# Galilean and Carrollian Hodge star operators

Marián Fecko
Department of Theoretical Physics,
Faculty of Mathematics, Physics and Informatics,
Comenius University in Bratislava,
Mlynská dolina F2, 84248 Bratislava, Slovakia e-mail:
<EMAIL_ADDRESS>
_The standard Hodge star operator is naturally associated with metric tensor
(and orientation). It is routinely used to concisely write down physics
equations on, say, Lorentzian spacetimes. On Galilean (Carrollian) spacetimes,
there is no canonical (nonsingular) metric tensor available. So, usual
construction of the Hodge star does not work. Here we propose analogs of the
Hodge star operator on Galilean (Carrollian) spacetimes. They may be used to
write down important physics equations, e.g. equations of Galilean
(Carrollian) electrodynamics._

Keywords: Galilean spacetime, Carrollian spacetime, Galilean electrodynamics,
Carrollian electrodynamics, Hodge star operator

###### Contents

  1. 1 Introduction - standard Hodge star operator
  2. 2 Galilean and Carrollian Hodge stars
  3. 3 Explicit formulae and some properties
    1. 3.1 General explicit formulae
    2. 3.2 Basic properties
    3. 3.3 Behaviour with respect to pull-back
  4. 4 An application: Electrodynamics
    1. 4.1 Standard electrodynamics
    2. 4.2 Galilei and Carroll electrodynamics
  5. 5 Summary and conclusions
  6. A Appendices
    1. A.1 Galilean and Carrollian manifolds
      1. A.1.1 Galilean manifold
      2. A.1.2 Carrollian manifold
      3. A.1.3 Top-degree forms and polyvectors
    2. A.2 Details of computation of the four analogs of Hodge star
      1. A.2.1 Hodge stars based on tensors $h^{ab}$/$\tilde{h}_{ab}$
      2. A.2.2 Hodge stars based on tensors $k_{ab}$/$\tilde{k}^{ab}$
    3. A.3 Explicit formulas for $1+3$

##  1 Introduction - standard Hodge star operator

The _Hodge star_ operator $*\equiv*_{g,o}$ is a well-known _canonical_ linear
mapping acting on differential forms on (pseudo)Riemannian oriented manifold
$(M,g,o)$,

$*:\Omega^{p}\to\Omega^{d-p},\hskip
28.45274pt\Omega^{p}\equiv\Omega^{p}(M,g,o)$ (1)

($d$ beeing the dimension of $M$; see e.g. [1]-[7]). In components (w.r.t. a
local frame field $e_{a}$), it is defined as

$(*\alpha)_{a\dots b}:=\frac{1}{p!}\alpha_{c\dots d}\ \omega^{c\dots d}_{\ \ \
\ a\dots b}\ ,\hskip 28.45274pt\omega^{c\dots d}_{\ \ \ \ a\dots
b}:=g^{ce}\dots g^{df}\omega_{e\dots fa\dots b}\ ,$ (2)

where $\omega\equiv\omega_{g,o}$ stands for the _metric volume form_ on
$(M,g,o)$.

The Hodge star $*$ is a remarkably effective tool for concise and coordinate-
free expression of important physics equations. For example, the Maxwell
equations may be written, on any (Lorentzian) spacetime, as

$dF=0\ ,\hskip 28.45274ptd*F=-J$ (3)

(here $J$ stands for $(d-1)$-form of current). Notice that no linear
connection is needed for that (no semicolons in local coordinates).

Recently, much research has been devoted to the study of physics in _Galilean_
and _Carrollian_ spacetimes, see e.g. [8]-[17] or a long lists of references
in recent PhD theses [18] and [19]. Now if we wanted to use Hodge star
operator on these spacetimes, a serious problem would immediately arise: There
is _no canonical_ metric tensor on either Galilean or Carrollian spacetime.
And since metric tensor $g$ turns out to be key element in construction of the
$*_{g}$, the routine procedure fails, there.

##  2 Galilean and Carrollian Hodge stars

From (2) we see that the role of the metric tensor $g$ in construction of the
“mixed” tensor $\omega^{c\dots d}_{\ \ \ \ a\dots b}$, i.e. in construction of
the (standard) Hodge star, is twofold: It enters the scene via

\- the _metric_ volume form $\omega_{g}$ and

\- the co _metric_ $g^{-1}$ (for raising indices).

Concequently, _if_ the two objects, _canonical_ volume form and
$\binom{2}{0}$-type tensor field (for raising indices) were also available on
Galilean/Carrollian spacetimes, one could also construct $\omega^{c\dots d}_{\
\ \ \ a\dots b}$ there, i.e. potentially useful _analogs_ of the Hodge star
were possible on _Galilean_ (_Carrollian_) spacetime as well.

More generally, analogs of the standard Hodge star could be defined on
Galilean and Carrollian spacetimes, if the necessary mixed tensor field
$\omega^{c\dots d}_{\ \ \ \ a\dots b}$ could be constructed using _canonical_
tensor fields _available_ there.

It turns out that such a construction is indeed possible.

Let us start with the fact (see Appendix A.1) that, on _both_ Galilean and
Carrollian spacetimes, _both_ (canonical) top-degree form and top-degree
polyvector

$\omega_{a\dots b}\ ,\hskip 56.9055pt\tilde{\omega}^{a\dots b}\ ,$ (4)

are available, see (65) and (70).

So we have at least two different possibilities for constructing the needed
_mixed_ tensor $\omega^{a\dots b}_{\ \ \ \ c\dots d}$: we

\- _either_ start from $\omega_{a\dots bc\dots d}$ and _raise_ (somehow) the
group $(a\dots b)$,

\- _or_ start from $\tilde{\omega}^{a\dots bc\dots d}$ and _lower_ (somehow)
the group $(c\dots d)$.

Since we start from _canonical_ tensors $\omega_{a\dots bc\dots d}$ and
$\tilde{\omega}^{a\dots bc\dots d}$, the resulting tensor $\omega^{a\dots
b}_{\ \ \ \ c\dots d}$ will be canonical as well provided that the
_procedures_ of raising and lowering indices will be canonical, too.

Now recall (see Sections (A.1.1) and (A.1.2)) what canonical tensors (with the
potential of raising and lowering indices) are available on the spacetimes of
our interest:

\- Galilean: $(M,\xi,h),\quad
h\in\binom{2}{0},\quad\xi\in\binom{0}{1}\quad\Rightarrow\quad
k:=\xi\otimes\xi\in\binom{0}{2}$,

\- Carrollian:
$(M,\tilde{\xi},\tilde{h}),\quad\tilde{h}\in\binom{0}{2},\quad\tilde{\xi}\in\binom{1}{0}\quad\Rightarrow\quad\tilde{k}:=\tilde{\xi}\otimes\tilde{\xi}\in\binom{2}{0}$.

So indeed, on both spacetimes of interest, there is a possibility to both
raise and lower indices in a _canonical_ way:

\- Galilean: raising with $h\in\binom{2}{0}$, lowering with
$k\in\binom{0}{2}$,

\- Carrollian: raising with $\tilde{k}\in\binom{2}{0}$, lowering with
$\tilde{h}\in\binom{0}{2}$.

This means that there are, on _both_ spacetimes of interest, as many as two
_analogs of the Hodge star_ operator (in the sense of the canonical mapping of
$p$-forms to $(d-p)$-forms on $d$-dimensional $M$), namely those based on
tensors $h^{ab}$/$\tilde{h}_{ab}$ and those based on tensors
$k_{ab}$/$\tilde{k}^{ab}$.

