Nonlinear Connections and Nearly Autoparallel Maps in General Relativity

8/14/2019 Nonlinear Connections and Nearly Autoparallel Maps in General Relativity

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General Relativity and Gravitation, Vol. 35, No. 5, May 2003 ( C 2003)

Nonlinear Connections and Nearly AutoparallelMaps in General Relativity

Heinz Dehnen 1 and Sergiu I. Vacaru 2

Received January 3, 2003

We apply the method of moving anholonomic frames with associated nonlinear con-nections to the (pseudo) Riemannian space geometry and examine the conditions whenlocally anisotropic structures (Finsler like and more general ones) could be modeled inthe general relativity theory and/or EinsteinCartanWeyl extensions [1]. New classesof solutions of the Einstein equations with generic local anisotropy are constructed.We formulate the theory of nearly autoparallel (na) maps generalizing the conformaltransforms and formulate the Einstein gravity theory on nabackgrounds provided witha set of namap invariant conditions and local conservation laws. There are illustratedsome examples when vacuum Einstein elds are generated by Finsler like metrics andchains of namaps.

KEY WORDS: Anholonomic frame; Finsler metric; EinsteinCartanWeyl theory.

1. INTRODUCTION

There have been constructed various classes of exact solutions of the Einsteinequations, in different dimensions, parametrized by offdiagonal metrics whichposses generic local anisotropy and depend on three and more variables [2]. Theydescribe anisotropic worm-hole and uxtube congurations, Diracwaves prop-agating selfconsistently with three dimensional solitons in anisotropic Taub NUTspacetimes, static black ellipsoid/torus solutions with polarizations of constantsand a number of another nonlinear gravitational and matter eld interactions and/or

1 Fachbereich Physik, Universit at Konstanz, Postfach M 677, D78457, Konstanz, Germany; e-mail:

[email protected] Centro Multidisciplinar de Astrosica - CENTRA, Departamento de Fisica, Instituto Supe-rior Tecnico, Av. Rovisco Pais 1, Lisboa, 1049-001, Portugal; e-mail: [email protected];sergiu [email protected]

807

0001-7701/03/0500-0807/0 C 2003 Plenum Publishing Corporation


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808 Dehnen and Vacaru

static congurations subjected to some anholonomic constraints. It should be em-phasized that such solutions can be generated even in the framework of the generalrelativity theory if the offdiagonal metrics and anholonomic frames are intro-duced into consideration. The tangent Lorentz symmetry may be preserved fora number of solutions which can be emphasized with respect to correspondinglyadapted tetrad (vierbein) bases. For certain models with reduction from higher tolower dimensions the Lorentz symmetry may be violated in the bulk of for someparticular dimensions.

With respect to anisotropic cosmological scenaria, we note that there are pos-sible effects when the anholonomic frames induce additional (to some anisotropicmatter distributions) local anisotropies which also may contribute to the cosmicbackground radiation. In order to compare the anisotropic effects of two differentorigins, it is necessary to perform a rigorous denition and analysis of fundamentaleld equations with respect to anholonomic bases: we have dealings with modelswith constrained dynamics and mixed holonomic and anholonomic variables.

Theories of strings, gravity and matter eld locally anisotropic interactions (inbrief, we shall usesuch terms as anisotropic gravity, anisotropic strings, anisotropicspacetime and so on) have been elaborated following low energy limits of (super)string theory when some anholonomic (super) frame and spinor structures withassociated nonlinear connections are emphasized [3]. Such effective models areinduced in a usual manner if the (super) frame (or, equivalently, vielbein) elds sat-isfy some anholonomy conditions. In other turn, there were developed alternativeapproaches to anisotropic spacetimes and gravity which are grounded on Finslergeometry and generalizations without any connection to modern (super) gravityand string theories [4, 5, 6, 7]. For instance, there is a subclass of models withlocal anisotropy with violations of the local Lorentz invariance which is extendedto some transforms in Finsler geometry [8]. This introduces non infrequently themisunderstanding that an unusual relativity is presented in all Finsler like theo-ries and lied to the misinterpretation that experimentally such Finsler spaces metrather stringent constrains [9].

A surprising result is that Finsler like metrics and their generalizations couldbe found as solutions of the Einstein equations in general relativity and higherdimension gravity models (see [2] and Sections 2 and 8 in this work). The pointis to model various type of locally anisotropic structures by using anholonomicframes on (pseudo) Riemannian spacetimes. This class of anisotropic spacetimesare compatible with the paradigm of the EinsteinLorentzPoincare relativity andphysical interpretation of experiments in such curved spacetimes haveto be adaptedwith respect to anholonomic frames of reference.

The problem of equivalence of spaces with generalized metrics and connec-tions was considered in a series of works by E. Cartan [5, 10] who developed anunied approach to the Riemannian, afne and projective connection spaces, tober bundles, Finsler and another type of curved spaces by using moving frames


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Nonlinear Connections and Nearly Autoparallel Maps in General Relativity 809

and differential forms. Paper [5] contains also the idea on nonlinear connection(Nconnection) associated to an anholonomic frame. The global denition of Nconnection is due to W. Barthel [11] and this concept was developed and appliedby R. Miron and M. Anastasiei [7] in their geometry of generalized Lagrangeand Finsler spaces modeled on vector and tangent bundles. The further geomet-ric extensions and applications in physics are connected with anholonomic (su-per) frames, metrics and connections in spinor spaces and superbundles providedwith Nconnection structures and the geometry of locally anisotropic strings andgravity [3].

