Some Useful Operators on Differential Forms on Galilean and Carrollian
Spacetimes

Some Useful Operators on Differential Forms
on Galilean and Carrollian Spacetimes

Marián FECKO

M. Fecko

Department of Theoretical Physics, Comenius University in Bratislava, Slovakia
<EMAIL_ADDRESS>
Received August 30, 2022, in final form April 11, 2023; Published online April
22, 2023

Differential forms on Lorentzian spacetimes is a well-established subject. On
Galilean and Carrollian spacetimes it does not seem to be quite so. This may
be due to the absence of Hodge star operator. There are, however, potentially
useful analogs of Hodge star operator also on the last two spacetimes, namely
intertwining operators between corresponding representations on forms. Their
use could perhaps make differential forms as attractive tool for physics on
Galilean and Carrollian spacetimes as forms on Lorentzian spacetimes
definitely proved to be.

Hodge star operator; Galilean spacetime; Carrollian spacetime; intertwining
operator

53Z05; 83C99; 22E70

##  1 Introduction and motivation

A lot of important mathematics routinely used in physics reduces to
manipulations with _differential forms_ on (pseudo)-_Riemannian_ manifolds
(see, e.g., [1, 4, 6, 10, 11, 18, 22]).

In particular, on 3-dimensional Euclidean space, $E^{3}$, the whole standard
vector calculus more or less boils down to simple manipulations with
combinations of just two operators on forms, the exterior derivative $d$ and
the _Hodge star_ operator $*$.

Similarly on Minkowski spacetime $E^{1,3}$, expression of, say, Maxwell
equations and its consistency condition (local conservation of electric
charge),

${\rm d}*F=-J,\qquad{\rm d}F=0,\qquad{\rm d}J=0,$

the wave operator $(\sim*{\rm d}*{\rm d})$, Lorentz gauge condition ($*{\rm
d}*A=0$) etc., again only use forms and the above mentioned two operators,
${\rm d}$ and $*$.

A lot of research is recently done on _Galilean_ and _Carrollian_ spacetimes,
see, e.g., [2, 3, 5, 8, 9, 14, 17, 23] or a long lists of references in recent
Ph.D. Theses [12] and [13]. What about using forms (together with the two
operators, ${\rm d}$ and $*$), there? It turns out, it cannot be done in a
straightforward way.

Let us illustrate it on the simplest versions of the two spacetimes, _Galilei_
and _Carroll_ spacetimes. (Galilean and Carrollian spacetimes are _locally_
Galilei and Carroll, respectively, in a similar way as usual curved
spacetimes, Lorentzian manifolds, are _locally_ Lorentz. See Appendices C and
D.) We can treat them as just $\mathbb{R}^{4}$ with global Cartesian
coordinates $(t,x,y,z)\equiv(t,\mathbf{r})$, where translations and rotations
act exactly as in Minkowski spacetime, but boosts act differently, namely

$\displaystyle\text{Galilei boost}\colon\ $ $\displaystyle
t^{\prime}=t,\qquad$ $\displaystyle\text{Carroll boost}\colon\ $
$\displaystyle t^{\prime}=t+\mathbf{v}\cdot\mathbf{r},$ (1.1)
$\displaystyle\mathbf{r}^{\prime}=\mathbf{r}+\mathbf{v}t,\qquad$
$\displaystyle\mathbf{r}^{\prime}=\mathbf{r}$ (1.2)

(Galilei and Carroll spacetimes may be treated as $c\to\infty$ and $c\to 0$
limits of the Minkowski spacetime, respectively.)

Now one can easily check that there is neither Galilei-invariant _metric
tensor_ on Galilei spacetime nor Carroll-invariant metric tensor on Carroll
spacetime (analogs of Minkowski metric tensor on Minkowski spacetime).111So
Hermann Minkowski could not construct useful spacetime metric prior to the
change of paradigm from Galilei relativity to Einstein relativity.

Recall, however, that the Hodge star operator _needs metric tensor_ (and
orientation) for its definition

$*\colon\ \Omega^{p}\to\Omega^{n-p},\qquad\Omega^{p}\equiv\Omega^{p}(M,g,o)$

and it behaves according to

$f^{*}*_{g,o}=*_{f^{*}g,f(o)}f^{*}$

w.r.t. diffeomorphisms; it therefore _commutes_ with $f^{*}$ for (orientation
preserving) isometries

$f^{*}g=g\Rightarrow f^{*}*_{g}=*_{g}f^{*}.$

In particular it is translation and rotation-invariant (in this sense) on
$E^{3}$ and Poincaré-invariant on Minkowski space.

The lack of Galilei/Carroll invariant metric tensors on Galilei/Carroll
spacetimes makes it impossible to construct Galilei/Carroll invariant Hodge
star operator in _standard_ way.

We can, however, try to find useful _analogs_ of the Hodge star. Namely to
find _directly_ operators

$\text{``}*\text{''}\colon\
\Omega^{p}\to\Omega^{q},\qquad\Omega^{p}\equiv\Omega^{p}\quad(\text{Galilei/Carroll})$

(we are ready to find such operators for _any pair_ of degrees rather than
just $p$ and $(n-p)$) acting on forms on Galilei/Carroll spacetime so that

$f^{*}\text{``}*\text{''}=\text{``}*\text{''}f^{*}\qquad\text{whenever $f$ is
Galilei/Carroll transformation}$ (1.3)

(i.e., boost from (1.1)–(1.2) or rotation).

Corresponding problem is addressed in Sections 2–4. Its solution is based on
observation, that Galilei/Carroll transformations on forms induce
_representations_ of Galilei/Carroll groups (and, consequently, of the Lie
algebras) on the _space of components_ of forms $\hat{\Lambda}^{p}$ and the
condition (1.3) is reflected in validity of the commutative diagram

$\begin{CD}\hat{\Lambda}^{p}@>{\hat{a}_{qp}}>{}>\hat{\Lambda}^{q}\\\
@V{\rho_{p}}V{}V@V{}V{\rho_{q}}V\\\
\hat{\Lambda}^{p}@>{}>{\hat{a}_{qp}}>\hat{\Lambda}^{q}\\\
\end{CD}\qquad\text{i.e.,}\quad\rho_{q}\circ\hat{a}_{qp}=\hat{a}_{qp}\circ\rho_{p}.$
(1.4)

So we actually look for _intertwining operators_ for some finite-dimensional
representations.

In what follows we will see that such operators do exist and that their list
depends heavily on the choice of particular spacetime (Galilei, Carroll,
Minkowski). In particular, in Minkowski spacetime, the _only_ (non-trivial)
intertwining operator turns out to be the Hodge star. In the Galilei and
Carroll spacetimes, on the contrary, there are operators

  1. –

acting similarly to (Minkowski) Hodge star (already known from [7]),

  2. –

with no similarity to (Minkowski) Hodge star.

So, in this sense, the structure of intertwining operators on forms is
slightly _richer_ in the Galilei and Carroll spacetimes than it is in the
Minkowski one.

Now the _standard_ Hodge star $*_{g,o}$ on a general $(M,g,o)$ happens to be
actually invariant even w.r.t.  _local_ isometries (it is _local Lorentz_
invariant on a general-relativistic spacetime). So its importance goes far
beyond isometries of $(M,g,o)$.

Therefore, from the perspective of usefulness in _Galilean and Carrollian_
spacetimes, we look for _analogs_ of Hodge star which enjoy the property of
_local_ Galilei/Carroll invariance. This can be, however, already achieved
(from results obtained in Sections 2–4) by standard techniques
($G$-structures, see Appendices C, D and E).

So the results obtained for Galilei/Carroll cases remain valid in _Galilean
and Carrollian_ spacetimes as well.

###  1.1 Structure of the paper

In Section 2, we first introduce standard parametrization of differential
forms in terms of their “1+3” decompositions. It is common for Minkowski,
Galilei and Carroll spacetimes. In this language, generators
$\rho_{p}=(S_{j},N_{j})$ of the representations of the corresponding three Lie
algebras on component spaces of $p$-forms are explicitly computed. Boosts
generators $N_{j}$ depend on particular spacetime.

Using the matrices of the generators, explicit matrix equations for the
intertwining operators $a_{qp}$ are written in Section 3. They are simplified
a lot due to rotations. Typically only a few free constants survive. The
constants are then further restricted by boosts.

Resulting nontrivial intertwining matrices are presented in Section 4. They
are also translated back into the original language of “1+3” decomposed
differential forms (as is convenient to see them in physics).

Section 5 provides summary and conclusions.

Appendix A presents detailed calculation of (more or less random) subset of
matrices $a_{qp}$.

In Appendix B, list of action of _Hodge star_ operators on forms is given.
They were computed in [7] by completely different method (appropriate
_modification_ of standard Hodge star _formula_ , using specific invariant
tensors present on Galilean and Carrollian spacetimes). They are useful for
comparison of the results.

Appendices C and D collect, for convenience of the reader, all information on
how we proceed, starting from Galilei and Carroll vector spaces, to Galilean
and Carrollian structure on manifolds (i.e., how to make each _tangent_ space
of $M$ a _Galilei and Carroll_ vector space). And how intertwining operators
are connected for the two situations.

Appendix E discusses in more detail why computations performed in Sections 2–4
(concerning Galilei and Carroll spacetimes) may be more or less directly used
in order to formulate corresponding statements for forms on Galilean and
Carrollian spacetimes.

Finally Appendix F provides explicit expressions for the $*$ operator on
general Galilean and Carrollian spacetimes.

##  2 Forms on Galilei, Carroll and Minkowski spacetime

###  2.1 Parametrization

In order to write down explicitly equations (1.4), we need to determine
explicit form of the matrices $\rho_{p}$ for

  1. –

all three $(1+3)$-dimensional spacetimes (Galilei, Carroll, Minkowski),

  2. –

all degrees $p=0,1,2,3,4$ of differential forms.

Let us start with a suitable coordinates and parametrization of forms.

In all three cases we have (global) coordinates
$\big{(}t,x^{i}\big{)}=(t,x,y,z)$ adapted to particular spacetime structures.
In these coordinates, we can use, in _all three_ cases, parametrization of
differential forms well-known from the Minkowski spacetime (see, e.g., [6,
Section 16.1]). That is, any $p$-form $\alpha$, $p=0,1,2,3,4$, may be first
uniquely decomposed as

$\alpha={\rm d}t\wedge\hat{s}+\hat{r},$ (2.1)

where the two hatted forms, $(p-1)$-form $\hat{s}$ and $p$-form $\hat{r}$,
respectively, are _spatial_ , meaning that they do not contain (“temporal”)
differential ${\rm d}t$ in its coordinate expression. Being spatial, $\hat{s}$
and $\hat{r}$ may be, depending on degree, expressed as

$f,\qquad\mathbf{a}\cdot{\rm d}\mathbf{r},\qquad\mathbf{a}\cdot{\rm
d}\mathbf{S},\qquad f{\rm d}V,$

where

$\displaystyle\mathbf{a}\cdot{\rm d}\mathbf{r}=a_{x}{\rm d}x+a_{y}{\rm
d}y+a_{z}{\rm d}z,$ $\displaystyle\mathbf{a}\cdot{\rm d}\mathbf{S}=a_{x}{\rm
d}S_{x}+a_{y}{\rm d}S_{y}+a_{z}{\rm d}S_{z}\equiv a_{x}{\rm d}y\wedge{\rm
d}z+a_{y}{\rm d}z\wedge{\rm d}x+a_{z}{\rm d}x\wedge{\rm d}y,$
$\displaystyle{\rm d}V={\rm d}x\wedge{\rm d}y\wedge{\rm d}z.$

So, the general decomposition formula (2.1) may be further specified, for the
five relevant degrees of differential forms, as follows:

$\displaystyle\Omega^{0}\colon\ \alpha=f,$ (2.2)
$\displaystyle\Omega^{1}\colon\ \alpha=f{\rm d}t+\mathbf{a}\cdot{\rm
d}\mathbf{r},$ (2.3) $\displaystyle\Omega^{2}\colon\ \alpha={\rm
d}t\wedge\mathbf{a}\cdot{\rm d}\mathbf{r}+\mathbf{b}\cdot{\rm d}\mathbf{S},$
(2.4) $\displaystyle\Omega^{3}\colon\ \alpha={\rm d}t\wedge\mathbf{a}\cdot{\rm
d}\mathbf{S}+f{\rm d}V,$ (2.5) $\displaystyle\Omega^{4}\colon\ \alpha=f{\rm
d}t\wedge{\rm d}V$ (2.6)

(Notice that the time coordinate $t$ itself may be present in _components_ of
the spatial parts, for example $a_{x}=a_{x}(t,x,y,z)$ in general.)

