text:
\section{Introduction}

The direct observation of gravitational waves from a merger of binary black holes marked the birth of an immensely exciting scientific era: gravitational radiation now provides a new window to observe the universe, complementing the electromagnetic signals traditionally used in astrophysics. Multi-messenger astronomy, however, requires a precise understanding and modelling of gravitational waves, which is made difficult by the nonlinear nature of general relativity. It is therefore of interest to find new, potentially measurable quantities sensitive to gravitational wave profiles. The goal of this paper is to describe one such observable affecting what is, perhaps, the simplest measuring device of astronomers: a spinning gyroscope.

A prominent manifestation of gravitational nonlinearity appears in so-called hereditary effects \cite{Blanchet:2013haa}, which depend on the entire past history of the system. In particular, {\it gravitational wave memory} \cite{Zeldovich:1974gvh,Braginsky:1985vlg,braginsky1987gravitational,Christodoulou:1991cr,Blanchet:1992br} is a permanent net change of certain metric components.
The simplest consequence of this behaviour is displacement memory, \ie the permanent change of distance between two nearby freely falling test masses after the passage of gravitational waves. Here we shall argue that gyroscopes similarly display an `orientation memory' sensitive to gravitomagnetic components of the radiating field.

Remarkably, memory is a {\it Newtonian} addition to the oscillatory waveforms emitted by bounded gravitational sources \cite{Wiseman:1991ss}. As a result, the effect may realistically be quite large for binary systems \cite{Mitman:2020bjf,Mitman:2020pbt,Mitman:2021xkq}, but its detection is hampered by the poor sensitivity of gravitational wave detectors at low frequencies. It is nevertheless conceivable that future experiments will be able to detect it: see \eg \cite{Favata:2010zu,Lasky:2016knh,Boersma:2020gxx}. One of the corollaries of this work is thus a novel proposal for the observation of memory---or at least a specific kind thereof.

Indeed, there exists by now a plethora of distinct manifestations of gravitational memory, ranging from standard displacement memory \cite{Zeldovich:1974gvh,Braginsky:1985vlg,braginsky1987gravitational} to the kick (velocity) memory effect caused by planar waves \cite{Zhang:2017rno,Zhang:2017geq} or compact sources \cite{Flanagan:2018yzh,Divakarla:2021xrd}, spin memory in a Sagnac interferometer \cite{Pasterski:2015tva,Mao:2018xcw}, and novel memory effects in modified theories of gravity \cite{Tahura:2020vsa,Hou:2020tnd,Seraj:2021qja,Tahura:2021hbk}. The unifying thread \cite{Strominger:2014pwa,Strominger:2015bla} underlying these seemingly unrelated observables is the existence of asymptotic Bondi-Metzner-Sachs (BMS) symmetries of gravity \cite{Bondi:1962px,Sachs2} and their various extensions \cite{Barnich:2009se,Barnich:2010eb,Campiglia:2014yka,Freidel:2021fxf}---an active field of research on its own. As we shall explain, gyroscopic memory is related to space-time symmetries as well: the precession rate turns out to coincide with the `covariant dual mass aspect' \cite{Freidel:2021qpz} that generates dual supertranslations \cite{Godazgar:2018qpq,Godazgar:2018dvh,Kol:2019nkc,Godazgar:2019dkh,Godazgar:2020gqd,Godazgar:2020kqd,Oliveri:2020xls,Kol:2020ucd} in an extended version of the standard BMS group. The resulting net change of orientation---the memory effect as such---stems from flux-balance equations for the angular momentum aspect, and is thus related to superrotation charges in the sense of \cite{Campiglia:2014yka,Compere:2018ylh,Compere:2019gft}. Remarkably, an additional nonlinear flux turns out to contribute; we show that it coincides with the local Hamiltonian generator of duality transformations in radiative phase space. As a bonus, the spin memory effect \cite{Pasterski:2015tva} is a special case of our considerations, since a Sagnac interferometer is but a gyroscope made of optical fibres.

To conclude this introduction, a comment is called for regarding the magnitude of precession as a function of the observer's distance to the source. While usual displacement memory is proportional to the inverse distance, gyroscopic memory (like spin memory) is proportional to its {\it square}: it is a weaker effect. This is not surprising, however, as even the best known precession effect in gravitational backgrounds with angular momentum---namely Lense-Thirring precession (see e.g. \cite[sec.\ 40.7]{Misner:1973prb})---involves the {\it cube} of the inverse distance. In that respect, our computation generalizes the Lense-Thirring effect to radiative space-times and `corrects' it by a dominant, overleading term at large distances.

The paper is organized as follows. Section \ref{se2} is devoted to a lightning review of the Bondi formalism for asymptotically flat gravitational fields, and introduces asymptotic geodesics therein. These are then used in section \ref{se3} to describe the kinematics of freely falling (or mildly accelerated) gyroscopes with respect to a natural choice of local tetrad---namely one that is tied to distant stars and obtained by suitably rotating a `source-oriented' frame where one vector points towards the origin of radiation. Finally, section \ref{se4} displays the precession rate and ensuing gyroscopic memory for freely falling observers, whereupon the result is related to dual asymptotic symmetries and flux-balance equations of gravitational data. Static observers are also briefly considered, and their gyroscopic memory effects are compared to earlier statements in the literature, such as asymptotic vorticity \cite{Herrera:2000uh,Herrera:2013poa} and spin memory \cite{Pasterski:2015tva}.

\section{Radiative asymptotically flat metrics}
\label{se2}

This preliminary section serves to set up our notation, review the Bondi formalism for asymptotically Minkowskian metrics, and derive geodesics near null infinity. These geodesics will eventually be the world lines of freely falling gyroscopes in section \ref{se3}.

\paragraph{Off-shell metric in Bondi gauge.} Consider a Lorentzian space-time manifold endowed with (retarded) Bondi coordinates $(u,r,\theta^a)$, where $u$ is retarded time, $r$ is a radius, and $\theta^a$ ($a=1,2$) are coordinates on a celestial sphere at future null infinity (see fig.\ \ref{fiZZ}). The manifold carries a metric $ds^2=g_{\mu\nu}dx^{\mu}dx^{\nu}$ whose components are taken to satisfy the Bondi gauge conditions
\beq
\label{bog}
g_{rr}=g_{ra}=0,
\qquad
\partial_r\det\left(r^{-2}g_{ab}\right)=0.
\ee
Any such metric is commonly written as
\beq
\label{e2}
ds^2
=
-e^{2\beta}(F d u^{2}+2 du\,dr)
+r^2\gamma_{ab}\big(d\theta^{a}-\frac{\,U^{a}}{r^2} du\big)\big(d\theta^{b}-\frac{\,U^{b}}{r^2}du\big)
\ee
in terms of a metric $\gamma_{ab}$ that will eventually include the effects of gravitational radiation, a vector field $U^a$ that will eventually carry information on angular momentum, and two functions $\beta,F$, the second of which will eventually sense the mass of the source. Off-shell, all these quantities are {\it a priori} arbitrary functions on space-time. To ensure that the metric \eqref{e2} is in fact asymptotically {\it flat}, one imposes the extra boundary condition
\beq
\label{BC}
\lim_{r\to \infty}\gamma_{ab}(u,r,\theta)
\equiv
q_{ab}(\theta),
\ee
where the right-hand side is the static, radius-independent metric of a unit sphere (with Ricci scalar $R[q]=2$) written in coordinates $\theta^a$. Eq.\ \eqref{BC} and the last condition in \eqref{bog} then imply that a sphere at constant $(u,r)$ has area $4\pi r^2$, so $r$ measures areal distance. No further boundary conditions are needed: in vacuum Einstein gravity, eq.\ \eqref{BC} turns out to fix the on-shell radial dependence of all other variables, as we recall below.

The gauge conditions \eqref{bog} imply that constant $u$ hypersurfaces are light-like, as $g^{uu}=g^{{\mu\nu}}\partial_\mu u\,\partial_\nu u=0$. Their normal vector $\boldsymbol{\ell}\equiv -g^{{\mu\nu}}\partial_\nu u\,\partial_\nu= e^{-2\beta}\partial_r$ satisfies $\boldsymbol{\ell}\cdot \nabla \boldsymbol{\ell}=0$, thus representing a congruence of outgoing affine null geodesics. Indeed, we are interested in asymptotically flat space-times with a gravitational source near the origin\footnote{We assume that the matter stress tensor has compact support in the bulk. Delocalized (\eg electromagnetic) sources would require, instead, further fall-offs for the stress tensor (see \eg \cite{Flanagan:2015pxa}).}---the initial center of mass of the source is assumed to be at $r=0$---so $\boldsymbol{\ell}$ is tangent to null rays emitted from the source, and integral curves of $\boldsymbol{\ell}$ eventually reach future null infinity (the region $r\to\infty$ where all other coordinates are kept finite). One should thus think of the large $r$ region as the typical location of detectors of gravitational waves. Indeed, it is the tangent vector $\boldsymbol{\ell}$ that will eventually be used in section \ref{se3} to build a source-oriented tetrad carried by observers far away from the source.

\begin{SCfigure}[2][t]
\centering
\includegraphics[width=.3\textwidth]{Penrose03.png}
\caption{A Penrose diagram of some asymptotically flat space-time, including Bondi coordinates $(u,r,\theta^a)$ in terms of which the (off-shell) metric takes the form \eqref{e2}. Throughout this work, we assume that some source of gravitational radiation is located near the origin ($r=0$). The resulting gravitational waves cross the world line of a gyroscope (red) and cause its orientation to change with respect to a tetrad built in section \ref{se3}. This change of orientation may be seen as a gravitomagnetic memory effect and is studied in detail in section \ref{se4}.}
\label{fiZZ}
\end{SCfigure}

