# Observing dynamic oscillatory behavior of triple points among black hole
thermodynamic phase transitions

Shao-Wen Wei1,2 111E-mail<EMAIL_ADDRESS>Yong-Qiang Wang1,2 222E-mail:
<EMAIL_ADDRESS>Yu-Xiao Liu1,2 333E-mail<EMAIL_ADDRESS>and Robert B.
Mann3 444E-mail<EMAIL_ADDRESS>1Lanzhou Center for Theoretical Physics,
Key Laboratory of Theoretical Physics of Gansu Province, School of Physical
Science and Technology, Lanzhou University, Lanzhou 730000, People’s Republic
of China
2Institute of Theoretical Physics $\&$ Research Center of Gravitation, Lanzhou
University, Lanzhou 730000, People’s Republic of China
3Department of Physics and Astronomy, University of Waterloo, Waterloo,
Ontario, Canada, N2L 3G1

###### Abstract

Understanding the dynamic process of black hole thermodynamic phase
transitions at a triple point is a huge challenge. In this letter, we carry
out the first investigation of dynamical phase behaviour at a black hole
triple point. By numerically solving the Smoluchowski equation near the triple
point for a six-dimensional charged Gauss-Bonnet anti-de Sitter black hole, we
find that initial small, intermediate, or large black holes can transit to the
other two coexistent phases at the triple point, indicating that thermodynamic
phase transitions can indeed occur dynamically. More significantly, we observe
characteristic weak and strong oscillatory behaviour in this dynamic process,
which can be understood from an investigation of the rate of first passage
from one phase to another. Our results further an understanding of the dynamic
process of black hole thermodynamic phase transitions.

###### pacs:

04.70.Dy, 68.35.Rh, 04.50.Kd, 02.50.-r.

Introduction Black holes are now widely believed to behave as thermodynamic
systems after the establishment of the four laws of black hole thermodynamics
Hawking ; Bekenstein ; Bardeen . As such, they can be expected to undergo
phase transitions, one of the first being the well-known Hawking-Page phase
transition Hawking:1982dh . Over the past decade it has been shown that black
holes exhibit an abundance of phase behaviour Kubiznak:2016qmn , upon
interpreting the cosmological constant in anti-de Sitter (AdS) space as the
pressure of the black hole system in extended phase space Kastor ; Kubiznak .
Known as black hole chemistry, small/large charged black hole phase
transitions take place, reminiscent of the liquid-gas phase transition of a
Van der Waals (VdW) fluid Kastor ; Kubiznak .

Recently studies of the dynamics of phase transitions have been carried out Li
; LiWang ; LiWang2  by solving a special case of the Fokker-Planck equation
known as the Smoluchowski equation (SE). A dynamic transition indeed occurs
between the unstable and stable black hole phases, and further investigation
confirmed that even between stable small and large black hole phases, the
dynamic process can also occur Weiwa . The SE has been also been used to study
analytically the evolution of the black hole mass distribution and
supermassive black hole merger rates Taniguchi ; Erickcek .

The discovery of black hole triple points Altamirano ; Wei2 ; Frassino:2014pha
strengthens the chemical interpretation of black hole thermodynamics. It
suggests that some black hole systems are similar to water, where solid,
liquid, and gas phases can coexist, with the triple point being a coexistence
phase of stable small, intermediate, and large black holes. Across the triple
point, the three phases can transition into one another. Consequently triple
points expose significant characteristic properties of the system. However the
dynamics of this transition process for black holes is still unknown.

In this paper, we aim to test and uncover such dynamic processes. In so doing
we shall discover important properties for further understanding black hole
thermodynamics and, by implication, their underlying degrees of freedom.

Figure 1: Phase structures near the triple point for the six dimensional
charged GB-AdS black holes with $Q$=0.2 and $\alpha$=1.06 in the $P$-$T$
diagram. “SBH”, “IBH”, and “LBH” are for the small, intermediate, and large
black holes.

Triple point in black hole systems In order to exhibit the rich structure of
the triple point, we start with the six-dimensional charged Gauss-Bonnet (GB)
AdS black holes Wei2 ; Frassino:2014pha . The corresponding equation of state
in the extended phase space reads

$P=\frac{T}{r_{\rm h}}-\frac{3}{4\pi r_{\rm h}^{2}}+\frac{2\alpha T}{r_{\rm
h}^{3}}-\frac{\alpha}{4\pi r_{\rm h}^{4}}+\frac{Q^{2}}{8\pi r_{\rm h}^{8}},$
(1)

where $P$ and $T$ is the pressure and temperature of the black hole system.
All quantities are measured in Planck units, with $\alpha$ the GB coupling
parameter, $Q$ denoting the black hole charge, and $r_{\rm h}$ the horizon
radius, characterizing the size of the black hole as either small,
intermediate, or large Kubiznak:2016qmn ; Boulware ; Cai2 ; Cvetic2 ; Cai .

