aainstitutetext: Department of Physics, King’s College London, The Strand,
London WC2R 2LS, UKbbinstitutetext: Centre for Particle Theory, Durham
University, South Road, Durham, DH1 3LE, UKccinstitutetext: Perimeter
Institute, 31 Caroline Street North, Waterloo, ON, N2L 2Y5, Canada

# Thermodynamics of Many Black Holes

Ruth Gregory b Zheng Liang Lim b Andrew Scoins<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We discuss the thermodynamics of an array of collinear black holes which may
be accelerating. We prove a general First Law, including variations in the
tensions of strings linking and accelerating the black holes. We analyse the
implications of the First Law in a number of instructive cases, including that
of the C-metric, and relate our findings to the previously obtained
thermodynamics of slowly accelerating black holes in anti-de Sitter spacetime.
The concept of thermodynamic length is found to be robust and a
Christoudoulou-Ruffini formula for the C-metric is shown.

##  1 Overview

Black hole thermodynamics is a rich subject, straddling both the classical and
quantum aspects of gravity. The thermodynamic charges of a black hole such as
entropy and temperature, while intrinsically quantum in nature, are related to
classical attributes such as horizon area and surface gravity
Bekenstein:1973ur ; Bekenstein:1974ax ; Hawking:1974sw ; Gibbons:1976 .
Indeed, it was considering the classical response of a black hole to infalling
matter that led Bardeen, Carter, and Hawking to make the link between black
hole variations and the First Law of thermodynamics in their seminal paper
Bardeen:1973gs .

More recently, our understanding of black hole thermodynamics and the
interpretation of the various parameters has also been improving. The first
law of thermodynamics in gravitational systems has been more comprehensively
understood as an extended thermodynamical law by including pressure in the
guise of variations in vacuum energy Teitelboim:1985dp ; Kastor:2009wy ;
Dolan:2010ha ; Dolan:2011xt ; Kubiznak:2012wp , and a more complete
understanding of the nature of “$M$” for the black hole has emerged as the
enthalpy of the system Kastor:2009wy ; see Kubiznak:2016qmn  for a review.

These attempts at understanding the First Law have largely considered single,
isolated, black holes, as in the Kerr-Newman family of solutions. However,
there are more complex, and therefore more interesting, multi-black hole
systems for which exact solutions are known. Such geometries are thus amenable
to thermodynamic analysis. For example, the Israel-Khan solution IsraelKhan
is an asymptotically flat geometry consisting of two black holes kept apart by
a “strut”—a conical defect with an angular excess—corresponding to a negative
tension cosmic string. More generally, one can sacrifice global asymptotic
flatness to remove the unphysical negative-tension defect by running a
positive tension cosmic string through the spacetime AFV ; AGK ; GKW . In
doing so, one retains local asymptotic flatness away from the core.
Generalising further, the accelerating black hole, encoded in the C-metric
Kinnersley:1970zw ; Plebanski:1976gy , consists of a black hole with a
protruding cosmic string Gregory:1995hd  (or an imbalance between antipodal
strings) that provides an accelerating force. In this case, not only is
asymptotic flatness lost near the string extending to spatial infinity, but a
non-compact acceleration horizon forms. Such systems beg the question: how
does one define thermodynamics for a geometry which is neither asymptotically
flat, an isolated black hole, nor (in the case of the Israel-Khan solution)
stable?

Early thermodynamic investigations of black holes with conical defects focused
on a fixed deficit threading the horizon AFV ; Martinez:1990sd ; Costa:2000kf
; Dutta:2005iy ; Herdeiro:2009vd , or a deficit “variation” during the capture
of a cosmic string Bonjour:1998rf . The thermodynamic consequences of a truly
varying deficit, however, were not worked out until recently. In particular,
an accelerating, asymptotically locally anti-de Sitter black hole has provided
a context within which one maintains excellent computational control. This is
owing both to one’s ability to accelerate a black hole without forming an
acceleration horizon, and the availability of the holographic dictionary
Skenderis:2005 . A fully general First Law was hence derived Appels:2016uha ;
Appels:2017xoe , accounting for a variation in a string’s tension $\mu$:

$\delta M=T\delta S-\lambda\delta\mu+\cdots\,.$ (1)

This tension comes paired with a conjugate thermodynamic potential $\lambda$,
christened the thermodynamic length of the string Appels:2017xoe . These
results were later generalised to accelerating black holes carrying rotational
and $U(1)$ gauge charge Anabalon:2018ydc ; Anabalon:2018qfv . Interestingly,
the expression for thermodynamic tension parallels that of the gravitational
tension of Kaluza-Klein black strings Traschen:2001pb ; Harmark:2004ch ;
Kastor:2012dt , a set-up with no conical deficits.

Some understanding of the origin of thermodynamic length has also arisen.
Considering a system of two black holes coupled by a strut, Krtouš and
Zelnikov Krtous:2019 have found a thermodynamic length corresponding to the
strut worldvolume evaluated at some fixed time. This has since been verified
for similarly coupled Kerr-Newman black holes Ramirez-Valdez:2020 ; Garcia-
Compean:2020 .

One should expect that if gravitational solutions are truly representatives of
a first law of thermodynamics in the classical limit, then one will find
common features no matter the number of black holes involved. We demonstrate
this here, by calculating variations of an array of collinear black holes –
connected by strings – which may be accelerated by external strings so as to
form an acceleration horizon. We allow all parameters in the solution to vary
and thereby prove a general First Law,

$\delta M=\sum_{I}T_{I}\delta S_{I}-\sum_{J}\lambda_{J}\delta\mu_{J}\,,$ (2)

wherein the temperatures $T_{I}$ and entropies $S_{I}$ of the compact black
hole horizons contribute together with the thermodynamic lengths $\lambda_{J}$
and tensions $\mu_{J}$ of the strings. We justify the quantities appearing in
(2), and consider its implications in a number of instructive cases, including
a triple black hole system and the C-metric geometry. A key feature of our
result is that the system behaves as a composite; the individual black holes
are not thermodynamically isolated, but each interacts with the other, a
variation of one having implications for all the rest.

Note also that the First Law (2) further supports the notion of $M$ as
enthalpy Kastor:2009wy , even though there is no cosmological constant present
here. The energy momentum of the conical deficit, or cosmic string, takes the
form of a worldsheet cosmological constant: the string has a tension equal in
magnitude and opposite in sign to its energy density. Thus, the
“$-\delta\mu_{J}$” term in (2) is in fact a “$+\delta p_{J}$” term, or
pressure term, for the cosmic string. That the First Law contains a
$\lambda\delta p$, rather than $p\delta\lambda$ is indicative that $M$ truly
represents an enthalpy, and not an internal energy as previously imagined.

The outline of the paper is as follows: In section 2, we review the
construction of black hole arrays and acceleration horizons in Weyl gauge Weyl
. In section 3 we formulate a First Law for such systems, justifying the
charges and potentials involved. Section 4 discusses implications of the
result via some instructive examples and contains a novel Christodoulou-
Ruffini mass formula Christodoulou:1972kt for the C-Metric.

##  2 Four dimensional Weyl metrics: black hole arrays

In this section we briefly review the multi-black hole solutions we will be
analysing. We will largely follow the presentation of Fay , with minor
notational changes. The main new result in this section is a discussion of the
determination of the acceleration scale for an array of accelerating black
holes in (33). The black holes are aligned along an axis, and are static in
the sense of possessing a time-like Killing isometry in the region between the
black hole and acceleration horizons. Though an Israel-Khan-like solution for
two rotating black holes is known MankoRuiz:2019 , solutions for three or more
Kerr black holes remain elusive. We want to consider an arbitrary number of
horizons and thus will sidestep any discussion of rotation. One expects that
rotational charges may be included in the obvious way, once an appropriate
geometry is written down.

With temporal and axial symmetry, the metric can be written in a block
diagonal (Weyl) form, with metric functions $\gamma,\nu,\text{ and }\alpha$
depending only on transverse coordinates $r$ and $z$:

$ds^{2}=e^{2\gamma}dt^{2}-e^{2(\nu-\gamma)}(dr^{2}+dz^{2})-\alpha^{2}e^{-2\gamma}d\phi^{2}\,.$
(3)

The Einstein equations are:

$\displaystyle\Delta\alpha$ $\displaystyle=$ $\displaystyle-8\pi G\alpha
e^{2(\nu-\gamma)}\left[T^{r}_{r}+T^{z}_{z}\right]$ (4)
$\displaystyle\Delta\gamma+\frac{\nabla\gamma\cdot\nabla\alpha}{\alpha}$
$\displaystyle=$ $\displaystyle 4\pi
Ge^{2(\nu-\gamma)}\left[T^{t}_{t}-T^{r}_{r}-T^{z}_{z}-T^{\phi}_{\phi}\right]$
(5) $\displaystyle\Delta\nu+(\nabla\gamma)^{2}$ $\displaystyle=$
$\displaystyle-8\pi Ge^{2(\nu-\gamma)}T^{\phi}_{\phi}$ (6)
$\displaystyle\frac{\partial_{\pm}^{2}\alpha}{\alpha}+2(\partial_{\pm}\gamma)^{2}-2\partial_{\pm}\nu\frac{\partial_{\pm}\alpha}{\alpha}$
$\displaystyle=$ $\displaystyle 8\pi G[T_{rr}-T_{zz}\pm 2iT_{rz}]$ (7)

where $T_{a}^{b}$ is the energy momentum tensor of any bulk matter, $\Delta$
is the two dimensional Laplacian
$(\partial_{r}^{2}+\partial_{z}^{2}=\partial_{+}\partial_{-})$, with
$\partial_{\pm}=\partial_{r}\mp i\partial_{z}$ the derivatives with respect to
the complex coordinates $(r\pm iz)/2$.