The rst purpose of this paper is to demonstrate that anholonomic frame struc-tures with associated Nconnections on (pseudo) Riemannian spacetimes display anew locally anisotropic picture of the Einstein gravity. Here is to be noted that theelaboration of models with locally anisotropic interactions is considered to entailgreat difculties because of the problems connected with the denition of conser-vation laws on spaces with local anisotropy. It will be recalled that, for instance,in special relativity the conservation laws of energymomentum type are denedby the global group of automorphisms (the Poincare group) of the fundamen-tal Minkowski spaces. For the (pseudo) Riemannian spaces one has only tangentspaces automorphisms and for particular cases there are symmetries generated byKilling vectors. No global or local automorphisms exist on generic anisotropicspaces and in result of this fact the formulation of anisotropic conservation laws issophisticate and full of ambiguities. Nevertheless, we shall prove that a variant of denition of energymomentum values for gravitational and matter elds locallyanisotropic interactions is possible if we introduce moving frames and considerthat anisotropies are effectively modeled on (pseudo) Riemannian spacetimes.

The second aim of this paper is to develop a necessary geometric background(the theory of nearly autoparallel maps, in brief, namaps, and tensor integralformalism on multispaces) for formulation and a detailed investigation of conser-vation laws on locally isotropic and/or anisotropic curved spaces. We shall adaptto anisotropic spacetimes the theory of namaps of generalized afne spaces,Einstein-Cartan and Einstein spaces, bre bundles and different subclasses of gen-eralizations of Finsler spaces, (see [12, 13, 14] and [15] as reviews of the secondauthors results published in some less accessible books and journals from formerURSS and Romania).

The problem of denition of the tensor integration as the inverse operation of covariant derivation was posed and studied by A. Mo or [16]. The tensorintegraland bitensor formalism turned out to consist in a new approach to formulationof conservation laws in general relativity [14, 15]. In order to extend the tensorintegral constructions we proposed to take into consideration nearly autoparalleland nearly geodesic [17, 15] maps (in brief, we write namaps, ngtheory andso on) which forms a subclass of local 11 maps of curved spaces with deforma-tion of the connection and metric structures. The third purpose of this work is to


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synthesize the results on namaps and multispace tensor integrals, to reformulatethem for anholonomic (pseudo) Riemannian spacetimes and to propose a variantof denition of conservation laws and energymomentum type values on locallyanisotropic spacetimes.

Our investigations are completed by giving some explicit examples of newsolutions of Einstein equations in general relativity which admit nearly autoparallelmaps and/or Finsler like structures.

The paper is organized as follows: Section 2 outlines the geometry of anholo-nomic frames with associated nonlinear connection structures. The general criteriawhen a Finsler like metric could be embedded into the Einstein gravity is formu-lated. Section 3 is devoted to the theory of nearly autoparallel (na) maps of locallyanisotropic spacetimes. The classication of namaps and corresponding invariantconditions are given in Section 4. In Section 5 we dene the nearly autoparalleltensorintegral on locally anisotropic multispaces. The problem of formulationof conservation laws on spaces with local anisotropy is studied in Section 6. Wepresent a denition of conservation laws for locally anisotropic gravitational eldson naimages of locally anisotropic spaces in Section 7. Some new classes of vacuum and nonvacuum solutions of the Einstein equations, induced by Finslerlike metrics, are constructed in Section 8. In Section 9 we illustrate how a classof vacuum Einstein elds with Finsler like structures can be mapped via chainsof natransforms to the at Minkowski spacetime. The results are outlined inSection 10.

2. ANHOLONOMIC FRAMES AND ANISOTROPIC METRICS

We reformulate the Einstein gravity theory with respect to anholonomicframes and associated nonlinear connections (Nconnection) modelling m dimen-sional local anisotropies in (pseido) Riemannian ( n + m)dimensional spacetimes.Previous approaches based on the moving frame method (tetrads or vierbeins, infour dimensions) considered in general relativity, metricafne and gauge grav-ity theories were developed without any relation to the Nconnections formal-ism (see, for instance, Refs. [18, 19, 20]). The Nconnection geometry was for-mally investigated in details in the framework of generalized Lagrange and Finslergeometries modeled on vector and tangent bundle spaces [7] and with applicationsto locally anisotropic spinor differential geometry and supergravity and super-strings theories [3]. It was proven in a series of works [2] that the Nconnectionmethod is an efcient method of constructing exact solutions of the Einstein equa-tions parametrized by offdiagonal matrices, with some emphasized anisotropicdirections and/or imposed anholonomic constraints, or in order to develop selfconsistent relativistic theories with generic anisotropy, anisotropic distributions of matter and eld interactions, kinetic and thermodynamic processes on (pseudo)Riemannian spacetimes.