Alternatively, we can uniquely identify forms from (2.2)–(2.6) just with
columns

$\displaystyle f,\qquad\left(\begin{matrix}f\\\
\mathbf{a}\end{matrix}\right),\qquad\left(\begin{matrix}\mathbf{a}\\\
\mathbf{b}\end{matrix}\right),\qquad\left(\begin{matrix}f\\\
\mathbf{a}\end{matrix}\right),\qquad f.$ (2.7)

Now we go on to study action of corresponding spacetime symmetry groups on
forms within this parametrization.

###  2.2 Generators of rotations

In the above mentioned coordinates, what is common for all three spacetimes is
action of

  1. –

time and space translation,

  2. –

spatial (3D) rotations

$\displaystyle t^{\prime}=t+t_{0},$ (2.8)
$\displaystyle\mathbf{r}^{\prime}=R\mathbf{r}+\mathbf{r}_{0},$ (2.9)

where $R$ denotes $3\times 3$ rotation matrix.

What is different, is action of boosts (see below, Section 2.3).

Concerning scrambling of components of forms, time and space _translations_
alone do _nothing_ (since ${\rm d}t_{0}$ and ${\rm d}\mathbf{r}_{0}$ vanish),
so we can forget about them and restrict to rotations and boosts.

Moreover, we can restrict to _infinitesimal_ rotations and boosts since even
if $\rho_{p}$ in the commutative diagram (1.4) were meant as operators
representing corresponding symmetry _Lie group_ , the same intertwining
operators enter the “infinitesimal version” of the commutative diagram, where
$\rho_{p}$ already denote operators of the _derived_ representation of the
corresponding symmetry _Lie algebra_.

For infinitesimal rotations about $j$-th axis, formulas (2.8)–(2.9) reduce to

$\left(\begin{matrix}t^{\prime}\\\
\mathbf{r}^{\prime}\end{matrix}\right)=\left(\begin{matrix}t\\\
\mathbf{r}\end{matrix}\right)+\epsilon\left(\begin{matrix}0&0\\\
0&l_{j}\end{matrix}\right)\left(\begin{matrix}t\\\
\mathbf{r}\end{matrix}\right),$ (2.10)

where

$(l_{j})_{ik}=\epsilon_{ijk}.$

Formula (2.10) induces action on coordinate basis of forms

$\displaystyle{\rm d}t^{\prime}={\rm d}t,$ (2.11) $\displaystyle{\rm
d}\mathbf{r}^{\prime}=(\mymathbb{1}+\epsilon l_{j}){\rm d}\mathbf{r},$ (2.12)
$\displaystyle{\rm d}\mathbf{S}^{\prime}=(\mymathbb{1}+\epsilon l_{j}){\rm
d}\mathbf{S},$ (2.13) $\displaystyle{\rm d}V^{\prime}={\rm d}V.$ (2.14)

From requirement

$\displaystyle f^{\prime}=f,$ (2.15) $\displaystyle f^{\prime}{\rm
d}t^{\prime}+\mathbf{a}^{\prime}\cdot{\rm d}\mathbf{r}^{\prime}=f{\rm
d}t+\mathbf{a}\cdot{\rm d}\mathbf{r},$ (2.16) $\displaystyle{\rm
d}t^{\prime}\wedge\mathbf{a}^{\prime}\cdot{\rm
d}\mathbf{r}^{\prime}+\mathbf{b}^{\prime}\cdot{\rm d}\mathbf{S}^{\prime}={\rm
d}t\wedge\mathbf{a}\cdot{\rm d}\mathbf{r}+\mathbf{b}\cdot{\rm d}\mathbf{S},$
(2.17) $\displaystyle{\rm d}t^{\prime}\wedge\mathbf{a}^{\prime}\cdot{\rm
d}\mathbf{S}^{\prime}+f^{\prime}{\rm d}V^{\prime}={\rm
d}t\wedge\mathbf{a}\cdot{\rm d}\mathbf{S}+f{\rm d}V,$ (2.18) $\displaystyle
f^{\prime}{\rm d}t^{\prime}\wedge{\rm d}V^{\prime}=f{\rm d}t\wedge{\rm d}V,$
(2.19)

we can compute, for each degree, primed components in terms of unprimed and
get

$\displaystyle\Omega^{0}\colon\ f^{\prime}=f,$ (2.20)
$\displaystyle\Omega^{1}\colon\ \left(\begin{matrix}f^{\prime}\\\
\mathbf{a}^{\prime}\end{matrix}\right)=\left(\begin{matrix}f\\\
\mathbf{a}\end{matrix}\right)+\epsilon\left(\begin{matrix}0&0\\\
0&l_{j}\end{matrix}\right)\left(\begin{matrix}f\\\
\mathbf{a}\end{matrix}\right),$ (2.21) $\displaystyle\Omega^{2}\colon\
\left(\begin{matrix}\mathbf{a}^{\prime}\\\
\mathbf{b}^{\prime}\end{matrix}\right)=\left(\begin{matrix}\mathbf{a}\\\
\mathbf{b}\end{matrix}\right)+\epsilon\left(\begin{matrix}l_{j}&0\\\
0&l_{j}\end{matrix}\right)\left(\begin{matrix}\mathbf{a}\\\
\mathbf{b}\end{matrix}\right),$ (2.22) $\displaystyle\Omega^{3}\colon\
\left(\begin{matrix}f^{\prime}\\\
\mathbf{a}^{\prime}\end{matrix}\right)=\left(\begin{matrix}f\\\
\mathbf{a}\end{matrix}\right)+\epsilon\left(\begin{matrix}0&0\\\
0&l_{j}\end{matrix}\right)\left(\begin{matrix}f\\\
\mathbf{a}\end{matrix}\right),$ (2.23) $\displaystyle\Omega^{4}\colon\
f^{\prime}=f.$ (2.24)

All the formulas may be written (for rotation about $j$-th axis) as
multiplication of the correspondent component column by matrix

$\mymathbb{1}+\epsilon S_{j},$

where

$\displaystyle\Omega^{0}\colon\ S_{j}=0,$ $\displaystyle\Omega^{1}\colon\
S_{j}=\left(\begin{matrix}0&0\\\ 0&l_{j}\end{matrix}\right),$
$\displaystyle\Omega^{2}\colon\ S_{j}=\left(\begin{matrix}l_{j}&0\\\
0&l_{j}\end{matrix}\right),$ $\displaystyle\Omega^{3}\colon\
S_{j}=\left(\begin{matrix}0&0\\\ 0&l_{j}\end{matrix}\right),$
$\displaystyle\Omega^{4}\colon\ S_{j}=0.$

###  2.3 Generators of boosts

Let us display all three kinds of infinitesimal boosts along $\mathbf{n}$:

$\displaystyle\text{Lorentz}\colon\ $ $\displaystyle
t^{\prime}=t+\epsilon\mathbf{n}\cdot\mathbf{r},$ (2.25)
$\displaystyle\mathbf{r}^{\prime}=\mathbf{r}+\epsilon\mathbf{n}t,$ (2.26)
$\displaystyle\text{Galilei}\colon\ $ $\displaystyle t^{\prime}=t,$ (2.27)
$\displaystyle\mathbf{r}^{\prime}=\mathbf{r}+\epsilon\mathbf{n}t,$ (2.28)
$\displaystyle\text{Carroll}\colon\ $ $\displaystyle
t^{\prime}=t+\epsilon\mathbf{n}\cdot\mathbf{r},$ (2.29)
$\displaystyle\mathbf{r}^{\prime}=\mathbf{r}.$ (2.30)

(As for the Carroll boosts, see [16, formula (9)], [5, formula (2.10)] or
Appendix C.2.) The following formulas conveniently cover all three cases at
once:

$\displaystyle
t^{\prime}=t+\epsilon\lambda_{1}\mathbf{n}\cdot\mathbf{r},\qquad\mathbf{r}^{\prime}=\mathbf{r}+\epsilon\lambda_{2}\mathbf{n}t.$

The three special cases clearly correspond to values

$\displaystyle\text{Lorentz}\colon\ $
$\displaystyle(\lambda_{1},\lambda_{2})=(1,1),$ (2.31)
$\displaystyle\text{Galilei}\colon\ $
$\displaystyle(\lambda_{1},\lambda_{2})=(0,1),$ (2.32)
$\displaystyle\text{Carroll}\colon\ $
$\displaystyle(\lambda_{1},\lambda_{2})=(1,0).$ (2.33)

Induced action on coordinate basis of forms comes out to be

$\displaystyle\Omega^{1}\colon\ $ $\displaystyle{\rm d}t^{\prime}={\rm
d}t+\epsilon\lambda_{1}\mathbf{n}\cdot{\rm d}\mathbf{r},$ (2.34)
$\displaystyle{\rm d}\mathbf{r}^{\prime}={\rm
d}\mathbf{r}+\epsilon\lambda_{2}\mathbf{n}{\rm d}t;$ (2.35)
$\displaystyle\Omega^{2}\colon\ $ $\displaystyle{\rm d}t^{\prime}\wedge{\rm
d}\mathbf{r}^{\prime}={\rm d}t\wedge{\rm
d}\mathbf{r}-\epsilon\lambda_{1}(\mathbf{n}\cdot\mathbf{l}){\rm d}\mathbf{S},$
(2.36) $\displaystyle{\rm d}\mathbf{S}^{\prime}={\rm
d}\mathbf{S}+\epsilon\lambda_{2}{\rm d}t\wedge(\mathbf{n}\cdot\mathbf{l}){\rm
d}\mathbf{r};$ (2.37) $\displaystyle\Omega^{3}\colon\ $ $\displaystyle{\rm
d}t^{\prime}\wedge{\rm d}\mathbf{S}^{\prime}={\rm d}t\wedge{\rm
d}\mathbf{S}+\epsilon\lambda_{1}\mathbf{n}{\rm d}V,$ (2.38) $\displaystyle{\rm
d}V^{\prime}={\rm d}V+\epsilon\lambda_{2}{\rm d}t\wedge(\mathbf{n}\cdot{\rm
d}\mathbf{S});$ (2.39) $\displaystyle\Omega^{4}\colon\ $ $\displaystyle{\rm
d}t^{\prime}\wedge{\rm d}V^{\prime}={\rm d}t\wedge{\rm d}V.$ (2.40)

Writing down again (2.15)–(2.19) we get boost (along $x_{j}$) analogs of
(2.20)–(2.24)

$\displaystyle\Omega^{0}\colon\ f^{\prime}=f,$ (2.41)
$\displaystyle\Omega^{1}\colon\ \left(\begin{matrix}f^{\prime}\\\
\mathbf{a}^{\prime}\end{matrix}\right)=\left(\begin{matrix}f\\\
\mathbf{a}\end{matrix}\right)+\epsilon\left(\begin{matrix}0&\lambda_{2}e_{j}^{\rm
T}\\\ \lambda_{1}e_{j}&0\end{matrix}\right)\left(\begin{matrix}f\\\
\mathbf{a}\end{matrix}\right),$ (2.42) $\displaystyle\Omega^{2}\colon\
\left(\begin{matrix}\mathbf{a}^{\prime}\\\
\mathbf{b}^{\prime}\end{matrix}\right)=\left(\begin{matrix}\mathbf{a}\\\
\mathbf{b}\end{matrix}\right)+\epsilon\left(\begin{matrix}0&-\lambda_{2}l_{j}\\\
\lambda_{1}l_{j}&0\end{matrix}\right)\left(\begin{matrix}\mathbf{a}\\\
\mathbf{b}\end{matrix}\right),$ (2.43) $\displaystyle\Omega^{3}\colon\
\left(\begin{matrix}f^{\prime}\\\
\mathbf{a}^{\prime}\end{matrix}\right)=\left(\begin{matrix}f\\\
\mathbf{a}\end{matrix}\right)+\epsilon\left(\begin{matrix}0&\lambda_{1}e_{j}^{\rm
T}\\\ \lambda_{2}e_{j}&0\end{matrix}\right)\left(\begin{matrix}f\\\
\mathbf{a}\end{matrix}\right),$ (2.44) $\displaystyle\Omega^{4}\colon\
f^{\prime}=f$ (2.45)

(where $e_{j}$ denotes unit vector along $x_{j}$). They can be written via
multiplication by

