misner strings present has been recently formulated in Kubiznak:2019yiu (see also Ballon:2019uha for the extension to the $U(1)$ charged case). The key step was to introduce a new pair of conjugate quantities $\psi-N$, allowing for the independent variations of the NUT parameter in the first law. For the uncharged Taub–NUT-AdS solution such a new first law then takes the following form:

[TABLE]

Here, $M$ stands for the mass of the solution, $T$ is the temperature of the ‘black hole horizon’, $S$ is the corresponding entropy given by the Bekenstein–Hawking area law,111In this paper we do not pursue the second proposal put forward in Kubiznak:2019yiu , where the entropy is identified with the ‘Noether charge’ Garfinkle:2000ms , which leads to some pathological features. $P$ is the cosmological constant pressure and $V$ is the conjugate thermodynamic volume,

[TABLE]

However, in Kubiznak:2019yiu no physical interpretation or geometrical meaning were given to the two new quantities $\psi$ and $N$. It is a purpose of this paper to amend this situation.

As already noted in Carlip:1999cy , at the positions of Misner strings there are Killing horizons.222Let us stress here that the corresponding Killing generators are different on the north and south pole axes and are different from the generator of the black hole horizon. In what follows we propose to identify the corresponding surface gravity with the potential $\psi$. In fact, for an asymmetric distribution of Misner strings the north pole and south pole surface gravities are different and yield thus two ‘independent’ potentials $\psi_{\pm}$. The corresponding conjugate quantities $N_{\pm}$, we call them the gravitational Misner charges, can be obtained by a Komar-type integration over the tubes surrounding the Misner string singularities. Such an integral is finite for the asymptotically flat Taub–NUTs but diverges in the AdS case where it has to be ‘renormalized’. We show that similar to the black hole volume, this renormalization can be achieved with the help of Killing co-potentials Kastor:2009wy . The prescription for $N_{\pm}$ (see (38) below) originates from a careful re-derivation of the generalized Smarr relation, taking into account the new boundaries induced by the presence of Misner strings. Interestingly, the tube integrals also modify the ‘standard prescription’ for the thermodynamic volume $V$, see (39) below. Of course, by construction the resultant quantities obey the generalized Smarr relation

[TABLE]

and (with a careful choice of the Killing co-potential) also the extended first law

[TABLE]

Such a first law not only accounts for possible asymmetric distribution of Misner strings, but also the ‘strength’ of the overall Misner string can be independently varied. In this sense it is of full cohomogeneity and reduces to (1) upon fixing the Misner string strengths.

Our paper is organized as follows. In the next section we review the basic facts about the Taub–NUT-AdS solution and discuss the Misner Killing horizons. The thermodynamic argument of Kubiznak:2019yiu leading to the introduction of the new thermodynamic conjugate pair $\psi-N$ is briefly summarized in Sec. III. Sec. IV is devoted to the derivation of the generalized Smarr relation; novel Komar-like formulae for the Misner charges $N_{\pm}$ and the volume $V$ are presented. In Sec. V such formulae are applied to the Taub–NUT-AdS spacetime to derive the generalized first law for an arbitrary distribution of Misner strings. Sec. VI remarks on a potential physical interpretation of Misner charges. Summary and final discussion are presented in Sec. VII. The appendix is devoted to the generalized first law for the dyonic Taub–NUT solutions.

II Geometry of Taub–NUT-AdS spacetime

The AdS Taub–NUT solution reads

[TABLE]

where $n$ stands for the NUT charge, $m$ for the mass parameter, and $l$ for the AdS radius,

[TABLE]

The above metric describes a situation where the Misner strings are ‘symmetrically distributed’: both the south $(\theta=\pi)$ and the north ($\theta=0)$ pole axis are ‘equally singular’. As we shall show in this paper, for formulating thermodynamics this is not required and an ‘arbitrary’ distribution of these defects, or strength of Misner strings, can be considered. The corresponding metric can be obtained by the following ‘large coordinate transformation’ Clement:2015cxa

[TABLE]

corresponding to ‘threading the spacetime with an overall Misner string’ (changing effectively the strengths of north pole and south pole Misner strings), upon which we obtain

