DIFFERENTIAL GEOMETRY AND APPLICATIONS Proc. Conf., Aug. 28 – Sept. 1, 1995, Brno, Czech Republic Masaryk University, Brno 1996, 573–602

CLASSICAL PARTICLE PHASE SPACE IN GENERAL RELATIVITY

JOSEF JANYŠKA AND MARCO MODUGNO

Abstract. The geometric structure of phase space in the Einstein general relativistic framework is studied. Namely, the jet space of time-like submanifolds of space-time is introduced and the associated connections, jet–contact and contact structure are analysed. Moreover, the electromagnetic field is incorporated in the gravitational structures. A rigorous mathematical treatment of units of measurement is included as well.

The above setting is regarded as a framework aimed at extending to the Einstein case a recent quantisation procedure in the Galilei case, based on jets, connections and co–symplectic forms. According to this purpose, a structural comparison between Einstein’s and Galilei’s frameworks is performed.

Introduction

Recently, a covariant geometric approach to general relativistic quantum mechanics based on jets, general connections and co–symplectic forms has been presented in a Galilei general relativistic background (i.e. in a curved space–time with absolute time). The case of a scalar particle has been studied in [4], [5]; later, this formulation has been extended to a spin particle in [1].

The authors started from Galilei’s general relativistic model just as first step toward Einstein’s case. Actually, several interesting techniques developed in the Galilei case can be fruitfully used also in the Einstein case, after a certain adjustment. This paper is aimed at transferring to the Einstein space–time the basic geometrical techniques developed in the Galilei case, in view of a covariant formulation of quantum mechanics in the Einstein framework by an approach analogous to [1], [4], [5].

The Galilei space–time is assumed to be a fibred manifold equipped with a vertical Riemannian metric and a fibre preserving linear connection. On the other

1991 Mathematics Subject Classification. 53C05, 53C15, 58A20, 70H40, 81S10, 83C10. Key words and phrases. General relativity, analytical mechanics, jets, jet–contact structure,

non-linear connections, contact structure, co–symplectic forms, quantisation.

This research has been supported by grant No. 201/93/2125 of GA CR, GNFM of CNR, MURST, University of Florence and Contract ERB CHRXCT 930096 of EEC.

This paper is in final form and no version of it will be submitted for publication elsewhere.

hand, the Einstein space time is assumed to be a time orientable Lorentzian manifold. A fundamental role in the classical and quantum theory is played by the classical phase space, namely by the first jets of sections in the Galilei case and by the first jets of time–like submanifolds in the Einstein case.

Essentially, the goal of the present paper is to show how typical geometrical constructions arising from the fibring in the Galilei case can be recovered through the Lorentzian metric in the Einstein case. The price we pay consists in the fact the several objects living on space–time in the Galilei case have their counterpart living on the jet space in the Einstein case. Actually, this fact encodes one of the main difficulties for a covariant quantum theory in the Einstein background.

Summarising the main results of the paper, we obtain the Einstein analogues of the differential of the absolute time function, of the tangent sequence of space– time fibred manifold, of its splitting over the jet space, of the first and second order connections on the jet bundle and of the co–symplectic form. These objects have a key role in the Galilei’s case, hence we expect they will be an essential tool for pursuing the programme toward a covariant quantum mechanics in Einstein’s case.

Our approach yields a structural comparison between Galilei’s and Einstein’s settings.

Throughout the paper all manifolds and maps are assumed to be smooth.

We shall be involved with non–linear connections on fibred manifolds. So, we recall a few basic notions and results (see [14]).

Let us consider a fibred manifold q : F → B. We denote a fibred chart of F by (x^{φ}, y^{i}) and the induced chart of T F by (x^{φ}, y^{i}, ẋ^{φ}, ẏ^{i}).

We recall that a (general) connection can be defined as a tangent valued form

C : F → T^{∗}B ⊗_{F} T F

projectable over id_{T B}. The connection C is also characterised by the vertical projection, or, equivalently, by the vertical valued form

ν_{C} : T F → V F , ν_{C} : F → T^{∗}F ⊗_{F} V F .

We have the coordinate expressions

C = d^{φ}⊗(∂_{φ} + C_{φ}^{i}∂_{i}) ,  ν_{C} = (d^{i} − C_{φ}^{i}d^{φ})⊗∂_{i} ,  C_{φ}^{i} ∈ C^{∞}(F) .

The curvature and the torsion of C can be expressed in a unified way by means of the Frölicher-Nijenhuis bracket [ , ]. The torsion requires the choice of a vertical valued 1–form

Φ : F → T^{∗}B ⊗_{F} V F .

Actually, in most cases, the geometrical structure of q : F → B exhibits a distinguished Φ; for instance, the standard torsion of linear connections on a manifold M are taken with respect to the basic form id_{M} : M → T^{∗}M ⊗_{M}T M. Curvature and torsion fulfill generalised first and second Bianchi’s identities, which are again expressed through the Frölicher-Nijenhuis bracket.

Thus, the curvature and the torsion with respect Φ are defined, respectively, as the 2–forms

R_{C} = \frac{1}{2}[C, C] : F → ∧^{2}T^{∗}B ⊗_{F} V F , T_{C} = [C, Φ] : F → ∧^{2}T^{∗}B ⊗_{F} V F ,

with coordinate expressions, respectively,

(0.1)  R_{C} = (∂_{λ}C_{μ}^{i} + C_{λ}^{j} ∂_{j}C_{μ}^{i}) d^{λ} ∧ d^{μ}⊗∂_{i} ,
(0.2)  T_{C} = (∂_{λ}Φ_{μ}^{i} + C_{λ}^{j} ∂_{j}Φ_{μ}^{i} − Φ_{μ}^{j} ∂_{j}C_{λ}^{i}) d^{λ} ∧ d^{μ}⊗∂_{i} .

We assume the following fundamental unit spaces [4]:

(1) the oriented 1–dimensional vector space 𝕋 over IR of time intervals,

(2) the positive 1–dimensional semi–vector space 𝕃 over IR^{+} of lengths,

(3) the positive 1–dimensional semi–vector space 𝕄 over IR^{+} of masses.

A time unit of measurment is defined to be an oriented basis of 𝕋 or its dual

u_{0} ∈ 𝕋 , u^{0} ∈ 𝕋^{∗} .

Further details about the mathematical formalism on units of measurement can be found in [1], [5].

We shall be involved with the following construction. Let M be a manifold and consider the vector bundle π_{M} : 𝕋^{∗} ⊗ T M → M. The projection T π_{M} : T (𝕋^{∗}⊗T M) → T M can be regarded as vector valued 1-form

σ : 𝕋^{∗}⊗T M → T^{∗}(𝕋^{∗}⊗T M) ⊗_{𝕋^{∗}⊗T M} T M ;

its coordinate expression is

σ = d^{φ}⊗∂_{φ} . (0.3)

1. Galilei’s case

In this section we briefly recall the basic geometric constructions on Galilei’s general relativistic space–time introduced in [4], [5], along with some additional results on the correspondence between space–time connections and connections on the phase space and the ‘universal’ electric and magnetic fields.

1.1. Gravitational structures

We start with the phase space and the gravitational structures in the Galilei framework.

Assumption G.1. We assume space–time to be a 4-dimensional oriented fibred manifold

t : E → T ,

over a 1-dimensional oriented affine space T (time), associated with the vector space 𝕋, equipped with a scaled Riemannian metric on the fibres, i.e. with a scaled vertical Riemannian metric,

g : E → 𝕃^{2}⊗V^{∗}E ⊗_{E} V^{∗}E .

We observe that g can be regarded as a non-scaled Riemannian metric on the vector bundle 𝕃^{∗}⊗V E → E.

The typical space–time coordinate charts, adapted to the fibring, to a time unit of measurement u_{0} and to the space–time orientation, will be denoted by (x^{0}, x^{i}). Throughout this section, the index 0 will refer to the base space, Latin indices i, j, p, … = 1, 2, 3 will refer to the fibres, while Greek indices λ, μ, φ, ⋯ = 0, 1, 2, 3 will refer both the base space and the fibres.

In coordinates

g = g_{ij}ď^{i}⊗ď^{j} ,  g_{ij} : E → 𝕃^{2}⊗IR ,  |g| ≡ det(g_{ij}) > 0 ,

where the check ‘ˇ ’ denotes vertical restriction.

A motion in E is defined to be a section s : T → E.

The phase space is defined to be the first jet space of sections π_{0}^{1} : JE ≡ J_{1}E → E. A relevant feature of the Galilei case is due to the affine structure of the bundle JE → E. A first consequence is the fibred isomorphism over JE

V_{E}JE ≃ JE ×_{E} 𝕋^{∗}⊗V E .

We deal with the natural complementary jet–contact maps

� : JE × 𝕋 → T E , ϑ : JE ×_{E} T E → V E ,

or equivalently

� : JE → 𝕋^{∗}⊗T E , ϑ : JE → T^{∗}E ⊗_{E} V E ,

which split the natural exact sequence

0 ⟶ V E ⟶ T E ⟶^{dt} E × 𝕋 ⟶ 0 , (1.1)

through the exact sequence over JE

0 ⟶ JE × 𝕋 ⟶^{�} JE ×_{E} T E ⟶^{ϑ} JE ×_{E} V E ⟶ 0 . (1.2)

We have the coordinate expressions

� = u^{0}⊗�_{0} = u^{0}⊗(∂_{0} + x_{0}^{i}∂_{i}) , ϑ = ϑ^{i}⊗∂_{i} = (d^{i} − x_{0}^{i}d^{0})⊗∂_{i} , (1.3)

where (x^{0}, x^{i}, x_{0}^{i}) is the induced coordinate chart on JE.

