††institutetext: School of Physics and Astronomy, China West Normal
University, 1 Shida Road, Nanchong, Sichuan 637002, People’s Republic of China

# Consistent mass formulae for higher even-dimensional Taub-NUT spacetimes and
their AdS counterparts

Di Wu, Shuang-Qing Wu∗***Corresponding author<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Currently, there is a great deal of interest in seeking of consistent
thermodynamics of the Lorentzian Taub-NUT spacetimes. Despite a lot of
“satisfactory” efforts have been made, all of these activities have been
confined to the four-dimensional cases, with the higher even-dimensional cases
remaining unexplored. The aim of this article is to fill the gap for the first
time. To the end of this subject, we first adopt our own idea that “The NUT
charge is a thermodynamical multi-hair” to investigate the consistent
thermodynamics of $D=6,8,10$ Lorentzian Taub-NUT spacetimes without a
cosmological constant. Similarly to the $D=4$ cases as did in our previous
works, we find that the first law and Bekenstein-Smarr mass formulas are
perfectly satisfied if we still assign the secondary hair: $J_{n}=Mn$ as a
conserved charge in both mass formulae. Turning to the cases with a nonzero
cosmological constant, our treatment continues to work very well and all the
results can be fairly generalized to the corresponding Taub-NUT AdS spacetimes
in higher even-dimensions, although we do not know how to define and introduce
a similar higher-dimensional version of the dual (magnetic) mass that is well
known in four dimensions.

###### Keywords:

Black Holes

##  1 Introduction

Recently, there has been a resurgence of great interest in exploring the
consistent thermodynamics of the Lorentzian Taub-NUT spacetimes PRD100-101501
; PRD105-124013 ; PRD100-064055 ; JHEP0719119 ; CQG36-194001 ; PLB798-134972 ;
JHEP0520084 ; PRD100-104016 ; PLB832-137264 ; PLB802-135270 ; IJMPD31-2250021
; JHEP0821152 ; PRD101-124011 ; PRD105-124034 ; PRD103-024052 ; JHEP0321039 ;
PRD106-024022 ; EPJP130-124 ; 2112.00780 ; 2208.05494 . In our opinion, these
current investigations of the first law of the NUT-charged spacetimes can be
categorized into three different schemes: (I) Retaining the mass unmodified
and introducing new global-like charges (secondary hairs) together with their
conjugate potentials PRD100-101501 ; PRD105-124013 ; (II) Keeping the mass
unchanged and including new nonglobal Misner charges and their conjugate
variables PRD100-064055 ; PLB798-134972 ; JHEP0719119 ; JHEP0520084 ;
CQG36-194001 ; PRD100-104016 ; and (III) Only modifying the mass by taking
account for the contribution of new nonglobal charges PRD101-124011 ;
PRD105-124034 . Note that in recent eprint 2208.05494 , the thermodynamic mass
that enters into the first law of the four-dimensional Taub-NUT spacetime is
the horizon mass PRD56-961 . Besides these, there is fewer interest
PRD100-064055 ; JHEP0821152  to consider the entropy as the Noether charge
CQG17-3317  that includes the horizon area and the contribution from the
Misner strings. However, all of the above-mentioned efforts are only
restricted to four-dimensional cases, leaving thermodynamics of the Lorentzian
Taub-NUT spacetimes in higher even-dimensions unexplored, which motivates the
subject of the present article.

In our previous papers PRD100-101501 ; PRD105-124013 , we have advocated a new
idea that “The NUT charge is a thermodynamical multi-hair” and put forward a
simple, systematic way to study the consistent thermodynamics of almost all of
the four-dimensional (dyonic) NUT-charged spacetimes. It should be emphasized
that, unlike all other attempts PRD100-064055 ; JHEP0719119 ; CQG36-194001 ;
PLB798-134972 ; JHEP0520084 ; PRD100-104016 ; PLB832-137264 ; PLB802-135270 ;
IJMPD31-2250021 ; JHEP0821152 ; PRD101-124011 ; PRD105-124034 ; 2112.00780 ;
2208.05494 , our scheme only relies on deriving firstly a new meaningful
Christodoulou-Ruffini-type squared-mass formula PRL25-1596 ; PRD4-3552
satisfied by the four-dimensional (dyonic) NUT-charged spacetimes, and the
only needed input in this derivation is to introduce the secondary hairs:
($J_{n}=mn$, $Q_{n}=qn$ and $P_{n}=pn$) as new conserved charges. Then the
consistent thermodynamic first law and Bekenstein-Smarr mass formulas of these
NUT-charged spacetimes can be deduced via some simple and purely algebraic
manipulations from this squared-mass formula, which can hardly be given by the
other papers as mentioned above. Subsequently, the usual Bekenstein-Hawking
one-quarter area-entropy relation can be naturally restored for the generic
NUT-charged spacetime (and all its extensions) without imposing any constraint
condition and with no need to assume ahead that the one-quarter area-entropy
relation should hold true. The advantage of our proposal that the NUT charge
acts as a thermodynamical multi-hair is that it can not only explicate the
rotation-like and electromagnetic charge-like characters, but also
simultaneously explain many other exotic properties. What is more, our
consistent mass formulae PRD100-101501 ; PRD105-124013  are unique, and all
expressions for thermodynamical quantities are exceedingly simple and
succinct. This is in contrast to all other works where not only can the
consistent first law of the NUTty dyonic spacetimes have the electric-type,
magnetic-type, mixed-type versions JHEP0719119 ; JHEP0520084 , and even many
other ones PRD100-104016 , but also the expressions of the related
thermodynamical variables are quite complicated.

In this work, we will continue to apply our proposal that “The NUT charge is a
thermodynamical multi-hair” to investigate consistent thermodynamics of the
$D=6,8,10$ Lorentzian Taub-NUT spacetimes without and with a cosmological
constant. Our paper is organized as follows. In sec. 2, we start with the
construction of a novel Christodoulou-Ruffini-like squared-mass formula of the
six-dimensional Lorentzian Taub-NUT solution by additionally including only
one secondary hair $J_{n}=Mn$, as did in refs. PRD100-101501 ; PRD105-124013 .
Using this squared-mass formula, both the differential and integral mass
formulae can be deduced through a simple mathematical manipulation. Then, the
procedure is extended to the six-dimensional Lorentzian Taub-NUT-AdS case. In
sec. 3, we proceed to discuss the cases of the eight-dimensional Lorentzian
Taub-NUT and Taub-NUT-AdS spacetimes, respectively. Then, in sec. 4, we extend
to investigate the cases of the ten-dimensional Taub-NUT spacetime and its AdS
extension. We find that our scheme in the $D=6,8,10$ cases works successfully
as in the four-dimensional case PRD100-101501 , and summarize in sec. 5 the
main results for the generic $(2k+2)$-dimensional Taub-NUT-AdS spacetimes.
Finally, we present our conclusions and outlooks in sec. 6.

##  2 6-dimensional Taub-NUT spacetime

As shown in ref. CQG19-2051  for the six-dimensional Taub-NUT spacetime, there
are two different choices for the base space, namely, $S^{2}\times{}S^{2}$ and
$\mathbb{CP}^{2}$. We start our investigation of the mass formulas in the case
of the $S^{2}\times S^{2}$ base space, but the same procedure is also
applicable to the case of the $\mathbb{CP}^{2}$ base space. Using
$S^{2}\times{}S^{2}$ as a base space, the metric of the six-dimensional
Lorentzian Taub-NUT solution has the form:

$\displaystyle ds_{6}^{2}$ $\displaystyle=$
$\displaystyle-f(r)\Big{(}dt+2n\sum_{i=1}^{2}\cos\theta_{i}{}d\phi_{i}\Big{)}^{2}+\frac{dr^{2}}{f(r)}+\big{(}r^{2}+n^{2}\big{)}\sum_{i=1}^{2}\big{(}d\theta_{i}^{2}+\sin^{2}\theta_{i}{}d\phi_{i}^{2}\big{)}\,,\quad$
(1)

where

$f(r)=\frac{r^{4}+6n^{2}r^{2}-3n^{4}-6mr}{3\big{(}r^{2}+n^{2}\big{)}^{2}}\,,$

in which $m$ and $n$ are the mass parameter and the NUT charge parameter,
respectively.