\paragraph{On-shell metrics.} The Bondi framework naturally suggests solving Einstein's equations perturbatively in $1/r$, \ie as an expansion near null infinity. Vacuum dynamics thus constrains the arbitrary functions of \eqref{e2} in the form of asymptotic expansions \cite{Flanagan:2015pxa}
\begin{align}
\label{fex}
F
&=
1-\frac{2 m}{r}-\frac{2F_2}{r^2}+\mathcal{O}(r^{-3}),\\
\label{bex}
\beta
&=
\frac{\beta_{2}}{r^2}+\mathcal{O}(r^{-3}),\\
\label{gex}
\gamma_{ab}
&=
q_{ab}+\frac{1}{r} C_{ab}+\frac{1}{r^{2}} D_{ab}+\mathcal{O}(r^{-3}),\\
\label{uex}
{U}^{a}
&=
\mathring{U}^{a}+\frac{1}{r}\left[-\frac{2}{3} L^{a}+\frac{1}{16} D^{a}(C_{bc} C^{bc})
+\frac{1}{2} C^{ab} D^{c} C_{bc}\right]+\mathcal{O}(r^{-2}),
\end{align}
where indices $a,b$ on celestial spheres are raised and lowered with the static metric \eqref{BC}, $D$ denotes the spherical Levi-Civita connection, and each coefficient of $1/r^n$ is some function of $(u,\theta)$, as follows. First, the subleading metric correction $C_{ab}$ in \eqref{gex} is the Bondi shear \cite{Christodoulou:1991cr}, measuring (as its name indicates) the shear of outgoing null geodesic congruences near infinity. It is a symmetric and traceless tensor that depends {\it freely} on $(u,\theta)$; in fact, it is the only genuinely free data of on-shell metrics, and it will be the key object contributing to precession in sections \ref{se3}--\ref{se4}. It determines all other subleading coefficients in \eqref{fex}--\eqref{uex} according to
\begin{align}
\label{e9}
F_2
&=
-\frac{1}{32}C^2
-\frac{1}{6}\left(D_{a} L^{a}\right)-\frac{1}{8}\left(D_{a} C^{a b}\right)\left(D_{d} C_{b}^{d}\right),\\
\label{e10}
\beta_2
&=
-\frac{1}{32}C^2,
\qquad
D_{ab}
=
\frac14 q_{ab}C^2,
\qquad
\mathring{U}^{a}
=
-\frac12 D_bC^{ab},
\end{align}
where $C^2\equiv C_{ab} C^{ab}$ for brevity. The only quantities that are {\it not} directly fixed by shear are the Bondi mass aspect $m(u,\theta)$ of eq.\ \eqref{fex}, and the angular momentum aspect $L^a(u,\theta)$ in \eqref{uex} (respectively measuring densities of energy and angular momentum on celestial spheres): their initial configuration at some time $u_0$ is arbitrary, but their time evolution is otherwise given by the news tensor $N_{ab}\equiv\partial_uC_{ab}$ according to the balance equations \cite{Flanagan:2015pxa}
\beq
\label{balance}
\dot{m}
=
\frac{1}{4}D_aD_bN^{ab}-\mathcal{P},
\qquad
\dot{L}_{a}
=
D_am+\frac{1}{2}D^bD_{[a}D^cC_{b]c}
-\mathcal{J}_a,
\ee
where the dot denotes partial derivatives with respect to retarded time $u$, and we have introduced local fluxes of energy and angular momentum:
\beq
\label{fluxes}
\mathcal{P}
\equiv
\frac{1}{8} N_{ab}N^{ab},
\qquad
\mathcal{J}_a
\equiv
-\frac{1}{4} D_b(N^{bc} C_{ac})-\frac{1}{2} D_bN^{bc}C_{ac}.
\ee
The balance equations \eqref{balance} will play an important role in section \ref{se4}, allowing us to relate gyroscopic memory to angular momentum fluxes. We stress once more that the time-dependence of news is arbitrary, unless a specific gravitational source has been chosen.\footnote{Suitably `generic' asymptotically flat initial data actually satisfy $N_{ab}\sim \mathcal{O}(|u|^{-3/2})$ at large $|u|$ \cite{christodoulou1993global}, but this will play no role for our purposes.}

To summarize, the on-shell form of the Bondi metric \eqref{e2} satisfies the asymptotics
\beq
\label{e13}
\begin{split}
ds^2
\sim&
-\left(1-\tfrac{2m}{r}-\tfrac{2F_2}{r^2}\right)du^2-2\left(1-\tfrac{C^2}{16r^2}\right)du\,dr
+\left(r^2q_{ab}+rC_{ab}+\tfrac{1}{4}q_{ab}C^2\right)d\theta^a\,d\theta^b\\
&+2\left(\tfrac{1}{2}D^bC_{ab}-\tfrac{1}{r}\left[-\tfrac{2}{3}L_a+\tfrac{1}{16}\partial_a(C^2)\right]
\right)du\,d\theta^a
\end{split}
\ee
at large $r$, where the omitted components $rr$ and $ra$ vanish identically owing to \eqref{bog}. The lead actor in this expression is the asymptotic shear $C_{ab}(u,\theta)$, which determines $F_2$ according to eq.\ \eqref{e9} and defines the news $N_{ab}$ that crucially affects the balance equations \eqref{balance} for energy and angular momentum. Subleading corrections to the metric components in \eqref{e13} turn out to be irrelevant in what follows, so we systematically neglect them.

\paragraph{Asymptotic geodesics.}
We conclude this section by writing down the large $r$ asymptotic solution of the geodesic flow equation ${\boldsymbol u}\cdot\nabla{\boldsymbol u}=0$. (This is a key preliminary for freely falling tetrads and their spin connections built in section \ref{se3}.) For definiteness, integration constants are fixed by demanding that freely falling observers have an $\mathcal{O}(r^{-2})$ radial velocity relative to the source at some reference time $u=u_0$. Thus, upon expressing proper velocity as
\beq
\label{e14}
{\boldsymbol u}
=
\gamma(\partial_u+v^r\partial_r+v^a\partial_a),
\ee
a mildly tedious but straightforward computation yields
\begin{align}
\label{geu}
\gamma
&=
1+\frac{m_0}{r}+\frac{\gamma_2}{r^2},
\qquad
\gamma_2
=
\int_{u_0}^u du' \Big(m+\frac{1}{16}C^2\Big),\\
\label{ger}
v^r
&=
\frac{m-m_0}{r}+\frac{1}{r^2}\Big(-\gamma_2-\frac{1}{6}D_aL^a-\frac{1}{8}(D\cdot C)^2\Big),\\
\label{gea}
v^a
&=
-\frac{1}{2r^2}D_bC^{ab}+\frac{1}{r^3}\Big(D^a\gamma_2-\frac23L^a+\frac12C^{ab}D^cC_{bc}\Big),
\end{align}
where $m_0=m(u_0,\theta)$ is the initial mass aspect and $(D\cdot C)^2\equiv D_bC^{ab}D^cC_{ac}$. Note the following elementary consistency check: in non-radiating backgrounds (say $C_{ab}=0$, hence $\dot m=0$), the leading radial velocity \eqref{ger} contains a term $-m_0u/r^2$ caused by the usual Newtonian gravitational acceleration $\dot{v}^r=-m_0/r^2$.

\section{Gyroscopes in local frames}
\label{se3}

This section is devoted to the kinematics of a gyroscope near null infinity in an asymptotically flat gravitational field whose metric takes the Bondi form \eqref{e2}. Accordingly, we start by reviewing the equations of motion of a spin vector relative to any local reference frame in terms of the associated spin connection. We then construct a frame tied to distant stars by first building a source-oriented tetrad, then performing suitable angle-dependent rotations. We end by displaying the spin connections associated with freely falling observers and static observers in Bondi coordinates. Note that most of the discussion for now will be performed off-shell: the metric \eqref{e2} need not satisfy Einstein's equations. This is all a key prerequisite for section \ref{se4}, where the spin connection gives rise to gyroscopic precession, hence to memory effects once gravitational dynamics is taken into account.

\subsection{Gyroscopic kinematics}
\label{segyk}

Consider a (small\footnote{This drastically simplifies the problem: the spin of large freely falling bodies generally satisfies intricate relativistic evolution equations \cite{Dixon:1970zza}, but these boil down to parallel transport for small systems.}) gyroscope with proper velocity ${\boldsymbol u}=u^\mu\partial_\mu$ and spin $\boldsymbol{S}=S^{\mu}\partial_\mu$ such that $\boldsymbol{S}\cdot{\boldsymbol u}=0$. When the gyroscope falls freely, its spin obeys the parallel transport equation ${\boldsymbol u}\cdot\nabla\boldsymbol{S}=0$. More generally, if external forces cause an acceleration $a^\mu=u^\nu\nabla_\nu u^\mu$, the spin vector obeys Fermi-Walker transport $({\boldsymbol u}\cdot\nabla\boldsymbol{S})^\mu=(u^\mu a^\nu-u^\nu a^\mu)S_\nu$ so as to remain orthogonal to velocity \cite[chap.\ 6]{Misner:1973prb}. Now let an observer carrying the gyroscope measure its orientation with respect to some local tetrad of vectors $\be{\mu}=\e{\mu}{\nu}\partial_\nu$ that are orthonormal in the sense that $\be{\mu}\cdot \be{\nu}=\eta_{\hat\mu\hat\nu}$. The spin vector can then be written with hatted components as $\boldsymbol{S}=S^{\hat\mu}\be{\mu}$, and the Fermi-Walker equation becomes
\beq
\label{e17}
\frac{\dd S^{\hat \mu}}{\dd\tau}
=
\big(-u^{\alpha}
\omega_\alpha^{\;\;\hat \mu \hat \nu}+u^{\hat \mu}a^{\hat \nu}\big)S_{\hat \nu}
\ee
where $u^{\hat \mu}=e^{\hat \mu}{}_\nu u^\nu$ are the components of proper velocity in the observer's frame, while
\beq
\label{e18}
\omega_{\mu}{}^{\hat\mu \hat\nu}
\equiv
e^{\hat\mu}{}_{\alpha} \nabla_{\mu} e^{\hat \nu\alpha }
=
e^{\hat\mu}{}_{\alpha}\big(\partial_{\mu} e^{\hat\nu\alpha }+\Gamma_{\sigma \mu}^{\alpha} e^{\hat\nu\sigma }\big)
\ee
is the spin connection one-form associated with the tetrad $\{\be{\mu}\}$. (Hatted indices are raised and lowered with $\eta_{\hat{\mu}\hat{\nu}}$, plain indices with $g_{{\mu\nu}}$.) Eq.\ \eqref{e17} states that the precession of gyroscopes is determined by the spin connection, so the latter is a key object.

In what follows, we always restrict attention to frames that are adapted to the observer in the sense that $\be{0}={\boldsymbol u}$ coincides with proper velocity. Then the spin vector $\boldsymbol{S}=S^{\hat i}\be{i}$ is purely spatial ($i=1,2,3$) and the precession equation \eqref{e17} becomes
\beq
\label{e19}
\frac{d S^{\hat{i}}}{d \tau}
=
\Omega^{\hat{i}}_{\;\hat{j}}  S^{\hat{j}},
\qquad
\Omega^{\hat{i}\hat{j}}
\equiv
-u^{\alpha}\omega_\alpha^{\;\;\hat i \hat j},
\ee
where the observer's acceleration no longer contributes. The antisymmetric tensor $\Omega_{\hat{i}\hat{j}}$ thus yields the gyroscope's precession rate. It can be dualized into a vector $\boldsymbol{\Omega}=\Omega^{\hat{i}}\be{i}\equiv-\tfrac{1}{2}\epsilon^{{\hat{i}}{\hat{j}}{\hat{k}}}\Omega_{\hat{j}\hat{k}}$, so that eq.\ \eqref{e19} reads $\dot\boldsymbol{S}=\boldsymbol{\Omega}\times\boldsymbol{S}$ in terms of the usual cross product.

Determining the gyroscope's motion thus requires that one chooses a velocity ${\boldsymbol u}$ and a local frame to compute the spin connection \eqref{e18}. Accordingly, we now build two tetrads: the first will be source-oriented (section \ref{sesof}), while the second is tied to distant stars (section \ref{sestar}). As for the observer's velocity, we shall consider two cases: free fall, and static locations at constant $(r,\theta)$ in Bondi coordinates. Both types of world lines are understood to live at large $r$, so asymptotic expansions near infinity will often be used.

