often overcome by turning on background magnetic fields [12]. Nonzero magnetic
fields can lead to new physical effects, including the appearance of new
vibrational magnetophonon modes [75, 76] or modification of the pinning
frequency [23]. Weak and strong background magnetic fields can be accounted
for in a hydrodynamic theory akin to the work of [77, 78]. We have included
the presence of weak electromagnetic fields in our hydrodynamic framework in
appendix A, but have not studied their physical effects. Accounting for strong
magnetic fields would be especially interesting and requires a non-trivial
extension of our formalism. We hope to return to these phenomena in the
future.

It would be interesting to use the techniques developed here to study
unidirectional charge density wave states, which can be thought of as the
smectic phase of an electronic liquid crystal. While a lot of work has been
done in this direction, we note that we are not aware of a complete
formulation of the corresponding hydrodynamic theory along the lines of [37,
38, 27]. Such a theory, which we expect to be closely related to superfluids
[79], will likely reveal new transport properties. Topological defects, in
this context, can be incorporated as vortices of a single phase field. A
careful analysis of these systems, including the interplay with translational
disorder and impurities, will be enlightening and shall be one of our main
goals moving forward.

Our approach did not assume any particular type of boost symmetry and, as
such, our results are valid for both Galilean and relativistic crystals, or
crystals with no boost symmetry at all. The case of relativistic boost
symmetry is interesting in its own right, given the large body of work on
holographic models of pinned charge density waves (see e.g. [38, 80, 81, 82,
83, 84, 85]), which could potentially be extended to accommodate the presence
of topological defects and plasticity. We have given a manifestly relativistic
formulation of pinned plastic crystals in appendix D. The construction
suggests that holographic models of dislocations and plasticity will require a
bi-metric theory in the bulk, wherein the additional metric will give rise to
a dynamical reference metric at the boundary. It would be interesting to
pursue this line of thought further and construct holographic models for the
solid-liquid phase transitions discussed here.

###### Acknowledgements.

We would like to thank Luca Delacretaz, Blaise Gouteraux, Maziyar Jalaal,
Alexander Krikun, Edan Lerner, and Jan Zaanen for various helpful discussions
and constructive comments on our manuscript. JA and AJ are partly supported by
the Netherlands Organization for Scientific Research (NWO) and by the Dutch
Institute for Emergent Phenomena (DIEP) cluster at the University of
Amsterdam. AJ is funded by the European Union’s Horizon 2020 research and
innovation programme under the Marie Skłodowska-Curie grant agreement NonEqbSK
No. 101027527. RL was supported, in part, by the cluster of excellence ct.qmat
(EXC 2147, project-id 390858490).

##  Appendix A Viscoplastic hydrodynamics with background sources

In this appendix, we revisit the framework of viscoplastic hydrodynamics while
paying careful attention to the background sources. In addition to extending
the applicability of the framework to plastic crystals on curved spacetime
backgrounds and in the presence of background electromagnetic fields, this
will also allow us to obtain the hydrodynamic correlation functions using
background variational methods. We also include the effects of pinning in the
following following the construction of [27].

###  A.1 Aristotelian geometry and conservation laws

We wish to derive the conservation laws for viscoplastic hydrodynamics that
served as the starting point of our discussion in section II.3 and section
V.1. To this end, let us couple our crystal to an Aristotelian spacetime
background, which is the appropriate geometric structure for systems without a
boost symmetry [42, 86, 40]; see also [87, 51, 88, 89, 90]. The spacetime
background includes a _clock form_ $n_{\mu}$ (coupled to the energy density
and flux) and a degenerate _spatial metric_ $h_{\mu\nu}$ (coupled to the
momentum density and the stress tensor). The covariant indices
$\mu,\nu,\ldots$ run over both the spatial indices $i,j,\ldots$ as well as the
time coordinate $t$. The spatial metric admits a null eigenvector $v^{\mu}$,
called the _frame velocity_ , satisfying $v^{\mu}h_{\mu\nu}=0$ and
$v^{\mu}n_{\mu}=1$. Using this, it is also convenient to define a
contravariant spatial metric $h^{\mu\nu}$ using
$h^{\mu\nu}h_{\nu\rho}=\delta^{\mu}_{\rho}-v^{\mu}n_{\rho}$ and
$h^{\mu\nu}n_{\nu}=0$. The flat space (Cartesian) limit is given by

$\displaystyle n_{\mu}\mathrm{d}x^{\mu}=\mathrm{d}t,\qquad
v^{\mu}\partial_{\mu}=\partial_{t},$ $\displaystyle
h_{\mu\nu}\mathrm{d}x^{\mu}\mathrm{d}x^{\nu}=\delta_{ij}\mathrm{d}x^{i}\mathrm{d}x^{j},$
$\displaystyle
h^{\mu\nu}\partial_{\mu}\otimes\partial_{\nu}=\delta^{ij}\partial_{i}\otimes\partial_{j}~{}.$
(89)

We also introduce a background gauge field $A_{\mu}$ to couple to the particle
density and flux.

We can define the covariant version of crystal frame fields as
$e^{I}_{\mu}=\partial_{\mu}\phi^{I}$. We can also define the covariant crystal
velocity $u^{\mu}_{\phi}$ such that $u_{\phi}^{\mu}e^{I}_{\mu}=0$ and
$u_{\phi}^{\mu}n_{\mu}=1$. The induced metric on the crystal space is defined
naturally as $h^{IJ}=h^{\mu\nu}e_{\mu}^{I}e_{\nu}^{J}$, which can be used to
define the strain tensor $\kappa_{IJ}$ same as (5c). The “inverse frame
fields” $e^{\mu}_{I}$ can be defined via
$e^{\mu}_{I}e^{I}_{\nu}=\delta^{\mu}_{\nu}-u^{\mu}_{\phi}n_{\nu}$ and
$n_{\mu}e^{\mu}_{I}=0$. It is worth noting that
$e^{\mu}_{I}=h^{\mu\nu}e_{\nu}^{J}h_{IJ}$.

Consider a crystal described by some effective action $S$, expressed as a
functional of the background fields as well as the dynamical crystal fields.
An infinitesimal variation of $S$ can be parametrized as

$\displaystyle\delta S$
$\displaystyle=\int\\!\mathrm{d}t\mathrm{d}^{d}x\sqrt{\gamma}\bigg{[}\left(\pi^{\mu}v^{\nu}+\frac{1}{2}\tau^{\mu\nu}\right)\\!\delta
h_{\mu\nu}-\epsilon^{\mu}\delta n_{\mu}+j^{\mu}\delta A_{\mu}$
$\displaystyle\qquad\qquad+K_{I}\delta\phi^{I}+\frac{1}{2}U^{IJ}\left(\delta\psi_{IJ}-e_{K}^{\mu}\partial_{\mu}\psi_{IJ}\delta\phi^{K}\right)$
$\displaystyle\qquad\qquad+\ell^{\prime}L_{I}\delta\Phi^{I}\bigg{]},$ (90)

with $\gamma$ denoting the determinant of
$\gamma_{\mu\nu}=h_{\mu\nu}+n_{\mu}n_{\nu}$. This parametrisation defines the
covariant momentum density $\pi^{\mu}$ (with $\pi^{\mu}n_{\mu}=0$), stress
tensor $\tau^{\mu\nu}$ (with $\tau^{\mu\nu}n_{\nu}=0$), energy current
$\epsilon^{\mu}$, and the particle number current $j^{\mu}$. See [42] for more
details. We have parametrised the variations with respect to $\delta\phi^{I}$
and $\delta\mathbb{h}_{IJ}=\ell\delta\psi_{IJ}$ such that the dual the
operators $K_{I}$ and $U^{IJ}$ transform covariantly under the
$\operatorname{Diff}(\phi)$ symmetry in (6). We have also allowed the action
to depend on a set of background crystal fields $\Phi^{I}$, explicitly
breaking the translation symmetry of the crystal and giving rise to the
physics of pinning; see section V or [27] for more discussion.

Taking the variations in (A.1) to be physical spacetime diffeomorphisms and
gauge transformations, we can obtain the respective conservation laws

$\displaystyle\left(\nabla_{\mu}+F^{n}_{\mu\lambda}v^{\lambda}\right)\epsilon^{\mu}$
$\displaystyle=-v^{\nu}\left(F_{\nu\mu}j^{\mu}-F^{n}_{\nu\mu}\epsilon^{\mu}\right)-\tau^{\mu\lambda}h_{\lambda\nu}\nabla_{\mu}v^{\nu}$
$\displaystyle\qquad-
K_{I}v^{\mu}e^{I}_{\mu}-\frac{1}{2}U^{IJ}u^{\mu}_{\phi}\partial_{\mu}\psi_{IJ}$
$\displaystyle\qquad-\ell^{\prime}L_{I}v^{\mu}\partial_{\mu}\Phi^{I}~{}~{},$
(91a)
$\displaystyle\left(\nabla_{\mu}+F^{n}_{\mu\lambda}v^{\lambda}\right)\left(v^{\mu}\pi^{\nu}+\tau^{\mu\nu}\right)$
$\displaystyle=h^{\nu\lambda}\left(F_{\lambda\mu}j^{\mu}-F^{n}_{\lambda\mu}\epsilon^{\mu}\right)-\pi^{\mu}\nabla_{\mu}v^{\nu}$
$\displaystyle\qquad+K_{I}h^{\nu\mu}e^{I}_{\mu}+\ell^{\prime}L_{I}h^{\nu\mu}\partial_{\mu}\Phi^{I}~{}~{},$
(91b)
$\displaystyle\left(\nabla_{\mu}+F^{n}_{\mu\lambda}v^{\lambda}\right)j^{\mu}=0~{}~{}.$
(91c)

Here $F_{\mu\nu}=2\partial_{[\mu}A_{\nu]}$ and
$F^{n}_{\mu\nu}=2\partial_{[\mu}n_{\nu]}$. The covariant derivative
$\nabla_{\mu}$ is defined with respect to the torsional Aristotelian
connection

$\Gamma^{\lambda}_{\mu\nu}=v^{\lambda}\partial_{\mu}n_{\nu}+\frac{1}{2}h^{\lambda\rho}\left(\partial_{\mu}h_{\nu\rho}+\partial_{\nu}h_{\mu\rho}-\partial_{\rho}h_{\mu\nu}\right).$
(92)

Restricting the background spacetime sources to their flat form in (89), the
conservation laws reduce to their flat space versions given in (21) and (63).