Our aim is to derive various mass formulae and to discuss consistent
thermodynamics of the six-dimensional Lorentzian Taub-NUT spacetime. To begin
with, let us present some known quantities that can be evaluated via the
standard method. First, the area and the surface gravity at the horizon are
easily computed as

$A_{h}=16\pi^{2}\big{(}r_{h}^{2}+n^{2}\big{)}^{2}=16\pi^{2}\mathcal{A}_{h}\,,\qquad\kappa=\frac{1}{2}f^{\prime}(r_{h})=\frac{1}{2r_{h}}\,,$
(2)

in which a reduced horizon area $\mathcal{A}_{h}=(r_{h}^{2}+n^{2})^{2}$ is
introduced just for briefness, and $r_{h}$ represents the greatest root of the
horizon equation: $r_{h}^{4}+6n^{2}r_{h}^{2}-3n^{4}-6mr_{h}=0$.

As for the global conserved charges ($M$ and $N$), the Komar mass is
divergent, while the Abbott-Deser (AD) mass NPB195-76  is finite. The AD mass
$M$ associated to the Killing vector $\partial_{t}$ and the NUT charge $N$
read

$M=8\pi{}m\,,\qquad N=8\pi{}n\,.$ (3)

###  2.1 Consistent mass formulas of the 6-dimensional Taub-NUT spacetime

In order to establish the first law which is reasonable and consistent in both
physical and mathematical senses, we employ the algebraic approach suggested
in refs. PRD100-101501 ; PRD105-124013 ; PLB608-251  to construct a meaningful
Christodoulou-Ruffini-type squared-mass formula. First, via reexpressing
$r_{h}=\sqrt{\mathcal{A}_{h}^{1/2}-n^{2}}$ in terms of the reduced horizon
area and substituting it into the equation:
$(r_{h}^{4}+6n^{2}r_{h}^{2}-3n^{4})^{2}=36m^{2}r_{h}^{2}$, we get the
following identity:

$m^{2}=\frac{1}{36\sqrt{\mathcal{A}_{h}}}\big{(}\mathcal{A}_{h}+4n^{2}\sqrt{\mathcal{A}_{h}}-8n^{4}\big{)}^{2}+\frac{m^{2}n^{2}}{\sqrt{\mathcal{A}_{h}}}\,,$
(4)

which can be alternatively converted to a quartic polynomial of
$\mathcal{A}_{h}$:

$\big{(}\mathcal{A}_{h}^{2}+36m^{2}n^{2}+64n^{8}\big{)}^{2}=16\big{(}9m^{2}+16n^{6}-2n^{2}\mathcal{A}_{h}\big{)}^{2}\mathcal{A}_{h}\,.$

Next, in addition to the conserved charges $M$ and $N$ given in eq. (3), only
one extra input that we need is to introduce the secondary hair
$J_{n}=Mn=8\pi{}mn$ as a thermodynamic independent variable. Then after
substituting $m=M/(8\pi)$, $n=N/(8\pi)$ and $\mathcal{A}=8\pi\mathcal{A}_{h}$
into eq. (4), one can arrive at an useful identity

$M^{2}=\frac{\sqrt{2\pi}}{18\sqrt{\mathcal{A}}}\Big{(}\mathcal{A}+\frac{N^{2}}{8\pi^{2}}\sqrt{2\pi\mathcal{A}}-\frac{N^{4}}{64\pi^{3}}\Big{)}^{2}+\frac{2\sqrt{2\pi}}{\sqrt{\mathcal{A}}}J_{n}^{2}\,,$
(5)

which is our new Christodoulou-Ruffini-like squared-mass formula for the six-
dimensional Taub-NUT spacetime. Alternatively, the above equation (5) can be
converted to a quartic polynomial of the area
$\mathcal{A}=\mathcal{A}(M,N,J_{n})$:

$\Big{(}\mathcal{A}^{2}+36J_{n}^{2}+\frac{N^{8}}{4096\pi^{6}}\Big{)}^{2}=\frac{\mathcal{A}}{8\pi^{3}}\Big{(}N^{2}\mathcal{A}-36\pi{}M^{2}-\frac{N^{6}}{64\pi^{3}}\Big{)}^{2}\,.$
(6)

Having finished this task, we are now in a position to obtain the differential
and integral mass formulae for the six-dimensional Taub-NUT spacetime. Since
the secondary hair $J_{n}$ will be treated as an independent variable, the
above squared-mass formula (5) can be regarded formally as a basic functional
relation: $M=M(\mathcal{A},N,J_{n})$. As did in refs. PRD100-101501 ;
PRD105-124013 ; PRD103-044014 ; PRD101-024057 ; PRD102-044007 ,
differentiating it with respect to the thermodynamical variables
($\mathcal{A},N,J_{n}$) yields their conjugate quantities, and subsequently we
can arrive at the differential and integral mass formulae with the conjugate
thermodynamic potentials given by the ordinary Maxwell relations.

For instance, differentiating the squared-mass formula (5) with respect to
$\mathcal{A}$ yields one-quarter of the surface gravity:

$\kappa=4\frac{\partial{}M}{\partial\mathcal{A}}\Big{|}_{(N,J_{n})}=\frac{1}{2r_{h}}\,,$
(7)

which is exactly the same one as given in eq. (2). Similarly, by
differentiating the squared-mass formula (5) with respect to the NUT charge
$N$ and the secondary hair $J_{n}$, then their conjugate gravito-magnetic
potential $\psi_{h}$ and quasi-angular momentum $\omega_{h}$ can be derived,
respectively, as follows:

$\psi_{h}=\frac{\partial{}M}{\partial{}N}\Big{|}_{(\mathcal{A},J_{n})}=\frac{4nr_{h}\big{(}r_{h}^{2}-3n^{2}\big{)}}{3\big{(}r_{h}^{2}+n^{2}\big{)}}\,,\qquad\omega_{h}=\frac{\partial{}M}{\partial{}J_{n}}\Big{|}_{(\mathcal{A},N)}=\frac{n}{r_{h}^{2}+n^{2}}\,.$
(8)

Now, one can check that both the differential and integral mass formulae are
completely fulfilled

$\displaystyle dM$ $\displaystyle=$
$\displaystyle(\kappa/4)d\mathcal{A}+\omega_{h}{}dJ_{n}+\psi_{h}{}dN\,,$ (9)
$\displaystyle 3M$ $\displaystyle=$
$\displaystyle\kappa\mathcal{A}+4\omega_{h}{}J_{n}+\psi_{h}{}N\,,$ (10)

among all the aforementioned thermodynamical conjugate pairs. Comparing these
mass formulae (9-10) with the standard ones, it is highly urged that the
following familiar identifications be made:

$S=\frac{A_{h}}{4}=\frac{\pi}{2}\mathcal{A}=4\pi^{2}\big{(}r_{h}^{2}+n^{2}\big{)}^{2}\,,\qquad
T=\frac{\kappa}{2\pi}=\frac{1}{4\pi{}r_{h}}\,,$ (11)

which naturally recovers the famous Bekenstein-Hawking one-quarter area-
entropy relation of the six-dimensional Taub-NUT spacetime, completely similar
to the $D=4$ cases.

###  2.2 Extension to the Taub-NUT-AdS6 spacetime

Now we will extend the above work to explore the Lorentzian Taub-NUT-AdS6
spacetime with a nonzero negative cosmological constant. The metric is still
given by eq. (1), but now we have

$f(r)=\frac{r^{4}+6n^{2}r^{2}-3n^{4}-6mr+3g^{2}\big{(}r^{6}+5n^{2}r^{4}+15n^{4}r^{2}-5n^{6}\big{)}}{3\big{(}r^{2}+n^{2}\big{)}^{2}}\,,$

where $l=1/g$ is the cosmological scale.

