Steps in Differential Geometry, Proceedings of the Colloquium
on Differential Geometry, 25–30 July, 2000, Debrecen, Hungary
ON HOLLAND’S FRAME FOR RANDERS SPACE AND ITS
APPLICATIONS IN PHYSICS
P. L. ANTONELLI AND I. BUCATARU
Abstract. From a rigorous Finsler geometric perspective, we re-examine Holland’s Randers space theory of motion of an electron in flat Minkowski spacetime permeated with an electromagnetic field. Holland’s theory was motivated by analogy with plastic deformation and dislocation in Bravais crystals,
through work of D. Bohm on Quantum Mechanics.
We develop the anholonomic geometry of a Randers space using Holland’s
frame and determine two Finsler connections, one of them the crystallographic
connection, the other just Cartan’s connection expressed in terms of anholonomy. Corresponding to these, we give two fully covariant versions of Holland’s
theory each of which covers the case of a curved Minkowski space-time. The
crystallographic theory with extra matter is most promising.
0. Introduction
The well-known approach of D. Bohm (see [BH]) to quantum mechanics casts
quantum theory into a classical form so that classical and quantum physics can be
more easily compared. As a result, the so-called “quantum potential” arises as a
“guide” to motion of particles in space-time as in de Broglie’s pilot-wave theory
[C]. P. Holland [H1 ], [H2 ] noted that a draw-back of this quantum potential
method is that it must derive from ad hoc use of Schrödinger’s equation and further
asked whether the converse procedure, “that of reformulating classical theory in a
manner which is more in accord with the spirit of quantum theory (particularly
in a way which emphasizes the unity of field and particle), might not suggest an
alternative starting point for the theoretical treatment of quantum effects”, [H1 ].
His differential geometric method is based on fundamental work of S. Amari on a
Finsler approach to crystal dislocation theory, [A].
Holland studies a unified formalism which uses a anholonomic frame (nonintegrable 1-form) on space-time, a sort of plastic deformation, arising from consideration of a charged particle moving in an external electromagnetic field in the
background space-time viewed as a strained medium. In fact, Ingarden [I] was first
Key words and phrases. Finsler connections, P. Holland’s Randers frame, anholonomy, plastic
deformation, electromagnetic theory.
The authors would like to thank Vivian Spak for her excellent typesetting of this manuscript.
The fist author has been partially supported by NSERC-7667.
The second author was a PIMS Postdoctoral Fellow.
39
40
P. L. ANTONELLI AND I. BUCATARU
to point out that the Lorentz force law, in this case, can be written as a geodesic
equation on a Finsler space called Randers space [R]. This results in geometrical entities which depend on the electromagnetic field (vector potential), particle
(velocity) and background space-time parameters, [H1 ]. The Finsler structure implies the existence of a global anholonomic (Holland) frame which in turn yields a
connection with torsion and vanishing Finsler curvatures. Holland shows that the
usual electromagnetic theory can be recovered from an averaging process applied
to the Holland frame and corresponding flat connection, [H1 ].
Unfortunately, Holland’s theory is not truly Finslerian as he presented it. Technically, his notion of covariant differential in Randers space (equation (19) in [H1 ]
and his local expression for the flat connection (equation (21) in [H1 ] appear to be
incorrect. In the present paper we use Finsler geometry to derive Holland’s frame
for an arbitrary Randers space; see [AIM], [M], [MR] and [R] for further references
on these Finsler spaces.
Holland’s idea has led us to find Holland type frames for Kropina spaces, as
well, and these results will be reported on elsewhere. The upshot of our work
so far is that it allows us to define intrinsic frames for C-reducible Finsler spaces
of dimension exceeding three. Previous workers have not succeeded in finding
parallel translation invariant frames because of the failure of the so-called strongly
non-Riemannian condition, [M], [MI]. We have found these for the crystallographic
connection, defined herein. They are conformally invariant.
The final remarks in this paper show how to obtain two “fully covariant” curved
space-time versions of Holland’s Theory via Theorem 3.3 or Theorem 3.4. Especially significant for the latter is the analogy with extra matter defects in crystal
lattices [GZ], [K1 ], [K2 ]. These may be considered to be solutions to problems
posed in [H1 ], [H2 ]. After preliminary material on Finsler geometry in Section 2
we introduce anholonomic frames, and the associated crystallographic connection.
In Section 3 we introduce Holland’s frame for Randers spaces and prove the main
Theorems 3.3 and 3.4 of our paper. In Section 4 we consider the original set-up of
Holland and rederive his results taking care to explain them in our notation.
1. Finsler Spaces and Finsler Connections
Let M be a real n-dimensional connected manifold of C ∞ -class and (T M, π,M )
its tangent bundle with zero section removed. Every local chart U, ϕ = (xi ) on
M induces a local chart π −1 (U ), φ = (xi , y i ) on T M . The kernel of the linear
mapping π∗ = T T M → T M is called the vertical distribution and is denoted
∂
by V T M . For every u ∈ T M , Ker π∗,u = Vu T M is spanned by ∂y
i |u . By a
nonlinear connection on T M we mean a regular n-dimensional distribution H :
u ∈ T M 7→ Hu T M which is supplementary to the vertical distribution i.e.
(1.1)
Tu T M = Hu T M ⊕ Vu T M,
∀ u ∈ T M.
∂
A basis for Tu T M adapted to the direct sum (1.1) has the form δxδ i |u = ∂x
i |u −
j
∂
∂
i
i
i
i
j
Ni (u) ∂yj |u , ∂yi |u . The dual basis of this is (dx , δy = dy + Nj dx ). These are
ON HOLLAND’S FRAME FOR RANDERS SPACE
41
the Berwald bases. The vector field mapping J : X(T M ) → X(T M ), defined by:
(1.2)
J=
∂
⊗ dxi ,
∂y i
is globally defined on T M and is called the almost tangent structure. It has the
properties:
1◦ . J 2 = 0;
2◦ . Im J = Ker J = V T M . See [MA] or [AIM] for more discussion.
A linear connection (Koszul connection) on T M is a map ∆ : (X, Y ) ∈ X(T M ) ×
X(T M ) 7→ ∆X Y ∈ X(T M ) for which we have:
1◦ . ∆f X+gY Z = f ∆X Z + g∆Y Z;
2◦ . ∆X f Y = X(f ) + f ∆X Y ;
∆X (Y + Z) = ∆X Y + ∆X Z.
Definition 1.1. A linear connection ∆ on T M is called a Finsler connection (or
an N -linear connection) if:
1◦ ∆ preserve by parallelism the horizontal distribution HT M ;
2◦ The almost tangent structure J is absolute parallel with respect to ∆.
For a Finsler connection ∆ it is immediate that ∆ preserves also the vertical dis∂
tribution. In the Berwald basis ( δxδ i , ∂y
i ) adapted to the decomposition (1.1), a
Finsler connection can be expressed as:

δ
∂

k δ
k ∂

∆ δ i j = Fji
 ∆ δxδ i j = Fji k ;

δx ∂y
δx
δx
∂y k
(1.3)

δ
∂

k δ
k ∂

∆ ∂
= Cji
;
∆ ∂ i j = Cji
.
j
k
i
∂y δx
∂y ∂y
δx
∂y k
k
Observe that under a change of induced coordinates on T M the functions Fji
transk
form like the coefficients of a linear connection on M and Cji as the components
k
of a (1, 2) tensor field on M . We will say that Cji
is a (1, 2)-type Finsler tensor
field. In general, a tensor field of (r, s)-type on T M is called Finsler tensor field
(or d-tensor field ) if under a change of induced coordinates on T M its components
transform like the components of a (r, s) type tensor on the base manifold M .
For X = X i δxδ i , a horizontal vector field, the absolute differential with respect
to the Finsler connection ∆ is given by:
(1.4)
i
i
∆X i = dX i + Fjk
X j dxk + Cjk
X j δy k = dX i + ωji X j ,
i
i
where ωji = Fjk
dxk + Cjk
δy k is the connection 1-form of ∆. The formula (1.4) can
be written in an equivalent form:
(1.4)0
i
∆X i = X|k
dxk + X i |k δy k .
42
P. L. ANTONELLI AND I. BUCATARU
i
Here, X|k
and X i |k are the horizontal and the vertical covariant derivatives of X i ,
respectively. These must satisfy
(1.5)
(1.5)0
∆
δ
δxk
Xi
δ
i δ
= X|k
;
δxi
δxi
i
X|k
=
δX i
i
+ Frk
Xr;
δxk
∆
∂
∂y k
Xi
δ
δ
= X i |k i ,
δxi
δx
X i |k =
i.e.
∂X i
i
+ Crk
Xr.
∂y k
Of course, the horizontal and vertical covariant derivatives with respect to a Finsler
connection ∆, can be defined in general for a Finsler tensor field. For example,
if Tji are the components of an (1, 1) Finsler tensor field then h and v-covariant
derivatives are given by:
(1.6)

δTji

i
i
r

T
=
+ Frk
Tjr − Fjk
Tri and

 j|k
δxk

∂Tji


i
r
 Tji |k =
+ Crk
Tjr − Cjk
Tri .
∂y k
For a Finsler connection ∆ one considers typically
T (X, Y ) = ∆X Y − ∆Y X − [X, Y ]
and
R(X, Y )Z = ∆X ∆Y Z − ∆Y ∆X Z − ∆[X,Y ] Z
the torsion and the curvature. It is well known [AIM], [MA] that in the basis
∂
( δxδ i , ∂y
i ) there are only five components of torsion and three components of curvature. The five components of torsion are:
(1.7)

δ
δ
δ
δ

k

hT
,
=: Tijk k = (Fji
− Fijk ) k ;


i δxj

δx
δx
δx

!


