text:
\section{Introduction}
\label{Intro}
Matter affects the dynamics of both electromagnetic and gravitational fields. For electromagnetism, the effective classical description in many materials is provided by the so-called Maxwell's equations in matter \cite{Jackson:1998nia}. In the effective description, the microscopic Maxwell's equations $\nabla_{\!\nu} F^{\mu\nu} = J^\mu$ are replaced by $\nabla_{\!\nu} H^{\mu\nu} = J^\mu_{\rm free}$, where the tensor $H^{\mu\nu}$ differs from $F^{\mu\nu}$ by substituting the electric field ${\bf E} \to {\bf D}$, and the magnetic field ${\bf B}\to {\bf H}$. The current $J^\mu_{\rm free}$ is the current of ``free'' charges, and the vectors ${\bf D}$, ${\bf H}$ are to be expressed in terms of ${\bf E}$, ${\bf B}$ through the so-called constitutive relations that are specific to a given material. The constitutive relations reflect the existence of both electric and magnetic polarization. As a simple example, consider the Maxwell action in flat space, $S_{\rm eff} = -\coeff14 \int d^4x \, F_{\mu\nu} F^{\mu\nu}$, or in terms of the electric and magnetic fields
\begin{equation}
  S_{\rm eff} = \frac{1}{2} \int\!\!d^4x \left( {\bf E}^2 - {\bf B}^2 \right).
\end{equation}
In the presence of matter, boost invariance is broken by the matter rest frame, and a more general action is allowed by rotation invariance:
\begin{equation}
\label{eq:emaction-2}
  S_{\rm eff} = \frac{1}{2} \int\!\!d^4x \left(\varepsilon_{\text e} {\bf E}^2 - \frac{{\bf B}^2}{\mu_{\text m}} \right).
\end{equation}
The coefficients $\varepsilon_{\text e}$ (electric permittivity) and $\mu_{\text m}$ (magnetic permeability) can in principle be calculated from the fundamental microscopic theory. The action gives rise to the constitutive relations ${\bf D} = \varepsilon_{\text e} {\bf E}$ and ${\bf B} = \mu_{\text m} {\bf H}$ (see e.g.~\cite{Kovtun:2016lfw} for a covariant discussion). This leads to the refractive index $\sqrt{\varepsilon_{\text e} \mu_{\text m}}$ for electromagnetic waves propagating through matter.

One can ask similar questions about gravity. While the naive notions of electric and magnetic  polarizations do not have a gravitational analogue, one can ask about the effective description of gravitational fields in matter. Just like in electromagnetism the definition of the macroscopic tensor $H^{\mu\nu}$ boils down to the structure of the macroscopic current~$J^\mu$, one can ask about the structure of the macroscopic energy-momentum tensor $T^{\mu\nu}$ in the Einstein's equations $G_{\mu\nu} = \kappa^2\, T_{\mu\nu}$, where $G_{\mu\nu}$ is the Einstein tensor, and $\kappa^2 \equiv 8\piG$, with $G$ the Newton's constant. As an example, consider the following effective action for equilibrium gravity-matter configurations,
\begin{equation}
\label{eq:S1}
  S_{\rm eff} = \int \!\!d^{d+1}x\,\sqrt{-g} \left[ \frac{1}{2\kappa^2} R + p(T) \right],
\end{equation}
where $R$ is the Ricci scalar, and $p(T)$ is the pressure as a function of temperature, given by the equation of state. As we will show shortly, this action gives rise to Einstein's equations sourced by the perfect fluid energy-momentum tensor, $G^{\mu\nu} = \kappa^2\left[(\epsilon{+}p) u^\mu u^\nu + p g^{\mu\nu} \right]$, where $\epsilon \equiv -p + T \partial p/\partial T$ is the energy density, derived from the pressure by the standard Euler relation of thermodynamics. However, in curved space, there is more to thermodynamics than just the equation of state. The perfect fluid model of matter ignores the fact that the energy-momentum tensor of macroscopic matter in general depends on curvature, which is true even for an ideal (quantum) gas of spinless particles. As an example, note that the action~(\ref{eq:S1}) is only an effective macroscopic action, and the coefficient in front of $R$ is not required to be constant due to the presence of matter. More generally, one can have
\begin{equation}
\label{eq:S2}
  S_{\rm eff} = \int \!\!d^{d+1}x\,\sqrt{-g} \left[ f(T) R + p(T) \right]\,,
\end{equation}
where $f(T)$ plays the role of a gravitational susceptibility. Just like the pressure $p(T)$, the susceptibility coefficient $f(T)$ can in principle be evaluated from the fundamental microscopic theory, and will depend on temperature. The vacuum contributions to $p(T)$ and $f(T)$ have to be matched to the cosmological constant and the Newton's constant respectively, $p(T) = (-2\Lambda)/2\kappa^2 + p_{\rm matter}(T)$, $f(T)=1/2\kappa^2 + f_{\rm matter}(T)$. As the temperature $T$ depends on the metric,%
\footnote{
For matter in equilibrium, one has $T=T_0/\sqrt{-g_{00}}$ in suitable coordinates, see e.g.~\cite{Landau:1980mil}.
}
the Einstein's equations obtained by varying (\ref{eq:S2}) with respect to the metric will be sourced by the energy-momentum tensor which is not of the perfect fluid form, due to $f_{\rm matter}(T)$.

In what follows, we will make the above intuition precise. Focusing on the simplest form of relativistic matter which is locally rotation-invariant (that is, a fluid), we will derive the Einstein equations which govern the effective gravitational fields. Among other things, we will explicitly evaluate $f_{\rm matter}(T)$ for an ideal gas of massive fermions at non-zero temperature and density. We will then consider an application of the formalism to spherical equilibria of gravitating matter. [Convention: We work in natural units $c = \hbar = 1$. The metric signature is mostly plus.]

\section{Einstein's equations in matter}
\label{EMEquations}
We consider a macroscopic system that has degrees of freedom which couple to the metric $g_{\mu\nu}$ and to an Abelian gauge field $A_\mu$. Our procedure for deriving the effective Einstein-Maxwell equations will be as follows. (I) We first consider the matter in thermal equilibrium, subject to external, time-independent $g_{\mu\nu}$ and $A_\mu$. The system is then characterized by the grand canonical partition function $Z$, and the generating functional $W[g,A]=-i\ln Z$. The generating functional (which is the grand canonical potential, up to a factor of $T$) is extensive in the thermodynamic limit, and can be written as an integral of a local density. (II)~We identify the equilibrium generating functional with the equilibrium effective action, $W[g,A] = S_{\rm eff}[g,A]$. With the fields $g$ and $A$ now dynamical, the equations of motion which follow from the equilibrium effective action will describe equilibrium configurations of the metric and the gauge field. (III) We then argue that for time dependent processes that are sufficiently slow, the equations of motion that describe equilibrium configurations may serve as a useful approximation to describe near-equilibrium configurations. The resulting Einstein-Maxwell equations will contain perfect fluid dynamics, electromagnetic polarization, and the gravitational susceptibilities that parametrize the response of the matter's pressure to curvature. The resulting equations clearly do not include any dissipative processes; the latter, parametrized by the dissipative transport coefficients such as the electrical conductivity and the viscosity, can be introduced in the constitutive relations {\it a~posteriori}. Step (III) is just the standard construction of fluid dynamics as an extension of local equilibrium thermodynamics. If we choose to keep only $g$ (but not $A$) dynamical, we have the Einstein (rather than Einstein-Maxwell) equations in matter, with $A$ as a background field which can couple for example to the baryon number current.

\subsection{The generating functional}
\label{EqGenFunc}
Consider matter in thermal equilibrium, described by the generating functional $W[g,A]$. The generating functional encodes equilibrium (zero-frequency) correlation functions of the energy-momentum tensor and the conserved $U(1)$ current which couple to the external metric $g_{\mu\nu}$ and the external gauge field $A_\mu$. In the context of relativistic hydrodynamics such equilibrium generating functionals were first discussed in \cite{Banerjee:2012iz, Jensen:2012jh}; the present discussion makes use of the setup in \cite{Jensen:2012jh, Jensen:2013kka, Hernandez:2017mch, Kovtun:2018dvd}.

Thermal equilibrium is characterized by a timelike Killing vector $V^\mu$, which in suitable coordinates (local rest frame of the fluid) takes the form $V^\mu = (1,{\bf 0})$. The matter velocity $u^\mu$, temperature $T$, and the chemical potential $\mu$ are defined via
\begin{equation}
\label{eq:uTmu}
  u^\mu = \frac{V^\mu}{\sqrt{-V^2}}\,,\
  T = \frac{1}{\beta_0 \sqrt{-V^2}}\,,\
  \mu = \frac{ V^\rho A_\rho + \Lambda_V}{\sqrt{-V^2}}\,,
\end{equation}
where the constant $\beta_0$ sets the normalization of temperature, and $\Lambda_V$ is a gauge function introduced to ensure the gauge invariance of the chemical potential. The fact that the system is in equilibrium is captured by the conditions
\begin{equation}
\label{eq:equil-cond}
  L_V g_{\mu\nu}=0\,,\quad L_V A_\mu + \partial_\mu \Lambda_V = 0\,,
\end{equation}
where $L_V$ denotes the Lie derivative with respect to the timelike vector $V^\mu$.

The generating functional is extensive in the thermodynamic limit, and can therefore be expressed as an integral of a local density,
\begin{equation}
\label{eq:W1}
  W[g,A] = \int\!\!d^{d+1}x\,\sqrt{-g}\; {\cal F}[g,A]\,,
\end{equation}
where $\mathcal{F}[g,A]$ is a local function of the external sources $g_{\mu\nu}, A_\mu$. The variation of the generating functional with respect to the sources defines the energy-momentum tensor and the conserved current of the fluid,
\begin{equation}
\label{defTJ}
T^{\mu\nu} = \frac{2}{\sqrt{-g}} \, \frac{\delta W}{\delta g_{\mu\nu}}\, , \quad J^\mu = \frac{1}{\sqrt{-g}} \, \frac{\delta W}{\delta A_{\mu}}.
\end{equation}
Provided that the system of interest does not have any anomalies, the generating functional is both diffeomorphism and gauge invariant. As a consequence of this we have the conservation laws
\begin{subequations}
\label{eq:TJ-cons-1}
\begin{align}
  & \nabla_{\!\mu} T^{\mu\nu} = F^{\nu\lambda}J_\lambda\,,\\
\label{eq:J-cons-1}
  & \nabla_{\!\mu} J^\mu = 0\,,
\end{align}
\end{subequations}
with $F_{\mu\nu} \equiv \partial_\mu A_\nu - \partial_\nu A_\mu$.

When the sources $g_{\mu\nu}, A_\mu$ vary on length scales much longer than the correlation length of the system, the density $\mathcal{F}[g,A]$ admits an expansion in terms of the derivatives of the external sources. Thus, finding the generating functional up to a given order in derivatives boils down to finding the diffeomorphism- and gauge-invariant objects made out of the sources $g_{\mu\nu}, A_\mu$, as well as the hydrodynamic variables of eq.\ \eqref{eq:uTmu}.

Being in thermal equilibrium implies certain constraints on the hydrodynamic variables, such as
\begin{equation}
u^\lambda \partial_\lambda T = 0, \quad u^\lambda \partial_\lambda \mu = 0,
\end{equation}
i.e.~the temperature and the chemical potential are time independent. The expansion $\nabla{\cdot}u$ and the shear tensor $\sigma^{\mu\nu} = \Delta^{\mu\alpha} \Delta^{\nu\beta} (\nabla_\alpha u_\beta + \nabla_\beta u_\alpha - \coeff23 \Delta_{\alpha\beta} \nabla{\cdot} u)$ both vanish in equilibrium, ensuring that there is no entropy production due to the bulk or shear viscosities in equilibrium. The spatial projector is $\Delta^{\mu\nu} \equiv g^{\mu\nu} + u^\mu u^\nu$.

