finish newest BDNK paper

tex/grhd.tex

@@ -4118,6 +4118,7 @@ with gravity}
     that only those first-order theories with a frame choice that satisfies
     \begin{align}
       u_\mu u_\nu T^{\mu\nu} = \epsilon \, ,
+      \label{eq:instable-frames}
     \end{align}
     are plagued by this problem. The physical interpretation of this condition is that
     all co-moving observes of the fluid observe the energy density as if it were in equilibrium,
@@ -4134,6 +4135,76 @@ with gravity}
     Lastly, they say a word about very recent development based on Carter's formalism
     and the variational principle. \detail{Only highlighting this, because Dumbser is involved
     in one of the papers.}
+  \item Discussion of results: The paper discusses BDNK theory that includes all relevant
+    viscous fluxes and they show that it satisfies conditions 1)-4). The results hold in
+    the full non-linear regime. They prove this rigorously and provide inequalities for
+    the viscosities that mark the validity regime of the theory.
+    They do this by avoiding frame choices in which \eqref{eq:instable-frames} hold
+    (note that MIS theories also adopt this choice). Not using a frame
+    in which \eqref{eq:instable-frames} should be seen as a natural step, because
+    other formulations also do not adopt the similar relation for pressure, but instead
+    allow for out-of-equilibrium corrections to the pressure too.
+    The main steps to derive the results are: determine the characterisitic speeds
+    (requires good understanding of the DOFs and geometric properties to come up with
+    the right factorizations), diagonalize the principle symbol (requires a good choice
+    for the first order reduction in order to keep the number of spurious modes, the
+    ones with zero speeds, to a minimum), establish local well-posedness
+    (this is the most technical part; requires introducing of pseudo-differential operators,
+    for which the others developed separate theorems), address stability by determining the signs of the
+    Fourier modes in linear perturbations.
+    They highlight that BDNK is derivable from kinetic theory, but also say that this does not
+    guarantee that the theory is necessarily physical (\eg{} Eckart and Landau and Lifshitz
+    are also derivable from kinetic theory).
+    They say that BDNK theory, unlike MIS theory, is capable of handling shocks,
+    which means that the Rankine-Hugoniot-type conditions can be derived in that theory.
+    There is work that investigates shocks for the conformally symmetric case by means
+    of numerical (Pandya et al.) and analytical (Freist\"uhler) computations.
+    The authors then also question how well \textit{any} hydro theory (including ideal hydro)
+    can even describe shocks. The argument is that any viscous theory is an effective theory
+    which comes with a particular validity regime in which it can be applied. However,
+    shocks are probably the most extreme regime in terms of gradients that appear in dynamical
+    situations. People have been studying shocks in ideal theories a lot, because the
+    equations allow one to do so in terms of weak (distributional) solutions and using
+    the Rankine-Hugoniot conditions. Because the latter is also available for BDNK theory,
+    one should be able to this there too. However, being able to study these scenarios
+    does not mean that the corresponding results are then necessarily physical, or accurate
+    in the sense that the results would significantly change when higher (second) order corrections
+    would be included.
+    There is only one universally understood definition of entropy for out-of-equilibrium
+    systems and that one results from the Boltzmann equations.
+    Againm, one should keep in mind that gradient expansions are effective theories and, thus,
+    it is only really necessary to construct a formula for entropy production which yields
+    non-negative values only within the regime of applicability of the theory. Note
+    that the Eckart and Landau and Lifshitz theories construct entropy currents such that
+    entropy produciton is non-negative in the presence of arbitrarily large gradients,
+    but as we know those theories are not causal nor stable. Furthermore, also MIS theory
+    tries to do the same.
+  \item Generalized Navie-Stokes theory: Here follows a standard presentation of the ideal
+    fluid equations and then a first-order gradient expansion written in the most general
+    frame. In this part they mention frequently that \cite{kovtun2019first} quite often
+    and say that its essentially the same system.
+    They then specify a frame: They assume that $J^\mu = N u^\mu$, because that allows
+    to use the identity $u^\lambda \nabla_\lambda n = -n \nabla_\lambda u^\lambda$. The
+    rest of the constitutive relations are given then in equation (10),(11).
+    Note that they avoid the condition \eqref{eq:instable-frames}, however, in the resulting
+    system they find it holds that $\EE ~ \epsilon + \OO(\partial^2)$.
+    Furthermore, in (12) they write out the corresponding EoMs, which is very nice.
+    They then also discuss entropy-production and give the corresponding off- and on-shell
+    expressions.
+  \item Causality: They briefly list the EoMs of the combined Einstein+hydro system
+    but only focus on the principle parts. Then the theorem for causality of the corresponding
+    EoMs is given.
+  \item Strong hyperbolicity and local well-posedness: This is a very technical section,
+    although even more technical stuff has already been moved to appendices. I skipped most
+    of it, but just note here that the proof of well-posedness works with Sobolev spaces.
+  \item Linear stability: \question{They state that in equilibrium $\beta_\mu = u_\mu/T$ must
+    be Killing. Why? They cite a Israel and Stewards with this statement.}
+    They say that linear stability analysis can be quite involved, in particular for
+    states with non-vanishing background velocity. To this end they prove a new theorem in this
+    section that allows to conclude linear stability for causal theories and
+    arbitrary background velocity. I did not study the proof.
+    Further below they also test the theorem with a toy problem and later on derive
+    conditions for the transport coefficients so that linear stability is ensured.
 \end{itemize}