To derive the configuration equations, we need to consider the coupling of the
action to background sources like (12). However, in the dynamical case, we can
also introduce an external momentum source $\Pi_{\mu}^{\text{ext}}$ for the
crystal velocity $u_{\phi}^{\mu}$ in addition to an external stress source
$T^{\mu\nu}_{\text{ext}}$ for the strain tensor
$\kappa_{\mu\nu}=e_{\mu}^{I}e_{\nu}^{J}\kappa_{IJ}$. We take the source action
to be

$S_{\text{source}}=\int\mathrm{d}t\mathrm{d}^{d}x\sqrt{\gamma}\bigg{[}T^{\mu\nu}_{\text{ext}}\kappa_{\mu\nu}+\Pi_{\mu}^{\text{ext}}u_{\phi}^{\mu}\bigg{]}.$
(93)

These couplings are, of course, manifestly invariant under spacetime
diffeomorphisms and gauge transformations. Variations of the source action can
be obtained to be

$\displaystyle\delta S_{\text{source}}$
$\displaystyle=\int\mathrm{d}t\mathrm{d}^{d}x\sqrt{\gamma}\Bigg{[}\kappa_{\mu\nu}\delta
T^{\mu\nu}_{\text{ext}}+u_{\phi}^{\mu}\delta\Pi_{\mu}^{\text{ext}}$
$\displaystyle\qquad+K_{I}^{\text{ext}}\delta\phi^{I}+\frac{1}{2}U^{IJ}_{\text{ext}}\left(\delta\psi_{IJ}-e_{K}^{\mu}\partial_{\mu}\psi_{IJ}\delta\phi^{K}\right)$
$\displaystyle-\left(\bar{T}^{\mu\rho}_{\text{ext}}h_{\rho\nu}u^{\nu}_{\phi}-T^{\rho\sigma}_{\text{ext}}\kappa_{\rho\sigma}v^{\mu}+\Pi_{\rho}^{\text{ext}}u_{\phi}^{\rho}(u^{\mu}_{\phi}-v^{\mu})\right)\delta
n_{\mu}$
$\displaystyle+\left(\bar{T}^{\mu\nu}_{\text{ext}}+\left(T^{\rho\sigma}_{\text{ext}}\kappa_{\rho\sigma}+\Pi_{\rho}^{\text{ext}}u_{\phi}^{\rho}\right)h^{\mu\nu}\right)\frac{1}{2}\delta
h_{\mu\nu}\Bigg{]},$ (94)

where
$\bar{T}^{\mu\nu}_{\text{ext}}=T^{\rho\sigma}_{\text{ext}}(\delta_{\rho}^{\mu}-n_{\rho}u^{\mu}_{\phi})(\delta_{\sigma}^{\nu}-n_{\sigma}u^{\nu}_{\phi})$
is the external stress tensor projected against the crystal velocity and we
have defined

$\displaystyle U^{IJ}_{\text{ext}}$ $\displaystyle=-\ell
T^{\mu\nu}_{\text{ext}}e_{\mu}^{I}e_{\nu}^{J},$ $\displaystyle
K_{I}^{\text{ext}}$
$\displaystyle=\left(\nabla_{\mu}+F^{n}_{\mu\lambda}v^{\lambda}\right)\left(T^{\mu\nu}_{\text{ext}}e_{\nu}^{J}\mathbb{h}_{JI}+\Pi_{\nu}^{\text{ext}}e^{\nu}_{I}u^{\mu}_{\phi}\right)$
$\displaystyle\qquad+\frac{1}{2}U^{JK}_{\text{ext}}e_{I}^{\mu}\partial_{\mu}\psi_{JK}~{}.$
(95)

The terms in the last two lines of (A.1) tell us how the definitions of the
conserved currents obtained by varying the total action $S+S_{\text{source}}$
with respect to the background fields gets contributions from the crystal
sources. These will play a role in the computation of correlation functions in
A.4. The terms in the second line tell us that the equations of motion for
$\phi^{I}$ and $\psi_{IJ}$ are given by (14). Finally, the terms in the first
line are the coupling terms for the strain and crystal velocity operators.

###  A.2 Covariant viscoplastic hydrodynamics

We now redo the second law analysis for viscoplastic hydrodynamics from
section II.3 and for pinned viscoplastic hydrodynamics from section V.1,
keeping track of all the background fields. We define the covariant fluid
velocity $u^{\mu}$, normalised as $u^{\mu}n_{\mu}=1$. We can isolate the
spatial components of the fluid velocity with respect to the background frame
velocity as $\vec{u}^{\mu}=u^{\mu}-v^{\mu}$, with
$\vec{u}_{\mu}=h_{\mu\nu}\vec{u}^{\nu}$. The thermodynamic relations take the
same form as (13c) and (62), i.e.

$\displaystyle\mathrm{d}\epsilon$
$\displaystyle=T\mathrm{d}s+\mu\mathrm{d}n+\frac{1}{2\rho}\mathrm{d}\vec{\pi}^{2}-\frac{1}{2}r_{IJ}\mathrm{d}h^{IJ}-\frac{1}{2}\mathbb{r}^{IJ}\mathrm{d}\mathbb{h}_{IJ}$
$\displaystyle\qquad+\ell^{\prime
2}m^{2}h_{IJ}(\phi^{J}-\Phi^{J})\,\mathrm{d}(\phi^{I}-\Phi^{I}),$
$\displaystyle\mathrm{d}p$
$\displaystyle=s\mathrm{d}T+n\mathrm{d}\mu+\frac{1}{2}\rho\mathrm{d}\vec{u}^{2}+\frac{1}{2}r_{IJ}\mathrm{d}h^{IJ}+\frac{1}{2}\mathbb{r}^{IJ}\mathrm{d}\mathbb{h}_{IJ}$
$\displaystyle\qquad-\ell^{\prime
2}m^{2}h_{IJ}(\phi^{J}-\Phi^{J})\,\mathrm{d}(\phi^{I}-\Phi^{I}),$
$\displaystyle\epsilon$ $\displaystyle=-p+Ts+\mu n+u^{\mu}\pi_{\mu}.$ (96)

The momentum density related to the fluid velocity as
$\pi_{\mu}=\rho\vec{u}^{\nu}$. We have also used
$\vec{\pi}^{2}=\pi_{\mu}\pi_{\nu}h^{\mu\nu}=\rho^{2}\vec{u}^{2}$.

The statement of the second law of thermodynamics in the covariant language is
that there must exist an entropy current $s^{\mu}$ such that

$\left(\nabla_{\mu}+F^{n}_{\mu\lambda}v^{\lambda}\right)s^{\mu}=\Delta\geq 0.$
(97)

We start with an appropriately covariantised parametrisation of the
constitutive relations (24) and (64), i.e.

$\displaystyle\epsilon^{\mu}$ $\displaystyle=\epsilon
u^{\mu}+p\,\vec{u}^{\mu}+r_{IJ}e^{I\mu}v^{\nu}e^{J}_{\nu}+{\cal
T}^{\mu\nu}\vec{u}_{\nu}+{\cal E}^{\mu},$ $\displaystyle\tau^{\mu\nu}$
$\displaystyle=\rho\vec{u}^{\mu}\vec{u}^{\nu}+p\,h^{\mu\nu}-r_{IJ}e^{I\mu}e^{J\nu}+{\cal
T}^{\mu\nu},$ $\displaystyle j^{\mu}$ $\displaystyle=nu^{\mu}+{\cal J}^{\mu},$
$\displaystyle K_{I}$
$\displaystyle=-\left(\nabla_{\mu}+F^{n}_{\mu\lambda}v^{\lambda}\right)\\!\left(r_{IJ}e^{J\mu}\right)+\frac{\ell}{2}\mathbb{r}^{JK}e_{I}^{\mu}\partial_{\mu}\psi_{JK}$
$\displaystyle\qquad-\ell^{\prime
2}m^{2}h_{IJ}\left(\phi^{J}-\Phi^{J}\right)+{\cal K}_{I},$ $\displaystyle
U^{IJ}$ $\displaystyle=\ell\mathbb{r}^{IJ}+{\cal U}^{IJ},$ $\displaystyle
L_{I}$
$\displaystyle=\ell^{\prime}m^{2}h_{IJ}\left(\phi^{J}-\Phi^{J}\right)+{\cal
L}_{I}~{}~{},$ (98)

where and $e^{I\mu}=h^{\mu\nu}e^{I}_{\nu}$. The covariant parametrisation of
the constitutive relations has been motivated from [42]. All the dissipative
corrections are defined to be transverse to $n_{\mu}$. Let us pretend that the
entropy current is given by the ideal form $su^{\mu}$. By a straight-forward
computation, we can find that

$\displaystyle
T\left(\nabla_{\mu}+F^{n}_{\mu\lambda}v^{\lambda}\right)(su^{\mu})$
$\displaystyle=-{\cal
E}^{\mu}\left(\frac{1}{T}\partial_{\mu}T-u^{\nu}F^{n}_{\nu\mu}\right)$
$\displaystyle\qquad-{\cal
T}^{\mu\nu}h_{\nu\rho}\left(\nabla_{\mu}u^{\rho}-u^{\rho}u^{\lambda}F^{n}_{\lambda\mu}\right)$
$\displaystyle\qquad-{\cal
J}^{\mu}\left(T\partial_{\mu}\frac{\mu}{T}+u^{\nu}F_{\nu\mu}\right)$
$\displaystyle\qquad-{\cal K}_{I}u^{\nu}e^{I}_{\nu}-\frac{1}{2}{\cal
U}^{IJ}u^{\mu}_{\phi}\partial_{\mu}\psi_{IJ}-\ell^{\prime}{\cal
L}_{I}u^{\mu}\partial_{\mu}\Phi^{I}$
$\displaystyle\qquad-T\left(\nabla_{\mu}+F^{n}_{\mu\lambda}v^{\lambda}\right)\\!\left(\frac{1}{T}{\cal
E}^{\mu}-\frac{\mu}{T}{\cal J}^{\mu}\right).$ (99)

From here, we can read out the corrected entropy current

$s^{\mu}=su^{\mu}+\frac{1}{T}{\cal E}^{\mu}-\frac{\mu}{T}{\cal J}^{\mu},$
(100)

while the dissipation rate is given as

$\displaystyle T\Delta$ $\displaystyle=-{\cal
E}^{\mu}\left(\frac{1}{T}\partial_{\mu}T-u^{\nu}F^{n}_{\nu\mu}\right)$
$\displaystyle\qquad-{\cal
T}^{\mu\nu}h_{\nu\rho}\left(\nabla_{\mu}u^{\rho}-u^{\rho}u^{\lambda}F^{n}_{\lambda\mu}\right)$
$\displaystyle\qquad-{\cal
J}^{\mu}\left(T\partial_{\mu}\frac{\mu}{T}+u^{\nu}F_{\nu\mu}\right)$
$\displaystyle\qquad-{\cal K}_{I}u^{\nu}e^{I}_{\nu}-\frac{1}{2}{\cal
U}^{IJ}u^{\mu}_{\phi}\partial_{\mu}\psi_{IJ}$
$\displaystyle\qquad-\ell^{\prime}{\cal L}_{I}u^{\mu}\partial_{\mu}\Phi^{I},$
(101)

which generalises (27) and (65) to curved spacetime backgrounds. The
discussion for one-derivative order constitutive relations in section II.4 and
V.1 can be generalised to curved background by simply replacing

$\displaystyle e^{Ii}\frac{1}{T}\partial_{i}T$ $\displaystyle\to
e^{I\mu}\left(\frac{1}{T}\partial_{\mu}T-u^{\nu}F^{n}_{\nu\mu}\right),$
$\displaystyle\partial_{k}u^{k}$
$\displaystyle\to\left(\nabla_{\mu}+F^{n}_{\mu\lambda}v^{\lambda}\right)u^{\mu},$
$\displaystyle 2e^{\langle I}_{i}e^{J\rangle}_{j}\partial^{i}u^{j}$
$\displaystyle\to 2e^{\langle
I\mu}e^{J\rangle\nu}h_{\nu\rho}\left(\nabla_{\mu}u^{\rho}-u^{\rho}u^{\lambda}F^{n}_{\lambda\mu}\right),$
$\displaystyle e^{Ii}T\partial_{i}\frac{\mu}{T}$ $\displaystyle\to
e^{I\mu}\left(T\partial_{\mu}\frac{\mu}{T}+u^{\nu}F_{\nu\mu}\right),$
$\displaystyle(\partial_{t}+u^{i}\partial_{i})\phi^{I}$ $\displaystyle\to
u^{\mu}\partial_{\mu}\phi^{I},$
$\displaystyle(\partial_{t}+u^{i}_{\phi}\partial_{i})\psi_{IJ}$
$\displaystyle\to u^{\mu}_{\phi}\partial_{\mu}\psi_{IJ},$
$\displaystyle(\partial_{t}+u^{i}\partial_{i})\Phi^{I}$ $\displaystyle\to
u^{\mu}\partial_{\mu}\Phi^{I},$ (102)

together with the parametrisation of the derivative corrections ${\cal
E}^{\mu}=e^{\mu}_{I}{\cal E}^{I}$, ${\cal
T}^{\mu\nu}=e^{\mu}_{I}e^{\nu}_{J}{\cal T}^{IJ}$, and ${\cal
J}^{\mu}=e^{\mu}_{I}{\cal J}^{I}$.