In order to foresee some important features of the resulting operators, just
recall (see again Sections (A.1.1) and (A.1.2)) how (matrices of) components
of the necessary canonical tensors look like (in _any adapted_ frame/coframe):

Galilean $\displaystyle:$ $\displaystyle
h^{ab}\leftrightarrow\left(\begin{matrix}h^{00}&h^{0i}\\\
h^{i0}&h^{ij}\end{matrix}\right)=\left(\begin{matrix}0&0\\\ 0&\
\delta^{ij}\end{matrix}\right),$ (6) $\displaystyle
k_{ab}\leftrightarrow\left(\begin{matrix}k_{00}&k_{0i}\\\
k_{i0}&k_{ij}\end{matrix}\right)=\left(\begin{matrix}1&0\\\
0&0\end{matrix}\right).$ Carrollian $\displaystyle:$
$\displaystyle\tilde{h}_{ab}\leftrightarrow\left(\begin{matrix}\tilde{h}_{00}&\tilde{h}_{0i}\\\
\tilde{h}_{i0}&\tilde{h}_{ij}\end{matrix}\right)=\left(\begin{matrix}0&0\\\
0&\ \delta_{ij}\end{matrix}\right),$ (8)
$\displaystyle\tilde{k}^{ab}\leftrightarrow\left(\begin{matrix}\tilde{k}^{00}&\tilde{k}^{0i}\\\
\tilde{k}^{i0}&\tilde{k}^{ij}\end{matrix}\right)=\left(\begin{matrix}1&0\\\
0&0\end{matrix}\right).$

Since the matrices $h^{ab}$ and $\tilde{h}_{ab}$ are typically much _less_
degenerate (their rank being $n$) than the matrices $k_{ab}$ and
$\tilde{k}^{ab}$ (rank $1$), the procedure of raising/lowering indices using
tensors $h^{ab}$/$\tilde{h}_{ab}$ is much less “lossy” than that using tensors
$k_{ab}$/$\tilde{k}^{ab}$. (Recall that raising and lowering indices is
perfectly “lossless” in the standard Riemannian case.) Put it differently,
analogs of Hodge star based on $h^{ab}$/$\tilde{h}_{ab}$ are _in general_
expected to be _more useful_ than those based on tensors
$k_{ab}$/$\tilde{k}^{ab}$.

All the operators are computed in detail in Appendix A.2 and the results are
summarized and discussed in Section 3.

##  3 Explicit formulae and some properties

###  3.1 General explicit formulae

In both Galilean and Carrollian spacetimes, exactly as is the case in
Lorentzian spacetime (see e.g. [25, 26] and Sec.16.1 in [7]), any $p$-form
$\alpha$ may be uniquely written, w.r.t. (local) adapted coframe field
$e^{a}=(e^{0},e^{i})$, as

$\alpha=e^{0}\wedge\hat{s}+\hat{r}\ ,$ (9)

where the two hatted forms, $(p-1)$-form $\hat{s}$ and $p$-form $\hat{r}$,
respectively, are _spatial_ , meaning that they do not contain the
(“temporal”) 1-form $e^{0}$.

When applying both versions of Hodge star operator $*$ onto $(1+n)$-decomposed
presentation (9) of $\alpha$, we get (in any dimension $d=1+n$) the following
results (see Appendix A.2 for more details).

First, Hodge stars based on (rank $n$) tensors $h^{ab}$ and $\tilde{h}_{ab}$,
respectively, act as follows: 111We display the results, just for comparison,
together with the Lorentzian case, based on standard metric $g_{ab}$. Notice
that all this is compatible with an observation that, naively, Galilean limit
($c\to\infty$) of the Lorentzian case may be written as $e^{0}\gg e^{i}$ and
for Carrollian limit ($c\to 0$) we have $e^{0}\ll e^{i}$.

$\displaystyle*(e^{0}\wedge\hat{s}+\hat{r})=$ $\displaystyle\
e^{0}\wedge\hat{*}\hat{r}+\hat{*}\hat{\eta}\hat{s}$
$\displaystyle\text{\emph{Lorentzian} Hodge star},$ (10)
$\displaystyle*(e^{0}\wedge\hat{s}+\hat{r})=$ $\displaystyle\
e^{0}\wedge\hat{*}\hat{r}$ $\displaystyle\text{\emph{Galilean} Hodge star},$
(11) $\displaystyle*(e^{0}\wedge\hat{s}+\hat{r})=$ $\displaystyle\hskip
49.79231pt\hat{*}\hat{\eta}\hat{s}$ $\displaystyle\text{\emph{Carrollian}
Hodge star}.$ (12)

(see e.g. [25, 26] for (10), (80) and (89) for (11) and (12)). Here $\hat{*}$
stands for standard “Euclidean” ($n$-dimensional) Hodge star (it only operates
on spatial forms) and $\hat{\eta}$ is the “main automorhism” on forms (just a
multiple by $(-1)^{p}$; see e.g. Sec.5.3 in [7]).

Notice that this type of Galilean $*$ vanishes for $p=d=1+n$ (top-degree
forms) and the Carrollian $*$ vanishes for $p=0$ (functions).

Second, there are Hodge stars based on (rank $1$) tensors $k_{ab}$ (for
Galilean case) and $\tilde{k}^{ab}$ (for Carrollian case). Their action is
displayed in (100) - (102) for Galilean case and (110) - (112) for Carrollian
case. The comparison shows the following:

_Galilean_ : The “Hodge star” $*$ based on $k_{ab}$

\- vanishes for all $p<n$ (unlike those given by (11)),

\- coincides for $p=n$ with (11),

\- does not vanish for $p=n+1$ (unlike the one given by (11)).

_Carrollian_ : The “Hodge star” $*$ based on $\tilde{k}^{ab}$

\- vanishes for all $p>1$ (unlike those given by (12)),

\- coincides for $p=1$ with (12),

\- does not vanish for $p=0$ (unlike the one given by (12)).

Now, it is clear that _zero_ operators from $p$-forms to $q$-forms are,
although being formally canonical, not very interesting, notably if there are
also _non-zero_ canonical operators available in the same situation.

So, from this point of view, we can conclude that we should

\- _in general_ choose (11) and (12) ,

i.e. star operators based on (less degenerate) tensors $h^{ab}$ and
$\tilde{h}_{ab}$, the _only exceptions_ being

\- _Galilean_ : $p=d=1+n$, when (102) is more useful (= non-zero) than (11),

\- _Carrollian_ : $p=0$, when (110) is more useful (= non-zero) than (12).

So the two _exceptional_ cases (which differ from (11) and (12)) read

$\displaystyle*(e^{0}\wedge\hat{\omega})=$ $\displaystyle\ 1$ Galilean Hodge
star for $\displaystyle p=1+n,$ (13) $\displaystyle*1=$ $\displaystyle\
e^{0}\wedge\hat{\omega}$ Carrollian Hodge star for $\displaystyle p=0.$ (14)

For even more explicit expressions of (10) - (12) and (13) - (14) for the
usual $1+3$ (rather than $1+n$) case, see Appendix A.3.

For a completely different approach (based on the concept of _intertwining
operator_) to finding formulas (125) - (139) from Appendix A.3, which also
provides an interesting additional insight, see paper [27].

###  3.2 Basic properties

Exactly as is the case for the standard Hodge star, both new Hodge stars map
linearly $p$-forms on $(d-p)$-forms and vice versa,

$\Omega^{p}\underset{*}{\overset{*}{\rightleftarrows}}\Omega^{d-p}\ .$ (15)

A straightforward consequence of (11) and (12) is a striking feature of the
new Hodge stars - they both square to _zero_ 222Except for the “extremal”
degrees $p=0$ and $p=d$ when, as is mentioned at the end of Section 3.1, we
prefer (102) to (11) and (110) to (12). In these exceptional cases it squares
to (plus or minus) the unit operator, exactly as we are accustomed to in the
standard case.,

$**=0\ ,\hskip 28.45274pt**:\Omega^{p}\to\Omega^{p}\ ,\hskip 28.45274pt0\neq
p\neq d\ .$ (16)

This is in sharp contrast with the standard Hodge operator $*_{g}$ from (2)
which, as is well-known, squares to (plus or minus) the _unit_ operator,

$*_{g}*_{g}=\pm\hat{1}\ ,\hskip 28.45274pt*_{g}*_{g}:\Omega^{p}\to\Omega^{p}\
,\hskip 28.45274pt\text{for \emph{any}}\ p\ .$ (17)

So, contrary to the standard case, neither the _Galilean_ -Hodge star nor the
_Carrollian_ -Hodge star are _isomorphisms_ between the two spaces of equal
dimensions (spaces of $p$-forms and $(d-p)$-forms, respectively; exceptions
are $p=0$ and $p=d$). Their (non-zero) kernels are

$\displaystyle*(e^{0}\wedge\hat{s})=$ $\displaystyle\ 0$ Galilean case
$\displaystyle{\text{\rm deg\,}}\hat{s}\neq n,$ (18) $\displaystyle*\
\hat{r}=$ $\displaystyle\ 0$ Carrollian case $\displaystyle{\text{\rm
deg\,}}\hat{r}\neq 0.$ (19)

One can also rephrase the statement as that we can _no longer_ speak of
Galilean-Hodge _duality_ as well as Carrollian-Hodge _duality_ (with the above
mentioned exceptions).