A connection K on the bundle T E → E can be expressed, equivalently, by a tangent valued form, or by a vertical valued form

K : T E → T^{∗}E ⊗_{T E} T T E , ν_{K} : T E → T^{∗}T E ⊗_{T E} T E ,

CLASSICAL PARTICLE PHASE SPACE 577

respectively, with coordinate expressions K = d^{φ}⊗(∂_{φ} + K_{φ}^{μ} ˙∂_{μ}) , ν_{K} = (ḋ^{μ} − K_{φ}^{μ}d^{φ})⊗∂_{μ} , (1.4) K_{φ}^{μ} ∈ C^{∞}(T E), where (x^{φ}, ẋ^{φ}) is the induced coordinate chart on T E.

The connection K is linear if and only if its coordinate expression is of the type K_{φ}^{λ} = K_{φ}^{λ}_{ψ}ẋ^{ψ} , K_{φ}^{λ}_{ψ} ∈ C^{∞}(E) . (1.5)

If the connection K is linear, then we can define the vertical projection ν_{Λ^{∗}⊗K} : T (𝕋^{∗}⊗T E) → 𝕋^{∗}⊗T E, where Λ is the canonical linear flat connection on 𝕋. For the sake of brevity, we shall use the same symbol and write ν_{K} ≡ ν_{Λ^{∗}⊗K}. In the induced coordinate chart (x^{φ}, ẋ_{0}^{φ}) on 𝕋^{∗}⊗T E, we have ν_{K} = ν_{Λ^{∗}⊗K} = u^{0}⊗(ḋ_{0}^{μ} − K_{φ}^{μ}_{ψ}ẋ_{0}^{ψ}d^{φ})⊗∂_{μ} . (1.6)

Analogously, a connection Γ on the affine bundle JE → E can be expressed, equivalently, by a tangent valued form, or by a vertical valued form

Γ : JE → T^{∗}E ⊗_{JE} T JE , ν_{Γ} : JE → 𝕋^{∗}⊗T^{∗}JE ⊗_{JE} V E ,

respectively, with coordinate expressions Γ = d^{φ}⊗(∂_{φ} + Γ_{φ}i0∂_{i}^{0}) , ν_{Γ} = u^{0}⊗(d_{0}^{i} − Γ_{φ}i0d^{φ})⊗∂_{i} , (1.7) Γ_{φ}i0 ∈ C^{∞}(JE).

The connection Γ is affine if and only if its coordinate expression is of the type

Γ_{φ}i0 = Γ_{φ}i00 jx_{0}^{j} + Γ_{φ}i00 0 , Γ_{φ}i00 α ∈ C^{∞}(E) .

Theorem 1.1. For any linear connection K on T E, the map

ν_{Γ} = ϑ ∘ ν_{K} ∘ T �

given by the following diagram

T JE u_{(π_{JE},T � )} w^{ν_{Γ}} V JE w^{≃} JE ×_{E} (𝕋^{∗}⊗V E)

JE ×_{E} T (𝕋^{∗}⊗T E)  w^{(id_{JE}×ν_{K})}  JE ×_{E} (𝕋^{∗}⊗T E) u^{(id_{JE}×ϑ)}

turns out to be a connection on the bundle JE → E with coordinate expression Γ_{φ}i0 = K_{φ}^{i}_{j}x_{0}^{j} + K_{φ}^{i}_{0} − x_{0}^{i}(K_{φ}^{0}_{j}x_{0}^{j} + K_{φ}^{0}_{0}) . (1.8)

Thus, we have obtained a map

χ : K ↦ Γ .

Proof. It can be proved in coordinates by using (1.3) and (1.4). QED A linear connection K is said to be time–preserving if ∇^{K}(dt) = 0; in coordi-

nates it reads K_{φψ}^{0} = 0.

Corollary 1.1. If K is time–preserving, then the corresponding Γ is affine. Moreover, the correspondence between time–preserving linear connections on T E and affine connections on JE is one–to–one, [5].

Corollary 1.2. If K is time–preserving and torsion free, then the corresponding Γ is torsion free, with respect to the scaled vertical valued form ϑ on JE.

The curvatures (see Introduction), of a connection K on T E and of a connection Γ on JE are, respectively, the 2–forms

R_{K} = \frac{1}{2}[K, K] : T E → ∧^{2}T^{∗}E ⊗_{E} T E , R_{Γ} = \frac{1}{2}[Γ, Γ] : JE → 𝕋^{∗} ∧^{2} T^{∗}E ⊗_{E} V E ,

whose coordinate expressions can be easily computed by (0.1).

Theorem 1.2. If Γ is the connection on JE induced by a linear connection K on T E, then the we have

R_{Γ} = ϑ∘R_{K} ∘� ,

according to the following commutative diagram

JE ×_{E} 𝕋^{∗}⊗T E w^{id_{JE}×R_{K}} JE ×_{E} (∧^{2}T^{∗}E ⊗_{E} 𝕋^{∗}⊗T E) u_{ϑ}

JE u^{(id_{JE}, � )}  w^{R_{Γ}}  ∧^{2}T^{∗}E ⊗_{E} 𝕋^{∗}⊗V E

i.e. in coordinates

(R_{Γ})_{λμ}i0 = (R_{K})_{λμ}^{i}_{j}x_{0}^{j} + (R_{K})_{λμ}^{i}_{0} − x_{0}^{i}((R_{K})_{λμ}^{0}_{j}x_{0}^{j} + (R_{K})_{λμ}^{0}_{0}) . Proof. It can be proved by using (1.8) and the coordinate expressions of R_{K}

and R_{Γ}.  QED

We observe that a connection K on T E and a Riemannian metric h on E would induce the scaled 2–form Ω(h, K) on 𝕋^{∗}⊗T E given by ν_{K} ¯∧σ, where σ is defined in the Introduction and ¯∧ denotes the wedge product followed by the contraction through the metric h. But, in our framework, we cannot define ν_{K} ¯∧σ since our metric g is only vertical; we need a projection of T E on V E. Actually, we can use the projection ϑ : JE ×_{E} T E → V E.

Thus, we set

Ω(g, K) : = (ϑ ∘ ν_{K})¯∧(ϑ ∘ σ) : JE ×_{E} 𝕋^{∗}⊗T E → 𝕋^{∗}⊗𝕃^{2}⊗∧^{2}T^{∗}(𝕋^{∗}⊗T E) .

In coordinates

Ω(g, K) = g_{ij}u^{0}⊗(ḋ_{0}^{i} − x_{0}^{i}ḋ_{0}^{0} − (K_{φ}^{i} − x_{0}^{i}K_{φ}^{0})d^{φ}) ∧ ϑ^{j} . (1.9)

On the other hand, a connection Γ on JE and the vertical metric g yield the natural contact scaled 2–form Ω(g, Γ) on JE, [5],

Ω(g, Γ) = ν_{Γ} ¯∧ϑ : JE → 𝕋^{∗}⊗𝕃^{2}⊗∧^{2}T^{∗}JE

with coordinate expression

Ω(g, Γ) = g_{ij}u^{0}⊗(d_{0}^{i} − Γ_{φ}i0d^{φ}) ∧ ϑ^{j} . (1.10)

Theorem 1.3. If Γ is the connection on JE associated with a linear connection K on T E, then

Ω(g, Γ) = (id_{JE}, � )^{∗}Ω(g, K) .

Proof. The pullback of the form Ω(g, K) with respect to (id_{JE}, � ) is the form

(id_{JE}, � )^{∗}Ω(g, K) : JE → 𝕋^{∗}⊗𝕃^{2}⊗∧^{2}T^{∗}JE .

The identification of Ω(g, Γ) and (id_{JE}, � )^{∗}Ω(g, K) follows from (1.3) (1.8), (1.9)

and (1.10).  QED

In [7] all 2–forms on JE generated naturally by a vertical Riemannian metric and by a time–preserving linear connection have been classified by using naturality methods of differential geometry, [8], [9]. It has been proved that all such forms are constant multiples of the form Ω(g, Γ).

The form Ω(g, Γ) is non-degenerate, namely

dt ∧ Ω(g, Γ) ∧ Ω(g, Γ) ∧ Ω(g, Γ) : JE → 𝕋^{∗2}⊗𝕃^{6}⊗∧^{7}T^{∗}JE

is a scaled volume form on JE.

Theorem 1.4. In the case when K is torsion free and time–preserving, the 2–form Ω(g, Γ) is closed if and only if (see [4], [5])

∇g = 0, R^{i}_{λ}^{j}_{μ} = R^{j}_{μ}^{i}_{λ} . QED

Assumption G.2. Space–time is assumed to be equipped with a torsion free time–preserving connection K^{♮}, called gravitational; moreover, we postulate the following condition as a field equation for g and K^{♮}

dΩ(g, Γ^{♮}) = 0 ,

where Γ^{♮} : = χ∘K^{♮} .

Thus, under assumptions (G.1) and (G.2), the form

Ω^{♮} : = Ω(g, Γ)

turns out to be “co–symplectic”. This form encodes the main information arising from the classical space–time background and is the source of the quantisation procedure presented in [4], [5].

We recall that a second order connection on E is defined as a section

γ : JE → J_{2}E ,

where J_{2}E is the space of second order jets of sections of E → T.

Moreover, we recall that �_{2} : J_{2}E ↪ 𝕋^{∗} ⊗T JE turns out to be exactly the fibred submanifold over JE, which makes the following diagram commutative

J_{2}E  w^{�_{2}} u_{π_{1}^{2}}  𝕋^{∗}⊗T JE u_{id⊗T π_{0}^{1}}

JE  w^{�} u_{π^{1}}  𝕋^{∗}⊗T E u_{id⊗T t}

T  w^{1_{𝕋}}  𝕋^{∗}⊗𝕋

where �_{2} denotes the second order jet–contact map.

