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Derivation of the Finslerian Gauge Field Equations.

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Annalen der Physik. 7. Folge, Band 41, Heft 3, 1984, S. 222-227
J. A. Barth, Leipzig
Derivation of the Finslerian Gauge Field Equations
Department of Theoretical Physics, Moscow State University, Moscow, USSR
A b s t r a c t . As is well known the simplest way of formulating the equations for the Yang-Mills
gauge fields consists in taking the Lagrangian t o be quadratic in the gauge tensor [l-51, whereas
the application of such an approach to the gravitational field yields equations which are of essentially
more complicated structure than the Einstein equations. On the other hand, in the gravitational
field theory the Lagrangian can be constructed to be of forms which may be both quadratic and linear
in the curvature tensor, whereas the latter possibility is absent in the current gauge field theories.
In previous work [6] it has been shown that the Finslerian structure of the space-time gives rise t o
certain gauge fields provided t h a t the internal symmetries may be regarded a s symmetries of a threedimensional Riemannian space. Continuing this work we show that appropriate equations for these
gauge fields can be formulated in both ways, namely on the basis of the quadratic Lagrangian or,
if a relevant generalization of the Palatini method is applied, on the basis of a Lagrangian linear in
the gauge field strength tensor. The latter possibility proves to result in equations which are similar
to the Einstein equations, a distinction being that the Finslerian Cartan .curvature tensor rather
than the Riemann curvature tensor enters the equations.
Ableitung von Eichfeldgleichungen im Finsler-Raum
I n h a l t s i i b e r s i c h t . Bekanntlich besteht die einfachste Moglichkeit der Formulierung der Gleichungen fur Yang-Mills Eichfelder darin, eine Lagrange-Funktion zu nehmen, die quadratisch im
Eichtensor ist [175], wahrend die Anwendung eines derartigen Zugangs auf das Gravitationsfeld
zu Gleichungen fuhrt, die eine wesentlich kompliziertere Struktur haben als die Einsteinschen Gleichungen. Andererseits kann in der Gravitationsfeldtheorie die Lagrange-Funktion aufgebaut werden
aus Formen, die sowohl quadratisch als auch linear im Kriimmungstensor sind, wahrend diese letztere Moglichkeit in den iiblichen Eichfeldtheorien nicht besteht. I n einer friiheren Arbeit [GI wurde
gezeigt, daB die Finder-Struktur der Raum-Zeit zu bestimmten Eichfeldern fuhrt, wenn man die
inneren Symmetrien als Symmetrien eines dreidimensionalen Riemannschen Raumes betrachtet. In
Fortsetzung dieser Arbeit zeigen wir, daI3 geeignete Gleichungen fur diese Eichfelder auf zwei Arten
formuliert werden kiinnen : entweder auf der Basis einer quadratischen Lagrange-Funktion oder, mit
einer entsprechenden Verallgemeinerung der Palatini-Methode, auf der Babis einer Lagrange-Funktion, die linear im Feldstarkentensor des Eichfeldes ist. E s stellt sich heraus, daB diese letztere Moglichkeit zu Gleichungen fiihrt, die ahnlich den Einstein-Gleichungen sind, wobei aber der Finslersche
Cartan Kriimmungstensor anstelle des Riemansschen Kriimmungstensors in die Gleichungen eingeht.
1. Finslerian Origin of Gauge Fields
The Riemannian metric is known to give rise to connection coefficients giving one
the possibility to formulate covariant equations for physical fields. It appears that the
Finslerian metric [7] yields, apart from the conventionally obtainable connection coefficients, the gauge fields, thereby giving a tool for formulating the gauge-covariant
G. S. ASANOV,Derivation of the Finslerian Gauge Field Equations
equations for physical fields exhibiting internal properties [6]. Such an approach is
suggested by the concept of the indicatrix which is an important constituent of a Pinsler
space. By the indicatrices of a n N-dimensional Finder space with the metric function
P ( x , y), where xi are the local coordinates of the background manifold and yi denote the
tangent vectors, one understands (N-1)-dimensional hypersurfaces which lie in the
tangent spaces to the Finder space and which are defined by the equation F ( x , y ) = 1,
where xi are considered t o be fixed and yi arbitrary (the spheres in the case of a Riemannian space). The indicatrix may be regarded as a tool for defining one-to-one
correspondence between the unit tangent vectors li = yi//p(x,y ) and N-1 functionally
independent scalars ua(x,y ) of the zero-degree homogeneity in yi, i.e., ua(x,ky) = ua(x,y)
a t m y k > 0. These ua serve as intrinsic coordinates in the indicatrix. This parametrical
representation li = ti(x, u ) (equation (5.8.2) of [7]) of the indicatrix enables one to cast
any ( 2 , y)-dependent field of the zero-degree homogeneity in yi into the parametrical
form by substituting ti(x,u ) for yi. For example, the parametrical representation of the
Finslerian metric tensor gii(x, y ) = -P P ( x , y)/ayiayiwill read bii(x, u ) = gii(x, t(x, u)).
