IJMMS 2003:27, 1693–1702
PII. S0161171203211145
http://ijmms.hindawi.com
© Hindawi Publishing Corp.
SYNGE-BEIL AND RIEMANN-JACOBI JET STRUCTURES
WITH APPLICATIONS TO PHYSICS
VLADIMIR BALAN
Received 6 November 2002
In the framework of geometrized first-order jet approach, we study the SyngeBeil generalized Lagrange jet structure, derive the canonic nonlinear and Cartan
connections, and infer the Einstein-Maxwell equations with sources; the classical
ansatz is emphasized as a particular case. The Lorentz-type equations are described and the attached Riemann-Jacobi structures for two certain uniparametric
cases are presented.
2000 Mathematics Subject Classification: 58A20, 58A30, 53B15, 53B50, 53C80.
1. Preliminaries. Let (T , hαβ ) and (M, γij ) be two Ꮿ∞ pseudo-Riemannian
manifolds of dimensions m and n, respectively. We denote by ζ = (E = j 1 (T ,
M), π , T × M) the first-order jet bundle of mappings ϕ : T → M, with the local
coordinates
α i A
t , x , y (α,i,A)∈I∗ ≡ y µ µ∈I .
(1.1)
Throughout the paper, we consider the sets
I = Ih ∪ Iv ,
Ih = Ih1 ∪ Ih2 ,
Iv = m + n + 1, N,
Ih1 = 1, m,
I∗ = Ih1 × Ih2 × Iv ,
Ih2 = m + 1, m + n,
N = m + n + mn;
(1.2)
the indices will implicitly take the values
α, β, . . . ∈ Ih1 ,
i, j, . . . ∈ Ih2 ,
A, B, . . . ∈ Iv ,
For A = m + n + n(i − m − 1) + α, we will denote A ≡
λ, µ, . . . ∈ I.
i
α
(1.3)
i
and y A ≡ x (α) =
∂x i /∂t α .
We endow E with the sub-Riemannian Synge-Beil metric (see [9])
g̃AB ≡ g̃
(αi )(βj )
= hαβ (t)gij (t, x, y),
(1.4)
where
gij (t, x, y) = γij (x) + εUi (t, x, y)Uj (t, x, y),
∀i, j ∈ Ih2 , ε ∈ {±1},
(1.5)
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VLADIMIR BALAN
and Ui (t, x, y) is a distinguished 1-form on E (see [1]). We call (E, g̃) the SyngeBeil (SB) jet model. The inverse of gij is g ij = γ ij − εΘU i U j , where U i = γ ij Uj ,
Θ = (1 + U∗ )−1 , and the star index denotes transvection with U i .
We remark the important particular Synge-Beil uniparametric (SBU) autonomous normalized case, where m = 1, s = t 1 = t, and h11 = 1, for which we can
use the Finsler-Lagrange tangent space notations from [5]. Shifting the indices
i not
left by one unit (hence, Ih = 1, n, Iv = n + 1, 2n), we have y A ≡ y (1) = y i . In
2
this case, considering
1/2
Ui = k 1 − n−2 (x, y)
yi ,
k > 0,
(1.6)
we encounter three important extensively studied cases.
(I) The Synge classical framework (see [10]), obtained for ε = 1 and k = 1,
where yi = γij y j , n(x, y) is the refraction index of relativistic optics (see
[7, 9]), and the direction y = X(x) is provided by a vector field X ∈ ᐄ(M).
(II) If the potentials Ui in (1.5) are 0-homogeneous relative to y, in the limit
case n → ∞ with ε = 1, k ∈ Ᏺ(M), we have
gij (x, y) = γij (x) + k · Ui (x, y)Uj (x, y),
and we may consider the Finsler fundamental function F =
(1.7)
√
L, where
L = gij (x, y)y i y j .
(1.8)
This is the relativistic Beil-type metric (see [3, 4]) with the two intensively studied subcases
Ui ∈
−1/2
−1 γjk v j v k
vi , sj (x)v j
vi ,
(1.9)
where s ∈ ᐄ∗ (M), and v ∈ X ∗ (T M) is 0-homogeneous in y.
(III) The generalized Lagrange model of relativistic optics studied by Miron
and Kawaguchi (see [6, 7]) is obtained as limit case n → ∞ with ε = 1, k = 1/c 2
(c = speed of light), where the metric is
gij (x, y) = γij (x) + c −2 · yi yj ,
∀i, j ∈ Ih2 .