###  A.3 Without energy sources

Let us record the version of the discussion above when the clock form
$n_{\mu}$ is fixed to its flat spacetime form. This will allow us to compute
correlations of all observables except the energy current, which we have
ignored in the core of our paper as we neglected temperature fluctuations.
Setting $n_{\mu}\mathrm{d}x^{\mu}=\mathrm{d}t$ means that

$\displaystyle v^{\mu}\partial_{\mu}$
$\displaystyle=\partial_{t}+v^{i}\partial_{i},$ $\displaystyle
h_{\mu\nu}\mathrm{d}x^{\mu}\mathrm{d}x^{\nu}$
$\displaystyle=\vec{v}^{2}\mathrm{d}t^{2}-2v_{i}\mathrm{d}x^{i}\mathrm{d}t+g_{ij}\mathrm{d}x^{i}\mathrm{d}x^{j},$
$\displaystyle h^{\mu\nu}\partial_{\mu}\otimes\partial_{\nu}$
$\displaystyle=g^{ij}\partial_{i}\otimes\partial_{j},$ (103)

where $\vec{v}^{2}=g_{ij}v^{i}v^{j}$. The spatial indices in the following are
raised/lowered using the spatial metric $g_{ij}$ and its inverse $g^{ij}$. The
coupling structure (A.1) gives rise to

$\displaystyle\delta S$
$\displaystyle=\int\\!\mathrm{d}t\mathrm{d}^{d}x\sqrt{g}\bigg{[}-\pi_{i}\delta
v^{i}+\frac{1}{2}\tau^{ij}\delta h_{ij}+n\delta A_{t}+j^{i}\delta A_{i}$
$\displaystyle\qquad\qquad+K_{I}\delta\phi^{I}+\frac{1}{2}U^{IJ}\left(\delta\psi_{IJ}-e_{K}^{k}\partial_{k}\psi_{IJ}\delta\phi^{K}\right)$
$\displaystyle\qquad\qquad+\ell^{\prime}L_{I}\delta\Phi^{I}\bigg{]}.$ (104)

Since only the spatial components of the strain tensor $\kappa_{ij}$ and
crystal velocity $u^{i}_{\phi}$ are independent, we can switch off the time-
components of their respective sources $T^{tt}_{\text{ext}}$,
$T^{ti}_{\text{ext}}$, $\Pi_{t}^{\text{ext}}$ for simplicity. The variation of
the source action in (A.1) then becomes

$\displaystyle\delta S_{\text{source}}$
$\displaystyle=\int\mathrm{d}t\mathrm{d}^{d}x\sqrt{\gamma}\Bigg{[}\kappa_{ij}\delta
T^{ij}_{\text{ext}}+u_{\phi}^{i}\delta\Pi_{i}^{\text{ext}}$
$\displaystyle\qquad+K_{I}^{\text{ext}}\delta\phi^{I}+\frac{1}{2}U^{IJ}_{\text{ext}}\left(\delta\psi_{IJ}-e_{K}^{k}\partial_{k}\psi_{IJ}\delta\phi^{K}\right)$
$\displaystyle+\left(T^{ij}_{\text{ext}}+\left(T^{kl}_{\text{ext}}\kappa_{kl}+\Pi_{k}^{\text{ext}}u_{\phi}^{k}\right)g^{ij}\right)\frac{1}{2}\delta
g_{ij}\Bigg{]},$ (105)

where the definitions of $U^{IJ}_{\text{ext}}$ and $K_{I}^{\text{ext}}$ become

$\displaystyle U^{IJ}_{\text{ext}}$ $\displaystyle=-\ell
T^{ij}_{\text{ext}}e_{i}^{I}e_{j}^{J},$ $\displaystyle K_{I}^{\text{ext}}$
$\displaystyle=\frac{1}{\sqrt{g}}\partial_{t}\left(\sqrt{g}\,\Pi_{i}^{\text{ext}}e^{i}_{I}\right)+\nabla_{i}\\!\left(T^{ij}_{\text{ext}}e_{j}^{J}\mathbb{h}_{JI}+\Pi_{k}^{\text{ext}}e^{k}_{I}u^{i}_{\phi}\right)$
$\displaystyle\qquad+\frac{1}{2}U^{JK}_{\text{ext}}e_{I}^{k}\partial_{k}\psi_{JK}~{}.$
(106)

The conservation equations, on the other hand, reduce to the non-covariant
form

$\displaystyle\frac{1}{\sqrt{g}}\partial_{t}(\sqrt{g}\,\epsilon)+\nabla_{i}\epsilon^{i}$
$\displaystyle=E_{i}\left(j^{i}-nv^{i}\right)-\tau^{ij}\left(\nabla_{i}v_{j}+{\textstyle\frac{1}{2}}\partial_{t}g_{ij}\right)$
$\displaystyle\qquad-
K_{I}\left(e^{I}_{t}+v^{i}e^{I}_{i}\right)-\frac{1}{2}U^{IJ}(\partial_{t}+u^{i}_{\phi}\partial_{i})\psi_{IJ}$
$\displaystyle\qquad-\ell^{\prime}L_{I}\left(\partial_{t}\Phi^{I}+v^{i}\partial_{i}\Phi^{I}\right)~{}~{},$
(107a)
$\displaystyle\frac{1}{\sqrt{g}}\partial_{t}(\sqrt{g}\,\pi_{i})+\nabla_{j}\left(v^{j}\pi_{i}+\tau^{j}_{~{}i}\right)$
$\displaystyle=E_{i}n+F_{ij}(j^{j}-nv^{j})-\pi_{k}\nabla_{i}v^{k}$
$\displaystyle\qquad+K_{I}e^{I}_{i}+\ell^{\prime}L_{I}\partial_{i}\Phi^{I}~{}~{},$
(107b)
$\displaystyle\frac{1}{\sqrt{g}}\partial_{t}(\sqrt{g}\,n)+\nabla_{i}j^{i}=0~{}~{}.$
(107c)

Here $E_{i}=F_{it}+F_{ij}v^{j}$ is the electric field with respect to the
background frame velocity $v^{i}$. Also, $\nabla_{i}$ is the spatial covariant
derivative with respect to $g_{ij}$. We have also identified the densities
$\epsilon=\epsilon^{\mu}n_{\mu}$ and $n=j^{\mu}n_{\mu}$. We can see that the
energy fluctuations decouple from the momentum and density fluctuations and
thus can be ignored in the isothermal limit.

The constitutive relations are given now given in their non-covariant form

$\displaystyle\epsilon^{i}$
$\displaystyle=\epsilon\,u^{i}+p\,\vec{u}^{i}+r_{IJ}e^{Ii}\left(e^{J}_{t}+v^{k}e^{J}_{k}\right)+{\cal
T}^{ij}\vec{u}_{j}+{\cal E}^{i},$ $\displaystyle\tau^{ij}$
$\displaystyle=\rho\vec{u}^{i}\vec{u}^{j}+p\,h^{ij}-r_{IJ}e^{Ii}e^{Jj}+{\cal
T}^{ij},$ $\displaystyle j^{i}$ $\displaystyle=nu^{i}+{\cal J}^{i},$
$\displaystyle K_{I}$
$\displaystyle=-\nabla_{i}\\!\left(r_{IJ}e^{Ji}\right)+\frac{\ell}{2}\mathbb{r}^{JK}e_{I}^{k}\partial_{k}\psi_{JK}$
$\displaystyle\qquad-\ell^{\prime
2}m^{2}h_{IJ}\left(\phi^{J}-\Phi^{J}\right)+{\cal K}_{I},$ $\displaystyle
U^{IJ}$ $\displaystyle=\ell\mathbb{r}^{IJ}+{\cal U}^{IJ},$ $\displaystyle
L_{I}$
$\displaystyle=\ell^{\prime}m^{2}h_{IJ}\left(\phi^{J}-\Phi^{J}\right)+{\cal
L}_{I}~{}~{}.$ (108)

Note that $u^{i}$ and $\vec{u}^{i}$ are generically different when the frame
velocity $v^{i}\neq 0$, i.e. $\vec{u}^{i}=u^{i}-v^{i}$. Both of these are
appropriate notions of fluid velocity. On the one hand, $u^{i}$ is aligned
parallel to the (ideal order) particle flux, while $\vec{u}^{i}$ is aligned
parallel to the flow of momentum $\pi_{i}=\rho\vec{u}_{i}$. The constitutive
relations in section II.4 and V.1 can be generalised to this particular
background by replacing

$\displaystyle\partial_{k}u^{k}$
$\displaystyle\to\nabla_{k}u^{k}+\frac{1}{2}g^{ij}\partial_{t}g_{ij},$
$\displaystyle 2e^{\langle I}_{i}e^{J\rangle}_{j}\partial^{i}u^{j}$
$\displaystyle\to 2e^{\langle Ii}e^{J\rangle
j}\left(\nabla_{i}u_{j}+\frac{1}{2}\partial_{t}g_{ij}\right),$ $\displaystyle
e^{Ii}T\partial_{i}\frac{\mu}{T}$ $\displaystyle\to
e^{Ii}\left(T\partial_{i}\frac{\mu}{T}-E_{i}-F_{ij}\vec{u}^{j}\right).$ (109)

###  A.4 Correlation functions via background variation

We can compute the retarded correlation functions of hydrodynamic observables
by inspecting how they respond to changes in the associated sources. We will
ignore the energy density and flux for clarity, although the procedure works
exactly the same as follows. First, we need the one point functions of various
operators in the presence of background sources. These can be obtained by
taking variations of the total onshell action
$S_{\text{tot}}=(S+S_{\text{source}})_{\text{onshell}}$ with respect to the
appropriate background fields. We find that

$\displaystyle\langle\pi_{i}\rangle$ $\displaystyle=-\frac{\delta
S_{\text{onshell}}}{\delta v^{i}}=\sqrt{g}\,\pi_{i},$
$\displaystyle\langle\tau^{ij}\rangle$ $\displaystyle=2\frac{\delta
S_{\text{onshell}}}{\delta
g_{ij}}=\sqrt{g}\Big{(}\tau^{ij}+T^{ij}_{\text{ext}}$ $\displaystyle\hskip
50.00008pt+T^{kl}_{\text{ext}}\kappa_{kl}g^{ij}+\Pi_{k}^{\text{ext}}u^{k}_{\phi}g^{ij}\Big{)},$
$\displaystyle\langle n\rangle$ $\displaystyle=\frac{\delta
S_{\text{onshell}}}{\delta A_{t}}=\sqrt{g}\,n,$ $\displaystyle\langle
j^{i}\rangle$ $\displaystyle=\frac{\delta S_{\text{onshell}}}{\delta
A_{i}}=\sqrt{g}\,j^{i},$ $\displaystyle\langle\kappa_{ij}\rangle$
$\displaystyle=\frac{\delta S_{\text{onshell}}}{\delta
T^{ij}_{\text{ext}}}=\sqrt{g}\,\kappa_{ij},$ $\displaystyle\langle
u^{i}_{\phi}\rangle$ $\displaystyle=\frac{\delta
S_{\text{onshell}}}{\delta\Pi_{i}^{\text{ext}}}=\sqrt{g}\,u^{i}_{\phi}.$ (110)

All the objects on the right should be understood as evaluated on the
solutions of the equations of motion, so they only depend on the background
sources. In the absence of sources, they reduce to the equilibrium values of
the respective operators. Note that for a Galilean system, even when the
crystal sources are absent, the particle flux $j^{i}$ is the same as the
momentum density $\pi_{i}$ only up to the frame velocity $v^{i}$ (in addition
to the metric factor). This difference manifests itself as slight differences
in their correlation functions.