Using the indicatrix projection factors ti = ati/aua and Uq = Faua/ayi, we can
introduce the metric tensor on the indicatrix as follows: gab(x,u ) = titabii. The projection factors will satisfy the reciprocity relations t:li = Ufya= 0, tiUq = d:, tiUf = hi
and also gab = hiitit6!g a b ~ , u ~= h,, where h,, = g,, - &,l, is the so-called angular
metric tensor, and 1%= Pgii. I n the following, the letters a,b, ... from the beginning of
the alphabet will relate to the parameters u' of the indicatrix and, accordingly, will be
specified over 1, 2, 3 and will be raised and lowered by means of the indicatrix metric
tensor gab, fib. On the other hand the letters i, j, .. . will serve as tensor indices associated with (local) coordinate systems xi of the background manifold, thus they will
range over 1, 2, 3, 4 and will be raised and lowered by means of the Finslerian metric
tensor gii, $i. The summation convention will be adopted. The tangent vectors will be
denoted by yi, instead of the designation &i used in [7].
Following the terminology used in the theory of gauge fields [l--61, we shall regard
the transformations ua = u"(i2) as global internal transformations, whereas their xdependent generalizations ua = ua(x,S ) as local gauge transformations or merely gauge
transformations. Using the projection factors of the Finslerian indicatrix, we can construct the gauge-covariant operator, t o be denoted as Di. To this end, we introduce the
following gauge fields
A aa
b.- - U;,it:,
Nf = ua
where the subscript I i denotes the Finslerian Cartan covariant derivative with respect
to xi. If wa(x,u ) is a vector under the gauge transformations, that is, &,(x, k) = u:wb
(2, u ( x , S ) ) , where ui = aub(x,u)/aiia,then the definition will read
D;w, = awa/axi - A;iwb f N%awu/aub.
It can be verified directly that the gauge fields under study possess the transformation
law u$fe - au2/axi - u C @= u$Afi and u f N ; audaxi = N f , where u$ =
auf/dii', which guarantees that our definition is a gauge vector, that is, D,Wb = utDiwa,
and that the objects
N:, = aN:/axi - aN:/axn N$aN:/aue - N;&v;/aue,
P ; ~=) a&~ ,,/axi
~ ~ - aA:i/axn
N:aA:,/auc - N;aAti/aue
A:,A$ - &N;n,
N:b = aN:/aUb A&,
Aiie = aAii/aue- qrfi,Ati - aqfie/axi q,!eA:i
- Njaqi,/auP - qi,aivyaue
Ann. Physik Leipzig 41 (1984)3
are gauge tensors, e.g., u $ F ~ , ) =
, ~ u~: F $ ) ~ ~ The
~ . notation dc is used for the indicatrix Christoffel symbols:
d, = (aga,lauc agcdJaua- aga,/aud).
The definition (2) can be extended to the case of other type geometric objects in the
conventional way, including the case of two-componentspinors which are treated usually
in the Riemannian approach (see, e.g., Sec. 12 of [8]). An important property of the
definition is that the operator Di appears to be metric, that is,
D,ti = ati/axi
+ r,@+ N$atiJa3= 0 ,
and also DiUT = Dd& = D,b" = 0. Here, rii denote the Finslerian Cartan connection coefficients. The identity (8) can be obtained by differentiating (7) with respect
to ?La.
Substituting (1) in (3)-(6), we find, after straightforward calculations, that the
gauge tensors are expressible explicitly through the Finslerian curvature tensors,
NZn = U!lrKjin,
FtlNin = t i UiRjin -F!1),in,
N'& = -@UiCTk,ry*,
Afaie . = -At aie- taket m un t pkam-
Here, cink = - agi,/ay" is the Cartan torsion tensor of a Finsler space; KIin is the
Rund curvature tensor, and Rkjinand Pknimare Cartan's curvature tensors (written
in the same notation as in the monograph [7]). Notice that Fflbin a",,NZ;, reduces
to the usual Yang-Mills gauge tensor in the particular case when the gauge field Atn
does not depend on the points ua o f the internal space.