(1.10)
Considering the general SB-jet case (1.5), we can fix a priori on E a nonlinear
connection N = {NµA }µ∈Ih , A∈Iv of coefficients
(αi )
Nβ
γ i
= − y (γ ) ,
αβ
(αi )
Nj
i k
= y (α) .
jk
(1.11)
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SYNGE-BEIL AND RIEMANN-JACOBI JET STRUCTURES
However, an open question (see [9]) addresses the physical significance of
choosing an alternative target-nonlinear connection coefficients provided by
the spray attached to the Lagrangian
L = g̃AB y A y B ,
(1.12)
(i)
(i)
given by Ñj α = Nj α + (ε/2)g ik ∂α (Uk Uj ).
The fixed nonlinear connection leads to a splitting T E = HE ⊕ V E, where
V E = Ker π∗ , and to the associated local adapted basis of vector fields [1, 9]
Ꮾ = δα ≡ ∂α − NαA δA , δi ≡ ∂i − NiA δA , δA ≡ ∂˙A =
where ∂α = ∂/∂t α , ∂i = ∂/∂x i , of dual basis
Ꮾ∗ = δα ≡ dt α , δi ≡ dx i ,
δA δy A = dy A + NαA dt α + NiA dx i
∂
∂y A
(α,i,A)∈I∗
(α,i,A)∈I∗
≡ δµ µ∈I ,
≡ δµ µ∈I .
(1.13)
(1.14)
For N fixed, a linear connection ∇ = {Lλµν }λ,µ,ν∈I in E has the adapted coefficients provided by δλ (∇δν δµ ) = Lλµν for all λ, µ, ν ∈ I; these split into 33 = 27
distinct subsets according to the three index subsets Ih1 , Ih2 , and Iv .
We endow E with the metric
G = hαβ (t)dt α ⊗ dt β + gij (t, x, y)dx i ⊗ dx j + g̃AB (t, x, y)δy A ⊗ δy B
h
g
(1.15)
g̃
with gij given in (1.5). The Cartan linear connection has the four essential sets
of coefficients
α
i
i
i
+ Ũjk
=
=
,
Lα
,
L
βγ
jk
βγ jk
ε U i Uj ;α − ΘU i U m Um Uj ;α
(1.16)
,
Lijα =
2
LijA ≡ Li k = γ im U j km − U j km ΘU i U m
(α)
(α )
j (α)
with
i
Ũjk
= γ im Ujkm − εΘ γjk∗ − Ujk∗ U i ,
ε δ{j Um Ui} − δm Ui Uj
Uijm =
,
2
ε δ {j Um Ui} − δ(m) Ui Uj
(α)
α
,
U i jm =
(α)
2
∂ {j γmi} − ∂m γij
,
γijm = |ij; m| =
2
(1.17)
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VLADIMIR BALAN
where we denote by α and k the natural covariant derivatives on (T , hαβ ) and
α
i
the Christoffel symbols of the metrics
(M, γij ), respectively, by βγ and jk
h and γ, respectively, and τ[i···j] = τi···j − τj···i , τ{i···j} = τi···j + τj···i .
The torsion and the curvature of ∇ adapted coefficients are given by
λ
,
δλ ᐀ δν , δµ = Tµν
λ
δλ ᏾ δν , δµ δρ = Rρµν
,
∀λ, µ, ν, ρ ∈ I.
(1.18)
In the Cartan connection case, the essential associated torsion coefficients are
(see [9])
i
= −Lijα ,
Tαj
T
T
(αi )
γ i
k = −δα Ljβ ,
β(γ )
T
(αi )
= δα [β Li k ,
j (γ])
(βj )(γk)
(αi )
γ i
k = −δα Ũjk ,
j (γ )
i
TjA
= LijA ,
i
(i)
δ
Tβγα = −ραβγ
y (δ) ,
l
(i)
i
Tjkα = ρjkl
y (α ) ,
(1.19)
(i)
Tβjα = 0,
δ
i
where ραβγ
and ρjkl
are the curvature components of the metrics h and γ
respectively. The nonholonomy coefficients ωλµν given by [δµ , δν ] = ωA
µν δA , for
λ
= Lλ[µν] + ωλµν , for all λ, µ, ν ∈ I, and
all µ, ν ∈ I, are related to torsion via Tµν
the essential curvature N-tensor fields (for explicit expressions, see [9]) are
λ
= δ[π Lλµ]ν + Lσ µ[ν Lλσ π ] + Lλµσ ωσ
Rµνπ
νπ .
