text:
\section{Introduction}
If  a black hole solution is known, it is very important to carry
out the stability analysis of the black hole. At the early stage of
studying the Schwarzschild black hole, a conventional method to
determine the stability is to solve the linearized Einstein equation
by choosing even-and odd-parity perturbations under the
Regge-Wheeler gauge for graviton, which leads to two Schr\"odinger
equations: Regge-Wheeler equation~\cite{Regge:1957td} and Zerilli
equation~\cite{Zerilli:1970se}. One may conclude that the
Schwarzschild black hole is stable because their potentials are
positive definite for the whole region outside the black hole,
implying that there is no exponentially growing
modes~\cite{Vishveshwara:1970cc,chan}. Equivalently, the stability
of a black hole depends on the sign of the imaginary part
$\omega_{\rm I}$  of their quasinormal frequencies
$\omega=\omega_{\rm R}+i \omega_{\rm I}$ when considering the time
dependence of $e^{-i\omega t}$~\cite{Berti:2009kk}. If $\omega_{\rm
I}$ is negative, the black hole is stable.  The real part
$\omega_{\rm R}$ has no bearing on stability properties.
Furthermore, the unstable condition of a black hole was suggested
to be $\omega_{\rm R}=0$ and $\omega_{\rm
I}\ge0$~\cite{Konoplya:2011qq}.

On the other hand, the stability analysis of the Schwarzschild
black hole obtained from  higher derivative gravity  is not an
easy task because it contains the second-order equation for a
massive graviton. A conventional stability method designed for a
graviton with two degrees of freedom (2 DOF) is not suitable for
studying the massive graviton (5 DOF)~\cite{Fierz:1939ix} which is
propagating on the black hole and de Sitter
spacetimes~\cite{Higuchi:1986py}. However, if one considers a
lower dimensional massive gravity, the situation is not so
complicated. Reminding that the three dimensional Einstein gravity
is a gauge theory, any propagating spin-2 mode belongs to massive
graviton which can be obtained from three-dimensional massive
gravity theories. Further, these theories are classified into
parity-even and parity-odd theories.

Recently, it was shown that the BTZ black
hole~\cite{Banados:1992wn,Banados:1992gq} is stable for all $\mu$
(Chern-Simons coupling constant) against the massive spin-2
perturbations in the topologically massive gravity
(TMG~\cite{Deser:1981wh}, parity-odd theory) by demanding
boundedness of the perturbation at the
horizon~\cite{Birmingham:2010gj}. On the other hand, it was
suggested that the BTZ black hole is stable for $m^2>1/2\ell^2$ in
new massive gravity (NMG, parity-even
theory)~\cite{Bergshoeff:2009hq} by computing quasinormal
frequencies and performing the $s$-mode
analysis~\cite{Myung:2011bn}.

Because of different parity, one uses  different stability analysis
for the BTZ black hole. Solving the first-order differential tensor
equation algebraically together with the boundary conditions, we
obtain all quasinormal frequencies of massive graviton for
parity-odd theories~\cite{Sachs:2008gt}: TMG  and generalized
massive gravity (GMG)~\cite{Bergshoeff:2009aq}.  If $\omega_{\rm I}
< 0$, the black hole seems to be  stable against the massive
graviton perturbation.

Given the parity-odd first order linearized equation, one obtains
its second-order linearized equation after squaring it, which
belongs to the parity-even theory, giving the ambiguity on sign of
the mass. Because of this ambiguity, someone prefers solving the
first-order equation directly~\cite{Sachs:2008gt}, instead of  the
second-order equation. Off-critical point, the parity-even gravity
theory usually provides the second-order linearized equation after
choosing the transverse-traceless gauge. It is known that ``solving
directly the second-order massive  equation" is a formidable task
for the BTZ black hole spacetimes. Fortunately, after choosing the
BTZ black string  perturbation for massive
graviton~\cite{Liu:2008ds}, the $s$-mode analysis may be performed
using two Schr\"odinger equations with frequency-dependent
potentials~\cite{Myung:2011bn}.

In this work, we study the massive graviton stability of the BTZ
black hole obtained from three-dimensional\footnote{It is well-known
that the  Einstein gravity in three dimensions has no propagating
degrees of freedom (DOF). This is  clearly shown  by counting a
massless graviton $h_{\mu\nu}$: $D(D-3)/2$. One has zero DOF for $D
=3$.  For a massive graviton,  it is changed into $ (D-2)(D + 1)/2$
which gives  2 DOF for $D = 3$. Thus,  massive generalizations of
the Einstein gravity~\cite{Bergshoeff:2009hq,Deser:1982vy}, allow
propagating degrees of freedom.  The three dimensional massive
gravity is regarded as a toy model of perturbative quantum gravity,
since we expect to have  less severe short-distance behavior than
four dimensional gravity with non-renormalizability.  We note that
if a black hole solution in three dimensional massive gravity is
found, a first issue is to examine its classical stability
properties as will be performed in the present paper.} massive
gravities which are classified into the parity-odd theories (TMG,
GMG) and parity-even gravity theories (NMG, six-derivative gravity
(SDG)~\cite{Bergshoeff:2012ev}). For the second-order
Schr\"{o}dinger  equation with an effective potential, we analyze
the massive graviton stability by checking if the potential is
positive for the whole range outside an event horizon.  However, we
should extend the stability condition  (\ref{flat-st}) for
asymptotically flat spacetimes to (\ref{xi-c}) for asymptotically
AdS spacetimes. In this case, the stability condition of $s$-mode is
given by  the generalized Breitenlohner-Freedman bound for spin-2
field. On the other hand, for a first-order massive graviton
equation, we perform the stability analysis by using the quasinormal
frequencies $(\omega=\omega_{\rm R}+i \omega_{\rm I})$. If
$\omega_{\rm I}$ is negative when considering the time dependence of
$e^{-i\omega t}$~\cite{Berti:2009kk}, the black hole is stable.
 Finally, we explore the black hole
stability connection between the parity-odd and parity-even
($s$-mode) massive gravity theories.

\section{Parity-even massive gravities}
In this work, we consider  the (non-rotating)
  BTZ black hole solution \cite{Banados:1992wn} which is a solution to all massive gravity theories. Its line element is
 given by
\begin{eqnarray}
ds_{{\rm BTZ}}^2&=&\bar{g}_{\mu\nu}dx^{\mu}dx^{\nu}\nonumber\\
 &=&-\left(-{\cal M}+\frac{r^2}{\ell^2}\right)dt^2+
 \left(-{\cal
 M}+\frac{r^2}{\ell^2}\right)^{-1}dr^2+r^2d\phi^2,\label{btz}
\end{eqnarray}
where ${\cal M}$ is the ADM mass given to be ${\cal M}=r_+^2/\ell^2$
with the horizon radius $r_+$ and AdS$_3$ curvature radius $\ell$.
Throughout the paper, the overbar denotes the background metric
\eqref{btz} for the BTZ black hole. The Ricci scalar, Ricci tensor,
Riemann tensor can be written in terms of the background metric
\eqref{btz} as follows:
\begin{eqnarray}
\bar{R}=6\Lambda,~~~~~\bar{R}_{\mu\nu}=2\Lambda
\bar{g}_{\mu\nu},~~~~~
\bar{R}_{\mu\nu\rho\sigma}=\Lambda(\bar{g}_{\mu\rho}\bar{g}_{\nu\sigma}
-\bar{g}_{\mu\sigma}\bar{g}_{\nu\rho}),
\end{eqnarray}
where $\Lambda=-1/\ell^2$. Also we adopt a notation of $(-,+,+)$ and
unit of $2\kappa^2=1$. For the perturbation around the BTZ black
hole
\begin{eqnarray}\label{pert}
g_{\mu\nu}=\bar{g}_{\mu\nu}+h_{\mu\nu},
\end{eqnarray} the
linearized Ricci tensor and Ricci scalar are given by
\begin{eqnarray}
\delta
R_{\mu\nu}(h)&=&\frac{1}{2}\Big(\bar{\nabla}_{\mu}\bar{\nabla}^{\rho}h_{\rho\nu}
+\bar{\nabla}_{\nu}\bar{\nabla}^{\rho}h_{\rho\mu}-\bar{\nabla}^2h_{\mu\nu}
-\bar{\nabla}_{\mu}\bar{\nabla}_{\nu}h\Big)+3\Lambda
h_{\mu\nu}-\Lambda \bar{g}_{\mu\nu}h,\label{lrmunu}\\
\delta
R(h)&=&\bar{\nabla}_{\alpha}\bar{\nabla}_{\beta}h^{\alpha\beta}-\bar{\nabla}^2h-2\Lambda
h\label{lr}.
\end{eqnarray}

Let us first introduce a three-dimensional massive gravity proposed
by Fierz and Pauli (FP) \cite{Fierz:1939ix}  whose action is given
by
\begin{eqnarray}\label{sfp}
S_{\rm FP}=S_{\rm bl}(h)-\frac{M_{\rm FP}^2}{4}\int
d^3x\sqrt{-g}\left(h_{\mu\nu}h^{\mu\nu}-h^2\right),
\end{eqnarray}
where $M_{\rm FP}^2$ is a mass parameter and $S_{\rm bl}(h)$ is the
bilinear form of the Einstein-Hilbert action with a cosmological
constant $\Lambda$. It is well known that the linearized Einstein
equation can be written as \cite{Higuchi:1986py}
\begin{eqnarray}\label{fpmeq}
\Big(\bar{\nabla}^{2}-2\Lambda-M_{\rm FP}^2\Big)h_{\mu\nu}^{\rm
FP}=0,
\end{eqnarray}
which is considered as a simplest equation for a massive graviton
with 2 DOF on the BTZ black hole spacetimes. In deriving this
equation, we have used the consistency condition of the linearized
Bianchi identity
\begin{equation} \label{tt-g} \bar{\nabla}_\mu
h^{\mu\nu}=0,~~h^{\mu}_{~\mu}=0,
\end{equation}
which is considered  as the transverse-traceless (TT)
gauge\footnote{We note that the action $S_{\rm FP}$ (\ref{sfp})
has no diffeomorphism invariance, while the TT gauge condition is
imposed only when considering diffeomorphism invariant actions of
$S_{\rm SDG},S_{\rm TMG}$, and $S_{\rm GMG}. $}.

 Now we  consider the parity-even
six-derivative gravity (SDG) \cite{Bergshoeff:2012ev}, whose action
is given by
\begin{eqnarray}\label{spet}
\hspace*{-1em}S_{{\rm SDG}}=\int d^3x\sqrt{-g} \Big[\sigma
R-2\lambda_S+\alpha R^2+\beta R_{\mu\nu}R^{\mu\nu}
+a_1\nabla_{\mu}R\nabla^{\mu}R+
a_2\nabla_{\rho}R_{\mu\nu}\nabla^{\rho}R^{\mu\nu}\Big],\label{pet}
\end{eqnarray}
where $\sigma=0,\pm1$ is a dimensionless parameter, $\lambda_S$ is a
cosmological parameter with mass dimension 2. Here parameters
$\alpha(\beta)$ have mass dimension $-2$ and $a_1(a_2)$ have $-4$.