First, we will employ the conformal completion method PRD73-104036  to
calculate the conserved mass $M$ of the Taub-NUT-AdS6 solution. This conformal
AMD mass can be evaluated via the integral in terms of the conformal Weyl
tensor over the spatial conformal boundary at infinity. The Taub-NUT-AdS6
spacetime is asymptotically local AdS, and admits an asymptotic boundary
5-metric that approaches to

$d\bar{s}_{5}^{2}=\lim_{r\to\infty}\frac{ds_{6}^{2}}{r^{2}}=-g^{2}\Big{(}dt+2n\sum_{i=1}^{2}\cos\theta_{i}{}d\phi_{i}\Big{)}^{2}+\sum_{i=1}^{2}\big{(}d\theta_{i}^{2}+\sin^{2}\theta_{i}{}d\phi_{i}^{2}\big{)}\,,$
(12)

with which one can define a normal vector:
$\hat{n}^{a}=-gr^{2}\delta_{r}^{a}$.

Note that the 5-volume form of the conformal boundary AdS metric (12) is
simply given by

$\mathbb{V}_{5}=g\sin\theta_{1}\sin\theta_{2}\,dt\wedge{}d\theta_{1}\wedge{}d\theta_{2}\wedge{}d\phi_{1}\wedge{}d\phi_{2}\,,$
(13)

then using the inner-product rule
$<\partial_{\mu},dx^{\mu}>=\delta_{\mu}^{\nu}$, we can obtain the area vector:
$d\Sigma_{t}=<\partial_{t},\mathbb{V}_{5}>=g\sin\theta_{1}\sin\theta_{2}\,d\theta_{1}\wedge{}d\theta_{2}\wedge{}d\phi_{1}\wedge{}d\phi_{2}$,
from which we can get its only non-vanishing component:

$dS_{t}=g\prod_{i=1}^{2}\sin\theta_{i}{}d\theta_{i}{}d\phi_{i}\,.$ (14)

Since the conserved charge associated with a unit Killing vector $\xi^{\nu}$
is defined as

$\mathcal{Q}[\xi]=\frac{1}{24\pi{}g^{3}}\int\big{(}r^{3}C^{t}_{~{}a{\nu}b}\hat{n}^{a}\hat{n}^{b}\xi^{\nu}dS_{t}\big{)}\big{|}_{r\to\infty}\,,$
(15)

where $C^{t}_{~{}a{\nu}b}$ is the Weyl conformal tensor, we can easily compute
the conformal mass as:

$M=\mathcal{Q}[\partial_{t}]=8\pi{}m\,.$ (16)

Unfortunately, since it is unclear to us how to define a dual (magnetic) mass
in the higher dimensional spacetime, we will not consider the dual mass here
and hereafter. The NUT charge will be simply taken as $N=8\pi{}n$ just like
the case without a cosmological constant.

Next, the surface gravity at the horizon which is specified by the largest
root of equation: $f(r_{h})=0$ can be evaluated as

$\kappa=\frac{1}{2}f^{\prime}(r_{h})=\frac{1+5g^{2}\big{(}r_{h}^{2}+n^{2}\big{)}}{2r_{h}}\,,$
(17)

while the horizon area reads: $A_{h}=16\pi^{2}\mathcal{A}_{h}$, in which the
reduced horizon area is still denoted as:
$\mathcal{A}_{h}=(r_{h}^{2}+n^{2})^{2}$.

Now we would like to derive a novel Christodoulou-Ruffini-like squared-mass
formula likewise the case without a cosmological constant. Accordingly,
inserting $r_{h}=\sqrt{\mathcal{A}_{h}^{1/2}-n^{2}}$ into the equation:
$[r_{h}^{4}+6n^{2}r_{h}^{2}-3n^{4}+3g^{2}(r_{h}^{6}+5n^{2}r_{h}^{4}+15n^{4}r_{h}^{2}-5n^{6})]^{2}=36m^{2}r_{h}^{2}$
yields

$m^{2}=\frac{1}{36\sqrt{\mathcal{A}_{h}}}\Big{[}\big{(}1+6g^{2}n^{2}\big{)}\big{(}\mathcal{A}_{h}+4n^{2}\sqrt{\mathcal{A}_{h}}-8n^{4}\big{)}+3g^{2}\mathcal{A}_{h}^{3/2}\Big{]}^{2}+\frac{m^{2}n^{2}}{\sqrt{\mathcal{A}_{h}}}\,,$
(18)

which can be converted to a sextic polynomial of $\mathcal{A}_{h}$:

$\displaystyle\Big{[}9g^{4}\mathcal{A}_{h}^{3}+\big{(}1+6g^{2}n^{2}\big{)}\big{(}1+30g^{2}n^{2}\big{)}\mathcal{A}_{h}^{2}+64n^{8}\big{(}1+6g^{2}n^{2}\big{)}^{2}+36m^{2}n^{2}\Big{]}^{2}$
$\displaystyle\quad=4\Big{[}18m^{2}+\big{(}1+6g^{2}n^{2}\big{)}\big{(}3g^{2}\mathcal{A}_{h}+24g^{2}n^{4}+4n^{2}\big{)}\big{(}8n^{4}-\mathcal{A}_{h}\big{)}\Big{]}^{2}\mathcal{A}_{h}\,.$

Finally, plugging $m=M/(8\pi)$, $n=N/(8\pi)$,
$\mathcal{A}=8\pi\mathcal{A}_{h}$ and $g^{2}=4\pi{}P/5$ into eq. (18), where
$P=(D-1)(D-2)g^{2}/(16\pi)$ is the generalized pressure PRD84-024037 , and
also introducing a secondary hair: $J_{n}=Mn$ as before, then after a little
algebra we obtain an useful identity:

$\displaystyle M^{2}$ $\displaystyle=$
$\displaystyle\frac{\sqrt{2\pi}}{18\sqrt{\mathcal{A}}}\bigg{[}\Big{(}1+\frac{3N^{2}}{40\pi}P\Big{)}\Big{(}\mathcal{A}+\frac{N^{2}}{8\pi^{2}}\sqrt{2\pi\mathcal{A}}-\frac{N^{4}}{64\pi^{3}}\Big{)}+\frac{3}{10\pi}(2\pi\mathcal{A})^{3/2}P\bigg{]}^{2}$
(19) $\displaystyle+\frac{2\sqrt{2\pi}}{\sqrt{\mathcal{A}}}J_{n}^{2}\,,$

which is nothing but the Christodoulou-Ruffini-like squared-mass formula for
the six-dimensional Taub-NUT-AdS spacetime. Eq. (19) consistently reduces to
eq. (5) obtained in the case of the six-dimensional Taub-NUT spacetime when
the generalized pressure $P$ is turned off.

The differentiation of the squared-mass formula (19) leads to the first law:

$dM=(\kappa/4)d\mathcal{A}+\omega_{h}{}dJ_{n}+\psi_{h}{}dN+VdP\,,$ (20)

where

$\displaystyle\kappa$ $\displaystyle=$ $\displaystyle
4\frac{\partial{}M}{\partial\mathcal{A}}\Big{|}_{(N,J_{n},P)}=\frac{1+5g^{2}\big{(}r_{h}^{2}+n^{2}\big{)}}{2r_{h}}\,,\qquad\omega_{h}=\frac{\partial{}M}{\partial{}J_{n}}\Big{|}_{(\mathcal{A},N,P)}=\frac{n}{r_{h}^{2}+n^{2}}\,,$
$\displaystyle\psi_{h}$ $\displaystyle=$
$\displaystyle\frac{\partial{}M}{\partial{}N}\Big{|}_{(\mathcal{A},J_{n},P)}=\frac{2nr_{h}\big{[}2r_{h}^{2}-6n^{2}+3g^{2}\big{(}r_{h}^{4}+10n^{2}r_{h}^{2}-15n^{4}\big{)}\big{]}}{3\big{(}r_{h}^{2}+n^{2}\big{)}}\,,$
$\displaystyle V$ $\displaystyle=$
$\displaystyle\frac{\partial{}M}{\partial{}P}\Big{|}_{(\mathcal{A},N,J_{n})}=\frac{16\pi^{2}r_{h}(r_{h}^{6}+5n^{2}r_{h}^{4}+15n^{4}r_{h}^{2}-5n^{6})}{5\big{(}r_{h}^{2}+n^{2}\big{)}}\,.$

When the NUT charge parameter $n$ vanishes, the thermodynamic volume reduces
to $V=16\pi^{2}r_{h}^{5}/5$.