This equation of state possesses a triple point, shown as a blue dot in Fig. 1
in the $P$-$T$ phase diagram, where $Q$=0.2 and $\alpha$=1.06. The other two
black dots signify critical points, where the phase transition is second
order. The diagram is quite similar to that of the phase structure of water,
except that coexistence curve of small and intermediate black holes has a
positive slope. The degenerate coexistence regions in the $P$-$T$ diagram are
better illustrated in a $T$-$r_{\rm h}$ phase diagram, shown in Fig. 2 for the
first time. The two critical points are located at the two peaks, and the
triple point becomes a horizontal line, along which these three black hole
coexistence phases can coexist. Correspondingly, these phases have the same
free energy, with the Gibbs free energy exhibiting double swallow tail
behaviour with both swallowtail intersections coinciding Wei2 ;
Frassino:2014pha . It is possible to construct two pair equal area regions on
each isothermal curve, but we will not pursue this as we are interested in the
dynamic process of the phase transition.

Figure 2: Phase structures near the triple point in the $T$-$r_{\rm h}$ space,
with $Q=0.2$ and $\alpha=1.06$.

Consider a canonical ensemble at temperature $T_{E}$ (and not the Hawking
temperature), composed of a series of black hole spacetimes (called the
landscape) with arbitrary horizon radius, and define a new Gibbs free energy

$\displaystyle G_{L}=H-T_{\rm E}S$ (2) $\displaystyle=\frac{2\pi r^{5}_{\rm
h}}{15}\left(4\pi P-\frac{5\pi T_{\rm E}}{r_{\rm h}}+\frac{5}{r_{\rm
h}^{2}}-\frac{20\pi\alpha T_{\rm E}}{r_{\rm h}^{3}}+\frac{5\alpha}{r_{\rm
h}^{4}}\right)+\frac{\pi Q^{2}}{9r_{\rm h}^{3}}$

where $H$ and $S$ are the respective enthalpy and entropy on the landscape Li
. This generalized off-shell free energy describes transient black hole
states, with $r_{\rm h}$ the order parameter, describing the underlying
microscopic degrees of freedom LiWang .

We illustrate $G_{L}$ at the triple point in Fig. 3 for an ensemble
temperature $T_{\rm E}$=0.108 with $Q$=0.2, $P$=0.018164, $\alpha$=1.080978,
$r_{\rm hs}$=0.667162, $r_{\rm hi}$=1.313823, $r_{\rm hl}$=1.884450, $r_{\rm
m1}$=0.905109, and $r_{\rm m2}$=1.626470. Each point on this curve denotes a
black hole state. However, not all of them are actual black hole solutions of
the Einstein GB equations. Only the local extrema correspond to actual black
holes, while the others are off-shell transient states Weiwa . The local
minima and maxima denote thermodynamic stable or unstable black holes.
Combining the fact that a stable thermodynamic system has the lowest free
energy, the black hole system will migrate to (or remain at) the well of the
deepest depth. For the parameter choice employed in Fig. 3, the three wells
have the same depth, indicating a triple point where the black hole system can
be in any arbitrary combination of these three states. More interestingly, the
two peaks have different heights, which govern the transit rate between the
various black hole states LiWang . Slightly changing the pressure or
temperature, we can obtain another triple point, with the general structure of
Fig. 3 holding, with the height of the peaks changing accordingly.

Figure 3: Behavior of the Gibbs free energy via landscape at the triple point
with $Q$=0.2, $T_{\rm E}$=0.108, and $P$=0.018164.