We remark that when choosing $a_1=a_2=0$, $\sigma=1,$ and
$8\alpha+3\beta=0$, the action (\ref{spet}) reduces to the NMG
action \cite{Bergshoeff:2009hq} as
\begin{eqnarray}\label{snmg}
S_{\rm NMG}=\int d^3x\sqrt{-g}\Bigg[R-2\lambda_S-\frac{1}{m^2}\left(
R_{\mu\nu}R^{\mu\nu}-\frac{3}{8}R^2\right)\Bigg],
\end{eqnarray}
where $m^2$ is a mass parameter with dimension $2$.

Varying the action  \eqref{pet} with respect to  $g^{\mu\nu}$ leads
to the  equation
\begin{eqnarray}\label{peteq}
\sigma\Big(R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R\Big)+\lambda_S
g_{\mu\nu}+E_{\mu\nu}+H_{\mu\nu}=0
\end{eqnarray}
with
\begin{eqnarray}
E_{\mu\nu}&=&\beta\Big[-\frac{1}{2}g_{\mu\nu}R_{\rho\sigma}R^{\rho\sigma}
+2R_{\mu\rho\nu\sigma}R^{\rho\sigma}+\nabla_{\gamma}\nabla^{\gamma}
R_{\mu\nu}+\frac{1}{2}g_{\mu\nu}\nabla_{\gamma}\nabla^{\gamma}R
-\nabla_{\mu}\nabla_{\nu}R\Big]\nonumber\\
&&+\alpha\Big[2RG_{\mu\nu}+2g_{\mu\nu}\nabla_{\gamma}\nabla^{\gamma}R
-2\nabla_{\mu}\nabla_{\nu}R\Big],\\
&&\nonumber\\
H_{\mu\nu}&=&a_1\Big[\nabla_{\mu}R\nabla_{\nu}R-2R_{\mu\nu}\nabla^2R
-\frac{1}{2}g_{\mu\nu}\nabla_{\rho}R\nabla^{\rho}R-2(g_{\mu\nu}\nabla^2
-\nabla_{\mu}\nabla_{\nu}\nabla^2)R\Big]\nonumber\\
&&a_2\Big[\nabla_{\mu}R_{\rho\sigma}\nabla_{\nu}R^{\rho\sigma}
-\frac{1}{2}g_{\mu\nu}\nabla_{\gamma}R_{\rho\sigma}\nabla^{\gamma}R^{\rho\sigma}
-\nabla^2R_{\mu\nu}-g_{\mu\nu}\nabla^{\rho}\nabla^{\sigma}\nabla^2R_{\rho\sigma}
\nonumber\\
&&+2\nabla^{\rho}\nabla_{(\mu}\nabla^2R_{\nu)\rho}
+2\nabla^{\rho}R_{\rho\sigma}\nabla_{(\mu}R_{\nu)}^{\sigma}
+2R_{\rho\sigma}\nabla^{\rho}\nabla_{(\mu}R_{\nu)}^{\sigma}
-2R_{\sigma(\mu}\nabla^2R_{\nu)}^{\sigma}\nonumber\\
&&-2\nabla_{\rho}R_{\sigma(\mu}\nabla_{\nu)}R^{\rho\sigma}
-2R_{\sigma(\mu}\nabla^{\rho}\nabla_{\nu)}R_{\rho}^{\sigma}\Big].
\end{eqnarray}
We note that the BTZ black hole solution (\ref{btz}) to Eq.
(\ref{peteq}) is allowed only when choosing
$\lambda_{S}=\sigma\Lambda-2(3\alpha+\beta)\Lambda^2$. Taking into
account the perturbation (\ref{pert}) and plugging  the TT gauge
(\ref{tt-g}) into  Eq.(\ref{peteq}), we obtain  the sixth-order
differential perturbation equation, which can be factored into three
pieces\footnote{In order to eliminate  scalar graviton, we require
three conditions  as \cite{Bergshoeff:2012ev}
\begin{eqnarray} a_1=-3a_2/8,~~\alpha=\Lambda
a_2/8-3\beta/8,~~-\sigma/2+3\Lambda^2
a_2/4-\Lambda\beta/4\neq0.\nonumber
\end{eqnarray}
}:
\begin{eqnarray}\label{petpeq}
\Big[\bar{\nabla}^2-2\Lambda\Big]
\Big[\bar{\nabla}^2-2\Lambda-M_+^2\Big]
\Big[\bar{\nabla}^2-2\Lambda-M_-^2\Big]h_{\mu\nu}=0.
\end{eqnarray}
Here the mass parameters $M^2_{\pm}$ denote
\begin{eqnarray}
M_{\pm}^2=\frac{\beta}{2a_2}-\Lambda\pm\frac{1}{2a_2}
\sqrt{10a_2^2\Lambda^2-6a_2\beta\Lambda+4a_2\sigma+\beta^2}.
\end{eqnarray}
In  Eq. (\ref{petpeq}), we read off two massive equations
\begin{eqnarray}\label{petmeq}
\Big[\bar{\nabla}^2-2\Lambda-M_+^2\Big]h_{\mu\nu}^{M_+}=0,~~~
\Big[\bar{\nabla}^2-2\Lambda-M_-^2\Big]h_{\mu\nu}^{M_-}=0
\end{eqnarray}
off-critical points ($M^2_+\not=M^2_-$). They describe 4 DOF for two
massive gravitons.

 Also, we note that for the NMG (\ref{snmg}), the
linearized equation is given by
\begin{eqnarray}\label{nmgmeq}
\Big[\bar{\nabla}^2-2\Lambda -M^2_{\rm NMG}\Big]h_{\mu\nu}^{\rm
NMG}=0, ~~M_{\rm NMG}^2~=~m^2-\frac{1}{2\ell^2}
\end{eqnarray}
off critical point ($m^2\neq1/2\ell^2$) and off-decoupling
limit\footnote{In the decoupling limit of $m^2\to0$, however, NMG
action (\ref{snmg}) reduces to massless NMG \cite{Deser:2009hb}
where  the fourth order equation appears, instead of the  second
order equation.} ($m^2\neq0$), which describes 2 DOF for a massive
graviton in three dimensional spacetimes.
\subsection{$s$-mode stability analysis}

We are now in a position to perform the stability analysis of
massive gravitons satisfying Eqs. (\ref{fpmeq}), (\ref{petmeq}), and
(\ref{nmgmeq}).  We propose that they  are propagating  on the BTZ
black hole background (\ref{btz}).   For this purpose, inspired by
the BTZ black string perturbations~\cite{Liu:2008ds},  we consider
the following two distinct (orthogonal) perturbations ansatz
\cite{Myung:2011bn}:\\
 the
 type I has two off-diagonal components $h_0$ and $h_1$
\begin{eqnarray}
h^I_{\mu\nu}=\left(
\begin{array}{ccc}
0 & 0 & h_0(r) \cr 0 & 0 & h_1(r) \cr h_0(r) & h_1(r) & 0
\end{array}
\right) e^{-i\omega t}e^{ik\phi} \,, \label{type1}
\end{eqnarray}
while for the type II, the metric tensor takes the form with four
components $H_0,~H_1,~H_2,$ and $H_3$ as
\begin{eqnarray}
h^{II}_{\mu\nu}=\left(
\begin{array}{ccc}
H_0(r)  & H_1(r) & 0  \cr H_1(r) & H_2(r) & 0 \cr 0 & 0 & H_3(r)
\end{array}
\right) e^{-i\omega t}e^{ik\phi} \,. \label{type2}
\end{eqnarray}
In this work, we focus on $s$-mode ($k=0$) case for simplicity.
Importantly,   Eqs. (\ref{fpmeq}), (\ref{nmgmeq}), and
(\ref{petmeq}) can be combined into a single massive equation
\begin{eqnarray}\label{meq}
\left(\bar{\nabla}^2-2\Lambda-M_{i}^2\right)h_{\mu\nu}^{M_i}=0,
\end{eqnarray}
where  $M_i^2(i=1,2,3,4)$ is  given by
\begin{eqnarray}
M_1^2=M_{{\rm FP}}^2,~ M_2^2=M_{\rm
NMG}^2,~M_3^2=M_{+}^2,~M_4^2=M_{-}^2.
\end{eqnarray}
For type I, plugging (\ref{type1}) into (\ref{meq}) and eliminating
$h_1(r)$ from $(t,\phi)$ and $(r,\phi)$ components of (\ref{meq})
lead to the Schr\"{o}dinger equation as
\begin{eqnarray}\label{schPhi}
\frac{d^2\Phi_{i}}{dr^{*2}}+\Big[\omega^2-V^{\rm
I}_{\Phi_i}\Big]\Phi_{i}=0,
\end{eqnarray}
where $r^*$ is the tortoise coordinate defined by the relation of
$dr^*=\ell^2dr/(r^2-r_+^2)$. Here, a new field $\Phi_i$ is defined
by $\Phi_i=h_0/\sqrt{r\{M_i^2(r^2-r_+^2)/\ell^2-\omega^2\}}$, and
$V^{\rm I}_{\Phi_i}$ is the $\omega$-dependent potential given by
\begin{eqnarray}\label{v1}
&&V^{\rm I}_{\Phi_i}(\omega,r)=\frac{r^2-r_+^2}{\ell^2}
\Bigg[M_{i}^2+\frac{15}{4\ell^2}-\frac{3r_+^2}{4\ell^2r^2}+
\frac{3M_i^4 r^2(r^2-r_+^2)} {\ell^6\left\{M_i^2(r^2-r_+^2)/\ell^2-\omega^2\right\}^2}\nonumber\\
&&\hspace{11em}
+\frac{2M_i^2(2r_+^2-3r^2)}{\ell^4\left\{M_i^2(r^2-r_+^2)/\ell^2-\omega^2\right\}}\Bigg].
\end{eqnarray}
We show that for $M_i^2\ge0$, all potentials $V^{\rm I}_{\Phi_i}$
are always positive definite for the whole range of $r_+\le
r\le\infty$. This may imply that for  $M_i^2\ge0$,  the BTZ black
hole  is stable against type I perturbation.

On the other hand, in type II case, substituting (\ref{type2}) into
(\ref{meq}) and after some manipulations, $(t,\phi)$ component of
(\ref{meq}) can be written as the other  Schr\"{o}dinger equation:
\begin{eqnarray} \label{schpsi}
\frac{d^2\Psi_i}{dr^{*2}}+[\omega^2-V^{\rm
II}_{\Psi_i}(\omega,r)]\Psi_i=0,
\end{eqnarray}
where $\Psi_i=H_1\sqrt{r(r^2-r_+^2)^2}/
\sqrt{M_i^2(r_+^2-r^2)\ell^2+(2r_+^2-r^2)+\omega^2\ell^4}$ and the
$\omega$-dependent potential $V^{\rm II}_{\Psi_i}$ is given by
\begin{eqnarray}\label{v2}
&&\hspace{-2em}V^{\rm
II}_{\Psi_i}(\omega,r)=\frac{r^2-r_+^2}{\ell^2}
\Bigg[M_i^2+\frac{7}{4\ell^2}-\frac{3r_+^2}{4\ell^2r^2}
 +\frac{3r^2(M_i^2+1/\ell^2)^2(r^2-r_+^2)} {\ell^6
 \left\{M_i^2(r_+^2-r^2)/\ell^2+(2r_+^2-r^2)/\ell^4+\omega^2\right\}^2}
\nonumber\\&&\hspace{8em}+\frac{4(M_i^2+1/\ell^2)(r^2-r_+^2)}
{\ell^4\left
\{M_i^2(r_+^2-r^2)/\ell^2+(2r_+^2-r^2)/\ell^4+\omega^2\right\}}\Bigg].