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The purpose of this Section is to analyze gravity theories with mixed holo-nomic and anholonomic variables and to demonstrate in which manner locallyanisotropic structures (Finsler like or more general ones) could be induced ingeneral relativity.

2.1. Anholonomy, Local Anisotropy, and Einstein Equations

In this work, spacetimes are modeled as smooth (i.e. class C ) (n + m)dimensional (pseudo) Riemannian manifolds V (n+ m) being Hausdorff, paracom-pact and connected. A spacetime V (n+ m) is enabled with the fundamental structuresof a symmetric metric g and of a linear, in general, nonsymmetric connection

(if we consider anholonomic frames, even the LeviCivita connection became

nonsymmetric) dening the covariant derivation D which is chosen to satisfy themetricity conditions D g = 0. The indices of geometrical objects are given withrespect to a frame vector eld = (i , a ) and its dual = (i , a ). For instance,a covariantcontravariant tensor Q is decomposed as

Q = Q ,

where is the tensor product. A holonomic frame structure on V (n+ m) could bestated by a local coordinate base

= / u

(1)of usual partial derivatives and the dual basis

d = du , (2)

of usual differentials. An arbitrary holonomic frame e could be related to a co-ordinate one by a local linear transform e = A , for which the matrix A

is

nondegenerate and there are satised the holonomy conditions,

e e e e = 0.

Let us consider a ( n + m)dimensional metric parametrized as

g =gi j + N ai N

b j h ab N

e j h ae

N ei h be h ab(3)

with respect to a local coordinate basis (2), du = (dx i , dya ), where the Greek indices run values 1 , 2, . . . , n + m , theLatin indices i , j, k , . . . from the middle of the alphabet run values 1 , 2, . . . , n and the Latin indices from the beginning of thealphabet, a , b, c, . . . run values 1 , 2, . . . , m . The coefcients gi j = gi j (u ), h ae =h ae (u )and N ai = N ai (u ) will be dened by a solution of the Einstein gravitationaleld equations. The local coordinated on V (n+ m) will be distinguished as u =( xi , ya ), or, in brief, u = ( x, y).


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The metric (3) can be rewritten in a block ( n n) (m m) form

g =gi j (u ) 0

0 h ab (u

)(4)

with respect to the anholonomic bases

= (i , a ) =

u = i =

xi

=

xi N bi (u

)

yb, a =

ya

(5)

and

= (d i , a ) = u = d i = dx i , a = ya = dya + N ak (u )dx k , (6)

where the coefcients N a j (u ) from (5) and (6) are treated as the components of

an associated nonlinear connection, Nconnection, structure [11, 7, 3].A frame (local basis) structure on V (n+ m) is characterized by its anholon-

omy coefcients w dened by some relations

= w . (7)

The rigorous mathematical denition of Nconnection is based on the for-malism of horizontal and vertical subbundles and on exact sequences in vectorbundles [11, 7]. In this work we introduce a Nconnection as a distribution whichfor every point u = ( x, y) V (n+ m) denes a local decomposition of the tangentspace

T u V (n+ m) = H u V V u V

into horizontal, H u V , and vertical (anisotropy), V u V , subspaces which is given bya set of coefcients N a j (u

).A Nconnection is characterized by its curvature

ai j =

N ai x j

N a j xi

+ N bi N a j yb

N b j N ai yb

. (8)

Here we note that the class of usual linear connections can be considered as aparticular case of Nconnections when

N a j ( x, y) =abj ( x) y

b .

The elongation (by Nconnection) of partial derivatives and differentials inthe adapted to the Nconnection operators (5) and (6) reects the fact that on the(pseudo) Riemannian spacetime V (n+ m) it is modeled a generic local anisotropycharacterized by anholonomy relations (7) when the anolonomy coefcients arecomputed as follows

w k i j = 0, wk a j = 0, w

k ia = 0, w

k ab = 0, w

cab = 0,

w ai j = ai j , w

ba j = a N

bi , w

bia = a N

bi .


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The frames (5) and (6) are locally adapted to the Nconnection structure, dene alocal anisotropy and are called adapted bases.

A Nconnection structure distinguishes the geometrical objects into horizon-tal and vertical components. Such objects are briey called dtensors, dmetricsand dconnections. Their components are dened with respect to an adapted basisof type (5), its dual (6), or their tensor products (dlinear or dafne transforms of such frames could also be considered). For instance, a covariant and contravariantdtensor Q , is expressed

Q = Q = Q i j i d

j + Q ia i a + Q b j b d

j + Q ba b a .

In this paper, as a locally anisotropic spacetime (in brief, anisotropic space-time) we shall consider a pseudoRiemannian spacetime provided with a metric of signature ( , + , + , + ) (a permutation of signs being also possible) and providedwith an anholonomic frame basis dened by an associated Nconnection structurewhen the coefcients of the metric and Nconnection are imposed to dene theEinstein equations.

A linear dconnection D on an anisotropic spacetime V (n+ m)

D = ( x, y) ,

is given by its hvcomponents, = L

i j k , L

abk , C

i j c , C

abc (9)

where

Dk j = Li j k i , Dk b = L

abk a , Dc j = C

i jci , Dc b = C

abc a .

