aainstitutetext: ETH Zurich, Zurichbbinstitutetext: University of Zurich,
Zurich

# Bosonic und Fermionic Quasinormal Modes of Rotating Black Holes Through
AdS/CFT

Jayant Rao b Shubhanshu Tiwari<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We study the holographic duals of bosonic and fermionic fields and compute
their Quasinormal Frequencies (QNF) by exploiting that they are massive
charged fields in $\mathrm{AdS_{2}\times\mathcal{K}}$. In particular, we
assess the stability of the scalar and fermionic operators in Myers-Perry’s
black holes with equal angular momenta. We compute the central charge and the
Hawking-entropy and introduce extremality parameters to determine the
properties of the dual CFT. Then, we briefly discuss how detection of Spin-2
fields can pave the way for more deep understanding of the holographic
principles. We end with a brief discussion of the implications onto Strong
Cosmic Censorship.

##  1 Introduction

Black hole solutions have been of most interest in the context of general
relativity and specifically also on the road to understanding how a possible
theory of quantum gravity may look like. The investigation of Quasinormal
Modes (QNM) has been of particular interest for a long time because it reveals
insights on black hole formation, ringdown as well as possible hints on
quantum effects Berti et al. (2009). Close to the horizon it can be shown that
scalar fields in near extremal black holes behave similar to charged scalar
fields subject to a $U(1)$ gauge field on $AdS_{2}$ Kunduri et al. (2007a).
Thus, it interesting to see what happens in the context of the holographic
duality principle to a charged scalar field close to the horizon.
One of the most active research topics in theoretical physics has been the
Holographic Duality hypothesis Maldacena (1999) which is best understood now
in the form of the AdS/CFT correspondence. Since it is of interest to
investigate higher-dimensional generalisations of black holes, it has become
very natural to consider the generalisations of the Kerr-Solution in higher
dimensions which is known as the Myers-Perry black hole. In order to exploit
some symmetry effects (but still retaining the main physical information) it
has also become advantageous to consider the 1-cohomogeneity solution with
equal angular momenta.
Using methods from the aforementioned holographic duality, it has become
particularly advantageous to consider the linear response formalism within
this theory and then determine the Quasinormal Frequencies which in turn of
course correspond to excited states of dual bulk operators. By separating the
variables of the different equations of motion, we identify the radial
asymptotic behaviour as well as the radial angular eigenvalues. Since we are
interested in well-defined dual correlators, we can make a well-defined ansatz
for the bulk fields and then use the retarded Green’s function in order to
find the quasinormal modes with the correct boundary conditions. We can then
understand using the conformal weights and from there the stability conditions
how the conformal Virasoro algebra must look like. We investigate scalars,
fermions and tensors in order to understand the stability of such fields and
if we can use the holographic dictionary to also make statements on the
classical predictions of General Relativity.
Finally, we also investigate tensorial modes close to the horizon and want to
understand how such (now detectable) higher spin perturbations look close to
the horizon and close to extremality. Specifically, it is interesting to see
how the concept of overtone modes comes to play with holography and how we can
understand more about the dual CFT from that in terms of the ringdown process.

##  2 Bosonic Fields

In this section, we aim to explain how to use the conjectured holographic
duality between $AdS_{2}$ and a boundary conformal field theory (CFT) in order
to find the Quasinormal frequencies (QNFs) of bosonic fields in the near-
horizon region (NH) for rotating black holes close to extremality. We then
apply our theory to explicitly determine the QNFs for specific solutions of
interest to us, foremost the 1-cohomogeneity solution to the Myers-Perry’s
black hole and secondly the Kerr solution. We are specifically interested in
understanding how the NH geometry can be understood in terms of a charged
scalar field in the near-horizon case. All of this is only valid inside the
near-extremal approximation of the black hole.

###  2.1 General Case

####  2.1.1 Preparations

Our studies are fundamentally motivated by the results obtained by H.Reall and
H.Kunduri in which they prove that the NH geometry of a black hole at
extremality can be understood to be locally isometric to
$AdS_{2}\times\mathcal{K}$. Here $\mathcal{K}$ refers to a compact topological
manifold. In explicit calculations, we one can directly see how this compact
space looks in detail, but we will in general use it to introduce a Kaluza-
Klein decomposition of the scalar field.
The general isometry to $AdS_{2}\times\mathcal{K}$ allows to write the metric
in form of a fibration of an $AdS_{2}$ metric, such that (in the case of a
rotating solution), we can write the most general form of the near-horizon
metric as Kunduri et al. (2007b):

$\displaystyle
ds^{2}=L(y)^{2}\left(-R^{2}dT^{2}+\frac{1}{R^{2}}dR^{2}\right)+$
$\displaystyle
g_{IJ}(y)(d\phi^{I}-k^{I}RdT)(d\phi^{J}-k^{J}RdT)+g_{AB}(y)dy^{A}dy^{B}.\qquad$
(1)

Here, $d\phi^{I}$ denotes the rotational Killing direction and $g_{AB}$
denotes the metric on the space $\mathcal{K}$. To remain in the most general
setting as possible we allow a mixing (i.e a non-trivial dependence) of the
$AdS_{2}$ metric and $\mathcal{K}$. The necessity for this will become obvious
in the case of the Kerr-solution.
In the first step, we introduce a massive scalar field $\Phi$ in this
background which obeys the Klein-Gordon equation. We decompose this into
Kaluza-Klein towers by making the necessary conditions for the space
$\mathcal{K}$. It is evident that by our symmetry assumptions it is reasonable
to factor $\Phi=\Xi(R,T)\mathcal{Y}(y^{A},\phi^{I})$.
Now, in the second step, we can write $\mathcal{Y}=e^{im^{I}\phi^{I}}Y(y^{A})$
and $\Xi=\chi(R)e^{i\omega t}$. The Klein-Gordon equation takes the form
$\nabla_{a}\nabla^{a}\Phi-\mu^{2}\Phi=0$. By the aforementioned decomposition,
we can now proceed to separate the equations in question. We use furthermore,
that the general covariant derivative in this space will lead us to the
equation for the angular part after separation:

$\nabla_{b}\nabla^{b}(L(y)^{2}\mathcal{Y})-\mu^{2}L^{2}(y)\mathcal{Y}-q^{2}L^{2}(y)\mathcal{Y}=\lambda\mathcal{Y}.$
(2)

Here $\lambda$ is the separation constant between the radial and angular part.
The constant $q$ which we will henceforth call charge $q$, can be understood
in terms of the mixing between the space $\mathcal{K}$ and the space $AdS_{2}$
through a charged field which is implemented as a vector potential $A=-RdT$
and a charge $q=m_{I}k^{I}$. When we expand 3.1.1 we are left with the
covariant derivative on $AdS_{2}$ as $\nabla_{2}$ and terms of the form
$(\nabla_{2}-iqA)\Phi$.
The third step is now about finding the radial equation. This is
straightforward due to the preparation we have made thus far. We arrive at

$-\frac{1}{R^{2}}\frac{\partial^{2}\Xi}{\partial
T^{2}}+\frac{\partial^{2}R\frac{\partial\Xi}{\partial R}}{\partial
R}-\frac{2iq}{R}\frac{\partial\Xi}{\partial T}+(q^{2}-\lambda-\mu^{2})\Xi=0.$
(3)

One example to show how general our approach is the Proca equation in this
background to illustrate the universality of our approach. They stem from a
Lagrangian density $\mathcal{L}$:

$\displaystyle\mathcal{L}=\frac{F_{ab}F^{ab}}{4}+\frac{\mu^{2}}{2}A^{a}A_{a}$
(4)

This gives rise to the Proca equation:

$\displaystyle\nabla_{a}F^{ab}=\mu^{2}A^{a}$ (5)

where $A=dF$. Although gauge invariance is not given, one can quickly see that
we have a Bianchi type $\nabla^{a}A_{a}=0$. Now let $l^{\mu}$ be any null
frame, then we make the ansatz Kunduri et al. (2007a) $A^{a}=\varphi_{1}Z^{a}$
where $Z^{a}$ are vector harmonics on $\mathcal{K}$. The FKKS-Method Frolov et
al. (2018) showed the separability of the Proca field on Kerr-AdS. If we write
$A_{i}=F_{ab}l^{a}m^{b}_{i}$ where $l^{a}$ is on $\mathrm{AdS_{2}}$ and
$m^{b}_{i}$ on $\mathcal{K}$, then we can get the equation:

$\left[\left(\nabla_{2}-iqA\right)^{2}+1-\lambda_{1}\right]\varphi_{1}=0$ (6)

By finding $\lambda_{1}$ on $\mathcal{K}$, we can find the effective mass of
the Proca field in this geometry.

####  2.1.2 Holographic Solution

We now aim to solve this problem with an approach using the $AdS/CFT$ duality.
We generically propose that that a one-dimensional conformal field theory is
exactly dual to the scalar field. This dual field lives on the boundary as
proposed by J.Maldacena. In order to do so, we aim to exploit the fact that
generically the setup leads to a charged $AdS_{2}$ space.
We wish to rewrite this in terms of more natural $AdS_{2}$ coordinates. For
this, we introduce $\zeta=\frac{1}{R}$. This transforms (3) into (after
inserting $\Xi=\chi(R)e^{i\omega t}$):

$\zeta^{2}\frac{\partial^{2}\chi}{\partial\zeta^{2}}-L^{2}(y)(q^{2}-\lambda-\mu^{2})\chi+2q\omega\zeta\chi+\zeta^{2}\omega^{2}\chi=0$
(7)

This can be explicitly resolved in terms of Whittaker functions
$W_{\kappa,\gamma},M_{\kappa,\gamma}$. In order to understand how the dual CFT
works, we need to compute the conformal scaling dimension at the boundary. For
this, we need to find the asymptotic behaviour at the boundary. This is at
$z\rightarrow 0$. There, we find for the Whittaker functions the result that,
after asymptotically expanding:

$\chi\sim z^{-\Delta_{\pm}}$ (8)

where $\Delta_{\pm}=\frac{1}{2}\pm\sqrt{q^{2}-\mu^{2}-\lambda+\frac{1}{4}}$ is
the conformal scaling weight close to the boundary.

We introduce close to the horizon, a parameter $\eta=1-\frac{r_{+}}{r_{-}}$
which measures how close to extremality, i.e ($r_{+}=r_{-}$) we are. We will
later explain why two horiozons suffice, but in general this is because of the
choice of de-Sitter space in our case. The connection to the extremal limit in
the $\mathrm{AdS_{2}/CFT_{1}}$ dual is made via the Hawking-temperature
$T_{H}$ close to extremality, which is zero at extremality. Thus, via the
surface gravity $\kappa$, we can write
$T_{H}=\frac{1}{2\pi}\frac{d\kappa}{d\eta}\eta$.
Finally, the quasinormal frequencies can be found canonically in a charged
$AdS_{2}$ as the poles of the retarded Green’s function in charged
$\mathrm{AdS/CFT}$ (see Faulkner et al. (2011)):

$\mathcal{G}_{R}=(4\pi
T)^{2\Delta}\frac{\Gamma(2\Delta)\Gamma(\frac{1}{2}+\Delta-\frac{i\omega}{2\pi
T}+iq_{\mathrm{AdS}}e_{d})\Gamma(1+\Delta-
iqe_{d})}{\Gamma(-2\Delta)\Gamma(\frac{1}{2}-\Delta-\frac{i\omega}{2\pi
T}+iq_{\mathrm{AdS}}e_{d})\Gamma(1-\Delta-iqe_{d})}.$ (9)

The poles are located within ($e_{d}=L^{2}(y)$):

$\displaystyle\delta\omega=2\pi
Ti(n-\frac{1}{2}+iq_{\mathrm{AdS_{2}}}L^{2}-\Delta)$ (10) $\displaystyle=2\pi
T\left(in-\frac{1}{2}+iq_{\mathrm{AdS_{2}}}L^{2}-\left(\frac{1}{2}\pm\sqrt{q^{2}-\mu^{2}-\lambda+\frac{1}{4}}\right)\right).$
(11)

At extremality, depending on the solution, we might have to also account for
superradiance. This will be discussed in the Myers-Perry’s solution.