Utilizing all the expressions obtained above, one can directly verify that the
Bekenstein-Smarr mass formula

$3M=\kappa\mathcal{A}+4\omega_{h}{}J_{n}+\psi_{h}{}N-2VP\,,$ (21)

is completely satisfied also. It is naturally suggested to identify
$S=A_{h}/4=4\pi^{2}\mathcal{A}_{h}$ and $T=\kappa/(2\pi)$, so that the
solution acts like a genuine black hole without breaking the classical one-
quarter area/entropy relation.

##  3 8-dimensional Taub-NUT spacetime

In this section, we will extend the above discussion to the case of the eight-
dimensional Taub-NUT spacetime, to which there are two different choices
CQG19-2051  for the base manifold, namely $S^{2}\times{}S^{2}\times{}S^{2}$
and $S^{2}\times{}\mathbb{CP}^{2}$. Likewise the six-dimensional case, we will
only consider the case where the base space is
$S^{2}\times{}S^{2}\times{}S^{2}$, so that the metric owns a $U(1)$ fibration
over $S^{2}\times{}S^{2}\times{}S^{2}$:

$ds_{8}^{2}=-f(r)\Big{(}dt+2n\sum_{i=1}^{3}\cos\theta_{i}{}d\phi_{i}\Big{)}^{2}+\frac{dr^{2}}{f(r)}+\big{(}r^{2}+n^{2}\big{)}\sum_{i=1}^{3}\big{(}d\theta_{i}^{2}+\sin^{2}\theta_{i}{}d\phi_{i}^{2}\big{)}\,,$
(22)

where

$f(r)=\frac{r^{6}+5n^{2}r^{4}+15n^{4}r^{2}-5n^{6}-10mr}{5\big{(}r^{2}+n^{2}\big{)}^{3}}\,.$

At the horizon which is the largest root of $f(r_{h})=0$, the area and the
surface gravity can be evaluated via the standard method as

$A_{h}=64\pi^{3}\big{(}r_{h}^{2}+n^{2}\big{)}^{3}=64\pi^{3}\mathcal{A}_{h}\,,\qquad\kappa=\frac{1}{2}f^{\prime}(r_{h})=\frac{1}{2r_{h}}\,,$
(23)

where we now denote the reduced horizon area:
$\mathcal{A}_{h}=(r_{h}^{2}+n^{2})^{3}$.

Similar to the six-dimensional case, the AD mass and the NUT charge can be
computed as:

$M=48\pi^{2}m\,,\qquad N=48\pi^{2}n\,.$ (24)

###  3.1 Consistent mass formulas of the 8-dimensional Taub-NUT spacetime

To derive our squared mass formula, we will adopt the same trick as did in the
last section, so we first express the positive root
$r_{h}=\sqrt{\mathcal{A}_{h}^{1/3}-n^{2}}$ in terms of the reduced horizon
area and substitute it into the equation:
$(r_{h}^{6}+5n^{2}r_{h}^{4}+15n^{4}r_{h}^{2}-5n^{6})^{2}=100m^{2}r_{h}^{2}$.
After some algebraic computations, one can obtain the following useful
identity:

$m^{2}=\frac{1}{100\mathcal{A}_{h}^{1/3}}\big{(}\mathcal{A}_{h}+2n^{2}\mathcal{A}_{h}^{2/3}+8n^{4}\mathcal{A}_{h}^{1/3}-16n^{6}\big{)}^{2}+\frac{m^{2}n^{2}}{\mathcal{A}_{h}^{1/3}}\,,$
(25)

which can also be converted into a polynomial of $\mathcal{A}_{h}$ after
eliminating the fractional powers. Due to its complexity, we shall omit it
here.

Subsequently, after inserting $m=M/(48\pi^{2})$, $n=N/(48\pi^{2})$ and
$\mathcal{A}=48\pi^{2}\mathcal{A}_{h}$ into eq. (25) and including only one
secondary hair: $J_{n}=Mn$ as before, we can obtain a novel squared-mass
formula:

$M^{2}=\frac{\big{(}6\pi^{2}\big{)}^{1/3}}{50\mathcal{A}^{1/3}}\Big{[}\mathcal{A}+\frac{N^{2}\big{(}6\pi^{2}\mathcal{A}^{2}\big{)}^{1/3}}{576\pi^{4}}+\frac{N^{4}(36\pi\mathcal{A})^{1/3}}{165888\pi^{7}}-\frac{N^{6}}{15925248\pi^{10}}\Big{]}^{2}+\frac{2\big{(}6\pi^{2}\big{)}^{1/3}}{\mathcal{A}^{1/3}}J_{n}^{2}\,.$
(26)

Now we employ a similar procedure as manipulated in the previous section,
i.e., viewing the secondary hair $J_{n}=Mn$ as an independent thermodynamical
variable, then performing the partial derivative of the above squared-mass
formula (26) with respect to one of its thermodynamical quantities
($\mathcal{A},N,J_{n}$) and simultaneously fixing the remaining ones,
respectively, and this will lead to their corresponding conjugate quantities.

First, differentiating the squared-mass formula (26) with respect to
$\mathcal{A}$ yields one-sixth of the surface gravity:

$\kappa=6\frac{\partial{}M}{\partial\mathcal{A}}\Big{|}_{(N,J_{n})}=\frac{1}{2r_{h}}\,,$
(27)

which coincides with the one given in eq. (23). Next, the potential $\psi_{h}$
and the quasi-angular momentum $\omega_{h}$, which are conjugate to $N$ and
$J_{n}$, respectively, are given by

$\psi_{h}=\frac{\partial{}M}{\partial{}N}\Big{|}_{(\mathcal{A},J_{n})}=\frac{2nr_{h}\big{(}r_{h}^{4}+10n^{2}r_{h}^{2}-15n^{4}\big{)}}{5\big{(}r_{h}^{2}+n^{2}\big{)}}\,,\qquad\omega_{h}=\frac{\partial{}M}{\partial{}J_{n}}\Big{|}_{(\mathcal{A},N)}=\frac{n}{r_{h}^{2}+n^{2}}\,.$
(28)

Using all the above thermodynamical conjugate pairs, it is easy to check that
both differential and integral mass formulas are completely obeyed

$\displaystyle dM$ $\displaystyle=$
$\displaystyle(\kappa/6)d\mathcal{A}+\omega_{h}{}dJ_{n}+\psi_{h}{}dN\,,$ (29)
$\displaystyle 5M$ $\displaystyle=$
$\displaystyle\kappa\mathcal{A}+6\omega_{h}{}J_{n}+\psi_{h}{}N\,.$ (30)

Then it is natural to recognize

$S=\frac{A_{h}}{4}=\frac{\pi}{3}\mathcal{A}=16\pi^{3}\big{(}r^{2}+n^{2}\big{)}^{3}\,,\quad
T=\frac{\kappa}{2\pi}=\frac{1}{4\pi{}r_{h}}\,,$ (31)

so that the eight-dimensional Taub-NUT solution behaves like a genuine black
hole without violating the beautiful one-quarter area/entropy law. Here we do
not require in advance that the first law should be obeyed in order to obtain
the consistent thermodynamical relations, rather it is just a very natural by-
product of the purely algebraic deduction.