In the absence of matter or a cosmological constant, these have a very elegant
solution: one simply fixes the conformal gauge freedom remaining in the metric
(3) by setting $\alpha\equiv r/K$, which is consistent with (4). Note, we
introduce the parameter $K$ here to maintain a $2\pi$ periodicity of the
$\phi-$coordinate; this will become relevant when we discuss conical sources.
With $\alpha\propto r$, (5) becomes a cylindrical Laplace equation for
$\gamma$ in vacuo, with solution

$\gamma=-2G\int\frac{S({\bf r}^{\prime})d^{3}{\bf r}^{\prime}}{|{\bf r}-{\bf
r}^{\prime}|}$ (8)

for a source with energy density $S({\bf r})$. Note then that the metric
component $\gamma$ is nothing but the Newtonian source of axial symmetry. In
turn, $\nu$ is determined from $\gamma$ via (7). Since the equation for
$\gamma$ is linear, its solutions can be superposed; the nonlinearity of
Einstein gravity shows up in the solution of $\nu$. Note that, since
regularity of the $r$-axis requires $\nu(0,z)=-\log K$, in general there will
be conical singularities when regular solutions for $\gamma$ are superposed.
These can be interpreted as strings or struts supporting the static sources in
equilibrium.

###  2.1 The Schwarzschild solution

As described in Fay , a black hole may be represented by a finite-length line
source111We make the gauge choice to centre the rod at $z=0$., $8\pi GS({\bf
r})=\delta(r)/r$ for $z\in[-m,m]$, yielding

$\gamma_{S}=-{1\over
2}\int_{-m}^{m}{dz^{\prime}\over[r^{2}+(z-z^{\prime})^{2}]^{1/2}}={\frac{1}{2}}\log\frac{R_{-}-z_{-}}{R_{+}-z_{+}}\,,$
(9)

where

$Z_{\pm}=z\mp m\qquad,\qquad R_{\pm}^{2}=r^{2}+Z_{\pm}^{2}\,.$ (10)

Integration of (7) then gives

$\nu_{S}=\frac{1}{2}\log\frac{E_{+-}}{2R_{+}R_{-}}\,,$ (11)

where

$E_{+-}=R_{+}R_{-}+Z_{+}Z_{-}+r^{2}\,.$ (12)

Although this does not look like the familiar Schwarzschild black hole, the
simple transformation

$z=(\rho-m)\cos\theta\quad,\quad r^{2}=\rho(\rho-2m)\sin^{2}\theta$ (13)

in fact returns the metric to its standard spherical form, with $2m=2GM_{S}$
representing the Schwarzschild radius.

Figure 1:  Rindler worldlines of observers with differing accelerations but
same horizon.

###  2.2 Rindler space

Interestingly, one can formally introduce an acceleration horizon by adding a
semi-infinite line source (SILM) EmparanReall:2002 , where $8\pi GS({\bf
r})=\delta(r)/r$ for $z>z_{0}$:

$\gamma_{R}=-{1\over
2}\int_{z_{0}}^{\infty}{dz^{\prime}\over[r^{2}+(z-z^{\prime})^{2}]^{1/2}}\to{\frac{1}{2}}\ln\frac{R_{0}-Z_{0}}{\ell_{\gamma}}\,,$
(14)

where $Z_{0}=(z-z_{0})$, $R_{0}=\sqrt{r^{2}+Z_{0}^{2}}$, and the infinite
integral has been regulated by the lengthscale $\ell_{\gamma}$. Solving for
$\nu$ yields the Rindler metric in Weyl coordinates:

$ds^{2}=\frac{(R_{0}-Z_{0})}{\ell_{\gamma}}dt^{2}-\frac{\ell_{\gamma}}{2R_{0}}[dr^{2}+dz^{2}]-\frac{\ell_{\gamma}r^{2}}{(R_{0}-Z_{0})}d\phi^{2}\,.$
(15)

Since Rindler spacetime is simply flat spacetime as observed by an
accelerating observer, we can transform (14) to Minkowski spacetime (in
cylindrical polars) via the transformation

$t=\frac{\ell_{\gamma}}{2}\log\left(\frac{\tau+\zeta}{\zeta-\tau}\right)\quad,\quad
r=\frac{\rho}{\ell_{\gamma}}\sqrt{\zeta^{2}-\tau^{2}}\quad,\quad
z-z_{0}=\frac{\tau^{2}+\rho^{2}-\zeta^{2}}{2\ell_{\gamma}}\,.$ (16)

The origin of Minkowski corresponds to $z=z_{0}$, $r=0$, (i.e. the start of
the SILM), as expected. The origin of the Weyl system corresponds to
$\zeta=\sqrt{2\ell_{\gamma}z_{0}}$, which gives a natural choice of gauge for
the Weyl system. Note that the values of $z_{0}$ and $\ell_{\gamma}$ are
independent from the perspective of solving the Einstein equations, the former
is a gauge choice—the origin of the $z$-coordinate—and the latter because the
same Rindler horizon can apply to observers with differing accelerations
$A=1/\ell_{\gamma}$; see figure 1. Interpreting the origin of the Weyl system
as the location of the accelerating observer, thus fixing the gauge, gives
$z_{0}=1/2A$ from $\zeta=1/A$.

###  2.3 Many black holes

Now we can consider superposing solutions for $\gamma$, to build up multi-
black hole solutions as described in Fay . We will briefly review these
solutions, using a slightly different notation to Fay that is more suited to
our argument. Each black hole is represented by a rod of length $2m_{I}$,
$I=1..N$, and acceleration is represented by a SILM as described above. We
will label the rod ends at $z_{i}$, where $i=1..n$ and $z_{1}<z_{2}<..$. If we
have an array of accelerating black holes, $n=2N+1$, and the SILM begins at
$z_{n}$, if we have an array of (non-accelerating) black holes, then $n=2N$ is
even. This arrangement is depicted in figure 2

Figure 2:  The source arrangement for the multi-black hole system of section
2.3. In the non-accelerating case, the point $z_{n}$, representing the start
of the SILM (thick red arrow), and the SILM itself are absent; its
neighbouring string (dashed black) instead extends to $z\rightarrow\infty$.

A natural generalisation of previous notation is

$\displaystyle Z_{i}$ $\displaystyle=z-z_{i}\,,$ $\displaystyle R^{2}_{i}$
$\displaystyle=r^{2}+Z_{i}^{2}\,,$ (17) $\displaystyle X_{i}$
$\displaystyle=R_{i}-Z_{i}\,,\qquad$ $\displaystyle E_{ij}$
$\displaystyle=R_{i}R_{j}+Z_{i}Z_{j}+r^{2}\,.$

The solution for $\gamma$ is simply the superposition of the general
potentials from (9), with $\nu$ then obtained by quadrature:

$\displaystyle\gamma$
$\displaystyle=\frac{1}{2}\sum_{i=1}^{n}(-1)^{i+1}\log\frac{X_{i}}{\ell_{\gamma}}\,,$
(18) $\displaystyle\nu$
$\displaystyle=\frac{1}{4}\sum_{i,j=1}^{n}(-1)^{i+j+1}\log\frac{E_{ij}}{\ell_{\nu}^{2}}+O_{n}\gamma\,.$

Here, the $\ell$’s are integration constants that cancel only if $n$ is even,
and $O_{n}$ acts as a “switch” for additional terms when $n$ is odd:

$O_{n}=n-2\left\lfloor\frac{n}{2}\right\rfloor=\begin{cases}1&n\;\text{odd}\\\
0&n\;\text{even}\end{cases}\,.$ (19)

As we move to the thermodynamics of the system, we will need the limit of
these functions as we approach the axis, $r\to 0$. We therefore conclude this
subsection by finding the behaviour of (18) as $r\to 0$, and discussing the
conical deficits on the axis. Noting that $R_{i}\to|Z_{i}|$ as $r\to 0$, we
see that

$X_{i}\sim|Z_{i}|-Z_{i}+\frac{r^{2}}{2|Z_{i}|}=\begin{cases}2|Z_{i}|&z<z_{i}\\\
\frac{r^{2}}{2Z_{i}}&z>z_{i}\end{cases}\,,$ (20)

hence

$\gamma\sim\frac{1}{2}\sum_{i=1}^{p}(-1)^{i+1}\log\frac{r^{2}}{2|Z_{i}|\ell_{\gamma}}+\sum_{i=p+1}^{n}(-1)^{i+1}\log\frac{2|Z_{i}|}{\ell_{\gamma}}\;\;\;:\;\;\;\;z\in(z_{p},z_{p+1})$
(21)

where $p=0$ if $z<z_{1}$ leaving only the second sum, and conversely the first
sum for $z>z_{n}$.