\subsection{Source-oriented frame}
\label{sesof}

Regardless of whether an observer traces a geodesic or not, let her proper velocity read
\beq
\label{e21}
{\boldsymbol u}
=
\dfrac{dx^\mu}{d\tau}\partial_{\mu}
=
\gamma(\partial_u+v^r\partial_r+v^a\partial_a)
\ee
in Bondi coordinates, where the `time dilation factor'
\beq
\gamma
=
\frac{du}{d\tau}
=
\left[%
e^{2\beta}(F+2v^r)-r^2\gamma_{ab}(v^a-U^a/r^2)(v^b-U^b/r^2)
\right]^{-1/2}
\ee
normalizes ${\boldsymbol u}$ with respect to the off-shell metric \eqref{e2}. The earlier geodesic velocity \eqref{e14} is a special case of \eqref{e21}, while the velocity of a static observer just reads ${\boldsymbol u}=\gamma\partial_u$. We now build a local tetrad $\{\be{0}{},\be{r}{},\be{a}{}\}$ in a way that is compatible with the observer so that $\be{0}{}={\boldsymbol u}$, and adapted to the source in that $\be{r}$ is aligned with emitted light rays. To obtain $\be{r}$, recall from section \ref{se2} that $\boldsymbol{\ell}=e^{-2\beta}\partial_r$ is tangent to outgoing null rays; the radial tetrad vector thus coincides with $\boldsymbol{\ell}$ up to the fact that its component along $\be{0}$ is projected out, \ie
\beq
\label{e23}
\be{r}
=
\frac{1}{\gamma}\boldsymbol{\ell}-{\boldsymbol u}.
\ee
The tetrad is completed by two more vectors $\be{a}$ whose orthogonality to $\be{0},\be{r}$ yields
\beq
\label{e24}
\be{a}
=
\frac{\E{a}{a}}{r}\left[\partial_a+\gamma_{ab}\Big(r^2 v^{b}-U^{b}\Big)\boldsymbol{\ell}\right]
\ee
for some linearly independent vectors $\E{a}{\!}$ tangent to the sphere. The remaining orthonormality conditions $\be{a}\cdot\be{b}=\delta_{\hat a\hat b}$ then imply that these $\E{a}{\!}$'s form an orthonormal dyad with respect to the time-dependent, radius-dependent metric $\gamma_{ab}(u,r,\theta)$:
\beq
\label{e25}
\gamma_{ab}\,\E{a}{a}\,\E{b}{b}=\delta_{\hat a\hat b}.
\ee
The choice of dyad is of course not unique; in fact we do not intend to fix it, so as to preserve general covariance on celestial spheres. A natural choice is nevertheless to pick some `distant stars' at infinity that happen to be oriented orthogonally to the direction of the source, and take these stars to fix the axes of the dyad. Note that, in any case, the dyad typically depends on time and radius: using the on-shell expressions \eqref{gex}--\eqref{e10} and the identity $C_{ac}C^{bc}=\frac12 C^2\delta_a^b$ in condition \eqref{e25}, one finds $\E{a}{a}\sim\barE{a}{b}(\delta_b^a-\frac{1}{2r}C_b{}^a+\frac{1}{16r^2}C^2\delta_b^a)$ where $\barE{a}{b}(\theta)$ is a time-independent dyad on the sphere such that $q_{ab}\barE{a}{a}\barE{b}{b}=\delta_{\hat a\hat b}$.

The set of vectors $\{\be{0},\be{r},\be{a}\}$ is thus a source-oriented tetrad; it is good enough for many applications, including radiative effects at order $\mathcal{O}(r^{-1})$, and one may compute the corresponding spin connection \eqref{e18} to deduce a precession rate \eqref{e19}. There are two issues here, however. The first is that gravitational waves turn out to cause precession at order $\mathcal{O}(r^{-2})$, so a more sensitive frame is needed to capture the effects we are seeking. The second is that source-oriented frames rotate continuously, regardless of the presence of radiation, simply to remain directed towards the source (see fig.\ \ref{fiXX}). As we now show, both of these problems are solved by a frame suitably `tied to distant stars' whose existence is guaranteed by the asymptotically flat nature of space-time.

\begin{SCfigure}[2][t]
\centering
\includegraphics[width=.5\textwidth]{Tetrad02.png}
\caption{An observer looks at a source at the origin. The observer's local frame (in black, on the far right) is initially built so that one of its axes coincides with the source's direction, and it also happens to point towards some distant star. As the observer moves, the source-oriented tetrad rotates so as to remain directed towards the source (red frame). Reorienting the tetrad so that its (black) axes point once again towards distant stars requires a (red) compensating rotation \eqref{e30}.}
\label{fiXX}
\end{SCfigure}

\subsection{Frame tied to distant stars}
\label{sestar}

There is an obvious drawback in the source-oriented frame. Indeed, one expects free gyroscopes in Minkowski space-time to experience no precession, but the spin connection \eqref{e18} of the origin-oriented frame in flat space ($F=1$, $\beta=0$, $U^a=0$, $\gamma_{ab}=q_{ab}$) reads
\beq
\label{e27}
\overline{\omega}_{{\hat a}{\hat r}}
=
\overline{\zeta}_{{\hat a} a}d\theta^a,
\qquad
\overline{\omega}_{{\hat a}{\hat b}}
=
\barE{a}{b}D_b\overline{\zeta}_{{\hat b} b}d\theta^a,
\ee
where bars denote quantities evaluated in a Minkowskian background. This is manifestly non-zero: a free gyroscope with angular velocity $v^a$ precesses at a rate $-v^a\overline{\omega}_{a{\hat i}{\hat j}}(\theta^a)$ with respect to the source-oriented frame, purely due to the fact that the tetrad needs to rotate continuously in order to keep pointing towards the origin. It would be much more convenient to devise a tetrad whose spin connection vanishes in flat space. In fact, in pure Minkowski space, the solution simply consists in using the Cartesian frame whose spin connection is indeed zero. However, the proper frame is not as obvious for generic asymptotically flat space-times, where one can at best define an {\it asymptotically} inertial structure. One thus needs to build a frame tied to `distant stars' whose distance from the observer is effectively infinite, contrary to the source, which is at large but finite distance.

In practice, a frame $\{\bff{\mu}\}$ tied to distant stars can be obtained from the source-oriented tetrad $\{\be{\mu}\}$ by a celestially local rotation cancelling the spurious time-dependent rotation of the $\be{\mu}$'s. Thus we declare that
\beq
\label{e28}
\bff{0}\equiv\be{0}={\boldsymbol u},
\qquad
\bff{i}\equiv R_{{\hat i}}{}^{{\hat j}}(\theta)\be{j},
\ee
where the rotation matrix $R_{{\hat i}}{}^{{\hat j}}(\theta)$ only depends on angular coordinates $\theta^a$, is independent of the dynamical bulk metric, and may be chosen to reduce to the identity at some reference point $\theta_0$ to be thought of as the initial angular position of the observer. The change of tetrad \eqref{e28} entails a modification $\omega\to\omega'$ of the spin connection \eqref{e18}, expressed in terms of one-forms by the standard transformation law
\beq
\label{e29}
\omega^{\prime}_{{\hat i}{\hat j}}
=
R_{{\hat i}}{}^{{\hat m}} R_{{\hat j}}{}^{{\hat n}}\omega_{{\hat m}{\hat n}}
+R_{{\hat i}}{}^{{\hat m}}dR_{{\hat j}{\hat m}}.
\ee
Accordingly, the rotation that yields a frame whose spin connection $\omega'$ vanishes in Minkowski space is given by a path-ordered exponential
\beq
\label{e30}
R(\theta)
=
P\exp\int_{\theta_0}^{\theta}\overline{\omega},
\ee
where $\bar\omega$ is given by \eqref{e27} and should be thought of as an $so(3)$-valued one-form. The curve connecting the reference point $\theta_0$ to $\theta$ is, in principle, arbitrary, and its choice affects the value of $R$ since the curvature of $S^2$ does not vanish. The choice is nevertheless ultimately irrelevant, as one merely has to pick {\it some} path for every pair $(\theta_0,\theta)$ in a suitable open neighbourhood of the observer's initial position.\footnote{There exists no {\it globally} well-defined rotation mapping the source-oriented tetrad on a frame tied to distant stars because the second homotopy group of $S^2$ is non-trivial. This is not an issue for our purposes, as we only need to cover some open subset of $S^2$ in which the observer's motion takes place.} We shall assume that one such choice has been made, in such a way that $R(\theta_0)=\mathbb{I}$ is the identity matrix.

The geometric justification of this construction goes as follows (see fig.\ \ref{fiXX}). Suppose the source-oriented frame, centred at $\theta_0$ at some arbitrary initial time $u_0$, points towards three stars at infinity, one of them aligned with the source's direction. Then let time flow, whereupon the source-oriented frame rotates in a time-dependent manner so as to keep pointing towards the source. As it does so, the observer's position on a celestial sphere also changes from $\theta_0$ to some $\theta$. Now let the observer perform a rotation \eqref{e30} to her frame at every step of her motion; doing so reorients the frame so that its vectors are aligned once more with the three stars chosen at time $u_0$, regardless of the (possibly dynamical) bulk metric or of other time-dependent quantities. Accordingly, we refer to the tetrad \eqref{e28} as being `tied to distant stars' at infinity.

The spin connection of a tetrad tied to distant stars is readily deduced from the transformation law \eqref{e29}. Since the rotation $R$ was chosen so as to annihilate the spin connection of Minkowski space ($R\overline{\omega}R^{-1}+RdR^{-1}=0$), one finds indeed
\beq
\label{e31}
\omega^{\prime}_{{\hat i}{\hat j}}
=
R_{{\hat i}}{}^{{\hat m}} R_{{\hat j}}{}^{{\hat n}}\big(\omega_{{\hat m}{\hat n}}-\overline{\omega}_{{\hat m}{\hat n}}\big),
\ee
where $\omega$ is the spin connection \eqref{e18} of the source-oriented frame built in section \ref{sesof}. The actual components of $\omega$ and $\omega'$ depend on one's choice of world line---either a geodesic or a static line at fixed $(r,\theta)$, as mentioned at the end of section \ref{segyk}. It is therefore time to display these results: using the velocity \eqref{e14}--\eqref{gea} of geodesics at large $r$ and the ensuing source-oriented frame \eqref{e23}--\eqref{e24}, the spin connection \eqref{e18} follows after a long and painful, but otherwise straightforward calculation. The rotation \eqref{e31} then cancels the spurious change of orientation due to the leading angular velocity in \eqref{gea}, and one eventually finds\\
\begin{minipage}{.04\textwidth}
\rotatebox{90}{(Free fall)~~~}
\end{minipage}
\begin{minipage}{.95\textwidth}
\vspace{-.2em}
\begin{align}
\label{e33}
\omega^{\prime}_{{\hat r}{\hat a}}
&\sim
\barE{a}{a}\left(\frac{1}{4r^2} N_{ab}D_cC^{bc}du
+\frac{1}{2r^2}D^bC_{ab}dr
+\frac{1}{2}N_{ab}dx^b\right),\\
\label{e34}
\omega^{\prime}_{{\hat a}{\hat b}}
&\sim
-\frac{1}{2r^2}\barE{a}{a}\barE{b}{b}\big(D_{[a}D^cC_{b]c}-\frac12 N_{c[a}C^c{}_{b]}\big)du+\mathcal{O}(r^{-3})dr+\mathcal{O}(r^{-1})dx^a
\end{align}
\end{minipage}\\[.7em]
up to subleading corrections in $1/r$ that are implicitly neglected. (Our convention for antisymmetrization is $A_{[a}B_{b]}\equiv\frac12 (A_aB_b-A_bB_a)$.) This is expressed here in terms of on-shell metric data appearing in eq.\ \eqref{e13}, specifically in terms of the asymptotic shear $C_{ab}$ and the news tensor $N_{ab}=\partial_uC_{ab}$. (Mass and angular momentum aspects do not contribute to the spin connection at this order in $1/r$; in particular, Lense-Thirring precession \cite[sec.\ 40.7]{Misner:1973prb} is hidden in subleading terms that are unimportant for us.)