$\mymathbb{1}+\epsilon N_{j},$

where

$\displaystyle\Omega^{0}\colon\ N_{j}=0,$ $\displaystyle\Omega^{1}\colon\
N_{j}=\left(\begin{matrix}0&\lambda_{2}e_{j}^{\rm T}\\\
\lambda_{1}e_{j}&0\end{matrix}\right),$ $\displaystyle\Omega^{2}\colon\
N_{j}=\left(\begin{matrix}0&-\lambda_{2}l_{j}\\\
\lambda_{1}l_{j}&0\end{matrix}\right),$ $\displaystyle\Omega^{3}\colon\
N_{j}=\left(\begin{matrix}0&\lambda_{1}e_{j}^{\rm T}\\\
\lambda_{2}e_{j}&0\end{matrix}\right),$ $\displaystyle\Omega^{4}\colon\
N_{j}=0.$

###  2.4 List of all generators $\boldsymbol{\rho_{p}}$

If we denote, in general,

$\rho_{p}=S_{j}\ \text{and}\ N_{j}\ \text{on components of}\
\text{$p$-forms},$

then the full list of generators on (suitably parametrized components of)
$p$-forms explicitly reads

$\displaystyle\rho_{0}\colon\ S_{j}=0,\qquad N_{j}=0;$ (2.46)
$\displaystyle\rho_{1}\colon\ S_{j}=\left(\begin{matrix}0&0\\\
0&l_{j}\end{matrix}\right),\qquad
N_{j}=\left(\begin{matrix}0&\lambda_{2}e_{j}^{\rm T}\\\
\lambda_{1}e_{j}&0\end{matrix}\right);$ (2.47) $\displaystyle\rho_{2}\colon\
S_{j}=\left(\begin{matrix}l_{j}&0\\\ 0&l_{j}\end{matrix}\right),\qquad
N_{j}=\left(\begin{matrix}0&-\lambda_{2}l_{j}\\\
\lambda_{1}l_{j}&0\end{matrix}\right);$ (2.48) $\displaystyle\rho_{3}\colon\
S_{j}=\left(\begin{matrix}0&0\\\ 0&l_{j}\end{matrix}\right),\qquad
N_{j}=\left(\begin{matrix}0&\lambda_{1}e_{j}^{\rm T}\\\
\lambda_{2}e_{j}&0\end{matrix}\right);$ (2.49) $\displaystyle\rho_{4}\colon\
S_{j}=0,\qquad N_{j}=0.$ (2.50)

Observe a simple rule, which can save some computations in the future:

  1. –

Interchanging
$(\lambda_{1},\lambda_{2})\leftrightarrow(\lambda_{2},\lambda_{1})$ leads to
interchanging $\rho_{1}\leftrightarrow\rho_{3}$

or, as a formula,

$\displaystyle\rho_{3}(\lambda_{1},\lambda_{2})=\rho_{1}(\lambda_{2},\lambda_{1}).$
(2.51)

##  3 Intertwining operators $\boldsymbol{a_{qp}}$

###  3.1 What intertwining operators we are interested in

On Minkowski, Galilei or Carroll spacetime, respectively, let

$a_{qp}\colon\ \Omega^{p}\to\Omega^{q}$

be a linear operator realized as a _matrix_ on _components_ of the forms.
(Notice the order of indices convention – mapping from $p$ to $q$ is denoted
as $a_{qp}$ rather than vice versa.)

Recall (see (2.7)) that the components of $p$-forms are parametrized, for
$p=0,\dots,4$, either by single objects or by pairs. Therefore, the matrices
$a_{qp}$ are necessarily (constant) _block_ matrices of (block) dimensions
$1\times 1$, $1\times 2$, $2\times 1$ or $2\times 2$. As an example, matrices
$a_{12}$ and $a_{02}$ are of the form

$\displaystyle a_{12}=\left(\begin{matrix}\mathbf{c}^{\rm T}&\mathbf{d}^{\rm
T}\\\ C&D\end{matrix}\right),\qquad a_{02}=\left(\begin{matrix}\mathbf{c}^{\rm
T}&\mathbf{d}^{\rm T}\end{matrix}\right)$ (3.1)

for $\mathbf{c}$, $\mathbf{d}$ (column) vectors (i.e., $\mathbf{c}^{\rm T}$,
$\mathbf{d}^{\rm T}$ row vectors) and $C$, $D$ $3\times 3$ matrices.

Within the spaces $\Omega^{p}$ and $\Omega^{q}$, the corresponding (Lorentz,
Galilei or Carroll) transformations induce (as generators) matrices $\rho_{p}$
and $\rho_{q}$, respectively, acting on components of forms and explicitly
given, for both rotations and boosts, by (2.46)–(2.50).

Now we want $a_{qp}$ to be _intertwining operator_ for the two
representations. So we require that standard commutative diagram holds:

$\begin{CD}\Omega^{p}@>{a_{qp}}>{}>\Omega^{q}\\\
@V{\rho_{p}}V{}V@V{}V{\rho_{q}}V\\\ \Omega^{p}@>{}>{a_{qp}}>\Omega^{q}\\\
\end{CD}\qquad\text{i.e.,}\qquad\rho_{q}\circ a_{qp}=a_{qp}\circ\rho_{p}.$
(3.2)

Thus, we are to solve equation (3.2) for each $q,p=0,\dots,4$ (and any $j$
hidden in $\rho_{p}$ and $\rho_{q}$).

Realize that, as is always the case for intertwining operators, solutions
constitute a _vector space_ , so that we are actually to find _a basis_ of the
space of solutions.

As an example, for $a_{12}$ and $a_{02}$, we are to solve

$\displaystyle\rho_{1}\circ a_{12}=a_{12}\circ\rho_{2},\qquad\rho_{0}\circ
a_{02}=a_{02}\circ\rho_{2}$

for

  1. –

matrices $a_{12}$ and $a_{02}$ (i.e., for unknown $\mathbf{c}$, $\mathbf{d}$,
$C$ and $D$ in (3.1)),

  2. –

$\rho_{0}$, $\rho_{1}$ and $\rho_{2}$ displayed in (2.46), (2.47) and (2.48).

Performing this for _rotation_ generators $\rho_{0}$, $\rho_{1}$ and
$\rho_{2}$ _alone_ (i.e., only for $S_{j}$ operators from (2.46), (2.47) and
(2.48)), we get

$\displaystyle a_{12}=\left(\begin{matrix}\mathbf{0}^{\rm T}&\mathbf{0}^{\rm
T}\\\ k_{1}\mymathbb{1}&k_{2}\mymathbb{1}\end{matrix}\right),\qquad
a_{02}=0\equiv\left(\begin{matrix}\mathbf{0}^{\rm T}&\mathbf{0}^{\rm
T}\end{matrix}\right).$ (3.3)

So, $a_{02}$ is completely fixed by rotations alone (it _vanishes_ – there is
_no non-vanishing_ intertwining $a_{02}$ _irrespective_ of the spacetime under
consideration) and in $a_{12}$ still two free constants survive.

Repeating the same procedure with $a_{12}$ (already with ansatz (3.3)) for
_boost_ generators $\rho_{1}$ and $\rho_{2}$ we get the following restrictions
on $k_{1}$, $k_{2}$ (see the details in Appendix A):

$k_{1}\lambda_{2}=k_{2}\lambda_{2}=k_{2}\lambda_{1}=0$

(notice that they already depend on $\lambda_{1}$, $\lambda_{2}$, i.e., on
whether we speak of Lorentz, Galilei or Carroll case). This should be analyzed
for all three cases (2.31)–(2.33) separately and the result is that the _only
non-vanishing_ operator is (constant multiple of)

$a_{12}=\left(\begin{matrix}\mathbf{0}^{\rm T}&\mathbf{0}^{\rm T}\\\
\mymathbb{1}&\mymathbb{0}\end{matrix}\right)\qquad\text{for \emph{Carroll}
spacetime}.$

When translated back into the language of complete forms (rather than just
their component column), it produces the following (intertwining) mapping
$\Omega^{2}\to\Omega^{1}$:

$a_{12}\colon\ {\rm d}t\wedge\mathbf{a}\cdot{\rm
d}\mathbf{r}+\mathbf{b}\cdot{\rm d}\mathbf{S}\mapsto\mathbf{a}\cdot{\rm
d}\mathbf{r}\qquad\text{for \emph{Carroll} spacetime}.$

In what follows we perform similar calculation for _all_ $a_{qp}$ (and _all
three_ spacetimes).

###  3.2 General structure of all $\boldsymbol{a_{qp}}$ (no invariance, yet)

A priori, just from the parametrization structure (2.7) of spaces $\Omega^{q}$
and $\Omega^{p}$ (with no intertwining properties, yet) the operators $a_{qp}$

$\displaystyle a_{00}\qquad$ $\displaystyle a_{01}\qquad$ $\displaystyle
a_{02}\qquad$ $\displaystyle a_{03}\qquad$ $\displaystyle a_{04}$
$\displaystyle a_{10}\qquad$ $\displaystyle a_{11}\qquad$ $\displaystyle
a_{12}\qquad$ $\displaystyle a_{13}\qquad$ $\displaystyle a_{14}$
$\displaystyle a_{20}\qquad$ $\displaystyle a_{21}\qquad$ $\displaystyle
a_{22}\qquad$ $\displaystyle a_{23\qquad}$ $\displaystyle a_{24}$
$\displaystyle a_{30}\qquad$ $\displaystyle a_{31}\qquad$ $\displaystyle
a_{32}\qquad$ $\displaystyle a_{33}\qquad$ $\displaystyle a_{34}$
$\displaystyle a_{40}\qquad$ $\displaystyle a_{41}\qquad$ $\displaystyle
a_{42}\qquad$ $\displaystyle a_{43}\qquad$ $\displaystyle a_{44}$

should be parametrized as follows (we already encountered $a_{12}$ and
$a_{02}$ in (3.1)):

$\displaystyle\quad k\qquad$
$\displaystyle\left(\begin{matrix}k&\mathbf{c}^{\rm
T}\end{matrix}\right)\qquad$ $\displaystyle\left(\begin{matrix}\mathbf{c}^{\rm
T}&\mathbf{d}^{\rm T}\end{matrix}\right)\qquad$
$\displaystyle\left(\begin{matrix}k&\mathbf{c}^{\rm
T}\end{matrix}\right)\qquad$ $\displaystyle\quad k$ (3.4)
$\displaystyle\left(\begin{matrix}k\\\ \mathbf{c}\end{matrix}\right)\qquad$
$\displaystyle\left(\begin{matrix}k&\mathbf{c}^{\rm T}\\\
\mathbf{d}&C\end{matrix}\right)\qquad$
$\displaystyle\left(\begin{matrix}\mathbf{c}^{\rm T}&\mathbf{d}^{\rm T}\\\
C&D\end{matrix}\right)\qquad$
$\displaystyle\left(\begin{matrix}k&\mathbf{c}^{\rm T}\\\
\mathbf{d}&C\end{matrix}\right)\qquad$ $\displaystyle\left(\begin{matrix}k\\\
\mathbf{c}\end{matrix}\right)$ (3.5)
$\displaystyle\left(\begin{matrix}\mathbf{c}\\\
\mathbf{d}\end{matrix}\right)\qquad$
$\displaystyle\left(\begin{matrix}\mathbf{c}&C\\\
\mathbf{d}&D\end{matrix}\right)\qquad$
$\displaystyle\left(\begin{matrix}C&D\\\ E&F\end{matrix}\right)\qquad$
$\displaystyle\left(\begin{matrix}\mathbf{c}&C\\\
\mathbf{d}&D\end{matrix}\right)\qquad$
$\displaystyle\left(\begin{matrix}\mathbf{c}\\\ \mathbf{d}\end{matrix}\right)$
(3.6) $\displaystyle\left(\begin{matrix}k\\\
\mathbf{c}\end{matrix}\right)\qquad$
$\displaystyle\left(\begin{matrix}k&\mathbf{c}^{\rm T}\\\
\mathbf{d}&C\end{matrix}\right)\qquad$
$\displaystyle\left(\begin{matrix}\mathbf{c}^{\rm T}&\mathbf{d}^{\rm T}\\\
C&D\end{matrix}\right)\qquad$
$\displaystyle\left(\begin{matrix}k&\mathbf{c}^{\rm T}\\\
\mathbf{d}&C\end{matrix}\right)\qquad$ $\displaystyle\left(\begin{matrix}k\\\
\mathbf{c}\end{matrix}\right)$ (3.7) $\displaystyle\quad k\qquad$
$\displaystyle\left(\begin{matrix}k&\mathbf{c}^{\rm
T}\end{matrix}\right)\qquad$ $\displaystyle\left(\begin{matrix}\mathbf{c}^{\rm
T}&\mathbf{d}^{\rm T}\end{matrix}\right)\qquad$
$\displaystyle\left(\begin{matrix}k&\mathbf{c}^{\rm
T}\end{matrix}\right)\qquad$ $\displaystyle\quad k.$ (3.8)

Notice the symmetry of the structure of the matrices: The pattern (type of
matrix) happens to be symmetric with respect to both

  1. –

reflection across the _central horizontal_ axis and

  2. –

reflection across the _central vertical_ axis.