For the derivative expansion of the generating functional, we follow the notation of~\cite{Kovtun:2018dvd}. In four spacetime dimensions, and up to second order in derivatives of the sources, the equilibrium generating functional takes the form
\begin{equation}
\label{eq:W2}
  W[g,A] = \int\!\!d^{4}x\,\sqrt{-g}\left[ p + \sum_{n=1}^9 f_n\, s_n^{(2)}\right] + \cdots
\end{equation}
where the dots denote terms that are cubic or higher order in derivatives. The pressure $p=p(T,\mu)$ is a function of temperature and chemical potential, given by the equation of state. There are no first-order invariants. The $s_n^{(2)}$ denote the nine independent second-order invariants appearing in the generating functional \cite{Banerjee:2012iz}, and the $f_n = f_n(T,\mu)$ denote the corresponding thermodynamic susceptibilities. Table \ref{table:2ndOrder} summarizes the nine invariants, along with their transformation properties under parity $\mathcal{P}$, charge conjugation $\mathcal{C}$, and time reversal $\mathcal{T}$. The notation in the table is as follows: the acceleration is $a^\mu \equiv u^\nu \nabla_\nu u^\mu$, the electric field is $E_\mu = F_{\mu\nu} u^\nu$, the magnetic field is $B^\mu \equiv \coeff12 \epsilon^{\mu\nu\alpha\beta} u_\nu F_{\alpha\beta}$, and the vorticity vector is $\Omega^\mu \equiv \epsilon^{\mu\nu\alpha\beta} u_\nu \nabla_{\!\alpha} u_\beta$. The pressure and the susceptibilities have to be determined from the underlying microscopic theory. See \cite{Romatschke:2009ng, Moore:2010bu, Moore:2012tc, Megias:2014mba, Buzzegoli:2017cqy, Kovtun:2018dvd, Shukla:2019shf} where some of these susceptibilities have been computed for free quantum fields.

\begin{table}
\begin{center}
\def1.2{1.2}
\setlength\tabcolsep{4pt}
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|}
 \hline
 \hline
 $n$ & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9\\
 \hline
 \hline
 $s^{(2)}_n$
 & $R$
 & $a^2$  %
 & $\Omega^2$
 & $B^2$
 & $B{\cdot}\Omega$
 & $E^2$
 & $E{\cdot}a$
 & $B{\cdot}E$
 & $B{\cdot}a$
 \\
 \hline
  $\mathcal{P}$  & $+$ & $+$ & $+$ & $+$ & $+$ & $+$ & $+$ & $-$ & $-$\\
  \hline
  $\mathcal{C}$  & $+$ & $+$ & $+$ & $+$ & $-$ & $+$ & $-$ & $+$ & $-$\\
  \hline
  $\mathcal{T}$  & $+$ & $+$ & $+$ & $+$ & $+$ & $+$ & $+$ & $-$ & $-$\\
  \hline
\end{tabular}
\end{center}
\caption{The nine second-order invariants appearing in the derivative expansion eq.\ \eqref{eq:W2}. $\mathcal{P}, \mathcal{C}, \mathcal{T}$ respectively denote the eigenvalues under parity, charge conjugation and time reversal of the corresponding invariant.}
\label{table:2ndOrder}
\end{table}

\subsection{The energy-momentum tensor}
We can now compute the energy-momentum tensor following from the variation of the generating functional eq.\ \eqref{eq:W2}. Isolating the Einstein tensor, we have
\begin{equation}
\label{eq:Tmatter-def}
  T^{\mu\nu} = -\frac{1}{\kappa_{\rm eff}^2} \left( R^{\mu\nu} - \coeff12 R g^{\mu\nu} \right) + T_{\rm m}^{\mu\nu}\,,
\end{equation}
which defines the ``matter'' contribution $T_{\rm m}^{\mu\nu}$, and $1/\kappa_{\rm eff}^2 \equiv 2f_1(T,\mu)$. We can decompose $T_{\rm m}^{\mu\nu}$ with respect to the fluid velocity $u^\mu$ as
\begin{equation}
\label{eq:Tmn}
  T_{\rm m}^{\mu\nu} = {\cal E}_{\rm m} u^\mu u^\nu + {\cal P}_{\rm m}\Delta^{\mu\nu} +{\cal Q}_{\rm m}^\mu u^\nu + {\cal Q}_{\rm m}^\nu u^\mu + {\cal T}_{\rm m}^{\mu\nu},
\end{equation}
where the ``matter'' energy density is ${\cal E}_{\rm m}\equiv u_\mu T_{\rm m}^{\mu\nu}u_\nu$, the pressure is ${\cal P}_{\rm m}\equiv \frac13 \Delta_{\mu\nu}T_{\rm m}^{\mu\nu}$, the energy flux ${\cal Q}_{\rm m}^\mu \equiv - \Delta^\mu_{\alpha} T_{\rm m}^{\alpha\beta}u_\beta$ is transverse to $u^\mu$, and the stress ${\cal T}_{\rm m}^{\mu\nu} \equiv T_{\rm m}^{\langle \mu\nu\rangle}$ is transverse to $u_\mu$, symmetric, and traceless. The brackets denote the symmetric transverse traceless part, $X_{\langle \mu\nu\rangle} \equiv \frac12 (\Delta_{\mu\alpha} \Delta_{\nu\beta} + \Delta_{\nu\alpha} \Delta_{\mu\beta} -\frac23 \Delta_{\mu\nu} \Delta_{\alpha\beta}) X^{\alpha\beta}$.

For matter that is not coupled to external electric or magnetic fields, the only relevant two-derivative invariants are $f_{1,2,3}$. The matter energy-momentum tensor eq.\ \eqref{eq:Tmn} coming from the variation of the generating functional eq.\ \eqref{eq:W2} then has the form
\begin{subequations}
\label{eq:cr-statics}
\begin{align}
&\mathcal{E}_{\rm m} = \epsilon + f_1' R + \big( 4f_1' + 2f_1'' - f_2 - f_2'\big) a^2 +\big( f_1'-f_2-3f_3+f_3'\big) \Omega^2- 2 \left(f_1'-f_2\right) u^\alpha R_{\alpha\beta} u^\beta , \label{stressel}\\
&\mathcal{P}_{\rm m} = p - \frac{1}{3} \left(2f_1' + 4 f_1'' - f_2 \right) a^2 - \frac{1}{3} (2f_1'+f_3) \Omega^2 + \frac{4}{3} f_1' u^\alpha R_{\alpha\beta} u^\beta , \label{stressP}\\
&\mathcal{Q}_{{\rm m}\,\mu} = \big(f_1'+2f_3'\big) \epsilon_{\mu\nu\alpha\beta} a^\nu u^\alpha \Omega^\beta + 4f_3 \Delta^\nu_\mu R_{\nu\sigma} u^\sigma, \label{stressQ}\\
&\mathcal{T}_{{\rm m}\,\mu\nu} = \left( 4 f_1' + 2 f_1'' - 2 f_2\right) a_{\langle \mu} a_{\nu \rangle} - \frac{1}{2} \big( f_1' - 4f_3\big) \Omega_{\langle\mu} \Omega_{\nu\rangle} + 2 f_1' u^\alpha R_{\alpha\langle \mu \nu \rangle \beta} u^\beta, \label{stressT}
\end{align}
\end{subequations}
with
\begin{equation}
\begin{split}
\label{defprime}
&f_n' = T \, \frac{\partial f_n}{\partial T} + \mu \, \frac{\partial f_n}{\partial \mu},\\
&f_n'' =  T^2 \, \frac{\partial^2 f_n}{\partial T^2}+ 2\mu T \,\frac{\partial^2 f_n}{\partial \mu \, \partial T}  + \mu^2 \, \frac{\partial^2 f_n}{\partial \mu^2},
\end{split}
\end{equation}
and $\epsilon = - p + T\, \partial p/\partial T + \mu \, \partial p/\partial \mu$ is the zeroth-order energy density.

\subsection{The effective Einstein's equations}
We now promote the equilibrium generating functional to be the equilibrium effective action for the metric, $W[g,A] = S_{\rm eff}[g,A]$. This makes the metric a dynamical variable whose equations of motion are determined by the variational principle $\delta_g S_{\rm eff}[g,A] = 0$, or simply $T^{\mu\nu}[g,A] = 0$. According to the definition of the ``matter'' contribution in eq.~(\ref{eq:Tmatter-def}), the equations of motion are
\begin{equation}
\label{eq:EE-1}
  R^{\mu\nu} - \coeff12 R g^{\mu\nu}  = \kappa_{\rm eff}^2 T_{\rm m}^{\mu\nu}\,,
\end{equation}
where $T^{\mu\nu}_{\rm m}$ contains the perfect fluid, as well as the curvature-dependent contributions. The coefficient $\kappa_{\rm eff}^2 = 1/(2f_1)$ should be interpreted as ($8\pi$ times) the effective Newton's constant which now depends on temperature and chemical potential due to the presence of matter. For a fluid not subject to external electric and magnetic fields, $T^{\mu\nu}_{\rm m}$ is given by eqs.~(\ref{eq:Tmn}), (\ref{eq:cr-statics}).

Equations \eqref{eq:EE-1} are the effective Einstein's equations in equilibrium matter, analogous to the Maxwell's equations in matter for electro- and magneto-statics. The susceptibilities $f_{1,2,3}(T,\mu)$ characterize the leading-order (in derivatives) equilibrium response of the energy-momentum tensor to curvature. We emphasize that the effective Einstein's equations \eqref{eq:EE-1} do not arise from any modification of Einstein's gravity, but are simply a reflection of the gravitational properties of normal matter.

We do not write down the three-derivative and higher-order terms in $T^{\mu\nu}_{\rm m}$ which would come with their own susceptibilities, as in the standard effective field theory. The non-equilibrium terms in the constitutive relations of $T^{\mu\nu}_{\rm m}$ do not follow from the equilibrium generating functional, and have to be added separately, as is done in standard relativistic hydrodynamics.

In practice, one expects the effect of the susceptibilities on the gravitational dynamics to be very small. Indeed,  $p = (-2\Lambda)/2\kappa^2 + p_{\rm m}(T,\mu)$, $f_1=\frac12 M_{\rm Pl}^2 + f_{1,{\rm m}}(T,\mu)$, where $M_{\rm Pl} =  1/\sqrt{8\pi G}$ is the Planck mass, and the subscript ``m'' denotes the matter contribution. For non-cosmological applications one usually sets $\Lambda=0$, while the  temperatures and densities are such that $T,\mu \ll M_{\rm Pl}$. On the other hand, the susceptibilities are more relevant for non-gravitational dynamics of relativistic fluids where $f_{1,2,3}$ show up as thermodynamic transport coefficients, see e.g.~\cite{Kovtun:2018dvd}. When studying matter in Anti-de Sitter space, one has to keep $\Lambda$ non-zero, and the gravitational susceptibilities lead to interesting phenomena, as we discuss below.

Let us now discuss the application of the above theory to non-rotating spherically symmetric equilibrium states of gravitating matter.

\section{Spherical equilibria}

We are interested in equilibrium states of gravitating matter. The simplest way to find such equilibria in classical general relativity is by solving the Einstein's equations
\begin{equation}
  G_{\mu\nu} = \kappa^2 T_{\mu\nu}\,,
\end{equation}
given a certain equation of state that determines the energy-momentum tensor $T^{\mu\nu}$ on the right-hand side. The problem has been explored extensively, starting with the treatment by Tolman many years ago~\cite{Tolman}.
Spherical equilibrium for a perfect fluid with the equation of state given by free fermions at zero temperature was studied by Oppenheimer and Volkoff~\cite{Oppenheimer:1939ne}, and is by now textbook material~\cite{Shapiro:1983du}. Let us briefly review the setup.