2. Generalized Gauge Field Equations
Unless the projection factors are used in the construction of the Lagrangian density
L for the gauge fields Atn(x,u) and Nk(x, u), the simplest choice of L will be quadratic
in the tensor (4),according to
L = JIgaegbdbiibmnFf~)ainF~~)cim,
where the Jacobians are defined as follows: J ( x , u ) = I det(bii))Ill2 and I ( x , u)=
I det(g,b)) Ill2. Taking into account the explicit form (4)of the tensor B'fl)ainand using
the representation (3), we get from (13) that
= ~ J I F $ $ , aLJaAIl,,,= ~JIN$F$,,
aLjaA:, = ~JIALF?;;,~
-~ J I A ~ F E ~ ,
aLJaN;,i = -~JI$,,F$$, aL/aN;,e = -4~1&~yl"li;"b,
aL/aN; = ~ J I F ~ ? ( ~ A : , , , /~ U&Ca N $ / a q .
Derivation of the Finslerian Gauge Field Equations
Here, Ain,i = aA:Jaxi, A:n,c= aA:Jauc, etc. Inserting these relations in the definition of the Euler-Lagrange derivative :
+ ac(aL/aA:n,c)- aL/aAb
E: = ai(aL/aN;,i)+ a,(aL/aNge)- auaNc
~ g=
n ai(aL/aA:n,i)
an 3
where the operator di stands for the total derivative with respect to xi at constant ua,
and d, for the total derivative with respect to ucat constant x;, we obtain the following
result after a simple calculation:
Egn/4JI = DiB’f$,
Et/4JI = $,$in/4JI + A : f l $ , .
Here the operator Di is used for the gauge covariant divergence:
ap i n
~ . ~ ai nd Fain
a (1)b - i ( 1 ) b f Aei ( 1 ) b - A&F$$
f r?iiFT$?*
I n writing the relation (14) we have used the fact that the contraction of the gauge tende f
sor (5) reads Nib = dbNq
+ Ati = dbNi + d, In I + N!db In I ,
where in the second
step the following identity has been taken into account: 0 = @Digd = @(digd - 2Adi
N$dcgab)= 2(di In I - A:i N;dc In I ) . We also have used the following
identity d, In J(x, u ) NPdb In J(x, u)= r;i(x, t ( x , u)),which can be verfied directly.
Following the techniques outlined in the preceding section it is possible to rewrite
the gauge field equations in terms of the proper Finslerian tensors. Indeed, taking into
account the representations (l), (10)-(12) we find, after a comparatively short calculation, that our equations (14) and (15) are equivdent to the following relations involving
the Cartan torsion tensor Ciikand the Cartan curvature tensors Rmnikand Pmnik:
tIU$biiEin/4JI = hP%lk(Ripnli - Cji,ryrRim),
ULE24JI = $,E,”i/4JI f RhiiPkni,P-l - 2ypR~W,,,ni~ryr.
Next, this Finslerian technique offers a simple possibility to use the projection factors of the indicatrix and construct the Lagrangian density W linear in the gauge tensor,
as is done in fact in the proper Einstein gravitational theory, namely,
W = JIt~t~li’$)in,
whereas in the usual gauge approaches the simplest gauge field Lagrangian density is
quadratic in the gauge tensor because the indices b, i of different geometrical nature
cannot be contracted. However, proceeding from such a Lagrangian density, one cannot
simply take the gauge fields A:,, and NE as the field variables (as has been done in
deriving equations (16)- (17)), for it will not furnish us with a reasonable set of equations
for the gauge fields. This notwithstanding, there exists one interesting possibility suggested by the so-called Palatini method of deriving the gravitational field equations
from the Einstein gravitational Lagrangian density (see, e.g., Sec. 3 of [9]). According
to this method, in the Lagrangian density both the Riemannian metric tensor and the
Christoffel symbols are treated as independent field variables, which results in the equations yielding the Einstein equations as well as the representation of the Christoffel
symbols through the Riemannian metric tensor.
