$\mathcal{W}=P+\mathcal{W}_{(1)}+\mathcal{W}_{(2)}$ (142)

where

$\displaystyle\begin{split}P=&p_{0}T^{d}+r_{M}T^{d-2}{M}_{(1)}+r_{m}T^{d-2}\delta_{q1}m_{(0)}+\mathcal{O}(\mathring{\nabla}^{4})\\\
\mathcal{W}_{(1)}=&x_{1}^{(1)}T^{d-1}\,\kappa\\\
\mathcal{W}_{(2)}=&\sum_{i=1}^{12}x_{i}^{(2)}T^{d-2}S_{(i)}^{c}\,.\end{split}$
(143)

Here, $p_{0}$, $r_{m}$, $r_{M}$ and the $x_{i}^{(j)}$’s are real numbers. The
Kronecker delta $\delta_{q\,1}$ ensures that the hydrostatic partition
function includes all possible scalars for generic values of $q$ and for $q=1$
Note that $\kappa$ is conformal invariant in the hydrostatic limit for any
value of $q$, but is conformal invariant in general only for $q=0$. Therefore,
later, when including non hydrostatic corrections, we will need to modify the
expressions associated with $x_{1}^{(1)}$ and $x^{(2)}_{5}$ so that they are
conformal invariant for all values of $q$.

The stress tensor $T_{h+}^{\mu\nu}$ and spin current $S_{h}^{\lambda\mu\nu}$
associated with (143) can be obtained by varying the generating function $W$
with respect to the vielbein and spin connection respectively, and adding non
hydrostatic terms to expressions derived from terms linear in $K_{\mu}{}^{ab}$
so that the resulting stress tensor is of the BR type. (Recall the discussion
around (66).) Instead of carrying out this variation explicitly, we can read
the result off of (60)–(61) and (63)–(65) by making the substitutions

$\displaystyle\begin{split}P_{0}&=p_{0}T^{d}\qquad\epsilon_{0}=(d-1)p_{0}T^{d}\qquad\chi^{(1)}_{1}=x^{(1)}_{1}T^{d-1}\\\
\chi^{(2)}_{1}&=x^{(2)}_{1}T^{d-2}\qquad\chi^{(2)}_{2}=x^{(2)}_{2}T^{d-2}\qquad\chi^{(2)}_{3}=\left(-2(1-q)^{2}qx^{(2)}_{3}-2(d-2)qx^{(2)}_{4}\right)T^{d-2}\\\
\chi^{(2)}_{4}&=qx^{(2)}_{4}T^{d-2}\qquad\chi^{(2)}_{5}=x^{(2)}_{5}T^{d-2}\quad\chi^{(2)}_{6}=\left((1-q)^{3}x^{(2)}_{3}+2(d-2)(q-1)x^{(2)}_{4}+4(d-2)^{2}x^{(2)}_{7}\right)T^{d-2}\\\
\chi^{(2)}_{7}&=x^{(2)}_{6}T^{d-2}\qquad\chi^{(2)}_{8}=(1-q)x^{(2)}_{4}T^{d-2}\qquad\chi^{(2)}_{9}=x^{(2)}_{7}T^{d-2}\\\
\chi^{(2)}_{10}&=x^{(2)}_{8}T^{d-2}\qquad\chi^{(2)}_{11}=x^{(2)}_{9}T^{d-2}\qquad\chi^{(2)}_{12}=x^{(2)}_{10}T^{d-2}\\\
\chi^{(2)}_{13}&=x^{(2)}_{11}T^{d-2}\qquad\chi^{(2)}_{14}=x^{(2)}_{12}T^{d-2}\qquad\rho_{m}=(q^{2}(1-q)x^{(2)}_{3}+r_{m}\delta_{q\,1})T^{d-2}\qquad\rho_{M}=r_{M}T^{d-2}\,.\end{split}$
(144)

The explicit form of the resulting constitutive relations can be found in
appendix B.

Weyl invariance of the generating function $W$ guarantees that, in the
hydrostatic limit, the stress tensor and spin current will satisfy (130) and
(131). Yet, a naive computation of the Weyl transformation of
$T_{h+}^{\mu\nu}$ (or $T_{h}^{\mu\nu}$) resulting from the constitutive
relations given in appendix B do not reproduce (131). This is due to the
inhomogenous transformation properties of $k_{\mu}$ and $\mathcal{K}_{V\,\mu}$
under Weyl rescalings together with the fact that we have dropped terms
quadratic in contorsion. Once we include the full hydrostatic constitutive
relations derived from $W$, quadratic in contorsion, into the stress tensor,
then (131) will be satisfied. Since the constitutive relations for the stress
tensor at second order in the contorsion tensor are somewhat long we will have
chosen not to list those terms here. The interested reader can find the terms
relevant for this computation in appendix C. We have also checked that
$T_{h+}^{\mu\nu}$ and $S_{h}^{\lambda\mu\nu}$ satisfy (130) and (131) outside
of hydrostatic equilibrium.

In order to construct a Weyl covariant non hydrostatic tensor
$\mathcal{T}_{nh}^{\mu\nu}$ (and from it the stress tensor $T_{nh}^{\mu\nu}$
using the prescription of (141)) and the Weyl covariant non hydrostatic spin
current $S_{nh}^{\lambda\mu\nu}$, we consider the non covariant constitutive
relations (78) and use table 7 to remove from them all non Weyl covariant
expressions. We find that in order for the non hydrostatic spin current to
transform covariantly under Weyl rescalings, we need to set
$\sigma_{1}=T^{d-2}s_{1}$ and $\sigma_{2}=0$ in (78), leading to

$S_{nhBR}^{\lambda\mu\nu}=2T^{d-2}s_{1}\sigma^{\lambda[\mu}u^{\nu]}\,.$ (145)

To construct the Weyl covariant tensor $\mathcal{T}_{nh}^{\mu\nu}$ from the
non Weyl covariant expression, $T_{nh}^{\mu\nu}$, of (78) we must not only set
$\zeta=0$ in (78) and scale the remaining transport coefficients by powers of
$T$, we must also replace the ringed derivatives in (78) with Weyl covariant
ones:

$\displaystyle\begin{split}\mathcal{T}_{nh}^{(\mu\nu)}&=-hT^{d-1}\sigma^{\mu\nu}-\mathring{\mathcal{D}}_{\lambda}\left(S_{nhBR}^{\mu\lambda\nu}-S_{nhBR}^{\lambda\nu\mu}\right)\,\\\
\mathcal{T}_{nh}^{[\mu\nu]}&=\mathring{\mathcal{D}}_{\lambda}S^{\lambda\mu\nu}{}_{nhBR}+s_{A}T^{d-2}A^{[\mu}u^{\nu]}+s_{m}T^{d-2}\hat{m}^{[\mu}u^{\nu]}+s_{M}T^{d-2}\hat{M}^{\mu\nu}\,.\end{split}$
(146)

Note that (140) is trivially satisfied since
$S_{nhBR\,\lambda}{}^{\lambda\mu}=0$.

We can now construct the non hydrostatic stress tensor using the prescription
of (141) with $\mathcal{T}^{\mu\nu}$ and $S^{\mu\nu\rho}$ replaced by
$\mathcal{T}_{nh}^{\mu\nu}$ and $S_{nh}^{\mu\nu\rho}$. The only missing terms
we need to include are corrections to the $x_{1}^{(1)}$ and $x^{(2)}_{5}$
terms once $q\neq 0$ and we are outside of hydrostatic equilibrium. Some trial
and error reveals that we need to add to the spin current a term of the form
$4x_{5}^{(2)}q\theta\Delta^{\lambda[\mu}u^{\nu]}$ and to the stress tensor
contributions of the form
$T^{d-1}qx^{(1)}_{1}\theta((d-1)u^{\mu}u^{\nu}+\Delta^{\mu\nu})$ and
$\frac{1}{2}x^{(2)}_{5}T^{d-2}q(\mathcal{K}_{V}^{\nu}u^{\mu}-\mathcal{K}_{V}^{\mu}u^{\nu}-4\kappa_{A}^{\mu\nu})\theta$.
The resulting constitutive relations are presented in appendix B. It is given
in a non Landau like frame since the latter will not have manifest conformal
symmetry.

The constitutive relations for the conformal fluid described in this section
will be modified in the presence of a charge current. The chemical potential
and external gauge field associated with the charge current allow for
additional tensor structures both in the non conformally invariant case and
the conformally invariant one. Since we have refrained from classifying the
constitutive relations for the charge current in the presence of an external
torsion field, we will only discuss the $q=0$ torsionless, conformal,
constitutive relations for a charged fluid with spin. (Recall that since
torsion does not transform homogenously under Weyl rescalings, a $q\neq 0$,
torsionless background can be transformed into a torsionfull one.)

Consider the hydrostatic generating function for a charged fluid, (124). When
$q=0$ the vectors $k^{\mu}$ and $\mathcal{K}_{V}^{\mu}$ transform homogenously
under Weyl rescalings, (see table 7) so all the scalars in table 5 will
contribute to $\mathcal{W}$, except for $\tilde{S}_{(1)}$. Thus, the
generating function for a conformal charged fluid will take the form

$\mathcal{W}=p_{0}T^{d}+T^{d-1}x_{1}^{(1)}\kappa+\sum_{i=1}^{2}T^{d-2}{x}_{i}^{(2)}{S}_{i}^{(c)}+\sum_{i=2}^{6}T^{d-2}\tilde{x}_{i}^{(2)}\tilde{S}_{i}\,,$
(147)

where $p_{0}$ the $x^{(2)}_{i}$, and the $\tilde{x}^{(2)}_{i}$ all depend on
the chemical potential to temperature ratio $\mu/T$. The explicit form of the
hydrostatic constitutive relations for a conformal charged fluid, obtained by
the variation of (147) can be found in appendix B.

For the non hydrostatic terms we note that
$\Delta^{\mu\nu}\left(\mathring{\nabla}_{\nu}\left(T\frac{\mu}{T}\right)-E_{\nu}\right)$
transforms homogenously under Weyl rescalings. Thus, the constitutive
relations for the stress tensor and current are similar to those for the
uncharged fluid, with the addition of the new terms associated with
$\Delta^{\mu\nu}\left(\mathring{\nabla}_{\nu}\left(T\frac{\mu}{T}\right)-E_{\nu}\right)$
(and the dependence of the transport coefficients on $\mu/T$). The full
constitutive relations for a conformally invariant charged fluid with spin can
be found in appendix B.

##  8 Summary and discussion

In this work we have constructed the most general constitutive relations for
an uncharged fluid with spin in the presence of external torsion and a charged
fluid with spin in the absence of torsion. As emphasized in the introduction,
one does not expect background torsion to play a role in, say, the dynamics of
heavy ion collisions. The reason we have included torsion in the constitutive
relations is that, without it, one would obtain incorrect Kubo formula for the
various transport coefficients. (This is similar to the need for including a
non trivial background metric in order to compute Kubo formula in flat space.)
Another reason we have included background torsion is that it might be
relevant as an effective theory for certain condensed matter systems with
dislocations; see e.g. deJuan:2009ldt ; Mesaros:2009az . We have also carried
out a brief analysis of the effect of charge and of conformal invariance. Our
main results have been collected in appendix B for ease of reference.

To summarize the essential features of our analysis, consider the equations of
motion for an uncharged fluid, given by (27). Using the constitutive relations
in appendix B, the leading order terms for these equations (in the absence of
torsion) take the form

$\displaystyle\begin{split}\frac{\partial s}{\partial
T}u^{\mu}\partial_{\mu}T+\frac{\partial P}{\partial T}\theta&=0\\\
A^{\mu}&=0\\\ \sigma_{m}\hat{m}^{\mu}&=0\\\
\sigma_{M}\hat{M}^{\mu\nu}&=0\,,\end{split}$ (148)

Where $s=\frac{\partial P}{\partial T}$ is the entropy density,
$\theta=\mathring{\nabla}_{\mu}u^{\mu}$,
$\hat{m}^{\mu}=\mu^{\mu\alpha}u_{\alpha}-a^{\mu}$,
$\hat{M}^{\mu}=\Delta^{\mu}{}_{\alpha}\Delta^{\nu}{}_{\beta}\mu^{\alpha\beta}+\Omega^{\mu\nu}$
and
$A^{\mu}=\Delta^{\mu\nu}\mathring{\nabla}_{\nu}T+Tu^{\alpha}\mathring{\nabla}_{\alpha}u^{\mu}$
(not to be confused with the gauge potential $A_{\mu}$ which sources the
charge current).

The first two equations in (148) are the standard equations for energy and
momentum conservation. The last two equations follow from spin current
conservation and, as long as $\sigma_{m}$ and $\sigma_{M}$ are non zero imply

$\hat{m}^{\mu}=0\qquad\hat{M}^{\mu\nu}=0\,.$ (149)

The relations (149) were derived previously in Becattini:2013fla  by appealing
to the Boltzmann equation. (See Becattini:2020sww ; Becattini:2020ngo  for
reviews on this subject.) In our work these relations were obtained as
dynamical equations in the absence of charge. In the presence of charge the
relations (148) are modified and as a result, the first equality in (149) is
replaced by (125). We do find that, charged or not, (149) are always valid in
hydrostatic equilibrium.

One of the unusual features of (149) is that the spin chemical potential is
determined algebraically from the acceleration and vorticity. This means,
among other things, that one can not impose initial conditions for the spin
chemical potential. Rather, it is determined algebraically from derivatives of
the velocity. In a previous paper, Gallegos:2021bzp , we have worked out the
constitutive relations for a conformal fluid with spin, in the absence of
torsion, in what we referred to as the dynamical spin limit. In the notation
of the current paper, this corresponds to setting $\sigma_{m}=0$ and
$\sigma_{M}=0$. If $\sigma_{m}$ and $\sigma_{M}$ vanish, then the leading
equations of motion for the spin current become second order equations (recall
that the spin chemical potential is counted as first order in derivatives)
involving derivatives of $\hat{m}^{\mu}$ and $\hat{M}^{\mu\nu}$. This implies
that the equations of motion for the spin chemical potential become first
order differential equations. For this reason we have referred to the
$\sigma_{m}=0$ and $\sigma_{M}=0$ limits as a dynamical spin limit. Note that,
at least in our current formalism, one can not simply take the limit
$\sigma_{m}\to 0$ or $\sigma_{M}\to 0$ of (148) in order to obtain the
dynamical spin limit since we are working in a frame where higher order
corrections to the antisymmetric components of the stress tensor vanish. As
discussed in section 4.4 this frame choice is only allowed for $\sigma_{m}\neq
0$ and $\sigma_{M}\neq 0$. Thus, in a sense, the work in Gallegos:2021bzp
treats a specialized limit of the constitutive relations for hydrodynamics
with spin which can not be captured by our current formulation. As of the time
of writing of this note, we could not find symmetry arguments which would
enforce $\sigma_{m}=0$ or $\sigma_{M}=0$.