###  2.2 1-Cohomogeneity Myers-Perry’s Solution

####  2.2.1 Solution in de-Sitter space

We are in the following going to apply what we learned about the holographic
approach to the Myers-Perry solution which exhibits an enhanced $U(n)$
symmetry (where $n$ is the number rotation axes) due to assuming equal angular
momenta $a_{i}=a$. In Boyer-Lindquist coordinates ($r,t,\phi,\theta$) this
looks like:

$ds^{2}=\frac{-f(r)}{h(r)}dt^{2}+\frac{1}{f(r)}dr^{2}+r^{2}h(r)(d\psi+\mathcal{A}-\Omega(r)dt)^{2}+r^{2}d\Sigma^{2}.$
(12)

Here the sphere $S^{N}$ is expressed as a fibration over $\mathbb{CP}^{N}$ and
the helping functions
$f(r)=1-\frac{r^{2}}{L^{2}}-\frac{2M}{r^{2N}}\left(1+\frac{a^{2}}{L^{2}}\right)+\frac{2Ma^{2}}{r^{2N+2}}$
and $h(r)=1+\frac{2Ma^{2}}{r^{2N+2}}$ Davey et al. (2022). This is a very
common choice of coordinates for the Myers-Perry’s solution Cardoso and Lemos
(2001).
By Descartes rule of signs, the function $f(r)$ admits either 1 or 3 roots.
Since a solution in de-Sitter space must admit at least 2 roots, $f(r)$ has 3
roots. The roots of the function $f(r)$ determine the locations of the
horizons, which we name $r_{+},r_{-},r_{c}$. In the following, we are
interested in the extremal case, namely when $r_{+}$ and $r_{-}$ coincide.
This is equivalent to the case for which the Hawking Temperature of the black
hole $T_{H}$ vanishes.
The NH metric around the extremal case $r_{+}=r_{-}$ can be re-expressed in
terms of

$\begin{split}r\rightarrow r_{+}+\epsilon R,t\rightarrow\frac{T}{\epsilon},\\\
\psi\rightarrow\Psi+\frac{\Omega(r_{+})T}{\epsilon}\end{split}$ (13)

where $\epsilon$ is an infinitesimal parameter. We can expand (12) in terms of
(13) and get the NH-expansion when dropping terms to first order in $\epsilon$
and dropping all terms proportional to $\epsilon$:

$ds^{2}=-\frac{f^{\prime\prime}(r_{+})}{2h(r_{+})}R^{2}dT^{2}+\frac{2}{f^{\prime\prime}(r_{+})}\frac{dR^{2}}{R^{2}}+r_{+}^{2}h(r_{+})\left(d\Psi+\mathcal{A}-R\Omega^{\prime}(r_{+})dT\right)^{2}+r_{+}^{2}d\Sigma^{2}$
(14)

When naming $L^{2}=\frac{2}{f^{\prime\prime}(r_{+})}$ and rescaling
$T,\Omega$:

$\Omega\rightarrow\Omega(r_{+})\sqrt{h(r_{+})}\\\ T\rightarrow
L^{2}\sqrt{h(r_{+})}T,$ (15)

one can obtain the $AdS_{2}$ form of the (13) metric:

$ds^{2}=L^{2}\left(-R^{2}dT^{2}+\frac{dR^{2}}{R^{2}}\right)+r_{+}^{2}h(r_{+})\left(d\Psi+\mathcal{A}-R\Omega
L^{2}dT\right)^{2}+r_{+}^{2}d\Sigma^{2}.$ (16)

When comparing the general approach to this specific example of a black-hole
spacetime, we notice that we need to rewrite the $d\phi^{I}$ term as
$d\Psi+\mathcal{A}$. Also, the fibration of $AdS_{2}$ with constant $L$ is
truly a constant and independent of the coordinates on $\mathcal{K}$. The
metric of this NH region can be written as proposed above as
$\mathrm{AdS_{2}}\times\mathcal{K}$ where $\mathcal{K}$ is a squashed
$(d-2)$-dimensional sphere with metric:

$ds^{2}=(d\Psi+\mathcal{A})^{2}+r_{+}^{2}d\Sigma^{2}.$ (17)

With these preparations, we can find the radial equation of a scalar field
$\Phi$ in this background. It is straightforward that the Klein-Gordon
equation in charged $\mathrm{AdS_{2}}$ looks like:

$R^{2}\chi^{\prime\prime}(R)+2R\chi^{\prime}(R)-\left[\frac{(\omega-m\Omega
L^{2}R)^{2}}{R^{2}}-L^{2}\left(\mu^{2}+\frac{\lambda}{r_{+}^{2}}+\frac{m^{2}}{r_{+}^{2}h(r_{+})}\right)\right]\chi(R)=0.\qquad$
(18)

By comparing expressions, we note down that $q=m\Omega L^{2}$ and $A=-RdT$ as
in the previous section. By comparing the differential equation to the
previous section, just by inserting into
$\Delta_{\pm}=\frac{1}{2}\pm\sqrt{q^{2}-\tilde{\mu}^{2}-\tilde{\lambda}+\frac{1}{4}}$
the frequencies, one can find $\delta\omega$. By using
$\tilde{\lambda}=\frac{\lambda}{r_{+}^{2}},\tilde{\mu}^{2}=\mu^{2}+\frac{m^{2}}{r_{+}^{2}h(r_{+})}$,
we arrive at:

$\displaystyle\delta\omega^{NH}_{n,m}=\left(i\left[n+\frac{1}{2}+iq+\frac{1}{2}\sqrt{4\Omega^{2}L^{4}m^{2}-4L^{2}\left(\mu^{2}+\frac{\lambda}{r_{+}^{2}}+\frac{m^{2}}{r_{+}^{2}h(r_{+})}\right)-1}\right]\frac{d\kappa}{d\eta}\right)\eta\qquad\qquad$
(19)

Since extremal rotating solutions are sourced by superradiance,
$\omega^{(0)}<m\Omega_{H}$. Close to extremality, we can write the expression
as:

$\displaystyle\omega^{NH}_{n,m}=m\Omega_{H}+\left(m\frac{d\Omega}{d\sigma}+i\left[n+\frac{1}{2}+iq+\frac{1}{2}\sqrt{4\Omega^{2}L^{4}m^{2}-4L^{2}\left(\mu^{2}+\frac{\lambda}{r_{+}^{2}}+\frac{m^{2}}{r_{+}^{2}h(r_{+})}\right)-1}\right]\frac{d\kappa}{d\eta}\right)\eta\qquad$
(20)

This matches the result by Davey et al. (2022).

###  2.3 Kerr-Solution

The above discussion can be analogously applied to the Kerr solution of black
holes. In this section, we can finally explain the NH geometry of the Kerr
solution. In a paper by Dias et al. (2022), the NH region was closely examined
in the classical context of GR. The NH geometry found can be locally described
as a warped $AdS_{3}\times\mathbb{CP}^{n}$. Locally, the warped $AdS_{3}$ is
that of $AdS_{2}$ fibred over $S^{1}$. This yields the local
$SL(2,\mathbb{C})\times U(1)$ symmetry. This symmetry gives rise to left-
moving $SL(2,\mathbb{C})$ and right-moving $U(1)$ Virasoro generators.
In order to find the NH modes as before, we can use the Poincare Patch of the
$AdS_{2}$ geometry. This can be expressed as:

$ds^{2}=\left(\frac{1+cos^{2}(\theta)}{2}\right)\left(-(1+y^{2})d\tau^{2}+\frac{dy^{2}}{1+y^{2}}+d\theta^{2}\right)+\frac{2sin^{2}(\theta)}{1+cos^{2}(\theta)}(d\phi+yd\tau)^{2}$
(22)

If we use that a scalar massless mode can be separated as
$\Phi=\psi(r,\theta)e^{im\phi+i\omega t}$, then since the local isometry was
$SL(2,\mathbb{C})\times U(1)$ (and $SL(2,\mathbb{C})$ is the isometry group of
$AdS_{2}$ space), we can separate $\psi=\mathcal{Y}(y)\mathcal{T}(\theta)$.
This gives us then two separate ODEs for the $\theta$ and $y$ contribution
respectively. If we use slightly shifted coordinates that are no longer
globally valid on $AdS_{2}$ but admit a coordinate singularity, in this case
Lupsasca et al. (2014):

$ds^{2}=\left(\frac{1+cos^{2}(\theta)}{2}\right)\left(-(y^{2})d\tau^{2}+\frac{dy^{2}}{y^{2}}+d\theta^{2}\right)+\frac{2sin^{2}(\theta)}{1+cos^{2}(\theta)}(d\phi+yd\tau)^{2}$
(23)

The massless KG equation for $\Phi$ is $\nabla_{a}\nabla^{a}\Phi=0$. After
inserting the ansatz for $\psi$, the ODE for $\mathcal{Y}$ reads:

$y^{2}\mathcal{Y}^{\prime\prime}+2y\mathcal{Y}^{\prime}+\left(\frac{(\omega+my)^{2}}{y^{2}}+m^{2}\right)\mathcal{Y}=0$
(24)

The Whittaker function as earlier allows us to describe its asymptotics by
polynomials close to the horizon. This allows to read off the conformal weight
as:

$\Delta_{\pm}=-\frac{1}{2}\pm\sqrt{K-2m^{2}+\frac{1}{4}}$ (25)

By reinserting into 9 and using the $\Gamma(-n)=0$ prescription, we can
identify the NH modes as for the MPdS case as:

$\omega=m\Omega_{H}+\left(m\frac{d\Omega}{d\eta}+i\left(n+\frac{1}{2}+im+\sqrt{K-2m^{2}+\frac{1}{4}}\right)\right)\eta$
(26)

This matches our result and especially now only requires a discussion of the
parameter $K$. This is a solution of the differential equation Dias et al.
(2022):

$\mathcal{T}^{\prime\prime}+\cot(\theta)\mathcal{T}^{\prime}+\left(K-\frac{m^{2}}{\sin^{2}(\theta)}+\frac{m^{2}}{4}\sin^{2}(\theta)\right)\mathcal{T}=0.$
(27)

If $m=0$, $\mathcal{T}$ will is an azimutally symmetric spherical harmonic
$Y_{l0}$ and, thus, $K=l(l+1)$. If, however, $m\neq 0$, then the solutions to
27 need to be found numerically. $K$ is then the eigenvalue of the
corresponding differential operator.