This is just a simple consequence of the symmetry of parametrizations (2.7)
w.r.t. left-right reflection over the center. So there are altogether 9
_types_ of matrices $a_{qp}$.

###  3.3 How $\boldsymbol{a_{qp}}$ are restricted by rotation invariance alone

After applying rotational invariance alone (i.e., property (3.2) with just
rotation generators $S_{j}$), the matrices (3.4)–(3.8) simplify a lot. In
short the simplification may be described as application of the following
rules: All

  1. –

scalars remain intact (all scalars are rotation invariant),

  2. –

vectors become zero (no rotation invariant vectors except for zero),

  3. –

matrices become multiples of unity (no other rotation invariant matrices).

What we get is the following, much simpler, parametrization:

$\displaystyle\quad k\qquad$
$\displaystyle\left(\begin{matrix}k&\mathbf{0}^{\rm
T}\end{matrix}\right)\qquad$ $\displaystyle\left(\begin{matrix}\mathbf{0}^{\rm
T}&\mathbf{0}^{\rm T}\end{matrix}\right)\qquad$
$\displaystyle\left(\begin{matrix}k&\mathbf{0}^{\rm
T}\end{matrix}\right)\qquad$ $\displaystyle\quad k$ (3.9)
$\displaystyle\left(\begin{matrix}k\\\ \mathbf{0}\end{matrix}\right)\qquad$
$\displaystyle\left(\begin{matrix}k_{1}&\mathbf{0}^{\rm T}\\\
\mathbf{0}&k_{2}\mymathbb{1}\end{matrix}\right)\qquad$
$\displaystyle\left(\begin{matrix}\mathbf{0}^{\rm T}&\mathbf{0}^{\rm T}\\\
k_{1}\mymathbb{1}&k_{2}\mymathbb{1}\end{matrix}\right)\qquad$
$\displaystyle\left(\begin{matrix}k_{1}&\mathbf{0}^{\rm T}\\\
\mathbf{0}&k_{2}\mymathbb{1}\end{matrix}\right)\qquad$
$\displaystyle\left(\begin{matrix}k\\\ \mathbf{0}\end{matrix}\right)$ (3.10)
$\displaystyle\left(\begin{matrix}\mathbf{0}\\\
\mathbf{0}\end{matrix}\right)\qquad$
$\displaystyle\left(\begin{matrix}\mathbf{0}&k_{1}\mymathbb{1}\\\
\mathbf{0}&k_{2}\mymathbb{1}\end{matrix}\right)\qquad$
$\displaystyle\left(\begin{matrix}k_{1}\mymathbb{1}&k_{2}\mymathbb{1}\\\
k_{3}\mymathbb{1}&k_{4}\mymathbb{1}\end{matrix}\right)\qquad$
$\displaystyle\left(\begin{matrix}\mathbf{0}&k_{1}\mymathbb{1}\\\
\mathbf{0}&k_{2}\mymathbb{1}\end{matrix}\right)\qquad$
$\displaystyle\left(\begin{matrix}\mathbf{0}\\\ \mathbf{0}\end{matrix}\right)$
(3.11) $\displaystyle\left(\begin{matrix}k\\\
\mathbf{0}\end{matrix}\right)\qquad$
$\displaystyle\left(\begin{matrix}k_{1}&\mathbf{0}^{\rm T}\\\
\mathbf{0}&k_{2}\mymathbb{1}\end{matrix}\right)\qquad$
$\displaystyle\left(\begin{matrix}\mathbf{0}^{\rm T}&\mathbf{0}^{\rm T}\\\
k_{1}\mymathbb{1}&k_{2}\mymathbb{1}\end{matrix}\right)\qquad$
$\displaystyle\left(\begin{matrix}k_{1}&\mathbf{0}^{\rm T}\\\
\mathbf{0}&k_{2}\mymathbb{1}\end{matrix}\right)\qquad$
$\displaystyle\left(\begin{matrix}k\\\ \mathbf{0}\end{matrix}\right)$ (3.12)
$\displaystyle\quad k\qquad$
$\displaystyle\left(\begin{matrix}k&\mathbf{0}^{\rm
T}\end{matrix}\right)\qquad$ $\displaystyle\left(\begin{matrix}\mathbf{0}^{\rm
T}&\mathbf{0}^{\rm T}\end{matrix}\right)\qquad$
$\displaystyle\left(\begin{matrix}k&\mathbf{0}^{\rm
T}\end{matrix}\right)\qquad$ $\displaystyle\quad k.$ (3.13)

###  3.4 Restrictions on $\boldsymbol{a_{qp}}$ added by boosts

There are further restrictions on operators $a_{qp}$ displayed in (3.9)–(3.13)
added by boosts. Since boost generators already depend on
$(\lambda_{1},\lambda_{2})$, these restrictions depend on the choice of
particular spacetime. So we get, in general, different set of operators
$a_{qp}$ for different spacetimes.

An example of detailed computations is presented in Appendix A.

Here we give the results (and comments).

##  4 Results – non-trivial operators $\boldsymbol{a_{qp}}$

###  4.1 Which operators $\boldsymbol{a_{qp}}$ are trivial

It is evident from the structure of (3.2) that

  1. –

_zero_ operator (for any pair of spaces, $\Omega^{q}\to\Omega^{p}$) as well as

  2. –

_unit_ operator within any space ($\hat{1}\colon\Omega^{p}\to\Omega^{p}$)

are solutions, i.e., intertwining operators.

Unit operators form part of those found by our computation; so, for example,
$a_{11}$ and $a_{33}$ from (A.3) are in a sense trivial and they may be
omitted from our further discussion. (And they clearly work in _any_
spacetime.)

In what follows we summarize all _non-trivial_ intertwining operators found in
our systematic computations. They depend, as we have seen, on particular
spacetime.

###  4.2 Non-trivial operators $\boldsymbol{a_{qp}}$ on Minkowski spacetime

On Minkowski spacetime we have found the following non-trivial operators:

$\displaystyle a_{40}=a_{04}=1,\qquad
a_{31}=a_{13}=\left(\begin{matrix}1&\mathbf{0}^{\rm T}\\\
\mathbf{0}&\mymathbb{1}\end{matrix}\right),\qquad
a_{22}=\left(\begin{matrix}\hphantom{-}\mymathbb{0}&\mymathbb{1}\\\
-\mymathbb{1}&\mymathbb{0}\end{matrix}\right)$

(see (A.4) and (A.7); we use brief notation $a_{22}$ ignoring its (trivial)
_unit_ part).

From the matrices displayed above we can read-off how the five operators
actually act on corresponding differential forms:

$\displaystyle a_{40}\colon\ f\mapsto f{\rm d}t\wedge{\rm d}V,$ (4.1)
$\displaystyle a_{31}\colon\ f{\rm d}t+\mathbf{a}\cdot{\rm
d}\mathbf{r}\mapsto{\rm d}t\wedge\mathbf{a}\cdot{\rm d}\mathbf{S}+f{\rm d}V,$
(4.2) $\displaystyle a_{22}\colon\ {\rm d}t\wedge\mathbf{a}\cdot{\rm
d}\mathbf{r}+\mathbf{b}\cdot{\rm d}\mathbf{S}\mapsto{\rm
d}t\wedge\mathbf{b}\cdot{\rm d}\mathbf{r}-\mathbf{a}\cdot{\rm d}\mathbf{S},$
(4.3) $\displaystyle a_{13}\colon\ {\rm d}t\wedge\mathbf{a}\cdot{\rm
d}\mathbf{S}+f{\rm d}V\mapsto f{\rm d}t+\mathbf{a}\cdot{\rm d}\mathbf{r},$
(4.4) $\displaystyle a_{04}\colon\ f{\rm d}t\wedge{\rm d}V\mapsto f.$ (4.5)

Now if we compare this with the action of standard (Minkowski) _Hodge star_
operator $*\equiv*_{\text{Min}}$ (see, e.g., [6, Section 16.1]) on forms

$\displaystyle*f=f{\rm d}t\wedge{\rm d}V,$ (4.6) $\displaystyle*(f{\rm
d}t+\mathbf{a}\cdot{\rm d}\mathbf{r})={\rm d}t\wedge\mathbf{a}\cdot{\rm
d}\mathbf{S}+f{\rm d}V,$ (4.7) $\displaystyle*({\rm
d}t\wedge\mathbf{a}\cdot{\rm d}\mathbf{r}+\mathbf{b}\cdot{\rm
d}\mathbf{S})={\rm d}t\wedge\mathbf{b}\cdot{\rm
d}\mathbf{r}-\mathbf{a}\cdot{\rm d}\mathbf{S},$ (4.8) $\displaystyle*({\rm
d}t\wedge\mathbf{a}\cdot{\rm d}\mathbf{S}+f{\rm d}V)=f{\rm
d}t+\mathbf{a}\cdot{\rm d}\mathbf{r},$ (4.9) $\displaystyle*(f{\rm
d}t\wedge{\rm d}V)=-f,$ (4.10)

wee see that

  1. –

we merely “rediscovered” _standard_ Hodge star in Minkowski spacetime,

  2. –

_nothing more_ than (a multiple of) the Hodge star is intertwining on forms,
there.

Invariance of the Hodge star is well known. So what we _actually_ learned
about the Minkowski case is that the Hodge star is in fact _the only_ relevant
operator with the property, there. (And this is fairly plausible result;
otherwise we certainly met the other operators in important physics
equations.)

As we will see in the next two paragraphs, the situation in Galilei as well as
Carroll spacetimes turns out to be _refreshingly different_.

###  4.3 Non-trivial operators $\boldsymbol{a_{qp}}$ on Galilei spacetime

On _Galilei_ spacetime we have found quite a lot of non-trivial operators.

First, the following five _analogs_ of Minkowski ones:

$\displaystyle a_{40}=a_{04}=1,$ (4.11) $\displaystyle
a_{31}=\left(\begin{matrix}0&\mathbf{0}^{\rm T}\\\
\mathbf{0}&\mymathbb{1}\end{matrix}\right),$ (4.12) $\displaystyle
a_{22}=\left(\begin{matrix}\mymathbb{0}&\mymathbb{1}\\\
\mymathbb{0}&\mymathbb{0}\end{matrix}\right),$ (4.13) $\displaystyle
a_{13}=\left(\begin{matrix}1&\mathbf{0}^{\rm T}\\\
\mathbf{0}&\mymathbb{0}\end{matrix}\right)$ (4.14)

(see (A.5) and (A.9)). On corresponding differential forms:

$\displaystyle a_{40}\colon\ f\mapsto f{\rm d}t\wedge{\rm d}V,$ (4.15)
$\displaystyle a_{31}\colon\ f{\rm d}t+\mathbf{a}\cdot{\rm
d}\mathbf{r}\mapsto{\rm d}t\wedge\mathbf{a}\cdot{\rm d}\mathbf{S},$ (4.16)
$\displaystyle a_{22}\colon\ {\rm d}t\wedge\mathbf{a}\cdot{\rm
d}\mathbf{r}+\mathbf{b}\cdot{\rm d}\mathbf{S}\mapsto{\rm
d}t\wedge\mathbf{b}\cdot{\rm d}\mathbf{r},$ (4.17) $\displaystyle
a_{13}\colon\ {\rm d}t\wedge\mathbf{a}\cdot{\rm d}\mathbf{S}+f{\rm d}V\mapsto
f{\rm d}t,$ (4.18) $\displaystyle a_{04}\colon\ f{\rm d}t\wedge{\rm d}V\mapsto
f.$ (4.19)

In _Minkowski_ case, see (4.1)–(4.5), we realized that _this type_ of
operators was nothing but the _Hodge star_ acting on various degrees of forms,
see (4.6)–(4.10). Here this is, in a sense, a similar story. One can introduce
“Galilei Hodge star” on Galilei spacetime (see Appendix B and for more
details, including motivation, see [7]) and it turns out that (4.15)–(4.19)
just reproduce the formulas valid for the Galilei Hodge star. (Actually there
are as many as two possibilities for the Galilei Hodge described in [7] and
the last operator, $a_{04}$, reproduces one of them (the other gives zero.) So
we can say again, like we did at the end of Section 4.2, that in (4.15)–(4.19)
we “merely rediscovered” _Galilei_ Hodge star on Galilei spacetime.