The authors of Li:2020eon ; Hongo:2021ona  have set $\sigma_{m}=0$ and treated
$\sigma_{M}$ as perturbatively small by using dynamical considerations; see
also Hongo:2022izs  for a perturbative computation of the latter in an
effective model for QCD. Thus, we expect their analysis to be valid only for
flows for which this approximation is valid. We have found that their
resulting constitutive relations and correlation functions match ours where a
comparison can be made. An earlier derivation of the constitutive relations
for the stress tensor and spin current can be found in Hattori:2019lfp . One
of the differences between the current work and that of Hattori:2019lfp  is
that the authors of Hattori:2019lfp  used the spin current itself as a degree
of freedom (replacing our $m^{\mu}$ and $M^{\mu\nu}$). If we only consider the
constitutive relations for the stress tensor then our expressions match theirs
once we remove the BR contributions.

One particular transport coefficient which is absent from the constitutive
relations described both in Li:2020eon ; Hongo:2021ona  and in Hattori:2019lfp
is the first order term $\chi_{1}^{(1)}$ which contributes to the spin current
in the form

$S^{\lambda\mu\nu}=\ldots+2\chi_{1}^{(1)}\Delta^{\lambda[\mu}u^{\nu]}$ (150)

and contributes to the stress tensor via a BR term, (171). One argument that
has been made for the vanishing of $\chi_{1}^{(1)}$ in Li:2020eon ;
Hongo:2021ona  is that such terms do not contribute to the fully antisymmetric
components of the spin current and that minimal coupling of fermions to
torsion generate a fully antisymmetric spin current. While true, it doesn’t
seem to us a necessary condition for setting $\chi_{1}^{(1)}=0$. Once the
theory is coupled to torsion, even perturbatively, all terms compatible with
the symmetry should be allowed and therefore $\chi_{1}^{(1)}$ may appear in
the constitutive relations. An extended discussion of the behavior of
$\chi_{1}^{(1)}$ under CPT transformations can be found towards the end of
this section.

The hydrostatic equations (53) (derived from (51)) imply that the order in
derivatives of the contorsion tensor is tied to the acceleration and vorticity
terms which are usually counted as first order in derivatives. Indeed, in this
work we have set the contorsion tensor (and therefore the spin chemical
potential) to be first order in derivatives. But other possibilities are
allowed. For instance, one could allow for $u^{\mu}K_{\mu}{}^{ab}$ to be first
order in derivatives, but other components of the contorsion tensor to be
zeorth order in derivatives. This would be similar to the derivative counting
of the electromagnetic field strength in magnetohydrodynamics
Hernandez:2017mch  where the magnetic field is counted as one order lower than
the electric field. In both the former, and the latter cases, the spin
chemical potential is counted as first order in derivatives.

In order to have a zeroth order in derivatives spin chemical potential one
would need the acceleration and vorticity to be zeroth order in derivatives
quantities. We point out that it is impossible to achieve hydrostatic
equilibrium (with real temperature and velocity field) in Minkowski space with
a non zero acceleration and vorticity. Indeed, in hydrostatic equilibrium the
acceleration and vorticity are given by appropriate derivatives of the
velocity field which in Minkowski space must take the form $(1,\vec{0})$.
Thus, in Minkwoski space, (and in the absence of torsion), the spin chemical
potential must vanish in equilbirium. Since unforced fluid dynamics will tend
to reach an equilibrium configuration, the spin chemcial potential will tend
to vanish over time. For this reason it is also physically sensible to count
the spin chemical potential as a first order in derivative quantity. This
observation may be contrasted with an analysis of the spin chemical potential
in some applications of heavy ion collisions (see, .e.g, DeGroot:1980dk ;
Becattini:2012tc ; Becattini:2013fla ; Florkowski:2017ruc  ) where a
background velocity (in four dimensions) of the form

$u_{\mu}dx^{\mu}=\gamma\left(-\Omega ydx+\Omega xdy\right)$ (151)

with $\gamma^{-1}=\sqrt{1-\Omega^{2}(x^{2}+y^{2})}$, is used, and leads to

$a_{\mu}dx^{\mu}=-\gamma^{2}\Omega^{2}(xdx+ydy)\qquad\Omega_{\mu\nu}dx^{\mu}\wedge
dx^{\nu}=-\gamma^{3}\Omega dx\wedge dy\,.$ (152)

This solution is not a hydrostatic solution since, in order to avoid complex
solutions, one would either need to insert a boundary or glue this solution to
a stationary one. Either way, (151) can, at best, be metastable and will
eventually dissipate.

To properly deal with solutions for which vorticity and acceleration can be
properly counted as zero order in derivative quantities one would need to
develop a formalism where arbitrary powers of vorticity and acceleration
appear at the “ideal” fluid level. (Linearly perturbing normal fluid dynamics
around the solution (151) is equivalent to an uncontrolled truncation of the
latter.) One approach to constructing a hydrostatic theory which incorporates
vorticity and acceleration as zero order quantities would be to place the
theory on a curved manifold where the magnetic components of the Riemann
curvature, c.f., Jensen:2013kka , are counted as zero order in derivative
quantities, similar to the procedure for magnetohydrodynamics mentioned above.
On top of that, one should then be able to add contorsion and then obtain a
zero order in derivative spin chemical potential. In Li:2020eon  elements of
such an analysis have been carried out.

| $T$ | $PT$ | $CPT$
---|---|---|---
$T$ | + | + | +
$\mu$ | + | + | -
$u^{0}$ | + | + | +
$u^{i}$ | - | - | +
$\mu^{0i}$ | + | - | -
$\mu^{ij}$ | - | - | -
$m^{0}$ | - | - | -
$m^{i}$ | + | - | -
$M^{0i}$ | + | - | -
$M^{ij}$ | - | - | -
$\theta$ | - | - | -
$a^{0}$ | - | - | -
$a^{i}$ | + | - | -
$\sigma^{00}$ | - | - | -
$\sigma^{0i}$ | + | - | -
$\sigma^{ij}$ | - | - | -
$\Omega^{0i}$ | + | - | -
$\Omega^{ij}$ | - | - | -
(a) CPT transformation properties of the hydrodynamic variables and some of
their derivatives.

| $T$ | $PT$ | $CPT$
---|---|---|---
$g^{00}$ | + | + | +
$g^{0i}$ | - | + | +
$g^{ij}$ | + | + | +
$K_{0}^{i0}$ | + | - | -
$K_{0}^{ij}$ | - | - | -
$K_{i}^{j0}$ | - | - | -
$K_{i}^{jk}$ | + | - | -
$A^{0}$ | + | + | -
$A^{i}$ | - | + | -
$\kappa$ | - | - | -
$K^{0i}$ | + | - | -
$K^{ij}$ | - | - | -
$\kappa_{A}^{0i}$ | + | - | -
$\kappa_{A}^{ij}$ | - | - | -
(b) Spurionic CPT transformation properties of sources and some sources
contracted with hydrodynamic variables.

Table 10:  CPT transformation properties of the hydrodynamic variables, some
of their derivatives, and spurionic transformation properties of the sources.

As we have emphasized time and again, the background torsion we turn on is
simply a means to identify a canonical spin current and stress tensor which
are useful in describing the dynamics of fluids with spin. At the end of the
day, if we are to describe a flat torsionless background we may set torsion to
zero. Once we do so, we expect to be able to use the standard hydrodynamic
description of the fluid at least for the temperature and velocity fields.
Satisfyingly, this is indeed the case. If one uses the improved stress tensor
$T_{\hbox{\tiny I}}^{\mu\nu}$ defined in (19) then energy momentum
conservation will reduce to the usual relativistic hydrodynamic equations of
motion for $T$ and $u^{\mu}$. In addition, one will find that the spin
chemical potential follows the acceleration and vorticity as in (149) (or its
modified version (125) in the presence of charge). Thus, in the absence of
torsion the standard rules of hydrodynamics apply and they are supplemented by
an additional algebraic constraint relating the spin chemical potential to the
other hydrodynamic variables. Similar conclusions were reported in Li:2020eon
.

Our result is also inline with the general perception that in the absence of
torsion, improvement terms (sometimes referred to as pseduo-gauge
transformations) should not affect physical observables. Indeed, in our
formalism, carrying out a BR type of transformation (in the absence of
torsion) will not modify the equations of motion of the fluid.

We end this section with a discussion of the CPT transformation properties of
the hydrodynamic variables $u^{\mu}$, $T$, $\mu^{ab}$ and $\mu$ (allowing for
a chemical potential) and the spurionic version of these transformations on
the sources $e^{a}{}_{\mu}$, $\omega_{\mu}^{ab}$ and $A_{\mu}$ (if a $U(1)$
charge current is present). In what follows, to keep our notation compact, we
will consider even dimensional theories where parity flips the sign of all
spatial directions. An analysis of CPT for odd dimensions is straightforward.

The vielbein is even under CPT (and so is the metric), the spin connection is
odd under CPT and the vector potential is also odd under it. To determine the
transformation properties of the hydrodynamic variables we may use the
hydrostatic relations (31). Since the timelike Killing vector $V^{\mu}$ is
invariant under CPT, we find that $T$ is even and $\mu^{ab}$ and $\mu$ are odd
under CPT. We have collected the CPT transformation rules of various
quantities under CPT in table 10.

One can use the CPT transformation properties in table 10 to constrain
properties of transport coefficients. For instance, a CPT even partition
function would imply that $\chi^{(1)}_{1}(\mu,T)$ depend on odd powers of
$\mu$. A full analysis of constraints on CPT transformation properties of
transport coefficients would require a Schwinger-Keldysh effective action type
of analysis, similar to the one carried out in, e.g., Haehl:2014zda ;
Crossley:2015evo ; Jensen:2018hse . We leave such a study for future work.

## Acknowledgements

We thank Francesco Becattini, Saso Grozdanov, Zohar Komargodski, Giorgio
Torrieri, Enrico Speranza, Toby Wiseman and Ho-Ung Yee. DG and UG are
partially supported by the Delta-Institute for Theoretical Physics (D-ITP)
funded by the Dutch Ministry of Education, Culture and Science (OCW). In
addition, DG is supported in part by CONACyT through the program Fomento,
Desarrollo y Vinculacion de Recursos Humanos de Alto Nivel. AY is supported in
part by an Israeli Science Foundation excellence center grant 2289/18 and a
Binational Science Foundation grant 2016324.

##  Appendix A A compendium of decompositions

In this work we have often decomposed various external sources and
hydrodynamic quantities with respect to the residual $SO(d-1)$ symmetry
associated with spatial rotations orthogonal to the velocity field. In this
appendix we have collected the decomposition of various quantities into
representations of the $SO(d-1)$ symmetry for ease of access. Our main tool
for constructing a decomposition is the projection

$\Delta^{\mu\nu}=g^{\mu\nu}+u^{\mu}u^{\nu}\,.$ (153)

We will also often use a symmetrized and antisymmetrized indices using round
or square brackets respectively,

$A_{[\mu\nu]}=\frac{1}{2}\left(A_{\mu\nu}-A_{\nu\mu}\right)\qquad
S_{(\mu\nu)}=\frac{1}{2}\left(S_{\mu\nu}+S_{\nu\mu}\right)\,.$ (154)

###  A.1 Hydrodynamic quantities

We often need the decomposition of the gradient of the velocity field

$\mathring{\nabla}_{\mu}u_{\nu}=\frac{1}{d-1}\theta\Delta_{\mu\nu}-u_{\mu}a_{\nu}+\sigma_{\mu\nu}+\Omega^{\mu\nu}\,,\\\
$ (155)

where

$\displaystyle\begin{split}\theta&=\mathring{\nabla}_{\mu}u^{\mu}\,,\\\
a^{\nu}&=u^{\mu}\mathring{\nabla}_{\mu}u^{\nu}\,,\\\
\Omega_{\mu\nu}&=\Delta^{\rho}_{\mu}\Delta^{\sigma}_{\nu}\mathring{\nabla}_{[\rho}u_{\sigma]}\,,\\\
\sigma^{\mu\nu}&=\Delta^{\mu\nu\rho\sigma}\mathring{\nabla}_{\rho}u_{\sigma}\,\end{split}$
(156)

with

$\Delta^{\mu\nu\rho\sigma}=\Delta^{\mu(\rho}\Delta^{\sigma)\nu}-\frac{1}{d-1}\Delta^{\mu\nu}\Delta^{\rho\sigma}\,.$
(157)

Similarly, the decomposition of the chemical potential is given by

$\mu^{ab}=u^{a}m^{b}-u^{b}m^{a}+M^{ab}$ (158)

so that

$\displaystyle\begin{split}m^{a}&=\mu^{ab}u_{b}\\\
M^{ab}&=\Delta^{a}{}_{c}\Delta^{b}{}_{d}\mu^{cd}\,.\end{split}$ (159)

###  A.2 The contorsion tensor

Similar to the hydrodynamic fields we can decompose the contorsion tensor into
components,

$\displaystyle K_{\mu}{}^{ab}=$ $\displaystyle-
u_{\mu}\left(u^{a}k^{b}-u^{b}k^{a}+K^{ab}\right)+\frac{2\kappa}{d-1}u^{[a}\Delta^{b]}{}_{\mu}+2u^{[a}\kappa_{S}{}^{b]}{}_{\mu}+2u^{[a}\kappa_{A}{}^{b]}{}_{\mu}$
(160)
$\displaystyle+\frac{1}{d-2}\Delta_{\mu}{}^{[a}\mathcal{K}_{V}{}^{b]}+\mathcal{K}_{T\,\mu}{}^{ab}+\mathcal{K}_{A}{}_{\mu}{}^{ab}\,,$
(161)

where

$\displaystyle\begin{split}k^{a}&=u^{\mu}u_{b}K_{\mu}{}^{ab}\,,\\\
K^{ab}&=\Delta^{a}{}_{c}\Delta^{b}{}_{d}u^{\mu}K_{\mu}{}^{cd}\,,\end{split}$
(162)

denote the decomposition of $u_{\mu}K^{\mu ab}$ into a vector and
antisymmetric tensor,

$\displaystyle\begin{split}\kappa&=e^{\mu}{}_{c}K_{\mu}{}^{cd}u_{d}\,,\\\
\kappa_{S}{}^{\mu\nu}&=\Delta^{\mu\nu\rho}_{\hphantom{\mu\nu\rho}c}K_{\rho}{}^{cd}u_{d}\,,\\\
\kappa_{A}{}^{\mu\nu}&=\left(\frac{\Delta^{\mu}{}_{c}\Delta^{\rho\nu}-\Delta^{\nu}{}_{c}\Delta^{\mu\rho}}{2}\right)K_{\rho}{}^{cd}u_{d}\,,\\\
\end{split}$ (163)

denote the decomposition of $\Delta^{\nu\mu}K_{\mu}{}^{ab}u_{b}$ into a
scalar, symmetric traceless tensor and antisymmetric tensor, and

$\displaystyle\begin{split}\mathcal{K}_{V}{}^{\mu}&=2\Delta^{\mu}{}_{d}\Delta^{\rho}{}_{c}K_{\rho}{}^{cd}\,,\\\
\mathcal{K}_{A}{}^{\lambda\rho\sigma}&=\Delta^{[\rho}{}_{c}\Delta^{\sigma}{}_{d}\Delta^{\lambda]\alpha}K_{\alpha}{}^{cd}\\\
\mathcal{K}_{T}{}_{\mu}{}^{\alpha\beta}&=\Delta^{\alpha}{}_{c}\Delta^{d}{}_{\beta}\Delta^{\nu}{}_{\mu}K_{\nu}{}^{cd}-\frac{1}{d-2}\Delta^{[\alpha}_{\mu}\mathcal{K}_{V}{}^{\beta]}-\mathcal{K}_{A}{}_{\mu}{}^{\alpha\beta}\end{split}$
(164)

denote the decomposition of
$\Delta^{\nu\mu}\Delta_{a}{}^{\alpha}\Delta_{b}{}^{\beta}K_{\mu}{}^{ab}$ into
a vector, an antisymmetric tensor, and a mixed traceless tensor,

$\mathcal{K}_{T}{}_{\mu}{}^{\mu\sigma}=0\,,\qquad\mathcal{K}_{T}{}^{[\rho\sigma\lambda]}=0\,,\qquad\mathcal{K}_{A}{}^{\rho\sigma\lambda}=\mathcal{K}_{A}{}^{[\rho\sigma\lambda]}\,.$
(165)

###  A.3 Miscellaneous quantities and definitions

Apart from the above decompositions, we often use the following quantities
which vanish in hydrostatic equilibrium,

$\displaystyle\begin{split}A^{\mu}&=\Delta^{\mu\nu}\mathring{\nabla}_{\nu}T+Ta^{\mu}\\\
\hat{m}^{\mu}&=m^{\mu}-k^{\mu}-a^{\mu}\\\
\hat{M}^{\mu\nu}&=M^{\mu\nu}-K^{\mu\nu}+\Omega^{\mu\nu}\end{split}$ (166)

The first entry was taken from table 2 and the last two entries were taken
from (54).