Next,

$E_{ij}\sim\begin{cases}2Z_{i}Z_{j}&z<\text{Min}[z_{i},z_{j}]\;,\;\;z>\text{Max}[z_{i},z_{j}]\\\
\frac{r^{2}(z_{i}-z_{j})^{2}}{2|Z_{i}Z_{j}|}&\quad\text{Min}[z_{i},z_{j}]<z<\text{Max}[z_{i},z_{j}]\end{cases}\,.$
(22)

Hence if we approach the axis at the $I\textsuperscript{th}$ black hole, for
which $z\in(z_{2I-1},z_{2I})$,

$\displaystyle\nu\sim$
$\displaystyle\frac{1}{4}\sum_{i,j=1}^{2I-1}(-1)^{i+j+1}\log\left(\frac{2Z_{i}Z_{j}}{\ell_{\nu}^{2}}\right)+\frac{1}{4}\sum_{i,j=2I}^{n}(-1)^{i+j+1}\log\left(\frac{2Z_{i}Z_{j}}{\ell_{\nu}^{2}}\right)$
(23)
$\displaystyle+\frac{1}{2}\sum_{i=1}^{2I-1}\sum_{j=2I}^{n}(-1)^{i+j+1}\log\left(\frac{r^{2}(z_{i}-z_{j})^{2}}{2|Z_{i}Z_{j}|\ell_{\nu}^{2}}\right)+O_{n}\gamma\,.$

Away from the black holes, writing
$\nu_{0}=\frac{1}{2}\log\left(\frac{\sqrt{2}\ell_{\nu}}{\ell_{\gamma}}\right)$,
we have:

$\displaystyle\nu(0,z)$ $\displaystyle=O_{n}\nu_{0}$ $\displaystyle z<z_{1}$
(24)
$\displaystyle=\sum_{i=1}^{2I}\sum_{j=2I+1}^{n}(-1)^{i+j+1}\log(z_{j}-z_{i})+O_{n}\nu_{0}$
$\displaystyle\quad z_{2I}<z<z_{2I+1}$
$\displaystyle=O_{n}\left[\sum_{i=1}^{n-1}(-1)^{i}\log(z_{n}-z_{i})+\nu_{0}\right]$
$\displaystyle z_{2N}<z<z_{n}\,.$

Notice that $z_{j}-z_{i+1}<z_{j}-z_{i}<z_{j+1}-z_{i}$, thus $\nu(0,z)<\nu_{0}$
for the first string tension, and (for accelerating black holes)
$\nu_{N}<\nu_{0}$.

We can now identify the conical structure on the axis. The axis will have a
conical defect if the circumference of circles of proper radius $\Delta r$
around it are not $2\pi\Delta r$. For small $r$, $\Delta r\sim
e^{\nu(0,z)-\gamma(0,z)}$ and the circumference is $2\pi re^{-\gamma(0,z)}/K$,
hence the deficit angle $\delta$ is

$\delta=2\pi\lim_{r\to 0}\left[1-\frac{e^{-\nu(0,z)}}{K}\right]\,,$ (25)

which is related to the cosmic string tension via $\delta=8\pi G\mu$. (24)
dictates how the deficit angle changes as we move between the black holes. The
tension between the $I\textsuperscript{th}$ and $(I+1)\textsuperscript{th}$
black hole is

$\mu_{I}=\frac{1}{4}\left(1-\frac{e^{-O_{n}\nu_{0}}}{K}\prod_{i=1}^{2I}\prod_{j=2I+1}^{n}(z_{j}-z_{i})^{(-1)^{i+j}}\right)\,.$
(26)

The final black hole has $\mu_{N}$ as the deficit for $z>z_{2N}$,

$\mu_{N}=\begin{cases}\frac{1}{4}\left(1-\frac{1}{K}\right)&n=2N\\\
\frac{1}{4}\left(1-\frac{e^{-\nu_{0}}}{K}\prod_{i=1}^{N}\frac{(z_{n}-z_{2i-1})}{(z_{n}-z_{2i})}\right)&n=2N+1\end{cases}\;,$
(27)

and for the incident tension, $z<z_{1}$, we have

$\mu_{0}=\frac{1}{4}\left(1-\frac{e^{-O_{n}\nu_{0}}}{K}\right)\,.$ (28)

We now see the interpretation of $K$. For the non-accelerating black hole
array, there is an ambient tension running through the system, as the deficit
outside the array ($z<z_{1}$ and $z>z_{2N}$) have the same conical deficit of

$\mu_{0}=\mu_{N}=\frac{1}{4}\left(1-\frac{1}{K}\right)\,.$ (29)

Equation (24) shows that $e^{\nu(0,z)}<1$ between the black holes. Hence, if
we did not insert the parameter $K$, instead retaining a $2\pi$ periodicity of
$\phi$ for $z<z_{1}$ and $z>z_{2N}$, the conical singularity between any two
of the black holes would be an excess $\delta<0$, corresponding to a negative
tension “cosmic strut” as in Krtous:2019 . Although one can consider such
systems Costa:2000kf ; Krtous:2019 ; Ramirez-Valdez:2020 ; Garcia-Compean:2020
, we prefer to keep physical sources. We therefore take $K$ large enough that
all the conical singularities are deficits and correspond, in principle, to
physical cosmic strings AGK ; GKW . Note however that if $K>1$, there is an
ambient conical deficit through the spacetime, irrespective of whether there
is acceleration.

For an accelerating black hole array, we follow the convention of
Anabalon:2018ydc ; RuthAndy that $K$ measures the ambient deficit, i.e.

$\mu_{0}+\mu_{N}=\frac{1}{2}\left(1-\frac{1}{K}\right)\,.$ (30)

This in turn allows us to determine $\nu_{0}$:

$e^{\nu_{0}}=\left(\frac{\sqrt{2}\ell_{\nu}}{\ell_{\gamma}}\right)^{1/2}=\frac{1}{2}\left(1+\prod_{i=1}^{N}\frac{(z_{n}-z_{2i-1})}{(z_{n}-z_{2i})}\right)\equiv\frac{1}{2}\left(1+V_{n}\right)\,.$
(31)

thus we have

$\mu_{0}=\frac{1}{4}\left(1-\frac{2}{(1+V_{n})K}\right)\quad,\qquad\mu_{N}=\frac{1}{2}\left(1-\frac{2V_{n}}{(1+V_{n})K}\right)\,.$
(32)

Note however that the choice of $K$ is not unique; this one, (32), corresponds
to the same normalisation as the standard C-metric, however, if one were
viewing the metric as a split cosmic string, then an alternate natural choice
might be to normalise the “initial” deficit. That is, we could choose
$\mu_{0}=\frac{1}{4}\left(1-\frac{1}{K}\right)$, in which case
$\mu_{N}=\frac{1}{4}\left(1-\frac{V_{n}}{K}\right)$.

Finally, we are left with the length scale $\ell_{\gamma}$, which is (only)
present in an accelerating array, This parameter represents the net
acceleration scale of the spacetime. We expect that for small accelerations
(large $z_{n}$) this should asymptote the Rindler value $\ell_{\gamma}\sim
2z_{n}$. Interpreting the acceleration as the overall mass of the composite
black hole system divided by the overall force measured by the differential
deficit, we are led to

$\ell_{\gamma}=\frac{M}{\mu_{0}-\mu_{N}}=\frac{V_{n}+1}{V_{n}-1}\sum_{1}^{N}(-1)^{k}z_{k}\;\;,$
(33)

where $M=\sum m_{I}/K$ is the total mass of the system (see section 3.1). We
see that $\ell_{\gamma}$ has the required large $z_{n}$ limit and a clear
physical interpretation in close analogy with its pure Rindler cousin from
section 2.2.

##  3 Thermodynamics of an array of black holes

We now derive a First Law for collinear black holes with varying positive
tension strings and a possible acceleration horizon, the solutions for which
were presented in section 2.3.