Dynamic processes at the triple point  Based on the behavior of the Gibbs free
energy $G_{L}$, we examine the dynamic properties of black hole phase
transitions at the triple point. Given an initial state in the black hole
system (e.g. a small black hole), at the triple point it will evolve via a
diffusion process to other states since it can undergo thermodynamic phase
transitions. We denote $\rho(r_{\rm h},t)$ as the probability distribution of
the system staying at a given black hole state. The SE Zwanzig

$\displaystyle\frac{\partial\rho(r,t)}{\partial t}=D\frac{\partial}{\partial
r}\left(e^{-\frac{\beta G_{\rm L}(r)}{k_{\rm B}T_{\rm
E}}}\frac{\partial}{\partial r}\left(e^{\frac{\beta G_{\rm L}(r)}{k_{\rm
B}T_{\rm E}}}\rho(r,t)\right)\right)$ (3)

governs this behaviour, where we have replaced $r_{\rm h}$ with $r$ for
simplicity. The diffusion coefficient $D=k_{\rm B}T_{\rm E}/\zeta$ with
$k_{\rm B}$ and $\zeta$ being the Boltzman constant and dissipation
coefficient. Without loss of generality, we set $k_{\rm B}$=$\zeta$=1 in the
following, and choose the initial state to be a Gaussian wave packet

$\displaystyle\rho(r,0)=\frac{1}{0.005\sqrt{\pi}}e^{-\frac{(r-r_{\rm\textsf{j}})^{2}}{0.005^{2}}}$
(4)

located at $r_{\rm\textsf{j}}$. Setting $r_{\rm\textsf{j}}$=$r_{\rm hs}$,
$r_{\rm hi}$, or $r_{\rm hl}$ respectively means the initial state is peaked
at a coexistent small, intermediate, or large black hole state. By imposing
reflective boundary conditions at $r=0$ and $r=\infty$ (sufficiently large
distance), we numerically solve the SE equation for each case, illustrating
the results in Fig. 4. Regardless of the choice of initial state, we observe
that the probability $\rho(r,t)$ leaks to other states, implying that there
are indeed phase transitions between these three coexistent states. This takes
place very quickly: within $t=80$ the system attains its final stationary
state, with the local maxima of $\rho(r,t)$ at the location of these three
coexistent black holes. More detailed study indicates that these maxima share
the same value $\rho(r_{\rm\textsf{k}},t)$=0.4801, for $\textsf{k}=$ hs, hi,
and hl, indicating that the black hole system settles into a combination of
these coexistent phases. Moreover, this specific value is independent of the
initial state, which can be understood from the SE. After a sufficiently long
time the system approaches stationarity; $\rho(r,t)$ will no longer change,
and so the left side of (3) vanishes. The probability distribution is then
only determined by $G_{L}$ in (2).

This dynamic phase transition process at the triple point is quite different
from that of a VdW type phase transition LiWang ; LiWang2 ; Weiwa . We now
examine the detailed evolution of each coexistent black hole phases, plotting
in Fig. 5 the behaviour of $\rho(r_{\rm\textsf{k}},t)$ for $\textsf{k}=$hs,
hi, and hl, when the initial Gaussian wave packet is respectively peaked at
these states. We see that the initially large value
$\rho(r_{\rm\textsf{k}},t)$ in each case rapidly decays to a stationary value,
whilst the other two states (initially zero) grow toward this value.

Considering first Fig. 5, the initial Gaussian wave packet $\rho(r,0)$ is
peaked at the small black hole phase with $\rho(r_{\rm hs},0)$ large, while
$\rho(r_{\rm hi},0)$ and $\rho(r_{\rm hl},0)$ are negligibly small. As $t$
increases $\rho(r_{\rm hs},t)$ decreases, and $\rho(r_{\rm hi},t)$ and
$\rho(r_{\rm hl},t)$ increase as expected, the latter more slowly since the
initial state must surmount two barriers. The transition rate from the small
to intermediate black hole state is higher than the total rate from
intermediate to small and large black hole states. Once $t\geq 40$, all of
them tend to 0.4801, where the final stationary state is achieved. During the
evolution, we observe an interesting phenomenon for $\rho(r_{\rm hi},t)$: it
increases to a maximum of 0.5338 at $t=3.6986$ and then decreases with time to
its stationary value of 0.4801. This presumably takes place because the
intermediate black hole state can transit to both small and large black hole
states, the latter much more easily due to the smaller barrier between the
intermediate and large black hole states, shown in Fig. 3. As the large black
hole state becomes more populated, it will transit back to the intermediate
one, reducing the rate of the latter. Since $\rho(r_{\rm hi},t)<\rho(r_{\rm
hs},t)$, we refer to this novel behaviour as a weak oscillatory phenomenon.

Figure 4: Probability distribution $\rho(r,t)$ governed by the SE equation
with $Q$=0.2, $T_{\rm E}$=0.108, and $P$=0.018164. The initial Gaussian wave
packet, respectively, located at (a) small black hole state, (b) intermediate
black hole state, and (c) large black hole state.

