Some Useful Operators on Differential Forms on Galilean and Carrollian Spacetimes

. 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.

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), d * F = −J, dF = 0, dJ = 0, the wave operator (∼ * d * d), Lorentz gauge condition ( * d * A = 0) etc., again only use forms and the above mentioned two operators, 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, 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 R 4 with global Cartesian coordinates (t, x, y, z) ≡ (t, r), where translations and rotations act exactly as in Minkowski spacetime, but boosts act differently, namely Galilei boost : t = t, Carroll boost : (Galilei and Carroll spacetimes may be treated as c → ∞ and c → 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). 1  Recall, however, that the Hodge star operator needs metric tensor (and orientation) for its definition * : Ω p → Ω n−p , Ω p ≡ Ω p (M, g, o) and it behaves according to w.r.t.diffeomorphisms; it therefore commutes with f * for (orientation preserving) isometries 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 " * " : Ω p → Ω q , Ω p ≡ Ω p (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 * " * " = " * "f * 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 Λp and the condition (1.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 -acting similarly to (Minkowski) Hodge star (already known from [7]), -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.

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 ρ 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
Let us start with a suitable coordinates and parametrization of forms.
In all three cases we have (global) coordinates t, x i = (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 α, p = 0, 1, 2, 3, 4, may be first uniquely decomposed as where the two hatted forms, (p − 1)-form ŝ and p-form r, respectively, are spatial, meaning that they do not contain ("temporal") differential dt in its coordinate expression.Being spatial, ŝ and r may be, depending on degree, expressed as where So, the general decomposition formula (2.1) may be further specified, for the five relevant degrees of differential forms, as follows: ) ) (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 Now we go on to study action of corresponding spacetime symmetry groups on forms within this parametrization.

Generators of rotations
In the above mentioned coordinates, what is common for all three spacetimes is action of -time and space translation, -spatial (3D) rotations where R denotes 3 × 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 dt 0 and dr 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 ρ 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 ρ 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 From requirement ) we can compute, for each degree, primed components in terms of unprimed and get ) ) ) ) All the formulas may be written (for rotation about j-th axis) as multiplication of the correspondent component column by matrix where Ω 0 : S j = 0, Ω 1 : S j = 0 0 0 l j , Ω 2 : S j = l j 0 0 l j , Ω 3 : S j = 0 0 0 l j , Ω 4 : S j = 0.

.33)
Induced action on coordinate basis of forms comes out to be ) ) (where e j denotes unit vector along x j ).They can be written via multiplication by where λ 2 e j 0 ,

List of all generators ρ p
If we denote, in general, ρ p = S j and N j on components of p-forms, then the full list of generators on (suitably parametrized components of) p-forms explicitly reads (2.47) (2.48) (2.49) Observe a simple rule, which can save some computations in the future: - or, as a formula, 3 Intertwining operators a qp

What intertwining operators we are interested in
On Minkowski, Galilei or Carroll spacetime, respectively, let 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, . . ., 4, either by single objects or by pairs.Therefore, the matrices a qp are necessarily (constant) block matrices of (block) dimensions 1 × 1, 1 × 2, 2 × 1 or 2 × 2. As an example, matrices a 12 and a 02 are of the form for c, d (column) vectors (i.e., c T , d T row vectors) and C, D 3 × 3 matrices.Within the spaces Ω p and Ω q , the corresponding (Lorentz, Galilei or Carroll) transformations induce (as generators) matrices ρ p and ρ 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: Thus, we are to solve equation (3.2) for each q, p = 0, . . ., 4 (and any j hidden in ρ p and ρ 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 Performing this for rotation generators ρ 0 , ρ 1 and ρ 2 alone (i.e., only for S j operators from (2.46), (2.47) and (2.48)), we get 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 ρ 1 and ρ 2 we get the following restrictions on k 1 , k 2 (see the details in Appendix A): (notice that they already depend on λ 1 , λ 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) When translated back into the language of complete forms (rather than just their component column), it produces the following (intertwining) mapping Ω 2 → Ω 1 : In what follows we perform similar calculation for all a qp (and all three spacetimes).
Notice the symmetry of the structure of the matrices: The pattern (type of matrix) happens to be symmetric with respect to both -reflection across the central horizontal axis and -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 .

How 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 -scalars remain intact (all scalars are rotation invariant), -vectors become zero (no rotation invariant vectors except for zero), -matrices become multiples of unity (no other rotation invariant matrices).
What we get is the following, much simpler, parametrization:

Restrictions on 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 (λ 1 , λ 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).In what follows we summarize all non-trivial intertwining operators found in our systematic computations.They depend, as we have seen, on particular spacetime.

Non-trivial operators a qp on Minkowski spacetime
On Minkowski spacetime we have found the following non-trivial operators: 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: wee see that -we merely "rediscovered" standard Hodge star in Minkowski spacetime, -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.