We hence showed that we can systematically find the charge of the scalar field
in the NH region and then employ the retarded Green’s function method by Son
and Starinets (2002) in order to determine the QNM frequencies. In the
extremal limit, the AdS/CFT conjecture hence remains an important tool that
gives many results which align with the classical predictions.

###  2.4 The CFT Dual

Since we are using the bulk/boundary duality, the aforementioned calculation
provides us with insights about the dual field theory and its constituents.
The metric is that of a warped $AdS_{3}$ space, which we want to find dual to
a two-dimensional $\mathrm{CFT_{2}}$. We work with the local symmetries of the
NH-region that are as $SL(2,\mathbb{C})_{L}\times\left(U(1)\times
U(1)\right)_{R}$ which then give rise to leftmoving and rightmoving
generators. The resulting leftmoving Virasoro generators $L_{n}$ (for
$n\in\mathbb{Z}$) can be used to span the set of conformal generators in this
case. We know that the dual is governed by the dilaton operator
$D=L_{0}+\bar{L}_{0}$ (left- and rightmoving) with eigenvalue
$\Delta=h_{L}+h_{R}$ of the dilaton operator. Hence, we expect the CFT dual to
the bulk field to propagate with propagator:

$\displaystyle G(x,y)\sim\frac{1}{|x-y|^{2\Delta}}.$ (28)

The main parameters that will determine the properties of the CFT itself are
determined by the Bekenstein-Hawking entropy. In the case of the Myers-Perry’s
solution, this means:

$\displaystyle
S_{H}=\frac{k_{B}A}{4l_{P}^{2}}\sim\frac{\pi}{2}\int_{0}^{\pi}d\theta\sqrt{r_{+}^{4}h(r_{+})L^{2}}.$
(29)

We can use the Chow et al. (2009) approach to find the left-moving temperature
and construct the Virasoro generators via the Hawking Bekenstein entropy. We
note (which is a well-known fact) that the entropy of the black-hole is not
zero at extremality whereas the Hawking temperature is. We first work in the
near-horizon regime of the MPdS solution. We recall that we derived
$T_{H}=\frac{1}{2\pi}\frac{d\kappa}{d\eta}\nu$ where
$\eta=1-\frac{r_{+}}{r_{-}}$ is a parameter leading to the extremality limit.
If we introduce $T_{H}^{\prime}=\frac{\partial T_{H}}{\partial
r_{+}}\big{|}_{r_{+}=r_{-}}$, we can define with
$\Omega_{H}^{\prime}=\frac{\partial\Omega_{H}}{\partial
r_{+}}\big{|}_{r_{+}=r_{-}}$, the left-moving temperature as
$T_{L}=\frac{T^{\prime}_{H}}{\Omega^{\prime}_{H}}$. We can use Cardy’s formula
to postulate that $S=\frac{\pi^{2}}{3}c_{L}T_{L}$. In order to find the
central charge, we now need $T_{L}$. This can be found in two ways. The more
geometric approach is to consider what happens to the dual CFT when going from
the zero-temperature case to the finite temperature one is to map it to a
torus with radii $\frac{1}{T_{L/R}}$. From this, we can derive that
$T_{L}=\frac{1}{2\pi\Omega L^{2}}$ and $c$ can then be read off. The other
approach would be to just explicitly calculate the derivatives of
$T_{H},\Omega_{H}$ and take the respective extremal limit. Thus in the (NH)
metric, we can find:

$c_{L}=\frac{3S}{T_{L}}=6r_{+}^{2}L^{3}\Omega\pi^{2}.$ (30)

In general the Frolov-Thorne temperature Perry and Rodriguez (2022), can be
found as in the explanation before and leads in $N$ dimension to the result
and find. In the case of the Myers-Perry solution (not only in NH space), we
can derive the extremal central charge as:

$T_{L}=\frac{1}{2\pi}\frac{f^{\prime\prime}(r_{+})}{2r_{+}h(r_{+})}\frac{(r^{2N+1})^{2}}{(2N+2)\frac{h(r_{+})}{r_{+}}+h^{\prime}(r_{+})}=\frac{f^{\prime\prime}(r_{+})(r_{+}^{2N+2})}{(2N+1)2h(r_{+})}.$
(31)

Here, one needs to use that the surface gravity vanishes at extremality. This
is given by $\kappa=\frac{|f^{\prime}(r_{+})|}{2r_{+}h(r_{+})}=0$. This fixes
the central charge again via Cardy’s formula.
The local $SL(2,\mathbb{R})$ geometry gives rise to Virasoro generators that
generate the conformal symmetries of the space. The construction in this case
is going to be similar to the Kerr-Case considered in Hartman et al. (2009).
The construction follows a well-known construction in literature of the
Asymptotic Symmetry Group. The local generators are constructed as in the
Kerr/CFT Strominger (2001) and will be omitted here. We note that the left
moving generators $L_{n}$ satisfy a Virasoro algebra with central extension
$c_{L}$ whereas the right-moving $\mathfrak{l}_{n}$ do not have a central
extension term, because the charge $c_{R}=0$ at extremality. The calculation
is straightforward because $T_{R}=0$. Close to extremality, $c_{R}\neq 0$
because the temperature $T_{R}$ in only $0$ in the strict extremality limit.
One of the interesting insights provided by the use of the AdS/CFT duality is
that of using knowledge about the dual CFT to say something about the
stability of the scalar field in this NH region. For this the Breitenloher-
Friedman bound is useful, as it tells us necessary criteria for the bulk field
to be renormalisable.
The Breitenloher-Friedmann bound states $m^{2}L^{2}>-\frac{d^{2}}{4}$. For
1-cohomogeneity solutions of the Myers-Perry’s black hole, we know:

$\Delta_{\pm}=\frac{1}{2}\pm\sqrt{q^{2}-\mu^{2}-\lambda+\frac{1}{4}}=\frac{1}{2}\pm\frac{1}{2}\sqrt{4\Omega^{2}L^{4}m^{2}-4L^{2}\left(\mu^{2}+\frac{\lambda}{r_{+}^{2}}+\frac{m^{2}}{r_{+}^{2}h(r_{+})}\right)-1}.$

Specifically, this means that we can find a stability bound on scalar
perturbations in an $AdS_{2}$ spacetime. Since the effective mass is
$\sqrt{q^{2}-\mu^{2}-\lambda+\frac{1}{4}}$, we get an imaginary mass if:

$q^{2}+\frac{1}{4}\leq\mu^{2}+\lambda.$ (32)

This is the $\mathrm{AdS_{2}}$-Breitenlohner Freedman bound in
$\mathrm{AdS_{2}}$ space.

##  3 Fermionic Modes

###  3.1 General Dirac Fields

####  3.1.1 Setup

In this section, we aim to explain how to understand fermionic perturbations
in the NH region of extremal black holes. For this, we need to first describe
the necessary semantic tools which we need in order to make our theory
rigorous. We work in the same background as before.

$\displaystyle
ds^{2}=L(y)^{2}\left(-R^{2}dT^{2}+\frac{1}{R^{2}}dR^{2}\right)+g_{IJ}(y)(d\phi^{I}-k^{I}RdT)(d\phi^{J}-k^{J}RdT)+g_{AB}(y)dy^{A}dy^{B}.\qquad$

Spinor fields live in representations of the Lorentz-group $SO(3,1)$ of half-
integer spin. Under a Lorentz transformation
$x_{\mu}=\Lambda^{\mu}_{\nu}x^{\nu}$, the transformations of the Lorentz group
act in blocks of generators $\mathcal{M}_{\alpha\beta}$, such that a finite
Lorentz transformation looks like
$\Lambda=\exp(i\theta_{\alpha\beta}\mathcal{M}^{\alpha\beta})$. The parameters
$\beta_{\alpha\beta}$ act as boost parameters and are totally antisymmetric by
construction of the generators of the Lorentz group. If $\Psi$ denotes a
spinor field, and $S[\Lambda]$ is the spinor representation of $SO(3,1)$,
under a finite Lorentz transformation the spinor transforms as
$\Psi\rightarrow S[\Lambda]\Psi(\Lambda^{-1}x)$.
If we wish to consider fields with higher spin than scalar fields, it is
central to introduce a vielbein basis of the background one works with. We
refer to a frame of covectors $e^{a}=e^{a}_{\mu}dx^{\mu}$. These satisfies the
orthonormality relation $g^{\mu\nu}e^{a}_{\mu}e^{b}_{\nu}=\eta^{ab}$, where
$\eta^{ab}$ is the Minkowski metric. Thus the vielbein frame endows the
spacetime manifold $\mathcal{M}$ with its Lorentzian structure. The dual basis
lowers the latin indices $a,b$. In order to be able to describe spinor fields,
we need to introduce the gamma matrices $\gamma^{a}$ which we will use in the
Weyl representation. We will be working here in a setup with at most 5
dimensions, and so the spinor dimension will be 4.
The last formal ingredient is the so called spin connection. This enables us
to consider quantum fields with higher spin $s$ (here for $s=1/2$). For this
we define a general covariant derivative:

$\mathcal{D_{\mu}}=\partial_{\mu}-\Gamma_{\mu}-iqA_{\mu}$ (33)

where $\Gamma_{\mu}$ is the spin connection. We define the spin connection via
$\gamma_{cab}=e^{\mu}_{c}\left(\partial_{\nu}e_{a\mu}-\Gamma_{\mu\nu}^{\beta}e_{a\beta}\right)e_{\beta}^{\nu}$.
Intuitively, the spin connection is the connection such that the frame
$e^{a}_{\mu}$ is covariantly conserved: $\nabla_{a}e^{c}_{\mu}=0$. Thus the
formal abstract definition: $\omega_{\mu ab}=e^{\nu}_{a}\nabla_{\mu}e_{b\nu}$
can be made as well. With this the expression for $\Gamma_{\mu}$ can be
written as:

$\Gamma_{\mu}=-\frac{1}{2}e^{c}_{\mu}\gamma_{cab}S^{ab}=-\frac{1}{2}e^{c}_{\mu}\lambda_{cab}\frac{1}{2}[\gamma^{a},\gamma^{b}].$
(34)

Here $S^{ab}$ is the spinor generator of Lorentz transformations. As a last
remark one can point out that the $e^{a}_{\mu}$ act as projectors between the
Minkowski latin indices and the greek indices in curved spacetime. Hence
$\gamma^{\mu}=e^{\mu}_{a}\gamma^{a}$ and
$\partial_{a}=e_{a}^{\mu}\partial_{\mu}$.