The most general rotationally invariant static metric can be written as
\begin{equation}
\label{eq:ss1}
  ds^2 = -A(r) dt^2 + B(r) dr^2 + r^2 d\Omega^2\,,
\end{equation}
where $d\Omega^2 = d\theta^2 + \sin^2\theta \, d\varphi^2$ is the line element on the unit two-sphere. The metric is assumed to be sourced by the perfect fluid energy-momentum tensor. In the above static coordinates, $T_{\mu}^{\ \, \nu} = {\rm diag} (-\epsilon, p, p, p)$, where the energy density $\epsilon$ and the pressure $p$ are functions of $r$.
The metric eq.\ (\ref{eq:ss1}) can be written in terms of the ``potential'' $\Phi(r)$ and the ``mass function'' $m(r)$ as
\begin{equation}
\label{eq:ss2}
  ds^2 = -e^{2\Phi(r)} dt^2 + \left(1-\frac{2G m(r)}{r} \right)^{-1} dr^2 + r^2 d\Omega^2.
\end{equation}
The $tt$ and $rr$ components of the Einstein's equations give, respectively
\begin{align}
\label{eq:eett}
  & \frac{dm}{dr} = 4\pi r^2 \epsilon\,,\\
\label{eq:eerr}
  & \frac{d\Phi}{dr} = \frac{G m}{r^2} \frac{1+ 4\pi r^3 p/m}{1- 2G m/r}\,.
\end{align}
Due to the spherical symmetry of the problem, the $\theta\th$ and $\varphi\vp$ components of the Einstein's equations carry the same information. The $\theta\th$ equation contains $d^2\Phi/dr^2$, which can be eliminated by combining it with the $r$ derivative of eq.\eqref{eq:eerr}. This gives the equation
\begin{align}
\label{eq:eethth}
  \frac{dp}{dr} = -\frac{G m}{r^2} \frac{\epsilon + p}{1- 2G m/r} \left(1+\frac{4\pi r^3 p}{m} \right).
\end{align}

We thus have three equations (\ref{eq:eett}), (\ref{eq:eerr}) and (\ref{eq:eethth}) for four functions $m(r)$, $\Phi(r)$, $\epsilon(r)$, and $p(r)$, which are the Tolman-Oppenheimer-Volkoff (TOV) equations. The conservation of the energy-momentum tensor is automatic thanks to $\nabla_{\!\mu} G^{\mu\nu} = 0$, and does not have to be imposed as an independent condition. The extra equation must therefore come from the equation of state.

If we can express $\epsilon=\epsilon(p)$ from the equation of state, then eqs.\ (\ref{eq:eett}) and (\ref{eq:eethth}) give two coupled equations for the two unknown functions $m(r)$ and $p(r)$. These can be solved for $m(r), p(r)$ with appropriate boundary conditions, and the resulting solutions inserted into eq.\ \eqref{eq:eerr} to determine the potential $\Phi(r)$.

However, the equation of state can not usually be written in the form $\epsilon=\epsilon(p)$. Namely, in the grand canonical ensemble we have $\epsilon=\epsilon(T,\mu)$, $p=p(T,\mu)$, where $T$ is the temperature and $\mu$ is the chemical potential (assuming one species of particles). In other words, $p$ is a function of two independent variables, while $\epsilon$ is just one combination of the two variables. So we have three equations, and four unknown functions $T,\mu,m,\Phi$. To address this issue, note that eqs.\ (\ref{eq:eerr}) and (\ref{eq:eethth}) imply $dp/dr=-(\epsilon+p)\,d\Phi/dr$. Extensivity in the grand canonical ensemble implies $\epsilon+p=T\frac{\partial p}{\partial T} + \mu\frac{\partial p}{\partial\mu}$, so that
$$
  \frac{\partial p}{\partial T} \left( \frac{dT}{dr} + T \frac{d\Phi}{dr} \right) +
  \frac{\partial p}{\partial \mu} \left( \frac{d\mu}{dr} + \mu \frac{d\Phi}{dr} \right) = 0\,.
$$
As $\partial p/\partial T$ and $\partial p/\partial\mu$ are independent functions, this is solved by
\begin{equation}
\label{eq:Tmu-equil}
  T(r) = \bar T e^{-\Phi(r)}\,,\quad \mu(r) = \bar\mu \, e^{-\Phi(r)}\,,
\end{equation}
where $\bar T$ and $\bar\mu$ are constants of integration. This is a particular manifestation of the general statement that in equilibrium both $T$ and $\mu$ must be proportional to $1/\sqrt{g_{00}}$~\cite{RevModPhys.21.531}. Thus both $T$ and $\mu$ are expressed in terms of $\Phi$, and we in fact have only two unknown functions $m(r)$ and $\Phi(r)$, for which we have the two equations \eqref{eq:eett} and \eqref{eq:eerr}. Alternatively, eq.~(\ref{eq:Tmu-equil}) implies that $\tau\equiv T/\mu$ is a constant that does not depend on $r$. Then, for a given equation of state $p(T,\mu) = p(\tau \mu, \mu)$, we have
\begin{subequations}%
\label{eq:TOV}
\begin{align}
  & \frac{dm}{dr} = 4\pi r^2 \left(-p+T \frac{\partial p}{\partial T} + \mu \frac{\partial p}{\partial\mu} \right),\\
  & \frac{d\mu}{dr} = -\mu \frac{G m}{r^2} \frac{1+ 4\pi r^3 p/m}{1- 2G m/r}.
\end{align}
\end{subequations}
At the origin $r = 0$ we set $m(0)=0$, and $\mu(0)=\mu_0$ to some constant which determines the pressure at the center. The equations~(\ref{eq:TOV}) are then integrated numerically starting from the origin. If there is a point $r=R$ where the pressure drops to zero, we have a star. The metric eq.\ (\ref{eq:ss2}) then describes the star for $r<R$, and has to be matched to the Schwarzschild metric at $R$, so that $m(R)=M$ is the total mass of the star. If the pressure never drops to zero, this means the equation of state does not allow for stars, and we have a spherical distribution of matter that fills the whole space.

Another way to look at the TOV equations is to note that for the general spherically symmetric metric eq.\ \eqref{eq:ss1}, the Einstein tensor is schematically of the form $G_{\mu}^{\ \, \nu} = {\rm diag}(X, Y, Z, Z)$, which has to be matched to $T_\mu^{\ \, \nu} = {\rm diag}(-\epsilon, p, p, p)$. The TOV equations arise by enforcing $Y=Z$. On the other hand, for matter subject to a radial gravitational field one in general expects $T_{r}^{\ \, r} \neq T_\theta^{\ \,\theta}$. This is because the $r$-dependent metric makes the radial and the angular directions inequivalent, and there is no symmetry that would enforce $T_{r}^{\ \, r} = T_\theta^{\ \,\theta}$, or $Y=Z$. This difference between $T_{r}^{\ \, r}$ and $T_\theta^{\ \,\theta}$ which is not captured by the perfect fluid model of matter is precisely what is described by the effective Einstein's equations \eqref{eq:EE-1} in a systematic way.

Let us now incorporate the effects of the gravitational susceptibilities. Substituting the metric eq.\ \eqref{eq:ss2} into the effective Einstein's equations \eqref{eq:EE-1}, with the matter energy-momentum tensor given by eqs.\ \eqref{eq:Tmn}, \eqref{eq:cr-statics}, the equations for the potential and the mass function are
\begin{subequations}
\label{eq:spheq-22}
\begin{align}
&r^2 \big[ \epsilon (f_1 - f_1') + p (f_2 - 2f_1')\big] + f_1 (2f_1' - f_2) \frac{2Gm}{r} +\big[ f_1(2f_1' + f_2) - 2(f_1'^{2} + f_1^2)\big] 2G \,\frac{dm}{dr} \nonumber\\
&+\big[f_1(4f_1' + 2f_1''-f_2-f_2') + f_1' (2f_1'' - 3f_2 + f_2') + f_2(f_2 - 2 f_1'')\big]r(r - 2Gm)\left(\frac{d\Phi}{dr}\right)^2 \nonumber\\
&+2(f_1'-f_1)(2f_1'-f_2)(r-2Gm) \, \frac{d\Phi}{dr}  = 0\, , \label{modEE1}\\
&r^3 p + 4f_1 Gm + (r-2Gm)\bigg[ 4(f_1' - f_1) + (2f_1' - f_2) r \, \frac{d\Phi}{dr}\bigg] r \, \frac{d\Phi}{dr} = 0\, . \label{modEE2}
\end{align}
\end{subequations}
Here $f_1 = 1/16\pi G + f_{\rm 1,m}(T,\mu)$, where $f_{\rm 1,m}(T,\mu)$ is the matter contribution. The susceptibility $f_3(T,\mu)$ does not contribute for a non-rotating configuration. The primes are defined by eq.~\eqref{defprime}, as before.
It is easy to see that on taking the zeroth order values for the susceptibilities, $f_1 = 1/16\pi G, f_2  = 0$ in eqs.\ \eqref{modEE1}, \eqref{modEE2} above, one recovers the standard TOV equations \eqref{eq:eett}, \eqref{eq:eerr}. The temperature and the chemical potential are still given by eq.~\eqref{eq:Tmu-equil}, thanks to the general definition \eqref{eq:uTmu}. Thus eqs.~\eqref{eq:spheq-22} are the leading-order modification of the TOV equations by the matter's response to curvature.

\section{Free fermions}

For a given microscopic theory, the susceptibilities $f_{1,2,3}(T,\mu)$ that appear in the effective Einstein's equations may be computed from equilibrium two-point functions of the energy-momentum tensor, as derived for example in~\cite{Kovtun:2018dvd}. In a non-interacting theory, the energy-momentum tensor is quadratic in the fundamental fields, and the susceptibilities are given by one-loop diagrams that can be readily evaluated in the Matsubara formalism~\cite{Romatschke:2009ng, Moore:2010bu, Moore:2012tc, Megias:2014mba, Buzzegoli:2017cqy, Kovtun:2018dvd, Shukla:2019shf}.

The loop integrals will contain ultraviolet divergences, reflecting the need to renormalize the susceptibilities. Regulating the ultraviolet divergences with a large-momentum cutoff, we have $f_{n} = f_{n,{\rm UV}} + f_{n,{\rm m}}(T,\mu)$, where the matter contributions $f_{n,{\rm m}}(T,\mu)$ do not depend on the cutoff and vanish as $T,\mu\to0$. The cutoff-dependent contributions $f_{n,{\rm UV}}$ renormalize the parameters of the effective action. Clearly, as $f_1$ multiplies the Ricci scalar in the effective action, $f_{1,{\rm UV}}$ renormalizes the vacuum Newton's constant, so that $f_1=1/2\kappa^2 + f_{\rm 1,m}(T,\mu)$. The effective Newton's constant that appears in the effective Einstein's equations (\ref{eq:EE-1}) is then determined by
\begin{equation}
\label{eq:km}
  \frac{1}{2\kappa_{\rm eff}^2} = \frac{1}{2\kappa^2} + f_{1,{\rm m}}(T,\mu)\,.
\end{equation}
The ultraviolet divergences in $f_2$, $f_3$, on the other hand, have no physical meaning in a Lorentz-invariant microscopic theory.%
\footnote{
  For Lorentz-violating matter that breaks the Lorentz invariance through a preferred timelike vector, $f_{2,{\rm UV}}$, $f_{3,{\rm UV}}$ would renormalize the corresponding coefficients in the gravitational effective action. Tuning the Lorentz breaking parameter to zero gets rid of the coefficients.
}
As the presence of the terms $a^2$, $\Omega^2$ in the effective action is thermal state-specific, so must be the susceptibilities $f_2$, $f_3$. Thus in the gravitational effective action we have $f_2 = f_{\rm 2,m}$, $f_3 = f_{\rm 3,m}$.