Now there are, however, in addition to (4.11)–(4.14), operators with _no
analogy_ to Minkowski situation. They act as follows:

$\Omega^{0}\overset{a_{{}_{10}}}{\to}\Omega^{1}\overset{a_{21}}{\to}\Omega^{2}\overset{a_{32}}{\to}\Omega^{3}\overset{a_{43}}{\to}\Omega^{4}.$

Here

$\displaystyle a_{10}=\left(\begin{matrix}1\\\
\mathbf{0}\end{matrix}\right),\qquad
a_{21}=\left(\begin{matrix}\mathbf{0}&\mymathbb{1}\\\
\mathbf{0}&\mymathbb{0}\end{matrix}\right),\qquad
a_{32}=\left(\begin{matrix}\mathbf{0}^{\rm T}&\mathbf{0}^{\rm T}\\\
\mymathbb{0}&\mymathbb{1}\end{matrix}\right),\qquad
a_{43}=\left(\begin{matrix}1&\mathbf{0}^{\rm T}\end{matrix}\right).$ (4.20)

So their actual action on corresponding differential forms reads

$\displaystyle a_{10}\colon\ f\mapsto f{\rm d}t,$ $\displaystyle a_{21}\colon\
f{\rm d}t+\mathbf{a}\cdot{\rm d}\mathbf{r}\mapsto{\rm
d}t\wedge\mathbf{a}\cdot{\rm d}\mathbf{r},$ $\displaystyle a_{32}\colon\ {\rm
d}t\wedge\mathbf{a}\cdot{\rm d}\mathbf{r}+\mathbf{b}\cdot{\rm
d}\mathbf{S}\mapsto{\rm d}t\wedge\mathbf{b}\cdot{\rm d}\mathbf{S},$
$\displaystyle a_{43}\colon\ {\rm d}t\wedge\mathbf{a}\cdot{\rm
d}\mathbf{S}+f{\rm d}V\mapsto f{\rm d}t\wedge{\rm d}V.$

We see that the rule how these operators work is nothing but application of
_exterior product_

$\displaystyle(\dots)\mapsto{\rm d}t\wedge(\dots)$ (4.21)

on the form under consideration. This is clearly

  1. –

an operator of degree $+1$ and it is also,

  2. –

_Galilei-invariant_ operation on forms,

since

${\rm d}t^{\prime}={\rm d}t$

for Galilei boosts (2.27) and (2.28). (Speak nothing of rotations and
translations.) Finally, it should be noted that there were still two more
operators found, namely

$\displaystyle a_{03}=\left(\begin{matrix}1&\mathbf{0}^{\rm
T}\end{matrix}\right),\qquad a_{14}=\left(\begin{matrix}1\\\
\mathbf{0}\end{matrix}\right),$

i.e., with action on forms

$\displaystyle a_{03}\colon\ {\rm d}t\wedge\mathbf{a}\cdot{\rm
d}\mathbf{S}+f{\rm d}V\mapsto f,\qquad a_{14}\colon\ f{\rm d}t\wedge{\rm
d}V\mapsto f{\rm d}t.$

Both of them are, however, just _compositions_ of operators already described
previously, namely

$\displaystyle a_{03}=a_{04}\circ a_{43},\qquad a_{14}=a_{10}\circ a_{04}.$

###  4.4 Non-trivial operators $\boldsymbol{a_{qp}}$ on Carroll spacetime

On _Carroll_ spacetime we have found quite a lot of non-trivial operators,
too.

First, the following five _analogs_ of Lorentzian ones:

$\displaystyle a_{40}=a_{04}=1,\qquad
a_{31}=\left(\begin{matrix}1&\mathbf{0}^{\rm T}\\\
\mathbf{0}&\mymathbb{0}\end{matrix}\right),\qquad
a_{22}=\left(\begin{matrix}\mymathbb{0}&\mymathbb{0}\\\
\mymathbb{1}&\mymathbb{0}\end{matrix}\right),\qquad
a_{13}=\left(\begin{matrix}0&\mathbf{0}^{\rm T}\\\
\mathbf{0}&\mymathbb{1}\end{matrix}\right).$

(see (A.6) and (A.8)). On corresponding differential forms:

$\displaystyle a_{40}\colon\ f\mapsto f{\rm d}t\wedge{\rm d}V,$ (4.22)
$\displaystyle a_{31}\colon\ f{\rm d}t+\mathbf{a}\cdot{\rm d}\mathbf{r}\mapsto
f{\rm d}V,$ (4.23) $\displaystyle a_{22}\colon\ {\rm
d}t\wedge\mathbf{a}\cdot{\rm d}\mathbf{r}+\mathbf{b}\cdot{\rm
d}\mathbf{S}\mapsto\mathbf{a}\cdot{\rm d}\mathbf{S},$ (4.24) $\displaystyle
a_{13}\colon\ {\rm d}t\wedge\mathbf{a}\cdot{\rm d}\mathbf{S}+f{\rm
d}V\mapsto\mathbf{a}\cdot{\rm d}\mathbf{r},$ (4.25) $\displaystyle
a_{04}\colon\ f{\rm d}t\wedge{\rm d}V\mapsto f.$ (4.26)

Here the story is similar to the Galilei case. One can introduce “Carroll
Hodge star” in Carroll spacetime (again see Appendix B and [7]) and it turns
out that (4.22)–(4.26) just reproduce the formulas valid for the Carroll Hodge
star.

So we can say again, like we did at the end of Sections 4.2 and 4.3, that in
(4.22)–(4.26) we “merely rediscovered” _Carroll_ Hodge star in Carroll
spacetime.

In addition, again, there are operators with _no analogy_ to Minkowski
situation (albeit with a _strong_ analogy to the _Galilei_ one). They act as
follows:

$\Omega^{0}\overset{a_{{}_{01}}}{\leftarrow}\Omega^{1}\overset{a_{12}}{\leftarrow}\Omega^{2}\overset{a_{23}}{\leftarrow}\Omega^{3}\overset{a_{34}}{\leftarrow}\Omega^{4}.$

Here

$\displaystyle a_{01}=\left(\begin{matrix}1&\mathbf{0}^{\rm
T}\end{matrix}\right),\qquad a_{12}=\left(\begin{matrix}\mathbf{0}^{\rm
T}&\mathbf{0}^{\rm T}\\\ \mymathbb{1}&\mymathbb{0}\end{matrix}\right),\qquad
a_{23}=\left(\begin{matrix}\mathbf{0}&\mymathbb{0}\\\
\mathbf{0}&\mymathbb{1}\end{matrix}\right),\qquad
a_{34}=\left(\begin{matrix}1\\\ \mathbf{0}\end{matrix}\right).$

On corresponding differential forms it is

$\displaystyle a_{01}\colon\ f{\rm d}t+\mathbf{a}\cdot{\rm d}\mathbf{r}\mapsto
f,$ $\displaystyle a_{12}\colon\ {\rm d}t\wedge\mathbf{a}\cdot{\rm
d}\mathbf{r}+\mathbf{b}\cdot{\rm d}\mathbf{S}\mapsto\mathbf{a}\cdot{\rm
d}\mathbf{r},$ $\displaystyle a_{23}\colon\ {\rm d}t\wedge\mathbf{a}\cdot{\rm
d}\mathbf{S}+f{\rm d}V\mapsto\mathbf{a}\cdot{\rm d}\mathbf{S},$ $\displaystyle
a_{34}\colon\ f{\rm d}t\wedge{\rm d}V\mapsto f{\rm d}V.$

We can see that the rule how these four operators work is nothing but
application of _interior product_

$\displaystyle(\dots)\mapsto i_{\partial_{t}}(\dots)$ (4.27)

on the form under consideration. It is clearly

  1. –

an operator of degree $-1$ and,

  2. –

it is also _Carroll-invariant_ operation on forms,

since

$\partial_{t^{\prime}}=\partial_{t}$

for Carroll boost (2.29) and (2.30). (Speak nothing of rotations and
translations.)

Finally, it should be noted that there were still two more operators found,
namely

$\displaystyle a_{30}=\left(\begin{matrix}1\\\
\mathbf{0}\end{matrix}\right),\qquad
a_{41}=\left(\begin{matrix}1&\mathbf{0}^{\rm T}\end{matrix}\right),$

i.e., with action on forms

$\displaystyle a_{30}\colon\ f\mapsto f{\rm d}V,\qquad a_{41}\colon\ f{\rm
d}t+\mathbf{a}\cdot{\rm d}\mathbf{r}\mapsto f{\rm d}t\wedge{\rm d}V.$

Both of them are, however, just _compositions_ of operators already described
previously, namely

$\displaystyle a_{30}=a_{34}\circ a_{40},\qquad a_{41}=a_{40}\circ a_{01}.$

###  4.5 A note on the degree $\boldsymbol{\pm 1}$ operators

In Galilean and Carrollian spacetimes, invariant tensor fields is a standard
knowledge.

In Galilei case, say, two of them are, within our notation (in adapted
coordinates), tensors

$\delta^{ij}\partial_{i}\otimes\partial_{j}\in\mathcal{T}^{2}_{0},\qquad{\rm
d}t\in\mathcal{T}^{0}_{1}.$

In Carroll case we have similarly

$\delta_{ij}{\rm d}x^{i}\otimes{\rm
d}x^{j}\in\mathcal{T}^{0}_{2},\qquad\partial_{t}\in\mathcal{T}^{1}_{0}.$

For example, for the Galilean case, we can already see them, to credit just
some fathers-founders, in (differently looking) component presentations as

$\displaystyle g^{\alpha\beta}t_{\beta}=0\qquad$ in 1963 paper [20],
$\displaystyle h^{ab}t_{b}=0\qquad$ in 1966 paper [21],
$\displaystyle\gamma^{\alpha\beta}\psi_{\beta}=0\qquad$ in 1972 paper [15],

(they all describe the same equation).

On the other side, among the recent occurrence, we can mention notation and
terminology

$\displaystyle\text{Galilean}\colon\ $ $\displaystyle\gamma\quad\text{spatial
cometric},$ $\displaystyle\tau\quad\text{clock one-form};$ (4.28)
$\displaystyle\text{Carrollian}\colon\ $ $\displaystyle h\quad\text{spatial
metric},$ $\displaystyle\xi\quad\text{Carrollian vector field},$ (4.29)

with

$\gamma(\tau,\ \cdot\ )=0,\qquad h(\xi,\ \cdot\ )=0$

from 2020 in [8].

Now a simple observation is that, just being aware of tensor fields $\tau$ and
$\xi$ from (4.28) and (4.29), it should be clear from the very beginning that
two _invariant_ degree $\pm 1$ _operators_ are available,

$\tau\wedge(\dots)\quad\text{on}\ \Omega\ (\text{Galilei}),\qquad
i_{\xi}(\dots)\quad\text{on}\ \Omega\ (\text{Carroll}).$

Notice that both may be regarded as _differentials_ in corresponding de Rham
complex.

Also notice that they have _no counterpart_ for forms on _Lorentzian_
spacetimes since there is neither invariant one-form nor invariant vector
field, there. We identify these interpretations of our “degree $\pm 1$
operators” in (4.21) and (4.27).

###  4.6 Invariant operators and invariant subspaces

In Galilei and Carroll spacetimes, some features of resulting invariant
operators differ from those in Minkowski spacetime. Namely

  1. –

analogs of Hodge star happen to be degenerate,

  2. –

there are operators connecting spaces which were not connected before.

There is a simple _formal_ reason for these features could appear. Namely, the
representations of Galilei and Carroll groups on components of forms possess
_nontrivial invariant_ subspaces.

Due to Schur’s lemma, any intertwining operator between two _irreducible_
representations is either zero or isomorphism. So, for irreducible
representations, _non-zero_ intertwining operators are necessarily
isomorphisms, and therefore they operate between spaces of equal dimensions.

Now representations of Lorentz group on the space of components of $p$-forms
_are irreducible_. Indeed, we see directly from explicit formulas (namely from
(2.7) and (2.41)–(2.45)) that the space is

  1. –

either 1-dimensional (0-forms and 4-forms),

  2. –

or spanned by _pairs_ ($(f,\mathbf{a})$ or $(\mathbf{a},\mathbf{b})$),

  3. –

the two elements within the pair are _independent_ for rotations,

  4. –

the two elements within the pair _need one another_ for boosts,

  5. –

so for rotations _and boosts_ they _need one another_ ,

  6. –

so the _whole_ representation space is needed,

  7. –

so the representation is irreducible.