That is the price to pay for replacing _Lorentzian_ spacetime with _Galilean_
and _Carrollian_ spacetimes, while still insisting that the concept of “Hodge
star operator” remains _canonical_.

###  3.3 Behaviour with respect to pull-back

From the component expression (2) for the _standard_ Hodge star operator one
can see that the corresponding component-free expression reads

$*_{g}\alpha\ \sim\ C\dots C(g^{-1}\otimes\dots\otimes
g^{-1}\otimes\omega_{g}\otimes\alpha)\ ,$ (20)

where $g^{-1}$ denotes cometric and $C$ stands for contraction.

Now, let

$f:M\to M$ (21)

be a diffeomorphism of $M$. Then, taking into account

$f^{*}(g^{-1})=(f^{*}g)^{-1}\ ,\hskip
28.45274ptf^{*}\omega_{g}=\omega_{f^{*}g}\ ,$ (22)

we get a well-known (and useful) property of the (standard) Hodge star:

$f^{*}(*_{g}\alpha)=*_{f^{*}g}(f^{*}\alpha)\ ,\hskip
28.45274pt\text{i.e.}\hskip 28.45274ptf^{*}\circ*_{g}=*_{f^{*}g}\circ f^{*}\
.$ (23)

Now, consider replacing the (peudo)Riemannian manifold $(M,g)$ with a Galilean
manifold $(M,h,\xi)$ or a Carrollian manifold $(M,\tilde{h},\tilde{\xi})$. The
_structure_ (20) of the Hodge star remains the same, we just make
substitutions of tensor fields (depending on particular version, out of four
possibilities, of the star, discussed in Section 2),

$\displaystyle\text{\emph{Galilean} case}\ :$ $\displaystyle 1.$
$\displaystyle g^{-1}$ $\displaystyle\mapsto h\ ,$ $\displaystyle\omega_{g}$
$\displaystyle\mapsto\omega_{h,\xi}\ ,$ (24) $\displaystyle 2.$ $\displaystyle
g^{-1}$ $\displaystyle\mapsto k\ ,$ $\displaystyle\omega_{g}$
$\displaystyle\mapsto{\tilde{\omega}}_{h,\xi}\ .$ (25)
$\displaystyle\text{\emph{Carrollian} case}\ :$ $\displaystyle 1.$
$\displaystyle g^{-1}$ $\displaystyle\mapsto\tilde{h}\ ,$
$\displaystyle\omega_{g}$
$\displaystyle\mapsto\tilde{\omega}_{\tilde{h},\tilde{\xi}}\ ,$ (26)
$\displaystyle 2.$ $\displaystyle g^{-1}$ $\displaystyle\mapsto\tilde{k}\ ,$
$\displaystyle\omega_{g}$ $\displaystyle\mapsto\omega_{\tilde{h},\tilde{\xi}}\
.$ (27)

(where $\omega_{h,\xi}$, $\tilde{\omega}_{h,\xi}$,
$\omega_{\tilde{h},\tilde{\xi}}$ and $\tilde{\omega}_{\tilde{h},\tilde{\xi}}$
are defined in (65) and $k$ and $\tilde{k}$ is introduced in Section 2). Then,
taking into account

$f^{*}\omega_{h,\xi}=\omega_{f^{*}h,f^{*}\xi}\hskip
28.45274ptf^{*}\tilde{\omega}_{\tilde{h},\tilde{\xi}}=\tilde{\omega}_{f^{*}\tilde{h},f^{*}\tilde{\xi}}\
,$ (28)

(and similarly for all other “derived” canonical tensor fields mentioned
above) we get corresponding useful property of the “new” Hodge stars:

$f^{*}\circ*_{h,\xi}=*_{f^{*}h,f^{*}\xi}\circ f^{*}\hskip
28.45274pt\text{and}\hskip
28.45274ptf^{*}\circ*_{\tilde{h},\tilde{\xi}}=*_{f^{*}\tilde{h},f^{*}\tilde{\xi}}\circ
f^{*}\ .$ (29)

(where $*_{h,\xi}$ denotes _any_ of the two Galilean stars and similarly
$*_{\tilde{h},\tilde{\xi}}$ denotes _any_ of the two Carrolian stars). In
particular, for _structure-preserving_ diffeomorphisms, i.e. such that

$\displaystyle\text{in \emph{Lorentzian} case},$ $\displaystyle f^{*}g$
$\displaystyle=\ g,$ (30) $\displaystyle\text{in \emph{Galilean} case},$
$\displaystyle f^{*}h$ $\displaystyle=\ h,\hskip 11.38092ptf^{*}\xi=\ \xi,$
(31) $\displaystyle\text{in \emph{Carrollian} case},$ $\displaystyle
f^{*}\tilde{h}$ $\displaystyle=\ \tilde{h},\hskip 11.38092ptf^{*}\tilde{\xi}=\
\tilde{\xi}.$ (32)

holds the _corresponding_ Hodge star _commutes_ with pull-back,

$f^{*}\circ*=*\circ f^{*}.$ (33)

So, as an example, for the simplest versions of the two non-lorentzian
spacetimes, _Galilei_ and _Carroll_ spacetimes, the corresponding star
operators commute with (translations, rotations plus)

$\displaystyle\text{Galilei boost}\colon\ $ $\displaystyle
t^{\prime}=t,\qquad$ $\displaystyle\text{Carroll boost}\colon\ $
$\displaystyle t^{\prime}=t+\mathbf{v}\cdot\mathbf{r}\ ,$ (34)
$\displaystyle\mathbf{r}^{\prime}=\mathbf{r}+\mathbf{v}t,\qquad$
$\displaystyle\mathbf{r}^{\prime}=\mathbf{r}\ .$ (35)

(As for the Carroll boost, see e.g. (9) in [24], II.10 in [8] or Appendix
A.1.2.) Put it differently, the new operators happen to be _invariant_ w.r.t.
Galilei and Carroll transformations, respectively.

More generally, the new “Hodge stars” are invariant w.r.t. _local Galilean_
and _local Carrollian_ transformations, respectively, in the same way as, say,
the standard Hodge star on a Lorentzian spacetime is known to be invariant
w.r.t. _local Lorentzian_ transformations.

This property makes the Galilean and Carrollian Hodge star operators
potentially interesting from the point of view of writing important _physics
equations_ in terms of _differential forms_ on corresponding _spacetimes_ (see
an example in Section 4.2).

##  4 An application: Electrodynamics

As a simple application of Galilean/Carrollian Hodge star operator in physics,
let us have just a brief look at Galilei and Carroll _electrodynamics_. In
these two versions of electrodynamics (particular limits of the standard
Minkowski one, see e.g. [8, 20]), the corresponding equations of motion for
the fields $\mathbf{E}$ and $\mathbf{B}$ are Galilei/Carroll (rather than
Poincaré) invariant.

###  4.1 Standard electrodynamics

In _Minkowski_ spacetime, the (source-less, for simplicity) Maxwell equations

$\displaystyle{\text{\rm div\,}}\mathbf{E}$ $\displaystyle=$ $\displaystyle\
0\ ,$ (36) $\displaystyle\text{\rm curl\,}\mathbf{B}-\partial_{t}\mathbf{E}$
$\displaystyle=$ $\displaystyle\ 0\ ,$ (37) $\displaystyle\text{\rm
curl\,}\mathbf{E}+\partial_{t}\mathbf{B}$ $\displaystyle=$ $\displaystyle\ 0\
,$ (38) $\displaystyle{\text{\rm div\,}}\mathbf{B}$ $\displaystyle=$
$\displaystyle\ 0\ ,$ (39)

may be neatly written in terms of 2-form of electromagnetic field,

$F=dt\wedge\mathbf{E}\cdot d\mathbf{r}-\mathbf{B}\cdot d\mathbf{S}\ ,$ (40)

as follows:

$\displaystyle d*F$ $\displaystyle=$ $\displaystyle\ 0\ ,\hskip
56.9055pt*=*_{\eta}\ \ \ \text{here},$ (41) $\displaystyle dF$
$\displaystyle=$ $\displaystyle\ 0\ .$ (42)

As already mentioned in Section 3.3, just because of properties of ($d$ and)
$*_{\eta}$, this way of presentation of the Maxwell equations makes their
_Poincaré_ invariance evident.