Using the short-hand notation for the operators and respective sources

$\displaystyle{\cal O}$
$\displaystyle=\Big{(}\pi_{i},\tau^{ij},n,j^{i},\kappa_{ij},u^{i}_{\phi}\Big{)},$
$\displaystyle J$
$\displaystyle=\Big{(}{-v^{i}},{\textstyle\frac{1}{2}}g_{ij},A_{t},A_{i},T^{ij}_{\text{ext}},\Pi_{i}^{\text{ext}}\Big{)},$
(111)

the retarded correlation functions can be read off as

$G^{R}_{{\cal O}_{1}{\cal O}_{2}}=-\frac{\delta\langle{\cal
O}_{1}\rangle}{\delta J_{2}}\bigg{|}_{\text{flat}}.$ (112)

This procedure has been used to compute the correlation functions reported in
sections III.5 and V.5.

If the microscopic description underlying the crystal under consideration is
invariant under time-reversal symmetry, the correlation functions in momentum
space satisfy the so-called Onsager reciprocal relations

$G^{R}_{{\cal O}_{1}{\cal
O}_{2}}(\omega,k)=\eta^{T}_{1}\eta_{2}^{T}G^{R}_{{\cal O}_{2}{\cal
O}_{1}}(\omega,-k),$ (113)

where $\eta^{T}=(-,+,+,-,+,-)$ are the time-reversal eigenvalues of the
respective operators. These lead to the relations between the primed and un-
primed off-diagonal dissipative coefficients in (30a) and (67a). These
constraints can be adapted to alternative discrete symmetries that the crystal
might possess, like CPT, by using the appropriate eigenvalues of operators in
(113).

##  Appendix B Material diagrams for total matter displacement

In this appendix, we continue the discussion initiated in section III.3
regarding the type of materials described by our framework. As we noted
towards the end of that section, the form of the rheology equations depend on
the observable being probed, e.g. displacements of just the crystalline matter
or displacements of the total matter including interstitials. In view of the
fact that the distinction between the crystalline and interstitial matter in a
plastic crystal is a bit artificial because the two are not individually
conserved, let us look at the rheology equations from the perspective of the
total matter evolution, crystal plus interstitials. To this end, we define the
shear tensor with respect to the particle flux

$\displaystyle K_{ij}$
$\displaystyle=\frac{1}{n}\partial_{(i}j_{j)}=\dot{\varepsilon}_{ij}+\frac{1}{n}\partial_{(i}j^{\Delta}_{j)}$
$\displaystyle=\dot{\varepsilon}_{ij}-D_{\Delta}^{\|}\partial_{i}\partial_{j}\kappa^{k}_{~{}k}$
$\displaystyle\qquad-2D_{\Delta}^{\perp}\\!\left(\partial_{(i}\partial^{k}\kappa_{j)k}{\,-\,}\partial_{i}\partial_{j}\kappa^{k}_{~{}k}\right).$
(114)

For a Galilean crystal, this object is the same as the fluid shear tensor
$\partial_{(i}u_{j)}$ and serves as the equivalent of $\dot{\varepsilon}_{ij}$
for total matter evolution. Note that these expressions are only exact
linearly and when both $\mu$ and $T$ are fixed to their equilibrium values. We
can now express the rheology equations (39) in terms of $K_{ij}$ instead of
$\dot{\varepsilon}_{ij}$. We find

$\displaystyle\tau_{ij}$
$\displaystyle=-\delta_{ij}\left(B\kappa^{k}_{~{}k}+\zeta K^{k}_{~{}k}\right)$
$\displaystyle\quad-2G\kappa_{\langle ij\rangle}-2\eta K_{\langle ij\rangle},$
(115a) $\displaystyle\dot{\kappa}_{ij}$
$\displaystyle=\frac{1}{d}\delta_{ij}\left(K^{k}_{~{}k}-\Omega_{B}\kappa^{k}_{~{}k}\right)$
$\displaystyle\quad+K_{\langle ij\rangle}-\Omega_{G}\kappa_{\langle
ij\rangle}~{}~{}$
$\displaystyle\quad+D_{\Delta}^{\|}\partial_{i}\partial_{j}\kappa^{k}_{~{}k}+2D_{\Delta}^{\perp}\\!\left(\partial_{i}\partial^{k}\kappa_{jk}{\,-\,}\partial_{i}\partial_{j}\kappa^{k}_{~{}k}\right),$
(115b)

where we have ignored subleading terms in the derivative expansion as well as
the thermodynamic pressure, which becomes a constant when $T$ and $\mu$ are
frozen.

Due to the presence of diffusive corrections, the rheology equations (115) do
not admit a neat split into shear and bulk sectors. Instead, to be able to
draw the respective material diagrams, we need to perform a wavevector
decomposition of $K_{ij}$ and $\tau_{ij}$ as follows

$\displaystyle K_{ij}$
$\displaystyle=\frac{k_{i}k_{j}}{k^{2}}K^{\|}+\frac{k_{(i}}{k}K_{j)}^{\perp},$
$\displaystyle\tau_{ij}$
$\displaystyle=\frac{k_{i}k_{j}}{k^{2}}\tau^{\|}+2\frac{k_{(i}}{k}\tau_{j)}^{\perp}+\left(\delta_{ij}-\frac{k_{i}k_{j}}{k^{2}}\right)\tau^{\mathsf{T}}+\tau^{\mathsf{TT}}_{ij}.$
(116)

The transverse vector components $K_{i}^{\perp}$, $\tau_{i}^{\perp}$ are
transverse to $k^{i}$, while $\tau^{\mathsf{TT}}_{ij}$ is transverse and
traceless. Note that $K_{ij}$ is a pure gradient and does not get purely
transverse contributions. In addition $K^{\|}$ and $\tau^{\|}$ are the
longitudinal parts of $K_{ij}$ and $\tau_{ij}$ respectively, while
$\tau^{\mathsf{T}}$ is the transverse trace of $\tau_{ij}$. Note that the
crystal strain tensor $\kappa_{ij}$ was already decomposed in this format in
(42a).

Given this wavevector decomposition, we find that the transverse traceless
sector decouples trivially with

$\displaystyle\tau_{ij}^{\mathsf{TT}}$
$\displaystyle=-2G\kappa_{ij}^{\mathsf{TT}},$
$\displaystyle\dot{\kappa}_{ij}^{\mathsf{TT}}$
$\displaystyle=-\Omega_{G}\kappa^{\mathsf{TT}}_{ij},$
$\displaystyle\implies\dot{\tau}_{ij}^{\mathsf{TT}}$
$\displaystyle+\Omega_{G}\tau^{\mathsf{TT}}_{ij}=0~{},$ (117)

and has no contribution to the matter distortion strain.

Next, we have the vector sector with

$\displaystyle\tau^{\perp}_{i}$ $\displaystyle=-G\kappa^{\perp}_{i}-\eta
K^{\perp}_{i}$ $\displaystyle\dot{\kappa}^{\perp}_{i}$
$\displaystyle=K^{\perp}_{i}-\left(\Omega_{G}+k^{2}D_{\phi}^{\perp}\right)\kappa^{\perp}_{i},$
$\displaystyle\implies\dot{\tau}^{\perp}_{i}$
$\displaystyle+\left(\Omega_{G}+k^{2}D_{\Delta}^{\perp}\right)\tau^{\perp}_{i}=$
$\displaystyle-\left(\lambda_{G}^{2}G+\eta\left(\Omega_{G}+k^{2}D_{\Delta}^{\perp}\right)\right)\dot{K}^{\perp}_{i}-\eta\ddot{K}^{\perp}_{i},$
(118)

which behaves like a “$k$-dependent” Jeffrey material, with the relaxation
rate $\Omega_{G}$ replaced with $\Omega_{G}+k^{2}D_{\Delta}^{\perp}$. We have
given the corresponding material diagram in figure 7(a). In detail, we can
label the spring and dashpot components in the top elastic arm of the diagram
by $E_{1}$ and $\upsilon_{1}$ respectively, while the dashpot in the bottom
arm by $\upsilon_{2}$. Using circuit rules, the diagram gives rise to the
stress-strain relations

$\dot{\tau}^{\perp}_{i}+\frac{E_{1}}{\upsilon_{1}}\tau^{\perp}_{i}=-E^{\perp}_{1}\left(1+\frac{\upsilon_{2}}{\upsilon_{1}}\right)K^{\perp}_{i}-\upsilon_{2}\dot{K}^{\perp}_{i}.$
(119)

We can compare this to (B) to read off

$E_{1}=G,\qquad\upsilon_{1}=\frac{G}{\Omega_{G}+k^{2}D^{\perp}_{\Delta}},\qquad\upsilon_{2}=\eta.$
(120)

This is the usual Jeffrey’s model with $k$-dependence in one of the dashpots.
One learns that the effect of Goldstone (interstitial) diffusion in the
rheology equation is that the dashpot component in the elastic arm of the
circuit representation gets a wavenumber dependence.

(a) Circuit representation of the vector sector (B). The dashpot component in
the lower arm measures $K_{i}^{\perp}$. The dashpot component in the top
elastic arm has $k$-dependence.

(b) Circuit representation of the scalar sector (121). The shared dashpot
component measures $K^{\parallel}$ and connects two non-parallel circuits. All
the other components have $k$-dependence.

Figure 7: Circuit representation of the rheology equations from the
perspective of total matter displacement.

Finally, the diffusive corrections non-trivially couple the two scalar
sectors. It is neater to work in a basis with isotropic trace stress
$\tau=\frac{1}{d}\tau^{\|}+\frac{d-1}{d}\tau^{\mathsf{T}}$ and scalar shear
stress $\tau_{G}=\frac{1}{2}\tau_{\|}-\frac{1}{2}\tau^{\mathsf{T}}$, and
similarly for the strain tensor $\kappa=\kappa^{\|}+(d-1)\kappa^{\mathsf{T}}$
and $\kappa_{G}=\kappa_{\|}-\kappa^{\mathsf{T}}$. In the trace sector, we find

$\displaystyle\tau$ $\displaystyle=-B\kappa-\zeta K^{\|},$
$\displaystyle\dot{\kappa}$
$\displaystyle=K^{\|}-\left(\Omega_{B}+D_{\Delta}^{\|}k^{2}-{\textstyle
2\frac{d-1}{d}}D_{\Delta}^{\perp}k^{2}\right)\kappa$
$\displaystyle\qquad-{\textstyle
2\frac{d-1}{d}}D_{\Delta}^{\perp}k^{2}\kappa_{G}.$ (121a) On the other hand,
in the scalar shear sector we have $\displaystyle\tau_{G}$
$\displaystyle=-G\kappa_{G}-\eta K^{\|},$ $\displaystyle\dot{\kappa}_{G}$
$\displaystyle=K^{\|}-\left(\textstyle\Omega_{G}+2\frac{d-1}{d}D_{\Delta}^{\perp}k^{2}\right)\kappa_{G}$
$\displaystyle\qquad-k^{2}\left(\textstyle
D_{\Delta}^{\|}-2\frac{d-1}{d}D_{\Delta}^{\perp}\right)\kappa~{}.$ (121b)

If we ignore the $k$-dependence in these equations for the moment, both these
sectors behave as $k$-independent Jeffrey materials, but with the shared
$K_{\|}$. This sharing is merely a manifestation of the fact that $K_{ij}$,
like $\dot{\varepsilon}_{ij}$, is a pure gradient and has only one scalar
component which couples to both the scalar components of the stress tensor
$\tau$ and $\tau_{G}$. Restoring the $k$-dependence leads to additional non-
trivial couplings between the two sectors due to the diffusive terms. After
eliminating $\kappa$ and $\kappa_{G}$, we arrive at “generalised Maxwell
materials” [91] in both the sectors, still sharing the same $K_{\|}$, to wit

$\displaystyle\ddot{\tau}+\Omega_{1}\dot{\tau}+\Omega_{2}\tau$
$\displaystyle=-\alpha_{1}K^{\|}-\alpha_{2}\dot{K}^{\|}-\zeta\ddot{K}^{\|},$
$\displaystyle\ddot{\tau}_{G}+\Omega_{1}\dot{\tau}_{G}+\Omega_{2}\tau_{G}$
$\displaystyle=-\alpha_{1}^{G}K^{\|}-\alpha_{2}^{G}\dot{K}^{\|}-\eta\ddot{K}^{\|}.$
(122)