Dimensionality of spaces of components of $p$-forms for $p=0,\dots,4$ is

$1\qquad 4\qquad 6\qquad 4\qquad 1$

so, just because of these _dimensions_ , non-trivial intertwining operators
are only possible between spaces $\Omega^{0}\leftrightarrow\Omega^{4}$,
$\Omega^{1}\leftrightarrow\Omega^{3}$ and within each space,
$\Omega^{p}\to\Omega^{p}$. This is exactly used by the (Minkowski) Hodge star
(the last possibility for $p=2$). And, as an example, non-zero operator
$\Omega^{2}\to\Omega^{3}$ _cannot_ be intertwining.

For Galilei and Carroll spacetimes, the situation is different.

There _are_ , contrary to Minkowski case, _nontrivial invariant_ sub-spaces
not only for rotations but also for Galilei (as well as Carroll) _boosts_.

Namely formulas for infinitesimal rotations and Galilei/Carroll boosts, based
on generators listed in Section 2.4, readily show that, in our language for
components of $p$-forms (2.7), there are invariant subspaces spanned on
elements

$\displaystyle\text{Galilei case}\colon\ $ $\displaystyle
f,\qquad\left(\begin{matrix}f\\\
\mathbf{0}\end{matrix}\right),\qquad\left(\begin{matrix}\mathbf{a}\\\
\mathbf{0}\end{matrix}\right),\qquad\left(\begin{matrix}0\\\
\mathbf{a}\end{matrix}\right),\qquad f,$ (4.30) $\displaystyle\text{Carroll
case}\colon\ $ $\displaystyle f,\qquad\left(\begin{matrix}0\\\
\mathbf{a}\end{matrix}\right),\qquad\left(\begin{matrix}\mathbf{0}\\\
\mathbf{b}\end{matrix}\right),\qquad\left(\begin{matrix}f\\\
\mathbf{0}\end{matrix}\right),\qquad f.$

So, we have _non-trivial_ invariant subspaces in $\Omega^{1}$, $\Omega^{2}$
and $\Omega^{3}$ and, therefore, representations $\rho_{1}$, $\rho_{2}$ and
$\rho_{3}$ are not irreducible.

Then, nothing (so simple as in the Lorentz case) excludes, e.g., a _non-zero
intertwining_ operator $\Omega^{2}\to\Omega^{3}$. And indeed, we have found,
in Galilei case, the operator

$a_{32}\colon\ \left(\begin{matrix}\mathbf{a}\\\
\mathbf{b}\end{matrix}\right)\mapsto\left(\begin{matrix}0\\\
\mathbf{b}\end{matrix}\right)$

(see (4.20)). Notice that

$\operatorname{Ker}a_{32}\leftrightarrow\left(\begin{matrix}\mathbf{a}\\\
\mathbf{0}\end{matrix}\right)\qquad\text{and}\qquad\operatorname{Im}a_{32}\leftrightarrow\left(\begin{matrix}0\\\
\mathbf{a}\end{matrix}\right).$

Kernel and image spaces of _any_ intertwining operator are known to be
invariant subspaces, and here we see that they coincide with particular
invariant subspaces displayed in (4.30).

Similarly, the new Galilei Hodge star (acting on 1-forms)

$a_{31}\colon\ \left(\begin{matrix}f\\\
\mathbf{a}\end{matrix}\right)\mapsto\left(\begin{matrix}0\\\
\mathbf{a}\end{matrix}\right)$

has no reason to be isomorphism. And it indeed seized its chance:

$\operatorname{Ker}a_{31}\leftrightarrow\left(\begin{matrix}f\\\
\mathbf{0}\end{matrix}\right)\qquad\text{and}\qquad\operatorname{Im}a_{31}\leftrightarrow\left(\begin{matrix}0\\\
\mathbf{a}\end{matrix}\right).$

Again, they coincide with particular invariant subspaces displayed in (4.30).
Similar facts are true for the new Carroll Hodge star.

##  5 Summary and conclusions

Hodge star operator plays a key role in mathematical physics, in particular it
is frequently used on Lorentzian spacetimes.

A lot of research is recently done on _Galilean_ and _Carrollian_ spacetimes.
It is therefore natural to address the question about the use of the operator
there. We see immediately that it cannot be constructed in a straightforward
way, since the Hodge star is (by definition) associated with metric tensor and
there is no (canonical) metric tensor on the two spacetimes.

One can, however, look for potentially useful _analogs_ of the star operator.

In [7], this is achieved by proper modification of the standard formula
(making use of suitable invariant tensors). Results are summarized in Appendix
B.

Here we follow another idea. The search is based on observation of
_intertwining_ (equivariance) property of the (original) Hodge (in the sense
explained in the text). So we address the question of finding intertwining
operators from $p$-forms to $q$-forms on $(1+3)$-dimensional
Galilean/Carrollian spacetimes.

The problem is first solved in Sections 2–4 for _Galilei_ and _Carroll_
spacetimes.

Then it is explained in Appendices C–F why these computations actually
guarantee that the corresponding results also apply on general _Galilean_ and
_Carrollian_ spacetimes (and what small changes are needed).

What we found may be summarized as follows:

  1. 1.

On _Lorentzian_ spacetime, the Hodge star is _the only_ intertwining operator.

  2. 2.

On _Galilean_ spacetime, there is

     * –

Galilean Hodge star from [7] plus,

     * –

degree $+1$ operator $\alpha\mapsto\xi\wedge\alpha$.

  3. 3.

On _Carrollian_ spacetime, there is

     * –

Carrollian Hodge star from [7] plus,

     * –

degree $-1$ operator $\alpha\mapsto i_{\tilde{\xi}}\wedge\alpha$.

So these (algebraic) operators, when combined with exterior derivative $d$,
which _is known_ to be intertwining (differential) operator as well, lead to
systems of _locally_ Lorentz/Galilei/Carroll _invariant equations_. As an
example, equations

${\rm d}*F=-J,\qquad{\rm d}F=0$

(mentioned in Section 1) correspond to Lorentzian/Galilean/Carrollian
electrodynamics depending on _which particular_ Hodge star is used; see more
in [7]).

##  Appendix A Details of solving equations for (some) $\boldsymbol{a_{qp}}$

Here we provide details of solving equations (3.2) for $a_{qp}$. Not all of
them (to save space), just a small sample to see a typical computation.

As mentioned in Section 3.3, the first step is taking _rotations_ generators
$S_{j}$ alone. This leads to simplification (3.9)–(3.13). Then we repeat,
starting already with these simplified expressions, the same step with
_boosts_ generators $N_{j}$ alone. Since they already depend on
$(\lambda_{1},\lambda_{2})$, results become dependent on particular spacetime.

Due to the symmetry of the pattern mentioned in Section 3.2 we can often
compute several operators at once (perform a single computation valid for a
particular group of operators, namely for the “orbit” of the symmetry of the
pattern).

_Nonzero_ results (including _unit_ operators) are highlighted by being
$\boxed{\text{boxed}}$.

###  A.1 Operators $\boldsymbol{a_{11}}$, $\boldsymbol{a_{33}}$,
$\boldsymbol{a_{13}}$ and $\boldsymbol{a_{31}}$

For $a_{11}$, $a_{33}$, $a_{13}$ and $a_{31}$, the equations read

$\displaystyle\rho_{1}\circ a_{11}=a_{11}\circ\rho_{1},\qquad$
$\displaystyle\rho_{1}\circ a_{13}=a_{13}\circ\rho_{3},$
$\displaystyle\rho_{3}\circ a_{33}=a_{33}\circ\rho_{3},\qquad$
$\displaystyle\rho_{3}\circ a_{31}=a_{31}\circ\rho_{1}.$

From this we see that

  1. –

we only need to really compute $a_{11}(\lambda_{1},\lambda_{2})$ and
$a_{13}(\lambda_{1},\lambda_{2})$,

  2. –

$a_{33}$ may be produced from $a_{11}$ by swapping
$\lambda_{1}\leftrightarrow\lambda_{2}$ (see (2.51)),

  3. –

$a_{31}$ may be produced from $a_{13}$ by swapping
$\lambda_{1}\leftrightarrow\lambda_{2}$ (see (2.51)).

The equation for $a_{11}$ explicitly reads

$\left(\begin{matrix}0&\lambda_{2}e_{j}^{\rm T}\\\
\lambda_{1}e_{j}&0\end{matrix}\right)\left(\begin{matrix}k_{1}&\mathbf{0}^{\rm
T}\\\
\mathbf{0}&k_{2}\mymathbb{1}\end{matrix}\right)=\left(\begin{matrix}k_{1}&\mathbf{0}^{\rm
T}\\\
\mathbf{0}&k_{2}\mymathbb{1}\end{matrix}\right)\left(\begin{matrix}0&\lambda_{2}e_{j}^{\rm
T}\\\ \lambda_{1}e_{j}&0\end{matrix}\right),$

i.e.,

$k_{2}\lambda_{2}=k_{1}\lambda_{2},\qquad k_{2}\lambda_{1}=k_{1}\lambda_{1}.$
(A.1)

Solution is

$k_{1}=k_{2}\qquad\text{for all three spacetimes}.$

And since the system (A.1) remains intact for swapping
$\lambda_{1}\leftrightarrow\lambda_{2}$, we get $a_{11}=a_{33}$.

The equation for $a_{13}$ explicitly reads

$\left(\begin{matrix}0&\lambda_{2}e_{j}^{\rm T}\\\
\lambda_{1}e_{j}&0\end{matrix}\right)\left(\begin{matrix}k_{1}&\mathbf{0}^{\rm
T}\\\
\mathbf{0}&k_{2}\mymathbb{1}\end{matrix}\right)=\left(\begin{matrix}k_{1}&\mathbf{0}^{\rm
T}\\\
\mathbf{0}&k_{2}\mymathbb{1}\end{matrix}\right)\left(\begin{matrix}0&\lambda_{1}e_{j}^{\rm
T}\\\ \lambda_{2}e_{j}&0\end{matrix}\right),$

i.e.,

$k_{2}\lambda_{2}=k_{1}\lambda_{1}.$ (A.2)

Here we have for $a_{13}$

  1. –

$\lambda_{1}=\lambda_{2}=1$ (Lorentz) $\Rightarrow$ ($k_{1}=k_{2}=$
arbitrary),

  2. –

$\lambda_{1}=0$ (Galilei) $\Rightarrow$ $k_{2}=0$ ($k_{1}$ arbitrary),

  3. –

$\lambda_{2}=0$ (Carroll) $\Rightarrow$ $k_{1}=0$ ($k_{2}$ arbitrary).

Swapping $\lambda_{1}\leftrightarrow\lambda_{2}$ in (A.2) produces swapping
$k_{1}\leftrightarrow k_{2}$ for solutions. So it produces as $a_{31}$
swapping Galilei $\leftrightarrow$ Carroll for the above conclusion for
$a_{13}$.

Altogether we come to conclusion:

$\displaystyle\boxed{a_{11}=a_{33}}=\left(\begin{matrix}1&\mathbf{0}^{\rm
T}\\\ \mathbf{0}&\mymathbb{1}\end{matrix}\right),\qquad\boxed{\text{Lorentz,
Galilei, Carroll}}$ (A.3)
$\displaystyle\boxed{a_{13}}=\left(\begin{matrix}1&\mathbf{0}^{\rm T}\\\
\mathbf{0}&\mymathbb{1}\end{matrix}\right)\qquad\boxed{\text{Lorentz}}$ (A.4)
$\displaystyle\phantom{\boxed{a_{13}}}{}=\left(\begin{matrix}1&\mathbf{0}^{\rm
T}\\\ \mathbf{0}&\mymathbb{0}\end{matrix}\right)\qquad\boxed{\text{Galilei}}$
(A.5)
$\displaystyle\phantom{\boxed{a_{13}}}{}=\left(\begin{matrix}0&\mathbf{0}^{\rm
T}\\\ \mathbf{0}&\mymathbb{1}\end{matrix}\right),\qquad\boxed{\text{Carroll}}$
(A.6) $\displaystyle\boxed{a_{31}}=\left(\begin{matrix}1&\mathbf{0}^{\rm T}\\\
\mathbf{0}&\mymathbb{1}\end{matrix}\right)\qquad\boxed{\text{Lorentz}}$ (A.7)
$\displaystyle\phantom{\boxed{a_{13}}}{}=\left(\begin{matrix}1&\mathbf{0}^{\rm
T}\\\ \mathbf{0}&\mymathbb{0}\end{matrix}\right)\qquad\boxed{\text{Carroll}}$
(A.8)
$\displaystyle\phantom{\boxed{a_{13}}}{}=\left(\begin{matrix}0&\mathbf{0}^{\rm
T}\\\ \mathbf{0}&\mymathbb{1}\end{matrix}\right).\qquad\boxed{\text{Galilei}}$
(A.9)

##  Appendix B Galilean and Carrollian Hodge star operators

In [7], Galilean and Carrollian “Hodge star” operators were proposed via
appropriate modification of the standard Hodge operator formula. In order to
compare the results obtained in this way with the intertwining operators
discussed in this paper, we list the results from [7], here. Just for the sake
of completeness we start with the “standard” (Minkowski) Hodge star (see [6,
Section 16.1]).