[TABLE]

We stress that the parameter $s$ is a new physical dimensionful parameter. Roughly speaking, whereas $n$ corresponds to the differential strength between the two Misner strings, $s$ governs their (overall) absolute magnitude.333The situation is somewhat analogous to that of the C-metric, see Anabalon:2018qfv , where the acceleration parameter $A$ corresponds to the difference between the two cosmic string tensions (causing the black hole to accelerate), and the conical deficit parameter $K$ tunes the (overall) absolute magnitude. In particular, the choice of $s=0$ recovers the symmetric distribution of strings, while $s=\pm n$ makes the south (north) axis regular.444According to Misner misner1963flatter , all such strings are unobservable provided the time is identified as $t\sim t+8\pi n$ . In what follows we shall not impose this condition and consider the time coordinate non-compactified. As shown in Clement:2015cxa , the (asymptotically flat) spacetime is geodesically complete for any value of $s$, but the requirement for the absence of closed timelike and null geodesics requires $|s/n|\leq 1$.

There is a Killing horizon in the spacetime, located at the largest root $r_{+}$ of $f(r_{+})=0$ and generated by the Killing vector

[TABLE]

We shall refer to this horizon as a black hole horizon. The associated temperature is simply given by

[TABLE]

while its area reads

[TABLE]

There are yet other Killing horizons present in the spacetime. Namely, when the Misner string is present, the north/south pole axis is a Killing horizon of the following Killing vector:

[TABLE]

Note that, contrary to $k$, these Killing vectors are not properly normalized at infinity and their norm is there $\theta$-dependent. The corresponding surface gravities can be calculated using the standard formula:

[TABLE]

which yields

[TABLE]

In what follows we shall associate with these the following ‘Misner potentials’:555The normalization of the surface gravities of the strings—dividing by $4\pi$ rather than $2\pi$—is motivated by the Euclidean thermodynamics. There, in the vicinity of the Misner string, the Euclidean time coordinate behaves like an angular coordinate on $\mathbb{S}^{3}$, and therefore has period $4\pi$ if regularity is enforced.

[TABLE]

Note that when $s=0$, the two coincide and we recover

[TABLE]

III Overview of thermodynamic argument for $s=0$

Before we proceed to identifying the Misner charges $N_{\pm}$ by Komar-type integration, let us briefly recapitulate the thermodynamic argument for formulating the thermodynamics of Taub–NUT solutions with $s=0$, as presented in Kubiznak:2019yiu .

The basic assumption in Kubiznak:2019yiu was to identify the temperature of the spacetime with the temperature of the black hole horizon, (11), and the entropy with the black hole horizon area according to the Bekenstein–Hawking area law

[TABLE]

The mass was calculated by the conformal method to yield

[TABLE]

The next step was to calculate the Euclidean action Emparan:1999pm :

[TABLE]

and to associate it with a free energy, ${\cal G}=I/\beta$, $\beta=1/T$. By assumption this was equated with

[TABLE]

where $N$ was the new NUT-related (Misner) charge and $\psi$ the corresponding thermodynamic conjugate potential. Specifically, the following result was obtained in Kubiznak:2019yiu for the metric (A):

[TABLE]

We note here that the same expression remains valid also for the more general metric (9) with $s\neq 0$.

Using the explicit expression for ${\cal G}$, (22), and the fact that all quantities apart from $\psi$ and $N$ on the r.h.s. of (21) are known, one can deduce that

[TABLE]

Requiring the first law (1) then fixes $\psi$ and $N$ uniquely (up to an overall reciprocal constant factor) and yields

[TABLE]

as well as the thermodynamic volume

[TABLE]

Obviously, the thermodynamic potential $\psi$ in (24) coincides with the potential $\psi$ introduced in (17), given by the ‘surface gravity’ of the symmetrically distributed Misner strings. In what follows we generalize this notion and consider $\psi_{\pm}$ in (16) to be the new thermodynamic potentials associated with the Taub–NUT solutions. The remaining task is to find a geometric prescription for the conjugate quantities $N_{\pm}$. Since for general $s$ the potentials $\psi_{\pm}$ are different so will be the conjugate quantities $N_{\pm}$. As we shall see, the new Misner charges $N_{\pm}$ can be obtained by the Komar-like integrals. In AdS some of the integrals have to be ‘renormalized’ Magnon:1985sc . We do this here with the help of Killing co-potentials Kastor:2009wy .