k

δNjk
δ
δ
∂
δN
∂

i
k

vT
,
=: Rij k =
−
;


i δxj
j
i
k

δx
∂y
δx
δx
∂y



∂
δ
k δ
hT
, j = Cji
;
i

∂y δx
δxk

!


k

∂N
∂
δ
∂
∂

j
k
k


 vT ∂y i , δxj =: Pij ∂y k = ∂y i − Fij ∂y k ;





∂
∂
∂

k ∂
k
k

vT
,
=: Sij
= Cji
− Cij

i
j
k
∂y ∂y
∂y
∂y k
(h) h-torsion
(v) h-torsion
(h) hv-torsion
(v) hv-torsion
(v) v-torsion.
ON HOLLAND’S FRAME FOR RANDERS SPACE
43
The three components of curvature are given by:

i
i
δFjh
δFjk

i
m i
m i
i
m


R
=
−
+ Fjk
Fmh − Fjh
Fmk + Cjm
Rkh

jkh
h
k

δx
δx





i
∂Fjk
i
i
m
i
(1.8)
− Cjk|h
+ Cjm
Pkh
Pjkh
=
h

∂y





i
i

∂Cjk
∂Cjr

m i
m i
i

 Sjkr
−
+ Cjk
Cmr − Cjr
Cmk
=
∂y r
∂y k
For a Finsler connection ∆ we have the following Ricci identity

i
i
i
i
m
m
X|k|r
− X|r|k
= X m Rmkr
− X|m
Tkr
− X i |m Rkr




i
i
i
m
m
X|k
|r − X i |r|k = X m Pmkr
− X|m
Ckr
− X i |m Pkr
(1.9)



 i
i
m
X |k |r − X i |r |k = X m Smkr
− X i |m Skr
.
2. The Parallel Displacement Determined by
an Anholonomic Finsler Frame
Let V be an open subset on T M and
(2.1)
Yα : u ∈ V → Yα (u) ⊂ Vu T M,
α = 1, n
be a vertical frame over V . Assume that V = T M . If we consider Yα (u) =
∂
Yαi (u) ∂y
Yαi (u) are the entries of a nonsingular matrix. Denote by (Yjα )
i |u , then
the components of the inverse of these matrix. This means that:
(2.2)
Yαi Yiβ = δαβ .
Yαi Yjα = δji ,
We call (Yαi ) an anholonomic Finsler frame. Naturally, every geometric object field
on T M can be expressed in anholonomic Finsler frames. For example, if Tji are the
components of a (1, 1) Finsler field, then the anholonomic components are given
by:
(2.3)
Tβα = Yiα Tji Yβj .
Consider the anholonomic Berwald basis:
(2.4)
δ
δ
= Yαi i
δxα
δx
and
∂
∂
= Yαi i .
∂y α
∂y
k
Let ∆ be a Finsler connection with the holonomic coefficients (Fijk , Cij
) given
by (1.3). As we have seen ∆ preserves, by parallelism, the horizontal and vertical
distribution, so in the anholonomic Berwald basis ( δxδα , ∂y∂α ) the Finsler connection
44
P. L. ANTONELLI AND I. BUCATARU
∆ can be expressed as:

δ
∂
δ
∂
γ
γ


 ∆ δxδα β = Fβα γ , ∆ δxδα β = Fβα γ
δx
δx
∂y
∂y
(2.5)
δ
δ
∂
∂

γ
γ

 ∆ ∂α β = Cβα
, ∆ ∂α β = Cβα
.
∂y
∂y
δx
δxγ
∂y
∂y γ
γ
γ
The functions (Fβα
, Cβα
) are called the anholonomic coefficients of the Finsler
connections, ∆.
Proposition 2.1. Let ∆ be a Finsler connection with the holonomic coefficients
k
(Fijk , Cij
) and (Yαi ) be an anholonomic Finsler frame. Then, the anholonomic
coefficients of the Finsler connection ∆ are given by:
!

k
δYjγ

δYα
γ
γ
γ

j
k
i
j
k
 Fαβ =
+ Yα Fji Yβ Yk = Yα − i + Fji Yk Yβi

i

δx
δx





γ
k
i
γ
j
i

= Yα|i Yβ Yk = −Yj|i Yα Yβ

(2.6)
!