(1.20)
Denoting by |α, |i, |A, and |λ the covariant derivations given by ∇δµ , for µ ∈
Ih1 , Ih2 , Iv , and I, respectively, the Ricci identities for X ∈ ᐄ(E) and θ ∈ ᐄ∗ (E)
are
λ
λ
σ
X|[µ|ν]
= Rσλ µν X σ − Tµν
X|σ
,
σ
σ
θσ + Tµν
θλ|σ ,
θλ|[µ|ν] = Rλµν
∀λ, µ, ν ∈ I.
(1.21)
ν
The adapted components of the Ricci tensor field are given by Rλµ = Rλµν
and
ν
the scalar of curvature is R ≡ Gµν Rλµν = Rh + Rg + Rv , where
γ
Rh = hαβ ραβγ ,
k
k
Rg = γ ij − εΘU i U j ρijk
+ Uijk
,
Rv = g̃ AB RAB ,
i
i
i
and Ujkl
= Ũj[k|l]
+ Ũ m j[k Ũml]
+ Lij
(1.22)
p
(m
α)
m
y (α) .
ρpkl
2. Einstein-Maxwell equations. Denoting by Eµν = Rµν +(1/2)RGµν the Einstein N-tensor field, the Einstein equations with sources
SYNGE-BEIL AND RIEMANN-JACOBI JET STRUCTURES
Eµν = κ ᐀µν ,
µ, ν ∈ I,
1697
(2.1)
split
1
Rαβ − Rhαβ = κ ᐀αβ ,
2
1
Rij − Rgij = κ ᐀ij ,
2
1
RAB − RgAB = κ ᐀AB ,
2
0 = ᐀αi ,
0 = ᐀αA ,
(2.2)
Riα = κ ᐀iα ,
RAα = κ ᐀Aα ,
RiA = κ ᐀iA ,
RAi = κ ᐀Ai ,
where ᐀ = ᐀µν δµ ⊗ δν ∈ ᐀20 (E) is the energy-momentum tensor field and κ is
the cosmological constant. They satisfy the conservation laws
µ
µ
Eν|µ = κ ᐀ν|µ ,
∀µ ∈ I = Ih1 ∪ Ih2 ∪ Iv ,
(2.3)
where the indices are raised by means of the metric G in (1.15).
We note that for Ui ≡ 0, the Einstein equations reduce to the classical ones
on (T × M, h + g). Also, if one considers the extended electromagnetic 2-form
F = FAµ δy A ∧ δy µ ,
(2.4)
√
then the Lagrangian density ᏸ = (L + F λµ Fλµ ) det h with L given in (1.12) provides by the Hilbert-Palatini variation the Einstein equations of the form (2.1)
with the energy-momentum tensor field
1
ρ
Tµν = Fµρ Fµ − Gµν F ρπ Fρπ ,
4
µ, ν ∈ I.
(2.5)
In the extended potential Miron-Tatoiu approach [8], the electromagnetic tensor field F has the essential components
k 1 αγ
,
h gik y ([γ )
|β]
2
m
1
k
k
≡ F i j = d [i j] = γ [ik Ũmj]
+ εU [i Ũmj]
Uk hαβ y ( β ) ,
(α)
2 (α)
m
1
1
≡ F i j = g̃ [i C y C j] = U(m)[ji] hαγ y ( γ ) ,
(α)(β) 2 ( α ) |( β ) 2 β
FAβ ≡ F
FAj
FAB
(αi )β
=
(2.6)
derived from the deflection tensor fields (detailed in [9])
A
dA
µ = δ ∇δµ Ꮿ,
µ ∈ I, A ∈ Iv ,
(2.7)
where Ꮿ = y A δA is the Liouville field, by lowering the indices and antisymmetrization; where the raising/lowering of the indices are assumed to be
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VLADIMIR BALAN
performed via the metric G. The extended electromagnetic tensor field (2.6)
satisfies the following theorem.