For concreteness, let us consider free Dirac fermions of mass $m_{\rm f}$ in 3+1 dimensions.
At $T=0$, the susceptibilities $f_n$ are only non-zero for $|\mu|>m_{\rm f}$, just like the pressure $p(\mu)$. Evaluating the one-loop diagrams, one finds~(see e.g.\ \cite{Shukla:2019shf})
\begin{align}
 & f_{\rm 1,m} = -\frac{m_{\rm f}^2}{48\pi^2} \left[\frac{|\mu|}{m_{\rm f}^2} \sqrt{\mu^2 {-} m_{\rm f}^2} - \ln \!\left(\frac{|\mu| + \sqrt{\mu^2 {-} m_{\rm f}^2}}{m_{\rm f}}\right) \right]\!, \\
 & f_{\rm 2,m} = \frac{|\mu|}{24\pi^2}\frac{(2m_{\rm f}^2 - 3\mu^2)}{\sqrt{\mu^2 - m_{\rm f}^2}} \,, \\
 & f_{\rm 3,m} = -\frac{|\mu|}{96\pi^2}\sqrt{\mu^2 - m_{\rm f}^2} \,.
\end{align}
At $\mu=0$, we have $f_{n,{\rm m}}(T) = T^2 h_n(m_{\rm f}/T)$ with dimensionless functions $h_n(x)$. Evaluating the one-loop diagrams, for $T\ll m_{\rm f}$ one finds asymptotic expansions
\begin{align}
\label{eq:h1smallT}
 & h_1(x) \sim -\frac{e^{-x}}{4\sqrt{2}} \left( \frac{x^{1/2}}{3} + \frac{x^{-1/2}}{8} - \frac{5x^{-3/2}}{128} + \dots \right)\!,\\
\label{eq:h2smallT}
 & h_2(x) \sim -\frac{e^{-x}}{4\sqrt{2}} \left( \frac{x^{5/2}}{3} + \frac{9 x^{3/2}}{8} + \frac{235x^{1/2}}{128} + \dots \right)\!, \\
\label{eq:h3smallT}
 & h_3(x) \sim -\frac{e^{-x}}{16\sqrt{2}} \left( \frac{x^{3/2}}{3} + \frac{5 x^{1/2}}{8}  + \frac{35 x^{-1/2}}{128}\dots \right).
\end{align}
At high temperatures, $h_1(0) = -1/144$, $h_2(0) = -1/24$, $h_3(0)=-1/288$ \cite{Kovtun:2018dvd}. The functions $h_n(x)$ are shown in Fig.~\ref{fig:f123T}.

\begin{figure*}[t]
\includegraphics[scale=0.425]{f1T.pdf}
\includegraphics[scale=0.425]{f2T.pdf}
\includegraphics[scale=0.425]{f3T.pdf}
\caption{
\label{fig:f123T}
The thermodynamic susceptibilities $f_{n,{\rm m}}(T)/T^2$ for a gas of free Dirac fermions of mass $m_{\rm f}$ at $\mu=0$. The blues lines are the exact values from the one-loop diagrams, the dashed lines are the low-temperature approximations of eqs.~(\ref{eq:h1smallT}), (\ref{eq:h2smallT}), (\ref{eq:h3smallT}).
}
\end{figure*}

At the leading order in the non-relativistic semiclassical limit $\frac{m_{\rm f}}{T}\gg \frac{m_{\rm f}-\mu}{T}\gg1$, the susceptibilities are proportional to the particle number density $n = (m_{\rm f} T/2\pi)^{3/2}e^{(\mu-m_{\rm f})/T}$. Restoring the factors of $\hbar$, we have
\begin{align}
   f_{\rm 1,m} \sim -\frac{\hbar^2}{12} \frac{n}{m_{\rm f}}\,,\ \ \ \ \
   f_{\rm 2,m} \sim -\frac{\hbar^2}{12} \frac{n\, m_{\rm f}}{T^2}\,,\ \ \ \ \
   f_{\rm 3,m} \sim -\frac{\hbar^2}{48} \frac{n}{T}\,.
\end{align}
At non-zero $T$ and $\mu$, the susceptibilities $f_{n,{\rm m}}$ are plotted in Fig.~\ref{fig:f123Tmu}.

\begin{figure*}[t]
\includegraphics[height=3.85cm]{f1Tmu-t.pdf}
\includegraphics[height=3.85cm]{f2Tmu-t.pdf}
\includegraphics[height=3.85cm]{f3Tmu-t.pdf}
\caption{
\label{fig:f123Tmu}
Thermodynamic susceptibilities $f_{n,{\rm m}}(T,\mu)/T^2$ for a gas of free Dirac fermions of mass $m_{\rm f}$, at various values of $\mu/m_{\rm f}$. The curves of different color correspond to the values $\mu/m_{\rm f} = 0, 0.4, 0.8, 0.9, 1.1, 1.2, 1.6, 2.0$, from top to bottom in each plot. As $T\to0$, all curves go to zero for $\mu<m_{\rm f}$, and diverge for $\mu>m_{\rm f}$.
}
\end{figure*}

\section{Matter in anti-de Sitter space}
Consider uncharged conformal matter at temperature $T$ in four dimensions. The pressure is $p_{\rm m}(T) = c_0 T^4$, while the gravitational susceptibilities are $f_{\rm 1,m}(T) = \frac16 f_{\rm 2,m}(T) = \varphi_0 T^2$, where the constants $c_0$ and $\varphi_0$ are dimensionless in the natural units $\hbar=c=1$. For the conformally coupled real massless scalar field $c_0 = \frac{\pi^2}{90}$, and $\varphi_0 = 0$.%
\footnote{
For a massless scalar field in 3+1 dimensions, $\varphi_0$ is proportional to $(1-6\xi)$, where $\xi$ is the coupling to curvature, $\xi=1/6$ for the conformal coupling~\cite{Kovtun:2018dvd}.
}
For free massless Dirac fermions $c_0 = \frac78 \frac{4\pi^2}{90}$, and $\varphi_0 = -\frac{1}{144}$. For the black-body radiation (gas of non-interacting photons) $c_0 = \frac{2\pi^2}{90}$, and $\varphi_0 = -\frac{1}{36}$. See Ref.~\cite{Kovtun:2018dvd} for a summary of the calculations of $f_{\rm 1,m}$ and $f_{\rm 2,m}$ in free theories. For ${\cal N}=4$ supersymmetric $SU(N_c)$ Yang-Mills theory at infinite coupling and large $N_c$, a holographic calculation~\cite{Baier:2007ix} gives $c_0 = \frac{\pi^2}{8} N_c^2$, $\varphi_0 = -\frac{1}{16}N_c^2$. While $c_0$ must be positive for positive pressure, we are not aware of a general argument that would fix the sign of $\varphi_0$.

Let us now study the effects of conformal matter on the anti-de Sitter space AdS$_4$, with the cosmological constant $\Lambda = -3/\ell^2<0$. Einstein's equations in matter are applicable when
\begin{align}
\label{eq:lll}
  l_{\rm Pl} \ll \lambda_{\rm th} \lesssim \ell\,,
\end{align}
where $l_{\rm Pl} = 1/M_{\rm Pl}$ is the Planck length, and $\lambda_{\rm th} = 1/T$ is the thermal wavelength. The first inequality is the applicability of classical gravity, and the second is the applicability of the derivative expansion for the matter's generating functional. Using $\alpha = \frac{c_0 T^4}{M_{\rm Pl}^2/\ell^2}$, and $\gamma = \varphi_0 T^2/M_{\rm Pl}^2$, Einstein's equations in matter are applicable when $|\gamma| \ll 1$ and $|\gamma| \lesssim \alpha$. We will see below that violating the inequalities (\ref{eq:lll}) will lead to quantitative criteria of the breakdown of hydrostatics, which are invisible for perfect-fluid matter.

The parameter $\alpha$ can be thought of as the ratio of the energy density in matter to the energy density in the cosmological constant. Thus the configuration is ``matter-dominated'' for $\alpha\gg1$ and ``cosmological constant-dominated'' for $\alpha\ll1$. A matter-dominated configuration implies $(T\ell)^2\gg M_{\rm Pl}^2/T^2 \gg 1$, and thus the effects of the gravitational susceptibilities are small. In fact, for $T\ell \gg 1$ the configuration would be on the high-temperature (black hole) side of the Hawking-Page phase transition~\cite{Hawking:1982dh}.  On the other hand, a configuration that is cosmological constant-dominated with $\alpha = c_0 (T\ell)^2 (T/M_{\rm Pl})^2 \ll 1$ can have $T\ell \sim 1$, and thus the effect of the gravitational susceptibilities can be noticeable. Note that the ``thermal AdS'' phase of the Hawking-Page transition exists for $T\ell\lesssim \frac{1}{\pi}$, while the perfect-fluid approximation in AdS is valid for $T\ell\gg 1$. Hence the low-temperature phase of the Hawking-Page transition can not be described by a perfect fluid in AdS. Perfect radiation fluids in AdS$_4$ were studied in Ref.~\cite{Page:1985em}. Einstein's equations in matter (\ref{eq:EE-1}) allow for a systematic calculation of the $1/(\ell T)^2$ corrections to the perfect-fluid approximation.

The static metric takes the form (\ref{eq:ss1}), and the equilibrium temperature is $T(r)=T_0/\sqrt{A(r)}$. Using $\rho\equiv r/\ell$, effective Einstein's equations (\ref{eq:EE-1}) give two equations for $A(\rho)$ and $B(\rho)$:
\begin{align}
\label{eq:ads-1}
  & \rho A' + A - AB(1{+}3\rho^2) - \alpha_0  \frac{\rho^2 B}{A} + \gamma_0 \left( \frac{(\rho A' - 2A)^2}{2A^2} - 2B \right) = 0\,, \\
\label{eq:ads-2}
  & \rho B' - B + B^2 (1{+}3\rho^2) - \alpha_0 \frac{3\rho^2 B^2}{A^2} \nonumber\\
  & + \frac{\gamma_0}{A^2} \left( \frac{5\rho^2 A'^2 B}{2A} -2A(\rho B' {+} B^2 {-} B) -4\rho BA' + \rho^2 A' B' -2\rho^2 A'' B \right) = 0 \,,
\end{align}
where $\alpha_0 \equiv c_0 (T_0\ell)^2 (T_0/M_{\rm Pl})^2$, and $\gamma_0 \equiv \varphi_0 T_0^2/M_{\rm Pl}^2$. Note that Eq.~(\ref{eq:ads-1}) is an algebraic equation for $B(r)$, which can be used to derive a second-order differential equation for $A(r)$. Without matter we have $\alpha_0 =\gamma_0 =0$, and Eqs.~(\ref{eq:ads-1}), (\ref{eq:ads-2}) are immediately solved to give
\begin{align}
  B(r) = \frac{1}{1 - \frac{a'}{r} + \frac{r^2}{\ell^2}}\,,\ \ \ \
  A(r) = a \left( 1 - \frac{a'}{r} + \frac{r^2}{\ell^2} \right) \,,
\end{align}
where the integration constant $a'$ determines the mass of the AdS-Schwarzschild black hole. We will set $a'=0$ and consider matter at temperature $T$ in AdS, without the black hole. The integration constant $a$ can be absorbed into a rescaling of time.

When $\gamma_0 =0$ and $\alpha_0 \neq0$, Eqs.~(\ref{eq:ads-1}), (\ref{eq:ads-2}) describe the backreaction of conformal matter, approximated as a perfect fluid, on AdS. The solution for $A(r)$ which is regular as $r\to0$ has the following expansions
\begin{align}
  & A(r\to0) = a + \left( a + \frac{\alpha_0}{a}\right) \frac{r^2}{\ell^2} + \frac{\alpha_0 (a^2 + \alpha_0)}{5a^3} \frac{r^4}{\ell^4} + \dots\,,\\
\label{eq:Ainf-1}
  & A(r\to\infty) = b_0 \frac{r^2}{\ell^2} + b_0 - b_1\frac{\ell}{r} + \dots\,,
\end{align}
with integration constants $a$, $b_0$, $b_1$. The constant $a$ determines the temperature at the center, $T(r=0) = T_0/\sqrt{a}$. The constant $b_0$ can be set to one by a normalization of time. The remaining constant $b_1$ determines the mass of the configuration. The full solution for $A(r)$ can be constructed numerically to determine the large-$r$ asymptotics of the metric and thus the mass of the configuration for different $\alpha_0$~\cite{Page:1985em}. For a solution that is regular at the center, the mass of the matter configuration (in units of $\ell/(2G) \sim M_{\rm Pl}^2 \ell$) is $m(a) = b_1/b_0$.

With both $\gamma_0$ and $\alpha_0$ nonzero, Eqs.~(\ref{eq:ads-1}), (\ref{eq:ads-2}) describe the backreaction of conformal matter, taking into account the leading $1/(T\ell)^2$ correction to the perfect fluid model of equilibrium matter. The solution for $A(r)$ which is regular as $r\to0$ has the following expansions
\begin{align}
\label{eq:A0-2}
   & A(r\to 0) = a + \frac{a^2 {+} \alpha_0 {-} 4\gamma_0 a}{a {-} 6\gamma_0} \frac{r^2}{\ell^2} +  \frac{(\alpha_0 {+} 2 a \gamma_0) (a^2 {+} \alpha_0 {-} 4a \gamma_0) (a{-}18\gamma_0) }{5a(a {-} 6\gamma_0)^3} \frac{r^4}{\ell^4} + \dots\,,\\
\label{eq:Ainf-2}
  & A(r\to\infty) = b_0 \frac{r^2}{\ell^2} + b_0 - b_1\frac{\ell}{r} + \dots\,.
\end{align}
The leading-order large-$r$ asymptotics are not modified compared to the perfect-fluid case, and the difference between Eqs.~(\ref{eq:Ainf-1}) and (\ref{eq:Ainf-2}) only comes at order $O(\ell^2/r^2)$ and higher.  The equation for $A(r)$ can then be integrated numerically starting with the initial condition (\ref{eq:A0-2}) in order to obtain the mass $m(a) = b_1/b_0$.