###  3.1 Deriving the thermodynamic parameters

First we need to derive these relevant thermodynamic parameters. For the
entropy of a given black hole, we compute the area of the relevant horizon

$S_{I}=\lim_{r\to
0}\,\frac{\pi}{2K}\int_{z_{2I-1}}^{z_{2I}}re^{\nu-2\gamma}dz=\frac{\pi
m_{I}}{K}\lim_{r\to 0}\,re^{\nu-2\gamma}\,.$ (34)

For the temperature, the standard techniques apply, yielding

$T_{I}=\lim_{r\to
0}\,\frac{1}{2\pi}\,\frac{e^{2\gamma}}{re^{\nu}}=\frac{m_{I}}{2KS_{I}}\,.$
(35)

The limit of $re^{\nu-2\gamma}$ as we approach the axis is given by (21),
(23), and using (31), we obtain:

$\displaystyle\log(re^{\nu-2\gamma})$ $\displaystyle\to\log
2+O_{n}\log\left(\frac{\ell_{\gamma}e^{\nu_{0}}}{2}\right)+\sum_{i=1}^{2I-1}\sum_{j=2I}^{n}(-1)^{i+j+1}\log|z_{j}-z_{i}|\,.$
(36)

The most challenging thermodynamic quantity to identify is the total mass.
This is in part due to the fact that external strings which extend to infinity
prevent global asymptotic flatness and thus render the ADM mass ADM:1959 ill-
defined. The presence of a non-compact acceleration horizon further
complicates matters. Some attempt has been made Dutta:2005iy  to redefine ADM
mass in the presence of a conical defect by calculating the mass relative to
conical Minkowski space, rather than pure Minkowski as one would in the usual
construction. However, such a construction gives undesirable results. In
particular, one would conclude that the mass of the C-metric is vanishing.
This is puzzling from the perspective of having no smooth transition to the
non-accelerating black hole. It is also counter to the intuition gained from
the slowly accelerating black hole in AdS, for which the mass (with an
appropriately normalised time coordinate) is $M\textsubscript{AdS}=m/K$. One
may be confident in the AdS calculation due to the holographic correspondence.

Although one may struggle to find a useful notion of ADM mass, the existence
of the $\partial_{t}$ isometry means that one still has a Komar construction
Komar:1963 at one’s disposal. Focusing first on the non-accelerating case,
Costa:2000kf  calculated the ADM mass for a system of collinear black holes
without external strings ($\mu_{0}=\mu_{N}=0$). One can compute the asymptotic
behaviour,

$\displaystyle e^{2\gamma}\sim
1-\frac{2(\Sigma_{I=1}^{N}m_{I})}{\tilde{r}}+\mathcal{O}(\tilde{r}^{-2})\,,\qquad$
$\displaystyle\nu\sim\mathcal{O}(\tilde{r}^{-2})\,,$ (37)

where $\tilde{r}$ is a suitable radial coordinate, and simply read off the
mass. As discussed above, when we have an ambient conical deficit the ADM mass
is undefined, but we may instead read off the Komar mass as
$M=\sum_{I=1}^{N}m_{I}/K$.

When an acceleration horizon is present, the situation requires more
explanation. We take $k=\partial_{t}$ as our Killing vector field generating
time translations. The normalisation of $k$ is implicit in the choice (33) of
$\ell_{\gamma}$; see the discussion given at the end of section 2. The
covector associated to $k$ is $k^{\flat}=e^{2\gamma}dt$. Taking the exterior
derivative and Hodge dual, we find

$\star
dk^{\flat}=\frac{r}{Ke^{2\gamma}}\left[(\partial_{r}e^{2\gamma})dz-(\partial_{z}e^{2\gamma})dr\right]\wedge
d\phi\,.$ (38)

The causal structure of the spacetime is now significantly more complicated
than in the non-accelerating case, but there is still a well defined spatial
infinity PavelInterpreting . To calculate the total mass, one could, in
principle, integrate this form over a two-surface there. That said, it is more
instructive to use Gauss’ law to rewrite the boundary integral as the sum of
integrals over each black hole horizon and a bulk integration:

$\frac{1}{8\pi}\int_{\infty}\star
dk^{\flat}=\frac{1}{8\pi}\sum_{I=1}^{N}\int_{\mathcal{H}_{I}}\star
dk^{\flat}+M\textsubscript{bulk}\,.$ (39)

The quantity on the left hand side is the total mass222There is a caveat here
that we have divided through to retain only the mass of objects on one side of
the acceleration horizon. $M$. From (21) and (23), we have the relevant
behaviour for the integrand on the right hand side of (39) near the
$I\textsuperscript{th}$ horizon $\mathcal{H}_{I}$,

$\displaystyle\partial_{r}e^{2\gamma}\sim\frac{r}{2\left|z_{2I+1}z_{2I}\right|}\,,$
$\displaystyle e^{-2\gamma}\sim\frac{4\left|z_{2I+1}z_{2I}\right|}{r^{2}}\,,$
(40)

making the integrand straightforward:

$\lim_{r\rightarrow 0}\left[\star
dk^{\flat}\right]_{z\in(z_{2I},z_{2I+1})}=\frac{2}{K}dz\wedge d\phi+\ldots\,.$
(41)

Hence we conclude that the integral over $\mathcal{H}_{I}$, which we interpret
as the mass of an individual black hole in the array, is

$M_{I}\equiv\frac{1}{8\pi}\int_{\mathcal{H}_{I}}\star
dk^{\flat}=\frac{m_{I}}{K}\,.$ (42)

Finally, we note that the volume integral $M\textsubscript{bulk}$ vanishes,
and that the strings themselves make no contribution to the above calculation.

The conclusion is that the total Komar mass is directly related to the rod
lengths of compact horizons. The same result for the mass of the solitary
accelerating black hole has been proposed in Anabalon:2018qfv , albeit with a
non-commital attitude to the normalisation of $k$. We also observe a clear
similarity with the holographically calculated mass of a slowly accelerating
black hole in AdS Anabalon:2018ydc .

###  3.2 The First Law of Thermodynamics

We now show how to derive equation (2), the first law of thermodynamics for an
array of collinear black holes. Consider a variation to the array. The
solution (18) describes a coupled system; any variation of one black hole will
impact on all the others. Therefore, we do not expect individual First Laws
for each black hole. Instead, it makes sense to consider a variation of the
total mass

$M=\sum_{I=1}^{N}\frac{m_{I}}{K}\,,$ (43)

as this is a state function of the complete system. Indeed, this is the
philosophy for the First Law derived in Krtous:2019 . Thus, to derive a First
Law, we must compute

$\delta M=\sum_{I=1}^{N}\frac{1}{K}\delta m_{I}-m_{I}\frac{\delta
K}{K^{2}}\,.$ (44)

We begin by computing the variation in entropies for the individual black
holes:

$\sum_{I=1}^{N}T_{I}\delta
S_{I}=\frac{1}{2}\sum_{I=1}^{N}\delta\left(\frac{m_{I}}{K}\right)+S_{\Sigma}+\frac{O_{n}M}{2}\left(\frac{\delta\ell_{\gamma}}{\ell_{\gamma}}+\delta\nu_{0}\right)\,,$
(45)

having replaced $\ell_{\nu}=e^{2\nu_{0}}\ell_{\gamma}/\sqrt{2}$, and where

$S_{\Sigma}=\sum_{I=1}^{N}\frac{m_{I}}{2K}\sum_{i=1}^{2I-1}\sum_{j=2I}^{2N}(-1)^{i+j+1}\frac{\delta(z_{j}-z_{i})}{z_{j}-z_{i}}+O_{n}\sum_{I=1}^{N}\frac{m_{I}}{2K}\sum_{i=1}^{2I-1}(-1)^{i}\frac{\delta(z_{n}-z_{i})}{z_{n}-z_{i}}\\\
\,.$ (46)

This contains part of what we need for a First Law, but has a rather messy
sum!

Now we turn to the cosmic strings. We write the thermodynamic lengths for the
strings as $\lambda_{I}=-e^{\nu_{I}}L_{I}$, and then vary the tensions in
(26), (27), and (28) to obtain the contribution to the First Law coming from
the tensions:

$-\sum_{I=0}^{N}\lambda_{I}\delta\mu_{I}=\sum_{I=0}^{N}\frac{L_{I}\delta
K}{4K^{2}}+O_{n}\sum_{I=0}^{N}\frac{L_{I}}{4K}\delta\nu_{0}+\mu_{\Sigma}\,,$
(47)

where

$\mu_{\Sigma}=\sum_{I=1}^{N-1}\frac{L_{I}}{4K}\sum_{i=1}^{2I}\sum_{j=2I+1}^{2N}(-1)^{i+j+1}\frac{\delta(z_{j}-z_{i})}{z_{j}-z_{i}}+O_{n}\sum_{I=1}^{N}\frac{L_{I}}{4K}\sum_{i=1}^{2I}(-1)^{i}\frac{\delta(z_{n}-z_{i})}{z_{n}-z_{i}}\,.$
(48)

Putting these two expressions together, we have:

$\displaystyle\sum_{I=1}^{N}$ $\displaystyle T_{I}\delta
S_{I}-\sum_{I=0}^{N}\lambda_{I}\delta\mu_{I}$ (49) $\displaystyle=\frac{\delta
M}{2}+S_{\Sigma}+\mu_{\Sigma}+\sum_{I=0}^{N}\frac{L_{I}\delta
K}{4K^{2}}+\frac{O_{n}}{4K}\sum_{I=0}^{N}(L_{I}+2m_{I})\delta\nu_{0}+2m_{I}\frac{\delta\ell_{\gamma}}{\ell_{\gamma}}\,.$