The computation is identical for world lines at constant $(r,\theta)$, save for the fact that proper velocity \eqref{e21} simplifies to ${\boldsymbol u}=\gamma\partial_u$ (instead of the more involved expressions \eqref{e14}--\eqref{gea}). Using once more the tetrad \eqref{e23}--\eqref{e24} and the source-oriented spin connection \eqref{e18}, the rotated spin connection \eqref{e31} for a static world line at constant $(r,\theta)$ reads\\
\begin{minipage}{.04\textwidth}
\rotatebox{90}{(Static)~~~}
\end{minipage}
\begin{minipage}{.95\textwidth}
\vspace{-.2em}
\begin{align}
\label{e35}
\omega^{\prime}_{{\hat r}{\hat a}}
&\sim
\barE{a}{a}\left(\frac{1}{2r} D^b N_{ab}du
+\mathcal{O}(r^{-3})dr
+\frac{1}{2}N_{ab}dx^b\right),\\
\label{e36}
\omega^{\prime}_{{\hat a}{\hat b}}
&\sim
-\frac{1}{2r^2}\barE{a}{a}\barE{b}{b}\big(D_{[a}D^cC_{b]c}-\frac12 N_{c[a}C^c{}_{b]}\big)du
+\mathcal{O}(r^{-3})dr
+\mathcal{O}(r^{-1})dx^a,
\end{align}
\end{minipage}\\[.7em]
again up to neglected subleading corrections. Note the identical $\hat{a}\hat{b}$ components of the spin connection \eqref{e36} and its free fall analogue \eqref{e34}. By contrast, the $\hat{r}\hat{a}$ components \eqref{e35} differ sharply from their geodesic cousin \eqref{e33}: the $u$ component of the former scales as $1/r$ at large $r$, while that of the latter goes as $1/r^2$. We return to this in section \ref{sestat}.

Eqs.\ \eqref{e33}--\eqref{e36} are crucial results for the remainder of this paper. They determine the precession rate \eqref{e19} according to $\Omega_{{\hat i}{\hat j}}=-u^{\mu}\omega^{\prime}_{\mu{\hat i}{\hat j}}$, where ${\boldsymbol u}$ is either the geodesic velocity \eqref{e14}--\eqref{gea} or the static velocity ${\boldsymbol u}=\gamma\partial_u$, respectively requiring the free fall connection \eqref{e33}--\eqref{e34} or its static analogue \eqref{e35}--\eqref{e36}. The next section will indeed be devoted to a detailed study of this precession rate and its implications in terms of gravitational memory. (Note that one can, in fact, use a simplified expression for the precession rate: observers initially at rest in Bondi coordinates have an angular velocity $v^a=\mathcal{O}(r^{-2})$, so the rotation matrix \eqref{e30} evaluated along the angular position of an observer satisfies $R=\mathbb{I}+\mathcal{O}(r^{-2})$. It follows that $\Omega_{{\hat i}{\hat j}}\sim-u^{\mu}(\omega_{\mu{\hat i}{\hat j}}-\overline{\omega}_{\mu{\hat i}{\hat j}})$ at large $r$.)

\section{Gravitational memory from gyroscopic precession}
\label{se4}

The spin connections \eqref{e33}--\eqref{e36} determine the angular velocity \eqref{e19} of a spinning gyroscope, located far away from a gravitational source, with respect to a frame whose axes are tied to distant stars. We now investigate this precession in depth, mostly focussing on freely falling observers. Precession then takes place in a plane orthogonal to the source's direction and turns out to be related to a suitable notion of dual mass aspect \cite{Freidel:2021qpz}. Bursts of gravitational radiation thus lead to net changes of orientation that may be interpreted as a memory effect. The latter turns out to involve a non-local superrotation charge, a flux term for angular momentum, and a duality generator, all owing to flux-balance equations for gravitational data. We conclude by turning our attention to {\it static}, accelerated observers whose leading precession stems from the $u$ component of \eqref{e35} and yields a memory effect that contains the same information as standard displacement memory \cite{Zeldovich:1974gvh,Braginsky:1985vlg,braginsky1987gravitational}. The notion of spin memory \cite{Pasterski:2015tva} is also recovered as a special case.

\subsection{Precession as a dual mass aspect}

Consider a freely falling observer with velocity \eqref{e14}--\eqref{gea}, carrying a gyroscope whose precession rate \eqref{e19} involves the spin connection \eqref{e33}--\eqref{e34}. From this one readily finds
\begin{empheq}[box=\othermathbox]{equation}
\label{s11s}
\Omega_{{\hat a}{\hat r}}
=
\mathcal{O}(r^{-3}),
\qquad
\Omega_{{\hat a}{\hat b}}
=
\frac{\epsilon_{{\hat a}{\hat b}}}{r^2}
\widetilde\mathcal{M}
+\mathcal{O}(r^{-3}),
\qquad
\widetilde\mathcal{M}\equiv \frac14 D_{a}D_b\widetilde C^{ab}-\frac18 N_{ab}\widetilde C^{ab},
\end{empheq}
where the dual of any symmetric tensor $X_{ab}$ on the celestial sphere is defined as $\widetilde X_{ab}\equiv\epsilon_{c(a}X_{b)}{}^{c}=\epsilon_{ca}X_{b}{}^{c}-\frac{1}{2}\epsilon_{ab}X_c{}^c$ \cite{Godazgar:2018dvh} (symmetrization may be dropped when $X$ is traceless). This states that, at leading order in $1/r$, the gyroscope's axis only rotates in the plane tangent to the celestial sphere at the observer's location. The precession frequency is set by the quantity $\widetilde\mathcal{M}$, dubbed `dual covariant mass aspect' \cite{Freidel:2021qpz} in analogy with the usual covariant mass aspect $m+\frac18 N_{ab} C^{ab}$ \cite{Compere:2018ylh,Donnay:2021wrk}.\footnote{This justifies the notation $\widetilde\mathcal{M}$, as $\mathcal{M}$ normally indicates the usual (non-dual) covariant mass aspect. We stress $\widetilde\mathcal{M}$ has nothing to do with mass as such: it is related, instead, to multipole moments of the gravitational angular momentum (see eq.\ \eqref{e42} where $C^-$ is written solely in terms of spin multipole moments, or eq.\ (4.8b) of \cite{Compere:2019gft}). Mass and spin are indeed mutually dual in general relativity.} (Indeed, the latter reduces to $\widetilde\mathcal{M}$ when the shear $C_{ab}$ is replaced by its dual $\widetilde C_{ab}$ and the mass aspect $m$ is replaced by $-\frac14\epsilon^{ab}D_bD^cC_{ac}$, \ie by the vorticity of the angular motion of the geodesic flow stemming from eq.\ \eqref{gea}.) Gyroscopic precession near null infinity thus provides an observational protocol for the dual covariant mass aspect, as the latter essentially coincides with the rate \eqref{s11s}.

It is convenient, for later use, to Hodge-decompose tensors on the sphere in terms of scalar quantities with definite parity. Accordingly, in what follows we write angular momentum and shear as
\beq
\label{e41}
L_a
\equiv
D_aL^++\epsilon_{ab}D^bL^-,
\qquad
C_{ab}
\equiv
D_{\langle a}D_{b\rangle} C^{+}+\epsilon_{c(a} D_{b)} D^{c} C^{-},
\ee
where superscripts $\pm$ refer to parity eigenvalues of the respective functions, while angular brackets denote the symmetric, trace-free projection $D_{\langle a} D_{b\rangle}\equiv D_{( a} D_{b)}-\frac{1}{2}q_{ab} D^2$. Using this, one finds that the linear term in the precession rate \eqref{s11s} has odd parity:
\beq
\label{e42}
\widetilde\mathcal{M}
=
\mathcal{D} C^--\frac18N_{ab}\widetilde C^{ab},
\qquad
\mathcal{D}\equiv\frac18D^2(D^2+2).
\ee
Moreover, the balance equation \eqref{balance} for angular momentum implies the relation $\mathcal{D} C^-=\dot L^-+\mathcal{J}^-$, thanks to which the dual mass aspect \eqref{e42} can finally be recast as
\beq
\label{e43}
\widetilde\mathcal{M}
=
\dot L^-+\mathcal{J}^-+ \widetilde{\mathcal{J}},
\qquad
\widetilde{\mathcal{J}}\equiv-\frac18N_{ab}\widetilde C^{ab}.
\ee
A corollary of this rewriting is the absence of precession in non-radiative space-times, where the angular momentum aspect is constant and fluxes vanish since they are proportional to news. We now exploit the decomposition \eqref{e43} to relate gyroscopic memory to certain symmetries of the gravitational phase space.

\subsection{Gyroscopic memory and gravitational symmetries}

The gyroscope carried by a freely falling observer precesses according to eq.\ \eqref{e19}, with a rate \eqref{s11s} that {\it vanishes} in the absence of radiation thanks to the rewriting \eqref{e43} of the dual mass aspect. It follows that any finite burst of gravitational radiation leaves a permanent imprint---a memory---of its passage on the gyroscope's orientation. To compute this effect, our starting point is the precession equation \eqref{e19}, whose solution can formally be written as a time-ordered exponential of $\Omega$ acting on the initial spin vector. Since the angular velocity \eqref{s11s} is only accurate up to order $\mathcal{O}(r^{-2})$, the expansion of the exponential is only reliable up to first order in $\Omega$. We conclude that the net change of orientation that accompanies gravitational waves is
\beq
\label{e45}
\Delta S^{{\hat r}}
=
\mathcal{O}(r^{-3}),
\qquad
\Delta S^{{\hat a}}
=
\frac{1}{r^2}\epsilon^{{\hat a}{\hat b}}S^{\text{initial}}_{{\hat b}}
\left[%
\Delta L^-+\int_{-\infty}^{+\infty} du\left(\mathcal{J}^-+\widetilde\mathcal{J}\right)
\right]+\mathcal{O}(r^{-3}),
\ee
where we used eqs.\ \eqref{s11s} and \eqref{e43}, while $\Delta L^-\equiv\int_{-\infty}^{+\infty}du\,\dot L^-$ is the net change of the parity-odd component of gravitational angular momentum.

Eq.\ \eqref{e45} is the sought-for memory effect. In order to relate it to gravitational symmetries, note from the Hodge decomposition \eqref{e41} that $L^-=D^{-2}(\epsilon^{ab}D_bL_a)$ where $D^{-2}$ is the inverse of the Laplacian on the sphere, \ie a suitable Green's function $G$. Accordingly, the term in square brackets in the memory formula \eqref{e45} can be written as
\begin{empheq}[box=\othermathbox]{equation}
\label{e47}
\int_{-\infty}^{+\infty}du\,\widetilde\mathcal{M}
=
8\pi\left(\Delta Q_Y+\mathcal{F}_Y+\widetilde\mathcal{F}\right),
\end{empheq}
where $Y^a(\theta)\equiv\epsilon^{ab}D_bG(\theta,\theta')$ is a divergence-free vector field on $S^2$ whose superrotation charge $Q_Y$ and flux $\mathcal{F}_Y$ are given by\footnote{There is a subtlety here: one can redefine the charge and flux \eqref{e48} by mapping $N_a\to \hat{N}_a=N_a+\alpha_a$ and $\mathcal{J}_a\to \hat{\mathcal{J}}_a=\mathcal{J}_a-\dot{\alpha}_a$ for any one-form $\alpha_a$, and using hatted quantities in \eqref{e48}. For the charge to match its Wald-Zoupas version \cite{Wald:1999wa}, one has to take $\alpha_a=-u D_{a} m-\frac{1}{16} D_{a}\left(C_{bc} C^{bc}\right)-\frac{1}{4} C_{ab} D_{c} C^{bc}$ \cite{Flanagan:2015pxa}. Fortunately, the sum $\Delta Q_{Y}+\mathcal{F}_Y$ in \eqref{e47} is unaffected by this ambiguity.}
\beq
\label{e48}
Q_{Y}
\equiv
\frac{1}{8\pi}\oint_{S^2}\sqrt{q}\,d^2\theta\,Y^a L_a,
\qquad
\mathcal{F}_Y
\equiv
\frac{1}{8\pi}\int du\oint_{S^2}\sqrt{q}\,d^2\theta\,Y^a\mathcal{J}_a,
\ee
while the last term of \eqref{e47} is a seemingly exotic flux contribution:
\beq
\widetilde\mathcal{F}
\equiv
-\frac{1}{64\pi}\int du\,N_{ab}\widetilde C^{ab}.
\ee
In fact, this quantity is closely related to gravitational electric-magnetic duality: the Hamiltonian generator of local duality transformations $\delta C_{ab}=\varepsilon(\theta)\widetilde C_{ab}$, $\delta N_{ab}=\varepsilon(\theta)\widetilde N_{ab}$ with respect to the radiative symplectic form \cite{Ashtekar:1981bq}
\beq
\frac{1}{8}\int du \oint\sqrt{q}\,d^2\theta\,\delta N^{ab} \wedge\delta C_{ab}
\ee
is nothing but $\oint\sqrt{q}\,d^2\theta\,\varepsilon\widetilde\mathcal{F}$. One can think of this extra flux $\widetilde\mathcal{F}$ as an analogue of the super-centre of mass flux affecting usual displacement memory for test masses having an initial relative velocity \cite{Nichols:2018qac}; it just happens to be a duality generator in the present case.