\end{eqnarray}
We note that for $M_i^2\ge0$, all  potential $V^{\rm II}_{\Psi_i}$
is always positive definite for the whole range of $r_+\le
r\le\infty$, which states  that the BTZ black hole is stable against
type-II perturbation.

Hence, if one applies  type I and II perturbations to the
parity-even massive gravities, the stability conditions of the BTZ
black hole seem to be
\begin{eqnarray} \label{flat-st}
M_{\rm FP}^2 \ge 0,~~~~ m^2\ge \frac{1}{2\ell^2}, ~~~~~
M_{\pm}^2\ge0
\end{eqnarray}
in FP, NMG, and SDG, respectively. However, these conditions are
suitable for asymptotically flat spacetimes. We remind the reader
that our spacetime is asymptotically anti de Sitter spacetimes.
Therefore, we have to point out what is  the stability condition of
a massive graviton propagating on the AdS$_3$ spacetimes.   To see
this explicitly, let us consider asymptotically AdS$_3$ spacetimes,
which corresponds to a large $r$ limit ($r^{*}\to0$) in
Eq.(\ref{btz}). In this limit, the potentials (\ref{v1}) and
(\ref{v2})  take the same form  when expressing them in terms of a
tortoise coordinate $r^*$
\begin{eqnarray}
V^{\rm I}_{\Phi_i},~V^{\rm
II}_{\Psi_i}\sim\frac{\xi}{r^{*2}},\end{eqnarray} where
\begin{eqnarray} \xi=\ell^2\left(M_i^2+\frac{3}{4\ell^2}\right).
\end{eqnarray}
 As $r^{*}$ approaches $0$, Eqs. (\ref{schPhi}) and (\ref{schpsi})
become one-dimensional Schr\"{o}dinger equation with an inverse
square potential of the strength $\xi$ and the energy $E=\omega^2$.
It is known~\cite{Case:1950an,Moon:2012dy} that in this case, if
$\xi$ satisfies the condition,
\begin{eqnarray} \label{xi-c}
\xi\ge-\frac{1}{4}~~\Rightarrow~~M_i^2\ge-\frac{1}{\ell^2},
\end{eqnarray}
the energy spectrum is always continuous and positive. It is worth
noting that the stability condition (\ref{xi-c}) is consistent with
the regularized condition at $r^{*}=0$. Importantly, the stability
condition (\ref{xi-c}) is exactly the same with the
Breitenlohner-Freedman (BF) bound \cite{Breitenlohner:1982bm} for a
massive spin-2 field in AdS$_3$
spacetimes~\cite{Gover:2008sw,Lu:2011qx}
\begin{eqnarray}
\hspace*{-2em}\Big[\nabla_{(\rm AdS)}^2-2\Lambda-M_{(\rm
AdS)}^2\Big]h_{\mu\nu}=0~~~\Rightarrow~~~M_{(\rm AdS)}^2\ge M_{\rm
BF}^2=-\frac{1}{\ell^2}.
\end{eqnarray}
Hence, we should extend the stability condition (\ref{flat-st}) for
asymptotically flat spacetimes to the stability condition
(\ref{xi-c}) for asymptotically AdS spacetimes.

Finally, we dictates the stability condition of the BTZ black hole
\begin{equation} \label{e-sc}
M_{\rm FP}^2\ge-\frac{1}{\ell^2},~~~~ m^2\ge -\frac{1}{2\ell^2},
~~~~{\rm and}~~ M_{\pm}^2\ge -\frac{1}{\ell^2}
\end{equation}
off-critical points ($m^2\neq1/2\ell^2,~M_{\pm}^2\neq0$) and
off-decoupling limit ($m^2\neq0$), when using the $s$-mode analysis
for the parity-even massive gravity theories.

\section{Parity-odd massive gravities}
A parity-odd massive gravity in three dimensions was first
introduced  by Deser, Jackiw, and Templeton \cite{Deser:1982vy}. The
TMG  action includes a gravitational Chern-Simons term, which
reveals parity-violation or `odd' parity. In this section, we
introduce two parity-odd massive gravities of TMG and GMG, and
investigate the stability analysis of the BTZ black hole in those
gravities.
\subsection{TMG}
The action of TMG  with a negative
cosmological constant is given by \cite{Deser:1982vy}
\begin{eqnarray}
S_{{\rm TMG}}=\int d^3x\sqrt{-g}\Big[R-2\Lambda +\frac{1}{2\mu}
\epsilon^{\lambda\mu\nu}\Gamma_{\lambda\sigma}^{\rho}
\Big(\partial_{\mu}\Gamma_{\rho\nu}^{\sigma}
+\frac{2}{3}\Gamma_{\mu\tau}^{\sigma}\Gamma_{\nu\rho}^{\tau}\Big)\Big],
\end{eqnarray}
where $\mu$ is a parameter with mass dimension 1. The Einstein
equation takes the form
\begin{eqnarray}\label{tmg}
R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R+\Lambda g_{\mu\nu}
+\frac{1}{\mu}C_{\mu\nu}=0,
\end{eqnarray}
where the Cotton tensor $C_{\mu\nu}$ is defined by
\begin{eqnarray}\label{cmunu}
C_{\mu\nu}\equiv\epsilon_{\mu}^{~\alpha\beta}\nabla_{\alpha}
(R_{\beta\nu}-\frac{1}{4}g_{\beta\nu}R).
\end{eqnarray}
Introducing the perturbation (\ref{btz})
and applying the TT gauge condition (\ref{tt-g}) to the linearized
equation of $(\ref{tmg})$, we arrive at
\begin{eqnarray}\label{tmgpeq}
\Big[\bar{\nabla}^2-2\Lambda\Big]\Big[h_{\mu\nu} +\frac{1}{\mu}
\epsilon_{\mu}^{~\alpha\beta}\bar{\nabla}_{\alpha}h_{\beta\nu}\Big]=0.
\end{eqnarray}
From (\ref{tmgpeq}), we read off the first-order differential equation for a
massive graviton
\begin{eqnarray}\label{tmgeq}
\epsilon_{\mu}^{~\alpha\beta}\bar{\nabla}_{\alpha}h_{\beta\nu} +\mu
h_{\mu\nu} =0.
\end{eqnarray}
One can easily check that squaring it [equivalently, by applying
the first-order operator
$\epsilon_{\sigma}^{~\rho\mu}\bar{\nabla}_\rho-\mu
\delta_{\sigma}^{\mu}$ to (\ref{tmgeq})] leads to the second-order
equation
\begin{eqnarray}\label{tmgeq2}
\Big[\bar{\nabla}^2-2\Lambda-M_{\rm TMG}^2\Big]h_{\sigma\nu}=0
\end{eqnarray}
with $M_{\rm TMG}^2=\mu^2-1/\ell^2$. Using the bound given by the
stable condition (\ref{e-sc}), we have
\begin{equation}
M_{\rm TMG}^2 \ge -\frac{1}{\ell^2} \to \mu^2 \ge 0 \to |\mu| \ge 0
\end{equation}
which is consistent with that obtained in~\cite{Birmingham:2010gj},
indicating that the BTZ black hole is stable for all $\mu$ against
the massive spin-2 perturbation in TMG by demanding boundedness of
the perturbation at the horizon. The latter condition eliminates
 modes  which are growing in time and obeying the generalized boundary
conditions at asymptotic infinity. At this stage, we emphasize that
the authors in~\cite{Birmingham:2010gj} have used not the
first-order equation (\ref{tmgeq}) itself but a second-order
hypergeometric equation obtained by transforming two first-order
equations when analyzing the stability of the BTZ black hole.

\subsection{GMG}
We consider the GMG action which
consists of NMG and gravitational Chern-Simons term as \cite{Bergshoeff:2009hq,Liu:2009pha}
\begin{eqnarray}
&&\hspace*{-2em}S_{{\rm GMG}}=\int d^3x\sqrt{-g}\Big[\sigma
R-2\lambda_G +\frac{1}{m^2}\left(
R_{\mu\nu}R^{\mu\nu}-\frac{3}{8}R^2\right)+\frac{1}{2\mu}
\epsilon^{\lambda\mu\nu}\Gamma_{\lambda\sigma}^{\rho}
\Big(\partial_{\mu}\Gamma_{\rho\nu}^{\sigma}
+\frac{2}{3}\Gamma_{\mu\tau}^{\sigma}\Gamma_{\nu\rho}^{\tau}\Big)\Big],
\nonumber\\
&&\label{sgmg}
\end{eqnarray}
where $\lambda_G$ is a cosmological parameter with mass dimension 2.
From the GMG action, one derives  the Einstein equation
\begin{eqnarray}\label{gmgeq}
\sigma G_{\mu\nu}+\lambda_G
g_{\mu\nu}+\frac{1}{2m^2}K_{\mu\nu}+\frac{1}{\mu}C_{\mu\nu}=0,
\end{eqnarray}
where $C_{\mu\nu}$ is given by Eq.(\ref{cmunu}) and $K_{\mu\nu}$
takes the form
\begin{eqnarray}\label{kmunu}
  K_{\mu\nu}&=&2\nabla^2R_{\mu\nu}-\frac{1}{2}\nabla_\mu \nabla_\nu R-\frac{1}{2}\nabla^2Rg_{\mu\nu}\nonumber\\
        &+&4R_{\mu\rho\nu\sigma}R^{\rho\sigma} -\frac{3}{2} R R_{\mu\nu}-R_{\rho\sigma}R^{\rho\sigma}g_{\mu\nu}
         +\frac{3}{8}{R}^2 g_{\mu\nu}.
\end{eqnarray} It is pointed out that the BTZ black
hole solution (\ref{btz}) is allowed only for
$\lambda_G=\Lambda^2/4m^2+\sigma\Lambda$. Using (\ref{pert}) and the
TT gauge condition (\ref{tt-g}), the linearized equation of
(\ref{gmgeq}) can be written as
\begin{eqnarray}
\Big[\bar{\nabla}^2-2\Lambda\Big] \Big[\bar{\nabla}^2h_{\mu\nu}
+\frac{m^2}{\mu}\epsilon_{\mu}^{~\alpha\beta}\bar{\nabla}_{\alpha}
h_{\beta\nu}+\left(\sigma
m^2-\frac{5}{2}\Lambda\right)h_{\mu\nu}\Big]=0.
\end{eqnarray}
Considering the above equation, we read off the second-order equation of the massive
graviton
\begin{eqnarray}\label{gmgpeq}
\bar{\nabla}^2h_{\mu\nu}
+\frac{m^2}{\mu}\epsilon_{\mu}^{~\alpha\beta}\bar{\nabla}_{\alpha}
h_{\beta\nu}+\left(\sigma m^2-\frac{5}{2}\Lambda\right)h_{\mu\nu}=0
\end{eqnarray}
which is further factorized into
\begin{eqnarray}
\Big[\delta_{\mu}^{\beta}+
\frac{1}{m_+}\epsilon_{\mu}^{~\rho\beta}\bar{\nabla}_{\rho}
\Big]\Big[\delta_{\beta}^{\gamma}
+\frac{1}{m_-}\epsilon_{\beta}^{~\sigma\gamma}\bar{\nabla}_{\sigma}
\Big]h_{\gamma\nu}=0.