Therefore, by considering the inclusion �_{2}, each second order connection γ can be characterised as a scaled vector field

γ : JE → 𝕋^{∗}⊗T JE ,

which makes the following diagram commutative

JE  w^{γ} u_{id}  𝕋^{∗}⊗T JE u_{id⊗T π_{0}^{1}}

JE  w^{�} u_{π^{1}}  𝕋^{∗}⊗T E u_{id⊗T t}

T  w^{1_{𝕋}}  𝕋^{∗}⊗𝕋

In particular, a second order connection turns out to be a (first order) connection on the fibred manifold JE → T.

Now, for each time–preserving connection K, the map [5],

γ : = � ⌟ Γ : JE → 𝕋^{∗}⊗T E

turns out to be a second order connection; moreover, it is uniquely characterised by

γ⌟ Ω(g, Γ) = 0 .

In particular, we are involved with the gravitational second order connection

γ : = � ⌟ Γ^{♮} ,

which provides the equation of inertial motion of particles.

1.2. Electromagnetic field and geometric structure So far, we have assumed on space–time the gravitational structure associated

with the vertical Riemannian metric g and the space–time connection K. Next, we introduce the electromagnetic field.

Assumption G.3. We assume the electromagnetic field to be a closed scaled 2-form on E

F : E → (𝕃^{1/2}⊗𝕄^{1/2})⊗∧^{2}T^{∗}E .

We denote the local potential of F by A : E → (𝕃^{1/2}⊗𝕄^{1/2})⊗T^{∗}E. Thus, by definition, we set

2dA = F .

We define the ‘universal’ electric and ‘universal’ magnetic fields to be the scaled forms on JE

E : = −� ⌟F : JE → (𝕋^{∗2}⊗𝕃^{1/2}⊗𝕄^{1/2})⊗T^{∗}E ,
B : = F + 2dt ∧ E : JE → (𝕃^{1/2}⊗𝕄^{1/2})⊗∧^{2}T^{∗}E .

Thus, we can write

F = −2dt ∧ E + B.

Hence, the electric and magnetic fields associated with an observer o : E → JE are defined as the pullback forms

o^{∗}E : E → (𝕋^{∗}⊗𝕃^{1/2}⊗𝕄^{1/2})⊗T^{∗}E , o^{∗}B : E → (𝕃^{1/2}⊗𝕄^{1/2})⊗∧^{2}T^{∗}E .

We stress that the universal electric field carries the full information of the electromagnetic field.

Remark 1.1. We have the coordinate expressions

F = 2F_{0j}d^{0} ∧ d^{j} + F_{ij}d^{i} ∧ d^{j} = −2u_{0}E_{j}d^{0} ∧ ϑ^{j} + B_{ij}ϑ^{i} ∧ ϑ^{j} ,

and

E = E_{j}d^{j} + E_{0}d^{0} = E_{i}ϑ^{i} , B = B_{ij}d^{i} ∧ d^{j} + 2B_{0j}d^{0} ∧ d^{j} = B_{ij}ϑ^{i} ∧ ϑ^{j} ,

where

E_{j} = −u^{0}(F_{0j} + F_{hj}x_{0}^{h}) E_{0} = −u^{0}F_{h0}x_{0}^{h} ,
B_{ij} = F_{ij} B_{0j} = −F_{hj}x_{0}^{h} .

Therefore, in a chart adapted to an observer o, the coordinate expressions of the observed electric and magnetic fields are

o^{∗}E = −u_{0}F_{0j}d^{j} o^{∗}B = F_{ij}d^{i} ∧ d^{j} .

Next, we show that the electromagnetic field can be naturally incorporated into the gravitational structures of the phase spase.

Namely, let us consider the following gravitational objects on the phase space JE:

Γ^{♮} : JE → T^{∗}E ⊗_{E} T JE , γ^{♮} : JE → 𝕋^{∗}⊗T JE ,

Ω^{♮} : JE → 𝕋^{∗}⊗𝕃^{2}⊗∧^{2}T^{∗}JE ,

which fulfill the following structural relations

γ^{♮} = � ⌟ Γ^{♮} ,  Ω^{♮} = ν_{Γ^{♮}} ¯∧ϑ ,  γ^{♮}⌟ Ω^{♮} = 0 ,  dΩ^{♮} = 0 .

We are looking for total objects obtained correcting the gravitational objects by an electromagnetic term, in such a way to preserve the above relations.

For this purpose we need a suitable coupling constant. So, we consider a particle with a given mass and charge

m ∈ 𝕄 , q ∈ 𝕋^{∗}⊗𝕃^{3/2}⊗𝕄^{1/2} ,

and refer to the coupling constant

\frac{q}{m} ∈ 𝕋^{∗}⊗𝕃^{3/2}⊗𝕄^{∗1/2} .

We start from the obvious coupling of the electromagnetic field with the gravitational co–symplectic 2–form on JE.

Accordingly, we define the total co–symplectic form to be

Ω : = Ω^{♮} + \frac{q}{2m}F : JE → 𝕋^{∗}⊗𝕃^{2}⊗∧^{2}T^{∗}JE .

Here, the factor \frac{1}{2} is chosen in such a way to recover the standard normalization in practical formulas.

Of course, we obtain

dΩ = 0 .

We have

dt ∧ Ω ∧ Ω ∧ Ω = dt ∧ Ω^{♮} ∧ Ω^{♮} ∧ Ω^{♮} .

So, the electromagnetic field does not contribute to the total volume form on the phase space.

Moreover, in [5] the following result has been proved.

Theorem 1.5. There is a unique torsion free affine connection Γ and a unique second order connection on the bundle JE → E

Γ = Γ^{♮} + Γ^{e} , γ = γ^{♮} + γ^{e} ,

such that

γ = � ⌟ Γ ,  Ω = ν_{Γ} ¯∧ϑ ,  γ⌟ Ω = 0 .

Namely, γ^{e} turns out to be the Lorentz force

γ^{e} = \frac{q}{m}g^{♯}∘Ě : JE → 𝕋^{∗2}⊗V E

and Γ^{e} the electromagnetic soldering form

Γ^{e} = −\frac{q}{2m}g^{♯}^{2}∘(F − 2dt ∧ E) : JE → T^{∗}E ⊗_{E} (𝕋^{∗}⊗V E) ,

where g^{♯}^{2} denotes the metric isomorphism on the second component after vertical restriction.

Moreover, the above objects fulfill the following equalities

γ^{e} = � ⌟ Γ^{e} , −Γ^{e} ¯∧ϑ = Ω^{e} .

We have the coordinate expressions

γ^{e} = −\frac{q}{m}(F_{0}^{i} + F_{h}^{i}x_{0}^{h})u^{0}⊗∂_{i}^{0} , Γ^{e} = −\frac{q}{2m}u^{0}((F_{h}^{i}x_{0}^{h} + 2F_{0}^{i})d^{0} + F_{j}^{i}d^{j})⊗∂_{i}^{0} ,

hence

Γ_{h}i00 k = Γ_{h}^{♮}i00 k ,  Γ_{0}i00 k = Γ_{0}^{♮}i00 k + \frac{q}{2m}u^{0}F^{i}_{k} ,  Γ_{0}i00 0 = Γ_{0}^{♮}i00 0 + \frac{q}{2m}u^{0}F^{i}_{0} .

Hence, the above total Γ induces a total space–time connection K.

2. Einstein’s case

In this section, we study the geometric structures of phase space in the Einstein general relativistic framework.

We start with the basic assumptions on space–time.

Asssumption E.1. We assume space–time to be a 4–dimensional oriented and time–oriented manifold M equipped with a scaled Lorentzian metric of signature (+ − −−)

g : M → 𝕃^{2}⊗T^{∗}M ⊗_{M} T^{∗}M .

Local coordinate charts on M will be denoted by (x^{φ}), φ = 0, 1, 2, 3. The coordinate expression of g is then

g = g_{φψ}d^{φ}⊗d^{ψ} , g_{φψ} : M → 𝕃^{2}⊗IR .

In what follows we shall use local coordinate charts such that the vector ∂_{0} is time– like and time oriented and ∂_{1}, ∂_{2}, ∂_{3} are space–like; hence g_{00} > 0, g_{11}, g_{22}, g_{33} < 0.

Latin indices i, j, p, … will span space–like coordinates, while Greek indices λ, μ, φ, … will span space–time coordinates.

Obviously, in the Einstein case we have no time fibring t, i.e. no absolute time T. However we still have a vector space 𝕋 which now describes the proper time intervals. We stress that this fact is in full agreement with the standard interpretation of general relativity.

Asssumption E.2. We assume the light velocity to be a positive element

c ∈ 𝕋^{∗}⊗𝕃 .

We observe that g and g/c^{2} can be regarded as non-scaled Lorentzian metrics on the vector bundles 𝕃^{∗}⊗T M → M and 𝕋^{∗}⊗T M → M, respectively.

2.1. Jets of 1–dimensional submanifolds

In order to describe velocities of motions in the Einstein case we need jets of submanifolds. Let us recall the basic notions (for further details see [15]).

A k–jet of 1–dimensional submanifolds of M at x ∈ M is defined to be an equivalence class of 1–dimensional submanifolds touching each other at x with a contact of order k. The k–jet of a 1-dimensional submanifold s ⊂ M at x ∈ s is denoted by j_{k}s(x). The set of all k–jets of 1–dimensional submanifolds of M can be equipped, in a natural way, with a smooth structure. The corresponding manifold is denoted by J_{k}(M, 1).

For p > q we have the canonical projection

π_{q}^{p} : J_{p}(M, 1) ⟶ J_{q}(M, 1) : j_{p}s(x) ↦ j_{q}s(x),

which makes J_{p}(M, 1) a bundle over J_{q}(M, 1); in particular, π_{0}^{k} : J_{k}(M, 1) → J_{0}(M, 1) = M is a bundle. We set

\underline{ϕ} = π_{0}^{k}(ϕ) , ϕ ∈ J_{k}(M, 1).