Figure 5: Behaviors of the probability $\rho(r,t)$ at the coexistence small,
intermediate, and large black hole states, when the initial Gaussian wave
packet is peaked at the coexistent (a) small (b) intermediate and (c) large
black hole states.

Corresponding behaviour is also observed when the initial wave packet is
peaked at the intermediate black hole state, shown in Fig. 5. Here more fine
structure is present. As expected, $\rho(r_{\rm hi},t)$ decreases whereas
$\rho(r_{\rm hs},t)$ and $\rho(r_{\rm hl},t)$ both increase, with the former
reaching a maximum of 0.5337 at $t=3.6987$. At early times, these probability
distributions exhibit interesting behaviour. When $0<t<1.3840$, we find that
the probability leaks from the intermediate to both small and large black hole
states, with $\rho(r_{\rm hl},t)$ increasing faster than $\rho(r_{\rm hs},t)$
due to the lower barrier height. This pattern reverses for $t>1.3840$. Once
$t>3.2995$, $\rho(r_{\rm hs},t)$ is dominant among the three probabilities,
indicating that the system has a large probability to stay at the coexistent
small black hole state. For a short time afterward $\rho(r_{\rm hs},t)$
continues to grow to its maximum, after which it decays to stationarity,
transiting back to the coexistent intermediate and large black hole states.
Since $\rho(r_{\rm hs},t)$ eventually dominates we refer to this as a strong
oscillatory phenomenon. In Fig. 5 we see that for all $t$, $\rho(r_{\rm
hl},t)>\rho(r_{\rm hi},t)>\rho(r_{\rm hs},t)$. However we also observe
additional (tiny) oscillatory behaviour (shown in the insets in Fig. 5 and 5)
when the initial wavepacket is initially peaked at the coexistent intermediate
or large black hole phases.

First passage event  We seek here more clues about this oscillatory phenomena
from the distribution of first passage times of the phase transition. This is
defined as the rate at which a given initial state first reaches an unstable
black hole phase, represented by the peak of the free energy LiWang . From
Fig. 3 we have three cases: small to intermediate (case I), intermediate to
small and large (case II), and large to intermediate (case III). For all cases
we impose reflective boundary condition at $r=0$ and $r=\infty$ (sufficiently
large $r$) and absorbing boundary conditions at the peaks $r_{\rm m1}$ and
$r_{\rm m2}$, accordingly, where the latter model a given coexistent state
first leaving the system. Note that such conditions do not preserve
normalization of the probability distribution. Using the SE, we can express
the first passage rate $F_{P}(t)$ for these three cases as

$\displaystyle F_{P1}(t)$ $\displaystyle=$
$\displaystyle-\frac{\partial\rho(r_{\rm m1},t)}{\partial r},$ (5)
$\displaystyle F_{P2}(t)$ $\displaystyle=$
$\displaystyle\frac{\partial\rho(r_{\rm m1},t)}{\partial
r}-\frac{\partial\rho(r_{\rm m2},t)}{\partial r},$ (6) $\displaystyle
F_{P3}(t)$ $\displaystyle=$ $\displaystyle\frac{\partial\rho(r_{\rm
m2},t)}{\partial r},$ (7)

where $F_{P2}(t)\neq-\left(F_{P1}(t)+F_{P3}(t)\right)$ since each case has
different boundary conditions.

The numerical results are given in Fig. 6. For each case, there is a single
peak, respectively at $t_{1}$=0.0876, $t_{2}$=2.3725, and $t_{3}$=0.1026,
which can be interpreted as the length of time at which the system remains in
its initial state for each case before first transiting to another state. For
cases I and III the system first transits to other states after a short time,
whereas for case II the system remains in the intermediate black hole state
somewhat longer. This expectation is borne out from the mean first passage
time

$\langle t\rangle=\int_{0}^{\infty}tF_{P}dt,$ (8)

for each case, where we find $\langle t_{1}\rangle$=0.86, $\langle
t_{2}\rangle$=2.37, $\langle t_{3}\rangle$=1.75 (where $0<t<40$).

For case I, the initial state is at the coexistent small black hole state. As
time increases, its probability $\rho(r_{\rm hs},t)$ decreases and leaks to
the intermediate black hole state. Since $\langle t_{2}\rangle$ is larger than
$\langle t_{1}\rangle$ and $\langle t_{3}\rangle$, the system will stay longer
at the intermediate black hole state. Then it will transit back to the
coexistent small or large black hole states. Consequently $\rho(r_{\rm hi},t)$
first increases then decreases with time, and thus weak oscillatory behavior
is present in Fig. 5.