###  3.2 Extension to the Taub-NUT-AdS8 spacetime

In this subsection, we would like to deal with the Lorentzian Taub-NUT-AdS8
spacetime with a nonzero cosmological constant. The metric is still given by
eq. (22), but now

$\displaystyle f(r)$ $\displaystyle=$
$\displaystyle\frac{1}{5\big{(}r^{2}+n^{2}\big{)}^{3}}\Big{[}r^{6}+5n^{2}r^{4}+15n^{4}r^{2}-5n^{6}-10mr$
$\displaystyle+g^{2}\big{(}5r^{8}+28n^{2}r^{6}+70n^{4}r^{4}+140n^{6}r^{2}-35n^{8}\big{)}\Big{]}\,,$

in which $l=1/g$ is the cosmological scale.

We now begin with the computation of the conserved charges of the Taub-NUT-
AdS8 solution. The NUT charge $N$ is the same one just like the case without a
cosmological constant, and the conformal completion method is adopted to
calculate its conserved mass. The conformal boundary 7-metric of the Taub-NUT-
AdS8 spacetime is given by

$d\bar{s}_{7}^{2}=\lim_{r\to\infty}\frac{ds_{8}^{2}}{r^{2}}=-g^{2}\Big{(}dt+2n\sum_{i=1}^{3}\cos\theta_{i}{}d\phi_{i}\Big{)}^{2}+\sum_{i=1}^{3}\big{(}d\theta_{i}^{2}+\sin^{2}\theta_{i}{}d\phi_{i}^{2}\big{)}\,,$
(32)

with which a normal vector: $\hat{n}^{a}=-gr^{2}\delta_{r}^{a}$ can be
defined.

The conserved charge $\mathcal{Q}[\xi]$ associated with the Killing vector
$\xi^{\nu}$ is defined by

$\mathcal{Q}[\xi]=\frac{1}{40\pi{}g^{3}}\int\,\big{(}r^{5}C^{t}_{~{}a{\nu}b}\hat{n}^{a}\hat{n}^{b}\xi^{\nu}dS_{t}\big{)}\big{|}_{r\to\infty}\,,$
(33)

where $C^{t}_{~{}a{\nu}b}$ is the Weyl conformal tensor and the only nonzero
component of the area vector on the conformal boundary is

$dS_{t}=g\prod_{i=1}^{3}\sin\theta_{i}{}d\theta_{i}{}d\phi_{i}\,.$ (34)

Then the conformal mass is easily evaluated as:

$M=\mathcal{Q}[\partial_{t}]=48\pi^{2}m\,.$ (35)

Next, we want to compute some thermodynamic quantities at the Killing horizon
that is determined by $f(r_{h})=0$. At the horizon, the surface gravity can be
obtained via the standard method as

$\kappa=\frac{1}{2}f^{\prime}(r_{h})=\frac{1+7g^{2}\big{(}r_{h}^{2}+n^{2}\big{)}}{2r_{h}}\,,$
(36)

while the horizon area is $A_{h}=64\pi^{3}\mathcal{A}_{h}$, with the reduced
horizon area still being denoted as $\mathcal{A}_{h}=(r_{h}^{2}+n^{2})^{3}$.

Then we substitute $r_{h}=\sqrt{\mathcal{A}_{h}^{1/3}-n^{2}}$ into the
equation:
$[r_{h}^{6}+5n^{2}r_{h}^{4}+15n^{4}r_{h}^{2}-5n^{6}+g^{2}(5r_{h}^{8}+28n^{2}r_{h}^{6}+70n^{4}r_{h}^{4}+140n^{6}r_{h}^{2}-35n^{8})]^{2}=100m^{2}r_{h}^{2}$
to get an identity:

$m^{2}=\frac{1}{100\mathcal{A}_{h}^{1/3}}\Big{[}\big{(}1+8g^{2}n^{2}\big{)}\big{(}\mathcal{A}_{h}+2n^{2}\mathcal{A}_{h}^{2/3}+8n^{4}\mathcal{A}_{h}^{1/3}-16n^{6}\big{)}+5g^{2}\mathcal{A}_{h}^{4/3}\Big{]}^{2}+\frac{m^{2}n^{2}}{\mathcal{A}_{h}^{1/3}}\,.$
(37)

Supposed that only the secondary hair $J_{n}=Mn$ is needed to be included as
before, then after inserting $m=M/(48\pi^{2})$, $n=N/(48\pi^{2})$,
$\mathcal{A}=48\pi^{2}\mathcal{A}_{h}$ and $g^{2}=8\pi{}P/21$ into eq. (37),
one can arrive at the following squared-mass formula:

$\displaystyle M^{2}$ $\displaystyle=$
$\displaystyle\frac{\big{(}6\pi^{2}\big{)}^{1/3}}{50\mathcal{A}^{1/3}}\bigg{\\{}\Big{(}1+\frac{N^{2}}{756\pi^{3}}P\Big{)}\Big{[}\mathcal{A}+\frac{N^{2}\big{(}6\pi^{2}\mathcal{A}^{2}\big{)}^{1/3}}{576\pi^{4}}+\frac{N^{4}(36\pi\mathcal{A})^{1/3}}{165888\pi^{7}}$
(38)
$\displaystyle-\frac{N^{6}}{15925248\pi^{10}}\Big{]}^{2}+\frac{10}{63}\big{(}36\pi\mathcal{A}^{4}\big{)}^{1/3}P\bigg{\\}}^{2}+\frac{2\big{(}6\pi^{2}\big{)}^{1/3}}{\mathcal{A}^{1/3}}J_{n}^{2}\,,$

in which $P$ is the generalized pressure. We point out that the squared-mass
formula (38) consistently reduces to eq. (26) when the cosmological constant
vanishes.

Similar to the strategy as did in the last subsection, one can view the mass
as an implicit function: $M=M(\mathcal{A},N,J_{n},P)$, and then
differentiating the squared-mass formula (38) with respect to its variables
leads to a new reasonable differential mass formula:

$dM=(\kappa/6)d\mathcal{A}+\omega_{h}{}dJ_{n}+\psi_{h}{}dN+VdP\,,$ (39)

where

$\displaystyle\kappa$ $\displaystyle=$ $\displaystyle
6\frac{\partial{}M}{\partial{}\mathcal{A}}\Big{|}_{(N,J_{n},P)}=\frac{1+7g^{2}\big{(}r_{h}^{2}+n^{2}\big{)}}{2r_{h}}\,,\qquad\omega_{h}=\frac{\partial{}M}{\partial{}J_{n}}\Big{|}_{(\mathcal{A},N,P)}=\frac{n}{r_{h}^{2}+n^{2}}\,,$
$\displaystyle\psi_{h}$ $\displaystyle=$
$\displaystyle\frac{\partial{}M}{\partial{}N}\Big{|}_{(\mathcal{A},J_{n},P)}=\frac{2nr_{h}\big{[}r_{h}^{4}+10n^{2}r_{h}^{2}-15n^{4}+4g^{2}\big{(}r_{h}^{6}+7n^{2}r_{h}^{4}+35n^{4}r_{h}^{2}-35n^{6}\big{)}\big{]}}{5\big{(}r_{h}^{2}+n^{2}\big{)}}\,,$
$\displaystyle V$ $\displaystyle=$
$\displaystyle\frac{\partial{}M}{\partial{}P}\Big{|}_{(\mathcal{A},N,J_{n})}=\frac{64\pi^{3}r_{h}(5r_{h}^{8}+28n^{2}r_{h}^{6}+70n^{4}r_{h}^{4}+140n^{6}r_{h}^{2}-35n^{8})}{35\big{(}r_{h}^{2}+n^{2}\big{)}}\,.$

At the same time, one can check that the integral mass formulas

$5M=\kappa\mathcal{A}+6\omega_{h}{}J_{n}+\psi_{h}{}N-2VP\,,$ (40)

is also automatically satisfied.