We stress that, unlike the case of jets of sections of a fibred manifold, the bundle J_{p}(M, 1) → J_{p−1}(M, 1) is not affine.

Given a 1–dimensional submanifold s ⊂ M and an integer k ≥ 0, we have the map

j_{k}s : s → J_{k}(M, 1) : x ↦ j_{k}s(x) .

Clearly j_{k}s(s) ⊂ J_{k}(M, 1) is a 1–dimensional submanifold.

We have the canonical fibred isomorphism over M of the first jet bundle with the Grassmannian bundle of dimension 1

J(M, 1) ≡ J_{1}(M, 1) ⟶ Grass(M, 1) : ϕ ↦ L_{ϕ},

where L_{ϕ} ⊂ T\underline{ϕ}M is the tangent space at \underline{ϕ} of 1–dimensional submanifolds generating ϕ.

A local chart on M is said to be divided if the set of its coordinate functions is divided into two subsets of 1 and (dim M − 1) elements. Our typical notation for a divided chart will be

(x^{0}, x^{i}), 1 ≤ i ≤ dim M − 1 .

A divided chart and a 1–dimensional submanifold s ⊂ M are said to be related if the submanifold s can be expressed locally by formulas of the type

x^{i} = s^{i}(x^{0}) ;

i.e. , more precisely x^{i}|s = s^{i} ∘ x^{0}|s , with s^{i} : IR → IR.

Every divided chart on M determines canonically a local fibred chart (x^{0}, x^{i}; x_{0}^{i}) on J(M, 1). We shall always refer to such charts.

Thus we can write

x_{0}^{i} ∘ j_{1}s = ∂_{0}s^{i} ≡ (Ds^{i}) ∘ (x^{0}|s) .

If ϕ ∈ J(M, 1), then the subspace L_{ϕ} ⊂ T\underline{ϕ}M is the span L_{ϕ} = ⟨∂_{0} + ∂_{0}s^{i}∂_{i}⟩ .

2.2. Phase space

Next we introduce the phase space as the velocity space of the manifold M and exhibit the ‘jet-contact structure’ induced by the Lorentz metric.

A motion in M is defined to be a 1–dimensional time–like submanifold s ⊂ M. Let us consider a motion s. By definition, for each x ∈ s, T_{x}s lies inside the

light cone. The 1-jet prolongation js ≡ j_{1}s : s → J(M, 1) is said to be the velocity of s. The length of a vector v ∈ T s is defined to be ∥v∥ = \sqrt{|g(v, v)|} ∈ 𝕃⊗IR.

Lemma 2.1. Given a motion s we have the canonical isomorphism

u_{s} : T s → s × 𝕋 : v ∈ T_{x}s ↦ (x, \frac{±∥v∥}{c}) ∈ s × 𝕋 ,

where we choose + or − according to the time orientation of v.

We define the phase space

UM ≡ U_{1}M ⊂ J(M, 1)

to be the subspace of all 1–jets of motions. In other words, ϕ = js(\underline{ϕ}) ∈ J(M, 1) belongs to UM if and only if L_{ϕ} = T\underline{ϕ}s lies inside the light cone.

We stress that the bundle UM → M is not affine (even in the Minkowski case); its fibres are just Riemannian manifolds. This fact encodes one of the main differences between the Galilei and the Einstein cases.

Lemma 2.2. For each motion ι_{s} : s ↪ M, there is a unique linear map over UM → M

�_{s} : UM ×_{s} T s → T M ,

which makes the following diagram commutative

UM ×_{s} T s  w^{�_{s}}  T M

T s^{4 4 4 47}^{(js∘π_{s},id_{T s})} ^{ȷ ȷ ȷȷ fi}^{T ι_{s}}

The above Lemmas, which are based on the Lorentzian structure of space–time, allow us to recover the jet–contact structure in the Einstein case.

Proposition 2.1. The above maps u_{s} and �_{s}, for all motions s, yield the fibred morphism over M

� : UM → 𝕋^{∗}⊗T M ,

with coordinate expression

� = c α �_{0} = c α (∂_{0} + x_{0}^{i}∂_{i}) , (2.1)

where

α = 1/∥�_{0}∥ = 1/\sqrt{g_{00} + 2g_{0j}x_{0}^{j} + g_{ij}x_{0}^{i}x_{0}^{j}} ∈ 𝕃^{∗} .

Remark 2.1. We have

g∘(� , � ) = c^{2} .

Thus, the map � makes UM ⊂ 𝕋^{∗}⊗T M a fibred submanifold over M.

If s is a motion, then � ∘js : s → 𝕋^{∗}⊗T M is the scaled vector field representing the velocity of s.

The metric allows us to recover the Einstein analogue of the Galilei form dt (but, indeed, not of t).

Proposition 2.2. We have the scaled 1-form

τ : = \frac{g^{♭}}{c^{2}} ∘� : UM → 𝕋⊗T^{∗}M ,

with coordinate expression

τ ≡ τ_{λ} d^{λ} = \frac{α}{c} (g_{0λ} + g_{iλ}x_{0}^{i})d^{λ} .

Remark 2.2. We have � ⌟ τ = 1 , i.e. , in coordinates,

c α (τ_{0} + τ_{h}x_{0}^{h}) = 1 . (2.2)

2.3. Orthogonal splitting

The Lorentzian metric yields the standard splitting of each space–time vector into the parallel and orthogonal components with respect to any given time–like direction. This standard construction allows us to recover the Einstein analogues of the Galilei horizontal and vertical bundles and of the corresponding exact tangent sequence. In this context we establish coordinate formulas which will be largely used in the sequel.

We consider the parallel and orthogonal subspaces, with respect to the map � , of the space–time tangent and cotangent bundles pullbacked over the phase space.

Namely we define the vector bundles over UM

T^{∥}M := {(ϕ, X) ∈ UM ×_{M} T M | X ∈ L_{ϕ}} ,
T^{⊥}M := {(ϕ, X) ∈ UM ×_{M} T M | X ∈ L_{ϕ}^{⊥}} ,

and

T_{∥}^{∗}M := {(ϕ, ω) ∈ UM ×_{M} T^{∗}M | ⟨ω, L_{ϕ}^{⊥}⟩ = 0} ,
T_{⊥}^{∗}M := {(ϕ, ω) ∈ UM ×_{M} T^{∗}M | ⟨ω, L_{ϕ}⟩ = 0} .

Remark 2.3. We have the mutually inverse linear fibred isomorphism over UM

(id, � ) : UM × 𝕋 → T^{∥}M (id, τ) : T^{∥}M → UM × 𝕋 .

Thus, the bundles

T^{∥}M ≃ UM × 𝕋 → UM , T^{⊥}M → UM

will play here, respectively, the roles played by the bundles T × 𝕋 → T, V E → E in the Galilei case.

Proposition 2.3. We have the linear fibred splittings over UM

UM ×_{M} T M = T^{∥}M ⊕_{UM} T^{⊥}M , UM ×_{M} T^{∗}M = T_{∥}^{∗}M ⊕_{UM} T_{⊥}^{∗}M .

Remark 2.4. By restriction of the metric, we obtain the scaled metrics

g_{∥} : UM → 𝕃^{2}⊗T_{∥}^{∗}M ⊗_{UM} T_{∥}^{∗}M , g_{⊥} : UM → 𝕃^{2}⊗T_{⊥}^{∗}M ⊗_{UM} T_{⊥}^{∗}M

and the linear fibred isomorphisms over UM

g_{∥}^{♭} : T^{∥}M → 𝕃^{2}⊗T_{∥}^{∗}M , g_{⊥}^{♭} : T^{⊥}M → 𝕃^{2}⊗T_{⊥}^{∗}M ,
g_{∥}^{♯} : T_{∥}^{∗}M → 𝕃^{∗2}⊗T^{∥}M , g_{⊥}^{♯} : T_{⊥}^{∗}M → 𝕃^{∗2}⊗T^{⊥}M .

Lemma 2.3. The following mutually dual local bases of vector fields and forms are adapted to the above splittings

�_{0} ≡ ∂_{0} + x_{0}^{i}∂_{i} ,  b_{i} ≡ ∂_{i} − cατ_{i}�_{0} = (δ_{i}^{j} − cατ_{i}x_{0}^{j})∂_{j} − cατ_{i}∂_{0} ,
λ^{0} ≡ d^{0} + cατ_{i}ϑ^{i} = cατ_{φ}d^{φ} ,  ϑ^{i} ≡ d^{i} − x_{0}^{i}d^{0} .

The inverse transition maps are

∂_{0} = cατ_{0}�_{0} − x_{0}^{i}b_{i} ,  ∂_{i} = b_{i} + cατ_{i}�_{0} ,
d^{0} = λ^{0} − cατ_{i}ϑ^{i} ,  d^{i} = (δ_{j}^{i} − cατ_{j}x_{0}^{i})ϑ^{j} + x_{0}^{i}λ^{0} .

Lemma 2.4. We have

g^{♭}∘�_{0} = \frac{1}{α^{2}} λ^{0}  g^{♭}∘b_{i} = g_{ij}^{⊥}ϑ^{j} = (g_{iμ} − c^{2}τ_{i}τ_{μ})d^{μ} ,
g^{♯}∘λ^{0} = α^{2}�_{0} ,  g^{♯}∘ϑ^{i} = g_{⊥}^{ij}b_{j} = (g^{iμ} − x_{0}^{i}g^{0μ})∂_{μ} ,

where we have introduced the mutually inverse matrices

g_{ij}^{⊥} : = b_{i} · b_{j} = g_{ij} − c^{2}τ_{i}τ_{j} ,
g_{⊥}^{ij} : = ϑ^{i} · ϑ^{j} = g^{ij} − g^{i0}x_{0}^{j} − g^{j0}x_{0}^{i} + g^{00}x_{0}^{i}x_{0}^{j} .