_Minkowski_ Hodge star:

$\displaystyle*f=f{\rm d}t\wedge{\rm d}V,$ $\displaystyle*(f{\rm
d}t+\mathbf{a}\cdot{\rm d}\mathbf{r})={\rm d}t\wedge\mathbf{a}\cdot{\rm
d}\mathbf{S}+f{\rm d}V,$ $\displaystyle*({\rm d}t\wedge\mathbf{a}\cdot{\rm
d}\mathbf{r}+\mathbf{b}\cdot{\rm d}\mathbf{S})={\rm
d}t\wedge\mathbf{b}\cdot{\rm d}\mathbf{r}-\mathbf{a}\cdot{\rm d}\mathbf{S},$
$\displaystyle*({\rm d}t\wedge\mathbf{a}\cdot{\rm d}\mathbf{S}+f{\rm
d}V)=f{\rm d}t+\mathbf{a}\cdot{\rm d}\mathbf{r},$ $\displaystyle*(f{\rm
d}t\wedge{\rm d}V)=-f.$

_Galilei_ Hodge star:

$\displaystyle*f=f{\rm d}t\wedge{\rm d}V,$ (B.1) $\displaystyle*(f{\rm
d}t+\mathbf{a}\cdot{\rm d}\mathbf{r})={\rm d}t\wedge\mathbf{a}\cdot{\rm
d}\mathbf{S},$ (B.2) $\displaystyle*({\rm d}t\wedge\mathbf{a}\cdot{\rm
d}\mathbf{r}+\mathbf{b}\cdot{\rm d}\mathbf{S})={\rm
d}t\wedge\mathbf{b}\cdot{\rm d}\mathbf{r},$ (B.3) $\displaystyle*({\rm
d}t\wedge\mathbf{a}\cdot{\rm d}\mathbf{S}+f{\rm d}V)=f{\rm d}t,$ (B.4)
$\displaystyle*(f{\rm d}t\wedge{\rm d}V)=-f.$ (B.5)

_Carroll_ Hodge star:

$\displaystyle*f=f{\rm d}t\wedge{\rm d}V,$ (B.6) $\displaystyle*(f{\rm
d}t+\mathbf{a}\cdot{\rm d}\mathbf{r})=f{\rm d}V,$ (B.7) $\displaystyle*({\rm
d}t\wedge\mathbf{a}\cdot{\rm d}\mathbf{r}+\mathbf{b}\cdot{\rm
d}\mathbf{S})=-\mathbf{a}\cdot{\rm d}\mathbf{S},$ (B.8) $\displaystyle*({\rm
d}t\wedge\mathbf{a}\cdot{\rm d}\mathbf{S}+f{\rm d}V)=\mathbf{a}\cdot{\rm
d}\mathbf{r},$ (B.9) $\displaystyle*(f{\rm d}t\wedge vV)=-f.$ (B.10)

In fact, for both new cases, Galilei as well as Carroll, there are as many as
_two_ formulas for each degree of forms and some of them lead to zero
operator. What is displayed here is always the _non-zero_ choice. See also
Appendix F.

##  Appendix C Galilei and Carroll vector spaces

###  C.1 Galilei vector space – basic facts

It is a triple $(V,\xi,h)$, where

  1. –

$V$ is an $(n+1)$-dimensional vector space,

  2. –

$\xi$ is a non-zero covector (i.e., a $\binom{0}{1}$-tensor) in $V$,

  3. –

$h$ is a rank-$n$ symmetric type-$\binom{2}{0}$-tensor in $V$,

  4. –

such that $h(\xi,\ \cdot\ )=0$.

We call a frame $E_{a}=(E_{0},E_{i})$, $i=1,\dots,n$ in $(V,\xi,h)$ and a
(dual) coframe $E^{a}=\big{(}E^{0},E^{i}\big{)}$ _adapted_ (or distinguished)
if

  1. –

$E^{0}=\xi$,

  2. –

$h=\delta^{ij}E_{i}\otimes E_{j}$,

so that, in (any) adapted frame, the components of the two tensors have
“canonical form”

$\xi_{a}\leftrightarrow\left(\begin{matrix}\xi_{0}\\\
\xi_{i}\end{matrix}\right)=\left(\begin{matrix}1\\\
0\end{matrix}\right),\qquad
h^{ab}\leftrightarrow\left(\begin{matrix}h^{00}&h^{0i}\\\
h^{i0}&h^{ij}\end{matrix}\right)=\left(\begin{matrix}0&0\\\
0&\delta^{ij}\end{matrix}\right)=\left(\begin{matrix}0&0\\\
0&\mathbb{I}\end{matrix}\right).$

The change-of-basis matrix $A$ between any pair $\hat{E}_{a}$, $E_{a}$ of
adapted frames, given by $\hat{E}_{a}=A_{a}^{b}E_{b}$, has the structure

$\displaystyle
A^{b}_{a}\leftrightarrow\left(\begin{matrix}A^{0}_{0}&A^{0}_{i}\\\
A^{i}_{0}&A^{i}_{j}\end{matrix}\right)=\left(\begin{matrix}1&0\\\
v^{i}&R^{i}_{j}\end{matrix}\right),\qquad\text{i.e.,}\quad
A\leftrightarrow\left(\begin{matrix}1&0\\\ v&R\end{matrix}\right),$ (C.1)

where $R$ is $n$-dimensional rotation matrix.

Such matrices form a Lie group $G$, subgroup of ${\rm GL}(n+1,\mathbb{R})$,
the (homogeneous) _Galilei group_ ($R$ parametrizes rotations and $v$ Galilei
boosts, respectively).

###  C.2 Carroll vector space – basic facts

It is a triple $\big{(}\tilde{V},\tilde{\xi},\tilde{h}\big{)}$, where

  1. –

$\tilde{V}$ is an $(n+1)$-dimensional vector space,

  2. –

$\tilde{\xi}$ is a non-zero vector (i.e., a $\binom{1}{0}$-tensor) in
$\tilde{V}$,

  3. –

$\tilde{h}$ is a rank-$n$ symmetric type-$\binom{0}{2}$-tensor in $\tilde{V}$,

  4. –

such that $\tilde{h}\big{(}\tilde{\xi},\ \cdot\ \big{)}=0$.

We call a frame $E_{a}=(E_{0},E_{i})$ and a (dual) coframe
$E^{a}=\big{(}E^{0},E^{i}\big{)}$, $i=1,\dots,n$ in
$\big{(}\tilde{V},\tilde{\xi},\tilde{h}\big{)}$ _adapted_ (or distinguished)
if

  1. –

$E_{0}=\tilde{\xi}$,

  2. –

$\tilde{h}=\delta_{ij}E^{i}\otimes E^{j}$,

so that, in (any) adapted frame, the components of the two tensors have
“canonical form”

$\tilde{\xi}^{a}\leftrightarrow\left(\begin{matrix}\xi^{0}\\\
\xi^{i}\end{matrix}\right)=\left(\begin{matrix}1\\\
0\end{matrix}\right),\qquad\tilde{h}_{ab}\leftrightarrow\left(\begin{matrix}\tilde{h}_{00}&\tilde{h}_{0i}\\\
\tilde{h}_{i0}&\tilde{h}_{ij}\end{matrix}\right)=\left(\begin{matrix}0&0\\\
0&\delta_{ij}\end{matrix}\right)=\left(\begin{matrix}0&0\\\
0&\mathbb{I}\end{matrix}\right).$

The change-of-basis matrix $A$ between any pair $\hat{E}_{a}$, $E_{a}$ of
adapted frames, given by $\hat{E}_{a}=A_{a}^{b}E_{b}$, has the structure

$\displaystyle
A^{b}_{a}\leftrightarrow\left(\begin{matrix}A^{0}_{0}&A^{0}_{i}\\\
A^{i}_{0}&A^{i}_{j}\end{matrix}\right)=\left(\begin{matrix}1&v_{i}\\\
0&R^{i}_{j}\end{matrix}\right),\qquad\text{i.e.,}\quad
A\leftrightarrow\left(\begin{matrix}1&v^{\rm T}\\\ 0&R\end{matrix}\right),$
(C.2)

where $R$ is $n$-dimensional rotation matrix.

Such matrices form a Lie group $G$, subgroup of ${\rm GL}(n+1,\mathbb{R})$,
the (homogeneous) _Carroll group_ ($R$ parametrizes rotations and $v$ Carroll
boosts, respectively).

It follows that _boosts_ formulas $x^{\prime a}=A^{a}_{b}x^{b}$ lead to
physically strange looking (yet well-known) expressions (2.29) and (2.30).

###  C.3 Forms on Galilei/Carroll vector space

In Appendices C.3 and C.4 common treatment of forms on both vector spaces is
given. The group $G$ is either Galilei group from Appendix C.1 or Carroll
group from Appendix C.2. So whenever matrix $A$ is present in a formula, we
mean _either_ (in Galilei case) the one from (C.1) _or_ (in Carroll case) the
one from (C.2).

There is a natural right (free) action of the group $G$ on the set
$\mathcal{E}$ of adapted frames

$R_{A}\colon\ \mathcal{E}\to\mathcal{E},\qquad(R_{A}E)_{a}:=A^{b}_{a}E_{b}$

and, consequently, on coframes $\mathcal{E}^{*}$
($E^{a}\mapsto\big{(}A^{-1}\big{)}^{a}_{b}E^{b}$). This induces representation
$\rho_{p}$ of $G$ on the space of $p$-forms on $V$,

$\displaystyle\rho_{p}\colon\ G\to\operatorname{Aut}\Lambda^{p}V^{*},$
$\displaystyle\rho_{p}(A)\left(\frac{1}{p!}\alpha_{a\dots
b}E^{a}\wedge\dots\wedge E^{b}\right):=\frac{1}{p!}\alpha_{a\dots
b}\big{(}A^{-1}\big{)}^{a}_{c}E^{c}\wedge\dots\wedge\big{(}A^{-1}\big{)}^{b}_{d}E^{d},$

and, consequently, on the (isomorphic) space $\hat{\Lambda}^{p}$ of
_components_ of $p$-forms on $V$

$\displaystyle\hat{\rho}_{p}\colon\ G\to\operatorname{Aut}\hat{\Lambda}^{p},$
(C.3) $\displaystyle\hat{\rho}_{p}(A)\colon\ \alpha_{a\dots b}\ \mapsto\
\big{(}A^{-1}\big{)}^{c}_{a}\cdots\big{(}A^{-1}\big{)}^{d}_{b}\alpha_{c\dots
d}.$

It is well known (see, e.g., [6, 19, 22]) that the space $\Lambda^{p}V^{*}$ of
$p$-forms in $V$ is canonically isomorphic to the space of _equivariant_ maps

$\displaystyle\alpha_{p}\leftrightarrow\hat{\alpha}_{p},\qquad\hat{\alpha}_{p}\colon\
\mathcal{E}\to\hat{\Lambda}^{p},\qquad\hat{\alpha}_{p}\circ
R_{A}=\hat{\rho}_{p}\big{(}A^{-1}\big{)}\circ\hat{\alpha}_{p}.$ (C.4)

###  C.4 Intertwining operators on forms on Galilei/Carroll vector space

Let $\hat{\alpha}_{p}$ be an equivariant map (C.4) corresponding to a $p$-form
$\alpha_{p}$ in $V$ and let

$\displaystyle\hat{a}_{qp}\colon\ \hat{\Lambda}^{p}\to\hat{\Lambda}^{q}$ (C.5)

be an _intertwining_ operator (equivariant linear mapping) for representations
$\hat{\rho}_{p}$ and $\hat{\rho}_{q}$, respectively, i.e., obeying

$\displaystyle\hat{\rho}_{q}(A)\circ\hat{a}_{qp}=\hat{a}_{qp}\circ\hat{\rho}_{p}(A).$
(C.6)

Then a simple check shows that mere _composition_ of maps

$\mathcal{E}\overset{\hat{\alpha}_{p}}{\to}\hat{\Lambda}^{p}\overset{\hat{a}_{qp}}{\to}\hat{\Lambda}^{q}$

produces an equivariant map corresponding to a $q$-form in $V$.