IV Misner gravitational charges & generalized Smarr

To find the Komar integral prescription for the Misner charges, we re-derive the generalized Smarr relation for the Taub–NUT-AdS spacetime with arbitrary $s$ parameter, taking carefully into account the new boundaries induced by the presence of Misner strings. For simplicity we concentrate on the static case in $d$ number of spacetime dimensions.

Consider a Killing vector $k=\partial_{t}$, a generator of the black hole horizon. Being a Killing vector, it satisfies the following two identities:

[TABLE]

The first one means Kastor:2009wy that (at least locally) there exists a Killing co-potential 2-form $\omega$, such that

[TABLE]

Of course, such $\omega$ is not defined uniquely, and one can always perform

[TABLE]

where $\nabla_{c}\nu^{ca}=0$. In an Einstein space,

[TABLE]

the second identity (26) then yields

[TABLE]

or in the language of differential forms

[TABLE]

Integrating over a $(d-1)$-dimensional hypersurface $\Sigma$ and using the Stokes’ theorem, we thus recover

[TABLE]

For the vanishing NUT parameter this is a standard formula used for deriving the Smarr relation for AdS black hole spacetimes Kastor:2009wy . Namely, choosing $\Sigma$ to be a $t=const$ hypersurface, the integral over $\partial\Sigma$ can be split into an integral over the horizon which gives $TS$ and $PV$ terms and the integral over a sphere at infinity which is related to the black hole mass $M$ Kastor:2009wy .

In the case of the Taub–NUT, however, the integration becomes more subtle. Namely, when the Misner strings are present, $(\nabla t)^{2}$ is not well defined on the axis misner1963flatter and one is forced to introduce a new tube-like boundary around the Misner string singularities. As illustrated in Fig. 1, the boundary $\partial\Sigma$ thus consists of the two Misner string tubes $T_{+}$ and $T_{-}$ located at $\theta=\epsilon$ and $\theta=\pi-\epsilon$ respectively, the black hole horizon $H$ at $r=r_{+}$, and the sphere $S_{\infty}$ at $r=\infty$. Taking into account the orientation of the boundaries we have

[TABLE]

The integral (32) thus splits into the following five contributions:

[TABLE]

Here, the integration over the Misner tubes is over $r\in[r_{+},\infty]$, $\phi\in[0,2\pi]$ (and other azimuthal angles in the case of $d>4$), and at fixed $t=const$. We have also introduced the notation

[TABLE]

splitting the integral into the $r\to\infty$ and $r=r_{+}$ parts.

The first two terms are standard and yield respectively the thermodynamic mass $M$ and the product of entropy and temperature:

[TABLE]

The arrangement of the other terms is novel. Namely, we define the gravitational Misner charges by

[TABLE]

Here, $\psi_{\pm}$ are the potentials given by the surface gravities of the corresponding Misner strings, the integral of Killing co-potential is only evaluated at $r=\infty$ and removes the divergence in the standard way of modified Komar integrals in AdS Magnon:1985sc ; Kastor:2009wy .

The final integral in (34) contains the Killing copotential evaluated on the horizon in the usual way plus the contributions from the Killing co-potential arising from the tube boundary evaluated at the horizon $r=r_{+}$. This integral yields the modified thermodynamic volume

[TABLE]

With these definitions we thus derived the following generalized Smarr relation:

[TABLE]

A word of warning is due here. As noted in Cvetic:2010jb the freedom in the choice of the Killing co-potential, (28), results in non-uniqueness of the prescription for mass $M$ and other quantities. “The best one can do is to choose such $\omega$ so that the integral (36) yields the true thermodynamic mass.” The ultimate check is the validity of the first law of thermodynamics (4).