In more detail, the assignment reads

$\displaystyle d*F$ $\displaystyle=0$ $\displaystyle\leftrightarrow$
$\displaystyle{\text{\rm div\,}}\mathbf{E}$ $\displaystyle=0\ ,$ (43)
$\displaystyle\text{\rm curl\,}\mathbf{B}-\partial_{t}\mathbf{E}$
$\displaystyle=0\ ,$ (44) $\displaystyle dF$ $\displaystyle=0$
$\displaystyle\leftrightarrow$ $\displaystyle\text{\rm
curl\,}\mathbf{E}+\partial_{t}\mathbf{B}$ $\displaystyle=0\ ,$ (45)
$\displaystyle{\text{\rm div\,}}\mathbf{B}$ $\displaystyle=0$ (46)

(see, e.g., Section 16.1 in [7]). Notice that each spacetime equation on the
l.h.s. corresponds to as many as _two_ spatial equations on the r.h.s.

###  4.2 Galilei and Carroll electrodynamics

Now it looks natural, in an effort to switch to Galilei (Carroll)
electrodynamics, to just repeat the complete story with _Galilei_ (_Carroll_)
Hodge star operator. So again, introduce (see (117)) the 2-form of
electromagnetic field exactly as in (40) and write down equations of motion à
la (41) and (42), the only change being the replacement of the Minkowski Hodge
star with the Galilei (Carroll) one (so with $*$ from (11) or (12); more
explicitly from (132) or (137)):

$\displaystyle d*F$ $\displaystyle=$ $\displaystyle\ 0\ ,\hskip
56.9055pt*=*_{h,\xi}\ \text{or}\ *_{\tilde{h},\tilde{\xi}}\ ,\ \text{here}\ ,$
(47) $\displaystyle dF$ $\displaystyle=$ $\displaystyle\ 0\ .$ (48)

Then it is clear that, again just because of the properties of $d$ and $*$,
this is a system of 1-st order partial differential equations for the fields
$\mathbf{E}$ and $\mathbf{B}$, which is _Galilei_ or _Carroll_ (rather than
Poincaré) _invariant_.

All the items listed above provide a promising signal that equations (47) -
(48) are probably closely associated with the desired Galilei (Carroll)
electrodynamics.

How the system (47) - (48) actually looks like in terms of the fields
$\mathbf{E}$ and $\mathbf{B}$?

First, it is clear that (48) looks _the same_ for all three cases (standard
Minkowski, Galilei as well as Carroll), namely it always corresponds to (45)
and (46). So, both the Faraday’s law and nonexistence of magnetic monopoles
hold in all three versions of electrodynamics.

What really makes a difference is equation (47). When (132) and (137) - rather
than (127) - is used for computation of $*$, we get

$\displaystyle d*F$ $\displaystyle=0$ $\displaystyle\leftrightarrow$
$\displaystyle\text{\rm curl\,}\mathbf{B}$ $\displaystyle=0$
$\displaystyle\text{in Galilei case}\ ,$ (49) $\displaystyle d*F$
$\displaystyle=0$ $\displaystyle\leftrightarrow$
$\displaystyle\partial_{t}\mathbf{E}$ $\displaystyle=0\ ,\ {\text{\rm
div\,}}\mathbf{E}=0$ $\displaystyle\text{in Carroll case}\ .$ (50)

So, what we get when _all_ spatial equations, resulting from

$d*F=0\ ,\hskip 42.67912ptdF=0\ ,$ (51)

are displayed side by side for _all three_ cases is

Minkowski Galilei Carroll (52) $\displaystyle{\text{\rm div\,}}\mathbf{E}$
$\displaystyle=0\ ,$ $\displaystyle{\text{\rm div\,}}\mathbf{E}$
$\displaystyle=0\ ,$ (53) $\displaystyle\text{\rm
curl\,}\mathbf{B}-\partial_{t}\mathbf{E}$ $\displaystyle=0\ ,$
$\displaystyle\text{\rm curl\,}\mathbf{B}\hskip 31.2982pt$ $\displaystyle=0\
,$ $\displaystyle\partial_{t}\mathbf{E}$ $\displaystyle=0\ ,$ (54)
$\displaystyle\text{\rm curl\,}\mathbf{E}+\partial_{t}\mathbf{B}$
$\displaystyle=0\ ,$ $\displaystyle\text{\rm
curl\,}\mathbf{E}+\partial_{t}\mathbf{B}$ $\displaystyle=0\ ,$
$\displaystyle\text{\rm curl\,}\mathbf{E}+\partial_{t}\mathbf{B}$
$\displaystyle=0\ ,$ (55) $\displaystyle{\text{\rm div\,}}\mathbf{B}$
$\displaystyle=0\ ,$ $\displaystyle{\text{\rm div\,}}\mathbf{B}$
$\displaystyle=0\ ,$ $\displaystyle{\text{\rm div\,}}\mathbf{B}$
$\displaystyle=0\ .$ (56)

Checking equations in Galilei and Carroll columns versus _standard_ Maxwell
equations (left column) shows that there are some objects _missing_ in Galilei
as well as Carroll versions of “Maxwell equations” (51), namely

\- _displacement current_ $\partial_{t}\mathbf{E}$ in Ampère’s law in the
_Galilei_ case,

\- “ _Ampère_ term” $\text{\rm curl\,}\mathbf{B}$ in Ampère’s law in the
_Carroll_ case,

\- equation (53), i.e. _Gauss’s law_ , in the _Galilei_ case.

Now, while the absence of the displacement current in Galilei electrodynamics
and the “basic Ampère term” in Carroll electrodynamics are well-known (so
expected and desired) facts (namely in the so-called “magnetic limit” version
of Galilei electrodynamics, see [20], and “electric limit” version of Carroll
electrodynamics, see [8]), the Gauss’ law is usually present in all three
versions of electrodynamics. In particular, the Galilei system usually
contains the equations from the central column plus the equation (43). (The
fact that the Gauss’ law is not present in the spacetime differential forms
version of the Galilei electrodynamics needs further study. We _do get_ , by
the way, the Gauss’ law with the help of Galilei _connection_.)

So, to summarize, (51) _do_ represent _correct_ equations in _all three_
versions of electrodynamics, albeit they do not provide, in the Galilei case,
the whole story.

[Technically, the reason why just a _single_ equation (49) corresponds to (47)
(so that, at the end of the day, one equation is missing), contrary to _two_
equations (43) and (44) in Minkowski case, looks to be simple: The Galilei
star is no longer an isomorphism on 2-forms, it _kills_ once and for all the
“electric” part of $F$ and then the action of $d$ only produces a single term.
Notice, however, that neither the _Carroll_ star is an isomorphism on 2-forms
(it kills once and for all the “magnetic” part of $F$). And here action of $d$
produces _two_ terms. So the number of resulting equations _alone_ actually
depends on more subtle _details_ of actions of the two star operators, rather
than on their general common property of not being isomorphism.]

##  5 Summary and conclusions

The Hodge star operator belongs to essential tools in applications of
differential forms on (pseudo)Riemannian manifolds. In particular, this is
true for standard Lorentzian spacetimes.

However, Galilean and Carrollian spacetimes are not (pseudo)Riemannian
manifolds. There is no (distinguished, non-degenerate) metric tensor available
on them. Consequently, there is no standard “full-fledged” Hodge star operator
on them.

Nevertheless, it turns out that one can _mimic_ the usual construction of the
Hodge star using specific (well-known) canonical tensors available. Since
algebra of canonical tensors depends on particular spacetime, particular
constructions and the resulting “Hodge stars” depend on particular spacetime
as well.

The new “Hodge stars” retain some important properties of the standard one,
while losing others.

In particular, they remain to be _invariant_ w.r.t. structure preserving
diffeomorphisms (in particular w.r.t. Galilei or Carroll transformations,
respectively). More generally, w.r.t. _local_ Galilean and _local_ Carrollian
transformations, respectively. This makes them potentially interesting from
the point of view of writing _physics equations_ in terms of differential
forms on Galilean and Carrollian spacetimes.