A metric on V (n+ m) with its coefcients parametrized as an ansatz (3) can bewritten in distinguished form (4), as a metric dtensor (in brief, dmetric), withrespect to an adapted base (6), i. e.

s 2 = g (u) = gi j ( x, y)dx i dx j + h ab ( x, y) ya yb . (10)

Some Nconnection, dconnection and dmetric structures are compatible if there are satised the conditions

D g = 0.

For instance, a canonical compatible dconnectionc

=c Li jk ,

c Labk ,c C i jc ,

c C abc

is dened by the coefcients of dmetric (10), gi j ( x, y) and h ab ( x, y) , and by thecoefcients of a Nconnection,

c

Li jk =

1

2 gi n

(k gn j + j gnk n g jk ),

c Labk = b N ak +

12

h ac k hbc hdc b N d i hdb c N d i , (11)


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cC i j c =12

g ik c g jk ,

cC abc = 12 had (c hdb + b hdc d hbc ) .

The coefcients of the canonical dconnection dene a generalization foranisotropic spacetimes of the well known Christoffel symbols. By a local lin-ear nondegenerate transform to a coordinate frame we obtain the coefcients of the usual (pseudo) Riemannian metric connection.

For a canonical dconnection (9) the components of canonical torsion,

T ( , ) = T ,

T = + w

are expressed via dtorsions

T i. jk = T i

jk = Li jk L

ik j , T

i ja = C

i. ja , T

ia j = C

i ja ,

T i. ja = 0, T a

.bc = Sa.bc = C

abc C

acb , (12)

T a.i j = ai j , T

a.bi = b N

ai L

a.bj , T

a.i b = T

a.bi

which reects the anholonomy of the corresponding adapted frame of reference onV (n+ m); such torsions are induced effectively. With respect to holonomic framesthe dtorsions vanish.

For simplicity, hereafter, we shall omit the up left index c and consider onlyconnections and dconnections dened by compatible metric and Nconnectioncoefcients.

Putting the nonvanishing coefcients (9) into the formula for curvature

R( , ) = R ,

R =

+

+

w

we compute the components of canonical dcurvatures

R.ih . j k = k Li.h j j L

i.hk + L

m.h j L

imk L

m.hk L

im j C

i.ha

a. jk ,

R.ab. j k = k La.bj j L

a.bk + L

c.bj L

a.ck L

c.bk L

a.cj C

a.bc

c. jk ,

P .i j.ka = k Li. j k + C

i. jb T

b.ka k C

i. ja + L

i.lk C

l. j a L

l. jk C

i.la L

c.ak C

i. jc ,

P .cb .ka = a Lc.bk + C

c.bd T

d .ka k C

c.ba + L

c.dk C

d .ba L

d .bk C

c.da L

d .ak C

c.bd ,

S.i j.bc = cC i. j b bC

i. jc + C

h. jbC

i.hc C

h. jcC

ihb ,

S.ab .cd = d C a.bc cC

a.bd + C

e.bc C

a.ed C

e.bd C

a.ec .


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The Ricci dtensor R = R has the components

Ri j = R.k i. jk , Ria = 2 Pia = P .k i.ka ,

(13) Rai = 1 Pai = P .ba .ib , Rab = S

.ca .bc

and, in general, this dtensor is non symmetric.We can compute the scalar curvature R = g R of a d-connection D ,

R = R + S, (14)

where R = g i j Ri j and S = h ab Sab .By introducing the values (13) and (14) into the usual Einstein equations

R 12

g R = k ,

written with respect to an adapted frame of reference, we obtain the system of eldequations distinguished by Nconnection structure [7]:

Ri j 12

( R + S)gi j = k i j ,(15)

Sab 1

2( R + S)h ab = k ab ,

1 Pai = k ai ,2 Pi a = k ia ,

where i j , ab , ai and ia are the components of the energymomentum dtensor eld which includes the cosmological constant terms and possiblecontributions of dtorsions and matter, and k is the coupling constant.

2.2. Finsler Like Metrics in Einstein Gravity

In this subsection, we follow the almost Hermitian model of Finsler geome-try [7] which in our case will be induced on a V 2n (pseudo) Riemannian spacetime.Contrary to standard Finsler constructions we shall admit metrics of nontrivialsignatures.

The locally anisotropic structure is modeled on the manifold T V =T V (n) \{ 0}, where \{ 0} means that there is eliminated the null crosssection of thebundle projection : T V (n) V (n) . There are considered dmetrics of type (10)

with identical ( n n)dimensional blocks for both base and ber components. OnT V (n) we can dene a natural almost complex structure C (a ) ,

C (a )(i ) = / yi and C (a )(/ yi ) = i ,


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where the elongated derivatives (5) and elongated differentials act on the bundleT V being adapted to a nontrivial Nconnection N = { N k j ( x, y)} in T V and C

2(a ) =

I . The pair ( s 2 , C (a )

) denes an almost Hermitian structure on T V with anassociate 2form

= h i j ( x, y)i dx j

and the triad K 2n = (T V , s2 , C (a )) is an almost Kahlerian space. We can verifythat the canonical dconnection (9) satises the conditions

c D X (s 2) = 0, c D X (C (a )) = 0

for any dvector X on T V (n) and has zero hhh and vvv torsions (where h and

v denote the horizontal and vertical components).The notion of Lagrange space [21, 7] was introduced as a generalization of

Finsler geometry in order to geometrize the fundamental concepts in mechanics. Aregular Lagrangian L( xi , yi ) on T V is dened by a continuity class C function L : T V (n) IR for which the matrix

h i j ( x, y) =12

2 L yi y j

(16)

has the rank n . A dmetric (10) with coefcients of form (16), a corresponding

canonical dconnection (9) and almost complex structure C (a ) denes an almostHermitian model of Lagrange geometry.