First, let us deal with the sums $S_{\Sigma}$ and $\mu_{\Sigma}$ in these
expressions. Observing that $m_{I}=(z_{2I}-z_{2I-1})/2$, we can rewrite the
entropy sum as

$S_{\Sigma}=\sum_{k=1}^{2N}\sum_{i=1}^{2[\frac{k+1}{2}]-1}\\!\\!\\!\\!\sum_{j=2[\frac{k+1}{2}]}^{2N}\frac{(-1)^{i+j+k+1}}{4K}z_{k}\frac{\delta(z_{j}-z_{i})}{z_{j}-z_{i}}+\frac{O_{n}}{4K}\sum_{k=1}^{2N}\sum_{i=1}^{2\left[\frac{k+1}{2}\right]-1}(-1)^{i+k}z_{k}\frac{\delta(z_{n}-z_{i})}{z_{n}-z_{i}}$
(50)

Generalising Krtous:2019 for the thermodynamic lengths of strings in between
horizons as $L_{I}=z_{2I+1}-z_{2I}$ (with the exception of $L_{0}$ and $L_{N}$
– see later) gives the tension sum as

$\displaystyle\mu_{\Sigma}$ $\displaystyle=\sum_{k=2}^{2N-1}\
\sum_{i=1}^{2[\frac{k}{2}]}\sum_{j=2[\frac{k}{2}]+1}^{2N}\frac{(-1)^{i+j+k}}{4K}z_{k}\frac{\delta(z_{j}-z_{i})}{z_{j}-z_{i}}$
(51) $\displaystyle+\frac{O_{n}}{4K}\sum_{k=1}^{2N-1}\
\sum_{i=1}^{2\left[\frac{k}{2}\right]}(-1)^{i+k+1}z_{k}\frac{\delta(z_{n}-z_{i})}{z_{n}-z_{i}}+\frac{L_{N}}{4K}\sum_{i=1}^{2N}(-1)^{i}\frac{\delta(z_{n}-z_{i})}{z_{n}-z_{i}}\,.$

We now see that many of the terms in $S_{\Sigma}$ are cancelled by terms in
$\mu_{\Sigma}$, leaving just $k=1,2N$ from the entropy sum, and intermediate
$i,j$ terms from each when $2[\frac{k+1}{2}]$ differs from $2[\frac{k}{2}]+1$:

$\displaystyle S_{\Sigma}+\mu_{\Sigma}$ (52)
$\displaystyle=\sum_{j=2}^{2N}\frac{(-1)^{j+1}}{4K}z_{1}\frac{\delta(z_{j}-z_{1})}{z_{j}-z_{1}}+\sum_{i=1}^{2N-1}\frac{(-1)^{i+1}}{4K}z_{2N}\frac{\delta(z_{2N}-z_{i})}{z_{2N}-z_{i}}$
$\displaystyle+\sum_{k=2}^{2N-1}\left[\sum_{j=k+1}^{2N}\frac{(-1)^{j+k}}{4K}z_{k}\frac{\delta(z_{j}-z_{k})}{z_{j}-z_{k}}+\sum_{i=1}^{k-1}\frac{(-1)^{i+k+1}}{4K}z_{k}\frac{\delta(z_{i}-z_{k})}{z_{i}-z_{k}}\right]$
$\displaystyle+\frac{O_{n}}{4K}\left(\sum_{i=1}^{2N-1}\left[(-1)^{i}(L_{N}+z_{2N})-z_{i}\right]\frac{\delta(z_{n}-z_{i})}{z_{n}-z_{i}}+L_{N}\frac{\delta(z_{n}-z_{2N})}{z_{n}-z_{2N}}\right)$
$\displaystyle=\sum_{j=2}^{2N}\sum_{i=1}^{j-1}\frac{(-1)^{i+j+1}}{4K}(\delta
z_{j}-\delta
z_{i})+\frac{O_{n}}{4K}\sum_{k=1}^{2N}(-1)^{k}(L_{N}+z_{2N}-z_{k})\frac{\delta(z_{n}-z_{k})}{z_{n}-z_{k}}\,.$

We now have to identify $L_{N}$ (and $L_{0}$). We write

$L_{N}=z_{c}-z_{2N}\qquad,\qquad L_{0}=z_{1}-z_{c}$ (53)

in keeping with the expressions for $L_{I}$, where $z_{c}$ is a normalisation,
similar to that of the SILM in $\gamma$, to be determined. We can therefore
reduce this combination to

$\displaystyle S_{\Sigma}+\mu_{\Sigma}$
$\displaystyle=\sum_{I=1}^{N}\frac{\delta
m_{I}}{2K}+\frac{O_{n}}{4K}\sum_{k=1}^{2N}(-1)^{k}\left(\delta(z_{n}-z_{k})+(z_{c}-z_{n})\frac{\delta(z_{n}-z_{k})}{z_{n}-z_{k}}\right)$
(54) $\displaystyle=\sum_{I=1}^{N}\frac{\delta
m_{I}}{2K}\left(1-O_{n}\right)+\frac{O_{n}}{4K}(z_{c}-z_{n})\sum_{k=1}^{2N}(-1)^{k}\frac{\delta(z_{n}-z_{k})}{z_{n}-z_{k}}$
$\displaystyle=\sum_{I=1}^{N}\frac{\delta
m_{I}}{2K}\left(1-O_{n}\right)-\frac{O_{n}}{4K}(z_{c}-z_{n})\frac{\delta
V_{n}}{V_{n}}\,.$

Having simplified $S_{\Sigma}+\mu_{\Sigma}$, we now turn to the rest of the
putative First Law, (49). We note that the sum of the thermodynamic lengths
can be related to the sum of the masses:

$\sum_{I=0}^{N}L_{I}=\sum_{I=1}^{N-1}(z_{2I+1}-z_{2I})+L_{N}+L_{0}=-2\sum_{I=1}^{N}m_{I}\,.$
(55)

Hence,

$\displaystyle\sum_{I=1}^{N}T_{I}\delta
S_{I}-\sum_{\mu^{\prime}s}\lambda_{I}\delta\mu_{I}$
$\displaystyle=\frac{\delta M}{2}+\sum_{I=0}^{N}\frac{L_{I}\delta
K}{4K^{2}}+\sum_{I=1}^{N}\frac{\delta m_{I}}{2K}$ (56)
$\displaystyle+\frac{O_{n}}{4K}\left[\sum_{I=1}^{N}2m_{I}\frac{\delta\ell_{\gamma}}{\ell_{\gamma}}-\sum_{I=1}^{N}2\delta
m_{I}-(z_{c}-z_{n})\frac{\delta V_{n}}{V_{n}}\right]$ $\displaystyle=\delta
M+\frac{O_{n}}{2K}\left[m_{TOT}\frac{\delta\ell_{\gamma}}{\ell_{\gamma}}-\delta
m_{TOT}-(z_{c}-z_{n})\frac{\delta V_{n}}{2V_{n}}\right]\,,$

where $m_{TOT}=\sum_{I}m_{I}$ is shorthand for the sum of the individual rod
lengthscales. Thus, we have derived the First Law (2) for a general array of
black holes, provided we identify

$z_{c}=z_{n}+\frac{2m_{TOT}\delta\ell_{\gamma}/\ell_{\gamma}-2\delta
m_{TOT}}{\delta V_{n}/V_{n}}=z_{n}-\frac{4V_{n}m_{TOT}}{V_{n}^{2}-1}\,,$ (57)

for the accelerating black hole. For the non-accelerating black hole, the
First Law is automatically satisfied and we set
$L_{0}=L_{N}=(z_{1}+z_{2N})/2$.

##  4 Exploring Multi-Black Hole Spacetimes

Having derived these expressions, it is interesting to explore some sample
accelerating and non-accelerating black hole arrays to gain an understanding
of the interdependency of black hole entropy, and to see how the strings
contribute to the thermodynamic system as well as cross-checking against known
results.

###  4.1 Non-accelerating arrays

We start by considering non-accelerating black holes. This includes the
Schwarzschild case as a basic cross-check of our results, and the two black
hole system which has already been considered in the literature Costa:2000kf ;
Krtous:2019 ; Ramirez-Valdez:2020 ; Garcia-Compean:2020 .

####  4.1.1 Schwarzschild with a string

As discussed in section 2, the Schwarzschild solution (with an axial conical
defect) has $n=2$, $N=1$, and $z_{2}-z_{1}=2m$. Conventionally, we set the
centre of the rod at the origin so that $z_{2}=-z_{1}=m$. From (34) and (35)
we find that the entropy and temperature are $S=4\pi m^{2}/K$ and $T=1/8\pi m$
respectively, as expected. For the cosmic string piercing the horizon, we have
$\mu_{0}=\mu_{1}=\frac{1}{4}\left(1-\frac{1}{K}\right)$, and
$\lambda_{0}=\lambda_{1}=m$ in agreement with Appels:2017xoe .