Let us apply this method to the Lagrangian density (18). We shall assume the gauge
fields Ain and N i as well as the projection factors t: to be independent field variables,
not postulating the gauge fields to be expressible through the projection factors according
to the Finslerian relations (1) or any other relations. The relationship between the gauge
fields and the projection factors should arise as a solution of resultant equations. The
Ann. Physik Leipzig 41 (1984) 3
metric tensor gat, of the internal space a s well as the associated Christoffel symbols dc
will be regarded as fixed auxiliary fields. Under these conditions, from the definition
(18) and the representations (3)-(4) it follows that the Euler-Lagrange derivatives of
W are equal to
Egn = di(JI(Pntj - Pit:)) + dc(JIN$(Pnti- Pit%))
- JI(-Pit:Agi - tent&
Pnt$4ti teittA$)
- JI(-(Pnti
- d,(JIN:(qV - q?))
- Pit%)ddb,, + (&
- &)
where tan = f " t ? and qy = t:tigeaq:c. Since the definition (18) does not involve the
derivatives of the projection factors, the Euler-Lagrange derivative of W with respect
to t: will be equal t o the negative of the partial derivative which is equal to
awlat; = JIt;(F$$,
because of the definition (18) and the relation aJlat; = -UgJ ensuing from the reciprocity relations for the projection factors.
The question arises if it is possibles to resolve the equations EEn = 0 for the gauge
field A:%? The presentation (19) may be rewritten as
+ N$d,ti + Nit:)
+ ti(ditan+ &ten + NpdcPn)- Pi(dit$ - A$: + N@&)
- tt(diPi + A:itei + Njd,Pi + Nipi),
Et"/JI = Pn(ditk - A$;
where the notation
Ni = -jj(di(JI)
+ d,(JIN$))
has been introduced. I n the right-hand side of (22) the expressions in parentheses strongly resemble the definition of the gauge covariant derivative of the projection factors.
I n the Finslerian approach such a derivative vanishes identically according to equation
(8). Therefore, if we assume the Finslerian relationship (1)between the gauge fields and
the projection factors and take into account the definition (23), then equations (22) will
reduce to merely
Etn/JI = (t""tb - Pit;) N;c.
Owing to the relation (11)from equation (24) follows that
= - 2Ciz1,.yr.
Thus, t h e F i n s l e r i a n g a u g e f i e l d s (1) y i e l d a s o l u t i o n of t h e e q u a t i o n s
E P = 0 if a n d o n l y if t h e F i n s l e r s p a c e s a t i s f i e s t h e c o n d i t i o n
c;,,,yr = 0.
The condition will obviously be satisfied if Cli = 0. The metric function possessing the
property CEi = 0 has been proposed in [lo]; the Finslerian metric function of such a
type has been treated in [ l l ] .
Let us turn to the construction of the object E,". The relations (20) can readily be
represented as follows:
E:/JI = -if nc Ean/ J I
(tznti- Pit:) A:ic.
Derivation of the Finslerian Gauge Field Equations
22 7
Because of the aforesaid in the Finslerian case with the condition (25) satisfied the first
term in the right-hand side of equation (26) vanishes, we shall have, similarly to (24),
E,nIJI = (tanti - t%;)
Substituting here the Finslerian representation (12) we find that
bknU$Et/JI = F-IP$,,, - ynC~,,,yr.
From these observations we may conclude that t h e Finslerian gauge f i e l d s (1)
yield a s o l u t i o n of t h e e q u a t i o n s Egn = 0 a n d E: = 0 if a n d only if t h e
Finsler space satisfies t h e conditions (25) and PAirn
= 0.
Finally, by virtue of the Finslerian representation (lo), the relation (21) may be
rewritten as
-trawlat: = hmihikRkiin
- -hrh”hikRkii,.
2J I
The relation (29) is completely analogous to the representation of the Einstein tensor,
the difference being merely in that the Cartan curvature tensor Rkjinenters the relation
instead of the Riemann curvature tensor, and that the Riemannian metric tensor is
replaced by the Finslerian angular metric tensor hi+ We postpone solving the equations
obtained to another paper. It may be expected that a unification of the approach proposed here with the techniques of Finslerian extension of relativity theory [ll- 141
will permit one to get a deeper insight into the relationship between the gravitational
field and internal symmetries of physical fields.
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(1983). Ed. BERTOTTI,
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Bei der Redaktion eingegangen am 2. Dezember 1983.
Anschr. d. Verf.: Dr. G. S. ASANOV
Department of Theoretical Physics
Moscow State University, 117234 Moscow, USSR
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