\end{eqnarray}
Here $m_{\pm}$ take the forms
\begin{eqnarray}\label{mpm}
m_{\pm}=\frac{m^2}{2\mu}\pm\sqrt{\frac{m^4}{4\mu^2}-\sigma
m^2-\frac{\Lambda}{2}}.
\end{eqnarray}
This  implies that two massive gravitons with mass $m_\pm$  are
described by two first-order equations, respectively,
\begin{eqnarray}\label{gmgpm}
\epsilon_{\mu}^{~\alpha\beta}\bar{\nabla}_{\alpha}h_{\beta\nu} +m_+
h_{\mu\nu}
=0,~~~\epsilon_{\mu}^{~\alpha\beta}\bar{\nabla}_{\alpha}h_{\beta\nu}
+m_- h_{\mu\nu} =0.
\end{eqnarray}
As squaring their first-order equations, acting two operations
[$\epsilon_{\sigma}^{~\rho\mu}
\bar{\nabla}_{\rho}-m_+\delta_{\sigma}^{\mu}$] and
[$\epsilon_{\sigma}^{~\rho\mu}
\bar{\nabla}_{\rho}-m_-\delta_{\sigma}^{\mu}$] on (\ref{gmgpm})
leads to two second-order equations
\begin{eqnarray}\label{gmgpm2}
\Big[\bar{\nabla}^2-2\Lambda-M_{\rm
GMG+}^2\Big]h_{\sigma\nu}=0,~~~\Big[\bar{\nabla}^2-2\Lambda-M_{\rm
GMG-}^2\Big]h_{\sigma\nu}=0,
\end{eqnarray}
where
\begin{equation}
M_{{\rm GMG}\pm}^2=m_{\pm}^2-\frac{1}{\ell^2}.
\end{equation}
 Using the bound given by the stable condition (\ref{e-sc}),
we have the mass bound
\begin{equation} \label{even-st}
M_{\rm GMG \pm}^2 \ge -\frac{1}{\ell^2} \to m_{\pm}^2 \ge 0.
\end{equation}

Consequently,  the $s$-mode stability condition  after squaring
their first-order equations is given by
\begin{equation} \label{s-sodd}
m^2_i \ge 0,
\end{equation}
where
\begin{equation}
m^2_1=\mu^2({\rm TMG}),~~~~m^2_2=m^2_+({\rm
GMG}),~~~~m^2_3=m^2_-({\rm GMG}).
\end{equation}
\subsection{Quasinormal mode analysis}
We note that (\ref{tmgeq}) and (\ref{gmgpm}) belong to the
first-order equation and they are parity-odd, while (\ref{tmgeq2})
and (\ref{gmgpm2}) are the second-order equation and are
parity-even. Furthermore, (\ref{tmgeq2}) and (\ref{gmgpm2}) have
ambiguities on the sign of mass. Hence, it would be better to use
(\ref{tmgeq}) and (\ref{gmgpm}) than (\ref{tmgeq2}) and
(\ref{gmgpm2}) when considering an another stability analysis for
the parity-odd theories.

In this section, we redo  the stability analysis of the massive
graviton for TMG and GMG by computing quasinormal frequencies. For
this purpose, we note that  the type I and II perturbations are
suitable for the second-order differential equations (\ref{tmgeq2})
and (\ref{gmgpm2}), while these are inappropriate for applying to
the first-order equations (\ref{tmgeq}) and  (\ref{gmgpm}) directly.
As will be shown in Appendix,  applying type I and II to
(\ref{tmgeq}) and (\ref{gmgpm}) leads to all null perturbations due
to the parity-oddness of their equations.

Therefore, we perform  the stability analysis with solving
(\ref{tmgeq}) and (\ref{gmgpm})
 to find quasinormal frequencies by following the approach developed in \cite{Sachs:2008gt}.
We first note  that (\ref{tmgeq}) and  (\ref{gmgpm}) can be
written as a single first-order equation
\begin{eqnarray}\label{first}
\epsilon_{\mu}^{~\alpha\beta}\bar{\nabla}_{\alpha}h_{\beta\nu} +m_i
h_{\mu\nu} =0,
\end{eqnarray}
where $m_i$ denote $\mu$ and $m_{\pm}$. For our purpose, we consider
a BTZ black hole metric with ${\cal M}=1$ and $\ell=1$ $ (r_+=1)$
given in global coordinates
\begin{eqnarray}
ds^2_{\rm gc}=\bar{g}_{\mu\nu}dx^{\mu}dx^{\nu}=-\sinh^2\rho dt^2+\cosh^2\rho
d\phi^2+d\rho^2,\label{btz1}
\end{eqnarray}
which is obtained by  replacing $r$ by $r=\cosh\rho$ in (\ref{btz}).
In what follows we use light-cone coordinates with $u=t+\phi$ and
$v=t-\phi$ as
\begin{eqnarray}\label{lc}
ds^2_{\rm
lc}=\bar{g}_{\mu\nu}dx^{\mu}dx^{\nu}=\frac{1}{4}du^2+\frac{1}{4}dv^2
-\frac{1}{2}\cosh2\rho dudv+d\rho^2.
\end{eqnarray}
The  metric (\ref{lc}) admits isometry group of
SL(2,R)$\times$SL(2,R), which allows us two sets of the Killing
vector fields, $L_{0,\pm1}$ and $\bar{L}_{0,\pm1}$ given as
\begin{eqnarray}
L_0&=&-\partial_u,\nonumber\\
L_{-1}&=&e^{-u}\left(-\frac{\cosh2\rho}{\sinh2\rho}\partial_u
-\frac{1}{\sinh2\rho}\partial_v-\frac{1}{2}\partial_{\rho}\right),\nonumber\\
L_{1}&=&e^{u}\left(-\frac{\cosh2\rho}{\sinh2\rho}\partial_u
-\frac{1}{\sinh2\rho}\partial_v+\frac{1}{2}\partial_{\rho}\right)
\label{L}
\end{eqnarray}
and $\bar{L}_{0,\pm1}$ are defined by
operation of  $u\leftrightarrow v$ in (\ref{L}). Three vector fields
$L_{0,\pm1}$ satisfy the SL(2,R) algebra
\begin{eqnarray}
[L_0,L_{\pm1}]=\mp L_{\pm1},~~~[L_1,L_{-1}]=2L_0.
\end{eqnarray}
Taking  the perturbation ansatz in terms of  $(u,v,\rho)$
\begin{eqnarray}
h_{\mu\nu}=e^{-i\omega t-ik\phi}\psi_{\mu\nu}(\rho) =e^{-ip_+ u-ip_-
v}\psi_{\mu\nu}(\rho),~~~p_{\pm}=\frac{1}{2}(\omega\pm
k),
\end{eqnarray}
the $s$-mode ($k=0$) solutions to the first-order equation
(\ref{first}) are  given by  right($r$)/left($l$) moving modes:
\begin{eqnarray}
h^{r}_{\mu\nu}=e^{-2h_r t}\psi_{\mu\nu}= e^{-2h_r
t}(\sinh\rho)^{-2h_r}\left(
\begin{array}{ccc}
1 & 0 & \frac{2}{\sinh2\rho} \cr 0 & 0 & 0 \cr \frac{2}{\sinh2\rho}
& 0 & \frac{4}{\sinh^22\rho}
\end{array}
\right)\,,~~~h_r=-\frac{1}{2}(m_i+1) \label{sol1}
\end{eqnarray}
for $p_-=-ih_r$ and
\begin{eqnarray}
h^{l}_{\mu\nu}=e^{-2h_lt}\psi_{\mu\nu}=
e^{-2h_lt}(\sinh\rho)^{-2h_l}\left(
\begin{array}{ccc}
0 & 0 & 0 \cr 0 & 1 & \frac{2}{\sinh2\rho} \cr 0 &
\frac{2}{\sinh2\rho} & \frac{4}{\sinh^22\rho}
\end{array}
\right)\,,~~~h_l=\frac{1}{2}(m_i-1) \label{sol2}
\end{eqnarray}
for $p_+=-ih_l$. Note that the solution (\ref{sol1}) satisfies the
chiral highest weight condition of $\bar{L}_1h_{\mu\nu}=0$ and
(\ref{sol2}) satisfies  the anti-chiral highest weight condition
of $L_1h_{\mu\nu}=0$ which are equivalent to  the transversality
condition of $\bar{\nabla}_{\mu}h^{\mu}_{~\nu}=0$. However,
requiring both conditions leads to null modes.  Considering
relations $p_\pm=\omega^{r/l}/2$, the corresponding quasinormal
frequencies, whose quasinormal modes satisfy the boundary
conditions: ingoing modes at the horizon and Dirichlet boundary
condition at infinity, can be written as
\begin{eqnarray}
&&\omega^r=-2ih_r=2i \Big(\frac{m_i}{2}+\frac{1}{2}\Big),\\
&&\omega^l=-2ih_l=2i \Big(-\frac{m_i}{2}+\frac{1}{2}\Big).
\end{eqnarray}
The complete tower of right- and left-moving quasinormal modes is
generated by acting $L_{-1}\bar{L}_{-1}$ on $h^{r/l}_{\mu\nu}$ $n$
times as
\begin{equation}
h^{(n),~ r/l}_{\mu\nu}
=\Big(L_{-1}\bar{L}_{-1}\big)^nh^{r/l}_{\mu\nu},
\end{equation}
which leads to their quasinormal frequencies with overtone number
$n$
\begin{eqnarray}
&&\omega^r_{n}=-2i(h_r+n),\\
&&\omega^l_{n}=-2i(h_l+n).
\end{eqnarray}

 Since the stability condition is determined  by two basic
quasinormal frequencies with $\omega^{r/l}_{\rm I}<0$ for
$\omega^{r/l}=\omega_{{\rm R}}^{r/l}+i\omega_{\rm I}^{r/l}$, it is
given by
\begin{eqnarray} \label{odd-s}
|m_i|>1/\ell,
\end{eqnarray}
where the AdS$_3$ curvature radius $\ell$ is restored for
convenience. It seems, however, that there is some discrepancy
between  (\ref{odd-s}) and (\ref{s-sodd}).

\section{Discussions}
In this work, we have established the stability of the massive
graviton around BTZ black hole in massive gravity theories which are
classified into the parity-even gravity theories (NMG, SDG) and the
parity-odd theories (TMG, GMG). For the parity-even massive
gravities, the stability conditions employed by the $s$-mode
analysis are exactly the same with the BF-bound, which corresponds
to $M_i^2\ge-1/\ell^2$ (\ref{xi-c}). For the parity-even gravity
theories, the $s$-mode analysis and the BF-bound based on the
second-order massive equation are consistent with the
Birmingham-Mokhtari-Sachs result requiring the boundedness of
perturbation at the horizon where a second-order hypergeometric
equation was used. These are given by the condition of $m^2_i \ge 0
 ~(|m_i|\ge0)$.

We stress  that  the stability analysis performed by the quasinormal
frequencies gave a condition of $|m_i|>1/\ell$, being different from
$|m_i|\ge0$.