∂Yjγ

∂Yαk
γ
γ
γ

j
k
i
j
k

Cαβ =
+ Yα Cji Yβ Yk = Yα − i + Cji Yk Yβi

i

∂y
∂y





γ
γ
= Yαk |i Yβi Yk = −Yj |i Yαj Yβi .
Conversely, if for a Finsler connection ∆ we know the anholonomic coefficients
α
α
(Fβγ
, Cβγ
) then the holonomic coefficients are given by:

α
δYj

β
β
δ α
k
k α
k


 Fji = Yα Yj|β Yi = Yα δxβ + Yj Fδβ Yi






δYαk
β
k
α
k
δ


= −Yα|β Yj Yi = − β + Yδ Fαβ Yjα Yiβ


δx
(2.6)0

∂Yjα

β
β

k
k α
k
δ α

C
=
Y
Y
|
Y
=
Y
+
Y
C

ji
α j β i
α
j δβ Yi
β

∂Y





∂Y k

δ


= −Yαk |β Yjα Yiβ = − αβ + Yδk Cαβ
Yjα Yiβ .
∂Y
γ
γ
Proof. The first formulae for Fαβ
and Cαβ
can be obtained if we compare (1.3)
and (2.5). For the second we have to take into account
Yαi
δYjα
δY i
= −Yjα α
;
β
δx
δxβ
Yαi
δYiβ
δY i
= −Yiβ αj
j
δx
δx
and
∂Yjα
∂Y i
∂Y β
∂Y i
= −Yjα βα ;
Yαi ij = −Yiβ αj
β
∂y
∂y
∂y
∂y
which follows from (2.2) by differentiation.
Yαi
ON HOLLAND’S FRAME FOR RANDERS SPACE
45
We then find that for any Finsler connection (1.3)
δ
δ
δ
∂
(a)
,
= Ωγαβ γ + Rγαβ γ
α
β
δx δx
δx
∂y
δ
∂
∂
δ
γ
(2.7)
(b)
,
= (Pγαβ + Fβα
) γ + Ωγα(β) γ
δxα ∂y β
∂y
δx
∂
∂
∂
γ
(c)
,
= ηαβ
.
∂y α ∂y β
∂y γ
See [MI] for more discussion. Here, Rγαβ are the anholonomic components of the
(1, 2) type tensor field Rijk given by (1.7), Ωγαβ and Ωγα(β) are the anholonomic
objects defined by
!
δYβi
δYαi
γ
γ
(a)
Ωαβ = Yi
− β
and
δxα
δx
(2.8)
∂ γ
(b)
Ωγα(β) = Yαi
Y
Yβj .
∂y j i
Also, the (v)-hv-torsion has components
∂Nki
γ j k
i
(2.8)
(c)
Pγαβ =
−
F
jk Yi Yα Yβ
∂y j
while the (v)-v-torsion is
(2.8)
(d)
γ
ηαβ
=
Yiγ
∂Yβi
∂Y i
− βα
α
∂y
∂y
!
.
The frame Yαi is holonomic if and only if there exists n functions ϕα on M for
α
α
i
which Yiα = ∂ϕ
∂xi , i.e. the 1-form Yi vdx is exact.
Proposition 2.2. The frame Yαi is holonomic if and only if Ωγαβ = 0 = Ωγα(β) .
Proof. Ωγα(β) = 0 implies Yαi is independent of y, then Ωγαβ = 0 implies Yαi are given
by gradients of n functions on M .
Remark. Yαi is holonomic if and only if δxδα , δxδβ and δxδα , ∂y∂β are in V T M , i.e.
iff these Lie brackets are vertical. Of course, 2.7(c) expresses the fact that the fibre
of T M is integrable.
Let X α be the anholonomic components of a Finsler vector field X. Then the
absolute differential of X with respect to the Finsler connection ∆ can be expressed
as:
(2.9)
α
α
∆X α = dX α + Fβγ
X β dxγ + Cβγ
X β δy γ .
(Here, the reader may note the difference with (19) in [H1 ].
46
P. L. ANTONELLI AND I. BUCATARU
Proposition 2.3. Given an anholonomic Finsler frame (Yαi ), there exists a unique
Finsler connection ∆ for which the given frame is h- and v-covariant constant. We
call this the Crystallographic connection.
γ
i
Proof. From (2.6) we have that Yα|j
= 0 is equivalent with Fαβ
= 0 ⇐⇒
(2.10)
δYjγ k
δYαk
=
Y .
δxi
δxi γ
= 0 which is also equivalent with:
k
Fji
= −Yjα
γ
Similarly, Yαi |j = 0 ⇐⇒ Cαβ
∂Yjγ k
∂Yαk
=
Y .
∂y i
∂y i γ
(The reader may note the difference between (2.10) and (21) in [H1 ]).
(2.10)0
k
Cji
= −Yjα
Proposition 2.4. For the Finsler connection given by Proposition (2.3), all three
components of curvature vanish.
i
= 0, Yαi |k = 0 and the Ricci identities (1.9) we obtain
Proof. According to Yα|k
i
i
i
= 0.
Rjk`
= Pjk`
= Sjk`
3. The Anholonomic Finsler Frame Determined by a Randers Metric
(The Holland Frame)
Let us consider a Finsler space F n = (M, α) on an n-dimensional manifold M .
This means that:
1◦ α : T M → R is of C ∞ -class and continuous on the zero section;
2◦ α is positively homogeneous with respect to y;
3◦ The matrix with the entries:
1 ∂ 2 α2
(3.1)
aij =
has a constant rank n on T M.
2 ∂y i ∂y j
It’s known that a Finsler space has a canonical nonlinear connection HT M , with
the local coefficients:
1 ∂
(3.2)
Nji =
(γ i y k y ` ),
2 ∂y j k`
i
where γk`
are the Christoffel symbols of second kind for the metric tensor aij . There
is also a Finsler connection (Cartan connection) with the holonomic coefficients
given by:

 F k = 1 akr δari + δarj − δaij


 ij
2
δxj
δxi
δxr
(3.3)


1 kr ∂ari
∂arj
∂aij
k

 Cij = a
+
−
.
2
∂y j
∂y i
∂y r
As is well-known, this connection is metrical (aij|k = 0 and aij |k = 0) and h and v
k
symmetric (Tijk = 0 and Sij
= 0) [AIM], [MA].
ON HOLLAND’S FRAME FOR RANDERS SPACE
47
Together with the Finsler space F n = (M, α) we shall consider a covector field
bi (x)dxi on M (or an open set V of M ). Then β(x, y) = bi (x)y i is a scalar function
on T M (or on π −1 (V )).
The function L : T M → R, defined by:
(3.4)
L(x, y) = α(x, y) + β(x, y)
is also the fundamental function of a Finsler space [AIM], [MR]. The pair (M, L)
is called a Randers space. Denote by:
(3.5)
gij =
1 ∂ 2 L2
2 ∂y i ∂y j
the fundamental tensor of the Randers space (M, L). Taking into account the
homogeneity of α and L we have:

∂α
∂α
1

pi := y i = aij j ; pi := aij pj =


i

α
∂y
∂y



∂L
∂L
1 i

ij
j
i
 ` := y = g
; `i := gij ` =
= pi + bi
L
∂y j
∂y i
(3.6)
α
L

α

`i = pi ;
`i `i = pi pi = 1; `i pi = L
; pi `i =



L
α



β
β

b i pi = ,
bi `i = .
α
L
The metric tensors (aij ) and (gij ) are related by:
β
L
aij + bi pj + pi bj + bi bj − pi pj
α
α
L
= (aij − pi pj ) + `i `j .
α
i
Let us consider now Xα (x, y) an arbitrary, but fixed anholonomic frame (but
it could be also holonomic). Denote by
(3.7)
gij =
aαβ (x, y) = Xαi (x, y)Xβj (x, y)aij (x, y),
the components of the metric (aij ) with respect to (Xαi ). If (Xiα ) are the components of the inverse matrix of (Xαi ), denote:
`α = Xiα `i ,
pα = Xiα pi ,
`α = Xαi `i
and pα = Xαi pi .
Consider (Yji ) the matrix with the entries:
r r
α
α i
Yji (x, y) =
δji − `i `j +
p pj .
L
L
Then this matrix is invertible and the components of its inverse are:
!
r
r
L
L
`i `j − pi pj .
(Y −1 )ij =
δji +
α
α
48
P. L. ANTONELLI AND I. BUCATARU
Theorem 3.1. For a Randers space (M, L) consider in the case
r r
α
α i
i
i
i
(3.8)
Yγ =
δ − ` `γ +
p pγ
L γ
L
defined on open set V in T M where
anholonomic Finsler frame.
L
α
L
α
>0:
∂
> 0. Then (Yγ = Yγi ∂y
i )γ=1,n is an
We call it, Holland’s frame of Randers space [H1 ], [H2 ]. It is p-homogeneous of
degree zero in y and a conformal invariant in the sense that L 7→ eφ(x) · L leaves
Yγi fixed.
Proof. Consider also:
r
Yjγ
(3.9)
=
L
α
r
δjγ
+
L γ
` `j − pγ pj
α
!
.
We have to check that Yγi Yjγ = δji and Yγi Yiβ = δγβ . Let us verify the former:
!
r
r
L i
α γ
γ
i
γ
i γ
i
` `γ − p pγ
δj − ` `j +