Metrics h i j ( x, y) of rank n and constant signature on T V which can not bedetermined as a second derivative of a Lagrangian are considered in the socalledgeneralized Lagrange geometry on T V (n) [7].

A subclass of metrics of type (16) consists from those where instead of aregular Lagrangian one considers a Finsler metric function F on V (n) denedas F : T V (n) IR having the properties that it is of class C on T V and onlycontinuous on the image of the null crosssection in T V (n) , the restriction of F on

T V is a positive function homogeneous of degree 1 with respect to the variables yi , i. e.

F ( x, y) = F ( x, y) , IR ,

and the quadratic form on IR 2 , with coefcients

h i j ( x, y) =12

2 F 2 / yi y j (17)

(see (10)) given on T M , is nondegenerate and positive denite in standard Finslercase.

Very different approaches to Finsler geometry, its generalizations and appli-cations are examined in a number of monographs [4, 5, 6, 7, 8, 3, 15] considering


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that for such geometries the usual (pseudor) Riemannian metric interval

ds = gi j ( x) dx i dx j

on a manifold M is changed into a nonlinear one

ds = F ( xi , dx j ) (18)

dened by the Finsler metric F (fundamental function) on T M (it should benoted an ambiguity in terminology used in monographs on Finsler geometry andon gravity theories with respect to such terms as Minkowski space, metric functionand so on).

Geometric spaces with a cumbersome variational calculus and a number

of curvatures, torsions and invariants connected with nonlinear metric intervalsof type (16) are considered as less suitable for purposes of the modern eld andparticle physics.

In our approach to generalized Finsler geometries in (super) string, gravityand gauge theories [3, 15] we advocated the idea that instead of usual geometricconstructions based on straightforward applications of derivatives of (17) fol-lowing from a nonlinear interval (18) one should consider dmetrics (10) withcoefcients of necessity determined via an almost Hermitian model of a Lagrange(16), Finsler geometry (17) and/or their extendedvariants. This way, synthezing the

moving frame method with the geometry of Nconnections, we can investigate in aunied manner a various class of higher and lower dimension gravitational modelswith generic, or induced, anisotropies on some anholonomic and/or KaluzaKleinspacetimes.

Now we analyze the possibility to include ndimensional Finsler metrics into2ndimensional (pseudo) Riemannian spaces and formulate the general criteriawhen a Finsler like metric could be imbedded into the Einstein theory.

Let consider on T V an ansatz of type (3) when

gi j =12

2

F 2/ y

i

y j

and h i j =12

2

F 2

/ yi

y j

i.e.

g =12

2 F 2

yi y j + N k i N

l j

2 F 2

yk yl N l j

2 F 2

yk yl

N k i2 F 2

yk yl2 F 2

yi y j

. (19)

A metric g of signature ( , + , . . . , + ) induced by two Finsler functions F andF (17) (as a particular case F = F ) is to be treated in the framework of generalrelativity theory if this metric is a solution of the Einstein equations on a 2 ndimensional (pseudo) Riemannian spacetime written with respect to a holonomicframe. Here we note that, in general, a Nconnection on a Finsler space, subjectedto the condition that the induced (pseudo) Riemannian metric is a solution of usual


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Einstein equations, does not coincide with the well known Cartans Nconnectionin Finsler geometry [5, 6]. In such cases we have to examine possible compatibledeformations of Nconnection structures [7].

We can also introduces ansatzs of type (3) with gi j and h i j induced by aLagrange quadratic form (16). In Section 8 we shall construct solutions of theEinstein equations following from an ansatz for a generalized Finsler metric,

g =12

2 F 2

yi y j + N k i N

l j i j N

l j kl

N k i kl i j, (20)

where the 2 2 matrix is induced by a Finsler metric via a transform

kl = (C T

)ki 2 F 2

yi y j (C ) jl , (21)

parametrized by a 2 2 matrix ( C )( xi , yk ) and its transposition ( C T )( xi , yk ). Ageneral approach to the geometry of spacetimes with generic local anisotropy canbe developed on imbeddings into corresponding KaluzaKlein theories and ade-quate modeling of locally anisotropic interactions with respect to anholonomic orholonomic frames and associated Nconnection structures. As a matter of princi-ple every type of Finsler, Lagrange or generalized Lagrange geometry could bemodeled on a corresponding KaluzaKlein spacetime.

3. NEARLY AUTOPARALLEL MAPS

The aim of this Section is to formulate the theory of nearly geodesic maps(in brief, ngmaps) [17] and nearly autoparallel maps (in brief, namaps) [de-ned for metric afne space [22] and for anisotropic (super) spaces [15, 22] for(pseudo) Riemannian spacetimes provided with anholonomic frame andassociatedNconnection structures.