The constants $a$ and $\alpha_0$ must be positive, while the sign of $\gamma_0$ is not a priori determined. The third term in the expansion (\ref{eq:A0-2}) vanishes for $\gamma_0 = -\alpha_0/(2a)$. In fact, for this value of $\gamma_0$, all the terms in the expansion (\ref{eq:A0-2}) except for the first two vanish identically, and one can easily check that the AdS space $A(\rho) = a(1 +\rho^2)$, $B(\rho) = (1+ \rho^2)^{-1}$ is an exact solution to Eqs.~(\ref{eq:ads-1}), (\ref{eq:ads-2}).%
\footnote{
Similarly, when $a^2 + \alpha_0 = 4\gamma_0 a$, one finds an exact solution $A(\rho) = a$, $B(\rho) = (1+2\rho^2)^{-1}$, which is not asymptotically AdS. This can happen for $\gamma_0^2>\alpha_0/4$, and at sufficiently high central temperature, $a<4\gamma_0$.
}

\begin{figure}
\includegraphics[width=0.48\textwidth]{Avsz.pdf}
\hspace{0.02\textwidth}
\includegraphics[width=0.48\textwidth]{mvsa.pdf}
\caption{
\label{fig:ads}
Left: The metric component $A(r)=-g_{00}(r)$ as a function of $z \equiv 1/(1+r/\ell)$ for $a=1$ and $\alpha_0=1$, obtained by solving Eqs.~(\ref{eq:ads-1}), (\ref{eq:ads-2}) numerically. From top to bottom, the curves correspond to $\gamma_0 = 0.1, 0, -0.1, -0.5, -1$. The mass is determined by the behaviour of the curve at small $z$. As the value of $\gamma_0$ falls below $-\alpha_0/(2a)$, the curve changes from concave to convex, and the mass turns negative. Right: the mass of the matter configuration at $\alpha_0{=}1$ as a function of the constant $a$ which determines the central temperature, $T(r{=}0)=T_0/\sqrt{a}$. From top to bottom at large $a$, the curves correspond to $\gamma_0 = 0.1, 0, -0.1, -0.5$. For the perfect fluid ($\gamma_0 = 0$) the mass is bounded from above by a number of order one~\cite{Page:1985em}. For $0<a<6\gamma_0$, the space is no longer asymptotically AdS, and the curve is cut off.
}
\end{figure}

The total mass is positive for $\gamma_0 > -\alpha_0/(2a)$, and negative for $\gamma_0 < -\alpha_0/(2a)$, as illustrated in Fig.~\ref{fig:ads}. Correspondingly, the energy density of matter $T^{00}_{\rm m}$ in the right-hand side of Eq.~(\ref{eq:EE-1}) turns from being positive everywhere in space to being negative everywhere in space, as $\gamma_0$ crosses the critical value $-\alpha_0/(2a)$. Thus we see that for matter with negative $\varphi_0$ (such as free photons), the mass of the static configuration of matter becomes negative for
\begin{equation}
\label{eq:Tl}
  T(r{=}0) < \frac{1}{\ell} \left( \frac{2|\varphi_0|}{c_0}\right)^{1/2} \,.
\end{equation}
Of course, one does expect on physical grounds that the macroscopic fluid description of matter should become physically inadequate when the temperature is sufficiently small, $T\ell \lesssim 1$. However, one has to go beyond the perfect-fluid approximation of matter in order to see a quantitative criterion for the breakdown of hydrostatics, Eq.~(\ref{eq:Tl}). This breakdown corresponds to the violation of the second inequality in Eq.~(\ref{eq:lll}).

Another point to note is that the small-$r$ expansion (\ref{eq:A0-2}) is really an expansion in $(r/\ell)/(a{-}6\gamma_0)$, and therefore its radius of convergence shrinks to zero as $\gamma_0$ approaches $a/6$ from below. For $\gamma_0 > a/6$, the equation for $A(\rho)$ can still be integrated numerically by using the expansion (\ref{eq:A0-2}) and starting at a sufficiently small value of $\rho$. The result appears to be a solution which has $b_0 = 0$, and therefore the space that is not asymptotically AdS. This happens for
\begin{equation}
\label{eq:TMpl}
  T(r{=}0) > \frac{M_{\rm Pl}}{(6\varphi_0)^{1/2}}\,,
\end{equation}
which we interpret as the breakdown of the classical description at high temperatures. The criterion (\ref{eq:TMpl}) arises for matter with $\varphi_0>0$, and corresponds to the violation of the first inequality in Eq.~(\ref{eq:lll}).

\section{Concluding Comments}
Starting with the equilibrium generating functional, we have set up a framework to describe classical gravitational fields with matter present. Equations (\ref{eq:EE-1}) with the energy-momentum tensor (\ref{eq:Tmn}), (\ref{eq:cr-statics}) are the gravitational analogues of the electro- and magneto-statics in matter. The temperature- and density-dependent correction to the Newton's constant is given by eq.~(\ref{eq:km}), with $f_{\rm 1,m}$ for Dirac fermions plotted in Fig.~\ref{fig:f123Tmu}, leftmost panel.

For normal quantum matter in flat space the effects of the gravitational susceptibilities will be small. Consider the example of the TOV equations. For the perfect fluid approximation, the Einstein's equations are schematically of the form $1/L^2 \simeq m^4/M_{\rm Pl}^2$ where $L$ is a length scale, with $L^2$ coming from the two derivatives in the Einstein tensor, $1/M_{\rm Pl}^2 = 8\pi G$, and $m$ is the energy scale of the matter (this gives $L\simeq M_{\rm Pl}/m^2$, an estimate of a neutron star size for $m\simeq 1 {\rm GeV}$). With the gravitational susceptibilities taken into account, schematically
$$
  \frac{1}{L^2} \simeq \frac{1}{M_{\rm Pl}^2} \left( m^4 + \frac{m^2}{L^2}\right)\,,
$$
and the corresponding effects are $(m/M_{\rm Pl})^2$ suppressed.
This is expected, as the gravitational susceptibilities $f_n$ are quantum-mechanical in nature -- for classical particles in external gravitational field the equilibrium partition function does not depend on the Riemann curvature.

The implications are more interesting in Anti-de Sitter space, where we have studied hydrostatic configurations of uncharged conformal matter, such as a gas of free photons. The perfect fluid model of matter is only physically applicable for $T\ell \gg1$, where $\ell$ is the AdS radius. However the Einstein equations coupled to perfect-fluid matter exhibit no pathologies even for $T\ell \lesssim 1$, and one might be tempted to treat matter in AdS as a perfect fluid even at low temperatures. The gravitational susceptibilities studied in this paper allow for a systematic study of the $1/(T\ell)^2$ corrections to the perfect-fluid approximation. For free photons and fermions (as well as for the strongly coupled supersymmetric Yang-Mills theory), the effect of the gravitational susceptibilities is to reduce the total mass of the hydrostatic configuration, and eventually to drive the mass negative at small $T\ell$, see Eq.~(\ref{eq:Tl}). We interpret the negative mass as the breakdown of macroscopic hydrostatics. For example, for black-body radiation in AdS$_4$, macroscopic hydrostatics explicitly breaks down for
\begin{equation}
  T(r{=}0) \ell < \frac{\sqrt{5/2}}{\pi}  \approx 0.5\,.
\end{equation}
For the strongly coupled supersymmetric Yang-Mills theory in AdS$_4$, hydrostatics breaks down for
\begin{equation}
  T(r{=}0) \ell < \frac{1}{\pi}\,.
\end{equation}
Curiously, the last expression formally coincides with the onset of the Hawking-Page transition. It is satisfying to see how basic thermal physics of gravitational susceptibilities ``restores justice'', and explicitly disallows for the macroscopic hydrostatics at small $T\ell$.

{\em Note added ---} After this paper was completed, we became aware of the preprint \cite{Baier:2019ciw} that uses a simplified version of the Einstein's equations in matter in a cosmological setting.

\acknowledgments
The work was supported in part by the NSERC of Canada.