Eqs.\ \eqref{s11s} and \eqref{e47} are our main results: they explicitly display the relation between precession velocity and dual mass, and recast the net change of orientation \eqref{e45} as a sum of three terms related to various symmetries of the gravitational radiative phase space. To our knowledge, eq.\ \eqref{e47} is actually the first appearance of a duality generator in a gravitational memory effect. The remainder of this work is devoted to a comparison between these considerations and established results in the literature.

\subsection{Static observers and spin memory}
\label{sestat}

We have so far focussed on freely falling observers whose spin connection is given by \eqref{e33}--\eqref{e34}, but most of the literature on gravitational memory relies on {\it static} observers, whose spin connection takes the form \eqref{e35}--\eqref{e36} instead. Accordingly, we now briefly discuss the effects of the two kinds of components of \eqref{e35}--\eqref{e36} and compare them to asymptotic vorticity \cite{Herrera:2000uh,Herrera:2013poa} and the spin memory effect \cite{Pasterski:2015tva}. Incidentally, the latter only involves transverse components \eqref{e36} which coincide with their freely falling cousin \eqref{e34}, so the comparison could equally well be carried out for freely falling observers, but the computation of \cite{Pasterski:2015tva} was carried out for static observers so we stick to that convention.

For static observers (with velocity ${\boldsymbol u}=\gamma\partial_u$), the dominant precession takes place in the ${\hat r}{\hat a}$ plane and stems from the $\mathcal{O}(r^{-1})$ term of eq.\ \eqref{e35}, yielding a precession velocity
\beq
\label{e49}
\Omega_{{\hat a}{\hat r}}
\sim
\frac{1}{2r}\barE{a}{a} D^b N_{ab}
\ee
first found in \cite{Herrera:2000uh,Herrera:2013poa} thanks to the vorticity of static world lines in Bondi coordinates. That the effect occurs already at order $\mathcal{O}(r^{-1})$ is no surprise, since world lines at constant $(r,\theta)$ have a non-zero proper acceleration $u^\nu\nabla_\nu u^\mu
\sim(\frac{\dot{m}}{r},-\frac{\dot{m}}{r},\frac{1}{2r^2}D_b N^{ab})$ at large $r$. The memory effect induced by \eqref{e49} is proportional to the parity-even variation of shear $\Delta C^+$ (recall the Hodge decomposition \eqref{e41}), so it contains the same information as standard displacement memory, determined by the balance equation for supertranslations \cite{Zeldovich:1974gvh,Braginsky:1985vlg,braginsky1987gravitational}.

We now turn to the spin connection's ${\hat a}{\hat b}$ component \eqref{e36} and show that it reproduces spin memory \cite{Pasterski:2015tva}, \ie the Sagnac effect caused by gravitational waves on an interferometer facing a source of gravitational waves. To see this, first recall the principle of Sagnac interferometers: when a light ray is split in two separate beams moving in opposite directions along some closed path $\mathcal{C}$ (say an optical fibre), the apparatus's angular velocity $\Omega$ causes a difference $\Delta T$ in the time it takes for each beam to orbit the path \cite{wang2004generalized}:\footnote{We momentarily reinstate the speed of light $c$.}
\beq
\label{e51}
\Delta T
=
\frac{4}{c^2}\int_{\text{Int}(\mathcal{C})}\Omega_{ij}\,dx^i\wedge dx^j.
\ee
Now consider a Sagnac interferometer located far away from a gravitational source, and assume that the closed path $\mathcal{C}$ lies in a plane transverse to the direction of wave propagation (\ie that $\mathcal{C}$ is tangent to the celestial sphere at some large radius). The precession rate \eqref{s11s} can then be plugged into \eqref{e51} and readily reproduces spin memory. Indeed, assuming that the spatial extent $\ell$ of the interferometer is much smaller than the typical gravitational wavelength $\lambda$, the second term of the dual mass in \eqref{s11s} is smaller by a factor $\ell/\lambda\ll1$ than the first one, yielding
\beq
\label{e53}
\Omega
\sim
\frac{1}{2}D_{[a}D^cC_{b]c}\,d\theta^a\wedge d\theta^b
=
\frac14 d(D^bC_{ab}d\theta^a)
\ee
which is an exact two-form. (The factor $1/r^2$ of \eqref{s11s} has disappeared in \eqref{e53} because the precession rate is written as a two-form in coordinate basis $d\theta^a$, noting that $e^{\hat a}\sim r\,\overline{\zeta}^{{\hat a}}{}_{a}\,d\theta^a$ at leading order.) It then follows from eq.\ \eqref{e51} that
\beq
\Delta T
=
\frac{4}{c^2} \int_{\rm{Int}(\mathcal{C})} \Omega
=
\frac{1}{c^2} \oint_{\mathcal{C}} d\theta^aD^b {C}_{ab},
\ee
which coincides as announced with spin memory \cite{Pasterski:2015tva}.

\newpage
\section*{Acknowledgements}

We are grateful to Glenn Barnich, Jorrit Bosma, Marc Geiller, Mahdi Godazgar, Gregory Kozyreff and Marios Petropoulos for illuminating interactions on (dual) asymptotic symmetries, precession, and memory effects. A.S.\ also wishes to thank Gilles Esposito-Far\`ese, Roberto Oliveri and Simone Speziale for crucial discussions on gyroscopes and memory. The work of A.S.\ is funded by the European Union's Horizon 2020 research and innovation programme under the Marie Sk{\l}odowska-Curie grant agreement No 801505. The work of B.O.\ is supported by the ANR grant {\it TopO} No.\ ANR-17-CE30-0013-01, and by the European Union's Horizon 2020 research and innovation programme under the Marie Sk{\l}odowska-Curie grant agreement No.\ 846244.

\addcontentsline{toc}{section}{References}
\providecommand{\href}[2]{#2}\begingroup

redpajama_set_name: RedPajamaArXiv


VENLO, Netherlands--(BUSINESS WIRE)--Cimpress N.V. (Nasdaq: CMPR), the world leader in mass customization, announced today that Sean Quinn, senior vice president and chief financial officer, will present at the Barclays Global Technology, Media and Telecommunications conference in San Francisco on Wednesday, December 9, 2015.
Cimpress' presentation is scheduled for 1:40 p.m. PST. A live audio webcast of the presentation will be available on the Investor Relations section of the company's website at ir.cimpress.com and an audio replay will be available until June 6, 2016.


Who doesn't love a party or event? There is no doubt that ATL Events does.
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Sylvia Parkinson has worked for her family's company for 13 years and said that not a day goes by that she isn't thankful for what she does.
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Parkinson said that her favorite thing about working for ATL is seeing the end result come together.
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Over the years ATL has put on very diverse and grand events. The company planned John Madden's son's wedding, Christina Aguilera's rehearsal dinner, and has worked with Sonoma County Farm Bureau on several events including Love of the Land, Crab Feed and Ag Days. She said her favorite Farm Bureau event is the Crab Feed because it provides a great opportunity to catch up with people and has a huge silent auction.
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This overview tour is a two-hour walk of approximately one mile. It includes the graves of Susan B. Anthony and Frederick Douglass, as well as topics of local history, horticulture, geology, architecture, monument symbolism, and more. Please note that in addition to paved roads the route includes some hills and uneven terrain, grassy areas, and dirt paths. We recommend comfortable walking shoes -- not open sandals. Please call our Tour Line: 585.461.3494 with any questions.


Description: Draw the jumping ramp of the bicycle,and change the slopes and position of the ramps to adjust the air time,speed and angle of your stunt bike. Previous jumps are marked with white lines, older ones appear as green lines. Good luck! Mouse - Drag to draw the line.


Great post! I saw this film for the first time last year and what a crazy ride! Thank you for joining the blogathon!
Nice write-up. I can't say I'll ever read the book, but I enjoy the movie. Look at Maria Ouspenskaya being so nice. Love the bombastic music of Erich Wolfgang Korngold. It gives me goosebumps. And Ann Sheridan and Ronald Reagan are wonderful. You're so right about those character actors. The creme de la creme.
Every time I see this film, I want to read the book and discover all the juicy details that were left out. I hope you do a follow up post once you finish the novel.
I've watched Kings Row several times, and during my homeschooled teenage years was tempted to read it, but then I thought, "If the movie is depressing, how much more so the book must be!". So that was that. But I'm sure it is a great read, since it was such a big hit. It sounds like the author was just tired of the whole town and vented his frustration in a novel. He probably never realized that he was immortalizing it!
Yes, I don't know who had the last laugh in this matter, Bellamann or Fulton, MO!
Didn't realize Reagan had been in anything so dark. Good review.
I've never seen this, but funnily enough I've just watched an earlier small town film directed by Sam Wood, Our Town, so am very intrigued to see this one too! Will aim to do so very soon. I enjoyed your posting a lot.
Thanks! The folks in Our Town are angelic compared to the folks in Kings Row!
Agree you did a great job and picked a really good adaptation with a personal connection. I like this movie a lot too, and that's fascinating about the juicy book! Too bad so much detail had to be cut out. Thanks so much for joining the blogathon!
You are most welcome. I am glad you all focused on books to screen, as when I run out of an idea of what book to read next, I've been leaning towards the books that were turned into movies.
Fulton is only a little over an hour away from me and I have been there many times. It's for that reason I've always been fascinated by King's Row. In fact, I think I've read some of the street names in the book were based on actual street names in Fulton, although I can't be certain of that! Anyway, I've always loved the movie and thought it was so cool an actual Missourian (Bob Cummings) starred in it. You did a great job of covering the book and the film!
Thanks! A few streets in Rolla also have some of the same names as streets in Fulton;streets named after types of trees. 🙂 I almost mentioned that Cummings was a MO native, too, but then thought it made the sentence too long and I edited that bit of fact out.
I love this movie and have often thought about checking out the novel, but then I think about so many things I haven't done. As soon as I hear Korngold's score for the film I get chills. A terrific story to write about for this blogathon.
Thanks, Patricia. When I ventured to youtube to see if perhaps someone had put Kings Row there, I discovered a fascinating song battle: seems John Williams main theme to Star Wars sounds very similar to Korngold's for Kings Row! Check out the audio comparisons people have posted to youtube, along with stills from the two movies.
Whoa! I can only imagine how upset residents were when they discovered this "fictional" book was about them!
P.S. A tiny change…It is Liz who hosts the fab blog Now Voyaging.
Oops! Thanks for that info…glad you enjoyed my post on Kings Row.