####  3.1.2 Dirac Equation in Curved Space

The action for a fermion coupled to a curved background can be written as:

$S_{D}=\int
d^{4}x\sqrt{-g}\left([\bar{\Psi}\gamma^{\mu}(\mathcal{D}_{\mu}\Psi)-(\mathcal{D}_{\mu}\bar{\Psi})\gamma^{\mu}\Psi]-\mu\bar{\Psi}\Psi\right)$
(35)

where $\bar{\Psi}$ is the Dirac adjoint. By varying the action w.r.t.
$\bar{\Psi}$, we obtain the Dirac equation
i$\gamma^{\mu}\mathcal{D}_{\mu}-\mu\Psi=0$. Since we chose the Weyl
representation of the four-spinor, the spinor field naturally decomposes to
two chiral left- and right-handed bispinors. We can use the projectors
$P_{\pm}=\frac{1+\gamma^{5}}{2}$ to project to either side.
We write:

$\displaystyle\gamma_{0}=\begin{bmatrix}0&iI_{2}\\\ iI_{2}&0\\\
\end{bmatrix},\gamma_{1}=\begin{bmatrix}0&i\sigma_{3}\\\ -i\sigma_{3}&0\\\
\end{bmatrix},\gamma_{2}=\begin{bmatrix}0&i\sigma_{1}\\\ -i\sigma_{1}&0\\\
\end{bmatrix},\gamma_{3}=\begin{bmatrix}0&i\sigma_{2}\\\ -i\sigma_{2}&0\\\
\end{bmatrix}$ (36)

We now aim to write (35) explicitly in this general background and then solve
for the frequencies in the same way we solved for the scalar mode frequencies.
We can construct a co-frame basis:

$\displaystyle e^{0}=-LRdT\qquad$ $\displaystyle e^{1}=\frac{L}{R}dR$
$\displaystyle e^{2}=e_{I}(d\phi^{I}-k^{I}RdT)\qquad$ $\displaystyle
e^{i}=r_{+}\hat{e}^{i}_{a}dx^{a}$

For the Myers-Perry’s solution we will use $d\phi^{I}=d\Psi+\mathcal{A}$. The
$\hat{e}^{i}_{a}$ represents the vielbein on the Fubini study metric
$\hat{g}_{ab}$ in the NH limit. When we lower indices, we then get:

$e_{0}=\frac{1}{LR}\left(\partial_{T}-\Omega
L^{2}R\partial_{\Psi}\right)\qquad e_{1}=\frac{R}{L}\partial_{R}\qquad
e_{2}=\frac{1}{r_{+}\sqrt{h(r_{+})}}\partial_{\Psi}$ (37)

and on $\hat{e}^{i}_{a}$, we need to use an orthogonalisation procedure:

$e_{i}=\frac{1}{r_{+}}\left(\hat{e}_{i}-\langle\mathcal{A},\hat{e}_{i}\partial_{\Psi}\rangle\right)$
(38)

where $\hat{e}_{i}$ is the dual basis to $\hat{e}^{i}$. The spin connection
$\omega$ stems from the principle bundle structure associated with the spin
group $\mathrm{Spin(n)}$. Thus, we can compute its coefficients using the
Cartan Structure Equation $de+\omega\wedge e=0$.

$\gamma^{\mu}\Gamma_{\mu}=e^{\mu}_{a}\gamma^{a}\Gamma_{\mu}=-\frac{1}{2}e^{\mu}_{a}\gamma^{a}\omega_{\mu(ab)}[\gamma^{a},\gamma^{b}]=-\frac{1}{2}e^{\mu}_{d}e^{c}_{\mu}\gamma^{d}\lambda_{abc}[\gamma^{a},\gamma^{b}].$
(39)

As well as $\gamma^{\mu}\partial_{\mu}$:

$\gamma^{\mu}\partial_{\mu}=e^{\mu}_{a}\gamma^{a}\partial_{\mu}=\gamma^{a}\partial_{a}.$
(40)

Thus, the Dirac equation looks like:

$\left(\gamma^{a}\partial_{a}\
-\frac{1}{2}e^{\mu}_{d}e^{c}_{\mu}\gamma^{d}\lambda_{abc}[\gamma^{a},\gamma^{b}]\right)\Psi=m\Psi.$
(41)

The spin connection coefficients are then:

$\displaystyle\omega_{01}=-\frac{e^{0}}{L}-\frac{r_{+}\sqrt{h(r_{+})}e_{2}}{2}\qquad\omega_{02}=\frac{r_{+}\sqrt{h(r_{+})}e_{1}}{2}\qquad$
$\displaystyle\omega_{0i}=0$

$\omega_{12}=\frac{r_{+}\sqrt{h(r_{+})}e_{0}}{2}\qquad\omega_{1i}=\frac{R}{Lr_{+}}e^{i}\qquad\omega_{2i}=\frac{\sqrt{h(r_{+})}}{r_{+}}\mathcal{J}_{ij}e^{j}$
(42)

$\omega_{ij}=-\frac{\sqrt{h(r_{+})}}{r_{+}}\mathcal{J}_{ij}e^{2}+\frac{1}{r_{+}}\hat{\omega}_{kij}e^{k}$

Here $\mathcal{J}=-\frac{1}{2}d\mathcal{A}$ is the Kähler-form on
$\mathbb{CP}^{n}$. We now write down $\Gamma_{a}$:

$\gamma^{0}\Gamma_{0}=\frac{1}{2}\gamma^{1}\left(\frac{1}{L}+\gamma^{0}\gamma^{2}\frac{r_{+}\sqrt{h(r_{+})}}{2}\right)\qquad\gamma^{1}\Gamma_{1}=-\frac{1}{2}\hat{\gamma}^{2}\left(\frac{r_{+}\sqrt{h(r_{+})}}{2}\right)$
(43)
$\gamma^{2}\Gamma_{2}=-\frac{1}{4}\hat{\gamma}^{2}\left(-\frac{r_{+}\sqrt{h(r_{+})}}{2}-\frac{\sqrt{h(r_{+})}}{r_{+}}\mathcal{J}_{ij}\right)$

We introduced the notation that
$\hat{\gamma}^{2}=\gamma^{0}\gamma^{1}\gamma^{2}$. By expressing the equation
in terms of bispinors $\Psi_{L},\Psi_{R}$:

$\Psi_{L}=\mathcal{N}\exp{(i\omega
T+im_{I}\phi^{I})}\begin{bmatrix}R_{1}(R)Y_{1}(y)\\\
R_{2}(R)Y_{2}(y)\end{bmatrix}$ (44)

the Dirac equation takes the form (for one component):

$\frac{-i}{RL}\left(i\omega-i\Omega
L^{2}Rm\right)(R_{1}Y_{1})-i\frac{R}{L}\left(\partial_{R}\right)(R_{1}Y_{1})+i\frac{1}{r_{+}\sqrt{h(r_{+})}}m(R_{2}Y_{2})+\frac{1}{r_{+}}\hat{\mathcal{D}}(R_{2}Y_{2})+\frac{1}{2L}R_{1}Y_{1}=0$
(45)

We are interested in spinor harmonics on $\mathbb{CP}^{n}$, hence:
$\hat{\mathcal{D}}Y_{1/2}=\lambda Y_{1/2}$. This gives us the radial equation
for $R_{1}$ (or $R_{2}$ respectively after separating), namely that it
follows:

$R\partial_{R}(\left(R\partial_{R}\right)R_{1})-\left[\frac{(\omega-\Omega
L^{2}mR)^{2}}{R^{2}}-\frac{i\omega}{R}+\frac{1}{4}-L^{2}\left(\mu^{2}+\frac{\lambda^{2}}{r_{+}^{2}}+\frac{m^{2}}{r_{+}^{2}h(r_{+})}\right)\right]=0$
(46)

This can be again written very abstractly as the charged scalar field equation
on $\mathrm{AdS_{2}}$ where we have called the scalar field $\chi_{1/2}$ to
show the explicit spin dependence. Here $A=-RdT$ is again the gauge field in
this background with charge $m\Omega L^{2}$. We can solve this with similar
methods as in Section 2.

$\left((\nabla_{2}-iqA)^{2}-\mu^{2}+\frac{1}{4}\right)\chi_{1/2}=\lambda\chi_{1/2}$
(47)

###  3.2 Holographic Methods

Before we can analyse the quasinormal behaviour on $AdS_{2}$, one needs to
impose the correct boundary conditions that will make the $AdS/CFT$ duality
physically reasonable. Since we are describing quantum fields on the
asymptotic boundary, and this boundary is described purely by its radial
characteristics, we are interested in the pure derivative terms on the
boundary that act as our boundary conditions. Precisely if we consider $\delta
S$ only, we neglect the boundary action $\delta S_{bdry}$. We vary our action
such that $\delta S=0$, but not $\delta\left(S+S_{bdry}\right)=0$.
In the context of the AdS/CFT duality we claim that a well-defined conformally
symmetric quantum field which dual to the Dirac spinor $\Psi$ resides on the
boundary. Since we are interested in the quasinormal modes of such fermionic
perturbations, we are interested in ingoing asymptotic boundary conditions. In
the following, we are going to work within these assumptions to understand the
field theoretic approach to this.
Specifically, we recall we found a $\Delta_{\pm}$ which corresponds to two
different conformal weights for scalar fields $\phi=\phi_{+}+\phi_{-}$.
Similarly, we wish to decompose $\Psi=\Psi_{+}+\Psi_{-}$. Since the behaviour
at the boundary is asymptomatically determined by its scaling for
$r\rightarrow\infty$, we are interested in the behaviour of the radial
$\gamma^{1}$ far away. Indeed it is advantageuos to propose the ansatz for the
boundary conditions:

$\Psi_{\pm}=\mathcal{N}\exp{(i\omega
T+im_{I}\phi^{I})}\begin{bmatrix}i(\chi_{1}(R)\pm i\chi_{2}(R))S_{1}(y)\\\
(\chi_{1}(R)\pm i\chi_{2}(R))S_{2}(y)\\\ (\chi_{1}(R)\pm
i\chi_{2}(R))S_{1}(y)\\\ -i(\chi_{1}(R)\pm i\chi_{2}(R))S_{2}(y)\end{bmatrix}$
(48)

This allows to differ between the different conformal dimensions associated to
the dual fields $\Delta_{pm}$. By separating and obtaining the conformal
scaling dimension of the field, we can then use the charged Green’s function
approach to explicitly find the quasinormal frequencies.

####  3.2.1 Finding Asymptotic Scaling

We conclude this section by briefly describing the parameter $\lambda$ which
we need for the conformal dimension. Then, we can explain the value of
$\Delta_{\Psi}$. The eigenvalue can be explicitly found as the spinor harmonic
on $\mathbb{CP}^{n}$. The strategy to find $\lambda$ follows the idea that
$\mathbb{CP}^{n}$ is a homogeneous space $SU(n+1)/(U(n))$. The explicit
computation now uses Lie theory SEIFARTH and SEMMELMANN  to find the
eigenvalues:

$\displaystyle\lambda^{2}=l^{2}+\frac{1}{2}l\left(3n-2k-1\right)+\frac{1}{2}(n-k)(n-1)$
(49)
$\displaystyle\beta^{2}=l^{2}+\frac{1}{2}l\left(3n-2k+1\right)+\frac{1}{2}(n-k)(n+1)$
(50)

A consistency check reveals for $N=1$, that $\lambda^{2}=\beta^{2}=l(l+1)$.
Thus, we can state the spinor conformal dimension $\hat{\Delta}_{\Psi}$:

$\displaystyle\hat{\Delta}_{\Psi}=\frac{1}{2}\pm\sqrt{\frac{1}{2}-L^{2}\left(\frac{\lambda}{r_{+}^{2}}+\frac{m^{2}}{r_{+}^{2}h(r_{+})}+\mu^{2}\right)+m^{2}\Omega^{2}L^{4}}$
(51)

Now, we can similarly find the QNF of the Dirac Modes as:

$\displaystyle\omega^{NH}_{n,m}=\left(i\left[n+\frac{1}{2}+iq+\sqrt{\frac{1}{2}-L^{2}\left(\frac{\lambda}{r_{+}^{2}}+\frac{m^{2}}{r_{+}^{2}h(r_{+})}+\mu^{2}\right)+m^{2}\Omega^{2}L^{4}}\right]\frac{d\kappa}{d\eta}\right)\eta\qquad$
(52)

Unlike scalars fermions do not obey a superradiant bound, which explains the
missing of these terms in 52.