The consistency of the above thermodynamic relations suggests that one should
restore the well-known Bekenstein-Hawking area/entropy relation
$S=A_{h}/4=16\pi^{3}\mathcal{A}_{h}$ and Hawking temperature
$T=\kappa/(2\pi)$, which means that the eight-dimensional Taub-NUT-AdS
spacetime should be regarded as a generic black hole.

It is worth to note that the thermodynamic quantities of the base space of
$S^{2}\times\mathbb{CP}^{2}$ are the same ones as those in the case of
$S^{2}\times{}S^{2}\times{}S^{2}$ base space, because the the expression of
the radial function $f(r)$ remains unchanged, and we will not repeat them
here.

##  4 10-dimensional Taub-NUT spacetime

Finally, we will turn to consider the 10-dimensional Taub-NUT spacetime and
its AdS counterpart. As shown in ref. CQG19-2051  for the 10-dimensional Taub-
NUT spacetime, there are three different choices for the base manifold, namely
$S^{2}\times{}S^{2}\times{}S^{2}\times{}S^{2}$,
$S^{2}\times{}S^{2}\times\mathbb{CP}^{2}$, and
$\mathbb{CP}^{2}\times\mathbb{CP}^{2}$. We will only consider the case in
which the metric possesses a $U(1)$ fibration over
$S^{2}\times{}S^{2}\times{}S^{2}\times{}S^{2}$:

$ds_{10}^{2}=-f(r)\Big{(}dt+2n\sum_{i=1}^{4}\cos\theta_{i}{}d\phi_{i}\Big{)}^{2}+\frac{dr^{2}}{f(r)}+(r^{2}+n^{2})\sum_{i=1}^{4}\big{(}d\theta_{i}^{2}+\sin^{2}\theta_{i}{}d\phi_{i}^{2}\big{)}\,,$
(41)

where

$f(r)=\frac{5r^{8}+28n^{2}r^{6}+70n^{4}r^{4}+140n^{6}r^{2}-35n^{8}-70mr}{35\big{(}r^{2}+n^{2}\big{)}^{4}}\,.$

At the horizon which is defined by the largest root of $f(r_{h})=0$, the
horizon area and the surface gravity can be obtained as

$A_{h}=256\pi^{4}\big{(}r_{h}^{2}+n^{2}\big{)}^{4}=256\pi^{4}\mathcal{A}_{h}\,,\qquad\kappa=\frac{1}{2}f^{\prime}(r_{h})=\frac{1}{2r_{h}}\,,$
(42)

where the reduced area is denoted as: $\mathcal{A}_{h}=(r_{h}^{2}+n^{2})^{4}$.

The expressions of the AD mass and the NUT charge can be similarly calculated
as

$M=256\pi^{3}m\,,\qquad N=256\pi^{3}n\,.$ (43)

###  4.1 Consistent mass formulas of the 10-dimensional Taub-NUT spacetime

Adopting the same strategy as did before, we insert
$r_{h}=\sqrt{\mathcal{A}_{h}^{1/4}-n^{2}}$ into the equation:
$(5r_{h}^{8}+28n^{2}r_{h}^{6}+70n^{4}r_{h}^{4}+140n^{6}r_{h}^{2}-35n^{8})^{2}=4900m^{2}r_{h}^{2}$,
and after some computations, we can get an useful identity:

$\displaystyle m^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{4900\mathcal{A}_{h}^{1/4}}\big{(}5\mathcal{A}_{h}+8n^{2}\mathcal{A}_{h}^{3/4}+16n^{4}\mathcal{A}_{h}^{1/2}+64n^{6}\mathcal{A}_{h}^{1/4}-128n^{8}\big{)}^{2}+\frac{m^{2}n^{2}}{\mathcal{A}_{h}^{1/4}}\,.$
(44)

After substituting $m=M/(256\pi^{3})$, $n=N/(256\pi^{3})$,
$\mathcal{A}=256\pi^{3}\mathcal{A}_{h}$ and the secondary hair $J_{n}=Mn$ into
eq. (44), one can obtain the following squared-mass formula:

$\displaystyle M^{2}$ $\displaystyle=$
$\displaystyle\frac{\pi^{3/4}}{49\mathcal{A}^{1/4}}\bigg{[}\mathcal{A}+\frac{N^{2}(\pi\mathcal{A})^{3/4}}{10240\pi^{6}}+\frac{N^{4}\sqrt{\pi\mathcal{A}}}{83886080\pi^{11}}+\frac{N^{6}(\pi\mathcal{A})^{1/4}}{343597383680\pi^{16}}$
(45)
$\displaystyle-\frac{N^{8}}{2814749767106560\pi^{21}}\bigg{]}^{2}+\frac{4\pi^{3/4}}{\mathcal{A}^{1/4}}J_{n}^{2}\,.$

In the following, the differential and integral mass formulae for the ten-
dimensional Taub-NUT spacetime will be derived under the assumption that the
entire set of thermodynamic quantities is: the mass $M$, the NUT charge $N$,
and the secondary hair $J_{n}=Mn$, which will also be viewed as an independent
variable. Differentiating the squared-mass formula (45) with respect to
$\mathcal{A}$ yields one-eighth of the surface gravity:

$\kappa=8\frac{\partial{}M}{\partial\mathcal{A}}\Big{|}_{(N,J_{n})}=\frac{1}{2r_{h}}\,,$
(46)

which is accordance with the one given in eq. (42). The gravito-magnetic
potential $\psi_{h}$ and the quasi-angular momentum $\omega_{h}$, which are
conjugate to $N$ and $J_{n}$, respectively, can be computed as

$\displaystyle\psi_{h}$ $\displaystyle=$
$\displaystyle\frac{\partial{}M}{\partial{}N}\Big{|}_{(\mathcal{A},J_{n})}=\frac{8nr_{h}\big{(}r_{h}^{6}+7n^{2}r_{h}^{4}+35n^{4}r_{h}^{2}-35n^{6}\big{)}}{35\big{(}r_{h}^{2}+n^{2}\big{)}}\,,$
(47) $\displaystyle\omega_{h}$ $\displaystyle=$
$\displaystyle\frac{\partial{}M}{\partial{}J_{n}}\Big{|}_{(\mathcal{A},N)}=\frac{n}{r_{h}^{2}+n^{2}}\,.$
(48)

One can readily verify that both the differential and integral mass formulae

$\displaystyle dM$ $\displaystyle=$
$\displaystyle(\kappa/8)d\mathcal{A}+\omega_{h}{}dJ_{n}+\psi_{h}{}dN\,,$ (49)
$\displaystyle 7M$ $\displaystyle=$
$\displaystyle\kappa\mathcal{A}+8\omega_{h}{}J_{n}+\psi_{h}{}N\,,$ (50)

are fully obeyed by using all the thermodynamical conjugate pairs given above.
It is natural to identify

$S=\frac{A_{h}}{4}=\frac{\pi}{4}\mathcal{A}=64\pi^{4}\big{(}r_{h}^{2}+n^{2}\big{)}^{4}\,,\quad
T=\frac{\kappa}{2\pi}=\frac{1}{4\pi{}r_{h}}\,,$ (51)

so that the ten-dimensional Taub-NUT solution acts like a true black hole
without violating the beautiful one-quarter area/entropy relation. Here, we do
not require ahead that the first law be obeyed to achieve consistent
thermodynamical connections, rather, it is a very natural by-product of purely
algebraic deduction.