The explicit expressions for the coefficients appearing here are given by

$\displaystyle\Omega_{1}$
$\displaystyle=\tilde{\Omega}_{B}+\tilde{\Omega}_{G},$
$\displaystyle\Omega_{2}$
$\displaystyle=\tilde{\Omega}_{B}\tilde{\Omega}_{G}-X_{B}X_{G},$
$\displaystyle\alpha_{1}$
$\displaystyle=\zeta(\tilde{\Omega}_{B}\tilde{\Omega}_{G}-X_{B}X_{G})+B(\tilde{\Omega}_{G}-X_{G})$
$\displaystyle\alpha_{2}$
$\displaystyle=\zeta(\tilde{\Omega}_{B}+\tilde{\Omega}_{G})+B,$
$\displaystyle\alpha_{1}^{G}$
$\displaystyle=\eta(\tilde{\Omega}_{B}\tilde{\Omega}_{G}-X_{B}X_{G})+G(\tilde{\Omega}_{B}-X_{B})$
$\displaystyle\alpha_{2}^{G}$
$\displaystyle=\eta(\tilde{\Omega}_{B}+\tilde{\Omega}_{G})+G,$ (123)

where we defined

$\displaystyle\tilde{\Omega}_{B}$
$\displaystyle=\Omega_{B}+\left(D_{\Delta}^{\|}-2\frac{d-1}{d}D_{\Delta}^{\perp}\right)k^{2}~{}~{},$
$\displaystyle\tilde{\Omega}_{G}$
$\displaystyle=\Omega_{G}+2\frac{d-1}{d}D_{\Delta}^{\perp}k^{2}~{}~{},$
$\displaystyle X_{B}$
$\displaystyle=\left(D_{\Delta}^{\|}-2\frac{d-1}{d}D_{\Delta}^{\perp}\right)k^{2}~{}~{},$
$\displaystyle X_{G}$
$\displaystyle=2\frac{d-1}{d}D_{\Delta}^{\perp}k^{2}~{}~{}.$ (124)

Note that all these coefficients have $k$-dependence. The rheology equations
(B) can be represented by the material diagram given in figure 7(b), involving
two circuits for $\tau$ and $\tau_{G}$ that share a dashpot measuring
$K^{\|}$. Focusing on the $\tau_{G}$ circuit, we can label the springs and
dashpots in the two elastic arms by $E_{1}$, $E_{2}$ and $\upsilon_{1}$,
$\upsilon_{2}$ respectively, while the dashpot shared with the $\tau$ circuit
with $\upsilon_{3}$. We find the stress-strain relations

$\displaystyle\ddot{\tau}$
$\displaystyle+\left(\frac{E_{1}}{\upsilon_{1}}+\frac{E_{2}}{\upsilon_{2}}\right)\dot{\tau}+\frac{E_{1}E_{2}}{\upsilon_{1}\upsilon_{2}}\tau$
$\displaystyle=-E_{1}E_{2}\left(\frac{1}{\upsilon_{1}}+\frac{1}{\upsilon_{2}}+\frac{\upsilon_{3}}{\upsilon_{1}\upsilon_{2}}\right)K^{\|}$
$\displaystyle\quad-\left(E_{1}+E_{2}+\upsilon_{3}\left(\frac{E_{1}}{\upsilon_{1}}+\frac{E_{2}}{\upsilon_{2}}\right)\right)\dot{K}^{\|}-\upsilon_{3}\ddot{K}^{\|}.$
(125)

Comparing this expression to (B), we can read off

$\upsilon_{3}=\eta,$ (126)

while the explicit solution for the remaining coefficients $E_{1}$, $E_{2}$,
$\upsilon_{1}$, $\upsilon_{2}$ can be also be obtained. The discussion for the
$\tau$ sector proceeds similarly.

##  Appendix C Comparison to Zippelius et. al

In this appendix, we compare our results to the work of [34]. Instead of the
grand canonical ensemble, the authors in [34] work in the canonical ensemble
controlled by the temperature $T$, interstitial density $n_{\Delta}$, and
momentum $\pi_{i}$. This is characterised by the free energy density

${\cal F}=-p+(\mu-\mu_{0})n+u^{i}\pi_{i},$ (127)

where we perform the canonical transformation with respect to $\mu-\mu_{0}$
instead of just $\mu$ to avoid generating a linear term in strain in the free
energy. Using the definition of interstitial density from (22a), we can check
that

$\displaystyle\mathrm{d}{\cal F}$
$\displaystyle=-s\mathrm{d}T+(\mu-\mu_{0})\mathrm{d}n_{\Delta}+u^{i}\mathrm{d}\pi_{i}$
$\displaystyle\qquad-\frac{1}{2}\left(r_{IJ}-(\mu-\mu_{0})m_{0}vh_{IJ}\right)\mathrm{d}h^{IJ}$
$\displaystyle\qquad-\frac{1}{2}\left({\mathbb{r}}^{IJ}-(\mu-\mu_{0})m_{0}v\mathbb{h}^{IJ}\right)\mathrm{d}\mathbb{h}_{IJ},$
(128)

where $\mathbb{h}^{IJ}$ is the inverse of $\mathbb{h}_{IJ}$. Plugging in the
thermodynamic objects from (19), and specialising to the isothermal limit, we
can linearly expand the differential of ${\cal F}$ to get

$\displaystyle\mathrm{d}{\cal F}$ $\displaystyle=\frac{1}{\chi}\delta
n_{\Delta}\mathrm{d}n_{\Delta}+\frac{1}{\rho}\pi^{i}\mathrm{d}\pi_{i}$
$\displaystyle\qquad-\frac{n+B\alpha_{m}}{\chi}\left(\kappa^{k}_{~{}k}\mathrm{d}n_{\Delta}+\delta
n_{\Delta}\mathrm{d}\kappa^{k}_{~{}k}\right)$
$\displaystyle~{}~{}+\left(B+\frac{(n+B\alpha_{m})^{2}}{\chi}\right)\kappa^{k}_{~{}k}\mathrm{d}\kappa^{k}_{~{}k}$
$\displaystyle~{}~{}+2G\kappa^{\langle ij\rangle}\mathrm{d}\kappa_{\langle
ij\rangle}.$ (129)

We have used that, linearly, $\delta
n=\chi\delta\mu+B\alpha_{m}\kappa^{k}_{~{}k}$, and $\delta n_{\Delta}=\delta
n+n\kappa^{k}_{~{}k}$. We have also take the constant $m_{0}$ in the
definition of interstitial density in (22a) to be the equilibrium number
density $n_{0}$. Comparing this to (2.4) of [34], we arrive at the
identification of thermodynamic coefficients

$\displaystyle\hat{n}_{0}=\rho,\qquad\hat{\chi}_{0}=\frac{\chi}{n^{2}},\qquad\hat{\gamma}_{0}=-\frac{n(n+B\alpha_{m})}{\chi},$
$\displaystyle\hat{\mu}_{0}=G,\qquad\hat{\lambda}_{0}+\hat{\mu}_{0}=B+\frac{(n+B\alpha_{m})^{2}}{\chi}.$
(130)

We have denoted the coefficients in [34] with a “hat”. The first relation
stems from the fact that [34] works with a Galilean system, wherein the
particle density is the same as the momentum susceptibility; we have taken the
“mass per unit particle” proportionality factor $m$ in [34] to be 1.

To compare the dissipative coefficients, we need to perform a redefinition of
the linearised crystal fields

$\delta\phi^{\prime i}=\delta\phi^{i}-au^{i},$ (131)

which results in a transformation of the strain tensor and crystal velocity as

$\displaystyle\kappa^{\prime}_{ij}$
$\displaystyle=\kappa_{ij}-a\partial_{(i}u_{j)},$ $\displaystyle u^{\prime
i}_{\phi}$ $\displaystyle=u^{i}_{\phi}-a\partial_{t}u^{i}$
$\displaystyle=u^{i}_{\phi}+a\frac{n+B\alpha_{m}}{\rho}\partial^{i}\mu-a\frac{B}{\rho}\partial^{i}\kappa^{k}_{~{}k}$
$\displaystyle\qquad-a\frac{2G}{\rho}\partial_{j}\kappa^{\langle
ij\rangle}+{\cal O}(\partial^{2}).$ (132)

The interstitial density and flux with respect to the transformed crystal
fields are given as

$\displaystyle n^{\prime}_{\Delta}$
$\displaystyle=n_{\Delta}-an\partial_{k}u^{k},$ $\displaystyle j^{\prime
i}_{\Delta}$ $\displaystyle=j^{i}_{\Delta}+an\partial_{t}u^{i}$
$\displaystyle=j^{i}_{\Delta}-an\frac{n+B\alpha_{m}}{\rho}\partial^{i}\mu+an\frac{B}{\rho}\partial^{i}\kappa^{k}_{~{}k}$
$\displaystyle\qquad+an\frac{2G}{\rho}\partial_{j}\kappa^{\langle
ij\rangle}+{\cal O}(\partial^{2}).$ (133)

To keep the thermodynamic relation $\delta
n=\chi\delta\mu+B\alpha_{m}\kappa^{k}_{~{}k}$ intact, we also need to
transform the chemical potential

$\delta\mu^{\prime}=\delta\mu+\frac{aB\alpha_{m}}{\chi}\partial_{i}u^{i}.$
(134)

The authors in [34] work with Galilean-invariant crystals, for which we need
to set $\rho=n$ and remove the dissipative transport coefficients $\sigma$ and
$\gamma_{n}$ in order to make $j^{i}=\pi^{i}$. Having done this, we can
compare the interstitial flux $j^{\prime i}_{\Delta}$ to the one obtained in
[34]. The interstitial flux in [34] does not have any contributions from the
derivatives of the strain tensor. This can be achieved by choosing

$a=\frac{\rho}{\sigma_{\phi}}~{},$ (135)

which leads to

$\displaystyle j^{\prime i}_{\Delta}$
$\displaystyle=-\frac{n^{2}}{\sigma_{\phi}}\partial^{i}\mu~{}.$ (136)

The evolution equation for the transformed strain tensor can be read off using
(34c), leading to

$\displaystyle\dot{\kappa}^{\prime}_{ij}$
$\displaystyle=\partial_{(i}u_{j)}+\frac{n}{\sigma_{\phi}}\partial_{i}\partial_{j}\delta\mu^{\prime}$
$\displaystyle\qquad-\frac{1}{d}\delta_{ij}\Omega_{B}\left(\kappa^{\prime
k}_{~{}k}-\alpha_{m}\delta\mu^{\prime}\right)-\Omega_{G}\kappa^{\prime}_{\langle
ij\rangle}~{}.$ (137)

We have set $\lambda_{B}=\lambda_{G}=1$ following our discussion in section
III.2. Finally, we can obtain the stress tensor in the transformed basis as

$\displaystyle\tau^{ij}$
$\displaystyle=\left(p+B\alpha_{m}\delta\mu^{\prime}\right)\delta^{ij}-B\delta^{ij}\kappa^{\prime
k}_{~{}k}-2G\kappa^{\prime\langle ij\rangle}$
$\displaystyle\qquad-\left(\zeta+\frac{nB}{\sigma_{\phi}}\left(1+\frac{\alpha_{m}}{\chi}(n+B\alpha_{m})\right)\right)\delta^{ij}\partial_{k}u^{k}$
$\displaystyle\qquad-2\left(\eta+\frac{nG}{\sigma_{\phi}}\right)\partial^{\langle
i}u^{j\rangle}.$ (138)

Before we read out the translation between the respective transport
coefficients, we need to consider that the discussion in [34] also includes a
dynamical angular part of the strain tensor that does not contribute to
plasticity. See our discussion following (60). To switch these effects off we
need to take the following limit of the coefficients in their work

$\displaystyle\hat{n}_{f}\hat{a}_{0}^{2}\to\infty,\qquad\hat{D}_{\perp},\hat{D}_{\|}\to
0,$
$\displaystyle\text{keeping}~{}~{}\hat{D}_{\perp}\hat{n}_{f}\hat{a}_{0}^{2},~{}\hat{D}_{\|}\hat{n}_{f}\hat{a}_{0}^{2}~{}~{}\text{finite}.$
(139)

With these in mind, we can directly compare (136), (137), (138) to section IV
D and V A of [34]. We find the mapping

$\displaystyle\hat{\eta}=\eta+\frac{nG}{\sigma_{\phi}},\qquad\hat{\zeta}=\zeta+\frac{nB}{\sigma_{\phi}}\left(1+\frac{\alpha_{m}}{\chi}(n+B\alpha_{m})\right),$
$\displaystyle\hat{\Gamma}=\frac{1}{\sigma_{\phi}},$
$\displaystyle\frac{\hat{n}_{f}\hat{a}_{0}^{2}}{T}\left(\hat{D}_{\|}+\hat{D}_{\perp}\right)=2\hat{\nu}_{0}=\frac{\Omega_{G}}{G},$
$\displaystyle\frac{\hat{n}_{f}\hat{a}_{0}^{2}}{T}2\hat{D}_{\perp}=\frac{\Omega_{B}}{B}.$
(140)

Using the mappings above, we can directly compare the linearised mode spectrum
reported in [34] to the one found in section III.4. The matching works
perfectly, except that [34] has an extra mode in the transverse sector
associated with the dynamics of the angular degree of freedom. This mode is
removed from the spectrum upon using the limit prescribed in (139).