###  3.3 Fermionic CFT Dual

We can use the aforementioned results to discuss the CFT dual to a fermionic
field theory close to the horizon. After having discussed the CFT dual to a
bosonic scalar field, we will need to understand the behaviour of fermions in
local $AdS_{2}$. This is particularly relevant if one checks the behaviour of
fermions in global $AdS_{2}$ for any possible supersymmetric extensions of the
boundary CFT. In general the results about the conformal dimension:

$\Delta_{\Psi}=\frac{d}{2}+|m_{\mathrm{eff}}|=\frac{1}{2}\pm\sqrt{\frac{1}{2}+\Lambda}.$
(54)

This obeys the $AdS_{2}$ Breitenlohner-Freedman bound to give:

$\Lambda\geq-\frac{1}{2}.$ (55)

Since certain combinations of the quantum parameters render the conformal
scaling dimension complex, we find that the effective mass in AdS is subject
to tachyonic instabilities. This sets mass bounds on the fermions in the bulk
and thus suggests that fermions behave differently from bulk bosons.

The propagator of fermions in charged $AdS_{2}$ space can be found as Faulkner
et al. (2011):

$\mathcal{G}_{R}=(4\pi
T)^{2\Delta}\frac{\Gamma(2\Delta)\Gamma(\frac{1}{2}+\Delta-\frac{i\omega}{2\pi
T}+iq_{\mathrm{AdS}}e_{d})\Gamma(1+\Delta-
iqe_{d})}{\Gamma(-2\Delta)\Gamma(\frac{1}{2}-\Delta-\frac{i\omega}{2\pi
T}+iq_{\mathrm{AdS}}e_{d})\Gamma(1-\Delta-
iqe_{d})}\times\frac{m-iqe_{d}-\Delta}{m-iqe_{d}+\Delta}$ (56)

When considering the near-extremal region, we need to consider a finite
dimensional generalisation of the CFT. We make a change of coordinates to
$t=\frac{1}{2\pi T}\exp(2\pi T\tau)$:

$\langle\mathcal{O}(\tau)\mathcal{O}(\tau^{\prime})\rangle=\left(\frac{d\tau}{dt}\right)^{1+\Delta}\left(\frac{d\tau^{\prime}}{dt^{\prime}}\right)^{1+\Delta}\langle\mathcal{O}(t)\mathcal{O}(t^{\prime})\rangle$
(57)

As Faulkner et al. (2011) writes about charged AdS space, the fermionic
perturbations close to the horizon of an extremal black hole behave as a Fermi
fluid. Exploiting this in the case of the rotating black hole, in the finite
temperature generalisation, the gauge field
$A=RdT=\frac{1}{\zeta}dT=\frac{1}{\zeta}d\tau+d\tanh(\zeta/\zeta_{0})$ where
$\zeta_{0}$ is such that $T=\frac{1}{2\pi\zeta_{0}}$. Then, we can write for
the two-point correlation function:

$\langle\mathcal{O}(\tau)\mathcal{O}(\tau^{\prime})\rangle=\left(\frac{\pi
T}{\sinh(\pi T(\tau-\tau^{\prime}))}\right)^{1+2\Delta}\exp(-2\pi iT(m\Omega
L^{2})(\tau-\tau^{\prime})).$ (58)

In the case of an oscillatory solution close to the horizon, we observe that
om the extremality region, the dicrete quasinormal modes of the retarded
Green’s function form a branch cut. If we are close to extremality, the poles
begin accumulating at $\omega=0$. This leads to (in the oscillatory case only)
to a Fermi Sea close to the horizon.

##  4 Discussion and Conclusion

In this article, we examined the Quasinormal Frequencies (QNFs) using methods
from the holographic duality principles and used them to understand bosonic as
well as fermionic modes in the near horizon region of extremal rotating black
holes. The renormalisability criteria provided us with stability criteria on
the effective masses of the dual boundary operators that allowed us to
describe these perturbations also in light of the dictionary between CFTs and
their bulk duals. We were able to compute the central charges of the theory
for the case of the Myers-Perry’s black hole with equal angular momenta and
the Hawking temperature as well as the Hawking entropy. With this, it was then
possible to describe the boundary CFT as an ultimately thermal theory in this
limit.

In the case of spin-2 fields, we wish to analyse the behaviour due to the
holographic dictionary that we have proposed thus far. If we start from the
Fierz-Pauli action, we can find the perturbation operator for the Myers-Perry
black hole via varying the action and rewriting the equation of motion in this
case. If we proceed to decompose the spin 2 field
$h_{\mu\nu}=\chi(r)\mathcal{Y}_{\mu\nu}$, we can find the angular perturbation
operator very similarly. Its eigenvalues are $\lambda_{nlm}$. We can extract
the frequencies as per earlier to get $\omega_{nlm}$.
The claim we make now is that the quantum dual system behaves very similarly
to the system which we found before. In general each of the operators dual to
one such eigenmode $\omega_{nlm}$ will be dual to the correlation function of
the two quantum operators on the boundary. When comparing the frequencies we
found, we see that the spectrum is very similar in structure to the fermionic
Matsubara frequencies and so the Fermi-Dirac distribution supported by a
chemical potential $\mu$ would be a good fit for the parameters.
One phenomenon that we wish to consider is the so-called thermal ringdown
process of the boundary CFT. In this case, for example in black hole mergers,
we would observe the superposition of many such normal modes with
eigenfrequencies $\omega_{nlm}$. If we can write the mode
$\tilde{h}_{lm}(\omega)$ in Fourier space as:

$\tilde{h}(\omega)=\frac{1}{2\pi}\int dth_{QNM,lm}(t)e^{i\omega
t}=\frac{i}{2\pi}\sum_{n}\frac{C_{lm}}{\omega-\omega_{lmn}}\exp({i\omega
t^{*}}).$ (59)

If we now claim, indeed, that the statistical behaviour of
$|\tilde{h}(\omega)|^{2}$ is governed by a Fermi Dirac distribution:

$\frac{1}{\exp\left(\frac{(\omega-m\Omega_{H})}{T}\right)+1}$ (60)

, for near extremal black holes with $T_{H}\sim\frac{d\kappa}{d\sigma}\sigma$,
then we can understand something about the thermal ringdown of the boundary
CFT. If during a merger process the black-hole rings, i.e we are led to
excited overtones, this is akin to excited states of the dual quantum
operators. Recent analysis of GW has shown this to be the case Oshita (2022).
What this additionally tells is us is the behaviour of the long time
correlation function of the quantum dual which we can understand as:

$\langle\mathcal{O}_{l,m}\mathcal{O}_{l,-m}\rangle=\sum_{n}\exp({-i\omega_{lmn}}\Delta
t).$ (61)

Here we take the ensemble average in
$\langle\mathcal{O}_{l,m}\mathcal{O}_{l,-m}\rangle$. Hence, this provides new
insights into the holographic thermalisation process as a whole.

In the case of the Myers-Perry’s solution this is akin to using the angular
eigenvalue for a spin-2 field. This can be derived from separation of the
equation of motion:

$\Box h_{ab}+2R_{acbd}h^{cd}=0.$ (62)

The eigenvalue is (Durkee and Reall (2011)):

$\frac{\lambda_{lm}}{L^{2}}=4(1-\sigma)\left(\frac{N}{r_{+}^{2}}+\frac{N+1}{l^{2}}\right)+\frac{4\kappa(\kappa+N)}{r_{+}^{2}}$
(63)

Hence, we can write the frequencies $\omega_{lmn}$:

$\omega_{lmn}=m\Omega_{H}+m\frac{d\Omega}{d\sigma}\sigma+i\left(\frac{1}{2}+n\pm\delta_{lm}\right)\frac{d\kappa}{d\sigma}\sigma$
(64)

where $\delta_{lm}$ is part of conformal dimension of this tensor
$\delta_{lm}=\sqrt{\frac{1}{4}+\lambda_{lm}}$ which we just abbreviate. We
quickly see since $\lambda_{lm}$ is non-negative, the AdS-BF is definitely
satisfied and allows a well-defined dual boundary operator. So if we wish to
use the above stipulation it can definitely be valid to write:

$\langle\mathcal{O}_{l,m}\mathcal{O}_{l,-m}\rangle=\sum_{n}\exp({-\delta\omega_{lmn}}\Delta
t)=\sum_{n}\exp\left[-i\left(i\frac{1}{2}+n\pm\delta_{lm}\right)\frac{d\kappa}{d\sigma}\sigma\Delta
t\right]$ (65)

Hence, the overtone modes of rotating black holes close to extremality can
provide with valuable insights about the boundary correlators. Explicitly,
what, would be interesting is to understand the predicted mass spectrum of
(65). This would allow us to examine the holographic duality principles in a
more realistic environment.
Another aspect that is interesting is the aspect of Strong Cosmic Censorship
in the formulation of Christodulo. For uncharged rotating black holes it has
been shown by Dafermos and Luk (2017) that this hypothesis fails to hold. For
near-extremal charged black holes there is a strong hint it might be violated.
This gives us the possibility to calculate the spectral gap $\alpha$ to
investigate the strongest version of it by Christodulo. Here $\alpha$ is the
minimum of $\Im(\omega)$. Close to extremality
$\kappa_{-}=-\frac{d\kappa}{d\eta}$ and we define
$\beta=-\frac{\alpha}{\kappa_{-}}$. For the MP-dS black hole, this means for
$\beta$:

$\beta=\left[\frac{1}{2}+\frac{1}{2}\sqrt{4\Omega^{2}L^{4}m^{2}-4L^{2}\left(\mu^{2}+\frac{\lambda}{r_{+}^{2}}+\frac{m^{2}}{r_{+}^{2}h(r_{+})}\right)-1}\right]$
(66)

If $\beta\geq\frac{1}{2}$, then the field is in $H^{1}_{loc}$ and then the
Christodulo version doesn’t hold for the near-horizon modes close to
extremality. This region is however, crucial because we are interested in the
value of $\beta$ in this region specifically. We see here in (66) that then a
priori $\beta\geq\frac{1}{2}$. We find that the Strong Cosmic Censorship
hypothesis not violated if the $\mathrm{AdS_{2}}$-BF bound is respected. Thus,
if the bound on $\beta$ should be respected, then we are requiring a violation
of the AdS-BF bound. In (66), we need choose $m$ large enough such that the
bound is violated. By finding only one family of such modes that satisfy the
$\beta$ bound, we satisfy the SCC criterion. This can then be explored
numerically.
We can similarly discuss this for the case of Dirac fermions. For scalar modes
it was proven that this holds by Christodoulou (2008). We believe it to be
plausible that since, we decomposed the spinor into its scalar ingredients,
for this to hold as well in this case. This leaves us with the same condition,
namely $\beta<\frac{1}{2}$ for the SCC to hold. For the Reissner-Nordström
case it was proven in Destounis (2019) that the perturbations lie in
$H_{loc}^{s}$ for $s<\frac{1}{2}+\frac{\alpha}{\kappa_{-}}$. Hence, the
definition of $\beta$ can be justified very analogously. We find for
$\beta_{Dirac}$:

$\beta_{Dirac}=\frac{1}{2}+\sqrt{\frac{1}{2}-L^{2}\left(\frac{\lambda}{r_{+}^{2}}+\frac{m^{2}}{r_{+}^{2}h(r_{+})}+\mu^{2}\right)+m^{2}\Omega^{2}L^{4}}.$
(67)

Again we would need as a necessary condition that the $\mathrm{AdS_{2}-BF}$ to
be violated. This again makes a striking connection for near-extremal black
holes and dual conformal structure.