 We may  interpret it by mentioning that the connection between potential and quasinormal frequencies
 condition is guaranteed  if  the second-order equation  is used  as  Schr\"odinger equation~\cite{Moon:2011sz,Berti:2009kk,Konoplya:2011qq}.
 We here have obtained the quasinormal frequencies by using the first-order equation.
Hence, the condition of $|m_i|>1/\ell$ based on quasinormal modes
does not comprise the stability condition $|m_i| \ge 0$  obtained by
solving the second-order equation.  In this case,  the unstable
quasinormal modes exist for $0\le|m_i|\le1/\ell$. It is, however,
pointed out that these unstable modes may  be truncated by requiring
the boundedness at the horizon when considering the generalized
boundary conditions at asymptotic infinity \cite{Birmingham:2010gj}.
Hence, it suggests that  the stability condition might   be extended
to comprise
\begin{eqnarray}
|m_i|\ge0.
\end{eqnarray}

Finally, we would like to mention that the stability of a black hole
in four-dimensional massive gravity is determined by the
Gregory-Laflamme instability of a five-dimensional  black string. It
turned out that the small Schwarzschild black hole in the dRGT
massive gravity~\cite{Babichev:2013una} and fourth-order
gravity~\cite{Myung:2013doa} is unstable against the metric and
Ricci tensor perturbations.  In the present work, the stability was
mainly determined by the asymptotes of black hole spacetimes. Hence,
it suggests for a future direction that the stability of the massive
graviton around a BTZ black hole  would be revisited   by using the
(in)stability of four-dimensional black string.

\section*{\bf Acknowledgments}

 This work was supported  by the National
Research Foundation of Korea (NRF) grant funded by the Korea
government (MEST) (No.2012-R1A1A2A10040499).

\section*{Appendix: Inappropriateness of type I and II for parity-odd theory}

We first substitute type I perturbation (\ref{type1}) into the TMG
equation (\ref{tmgeq}). It turns out that  $(t,\phi)$ and
$(r,\phi)$ components of  (\ref{tmgeq}) yield just $\mu h_0(r)=0$
and $\mu h_1(r)=0$, which implies that type I perturbation becomes
null unless $\mu=0$.  Similarly, for type II perturbation
(\ref{type2}),  the components $(t,t), (r,r), (\phi,r),$ and
$(\phi,\phi)$ of
 (\ref{tmgeq}) are given by
\begin{eqnarray}
\mu H_0(r)~=~0, ~~~\mu H_2(r)~=~0,~~~\mu H_1(r)&=&0,~~~\mu
H_3(r)~=~0,\nonumber
\end{eqnarray}
which leads to all null components for the type II perturbation of
 $H_0=H_1=H_2=H_3=0$ unless $\mu=0$.

For GMG,  applying type I perturbation (\ref{type1}) to
(\ref{gmgpeq}), we find that the corresponding solution to $(t,r)$
and $(r,r)$ components of Eq.(\ref{gmgpeq}) is given by
$h_0(r)=h_1(r)=0$.   In type II case (\ref{type2}), we note  that
$H_0(r)$, $H_2(r)$, and $H_3(r)$ can be expressed in terms of
$H_1(r)$ by considering the traceless condition, $(\phi,t)$
component, and $(\phi,r)$ component in (\ref{gmgpeq}), respectively:
\begin{eqnarray} \label{adx}
&&H_0(r)=\frac{{\cal M}-r^2/\ell^2}{\omega r}\Big[({\cal
M}-3r^2/\ell^2)H_1(r)+r({\cal M}-r^2/\ell^2)H_1^{\prime}(r)\Big],\nonumber\\
&&H_2(r)=\frac{1}{\omega({\cal M}-r^2/\ell^2)}\Big[({\cal
M}-r^2/\ell^2)H_1^{\prime}(r)-2rH_1(r)/\ell^2\Big],\nonumber \\
&&H_3(r)=-\frac{r({\cal M}-r^2/\ell^2)}{\omega}H_1(r).\nonumber\\
\end{eqnarray}
Substituting (\ref{adx}) into $(t,r)$ and $ (t,\phi)$ components
of (\ref{gmgpeq}), we find  $H_1(r)=0$. In this case, it yields
$H_0(r)=H_2(r)=H_3(r)=0$ when using (\ref{adx}) again. As a result
type II perturbation becomes null for parity-odd gravity theories.

This proves the inappropriateness of type I and II for parity-odd
theory.

\newpage

redpajama_set_name: RedPajamaArXiv


text:
Shotts at the World Pipe Band Championships
by Megan Canning | Oct 4, 2013
The build-up to the 2013 World Pipe Band Championships was an intense week for the band, with practices taking place at the Shotts band hall, as well as outside West Brewery at Glasgow Green on the Thursday and Friday before the competition. Band members also attended events during Glasgow's Piping Live! festival, in which band members performed and socialised.
With the Worlds approaching, the change in format to create a new two day event was an unknown quantity. The new two day set-up would require a qualifier on the Saturday the 17th of August where each band would perform both an MSR and Medley, and then a 12 band final on Sunday the 18th where the top bands to make it through would play again with a second MSR and Medley. Grade 1 was split into two qualifying heats, and the winner was determined from the Sunday final.
For Shotts and Dykehead Caledonia, the goal was to get through the Saturday qualifying round, but also to preserve the instruments for the possibility of four performances. The weather would be the biggest variable and remained the biggest worry. As Saturday morning approached, the forecast of heavy rain looked foreboding. Shotts was drawn in the first heat, and would play the MSR in the morning with the Medley in the afternoon. Conditions were terrible all morning, with several bands having to both play and prepare in the pouring rain. Shotts initially enjoyed a brief dry spell in which to tune up, but was soon plagued by rain again in final tuning which persisted through the whole of the band's qualifying MSR performance. After drying out instruments and uniforms, the band returned again to play its Medley in the afternoon. Unlike the morning, the band tuned up and competed in dry and sunny conditions, and was very pleased with its afternoon performance. You can listen to the band's qualifying medley performance here below:
The band made it through to the Sunday final, and the Shotts drum corps had won the drumming in both the qualifying MSR and Medley events. The band now had to do it all over again. On Sunday morning, rain fell intermittently and the band prepared in the same way as it had the day before. Unlike other bands in the early part of the morning, Shotts' tune-up, including final tuning, had remained relatively dry and the band progressed to the line. However, as the band began to play its MSR, a torrential downpour of biblical standards swept across Glasgow Green. While the rain had been bad throughout the weekend, nothing had come down as rapidly or as overwhelmingly as this particular downpour, and band members could see the crowd ducking for cover as members played on. Even the judges struggled to continue, as the severe wind which had brought the downpour played havoc with capes and score sheets. By the time the band finished its MSR performance it received a huge applause from the audience in collective appreciation for surviving the weather, and emerged soaked, but ready for the challenge of the afternoon.
Shotts perform in the most difficult of conditions
After the soaking in the MSR performance, the band had a lot of work to do to turn around quickly for the Medley. The afternoon tune-up was essentially starting from scratch, with changeable conditions which continued to test the band under pressure. As the band presented its medley in its final performance of the weekend, direct sunshine emerged towards the end of the performance which presented an opposite but equally difficult challenge for the pipes. All in all, like every band on the park that weekend, Shotts and Dykehead Caledonia had done what it could to brave the conditions, perform its best under difficult circumstances, and carry on. The band finished 8th overall, and felt proud of its overall presentation across a difficult weekend. You can watch the BBC's official coverage of the band's final Medley here:

redpajama_set_name: RedPajamaCommonCrawl


On Saturday May 10, 2014 Africa Access highlighted great books at the Book Fair. Author Mubina Kirmani was on hand to share her 2014 CABA Best Book, Bundle of Secrets. Children made Anansi spiders, got spider tattoos and explored CABA winners.


text:
Q: Using a Wordpress plugin feature elsewhere within theme files I'm using the ultimate member online plugin, which shows members' online status.
When enabled to show on a user's UM profile page, it looks like this, for example:
When creating posts, it is common for data pertaining to the author to display somewhere within the post. For example:

What I'd like to accomplish is to have online status (shown in first image, above) also be displayed within this author data, shown within a post.
I've poked around in theme/plugin files, and I'm CLOSE!

*

*I found where I need to edit, within the theme files (wp-content/themes/plum/inc/template-tags.php file)

*I've narrowed the plugin files and content within them down, to what i believe would be the appropriate code to insert into theme files. Screenshots and copied [relevant] text can be found here:

*Ive been playing around, reading the web, creating a post on the official wp forums (as well as searching within them), and this is the best I've got so far:
On the surface, it looks like the problem is solved. However, that user ("testAuthor1"), is offline (and the greenlight there in that very last imgur link displays the user as online).

redpajama_set_name: RedPajamaStackExchange


text:
NC State, UNC serve as warning signs
Jessica Mark
The recent decisions by UNC-Chapel Hill and North Carolina State to shift all undergraduate classes online has many Wake Forest students wondering whether their university will soon follow suit.
UNC's reopening guidelines included all the basic precautions: social distancing, masks, the self-monitoring of symptoms and size limits on gatherings. A week after classes began, however, UNC's quarantine dorm was almost filled to capacity. Outbreaks grew, and UNC had to revise its plan to keep students safe. With over 500 students in quarantine and isolation, NC State took the same measure on Aug. 20.
In the aftermath of these events, President Nathan Hatch sent a message to Wake Forest students on Friday, Aug. 21, warning them about the high stakes under which the school is operating this semester.
"Our success relies on each person doing their part," Hatch said. "Please remember as we go into the weekend, one large gathering, one party that infects a large number of people, can close us down."
The university's Our Way Forward plan underscores how the entire community has a responsibility to adhere to the guidelines outlined on the site. That includes students wearing masks and social distancing, but also includes the university's administration embracing the reality that the college experience transcends the classroom. The student experience at Wake Forest is just as much a social experience, where one feels a sense of community and inclusion, as it is an academic experience.
Hatch's cautionary words capture a very small part of the reality on campus and, like the initial plan at UNC-Chapel Hill, will likely not be sufficient to protect against outbreaks.
Freshmen came to campus last week yearning to make new friends. Upperclassmen reunited after having their spring semester cut short. The temptation to socialize and gather in groups is high. In addition, as both students and administrators know, the first week of classes is notorious for parties as the intensity of classes has not yet kicked in.
The challenges facing the Wake Forest community are exacerbated by actions taken to de-densify residence halls. With an increased amount of students living off-campus, the university has less control over student activity.
Students at other universities have taken an activist approach against their school's reopening plans. At UNC, students, faculty and campus workers gathered to protest the university's reopening. The protesters began with a moment of silence for those who passed away from COVID-19, and they then laid on the quad for over 30 minutes. Lying down was intended to send a message of accountability: when the Chapel Hill community became infected with COVID-19, the university would be accountable.
The reality of living on campus through the current pandemic is clear: there is risk. The extent of that risk will likely hinge on collective efforts to work together and be transparent about what is taking place on campus, but there is no way to completely shield students from the spread of the virus at Wake Forest.
With multiple college campuses in close proximity to Wake Forest having to make dramatic changes to their semester plans, the question for the Reynolda campus community is obvious — will the university go remote? Will the current approach to the virus be sufficient, or will it result in outbreaks even greater than those seen elsewhere?