588 JOSEF JANYŠKA AND MARCO MODUGNO

We shall be involved with further useful identies.

Lemma 2.5. We have

(2.3)  (g^{iμ} − x_{0}^{i}g^{0μ})τ_{μ} = 0
(2.4)  (g_{iμ} − c^{2}τ_{i}τ_{μ})d^{μ} = (g_{ij} − c^{2}τ_{i}τ_{j})ϑ^{j}
(2.5)  (g^{iμ} − x_{0}^{i}g^{0μ})(g_{iν} − c^{2}τ_{i}τ_{ν}) = δ_{ν}^{μ} − c^{2}τ_{ν}τ^{μ} .

Proof. Formula (2.3) follows from

g∘(ϑ^{i}, τ) = 0 .

Formula (2.4) follows from (2.2), which gives

−(g_{ij} − c^{2}τ_{i}τ_{j})x_{0}^{j} = (g_{i0} − c^{2}τ_{i}τ_{0}) .

Formula (2.5) follows from

(g^{iμ} − x_{0}^{i}g^{0μ})(g_{iν} − c^{2}τ_{i}τ_{ν}) = g^{iμ}g_{iν} − c^{2}g^{iμ}τ_{i}τ_{ν} − g^{0μ}g_{iν}x_{0}^{i} + c^{2}g^{0μ}τ_{i}τ_{ν}x_{0}^{i} = = g^{iμ}g_{iν} − c^{2}g^{iμ}τ_{i}τ_{ν} − g^{0μ}g_{iν}x_{0}^{i} + g^{0μ}(g_{0ν} + g_{jν}x_{0}^{j}) − c^{2}g^{0μ}τ_{0}τ_{ν} = δ_{ν}^{μ} − c^{2}τ_{ν}τ^{μ} .

QED

The parallel and the orthogonal projections induced by the Lorentzian metric will provide the Einstein analogues of the exact tangent sequence (1.1) and of its splitting over the jet space (1.2) in the Galilei case, respectively.

Proposition 2.4. We have the linear exact sequence over UM

0 → T^{⊥}M → UM ×_{M} T M ⟶^{λ} T^{∥}M → 0 ,

and its splitting

0 → T^{∥}M ⟶ UM ×_{M} T M ⟶^{ϑ} T^{⊥}M → 0 ,

where the parallel and orthogonal projections

λ = τ ⊗� : UM ×_{M} T M → T^{∥}M , ϑ = 1_{M} − τ ⊗� : UM ×_{M} T M → T^{⊥}M

have the coordinate expressions λ = λ^{0}⊗�_{0} = c^{2}τ_{ν}τ^{μ}d^{ν} ⊗∂_{μ} = cατ_{ν}(δ_{0}^{μ} + δ_{i}^{μ}x_{0}^{i})d^{ν} ⊗∂_{μ} (2.6) and (2.7) ϑ = ϑ^{i}⊗b_{i} = (δ_{ν}^{μ} − c^{2}τ_{ν}τ^{μ})d^{ν} ⊗∂_{μ}

= (δ_{ν}^{μ} − cατ_{ν}(δ_{0}^{μ} + δ_{i}^{μ}x_{0}^{i}))d^{ν} ⊗∂_{μ} = (g^{iμ} − x_{0}^{i}g^{0μ})(g_{iν} − c^{2}τ_{i}τ_{ν})d^{ν} ⊗∂_{μ} .

Corollary 2.1. We have the linear exact sequence over UM

0 → T_{∥}^{∗}M → UM ×_{M} T^{∗}M ⟶^{ϑ^{∗}} T_{⊥}^{∗}M → 0 ,

and its splitting

0 → T_{⊥}^{∗}M ⟶ UM ×_{M} T^{∗}M ⟶^{λ^{∗}} T_{∥}^{∗}M → 0 ,

where the parallel and orthogonal projections

λ^{∗} = � ⊗τ : UM ×_{M} T^{∗}M → T_{∥}^{∗}M , ϑ^{∗} = 1_{M}^{∗} − � ⊗τ : UM ×_{M} T^{∗}M → T_{⊥}^{∗}M

have the coordinate expressions

λ^{∗} = �_{0}⊗λ^{0} , ϑ^{∗} = b_{i}⊗ϑ^{i} .

The physical interpretation of our maps can be achieved by reading the splitting of velocity of motions in our notation.

An observer is defined to be a section

o : M → UM .

Then � ∘ o : M → 𝕋^{∗}⊗T M is the scaled vector field representing the velocity of the observer o.

By using the parallel and the orthogonal projections associated with the observer o, we obtain the following splitting.

Proposition 2.5. We have

� ∘js = (� ∘js)^{∥} + (� ∘js)^{⊥} = δ (� ∘o|s + β̄) ,

where

δ = (� ∘js)⌟(τ ∘o|s) = \frac{g(� ∘o|s, � ∘js)}{c^{2}} = \frac{c}{\sqrt{c^{2} − ∥β̄∥^{2}}} : s → IR

and

β̄ = \frac{c^{2}((� ∘js)⌟ (ϑ∘o|s))}{g(� ∘o, � ∘js)} : s → 𝕋^{∗}⊗T^{⊥}M .

The vector β̄ ∈ 𝕋^{∗} ⊗ L_{o}^{⊥} can be interpreted as the velocity of the motion s observed by the observer o. We have

δ > 1 , ∥β̄∥ = \frac{c \sqrt{δ^{2} − 1}}{δ} < c ,

which express, respectively, the time dilatation and the fact that the observed velocity of a motion is smaller than c.

2.4. Vertical bundle of the phase space

We have noticed that in the Einstein case the fibres of the phase space are just Riemannian manifolds; in spite of this fact, the Lorentz metric allows us to recover important technical results, which in the Galilei case follow from the affine structure of the phase space.

We shall be involved with the vertical tangent bundle V UM of π_{0}^{1} : UM → M.

Lemma 2.6. The vertical prolongation of �

V � : V UM → V (𝕋^{∗}⊗T M) ≃ (𝕋^{∗}⊗T M) ×_{M} (𝕋^{∗}⊗T M)

factorises through a fibred isomorphism over UM

v^{⊥} : V UM → 𝕋^{∗}⊗T^{⊥}M ,

according to the commutative diagram

\begin{matrix}V UM w^{V �}_{[ [ [[]}_{v^{⊥}} \\ (𝕋^{∗}⊗T M) ×_{M} (𝕋^{∗}⊗T M) \end{matrix} ; 𝕋^{∗}⊗T^{⊥}M^{h h hh j}

We have the coordinate expressions

v^{⊥} = c α d_{0}^{i}⊗b_{i} , v^{⊥}^{−1} = \frac{1}{c α}ϑ^{i}⊗∂_{i}^{0} . (2.8)

Proof. Let ϕ ∈ UM and X ∈ V_{ϕ}UM. Then X can be expressed as X = dσ(0), where σ : IR → U_{1\underline{ϕ}}M, is a vertical curve such that σ(0) = ϕ. In virtue of the definition of � , we have c^{2} = g∘(� ∘σ, � ∘σ); hence, by taking into account that σ is vertical, we can write

0 = g(T (� ∘σ)(0), (� ∘σ)(0)) = g((V � ∘dσ)(0), (� ∘σ)(0)) = g(V � (X), � (ϕ)) .

Eventually, the coordinate expression (2.8) follows from (2.1).  QED

The isomorphism V UM ≃ 𝕋^{∗}⊗T^{⊥}M is Einstein the analogue of V JE ≃ JE ×_{E} 𝕋^{∗}⊗V E.

The map v^{⊥}^{−1} can be regarded as a scaled soldering 1-form

Φ : = v^{⊥}^{−1} : UM → 𝕋⊗T^{∗}M ⊗_{UM} V UM .

This form is the Einstein analogue of ϑ : JE → 𝕋⊗T^{∗}E⊗_{JE}V JE.

2.5. Connections

In the Einstein case we can recover the mapping from space–time connections to connections on the phase space. However, we do not know the analogue of Corollary 1.1. In the Galilei case the connection on the phase space is affine; on the contrary, in the Einstein case, the connection on the phase space is highly non linear. However, our formalism allows us to handle also this case without problems.

For a connection K on T M we can repeat the considerations of the Galilei case.

A connection Γ on UM can be represented, equivalently, by a tangent valued form, or by a vector valued form

Γ : UM → T^{∗}M ⊗_{UM} T UM , v^{⊥}∘ν_{Γ} : UM → T^{∗}UM ⊗_{UM} (𝕋^{∗}⊗T^{⊥}M) ,

with coordinate expressions

Γ = d^{φ}⊗(∂_{φ} + Γ_{φ}i0∂_{i}^{0}) , v^{⊥}∘ν_{Γ} = cα(d_{0}^{i} − Γ_{φ}_{0}^{i}d^{φ})⊗b_{i} , (2.9)

Γ_{φ}i0 ∈ C^{∞}(UM).

We can define the torsion of Γ as the scaled vertical valued 2-form (see Introduction)

[Γ, Φ] : UM → 𝕋⊗∧^{2}T^{∗}M ⊗_{UM} V UM .

Theorem 2.1. For any linear connection K on T M the map

v^{⊥}∘ν_{Γ} = ϑ ∘ ν_{K} ∘ T � (2.10)

given by the following diagram

T UM u_{(π_{UM} ,T � )}  w^{ν_{Γ}}  V UM  w^{v^{⊥}}  𝕋^{∗}⊗T^{⊥}M

UM ×_{M} T (𝕋^{∗}⊗T M)  w^{(id_{UM} ×ν_{K})}  UM ×_{E} (𝕋^{∗}⊗T M) u^{ϑ}

turns out to be a connection on the bundle UM → M with coordinate expression

Γ_{φ}i0 = K_{φ}^{i}_{j}x_{0}^{j} + K_{φ}^{i}_{0} − x_{0}^{i}(K_{φ}^{0}_{j}x_{0}^{j} + K_{φ}^{0}_{0}) . (2.11)

Thus, we have obtained a map

χ : K ↦ Γ .