##  Appendix D Relativistic plastic hydrodynamics

In this appendix, we specialise the results from the bulk of the paper to
plastic crystals with relativistic boost symmetry. In principle, our boost-
agnostic construction covers relativistic systems as a special case (see e.g.
[42]). However, it is neater to consider this case separately manifesting the
full spacetime covariance at each step. This will allow us to directly compare
with the work of [36] and is also more relevant for holographic models of
crystals such as [80, 81, 82, 83, 84].

###  D.1 Relativistic conservation laws

Throughout this appendix, we will keep the relativistic background metric
$g_{\mu\nu}$ turned on, as well as the background gauge field $A_{\mu}$. The
covariant indices $\mu,\nu,\ldots$ run over both space and time. We will use
$g_{\mu\nu}$ to raise/lower the covariant indices, while the covariant
derivative $\nabla_{\mu}$ shall also be defined with respect to $g_{\mu\nu}$;
these conventions differ from the boost-agnostic discussion in appendix A. The
fluid velocity $u^{\mu}$ (normalised as $u^{\mu}u_{\mu}=-1$), temperature $T$,
and chemical potential $\mu$ denote the respective relativistic notions,
related to the boost-agnostic definitions used in the rest of the paper via
Lorentz factors; see [42].

In a relativistic crystal, we can combine the space and time derivatives of
the crystal fields $\phi^{I}$ into the covariant crystal frame fields
$e^{I}_{\mu}=\partial_{\mu}\phi^{I}$; see [37, 38]. The crystal velocity
$u^{\mu}_{\phi}$ can be defined as the unique time-like vector transverse to
the frame fields, i.e. $u^{\mu}_{\phi}e^{I}_{\mu}=0$. We normalise it so that
$u^{\mu}_{\phi}u^{\nu}_{\phi}g_{\mu\nu}=-1$. In the relativistic context, it
is also more useful to define the induced inverse crystal metric $h^{IJ}$
covariantly using the full relativistic metric $g_{\mu\nu}$, i.e. we take
$h^{IJ}=g^{\mu\nu}e^{I}_{\mu}e^{I}_{\nu}$. The induced metric $h_{IJ}$, the
reference metric $\mathbb{h}_{IJ}$, and the strain tensor $\kappa_{IJ}$ are
defined in the same way as our discussion in section II.1. We can also use
this to define an inverse frame field $e_{I}^{\mu}=h_{IJ}e^{I\mu}$; note that
this is _not_ the inverse of $e^{I}_{\mu}$, which is not a square anymore. In
particular, we have $e^{I}_{\mu}e_{J}^{\mu}=\delta^{I}_{J}$ but
$e^{I}_{\mu}e^{\nu}_{I}=\delta^{\mu}_{\nu}+u^{\mu}_{\phi}u^{\phi}_{\nu}$.

In the relativistic case, the coupling of background sources to currents takes
a familiar form

$\displaystyle\delta S$
$\displaystyle=\int\mathrm{d}^{d+1}x\sqrt{-g}\bigg{[}\frac{1}{2}T^{\mu\nu}\delta
g_{\mu\nu}+J^{\mu}\delta A_{\mu}$
$\displaystyle\qquad\qquad+K_{I}\delta\phi^{I}+\frac{1}{2}U^{IJ}\left(\delta\psi_{IJ}-e^{\mu}_{K}\partial_{\mu}\psi_{IJ}\delta\phi^{K}\right)$
$\displaystyle\qquad\qquad+\ell^{\prime}L_{I}\delta\Phi^{I}\bigg{]}.$ (141)

From these, we can obtain the relativistic version of the energy-momentum and
particle conservation laws

$\displaystyle\nabla_{\mu}T^{\mu\nu}$
$\displaystyle=K_{I}e^{I\nu}-\frac{1}{2}U^{IJ}u^{\mu}_{\phi}u^{\nu}_{\phi}\partial_{\nu}\psi_{IJ}+F^{\nu\rho}J_{\rho}$
$\displaystyle\qquad+\ell^{\prime}L_{I}\nabla^{\nu}\Phi^{I}~{}~{},$ (142a)
$\displaystyle\nabla_{\mu}J^{\mu}$ $\displaystyle=0~{}~{}.$ (142b)

On the other hand, the crystal sources can be defined via the source action

$S_{\text{source}}=\int\mathrm{d}^{d+1}x\sqrt{-g}\,\bigg{[}T^{\mu\nu}_{\text{ext}}\kappa_{\mu\nu}+\Pi_{\mu}^{\text{ext}}u^{\mu}_{\phi}\bigg{]}.$
(143)

Performing a variation of the source action, we find

$\displaystyle\delta S_{\text{source}}$
$\displaystyle=\int\mathrm{d}^{d+1}x\sqrt{-g}\,\Bigg{[}\kappa_{\mu\nu}\delta
T^{\mu\nu}_{\text{ext}}+u^{\mu}_{\phi}\delta\Pi_{\mu}^{\text{ext}}$
$\displaystyle+K^{\text{ext}}_{I}\delta\phi^{I}+\frac{1}{2}U^{IJ}_{\text{ext}}\left(\delta\psi_{IJ}-e^{\mu}_{K}\partial_{\mu}\psi_{IJ}\delta\phi^{K}\right)$
$\displaystyle+\frac{1}{2}\left(\bar{T}^{\mu\nu}_{\text{ext}}+T^{\rho\sigma}_{\text{ext}}\kappa_{\rho\sigma}g^{\mu\nu}+\Pi_{\rho}^{\text{ext}}u^{\rho}_{\phi}P^{\mu\nu}_{\phi}\right)\delta
g_{\mu\nu}\Bigg{]},$ (144)

where
$\bar{T}^{\mu\nu}_{\text{ext}}=P_{\phi}^{\mu\rho}P_{\phi}^{\nu\sigma}T_{\rho\sigma}^{\text{ext}}$
is defined to be transverse to $u^{\mu}_{\phi}$, with the projector operator
$P^{\mu\nu}_{\phi}=g^{\mu\nu}+u^{\mu}_{\phi}u^{\nu}_{\phi}$. We have also
defined

$\displaystyle U^{IJ}_{\text{ext}}$ $\displaystyle=-\ell
e^{I}_{\mu}e^{J}_{\nu}T^{\mu\nu}_{\text{ext}},$ $\displaystyle
K^{\text{ext}}_{I}$
$\displaystyle=\nabla_{\mu}\left(T^{\mu\nu}_{\text{ext}}e^{J}_{\nu}\mathbb{h}_{IJ}+T^{\rho\nu}_{\text{ext}}u^{\phi}_{\rho}e_{I\nu}u^{\mu}_{\phi}+\Pi^{\text{ext}}_{\nu}e^{\nu}_{I}u^{\mu}_{\phi}\right)$
$\displaystyle\qquad+\frac{1}{2}U^{JK}_{\text{ext}}e^{\mu}_{I}\partial_{\mu}\mathbb{h}_{JK},$
(145)

which result in the configuration equations for $\phi^{I}$ and $\psi_{IJ}$ as
in (20).

###  D.2 Relativistic viscoplastic hydrodynamics

The equation of state for the thermodynamic pressure $p$ can be expressed in
terms of the relativistic temperature $T$ and chemical potential $\mu$. The
thermodynamics takes the appropriate form

$\displaystyle\mathrm{d}\epsilon$
$\displaystyle=T\mathrm{d}s+\mu\mathrm{d}n-\frac{1}{2}r_{IJ}\mathrm{d}h^{IJ}-\frac{1}{2}\mathbb{r}^{IJ}\mathrm{d}\mathbb{h}_{IJ}$
$\displaystyle\qquad+\ell^{\prime
2}m^{2}h_{IJ}(\phi^{J}-\Phi^{J})\,\mathrm{d}(\phi^{I}-\Phi^{I}),$
$\displaystyle\mathrm{d}p$
$\displaystyle=s\mathrm{d}T+n\mathrm{d}\mu+\frac{1}{2}r_{IJ}\mathrm{d}h^{IJ}+\frac{1}{2}\mathbb{r}^{IJ}\mathrm{d}\mathbb{h}_{IJ}$
$\displaystyle\qquad-\ell^{\prime
2}m^{2}h_{IJ}(\phi^{J}-\Phi^{J})\,\mathrm{d}(\phi^{I}-\Phi^{I}),$
$\displaystyle\epsilon$ $\displaystyle=Ts+\mu n-p.$ (146)

Note that there is no explicit momentum term in the thermodynamic relations
due to Lorentz-invariance. Momentum appears implicitly through the Lorentz
factors contained within the “proper” relativistic densities.

We now consider the hydrodynamic constitutive equations for $T^{\mu\nu}$,
$J^{\mu}$, $K_{I}$, and $U^{IJ}$. With hindsight from our boost-agnostic
calculation, we choose the parametrisation of the constitutive relations as

$\displaystyle T^{\mu\nu}$
$\displaystyle=(\epsilon+p)u^{\mu}u^{\nu}+pg^{\mu\nu}-r_{IJ}e^{I\mu}e^{J\nu}+\mathcal{T}^{\mu\nu},$
$\displaystyle J^{\mu}$ $\displaystyle=nu^{\mu}+\mathcal{J}^{\mu}~{}~{},$
$\displaystyle K_{I}$
$\displaystyle=-\nabla_{\mu}\\!\left(r_{IJ}e^{J\mu}\right)+\frac{\ell}{2}\mathbb{r}^{JK}e^{\mu}_{I}\partial_{\mu}\psi_{JK}$
$\displaystyle\qquad-\ell^{\prime
2}m^{2}h_{IJ}\left(\phi^{J}-\Phi^{J}\right)+\mathcal{K}_{I}~{}~{},$
$\displaystyle U^{IJ}$
$\displaystyle=\ell\mathbb{r}^{IJ}-\frac{1}{u_{\mu}u^{\mu}_{\phi}}\mathcal{U}^{IJ}~{}~{},$
$\displaystyle L_{I}$
$\displaystyle=\ell^{\prime}m^{2}h_{IJ}\left(\phi^{J}-\Phi^{J}\right)+{\cal
L}_{I}~{}~{},$ (147)

where ${\cal T}^{\mu\nu}$, ${\cal J}^{\mu}$, ${\cal K}_{I}$, ${\cal U}^{IJ}$,
and ${\cal L}_{I}$ denote the respective dissipative corrections. We shall
work in the Landau frame [45, 44], which is to say that we choose the
direction of the fluid velocity so that there is no dissipation in the energy
density, energy flux/momentum density, or particle density in the comoving
frame of the fluid. This amounts to choosing