###  4.2 Extension to the Taub-NUT-AdS10 spacetime

Finally we would like to tackle with the Lorentzian Taub-NUT-AdS10 spacetime
with a nonzero cosmological constant. The metric is still given by eq. (41),
and now we have

$\displaystyle f(r)$ $\displaystyle=$
$\displaystyle\frac{1}{35\big{(}r^{2}+n^{2}\big{)}^{4}}\Big{[}5r^{8}+28n^{2}r^{6}+70n^{4}r^{4}+140n^{6}r^{2}-35n^{8}-70mr$
(52)
$\displaystyle+5g^{2}\big{(}7r^{10}+45n^{2}r^{8}+126n^{4}r^{6}+210n^{6}r^{4}+315n^{8}r^{2}-63n^{10}\big{)}\Big{]}\,,$

where $l=1/g$ is the cosmological scale.

Let us start by calculating the conserved charges (primary hairs) of the Taub-
NUT-AdS10 solution, following the same steps as done in secs. 2.2 and 3.2. The
NUT charge $N$ is simply given as the same one as that in the absence of a
cosmological constant. The conserved mass will be also calculated via the
conformal completion approach, with which the conformal boundary 9-metric of
the ten-dimensional Taub-NUT-AdS spacetime being given by

$\displaystyle d\bar{s}_{9}^{2}$ $\displaystyle=$
$\displaystyle\lim_{r\to\infty}\frac{ds_{10}^{2}}{r^{2}}=-g^{2}\Big{(}dt+2n\sum_{i=1}^{4}\cos\theta_{i}{}d\phi_{i}\Big{)}^{2}+\sum_{i=1}^{4}\big{(}d\theta_{i}^{2}+\sin^{2}\theta_{i}{}d\phi_{i}^{2}\big{)}\,,$
(53)

together with a normal vector: $\hat{n}^{a}=-gr^{2}\delta_{r}^{a}$.

The conserved charge $\mathcal{Q}[\xi]$ associated with the Killing vector
$\xi^{\nu}$ is defined by

$\mathcal{Q}[\xi]=\frac{1}{56\pi{}g^{3}}\int\big{(}r^{7}C^{t}_{~{}a{\nu}b}\hat{n}^{a}\hat{n}^{b}\xi^{\nu}dS_{t}\big{)}\big{|}_{r\to\infty}\,,$
(54)

where $C^{t}_{~{}a{\nu}b}$ is the Weyl conformal tensor and the only nonzero
component of the area vector associated with the conformal boundary is

$\displaystyle dS_{t}$ $\displaystyle=$ $\displaystyle
g\prod_{i=1}^{4}\sin\theta_{i}{}d\theta_{i}{}d\phi_{i}\,.$ (55)

Then the conformal mass can be simply computed as:

$M=\mathcal{Q}[\partial_{t}]=256\pi^{3}m\,.$ (56)

Below, we will evaluate some thermodynamic quantities related to the Killing
horizon which is specified by $f(r_{h})=0$. The surface gravity at the horizon
is easily obtained via the standard method as

$\kappa=\frac{1}{2}f^{\prime}(r_{h})=\frac{1+9g^{2}\big{(}r_{h}^{2}+n^{2}\big{)}}{2r_{h}}\,,$
(57)

and the event horizon area still reads $A_{h}=256\pi^{4}\mathcal{A}_{h}$, in
which the reduced horizon area is $\mathcal{A}_{h}=(r_{h}^{2}+n^{2})^{4}$.

Now it is a position to derive a novel squared-mass formula. Inserting
$r_{h}=\sqrt{\mathcal{A}_{h}^{1/4}-n^{2}}$ into the equation:
$[5r_{h}^{8}+28n^{2}r_{h}^{6}+70n^{4}r_{h}^{4}+140n^{6}r_{h}^{2}-35n^{8}+5g^{2}(7r_{h}^{10}+45n^{2}r_{h}^{8}+126n^{4}r_{h}^{6}+210n^{6}r_{h}^{4}+315n^{8}r_{h}^{2}-63n^{10})]^{2}=4900m^{2}r_{h}^{2}$,
and after a little algebra, we can obtain a useful identity:

$\displaystyle m^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{4900\mathcal{A}_{h}^{1/4}}\Big{[}\big{(}1+10g^{2}n^{2}\big{)}\big{(}5\mathcal{A}_{h}+8n^{2}\mathcal{A}_{h}^{3/4}+16n^{4}\mathcal{A}_{h}^{1/2}+64n^{6}\mathcal{A}_{h}^{1/4}-128n^{8}\big{)}$
(58)
$\displaystyle+35g^{2}\mathcal{A}_{h}^{5/4}\Big{]}^{2}+\frac{m^{2}n^{2}}{\mathcal{A}_{h}^{1/4}}\,.$

Then after plugging $m=M/(256\pi^{3})$, $n=N/(256\pi^{3})$,
$\mathcal{A}=256\pi^{3}\mathcal{A}_{h}$, and $g^{2}=2\pi{}P/9$ into eq. (58),
where $P$ is the generalized pressure, and the secondary hair: $J_{n}=Mn$, one
can get the following identity:

$\displaystyle M^{2}$ $\displaystyle=$
$\displaystyle\frac{\pi^{3/4}}{49\mathcal{A}^{1/4}}\bigg{\\{}\Big{(}1+\frac{5N^{2}}{147456\pi^{5}}P\Big{)}\Big{[}\mathcal{A}+\frac{N^{2}(\pi\mathcal{A})^{3/4}}{10240\pi^{6}}+\frac{N^{4}\sqrt{\pi\mathcal{A}}}{83886080\pi^{11}}+\frac{N^{6}(\pi\mathcal{A})^{1/4}}{343597383680\pi^{16}}$
(59)
$\displaystyle-\frac{N^{8}}{2814749767106560\pi^{21}}\Big{]}+\frac{7}{18\pi}(\pi\mathcal{A})^{5/4}P\bigg{\\}}^{2}+\frac{4\pi^{3/4}}{\mathcal{A}^{1/4}}J_{n}^{2}\,,$

which is the Christodoulou-Ruffini-like squared-mass formula for the ten-
dimensional Taub-NUT-AdS spacetime. We again point out that this squared-mass
formula consistently reduces to the one obtained in eq. (45) when the
generalized pressure $P$ is turned off.

Now, as did before, one can regard the mass $M$ as an elementary function:
$M=M(\mathcal{A},N,J_{n},P)$, and then after differentiating the squared-mass
formula (59) with respect to its variables, one can obtain a reasonable
differential mass formula:

$dM=(\kappa/8)d\mathcal{A}+\omega_{h}{}dJ_{n}+\psi_{h}{}dN+VdP\,,$ (60)

where

$\displaystyle\kappa$ $\displaystyle=$ $\displaystyle
8\frac{\partial{}M}{\partial\mathcal{A}}\Big{|}_{(N,J_{n},P)}=\frac{1+9g^{2}\big{(}r_{h}^{2}+n^{2}\big{)}}{2r_{h}}\,,\qquad\omega_{h}=\frac{\partial{}M}{\partial{}J_{n}}\Big{|}_{(\mathcal{A},N,P)}=\frac{n}{r_{h}^{2}+n^{2}}\,,$
$\displaystyle\psi_{h}$ $\displaystyle=$
$\displaystyle\frac{\partial{}M}{\partial{}N}\Big{|}_{(\mathcal{A},J_{n},P)}=\frac{2nr_{h}}{35\big{(}r_{h}^{2}+n^{2}\big{)}}\Big{[}4\big{(}r_{h}^{6}+7n^{2}r_{h}^{4}+35n^{4}r_{h}^{2}-35n^{6}\big{)}$
$\displaystyle\qquad\qquad+5g^{2}\big{(}5r_{h}^{8}+36n^{2}r_{h}^{6}+126n^{4}r_{h}^{4}+420n^{6}r_{h}^{2}-315n^{8}\big{)}\Big{]}\,,$
$\displaystyle V$ $\displaystyle=$
$\displaystyle\frac{\partial{}M}{\partial{}P}\Big{|}_{(\mathcal{A},N,J_{n})}=\frac{256\pi^{4}r_{h}\big{(}7r_{h}^{10}+45n^{2}r_{h}^{8}+126n^{4}r_{h}^{6}+210n^{6}r_{h}^{4}+315n^{8}r_{h}^{2}-63n^{10})}{63\big{(}r_{h}^{2}+n^{2}\big{)}}\,.$

In the meanwhile, one can easily verify that the Bekenstein-Smarr mass formula

$7M=\kappa\mathcal{A}+8\omega_{h}{}J_{n}+\psi_{h}{}N-2VP\,,$ (61)

is completely satisfied also.