text:
Woman Who Was Killed in SUV Crash on I-8 in Alpine ID'd
The driver of an SUV died Friday after the vehicle went down a highway embankment near Alpine. Photo via OnScene.TV.
Authorities have publicly identified a woman who was killed last week when the SUV she was driving crashed on Interstate 8 in her hometown of Alpine.
Bonnie Roth, 35, lost control of the Chevrolet Tahoe for unknown reasons about 8:30 a.m. Friday while she and two passengers, girls ages 3 and 9, were traveling to the west through the rural eastern San Diego County town, according to the county Medical Examiner's Office and California Highway Patrol.
The vehicle veered into an unpaved center median west of Tavern Road, overturned, tumbled down an embankment and came to rest upright on the eastbound side of the freeway.
Roth was ejected from the vehicle and died at the scene, CHP public- affairs Officer Travis Garrow said.
Paramedics took the children — who had been riding in the back seat of the SUV, the younger one secured in a car seat and the other wearing a seat belt — to Rady Children's Hospital in San Diego for treatment of non-life- threatening injuries.
Officials have not disclosed the relationship between Roth and the girls.
Chris Harris on his Mercedes SL 500 R129
'Laguna Beach' Alum Kristin Cavallari Says Lauren Conrad Feud Was 'So Messed Up'

redpajama_set_name: RedPajamaCommonCrawl


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The Best Locations and Small Events to Attend During Paris Design Week 2018 ⇒ Paris Design Week isn't made just of Maison et Objet, there will be plenty of events and locations around the city for you to attend. Four of the city's most vibrant districts will celebrate the creative industry: Saint-Germain-des-Prés, Les Halles-Marais-Bastille, Opéra-Concorde-Étoile and Barbès-Stalingrad-Saint Ouen. And for each of them, a special evening filled with exciting private views and events, bringing together the entire design community.
The Covet Group opened last year a new luxury design space/showroom, Covet Paris. This is the fourth luxury design space of its kind, following similar projects of the Covet Group in New York, London and Oporto.
This is an opulent palace of curated design that's home to several high-end brands, such as Boca do Lobo, LUXXU, BRABBU, Essential Home, DelightFull, Circu and Maison Valentina. The showroom is open for visits and it is located at 154 Rue des Rosiers, 93400 Saint-Ouen, Paris-France. Durig Paris Design Week, they will host numerous activities, so you should book a visit.
During Paris Design Week, Italian brand De Padova, which has recently opened a showroom in Rue de la Chaise, in the former Boffi Bains showroom, will present its new products. This year, the brand has chosen to focus on sofas, with two distinct new propositions Blendy and Erie, respectively designed by Omi Tahara and Elisa Ossino. Using round enveloping lines to counterbalance the thinness of the structure, Tahara and Ossino have successfully translated the spirit of the brand's late founders, Magdalena De Padova and Vico Magistretti.
When it comes to mastering the art of colour, Atelier Mériguet-Carrère truly becomes the top reference. Labelled Enterprise du Patrimoine Vivant by the French Ministry for the Economy, the brand has delved into its archives and pulled out a palette of 144 colours that you should definitely check out.
A similar approach was taken by the Disderot showroom in Passage Saint-Paul. The exhibition will honour the brand's models, including some never shown before, from the postwar years. A time that today's generations have not experienced, but that still sparks a lot of interest from antique lovers. (Re)discover the works of Abraham & Rol, RenéJean Caillette, Pierre Disderot, Roger Fatus, Michel Mortier, Joseph-André Motte, Pierre Paulin, Alain Richard, Olivier Mourgue and Pierre Guariche, that continue to be produced today with the greatest respect for the originals and the savoir-faire of Ateliers Pierre Diderot.


text:
Q: CF How to list all the files in the root directory of my project So, by now I have managed to put one of my applications on CF but I can't access the infrastructure due to the corporate model of my organization and hence its not possible for me to check the pushed code-base by accessing the server itself,
Hence, I was wondering whether there is an option to display the project tree/structure as it is present on the server instance ?
I have tried the following command but the same doesn't seem to work,
files - Print out a list of files in a directory or the contents of a specific file of an app running on the DEA backend
Directory
Output to the command present in the CF Documentation

Also, I see that it has been written that this works for a DEA backend
  , what does DEA stands for ?
I am running a nodejs app should this command work for me ?

Why I need the directory structure ?

I have setup a node app which serves the html and css pages via
  webpack build tool. So basically the html pages have the injected css
  & js file path at runtime. On my local machine I do have the access to
  this generated folder and I also find that the referenced css and js
  executes but the same is not working on my deployed app hence I want
  to understand what exactly is the folder structure there for the
  deployed app.

UPDATE

One strange thing , today I also tried to deploy the same
  project on my own pivotal cloud services instance using CF and to my
  surprise it just worked like charm as it were working locally on my
  machine i.e. no path issues which I wanted to rectify by having a look
  at the folder structure. Maybe it is something related to my server
  configuration as we are not using PWS with CF and AWS at the Iaas
  layer.
We here have a concept of Endpoints registration in the developed tool
  where the hosted apps/components are configured . Possibly that is
  causing this issue, once resolved I will share the reason and fix.

SOLUTION

So, it didn't work for me because I was behind a proxy and hence I
  eventually ended up creating an SSH connection string and an OTP.
  Then, I used PUTTY with the same proxy settings and ssh
  url:port to access the directory structure.

A: If your PCF platform Ops team has enabled the cf ssh access then you can run cf ssh <app-name> to ssh into the app container. You can then see how the files are laid out.
DEA stands for Droplet Execution Agent. DEA is the container orchestration engine in OSS CF. On PCF, DEA has been replaced with DIEGO container orchestration engine.

redpajama_set_name: RedPajamaStackExchange


text:
Privacy Noticemovejbadmin2019-05-29T07:35:36-04:00
Privacy Policy of JB Moving Services Inc.
JB Moving Services Inc. operates the https://movejb.com website, which provides the SERVICE.
This page is used to inform website visitors regarding our policies with the collection, use, and disclosure of Personal Information if anyone decided to use our Service, the JB Moving Services Inc. website.
The terms used in this Privacy Policy have the same meanings as in our Terms and Conditions, which is accessible at https://movejb.com, unless otherwise defined in this Privacy Policy.