Our geometric arena consists from pairs of open regions ( U , U ) of two locally

anisotropic spacetimes, U V (n+ m)

and U V (n+ m)

, and necessary 11 local maps f : U U given by some functions f (u) of smoothly class C r (U ), (r > 2, orr = for analytic functions) and their inverse functions f (u) with correspondingnonzero Jacobians in every point uU and uU .

We consider that two open regions U and U are attributed to a commonfor fmap coordinate system if this map is realized on the principle of coordinateequality q (u ) q(u ) for every point qU and its fimage qU . We note thatall calculations included in this work will be local in nature and taken to referto open subsets of mappings of type U f U . For simplicity, we suppose

that in a xed common coordinate system for U and U the spacetimes V (n+ m)

andV (n+ m) are characterized by a common Nconnection structure, when

N a j (u) = N a j (u) = N

a j (u),


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which leads to the possibility to establish common local bases, adapted to a givenNconnection, on both regions U and U . We consider that on V (n+ m) it is deneda linear dconnection structure with components . . On the space V

(n+ m) thelinear dconnection is considered to be a general one with torsion

T . =.

. + w

.

and nonmetricity

K = D g .

As a particular case we can consider maps to (pseudo) Riemannian spacetimes,when K = 0.

Geometrical objects on V (n+ m) are specied by underlined symbols (for ex-ample, A , B ) or by underlined indices (for example, Aa , Bab ).

For our purposes it is convenient to introduce auxiliary symmetric dconnec-tions, . =

. on V

(n+ m) and .

= .

on V (n+ m) dened, correspondingly,as

. =

. + T

. and

. =

.

+ T . .

We are interested in denition of local 11 maps from U to U characterizedby symmetric, P

. , and antisymmetric, Q

. , deformations:

.

= . + P. (22)

and

T . = T

. + Q. . (23)

The auxiliary linear covariant derivations induced by . and .

are denoted

respectively as ( ) D and ( ) D.Curves on U are parametrized

u = u () = ( xi (), yi ()), 1 < < 2 ,

where the corresponding tangent vector elds are dened

v =du

d =

dx i ()d

,dy j ()

d .

Definition 1. A curve l is called auto parallel, aparallel, on V (n+ m) if its tangent vector eld v satises the aparallel equations

v Dv = v ( ) D v = ()v , (24)

where () is a scalar function on V (n+ m) .


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Let a curve l be given in parametric form u = u (), 1 < < 2 withthe tangent vector eld v = du

d = 0. We suppose that a 2dimensional distribu-tion E

2(l ) is dened along l , i.e. in every point ul a 2-dimensional vector space

E 2(l ) is xed. The introduced distribution E 2(l ) is coplanar along l if everyvector p (u b(0) ) E 2(l ), u

(0)l rests contained in the same distribution after parallel

transports along l , i.e. p (u ()) E 2(l ).

Definition 2. A curve l is called nearly autoparallel, or, in brief, naparallel,on the spacetime V (n+ m) if a coplanar along l distribution E 2(l ) containing thetangent to l vector eld v () , i.e. v () E 2(l ), is dened.

We can dene nearly autoparallel maps of anisotropic spacetimes as an

anisotropic generalization of the constructions for the locally isotropic spaces(see ng[17] and namaps [12, 15]):

Definition 3. Nearly autoparallel maps, namaps, of locally anisotropic space-times are dened as local 11 mappings V (n+ m) V (n+ m) which change everyaparallel on V (n) into a naparallel on V (n+ m) .

Now we formulate the general conditions when some deformations (22) and(23) characterize na-maps:

Let an a-parallel lU is given by some functions u = u ( )(), v = du

d ,

1 < < 2 , satisfying the equations (24). We suppose that to this aparallelcorresponds a naparallel l U given by the same parameterization in a commonfor a chosen namap coordinate system on U and U . This condition holds for thevectors v(1) = v Dv

and v(2) = v Dv(1) satisfying the equality

v(2) = a ()v + b()v(1) (25)

for some scalar functions a () and b() (see Denitions 2 and 3). Putting thesplittings (22) and (23) into the expressions for v(1) and v

(2) from (25) we obtain:

v v v D P . + P . P . + Q . P . = bv v P . + a v , (26)

where

b(, v ) = b 3, and a (, v ) = a + b vb b 2 (27)

are called the deformation parameters of namaps.The algebraic equations for the deformation of torsion Q . should be written

as the compatibility conditions for a given nonmetricity tensor K on V (n+ m)

(or as the metricity conditions if the dconnection D is required to be metric):

D G P . ( G ) K = Q. ( G ) , (28)

where ( ) denotes the symmetrical alternation.


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So, we have proved this

Theorem 1. The namaps from an anisotropic spacetime V (n+ m) to an

anisotropic spacetime V (n+ m)

with a xed common nonlinear connection struc-ture, N a j (u) = N

a j (u), and given dconnections,

. on V

(n+ m) and . onV (n+ m) , are locally parametrized by the solutions of equations (26) and (28) for every point u and direction v on U V (n+ m) .