Also, we decompose the external field strength, $F^{\mu\nu}$, into a magnetic
component, $B_{\mu\nu}$ and an electric component, $E_{\mu}$,

$E^{\mu}=F^{\mu\nu}u_{\nu}\qquad
B_{\mu\nu}=\Delta_{\mu\rho}\Delta_{\nu\sigma}F^{\rho\sigma}\,.$ (167)

Finally, when considering conformal theories, we use a Weyl covariant
connection similar to that discussed in Loganayagam:2008is ,

$\mathcal{A}_{\mu}=a_{\mu}-\frac{\theta}{d-1}u_{\mu}\,.$ (168)

(See (138).)

###  A.4 Scalars, vectors and tensors

Often, we need to classify various scalar, vector and tensor structures. We
have collected these in the tables below.

Order two hydrostatic scalars
---
$S_{(1)}=M_{\mu\nu}\kappa_{A}^{\mu\nu}$ | $S_{(2)}=M_{\mu\nu}K^{\mu\nu}$ | $S_{(3)}=m_{\mu}k^{\mu}$ | $S_{(4)}=m_{\mu}\mathcal{K}_{V}^{\mu}$ | $S_{(5)}=\kappa^{2}$
$S_{(6)}=k\cdot k$ | $S_{(7)}=K_{\mu\nu}K^{\mu\nu}$ | $S_{(8)}=k\cdot\mathcal{K}_{V}$ | $S_{(9)}=\mathcal{K}_{V}\cdot\mathcal{K}_{V}$ | $S_{(10)}=K_{\mu\nu}\kappa_{A}{}^{\mu\nu}$
$S_{(11)}=\kappa_{A\,\mu\nu}\kappa_{A}{}^{\mu\nu}$ | $S_{(12)}=\kappa_{S\,\mu\nu}\kappa_{S}{}^{\mu\nu}$ | $S_{(13)}=\mathcal{K}_{A\,\mu\nu\rho}\mathcal{K}_{A}{}^{\mu\nu\rho}$ | $S_{(14)}=\mathcal{K}_{T\,\mu\nu\rho}\mathcal{K}_{T}{}^{\mu\nu\rho}$ |
Table 11: Summary of second order independent inequivalent hydrostatic scalars
which may contribute to the equations of motion up to second order in
derivatives. This information can also be found in table 1. Second order Weyl
covariant scalars
---
$S_{(1)}^{c}=S_{(1)}$, $S_{(2)}^{c}=S_{(2)}$,
$S_{(3)}^{c}=(1-q)((1-q)k+qm)^{2}$,
$S_{(4)}^{c}=((1-q)k+qm)\cdot(\mathcal{K}_{V}-2(d-2)k)$
$S_{(5)}^{c}=\kappa^{2}$, $S_{(6)}^{c}=S_{(7)}$,
$S_{(7)}^{c}=(\mathcal{K}_{V}-2(d-2)k)^{2}$, $S_{(8)}^{c}=S_{(10)}$,
$S_{(9)}^{c}=S_{(11)}$, $S_{(10)}^{c}=S_{(12)}$, $S_{(11)}^{c}=S_{(13)}$,
$S_{(12)}^{c}=S_{(14)}$
Table 12: Summary of second order Weyl covariant hydrostatic scalars which may
contribute to the equations of motion up to second order in derivatives. This
information can also be found in table 9.  Order two hydrostatic scalars in
the presence of charge
---
$\tilde{S}_{(1)}=E_{\mu}m^{\mu}$ | $\tilde{S}_{(2)}=E_{\mu}k^{\mu}$ | $\tilde{S}_{(3)}=E_{\mu}\left(\mathcal{K}_{V}\right)^{\mu}$
$\tilde{S}_{(4)}=B_{\mu\nu}M^{\mu\nu}$ | $\tilde{S}_{(5)}=B_{\mu\nu}K^{\mu\nu}$ | $\tilde{S}_{(6)}=B_{\mu\nu}\kappa_{A}{}^{\mu\nu}$
Table 13: Second order hydrostatic scalars which are associated with a $U(1)$
charge which may contribute to the equations of motion up to second order in
derivatives. This information can also be found in table 5.  First order non
hydrostatic independent scalars
---
$\theta$ |  |
First order non hydrostatic independent vectors
$T\mathring{\nabla}_{\mu}\left(\frac{\mu}{T}\right)-E_{\mu}$ |  |
First order non hydrostatic independent symmetric tensors
$\sigma^{\mu\nu}$ |  |
Table 14: Independent non hydrostatic data. The entries of this table are
collected from table 3 and the discussion around table 6.

##  Appendix B The constitutive relations

For ease of reference, we have collected the full set of constitutive
relations for a fluid with a spin current. We will first present our most
general result for the constitutive relations of the stress tensor and spin
current in a curved torsionfull background relevant when expanding the
equations of motion to second order in derivatives and keeping terms linear in
torsion. We will then discuss various modifications of this result when we go
to a flat spacetime and set the torsion to zero, take the conformal limit or
consider the effect of charge. At appropriate places we refer to the main text
where the computations have been carried out.

The constitutive relations on a curved torsionfull background are given by

$\displaystyle\begin{split}T^{\mu\nu}&=T_{id}^{\mu\nu}+T_{h+}^{\mu\nu}+T_{nh}^{\mu\nu}\\\
S^{\lambda\mu\nu}&=S_{id}^{\lambda\mu\nu}+S_{h}^{\lambda\mu\nu}+S_{nh}^{\lambda\mu\nu}\,.\end{split}$
(169)

Here, the subscript $id$ refers to the ideal components of the fluid,
generated from the pressure term in the generating function. The subscripts
$h$ and $h+$ refer to components coming from higher order contributions to the
hydrostatic generating function, with the $+$ denoting the BR completion of
the hydrostatic terms as discussed at length at the beginning of section 4.
The subscript $nh$ refers to non hydrostatic terms. We further decompose these
two terms into BR terms and non BR terms,

$\displaystyle\begin{split}T_{h+}^{\mu\nu}&=T_{h+BR}^{\mu\nu}+T_{h\,nBR}^{\mu\nu}\\\
T_{nh}^{\mu\nu}&=T_{nh\,BR}^{\mu\nu}+T_{nh\,nBR}^{\mu\nu}\\\
S_{h}^{\lambda\mu\nu}&=S_{h\,BR}^{\lambda\mu\nu}+S_{h\,nBR}^{\lambda\mu\nu}\\\
S_{nh}^{\lambda\mu\nu}&=S_{nh\,BR}^{\lambda\mu\nu}+S_{nh\,nBR}^{\lambda\mu\nu}\,,\end{split}$
(170)

with

$\displaystyle\begin{split}T_{h+BR}^{\mu\nu}&=\frac{1}{2}\mathring{\nabla}_{\lambda}\left(S_{h\,BR}^{\lambda\mu\nu}-S_{h\,BR}^{\mu\lambda\nu}-S_{h\,BR}^{\nu\lambda\mu}\right)\\\
T_{nh\,BR}^{\mu\nu}&=\frac{1}{2}\mathring{\nabla}_{\lambda}\left(S_{nh\,BR}^{\lambda\mu\nu}-S_{nh\,BR}^{\mu\lambda\nu}-S_{nh\,BR}^{\nu\lambda\mu}\right)\,.\end{split}$
(171)

###  B.1 Generic, uncharged fluid in a torsionfull, curved background.

The explicit form of the constitutive relations for the ideal components is
given by

$\displaystyle\begin{split}T_{id}^{\mu\nu}=&\left(\epsilon_{0}+(\rho_{m}+T\rho_{m}^{\prime})m_{\alpha}m^{\alpha}+(\rho_{M}+T\rho_{M}^{\prime})M_{\alpha\beta}M^{\alpha\beta}\right)u^{\mu}u^{\nu}\\\
&+(P_{0}+\rho_{m}m_{\alpha}m^{\alpha}+\rho_{M}M^{\alpha\beta}M_{\alpha\beta})\Delta^{\mu\nu}\\\
&+u^{\mu}m_{\alpha}M^{\alpha\nu}\left(2\rho_{m}-4\rho_{M}\right)+\mathcal{O}(\nabla^{4})\\\
S_{id}^{\lambda\mu\nu}=&u^{\lambda}\left(4\rho_{m}m^{[\mu}u^{\nu]}-4\rho_{M}M^{\mu\nu}\right)+\mathcal{O}(\nabla^{4})\,,\end{split}$
(172)

where

$\epsilon_{0}=T\frac{\partial P_{0}}{\partial T}-P_{0}\,.$ (173)

(Taken from (60) and (61).) Here $\mathcal{O}(\nabla^{4})$ denotes terms which
are fourth order in derivatives but do not contain any explicit derivatives of
the hydrodynamic fields.

The explicit form of the hydrostatic components of the stress tensor and
current are given by

$\displaystyle\begin{split}S_{h\,BR}{}^{\lambda\mu\nu}=&2\chi^{(1)}_{1}\Delta^{\lambda[\mu}u^{\nu]}\\\
&-2\chi_{1}^{(2)}M^{\lambda[\mu}u^{\nu]}+2\chi_{2}^{(2)}u^{\lambda}M^{\mu\nu}-2\chi_{3}^{(2)}u^{\lambda}u^{[\mu}m^{\nu]}+4\chi_{4}^{(2)}\Delta^{\lambda[\mu}m^{\nu]}\\\
&+4\chi_{5}^{(2)}\kappa\Delta^{\lambda[\mu}u^{\nu]}+2u^{\lambda}\left(2\chi^{(2)}_{6}k^{[\mu}u^{\nu]}+2\chi^{(2)}_{7}K^{\mu\nu}+\chi^{(2)}_{10}\kappa_{A}{}^{\mu\nu}\right)\\\
&+2u^{\lambda}\chi^{(2)}_{8}\mathcal{K}_{V}{}^{[\mu}u^{\nu]}+4\Delta^{\lambda[\mu}\left(\chi^{(2)}_{8}k^{\nu]}+2\chi^{(2)}_{9}\mathcal{K}_{V}{}^{\nu]}\right)-2\chi^{(2)}_{10}K^{\lambda[\mu}u^{\nu]}\\\
&-4\chi^{(2)}_{11}\kappa_{A}{}^{\lambda[\mu}u^{\nu]}+4\chi^{(2)}_{12}\kappa_{S}{}^{\lambda[\mu}u^{\nu]}\
+4\chi^{(2)}_{13}\mathcal{K}_{A}{}^{\lambda\mu\nu}+4\chi^{(2)}_{14}\mathcal{K}_{T}{}^{\mu\nu\lambda}\\\
S_{h\,nBR}{}^{\lambda\mu\nu}=&+2u^{\lambda}\left(\chi^{(2)}_{1}\kappa_{A}{}^{\mu\nu}+\chi_{2}^{(2)}K^{\mu\nu}-\chi^{(2)}_{3}u^{[\mu}k^{\nu]}\right)-2\chi_{4}^{(2)}u^{\lambda}u^{[\mu}\mathcal{K}_{V}{}^{\nu]}+\mathcal{O}(\nabla^{2})\\\
\end{split}$ (174a) and
$\displaystyle\begin{split}T_{h\,nBR}^{\mu\nu}=&\left(T\frac{\partial\chi^{(1)}_{1}}{\partial
T}-\chi^{(1)}_{1}\right)\kappa
u^{\mu}u^{\nu}+\chi^{(1)}_{1}\frac{d-2}{d-1}\kappa\Delta^{\mu\nu}\\\
&+2\chi^{(1)}_{1}u^{(\mu}k^{\nu)}+\frac{1}{2}\chi^{(1)}_{1}u^{\mu}\mathcal{K}_{V}{}^{\nu}-\chi^{(1)}_{1}\kappa_{S}{}^{\mu\nu}-\chi^{(1)}_{1}\kappa_{A}{}^{\mu\nu}\\\
&+\chi_{1}^{(2)}\left(\mathcal{K}_{T}{}^{\alpha\beta[\mu}+\mathcal{K}_{A}{}^{\alpha\beta[\mu}\right)u^{\nu]}M_{\alpha\beta}+\frac{\chi^{(2)}_{1}\kappa
M^{\mu\nu}}{d-1}\\\
&+\frac{\chi_{1}^{(2)}}{2(d-2)}u^{[\mu}M^{\nu]\beta}\mathcal{K}_{V\,\beta}-2\chi^{(2)}_{1}u^{[\mu}\kappa_{A}{}^{\nu]\beta}m_{\beta}+\chi^{(2)}_{1}M^{\alpha[\mu}\left(\kappa_{A}{}^{\nu]}{}_{\alpha}-\kappa_{S}{}^{\nu]}{}_{\alpha}\right)\\\
&-2\chi^{(2)}_{2}u^{[\mu}\left(M^{\nu]\alpha}k_{\alpha}+K^{\nu]\alpha}m_{\alpha}\right)-\chi^{(2)}_{3}u^{[\mu}\left(M^{\nu]\alpha}k_{\alpha}+K^{\nu]\alpha}m_{\alpha}\right)\\\
&+2\chi^{(2)}_{4}\mathcal{K}_{A}{}^{\mu\nu\alpha}m_{\alpha}+2\chi^{(2)}_{4}\mathcal{K}_{T}{}^{[\mu\nu]\alpha}m_{\alpha}+\frac{\chi^{(2)}_{4}}{(d-2)}\mathcal{K}_{V}^{[\mu}m^{\nu]}\\\
&-\chi^{(2)}_{4}u^{[\mu}M^{\nu]\alpha}\mathcal{K}_{V\,\alpha}+2\chi^{(2)}_{4}u^{[\mu}\left(\kappa_{A}{}^{\nu]\alpha}-\kappa_{S}{}^{\nu]\alpha}\right)m_{\alpha}+\frac{2\chi^{(2)}_{4}(d-2)}{(d-1)}\kappa
u^{[\mu}m^{\nu]}\,.\end{split}$ (174b)

The non hydrostatic components of the stress tensor and current are given by

$\displaystyle\begin{split}S_{nh\,nBR}^{\lambda\mu\nu}&=0\,,\\\
S_{nh\,BR}^{\lambda\mu\nu}&=2\sigma_{1}\sigma^{\lambda[\mu}u^{\nu]}+2\sigma_{2}\theta\Delta^{\lambda[\mu}u^{\nu]}\,,\\\
T_{nh\,nBR}^{\mu\nu}&=-\zeta\theta\Delta^{\mu\nu}-\eta\sigma^{\mu\nu}+\sigma_{A}A^{[\mu}u^{\nu]}+\sigma_{m}\hat{m}^{[\mu}u^{\nu]}+\sigma_{M}\hat{M}^{\mu\nu}\,.\end{split}$
(175)

(Taken from (78).)