Proof. It can be proved in coordinates by using (1.6), (2.1) and (2.9). QED

Corollary 2.2. The metric g yields the gravitational connections on T M and UM

K^{♮} : = κ , v^{⊥}∘ν_{Γ^{♮}} : = ϑ∘ν_{K^{♮}} ∘T � ,

where κ is the Levi–Civita connection with the Christoffel symbols

κ_{φψ}^{σ} = −\frac{g^{στ}}{2} (∂_{φ}g_{τψ} + ∂_{ψ}g_{τφ} − ∂_{τ} g_{φψ}) . (2.12)

2.6. Curvature

In the Einstein case the curvatures fulfill relations analogous to the Galilei case.

The curvatures (see Introduction) of a connection K on T M and of a connection Γ on UM are, respectively, the 2–forms

R_{K} = \frac{1}{2}[K, K]:T M → ∧^{2}T^{∗}M ⊗_{M} T M, R_{Γ} = \frac{1}{2}[Γ, Γ]:UM → ∧^{2}T^{∗}M ⊗_{UM} V UM,

whose coordinate expressions can be easily computed by (0.1).

Moreover, we obtain

v^{⊥}∘R_{Γ} : UM → ∧^{2}T^{∗}M ⊗_{UM} (𝕋^{∗}⊗T^{⊥}M) ,

with coordinate expression

v^{⊥}∘R_{Γ} = c α (R_{Γ})_{λμ}i0 d^{λ} ∧ d^{μ}⊗b_{i} .

Theorem 2.2. If Γ is the connection on UM induced by a linear connection K on T M, then the following diagram commutes

UM ×_{M} 𝕋^{∗}⊗T M w^{id_{UM} ×R_{K}} UM ×_{M} (∧^{2}T^{∗}M ⊗_{M} 𝕋^{∗}⊗T M) u_{id_{∧^{2}T^{∗}M} ⊗ϑ}

UM u^{(id_{UM} , � )}  w^{v^{⊥}∘R_{Γ}}  ∧^{2}T^{∗}M ⊗_{UM} (𝕋^{∗}⊗T^{⊥}M)

i.e.,

R_{Γ} = ϑ∘R_{K} ∘ � , (2.13)

i.e. in coordinates

(R_{Γ})_{λμ}i0 = (R_{K})_{λμ}^{i}_{j}x_{0}^{j} + (R_{K})_{λμ}^{i}_{0} − x_{0}^{i}((R_{K})_{λμ}^{0}_{j}x_{0}^{j} + (R_{K})_{λμ}^{0}_{0}) .

Proof. It can be proved in coordinates by using (0.1), (2.1), (2.7) and (2.11).

2.7. Second order connection In the Einstein case we can also recover the analogues of the Galilei second

order jet–contact structure and connection. This result is interesting because the second order connection provides the equation of inertial motion of particles.

Let us denote by U_{2}M ⊂ J_{2}(M, 1) the subspace of all 2–jets of motions.

Lemma 2.7. For each motion ι_{s} : s ↪ M, there is a unique linear map over U_{2}M → UM

�_{s2} : U_{2}M ×_{s} T s → T UM

such that the following diagram commutes

U_{2}M ×_{s} T s  w^{�_{s2}}  T UM

T s^{4 4 4 47}^{(j_{2}s∘π_{s},id_{T s})} ^{ȷ ȷ ȷȷ fi}^{T js}

Proposition 2.6. The maps u_{s} and �_{s2} yield the fibred inclusion over UM

�_{2} : U_{2}M → 𝕋^{∗}⊗T UM ,

with coordinate expression

�_{2} = c α (∂_{0} + x_{0}^{i}∂_{i} + x_{00}^{i}∂_{i}^{0}) .

Corollary 2.3. Thus U_{2}M ⊂ 𝕋^{∗} ⊗ T UM turns out to be exactly the fibred submanifold over UM, which makes the following diagram commutative

U_{2}M  w^{�_{2}} u_{π_{1}^{2}}  𝕋^{∗}⊗T UM u_{(π_{UM} ,id⊗T π_{0}^{1})}

UM  w^{(id,� )} u_{π^{1}}  UM ×_{M} (𝕋^{∗}⊗T M) u_{id⊗τ}

M  w^{1_{𝕋}}  𝕋^{∗}⊗𝕋

A second order connection on M is defined to be a section

γ : UM → U_{2}M .

Corollary 2.4. By considering the inclusion �_{2}, every second order connections γ can be characterised as a scaled vector field

γ : UM → 𝕋^{∗}⊗T UM ,

which makes the following diagram commutative

UM  w^{γ} u_{id}  𝕋^{∗}⊗T UM u_{(π_{UM} ,id⊗T π_{0}^{1})}

UM  w^{(id,� )} u_{π^{1}}  UM ×_{M} (𝕋^{∗}⊗T M) u_{id⊗τ}

M  w^{1_{𝕋}}  𝕋^{∗}⊗𝕋

The above result is fully analogous to the Galilei case; however, now we have no fibring of UM over time, hence we cannot properly say that γ is a connection on UM over time.

The coordinate expression of a second order connection is of the type

γ = c α (∂_{0} + x_{0}^{i}∂_{i} + γ_{0}i0∂_{i}^{0}) , γ_{0}i0 ∈ C^{∞}(UM) . (2.14)

Theorem 2.3. If Γ is a connection on UM, then

γ : = � ⌟ Γ : UM → 𝕋^{∗}⊗T UM

is a second order connection with coordinate expression (2.14), where

γ_{0}i0 = Γ_{0}i0 + Γ_{j}i0x_{0}^{j} .
Proof. It follows from (2.1) and (2.9).  QED

Corollary 2.5. The metric g yields the second order gravitational connection

γ^{♮} : = � ⌟ Γ^{♮} .

2.8. Distinguished 2–forms Next, we study the 2-forms induced on the phase space by the connections.

Again, in the Einstein case the Lorentzian metric allows us to recover objects and relations which in the Galilei case are provided by the fibring over time.

Let us consider a linear connection K and recall the canonical form σ (see (0.3)). Then, we define the 2–form

Ω(g, K) : = ν_{K} ¯∧σ : 𝕋^{∗}⊗T M → 𝕋^{∗}⊗𝕃^{2}⊗∧^{2}T^{∗}(𝕋^{∗}⊗T M)

with coordinate expression (see (1.5), (0.3))

Ω(g, K) = ν_{K} ¯∧σ = g_{λμ}u^{0}⊗(ḋ_{0}^{λ} − K_{φ}^{λ}_{ψ}ẋ_{0}^{ψ}d^{φ}) ∧ d^{μ} .

Remark 2.5. Let us consider the maps

π_{1} : UM ⊗_{M} (𝕋^{∗}⊗T M) → UM , π_{2} : UM ⊗_{M} (𝕋^{∗}⊗T M) → 𝕋^{∗}⊗T M .

Then, we can write on UM ⊗_{M} (𝕋^{∗}⊗T M)

σ ≃ σ∘π_{2} = (λ + ϑ)∘π_{1} ≃ λ + ϑ .

CLASSICAL PARTICLE PHASE SPACE 595

Therefore, we obtain the splitting of Ω(g, K) over UM ⊗_{M} (𝕋^{∗}⊗T M) into parallel and orthogonal components

Ω(g, K) ≡ Ω^{∥}(g, K) + Ω^{⊥}(g, K) : = ν_{K} ¯∧λ + ν_{K} ¯∧ϑ ,

where

Ω^{∥}(g, K) , Ω^{⊥}(g, K) : UM ×_{M} (𝕋^{∗}⊗T M) → 𝕋^{∗}⊗𝕃^{2}⊗∧^{2}T^{∗}(𝕋^{∗}⊗T M)

have the coordinate expressions

Ω^{∥}(g, K) = c^{2}τ_{λ}τ_{μ}u^{0}⊗(ḋ_{0}^{λ} − K_{φ}^{λ}_{ψ}ẋ_{0}^{ψ}d^{φ}) ∧ d^{μ} ,
Ω^{⊥}(g, K) = (g_{λμ} − c^{2}τ_{λ}τ_{μ})u^{0}⊗(ḋ_{0}^{λ} − K_{φ}^{λ}_{ψ}ẋ_{0}^{ψ}d^{φ}) ∧ d^{μ} .

The above forms on the tangent space of space–time induce via pullback analogous forms on the phase space.

Remark 2.6. We obtain on UM the 2-form ω(g, K) = ω^{∥}(g, K) + ω^{⊥}(g, K) : = �^{∗}Ω(g, K) : UM → 𝕋^{∗}⊗𝕃^{2}⊗∧^{2}T^{∗}UM where

ω^{∥}(g, K) : = (id_{UM}, � )^{∗}Ω^{∥}(g, K) , ω^{⊥}(g, K) : = (id_{UM}, � )^{∗}Ω^{⊥}(g, K) .