Rift is still getting Alpha Patchnotes, find the latest below.
* You can now link locations in chat! Just like other links, shift-right-click your map or minimap while entering chat text. Clicking someone's location link in chat will place a waypoint on your own map.
* You can now swim up using the jump key while also in mouselook mode (default: right mouse button to look around).
* Added a custom error if you cannot use a questgiving item.
* Raid groups roaming across the lands to destroy non-Rift/Invasion content will see lowered experience rewards.
* Fixed a bug during Major Rifts where low ranking contributors wouldn't get Planarite. All contributors should receive at least some Planarite.
* Reduced the amount of zone event participation required to earn the rare Sourceshards.
* Experience for appropriately sized groups relative to the difficulty of Rift, Invasion, and Foothold content has been returned to higher amounts as seen previously in Beta.
* Experience for Rift, Invasion, and Foothold content now scales down from these increased levels when very oversized groups are participating – with the net result being that very large groups should see experience values similar to the amounts granted prior to this update. Smaller groups will see increased experience reward.
* Added additional fail-safes to ensure Colossi bosses appear at the end of events.
* Links in the pages of the book 'Clever Kehmeh' have been fixed so all seven pages are readable.
* Anthousa Mona and Kaspar Massi have updated character models. Kaspar also has a new voice!
* The necklace rewards for the Epic Cult Saga have had their power increased.
* Quests for Ancient Wardstones are now labeled as such in your quest journal.
* After turning in quests at some Ancient Wardstones, a Rare Planar Goods Merchant might appear so you can purchase sweet gear with the currency you just earned.
* Abilities and other effects that increase the duration of other effects now do so before that duration calculation is modified by diminishing returns.
* You can no longer cycle between ranged and melee weapons while casting an ability.
* Clerics: Fixed a bug causing some procced heals to consume effects that modify next heal.
* Ashen Defense: Corrected the debuff tooltip to match the ability description.
* Expose Weakness: Fixed a bug so that the charges of this ability are only consumed by physical damage effects. Also fixed so this correctly adds extra damage.
* Poison Malice: Removed from global cooldown. Now costs 25 Energy to activate.
* Deafening Music: This ability now updates the tooltip damage numbers for Coda of Fury and Coda of Wrath.
* Virtuoso: Removed from global cooldown. Now costs 25 Energy to activate.
* Coda of Cowardice: Debuff should now be in the same stacking group as the Archon's Ashen Defense and the Shaman's Vengeance of the Piercing Cold.
* Dancing Steel: Fixed so that it includes immunity to knockback when active.
* Blade Tempo, Blade and Soul Parity, Dualism, Double Coup: Removed from global cooldown. These now cost 25 Energy to activate.
* Entropic Veil: Will no longer be overwritten by less effective buffs.
* Satyr: Reduced size of the model; it's now approximate in size to a Bahmi.
* Icy Carapace: Fixed an issue with the ability description and debuff tooltip. They now correctly show damage values.
* Fixed a number of ability tooltip issues.
* Soul Drain: No longer applies Clinging Spirit without the caster having spent any points on that ability. Changed tooltip to better indicate that this is a player-based area of effect ability.
* Bolt of Judgment: Fixed the Spell Power contribution on ranks 1-9.
* Clinging Spirit: Fixed the tooltip to reflect that this only increases magical damage taken.
* Life and Death Concord: Description now properly indicates that the effect may not occur again for 6 seconds, rather than 8.
* Righteous Mandate: No longer blocks itself from being recast on the same target.
* Doctrine of Loyalty: Rank 5 now properly removes a Conviction.
* Hit and Run, Controlled Fire, Close Quarters Combat, and Fire and Forget: Removed from global cooldown. Now cost 25 Energy to activate.
* Quick Reload: Removed from global cooldown.
* Retreat: Should now put you 20 meters back from your current location at use.
* Lightning Fury: Fixed a tooltip error so it properly indicates that this reduces armor by a fixed value, instead of a percentage. Tooltip also now updates when spending points in Collateral Damage.
* Fan Out: Tooltip now updates when you add points into Collateral Damage.
* On the Double, Gateway: The additional effect making you immune to Snares and Roots (when you've purchased Gateway) is now displayed on On the Double.
* Skeletal Zealot: Fixed ranks 1, 2, 4, and 5 to properly respawn if it was alive when the owner died in a Warfront.
* Fell Blades: Healing debuff should now be in the same stacking group as the Druid's Spiteful Spirit.
* Ebon Fury: Taken off of global cooldown. Now costs 25 energy to activate.
* Balance of Power, Tip the Balance: Fixed an issue where these were being additionally affected by Attack Power, making the heal amount greater than it should have been.
* Searing Transfusion: Will no longer consume effects that don't actually modify it, such as Flashover, Enflamed Rejuvenation, or Greater Empowerment.
* Life of the Era: Fixed the tooltip to clear up confusion with the radius appearing to be the buff value.
* Fiery Blessing: No longer triggers the global cooldown, as the tooltip already claimed it did.
* Heat Wave, Internalize Charge: Will no longer be overwritten by less effective buffs.
* Withering Flames: Increased damage.
* Fleeting Instinct: Fixed so that it doesn't cause stealth to break when active.
* Master Huntsman: Fixed a spelling error in ability description.
* Track Beasts, Track Humanoids: Updated the tooltips to indicate that only one tracking ability can be active at a time.
* Crest of the Abyss: Now properly applies its armor buff.
* Boosted Recovery: The incoming healing bonus has been reduced to 2-6%, down from 10-30%, to be in line with the Warlord ability Figurehead.
* Ruthless Stalker, Rift Barrier, Freedom of Movement, Hasted Time: No longer causes Stealth to break when using Shadow Shift, Shadow Stalk, Shadow Warp, and Flashback.
* Stalker Phase: No longer causes Stealth to break when using Shadow Stalk.
* Track Air Rifts: Updated the tooltip to indicate that only one tracking ability can be active at a time.
* Carpet Bombing: Taken off of global cooldown. Now costs 25 Energy to activate.
* Rapid Setup: Removed from global cooldown.
* Annihilation Bomb: Should now deal damage if the enemy is immune to stuns instead of failing both damage and stun components.
* Track Constructs: Updated the tooltip to indicate that only one tracking ability can be active at a time.
* Frozen Earth: No longer stacks with other snares.
* Courage of the Bear: Fixed the tooltip to include restriction of only maintaining one Courage at a time.
* Bitter Wind: Tooltip now explains that this ability does not trigger a global cooldown.
* Lightning Hammer: Fixed Rank 7′s tooltip.
* Endless Winter: Will now show the icon with Show Ability Icons option enabled.
* Brutalize: Updated the icon to actually look like a bleed effect instead of using the same icon as Massive Blow.
* Arctic Blast: Now has a 20 second cooldown.
* Improved Electrocute: Now properly affects the cooldown of Electrocute.
* Echoes of Ascension: Now also restores 20% mana.
* Blessing of the Stone, Creeping Remedy, Little Off the Top: All of these will now correctly appear with Show Ability Icons setting enabled.
* Discharge: To better fit with desired usage, Discharge has a few new changes. Cooldown has been changed to 8 seconds. Damage reduced to come in line with other large-hit abilities. The Damage effect is now increased by Attack Power, to scale better with characters and gear.
* Singularity: Tooltip now displays the correct durations for each Attack Point.
* Rift Summon: Fixed tooltip.
* Quality Care: Clarified the tooltip on this ability to help explain it better.
* Healing Cataract: Fixed a bug causing the mana cost to be calculated as a single-target heal rather than a raid heal.
* Crushing Wave: Range reduced to 20 meters.
* Overflow: Now with more actual heals!
* Intercept: Can now be used on allies that are behind the Warlord.
* Aspect of the Elements: Power cost changed to match with other Aspects.
* Empowered Strike: Threat generation has been increased to match Leader's Mark.
* Podoxavus is no longer quite so spammy on death.
* There is a new porticulum at Knight's Stand to make travel around Freemarch easier. We also added a path from one side of Smith's Haven to the other.
* Washed Away: Reduced Mayor Jerrin's respawn time.
* I Heard a Storm Was Coming: Reduced the Fathomlord's respawn time.
* No Mercy for the Merciless: Reduced Iriri the Merciless' respawn time.
* Burning Down the Hill: Increased the number of tents that can be set on fire, and decreased the respawn of kerosene urns. Decreased general frustration level by 80%.
* Fixed a missing quest – Protector of Your Weak – in the Guardian quest flow.
* Removed the obsolete reagent vendor. These were not the items you were looking for.
* Hammerlord Post has more generally 'interesting' things going on in it.
* Polish to loot tables for Rift events.
* The Living and the Dead: Now with voiceover!
* In the Blighted Pit, characters with pets should no longer aggro Gnarlbone while he is still behind his energy field.
* Rift – Eaters of the Dead: The Sable Broodmother should now be killable.
* The Uncovered Grave that can occasionally be found in Treant's Grove is now home to some Undead who don't particularly like their crypt being disturbed.
* Death faction NPCs are now friendly with their buddies, the Death Rift NPCs.
* Rift – Deepstrike Summons: Skarn in this Rift event no longer have extra health and damage boosts.
* Water, Life, Death, and Air Raid Rifts are now available.
* Fixed Raid readycheck behavior. Indicators will now clear properly when leaving a group. Failing a readycheck now has a sound effect.
* Fixed a bug with parties disbanding and reforming inside an instance not being properly joined to that instance.
* Iceclaw Skuttler's Shell Shock no longer crits, and has reduced damage.
* Rorf's Infected Wound should now properly remove when the target reaches full health.
* Flesheater Autoch's Wounding Charge now deals less damage.
* A number if item drops in this zone were set to an incorrect level. These have been corrected and items upgraded to match with their new, proper level requirement.
* Eliam the Corrupted: Fixed a bug where Wretched Souls were not casting Wretched Union during this encounter.
* Whitefall Steppes: If you manage to get the other team's Sourcestone into your own starting area, it will return to its spawn point.
* Rewards from Warfront quests no longer have a required level to use.
* Embrace the End: Quest now rewards The Moribund notoriety, instead of Eldritch Seekers.
* Quests that require winning 5 matches of a given Warfront no longer give multiple updates for each member of the raid who is on the quest.
* Expert Dungeons now drop recipes for Apothecary and Runecrafting.
* Parchment and Vellum recipes are once again available from any Artificer Trainer.
* Planar faction merchants now have new Runecrafting recipes. These include Planar resistance runes and, in some cases, unique-flagged Planar runes.
* Recipes purchased from Planar faction merchants can no longer be resold for silver.
* The Runecrafting Artisan Store now offers some new Master level recipes. Pure stat runes for weapons and three unique Planar runes have recipes for sale.
* The Apothecary recipes offered by the Arcane Hand, Dragonslayer Covenant, Order of Mathos, and Icewatch have been changed. Heroic Manasurge Vial, Heroic Fortified Vial, and Heroic Enduring Vials have been removed. New recipes for Heroic Warding Potion, Heroic Philter of Power, Sheepish Concoction, and Exalted Bottle of Tricks have been added in their place.
* Removed faction requirement to purchase Runecrafting and Apothecary Artisan recipe scrolls.
* Fixed an issue where the Light Seafoam Dye recipe scroll required 90 skill to use instead of 110, which was the recipe's actual requirement to craft.
* The Favor quartermasters in Meridian and Sanctum now offer Runecrafting recipes for Valorous Runes.
* Fixed an issue where Rugged Fabric was not combining properly into Processed Cloth.
* Updated Riding Chaps, Insoles, Padding, and Whetstones. All of these temporary items now have a duration of four hours, rather than 5-30 minutes.
* Recipe for Ceremonial Visage now requires Stillmoor-appropriate drops instead of Iron Pine drops.
* Soft Tissue Boots can now be Runebroken.
* Renamed Icebound Tear to Icebound Heart.
* All items sold from Vendors can no longer be Salvaged or broken down with Runebreaking.
* Fixed a bug causing some items to reset their cooldowns when teleporting or logging out and in.
* Potions that grant Charge now display what they're modifying in the combat log.
* Iron Claw Shooter: Fixed so it boosts Attack Power instead of Spell Power.
* Amphibious Brew: Corrected duration to 8 minutes.
* The PvP reward, Diminishing Helm: This was improperly set as a breastplate. Fixed it to go on your head instead.
* Changed overhead PvP Quest icons to be color coded based on faction.
* Added new PvP mouse cursors.
* Fixed missing projectile effect on Flame Spear.
* Improved lighting for more natural-looking character skin.
* Updated a bunch of Soul ability sounds.
* The 'UI Volume' slider no longer affects non-environmental background noise anymore.
* Additional volume mix tuning for relative loudness/quietness of audio.
* The camera should work better with underwater clipping, making it so you aren't seeing above and below water at the same time.
* When replying to a tell, you can now hit Tab to cycle through names of people you've recently received /tells from.
* The macro system now supports the /suppressmacrofailures command, which will block the "Can only cast one ability per macro" and other messages generated by any commands following the /suppressmacrofailures entry.
* A macro's #show will now work after training up ranks in a #show'n ability.
* Fixed mousebutton 3, 4, and 5 + a keyboard modifier key (SHIFT, ALT, CTRL) not working very well when bound to an action bar.
* Also fixed mousebutton 3, 4, and 5 hiding the cursor.
* /wink correctly prints the wink emote text, rather than /flirt's.
* Substitution tokens such as %t, %R, etc, now work in /tweets.
* You can now set how many quests to track on-screen at once (up to 10) by changing the setting in the Interface -> Display category.
* The mouse cursor should now correctly change to the glowing or greyed out gauntlet to indicate whether you can cast a spell on a ground target, Salvage, or Runebreak.
* The Auction House now properly saves and displays the duration of the last posted auction.
* Planar Augments are now removed from the Crafting display after they've been used to create an item.
* Moved quest log indicators to the correct location for quests that group members are also on.
* Friend login/logout messages are now displayed in the chat window. Added an option under Social settings to toggle these notifications.
* Edit Layout mode: It's now possible to enter decimal points into the scale/alpha setting boxes.
* Additional Edit Layout improvements.
* Mount icons on toolbars will now properly grey out when inside instances.
* We also display a better error message when you're blocked from mounting in an area, rather than 'Invalid Target'.
* Added support for additional Turkish chat characters.