Although circumstances may seem discouraging, there is a potential silver lining: Wake Forest has the ability to learn from the experiences of other schools. In fact, on Tuesday, the university implemented new safety measures. These measures include surveillance testing and a campus operating color-coded status system. There are five colors included in the status system: Green, Yellow, Orange, Red and Purple. Each color indicates a different level of viral presence in the Wake Forest community. Depending on the color/operating level of the university, different restrictions will be implemented (to learn more about the dashboard, turn to page 1). As of Aug. 26, the university's operating level is Yellow, which signifies the campus will be operating under "new normal" restrictions.
The WFU administration also released a COVID-19 dashboard that sets forth the amount of positive, laboratory-verified test results that have been communicated to the university. As of Aug. 26, the dashboard confirms 11 cases, with four of those over the last five days. The university was not clear about how many tests were performed in order to confirm these positive cases. In a message to the campus community on Aug. 25, Hatch explained the purpose behind the new initiatives.
"These three measures will aid our ability to be prepared, stay informed and remain healthy," Hatch said.
However, in the midst of new school closures throughout North Carolina, the question still remains: will these measures be enough to keep all Wake Forest students safe?
WFU updates COVID policies for Spring 2022
Successful booster shot clinic
Booster shots come to campus
COVID-19 cases rise above 100
In Charleston, South Carolina, longing for normalcy reflects nationwide struggle
Vaccine will be required for Fall 2021
WFU vaccinates Group Four students
University returns to Yellow after six months
Wake Forest mishandled hundreds of COVID-19 tests


text:
Unleashing the Immune System's Cancer-Killing Potential
Greg Breining , special to AAMCNews
After years of failing to live up to its promise, immunotherapy is saving tens of thousands of lives today.
Two cytotoxic T cells (red) attack a cancer cell (white).
Rita Elena Serda, Duncan Comprehensive Cancer Center at Baylor College of Medicine, National Cancer Institute, NIH
For years, immunotherapy—a treatment intended to bolster the immune system to eradicate cancer—was an exciting idea that had not lived up to its promise.
Until recently, immunotherapy was "not very successful at all," said James P. Allison, PhD, chair of the Department of Immunology at The University of Texas MD Anderson Cancer Center in Houston. "That was one of the reasons it was kind of held in disrepute."
But attitudes changed in the last decade because of the success of clinical trials of monoclonal antibodies, such as ipilimumab (trade name Yervoy), which was approved by the Food and Drug Administration (FDA) in 2011 for the treatment of melanoma. The development of ipilimumab was based on Allison's work while he was director of the Cancer Research Laboratory at the University of California, Berkeley.
"We're past the stage where we say, 'Oh, this next thing will take it over the top,' because 'over the top' is where we are now. There are tens of thousands of people alive who wouldn't be otherwise."
James P. Allison, PhD
Fast forward to August 2017. The FDA approved a cancer therapy, developed by researchers at the University of Pennsylvania, that genetically engineers a patient's T cells to more effectively attack leukemia cells and, in some patients, eradicate the disease. The technology, chimeric antigen receptor (CAR) T, is approved specifically for the treatment of refractory B-cell acute lymphocytic leukemia. This therapy is also showing promise in treating chronic lymphocytic leukemia and non-Hodgkin lymphoma.
"We're past the stage where we say, 'Oh, this next thing will take it over the top,' because 'over the top' is where we are now," said Allison, also a primary inventor of a kind of immunotherapy known as checkpoint blockade. "There are tens of thousands of people alive who wouldn't be otherwise."
Other researchers agree that immunotherapy has turned the corner.
The National Institutes of Health (NIH) is the driving force for biomedical research and discovery in this country, says John Burklow of NIH.
"I think it's been dramatically successful. Patients are achieving complete regression of their cancers. And there's a significant percentage of these patients who have very long remissions and are probably cured," says Gerald P. Linette, MD, PhD, chief medical officer for Cancer Immunotherapy at Penn Medicine and professor of medicine at the Hospital of the University of Pennsylvania in Philadelphia.
"Historically our role [in academic medicine] has been more in early discovery or basic research—getting funding from the NIH to study a very fundamental aspect related to the immune system," said Linette. "But I would say at Penn and a few other places around the country that have the infrastructure, we can actually take it one step further and do real translational research … we take a good idea from the lab and because of the facilities and expertise can actually develop our own treatments, independent of industry."
Many of us live with mutations that lead to cancer. But tumors never develop because our ever-watchful immune systems identify the mutations, mobilize antibodies and leucocytes, and clear the malignant cells.
Sometimes, however, our immune systems are overwhelmed or fail to respond. Immunotherapy is designed to help the immune system by revving up its response or making the cancer cells more vulnerable.
A time-tested precursor to modern immunotherapy is bone marrow transplantation, developed first in the 1950s at the Fred Hutchinson Cancer Research Center in Seattle, Wash. Cancer vaccines such as BCG (bacille Calmette-Guerin) and interferon alfa and interleukin-2 have been used to stimulate immune systems. But today, two newer forms of immunotherapy are generating the most excitement: checkpoint inhibitors and CAR T- cells.
Checkpoint inhibitors, said Allison, "take the brakes off the T cells," the specialized class of white blood cell that intercept and destroy intruders such as viruses and mutant cells. Cancer cells often defend themselves by displaying surface molecules that latch onto a molecular receptor on the T cells. Blocking this "checkpoint" turns the T cells off and allows tumor cells to avoid immune destruction.
Researchers are finding checkpoint inhibitors to block these interactions and allow the T cells to continue their attack. The FDA approved this therapy, referred to as "checkpoint blockade" for melanoma in 2011. This is the treatment former president Jimmy Carter received for melanoma that had metastasized. After having surgery and radiation therapy as well, Carter announced in 2015 that his cancer was undetectable.
"There's still a lot of basic research and laboratory research that needs to take place to fulfill the promise and potential of cancer immunotherapy. We need more [financial] investment combined with higher rates of clinical trial participation."
Gerald P. Linette, MD, PhD
Alison conducted a randomized placebo-controlled trial with patients in late-stage melanoma. "Of the patients who were treated, about 20% of them are alive four years after they were treated," reported Allison, a pioneer in developing checkpoint inhibitors. Two prominent checkpoints on T cells are PD-1 and CTLA-4. "If you give patients the two drugs, one targeting CTLA-4 and the other targeting PD-1, about 60% [of melanoma patients] survive at least two years later," he continued. "They're basically cured."
New approvals of checkpoint blockades followed for treatment of lung cancer, bladder cancer, some types of colon cancer, kidney cancer, and Hodgkin lymphoma, said Allison. In May 2017, the FDA approved a PD-1 blockade (with the monoclonal antibody pembrolizumab, trade name Keytruda) for several cancers—including colorectal, endometrial, and gastrointestinal—with a common biomarker. This is the first time an immunotherapy treatment was based on a common biomarker for different kinds of cancer.
With CAR T, another remarkable immunotherapy, T cells are drawn from the patient and engineered to produce chimeric antigen receptors on their surface. The CAR proteins allow T cells to recognize an antigen on tumor cells. The engineered cells are multiplied to the millions and infused back into the patient. The cells attack the cancer and remain in the body long after treatment to continue their attack on malignant cells.
National Cancer Institute Collaborates With Academic Medicine
Researchers stress that many advances in immunotherapy would not have been possible without funding from the National Institutes of Health (NIH) and particularly the National Cancer Institute. In fact, medical schools sometimes collaborate directly with the NIH.
Nicholas P. Restifo, MD, director of the Center for Cell-based Therapy at the National Cancer Institute, coauthored a study recently published in Nature showing that more than 100 protein-encoding genes in melanoma cells play a role in allowing the immune system to do its work. If some of these genes are missing through mutations in the cancer cells, the T cells struggle to identify or destroy the cancer.
Restifo combed through the human genome, using CRISPR-Cas9 as a gene-editing "scissors" to systematically "knock out" genes, by mimicking mutations, in 123,000 locations to determine which genes were vital to immune system targeting.
Though Restifo's NIH lab devised and ran the study, the work couldn't have been done without the help of researchers at the Broad Institute of MIT and Harvard, he said. "[They] are innovators in CRISPR, so they are really good at creating those scissors we were talking about." Likewise, researchers at the New York Genome Center, a consortium of medical universities and laboratories, are specialists in analyzing the DNA, he added. "It takes a lot of different experts to collaborate."
Immunotherapy may have advantages other therapies don't.
"[With immunotherapy] you're not attacking the tumor directly, so unlike chemotherapy or the small molecules [that penetrate the cancer cell membrane], you don't really need to know what is causing the tumor," said Allison. And once you've got the cells—genetically modified T-cells or checkpoint blockade—"you've got memory," he continued. "Once you get an immune response started, the immune system itself can take care of new antigens that appear [for the rest of a person's life]."
Thanks to academic medicine
Academic institutions are in a good position to study the whole panoply of problems and come up with novel solutions, noted Ross McKinney Jr., MD, AAMC chief scientific officer. "These are difficult, complicated problems that require a variety of different strategies."
"Some of our institutions have been the real pioneers in the immunotherapy world," McKinney observed. "That's the role of the academic medical center—to generate the idea, to demonstrate that it has potential, and then, because we are not in the business of selling drugs and therapies, we license or hand it off to corporate partners that are used to doing that kind of work."
Despite rapid recent advances in immunotherapy, Linette noted, "We're nowhere near close to successfully treating all patients that walk through the door. There's still a lot of basic research and laboratory research that needs to take place to fulfill the promise and potential of cancer immunotherapy [for people with all cancers]. We need more [financial] investment combined with higher rates of clinical trial participation."
Greg Breining, special to AAMCNews
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The recent death of 60-year-old Gregory Murrey after falling over an upper deck railing at Turner Field has raised awareness of one of dangers of the modern large-capacity ballparks. While fans have died from falls in the past, it's really been in the last 50 years that there has been a spike in such fatalities. And with good reason: today's mega-stadiums are much larger and higher than those in earlier decades. What follows is a list of these fatalities from 1969 to the present, including suicides.


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Shop Vinyl Augustus Pablo – Thriller (RSD
Augustus Pablo – Thriller (RSD
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Thriller is the 1975 album from Augustus Pablo, the reggae visionary who popularized the use of the melodica in the genre. As the founder of multiple labels, he was a crucial part of invigorating the burgeoning dub reggae scene in it's early days. The record's warm, mellow sound makes for a breezy and hypnotic listen, perfectly paired with a jazz cigarette (for those that partake). ORG Music will reissue this lost gem for the first time since the 70's on transparent red vinyl, limited to a run of 2,000 copies, exclusively for RSD Black Friday.
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Hissing Tiles – Boychoir
Boychoir is the band's second LP, and first for the experimental label Whited Sepulchre (Midwife, Planning For Burial, Forest Management). Recorded at Cincinnati's historic Ultrasuede [...]
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Thriller is the 1975 album from Augustus Pablo, the reggae visionary who popularized the use of the melodica in the genre. As the founder of multiple labels, he was a […]
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Andreas Rudolff Bodenstein von Karlstadt, född omkring 1480 i Karlstadt, inte långt från Würzburg, död 24 december 1551 i Basel, var en tysk teolog, och en av reformationens märkligaste personligheter.