Non-trivial operators 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: ) ) ) (see (A.5) and (A.9)).On corresponding differential forms: ) Here So their actual action on corresponding differential forms reads We see that the rule how these operators work is nothing but application of exterior product on the form under consideration.This is clearly -an operator of degree +1 and it is also, -Galilei-invariant operation on forms, since dt = dt 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 i.e., with action on forms Both of them are, however, just compositions of operators already described previously, namely a 03 = a 04 • a 43 , a 14 = a 10 • a 04 .

Non-trivial operators 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: (see (A.6) and (A.8)).On corresponding differential forms: 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: Here On corresponding differential forms it is We can see that the rule how these four operators work is nothing but application of interior product Finally, it should be noted that there were still two more operators found, namely i.e., with action on forms Both of them are, however, just compositions of operators already described previously, namely a 30 = a 34 • a 40 , a 41 = a 40 • a 01 .

A note on the degree ±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 In Carroll case we have similarly For example, for the Galilean case, we can already see them, to credit just some fathers-founders, in (differently looking) component presentations as h ab t b = 0 in 1966 paper [21], γ αβ ψ β = 0 in 1972 paper [15], (they all describe the same equation).
On the other side, among the recent occurrence, we can mention notation and terminology Galilean : γ spatial cometric, from 2020 in [8].Now a simple observation is that, just being aware of tensor fields τ and ξ from (4.28) and ( 4.29), it should be clear from the very beginning that two invariant degree ±1 operators are available, τ ∧ (. . . ) on Ω (Galilei), i ξ (. . . ) on Ω (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 ±1 operators" in (4.21) and (4.27).

Invariant operators and invariant subspaces
In Galilei and Carroll spacetimes, some features of resulting invariant operators differ from those in Minkowski spacetime.Namely -analogs of Hodge star happen to be degenerate, -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 -either 1-dimensional (0-forms and 4-forms), -or spanned by pairs ((f, a) or (a, b)), -the two elements within the pair are independent for rotations, -the two elements within the pair need one another for boosts, -so for rotations and boosts they need one another, -so the whole representation space is needed, -so the representation is irreducible.Dimensionality of spaces of components of p-forms for p = 0, . . ., 4 is so, just because of these dimensions, non-trivial intertwining operators are only possible between spaces Ω 0 ↔ Ω 4 , Ω 1 ↔ Ω 3 and within each space, Ω p → Ω p .This is exactly used by the (Minkowski) Hodge star (the last possibility for p = 2).And, as an example, non-zero operator Ω 2 → Ω 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 Carroll case : So, we have non-trivial invariant subspaces in Ω 1 , Ω 2 and Ω 3 and, therefore, representations ρ 1 , ρ 2 and ρ 3 are not irreducible.Then, nothing (so simple as in the Lorentz case) excludes, e.g., a non-zero intertwining operator Ω 2 → Ω 3 .And indeed, we have found, in Galilei case, the operator 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) has no reason to be isomorphism.And it indeed seized its chance: Again, they coincide with particular invariant subspaces displayed in (4.30).Similar facts are true for the new Carroll Hodge star.

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. On Lorentzian spacetime, the Hodge star is the only intertwining operator.
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 d * F = −J, dF = 0 (mentioned in Section 1) correspond to Lorentzian/Galilean/Carrollian electrodynamics depending on which particular Hodge star is used; see more in [7]).
A Details of solving equations for (some) 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 (λ 1 , λ 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 .
The equation for a 11 explicitly reads i.e., Solution is for all three spacetimes.And since the system (A.1) remains intact for swapping λ 1 ↔ λ 2 , we get a 11 = a 33 .The equation for a 13 explicitly reads 0 λ 2 e T j λ 1 e j 0 i.e., Here we have for a 13 Swapping λ 1 ↔ λ 2 in (A.2) produces swapping k 1 ↔ k 2 for solutions.So it produces as a 31 swapping Galilei ↔ Carroll for the above conclusion for a 13 .Altogether we come to conclusion: -ξ is a non-zero covector (i.e., a 0 1 -tensor) in V , -h is a rank-n symmetric type-2 0 -tensor in V , -such that h(ξ, • ) = 0.
We call a frame E a = (E 0 , E i ), i = 1, . . ., n in (V, ξ, h) and a (dual) coframe E a = E 0 , E i adapted (or distinguished) if so that, in (any) adapted frame, the components of the two tensors have "canonical form" The change-of-basis matrix A between any pair Êa , E a of adapted frames, given by Êa = A b a E b , has the structure where R is n-dimensional rotation matrix.Such matrices form a Lie group G, subgroup of GL(n + 1, R), the (homogeneous) Galilei group (R parametrizes rotations and v Galilei boosts, respectively).
We call a frame E a = (E 0 , E i ) and a (dual so that, in (any) adapted frame, the components of the two tensors have "canonical form" The change-of-basis matrix A between any pair Êa , E a of adapted frames, given by Êa = A b a E b , has the structure where R is n-dimensional rotation matrix.Such matrices form a Lie group G, subgroup of GL(n + 1, R), the (homogeneous) Carroll group (R parametrizes rotations and v Carroll boosts, respectively).
It follows that boosts formulas x 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 E of adapted frames and, consequently, on coframes E * E a → A −1 a b E b .This induces representation ρ p of G on the space of p-forms on V , and, consequently, on the (isomorphic) space Λp of components of p-forms on V ρp : It is well known (see, e.g., [6,19,22]) that the space Λ p V * of p-forms in V is canonically isomorphic to the space of equivariant maps 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, ξ, h), each tangent space T m M is endowed with the structure of Galilei vector space described in Appendix C.1.
Here points in the fiber over m ∈ 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 :