###  B.2 Conformal uncharged fluid in a torsionfull, curved background.

The explicit form of the constitutive relations for the ideal components of an
uncharged conformal fluid is given by

$\displaystyle\begin{split}T_{id}^{\mu\nu}=&\left(p_{0}(d-1)T^{d}+\delta_{q\,1}r_{m}(d-1)T^{d-2}m_{\alpha}m^{\alpha}+r_{M}(d-1)T^{d-2}M_{\alpha\beta}M^{\alpha\beta}\right)u^{\mu}u^{\nu}\\\
&+(p_{0}T^{d}+\delta_{q\,1}r_{m}T^{d-2}m_{\alpha}m^{\alpha}+r_{M}T^{d-2}M^{\alpha\beta}M_{\alpha\beta})\Delta^{\mu\nu}\\\
&+u^{\mu}m_{\alpha}M^{\alpha\nu}T^{d-2}\left(2\delta_{q\,1}r_{m}-4r_{M}\right)+\mathcal{O}(\nabla^{4})\\\
S_{id}^{\lambda\mu\nu}=&u^{\lambda}T^{d-2}\left(4\delta_{q\,1}r_{m}m^{[\mu}u^{\nu]}-4r_{M}M^{\mu\nu}\right)+\mathcal{O}(\nabla^{4})\,,\end{split}$
(176)

where $p_{0}$, $r_{m}$ and $r_{M}$ are real constants. (Taken by inserting
(144) into (60).)

The explicit form of the hydrostatic components of the stress tensor and
current are given by

$\displaystyle\begin{split}T^{-(d-2)}S_{h\,BR}{}^{\lambda\mu\nu}=&2x^{(1)}_{1}T\delta_{q\,0}\Delta^{\lambda[\mu}u^{\nu]}\\\
&-2x_{1}^{(2)}M^{\lambda[\mu}u^{\nu]}+2x_{2}^{(2)}u^{\lambda}M^{\mu\nu}-4(1-q)^{2}x_{3}^{(2)}u^{\lambda}(qu^{[\mu}u^{\nu]}+(q-1)k^{[\mu}u^{\nu]})\\\
&+x_{4}^{(2)}\Bigg{(}u^{\lambda}\left(4(d-2)qu^{[\mu}m^{\nu]}+8(d-2)(q-1)k^{[\mu}u^{\nu]}-2(q-1)K_{V}^{[\mu}u^{\nu]}\right)\\\
&+4(1-q)\Delta^{\lambda[\mu}k^{\nu]}+4q\Delta^{\lambda[\mu}m^{\nu]}\Bigg{)}\\\
&+4x_{5}^{(2)}\kappa\Delta^{\lambda[\mu}u^{\nu]}+4x_{6}^{(2)}u^{\lambda}K^{\mu\nu}\\\
&+x_{7}^{(2)}\left(u^{\lambda}\left(16(d-2)^{2}k^{[\mu}u^{\nu]}-8(d-2)K_{V}^{\mu}u^{\nu]}\right)+8\Delta^{\lambda[\mu}K_{V}^{\nu]}-16(d-2)\Delta^{\lambda[\mu}k^{\nu]}\right)\\\
&+2x^{(2)}_{8}\left(u^{\lambda}\kappa_{A}^{\mu\nu}-K^{\lambda[\mu}u^{\nu]}\right)-4x^{(2)}_{9}\kappa_{A}^{\lambda[\mu}u^{\nu]}+4x^{(2)}_{10}\kappa_{S}^{\lambda[\mu}u^{\nu]}+4x^{(2)}_{11}K_{A}^{\lambda\mu\nu}+4x^{(2)}_{12}K_{T}^{\mu\nu\lambda}\\\
T^{-(d-2)}S_{h\,nBR}{}^{\lambda\mu\nu}=&2x^{(2)}_{1}u^{\lambda}\kappa_{A}{}^{\mu\nu}+2x_{2}^{(2)}u^{\lambda}K^{\mu\nu}+4q(1-q)x^{(2)}_{3}u^{\lambda}\left((q-1)u^{[\mu}k^{\nu]}+qm^{[\mu}u^{\nu]}\right)\\\
&+2qx^{(2)}_{4}u^{\lambda}\left(2(d-2)u^{[\mu}k^{\nu]}-u^{[\mu}K_{V}^{\nu]}\right)\end{split}$
(177)

and

$\displaystyle\begin{split}T^{-(d-2)}T_{h\,nBR}^{\mu\nu}=&(d-2)Tx^{(1)}_{1}\kappa
u^{\mu}u^{\nu}+Tx^{(1)}_{1}\frac{d-2}{d-1}\kappa\Delta^{\mu\nu}+2Tx^{(1)}_{1}u^{(\mu}k^{\nu)}\\\
&+\frac{1}{2}Tx^{(1)}_{1}u^{\mu}\mathcal{K}_{V}{}^{\nu}-Tx^{(1)}_{1}\kappa_{S}{}^{\mu\nu}-Tx^{(1)}_{1}\kappa_{A}{}^{\mu\nu}\\\
&+x^{(2)}_{1}\Bigg{(}\frac{\kappa
M^{\mu\nu}}{d-1}+\frac{1}{2(d-2)}u^{[\mu}M^{\nu]}{}_{\beta}K_{V}^{\beta}+\left(K_{A}^{\alpha\beta[\mu}u^{\nu]}+K_{T}^{\alpha\beta[\mu}u^{\nu]}\right)M_{\alpha\beta}\\\
&-u^{[\mu}\kappa_{A}^{\nu]\beta}m_{\beta}+M_{\alpha}{}^{[\mu}\kappa_{A}^{\nu]}{}_{\alpha}-M_{\alpha}{}^{[\mu}\kappa_{S}^{\nu]}{}_{\alpha}\Bigg{)}-2x^{(2)}_{2}\left(u^{[\mu}K^{\nu]\alpha}m_{\alpha}+u^{[\mu}M^{\nu]\alpha}k_{\alpha}\right)\\\
&+(1-q)x^{(2)}_{3}\Bigg{(}2(q-1)u^{[\mu}K^{\nu]\alpha}m_{\alpha}-2qu^{[\mu}M^{\nu]}{}_{\alpha}m^{\alpha}+2(q-1)u^{[\mu}M^{\nu]}{}_{\alpha}k^{\alpha}\Big{)}\\\
&+qx^{(2)}_{4}\Bigg{(}2K_{A}^{[\mu\nu]\alpha}m_{\alpha}+2K_{T}^{[\mu\nu]\alpha}m_{\alpha}+\frac{K_{V}^{[\mu}m^{\nu]}}{d-2}+2(d-2)u^{[\mu}K^{\nu]\alpha}m_{\alpha}\\\
&+\frac{2(d-2)}{d-1}\kappa
u^{[\mu}m^{\nu]}+2(d-2)u^{[\mu}M^{\nu]\alpha}k_{\alpha}-u^{[\mu}M^{\nu]\alpha}K_{V\,\alpha}+u^{[\mu}\kappa_{A}^{\nu]\alpha}m_{\alpha}-u^{[\mu}\kappa_{S}^{\nu]\alpha}m_{\alpha}\Bigg{)}\,,\end{split}$
(178)

where the $\chi^{(i)}_{j}$’s are real constants. (Taken by inserting (144)
into (65).) The non hydrostatic components of the stress tensor and spin
current are given by

$\displaystyle\begin{split}S_{nh\,nBR}^{\lambda\mu\nu}=&0\,,\\\
S_{nh\,BR}^{\lambda\mu\nu}=&4x_{5}^{(2)}q\theta\Delta^{\lambda[\mu}u^{\nu]}+2T^{d-2}s_{1}\sigma^{\lambda[\mu}u^{\nu]}\,,\\\
T_{nh\,nBR}^{\mu\nu}=&-hT^{d-1}\sigma^{\mu\nu}+s_{A}T^{d-2}A^{[\mu}u^{\nu]}+s_{m}T^{d-2}\hat{m}^{[\mu}u^{\nu]}+s_{M}T^{d-2}\hat{M}^{\mu\nu}\\\
&+qT^{d-2}s_{1}\left(\frac{\theta\sigma^{\mu\nu}}{d-1}-\sigma^{\mu\rho}a_{\rho}u^{\nu}\right)+T^{d-1}qx^{(1)}_{1}\theta((d-1)u^{\mu}u^{\nu}+\Delta^{\mu\nu})\\\
&+\frac{1}{2}x^{(2)}_{5}T^{d-2}q(\mathcal{K}_{V}^{\nu}u^{\mu}-\mathcal{K}_{V}^{\mu}u^{\nu}-4\kappa_{A}^{\mu\nu})\theta\end{split}$
(179)

where $s_{1}$, $s_{A}$, $s_{m}$ and $s_{M}$ are real constants. (Taken from
(145), (146) and by using

$\frac{1}{2}\mathring{\mathcal{D}}_{\lambda}\left(S^{\lambda\mu\nu}-S^{\mu\lambda\nu}-S^{\nu\lambda\mu}\right)=\frac{1}{2}\mathring{\nabla}_{\lambda}\left(S^{\lambda\mu\nu}-S^{\mu\lambda\nu}-S^{\nu\lambda\mu}\right)+S^{\mu\nu\rho}\mathcal{A}_{\rho}-S_{\rho}{}^{\rho\mu}\mathcal{A}^{\nu}$
(180)

in implementing the prescription of (141) as discussed below (146) .) Note
that the last term on the right hand side of $T^{\mu\nu}_{nh\,nBR}$ does not
satisfy the Landau frame condition and may be removed by a frame
transformation. Such a frame transformation will induce a non standard scaling
relation for the hydrodynamic variables which will ensure conformal
invariance.

###  B.3 Generic charged fluid in a torsionless background.

In a torsionless background many of the previous constitutive relations
simplify. In addition, there are additional contributions to the ideal and non
ideal constitutive relations coming from the background electromagnetic field
and the charge chemical potential. The explicit form of the constitutive
relations for the ideal components is given by

$\displaystyle\begin{split}T_{id}^{\mu\nu}=&\left(\epsilon_{0}+\left(\rho_{m}+T\frac{\partial\rho_{m}}{\partial
T}+\mu\frac{\partial\rho_{m}}{\partial\mu}\right)m_{\alpha}m^{\alpha}+\left(\rho_{M}+T\frac{\partial\rho_{M}}{\partial
T}+\mu\frac{\partial\rho_{M}}{\partial\mu}\right)M_{\alpha\beta}M^{\alpha\beta}\right)u^{\mu}u^{\nu}\\\
&+(P_{0}+\rho_{m}m_{\alpha}m^{\alpha}+\rho_{M}M^{\alpha\beta}M_{\alpha\beta})\Delta^{\mu\nu}\\\
&+u^{\mu}m_{\alpha}M^{\alpha\nu}\left(2\rho_{m}-4\rho_{M}\right)+\mathcal{O}(\nabla^{4})\\\
S_{id}^{\lambda\mu\nu}=&u^{\lambda}\left(4\rho_{m}m^{[\mu}u^{\nu]}-4\rho_{M}M^{\mu\nu}\right)+\mathcal{O}(\nabla^{4})\,,\end{split}$
(181)

where now

$\epsilon_{0}=T\frac{\partial P_{0}}{\partial T}+\mu\frac{\partial
P_{0}}{\partial\mu}-P_{0}\,.$ (182)

(Obtained by varying the pressure term in the generating function given in
(124).)

The explicit form of the hydrostatic components of the stress tensor and
current are given by

$\displaystyle\begin{split}S_{h\,BR}{}^{\lambda\mu\nu}=&2\chi^{(1)}_{1}\Delta^{\lambda[\mu}u^{\nu]}\\\
&-2\chi_{1}^{(2)}M^{\lambda[\mu}u^{\nu]}+2\chi_{2}^{(2)}u^{\lambda}M^{\mu\nu}-2\chi_{3}^{(2)}u^{\lambda}u^{[\mu}m^{\nu]}+4\chi_{4}^{(2)}\Delta^{\lambda[\mu}m^{\nu]}\\\
&-2\tilde{\chi}^{(2)}_{2}u^{\lambda}u^{[\mu}E^{\nu]}+4\tilde{\chi}^{(2)}_{3}\Delta^{\lambda[\mu}E^{\nu]}+2\tilde{\chi}^{(2)}_{5}u^{\lambda}B^{\mu\nu}-2\tilde{\chi}^{(2)}_{6}B^{\lambda[\mu}u^{\nu]}\\\
S_{h\,nBR}{}^{\lambda\mu\nu}=&2u^{\lambda}\left(-2\tilde{\chi}^{(2)}_{1}u^{[\mu}E^{\nu]}+\tilde{\chi}^{(2)}_{4}B^{\mu\nu}\right)\\\
\end{split}$ (183)

and

$\displaystyle\begin{split}T_{h\,nBR}^{\mu\nu}=&0\,.\end{split}$ (184)

(Obtained by varying the non pressure terms in the generating function given
in (124).) The non hydrostatic components of the stress tensor and current are
given by

$\displaystyle\begin{split}S_{nh\,nBR}^{\lambda\mu\nu}&=0\,,\\\
S_{nh\,BR}^{\lambda\mu\nu}&=2\sigma_{1}\sigma^{\lambda[\mu}u^{\nu]}+2\sigma_{2}\theta\Delta^{\lambda[\mu}u^{\nu]}+2\sigma_{3}u^{\lambda}u^{[\mu}\Delta^{\nu]\rho}\left(\mathring{\nabla}_{\rho}\left(\frac{\mu}{T}\right)-E_{\rho}\right)\,,\\\
T_{nh\,nBR}^{\mu\nu}&=-\zeta\theta\Delta^{\mu\nu}-\eta\sigma^{\mu\nu}+\sigma_{m}\left(\hat{m}^{[\mu}u^{\nu]}+\tilde{\sigma}_{A}A^{[\mu}u^{\nu]}-\tilde{\sigma}_{e}u^{[\mu}\Delta^{\nu]\rho}\left(\mathring{\nabla}_{\rho}\left(\frac{\mu}{T}\right)-E_{\rho}\right)\right)+\sigma_{M}\hat{M}^{\mu\nu}\,.\end{split}$
(185)

(Obtained using the entries in table 6.) In writing the last equality we have
replaced what we referred to in the uncharged case as $\sigma_{A}$ with
$\sigma_{m}\tilde{\sigma_{A}}$ since the resulting equations of motion take a
simpler form. Recall that if $\sigma_{m}=0$ we are in the dynamical spin limit
and the constitutive relations change drastically. See Gallegos:2021bzp  or
section 8 for an extended discussion.