p pj
Yγ Yj = δγ +
α
L
r
r
r
L i
L i
α i
` `j + `i `γ pγ pj
` `j −
= δji − `i `j +
p pj +
α
α
L
r
α i
− pi pj + pi pγ `γ `j −
p pj = δji .
L
Theorem 3.2. With respect to Holland’s frame the holonomic components of the
Finsler metric tensor (aαβ ) is the Randers metric (gij ), that is:
gij = Yiα Yjβ aαβ .
(3.10)
Proof. We have
!
L β
β
` `j − p pj aαβ
+
=
α
r
r
L
L
=
aαj + pα `j −
pα pj
α
α
!
!
r
r
r
r
L
L
L
L
=
δiα +
`α `i − pα pi ·
aαj + pα `j −
pα pj
α
α
α
α
r
r
L
L
L
L α L
= aij +
pi `j − pi pj + ·
pj `i
α
α
α
α L α
r
r
L α
L
Lα
L
L
L
+ · `i `j − ·
`i pj − pj pi −
pi `j + pi pj
α L
α
αL
α
α
α
r
Yjβ aαβ
Yiα Yjβ aαβ
L
α
r
δjβ
ON HOLLAND’S FRAME FOR RANDERS SPACE
=
49
L
(aij − pi pj ) + `i `j = gij .
α
Corollary. We have
Yiα pα = `i ,
Yαi pα = `i .
Theorem 3.3. Consider the Randers space (M, L) and Holland’s Frame (Yαi ).
Then there exists a unique Finsler connection ∆ (Cartan connection) with local
α
α
coefficients (Fβγ
, Cβγ
) for which
(1) aαβ|γ = 0 and aαβ |γ = 0
(2) the anholonomic components of the (h) h-torsion (v) v-torsion are
α
α
α
τβγ
:= Fγβ
− Fβγ
= −Ωα
βγ
and
α
α
α
Σα
βγ := Cγβ − Cβγ = −ηβγ .
Proof. Let us consider ∆ the Cartan connection of the Randers space (M, L). Dei
i
note by (Fjk
, Cjk
) it’s holonomic coefficients. This is the unique Finsler connection
which satisfies Matsumoto’s Axioms:
(a) gij|k = 0 and gij |k = 0;
i
i
(b) Tjk
= 0 and Sjk
= 0.
α
α
Consider (Fβγ
, Cβγ
) the anholonomic coefficients of this Cartan connection (2.6).
All we have to prove is that (1) and (a) are equivalent and also (2) and (b). First
we prove that
gij|k = Ykα Yjβ aαβ|γ Ykγ
(3.12)
and
gij |k = Yiα Yjβ aαβ |γ Ykγ .
Let us start with the (RHS) of (3.12)1 ,
δaαβ
δ
δ
α β γ
aαβ|γ Yiα Yjβ Ykγ =
−
a
F
−
a
F
δβ
δα
αγ
βγ Yi Yj Yk
δxγ
δaαβ α β
δ
δ
= Ykγ
Y Y − aδβ Yjβ Fαγ
Ykγ Yiα − aδα Yiα Fβγ
Ykγ Yjβ
δxγ i j
δaαβ α β
δ
δ
=
Y Y − gj` Yδ` Fαγ
Ykγ Yiα − gi` Yδ` Fβγ
Ykγ Yjβ
δxk i j
`
δ
δYα
m `
α
+
Y
F
= k (g`m Yα` Yβm )Yiα Yjβ − gj`
α
mk Yi
δx
δxk
!
δYβ`
m `
− gi`
+ Yβ Fmk Yjβ
δxk
=
δYβm β
δgij
δYα` α
`
`
−
g
F
−
g
F
+
g
Y
+
g
Y
j`
i`
im
`j
ik
jk
δxk
δxk i
δxk j
50
P. L. ANTONELLI AND I. BUCATARU
− gj`
δYβ` β
δYα` α
Y
−
g
Y = gij|k .
i`
δxk i
δxk j
We have used: aδβ Yjβ = gj` Yδ` and aδα Yiα = gi` Yδ` according to (3.10) and
δYα`
`
+ Yαm Fmk
according to (2.6).
δxk
A similar formula works for v-covariant derivative. We also have, as in [MI],
 α
α
i
α j k