Both new “Hodge stars”, however, cease to be linear _isomorphisms_ (in
general). One can no longer speak of Hodge _duality_ (modulo exceptions).

In the three spacetimes under consideration (Lorentzian, Galilean and
Carrollian) one can express differential forms in terms of a _pair_ of
_spatial_ forms. Then the three Hodge star operators act according to (10) -
(12) or (13) - (14).

For example, on Minkowski, Galilei and Carroll $(1+3$)-dimensional spacetimes,
the action on the 2-form $F$ of electromagnetic field reads

$\displaystyle*_{M}(dt\wedge\mathbf{E}\cdot d\mathbf{r}-\mathbf{B}\cdot
d\mathbf{S})$ $\displaystyle=$ $\displaystyle dt\wedge(-\mathbf{B})\cdot
d\mathbf{r}-\mathbf{E}\cdot d\mathbf{S}\ ,$ (57)
$\displaystyle*_{G}(dt\wedge\mathbf{E}\cdot d\mathbf{r}-\mathbf{B}\cdot
d\mathbf{S})$ $\displaystyle=$ $\displaystyle dt\wedge(-\mathbf{B})\cdot
d\mathbf{r}\ ,$ (58) $\displaystyle*_{C}(dt\wedge\mathbf{E}\cdot
d\mathbf{r}-\mathbf{B}\cdot d\mathbf{S})$ $\displaystyle=$
$\displaystyle\hskip 71.13188pt-\mathbf{E}\cdot d\mathbf{S}\ .$ (59)

Consequently, single (universal) _manifestly invariant_ equation on spacetime

$d*F=0\ ,$ (60)

has substantially different 3-dimensional content (expression in terms of
fields $\mathbf{E}$ and $\mathbf{B}$), depending on _which_ particular Hodge
star operator ($*_{M}$, $*_{G}$ or $*_{C}$) is used (see (53) and (54).

##  Appendix A Appendices

###  A.1 Galilean and Carrollian manifolds

Here, for convenience of the reader, basic definitions and facts are
collected. In order to learn more details, one should consult references at
the end of the paper, e.g., [8]-[17] and [21]-[24].

####  A.1.1 Galilean manifold

Oriented Galilean manifold is a 4-tuple $(M,\xi,h,o)$, where

\- $(M,o)$ is an $d$-dimensional oriented manifold, $d=1+n$,

\- $\xi$ is an everywhere non-zero covector (i.e. a $\binom{0}{1}$-tensor)
field on $M$,

\- $h$ is an everywhere rank-$n$ symmetric type-$\binom{2}{0}$-tensor field on
$M$,

\- such that $h(\xi,\ \cdot\ )=0$. We call a (local) right-handed frame field
$e_{a}=(e_{0},e_{i})$ and the (dual) coframe field $e^{a}=(e^{0},e^{i})$,
$i=1,\dots,n$ on $(M,\xi,h,o)$ _adapted_ (or distinguished) if

\- $e^{0}=\xi$,

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

so that, in (any) adapted frame field, the components of the two tensor fields
have "canonical form"

$\xi_{a}\leftrightarrow\left(\begin{matrix}\xi_{0}\\\
\xi_{i}\end{matrix}\right)=\left(\begin{matrix}1\\\ 0\end{matrix}\right)\
,\hskip 28.45274pth^{ab}\leftrightarrow\left(\begin{matrix}h^{00}&h^{0i}\\\
h^{i0}&h^{ij}\end{matrix}\right)=\left(\begin{matrix}0&0\\\ 0&\
\delta^{ij}\end{matrix}\right)=\left(\begin{matrix}0&0\\\ 0&\
\mathbb{I}\end{matrix}\right)\ .$ (61)

The (point-dependent) change-of-basis matrix $A$ between any pair
$\hat{e}_{a}$, $e_{a}$ of adapted frame fields, given by
$\hat{e}_{a}=A_{a}^{b}e_{b}$, has the structure

$A^{b}_{a}\leftrightarrow\left(\begin{matrix}A^{0}_{0}&A^{0}_{i}\\\
A^{i}_{0}&A^{i}_{j}\end{matrix}\right)=\left(\begin{matrix}1&0\\\ v^{i}&\
R^{i}_{j}\end{matrix}\right)\hskip 14.22636pt\text{i.e.}\hskip
14.22636ptA\leftrightarrow\left(\begin{matrix}1&0\\\ v&\ R\end{matrix}\right)$
(62)

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

In each point, such matrices form a Lie group $G$, subgroup of
$GL(d,\mathbb{R})$, the (homogeneous) _Galilei group_ ($R$ parametrizes
rotations and $v$ boosts, respectively).

####  A.1.2 Carrollian manifold

Oriented Carrollian manifold is a 4-tuple $(M,\tilde{\xi},\tilde{h},o)$, where

\- $(M,o)$ is an $d$-dimensional oriented manifold, $d=1+n$,

\- $\tilde{\xi}$ is an everywhere non-zero vector (i.e. a
$\binom{1}{0}$-tensor) field on $M$,

\- $\tilde{h}$ is an everywhere rank-$n$ symmetric type-$\binom{0}{2}$-tensor
field on $M$,

\- such that $\tilde{h}(\tilde{\xi},\ \cdot\ )=0$. We call a (local) right-
handed frame field $e_{a}=(e_{0},e_{i})$ and the (dual) coframe field
$e^{a}=(e^{0},e^{i})$, $i=1,\dots,n$ on $(M,\tilde{\xi},\tilde{h},o)$
_adapted_ (or distinguished) if

\- $e_{0}=\tilde{\xi}$,

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

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

$\xi^{a}\leftrightarrow\left(\begin{matrix}\xi^{0}\\\
\xi^{i}\end{matrix}\right)=\left(\begin{matrix}1\\\ 0\end{matrix}\right)\
,\hskip 28.45274pth_{ab}\leftrightarrow\left(\begin{matrix}h_{00}&h_{0i}\\\
h_{i0}&h_{ij}\end{matrix}\right)=\left(\begin{matrix}0&0\\\ 0&\
\delta_{ij}\end{matrix}\right)=\left(\begin{matrix}0&0\\\ 0&\
\mathbb{I}\end{matrix}\right)$ (63)

The (point-dependent) change-of-basis matrix $A$ between any pair
$\hat{e}_{a}$, $e_{a}$ of adapted local frame fields, given by
$\hat{e}_{a}=A_{a}^{b}e_{b}$, has the structure

$A^{b}_{a}\leftrightarrow\left(\begin{matrix}A^{0}_{0}&A^{0}_{i}\\\
A^{i}_{0}&A^{i}_{j}\end{matrix}\right)=\left(\begin{matrix}1&v_{i}\\\ 0&\
R^{i}_{j}\end{matrix}\right)\ ,\hskip 14.22636pt\text{i.e.}\hskip
14.22636ptA\leftrightarrow\left(\begin{matrix}1&v^{T}\\\ 0&\
R\end{matrix}\right)$ (64)

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

In each point, such matrices form a Lie group $G$, subgroup of
$GL(d,\mathbb{R})$, the (homogeneous) _Carroll group_ ($R$ parametrizes
rotations and $v$ boosts, respectively).

####  A.1.3 Top-degree forms and polyvectors

Consider, on a patch $\mathcal{O}$ of a $(1+n)$-dimensional manifold $M$,
_arbitrary local_ frame field $e_{a}=(e_{0},e_{i})$ and the dual coframe field
$e^{a}=(e^{0},e^{i})$. In terms of the frame/coframe fields, we introduce
(local, everywhere non-vanishing) top-degree form $\omega(e)$ and top-degree
polyvector $\tilde{\omega}(e)$

$\omega(e):=e^{0}\wedge e^{1}\wedge\dots\wedge e^{n}\ ,\hskip
42.67912pt\tilde{\omega}(e):=e_{0}\wedge e_{1}\wedge\dots\wedge e_{n}\ .$ (65)

Introduce, similarly, a primed frame/coframe on primed $\mathcal{O}^{\prime}$.
Then, on the overlap $\mathcal{O}\cap\mathcal{O}^{\prime}$, one has

$e^{\prime}_{a}=A_{a}^{b}e_{b}\ ,\hskip 28.45274pte^{\prime
a}=(A^{-1})^{a}_{b}e^{b}$ (66)

(where $A\equiv A(x)$ is a unique position-dependent regular matrix) and we
standardly get

$\omega(e^{\prime})=(\det A)^{-1}\ \omega(e)\ ,\hskip
28.45274pt\tilde{\omega}(e^{\prime})=(\det A)\ \tilde{\omega}(e)\ .$ (67)

Now consider restriction of the above situation to

\- oriented _Galilean/Carrollian_ manifold,

\- _adapted_ frame/coframe fields.