In addition to the energy momentum tensor and spin current we must also
specify the constitutive relations for the charge current, $J^{\mu}$. Using
(120) and the last entry in table 6 we find

$J^{\mu}=\frac{\partial
P_{0}}{\partial\mu}u^{\mu}+\sigma_{E}\Delta^{\mu\rho}\left(T\mathring{\nabla}_{\rho}\left(\frac{\mu}{T}\right)-E_{\rho}\right)$
(186)

where we are working in a (Landau) frame where $J^{\mu}u_{\mu}=-\frac{\partial
P_{0}}{\partial\mu}$.

###  B.4 Conformal charged fluids in a torsionless background

The constitutive relations for a conformal charged fluid (with $q=0$) in a
flat torsionless background are of the form

$\displaystyle\begin{split}T_{id}^{\mu\nu}=&\left(p_{0}(d-1)T^{d}+r_{M}(d-1)T^{d-2}M_{\alpha\beta}M^{\alpha\beta}\right)u^{\mu}u^{\nu}\\\
&+(p_{0}T^{d}+r_{M}T^{d-2}M^{\alpha\beta}M_{\alpha\beta})\Delta^{\mu\nu}-4u^{\mu}m_{\alpha}M^{\alpha\nu}T^{d-2}r_{M}+\mathcal{O}(\nabla^{4})\\\
S_{id}^{\lambda\mu\nu}=&-4u^{\lambda}T^{d-2}r_{M}M^{\mu\nu}+\mathcal{O}(\nabla^{4})\,,\end{split}$
(187)

where now $p_{0}$, $r_{m}$, and $r_{M}$ are functions of $mu/T$. (This
relation can be obtained from (181) together with the constraint coming from
the requirement that the $m_{\alpha}m^{\alpha}$ term be conformal invariant
and that $P_{0}=T^{d}p_{0}(\mu/T)$and $\rho_{M}=T^{d-2}r_{m}(\mu/T)$.) The
explicit form of the hydrostatic components of the stress tensor and current
are given by

$\displaystyle\begin{split}T^{-(d-2)}S_{h\,BR}{}^{\lambda\mu\nu}=&2x^{(1)}_{1}T\Delta^{\lambda[\mu}u^{\nu]}-2x_{1}^{(2)}M^{\lambda[\mu}u^{\nu]}+2x_{2}^{(2)}u^{\lambda}M^{\mu\nu}\\\
&-2\tilde{x}^{(2)}_{2}u^{\lambda}u^{[\mu}E^{\nu]}+4\tilde{x}^{(2)}_{3}\Delta^{\lambda[\mu}E^{\nu]}+2\tilde{x}^{(2)}_{5}u^{\lambda}B^{\mu\nu}-2\tilde{x}^{(2)}_{6}B^{\lambda[\mu}u^{\nu]}\\\
T^{-(d-2)}S_{h\,nBR}{}^{\lambda\mu\nu}=&2u^{\lambda}\left(-2\tilde{x}^{(2)}_{1}u^{[\mu}E^{\nu]}+\tilde{x}^{(2)}_{4}B^{\mu\nu}\right)\end{split}$
(188)

and

$\displaystyle\begin{split}T_{h\,nBR}^{\mu\nu}=&0\,.\end{split}$ (189)

(Obtained by removing non conformal terms from (183) (see (147)) and setting
$\chi^{(1)}_{1}=T^{d-1}x_{1}^{(1)}(\mu/T)$ and
$\chi^{(2)}_{i}=T^{d-1}x_{i}^{(2)}(\mu/T)$.) The non hydrostatic components of
the stress tensor and spin current are given by

$\displaystyle\begin{split}S_{nh\,nBR}^{\lambda\mu\nu}=&0\,,\\\
S_{nh\,BR}^{\lambda\mu\nu}=&2T^{d-2}s_{1}\sigma^{\lambda[\mu}u^{\nu]}+2T^{d-2}s_{3}u^{\lambda}u^{[\mu}\Delta^{\nu]\rho}\left(T\mathring{\nabla}_{\rho}\left(\frac{\mu}{T}\right)-E_{\rho}\right)\,,\\\
T_{nh\,nBR}^{\mu\nu}=&-hT^{d-1}\sigma^{\mu\nu}+s_{m}T^{d-2}\left(\hat{m}^{[\mu}u^{\nu]}+\tilde{s}_{A}A^{[\mu}u^{\nu]}-\tilde{s}_{e}u^{[\mu}\Delta^{\nu]\rho}\left(\mathring{\nabla}_{\rho}\left(\frac{\mu}{T}\right)-E_{\rho}\right)\right)+s_{M}T^{d-2}\hat{M}^{\mu\nu}\,.\end{split}$
(190)

(Obtained by removing non conformal terms from (185) ($\sigma_{2}=0$ and
$\zeta=0$ coming from the requirement of Weyl scaling, c.f., (131)) and
scaling $\sigma_{1}$, $\sigma_{3}$, $\eta$, $\sigma_{m}$, $\sigma_{M}$,
$\tilde{\sigma}_{A}$ and $\tilde{\sigma}_{e}$ by appropriate powers of the
temperature multiplied by a function of the dimensionless ratio $\mu/T$. Tthe
charge current is given by

$J^{\mu}=p_{0}^{\prime}T^{d-1}u^{\mu}+s_{E}T^{d-2}\Delta^{\mu\rho}\left(T\mathring{\nabla}_{\rho}\left(\frac{\mu}{T}\right)-E_{\rho}\right)$
(191)

(Obtained by scaling (186).)

##  Appendix C Contributions to the constitutive relations which are quadratic
in contorsion

As explained in the main text, in order to test conformal invariance (at non
zero $q$) of the constitutive relations, beyond the hydrostatic limit, we need
to know the full contributions of the scalars $S^{c}{}_{(1)}$ to
$S^{c}_{(12)}$ to the constitutive relations, including the symmetric
components of the stress tensor and contributions which are quadratic in the
contorsion. In this appendix we have collected the important compoents of
these contributions to the stress tensor for completeness. We denote the
components of the stress tensor associated with $x^{(2)}_{i}$, which are
second order in the contorsion, are not of the BR type, and also do not
transform homogenously under Weyl rescalings by $\bar{T}_{{(i)}}^{\mu\nu}$. We
have omitted an overall factor of $x^{(2)}_{i}T^{d-2}$ from these terms.

$\displaystyle\begin{split}\bar{T}_{(3)}^{[\mu\nu]}=&-2(1-q)^{3}k_{\alpha}K^{\alpha[\mu}u^{\nu]}\\\
\bar{T}_{(4)}^{[\mu\nu]}=&(1-q)\Bigg{(}2\left(K_{A}^{[\mu\nu]\alpha}+K_{T}^{[\mu\nu]\alpha}+(d-2)K^{\alpha[\mu}u^{\nu]}\right)k_{\alpha}+\frac{1}{(d-2)}K_{V}^{[\mu}k^{\nu]}+\frac{2(d-2)}{d-1}\kappa
u^{[\mu}k^{\nu]}\\\
&-k_{\alpha}\left(\kappa_{A}^{\alpha[\mu}u^{\nu]}+\kappa_{S}^{\alpha[\mu}u^{\nu]}\right)\Bigg{)}\\\
\bar{T}_{(5)}^{[\mu\nu]}=&-\delta_{q\,0}\kappa\left(K_{V}^{[\mu}u^{\nu]}+2\kappa_{A}^{\mu\nu}\right)\\\
\bar{T}_{(6)}^{[\mu\nu]}=&-4u^{[\mu}K^{\nu]}{}_{\alpha}k^{\alpha}\\\
\bar{T}_{(7)}^{[\mu\nu]}=&-4K_{V}^{[\mu}k^{\nu]}+\frac{8(d-2)^{2}}{d-1}\kappa
k^{[\mu}u^{\nu]}-\frac{4(d-2)}{d-1}\kappa K_{V}^{[\mu}u^{\nu]}\\\
\bar{T}_{(8)}^{[\mu\nu]}=&\frac{1}{2(d-1)}K^{\mu\nu}\kappa+k_{\alpha}K^{\alpha[\mu}u^{\nu]}-2k_{\alpha}\kappa_{A}^{\alpha[\mu}u^{\nu]}\\\
\bar{T}_{(9)}^{[\mu\nu]}=&\frac{2}{d-1}\kappa\kappa_{A}^{\mu\nu}\kappa+\frac{1}{(d-2)}u^{[\mu}\kappa_{A}{}^{\nu]}{}_{\alpha}\mathcal{K}_{V}{}^{\alpha}\\\
\bar{T}_{(10)}^{[\mu\nu]}=&-\frac{1}{(d-2)}u^{[\mu}\kappa_{S}{}^{\nu]}{}_{\alpha}\mathcal{K}_{V}{}^{\alpha}\\\
\bar{T}_{(11)}^{[\mu\nu]}=&-\frac{2}{d-2}\mathcal{K}_{A}^{\mu\nu\alpha}\mathcal{K}_{V\,\alpha}\\\
\bar{T}_{(12)}^{[\mu\nu]}=&-\frac{1}{2(d-2)}\mathcal{K}_{T}^{[\mu\nu]\alpha}\mathcal{K}_{V\,\alpha}\,.\end{split}$
(192)

Note that when $q=0$ the non homogenous transformation of $\tilde{T}_{(5)}$
vanishes as it should. We have, nevertheless, included the explicit form of
$\tilde{T}_{(5)}$ for completeness.