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 Here (W, ρ) is a representation module of G.The corresponding quantity (described by Φ) is called quantity (on M ) of type ρ. 2In particular, for representation ρp in Λp from (C.3), the corresponding quantities of type ρp are nothing but p-forms on the Galilean manifold (M, ξ, h).So, we can treat elements of Ω p (M ) as mappings Φ p : P → Λp , ρp , Φ p (eA) = ρp A −1 Φ p (e). (D.1)

D.3 Intertwining operators on forms on Galilean/Carrollian manifold
Let Φ p be an equivariant map (D.1) corresponding to a p-form α p on M and let âqp be an operator from (C.5).Then a simple check shows that mere composition of maps produces an equivariant map corresponding to a q-form on M .So we can reconstruct corresponding operators on forms on a Galilean manifold (mappings Ω p (M ) → Ω q (M ), transforming p-forms to q-forms on M ) once we know matrices âqp , intertwining operators between the representations ρp and ρq .
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, ξ, h)). 3 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 ↔ 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 → T v V reads w → γ(0) for γ(t) = v + wt.) Consequently, whenever we have a type p q tensor B in V , we can canonically associate with it a type p q tensor field b on V (as a manifold).The correspondence reads In the language of manifolds, we get from (E.3) in a standard way where L ξ is the Lie derivative.The specific structure of ξ (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 ρ 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) Êa = A −1 a b E b (with A = I+ 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 dx a ≡ (dt, dx i ) becomes an adapted coframe E a ≡ E 0 , E i .Put it differently, the operators listed in Section 2.4 are exactly those operators mentioned in (C.3) for which intertwining operators âqp are computed via solving (C.6).More precisely, they are generators of the derived representation ρ q of the Lie algebra G of the group G, given standardly by ρq (I + C) .= 1 + ρ q (C).
Since, however, solutions âqp of (C.6) do not change when ρq (A) is replaced by ρ 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.provide valid expressions of our Hodge star operator on forms on any Galilean/Carrollian spacetime.

e
a ↔ dx a , e a ↔ ∂ a , (E.1) i.e., B ≡ B ...b a... e a ⊗ • • • ⊗ e b ↔ B ...b a... dx a ⊗ • • • ⊗ ∂ b ≡ b (components B ...b a... are the same in both expressions and constant).Now if a linear mapping A : V → V preserves (say) a type 0 2 tensor B (bilinear form) in the usual sense of linear algebra,B(Au, Aw) = B(u, w), i.e., A c a B cd A d b = B ab (E.2)then, on V treated as a manifold, we speak of a smooth mappingf ≡ f A : V → V in coordinates x a → A a b x band the tensor field b is preserved in the sense of pull-backf * 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 = I + C, B(Cu, w) + B(u, Cw) = 0, i.e., C c a B cb + B ac C c b = 0.

F 1 + a 2 e 2 + a 3 e 3 ,a • dS → a 1 dS 1 + a 2 dS 2 + a 3 dS 3 ≡ a 1 e 2 ∧ e 3 + a 2 e 3 ∧ e 2 + a 3 e 1 ∧ e 3 ,
Explicit expressions of * -operator on Galilean/Carrollian spacetimesDue 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 âqp .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 dx a ≡ dt, dx i from Galilei/Carroll spacetime with a general adapted (possibly local and non-holonomic) coframe field e a ≡ e 0 , e i 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, ξ, h) and a Carrollian manifold M, ξ, h , by the properties Galilean frame/coframe field : e 0 = ξ, h = δ ij e i ⊗ e j , Carrollian frame/coframe field : e 0 = ξ, h = δ ij e i ⊗ e j .Then the above mentioned formulas (B.1)-B.10)with substitutions dt → e 0 , a • dr → a 1 e dV → e 1 ∧ e 2 ∧ e 3 3.2 General structure of all a qp (no invariance, yet)A priori, just from the parametrization structure (2.7) of spaces Ω q and Ω p (with no intertwining properties, yet) the operators a qp Which operators a qp are trivialIt is evident from the structure of (3.2) that -zero operator (for any pair of spaces, Ω q → Ω p ) as well as -unit operator within any space ( 1 : Ω p → Ω 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.) 19)roduce the formulas valid for the Galilei Hodge star.(Actuallythere 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.)Sowecan 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: .17) a 13 : dt ∧ a • dS + f dV → f dt, (4.18) a 04 : f dt ∧ dV → 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