So we can reconstruct corresponding operators _on forms_ (transforming
$p$-forms to $q$-forms)

$\displaystyle\hat{\alpha}_{p}\mapsto\hat{\beta}_{q}:=\hat{a}_{qp}\circ\hat{\alpha}_{p}$
(C.7)

once we know intertwining operators $\hat{a}_{qp}$ between the representations
$\hat{\rho}_{p}$ and $\hat{\rho}_{q}$.

##  Appendix D Galilean and Carrollian manifolds

###  D.1 Galilean/Carrollian structure

Here, again, the formal procedure is the same for both Galilean an Carrollian
cases. So when we write, say, “Galilean”, one can replace it by “Carrollian”
as well (simply imagine matrix $A$ being either from the former or from the
latter group).

On _Galilean manifold_ $(M,\xi,h)$, each _tangent space_ $T_{m}M$ is endowed
with the structure of _Galilei_ vector space described in Appendix C.1.

So the _Galilean structure_ on $M$ may be (standardly) described as a
$G$_-structure_ (see, e.g., [6, 15, 19, 22]), i.e., in terms of a principal
$G$-bundle

$\pi\colon\ P\to M.$

Here points in the fiber over $m\in M$ refer to _adapted_ frames in $T_{m}M$.
So the principal bundle may be regarded as a _restriction_ of the frame bundle
$LM$ to the sub-bundle of adapted frames.

There is a natural right (free, vertical and in each fibre transitive) action
of the group $G$ (homogeneous Galilei group) on $P$

$R_{A}\colon\ P\to P,\qquad\pi\circ R_{A}=\pi.$

###  D.2 Forms on Galilean/Carrollian manifold

In general, _geometric quantities_ on $M$ may be described either as sections
of _associated_ bundles or, equivalently (see, e.g., [6, 22]), as
_equivariant_ functions on the _principal_ bundle $P$, i.e., as mappings

$\Phi\colon\
P\to(W,\hat{\rho}),\qquad\Phi(eA)=\hat{\rho}\big{(}A^{-1}\big{)}\Phi(e),\qquad\text{i.e.,}\quad
R_{A}^{*}\Phi=\hat{\rho}\big{(}A^{-1}\big{)}\circ\Phi.$

Here $(W,\hat{\rho})$ is a representation module of $G$. The corresponding
quantity (described by $\Phi$) is called _quantity_ (on $M$) _of type_
$\hat{\rho}$.222$\Phi(e)\in W$ plays the role of “components” of the quantity
under consideration w.r.t. the frame $e\in P$. The “actual” (frame-dependent)
quantity $\varphi$ living on $M$ is computed from $\Phi$ by pull-back
$\varphi=\sigma^{*}\Phi$ w.r.t. a local section $\sigma$, i.e., a choice of
local frame field on $M$.

In particular, for representation $\hat{\rho}_{p}$ in $\hat{\Lambda}^{p}$ from
(C.3), the corresponding quantities of type $\hat{\rho}_{p}$ are nothing but
$p$_-forms_ on the Galilean manifold $(M,\xi,h)$. So, we can treat elements of
$\Omega^{p}(M)$ as mappings

$\displaystyle\Phi_{p}\colon\
P\to\big{(}\hat{\Lambda}^{p},\hat{\rho}_{p}\big{)},\qquad\Phi_{p}(eA)=\hat{\rho}_{p}\big{(}A^{-1}\big{)}\Phi_{p}(e).$
(D.1)

###  D.3 Intertwining operators on forms on Galilean/Carrollian manifold

Let $\Phi_{p}$ be an equivariant map (D.1) corresponding to a $p$-form
$\alpha_{p}$ on $M$ and let $\hat{a}_{qp}$ be an operator from (C.5).

Then a simple check shows that mere _composition_ of maps

$P\overset{\Phi_{p}}{\to}\hat{\Lambda}^{p}\overset{\hat{a}_{qp}}{\to}\hat{\Lambda}^{q}$

produces an equivariant map corresponding to a $q$-form on $M$.

So we can reconstruct corresponding operators on _forms_ on a _Galilean
manifold_ (mappings $\Omega^{p}(M)\to\Omega^{q}(M)$, transforming $p$-forms to
$q$-forms on $M$)

$\Phi_{p}\mapsto\hat{a}_{qp}\circ\Phi_{p}$ (D.2)

once we know matrices $\hat{a}_{qp}$, intertwining operators between the
representations $\hat{\rho}_{p}$ and $\hat{\rho}_{q}$.

In this way, a simple result from linear algebra (of Galilei _vector space_)
eventually becomes a much more interesting result concerning operators acting
on _differential_ forms on _any_ _Galilean manifold_ $(M,\xi,h)$).333The
composition in (D.2) simply means that _components_ (w.r.t.  _any adapted_
local frame) of differential forms are scrambled by (constant) matrices
$\hat{a}_{qp}$.

##  Appendix E Galilei/Carroll spacetime versus vector space computations

Each (finite dimensional) vector space $V$ may be regarded as a manifold
(components $x^{a}$ of vectors w.r.t. a frame $e_{a}$ may be used as global
coordinates, $v\leftrightarrow x^{a}$).

The linear structure of the space enables one to identify each tangent space
$T_{v}V$ with the space $V$ itself. (Canonical isomorphism $V\to T_{v}V$ reads
$w\mapsto\dot{\gamma}(0)$ for $\gamma(t)=v+wt$.)

Consequently, whenever we have a type $\binom{p}{q}$ tensor $B$ in $V$, we can
canonically associate with it a type $\binom{p}{q}$ tensor _field_ $b$ on $V$
(as a manifold). The correspondence reads

$e^{a}\leftrightarrow{\rm d}x^{a},\qquad e_{a}\leftrightarrow\partial_{a},$
(E.1)

i.e.,

$B\equiv B_{a\dots}^{\ \dots b}e^{a}\otimes\dots\otimes e_{b}\leftrightarrow
B_{a\dots}^{\ \dots b}{\rm d}x^{a}\otimes\dots\otimes\partial_{b}\equiv b$

(components $B_{a\dots}^{\ \dots b}$ are the same in both expressions and
_constant_).

Now if a linear mapping $A\colon V\to V$ preserves (say) a type $\binom{0}{2}$
tensor $B$ (bilinear form) in the usual sense of linear algebra,

$\displaystyle B(Au,Aw)=B(u,w),\qquad\text{i.e.,}\quad
A_{a}^{c}B_{cd}A^{d}_{b}=B_{ab}$ (E.2)

then, on $V$ treated as a manifold, we speak of a smooth mapping

$f\equiv f_{A}\colon\ V\to V\qquad\text{in coordinates}\quad x^{a}\mapsto
A^{a}_{b}x^{b}$

and the tensor field $b$ is preserved in the sense of _pull-back_

$\displaystyle f_{A}^{*}b=b,$ (E.3)

i.e., as we usually understand the concept of “preserving a geometrical
quantity” on manifolds.

Preserving of $B$ may also be treated _infinitesimally_. In linear algebra, we
get from (E.2), for operators of the form $A=\mathbb{I}+\epsilon C$,

$B(Cu,w)+B(u,Cw)=0,\qquad\text{i.e.,}\quad C_{a}^{c}B_{cb}+B_{ac}C^{c}_{b}=0.$

In the language of manifolds, we get from (E.3) in a standard way

$\mathcal{L}_{\xi}b=0,\qquad\xi=x^{b}C_{b}^{a}\partial_{a},$

where $L_{\xi}$ is the _Lie derivative_. The specific structure of $\xi$
(_linear_ vector field, matrix $C$ is constant) is dictated by the fact that
the corresponding flow is to consist of linear transformations.

So, summing up, the computations of the type mentioned above may be either
realized in the language of linear algebra or in the language of (“linear”)
differential geometry.

Now notice that our computations in Sections 2.2 and 2.3, leading to explicit
expressions (2.46)–(2.50) for representations generators $\rho_{p}$ listed in
Section 2.4, are actually performed just within the above mentioned “linear
differential geometry”: Formulas like (2.11)–(2.12) for rotations or
(2.34)–(2.35) for boosts become exactly (infinitesimal versions of)
$\hat{E}^{a}=\big{(}A^{-1}\big{)}^{a}_{b}E^{b}$ (with $A=\mathbb{I}+\epsilon
C$) in the language of Appendix C.1 _once they are translated_ (via vocabulary
(E.1)) into the language of the Galilei _vector space_. (Standard coordinate
coframe ${\rm d}x^{a}\equiv({\rm d}t,{\rm d}x^{i})$ becomes an _adapted_
coframe $E^{a}\equiv\big{(}E^{0},E^{i}\big{)}$.)

Put it differently, the operators listed in Section 2.4 are exactly those
operators mentioned in (C.3) for which intertwining operators $\hat{a}_{qp}$
are computed via solving (C.6). More precisely, they are generators of the
_derived_ representation $\hat{\rho}^{\prime}_{q}$ of the Lie algebra
$\mathcal{G}$ of the group $G$, given standardly by

$\hat{\rho}_{q}(\mathbb{I}+\epsilon
C)\doteq\hat{1}+\epsilon\hat{\rho}^{\prime}_{q}(C).$

Since, however, solutions $\hat{a}_{qp}$ of (C.6) do not change when
$\hat{\rho}_{q}(A)$ is replaced by $\hat{\rho}^{\prime}_{q}(C)$, our
intertwining operators listed (in the form of _matrices_) in Sections 4.2–4.4
are exactly those mentioned in equation (C.6).

Then, however, (C.7) describes the corresponding operators on forms (still in
the Galilei vector space) and finally (D.2) describes the operators on _forms_
on _Galilean manifold_.

##  Appendix F Explicit expressions of $\boldsymbol{*}$-operator on
Galilean/Carrollian
spacetimes

Due to (D.2), our Hodge star operator, when acting on forms on _any_
Galilean/Carrollian spacetime, scrambles components of forms w.r.t.  _any
adapted_ frame by _the same_ matrices, namely by $\hat{a}_{qp}$. Since we know
how these matrices look like for forms on special Galilean/Carrollian
spacetime, the Galilei/Carroll spacetime (see (B.1)–B.10)), we are simply to
replace our particular adapted (global, holonomic) coframe field ${\rm
d}x^{a}\equiv\big{(}{\rm d}t,{\rm d}x^{i}\big{)}$ from Galilei/Carroll
spacetime with a _general adapted_ (possibly local and non-holonomic) _coframe
field_ $e^{a}\equiv\big{(}e^{0},e^{i}\big{)}$ living on the
Galilean/Carrollian spacetime under consideration. Recall that they are
defined (see Appendices C.1 and C.2), for a Galilean manifold $(M,\xi,h)$ and
a Carrollian manifold $\big{(}M,\tilde{\xi},\tilde{h}\big{)}$, by the
properties

$\displaystyle\text{Galilean frame/coframe field}\colon\ $ $\displaystyle
e^{0}=\xi,\qquad h=\delta^{ij}e_{i}\otimes e_{j},$
$\displaystyle\text{Carrollian frame/coframe field}\colon\ $ $\displaystyle
e_{0}=\tilde{\xi},\qquad\tilde{h}=\delta_{ij}e^{i}\otimes e^{j}.$

Then the above mentioned formulas (B.1)–B.10) with _substitutions_

$\displaystyle{\rm d}t\mapsto e^{0},$ $\displaystyle\mathbf{a}\cdot{\rm
d}\mathbf{r}\mapsto a_{1}e^{1}+a_{2}e^{2}+a_{3}e^{3},$
$\displaystyle\mathbf{a}\cdot{\rm d}\mathbf{S}\mapsto a_{1}{\rm
d}S_{1}+a_{2}{\rm d}S_{2}+a_{3}{\rm d}S_{3}\equiv a_{1}e^{2}\wedge
e^{3}+a_{2}e^{3}\wedge e^{2}+a_{3}e^{1}\wedge e^{3},$ $\displaystyle{\rm
d}V\mapsto e^{1}\wedge e^{2}\wedge e^{3}$

provide valid expressions of our Hodge star operator on forms on _any_
Galilean/Carrollian spacetime.

And just for the sake of completeness, let’s state how the degree $\pm 1$
operators look like:

$\xi\wedge(\dots)\quad\text{on}\ \Omega\ (\text{Galilei}),\qquad
i_{\tilde{\xi}}(\dots)\quad\text{on}\ \Omega\ (\text{Carroll}).$

### Acknowledgements

The author acknowledges support from grant VEGA 1/0703/20 and sincerely thanks
the anonymous referees whose remarks contributed to the quality of the paper.

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