## References

  * (1) STAR collaboration, _Global $\Lambda$ hyperon polarization in nuclear collisions: evidence for the most vortical fluid_, _Nature_ 548 (2017) 62 [1701.06657].
  * (2) STAR collaboration, _Global polarization of $\Lambda$ hyperons in Au+Au collisions at $\sqrt{s_{{}_{NN}}}$ = 200 GeV_, _Phys. Rev. C_ 98 (2018) 014910 [1805.04400].
  * (3) R. Takahashi et al., _Spin hydrodynamic generation_ , _Nature_ 12 (2016) 52.
  * (4) S. A. Voloshin, _Polarized secondary particles in unpolarized high energy hadron-hadron collisions?_ , nucl-th/0410089.
  * (5) N. Armesto, N. Borghini, S. Jeon and U. A. Wiedemann, eds., _Proceedings, Workshop on Heavy Ion Collisions at the LHC: Last Call for Predictions: Geneva, Switzerland, May 14 - June 8, 2007_ , vol. 35, 2008.  10.1088/0954-3899/35/5/054001.
  * (6) F. Becattini, F. Piccinini and J. Rizzo, _Angular momentum conservation in heavy ion collisions at very high energy_ , _Phys. Rev. C_ 77 (2008) 024906 [0711.1253].
  * (7) B. Betz, M. Gyulassy and G. Torrieri, _Polarization probes of vorticity in heavy ion collisions_ , _Phys. Rev. C_ 76 (2007) 044901 [0708.0035].
  * (8) F. Becattini, V. Chandra, L. Del Zanna and E. Grossi, _Relativistic distribution function for particles with spin at local thermodynamical equilibrium_ , _Annals Phys._ 338 (2013) 32 [1303.3431].
  * (9) W. Florkowski, B. Friman, A. Jaiswal, R. Ryblewski and E. Speranza, _Spin-dependent distribution functions for relativistic hydrodynamics of spin-1/2 particles_ , _Phys. Rev. D_ 97 (2018) 116017 [1712.07676].
  * (10) W. Florkowski, B. Friman, A. Jaiswal and E. Speranza, _Relativistic fluid dynamics with spin_ , _Phys. Rev. C_ 97 (2018) 041901 [1705.00587].
  * (11) F. Becattini, W. Florkowski and E. Speranza, _Spin tensor and its role in non-equilibrium thermodynamics_ , _Phys. Lett. B_ 789 (2019) 419 [1807.10994].
  * (12) F. Becattini and M. A. Lisa, _Polarization and Vorticity in the Quark–Gluon Plasma_ , _Ann. Rev. Nucl. Part. Sci._ 70 (2020) 395 [2003.03640].
  * (13) F. Becattini, _Polarization in relativistic fluids: a quantum field theoretical derivation_ , 4, 2020, 2004.04050.
  * (14) F. Becattini, _Does the spin tensor play any role in non-gravitational physics?_ , _Nucl. Phys. A_ 1005 (2021) 121833 [2003.01406].
  * (15) R. Singh, G. Sophys and R. Ryblewski, _Spin polarization dynamics in the Gubser-expanding background_ , _Phys. Rev. D_ 103 (2021) 074024 [2011.14907].
  * (16) J. L. Mañes, M. Valle and M. A. Vázquez-Mozo, _Chiral torsional effects in anomalous fluids in thermal equilibrium_ , _JHEP_ 05 (2021) 209 [2012.08449].
  * (17) S. Bhadury, J. Bhatt, A. Jaiswal and A. Kumar, _New Developments in Relativistic Fluid Dynamics with spin_ , 2101.11964.
  * (18) W. Florkowski and R. Ryblewski, _On the interpretation of $\Lambda$ spin polarization measurements_, 2102.02890.
  * (19) S. Floerchinger and E. Grossi, _Conserved and non-conserved Noether currents from the quantum effective action_ , 2102.11098.
  * (20) R. Singh, M. Shokri and R. Ryblewski, _Spin polarization dynamics in the Bjorken-expanding resistive MHD background_ , 2103.02592.
  * (21) A. Das, W. Florkowski, R. Ryblewski and R. Singh, _Pseudo-gauge dependence of quantum fluctuations of energy in a hot relativistic gas of fermions_ , 2103.01013.
  * (22) N. Weickgenannt, E. Speranza, X.-l. Sheng, Q. Wang and D. H. Rischke, _Derivation of the nonlocal collision term in the relativistic Boltzmann equation for massive spin-1/2 particles from quantum field theory_ , 2103.04896.
  * (23) F. Becattini, M. Buzzegoli, A. Palermo, G. Inghirami and I. Karpenko, _Local polarization and isothermal local equilibrium in relativistic heavy ion collisions_ , 2103.14621.
  * (24) E. Speranza, F. S. Bemfica, M. M. Disconzi and J. Noronha, _Challenges in Solving Chiral Hydrodynamics_ , 2104.02110.
  * (25) K. J. Gonçalves and G. Torrieri, _Vector resonances spin alignement as a probe of spin hydrodynamics and freeze-out_ , 2104.12941.
  * (26) D. She, A. Huang, D. Hou and J. Liao, _Relativistic Viscous Hydrodynamics with Angular Momentum_ , 2105.04060.
  * (27) C. Yi, S. Pu and D.-L. Yang, _Reexamination of local spin polarization beyond global equilibrium in relativistic heavy ion collisions_ , _Phys. Rev. C_ 104 (2021) 064901 [2106.00238].
  * (28) D.-L. Wang, S. Fang and S. Pu, _Analytic solutions of relativistic dissipative spin hydrodynamics with Bjorken expansion_ , 2107.11726.
  * (29) M. Hongo, X.-G. Huang, M. Kaminski, M. Stephanov and H.-U. Yee, _Relativistic spin hydrodynamics with torsion and linear response theory for spin relaxation_ , _JHEP_ 11 (2021) 150 [2107.14231].
  * (30) S. Lin, _Quantum Kinetic Theory for Quantum Electrodynamics_ , 2109.00184.
  * (31) Z. Chen and S. Lin, _Quantum Kinetic Theory with Vector and Axial Gauge Fields_ , 2109.08440.
  * (32) M. Buzzegoli, _Pseudo-gauge dependence of the spin polarization and of the axial vortical effect_ , 2109.12084.
  * (33) Y.-C. Liu and X.-G. Huang, _Spin Polarization Formula for Dirac Fermions at Local Equilibrium_ , 2109.15301.
  * (34) M. Valle and M. A. Vazquez-Mozo, _On Nieh-Yan Transport_ , 2112.02003.
  * (35) L. Dong and S. Lin, _Dilepton Helical Production in a Vortical Quark-Gluon Plasma_ , 2112.07153.
  * (36) E. Sonin, _Spin currents and spin superfluidity_ , _Advances in Physics_ 59 (2010) 181 [https://doi.org/10.1080/00018731003739943].
  * (37) J. König, M. C. Bønsager and A. H. MacDonald, _Dissipationless spin transport in thin film ferromagnets_ , _Phys. Rev. Lett._ 87 (2001) 187202.
  * (38) W. Chen and M. Sigrist, _Dissipationless multiferroic magnonics_ , _Phys. Rev. Lett._ 114 (2015) 157203.
  * (39) S. Takei and Y. Tserkovnyak, _Superfluid spin transport through easy-plane ferromagnetic insulators_ , _Phys. Rev. Lett._ 112 (2014) 227201.
  * (40) S. Takei, B. I. Halperin, A. Yacoby and Y. Tserkovnyak, _Superfluid spin transport through antiferromagnetic insulators_ , _Phys. Rev. B_ 90 (2014) 094408.
  * (41) M. Hongo, X.-G. Huang, M. Kaminski, M. Stephanov and H.-U. Yee, _Spin relaxation rate for heavy quarks in weakly coupled QCD plasma_ , 2201.12390.
  * (42) A. D. Gallegos, U. Gürsoy and A. Yarom, _Hydrodynamics of spin currents_ , 2101.04759.
  * (43) F. Belinfante, _On the current and the density of the electric charge, the energy, the linear momentum and the angular momentum of arbitrary fields_ , _Physica_ 7 (1940) 449 .
  * (44) L. Rosenfeld, _Sur le tenseur d’impulsion-energie, acad_ , _Roy. Belg. Memoirs de Classes de Science_ 18 (1940) .
  * (45) K. Jensen, M. Kaminski, P. Kovtun, R. Meyer, A. Ritz and A. Yarom, _Towards hydrodynamics without an entropy current_ , _Phys. Rev. Lett._ 109 (2012) 101601 [1203.3556].
  * (46) N. Banerjee, J. Bhattacharya, S. Bhattacharyya, S. Jain, S. Minwalla and T. Sharma, _Constraints on Fluid Dynamics from Equilibrium Partition Functions_ , _JHEP_ 09 (2012) 046 [1203.3544].
  * (47) S. Jain and T. Sharma, _Anomalous charged fluids in 1+1d from equilibrium partition function_ , _JHEP_ 01 (2013) 039 [1203.5308].
  * (48) K. Jensen, _Triangle Anomalies, Thermodynamics, and Hydrodynamics_ , _Phys. Rev. D_ 85 (2012) 125017 [1203.3599].
  * (49) M. Valle, _Hydrodynamics in 1+1 dimensions with gravitational anomalies_ , _JHEP_ 08 (2012) 113 [1206.1538].
  * (50) S. Bhattacharyya, S. Jain, S. Minwalla and T. Sharma, _Constraints on Superfluid Hydrodynamics from Equilibrium Partition Functions_ , _JHEP_ 01 (2013) 040 [1206.6106].
  * (51) N. Banerjee, S. Dutta, S. Jain, R. Loganayagam and T. Sharma, _Constraints on Anomalous Fluid in Arbitrary Dimensions_ , _JHEP_ 03 (2013) 048 [1206.6499].
  * (52) K. Jensen, R. Loganayagam and A. Yarom, _Thermodynamics, gravitational anomalies and cones_ , _JHEP_ 02 (2013) 088 [1207.5824].
  * (53) J. Bhattacharya, S. Bhattacharyya and M. Rangamani, _Non-dissipative hydrodynamics: Effective actions versus entropy current_ , _JHEP_ 02 (2013) 153 [1211.1020].
  * (54) C. Eling, Y. Oz, S. Theisen and S. Yankielowicz, _Conformal Anomalies in Hydrodynamics_ , _JHEP_ 05 (2013) 037 [1301.3170].
  * (55) J. Armas, _How Fluids Bend: the Elastic Expansion for Higher-Dimensional Black Holes_ , _JHEP_ 09 (2013) 073 [1304.7773].
  * (56) K. Jensen, P. Kovtun and A. Ritz, _Chiral conductivities and effective field theory_ , _JHEP_ 10 (2013) 186 [1307.3234].
  * (57) S. Chapman, C. Hoyos and Y. Oz, _Superfluid Kubo Formulas from Partition Function_ , _JHEP_ 04 (2014) 186 [1310.2247].
  * (58) J. L. Mañes and M. Valle, _Parity odd equilibrium partition function in 2 + 1 dimensions_ , _JHEP_ 11 (2013) 178 [1310.2113].
  * (59) K. Jensen, R. Loganayagam and A. Yarom, _Anomaly inflow and thermal equilibrium_ , _JHEP_ 05 (2014) 134 [1310.7024].
  * (60) K. Jensen, R. Loganayagam and A. Yarom, _Chern-Simons terms from thermal circles and anomalies_ , _JHEP_ 05 (2014) 110 [1311.2935].
  * (61) S. Bhattacharyya, _Entropy current and equilibrium partition function in fluid dynamics_ , _JHEP_ 08 (2014) 165 [1312.0220].
  * (62) J. Armas, _(Non)-Dissipative Hydrodynamics on Embedded Surfaces_ , _JHEP_ 09 (2014) 047 [1312.0597].
  * (63) S. Bhattacharyya, _Entropy Current from Partition Function: One Example_ , _JHEP_ 07 (2014) 139 [1403.7639].
  * (64) E. Megias and M. Valle, _Second-order partition function of a non-interacting chiral fluid in 3+1 dimensions_ , _JHEP_ 11 (2014) 005 [1408.0165].
  * (65) K. Jensen, _Aspects of hot Galilean field theory_ , _JHEP_ 04 (2015) 123 [1411.7024].
  * (66) L. Di Pietro and Z. Komargodski, _Cardy formulae for SUSY theories in $d=$ 4 and $d=$ 6_, _JHEP_ 12 (2014) 031 [1407.6061].
  * (67) M. Harder, P. Kovtun and A. Ritz, _On thermal fluctuations and the generating functional in relativistic hydrodynamics_ , _JHEP_ 07 (2015) 025 [1502.03076].
  * (68) M. Valle, _Torsional response of relativistic fermions in $2+1$ dimensions_, _JHEP_ 07 (2015) 006 [1503.04020].
  * (69) N. Banerjee, S. Dutta and A. Jain, _Equilibrium partition function for nonrelativistic fluids_ , _Phys. Rev. D_ 92 (2015) 081701 [1505.05677].
  * (70) J. Armas, J. Bhattacharya, A. Jain and N. Kundu, _On the surface of superfluids_ , _JHEP_ 06 (2017) 090 [1612.08088].
  * (71) J. Hernandez and P. Kovtun, _Relativistic magnetohydrodynamics_ , _JHEP_ 05 (2017) 001 [1703.08757].
  * (72) J. Armas and A. Jain, _One-form superfluids \ & magnetohydrodynamics_, _JHEP_ 01 (2020) 041 [1811.04913].
  * (73) P. Kovtun and A. Shukla, _Einstein’s equations in matter_ , _Phys. Rev. D_ 101 (2020) 104051 [1907.04976].
  * (74) P. Kovtun, _First-order relativistic hydrodynamics is stable_ , _JHEP_ 10 (2019) 034 [1907.08191].
  * (75) J. Armas, J. Hartong, E. Have, B. F. Nielsen and N. A. Obers, _Newton-Cartan Submanifolds and Fluid Membranes_ , _Phys. Rev. E_ 101 (2020) 062803 [1912.01613].
  * (76) J. Armas and A. Jain, _Effective field theory for hydrodynamics without boosts_ , 2010.15782.
  * (77) J. de Boer, J. Hartong, N. A. Obers, W. Sybesma and S. Vandoren, _Perfect Fluids_ , _SciPost Phys._ 5 (2018) 003 [1710.04708].
  * (78) S. Li, M. A. Stephanov and H.-U. Yee, _Non-dissipative second-order transport, spin, and pseudo-gauge transformations in hydrodynamics_ , 2011.12318.
  * (79) J. Bhattacharya, S. Bhattacharyya, S. Minwalla and A. Yarom, _A Theory of first order dissipative superfluid dynamics_ , _JHEP_ 05 (2014) 147 [1105.3733].
  * (80) S. Bhattacharyya, _Constraints on the second order transport coefficients of an uncharged fluid_ , _JHEP_ 07 (2012) 104 [1201.4654].
  * (81) L. Landau and E. Lifshitz, _Fluid Mechanics: Volume 6_ , no. 6. Elsevier Science, 2013.
  * (82) P. Glorioso and H. Liu, _The second law of thermodynamics from symmetry and unitarity_ , 1612.07705.
  * (83) K. Jensen, R. Marjieh, N. Pinzani-Fokeeva and A. Yarom, _An entropy current in superspace_ , _JHEP_ 01 (2019) 061 [1803.07070].
  * (84) F. M. Haehl, R. Loganayagam and M. Rangamani, _Inflow Mechanism for Hydrodynamic Entropy_ , _Phys. Rev. Lett._ 121 (2018) 051602 [1803.08490].
  * (85) K. Fukushima and S. Pu, _Spin hydrodynamics and symmetric energy-momentum tensors – A current induced by the spin vorticity –_ , _Phys. Lett. B_ 817 (2021) 136346 [2010.01608].
  * (86) K. Jensen, R. Marjieh, N. Pinzani-Fokeeva and A. Yarom, _A panoply of Schwinger-Keldysh transport_ , _SciPost Phys._ 5 (2018) 053 [1804.04654].
  * (87) A. Iorio, L. O’Raifeartaigh, I. Sachs and C. Wiesendanger, _Weyl gauging and conformal invariance_ , _Nucl. Phys. B_ 495 (1997) 433 [hep-th/9607110].
  * (88) I. Buchbinder and I. Shapiro, _On the renormalization of models of quantum field theory in an external gravitational field with torsion_ , _Phys. Lett. B_ 151 (1985) 263.
  * (89) I. Shapiro, _Physical aspects of the space-time torsion_ , _Phys. Rept._ 357 (2002) 113 [hep-th/0103093].
  * (90) R. Loganayagam, _Entropy Current in Conformal Hydrodynamics_ , _JHEP_ 05 (2008) 087 [0801.3701].
  * (91) F. de Juan, A. Cortijo and M. A. H. Vozmediano, _Dislocations and torsion in graphene and related systems_ , _Nucl. Phys. B_ 828 (2010) 625 [0909.4068].
  * (92) A. Mesaros, D. Sadri and J. Zaanen, _Parallel Transport of Electrons in Graphene Parallels Gravity_ , _Phys. Rev. B_ 82 (2010) 073405 [0909.2703].
  * (93) K. Hattori, M. Hongo, X.-G. Huang, M. Matsuo and H. Taya, _Fate of spin polarization in a relativistic fluid: An entropy-current analysis_ , _Phys. Lett. B_ 795 (2019) 100 [1901.06615].
  * (94) S. R. De Groot, _Relativistic Kinetic Theory. Principles and Applications_. 1980.
  * (95) F. Becattini, _Covariant statistical mechanics and the stress-energy tensor_ , _Phys. Rev. Lett._ 108 (2012) 244502 [1201.5278].
  * (96) F. M. Haehl, R. Loganayagam and M. Rangamani, _The eightfold way to dissipation_ , _Phys. Rev. Lett._ 114 (2015) 201601 [1412.1090].
  * (97) M. Crossley, P. Glorioso and H. Liu, _Effective field theory of dissipative fluids_ , _JHEP_ 09 (2017) 095 [1511.03646].


# Covering a compact space by fixed-radius or growing random balls

David J. Aldous Department of Statistics, 367 Evans Hall # 3860, U.C. Berkeley
CA 94720<EMAIL_ADDRESS>www.stat.berkeley.edu/users/aldous.

###### Abstract

Simple random coverage models, well studied in Euclidean space, can also be
defined on a general compact metric space. By analogy with the geometric
models, and with the discrete coupon collector’s problem and with cover times
for finite Markov chains, one expects a “weak concentration” bound for the
distribution of the cover time to hold under minimal assumptions. We give two
such results, one for random fixed-radius balls and the other for sequentially
arriving randomly-centered and deterministically growing balls. Each is in
fact a simple application of a different more general bound, the former
concerning coverage by i.i.d. random sets with arbitrary distribution, and the
latter concerning hitting times for Markov chains with a strong monotonicity
property. The growth model seems generally more tractable, and we record some
basic results and open problems for that model.

##  1 Introduction

Analogs of the classical coupon collector’s problem have been extensively
studied in several different contexts. One context is geometric: covering by
(for instance) random balls in Euclidean space [14, 19]. Another context
involves the time for an irreducible finite-state Markov chain to visit every
state. Systematic study of that cover time $C_{MC}$, particularly for the case
of random walks on graphs, started in the 1980s [1]. In any context, study of
the expectation of the cover time (or more refined study of exact limit
rescaled distributions) necessarily depends on the specifics of a model, and
has been carried out via explicit calculations for many models. However one
expects that the “weak concentration” property of the coupon collector time
$T_{n}$ (that s.d.$(T_{n})/\mathbb{E}T_{n}\to 0$ as $n\to\infty$) should
extend quite generally to other cover time contexts, and should hold under
minimal assumptions even when one cannot calculate the expectation explicitly.
Indeed this is known to be true in the Markov chain context (see section 6).
The purpose of this article is to study one analog of geometric covering, in
which the Euclidean space is replaced by a metric space.

Another part of our purpose is to spotlight two different general methods
(known, but apparently not well known) for showing weak concentration in
general settings without calculating the expectation of the cover time111Other
than its order of magnitude.. In each of sections 2 and 3 we specify a model
(fixed-radius or growing random balls), recall the relevant general method,
and show that a concentration bound is obtained very easily using that method.
The growth model seems worthy of further study: we give some more basic
results in section 4 and pose some challenging open problems. The special case
of the circle is outlined in section 5. Further discussion of models and
methodology is deferred to section 6.