We find that $F_{P}(t)$ is governed not only by the barrier height, as
discussed previously Li ; LiWang , but also by the barrier width. An
inspection of Fig. 3 indicates that the difference in barrier heights $\Delta
G_{L}=G_{m1}-G_{m2}\simeq 0.001$, whereas the difference in barrier widths is
$\Delta r_{m1}$-$\Delta r_{m2}\simeq$0.02, 20 times larger than $\Delta
G_{L}$. These larger barrier width values will dominate over the small barrier
height difference, with smaller values of $\Delta r$ leading to a peak in
$F_{P}(t)$ at small $t$. Since $\Delta r_{m1}=0.2379$ (case I) is slightly
smaller than $\Delta r_{m2}=0.2580$ (case III), we have $t_{1}<t_{3}$. For
case II absorbing boundary conditions are imposed at both sides, and so the
intermediate black hole can transit to large and small black holes through the
right and left boundaries respectively. Since the right barrier is lower than
the left one and closer to the well, the peak of the second part of
$F_{P2}(t)$ corresponds to a small time, exactly the situation shown in Fig.
6.

In order to understand the strong oscillatory behavior shown in Fig. 5, we,
respectively, plot the two parts of $F_{P2}(t)$ with red solid and blue dashed
curves in Fig. 6. The first and second parts denote the transitions to the
small and large black hole states. Their peaks are at $t_{i1}$=0.2568 and
$t_{i2}$=0.1507, and we find that the lower barrier corresponds to a higher
peak at shorter time. However, we find from (8) that $\langle
t_{i1}\rangle=0.28<\langle t_{i2}\rangle=0.31$. Hence $\rho(r_{\rm hs},t)$
will at some time become larger than $\rho(r_{\rm hl},t)$. Thus strong
oscillatory behavior is exhibited when the initial state is peaked at the
coexistent intermediate black hole state. The fact that $t_{i1}>t_{i2}$
indicates that $\rho(r_{\rm hl},t)$ increases faster than $\rho(r_{\rm hs},t)$
at early transition times.

Moreover, apart from the coexistence with the small black hole state, we find
that the Gibbs free energy $G_{\rm L}$ is approximately symmetric about
$r_{\rm m2}$. Hence $\rho(r_{\rm hi},t)$ and $\rho(r_{\rm hl},t)$ exhibit
similar behavior when the initial wave packet is located at coexisting
intermediate or large black hole states. From Figs. 5 and 5, we observe
$\rho(r_{\rm hi},t)$ and $\rho(r_{\rm hl},t)$ share approximately the similar
tiny oscillatory behavior with time, as shown in the insets. The presence of
these tiny oscillations is due to coexistence with the small black hole state.

Figure 6: (a) First passage time rate for case I, II, and III. (b) The first
and second parts of the first passage time for $F_{P2}(t)$.

Summary We have carried out the first investigation of dynamical phase
behaviour at a black hole triple point. The degenerate triple point in a
$P$-$T$ phase diagram becomes a horizontal line in the $T$-$r_{\rm h}$ phase
diagram, with each coexistence region explicitly displayed. On the free energy
landscape, with $r_{\rm h}$ as the order parameter, the characteristic pattern
of the triple point is that of three potential wells of the same depth, with
the coexistent black hole phases located at the minima of the wells.

Transitions between these different phases are quite different from that of
first order VdW type phase transitions, and can be studied dynamically via the
Smoluchowski equation. We find that, regardless of the initial phase in the
black hole system, each phase can transit into the others, attaining
stationarity of all three phases after a short time. More significantly, we
observe both weak oscillatory behaviour, in which the probability distribution
of one state attains a maximal value before decaying to stationarity, and
strong oscillatory behaviour, in which this maximum dominates over the other
two states. A study of the mean first passage time indicates that if the
system is initially in the intermediate black hole state, it takes longer to
transit to one of the other states than for the other two cases.

Our results indicate that black holes have interesting dynamical phase
behaviour whose relationship to their underlying degrees of freedom remains to
be understood. It would be interesting to apply our approach to other black
hole systems with various types of phase transitions, including angular
momenta Altamirano:2014tva , hair Giribet ; Hennigar ; Dykaar:2017mba , and
acceleration Anabalon:2018ydc ; Anabalon:2018qfv , where more interesting
dynamic phenomena should be observed.