Then the change-of-basis matrices $A$ have the form (62) or (64), depending on
whether we speak of Galilean or Carrollian manifold. In both cases, however,
we have clearly

$\det A=1$ (68)

and, therefore, on the overlap $\mathcal{O}\cap\mathcal{O}^{\prime}$,

$\omega(e^{\prime})=\omega(e),\hskip
28.45274pt\tilde{\omega}(e^{\prime})=\tilde{\omega}(e).$ (69)

This means that both objects, $\omega$ and $\tilde{\omega}$, are well-defined
on the union $\mathcal{O}\cup\mathcal{O}^{\prime}$ and, consequently, that
_canonical global_ objects are actually defined in terms of (local)
expressions (65).

Put another way,

\- on _both_ Galilean _and_ Carrollian (oriented) manifolds ,

\- _both_ canonical _volume form_ and canonical _top-degree polyvector_ exist.
333It is clear, that _any constant multiples_ of the tensor fields discussed
above are canonical as well.

In components (w.r.t. arbitrary local _adapted_ frame/coframe fields) this may
be written in terms of the Levi-Civita symbols as

$\omega_{a\dots b}=\epsilon_{a\dots b}\ ,\hskip
56.9055pt{\tilde{\omega}}^{a\dots b}=\epsilon^{a\dots b}\ .$ (70)

###  A.2 Details of computation of the four analogs of Hodge star

Here, detailed calculations of (analogs of) Hodge star operators, leading to
results collected in Section 3.1, are shown.

Component definition (2) is equivalent to an expression for action of $*$ on
_basis_ $p$-forms (on $d$-dimensional manifold)

$*(e^{a}\wedge\dots\wedge e^{b})=\frac{1}{(d-p)!}\omega^{a\dots b}_{\hskip
17.07182ptc\dots d}\ e^{c}\wedge\dots\wedge e^{d}.$ (71)

What we need to calculate from (71) is action of $*$ on a _general decomposed_
$p$-form, i.e.

$*\alpha=*(e^{0}\wedge\hat{s}+\hat{r})=\ ?\hskip
28.45274pt\hat{s}=\hat{s}_{p-1}\ ,\ \hat{r}=\hat{r}_{p}$ (72)

In _Lorentzian_ case, the answer reads

$*(e^{0}\wedge\hat{s}+\hat{r})=e^{0}\wedge\hat{*}\hat{r}+\hat{*}\hat{\eta}\hat{s}$
(73)

(see e.g. [25, 26] or, in the simplest case, Section 16.1. in [7]), where
$\hat{*}$ is the Euclidean Hodge star and $\hat{\eta}$ is the main
automorphism on forms, i.e. just multiplication by $(-1)^{p}$. What we want
here are similar formulas for all four new cases, Galilean versus Carrollian
case and $h^{ab}$/$\tilde{h}_{ab}$ versus $k_{ab}$/$\tilde{k}^{ab}$ case.

Because of specific features of the tensors
$h^{ab},\tilde{h}_{ab},k_{ab},\tilde{k}^{ab}$, it is convenient to discuss the
two parts ($e^{0}\wedge\hat{s}$ and $\hat{r}$) separately, i.e. to compute

$*(e^{0}\wedge e^{i}\wedge\dots\wedge e^{j}),\hskip
42.67912pt*(e^{i}\wedge\dots\wedge e^{j}),$ (74)

and then combine the two results into the answer to (72).

Note: In what follows, we multiply the canonical tensors
$h^{ab},\tilde{h}_{ab},k_{ab},\tilde{k}^{ab}$ by a _free parameter_ $\lambda$
(and sometimes also $\omega$ and $\tilde{\omega}$ by $\mu$; the tensors
clearly remain to be canonical) and choose particular value of $\lambda$ (and
$\mu$) at the end of computation so that the result resembles the standard
Lorentzian case _as much as possible_.

####  A.2.1 Hodge stars based on tensors $h^{ab}$/$\tilde{h}_{ab}$

Galilean case \- based on $h^{ab}$: Here, w.r.t. an _adapted_ frame (see (61)
and (70)),

$\omega^{a\dots b}_{\hskip 17.07182ptc\dots d}:=h^{a\hat{a}}\dots
h^{b\hat{b}}\omega_{\hat{a}\dots\hat{b}c\dots d}=h^{a\hat{a}}\dots
h^{b\hat{b}}\epsilon_{\hat{a}\dots\hat{b}c\dots d}\ .$ (75)

From $h^{00}=h^{0i}=0$ we can immediately deduce that

$\omega^{0i\dots j}_{\hskip 19.91684ptk\dots l}=0,\hskip
28.45274pt\text{i.e.}\hskip 28.45274pt*(e^{0}\wedge\hat{s})=0.$ (76)

On strictly _spatial_ $p$-form $\hat{r}$ it is more interesting. For
$h^{ij}=\lambda\delta^{ij}$ we get from (71) on _basis_ spatial $p$-form

$\begin{array}[]{rcl}*(\underbrace{e^{i}\wedge\dots\wedge
e^{j}}_{p})&=&\frac{1}{(1+n-p)!}\omega^{i\dots j}_{\hskip 17.07182ptc\dots d}\
e^{c}\wedge\dots\wedge e^{d}\\\ &=&\frac{(1+n-p)}{(1+n-p)!}\omega^{i\dots
j}_{\hskip 17.07182pt0k\dots l}\ e^{0}\wedge e^{k}\wedge\dots\wedge e^{l}\\\
&=&\lambda^{p}e^{0}\wedge\frac{1}{(n-p)!}\delta^{i\hat{i}}\dots\delta^{j\hat{j}}\omega_{\hat{i}\dots\hat{j}0k\dots
l}\ e^{k}\wedge\dots\wedge e^{l}\\\
&=&\lambda^{p}e^{0}\wedge\frac{1}{(n-p)!}\epsilon_{i\dots j0k\dots l}\
e^{k}\wedge\dots\wedge e^{l}\\\
&=&(-\lambda)^{p}e^{0}\wedge\frac{1}{(n-p)!}\epsilon_{0i\dots jk\dots l}\
e^{k}\wedge\dots\wedge e^{l}\\\
&=&(-\lambda)^{p}e^{0}\wedge\frac{1}{(n-p)!}\epsilon_{i\dots jk\dots l}\
e^{k}\wedge\dots\wedge e^{l}\\\
&=&(-\lambda)^{p}e^{0}\wedge\hat{*}\left(e^{i}\wedge\dots\wedge
e^{j}\right),\end{array}$

i.e., on _complete_ spatial $p$-form $\hat{r}$,

$*\hat{r}=(-\lambda)^{p}e^{0}\wedge\hat{*}\hat{r}.$ (77)

Combining with (76), we have

$*(e^{0}\wedge\hat{s}+\hat{r})=e^{0}\wedge(-\lambda)^{p}\hat{*}\hat{r}.$ (78)

In order to make this formula as similar as possible to (73), we have to
choose

$\lambda=-1\ ,\hskip 28.45274pt\text{i.e.}\hskip
28.45274pth^{ij}=-\delta^{ij}.$ (79)

For this fixation of the arbitrary constant, we get

$*(e^{0}\wedge\hat{s}+\hat{r})=e^{0}\wedge\hat{*}\hat{r}.$ (80)

Carrollian case \- based on $\tilde{h}_{ab}$:

Here

$\omega^{a\dots b}_{\hskip 17.07182ptc\dots d}:=\tilde{\omega}^{a\dots
b\hat{c}\dots\hat{d}}\tilde{h}_{c\hat{c}}\dots\tilde{h}_{d\hat{d}}=\epsilon_{a\dots
b\hat{c}\dots\hat{d}}\tilde{h}_{c\hat{c}}\dots\tilde{h}_{d\hat{d}}=\epsilon_{a\dots
bk\dots l}\tilde{h}_{ck}\dots\tilde{h}_{dl}\ .$ (81)