####  4.1.2 Two black holes

The First Law for the two black hole system, with $K=1$, was explored in
Krtous:2019 . This value of $K$ means that there are no strings running to
infinity, but instead the black holes are held apart by a negative tension
strut. Nevertheless, for larger $K$,

$K\geq\frac{D^{2}-(m_{2}-m_{1})^{2}}{D^{2}-(m_{1}+m_{2})^{2}}$ (58)

where $D$ is the distance between the centres of the two rods, we find results
harmonious with their conclusions: the First Law holds with the thermodynamic
length of the defect connecting the black holes given by the worldsheet volume
of the string per unit time. The thermodynamic lengths of the semi-infinite
strings are now

$\lambda_{0}=\lambda_{2}=\frac{z_{4}-z_{1}}{2}=\frac{D}{2}+\frac{m_{1}+m_{2}}{2}$
(59)

i.e. the system responds to the average mass, and the length between the black
holes. Note that $\lambda_{1}=-(z_{3}-z_{2})e^{\nu_{1}}$ also has a factor of
the separation that is important for consistency in varying the net conical
deficit of the system. We discuss this in more detail below for three black
holes.

####  4.1.3 Three black holes

The three black hole system has rods on the intervals $(z_{1},z_{2})$,
$(z_{3},z_{4})$, and $(z_{5},z_{6})$; see figure 3.

Figure 3: The source arrangement for three (non-accelerating) black holes.

We are interested in exploring how the locations of the sources affect entropy
and tension, and how a perturbation of one black hole impacts on the others.
Hence, we consider a set-up in which the two outer black holes have equal mass
and spacing from the middle black hole, which is centred around the origin:
$z_{6}-z_{5}=z_{2}-z_{1}=2m_{0}$, and $z_{6}=-z_{1}=z_{0}$, $z_{4}=-z_{3}=m$.
The entropies and tensions then become:

$\displaystyle S_{1}$ $\displaystyle=\frac{4\pi
m_{0}^{2}}{K}\frac{(z_{0}+m_{0})(z_{0}+m_{0}+m)}{z_{0}(z_{0}+m_{0}-m)}=S_{3}$
(60) $\displaystyle S_{2}$ $\displaystyle=\frac{4\pi
m^{2}}{K}\frac{(z_{0}^{2}-m_{0}^{2})(z_{0}+m_{0}+m)^{2}}{z_{0}^{2}(z_{0}-m_{0}+m)^{2}}$
$\displaystyle\mu_{0}$ $\displaystyle=\frac{1}{4}\left(1-\frac{1}{K}\right)$
$\displaystyle\mu_{1}$
$\displaystyle=\frac{1}{4}\left(1-\frac{z_{0}^{2}(z_{0}^{2}-(m_{0}-m)^{2})}{(z_{0}^{2}-m_{0}^{2})(z_{0}^{2}-(m_{0}+m)^{2})K}\right)=\mu_{2}\,.$

It is easy to see that $\mu_{1}<\mu_{0}$. This is to be expected: in order to
retain equilibrium, additional force must be applied on the outer black holes
to counterbalance their attraction of the middle one.

For the thermodynamic lengths we have:

$\displaystyle\lambda_{0}$
$\displaystyle=(z_{6}-z_{1})/2=(z_{0}+m_{0})=\lambda_{3}\,,$ (61)
$\displaystyle\lambda_{1}$
$\displaystyle=-(z_{0}-m_{0}-m)\frac{(z_{0}^{2}-m_{0}^{2})(z_{0}^{2}-(m_{0}+m)^{2})}{z_{0}^{2}(z_{0}^{2}-(m_{0}-m)^{2})}=\lambda_{2}\,.$

Thus the thermodynamic length of the ambient deficit—that is, the total from
both string $1$ and $4$—is the distance from the north pole of the topmost
black hole to the south pole of the bottom-most black hole. The length
associated to the intermediate strings is minus the distance between the
horizons of adjacent black holes (see figure 4).

Figure 4: The variation of tensions and thermodynamic lengths of the three
black hole system: Left: for equal masses as a function of black hole
separation, and Right: for fixed black hole separation but varying the mass of
the outer black hole. On the left, the tension is set by taking the minimal
value consistent with zero tension between the black holes at minimum
separation, $z_{min}=3$, giving an ambient tension of $41/324$. On the right
the separation is set at $z_{0}=10$, and the outer black hole mass varies from
zero to $8.8$, which is very close to the merger limit of maximal tension,
$\mu_{0}\sim 0.244$.

We have found an interesting phenomenon where the thermodynamic lengths of the
outer strings are positive whereas those of interior strings are negative.
This is puzzling from the perspective of the individual black holes. However,
upon taking the system as a composite it makes sense: if we alter the overall
tension, we must account for the contributions from both inner and outer
cosmic strings. The negative contribution from the interior lengths then
counteracts the positive contribution from the outer lengths. Explicitly,
first set up the three black holes so that there is no deficit between the
central and outer black holes. That is, $K$ takes the value

$K_{0}=\frac{z_{0}^{2}(z_{0}^{2}-(m_{0}-m)^{2})}{(z_{0}^{2}-m_{0}^{2})(z_{0}^{2}-(m_{0}+m)^{2})}\,.$
(62)

We now “add” a cosmic string to the system by increasing $K$ to $K_{0}+K_{1}$,
so that

$\delta\mu_{0}=\frac{K_{1}}{4K_{0}(K_{0}+K_{1})}\qquad,\qquad\delta\mu_{1}=\frac{K_{1}}{4(K_{0}+K_{1})}=K_{0}\delta\mu_{0}\,.$
(63)

Note that the tension of the ambient cosmic string through the whole spacetime
increases from $\mu_{0}=(1-1/K_{0})/4$ to $\mu_{0}+\delta\mu_{0}$. However,
the region between the black holes, which initially had no deficit, now
exhibits a cosmic string with tension $K_{0}\delta\mu_{0}$, i.e., a slightly
greater tension than the increase in ambient string tension. Now let us look
at the overall change in energy:

$\displaystyle\sum_{I=0}^{3}\lambda_{I}\delta\mu_{I}$
$\displaystyle=\lambda_{0}\delta\mu_{0}+\lambda_{1}\delta\mu_{1}+\lambda_{2}\delta\mu_{2}+\lambda_{3}\delta\mu_{3}$
(64)
$\displaystyle=2(z_{0}+m_{0})\frac{K_{1}}{4K_{0}(K_{0}+K_{1})}-2\frac{(z_{0}-m_{0}-m)}{K_{0}}\frac{K_{1}}{4(K_{0}+K_{1})}$
$\displaystyle=\frac{(4m_{0}+2m)K_{1}}{4K_{0}(K_{0}+K_{1})}\,.$

This is the total length of string captured by the black holes multiplied by
the tension. We conclude therefore that the thermodynamic lengths really do
behave in concert, combining in such a way that the overall modification of
tension has a sensible impact on the overall thermodynamics of the system.

Figure 5: The entropy of the outer (Left) and middle (Right) black holes for
equal masses as a function of black hole separation. The black holes all have
unit mass, with an ambient tension of 41/324. The tension is set by taking the
minimal value consistent with no strut between the black holes at minimum
separation, here $z_{min}=3$.

Turning to the entropies, one sees that $S_{2}/m^{2}>S_{1}/m_{0}^{2}$.
Essentially, this is saying that the inner black hole has a higher entropy in
units of its mass (squared) than the outer ones. We understand this from the
impact of the conical deficits: entropy is decreased in general by having a
conical deficit, as part of the horizon is “cut out”, leaving a rugby, as
opposed to soccer, ball shape. We would expect that the entropy of the middle
black hole would be relatively higher, as the deficit running through this
black hole is less than the deficit emerging from the outer poles of the outer
black holes.

The picture is a little more subtle than this broad brush expectation however;
the central black hole has a uniform tension, $\mu_{1}$, running through it,
so naively, we might expect that the entropy might be tracked by $4\pi
m^{2}(1-4\mu_{1})$, but in fact the entropy is higher than this. For the outer
black holes, we might expect the entropy to be tracked by the average tension
between the poles, but again, it is higher. Indeed, the entropy is higher even
than the Schwarzschild entropy for a range of separation values $z_{0}$; see
figure 5.