_Acknowledgements_.—This work was supported by the National Natural Science
Foundation of China (Grants No. 12075103, No. 11675064, No. 11875151, and No.
12047501) and the Natural Sciences and Engineering Research Council of Canada.

## References

  * (1) S. W. Hawking, Commun. Math. Phys. 43, 199 (1975).
  * (2) J. D. Bekenstein, Phys. Rev. D 7, 2333 (1973).
  * (3) J. M. Bardeen, B. Carter, and S. Hawking, Commun. Math. Phys.31, 161 (1973).
  * (4) S. W. Hawking and D. N. Page, Commun. Math. Phys. 87, 577 (1983)
  * (5) D. Kubiznak, R. B. Mann and M. Teo, Class. Quant. Grav. 34, no.6, 063001 (2017), [arXiv:1608.06147 [hep-th]].
  * (6) D. Kastor, S. Ray, and J. Traschen, Class. Quant. Grav. 26, 195011 (2009), [arXiv:0904.2765 [hep-th]].
  * (7) D. Kubiznak and R. B. Mann, J. High Energy Phys. 1207, 033 (2012), [arXiv:1205.0559 [hep-th]].
  * (8) R. Li, and J. Wang, Phys. Rev. D 102, 024085 (2020).
  * (9) R. Li, K. Zhang, and J. Wang, J. High Energy Phys. 2010, 090 (2020), [arXiv:2008.00495 [hep-th]].
  * (10) R. Li and J. Wang, [arXiv:2012.05424 [gr-qc]].
  * (11) S.-W. Wei, Y.-X. Liu, and Y.-Q. Wang, [arXiv:2009.05215 [gr-qc]].
  * (12) H. Mouri and Y. Taniguchi, Astrophys. J. Lett. 566, L17 (2002), [astro-ph/0201102 [astro-ph]].
  * (13) A. L. Erickcek, M. Kamionkowski, and A. J. Benson, Mon. Not. Roy. Astron. Soc. 371, 1992 (2006), [astro-ph/0604281 [astro-ph]].
  * (14) N. Altamirano, D. Kubiznak, R. B. Mann, and Z. Sherkatghanad, Class. Quant. Grav. 31,042001 (2014), [arXiv:1308.2672 [hep-th]].
  * (15) S.-W. Wei and Y.-X. Liu, Phys. Rev. D 90, 044057 (2014), [arXiv:1402.2837[hep-th]].
  * (16) A. M. Frassino, D. Kubiznak, R. B. Mann and F. Simovic, J. High Energy Phys. 09, 080 (2014), [arXiv:1406.7015 [hep-th]].
  * (17) D. G. Boulware and S. Deser, Phys. Rev. Lett. 55, 2656 (1985).
  * (18) R. G. Cai, Phys. Rev. D 65, 084014 (2002), [arXiv:hep-th/0109133].
  * (19) M. Cvetic, S. Nojiri, and S. D. Odintsov, Nucl. Phys. B 628, 295 (2002), [arXiv:hep-th/0112045].
  * (20) R.-G. Cai, L.-M. Cao, L. Li, and R.-Q. Yang, J. High Energy Phys. 1309, 005 (2013), [arXiv:1306.6233[gr-qc]].
  * (21) R. Zwanzig, Nonequilibrium Statistical Mechanics, Oxford University Press (2001).
  * (22) N. Altamirano, D. Kubiznak, R. B. Mann and Z. Sherkatghanad, Galaxies 2, 89 (2014), [arXiv:1401.2586 [hep-th]].
  * (23) G. Giribet, M. Leoni, J. Oliva, and S. Ray, Phys. Rev. D 89, 085040 (2014), [arXiv:1401.4987 [hep-th]].
  * (24) R. A. Hennigar, E. Tjoa, and R. B. Mann, J. High Energ. Phys. 1702, 070 (2017), [arXiv:1612.06852 [hep-th]].
  * (25) H. Dykaar, R. A. Hennigar and R. B. Mann, J. High Energy Phys. 1705, 045 (2017), [arXiv:1703.01633 [hep-th]].
  * (26) A. Anabalon, M. Appels, R. Gregory, D. Kubiznak, R. B. Mann, and A. Ovgun, Phys. Rev. D 98, 104038 (2018), [arXiv:1805.02687 [hep-th]]
  * (27) A. Anabalon, F. Gray, R. Gregory, D. Kubiznak, and R. B. Mann, J. High Energy Phys. 1904, 096 (2019), [arXiv:1811.04936 [hep-th]].