 τβγ + Ωβγ = Tjk Yi Yβ Yγ




 C α + Ωα = C i Y α Y j Y k
(equivalent to (2.6)2 )
δ
Fαβ
Yδ` Ykβ =
βγ
β(γ)
jk i
γ
β

α

X


α
i

+ηβγ
= Sjk
Yiα Yβj Yγk .


βγ
Using these formulae we have that (2) and (b) are equivalent.
Remark. From the second formula of (3.13) we can determine the vertical anholonomic coefficients of the Cartan connection as:
α
i
α j k
Cβγ
= −Ωα
β(γ) + Cjk Yi Yβ Yγ ,
where
i
Cjk
=
∂ 3 L2
1 i`
g
.
`
4
∂y ∂y j ∂y k
Theorem 3.4. Consider the Randers space (M, L) with Holland’s Frame. The
crystallographic connection satisfies:
δaαβ
1) gij|k = Yiα Yjβ Ykγ
δxγ
∂a
αβ
gij |k = Yiα Yjβ Ykγ
.
∂y γ
2) The holonomic components of the (h) h-torsion, (v) v-torsion and (h) hvtorsion are:
i
i β γ
Tjk
= Ωα
βγ Yα Yj Yk ;
i
i β γ
Cjk
= Ωα
β(γ) Yα Yj Yk ,
i
α
Sjk
= ηβγ
Yαi Yjβ Ykγ
and
respectively.
i
3) This connection is flat and Yα|j
= 0, Yαi
j
= 0.
Proof. According to Propositions 2.3 and 2.4, the anholonomic components for
δa
α
α
the crystallographic connection are Fβγ
= Cβγ
= 0. Then aαβ|γ = δxαβ
and
γ
∂a
aαβ |γ = ∂yαβ
γ . As (3.12) holds also for this connection we have that (1) is verified.
Pα
α
α
α
Since Fβγ
= Cβγ
= 0 then, βγ = τβγ
= 0 and (3.13) gives (2). Proposition 2.4
gives the condition (3).
Remark. Since the Holland frame is a conformal invariant it follows from (2.8)b
α
and (2.8)d that the anholonomic objects Ωα
β(γ) and ηβγ are conformal invariants.
ON HOLLAND’S FRAME FOR RANDERS SPACE
51
i
Also from Theorem 3.4, 2) we have that the (v) v-torsion Sjk
and (h) hv-torsion
i
Cjk of the crystallographic connection are conformal invariant.
4. Anholonomic Geometry of Flat Riemannian Space Versus Randers
Space
Let (aij ) be a regular Riemannian metric possibly with non-positive-definite
i
signature. Suppose (M, aij ) is a flat space. Denote by (γjk
) the Christoffel symbols
of second kind of the Riemannian metric.
Since the Riemannian space (M, aij ) is flat there exists a frame Xαi (x) (holonomic or not) on the base manifold such that aαβ = Xαi (x)Xβj (x)aij (x) are constants. With respect to this frame the coefficients of Levi-Civita connection are
α
γβγ
= 0.
p
p
Then α(x, y) =
aij (x)y i y j =
aαβ y α y β is the fundamental function of a
Finsler metric.
If we assume that the manifold M is endowed
with a covector field b = bi (x)dxi ,
p
then L : T M → R, given by L(x, y) = aij (x)y i y j + bi (x)y i is the fundamental
function of a Randers space.
Theorem 4.1. For the Randers space (M, L) above with Holland’s Frame, the
Cartan connection is the unique connection which in anholonomic coordinates satisfies:
1) aαβ|γ = 0 and aαβ |γ = 0;
Pα
α
α
2) τβγ
= −Ωα
βγ and
βγ = −ηβγ .
The anholonomic coefficients of the Cartan connection are given by
(4.1)
α
Fβγ
=
1 αδ
a (aβε Ωεδγ + aγε Ωεδβ − aδε Ωεβγ )
2
α
Cβγ
=
1 αδ
ε
ε
ε
a (aβε ηδγ
+ aγε ηδβ
− aδε ηβγ
).
2
Proof. According to Theorem 3.3 the Cartan connection is the unique Finsler connection of the Randers space (M, L) for which (1) and (2) hold.
¿From (1) and (2) by some standard computation we can get:
1
δaδβ
δαδγ
δaβγ
1
α
Fβγ
= aαδ
+
−
+ aαδ (aβε Ωεδγ + αγε Ωεδβ − aδε Ωεβγ )
2
δxγ
δxβ
δxδ
2
and
α
Cβγ
1
= aαδ
2
∂aδβ
∂aδγ
∂aβγ
+
−
∂y γ
∂y β
∂y δ
1
ε
ε
ε
+ aαδ (aβε ηδγ
+ aγε ηδβ
− aδε ηβγ
).
2
Taking into account that (aαβ ) are constants we get (4.1).
Denote now by “s” the arc length with respect the Riemannian metric, i.e.
ds = (aij (x)dxi dxj )1/2 = (aαβ dxα dxβ )1/2 and by “S” the arc length with respect
52
P. L. ANTONELLI AND I. BUCATARU
to Finsler metric, i.e.
(4.2)
dS = (aij (x)dxi dxj )1/2 + bi (x)dxi = (gij (x, y)dxi dxj )1/2 .
It is known that if we perform a variation of (4.2) we get the Lorentz force equation:
(4.3)
i
where γjk
j
k
j
d2 xi
i dx dx
i dx
+
γ
=
F
,
jk
j
ds2
ds ds
ds
are the Christoffel symbols of the Riemannian metric (aij ) and Fji =
∂b
∂bk
j
aik ( ∂x
j − ∂xk ) is the electro-magnetic tensor field. But the equation (4.3) has an
equivalent form:
j
k
d2 xi
i dx dx
+ Fjk
= 0,
2
dS
dS dS
i
where Fjk
are coefficients of the Cartan connection of the Randers space (M, L). In
anholonomic coordinates given by Holland’s Frame, the equation (4.3)0 becomes:
(4.3)0
β
γ
d2 xα
α dx dx
+ Fβγ
= 0.
2
dS
dS dS
If we take into account the Theorem (4.1), then the Lorentz equation (4.3) can be
expressed anholonomically as:
(4.4)
(4.5)
d2 xα
dxβ dxγ
+ aαδ (aβε Ωεδγ )
= 0.
2
dS
dS dS
Theorem 4.2. Let (M, L) be a Randers space
Pαas above withα Holland’s Frame and
α
= 0. Moreover, this
crystallographic connection. Then τβγ