Similarly, we can track what happens to the entropy of one black hole as a
result of changing the mass of the others. For example, keeping the central
black hole at unit mass, and keeping the other two black holes at a given
distance, we can see how the entropy of the central black hole

$S\textsubscript{central}=\frac{4\pi}{K}\frac{(1-m^{2}/z_{0}^{2})(1+m/z_{0}+1/z_{0})^{2}}{(1-m/z_{0}+1/z_{0})^{2}}$
(65)

alters as we change the mass of the outer black holes. The mass of the outer
hole $m_{0}$ can range from zero to $z_{0}-1$, however at this point the
horizons merge and to maintain a non-negative tension between the black holes
we would have to have a maximal deficit of $2\pi$. Instead, we choose a
maximal mass $m_{\text{max}}$, and set $K$ so that at the maximal mass there
is no deficit between the black holes:

$K=K_{c}\equiv\frac{z_{0}^{2}(z_{0}^{2}-(m_{\text{max}}-1)^{2})}{(z_{0}^{2}-m_{\text{max}}^{2})(z_{0}^{2}-(m_{\text{max}}+1)^{2})}\,.$
(66)

Figure 6 shows the variation of the entropy of the central black hole for a
separation $z_{0}=10$, and a mass range up to $m_{\text{max}}=8.8$. This is
very close to the merger limit, giving a large external tension $\mu_{0}\sim
0.244$, so a deficit angle of $\delta/(2\pi)\sim 0.977$. As before, the
entropy is normalised by the entropy of a single black hole in a spacetime
with both this ambient deficit ($S_{K}=4\pi/K$) as well as that of a black
hole with a cosmic string of tension $\mu_{1}$ running through
($S_{\mu}=4\pi(1-4\mu_{1})$).

Figure 6: The entropies of the outer (Left) and central (Right) black holes as
a function of the mass $m_{0}$ of the outer black holes. The mass of the
central black hole is fixed at $1$, and the outer black holes have the same
mass $m_{0}$.

We now see a more nuanced behaviour. Initially, at $m=0$, the spacetime is
precisely that of a single black hole of unit mass pierced by a cosmic string
of tension $\mu_{0}=\mu_{1}=(1-1/K)/4$. As we switch on the black hole mass at
$z_{0}$, $\mu_{1}$ decreases, and this results in an increase in entropy, but
this is over and above what we would expect simply from a drop in $\mu_{1}$.
This comes primarily from the $m$ dependence in (65). As we increase the mass
further however, while the function $S_{2}/S_{K}$ continues to grow, the ratio
$S_{2}/S_{\mu}$, of the entropy to that of a black hole with the $\mu_{1}$
cosmic string starts to drop, eventually becoming less than one. We can
understand this as being a consequence of the very large deficit in the
majority of the spacetime, even though locally, at the central black hole,
there is no cosmic string. The outer black holes are very close (within a
Schwarzschild radius) to the central black hole, thus the geometry is strongly
distorted there.

###  4.2 Accelerating arrays

Now let us consider accelerating black hole arrays of the type depicted in
figure 2. The main difference with the non-accelerating array is that we have
chosen the parameter $K$ to represent the ambient tension, so that the
asymptotic tension in principle varies with the locations of the rod ends. The
expressions for entropy, temperature, tension and thermodynamic length are
readily worked out from (34) and (35), though are not particularly
illuminating. However, we can intuit the general behaviour as we vary the
black hole masses and positions.

First, note that $\mu_{0}>\mu_{N}$. We expect this because since the black
holes are accelerating there must be an imbalance between the tension of the
string coming in from infinity and that of the string exiting through the
acceleration horizon. Next, as we increase the first black hole mass $m_{1}$,
the first tension $\mu_{1}$ will drop, as more of the pulling power of the
string will be used to accelerate the increased mass. Whether the subsequent
string tensions increase or decrease depends on the masses of the individual
black holes: the second black hole will be attracted to the first (and third,
if present) which provides an additional attractive force over and above that
of the cosmic string. Typically, if the black holes are well separated
relative to their size, the string tensions will cascade down in magnitude as
one moves along the array, but for large black holes, this need not be the
case (see the two black hole case below).

####  4.2.1 The C-metric

It is worth briefly checking the C-metric results, first proposed in
Anabalon:2018qfv . The C-metric has a single horizon and a SILM so we have
$n=3$ and $N=1$. The metric in Weyl form is

$ds^{2}=\frac{X_{1}X_{3}}{\ell_{\gamma}X_{2}}dt^{2}-\frac{\ell_{\gamma}E_{12}E_{23}}{4R_{1}R_{2}R_{3}E_{13}}\left(\frac{z_{3}-z_{0}}{z_{3}-z_{2}}\right)^{2}[dr^{2}+dz^{2}]-r^{2}\frac{\ell_{\gamma}X_{2}}{X_{1}X_{3}}\frac{d\phi^{2}}{K^{2}}\,,$
(67)

where $z_{0}=(z_{1}+z_{2})/2$ is the centre of the black hole rod, and we have
replaced $V_{3}=(z_{3}-z_{1})/(z_{2}-z_{2})$. Here,
$\ell_{\gamma}=2(z_{3}-z_{0})$ is shown to be the reciprocal of the
acceleration of a small black hole in appendix A, where we also note the
transformation between this metric and the more familiar spherical
coordinates.

Turning to the thermodynamics, we compute $z_{c}$ as

$z_{c}=z_{3}-\frac{(z_{3}-z_{2})(z_{3}-z_{1})}{(z_{3}-z_{0})}=\frac{(z_{3}z_{0}-z_{1}z_{2})}{(z_{3}-z_{0})}\,.$
(68)

Meanwhile, the entropy and thermodynamic lengths are

$\displaystyle S$ $\displaystyle=\frac{4\pi
m^{2}}{K}\frac{(z_{3}-z_{0})^{2}}{(z_{3}-z_{2})(z_{3}-z_{1})}$
$\displaystyle\to$ $\displaystyle\frac{4\pi m^{2}}{(1-4m^{2}A^{2})}$ (69)
$\displaystyle\lambda_{0}$
$\displaystyle=e^{\nu_{0}}(z_{c}-z_{1})=m\frac{(z_{3}-z_{1})}{(z_{3}-z_{2})}$
$\displaystyle\to$ $\displaystyle\frac{m(1+2mA)}{(1-2mA)}$
$\displaystyle\lambda_{1}$
$\displaystyle=e^{\nu_{1}}(z_{2}-z_{c})=m\frac{(z_{3}-z_{2})}{(z_{3}-z_{1})}$
$\displaystyle\to$ $\displaystyle\frac{m(1-2mA)}{(1+2mA)}$

in agreement with the parameters proposed in Anabalon:2018qfv .

It is also straightforward to write down a Christodoulou-Ruffini-like formula
Christodoulou:1972kt for the C-metric. Following RuthAndy , define a quantity
$\Delta$ characterising the average tension emerging from the black hole
horizon, and a quantity $C$ characterising the tension differential:

$\displaystyle\Delta$ $\displaystyle=1-2(\mu_{0}+\mu_{1})=\frac{1}{K}\,,$ (70)
$\displaystyle C$
$\displaystyle=\frac{\mu_{0}-\mu_{1}}{\Delta}=\frac{z_{2}-z_{1}}{4\ell_{\gamma}}\rightarrow
mA\,.$

Then one finds that

$M^{2}=\frac{\Delta S}{4\pi}\left(1-4C^{2}\right)\,.$ (71)

Increasing the acceleration of the black hole while maintaining a constant
ambient deficit removes energy from the black hole. This result is not as
unsettling as it may first appear, as energy may be lost both across the
acceleration horizon and as gravitational radiation at future infinity
Podolsky:2003gm .

####  4.2.2 Two accelerating black holes

As a less trivial example, we present results for the two accelerating black
hole system, first explored in Fay . We have

$\displaystyle\ell_{\gamma}$
$\displaystyle=2z_{5}-\frac{(z_{4}^{2}-z_{3}^{2}+z_{2}^{2}-z_{1}^{2})z_{5}-(z_{4}-z_{3}+z_{2}-z_{1})(z_{1}z_{3}+z_{2}z_{4})}{(z_{4}-z_{3}+z_{2}-z_{1})z_{5}+z_{1}z_{3}-z_{2}z_{4}}$
(72) $\displaystyle\sim
2(z_{5}-z\textsubscript{com})+\mathcal{O}\left(z_{5}^{-1}\right)\,,$

where
$z\textsubscript{com}=\frac{z_{4}^{2}-z_{3}^{2}+z_{2}^{2}-z_{1}^{2}}{2(z_{4}-z_{3}+z_{2}-z_{1})}$
is the centre of mass of the pair of black holes (this formula generalises to
any number of accelerating black holes).