Let us also note that integrals similar to (38), renormailzed ‘by hand’ instead of using the Killing co-potential, were considered in Garfinkle:2000ms . In that case, however, they were interpreted as Misner string contributions to the total Noether charge entropy Garfinkle:2000ms rather than independent Misner charges $N_{\pm}$.

V First law with variable Misner string strengths

Let us now illustrate the above definitions on the $(d=4$) Taub–NUT-AdS spacetime (9) with arbitrary distribution of Misner strings. Such a metric admits a hidden symmetry encoded in the principal Killing–Yano tensor $h=db$, where Kubiznak:2007kh

[TABLE]

Such a tensor underlies many remarkable properties of the spacetime Frolov:2017kze . In particular, it gives the following Killing co-potential for the Killing vector $k=\partial_{t}$:

[TABLE]

It can easily be shown that indeed $\nabla\cdot\omega=k$. Since $\sqrt{-g}=(r^{2}+n^{2})\sin\theta$, the Hodge dual is

[TABLE]

At the same time we have

[TABLE]

It is now easy to show that the first integral (36) simply yields the mass $m$, while (37) yields $TS=\frac{1}{4}f^{\prime}(r_{+})(r_{+}^{2}+n^{2})$, all consistent with

[TABLE]

The integrals (38) over the Misner tubes are a bit more interesting. We find

[TABLE]

and the Killing co-potential part indeed renormalizes the first integral. Upon using $\psi_{\pm}$ in (16), we thus find

[TABLE]

Finally, considering the volume integral (39), we find

[TABLE]

and therefore

[TABLE]

in accord with (25).

It is then easy to verify that the obtained quantities satisfy the desired first law (4),

[TABLE]

and by construction also the generalized Smarr relation (40). Note that the obtained first law is of full cohomogeneity as all the parameters $\{r_{+},n,s,l\}$ may now be varied. In other words, processes involving redistributing the Misner strings are allowed and described by the obtained first law.

Note also that for $s=0$ (symmetric distribution of Misner strings) the two $\psi_{\pm}$ coincide and yield the same $\psi$, (17). In this case (and this case only) we can define new Misner charge

[TABLE]

and write the first law in the form (1) presented in Kubiznak:2019yiu .

Note also that for $s=+n$, the Misner string disappears on the south pole and (50) reduces to

[TABLE]

and analogously for $s=-n$.

VI Some remarks on Noether charges

The above derived extended first law (50) contains two new terms

[TABLE]

where the Misner potentials $\psi_{\pm}$ are given by the surface gravity of the Killing horizons associated with the Misner strings and $N_{\pm}$ are the conjugate gravitational Misner charges. It remains to be seen if (in Lorentzian signature) $\psi_{\pm}$ can be interpreted as a temperature. If this were the case, then the Lorentzian Taub-NUT solution would be somewhat analogous to de Sitter black holes, e.g. Dolan:2013ft , constituting in general a multi-temperature system out of equilibrium. If such an interpretation were possible then the statement of the first law would be commensurate with statement that $N_{+}$ and $N_{-}$ are the entropies associated with the Misner strings.

In the spirit of de Sitter analogy, one could then define a ‘total entropy’

[TABLE]

Note that this is different from what was in Kubiznak:2019yiu called the ‘Noether charge entropy’, $S_{\mbox{\tiny NC}}$, the difference between the two being given by the surface gravity factors:

[TABLE]

Obviously, the two coincide only when the ‘equilibrium’ is restored, imposing $\psi_{+}=\psi_{-}=T$, a situation studied in Garfinkle:2000ms .

Let us note, however, that provided $\psi_{\pm}$ truly correspond to temperatures, then it would seem more natural to define the entropy of each of the Misner horizons as Noether charge associated to $k_{\pm}$ (the generators of the horizons) normalized by the surface gravity factors, rather than $k=\partial_{t}$,

[TABLE]

specializing for the moment to the asymptotically flat case where the Killing co-potentials do not appear. This procedure is equivalent to the Lorentzian version of the argument in Carlip:1999cy and yields an entropy proportional to the (infinite) area of the strings:

[TABLE]

where $R$ is the radial cutoff. Using the fact that $k_{\pm}$, (13), can be written as