GENERAL GAMEPLAY* You can now link locations in chat! Just like other links, shift-right-click your map or minimap while entering chat text. Clicking someone's location link in chat will place a waypoint on your own map.* You can now swim up using the jump key while also in mouselook mode (default: right mouse button to look around).* Added a custom error if you cannot use a questgiving item.* Raid groups roaming across the lands to destroy non-Rift/Invasion content will see lowered experience rewards.
RIFTS AND INVASIONS* Fixed a bug during Major Rifts where low ranking contributors wouldn't get Planarite. All contributors should receive at least some Planarite.* Reduced the amount of zone event participation required to earn the rare Sourceshards.* Experience for appropriately sized groups relative to the difficulty of Rift, Invasion, and Foothold content has been returned to higher amounts as seen previously in Beta. * Experience for Rift, Invasion, and Foothold content now scales down from these increased levels when very oversized groups are participating – with the net result being that very large groups should see experience values similar to the amounts granted prior to this update. Smaller groups will see increased experience reward.* Added additional fail-safes to ensure Colossi bosses appear at the end of events.
CULT SAGA* Links in the pages of the book 'Clever Kehmeh' have been fixed so all seven pages are readable.* Anthousa Mona and Kaspar Massi have updated character models. Kaspar also has a new voice!* The necklace rewards for the Epic Cult Saga have had their power increased.
ANCIENT WARDSTONES* Quests for Ancient Wardstones are now labeled as such in your quest journal. * After turning in quests at some Ancient Wardstones, a Rare Planar Goods Merchant might appear so you can purchase sweet gear with the currency you just earned.
COMBAT* Abilities and other effects that increase the duration of other effects now do so before that duration calculation is modified by diminishing returns. * You can no longer cycle between ranged and melee weapons while casting an ability.
SOULS* Clerics: Fixed a bug causing some procced heals to consume effects that modify next heal.
ARCHON* Ashen Defense: Corrected the debuff tooltip to match the ability description.
ASSASSIN* Expose Weakness: Fixed a bug so that the charges of this ability are only consumed by physical damage effects. Also fixed so this correctly adds extra damage.* Poison Malice: Removed from global cooldown. Now costs 25 Energy to activate.
BARD* Deafening Music: This ability now updates the tooltip damage numbers for Coda of Fury and Coda of Wrath.* Virtuoso: Removed from global cooldown. Now costs 25 Energy to activate.* Coda of Cowardice: Debuff should now be in the same stacking group as the Archon's Ashen Defense and the Shaman's Vengeance of the Piercing Cold.
BLADEDANCER* Dancing Steel: Fixed so that it includes immunity to knockback when active.* Blade Tempo, Blade and Soul Parity, Dualism, Double Coup: Removed from global cooldown. These now cost 25 Energy to activate.
CHLOROMANCER* Entropic Veil: Will no longer be overwritten by less effective buffs.
DRUID* Satyr: Reduced size of the model; it's now approximate in size to a Bahmi.
ELEMENTALIST* Icy Carapace: Fixed an issue with the ability description and debuff tooltip. They now correctly show damage values.
INFILTRATOR [PvP]* Fixed a number of ability tooltip issues.
INQUISITOR* Soul Drain: No longer applies Clinging Spirit without the caster having spent any points on that ability. Changed tooltip to better indicate that this is a player-based area of effect ability.* Bolt of Judgment: Fixed the Spell Power contribution on ranks 1-9.* Clinging Spirit: Fixed the tooltip to reflect that this only increases magical damage taken.* Life and Death Concord: Description now properly indicates that the effect may not occur again for 6 seconds, rather than 8.
JUSTICAR* Righteous Mandate: No longer blocks itself from being recast on the same target.* Doctrine of Loyalty: Rank 5 now properly removes a Conviction.
MARKSMAN* Hit and Run, Controlled Fire, Close Quarters Combat, and Fire and Forget: Removed from global cooldown. Now cost 25 Energy to activate.* Quick Reload: Removed from global cooldown.* Retreat: Should now put you 20 meters back from your current location at use.* Lightning Fury: Fixed a tooltip error so it properly indicates that this reduces armor by a fixed value, instead of a percentage. Tooltip also now updates when spending points in Collateral Damage.* Fan Out: Tooltip now updates when you add points into Collateral Damage.* On the Double, Gateway: The additional effect making you immune to Snares and Roots (when you've purchased Gateway) is now displayed on On the Double.
NECROMANCER* Skeletal Zealot: Fixed ranks 1, 2, 4, and 5 to properly respawn if it was alive when the owner died in a Warfront.
NIGHTBLADE* Fell Blades: Healing debuff should now be in the same stacking group as the Druid's Spiteful Spirit.* Ebon Fury: Taken off of global cooldown. Now costs 25 energy to activate.
PALADIN* Balance of Power, Tip the Balance: Fixed an issue where these were being additionally affected by Attack Power, making the heal amount greater than it should have been.
PURIFIER* Searing Transfusion: Will no longer consume effects that don't actually modify it, such as Flashover, Enflamed Rejuvenation, or Greater Empowerment.* Life of the Era: Fixed the tooltip to clear up confusion with the radius appearing to be the buff value.* Fiery Blessing: No longer triggers the global cooldown, as the tooltip already claimed it did.
PYROMANCER* Heat Wave, Internalize Charge: Will no longer be overwritten by less effective buffs.* Withering Flames: Increased damage.
RANGER* Fleeting Instinct: Fixed so that it doesn't cause stealth to break when active.* Master Huntsman: Fixed a spelling error in ability description.* Track Beasts, Track Humanoids: Updated the tooltips to indicate that only one tracking ability can be active at a time.
REAVER* Crest of the Abyss: Now properly applies its armor buff.
RIFTSTALKER* Boosted Recovery: The incoming healing bonus has been reduced to 2-6%, down from 10-30%, to be in line with the Warlord ability Figurehead.* Ruthless Stalker, Rift Barrier, Freedom of Movement, Hasted Time: No longer causes Stealth to break when using Shadow Shift, Shadow Stalk, Shadow Warp, and Flashback.* Stalker Phase: No longer causes Stealth to break when using Shadow Stalk.* Track Air Rifts: Updated the tooltip to indicate that only one tracking ability can be active at a time.
SABOTEUR* Carpet Bombing: Taken off of global cooldown. Now costs 25 Energy to activate.* Rapid Setup: Removed from global cooldown.* Annihilation Bomb: Should now deal damage if the enemy is immune to stuns instead of failing both damage and stun components.* Track Constructs: Updated the tooltip to indicate that only one tracking ability can be active at a time.
SHAMAN* Frozen Earth: No longer stacks with other snares.* Courage of the Bear: Fixed the tooltip to include restriction of only maintaining one Courage at a time.* Bitter Wind: Tooltip now explains that this ability does not trigger a global cooldown.* Lightning Hammer: Fixed Rank 7′s tooltip.* Endless Winter: Will now show the icon with Show Ability Icons option enabled.* Brutalize: Updated the icon to actually look like a bleed effect instead of using the same icon as Massive Blow.
STORMCALLER* Arctic Blast: Now has a 20 second cooldown.* Improved Electrocute: Now properly affects the cooldown of Electrocute.
TEMPLAR [PvP]* Echoes of Ascension: Now also restores 20% mana.* Blessing of the Stone, Creeping Remedy, Little Off the Top: All of these will now correctly appear with Show Ability Icons setting enabled.
VOID KNIGHT* Discharge: To better fit with desired usage, Discharge has a few new changes. Cooldown has been changed to 8 seconds. Damage reduced to come in line with other large-hit abilities. The Damage effect is now increased by Attack Power, to scale better with characters and gear.* Singularity: Tooltip now displays the correct durations for each Attack Point.* Rift Summon: Fixed tooltip.* Quality Care: Clarified the tooltip on this ability to help explain it better.
WARDEN* Healing Cataract: Fixed a bug causing the mana cost to be calculated as a single-target heal rather than a raid heal.* Crushing Wave: Range reduced to 20 meters.* Overflow: Now with more actual heals!
WARLORD* Intercept: Can now be used on allies that are behind the Warlord.* Aspect of the Elements: Power cost changed to match with other Aspects.* Empowered Strike: Threat generation has been increased to match Leader's Mark.
DROUGHTLANDS* Podoxavus is no longer quite so spammy on death.
FREEMARCH* There is a new porticulum at Knight's Stand to make travel around Freemarch easier. We also added a path from one side of Smith's Haven to the other.* Washed Away: Reduced Mayor Jerrin's respawn time.* I Heard a Storm Was Coming: Reduced the Fathomlord's respawn time.* No Mercy for the Merciless: Reduced Iriri the Merciless' respawn time.* Burning Down the Hill: Increased the number of tents that can be set on fire, and decreased the respawn of kerosene urns. Decreased general frustration level by 80%.
IRON PINE PEAKS* Fixed a missing quest – Protector of Your Weak – in the Guardian quest flow.
MERIDIAN* Removed the obsolete reagent vendor. These were not the items you were looking for.
MOONSHADE HIGHLANDS* Hammerlord Post has more generally 'interesting' things going on in it.* Polish to loot tables for Rift events.
SCARLET GORGE* The Living and the Dead: Now with voiceover!
SCARWOOD REACH* In the Blighted Pit, characters with pets should no longer aggro Gnarlbone while he is still behind his energy field.* Rift – Eaters of the Dead: The Sable Broodmother should now be killable.
SILVERWOOD* The Uncovered Grave that can occasionally be found in Treant's Grove is now home to some Undead who don't particularly like their crypt being disturbed.
STILLMOOR* Death faction NPCs are now friendly with their buddies, the Death Rift NPCs.
STONEFIELD* Rift – Deepstrike Summons: Skarn in this Rift event no longer have extra health and damage boosts.
DUNGEONS AND RAIDS* Water, Life, Death, and Air Raid Rifts are now available.* Fixed Raid readycheck behavior. Indicators will now clear properly when leaving a group. Failing a readycheck now has a sound effect.* Fixed a bug with parties disbanding and reforming inside an instance not being properly joined to that instance.
ABYSSAL PRECIPICE* Iceclaw Skuttler's Shell Shock no longer crits, and has reduced damage.
FALL OF LANTERN HOOK* Rorf's Infected Wound should now properly remove when the target reaches full health.
KING'S BREACH* Flesheater Autoch's Wounding Charge now deals less damage.
RUNIC DESCENT* A number if item drops in this zone were set to an incorrect level. These have been corrected and items upgraded to match with their new, proper level requirement.* Eliam the Corrupted: Fixed a bug where Wretched Souls were not casting Wretched Union during this encounter.
PVP/WARFRONTS* Whitefall Steppes: If you manage to get the other team's Sourcestone into your own starting area, it will return to its spawn point.* Rewards from Warfront quests no longer have a required level to use.* Embrace the End: Quest now rewards The Moribund notoriety, instead of Eldritch Seekers.* Quests that require winning 5 matches of a given Warfront no longer give multiple updates for each member of the raid who is on the quest.
CRAFTING* Expert Dungeons now drop recipes for Apothecary and Runecrafting.* Parchment and Vellum recipes are once again available from any Artificer Trainer.* Planar faction merchants now have new Runecrafting recipes. These include Planar resistance runes and, in some cases, unique-flagged Planar runes.* Recipes purchased from Planar faction merchants can no longer be resold for silver.* The Runecrafting Artisan Store now offers some new Master level recipes. Pure stat runes for weapons and three unique Planar runes have recipes for sale.* The Apothecary recipes offered by the Arcane Hand, Dragonslayer Covenant, Order of Mathos, and Icewatch have been changed. Heroic Manasurge Vial, Heroic Fortified Vial, and Heroic Enduring Vials have been removed. New recipes for Heroic Warding Potion, Heroic Philter of Power, Sheepish Concoction, and Exalted Bottle of Tricks have been added in their place.* Removed faction requirement to purchase Runecrafting and Apothecary Artisan recipe scrolls.* Fixed an issue where the Light Seafoam Dye recipe scroll required 90 skill to use instead of 110, which was the recipe's actual requirement to craft.* The Favor quartermasters in Meridian and Sanctum now offer Runecrafting recipes for Valorous Runes.* Fixed an issue where Rugged Fabric was not combining properly into Processed Cloth.* Updated Riding Chaps, Insoles, Padding, and Whetstones. All of these temporary items now have a duration of four hours, rather than 5-30 minutes.* Recipe for Ceremonial Visage now requires Stillmoor-appropriate drops instead of Iron Pine drops.* Soft Tissue Boots can now be Runebroken.* Werewolves can now be Butchered.* Renamed Icebound Tear to Icebound Heart.* All items sold from Vendors can no longer be Salvaged or broken down with Runebreaking.
ITEMS* Fixed a bug causing some items to reset their cooldowns when teleporting or logging out and in.* Potions that grant Charge now display what they're modifying in the combat log.* Iron Claw Shooter: Fixed so it boosts Attack Power instead of Spell Power.* Amphibious Brew: Corrected duration to 8 minutes.* The PvP reward, Diminishing Helm: This was improperly set as a breastplate. Fixed it to go on your head instead.
GRAPHICS AND AUDIO* Changed overhead PvP Quest icons to be color coded based on faction.* Added new PvP mouse cursors.* Fixed missing projectile effect on Flame Spear.* Improved lighting for more natural-looking character skin.* Updated a bunch of Soul ability sounds.* The 'UI Volume' slider no longer affects non-environmental background noise anymore.* Additional volume mix tuning for relative loudness/quietness of audio.* The camera should work better with underwater clipping, making it so you aren't seeing above and below water at the same time.
COMMANDS AND MACROS* When replying to a tell, you can now hit Tab to cycle through names of people you've recently received /tells from.* The macro system now supports the /suppressmacrofailures command, which will block the "Can only cast one ability per macro" and other messages generated by any commands following the /suppressmacrofailures entry.* A macro's #show will now work after training up ranks in a #show'n ability.* Fixed mousebutton 3, 4, and 5 + a keyboard modifier key (SHIFT, ALT, CTRL) not working very well when bound to an action bar.* Also fixed mousebutton 3, 4, and 5 hiding the cursor.* /wink correctly prints the wink emote text, rather than /flirt's.* Substitution tokens such as %t, %R, etc, now work in /tweets.
UI/SETTINGS* You can now set how many quests to track on-screen at once (up to 10) by changing the setting in the Interface -> Display category.* The mouse cursor should now correctly change to the glowing or greyed out gauntlet to indicate whether you can cast a spell on a ground target, Salvage, or Runebreak. * The Auction House now properly saves and displays the duration of the last posted auction.* Planar Augments are now removed from the Crafting display after they've been used to create an item.* Moved quest log indicators to the correct location for quests that group members are also on.* Friend login/logout messages are now displayed in the chat window. Added an option under Social settings to toggle these notifications.* Edit Layout mode: It's now possible to enter decimal points into the scale/alpha setting boxes.* Additional Edit Layout improvements.* Mount icons on toolbars will now properly grey out when inside instances.* We also display a better error message when you're blocked from mounting in an area, rather than 'Invalid Target'.* Added support for additional Turkish chat characters.