##  2 Covering with fixed radius random balls

Here we indicate how a concentration result for covering, obtainable on
Euclidean space in sharp form by explicit calculation [14], can be extended to
weak bounds in a very general setting. Take a compact metric space $(S,\rho)$.
Let $\mu$ be a probability measure on $S$ with full support, and for $r>0$
define

$\eta(r):=\inf_{s}\mu(\mathrm{ball}(s,r))>0$

where $\mathrm{ball}(s,r)=\\{s^{\prime}:\rho(s,s^{\prime})\leq r\\}$. Write
$\sigma_{1},\sigma_{2},\ldots$ for i.i.d. random points of $S$ from
distribution $\mu$. For fixed $r_{0}>0$ consider the random subset

$\mathcal{R}_{n}=\mathcal{R}_{n}^{(r_{0})}:=\cup_{1\leq i\leq
n}\mathrm{ball}(\sigma_{i},r_{0}).$

We call this the fixed-radius model. Consider the cover time

$C=C^{(r_{0})}:=\min\\{n:\mathcal{R}_{n}=S\\}$ (1)

for which compactness easily implies $\mathbb{E}C<\infty$. The probability
that a given point $s$ is in $\mathrm{ball}(\sigma_{i},r_{0})$ equals
$\mu(\mathrm{ball}(s,r_{0}))$, and so the mean time until point $s$ is covered
equals $1/\mu(\mathrm{ball}(s,r_{0}))$, which is at most $1/\eta(r_{0})$. So
to obtain a concentration result for $C$ a natural assumption is that
$\mathbb{E}C\gg 1/\eta(r_{0})$, in other words that $\eta(r_{0})\mathbb{E}C$
is large. Our result below is of that general form, but also involves the
dimension-related quantity $d(r)$ defined as the smallest integer such that

each ball of radius $r$ can be covered by $d(r)$ balls of radius $r/2$.  (2)

###### Proposition 1

In the fixed-radius model, for the cover time $C$ at (1),

$\mathrm{var}\left(\frac{C}{\mathbb{E}C}\right)\leq\kappa\,\frac{d(r_{0})}{\eta(r_{0}/2)\mathbb{E}C}$

for the absolute constant $\kappa$ stated in Proposition 2 below.

We will derive Proposition 1 from a known general result, discussed as
Proposition 2 below.

###  2.1 The random subset cover bound

Here we copy the setup and result directly from [5]. Let $S_{0}$ be a finite
set. Let $\mathcal{Y}$ be a random subset of $S_{0}$, whose distribution is
arbitrary subject to the requirement

${\mathbb{P}}(s\in\mathcal{Y})>0$ for each $s\in S_{0}$. (3)

Let $\mathcal{Y}_{1},\mathcal{Y}_{2},\ldots$ be independent random subsets
distributed as $\mathcal{Y}$. Let $\mathcal{R}_{n}$ be the range of this
process: $\mathcal{R}_{n}=\cup_{i\leq n}\mathcal{Y}_{i}$ and let $C_{set}$ be
the cover time

$C_{set}:=\min\\{n:\mathcal{R}_{n}=S_{0}\\}.$

Note $\mathbb{E}C_{set}<\infty$ by (3) and finiteness of $S_{0}$. For any non-
random subset $B\subset S_{0}$ let $c(B)$ be the mean cover time of $B$:

$c(B):=\mathbb{E}C(B);\quad C(B):=\min\\{n:\mathcal{R}_{n}\supseteq B\\}.$

Our bound involves the terminal set

$\mathcal{T}:=S_{0}\setminus\mathcal{R}_{C_{set}-1}$

that is the last uncovered portion of $S_{0}$.

######  Proposition 2 ([5] Theorem 1)

$\mathrm{var}\left(\frac{C_{set}}{\mathbb{E}C_{set}}\right)\leq\kappa\,\frac{\mathbb{E}c(\mathcal{T})}{\mathbb{E}C_{set}}$
for an absolute constant $\kappa$.

Though stated in [5] for a finite state space $S_{0}$, Proposition 2 extends
to continuous space, in particular our compact metric space $S$, with
unchanged proof, except that now we need to replace assumption (3) by the
assumption $\mathbb{E}C_{set}<\infty$.

Of course it may be difficult to analyze $\mathcal{T}$, and so one does not
expect to obtain sharp bounds on specific models in this way. But Proposition
2 may be useful in obtaining order of magnitude bounds in general settings. In
particular if there is some geometric or metric structure on the set and if
the random subsets $\mathcal{Y}$ are small in diameter, then $\mathcal{T}$
must be small in diameter, so one needs only to bound $c(B)$ as a function of
the diameter of $B$. The next section gives a simple illustration of that
method.

###  2.2 Proof of Proposition 1

In the notation of Proposition 2, the terminal set $\mathcal{T}$ is such that
$\mathcal{T}\subset\mathrm{ball}(s,r_{0})$ for some $s\in S$, so

$c(\mathcal{T})\leq\sup_{s}\mathbb{E}C(\mathrm{ball}(s,r_{0})).$

The mean time until one of the random centers $\sigma$ falls in a given ball
of radius $r_{0}/2$ is at most $1/\eta(r_{0}/2)$. Note that a ball of radius
$r_{0}/2$ is covered by any ball of radius $r_{0}$ whose center is in the
former ball. So from the definition of dimension $d$, for each $s$ there are
$d$ points $s_{1},\ldots,s_{d}$ such that $\mathrm{ball}(s,r_{0})$ is covered
whenever each of $(\mathrm{ball}(s_{i},r_{0}/2),1\leq i\leq d)$ contains at
least one of the random centers $\sigma$, and so

$\sup_{s}\mathbb{E}C(\mathrm{ball}(s,r_{0}))\leq d/\eta(r_{0}/2).$

The result follows from Proposition 2.

##  3 The growth model

Consider as before a compact metric space $(S,\rho)$, a probability measure
$\mu$ on $S$, but now introduce two rates $0<\lambda<\infty$ and $0<v<\infty$.
Write $0<\tau_{1}<\tau_{2}<\ldots$ for the times of a rate-$\lambda$ Poisson
process, and write $\sigma_{1},\sigma_{2},\ldots$ for i.i.d. random points of
$S$ from distribution $\mu$. The verbal description

> seeds arrive at times of a Poisson process at i.i.d. random positions, and
> then create balls whose radius grows at rate $v$

is formalized as the set-valued growth process

$\mathcal{X}(t):=\cup_{i:\tau_{i}\leq t}\
\mathrm{ball}\,(\sigma_{i},v(t-\tau_{i})).$ (4)

We study the cover time

$C:=\min\\{t:\ \mathcal{X}(t)=S\\}$

which is finite because $\mathbb{E}\tau_{1}=1/\lambda$ and so

$1/\lambda\leq\mathbb{E}C\leq 1/\lambda+\Delta/v$ (5)

where $\Delta$ is the diameter of $S$. To obtain a concentration bound it is
natural to require that $\mathbb{E}C$ is large relative to the maximum
expected time to cover any given single point, that is relative to

$c^{*}:=\max_{s\in S}\mathbb{E}C(s);\quad C(s):=\min\\{t:\
s\in\mathcal{X}(t)\\}.$

It turns out this is the only requirement.

###### Proposition 3

In the growth model (4),
$\mathrm{var}\left(\frac{C}{\mathbb{E}C}\right)\leq\frac{c^{*}}{\mathbb{E}C}$.

We will derive Proposition 3 from a known general result, discussed as
Proposition 4 below. Note that the expectation of the number of balls covering
$v$ at time $t$ equals $\int_{0}^{t}\mu(\mathrm{ball}(s,vu))\ \lambda du$ and
so from the Poisson property

${\mathbb{P}}(C(s)>t)=\exp\left(-\int_{0}^{t}\mu(\mathrm{ball}(s,vu))\ \lambda
du\right)$ (6)

from which we can in principle obtain a formula for $\mathbb{E}C(s)$.

###  3.1 A monotonicity bound for Markov chains

Here we copy the setup and result directly from [7]. The setting there is a
continuous-time Markov chain $(X_{t})$ on a finite state space $\Sigma$, where
we study the hitting time

$T:=\inf\\{t:\ X_{t}\in\Sigma_{0}\\}$ (7)

for a fixed subset $\Sigma_{0}\subset\Sigma$. Assume

$h(x):=\mathbb{E}_{x}T<\infty\mbox{ for each }x\in\Sigma$ (8)

which holds in the finite case under the natural “reachability” condition.
Assume also a rather strong “monotonicity” condition:

$h(x^{\prime})\leq h(x)\mbox{ whenever $x\to x^{\prime}$ is a possible
transition}.$ (9)

######  Proposition 4 ([7])

Under conditions (8, 9), for any initial state,

$\frac{\mathrm{var}\ T}{\mathbb{E}T}\leq\max\\{h(x)-h(x^{\prime}):\ x\to
x^{\prime}\mbox{ a possible transition}\\}.$

Though stated in [7] for a finite state space $\Sigma$, Proposition 4 extends
to continuous space with essentially unchanged proof.

###  3.2 Proof of Proposition 3

The cover time $C$ for our growth model $\mathcal{X}(t)$ at (4) is of the form
in Proposition 4; the state space is the space of compact subsets $x$ of the
compact metric space $S$. The only discontinuities of $h(\mathcal{X}(t))$ are
at a time $\tau$ when a new seed arrives at a point $\sigma$, at which time
there is a transition $x\to x\cup\\{\sigma\\}$ of $\mathcal{X}(t)$. To apply
Proposition 4 to prove Proposition 3 it is enough to show that, for each pair
$(x,\sigma)$,

$h(x)-h(x\cup\\{\sigma\\})\leq\mathbb{E}C(\sigma).$ (10)

But this holds by considering the natural coupling
$(\mathcal{X}(t),\mathcal{X}^{\prime}(t)=\mathcal{X}(t)\cup\mathrm{ball}(\sigma,vt),t\geq
0)$ of the growth processes with
$\mathcal{X}(0)=x,\mathcal{X}^{\prime}(0)=x\cup\\{\sigma\\}$. In this
coupling, for the time $C^{*}(\sigma)$ at which $\sigma$ is reached by a ball
of $\mathcal{X}(\cdot)$ whose seed arrived after time $0$, we have (by the
triangle inequality on $S$) that
$\mathcal{X}(C^{*}(\sigma)+t)\supseteq\mathcal{X}^{\prime}(t)$, and so the
cover times for these two processes differ by at most $C^{*}(\sigma)$. But
this $C^{*}(\sigma)$ is distributed as $C(\sigma)$ for the growth process
started at the empty set, establishing (10).

##  4 Further analysis of the general growth model

Comparing the statements of Propositions 1 and 3 suggests that the growth
model is more tractable for the study of covering. Intuitively this is because
the behavior of the growth model is “smoother” in that it does not rely on the
detailed geometry of the space $(S,\rho)$ at the given distance $r_{0}$. In
this section we record some simple observations and then pose some open
problems.

We can “standardize” the growth model by choosing time and distance units to
make $\lambda=v=1$. With this standardization we have a relationship between
the diameter $\Delta$ and $\mathbb{E}C$.

###### Proposition 5

In the standardized growth model on a space $(S,\rho)$,
(a) $\mathbb{E}C\leq 1+\Delta$.
(b) If $S$ is connected then $\Delta\leq\kappa_{1}(\mathbb{E}C)^{2}$ for an
absolute constant $\kappa_{1}$.

Proof.  Part (a) is (5). For (b), at time $t$ the sum of diameters of balls is
at most

$D(t):=2\sum_{i}(t-\tau_{i})^{+}.$

By connectedness we must have

$\Delta\leq D(C).$

We can rewrite $D(t)$ in terms of the Poisson counting process $(N(t))$ as
$D(t)=2\int_{0}^{t}N(u)du$ and then

$\Delta\leq\mathbb{E}D(C)=2\int_{0}^{\infty}\mathbb{E}[N(t)1_{(t\leq C)}]\
dt.$

Using the Cauchy-Schwarz inequality

$\Delta\leq 2\int_{0}^{\infty}(t^{2}+t)^{1/2}\ \sqrt{{\mathbb{P}}(C\geq t)}\
dt.$ (11)

Now the obvious submultiplicative property of the cover time $C$, that is

${\mathbb{P}}(C\geq t_{1}+t_{2})\leq{\mathbb{P}}(C\geq t_{1})\
{\mathbb{P}}(C\geq t_{2})$

combined with Markov’s inequality ${\mathbb{P}}(C\geq e\mathbb{E}C)\leq
e^{-1}$ implies an exponential tail bound

${\mathbb{P}}(C\geq t)\leq\exp(1-{\textstyle\frac{t}{e\mathbb{E}C}})$ (12)

and the result follows from (11) and straightforward calculus bounds (note
$\mathbb{E}C\geq 1$ from (5)).

Continuing with this standardization, consider a sequence of connected compact
metric spaces $S=S^{(n)}$ and probability distributions $\mu=\mu^{(n)}$.
Proposition 3 implies that as $n\to\infty$

$\mbox{ if }{\textstyle\frac{c^{*}}{\mathbb{E}C}}\to 0\mbox{ then
}{\textstyle\frac{C}{\mathbb{E}C}}\to 1\mbox{ in }L^{2}.$ (13)

Can we relate the hypothesis $c^{*}/\mathbb{E}C\to 0$ to other aspects of the
spaces? Recall that $c^{*}$ is in principle directly calculable from (6),
whereas determining whether $\mathbb{E}C$ is of the same order, or larger
order, than $c^{*}$ requires some more detailed knowledge of the space $S$.

If the diameters $\Delta^{(n)}$ are bounded (as $n$ increases) then by
Proposition 5 the mean cover times $\mathbb{E}C^{(n)}$ are bounded; because
${\mathbb{P}}(C^{(n)}>t)\geq\exp(-t)$ the conclusion (and hence the
assumption) of (13) is false. So we need study only the case
$\Delta^{(n)}\to\infty$. Here is a simple example to show that the conclusion
of (13) is not always true.

Example. Take $S^{(n)}$ to be the real line segment $[0,n]$ and
$\mu^{(n)}(0)=1-1/n$ and $\mu^{(n)}(n)=1/n$. One easily sees that

$n^{-1}C^{(n)}\to_{d}\min(1,{\textstyle\frac{1}{2}}(1+\xi))$

where $\xi$ has Exponential(1) distribution.

In an opposite direction, we note a simple upper bound on $\mathbb{E}C/c^{*}$,
that is a lower bound on $c^{*}/\mathbb{E}C$, in terms of the covering number

$\mathrm{cov}(r):=\mbox{ minimum number of radius $r$ balls that cover $S$ }.$
(14)

###### Proposition 6

In the standardized growth model,

$\frac{\mathbb{E}C}{c^{*}}\leq\min_{a>0}[a+e(e+\log\mathrm{cov}(ac^{*}))].$

Proof.  As at (12) the submultiplicative property of $C(s)$ implies
${\mathbb{P}}(C(s)\geq t)\leq\exp(1-{\textstyle\frac{t}{e\mathbb{E}C(s)}})$.
Applying this to the centers $(s_{i})$ of $\mathrm{cov}(r)$ covering radius
$r$ balls,

${\mathbb{P}}(\max_{i}C(s_{i})\geq t)\leq e\
\mathrm{cov}(r)\exp(-{\textstyle\frac{t}{ec^{*}}}).$

Setting $t_{0}:=ec^{*}\log\mathrm{cov}(r)$,

$\mathbb{E}[\max_{i}C(s_{i})]=\int_{0}^{\infty}{\mathbb{P}}(\max_{i}C(s_{i})\geq
t)\;dt\leq t_{0}+e\cdot ec^{*}.$

Because $C\leq r+\max_{i}C(s_{i})$ we have

$\mathbb{E}C\leq r+ec^{*}(e+\log\mathrm{cov}(r)).$

Setting $r=ac^{*}$ gives the stated bound.

###  4.1 The minimizing seed distribution

For the standardized growth model on connected compact $(S,\rho)$, take two
points $s_{1},s_{2}$ which are diametrically opposite, that is
$\rho(s_{1},s_{2})=\Delta$. Then the maximum of $\mathbb{E}_{\mu}C$ over $\mu$
equals $1+\Delta$, attained by the measure $\mu$ degenerate at $s_{1}$. But
what can we say about the minimum of $\mathbb{E}_{\mu}C$ over $\mu$?