[TABLE]

(that is surface gravities of Misner strings are at the same time their angular velocities) yields the following relation between $\hat{N}_{\pm}$ and $N_{\pm}$:

[TABLE]

It is somewhat interesting that the first law does not make use of this entropy, but rather only a “portion” of it encoded in $N_{\pm}$. Presumably this has to do with the fact that the first law is constructed for an observer whose asymptotic motion aligns with the generator of the black hole horizon, and not the generators of the string horizons. Note also that whereas $N_{\pm}$ are finite their two contributions in (59) both diverge (unless ‘renormalized’ by for example the background substraction procedure).

VII Summary and discussion

As shown recently, the thermodynamics of the Lorentzian Taub–NUT-AdS spacetimes with the Misner strings present can be consistently formulated Kubiznak:2019yiu . The key ingredient is to introduce a new pair of conjugate quantities $\psi-N$. However, in Kubiznak:2019yiu no physical intuition or geometrical interpretation for these quantities was offered and the entire construction based on thermodynamic considerations, namely that the full cohomogeneity first law where the NUT parameter is independently varied should be valid.

In this paper we have extended these results in a number of ways. In particular, we have identified the new potential $\psi$ with the surface gravity of the Misner string Killing horizon that is present in the spacetime. For a symmetric distribution of Misner strings (the case of $s=0$ studied in Kubiznak:2019yiu ) such a surface gravity is the same on the north and south pole axes. However, for a general Taub–NUT spacetime the strengths of the two Misner strings will be different and so will be their corresponding surface gravities, leading thus to two independent potentials $\psi_{\pm}$. We have shown that the corresponding thermodynamically conjugate charges $N_{\pm}$ can then be obtained by a Komar-type integration over the tubes surrounding the Misner string singularities (38). This establishes the geometrical meaning of $N_{\pm}$ (though the physical meaning, for example an interpretation in terms of entropy, remains to be seen). Similar tube contributions also modify the ‘standard formula’ Kastor:2009wy ; Cvetic:2010jb for the geometric black hole volume $V$, see (39). To renormalize the necessary divergences present in the AdS case we have used the method of Killing co-potentials Kastor:2009wy . The resultant charges are then automatically finite and by construction satisfy the generalized Smarr relation (40).

We have also extended the first law (1) presented in Kubiznak:2019yiu to a more general law (4), allowing for the asymmetric distribution of Misner strings. In fact, the parameter $s$ (governing this distribution) can now be freely varied in (4). Such a law thus allows for variable Misner string strengths and in that sense is of higher cohomogeneity than (1). The variable string strengths can perhaps be realized in a process of capture of a Misner string by a black hole or an idealized axisymmetric merger of two Taub–NUT solutions.

The fact that we have explicitly shown that the thermodynamic volume receives contributions from the Misner strings raises further interesting questions concerning the reverse isoperimetric inequality Cvetic:2010jb .666Recall that it was conjectured in Cvetic:2010jb that for a black hole with area ${\cal A}$ and thermodynamic volume ${\cal V}$ the ratio

${\cal R}=\left(\frac{(d-1){\cal V}}{{\cal A}_{d-2}}\right)^{1/(d-1)}\left(\frac{{\cal A}_{d-2}}{{\cal A}}\right)^{1/(d-2)},$

satisfies ${\cal R}\geq 1$. Namely, it is natural to wonder whether the ‘area’ that appears in this formula is simply the area of the black hole horizon while the volume is the total thermodynamic volume of the spacetime, or whether the area should receive additional contributions from the Misner strings. Yet another possibility is that the horizon and Misner strings should be considered separately in the context of this inequality, similar to the event and cosmological horizons for de Sitter black holes Dolan:2013ft . If this is the case, the reverse isoperimetric inequality would continue to hold for the horizon, while, since the ‘area’ of the Misner strings is infinite, it would seem to imply that it is the isoperimetric inequality that holds for them, analogous to the volume between the black hole and cosmological horizons in the de Sitter case.