We call (26) and (28) the basic equations for namaps of locally anisotropicspacetimes. They generalize the corresponding Sinyukovs equations [17] whichwere introduced for isotropic spaces provided with symmetric afne connectionstructure, hold for generalized Finsler metrics modeled on vector and tangent

bundle spaces and consist a particular caseof the namaps of (super) vector bundlesprovided with Nconnection structures [15, 12].

4. CLASSIFICATION OF N aMAPS

Namaps are classied on possible polynomial parametrizations on variablesv of deformations parameters a and b , see formulas (26) and (27)).

Theorem 2. There are four classes of namaps characterized by correspondingdeformation parameters and tensors and basic equations:

1. for na (0) maps, (0) maps,

P (u) = ( )

( is Kronecker symbol and = (u) is a covariant vector deld);2. for na (1) maps

a (u , v ) = a (u)v v , b(u , v ) = b (u)v

and P . (u) is the solution of equations

D( P . ) + P ( P

. ) P

( Q

. ) = b( P

. ) + a (

) ; (29)

3. for na (2) maps

a (u , v ) = a (u)v , b(u , v ) =b v v

(u)v, v = 0,

(30)P . (u) = (

) + ( F

)

and F (u) is the solution of equations

D( F ) + F F

( ) Q

. ( F

) = ( F

) + (

) (31)

( (u), (u), (u), (u) are covariant dvectors);


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4. for na (3) maps

b(u, v

)=

v v v

v v ,

(32)P . (u) = (

) +

,

where is the solution of equations

D = + + Q . , (33)

(u), (u), (u), (u) and (u) are dtensors.

Proof. We sketch the proof respectively for every point in the theorem:

1. It is easy to verify that aparallel equations (24) on V (n+ m) transform intosimilar ones on V (n+ m) if and only if deformations (22) with deformationdtensors of type P (u) = ( ) are considered.

2. Using corresponding to na (1) maps parametrizations of a (u , v )and b(u , v )(see conditions of the theorem) for arbitrary v = 0 on U V (n+ m) andafter a redenition of deformation parameters we obtain that equations(29) hold if and only if P satises (22).

3. In a similar manner we obtain basic na (2) map equations (31) from (26)

by considering na (2) parametrizations of deformation parameters and dtensor.4. For na (3) maps we mast take into consideration deformations of torsion

(23) and introduce na (3) parametrizations for b(u , v )and P into the ba-sic naequations (26). The resulting equations, for na (3) maps, are equiva-lent to equations (33) (with a corresponding redenition of deformation pa-rameters).

We point out that for (0) -maps we have not differential equations on P . (inthe isotropic case one considers a rst order system of differential equations onmetric [17]; we omit constructions with deformation of metric in this Section).

To formulate invariant conditions for reciprocal namaps (when every a-parallel on V (n+ m) is also transformed into naparallel on V (n+ m)) it is convenientto introduce into consideration the curvature and Ricci tensors dened for auxiliaryconnection . :

r .. = [

. ] +

. [

. ] +

w

and, respectively, r = r . . , where [ ] denotes antisymmetric alternation of

indices, and to dene the values:(0) T . =

. T

.

1(n + m + 1)

(. )

( T

. ) ,


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(0) W = r +

1n + m + 1

( w

) r [ ] +

r [ ]

r [ ] 1(n + m + 1)2 2 w [ ] [ w ]

+ 2

w

w

2

w

w

,

(3) T . =

. +

1n + m

q . +

( ) D q

+1

n+

m

1

q . +

( ) D q + ( ) D q

1

n + m

q . + ( ) D q

+1

n + m 1q . +

( ) D q ,

(3) W

. = .. + q . . +

q p

q p

q p[ ] ,

(n + m 2) p = q . . +1

n + m . . q

. .

+ q + q . . + q

. . ,

where q = = 1,

= r +

12

( ) + w

and = .For similar values on V (n+ m) we write, for instance,

= r 12

( ) w

and note that (0) T . ,(0) W . , T

. , T

. , W

. , W

. ,

(3) T . ,(3) W . are

given, correspondingly, by auxiliary connections . ,

. =

. + F

( ) D( F ) ,

. =

. + F D( F

) ,

. =

. + ( F ) ,

. =

. + (

) ,

where = F .


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Theorem 3. Four classes of reciprocal namaps of locally anisotropic space-times are characterized by corresponding invariant criterions:

1. for amaps(0) T . =

(0) T . ,(34)

(0) W . =(0) W . ;

2. for na (1) maps

3 ( ) D P . + P. P

. = r

.(. ) r

.(. )

+ T . ( P . ) + Q

. ( P

. ) + b( P

. ) +

( a ) ; (35)

3. for na (2) maps

T . = T . ,

(36)W . = W

. ;

4. for na (3) maps(3) T . =

(3) T . ,(37)

(3)

W

. =(3)

W

. .Proof.

1. Let us prove that the ainvariant conditions (34) hold. Deformations of dconnections of type

(0)

= + ( ) (38)

dene aapplications. Contracting indices and we can write

=1

m + n + 1

. (39)

Introducing the dvector into previous relation and expressing

= T

+

and similarly for underlined values we obtain the rst invariant conditionsfrom (34). Putting deformation (38) into the formulas for r and r =r we obtain respectively the relations

r r =

[ ] + [

] +

( )w

(40)

and

r r = [ ] + (n + m 1) + w

+ w , (41)


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where

= ( ) D .