In coordinates (2.15) ω(g, K) =

= c α ((g_{iμ}−c^{2} τ_{i}τ_{μ})d_{0}^{i}−(\frac{1}{2} c α ∂_{φ}(α^{−2})τ_{μ}+g_{λμ}(K_{φ}^{λ}_{0}+K_{φ}^{λ}_{j}x_{0}^{j}))d^{φ})∧d^{μ} .

and

(2.16) ω^{∥}(g, K) = −c α (\frac{1}{2} c α ∂_{φ}(α^{−2})τ_{μ} + c^{2}τ_{λ}τ_{μ}(K_{φ}^{λ}_{0} + K_{φ}^{λ}_{j}x_{0}^{j}))d^{φ} ∧ d^{μ}

(2.17) ω^{⊥}(g, K) =

= c α (g_{iμ}−c^{2}τ_{i}τ_{μ})(d_{0}^{i}−(K_{φ}^{i}_{0}+K_{φ}^{i}_{j}x_{0}^{j}−x_{0}^{i}(K_{φ}^{0}_{0}+K_{φ}^{0}_{j}x_{0}^{j}))d^{φ})∧d^{μ} . On the other hand a connection Γ on UM and the metric g yield the scaled

2–form on UM

Ω(g, Γ) : = (v^{⊥}∘ν_{Γ})¯∧ϑ : UM → 𝕋^{∗}⊗𝕃^{2}⊗∧^{2}T^{∗}UM ,

with coordinate expression

(2.18)  Ω(g, Γ) = c α (g_{iμ} − c^{2}τ_{i}τ_{μ}) (d_{0}^{i} − Γ_{φ}i0d^{φ}) ∧ d^{μ}

= c α (g_{ij} − c^{2}τ_{i}τ_{j}) (d_{0}^{i} − Γ_{φ}i0d^{φ}) ∧ ϑ^{j} (2.19) = c α (g_{ij} − c^{2}τ_{i}τ_{j}) (d_{0}^{i} − γ_{0}i0d^{0} − Γ_{h}i0ϑ^{h}) ∧ ϑ^{j} , (2.20) where γ : = � ⌟ Γ.

596 JOSEF JANYŠKA AND MARCO MODUGNO

Theorem 2.4. If Γ is the connection on UM induced by a linear connection K on T M then Ω(g, Γ) = ω^{⊥}(g, K) . (2.21)

Proof. It can be proved by using (2.11), (2.17) and (2.19).  QED

Theorem 2.5. There is a unique second order connection γ such that

γ⌟ Ω(g, Γ) = 0 .

Namely,

γ = � ⌟ Γ : UM → 𝕋^{∗}⊗T UM .

Proof. It follows from (2.1), (2.14) and (2.20). QED 2.9. Distinguished metric 2–forms Eventually, we specialise the results of the above section by referring to the

gravitational connections K^{♮} ≡ κ and Γ^{♮} induced by g. We analyse the contact structure on UM via the form c^{2}τ. This structure is expected to be important for our covariant quantisation programme in the Einstein background.

We start by studying the gravitational 2–form Ω(g, κ) on 𝕋^{∗}⊗T M.

Remark 2.7. The metric g yields the scaled Liouville 1–form

θ : 𝕋^{∗}⊗T M → 𝕋^{∗}⊗𝕃^{2}⊗T^{∗}(𝕋^{∗}⊗T M)

defined by

θ(X) : = g(T π_{M}(X) , π_{𝕋^{∗}⊗T M}(X)) , ∀X ∈ T (𝕋^{∗}⊗T M) .

In coordinates

θ = g_{μλ}ẋ_{0}^{λ}u^{0}⊗d^{μ} .

Proposition 2.7. The 2–form Ω(g) : = dθ : 𝕋^{∗}⊗T M → 𝕋^{∗}⊗𝕃^{2}⊗∧^{2}T^{∗}(𝕋^{∗}⊗T M) , (2.22) with coordinate expression

Ω(g) = u^{0}⊗(∂_{φ}g_{μλ}ẋ_{0}^{λ}d^{φ} + g_{μλ}ḋ_{0}^{λ}) ∧ d^{μ} ,

is a symplectic form on 𝕋^{∗}⊗T M and Ω(g) = Ω(g, κ) . (2.23)

Next, we study the gravitational 2–form on UM

Ω^{♮} : = Ω(g, Γ^{♮}) .

Lemma 2.8. We have ω^{∥}(g, κ) = 0 . (2.24)

Proof. It can be proved in coordinates, by using (2.16) and (2.12).  QED

Proposition 2.8. We obtain

ω(g) : = �^{∗}Ω(g) = ω(g, κ) .

Proof. It follows from (2.21), (2.24), (2.22).

Lemma 2.9. The form

η : = c^{2}τ ∧ ω(g) ∧ ω(g) ∧ ω(g) : UM → 𝕋^{∗4}⊗𝕃^{8}⊗∧^{7}T^{∗}UM ,

with coordinate expression

c^{2}τ ∧ ω(g) ∧ ω(g) ∧ ω(g) = 6 c^{4}α^{4} |g| d^{0} ∧ d^{1} ∧ d^{2} ∧ d^{3} ∧ d_{0}^{1} ∧ d_{0}^{2} ∧ d_{0}^{3} ,

is a scaled volume form on UM.

Lemma 2.10. We obtain �^{∗}θ = c^{2}τ .

Theorem 2.6. The form of UM

Ω^{♮} : = Ω(g, Γ^{♮}) = ω(g, κ) = ω(g) = c^{2} dτ .

is the scaled contact 2–form generated by the contact 1–form c^{2}τ.

Indeed, the 2–form Ω^{♮} is our candidate for the quantisation programme in the Einstein case along the lines of [4], [5]. The fact that this form is closed is interesting for our aim. We stress that the form Ω^{♮} has a natural global potential; an analogous result is not true in the Galilei case.

2.10. Electromagnetic field and geometric structure So far, we have assumed on space–time the gravitational structure associated

with the Lorentzian metric g. Next, we introduce the electromagnetic field.

Assupption E.3. We assume the electromagnetic field to be a closed scaled 2-form on M

F : M → (𝕋^{∗}⊗𝕃^{3/2}⊗𝕄^{1/2})⊗∧^{2}T^{∗}M .

We denote the local potential of F by A : M → (𝕋^{∗}⊗𝕃^{3/2}⊗𝕄^{1/2})⊗T^{∗}M. Thus, by definition, we set

2dA = F .

We define the ‘universal’ electric and ‘universal’ magnetic fields to be the scaled forms on UM

E : = −� ⌟F : UM → (𝕋^{∗2}⊗𝕃^{3/2}⊗𝕄^{1/2})⊗T_{⊥}^{∗}M ,
B : = F + 2τ ∧ E : UM → (𝕋^{∗}⊗𝕃^{3/2}⊗𝕄^{1/2})⊗∧^{2}T_{⊥}^{∗}M .

Thus, we can write

F = −2τ ∧ E + B.

Hence, the electric and magnetic fields associated with an observer o : M → UM are defined as the pullback forms

o^{∗}E : M → (𝕋^{∗2}⊗𝕃^{3/2}⊗𝕄^{1/2})⊗T^{∗}M , o^{∗}B : M → (𝕋^{∗}⊗𝕃^{3/2}⊗𝕄^{1/2})⊗∧^{2}T^{∗}M .

We stress that the universal electric field carries the full information of the electromagnetic field.

Remark 2.8. We have the coordinate expressions

F = 2F_{0j}d^{0} ∧ d^{j} + F_{ij}d^{i} ∧ d^{j} = −\frac{2}{cα}E_{j}λ^{0} ∧ ϑ^{j} + B_{ij}ϑ^{i} ∧ ϑ^{j} ,

and

E = E_{i}ϑ^{i} = E_{j}d^{j} + E_{0}d^{0} , B = B_{ij}ϑ^{i} ∧ ϑ^{j} = B_{ij}d^{i} ∧ d^{j} + 2B_{0j}d^{0} ∧ d^{j} ,

where

E_{j} = −cα(F_{0j} + F_{hj}x_{0}^{h}) E_{0} = −cαF_{h0}x_{0}^{h} ,
B_{ij} = F_{ij} + τ_{i}E_{j} − τ_{j}E_{i} B_{0j} = −B_{hj}x_{0}^{h} .

Therefore, in a chart adapted to an observer o, the coordinate expressions of the observed electric and magnetic fields are

o^{∗}E = −\frac{c}{\sqrt{g_{00}}} F_{0j}d^{j}

o^{∗}B = (F_{ij} − \frac{1}{g_{00}} (g_{i0}F_{0j} − g_{j0}F_{0i}))d^{i} ∧ d^{j} .

Next, we show that the electromagnetic field can be naturally incorporated into the gravitational structures of the phase space.

Namely, let us consider the following gravitational objects on the phase space UM:

Γ^{♮} : UM → T^{∗}M ⊗_{UM} T UM ,  γ^{♮} : UM → 𝕋^{∗}⊗T UM ,
c^{2}τ : UM → 𝕋^{∗}⊗𝕃^{2}⊗T^{∗}UM ,  Ω^{♮} : UM → 𝕋^{∗}⊗𝕃^{2}⊗∧^{2}T^{∗}UM ,

which fulfill the following structural relations

γ^{♮} = � ⌟ Γ^{♮} ,  Ω^{♮} = ν_{Γ^{♮}} ¯∧ϑ ,  γ^{♮}⌟ Ω^{♮} = 0 ,  c^{2}dτ^{♮} = Ω^{♮} .

We are looking for total objects obtained correcting the gravitational objects by an electromagnetic term, in such a way to preserve the above relations.

For this purpose we need a suitable coupling constant. So, we consider a particle with a given mass and charge

m ∈ 𝕄 , q ∈ 𝕋^{∗}⊗𝕃^{3/2}⊗𝕄^{1/2} ,

and refer to the coupling constant

\frac{q}{m} ∈ 𝕋^{∗}⊗𝕃^{3/2}⊗𝕄^{∗1/2}

We start from the obvious coupling of the electromagnetic field with the gravitational contact 2-form on UM.

Accordingly, we define the total 2-form to be

Ω : = Ω^{♮} + \frac{q}{2mc}F : UM → 𝕋^{∗}⊗𝕃^{2}⊗∧^{2}T^{∗}UM .

Here, the factor \frac{1}{2} is chosen in such a way to recover the standard normalization in practical formulas.

Of course, we obtain

dΩ = 0 .