${\cal T}^{\mu\nu}u_{\nu}={\cal J}^{\mu}u_{\mu}=0.$ (148)

These constitutive relations must satisfy the second law of thermodynamics,
i.e. there must exist an entropy current $S^{\mu}$ such that

$\nabla_{\mu}S^{\mu}=\Delta\geq 0,$ (149)

on solutions of the equations of motion. We choose the entropy current to take
the canonical form

$\displaystyle S^{\mu}$
$\displaystyle=su^{\mu}-\frac{\mu}{T}\mathcal{J}^{\mu}~{}~{}.$ (150)

Going through with the entropy divergence calculation as in appendix A, we
find that the relativistic dissipation rate is given by

$\displaystyle T\Delta$ $\displaystyle=-{\cal
T}^{\mu\nu}\nabla_{\mu}u_{\nu}-{\cal
J}^{\mu}\left(T\partial_{\mu}\frac{\mu}{T}+u^{\nu}F_{\nu\mu}\right)$
$\displaystyle\qquad-{\cal K}_{I}u^{\mu}e^{I}_{\mu}-\frac{1}{2}{\cal
U}^{IJ}u^{\mu}_{\phi}\partial_{\mu}\psi_{IJ}$
$\displaystyle\qquad-\ell^{\prime}{\cal L}_{I}u^{\mu}\partial_{\mu}\Phi^{I}.$
(151)

Up to first order in derivatives, we can work out the hydrodynamic
constitutive relations as in II.4. First, in the vector sector we have [37,
38]

$\begin{pmatrix}\mathcal{J}^{I}\\\ {\cal K}^{I}\\\ {\cal
L}^{I}\end{pmatrix}=-\begin{pmatrix}\sigma_{n}&\gamma_{n\phi}&\gamma_{n\Phi}\\\
\gamma^{\prime}_{n\phi}&\sigma_{\phi}&\sigma_{\times}\\\
\gamma^{\prime}_{n\Phi}&\sigma^{\prime}_{\times}&\sigma_{\Phi}\end{pmatrix}\begin{pmatrix}P^{I\mu}(T\partial_{\mu}\frac{\mu}{T}+u^{\nu}F_{\nu\mu})\\\
u^{\mu}e^{I}_{\mu}\\\
\ell^{\prime}u^{\mu}\partial_{\mu}\Phi^{I}\end{pmatrix},$ (152a) where we have
chosen the parametrisation of dissipative corrections ${\cal
J}^{\mu}=P^{\mu}_{I}{\cal J}^{I}$. We have defined the projected frame fields
as $P^{I}_{\mu}=e^{I}_{\mu}+u_{\mu}u^{\nu}e^{I}_{\nu}$. As before, we get
plasticity effects in the scalar and symmetric-traceless tensor sector
$\displaystyle\begin{pmatrix}{\cal T}^{IJ}\\\ {\cal U}^{IJ}\end{pmatrix}$
$\displaystyle=-h^{IJ}\begin{pmatrix}\zeta_{\tau}&\zeta_{\tau\mathbb{h}}\\\
\zeta^{\prime}_{\tau\mathbb{h}}&\zeta_{\mathbb{h}}\end{pmatrix}\begin{pmatrix}P_{I}^{\mu}P^{I}_{\nu}\nabla_{\mu}u^{\nu}\\\
\frac{1}{2}h^{KL}u^{\mu}_{\phi}\partial_{\mu}\psi_{KL}\end{pmatrix}$
$\displaystyle-\begin{pmatrix}\eta_{\tau}&\eta_{\tau\mathbb{h}}\\\
\eta^{\prime}_{\tau\mathbb{h}}&\eta_{\mathbb{h}}\end{pmatrix}\begin{pmatrix}2P^{\langle
I}_{\mu}P^{J\rangle}_{\nu}\nabla^{\mu}u^{\nu}\\\ h^{K\langle I}h^{J\rangle
L}u^{\mu}_{\phi}\partial_{i}\psi_{KL}\end{pmatrix},$ (152b)

where we have again defined ${\cal T}^{\mu\nu}=P^{\mu}_{I}P^{\nu}_{J}{\cal
T}^{IJ}$.

We get the same constraints as sections II.4 and V.5 for Onsager’s relations,
i.e.

$\displaystyle\gamma^{\prime}_{n\phi}=-\gamma_{n\phi},\qquad\eta^{\prime}_{\tau\mathbb{h}}=\eta_{\tau\mathbb{h}},\qquad\zeta^{\prime}_{\tau\mathbb{h}}=\zeta_{\tau\mathbb{h}},$
$\displaystyle\gamma^{\prime}_{n\Phi}=-\gamma_{n\Phi},\qquad\sigma^{\prime}_{\times}=\sigma_{\times}.$
(153a) On the other hand, demanding the entropy production to be sign-
definite, we get the positivity constraints $\displaystyle\eta_{\tau}\geq
0,\qquad\zeta_{\tau}\geq 0,\qquad\sigma_{\phi}\geq 0,$
$\displaystyle\sigma_{n}\geq
0,\qquad\eta_{\tau}\eta_{\mathbb{h}}\geq\eta_{\tau\mathbb{h}}^{2},\qquad\zeta_{\tau}\zeta_{\mathbb{h}}\geq\zeta_{\tau\mathbb{h}}^{2},$
$\displaystyle\sigma_{\Phi}\sigma_{\phi}\geq\sigma_{\times}^{2}.$ (153b)

###  D.3 Modes and correlators

Let us now specialise to the isothermal regime like the bulk of the paper. In
the relativistic context, this decouples the $u_{\mu}$ projection of the
energy-momentum conservation equation (142a). Following through with the
computation, we find the exact same modes as discussed in sections III.4 and
V.4, only the momentum susceptibility $\rho$ should be replaced with its
relativistic version $\epsilon+p$ (where the speed of light $c$ has been set
to 1). The same holds for all the correlation functions discussed in sections
III.5 and V.5.

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Cooling the Earth with $CO_{2}$ filled containers in space

Orfeu Bertolami1,2, and Clovis Jacinto de Matos3

1 Departamento de Física e Astronomia, Faculdade de Ciências, Universidade do
Porto,

Rua do Campo Alegre s/n, 4169-007 Porto, Portugal

2 Centro de Física das Universidades do Minho e do Porto, Rua do Campo Alegre
s/n, 4169-007 Porto, Portugal

3 European Space Agency - ESA Headquarters, 75007 Paris, France

###### Abstract

We argue that geostationary (GEO) reflective containers filled with $CO_{2}$
could be used as shading devices to selectively cool areas on Earth’s surface.
This proposal would be an interesting addition to the recently discussed
suggestion of dumping $CO_{2}$ to space through the well of a space lift [1].
We also explore the possibility of producing propellants in GEO out of
greenhouse gases expelled from the space lift. Finally, we discuss the much
less effective idea of filtering the most prominent infrared bands of the
incoming solar radiation using the $CO_{2}$ wrapped in transparent vessels.

E-mail addresses<EMAIL_ADDRESS><EMAIL_ADDRESS>
##  1 Introduction

In a recent publication [1] it was proposed that $CO_{2}$ could be transported
to space by some suitable adaptations of the space lift. It was shown that the
well of a geostationary orbital lift or space elevator as it is usually
referred to, could be used for dumping greenhouse gases into space. Naturally,
it was assumed that the known requirements to build a stable orbital lift are
satisfied, and it was discussed how to use this infrastructure to dump
greenhouse gases away from Earth’s atmosphere.

Indeed, based on recent advances in material science, more specifically, in
the development of carbon nanotubes and macro-scale single crystal graphene,
it has been pointed out that the space lift could be a reality in a
foreseeable future (see Ref. [2] for a recent assessment). In Ref. [1], the
use of the space lift well was considered to dump the excess of anthropogenic
atmospheric $CO_{2}$ into space, but of course, the concept could also be
considered for other greenhouse gases (like for example $CH_{4}$).

The typical dimensions of the orbital lift are its vertical extension,
$r_{G}\simeq 35786$ km, and an assumed constant cross-sectional area, $A=\pi
r^{2}$, where $r$ is the radius of the well. The anchor of the orbital lift
could be a geostationary satellite in an equatorial plane orbit. The idea of
Ref. [1] is to inject $CO_{2}$ into the well of the orbital lift and generate
an upward flow so to dump $CO_{2}$ into space. Natural conditions do not allow
for any effective upwards flow as Earth’s escape velocity is much greater than
the typical average velocities of the $CO_{2}$ molecules in the air.
Therefore, conditions for an upward flow must be engineered. The steps to
create an upward flow of $CO_{2}$ molecules are detailed in Ref. [1] and, in
its simplest form, involve the following steps: i) separation of the $CO_{2}$
in the air and its injection into the well, which has been emptied of its
initial content; ii) ionisation of the $CO_{2}$ in the well via soft X-ray
irradiation; iii) acceleration of the charged $CO_{2}$ through an electric
field along the vertical axis of the orbital lift.

Under reasonable assumptions, it was shown that the outward flow of $CO_{2}$
is given by, $\Phi=j\pi r^{2}$, where $j=\rho v_{f}$, $\rho$ and $v_{f}$ being
the $CO_{2}$ density and $v_{f}$ the final velocity at the upper end of the
well. For $\rho=4\times 10^{-4}~{}kg/m^{3}$ and $r=15~{}m$ one gets:
$\Phi_{1}=4.2~{}ton/s$ for the scenario where the first and last sections of
the well are under the effect of the electric field and a middle section with
no electric field; $\Phi_{2}=3.4~{}ton/s$ for the scenario where ionised
$CO_{2}$ is accelerated by an electric field in three sections of the well
with two intermediate sections with no electric field [1]. These flows
correspond to less than $2\%$ of the anthropogenic $CO_{2}$ yield.

Improvements on the estimated yields are possible and were discussed in Ref.
[1]. However, the point we would like to make in the present work is that the
$CO_{2}$ flow could be used to fill reflective or transparent containers that
would reflect or filter part of the incoming solar radiation at the
wavelengths, $4.3~{}\mu m$ and $15~{}\mu m,$ making it somewhat ”cooler”. For
sure, in what concerns filtering, the individual $CO_{2}$ molecules may re-
emit the absorbed radiation after a while, however, the absorbed radiation can
be dissipated and re-emission avoided if the concentration of $CO_{2}$ is
sufficiently high, the greenhouse effect. We believe that, even though most of
the greenhouse effect is due to the absorption of these infrared wavelengths
when solar radiation is reflected back to space by the ground on Earth, the
depletion of these wavelengths in the incoming radiation may have an
attenuating effect on the net greenhouse effect.

Before we discuss the details of the present proposal let us point out that
the thrust due to the injection of gas into space that is transmitted on the
structure of the space lift can be estimated to be about $6.3\times
10^{7}~{}N$ for the first scenario discussed above. However, this undesirable
effect can be avoided by adopting the simple solution of directing a symmetric
ejection of the $CO_{2}$ along a direction perpendicular to the axis of the
well. The desired cancellation can be achieved by a radially symmetric set of
nozzles perpendicular to the axis of the lift at the top end of the well. In
what follows, we shall consider that the delivered $CO_{2}$ gas at the top of
the well is used to fill the reflective or infrared absorbing vessels.

##  2 Reflective $CO_{2}$ filled containers

The main idea here is to use the $CO_{2}$ ejected by the nozzles at the top of
the space lift to fill vessels wrapped with reflective material so to redirect
back to space the incoming solar radiation. The shade provided by the vessels
will depend on their number and configuration. A sufficiently large set of
vessels can cover a considerable area. The vessels can be endowed with
additional features, but their main property is their reflective power, which
can be quite high.