= 0, βγ = 0 and Cβγ
i
Finsler connection is flat and metric and Yα|j
= 0, Yαi j = 0.
Proof. As (aαβ ) are constant we have by Theorem 3.4(1) that the crystallographic
connection is metric. Flatness follows from Proposition 2.4 and the parallel translation invariance of Yαi from Proposition 2.3.
Remark. As for the crystallographic connection, we have the anholonomic coeffiα
α
cients Fβγ
= Cβγ
= 0, and the induced absolute differential is:
(4.6)
∆X α = dX α .
Consequently, the geodesic equations for this connection are
d2 xα
dS 2
= 0.
If the Riemannian metric is positive definite we can choose the frame Xαi (x)
such that
Xαi (x)Xβj (x)aij (x) = δαβ .
In this case according with (3.10) we have:
gij Yαi Yβj = δαβ
∂
and this means exactly that Yα = Yαi ∂y
is an orthogonal frame.
i
α=1,n
ON HOLLAND’S FRAME FOR RANDERS SPACE
53
Final Remark. Consider the absolute differential with holonomic coefficients (2.9),
(2.10) and (2.10)0 in the present paper and compare them to (19) and (21) in [H1 ].
For the case of Theorem 4.2 these two sets are identical. However, introduction
of matter curves the flat Minkowski’s 4-space of special relativity. Holland claims
that a fully covariant theory (i.e. a curved space version of Theorem 4.2) is still
possible. Theorem 3.3 is one way to complete his program. In this case the Cartan
torsion tensor plays a crucial role and the three curvature tensors will not vanish.
Unfortunately, the Holland frame is not invariant under parallel translation.
Theorem 3.4 also completes Holland’s program. Indeed, not only is Holland’s
frame invariant, but equation (1) in the statement of Theorem 3.4 has the character
of “extra matter” caused by point defects [K2 ]. Point defects in Bravais crystals are
accounted for by non-metricity. Thus, matter in Minkowski 4-space is expressed
according to defect theory via (1). It is obvious that this description is in the spirit
of Holland’s original idea, based on the geometry of defects in crystal lattices, [H1 ],
[H2 ].
References
[A]
Amari, S., A theory of deformations and stresses of Ferro magnetic substances by Finsler
geometry, RAAG Memoirs of the Unifying Study of Basic Problems in Engineering and
Physical Sciences by Means of Geometry (K. Kondo, ed.), vol 3, Gakujutsu Bunken, FukyuKai, Tokyo, 1962, pp. 193–214.
[AIM] Antonelli, P.L., Ingarden, R.S. and Matsumoto, M., The Theory of Sprays and Finsler
Spaces with Applications in Physics and Biology, Kluwer Academic Press, Dordrecht 1993,
pp. 350.
[BH] Bohm, D. and Hiley, B.J., The Undivided Universe, Routledge, London and New York
1993.
[C]
Cushing, James, T., Quantum Mechanics, Univ. of Chicago Press 1994.
[GZ] Günther, H. and Zorawski, M., On the geometry of point defects and dislocations, Ann.
der Phys. 7, (1985), 41–46.
[H1 ] Holland, P.R., Electromagnetism, particles and anholonomy, Phys. Lett. 91, (1982), 275–
278.
[H2 ] Holland, P.R. and Philippidis, C., Anholonomic deformations in the ether: a significance
for the electrodynamic potentials, Quantum Implications, (B.J. Hiley and F.D. Peat, eds.),
Routledge and Kegan Paul, London and New York , 1987, pp. 295–311.
[I]
Ingarden, R.S., Differential geometry and physics, Tensor, N.S. 30, (1976), 201–209.
[K1 ] Kröner, E., The continuized crystal – a bridge between micro- and macromechanics, Z.
angew. Math. Mech. 66, (1986), 5, 284–292.
[K2 ] Kröner, E., The differential geometry of elementary point and line defects in Bravais
crystals, Int. J. Theor. Phys. 29, (1990), 1219–1237.
[M]
Matsumoto, M., Theory of Finsler Spaces with (α, β)-metric, Rep. Math. Phys. 31, (1992),
43–83.
[MA] Miron, R. and Anastasiei, M., The Geometry of Lagrange Spaces: Theory and Applications, Kluwer Academic Press, Dordrecht, 1994.
[MI] Miron, R. and Izumi, H., Invariant frame in generalized metric spaces, it Tensor, N.S. 42,
(1985), 272–282.
[MR] Miron, R., General Randers spaces, in Lagrange and Finsler Geometry. Application to
Physics and Biology, (P. Antonelli and R. Miron, eds.), Kluwer Academic Press, Dordrecht
(1996), 123–140.
54
[R]
P. L. ANTONELLI AND I. BUCATARU
Randers, G., On an asymmetric metric in the four-spaces of general relativity, Phys. Rev.
59, (1941), 195–199.
Department of Mathematical Sciences, University of Alberta, Edmonton, Alberta,
Canada, T6G 2G1.
E-mail address: pa2@gpu.srv.ualberta.ca
Faculty of Mathematics, “Al.I.Cuza” University, Iaşi, Iaşi 6600, Romania
E-mail address: bucataru@uaic.ro