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sec_conc_4

# Experimental study of second sound quench detection for superconducting
cavities

J. Plouin<EMAIL_ADDRESS>B. Baudouy  A. Four  J.P. Charrier  L.
Maurice  J. Novo  CEA Paris-Saclay, Université Paris-Saclay, Gif-sur-Yvette,
France  B. J. Peters  Karlsruher Institut fur Technologie Postfach 3640 76021
Karlsruhe, Germany  K. Liao  Imperial College London and Ideabatic LTD, SW7
2AZ, 32 Lingrey Court, Cambridge, 10 CB2 9JA, United Kingdom

(Received 28 February 2019; published 26 August 2019)

###### Abstract

Superconducting RF cavities are used in particle accelerators to provide
energy to the particle beam. Such cavities are mostly fabricated in niobium
and often operated in superfluid helium. One of their limits of operation is
the appearance of a local quench, initiated by a local field enhancement due
to a defect, which leads to a normal conducting transition of the cavity.
Localizing the quench area can be achieved with temperature mapping systems.
Another method is the use of second sound wave propagation in superfluid
helium. Measuring the time of propagation of these waves from quench location
to special sensors, called Oscillating Superleak Transducers (OSTs), and using
their well-known velocity should allow trilateration. However, most of
experimental measurements on cavities show “premature signals”, i.e. the
second sound signals arrive earlier on the OSTs than expected. This paper
presents several quench experiments on cavities equipped with OSTs and
temperature mapping quench detection systems. Two hypotheses can explain the
observed premature signals. The first one assesses faster propagation in
helium. An experimental setup has been developed for testing this hypothesis,
where second sound is created by a localized heater in a controlled
environment up to 4.3 kW/cm2 and 2.8 J. Premature signals could not be
verified in this setup. A second hypothesis based on a simple model including
several processes in niobium and second sound propagation in helium is
discussed. The model improves significantly the prediction of the times of
arrival of the second sound waves. The overall study shows that the processes
in niobium play a prominent role in the second sound detection for
superconducting cavities.

Keywords

Superconducting cavity, Quench, Superfluid helium, Second sound

Published in Physical Review Accelerators and Beams 22, 083202 (2019)
doi: 10.1103/PhysRevAccelBeams.22.083202

##  I Introduction

Radio-frequency cavities are state of the art technology in high energy
particle accelerators to increase the momentum of charged particles. The
resonating electromagnetic field causes coulomb heating on the inner surfaces
of these cavities. Normal-conducting cavities are usually made of copper. The
generated heat can be greatly deceased by using a superconducting cavity in
the Meissner state. The element with the highest lower critical field is
niobium, which is thus mostly used for this purpose Padam2008 . To generate
high accelerating gradients, the cavities are operated close to their
operational limit. A number of different processes are known to limit the
operation of superconducting cavities Padam2008 . The limiting process that
normally occurs for very well performing cavities is the localized quench
where a local increase of the electromagnetic field due to a defect drives a
local area to normal-conducting transition, and due to the increased heating
to the global normal-conducting state Geng2009 . Even small imperfections on
the inner cavity surface can cause this quench. These spots can be treated
once their locations are detected. A number of methods has been applied for
this Iwashita2014 ; Navitski2014 .

Since the 1980’s temperature sensors are used to detect the temperature
increase on the outer cavity surface Conway2017 . These temperature mapping
systems are based on temperature-dependent carbon sensors, mostly Allen-
Bradley resistors Padam1987 . In addition to being sensitive for temperature
measurements in both normal (He I) and superfluid (He II) helium, their
geometry can be customized to match the topology of the superconducting
cavity. Some tests were made with temperature sensors that were glued to the
cavities surface, but mechanical systems that press the sensor to the cavity
surface are more common. The glued sensors are normally not reusable and a
high number of sensors are presented and thus read-out systems are needed. The
mechanical systems require custom-fit frame for each cavity geometry, but for
elliptical cavities they can be mounted on a rotating frame, that allows
moving them around the cavity in order to locate the quench spot. Another
method for the detection is the utilization of the special heat transport
properties in He II. In superfluid helium, heat is transported as a wave,
called second sound. This wave travels with a known, temperature-dependent
velocity. If at least three sensors, capable of detecting second sound with
precise time information and known positions relative to the cavity were
mounted, trilateration of the quench spot is possible. Conway et al.
introduced this method and utilized oscillating superleak transducers (OSTs)
for the detection of second sound waves Conway2008 .

For the last ten years, quench localization with 2nd sound detection has been
developed in several labs. It was found that this method is cheaper and easier
to implement than temperature mapping systems; however, nearly all these
laboratories have observed the same kind of premature signals: the second
sound signal arrives on the OSTs earlier than expected Maximenko2011 ;
Liao2012 ; Plouin2013 ; Tama2016 ; Junginger2015 ; Koettig2015 . Efforts have
been pursued to explain these premature signals and hypothesis have been
envisaged. Eichhorn et al. Eichhorn2015  have supposed that the second sound
wave could be generated before the RF system can detect the quench; they have
developed a corresponding model, which could explain some tens of µs error,
much lower than experimentally observed. Liu Liu2012  and Eichhorn
Eichhorn2015b  have considered the heat diffusion in niobium, in addition to
the second sound propagation, which could cause premature signals, but their
models have not been systematically compared to experimental results. The
possibility of other effects beside second sound propagation in He II has also
been considered, and experiments have been carried out with local heaters as a
source for the heat propagation. Liao et al. Liao2011  have generated heat
flux up to 200 W/cm2, and did not observe premature signal, while Junginger et
al. Junginger2015 , with heat flux up to 350 W/cm2 observed 10% faster signals
in particular conditions. Very recently, a new technique using flow
visualization in superfluid helium has been proposed by Bao et al. to detect
quench-spot in RF cavities Bao2019 . A prof-of-concept experiment with a
miniature heater is presented for heat flows below 300 W/m2. A ideal boiling
zone model is proposed to explain the premature signal on the data obtained by
Junginger2015  and (Koettig2015, ), but no test with a RF cavity has been
performed yet.

Up to now, none of these hypothesis and associated models could properly
describe premature signals measured during cavity tests.

The present paper is a contribution to these efforts. We performed five tests
on two single-cell superconducting cavities, with variation of the helium bath
temperature for one of them, and always observed premature signals. In order
to study the pure heat propagation in He II, we carried out experiments with a
localized heater as a source for the second sound wave. We reached values of
surface power (4.3 kW/cm2) and global energy (2.8 J) that were never achieved
in similar experiments, for these experiments no premature signal was
observed.

Figure 1: 1.3 GHz single-cell cavity equipped with temperature mapping system
and OSTs.

##  II Tests on Cavities

###  II.1 Setup and cavities test description

At CEA Paris-Saclay, superconducting cavities are usually tested in superfluid
helium in a dedicated vertical cryostat on the Supratech platform Cenni2017 ;
Baudouy2010 . For several single-cell cavities such tests could be realized
with second sound sensors for quench localization. During those tests, each
cavity was equipped with a temperature mapping system, and with four OSTs
facing the cavity in the equator plane, placed at around 6.5 cm from the
cavity. All these elements can be seen in Figure 1. Two Tesla shape single-
cell cavities, C1-21 Eozenou2010 ; Eozenou2012  and 1AC3 Eozenou2014  have
undergone these tests; they will be named respectively A and B. These cavities
have the same elliptical geometry, and thus identical RF parameters: frequency
f = 1.3 GHz, shunt impedance r/Q = 104 and geometrical factor G = 283 . They
were supplied by two different manufacturers, from niobium plates with RRR
values between 250 and 300, and both of them have undergone several chemical
and electropolishing treatments.

Five tests where localized quench was reached without electron emission could
be realized: two tests with Cavity A and three tests with Cavity B. The
$Q_{0}$ vs. $E_{acc}$ curves are shown in Figure 2, $Q_{0}$ being the cavity
quality factor and $E_{acc}$ the accelerating field. The accelerating field is
increased by injecting RF power in the cavity, until the cavity quenches,
which means that the injected power is entirely reflected. The quench
appearance is indicated by a vertical arrow for each test, which thus
determines the maximum accelerating field allowed in the cavity ($E_{max}$).
The test results are summarized in Table 1. The differences in the
$Q_{0}$/$E_{acc}$ curve for Cavity B from one test to the another is due to
the intermediate chemical/electropolishing treatments. The helium bath
temperature was about 1.5 K for Cavity A and about 1.6 K for Cavity B.

Figure 2: $Q_{0}/E_{acc}$ curves of the tested single-cell cavities. Vertical arrows show quench appearance Table 1:  Cavity tests summary | Cavity name | $E_{max}$
---|---|---
test 1 | A | 30 MV/m
test 2 | A | 31 MV/m
test 3 | B | 34 MV/m
test 4 | B | 35 MV/m
test 5 | B | 37 MV/m

###  II.2 Quench localization with temperature mapping system

Initially the quench position was localized by the temperature mapping system,
in order to have a reference for comparison with the OSTs measurements. This
system consists of 17 thermometers fixed on a structure able to rotate around
the cavity axis Berry2003 . Each thermometer is constituted of a carbon
resistor placed inside a copper support which is in contact with the cavity
surface. The aim of this measurement is to observe the temperature elevation
when the quench occurs. The system works as follows: at each azimuthal
position, the 17 thermometers are successively monitored. The duration of a
single readout is long compared to the quench dynamics, so the following
method is used: the sequence is repeated about twenty times while the cavity
is forced to quench. For each thermometer the maximum value of this series of
measurements is used, because it corresponds to the closest in time to the
quench occurrence. As an example the temperature array is shown in Figure 3
around the quench location (top), and is projected on the cavity surface
(bottom). The resolution is about 3 degrees in latitude and 1 cm in longitude
(distance between two sensors).

Figure 3: Top: Thermal cartography of Cavity B during a test. Azimuthal angle
gives the temperature mapping position around the cavity; thermometer number
can be read in Figure 1. Bottom: Cartography projection on cavity surface
(left) and identification of the quench spot on the cavity marked by $\oplus$
symbol (right).

This temperature mapping system is used to produce a map showing the measured
temperature with respect to the bath. As the mapping system itself is immersed
in the bath, the temperature given by the carbon resistors is different from
the surface cavity temperature. A temperature difference equal to zero means
that locally the cavity surface is at the helium bath temperature. This
temperature mapping system is efficient to achieve a localization within a
precision of a less than one centimeter in each direction. According to the
temperature maps, the quench location of Cavity A always lands at the equator
and that of Cavity B at the bottom half of the cavity (see Figure 3).