redpajama_set_name: RedPajamaArXiv


text:
Trout Species of North America
Through watercolor paintings, Prosek has dedicated his unique talent to bringing life trout species of North America, several of which are becoming critically endangered fish.
By James Prosek
| May 2013
James Prosek profiles over 100 varieties of trout through watercolor paintings in "Trout of the World." Learn about about the diverse variety of trout species of North America.
Cover Courtesy Abram Books
Adirondack Brook Trout, St. Regis river, West Branch, New York state; Salvelinus fontinalis
Illustration Courtesy James Prosek
Blueback Trout, Rainbow Lake, Maine; Salvelinus alpinus oguassa
Gila Trout, New Mexico; Oncorhynchus gilae gilae
Colorado River Cutthroat Trout, Uinta Mountains, Utah; Oncorhynchus clarki pleuriticus
Bull Trout, Hood River, Oregon; Salvelinus confluentus
Rio Grande Cutthroat Trout, New Mexico; Oncorhynchus clarki virginalis
Humboldt Cutthroat Trout, Mary's River Basin, Nevada; Oncorhynchus clarki bouvieri
Mexican Golden Trout, Río Verde, Chihuahua, Mexico; Oncorhynchus chrysogaster
Golden Trout, Volcano Creek, California; Oncorhynchus mykiss aguabonita
Yellowstone Cutthroat Trout, spawning male, Yellowstone Lake Tributary, Wyoming; Oncorhynchus clarki bouvieri
Prosek has a passion for the trout species and conveys their beauty with the stroke of his brush in Trout of the World (Abrams Books, 2013), a collection of 100 gorgeous watercolor paintings. The text provides an historical profile of each fish along with personal reflections by the author. Prosek savors the beauty of various fishing spots, along with the fate of the world of trout, and also contemplates man's role in extinction of animals. A must for the library of the recreational fisherman as well as those interested in ichthyology. The following is an excerpt on trout species of North America from the chapter "Selected Diversity of North American Trouts".
In September 2000, driving east from California to Colorado, I decided to stop in the desert to try to catch a native trout called the Humboldt cutthroat. I took Route 50 around Lake Tahoe and then into the desert wasteland, where the air hovered at around 100 degrees. I spent the night in Elko, Nevada, lights flashing from casinos, the vibration of trucks on Route 80 shaking the water in the glass on my bedside table.
The next day I drove along the roads outside Elko toward the upper tributaries of the Mary's River. Rabbitbrush was in flower along the dirt trails, mountains formed up ahead, and a plume of dust behind me obscured what had come before. Antelope congregated around a water hole. A golden eagle flew across the road. I hit a whip-poor-will with my car and stopped to look for it and lament its death.
I was following a publication put out by the state of Nevada in the 1970s, a survey of remaining populations of the native cutthroat trout in these upper tributaries. It looked as if the stream with the greatest abundance of fish was one called Wildcat Creek. I wondered if, some thirty years later, the fish were still there. I followed a topographical map to find it. After turning onto a smaller dirt road on the other side of a cattle guard, I slowed to speak with a man standing by his truck — the only person I'd seen for miles. His name was Bill Gibbs, and he owned the ranch through which Wildcat Creek ran. He was watching yearling cattle that would soon be brought along Route 80 to feedlots in Nebraska. I asked him if he knew about the trout in Wildcat Creek. Would he mind if I fished it? He said that the summer before, a wildfire had burned on the property and gone right over the creek, and he wasn't sure if the trout had survived. He pointed to where the creek was. It would be about a mile walk from the road, and I was welcome to fish it.
When I saw the creek after a walk through the hot desert country, it looked like a creek in Hades. I half expected to see a fallen angel crouched beside it. The willows along the bank had been charred black, but there was a small intermittent blue ribbon of water coursing through. It resembled a sliver of sky. And when I peered into the water through the tangle of burned brush, I saw fish, remarkable luminous trout, and not so small, considering that the creek was no more than three feet across in most places. I used only the top segment of my three-piece fly-fishing rod to dap a fly into small pools that I approached quietly and carefully.
In short order I caught one. Then another, and another. I carefully laid them on my hat to take pictures and then let them go. I made sketches with graphite and colored pencil that might help me remember the pink, purple, and ochre hues that I would try to recreate in the studio. The trout I released were so eager to bite that I probably could have caught each one a second time. I was covered in dust from crawling along the bank, and charcoal lines were drawn on my clothes from the burned limbs of sagebrush and willow and aspen that I'd pushed through as I walked. That evolution could produce such a fish, able to survive what nature had thrown at it for millennia, was remarkable.
We sometimes, correctly, think of nature as fragile. I have described the trout this way, too, and it is indeed vulnerable to rapid alterations of its habitat by humans, from pollution, climate change, or drawing water for irrigation. But all I could think of at that moment was the resilience of nature, the relentlessness of these fish in holding on in an ecosystem that most would consider hostile. I walked back, parched and humbled, to my car, conflicted about having disturbed this place but elated over what I had witnessed.
This, of course, was a personal experience that went beyond what a single name or even theory can impart. People called the fish in these upper tributaries the Humboldt cutthroat trout, related to a fish called the Lahontan cutthroat trout. Individuals argued over whether the Humboldt cutthroat should be given a subspecies name or not. Is it a subspecies? Is it distinct enough a population to be given species status? The question What is that? did not seem the most relevant one to ask. In the desert the only clear answer to me at that moment was:
It is. You can argue over what category an individual fish fits into and never come up with a definitive answer—this kind of argument might never be resolved. But one truth you cannot deny is that it exists.
I believe that in the future we will no longer need to answer the question What category does that fish fit into—what family, order, genera, or even species? Rather, the relevant questions will be: What is it related to? What is its position in the history of life? How long ago did X separate from Y? I am not a taxonomist, one who spends his life trying to make sense of evolutionary history and then classify accordingly, though I admire that pursuit. My job is easy in comparison. I have set out to create a visual record of biological diversity. Instead of asking What is that? I am simply saying Here it is.
Since the publication of the first edition of this book, great changes have taken place in the communication and collection, retention, and dissemination of information about trout and other creatures on the planet. Not long ago, even as recently as the year 2000, my friends and I planned expeditions by handwriting letters to knowledgeable people in far-off lands and occasionally trying to coordinate a phone call. At that time, if you wanted to learn about a fish like the Humboldt cutthroat trout, there was very little written information and few images available.
You had to go see it yourself.
The Internet has changed this. Personal blogs and Web sites kept by the trout-obsessed, mostly amateur anglers, contain hundreds of photos of native trout, often geo-referenced to the place where they were caught. Such aggregations of information are valuable growing catalogs of biological diversity that—in my opinion, as they accumulate with new knowledge of genetic relationships, behavior, and habitat requirements—will help the importance of certain taxonomic categories fade away. The arguments over what to call fish, I feel, will lose out to an interconnected, complex, reticulate story of life.
More than sixty of the fish in this book represent diversity within a single species—the brown trout, known by the Linnaean binomial Salmo trutta. By painting and presenting them here, I hope I have conveyed that the word species—in the way it is used in public, conservationist, and even scientific circles—is insufficient to communicate biological reality.
Darwin expressed his frustration with the word species and questioned where to draw lines amid infinite beauty and diversity.
"It is really laughable," he wrote, "to see what different ideas are prominent in various naturalists' minds, when they speak of 'species' . . . It all comes, I believe, from trying to define the undefinable."
In Origin of Species, he wrote, "I look at the term 'species' as one arbitrarily given, for the sake of convenience, to a set of individuals closely resembling each other."
Such is the source of my preoccupation—you could even say obsession—with our legacy of naming nature, which started with Adam's task to name all the creatures in the garden (as the story goes) but is really as old as language itself. Naming is a necessary act in communicating about the natural world, and an essential part of being human, but it also has great limitations in expressing the real nature of reality—which Darwin exposed to us with his account of the ever-evolving origins of species. Ours is a fluid, continuous, chaotic, and ever-changing world.
One hundred and twenty watercolors hardly do justice to the diversity of trout of the world. This book is, of course, a fiction, as you could never build an aquarium large enough or nuanced enough to house all these fish, but here, within these few pages, we can gaze at a fraction of their beauty and see just what we are talking about when we warn that we may lose them. Unless there is an international concerted effort to save trout, we will be the last generation to see many of these fish alive in their native habitats. Their disappearance would not only be a physical loss but a loss to the human imagination.
Reprinted with permission by Trout of the World by James Prosek, published by Abrams Books, 2013.

redpajama_set_name: RedPajamaCommonCrawl


text:
Close Quarters, Season 1, Episode 1 – 32m
Nate Boyer interviews Johnny Walker, a former Iraqi interpreter who was attached to the Nacy SEALs and regarded as an honorary SEAL and brother to the men he fought alongside with. He recounts his incredible journey through his book, Codename: Johnny Walker, which is now in works to become a feature film.
Randy Couture
Nate Boyer goes face-to-face with MMA champion and actor, Randy Couture as they discuss topics from his early days in wrestling to his enlistment in the Army, to his UFC championships.
Florent Groberg
Nate Boyer interviews Medal of Honor recipient Florent Groberg, who risked death by tackling a suicide bomber to the ground in order to protect the members of his team.
Hank Hughes
Nate Boyer sits down with Academy Award-nominated filmmaker, Hank Hughes, who is an Army veteran that was nominated for his AFI student short film "Day One".


Do you ever meet someone and just feel a sense of peace and calm when you leave? Well take that and then multiply it by two. That is Jeff & Jessica for me. From our first meeting to the engagement session to sharing beers before the month of the wedding, I always left just feeling genuinely happy and in a better mood. Conversations with them are easy, they have giant, caring personalities for others, and together just an overall "it'll all work out" kind of attitude. When you get to hang out with such amazing people, it completely takes the "work" part out of photographing their wedding. It just feels right.
Jeff & Jessica had a beautiful September ceremony at the Loring Pasta Bar in Minneapolis. I was obsessed with their mini rooftop deck & greenery and the lighting in the main room (the shadows & contrast were AMAZING). Guests then headed over to the Varsity Theater for cocktails, dinner, and dancing. This venue is so cool and unique and all of Jeff & Jess's cinema themed touches seriously completed the space. They didn't have a wedding party, which gave me opportunity to put more focus on them as a couple and I'm seriously swooning over their images.
Jeff & Jess, Thank you again for inviting me into your lives & giving me the opportunity to be apart of your special day!


Adam Hochschild, a popular historian whose King Leopold's Ghost: A Story of Greed, Terror, and Heroism in Colonial Africa spelled out the full story of the Belgian debacle that inspired Joseph Conrad's Heart of Darkness, has written a powerful new book about the loose coalition of pacifists and activists that fought bitterly against England's participation in the Boer Wars and World War One a century ago. The book is called To End All Wars: A Story of Loyalty and Rebellion, 1914-1918 .
Where today we might see mindless killing, many of those who presided over the war's battles saw only nobility and heroism. "They advanced in line after line," recorded one British general of his men in action on that fateful July 1, 1916, at the Somme, writing in the stilted third-person usage of official reports, " ... and not a man shirked going through the extremely heavy barrage, or facing the machine-gun and rifle fire that finally wiped them out ... He saw the lines which advanced in such admirable order melting away under the fire. Yet not a man wavered, broke the ranks or attempted to come back. He has never seen, indeed could never have imagined, such a magnificent display of gallantry, discipline and determination. The reports that he had had from the very few survivors of this marvellous advance bear out what he saw with his own eyes, viz, that hardly a man of ours got to the German front line."
What was in the minds of such generals? How could they feel such a slaughter to be admirable or magnificent, worth more than the lives of their own sons? We can ask the same question of those who are quick to advocate military confrontation today, when, as in 1914, wars so often have unintended consequences.
The book begins in South Africa with the Boer Wars, which set up the expansionist tensions between England and Germany that erupted in the First World War. The pacifist movement that made England's path to war more difficult included many leading suffragettes (Charlotte Despard, Emily Hobhouse, the Pankhurst family) and socialists (Keir Hardie, Bertrand Russell). Hochschild's method in To End All Wars is to recover the human stories behind this movement, and especially to highlight the many interpersonal coincidences and contradictions -- family relationships, love affairs -- that complicated this public debate. Hochschild's portraits are vivid and his stories are gripping, but my favorite thing about his book is simply that it exists.
A year ago, kicking off this Philosophy Weekend series, I asked whether pacifism was in a coma, or whether it just seemed that way. I never doubt that the ethical question of war is the premium philosophical question of our time, and I'm sure that the human race will eventually manage to rid itself of the plague of militarism (and the all-powerful military profit motive) that pervades our world today. What we need from our writers, academics and historians today is a serious study of war. Not a record of battles, not a celebration of glory and sacrifice, but an examination of the nature of war itself, and of the intellectual and cultural threads in society that suppress common sense and enable this disease.
Any committed pacifist must feel lonely and isolated, but To End All Wars is the latest example of a major publication by a notable, award-winning author that tackles the question of pacifism directly, and with serious intent. Other recent examples include Nonviolence: The History of a Dangerous Idea by Mark Kurlansky and the groundbreaking, still controversial Human Smoke: The Beginnings of World War II, the End of Civilization by Nicholson Baker.
Is something good brewing here? Perhaps a certain awareness is spreading? I hope so. If you'd like to learn more about To End All Wars, here's a probing 40-minute Bat Segundo interview with the author, and here's an appreciative article by Christopher Hitchens that recently appeared on the cover of the New York Times Book Review.
This article is part of the series Philosophy Weekend. The next post in the series is Philosophy Weekend: Ayn Rand in the News (Paul Ryan, Adam Curtis). The previous post in the series is Philosophy Weekend: In Transit.
12 Responses to "Philosophy Weekend: Adam Hochschild and the Serious Study of War"
Oh I am so excited to hear about this. I just picked up King Leopold's Ghosts to read on the Kindle FINALLY after it being on my wish list for what seems like ever. I really want to pick this one up now as well. Sounds like a great book. Thanks!
I think it was Winston Churchill who said, "History is written by the victors." I'm amazed by the ever-growing revelations about people who opposed WWI and WW2. We were brought up in this country to believe so strongly in "the good war" but there are no good wars.
On that same note, it was years after the 1979 hostage crisis before I finally learned that in 1953, the secret services of the United States and England orchestrated the overthrow of a democratically elected Prime Minister in Iran, because he didn't want to give us cheap oil and because we were afraid he would take sides with Russia. As it turns out, gas prices went up anyway and we are more-or-less friends with Russia.
I don't understand why you keep saying pacifists must feel lonely.
Well, TKG, they tend to get laughed at a lot.
War and poverty are like religion to us. Sacred to the species, after all these thousands of years. It would be immoral weakness and heresy for us to give them up, especially war. Most dishonorable. Pacifism is foolish blasphemy.
"Well, TKG, they tend to get laughed at a lot."
It depends on how pacifism is defined.
One of the interesting things about this ongoing long term discussion has been defining just what pacifism is. From this discussion I have learned that pacifism is not never ever using violence under any circumstances. I think that the extremes who say never ever is violence justified do get ridiculed "a lot" by certain groups. But, at the same time, even the extremes have a huge contingent and say. In fact I'd ostensibly lump the pre-elected Obama as being in or close to this group and say that he had the support of this group.
I'd go as far as to say that the extreme or near extreme view is so prevalent that they were able to turn the 2008 election in their support and enthusiasm for Obama.
So there's no need to feel lonely.
But on the broader level, I learned that pacifism is not that violence is never justified, but that it is that violence is not justified to settle disputes or gain .
This latter -- original intent -- pacifism is shared by virtually every American regardless of party or ideology. It is the common western belief.
In this discussion of this post, this reflects the futility of WWI. This was a war simply over disputes. So many wars were of this nature. One example of the many I can think of was the 1895 war between China and Japan. Japan wanted territory that China had claim to, so when China did not agree, Japan started a war. The war ended with the Treaty of Shimenoseki where what Japan wanted was agreed to.
Today, everyone from the biggest peace and love hippy to strongest hawk believes that war for such reasons is unjust.
We are all pacifists in that we all agree on the principle that disputes are not to be settled by violence nor is gain to be achieved by violence.
When the use of violence comes in with regard to defense, survival or protection of innocents, what justifies the use of violence is the biggest disagreement. For example Obama as Senator spoke out against and never supported the post-911 Iraq invasion to oust the Hussein regime and provide a stable Iraq. But he does support and is carrying out violence in Libya to depose another dictator.
So that is the question of what rises to the level of justified violence or force.
Most political arguments revolve around that question and differences in opinion over what is justified. For example, Bosnia was OK and Libya is OK but Iraq was not OK for many people.
Now, as for the pacifists you think are laughed at and lonely who fall in to more extreme views where violence or force is almost never or never justified, I think that they do get ridicule at times, but surely not universally.
Not fighting the Nazis would be seen as extreme in my view and by most, I think. I don't think you fall in to that category, by your own admission, Levi.
But I do think it is laughable to think that European Jews simply needed immigration papers is laughable. Due to the folly of it.
But rest assured, most everyone does hold the pacifist view. It has won out over the course of the last century.
That makes sense, TKG -- I see where you're coming from and I'm glad you think the topic is at least worth serious discussion.
Personally, I'm not interested in what you describe as extreme pacifism. If somebody comes into my house and points a gun at me, and I've got a baseball bat in my hand, I will be swinging that bat. I don't even think there's much use debating these types of hypothetical or symbolic situations, because we've got enough real and serious situations in the world to deal with.
When I call myself a pacifist, I simply mean that I believe the practice of war between nations to be fraudulent, disastrous, self-perpetuating and unnecessary. I believe that it is not only possible but highly likely that future generations will manage to cure this societal disease, and I look forward to that day coming soon. This is the belief that gets me laughed at a lot. Regardless, I'm quite sure I'm right.
When mankind wearies of war after centuries of battling, it only makes sense to put down the arms and try a more peaceful path of understanding and compassion. Perhaps we may be fortunate enough in this millennium to see at least our world's leaders make the effort. We simply can no longer afford to kill people and blow things up without repercussions... after all, it is we the people who pay the price.
"I believe that it is not only possible but highly likely that future generations will manage to cure this societal disease, and I look forward to that day coming soon. This is the belief that gets me laughed at a lot."
Hi Levi, this clarifies your loneliness remark.
I'd say you are not lonely in your pacifism, you are lonely in your optimism.
I agree that many people are pessimistic about human nature and probably do scoff.
But I agree with you that it is inevitable. But the time frame is centuries.
TKG, you said WWI was "a war simply over disputes."
The thing is, most of us tend to draw an imaginary line, somewhere between antiquity and the present day, when we think about war. Few people get excited over the Huns invading Europe in 370 AD, or Julius Ceasar and the Gallic Wars, or the War of the Roses, as if those were somehow different than modern wars. Here's the difference: the Media is more prevalent, so various nations have to pretend they are doing things for the "right" reasons.
"...various nations have to pretend they are doing things for the "right" reasons.".
Yes, that is the debate. Is a good reason to use force really just a subterfuge to use force for gain?
If only we could get clean hands again and then keep them clean.
If the United States had clean hands, then at least when someone attacked us, we could honestly say it was unprovoked. I'm not saying that the people in the Twin Towers deserved to die - my, God, far from it - but I'm saying that if we had bargained with all the Middle Eastern countries for oil, just as if they were Great Britain or France, in other words, treated them like we would want to be treated, we might never have had the hostage crisis in the 1970s or any of the other tragedies right up through 9-11.