Intuitively this should be related to the covering numbers $\mathrm{cov}(r)$
at (14). And indeed there is a simple upper bound in terms of the covering
numbers. Given $r$, consider $\mu$ uniform on the centers $(s_{i},1\leq
i\leq\mathrm{cov}(r))$ of the covering radius-$r$ balls. Then $C\leq
r+\tau_{\mathrm{cov}(r)}$ where $\tau_{n}$ is the elementary coupon collector
time with $\mathbb{E}\tau_{n}=n(1+1/2+\ldots+1/n)\leq(1+\log n)n$. So we have
established

###### Proposition 7

In the standardized growth model,

$\min_{\mu}\mathbb{E}_{\mu}C\leq\min_{r>0}[r+\mathrm{cov}(r)(1+\log\mathrm{cov}(r))].$

For a bound in the opposite direction, observe first that for the Poisson
counting process $(N(t),0\leq t<\infty)$ of seed arrival times,

###### Lemma 8

If $t_{0}$ and $c_{0}$ are such that
${\mathbb{P}}(C>c_{0})+{\mathbb{P}}(N(c_{0})>t_{0})<1$ then
$\mathrm{cov}(c_{0})\leq t_{0}$.

Proof.  The assumption implies that the event $\\{C\leq c_{0},N(c_{0})\leq
t_{0}\\}$ has non-zero probability; on that event we have

$\mathrm{cov}(c_{0})\leq N(C)\leq N(c_{0})\leq t_{0}.$

Applying Lemma 8 with $c_{0}=3\mathbb{E}C$ and $t_{0}=3c_{0}$ gives
$\mathrm{cov}(3\mathbb{E}C)\leq 9\mathbb{E}C$. This is true for any $\mu$ and
so

###### Proposition 9

In the standardized growth model,

$\min_{\mu}\mathbb{E}_{\mu}C\geq\min\\{r:\mathrm{cov}(3r)\leq 9r\\}.$

Roughly speaking, if the space is $d$-dimensional in the sense that
$\mathrm{cov}(r)\asymp(A/r)^{d}$ for fixed large $A$, then Propositions 7 and
9 imply that $\min_{\mu}\mathbb{E}_{\mu}C$ is between orders
$A^{\frac{d}{d+1}}$ and $A^{\frac{d}{d+1}}\log A$.

###  4.2 Open problems for the general growth model

  * •

As mentioned above, can we find easily checkable conditions to ensure that
$c^{*}/\mathbb{E}C\to 0$?

  * •

Can one improve the upper and lower bounds on $\min_{\mu}\mathbb{E}_{\mu}C$
above? In particular, can $\min_{\mu}\mathbb{E}_{\mu}C$ be more sharply
related to some measure of entropy of the metric space (see e.g. [18] for
possible notions of entropy)?

  * •

For $\mu$ attaining the minimum $\min_{\mu}\mathbb{E}_{\mu}C$, do we always
have weak concentration? That is, is there a function $\psi(\Delta)\downarrow
0$ as $\Delta\uparrow\infty$ such that on every connected compact metric
space, for the standardized growth model,

$\mathrm{var}_{\mu}\left(\frac{C}{\mathbb{E}_{\mu}C}\right)\leq\psi(\Delta)$

for the minimizing $\mu$?

  * •

Is there an effective algorithmic procedure for finding a minimizing $\mu$?
This seems loosely similar to the well-studied k-median problem [8].

  * •

If $S$ is a compact group, with a metric invariant under the group action,
then is the uniform (Haar) measure the minimizing measure?

Regarding the final problem above, it can be shown that, on the circle of
integer circumference $L$, for the fixed-radius model with $r=1/2$, the mean
cover time for seed distribution $\mu$ uniform on $L$ evenly-spaced points is
smaller than that for $\mu$ uniform on the circle (the discrete analog is
noted in [12] Example 4.1). We do not know if this type of example is a
counter-example in the growth model; if so, replace by an asymptotic
($\Delta\to\infty$) conjecture.

##  5 The growth process on the circle

Here we consider the standardized growth model on the circle $S$ of
circumference $L$, with uniform distribution $\mu$. What is the $L\to\infty$
limit distribution of the cover time $C(L)$? We will treat this as another
example where the Poisson clumping heuristic (PCH) [2] gives a recipe for
calculating explicitly the limit distribution; the method is heuristic in the
sense of not justifying the approximations, but would provide a template for
making a rigorous proof.

###  5.1 The calculation

Consider an interval $A\subset S$ of length $a\ll L$. The number $N_{A}(t)$ of
balls intersecting $A$ at time $t$ has Poisson distribution with

$\mathbb{E}N_{A}(t)=\int_{0}^{t}\min(a+2u,L)\ L^{-1}du.$

We will use this only when $a$ and $t$ are order $L^{1/2+o(1)}$ and so we can
ignore the truncation. This gives (the equalities below are really
approximations)

$\mathbb{E}N_{A}(t)=(at+t^{2})/L$
${\mathbb{P}}(N_{A}(t)=0)=\exp(-(at+t^{2})/L).$ (15)

The probability $p(t)$ that a specified point $s\in S$ is not covered at time
$t$ is the case $a=0$, so

$p(t)=\exp(-t^{2}/L).$ (16)

As $t$ approaches $C(L)$ the uncovered region is a union of intervals of
lengths small relative to $L$. The PCH asserts that, as a good approximation
which gives correct asymptotics, one can assume these intervals have i.i.d.
lengths (with some distribution $\Lambda(t)$) and their centers are as a
Poisson process (of some rate $\lambda(t)$ per unit length); these quantities
are related by

$p(t)=\lambda(t)\mathbb{E}\Lambda(t).$

The number of uncovered intervals therefore has Poisson distribution with mean
$L\lambda(t)$ and so

${\mathbb{P}}(C(L)\leq t)=\exp(-L\lambda(t)).$

In this example it is easy to ascertain the distribution of $\Lambda(t)$.
Given that a point $s$ is uncovered, the conditional probability that the
interval $[s-a_{1},s+a_{2}]$ is uncovered (that is, is a subset of the
uncovered interval containing $s$) equals, by (15), $\exp(-(a_{1}+a_{2})t/L)$,
and therefore the whole uncovered interval $[s-A_{1},s+A_{2}]$ is such that
$A_{1}$ and $A_{2}$ are independent with Exponential($t/L$) distribution. This
length $A_{1}+A_{2}$ is the size-biased distribution of $\Lambda(t)$, so its
un-size-biased distribution is just Exponential($t/L$), with expectation

$\mathbb{E}\Lambda(t)=L/t.$

Combining the displayed equations above gives us

${\mathbb{P}}(C(L)\leq t)\approx\exp(-te^{-t^{2}/L})$ (17)

where we are now acknowledging that this is an approximation222For large $L$,
and $t$ not in the tails., expected to lead to the correct asymptotics.

###  5.2 Asymptotics

And indeed (17) corresponds, as one expects from general extreme value theory
[20], to a limit result of the form

$(C(L)-t_{0}(L))/\sigma(L)\to_{d}\zeta,\quad{\mathbb{P}}(\zeta\leq
x)=\exp(-e^{-x}),-\infty<x<\infty.$ (18)

To make this explicit, define $G(y)$ to be the inverse function of
$y=x\exp(-x^{2})$ for large $x$ and small $y$ and then define

$t_{0}(L):=L^{1/2}G(L^{-1/2})$ (19)

so that (17) becomes

${\mathbb{P}}(C(L)\leq t_{0}(L))\approx\exp(-1).$

One can now calculate from (17) that for fixed $x$

${\mathbb{P}}\left(C(L)\leq
t_{0}+{\textstyle\frac{xL^{1/2}}{2G(L^{-1/2})}}\right)\approx\exp(-e^{-x})$

corresponding to (18) with $t_{0}(L)$ defined by (19) and $\sigma(L)$ defined
by

$\sigma(L):={\textstyle\frac{L^{1/2}}{2G(L^{-1/2})}}.$

The function $G(L^{-1/2})$ is slowly varying, roughly as $\sqrt{\log L}$.

For comparison with the general result of Proposition 3, note that from (16)

$c^{*}=\mathbb{E}C(s)=\int_{0}^{\infty}\exp(-t^{2}/L)\
dt={\textstyle\frac{1}{2}}\pi^{1/2}L^{1/2}$

and so

$\mathrm{var}\left(\frac{C}{\mathbb{E}C}\right)\asymp\frac{1}{G^{4}(L^{-1/2})},\quad\frac{c^{*}}{\mathbb{E}C}\asymp\frac{1}{G(L^{-1/2})}.$

The right side is the upper bound (Proposition 3) and the left side is the
correct order of magnitude.

##  6 Discussion

###  6.1 Comments on the two models

The fixed-radius model is a natural generalization of covering Euclidean space
with balls, though this generalization to metric spaces has apparently has not
been studied before. The growth model in our simple form has also apparently
not been studied, though it can be regarded as an extremely basic model for
the spread of information or the spread of an epidemic, a field with a huge
literature studying models on graphs or Euclidean space [11, 16, 21]. A
related growth model in two dimensions, where seeds arrive (instead of as a
constant-rate process) as a Poisson process whose rate is the current occupied
area, is studied in [6, 9].

###  6.2 The growth model on other spaces

In addition to the open problems in section 4.2, there is much scope for
further study of the growth model on specific spaces. As well as other
classical compact spaces familiar from analysis, one can consider a finite
graph with edge lengths, with the metric of shortest route length. Moreover
there are random metric spaces of contemporary interest in probability, such
as the “mean-field model of distance” [4], the Brownian CRT [13], or the
Brownian map [17].

###  6.3 Other uses of the two general bounds

We have used two general methods – the random subset cover bound (Proposition
2) and the monotonicity bound (Proposition 4) – which are in principle
applicable in very general covering-like contexts to establish weak
concentration bounds in general settings without calculating the expectation
of the covering time. We provide some history of these methods below, and
speculate that there may be other applications not yet explored.

The random subset cover bound, Proposition 2, for general i.i.d. random
subsets of a set, was given in [5] as part of the proof of a weak
concentration bound for the Markov chain cover time $C_{MC}$. In the Markov
chain context, the i.i.d. subsets arise as excursions from a given state. In
the result, the essential condition is that the maximum mean hitting time to
any single state is $o(\mathbb{E}C_{MC})$. In that sense the bound is closely
analogous to the bounds in this article. In the 30 years since [5], study of
random walk cover times has entered a more sophisticated phase based on the
discovery [10] of its connection with Gaussian free fields and Talagrand’s
theory of majorizing measures. In contrast, the program of using general
results for i.i.d. random subsets as part of analysis of specific contexts
within covering seems not to have been developed until the recent work [12].
That paper discusses known results in combinatorial settings, develops new
general results and applies them to several topics: connectivity in random
graphs; covering a square with random discs; covering the edges of a graph by
spanning trees, and matroids by bases; and random $k$-SAT.

The monotonicity bound, Proposition 4, was given in [7] as a tool for
establishing weak concentration for first passage percolation times on general
graphs. It was also used [3] for weak concentration of the time of emergence
of the giant component in bond percolation on general graphs. Both contexts
involve hitting time of an increasing set-valued Markov process, as does our
application in section 3.2.

## References

  * [1] David Aldous.  An introduction to covering problems for random walks on graphs.  J. Theoret. Probab., 2(1):87–89, 1989.
  * [2] David Aldous.  Probability approximations via the Poisson clumping heuristic, volume 77 of Applied Mathematical Sciences.  Springer-Verlag, New York, 1989.
  * [3] David Aldous.  The incipient giant component in bond percolation on general finite weighted graphs.  Electron. Commun. Probab., 21:Paper No. 68, 9, 2016.
  * [4] David Aldous and J. Michael Steele.  The objective method: probabilistic combinatorial optimization and local weak convergence.  In Probability on discrete structures, volume 110 of Encyclopaedia Math. Sci., pages 1–72. Springer, Berlin, 2004.
  * [5] David J. Aldous.  Threshold limits for cover times.  J. Theoret. Probab., 4(1):197–211, 1991.
  * [6] David J. Aldous.  When knowing early matters: gossip, percolation and Nash equilibria.  In Prokhorov and contemporary probability theory, volume 33 of Springer Proc. Math. Stat., pages 3–27. Springer, Heidelberg, 2013.
  * [7] David J. Aldous.  Weak concentration for first passage percolation times on graphs and general increasing set-valued processes.  ALEA Lat. Am. J. Probab. Math. Stat., 13(2):925–940, 2016.
  * [8] Moses Charikar, Sudipto Guha, Éva Tardos, and David B. Shmoys.  A constant-factor approximation algorithm for the $k$-median problem.  volume 65, pages 129–149. 2002.  Special issue on STOC, 1999 (Atlanta, GA).
  * [9] Shirshendu Chatterjee and Rick Durrett.  Asymptotic behavior of Aldous’ gossip process.  Ann. Appl. Probab., 21(6):2447–2482, 2011.
  * [10] Jian Ding, James R. Lee, and Yuval Peres.  Cover times, blanket times, and majorizing measures.  Ann. of Math. (2), 175(3):1409–1471, 2012.
  * [11] Moez Draief and Laurent Massoulié.  Epidemics and rumours in complex networks, volume 369 of London Mathematical Society Lecture Note Series.  Cambridge University Press, Cambridge, 2010.
  * [12] Victor Falgas-Ravry, Joel Larsson, and Klas Markström.  Speed and concentration of the covering time for structured coupon collectors.  Adv. in Appl. Probab., 52(2):433–462, 2020.
  * [13] Christina Goldschmidt.  Scaling limits of random trees and random graphs.  In Random graphs, phase transitions, and the Gaussian free field, volume 304 of Springer Proc. Math. Stat., pages 1–33. Springer, Cham, [2020] ©2020.
  * [14] Peter Hall.  Introduction to the theory of coverage processes.  Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics. John Wiley & Sons, Inc., New York, 1988.
  * [15] Svante Janson.  One, two and three times $\log n/n$ for paths in a complete graph with random weights.  volume 8, pages 347–361. 1999.  Random graphs and combinatorial structures (Oberwolfach, 1997).
  * [16] István Z. Kiss, Joel C. Miller, and Péter L. Simon.  Mathematics of epidemics on networks, volume 46 of Interdisciplinary Applied Mathematics.  Springer, Cham, 2017.  From exact to approximate models.
  * [17] Jean-François Le Gall.  Uniqueness and universality of the Brownian map.  Ann. Probab., 41(4):2880–2960, 2013.
  * [18] Tom Leinster and Emily Roff.  The maximum entropy of a metric space.  arXiv:1908.11184v3, 2020.
  * [19] Mathew D. Penrose.  Random euclidean coverage from within, 2021.  arXiv 2101.06306.
  * [20] Sidney I. Resnick.  Extreme values, regular variation, and point processes, volume 4 of Applied Probability. A Series of the Applied Probability Trust.  Springer-Verlag, New York, 1987.
  * [21] Steven Riley, Ken Eames, Valerie Isham, Denis Mollison, and PieterTrapman.  Five challenges for spatial epidemic models.  Epidemics, 10:68–71, 2015.