Theorem 2.1. The 2-form F is subject to the two sets of the Maxwell extended
equations with sources
1
([εk )
S hαε gik y |β]|γ ,
2 αβγ
αβγ
1
,
d [i β|j] + y(m) Lm [iβ
F i j|β =
− d i m Lm
jβ
(
)
)
(α)
(
α
|j]
α
α
2
1
m
F i j |γ =
d [i
j] + y m ∂ [j Li]γ
(α) (β)
(α)(β)
2 ( α )γ|( β )
γ
− d [i m + Lk [i(m) y k Lm
δβ ,
j]δ
( α )( ε )
(
)
ε
α
l
1 p
m
S F i j|k = − S d i m + Li m y(p) ρjkl
y (β) ,
(
(
)
)(
)
α
(
)
α
α
β
β
2 ijk
ijk
S F i j| k + F i j |k = 0,
F i j | k ≡ 0,
(α) (β)
(α)(β) (γ )
(α)(β)
ijk
S F
g̃ BC FBα|C = −4π Jα ,
(αi )β|γ
=
g̃ BC FBi|C = −4π Ji ,
Gλµ FAλ|µ = 4π JA ,
(2.8)
(2.9)
where we denoted by J = Jµ δµ ∈ ᐄ∗ (E) the adapted dual electric current and
by S the cyclic summation of the corresponding indices below.
The last Maxwell equations in (2.8) were first derived in [9]. Note that in the
SBU case with Ui ≡ 0, the equations above provide in particular the classical
Maxwell equations with sources
S Fij;k = 0,
ijk
γ ij Fik;j = −4π Jk .
(2.10)
3. Extended Lorentz equations. The extended Lorentz equations associated
µ
to the SB model are Gνρ (∇ᐂρ /ds) = FAν ᐂA [2]; denoting FA = Gµν FAν , for all
µ ∈ I; they can be rewritten as
∇ᐂµ
µ
= FA ᐂ A ,
ds
(3.1)
where
ᐂµ
µ∈I
≡
ᐂ = ᐂµ δµ ,
dy A
dt α dx i δy A
dt β
dx j
,
,
=
+ NβA
+ NjA
ds ds ds
ds
ds
ds
(3.2)
(α,i,A)∈I∗
is the covariant velocity along the trajectory of the moving test-particle
c : J ⊂ R → E,
c(s) = t(s), x(s), y(s) ,
µ
∀s ∈ J.
(3.3)
We have denoted ∇ᐂµ /ds = δᐂµ /ds + Lνρ ᐂν ᐂρ , for all µ ∈ I, and will further use the dot notation for expressing the s-derivation. The explicit Lorentz
SYNGE-BEIL AND RIEMANN-JACOBI JET STRUCTURES
1699
equations were described in [2]. In the SBU case with Ui dependent on x only,
the electromagnetic tensors (2.6) have the components
FAα = 0,
FAi ≡ g ij F
m
(k1)j
l
= g ij g [kl Ũmj]
y ( 1 ),
FAB = 0,
(3.4)
and the Lorentz equations (3.1) reduce to [2]
α
ẗ + ṫ β ṫ γ = 0,
βγ α
i
m
k
l
ẍ i + ẋ j ẋ k = g ij g [kl Ũmj]
y ( 1 ) ᐂ( 1 ) ,
jk
(3.5)
ᐂ̇A = 0,
and hence are characterized by constant vertical adapted velocity vector field.
4. Riemann-Jacobi structure and energy-dependent Lagrangians. We further consider in the SBU framework two particular cases.
(I) The Miron-Kawaguchi generalized Lagrange case, for g given by
gij (x, y) = γij (x) + k · yi yj
(4.1)
with k ∈ Ᏺ(M), where the Lagrangian (1.12) becomes L̃ = y0 (1 + ky0 ) and
the null index denotes transvection with y. In this case, the Legendre transformation is given locally by (x, y) ∈ T M → (x, p) ∈ T ∗ M, pi = ∂L/∂y i =
2yi (1 + 2ky0 ), i = 1, n, and is a local diffeomorphism on the set
D = (x, y) | y = 0, 1 + 2ky0 = 0 ⊂ T M.
(4.2)
The associated Hamiltonian is
H ≡ yi
∂L
− L = y0 1 + 3ky0
i
∂y
(4.3)
and the local Riemann-Jacobi structure (T U, ĝ) provided by the directional
variables U i = y i is defined by the scaled metric
1
3
y0 + 2ky02 δij ,
ĝij ≡ H + U∗ δij =
2
2
(4.4)
where U∗ = δij U i U j .