text:
Community Voices is generously supported by The Minneapolis Foundation; learn why.
Remembering my father, and the struggle for civil rights in 2017
This past fall, in the days following the election, I began thinking a great deal about the social struggles in our country, and my family's connection to them.
By David Schlesinger
Courtesy of the author
The thank-you note
On a snowy afternoon in the early spring of 2003, at the age of 24, I stood in my father's law office in downtown St. Paul and squinted to make out a framed thank-you note. The handwritten envelope was addressed upside down and lacked a street address. The note itself read:
Dear Mr. bob how are you fine I hope and here is a $1.00 bill from my heart, I wish I was able to work and pay you for the good deed you done me a great favor and I never will stop thanking you for it and this is only a pkg. cigerettes …
Although I asked him many questions before he died, I never asked him directly about the story behind that particular note, with its awkward syntax and funny envelope.
A searing sock-drawer discovery
Way before that, sometime in the early '90s, while snooping in my father's sock drawer as a kid of about 12, I came upon a faded copy of an obscure magazine called SAGA. It was more than 20 years old. For a while, it was a mystery to me why my father kept it there. I soon discovered that it contained an article about the Lawyers' Committee for Civil Rights Under Law's office in Cairo, Illinois, where he had worked in the early '70s. About my father, the article said:
Also on the Lawyers Committee is Bob Schlesinger, 24, and fresh out of Indiana law school. He is typical of the new breed of liberal, courageous lawyers … long of hair, long on guts. He says he expects to be in jail within a few months because he is not only engaged in work upon which the establishment frowns, but is also a conscientious objector to the Vietnam War, despite the fact that he is a second lieutenant in the reserves. "I just won't go," he declares.
As dated as that seems now, discovering it as a kid was searing. Not only was being a lawyer cool, but my dad was apparently once cool, too. And he had played some role in the civil rights and anti-war movements.
In my early teens, my dad told me about his years as a young lawyer in Cairo. He explained that Cairo, which is located on the confluence of the Mississippi and Ohio rivers, is the southernmost city in Illinois. And he described Cairo of that time as a hotbed of social struggle and violence, with opposing organizations of protesters for civil rights for African-Americans and demonstrators opposed to those rights. He told me about how he brought several cases to improve prison conditions and the Lawyers' Committee's strategy of demanding a jury trial on every petty offense with which their minority clients had been charged.
Scrapbook shows violence, protests, colleagues
My dad had a scrapbook from his time in Cairo. It contained a newspaper article about how someone attacked the Lawyers' Committee office by throwing a brick through the window, and then throwing a bomb into the office. The bomb contained five sticks of dynamite, but the fuse went out before it exploded. The scrapbook also contained photos of his colleagues, including other lawyers, and his friend Carol, a reporter. There were also photos of protesters from both sides and an image of my dad sitting next to an unknown man, with a look of utter determination in his eyes. You could also see from the photo that he had pretty darn long hair for a lawyer.
Bob Schlesinger in Cairo, Illinois.
Something else stood out for me about my dad's time in Cairo. He told me how one day he drove outside of town with one of his colleagues and watched a huge flock of migrating birds taking off from the river. It was silent except for the flapping of wings, the water on the river, and the highway in the distance. They had a moment of quiet in a time of so much turmoil.
My dad was also involved in litigation as a plaintiff himself during his time in Cairo. Just as SAGA magazine said, he had attempted to become a conscientious objector after joining the military through ROTC. At the time, in order to obtain status as a conscientious objector, a person needed to have a religious objection to killing under any circumstances. The problem was that my dad didn't conceive of his objection as religious; for him it was philosophical. When his conscientious objector application was denied, he brought his own civil rights case against the Army in federal court in Washington, D.C. He ultimately received conscientious objector status and an honorable discharge. When his lawyer called him with the news, my father asked what happened. The lawyer reminded him of the Muhammed Ali case that had just been decided by the Supreme Court two weeks before and said he had written a letter to the judge, stating, "Schlesinger is just like Mohammed Ali except he's not black and he's never been heavyweight champion of the world."
An important uncle
David Schlesinger
While my dad's work as a civil rights lawyer occurred within the historical context of the Vietnam War and the civil rights movement, it also occurred within a particular, but related, personal context. My father was the namesake of his uncle, who had been killed as the U.S. Army drove into Germany in World War II. When my grandfather found out about his brother Bob's death on March 29, 1945, from his own military post in Ashville, North Carolina, he wrote his parents a letter, observing that they had lost the person with, "the nicest personality and the most unselfish attitude of anyone in the family (except mother)" but that, "[i]t all goes to show that people should be prepared to die, if necessary, to protect the national security."
Shortly after the end of World War II, the Army sent my great-grandparents a photo of that first Bob Schlesinger's grave in Belgium's Henri-Chapelle Cemetery. The grave had been marked with a cross. My great-grandfather engaged in a letter-writing campaign to have the Army replace the cross with a Star of David, something that did not happen for several years.
It was an emotionally complex thing for my father to avoid Vietnam when the man he was named after had lost his life in World War II. While my father had many of the human qualities of his uncle, the two had different views about the ethics and utility of war. At the same time, both struggled for our country in their own way, and in their own time.
By the time I discovered the SAGA magazine, my dad was practicing as a solo family lawyer. Although he was working in an arguably mundane area of law compared to civil rights, his dedication to his clients had not lessoned. And although family law was unlike civil rights litigation in that it was not tied up as directly with larger social struggles, my dad impressed upon me that a custody battle, for example, concerns a person's children, the most important relationship in a person's life.
Civil rights work in another form
Despite his step back from civil rights litigation, as I grew up, my father's civil rights work continued in another form. My family was part of a reform Jewish congregation. When one of the rabbis came out of the closet as a lesbian, this news was not received entirely well. My parents and a handful of other members of the congregation resigned their membership and formed a new synagogue around the rabbi who had come out. That synagogue – Shir Tikvah in Minneapolis – is thriving to this day.
As a kid, all of this was percolating within me, and by the time I went to college, I knew I wanted to become a lawyer. Although I couldn't have articulated it at the time, I believed that by doing so, I would be able to engage in these larger social struggles that were wrapped up with the life of my family, and of our country.
My dad came to visit me for a weekend when I was in my junior year of college in Fredericksburg, Virginia. I remember hiking with him in the Blue Ridge Mountains, and telling him that I wanted to go to law school. He felt privileged to be a lawyer himself, and was excited that I was interested in following in his footsteps.
After college I moved back to Minnesota and worked on Sen. Paul Wellstone's 2002 re-election campaign. In the months preceding the election in November, I probably had the same determined look in my eye that my dad had back in Cairo. But then, in September, it became clear that my father was seriously ill, and that he probably had amyotrophic lateral sclerosis (ALS or Lou Gehrig's disease). A few weeks after that probable diagnosis, on Oct. 25, 2002, Wellstone's small plane crashed in northern Minnesota, killing the senator and seven others. I knew at the time Wellstone's plane crashed that I was likely to lose my dad as well.
The feeling of that foggy, rainy day in October stayed with me as I started law school. Though he was walking with a cane, my dad was with me when I attended admitted students events at the University of Minnesota in the late summer of 2003. During law school, I spent as much time with my dad as I could. I visited him in his office and looked, a bit bemused, at that funny thank you note. And in the fall of 2004, I walked near a lake as he buzzed along next to me in his electric wheelchair. He pointed out a particular bird, and as he struggled to enunciate the words, I remembered the story about watching the birds fly off the water in Cairo, a memory of another moment of relative peace.
My dad passed away on June 29, 2005. Less than a year later, I graduated from law school. Since then, I have been practicing employment law on behalf of employees, litigating cases under the Civil Rights Act of 1964. The thank-you note from Cairo hangs now on my office wall. It is a reminder that even if the remuneration is sometimes small, I have the opportunity in my work to make tangible improvements in people's lives.
A question answered
That note has also been a reminder to me of the mystery of unanswered questions. However, recently, something interesting happened in that regard. This past fall, in the days following the election, I began thinking a great deal about the social struggles in our country, and my family's connection to them. The demonstrators from Ferguson, Missouri, reminded me of protesters struggling for civil rights in Cairo. And in Khizr Khan's words about his Muslim son who died for our county, I heard an echo of my great grandfather's letters about how the Star of David should mark the grave of his son.
I began digging into the past, trying to get a sense of where we stand now. I emailed Carol, my father's friend from his time in Cairo, I hoping to ask about her recollections. To my surprise, she told me that she had just finished writing a memoir about that time.
In the final pages of her book, Carol recalled an elderly African-American woman – with whom Carol lived – who believed, probably inaccurately, that people were stealing from her. When the woman wrote the police about this, they took her from her home and institutionalized her against her will, ostensibly because of concerns about her mental health. Based upon legal advice from my father, Carol was able to secure the woman's release. It was that woman who wrote my father the thank-you note that hung on his wall.
Although it has been more than 10 years since he died, in this way I have learned a little bit more about my father. However, had I asked him directly about the thank-you note, I doubt he would have told me the story. He believed that his actions in that instance were simply part and parcel with what lawyers do. Although I wish I could talk to him about how to react to the election of Donald Trump in the context of my work, I think he'd advise me to be guided by the history of the struggle for civil rights, and of the ability we all have to improve people's lives through that struggle. He would also remind me that, while that struggle is ongoing, you can still drive out of town, watch the birds fly off the water, and hear the flapping of wings in the quiet near the river.
David Schlesinger is a partner at Nichols Kaster, PLLP, where he represents victims of illegal workplace conduct. He lives in Minneapolis with his wife, Marina.
WANT TO ADD YOUR VOICE?
If you're interested in joining the discussion, add your voice to the Comment section below — or consider writing a letter or a longer-form Community Voices commentary. (For more information about Community Voices, email Susan Albright at salbright@minnpost.com.)
Interesting: Omnibus critic Marty now leads committee that crafts the massive bills
By Peter Callaghan | Staff Writer
MNopedia
The Beatles' one and only concert in Minnesota
By Britt Aamodt
More Community Voices articles

Remembering my father, and the struggle for civil rights in 2017

By David Schlesinger

Jan. 18, 2017

Courtesy of the author

The thank-you note

On a snowy afternoon in the early spring of 2003, at the age of 24, I stood in my father's law office in downtown St. Paul and squinted to make out a framed thank-you note. The handwritten envelope was addressed upside down and lacked a street address. The note itself read:

Dear Mr. bob how are you fine I hope and here is a $1.00 bill from my heart, I wish I was able to work and pay you for the good deed you done me a great favor and I never will stop thanking you for it and this is only a pkg. cigerettes ...