Comparing our new mass formulae as displayed in eqs. (60)-(61) with the
familiar standard ones, it is strongly suggested that one should make the
familiar identifications $S=A_{h}/4=64\pi^{4}\mathcal{A}_{h}$ and
$T=\kappa/(2\pi)$, which restores the famous Bekenstein-Hawking one-quarter
area-entropy relation of the ten-dimensional Taub-NUT-AdS spacetime in a very
pleasing way, so that the solution behaves like a genuine black hole.

Here, we also point out that thermodynamic quantities in the cases of
$S^{2}\times{}S^{2}\times\mathbb{CP}^{2}$ and
$\mathbb{CP}^{2}\times\mathbb{CP}^{2}$ base space should be the same ones as
those in the case of $S^{2}\times{}S^{2}\times{}S^{2}\times{}S^{2}$ base
manifold since the expression of the radial function $f(r)$ remains unchanged,
so we will not present them.

##  5 Summary: general ($2k+2$)-dimensional cases

To summarize, we have established the consistent thermodynamic first law and
Bekenstein-Smarr mass formula for the generic $D=(2k+2)$ Lorentzian Taub-NUT
(AdS) spacetimes whose metrics are compactly written as

$ds_{D}^{2}=-f(r)\Big{(}dt+2n\sum_{i=1}^{k}\cos\theta_{i}{}d\phi_{i}\Big{)}^{2}+\frac{dr^{2}}{f(r)}+(r^{2}+n^{2})\sum_{i=1}^{k}\big{(}d\theta_{i}^{2}+\sin^{2}\theta_{i}{}d\phi_{i}^{2}\big{)}\,,$
(62)

with the radial function being

$f(r)=\frac{r}{\big{(}r^{2}+n^{2}\big{)}^{k}}\bigg{\\{}\int^{r}\big{[}1+(2k+1)g^{2}\big{(}x^{2}+n^{2}\big{)}\big{]}\frac{\big{(}x^{2}+n^{2}\big{)}^{k}}{x^{2}}dx-2m\bigg{\\}}\,.$

These higher even-dimensional Taub-NUT-AdS spacetimes are shown to be subject
to the traditional forms of the first law and the Bekenstein-Smarr mass
formula as follows

$\displaystyle dM$ $\displaystyle=$ $\displaystyle
TdS+\omega_{h}{}dJ_{n}+\psi_{h}{}dN+VdP\,,$ (63) $\displaystyle(D-3)M$
$\displaystyle=$ $\displaystyle(D-2)(TS+\omega_{h}{}J_{n})+\psi_{h}{}N-2VP\,,$
(64)

provided that a new secondary hair: $J_{n}=Mn$ is included just like in the
case of their four-dimensional cousins PRD100-101501 ; PRD105-124013 .

The thermodynamical quantities that enter the above differential and integral
mass formulae are given below

$\displaystyle M$ $\displaystyle=$ $\displaystyle k(4\pi)^{k-1}m\,,\qquad
N=k(4\pi)^{k-1}n\,,\qquad J_{n}=k(4\pi)^{k-1}mn\,,$ $\displaystyle S$
$\displaystyle=$
$\displaystyle\frac{1}{4}\big{[}4\pi\big{(}r_{h}^{2}+n^{2}\big{)}\big{]}^{k}\,,\qquad
T=\frac{f^{\prime}(r_{h})}{4\pi}=\frac{1+(2k+1)g^{2}\big{(}r_{h}^{2}+n^{2}\big{)}}{4\pi{}r_{h}}\,,$
$\displaystyle\omega_{h}$ $\displaystyle=$
$\displaystyle\frac{n}{r_{h}^{2}+n^{2}}\,,\qquad
P=\frac{k(2k+1)}{8\pi}g^{2}\,,\qquad
V=\frac{(4\pi)^{k}r_{h}^{2}}{r_{h}^{2}+n^{2}}\int^{r_{h}}\frac{\big{(}x^{2}+n^{2}\big{)}^{k+1}}{x^{2}}dx\,,$
$\displaystyle\psi_{h}$ $\displaystyle=$
$\displaystyle-\frac{1+(2k+1)g^{2}\big{(}r_{h}^{2}+n^{2}\big{)}}{2nr_{h}}\big{(}r_{h}^{2}+n^{2}\big{)}^{k}+\frac{(2k-1)r_{h}^{2}-n^{2}}{2n\big{(}r_{h}^{2}+n^{2}\big{)}}\int^{r_{h}}\frac{\big{(}x^{2}+n^{2}\big{)}^{k}}{x^{2}}dx$
$\displaystyle+(2k+1)g^{2}\frac{(2k+1)r_{h}^{2}-n^{2}}{2n\big{(}r_{h}^{2}+n^{2}\big{)}}\int^{r_{h}}\frac{\big{(}x^{2}+n^{2}\big{)}^{k+1}}{x^{2}}dx\,.$

By the way, the following identity must be used to verify that both mass
formulae are indeed fulfilled:

$m=\int^{r_{h}}\big{[}1+(2k+1)g^{2}\big{(}x^{2}+n^{2}\big{)}\big{]}\frac{\big{(}x^{2}+n^{2}\big{)}^{k}}{2x^{2}}dx\,.$
(65)

##  6 Conclusions and outlooks

In this paper, we have successfully achieved the consistent first law and
Bekenstein-Smarr mass formula for the six-, eight-, and ten-dimensional
Lorentzian Taub-NUT (AdS) spacetimes. Similar to the cases of the four-
dimensional Lorentzian Taub-NUT (AdS) solutions, as did in our previous works
PRD100-101501 ; PRD105-124013 , we also import only one secondary hair:
$J_{n}=Mn$ here. A key rudiment of this work is to deduce a reasonable
Christodoulou-Ruffini-like squared-mass formula for each dimension, from which
the thermodynamical first law and Bekenstein-Smarr mass formula can be derived
via simple differentiations with respect to its thermodynamic variables, and
the resultant thermodynamical conjugate pairs meet their standard forms of the
differential and integral mass formulae. All the results obtained in this
paper resembles to the cases of the four-dimensional Lorentzian Taub-NUT-AdS
spacetime, however there is an exception in that the notion of a dual
(magnetic) mass in higher dimensions is currently unclear to be defined. Once
an appropriate definition for it is proposed, our present work might be
modified accordingly via the further inclusion of it.

Our study in this paper demonstrated that our idea “The NUT charge is a
thermodynamical multi-hair” has a universal applicability, and our method is
effective and systematical. A natural question is: whether it is applicable to
deal with the charged versions of the higher even-dimensional Taub-NUT
spacetimes CQG23-2849 ; PRD73-124039 . A preliminary research shows that only
including one secondary hair $J_{n}=Mn$ is not sufficient to resolve the
consistency of the first law and integral mass formula, so at least one more
charge should be added into them. Another related issue is: whether the
present work can be extended to treat thermodynamics of the higher even-
dimensional multi-NUTty spacetimes PLB593-218 ; CQG21-2937 ; PLB634-448 ,
since the solutions studied in this paper can be viewed as a special equal-NUT
case of these more general spacetimes with multi NUT parameters. We hope to
report the related work soon.

###### Acknowledgements.

This work is supported by the National Natural Science Foundation of China
(NSFC) under Grant No. 12205243, No. 11675130, and by the Doctoral Research
Initiation Project of China West Normal University under Grant No. 21E028.

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