However, while Ω^{♮} has a natural and global potential, the potential of Ω is defined up to a gauge and in general exists only locally. Indeed, we obtain locally

d(c^{2}τ) ≡ d(c^{2}τ^{♮} + \frac{q}{mc}A) = Ω .

We can write locally

(c^{2}τ^{♮} + \frac{q}{mc}A) ∧ Ω ∧ Ω ∧ Ω = (c^{2}τ^{♮} + \frac{q}{mc}A) ∧ Ω^{♮} ∧ Ω^{♮} ∧ Ω^{♮} .

This 7–form needs not to be a local volume form; for instance, if A = −g_{0μ}d^{μ}, then this 7–form vanishes. Hence, Ω needs not to be locally a contact 2–form.

Next, we look for the total second order connection on UM.

Lemma 2.11. A section γ : UM → 𝕋^{∗}⊗T UM is a second order connection if and only if γ − γ^{♮} is a section γ − γ^{♮} : UM → 𝕋^{∗}⊗V UM.

Theorem 2.7. There is a unique second order connection γ such that

γ⌟ Ω = 0 .

Namely, γ is given by γ = γ^{♮} + γ^{e} , where

γ^{e} = \frac{q}{mc}v^{⊥}^{−1}∘g^{♯}∘E ,

with coordinate expression

γ^{e} = −\frac{q}{mc}(g^{iμ} − x_{0}^{i}g^{0μ})(F_{0μ} + F_{jμ}x_{0}^{j})∂_{i}^{0} = \frac{q}{mc^{2}α}g_{⊥}^{ij}E_{j}∂_{i}^{0} .

Proof. In virtue of the expressions of Ω^{♮}, γ^{♮} and F, we have

γ^{♮}⌟ F = � ⌟ F , γ^{e}⌟ Ω^{♮} = g^{♭}∘v^{⊥}∘γ^{e} .

Then, we can write

0 = γ⌟ Ω = γ^{♮}⌟ Ω^{♮} + \frac{q}{2mc}γ^{♮}⌟ F + γ^{e}⌟ Ω^{♮} + \frac{q}{2mc}γ^{e}⌟ F

= 0 + \frac{q}{2mc}� ⌟ F + \frac{1}{2}g^{♭}∘v^{⊥}∘γ^{e} + 0 . QED

600 JOSEF JANYŠKA AND MARCO MODUGNO

Remark 2.9. Thus, we have recovered the Lorentz force

f : = −\frac{q}{mc} ∘g^{♯}∘E : UM → 𝕋^{∗}⊗T^{⊥}M ⊂ UM ×_{M} (𝕋^{∗2}⊗T M) ,

with coordinate expression

f = −\frac{qα}{m}(g^{iμ} − x_{0}^{i}g^{0μ})(F_{0μ} + F_{jμ}x_{0}^{j})b_{i} = \frac{q}{mc}g_{⊥}^{ij}E_{j}b_{i} ,

by a natural procedure as a byproduct of the coupling between gravitational and electromagnetic field.

Remark 2.10. The law of motion

∇_{γ}js : = j_{2}s − γ∘js = 0

for the unknown motion s ⊂ M reads

v^{⊥}∘∇_{γ^{♮}}js = f ∘js .

Eventually, we look for the total connection on UM. Let us set

g^{♯}^{2} : T^{∗}M ⊗_{M} T^{∗}M → 𝕃^{∗2}⊗T^{∗}M ⊗_{M} T M : α⊗β ↦ α⊗g^{♯}(β) .

Lemma 2.12. We have the following sections

Γ_{E}^{e} : = −\frac{q}{2mc}v^{⊥}^{−1}∘ϑ∘g^{♯}^{2}∘(2τ ∧ E) : UM → T_{∥}^{∗}M ⊗_{M} V UM ,
Γ_{B}^{e} : = −\frac{q}{2mc}v^{⊥}^{−1}∘g^{♯}^{2}∘B : UM → T_{⊥}^{∗}M ⊗_{M} V UM ,

with coordinate expressions

Γ_{E}^{e} = \frac{q}{2mc^{2}α}g_{⊥}^{ij}E_{j}τ ⊗∂_{i}^{0} , Γ_{B}^{e} = −\frac{q}{2mc^{2}α}g_{⊥}^{ij}B_{hj}ϑ^{h}⊗∂_{i}^{0} .

Moreover, we obtain

� ⌟ Γ_{E}^{e} = γ^{e} , −(v^{⊥}∘Γ_{E}^{e})¯∧ϑ = −\frac{q}{2mc}2τ ∧ E ,

� ⌟ Γ_{B}^{e} = 0 , −(v^{⊥}∘Γ_{B}^{e})¯∧ϑ = \frac{q}{2mc}B .

Proof. It follows by a computation in coordinates, by using the base (λ^{0}, ϑ^{i}), Lemma (2.3), and Lemma (2.4). QED

Let us consider a section

H : UM → T^{∗}M ⊗_{UM} V UM .

We define the map

A(v^{⊥}∘H) : UM → 𝕋^{∗}⊗∧^{2}T_{⊥}^{∗}M

by means of the composition

UM ⟶^{H} T^{∗}M ⊗_{UM} V UM ⟶^{ϑ^{∗}⊗(g^{♭}∘v^{⊥})} 𝕋^{∗}⊗T_{⊥}^{∗}M ⊗_{UM} T_{⊥}^{∗}M ⟶^{Alt} 𝕋^{∗}⊗∧^{2}T_{⊥}^{∗}M.

Lemma 2.13. The following conditions are equivalent

� ⌟ H = 0 , (v^{⊥}⌟ H)¯∧ϑ = 0 ;

H : UM → T_{⊥}^{∗}M ⊗_{UM} V UM ⊂ T^{∗}M ⊗_{UM} V UM , A(v^{⊥}∘H) = 0 ;

H = H_{j}^{i}ϑ^{j} ⊗∂_{i}^{0} , g_{ih}^{⊥}H_{j}^{h} = g_{ih}^{⊥}H_{j}^{h} .

Proof. It follows by a computation in coordinates, by using the base (λ^{0}, ϑ^{i}).

QED

Theorem 2.8. There is a unique connection Γ : UM → T^{∗}M ⊗_{UM} T UM, such

that

� ⌟ Γ = γ ,  (v^{⊥}∘ν_{Γ})¯∧ϑ = Ω ,  A(v^{⊥}∘(Γ − Γ^{♮})) = 0 .

Namely, Γ = Γ^{♮} + Γ^{e} , where

Γ^{e} = Γ_{E}^{e} + Γ_{B}^{e} = −\frac{q}{2mc}v^{⊥}^{−1}∘g^{♯}^{2}∘(F + 4τ ∧ E) ,

i.e. , in coordinates

Γ^{e} = \frac{q}{2mc^{2}α}g_{⊥}^{ij}(E_{j}τ − B_{hj}ϑ^{h})⊗∂_{i}^{0} .

Proof. It follows from the above Lemmas and from the expressions of γ and

Ω.  QED

References

1. D. Canarutto, A. Jadczyk, M. Modugno: Quantum mechanics of a spin particle in a curved space–time with absolute time, Rep. on Math. Phys., 36, 1 (1995), 95–140.

2. D. Canarutto, M. Modugno: General relativistic dynamics and structures observed by a frame of references, Rend. Sem. Mat. Univers. Politecn. Torino., 41 (1983), 65–93.

3. J. W. Gray: Some global properties of contact structures, Ann. of Math., 69, 2, (1959), 421-450.

4. A. Jadczyk, M. Modugno: An outline of a new geometrical approach to Galilei general relativistic quantum mechanics, in C.N. Yang, M.L. Ge and X. W. Zhou editors, Proc. XXI Int. Conf. on Differential geometrical methods in theoretical physics, Tianjin 5-9 June 1992, 543-556, Singapore, 1992, World Scientific.

5. A. Jadczyk, M. Modugno: Galilei general relativistic quantum mechanics, book manuscript, 1994.

6. J. Janyška: Natural 2-forms on the tangent bundle of a Riemannian manifold, Proceedings of the Winter School Geometry and Topology (Srní, 1992), Supplemento ai Rendiconti del Circolo Matematico di Palermo, Serie II 32 (1993), 165–174.

7. J. Janyška: Remarks on symplectic and contact 2–forms in relativistic theories, Bollettino U.M.I. (7) 9–B (1995), 587–616.

8. I. Kolář, P. Michor, J. Slovák: Natural operations in differential geometry, Springer-Verlag, 1993.

9. D. Krupka, J. Janyška: Lectures on Differential Invariants, Folia Fac. Sci. Nat. Univ. Purkynianae Brunensis, Brno, 1990.

10. P. Libermann: Legendre foliations on contact manifolds, Diff. Geom. Appl. 1 (1991), 57–76.

11. P. Libermann, Ch. M. Marle: Symplectic Geometry and Analytical Mechanics, Reidel Publ., Dordrecht, 1987.

12. L. Mangiarotti: Mechanics on a Galilean manifold, Riv. Mat. Univ. Parma (4) 5 (1979), 1–15.

13. M. Modugno: Formulation of analytical mechanics in general relativity, Ann. Inst. Henri Poincaré, Section A: Physique théorique XXI (1974), 147–174.

14. M. Modugno: Torsion and Ricci tensor for non linear connections, Diff. Geom. Appl., 1, 2 (1991), 177-192.

15. M. Modugno, A. M. Vinogradov: Some variations on the notions of connection, Annali di Matematica pura ed Applicata (IV), Vol. CLXVII (1994), 33–71.

Department of Mathematics, Masaryk University Janáčkovo nám 2a, 662 95 Brno, Czech Republic

Department of Applied Mathematics “G. Sansone” Via S. Marta 3, 50139 Florence, Italy

E–mail addresses : janyska@math.muni.cz, modugno@ingfi1.ing.unifi.it