###  II.3 OST measurements at 1.7 K

An Oscillating Superleak Transducer (OST) is a sensitive sensor to detect
second sound. The idea for this device originated in 1968 Williams1968 . It
consists of a capacitor like setup with one flexible membrane and a solid back
electrode. The membrane is flexible and has nanometer-sized pores. For the
shown measurements a metal-coated membrane with holes of about 100 nm were
used. According to the two-fluid theory the superfluid component of the helium
can pass through these holes while the normal fluid component cannot
Donnelly2009 . Thus, the arrival of a second sound wave, that has different
amounts of the two components in comparison to the helium between the two
plates Donnelly2009 , applies a pressure on the flexible membrane, decreasing
the distance between the two membrane plates. The two sides are charged with
120 V via a weakly coupled connection. For fast processes, such as a second
sound arrival, the charge on the OST stays nearly constant. A decrease of the
distance of the capacitor plates thus causes a drop in voltage of the membrane
Quadt2012 . Each OST has its own electronic circuit which provides the
polarization and amplification of the signal. For all the measurements
presented in this paper, the same set of OSTs, with diameter 20 mm, was used.
This set was provided by Cornell University Conway2009 .

Figure 4: Oscilloscope signals from OSTs during cavity quench. Vertical arrows
show 2nd sound wave arrival on OST.

For each test, the signals from OSTs during a series of quenches were
measured. The results were recorded by an oscilloscope that was triggered by
the reflected RF power from the cavity, indicating a quench. A typical
oscilloscope signal is shown in Figure 4. The blue curve shows the RF signal.
As explained previously, the signals from OSTs always start with a sharp
negative decrease. The times of arrival of the wave on each OST after quench
trigger can be measured with a precision better than 100 µs, and are indicated
by vertical arrows on the figure. The OSTs are ordered by their distance from
the quench spot, starting with the 1st OST as the closest. During each test,
the two nearest OSTs (see Figure 4) were nearly facing the quench location,
and they presented a clear signal while the two others were hidden by the
cavity. The signal of the furthest one was not detectable and is not shown on
the figure.

The times of arrival of the second sound signal on the two nearest OSTs are
represented in Figure 5 together with the corresponding times of arrival for
the following hypothesis: a pure second sound wave is emitted by a point-like
heat source situated at the quench location. These times are thus equal to the
distance between OST and quench location (given by temperature mapping system)
divided by 20 m/s, which is the second sound velocity at 1.7 K Donnelly . In
the following, we will refer to this calculation approach as the ”direct line
of sight”.

Figure 5: Times of arrival of the 2nd sound signal on the two closest OSTs
from the quench location at 1.7 K: as measured during experiment (Red) and as
calculated with in direct line of sight (Blue).

The difference between measured and calculated times of arrival shows that it
is not possible to consider the quench as a punctual emission of a 2nd sound
wave: the heat wave reaches the OSTs faster than expected by the direct line
of sight propagation.

Such premature signals have been experimentally observed by nearly all the
laboratories which have used second sound sensors to localize a quench in a
superconducting cavity Maximenko2011 ; Plouin2013 ; Tama2016 ; Eichhorn2015 ;
Liu2012 . Different hypothesis have been envisaged to explain this difference,
and will be discussed later in this paper.

###  II.4 OST Measurements with varying helium temperature

During test 4 on cavity B, the helium bath temperature was varied while
measuring the evolution of the OST signals during cavity quenches. The scope
signals are shown on the Figure 6 for four temperature values (1.71 K, 1.9 K,
2.03 K and 2.1 K), and for the three closest OSTs.

Figure 6: Oscilloscope signals from OSTs during cavity quench at different
temperatures. Horizontal: 4 ms/division. Vertical. RF Signal: 10m V/div; 2nd
and 3rd OST : 100 mV/div; 1st OST: 1 V/div. (1.71 K), 500mV/div. (1.9, 2.03
K), 100 mV/div (2.1 K).

The dotted vertical arrows show for each OST the time position taken into
account for the arrival of the 2nd sound wave. While this position always
stays easily detectable for the 1st OST (the closest to the quench location),
one can see that for the other OSTs, the signal becomes less evident to detect
with increasing temperature, and is even undetectable at 2.1 K for the 3rd
OST.

Figure 7: Recommended values of the second sound velocity at different
temperatures and values relevant for presented experiments, according to
Donnelly . Figure 8: Time of arrival on first (red) and second (green) OST
with varying temperature: experimental data (points with error bars) and
calculated data from the direct line of sight propagation (solid line). The
shaded area represents the error on calculation due to geometrical
measurements uncertainties.

The experimental results (see Figure 8) show how the time of arrival increases
with temperature, which is consistent with the evolution of second sound
velocity, decreasing with temperature in the [1.7-2.1 K] range, see Fig.7. For
the two closest OSTs, the experimental data are compared to the corresponding
times of arrival from the direct line of sight propagation, using the second
sound velocity values between 1.7 and 2.1 K Donnelly . These results show that
the experimental data has the same behavior as the model data when temperature
is growing. However, in the whole temperature range, the heat wave reaches the
OSTs faster than expected by the direct line of sight propagation. The error
bars indicated in Figure 8 take into account the uncertainty on the time of
arrival measurement ($\pm$ 0.05 ms, error bars on points) and the uncertainty
on the distance measurement between the quench spot on the OSTs ($\pm$ 2 mm,
shaded area on lines).

###  II.5 Conclusions on OSTs measurement during cavity tests

The first approach considered for the quench localization with OSTs was based
on the hypothesis of an instantaneously emitted heat wave by a point like
source (the quench location) which travels from this source to the OSTs with
the second sound velocity in superfluid helium. This hypothesis is not
compatible with the experimental results during cavity quenches in the
temperature range [1.7 -2.1 K].

Figure 9: 2D example of trilateration, for direct line of sight propagation
for test 1 (left) and test 2 (right). Cavity is represented seen from above by
a grey disk bordered by dark circle corresponding to its equator. Blue plain
circles have radii equal to the time propagation for corresponding OST
multiplied by 20 m/s. Dashed circles have radii equal to the time propagation
for corresponding OST multiplied by 28 m/s. OSTs are represented by small dark
rectangles, and quench position detected by temperature mapping system is in
red.

Thus, a quench location was unable to be determined with the standard
trilateration method. This is illustrated in Figure 9 for two cases (test 1
and 2) where the problem can be represented in 2D because quench occurs on the
equator (and OSTs are in the equator plane). The blue plain circles have radii
equal to the time propagation to corresponding OST multiplied by 20 m/s, the
second sound velocity at test temperature. The circles do not intersect,
showing that the localization is not possible. When the propagation velocity
is arbitrarily increased, up to 28 m/s in those cases, it becomes possible to
have circles intersection located on cavity equator (dashed circles). However
it does not fit with the quench position measured by temperature mapping
system, identified by a red region on equator as shown on the Figure 9. This
situation becomes even worse if more than two OSTs are used.

To be able to allow more information sources and account for measurement
errors, DESY developed an elegant method Tama2016 . They include the
information, that the heat spot will lie on the surface of the cavity. From
each surface point they calculate the length of the shortest path in helium to
each OST. Together with the second sound velocity and the OST signal they
obtain an expected distance the second sound has travelled in helium. The
distance between this previous calculated propagation length $s_{z,\phi}$ and
expected distance $d$ can be used for every spot on the cavity surface and
$n$-OSTs to determine the root mean square deviation ${RSME}_{z,\phi}$. The
area where this deviation is smallest, can be seen as the area where the
second sound was triggered.

${RMSE}_{z,\phi}=\sqrt{\frac{\sum_{i=1}^{n}(s_{z,\phi}-d_{i})^{2}}{n}}$ (1)

This method is used today by DESY to localize the quench position, and
comparison with temperature method shows that it works within a few
millimeters, if used in the right conditions. In practice, this method helps
to localize a small area which can then be examined with some imaging system
to find and cure the defect. However, the calculation results given by this
method still lead to premature signals.

Discarding the first hypothesis of pure second sound direct line of sight
propagation, two new hypothesis can then been envisaged:

  * •

Considering another phenomena in helium making the overall thermal signal
traveling faster than a pure second sound wave

  * •

Considering a part of thermal signal diffusing in niobium making the overall
thermal signal traveling faster than a pure second sound wave

##  III Propagation in Helium

In this section the validity of the first hypothesis is explored.

###  III.1 Tests with a localized heater in liquid helium

A number of ideas where brought forward to explain that the information of the
heat input into helium II could be transported faster than the second sound
velocity. Temperley theoretically predicted a faster propagation for high heat
inputs Temperley1951 . The theory was refined by Khalatnikov Khalatnikov1952
and later experimentally proven by Dessler and Fairbank Dessler1956 .
Additionally the appearance of boiling effects could also lead to a faster
thermal transport. Either by the emission of a first sound wave, which is
approximately ten times faster than second sound at the temperatures in
question Donnelly , or by effects concerning the gas bubble itself. Although
the signals in the cavity tests arrive too late to be seen as pure first sound
waves, an unproven theory by Dessler predicted an interaction between first
and second sound, which could explain a faster arrival of a second sound wave
Dessler1959 . The expansion of a gas bubble at the heating spot could also be
faster than the second sound.

Since the premature signals were observed experimentally, an attempt was made
to replicate them in a controlled environment. In order to exclude effects by
the cavity itself, the heating was carried out by a small heater, whose output
power and energy are regulated.

###  III.2 Heat pulse generation

The heat pulses are generated with a 50 RuO2-ZnO thick film resistor by the
company Bourns, which is specified to accept up to 40 W in DC at room
temperature. In the following it was used at cryogenic temperatures for pulsed
heat loads of more than 1 kW. This resistor has a surface of 34 mm2. The
pulses are generated by a synthesizer at 200 MHz, driven by a function
generator and are amplified by a RF amplifier able to provide more than 1 kW,
with a gain around 64 dB (Bruker BLAH 1000). A directional coupler is used to
control the signal injected in the resistor and to measure the reflected power
(which increases abruptly if the resistor burns). The whole setup is
schematized in Figure 10.

Figure 10: Experimental setup for the generation and control of the heat
pulse.

The heater and sensor positions are shown in Figure 11. Two OSTs have been
placed facing the heater.

Figure 11: Picture of the experimental setup.

###  III.3 Variation of the heat pulse power

For a set of temperatures in the range [1.3 -2.1 K] the influence of the heat
pulse power level on the OSTs signals was observed. The pulse duration is kept
equal to 100 µs, and the power is increased from about 150 W to 1500 W,
corresponding to a surface power from about 450 to 4500 W/cm2. A second sound
signal time of arrival has been measured for each OST, and the corresponding
estimated distance from the heater to the OST has been calculated using the
literature second sound velocities Donnelly . These calculated distances have
been plotted versus heat pulse power (Figure 12), for both OSTs and for the
whole set of temperatures. The results show clearly that the heat pulse power
level does not have any influence on these calculated distances. For both
OSTs, the mean calculated value has been plotted (dotted line), as well as the
distance measured on the setup (solid line). As shown in Figure 11, these
measured distances, resp. 54 mm and 133 mm, are from the heater to the closest
point of each OST. In both cases, the mean calculated distances are around 3
mm lower than expected. This difference could be compatible with the
inaccuracy of the distances measurements, due to the flexibility of the rods
holding the OSTs. Indeed, it was observed that these rods were slightly
deformed during the introduction of the setup in the cryostat.