Although I asked him many questions before he died, I never asked him directly about the story behind that particular note, with its awkward syntax and funny envelope.

A searing sock-drawer discovery

Way before that, sometime in the early '90s, while snooping in my father's sock drawer as a kid of about 12, I came upon a faded copy of an obscure magazine called SAGA. It was more than 20 years old. For a while, it was a mystery to me why my father kept it there. I soon discovered that it contained an article about the Lawyers' Committee for Civil Rights Under Law's office in Cairo, Illinois, where he had worked in the early '70s. About my father, the article said:

Also on the Lawyers Committee is Bob Schlesinger, 24, and fresh out of Indiana law school. He is typical of the new breed of liberal, courageous lawyers ... long of hair, long on guts. He says he expects to be in jail within a few months because he is not only engaged in work upon which the establishment frowns, but is also a conscientious objector to the Vietnam War, despite the fact that he is a second lieutenant in the reserves. "I just won't go," he declares.

As dated as that seems now, discovering it as a kid was searing. Not only was being a lawyer cool, but my dad was apparently once cool, too. And he had played some role in the civil rights and anti-war movements.

In my early teens, my dad told me about his years as a young lawyer in Cairo. He explained that Cairo, which is located on the confluence of the Mississippi and Ohio rivers, is the southernmost city in Illinois. And he described Cairo of that time as a hotbed of social struggle and violence, with opposing organizations of protesters for civil rights for African-Americans and demonstrators opposed to those rights. He told me about how he brought several cases to improve prison conditions and the Lawyers' Committee's strategy of demanding a jury trial on every petty offense with which their minority clients had been charged.

Scrapbook shows violence, protests, colleagues

My dad had a scrapbook from his time in Cairo. It contained a newspaper article about how someone attacked the Lawyers' Committee office by throwing a brick through the window, and then throwing a bomb into the office. The bomb contained five sticks of dynamite, but the fuse went out before it exploded. The scrapbook also contained photos of his colleagues, including other lawyers, and his friend Carol, a reporter. There were also photos of protesters from both sides and an image of my dad sitting next to an unknown man, with a look of utter determination in his eyes. You could also see from the photo that he had pretty darn long hair for a lawyer.

Courtesy of the author

Bob Schlesinger in Cairo, Illinois.

Something else stood out for me about my dad's time in Cairo. He told me how one day he drove outside of town with one of his colleagues and watched a huge flock of migrating birds taking off from the river. It was silent except for the flapping of wings, the water on the river, and the highway in the distance. They had a moment of quiet in a time of so much turmoil.

My dad was also involved in litigation as a plaintiff himself during his time in Cairo. Just as SAGA magazine said, he had attempted to become a conscientious objector after joining the military through ROTC. At the time, in order to obtain status as a conscientious objector, a person needed to have a religious objection to killing under any circumstances. The problem was that my dad didn't conceive of his objection as religious; for him it was philosophical. When his conscientious objector application was denied, he brought his own civil rights case against the Army in federal court in Washington, D.C. He ultimately received conscientious objector status and an honorable discharge. When his lawyer called him with the news, my father asked what happened. The lawyer reminded him of the Muhammed Ali case that had just been decided by the Supreme Court two weeks before and said he had written a letter to the judge, stating, "Schlesinger is just like Mohammed Ali except he's not black and he's never been heavyweight champion of the world."

An important uncle

David Schlesinger

While my dad's work as a civil rights lawyer occurred within the historical context of the Vietnam War and the civil rights movement, it also occurred within a particular, but related, personal context. My father was the namesake of his uncle, who had been killed as the U.S. Army drove into Germany in World War II. When my grandfather found out about his brother Bob's death on March 29, 1945, from his own military post in Ashville, North Carolina, he wrote his parents a letter, observing that they had lost the person with, "the nicest personality and the most unselfish attitude of anyone in the family (except mother)" but that, "t all goes to show that people should be prepared to die, if necessary, to protect the national security."

Shortly after the end of World War II, the Army sent my great-grandparents a photo of that first Bob Schlesinger's grave in Belgium's Henri-Chapelle Cemetery. The grave had been marked with a cross. My great-grandfather engaged in a letter-writing campaign to have the Army replace the cross with a Star of David, something that did not happen for several years.

It was an emotionally complex thing for my father to avoid Vietnam when the man he was named after had lost his life in World War II. While my father had many of the human qualities of his uncle, the two had different views about the ethics and utility of war. At the same time, both struggled for our country in their own way, and in their own time.

By the time I discovered the SAGA magazine, my dad was practicing as a solo family lawyer. Although he was working in an arguably mundane area of law compared to civil rights, his dedication to his clients had not lessoned. And although family law was unlike civil rights litigation in that it was not tied up as directly with larger social struggles, my dad impressed upon me that a custody battle, for example, concerns a person's children, the most important relationship in a person's life.

Civil rights work in another form

Despite his step back from civil rights litigation, as I grew up, my father's civil rights work continued in another form. My family was part of a reform Jewish congregation. When one of the rabbis came out of the closet as a lesbian, this news was not received entirely well. My parents and a handful of other members of the congregation resigned their membership and formed a new synagogue around the rabbi who had come out. That synagogue – Shir Tikvah in Minneapolis – is thriving to this day.

As a kid, all of this was percolating within me, and by the time I went to college, I knew I wanted to become a lawyer. Although I couldn't have articulated it at the time, I believed that by doing so, I would be able to engage in these larger social struggles that were wrapped up with the life of my family, and of our country.

My dad came to visit me for a weekend when I was in my junior year of college in Fredericksburg, Virginia. I remember hiking with him in the Blue Ridge Mountains, and telling him that I wanted to go to law school. He felt privileged to be a lawyer himself, and was excited that I was interested in following in his footsteps.

After college I moved back to Minnesota and worked on Sen. Paul Wellstone's 2002 re-election campaign. In the months preceding the election in November, I probably had the same determined look in my eye that my dad had back in Cairo. But then, in September, it became clear that my father was seriously ill, and that he probably had amyotrophic lateral sclerosis (ALS or Lou Gehrig's disease). A few weeks after that probable diagnosis, on Oct. 25, 2002, Wellstone's small plane crashed in northern Minnesota, killing the senator and seven others. I knew at the time Wellstone's plane crashed that I was likely to lose my dad as well.

The feeling of that foggy, rainy day in October stayed with me as I started law school. Though he was walking with a cane, my dad was with me when I attended admitted students events at the University of Minnesota in the late summer of 2003. During law school, I spent as much time with my dad as I could. I visited him in his office and looked, a bit bemused, at that funny thank you note. And in the fall of 2004, I walked near a lake as he buzzed along next to me in his electric wheelchair. He pointed out a particular bird, and as he struggled to enunciate the words, I remembered the story about watching the birds fly off the water in Cairo, a memory of another moment of relative peace.

My dad passed away on June 29, 2005. Less than a year later, I graduated from law school. Since then, I have been practicing employment law on behalf of employees, litigating cases under the Civil Rights Act of 1964. The thank-you note from Cairo hangs now on my office wall. It is a reminder that even if the remuneration is sometimes small, I have the opportunity in my work to make tangible improvements in people's lives.

A question answered

That note has also been a reminder to me of the mystery of unanswered questions. However, recently, something interesting happened in that regard. This past fall, in the days following the election, I began thinking a great deal about the social struggles in our country, and my family's connection to them. The demonstrators from Ferguson, Missouri, reminded me of protesters struggling for civil rights in Cairo. And in Khizr Khan's words about his Muslim son who died for our county, I heard an echo of my great grandfather's letters about how the Star of David should mark the grave of his son.

I began digging into the past, trying to get a sense of where we stand now. I emailed Carol, my father's friend from his time in Cairo, I hoping to ask about her recollections. To my surprise, she told me that she had just finished writing a memoir about that time.

In the final pages of her book, Carol recalled an elderly African-American woman – with whom Carol lived – who believed, probably inaccurately, that people were stealing from her. When the woman wrote the police about this, they took her from her home and institutionalized her against her will, ostensibly because of concerns about her mental health. Based upon legal advice from my father, Carol was able to secure the woman's release. It was that woman who wrote my father the thank-you note that hung on his wall.

Although it has been more than 10 years since he died, in this way I have learned a little bit more about my father. However, had I asked him directly about the thank-you note, I doubt he would have told me the story. He believed that his actions in that instance were simply part and parcel with what lawyers do. Although I wish I could talk to him about how to react to the election of Donald Trump in the context of my work, I think he'd advise me to be guided by the history of the struggle for civil rights, and of the ability we all have to improve people's lives through that struggle. He would also remind me that, while that struggle is ongoing, you can still drive out of town, watch the birds fly off the water, and hear the flapping of wings in the quiet near the river.

David Schlesinger is a partner at Nichols Kaster, PLLP, where he represents victims of illegal workplace conduct. He lives in Minneapolis with his wife, Marina.

WANT TO ADD YOUR VOICE?

If you're interested in joining the discussion, add your voice to the Comment section below — or consider writing a letter or a longer-form Community Voices commentary. (For more information about Community Voices, email Susan Albright at salbright@minnpost.com.)

This article first appeared on MinnPost and is republished here under a Creative Commons license.


text:
Rui Manuel Magalhães Casaca (born 18 October 1959) is a Portuguese former professional footballer who played as a central midfielder, and is the director of football at S.C. Braga.

Football career
Born in Marco de Canaveses, Porto District, Casaca started playing in the lower leagues with Merelinense F.C. and C.A. Valdevez, helping the latter team promote to the Segunda Liga in 1981. Starting in the following year and until his retirement, he competed solely in the Primeira Liga, with Rio Ave F.C. and Boavista FC.

In the 1985–86 season, Casaca scored a career-best four goals from 28 appearances to help Boavista finish in fifth position, with the subsequent qualification to the UEFA Cup. He retired in June 1994 at the age of 34, amassing top division totals of 305 matches and 18 goals, and immediately joined his main club's coaching staff; in January 1997 he was named interim manager, collecting one point in his two games in charge.

From 1997 to 2006, Casaca acted as director of football of Boavista and C.F. Os Belenenses, also being assistant to manager Manuel José at both sides. On 31 July 2007, in the same capacity, he returned to S.C. Braga where he had already occupied the position, and went on to remain there for several years.

Honours

Player
Boavista
Taça de Portugal: 1991–92
Supertaça Cândido de Oliveira: 1992

References

External links

1959 births
Living people
People from Marco de Canaveses
Portuguese footballers
Association football midfielders
Primeira Liga players
Liga Portugal 2 players
Merelinense F.C. players
Rio Ave F.C. players
Boavista F.C. players
Portuguese football managers
Primeira Liga managers
Boavista F.C. managers
Sportspeople from Porto District

redpajama_set_name: RedPajamaWikipedia