(II) The flat local Lagrange space with potential energy-dependence endowed with Riemann-Jacobi generalized Lagrange metric, with the square-type
Lagrangian (see [11])
L̂ =
1
1 ij δ yi − Ui yj − Uj = y0 + U0 + f ,
2
2
(4.5)
where f = (1/2)U∗ and the indices are raised/lowered by means of the Kronecker flat Euclidean metric. Here, the Hamiltonian is H = (1/2)y0 − f and
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VLADIMIR BALAN
the Legendre transformation provides momenta as potential shifts of direction pi = yi − Ui . For obtaining the h-paths associated to the Kern nonlinear
connection [5, Theorem 7.4.1, page 113], we apply for gij = δij , a = c = 1/2,
and b = 1 the following lemma.
Lemma 4.1. Let (M, γij ) be a (pseudo-)Riemannian space. Then
(a) the spray and the Kern nonlinear connection of the Lagrangian
L = ay0 + bU0 + cU∗ ,
a, b, c ∈ R,
(4.6)
with Ui ∈ ᐄ∗ (M), for raising/lowering performed using the metric γij
and U∗ = Ui U i , are, respectively, given by
γ ij ∂˙j ∂k L · y k − ∂j L )
4
a i
1 ia k j
= γ00 +
b γ y U ∂[k γa]j − yk ∂a U k + y k ∂k U i
2
4
− cγ ia U j U k ∂a γjk + 2Uj ∂a U j ,
Gi (x, y)≡
(4.7)
i
where γjk
are the Christoffel symbols of γij and
Nji (x, y) ≡
∂Gi
b ia i
γ ∂ [j γa]k · γjk · ∂a U k + ∂j U i ;
= aγj0
+
∂y j
4
(4.8)
(b) the Euler-Lagrange equations associated to L are the spray equations
ẍ i + 2Gi (x, ẋ) = 0,
y i = ẋ i ,
i = 1, n,
(4.9)
and the h-paths are given by
ẍ i + Nji (x, ẋ)ẋ j = 0,
y i = ẋ i ,
i = 1, n,
(4.10)
with the spray and nonlinear connection determined above.
Then the Kern spray for gij = (1/2) · (Hessy L)ij = δij is
Gi =
δij ∂ [j U k k] + Uk ∂j U k
4
,
(4.11)
and denoting Ωjk = ∂ [j Uk] , this is rewritten as Gi = δij (Ωj0 + ∂j f ). The associated nonlinear connection is then Nji = (1/4)Ωji , and its autoparallel curves
(the h-paths) satisfy
ẍ i =
Then we have the following theorem.
1 i j
Ω ẋ .
4 j
(4.12)
SYNGE-BEIL AND RIEMANN-JACOBI JET STRUCTURES
1701
Theorem 4.2. The h-paths described by (4.12) are as well:
(a) the extended Lorentz curves (see [2]) particularized to the almost Riemann
Lagrange special (ARLS) jet case associated to the flat metric δij and to
the potential Ui /2;
(b) the solutions of the Lorentz-Udriste force law (see [11]) of the RiemannJacobi-Lagrange structure (M = Rn , ĝij , 4Nji ), where ĝ is the RiemannJacobi metric
ĝij = (H + f )δij =
y0
δij ;
2
(4.13)
(c) the stationary curves (see [5]) of the reduced Lagrangian L̂ = (1/2)y0 +
U0 .
Proof. For (a) and (b), the Riemann-Jacobi and the flat metrics have null
Christoffel symbols; for (c), we apply the lemma for gij = δij , a = 1/2, b = 1,
and c = 0.
Acknowledgment. The present paper was partially supported by Grant
CNCSIS MEN 33784 (182)/23.07.2002.
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[3]
[4]
[5]
[6]
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1702
[10]
[11]
VLADIMIR BALAN
J. L. Synge, Relativity: The General Theory, Series in Physics, North-Holland Publishing, Amsterdam, 1960.
C. Udrişte, Geometric Dynamics, Mathematics and Its Applications, vol. 513,
Kluwer Academic Publishers, Dordrecht, 2000.
Vladimir Balan: Department Mathematics I, Politehnica University of Bucharest,
Splaiul Independenţei 313, RO-77206 Bucharest, Romania
E-mail address: vbalan@mathem.pub.ro