Placing the two black holes at $\pm z_{b}$ fixes the gauge, and we can see how
the string tensions and black hole entropies react to changes in black hole
mass and distance to the horizon (without loss of generality we can keep
$z_{b}$ fixed as a choice of scale). Writing

$S_{0}=\frac{(m_{1}+m_{2})(m_{1}+m_{2}+2z_{b})(m_{1}m_{2}-z_{b}^{2}+z_{5}^{2})^{2}}{(z_{5}+z_{b}+m_{1})(z_{5}-z_{b}-m_{2})(z_{5}(m_{2}+m_{1})+z_{b}(m_{2}-m_{1}))}\,,$
(73)

the entropies are

$\displaystyle S_{1}=\frac{4\pi
m_{1}^{2}}{K}\frac{S_{0}}{(2z_{b}+m_{1}-m_{2})(z_{5}+z_{b}-m_{1})}\,,$ (74)
$\displaystyle S_{2}=\frac{4\pi
m_{2}^{2}}{K}\frac{S_{0}}{(2z_{b}+m_{2}-m_{1})(z_{5}-z_{b}+m_{2})}\,.$

We can quickly see that if $m_{1}=m_{2}$, the entropy of the first black hole
will always be less than that of the second, which would be expected as the
mean deficit through the first black hole is greater than that through the
second. However, normalising the entropies with respect to their reference
$S_{K}=4\pi m_{I}^{2}/K$, we can see that the multiplicative factors in (74)
show that both initially decrease as $m_{2}$ increases from zero before
turning, although $S_{2}/S_{K}$ shows a sharper decrease and eventually drops
below $S_{1}/S_{K}$. Again, this behaviour is easy to see from the ratios in
(74). Figure 7 shows this behaviour with varying $m_{I}$.

Figure 7: The variation of entropies and thermodynamic lengths as a function
of mass in a two accelerating black hole set-up. The outgoing tension is fixed
at $\mu_{0}=1/8$, and the displacement of each rod from the origin at
$z_{b}=5$. One mass is fixed at unity, with the other mass varying from zero
to $4$. The upper plots show how the entropies, normalized by $4\pi
m_{I}^{2}/K$, vary, and the lower plots the thermodynamic length. Note that
$K$ varies as $m_{I}$ varies, in order to keep $\mu_{0}$ fixed.

In order to compare the impact of varying the masses of the black holes and
their separation, we first fix the outgoing tension at $z\to-\infty$, so that
we are comparing the same conical asymptotics. Figure 7 shows the effect of
varying the mass of the inner and outer black hole respectively on the
entropies and thermodynamic lengths. In each case, we fix one of the masses at
unity and vary the other. In both cases, varying the mass of the black hole
closer to the acceleration horizon ($m_{2}$) causes a “crossover” behaviour.

Figure 8 shows how the entropy, length (and tension) are affected by moving
the black holes apart. As before, the outgoing tension is fixed at $1/8$, and
both black hole masses are fixed at $m_{1}=m_{2}=1$; the acceleration horizon
is at $z_{5}=12$. The normalised entropy of the black hole closer to the
acceleration horizon increases as the black holes are moved apart, whereas the
entropy of the other black hole decreases sharply. We can get a rough
understanding of this by looking at the string tensions; the tension between
the second black hole and the acceleration horizon drops off sharply at large
separation, meaning that less of the angular direction is cut out by the
deficit, thus increasing entropy. The tension between the black holes,
$\mu_{1}$, in contrast increases, leading to an expectation that the first
entropy will decrease. While these statements are broadly true – note that we
have already normalised the $K$ factor out of the entropy, indicating that the
effect of this geometry is magnified. As expected, the thermodynamic lengths
exhibit a scaling with increasing separation, with the intermediate length
$\lambda_{1}$ negative and decreasing to compensate the increase in
$\lambda_{0}$.

Figure 8: The variation of thermodynamic parameters for the double
accelerating black hole set-up where the two black holes have equal mass
$m_{1}=m_{2}=1$ and the distance between them $2z_{b}$ is varied. The
acceleration horizon is fixed at $z_{5}=12$.

##  5 Conclusions

To sum up: we have proven a thermodynamic First Law for a composite system of
black holes, both accelerating and isolated. We have allowed the varying of
the tensions of the cosmic strings along the axis that are necessary for
maintaining the equilibrium configuration. As with the accelerating AdS black
hole thermodynamics previously developed, these strings have a corresponding
potential, the thermodynamic length, which has a direct specification in terms
of the Weyl coordinate parametrising the axis of symmetry of the black hole
array.

We have presented a range of accelerating and non-accelerating black hole
systems to illustrate the various facets of the thermodynamic parameters. The
main point is that the black holes form a fully composite thermodynamical
system—the variation of one black hole affects all the others. We also see how
the tensions and lengths in a composite system collude in such a way that the
overall picture makes intuitive sense, whereas the individual black hole
contributions may be less transparent.

Our findings, that the thermodynamic lengths between compact horizons is
related to the proper distance along the axis, are in agreement with previous
results Krtous:2019 ; Ramirez-Valdez:2020 ; Garcia-Compean:2020 . However, in
our construction there are also semi-infinite strings for which this proper
distance would be infinite, yet this is not what we would expect
thermodynamically. The thermodynamic length represents the contribution to the
enthalpy from the tension (negative pressure) of the cosmic string inside the
black hole, thus should be finite. We take this into account via a
renormalisation process, the $z_{c}$, similar to the renormalising of the
metric coefficients.

In fact, the result for $\lambda$ derived in Krtous:2019 , can actually be
understood in terms of the covariant-phase space formalism Harlow:2019 , much
as black hole entropy and temperature were interpreted by Iyer and Wald Wald ;
IyerWald1 . In this construction, the idea is that, on shell, variations the
action consist solely of boundary data. Taking this variation to be the action
of some Killing vector field corresponding to time translation, one can find a
quantity which vanishes when integrated over a Cauchy slice. Taking the
variation of this quantity, and splitting the integral up into boundary pieces
via Gauss’ law, one obtains the First Law. The contribution from infinity
gives the variation in mass and the contribution at the horizon gives $T\delta
S$. When strings are present, one must also consider the contribution from a
new surface: a “tube” which encases the string333Similar ideas have been
applied to thermodynamic investigations of black holes possessing Misner
strings NUT1 ; NUT2 .. It is precisely this contribution which provides
$\lambda\delta\mu$. From this perspective, the thermodynamic lengths
calculated in Anabalon:2018ydc ; Anabalon:2018qfv for the AdS C-metric may be
seen as renormalised worldvolumes (per unit time) of the infinite proper
length strings. The sense in which the external strings are renormalised in
the asymptotically flat case is less clear and would be interesting to
understand.

While the system of many black holes is not stable, it is nonetheless
interesting that it too displays sensible thermodynamic properties, further
supporting the inclusion of cosmic strings in the thermodynamic picture.

###### Acknowledgements.

This work was supported in part by the STFC [Consolidated Grant ST/P000371/1
-RG, DTG - AS], and by the Perimeter Institute for Theoretical Physics (RG).
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 Coordinate systems for the C-metric

We collect the transformation formulae between the standard C-metric
(expressed in Hong-Teo form Hong:2003gx ) and the Weyl form of section 2. The
C-metric in Weyl form is

$ds^{2}=\frac{X_{1}X_{3}}{\ell_{\gamma}X_{2}}dt^{2}-\frac{\ell_{\gamma}E_{12}E_{23}}{4R_{1}R_{2}R_{3}E_{13}}\left(\frac{V_{3}+1}{2}\right)^{2}[dr^{2}+dz^{2}]-r^{2}\frac{\ell_{\gamma}X_{2}}{X_{1}X_{3}}\frac{d\phi^{2}}{K^{2}}\,,$
(75)

where $\ell_{\gamma}=z_{3}-z_{0}=z_{3}-(z_{1}+z_{2})/2$ is the $z$-distance to
the centre of the black hole rod.

Now let $m=(z_{2}-z_{1})/2$ and $A=1/\ell_{\gamma}$. Define

$\displaystyle
r={\bar{r}}\sin\theta\frac{\sqrt{f({\bar{r}})g(\theta)}}{(1+A{\bar{r}}\cos\theta)^{2}}\,,\quad
z-z_{0}={\bar{r}}\frac{(A{\bar{r}}+\cos\theta)(1-m/{\bar{r}}+mA\cos\theta)}{(1+A{\bar{r}}\cos\theta)^{2}}\,,$
(76)

where

$f(R)=(1-A^{2}{\bar{r}}^{2})\left(1-\frac{2m}{{\bar{r}}}\right)\,,\qquad
g(\theta)=(1+2mA\cos\theta)\,.$ (77)

Then the Weyl metric (75) transforms to the C-metric in Hong-Teo coords
Hong:2003gx , rather than the standard Kinnersley-Walker coordinates discussed
in Fay :

$ds^{2}=\frac{1}{(1+A\bar{r}\cos\theta)^{2}}\left[\bar{f}(\bar{r})dt^{2}-\frac{d\bar{r}^{2}}{\bar{f}(\bar{r})}-\bar{r}^{2}\left(\frac{d\theta^{2}}{\bar{g}(\theta)}-\bar{g}(\theta)\sin^{2}\theta
d\phi^{2}\right)\right]\,.$ (78)

Here we see the direct interpretation of $\ell_{\gamma}$ as the acceleration
length scale; for small $m$, $A=1/\ell_{\gamma}$ corresponds to the magnitude
of the four-acceleration of the black hole PavelInterpreting .

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