From $\tilde{h}_{00}=\tilde{h}_{0i}=0$ (see (8)) we can immediately deduce
that

$\omega^{i\dots j}_{\hskip 19.91684ptc\dots d}=0,\hskip
28.45274pt\text{i.e.}\hskip 28.45274pt*\hat{r}=0.$ (82)

On _combined_ $p$-form $e^{0}\wedge\hat{s}$ it is more interesting. For
$\tilde{h}_{ij}=\lambda\delta_{ij}$ we get (see (71)) on _basis_ combined
$p$-form

$\begin{array}[]{rcl}*(e^{0}\wedge\overbrace{e^{i}\wedge\dots\wedge
e^{j}}^{p-1})&=&\frac{1}{(1+n-p)!}\omega^{0i\dots j}_{\hskip 17.07182ptc\dots
d}\ e^{c}\wedge\dots\wedge e^{d}\\\ &=&\frac{1}{(1+n-p)!}\omega^{0i\dots
j}_{\hskip 17.07182ptk\dots l}\ e^{k}\wedge\dots\wedge e^{l}\\\
&=&\frac{\lambda^{1+n-p}}{(1+n-p)!}\epsilon_{0i\dots jk\dots l}\
e^{k}\wedge\dots\wedge e^{l}\\\ &=&\lambda^{1+n-p}\
\frac{1}{(n-(p-1))!}\epsilon_{i\dots jk\dots l}\ e^{k}\wedge\dots\wedge
e^{l}\\\ &=&\lambda^{n-(p-1)}\ \hat{*}(e^{i}\wedge\dots\wedge
e^{j}),\end{array}$

i.e.

$*(e^{0}\wedge\hat{s})=\lambda^{n-(p-1)}\ \hat{*}\hat{s}.$ (83)

Combining with (82), we have

$*(e^{0}\wedge\hat{s}+\hat{r})=\lambda^{n-(p-1)}\ \hat{*}\hat{s}.$ (84)

In order to make this formula as similar as possible to (73), we need to
fulfill

$\lambda^{n-(p-1)}=(-1)^{p-1}.$ (85)

This is not possible for any choice of $\lambda$. But we can use freedom in
_both_ $\tilde{\omega}$ _and_ $\tilde{h}$ and take, from the very beginning,

$\tilde{\omega}^{0i\dots jk\dots l}=\mu(n)\epsilon_{0i\dots jk\dots l},\hskip
28.45274pt\tilde{h}_{ij}=\lambda(n,p)\delta_{ij}.$ (86)

Then we need to fulfill (instead of (85))

$\mu\lambda^{n-(p-1)}\equiv(\mu\lambda^{n})\lambda^{-(p-1)}=(-1)^{p-1},$ (87)

with solution

$\mu=(-1)^{n}\hskip 28.45274pt\lambda=-1.$ (88)

Then we get from (84) the final expression

$*(e^{0}\wedge\hat{s}+\hat{r})=\hat{*}\hat{\eta}\hat{s}.$ (89)

####  A.2.2 Hodge stars based on tensors $k_{ab}$/$\tilde{k}^{ab}$

Galilean case \- based on $k_{ab}$:

Here

$\omega^{a\dots b}_{\hskip 17.07182ptc\dots d}:={\tilde{\omega}}^{a\dots
b\hat{c}\dots\hat{d}}k_{c\hat{c}}\dots k_{d\hat{d}}={\epsilon}_{a\dots
b\hat{c}\dots\hat{d}}k_{c\hat{c}}\dots k_{d\hat{d}}.$ (90)

From $k_{00}=\lambda$ and all other possibilities vanishing (see (6)) we have

$\omega^{a\dots b}_{\hskip 17.07182ptc\dots d}={\epsilon}_{a\dots b0\dots
0}k_{c0}\dots k_{d0},$ (91)

and, consequently,

$*(e^{a}\wedge\dots\wedge
e^{b})=\frac{1}{(1+n-p)!}\lambda^{1+n-p}{\epsilon}_{a\dots b0\dots
0}\overbrace{e^{0}\wedge\dots\wedge e^{0}}^{1+n-p}.$ (92)

So there are just two _non-zero_ cases (values of $p$): 1\. $1+n-p=0$ (i.e.
$p=1+n$) and then

$*(e^{0}\wedge e^{1}\wedge\dots\wedge e^{n})=\epsilon_{01\dots n}=1,$ (93)

_irrespective_ of the choice of $\lambda$. 2\. $1+n-p=1$ (i.e. $p=n$) and then

$*(e^{1}\wedge\dots\wedge e^{n})=\lambda\epsilon_{1\dots
n0}e^{0}=(-1)^{n}\lambda e^{0}.$ (94)

Here we can obtain, by appropriate choice of $\lambda$, coincidence with
formula (80). Indeed, since (in $n$-dimensional _Euclidean_ space)

$\hat{*}(e^{1}\wedge\dots\wedge e^{n})=\text{\rm sgn\,}\hat{g}=1,$ (95)

we can rewrite (94) as

$*(e^{1}\wedge\dots\wedge e^{n})=(-1)^{n}\lambda
e^{0}\wedge\hat{*}(e^{1}\wedge\dots\wedge e^{n}),$ (96)

and this becomes, for the choice $\lambda=(-1)^{n}$,

$*(e^{1}\wedge\dots\wedge e^{n})=e^{0}\wedge\hat{*}(e^{1}\wedge\dots\wedge
e^{n}),$ (97)

coinciding with what (80) says for this case.

If we denote the “spatial volume element” as

$\hat{\omega}:=e^{1}\wedge\dots\wedge e^{n},$ (98)

we can summarize (93) and (97) as

$*\ \hat{\omega}=e^{0}\wedge\hat{*}\hat{\omega}\equiv e^{0}\hskip
28.45274pt*(e^{0}\wedge\hat{\omega})=1,$ (99)

or, make complete summarizing as

$\displaystyle*(e^{0}\wedge\hat{s}+\hat{r})=$ $\displaystyle\ 0$ for
$\displaystyle p<n,$ (100) $\displaystyle*\hat{\omega}=$ $\displaystyle\
e^{0}\hskip 28.45274pt*(e^{0}\wedge\hat{s})=0$ for $\displaystyle p=n,$ (101)
$\displaystyle*(e^{0}\wedge\hat{\omega})=$ $\displaystyle\ 1$ for
$\displaystyle p=1+n.$ (102)

Carrollian case \- based on $\tilde{k}^{ab}$:

Here

$\omega^{a\dots b}_{\hskip 17.07182ptc\dots
d}:=\tilde{k}^{a\hat{a}}\dots\tilde{k}^{b\hat{b}}\omega_{\hat{a}\dots\hat{b}c\dots
d}=\tilde{k}^{a\hat{a}}\dots\tilde{k}^{b\hat{b}}\epsilon_{\hat{a}\dots\hat{b}c\dots
d}.$ (103)

From $\tilde{k}^{00}=\lambda$ and all other possibilities vanishing (see (8))
we have

$\omega^{a\dots b}_{\hskip 17.07182ptc\dots
d}=\tilde{k}^{a0}\dots\tilde{k}^{b0}\epsilon_{0\dots 0c\dots d},$ (104)

and, consequently,

$*(e^{a}\wedge\dots\wedge
e^{b})=\frac{1}{(1+n-p)!}\tilde{k}^{a0}\dots\tilde{k}^{b0}\epsilon_{0\dots
0c\dots d}\overbrace{e^{c}\wedge\dots\wedge e^{d}}^{1+n-p}.$ (105)

So there are just two _non-zero_ cases (values of $p$): 1\. $p=0$, and then

$*1\ =\ \frac{1}{(1+n)!}\epsilon_{c\dots d}\overbrace{e^{c}\wedge\dots\wedge
e^{d}}^{1+n}\ =\ e^{0}\wedge e^{1}\wedge\dots\wedge e^{n}\ \equiv\
e^{0}\wedge\hat{\omega},$ (106)

_irrespective_ of the choice of $\lambda$. 2\. $p=1$, and then

$*e^{0}\ =\ \frac{1}{n!}\tilde{k}^{00}\epsilon_{0i\dots
j}\overbrace{e^{i}\wedge\dots\wedge e^{j}}^{n}\ =\ \lambda
e^{1}\wedge\dots\wedge e^{n}\ \equiv\ \lambda\hat{\omega}.$ (107)