Even though we consider the vessels to be as simple as possible, they can
themselves be endowed with nozzles so to use their $CO_{2}$ content for
propulsion. Once the vessels acquire their final volume, they could be
attached to each other, transported away from the space lift and cover a patch
over an area where cooling is most needed (like e.g. at the poles). As for the
vessels themselves they should be wrapped with a reflective material that
ensures gas tightness, is resistant to low/high temperatures and to the
interplanetary radiation. The backbone of its structure should be light,
resistant and paramagnetic. The properties for these structures can be
encountered, for instance, in aluminium alloys widely used in the aerospace
engineering. However, as the space lift itself might be built with carbon
nanotubes, this material could, in principle, be used for the backbone of the
vessels too. The flux of $CO_{2}$ at the end of the space lift is sufficient
to fill a great number of vessels and thus, in the theory, there is no limit
to the number of vessels to be attached together and to the shading area they
can provide.

##  3 Reflective $CO_{2}$ filled containers in the Inner Lagrange Point (L1)

The shadow provided by the reflective containers in GEO discussed above is
essentially of the same order of magnitude of the area covered by the ensemble
of containers. In order to shade larger areas it would require either quite
large reflecting surfaces in GEO or to move the mirror-like structure away
from Earth, beyond GEO orbital radius [3]. Alternatively, locating a mirror of
about 9.4 million km2 at the Lagrange Point (L1) between the Sun and Earth at
a distance of about $1.5\times 10^{9}~{}m$ from the Earth centre, one could
reduce the solar radiation over the Earth disk by about 1.8$\%$, which would
suffice to relief the global warming situation [4, 5]. On the other hand,
storing $CO_{2}$ and also $CH_{4}$ in GEO, through submitting $CH_{4}$ to
similar manipulations using the space lift as described in Ref. [1] for the
$CO_{2}$, would open new interesting avenues to fuel future space missions,
including a mission to drive reflective $CO_{2}$ containers to L1 as suggested
in Ref. [6].

In GEO, solar energy could be used to power the pyrolysis of methane, to
produce hydrogen,

$CH_{4}\,\xrightarrow{\text{{heat}}}\,C\,+\,2H_{2}$ (1)

and to sustain the Sabatier reaction of $CO_{2}$ to produce methane and water.

$CO_{2}\,+\,4H_{2}\,\xrightarrow[\text{pressure and
catalyst}]{\text{{heat}}}\,CH_{4}\,+\,2H_{2}O$ (2)

The electrolysis of water would then allow producing $H_{2}$ and $O_{2}$:

$2H_{2}O\,\xrightarrow{\text{{electric current}}}\,O_{2}\,+\,2H_{2}$ (3)

Hence, the $CO_{2}$ and $CH_{4}$ greenhouse gases dumped to space by the well
of the space lift could allow for yielding and storing important quantities of
oxygen, hydrogen and methane propellants directly at GEO. The resulting
infrastructure could fuel all types of space missions and, in particular, a
mission to place $CO_{2}$ filled containers to L1.

In fact, different spacecraft components can, in principle, be brought to
assemblage at GEO, so that spacecraft could be assembled and fueled with
propellants produced and stored at GEO. Thus, the infrastructure provided by
the space lift, and its well, can support the entire logistics of beyond GEO
space missions.

The concept proposed here opens the possibility to carry out missions with a
challenging high $\Delta v$ cost such as, for instance, Polar orbit ones. This
could allow for creating a significant surface to screen Earth’s polar caps so
to reduce their current melting rate.

##  4 Refractive $CO_{2}$ filled containers

Let us now discuss the main features of the refractive $CO_{2}$ filled
containers. As the reflective containers, the refractive containers are filled
with $CO_{2}$ at the top of the space lift. The vessels are essentially
transparent and we consider the simplest possible realization of the idea. The
most relevant property of the vessels is that they absorb radiation in the
relevant infrared wavelengths $4.3~{}\mu m$ and $15~{}\mu m$. The absorption
effect can be estimated from the radiative equation by the intensity
attenuation after crossing a medium with density, $\rho$, and a length, $L$
[7]:

$I_{\nu}(L)=I_{0}e^{-\tau_{\nu}},$ (4)

where the optical depth of the medium at those specific wavelengths,
$\tau_{\nu}$, is given in terms of an integral over a given length:

$\tau_{\nu}=\int_{0}^{L}k_{\nu}\rho ds,$ (5)

$k_{\nu}$ being the constant that characterises the medium at a given
wavelength.

Assuming that the gas density is uniform, then,

$\tau_{\nu}=k_{\nu}\rho L.$ (6)

The optical depth can also be expressed in terms of microscopic properties
through the number density of the absorbing molecules, $n$, and the cross
section of the absorbing process, $\sigma$:

$\tau_{\nu}=n\sigma_{\nu}L.$ (7)

It is easy to show that if the $CO_{2}$ behaves in the vessels as an ideal
gas, then:

$\sigma_{\nu}=k_{\nu}{\mu\over N_{A}},$ (8)

where $\mu$ is molecular weight of the $CO_{2}$ and $N_{A}$ Avogadro’s number.

The necessary data to estimate the $CO_{2}$ absorption under realistic
conditions can be found, for instance, in Ref. [8], in terms of the $CO_{2}$
concentration. For typical values, that is $400~{}ppm$, which under PNT
conditions corresponds to $\rho_{CO_{2}}=5.16\times 10^{-4}~{}kgm^{-3}$, one
obtains [8]: $k\equiv k_{\nu}\rho\simeq 1.0\times 10^{-1}~{}m^{-1}$, implying
that $k_{\nu}\simeq 1.93\times 10^{2}~{}m^{2}~{}kg^{-1}$.

The conditions to avoid that the absorbed energy is re-emitted is that the
excess energy due to the radiation is thermalised. Assuming a reasonable
attenuation figure, say $10\%$ of the incoming radiation on the wavelengths
$4.3~{}\mu m$ and $15~{}\mu m$ implies from Eq. (4) and (6) that
$\tau_{\nu}\simeq 10^{-1}$ and with data obtained from Ref. [8], one obtains
that $10^{-1}~{}m^{-1}<k<6\times 10^{-1}~{}m^{-1}$. Still assuming the
$CO_{2}$ in the vessels behaves as an ideal gas, then for the density figure
corresponding to $400~{}ppm$, one obtains the relations between pressure and
temperature:

$\left({p\over 1~{}atm}\right)=2.58\times 10^{-4}\left({T\over
273~{}K}\right),$ (9)

As the $CO_{2}$ freezing point is $T=194.65~{}K$, we consider, for instance,
$T\simeq 200~{}K$ and hence:

$\left({p\over 1~{}atm}\right)=1.89\times 10^{-4}.$ (10)

To evaluate the volume of the vessel, we consider a width, $L=1~{}m$, and then
the area facing the sun should be $A=10^{8}~{}m^{2}$, meaning that for, say,
$100$ vessels, the area facing the sun of each vessel should be
$10^{6}~{}m^{2}$, that is, for vessels that have a paving stone shape with a
square face, their length should be $1~{}km$.

Once the set of vessels are positioned, their $CO_{2}$ content will absorb the
incoming infrared radiation and heat up the contained gas, which will increase
its pressure. Thus, the vessels should be built with a material that might
allow for an increase in volume of an order of magnitude or so. Once the
pressure reaches a certain level, a built in set of symmetrically distributed
valves would allow for the release of gas into space.

Notice that the above considerations are equivalent to the discussion of the
greenhouse effect on a layer of atmosphere where one admits for the incoming
solar radiation a strong transmission and weak absorbing coefficient,
$\tau_{s}\simeq 0.9$ [9].

The infrared absorbing structures that we propose can be an interesting
addition to the arsenal of geoengineering space devices to be built in order
to face the ongoing climate change crisis and are a logical step forward on
the use of the space lift to dump greenhouse gases into space discussed in
Ref. [1]. Of course, most of the greenhouse effect is due to the atmospheric
absorption of the solar radiation reflected by Earth’s surface. In rough
terms, the main contributors to the reflection are soil, snow, vegetation and
water. The last three components do not reflect any incoming radiation in the
$4.3~{}\mu m$ and $15~{}\mu m$ wavelengths [10], meaning that soil reflection
is the main component. It is still to be thoroughly studied the effect of
depleting the incoming solar radiation from the main wavelengths that cause
the greenhouse effect, but it is reasonable to think that it might have a
beneficial impact.

##  5 Conclusions and Discussion

It is consensual that anthropogenic climate change due to the accumulation of
greenhouse gases in the atmosphere is a major civilizational challenge. In
order to face the disruptive effects on the patterns of the climate that are
already observed, one needs to embark on deep changes on the tenets of the
consumption society powered by cheap fossil fuels and built upon the mistaken
assumptions about Earth’s resources and its capability to admit an indefinite
dump of waste. Naturally, any solution to the problem involves drastic
reduction on emissions and radical socio-economic changes.

Given the structural nature and the depth of the changes that must be
implemented, geoengineering proposals have been put forward in order to buy us
some time. Indeed, severall proposals have been advanced, some of which
controverrsial, such as for instance, ocean fertilisation and alkalinity
enhancement, albedo enhancement through passive daytime radiative cooling [11,
12], the use of sky-facing thermally-emissive surfaces to radiate heat back
into space [13, 14], stratospheric aerosol injection (SAI), the so-called
“Budyko blancket” [15, 16, 17, 18], cloud brightening or a large set of
mirrors in the sky to reflect back into space a fraction of the incoming solar
radiation (see Ref. [18] for a review). Other set of ideas involve space
reflectors such as a space mirror [3, 4] or a myriad of reflecting bubbles
[6]. Our proposal of reflective vessels filled with $CO_{2}$ can be seen as a
realization of the latter. Of course, geoengineering proposals involve
necessarily some degree of negative effects.

Thus, in principle, any device that reduces and reflects back to space the
incoming solar radiation might be useful to reduce the amount of infrared
radiation trapped in the atmosphere. In the present work we have considered an
hypothetical device that involves containers filled with $CO_{2}$ gas. The
vessels can be reflective or transparent so to filter the two main absorbing
wavelength of the $CO_{2}$. The first possibility is much more promising. The
device can be coupled with the space lift proposed in a previous study [1]. In
the case of reflective vessel, an all spectrum shade is provided. In the case
of the refractive vessels, we have shown that an infrared ”shadow” can be
created by a set of vessels in a geostationary orbit with the net effect of
attenuating the incoming solar radiation in the $4.3~{}\mu m$ and $15~{}\mu m$
wavelengths if the vessels were filled with $CO_{2}$. The use of other
greenhouse gases, such as for instance methane, could extend the “blanket”
effect for other wavelengths in the infrared.

The possibility to produce and store rocket propellants ($H_{2}$, $O_{2}$,
$CH_{4}$) in GEO out of $CO_{2}$ and $CH_{4}$ greenhouse gases pumped from the
Earth atmosphere could also support space missions to transport the
reflective/refractive vessels to polar orbits, or to the Inner L1 Lagrange
point. At L1, it becomes possible to shade the entire Earth disk. As a bonus,
having a propellant refuelling station in GEO would unlock many possibilities
for new innovative space exploitation/exploration missions.

Given the urgency to avoid that the continuous climbing of the concentration
of greenhouse gases drive the Earth System (ES) to a Hot House Earth State
[22], even the most apparently unrealistic geoengineering proposals are
welcome to partially fix the problem. It should be pointed out that the most
pessimistic predictions about a possible collapse of the great regulatory
ecosystems [22] are in agreement with theoretical analyses based on a physical
model of the Earth System and the Hot House Earth State as an inevitable
outcome given the present intensity of human activities (see e.g. Refs. [23,
24, 25, 26, 27, 28]).

For sure, the magnitude of the problem asks, first of all, for a deep change
on the assumptions of the neoliberal market economy based on the wrong
assumption that a finite planet can afford an indefinite period of economical
growth without an inevitable collapse. It is becoming increasingly evident
that no sustainable future is possible without an encompassing plan for global
economic deceleration and a drastic reduction in the use of fossil fuels.

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