Biografi
Efter att ha studerat i Erfurt och Köln kom Karlstadt till Wittenberg, där hans filosofiska lärarverksamhet snart förskaffade honom stort anseende. Han erhöll där 1508 en kyrklig tjänst och blev 1510 teologie doktor; 1515 gjorde han en studieresa till Rom, där han förvärvade juridisk doktorsgrad. Karlstadt trädde nu i opposition mot den förr av honom hyllade skolastiska teologin, vilken han bekämpade på den augustinska nådeslärans grundval. Hans offentliga disputation med Johann Eck i Leipzig bidrog till att ytterligare klargöra för honom hans åskådnings oförenlighet med den gamla kyrkoläran; och sedan han 1520 blivit bannlyst på samma gång som Luther, blev brytningen med Rom fullständig.

Efter en tids reformatorisk verksamhet i Danmark återvände Karlstadt under Luthers Wartburgstid till Wittenberg och tog där ledningen över den reformatoriska rörelsen. Han genomförde då med rådets bifall en del kyrkliga reformer, nattvardens utdelande under bröd och vin, avskaffandet av hostians upplyftande och så vidare; därjämte angrep han häftigt kyrkobildernas bruk, på grund av att de lätt föranledde återfall i katolska tankegångar. Karlstadts kyrkoreformer stötte emellertid på motstånd, bland annat hos Fredrik den vise, och sedan oroligheter utbrutit i Wittenberg och Luther 1522 återkommit, avskaffades Karlstadts reformer.

Han miste nu allt inflytande i staden och ägnade sig åt skriftställarverksamhet samt sedan mitten av 1523 åt prästerlig tjänst i Orlamünde. Där fortsatte han sin i puritansk riktning gående kyrkoreformatoriska verksamhet, dock utan att gå så långt som den bekante Thomas Müntzer, vilken av Karlstadt manades till ett mera återhållsamt tillvägagångssätt. Sedan Karlstadt emellertid brutit med Wittenberg och Luther förgäves sökt hejda de oroligheter som utbrutit i Orlamünde, blev Karlstadt i september 1524 utvisad från landet. 1525 skrev Luther sin mot Karlstadt och "svärmeandarna" riktade skrift Wider die himmlischen propheten, på vilken Karlstadt svarade med Anzeig etlicher hauptartikel christlicher lehre, den bland Karlstadts talrika publikationer som mest belyser hans åskådning. Denna period av Karlstadts liv var rik på svårigheter och umbäranden. Han fick visserligen genom Luthers förmedling 1526 tillstånd att återvända till Sachsen, men tvingades återigen 1528 att bege sig bort. Efter växlande öden kom han 1530 till Zwingli i Schweiz och blev 1534 predikant och professor i Basel.

Karlstadt och hans ställning till Luther har blivit mycket olika bedömda. Under början av 1900-talet började man efter att under en lång tid sett honom som en bråkstake, se honom som föregångaren till en nyare tids innerligare personlighetsreligion och omvärderade något hans insats som representant för de mystiska riktningarna under reformationstiden.

Källor

Tyska professorer i teologi
Personer verksamma vid Wittenbergs universitet
Reformatorer
Teologer under 1500-talet
Tyska forskare under 1500-talet
Personer från Karlstadt
Födda 1480-talet
Avlidna 1551
Män
Ugglan
Alumner från Kölns universitet

redpajama_set_name: RedPajamaWikipedia


Now, all we need is to hear from you about what you think would be the best topics to present that day for an audience that is made up of patients, loved ones, nurses, and mental health professionals. Are there any issues you are interested in hearing more about, any burning questions about PNES?
Introduction to Psychogenic non-epileptic Seizures (PNES) –Lorna Myers, Ph.D.
How are psychogenic seizures diagnosed- Christos Lambrakis, M.D.
The neuropsychology of PNES-Robert Trobliger, Ph.D.
Maintaining your quality of life while living with PNES-Marcela Bonafina, Ph.D.
Caring for the Caretaker and local available resources-Ruifan Zeng, Ph.D.
We might repeat a couple because this is useful for newcomers who haven't attended previous conferences but we want to add new topics. So, please help. Type in any ideas you may have and we will seriously consider them.
By the way, registration will open soon. As soon as that happens, I will announce it. Also, we are seeing about getting educational credits for psychologists and about perhaps having the event filmed so it can go online.
Veteran Affairs and PNES / PTSD for one.
I'll keep going through my resources and get back with you.
1. Best ways to find treatment or a doctor that will understand how to treat your symptoms.
2. Maybe why most doctors don't believe symptoms of patients or how to address a doctor or nurse that says a patient is faking their symptoms.
How to deal with PNES in a school setting. Resources for schools. Education for schools.
Yes, please! Resources for schools & also for the parents of children with PNES!
Research medicinal cannabis affects on combined diagnosis of PNES and mental health conditions.
-handling communication in an interview. What do you disclose?
-communication with your main caregiver..anger, frustrations build. How can both parties handle this?
Ways to educate those closest to you.
How about turning it around and having a topic on PNES mimmics? For examaple, I was told by two neurologists and a psychiatrist – in a very non-compassionate way – that I had PNES and needed to be hospitalized as a psychiatric inpatient. When I went for a second opinion, the neurologist didn't even look at more than 11 hours of recording before "confirming" PNES. I went through eight months of extremely distressing "PNES" episodes, four neurologists (none of whom would discuss specifics about my tests with me), 12 weeks of PNES-based counseling, and having to justify a third opinion to my insurance company.
I have hyperekplexia, or excessive startle response. This was determined through a combination of EEG, EMg, and video monitoring, all of which could have been done by/at any of the previous testing centers. With the proper dose of clonazepam, my episodes disappeared.
Please consider this as an important topic.
read my article in International Journal of Epilepsy, efirst. published 8-14-18. A Proposed Etiology of Psychogenic Nonepileptic Seizures. You are right. PNES diagnosis is wrong.
I hope to attend! I live in Texas but my family is in CT so I will try to find a way to commute. I would like to hear more about physiological non-epileptic seizures. I had them due to a medication adverse reaction and I haven't found much information on them. None of my doctors considered that cause as a possibility and my neuropsychologist had never heard of them.
Lorna Myers, thanks so much for the post.Really thank you! Great.
PNES and PTSD and those with onset on vocal tics.
Although I won't be able to attend. I would like to hear considerations or thoughts of having a specialized in patient facility for those who can or have the ability to seek a form of treatment that could help train a person into thinking and feeling ways to help cope and minimize their PNES.
Also having different methods of therapies that may be helpful.


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Classic idea clarifies weird quantum action in ultracold gas
September 17th, 2020 Posted by Sonia Fernandez-UCSB
Researchers have used classical concepts to decipher strange quantum behaviors in an ultracold gas.
There they were, in all their weird quantum glory: ultracold lithium atoms in the optical trap. Held by lasers in a regular, lattice formation and "driven" by pulses of energy, these atoms were doing crazy things.
"A lot of funny things happen when you shake a quantum system."
"It was a bit bizarre," says David Weld, an associate professor who runs a lab in the physics department at the University of California, Santa Barbara. "Atoms would get pumped in one direction. Sometimes they would get pumped in another direction. Sometimes they would tear apart and make these structures that looked like DNA."
A link between classical and quantum physics?
These new and unexpected behaviors were the results of an experiment the researchers conducted to push the boundaries of our knowledge of the quantum world. The outcomes? New directions in the field of dynamical quantum engineering, and a tantalizing path toward a link between classical and quantum physics.
"A lot of funny things happen when you shake a quantum system," says Weld, whose lab creates "artificial solids"—low-dimensional lattices of light and ultracold atoms—to simulate the behavior of quantum mechanical particles in more densely packed true solids when subjected to driving forces.
The recent experiments were the latest in a line of reasoning that stretches back to 1929, when physicist and Nobel Laureate Felix Bloch first predicted that within the confines of a periodic quantum structure, a quantum particle under a constant force will oscillate.
"They actually slosh back and forth, which is a consequence of the wave nature of matter," Weld says. While these position-space Bloch oscillations were predicted almost a century ago, they were directly observed only relatively recently; in fact Weld's group was the first to see them in 2018, with a method that made these often rapid, infinitesimal sloshings large and slow, and easy to see.
A decade ago, other experiments added a time dependency to the Bloch oscillating system by subjecting it to an additional, periodic force, and found even more intense activity. Researchers discovered oscillations on top of oscillations—super Bloch oscillations.
Then things got weird…
For this study, the researchers took the system another step further, by modifying the space in which these atoms interact.
"We're actually changing the lattice," says Weld, by way of varying laser intensities and external magnetic forces that not only added a time dependency but also curved the lattice, creating an inhomogenous force field. Their method of creating large, slow oscillations, he adds, "gave us the opportunity to look at what happens when you have a Bloch oscillating system in an inhomogenous environment."
This is when things got weird. The atoms shot back and forth, sometimes spreading apart, other times creating patterns in response to the pulses of energy pushing on the lattice in various ways.
"We could follow their progress with numerics if we worked hard at it," Weld says. "But it was a little bit hard to understand why they do one thing and not the other."
It was insight from lead author Alec Cao, who is beginning his fourth year at UC Santa Barbara's College of Creative Studies, that led to a way of deciphering the strange behavior.
"When we investigated the dynamics for all times at once, we just saw a mess because there was no underlying symmetry, making the physics challenging to interpret," says Cao,.
To draw out the symmetry, the researchers simplified this seemingly chaotic behavior by eliminating a dimension (in this case, time) by utilizing a mathematical technique initially developed to observe classical nonlinear dynamics called a Poincaré section.
"In our experiment, a time interval is set by how we periodically modify the lattice in time," Cao says. "When we chucked out all the 'in-between' times and looked at the behavior once every period, structure and beauty emerged in the shapes of the trajectories because we were properly respecting the symmetry of the physical system."
Observing the system only at periods based on this time interval yielded something like a stop-motion representation of these atoms' complicated yet cyclical movements.
"What Alec figured is that these paths—these Poincaré orbits—tell us exactly why in some regimes of driving the atoms get pumped, while in other regimes of driving the atoms spread out and break up the wave function," Weld adds.
A decades-long quest
One direction the researchers could take from here, he says, is to use this knowledge to engineer quantum systems to have new behaviors through driving, with applications in burgeoning fields such as topological quantum computing.
"But another direction we can take is looking at whether we can study the emergence of quantum chaos as we start to do things like add interactions to a driven system like this," Weld says.
It's no small feat. Physicists for decades have been trying to find links between classical and quantum physics—a common math that might explain concepts in one field that seem to have no analog in the other, such as classical chaos, the language for which does not exist in quantum mechanics.
"You've probably heard of the butterfly effect—a butterfly flapping its wings in the Caribbean can cause a typhoon somewhere across the world," says Weld.
"That's actually a feature of classical chaotic systems, which have a sensitive dependence on initial conditions. That feature is actually very hard to reproduce in quantum systems—it's puzzling to come up with the same explanation in quantum systems. So this is maybe a small piece of that body of research."
The research appears in Physical Review Research.