Let us finally mention that the thermodynamic charges $N_{\pm}$ as well as the volume $V$ can also be obtained by pure thermodynamic considerations, from the Euclidean action (74), once the potentials $\psi_{\pm}$ have been identified. Namely, as noted in Sec. III, for general $s$ the free energy ${\cal G}$ corresponding to the Euclidean action (74) remains given by (22). Using (16) for $\psi_{\pm}$, (11) for $T$, and (2) for $P$, one can explicitly write

[TABLE]

The conjugate thermodynamic quantities are then given by

[TABLE]

By construction these satisfy the first law (4). The Smarr relation (40) is then equivalent to the relation ${\cal G}=M-TS-\psi_{+}N_{+}-\psi_{-}N_{-}$. Of course, one can easily check that the thermodynamic quantities (61) coincide with the geometric quantities derived in the previous section. This approach is similar to that used in Bueno:2018uoy where it was found that, in the absence of Misner strings, such terms are necessary for consistent thermodynamics of Taub solutions with toroidal base spaces.

Acknowledgements

We are grateful to Robert Mann for discussions. This work was supported in part by the Natural Sciences and Engineering Research Council of Canada. R.H. is grateful to the Banting Postdoctoral Fellowship programme. A.B., F.G., and D.K. acknowledge the Perimeter Institute for Theoretical Physics for their support. Research at Perimeter Institute is supported by the Government of Canada through the Department of Innovation, Science and Economic Development Canada and by the Province of Ontario through the Ministry of Research, Innovation and Science.

Appendix A Generalized first law for the dyonic Taub–NUTs

In this appendix we extend the recent results on thermodynamics of dyonic Taub–NUTs of the Einstein–Maxwell theory Ballon:2019uha , to account for asymmetric distributions of Misner strings and their potential variable strengths. To this purpose we shall not generalize the Komar integration to the presence of matter but rather take a short cut and use the Euclidean action.

The dyonic Taub–NUT–AdS solution with arbitrary parameter $s$ reads

[TABLE]

with

[TABLE]

where $e$ and $g$ are the electric and magnetic parameters, while other parameters assume the meaning as before.

The black hole horizon, located at $f(r_{+})=0$, is generated by $k=\partial_{t}$ and has the following temperature and entropy

[TABLE]

As in the uncharged case, the Misner string horizons are generated by $\xi_{\pm}=\partial_{t}-\frac{1}{2(n\pm s)}\partial_{\phi}$ , and give rise to the following Misner potentials:

[TABLE]

The parameters $e$ and $g$ are related to the electric $q$ and magnetic $q_{m}$ charges by the Gauss’s law:

[TABLE]

where $F=dA$ is the Maxwell field strength. By performing the integration over a sphere of radius $r$, we find

[TABLE]

which, contrary to common wisdom and known examples of dyonic solutions depends on the radius of the sphere. Remarkably, as found in Ballon:2019uha these charges enter the first law in a way that one of these is evaluated at infinity and the other on the horizon. For the purpose of this appendix we take

[TABLE]

The electrostatic potential is

[TABLE]

Employing the ‘electromagnetic duality’

[TABLE]

which relates $q\leftrightarrow q_{m}$, the corresponding magnetic potential is

[TABLE]

Turning now towards the action

[TABLE]

one finds

[TABLE]

The action as written corresponds to the grand-canonical (fixed electric potential and fixed magnetic charge) ensemble and so is a function of the following quantities:

[TABLE]

The corresponding thermodynamic quantities are then given by

[TABLE]

. This yields $S$ in (65), $Q$ in (69), $\phi_{m}$ in (73), and $V$ in (25). We also find the following Misner charges:

[TABLE]

Identifying $M=m$, this yields the following full cohomogeneity first law:

[TABLE]

where all 6 parameters of the solution: $\{n,m,e,g,s,l\}$ are independently varied. The relation ${\cal G}=M-TS-\phi Q-\psi_{+}N_{+}-\psi_{-}N_{-}$ is then equivalent to the Smarr relation

[TABLE]

The first law with $Q\to Q^{(+)}$ and $Q_{m}^{(+)}\to Q_{m}$ could be obtained analogously and so could be the special (constrained) subcases studied in Ballon:2019uha .