Here we can obtain, by the choice $\lambda=1$, coincidence with formula (89).
So

$*e^{0}=\hat{\omega}\ .$ (108)

We can summarize (106) and (97) as

$*1=e^{0}\wedge\hat{\omega},\hskip 28.45274pt*e^{0}=\hat{\omega},$ (109)

or, make complete summarizing as

$\displaystyle*1=$ $\displaystyle\ e^{0}\wedge\hat{\omega}$ for $\displaystyle
p=0,$ (110) $\displaystyle*e^{0}=$ $\displaystyle\ \hat{\omega}\hskip
28.45274pt*e^{i}=0$ for $\displaystyle p=1,$ (111)
$\displaystyle*(e^{0}\wedge\hat{s}+\hat{r})=$ $\displaystyle\ 0$ for
$\displaystyle p>1.$ (112)

###  A.3 Explicit formulas for $1+3$

In the usual $1+3$ dimensional Minkowski, Galilei and Carroll spacetimes, we
can use standard Cartesian coordinate basis as an adapted frame

$(e^{0},e^{i})\ =\ (dt,dx^{i})\ =\ (dt,dx,dy,dz).$ (113)

Then the general decomposition formula from (9) reads

$\alpha=dt\wedge\hat{s}+\hat{r},$ (114)

and may be further specified, for the five relevant degrees of differential
forms, as follows:

$\displaystyle\Omega^{0}:\ \ \ \alpha$ $\displaystyle=$ $\displaystyle\ f,$
(115) $\displaystyle\Omega^{1}:\ \ \ \alpha$ $\displaystyle=$ $\displaystyle\
fdt+\mathbf{a}\cdot d\mathbf{r},$ (116) $\displaystyle\Omega^{2}:\ \ \ \alpha$
$\displaystyle=$ $\displaystyle\ dt\wedge\mathbf{a}\cdot
d\mathbf{r}+\mathbf{b}\cdot d\mathbf{S},$ (117) $\displaystyle\Omega^{3}:\ \ \
\alpha$ $\displaystyle=$ $\displaystyle\ dt\wedge\mathbf{a}\cdot
d\mathbf{S}+fdV,$ (118) $\displaystyle\Omega^{4}:\ \ \ \alpha$
$\displaystyle=$ $\displaystyle\ fdt\wedge dV.$ (119)

where

$\displaystyle\mathbf{a}\cdot d\mathbf{r}$ $\displaystyle=$ $\displaystyle\
a_{x}dx+a_{y}dy+a_{z}dz,$ (120) $\displaystyle\mathbf{a}\cdot d\mathbf{S}$
$\displaystyle=$ $\displaystyle\ a_{x}dS_{x}+a_{y}dS_{y}+a_{z}dS_{z},$ (121)
$\displaystyle\equiv$ $\displaystyle\ a_{x}dy\wedge dz+a_{y}dz\wedge
dx+a_{z}dx\wedge dy,$ (122) $\displaystyle dV$ $\displaystyle=$
$\displaystyle\ dx\wedge dy\wedge dz.$ (123)

So, concerning this presentation of forms alone, there is no difference
between the three spacetimes (for the Minkowski case, see e.g. Sec.16.1 in
[7]).

Due to (10) - (12) and well-known Euclidean 3D-results,

$\hat{*}f=fdV,\hskip 22.76228pt\hat{*}(\mathbf{a}\cdot
d\mathbf{r})=\mathbf{a}\cdot d\mathbf{S},\hskip
22.76228pt\hat{*}(\mathbf{a}\cdot d\mathbf{S})=\mathbf{a}\cdot
d\mathbf{r},\hskip 22.76228pt\hat{*}(fdV)=f$ (124)

(see e.g. Sec.8.5 in [7]) we get, in this language, the following explicit
results for the action of the three Hodge star operators: 444The signs in the
Galilei and Carroll cases were chosen so that the results resembled the
standard Minkowski results as much as possible. _Minkowski_ Hodge star:

$\displaystyle*f$ $\displaystyle=$ $\displaystyle\ fdt\wedge dV,$ (125)
$\displaystyle*(fdt+\mathbf{a}\cdot d\mathbf{r})$ $\displaystyle=$
$\displaystyle\ dt\wedge\mathbf{a}\cdot d\mathbf{S}+fdV,$ (126)
$\displaystyle*(dt\wedge\mathbf{a}\cdot d\mathbf{r}+\mathbf{b}\cdot
d\mathbf{S})$ $\displaystyle=$ $\displaystyle\ dt\wedge\mathbf{b}\cdot
d\mathbf{r}-\mathbf{a}\cdot d\mathbf{S},$ (127)
$\displaystyle*(dt\wedge\mathbf{a}\cdot d\mathbf{S}+fdV)$ $\displaystyle=$
$\displaystyle\ fdt+\mathbf{a}\cdot d\mathbf{r},$ (128)
$\displaystyle*(fdt\wedge dV)$ $\displaystyle=$ $\displaystyle\ -f.$ (129)

_Galilei_ Hodge star:

$\displaystyle*f$ $\displaystyle=$ $\displaystyle\ fdt\wedge dV$ (130)
$\displaystyle*(fdt+\mathbf{a}\cdot d\mathbf{r})$ $\displaystyle=$
$\displaystyle\ dt\wedge\mathbf{a}\cdot d\mathbf{S},$ (131)
$\displaystyle*(dt\wedge\mathbf{a}\cdot d\mathbf{r}+\mathbf{b}\cdot
d\mathbf{S})$ $\displaystyle=$ $\displaystyle\ dt\wedge\mathbf{b}\cdot
d\mathbf{r},$ (132) $\displaystyle*(dt\wedge\mathbf{a}\cdot d\mathbf{S}+fdV)$
$\displaystyle=$ $\displaystyle\ fdt,$ (133) $\displaystyle*(fdt\wedge dV)$
$\displaystyle=$ $\displaystyle\ -f.$ (134)

_Carroll_ Hodge star:

$\displaystyle*f$ $\displaystyle=$ $\displaystyle\ fdt\wedge dV,$ (135)
$\displaystyle*(fdt+\mathbf{a}\cdot d\mathbf{r})$ $\displaystyle=$
$\displaystyle\ fdV,$ (136) $\displaystyle*(dt\wedge\mathbf{a}\cdot
d\mathbf{r}+\mathbf{b}\cdot d\mathbf{S})$ $\displaystyle=$ $\displaystyle\
-\mathbf{a}\cdot d\mathbf{S},$ (137) $\displaystyle*(dt\wedge\mathbf{a}\cdot
d\mathbf{S}+fdV)$ $\displaystyle=$ $\displaystyle\ \mathbf{a}\cdot
d\mathbf{r},$ (138) $\displaystyle*(fdt\wedge dV)$ $\displaystyle=$
$\displaystyle\ -f.$ (139)

Notice that in expressions (134) and (135), Hodge stars based on tensors
$k_{ab}$/$\tilde{k}^{ab}$ were used (i.e. (102) and (110) rather than (11) and
(12), leading to zero result). See the discussion at the end of Section 3.1.

As for general Galilean and Carrollian (rather than Galilei and Carroll)
spacetimes, the same formulae are true if we replace interpretation of the
l.h.s. of (120) - (123) with more general expressions

$\displaystyle\mathbf{a}\cdot d\mathbf{r}$ $\displaystyle=$ $\displaystyle\
a_{1}e^{1}+a_{2}e^{2}+a_{3}e^{3},$ (140) $\displaystyle\mathbf{a}\cdot
d\mathbf{S}$ $\displaystyle=$ $\displaystyle\ a_{1}e^{2}\wedge
e^{3}+a_{2}e^{3}\wedge e^{1}+a_{3}e^{1}\wedge e^{2},$ (141) $\displaystyle dV$
$\displaystyle=$ $\displaystyle\ e^{1}\wedge e^{2}\wedge e^{3}.$ (142)

where $e^{a}\equiv(e^{0},e^{i})$, $i=1,2,3$, represent any _adapted_ coframe
fields.

## Acknowledgement

The author acknowledges support from grants VEGA 1/0703/20 and VEGA 1/0719/23.

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