Figure 12: Distances from heater to OSTs versus heat pulse power, calculated
from times of arrival and recommended second sound velocity. For each OST, the
average calculated values are shown (dashed lines) as well as the measured
distances (solid lines).

However this distance difference can also be explained by a second sound
velocity difference. This is shown in Figure 13 where the measured distances
(54 mm and 133 mm) are divided by the times of arrivals of the OST signals ;
the difference between this apparent second sound velocity and the literature
values Donnelly  is plotted versus the heat flux at heater. One can observe
that the apparent velocity is between 0.1 and 2 ms higher than expected, and
that the difference is higher for OST1 than for OST2. This results can be
compared to the heat pulse experiments described in references Junginger2015
and Koettig2015 . The authors explore a lower level of heat flux than us:
[0-700 W/cm2] instead of [450-4500 W/cm2]. Their results also present a higher
than expected apparent velocity with a difference up to 1.5 m/s. In their
measurements the velocity difference is dependent on the heating power below
200 W/cm2. At higher heating powers the velocity difference remained constant.
While this observation might explain slightly premature signals, this effect
is too weak to explain the significant deviations that were observed in tests
with cavities.

Figure 13: Difference between the apparent second sound velocity and the
literature values versus heat flux at heater. Values are given for OST 1
(squares) and OST 2 (circles), and for five sets of temperature: 1.35 K (red),
1.41 K (orange), 1.7 K (green), 1.88 K (blue) and 2.1 K (purple).

###  III.4 Variation of the heat pulse energy

It is possible that the heat pulse energy, rather than the pulse power, could
have an influence on the heat wave propagation. Experiments have been carried
out at 1.7 K, where the pulse power was varied up to 1.5 kW, the pulse
duration up to nearly 2 s and thus the pulse energy up to nearly 2.8 J. The
results are presented in Figure 14, and show clearly that the pulse energy
does not have any effect on the heat wave propagation time.

Figure 14: Distances from heater to OSTs versus heat pulse energy, calculated
from times of arrival and recommended second sound velocity. For each OST the
measured distances is shown (solid lines).

###  III.5 Conclusions on heater experiments

These experiments demonstrate that the heat pulse up to 1.5 kW and 2.8 J
created heat waves fully compatible with the standard second sound waves in
superfluid helium.

Thus the hypothesis of a heat wave propagating significantly faster in helium
than the second sound waves has to be discarded, at least until the power and
energy reached in these experiments. By comparison one can estimate the
overall energy dissipated during a cavity quench, which corresponds to the
energy stored in the cavity just before quench occurrence ($W_{stored}$). This
quantity is related to the accelerating field reached in the cavity $E_{acc}$,
the cavity frequency and some geometric parameters: the coupling coefficient
$r/Q$ and the accelerating length $L_{acc}$Padam2008 .

$W_{stored}=\frac{(E_{acc}L_{acc})^{2}}{\omega*r/Q}=0.0141*E_{acc}^{2}$ (2)

All of the 1.3 GHz cavities tested have an $r/Q$ coefficient equal to 115 and
an accelerating length of 0.115 m. The maximum accelerating field varies from
30 to 37 MV/m (see Figure 2) and thus the maximal energy from 12.7 to 19.3 J.
The time to dissipate this energy during the quench was not measured but
assuming a typical quench duration in the order of one millisecond, one will
obtain an average heating power of 12.7 to 19.3 kW.

So the power and the energy dissipated in helium during a cavity quench is
less than one order of magnitude higher than the maximal energy dissipated
during the heater experiments. In contrast to the tests in the cavity no heat
transfer through niobium did hinder the cooling. Taking this into account, the
heating powers in the tests were even closer to the surface heat flux in the
performed cavity tests. Even if the presented results cannot be directly
transposed to cavity tests, at least they do not provide any argument to
support the hypothesis of a heat wave propagating faster in helium than the
standard second sound wave.

As additional information, during these experiments and for the highest levels
of pulse energy, a cracking sound was audible, even if not recorded or
measured. We identified this sound as the signature of a boiling as the
highest surface heat flux were much higher than the reported literature data
for the onset of boiling in superfluid helium VanSciver2012 . Actually,
boiling phenomena in similar conditions have already been observed and
recorded Peters2014 . Thus it seems that heat wave propagation is still
compatible with the second sound velocity even if boiling in helium occurs.

##  IV Propagation in Niobium

As the origin of the premature signals could not be found to be caused by
effects within the liquid helium, an influence of the cavity itself seems
likely. Two effects come into mind to explain this. First, the quench
propagation within in the cavity could be the reason. Second the heat transfer
in the niobium itself could also cause this.

The quench of a cavity is generally caused by a local increase of the magnetic
field on a small defect situated on the inner surface of the cavity. This spot
is heated very rapidly, which causes the neighboring area to quench as well.
This process continues until the heat input by the stored energy in the cavity
decreases below the value of cooling by the surrounding niobium or helium. So
the heat source on the outer surface of the cavity is rather an area, with a
size depending on the dynamic heating and the dynamic heat diffusion in
niobium and superfluid helium. Moreover, one can consider that the temperature
increase due to the heat flux diffuses faster in the niobium than in helium,
both due to the additional heating as well as the high thermal diffusivity of
superconducting niobium. While both processes will certainly play a role, it
is likely, that the rapid signal propagation will be dominated by the
occurring quench phenomena.

###  IV.1 Probing fast signal propagation in niobium

The signals arriving on an OST are the combination of second sound waves
arriving directly from the quench position, and second sound emitted from
other positions on the cavity. The very first signal arriving on the OST will
be detected, but it could be ”emitted” by one of these positions rather than
by the quench point itself. This situation has been discussed by Eichhorn et
al. in a 2D case Eichhorn2015b . It is illustrated in 3D in Figure 15, with an
isotropic spread of the heat flux in the niobium. The heat spot limit is
represented by the dotted points, each colored set corresponding to a given
time after the beginning of the quench at the niobium surface. All the points
of one color are situated at the same distance, on the cavity surface, from
the quench point. Figure 16 shows the different signals arriving on an OST,
represented by a dark dot. The yellow line corresponds to a signal arriving
directly from the quench location (large pink point) while the red and green
lines show the signals arriving from two different points on the cavity
surface.

Figure 15: Limit of the quench spot at different times (each time corresponds
to a color). The large pink point is the quench position and the large dark
points represent OSTs. Figure 16: Trajectory of the heat flux signal from
quench spot (large pink point) to OST (large dark point) for different
combination of heat transport in niobium and helium.

To obtain the time of arrival, a minimization of the propagation in helium and
in niobium has to be performed. This was implemented in Mathematica code. The
velocity of the heat signal was fixed to 20 m/s in helium and to 70 m/s in
niobium. Using a constant propagation velocity in niobium does not capture the
underlying physical processes. These are highly dynamic and include many
influencing factors, mainly the quench propagation and the thermal diffusion.
Using a velocity to describe the processes in niobium was originally meant as
an example calculation Peters2014 . It is based on simulated results in
Liu2012 . This approach was shown to match real systems surprisingly well as
shown in Eichhorn2015b .This is the reason, why the signal propagation in
niobium is represented by this velocity.

This minimization was applied to the geometry of the measured cavities. The
results are given in Figure 17, together with a summary of the data from
Figure 5 giving results within direct line of sight propagation.

Figure 17: Difference between calculated and measured times of arrival of the
2nd sound signal on the two closest OSTs from the quench location at 1.7 K.
Calculation with direct line of sight hypothesis (blue) and calculation with
the mixed niobium/helium heat propagation model (green).

It can be seen that the calculation from this niobium-helium transport model
reduces the error quite considerably for all measurements. It fits
particularly well for cavity A with the measurements for the first OST and
with an error less than 10% for the second OST. In this case this model could
describe the measured times of arrival. However, this model does not work as
good for cavity B, where the difference between measurements and calculation
is still quite large.

Moreover this niobium-helium transport model has also been used for
calculation with varying temperature and compared to the experimental results
for test 4 with cavity B. Results are shown in Figure 18 and Figure 19. In the
whole temperature range of investigation this model reduces the error compared
to the direct line of sight propagation calculations, but the differences
keeps quite large: around 30% for the 1st OST and around -10% for the 2nd OST.

Figure 18: Time of arrival on first (red) and second (green) OST with varying
temperature for Test 4 (cavity B). Experimental data (points with error bars),
calculated data with direct line of sight propagation (solid line) and with
niobium-helium transport model (dashed line). The shaded area represent the
error on calculation due to geometrical measurements uncertainties.

Figure 19: Difference between calculated and measured times of arrival of the
2nd sound signal on the two closest OSTs from the quench location with varying
temperature for Test 4. Calculation with direct line of sight propagation
(blue) and calculation with the mixed niobium/helium heat propagation model
(green). At 2.1 K the reference measured value is the average between the two
real measured values shown in Figure

18

The difference between cavity A and cavity B could be explained by a number of
reasons. The quench was located at different spots on the cavity. While in
cavity A, the spot was on the equator, which is the region with the highest
magnetic field on the inside, the spot on cavity B was in a lower field region
(Figure 3). The different spots will cause different patterns of quench
propagation in the cavity. Another reason could be the different accelerating
field in the cavities during the quench. In cavity B it is considerably
higher, corresponding to an increase of 30% of the dissipated energy for 34
MV/m and 50% for 37 MV/m. These differences will lead to different average
velocity of the temperature signal in niobium. As this velocity was just based
on a number of assumptions, it is not surprising that the model is not an
exact fit. Nevertheless this model gives an improved prediction of the time of
arrival of the temperature signal. From these calculation it seems likely that
the fast signal propagation in the niobium causes the premature signals.

##  V Conclusion

In this paper the surprising effect of premature second sound signals in
quench localization studies has been described and experimentally shown. This
was confirmed by the combination of OST and temperature mapping measurements
on two different cavities. These measurements allowed to test different ideas
on the origin of this effect. The pure second sound propagation via the direct
line of sight from a localized quench spot was used as a starting point.

Two hypotheses were suggested to explain the early arrival of the second sound
signal. A faster signal propagation in helium was probed in a dedicated
experiment, where a heater with input power up to $1.5$ kW, flux power to
$4.3$ kW/cm2 and energy $2.8$ J was used to simulate the quench spot. Such
amounts of power of energy had never been reached in similar measurements.
However, no faster signal propagation could be found under these conditions,
which are lower than in actual cavities, but in a relevant regime. Propagation
in niobium was probed by a model, which was used to evaluate the quench tests
with the cavities. This model was based on a number of assumptions. Although
these assumptions were quite broad, a considerable improvement on the
prediction of the time of arrival for the second sound at the OSTs could be
obtained. It can be concluded, that while a fast signal propagation in helium
seems very unlikely, the processes in the niobium are likely to induce the
described effect. It is assumed that the quench propagation on the inner
cavity wall and the thermal diffusion in the niobium are the main cause, in
the case of quenching cavities.

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