Source: UC Santa Barbara
Original Study DOI: 10.1103/PhysRevResearch.2.032032
Quantum cosmology explains anomalies beyond Einstein's physics
Quantum weirdness rewrites rules about heat in empty space
Team hits major milestone: quantum supremacy


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← What Communists Can Learn From Native Struggle: Sogorea Te and Primitive Accumulation
Behold! The Urban Miner →
Queer Liberation and Class Struggle Case Study: The Welsh Miner's Strike
Can you imagine a 100% male industry of miners acting in solidarity with communities of gays and lesbians? Can you imagine them dancing in queer spaces together, learning about each other over a beer? Can you imagine these men marching behind members of the queer community and under a queer banner in a parade? Seems hard to imagine . . .
Identity politics has long maintained that differences of identity along lines of race, gender, ability, and sexuality must be respected and tolerated. This has often been counterposed to what is characterized as the "class reductionist" approach of class unity above all else. The history of race riots, domestic violence, and macho heterosexism within the proletarian movement is all too real, and has provided the material basis for a form of postmodern politics in the 1980s towards today which has defensively fetishized forms of social difference.
Within this context, the communist movement has sought in various ways to reconcile the contradictions and move towards a higher plane of political unity. Unfortunately, often times these moves have ended up reifying differences – bowing to forms of sectoralism which keep differences static, with each "identity group" staying safe within its own silo – or attempts to paper over the real differences and antagonisms which exist in society and amongst the proletariat in particular, which amount to reifying the differences from another angle.
As a young generation of communists coming up in this context, we are seeking to carve out a space which can account for difference while also aiming towards the pedagogical development of radical understanding and unity amongst people of different backgrounds and identities. We seek out the common class interest amongst the proletariat in ways which facilitate the learning process amongst all its sectors – where male identified workers are able to break down the patriarchal values instilled in them since birth; where women workers are able to assert themselves amongst men and feel confident that they will not be silenced by a culture of machismo; where trans workers are able to occupy space with their fellow cis-workers and engage in radical dialogue that facilitates understanding between them, along with creating the space to strategize about taking down the boss.
This is a process of communist universalism, in which the common class interest amongst workers is sought neither by disregarding difference nor fetishizing it into sectoralism. It is a communist process which allows the sectors to commune with one another and seek the way forward through solidarity.
Some critics say that "privileged identities" such as straight white men are not willing to be in solidarity with "underprivileged" ones such as lesbians, and that communist universalism based on class conscious unity is a utopian dream at best, and dangerously misleading at worst. AS has been sympathetic to criticisms of reductionist logics, of trends within communism to squash the very real divisions and horizontal violences that accompany them. Since our inception as a collective, we have understood the basis of racial, industrial, sexual, gender, national, and in general, sectoral antagonisms as constituting the capitalist division of labor. We aspire to help build a communist movement that becomes victorious over the capitalist mode of production which depends upon a fractured labor process, split labor markets, the manipulation of social control buffer strata . . . all the tricks in the book to subvert the tendency toward unity that the proletariat historically gravitates toward.
We know that it is possible for the impulse of the working class to unite to ultimately triumph, because we see historical episodes of it, reaching as far back as the multi-gendered revolutionary resistance of European heretics in the late middle ages, to black/white/brown unity in resisting forced labor in colonial American history, to the central role played by women in the Mexican and Russian revolutions as they fought revolutionary wars for their class arm in arm with male counterparts. The list goes on and on.
Here's one chapter in that limitless story of a class in the making, in the process of struggling to unite and become conscious of itself and its incredible depth of diverse aspects, its wells of human variety and creativity that capital subsumes along with all dimensions of social life to fuel the production of value. British gays and lesbians organizing to support blue collar community of (presumably) straight male industrial workers. This may sound strange to our post-modern ears, but it gets stranger . . . the blue collar families embracing the queer movement's hand and promoting them to visibly represent their struggle!
The backdrop for this stranger than fiction episode is the ascendance of the neoliberal regime in Britain, where that country's first woman prime minister, Margaret Thatcher, and her bourgeois cronies sought to squash the organized labor movement in order to impose a program of austerity in the face of a long drawn out economic crisis. The keystone in this project was the privatization of the coal industry, nationalized in the 1940s, and a bastion of proletarian power crystalized in class-conscious communities and formally represented by industrial unions.
Below is an interview with two of the militants of the organization that is the subjects of this post. On the scene at that time in Britain, were collectives of gay and lesbian solidarity committees that organized amongst the queer communities in London in order to provide material aid the tens of thousands of striking miners. While being concerned with their particular oppression, these militants saw their fate bound up with the rest of their class and describe their work as
"…a double-issue campaign, because its a two-sided sword. The whole political idea, the fact that it is organized by lesbians and gay men who are taking gay liberation as one of the aims of the group, has been to take the ideas of gay liberation and lesbian liberation into the organized working class, the labor movement."
And therein we see the synthesis of a sensitivity to the particular given its deepest manifestation through positively contributing to the whole, and therefore enriching the universality of class struggle praxis. Although they might not consider themselves marxist, by building an integrated queer-labor struggle and offering solidarity across sexual lines, the campaign of Lesbians and Gays Support the Miners is a moving moment in a process of budding communist universalism. Its time for radicals, militants, oppressed, communists, and activists of 2011 to take the scattered seeds from these defunct buds and plant a garden of cross-bred hybrid synthesis of revolutionarily integrated projects!
Click Here to Download the Radical America issue with the article (pages 39-49)
This entry was posted in Analysis/Theory, History and tagged class struggle, Marxism, queer. Bookmark the permalink.
3 responses to "Queer Liberation and Class Struggle Case Study: The Welsh Miner's Strike"
Advance The Struggle | June 7, 2011 at 11:38 pm | Reply
AS also recommends reading this article by one of our comrades in Seattle on queer liberation and class struggle:
http://gatheringforces.org/2010/01/08/queer-liberation-is-class-struggle/
Rev | June 11, 2011 at 1:38 am | Reply
Hmm this piece was interesting because it explores the important question of how queers participate in the struggle of the international proletariat to overthrow the international bourgeoisie, while simultaneously participating in the struggle of the queer proletariat to overthrow the patriarchal and heteronormative institutions of oppression that exist within the proletariat.
What do people think is the main question that this piece is trying to answer? What do people think are the goals?
If those are the goals, then perhaps communists should do both. Where workers fight the boss we should organize support, and we should additionally discuss with the workers and the public the necessity of taking over the workplace and smashing the state. And if we are organized as queers or communists or nurses to do so then we should make it clear that we, as "queers" or as "communists" or as "nurses" , support such and such strike because this is a necessary step towards the reappropriation of the life activity of the whole international proletariat from the bourgeoisie that enforces the alienation of our life activity into social institutions that oppress and exploit us, and as queers or nurses we share that common goal with x group of struggling proletariat. We should also point out the ways in which the capitalist class and their conscious exploitative activities specifically oppress us a queers, communists, or nurses.
Where workers repress other workers for their sexuality, their race, their place in the division of labor, or the gender we should confront them openly and effectively and explain to them how that repression is reactionary. We should explain concretely how, for example, by making homosexuals invisible or hating people for their homosexuality they are oppressing the very people who grow their food, who built their home, who teach them at school, who make their goods on the assembly line, andwho drive their buses. They are oppressing the very people who suffer all of the same hardship, exploitation, state violence, imprisonment, expropriation, dislocation, and forced migration that they do. And they are oppressing their comrades in the historical struggle to smash and and supersede the capitalist class that benefits from the enslavement of the vast majority of humanity to the interests of value accumulation.
Where confrontation and explanation is ineffective, revolutionary queers, or communists, or nurses, should militantly defend themselves from repressive sectors of the working class. If the media lies about who we are we must tell the truth. If workers are violent towards us we must defend ourselves.
Where queers find their fellow queers unwilling to participate in the international revolutionary struggle, or communists find their fellow communists shirking their duties in favor of sectarian or dogmatic blabbering, or nurses find their fellow nurses indifferent towards a health care system based on profit and murder, we must try to reason with our would be comrades. We must not accept assimilationism into the cultural, material and ideological pillars of capital. We must not accept separatist isolation from our brothers and sisters in our enslavement by capital and our struggle for emancipation. Rather we must stand up for ourselves and our ideas and stand together with all others who have the consciousness and the will to do so, because our challenge is the greatest one that history has to offer.
Those are some of my thoughts upon reading this post. How do people think capital benefits from violence against queers or making queers invisible in our society?
JOMO | June 13, 2011 at 10:13 pm | Reply
Rev,
One of the things that come to mind about the role of repression around sexuality and gender under capitalism is the need for capital to push through the inevitability of conforming our lives to capitalist norms. I think queer liberation is class struggle to the extent that class struggle is toward communism, a society of unalienated labor, joy, freedom and creativity. Queer lib and gender binary smashing reveal how our humanity is flexible and creative and bending. In our capitalist societies, our formally-free freedoms are made to fit the expectations that capitalism requires of us: the freedom to work or be broke and hungry, the freedom to get a mad student loan w high interest rates or get the worst jobs, the freedom to go through the legal immigration process or be condemned to a life of criminalization, the freedom to take out a credit card loan or be miserable judging your life in relation to everything you see on TV…our formal freedoms need to coincide with the norms that benefit and perpetuate capitalism. Sexuality and gender presentation strikes at the very core of how we perceive ourselves, our capacities, our emotions. If capitalism says: Go be free and smash the gender binary, transgress heteronormativity all you want! Then, you are saying people can e free on a very deep level, that people can confront very deep seated social norms. If this level of human depth is being confronted, so many other things seem like they can similarly be confronted. Honestly, if you can confront the fact that you no longer identify with the body that you were born with, and that your self expression and your body can indeed be your own creation and work of art, then, how much harder is it to confront other issues? I believe that those people in our society, who have undergone the experiences of transitioning, coming out, disabilities — experiences that require them to confront a very deep part of themselves for their own well being and survival, have the capacity for revolution. This is way above and beyond the ways that capitalism will allow us to know our own strength and freedom. So, the whole gay marriage stuff, is a way to curtail our own belief in our power. They give us some, hoping we will forget the potentials that our freedom to love and change ourselves, can bring us.
I have felt in my own experience as a genderqueer person, that the gender binary and sexuality are one of those things that run so deep in maintaining capitalist society, but yet are one of the hardest things to pinpoint exactly how. But after mulling over this question over the years, I think that it is not because gender and sexuality are not integral to capitalism, but that the models, theories and visions of liberation and communism do not yet analyze exactly how deeply we are shaped by capitalism, and the visions offered do not discuss freedom in a way that feels personal and real and inspiring. The models of state capitalism with authoritarian dictatorships that have historically passed for communism, is a case in point. Any inspiring vision of communism needs to have a positive vision for trans liberation and queer liberation, otherwise it does not address human freedom and creativity as part of the vision for class struggle.
This is not to say that I disagree with the importance of understanding the role of the nuclear family and how it is both a product and perpetuation of capitalism, or of the importance of understanding how divisions in the working class around gender, sexuality, race, etc are important ways that capitalism perpetuates itself. I think these are important points that are part of a holistic understanding of understanding queer and trans struggles today.
I think what's so hard, and yet also so inspiring about this case of the Gay and Lesbian solidarity with the mine workers, is the level of vulnerability and risk-taking that both communities are taking on with one another. The need for dialogue has been monopolized by the liberals, and made to be very pathetic and text-bookish. But the level of dialogue that is presented in this case is one that is very daring, that is done through action, that has consequences. And we need to reclaim this kind of dialogue and relationship building. One that has stakes involved.
Thanks A/S for putting up this piece!!