Putting (38) into (40) and (41) we can express [ ] as

[ ] =1

n + m + 1r [ ] +

2n + m + 1

w [ ]

1

n + m + 1

[w ]

1n + m + 1

r [ ]

+2

n + m + 1 w

[ ] 1

n + m + 1 [ w

] . (42)

To simplify our consideration we can choose an atransform, parametriz-ed by corresponding vector from (38), (or x a local coordinate cart)the antisymmetrized relations (38) to be satised by dtensor

=1

n + m + 1r +

2n + m + 1

w 1

n + m + 1

w

r 2

n + m + 1

w

+1

n + m + 1

w

(43)

Introducing expressions (38), (42) and (43) into deformation of curvature(39) we obtain the second condition from (34) of a-map invariance:

(0) W =(0) W ,

where the Weyl dtensor on V (n+ m) is dened as

(0) W = r +

1n + m + 1

( w

) r [ ] +

r [ ]

r [ ]

1

(n + m + 1)2 2

w [ ] [

w ]

+ 2

w w

2

w w

.

2. To obtain na (1) invariant conditions we rewrite na (1) equations (29) as toconsider in explicit form covariant derivation ( ) D and deformations (22)and (23):

2 ( ) D P + ( ) D P + ( ) D P + P P + P P

+ P P = T ( P ) + H ( P ) + b( P ) + a ( ) . (44)


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Alternating the rst two indices in (44) we have

2 r ( ) r

( ) = 2 ( ) D P + ( ) D P 2( ) D P

+ P P + P P 2 P P .

Substituting the last expression from (44) and rescaling the deformationparameters and dtensors we obtain the conditions (29).

3. Now we prove the invariant conditions for na (0) maps satisfying condi-tions

= 0 and F F = 0

Let dene the auxiliary dconnection

=

( ) =

+ ( F ) (45)

and write

D = ( ) D F + F

,

where = F , or, as a consequence from the last equality,

( F ) = F

( ) D( F ) D( F ) + (

) .

Introducing the auxiliary connections

=

+ F

( ) D( F ) , =

+ F

D( F )

we can express the deformation (45) in a form characteristic for amaps:

= + (

) . (46)

Now its obvious that na (2) invariant conditions (46) are equivalent withainvariant conditions (34) written for dconnection (46). As a matterof principle we can write formulas for such na (2) invariants in terms of underlined and nonunderlined values by expressing consequentlyall used auxiliary connections as deformations of prime connections onV (n+ m) and nal connections on V (n+ m) . We omit such tedious calcula-tions in this work.

4. Finally, we prove the last statement, for na (3) maps, of this theorem. Let

q = e = 1, (47)

where is contained in

= + ( ) +

. (48)

Acting with operator ( ) D on (46) we write

( ) D q =( ) D q ( q ) e . (49)


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Contracting (48) with we can express

e = ( ) D q ( ) D q q q e .

Putting the last formula in (47) contracted on indices and we obtain

(n + m) =

+ e q + e ( ) D

( ) D . (50)

From these relations, taking into consideration (46), we have

(n + m 1) =

+ e

( ) D q ( ) D q

Using the equalities and identities (48) and (49) we can express the defor-mations (47) as the rst na (3) invariant conditions from (37).To prove the second class of na (3) invariant conditions we introduce twoadditional dtensors:

= r +

12

( ) + w

and

= r 12

( ) w . (51)

Using deformation (47) and (50) we write relation

=

= [ ] [ ]

, (52)

where

= ( ) D + ( + ) ,

and

= ( ) D[ ] + [ ] [ ] .

Let multiply (51) on q and write (taking into account relations (46)) therelation

e = q + [ q ] [ ]q . (53)

The next step is to express trough dobjects on V (n+ m) . To do this wecontract indices and in (51) and obtain

(n + m) [ ] = + eq e [ ] .

Then contracting indices and in (51) and using (52) we write

(n + m 2) = eq + [ ] + e( q ( q )),

(54)


22/44


23/44


systems for namaps, in [15, 12]). We can also formulate the Cauchy problemfor naequations on V (n+ m) and choose deformation parameters (27) as to makeinvolute mentioned equations for the case of maps to a given background spaceV (n+ m) . If a solution, for example, of na (1) map equations exists, we say thatthe anisotropic spacetime V (n+ m) is na (1) projective to the anisotropic spacetimeV (n+ m) . In general, we have to introduce chains of namaps in order to obtaininvolute systems of equations for maps (superpositions of na-maps) from V (n+ m)

to V (n+ m) :

U ng < i1> U 1

ng < i2>

ng < ik 1> U k 1

ng < ik > U k = U ,

where

U V (n+ m) , U 1 V (n+ m)1 , . . . , U k 1 V (n+ m)k 1 , U k V (n+ m)k , U V (n+ m)

with corresponding splitting of auxiliary symmetric connections

.

= < i1> P. + < i2> P

. + + < ik > P

.

and torsion

T . = T

. + < i1> Q. + < i2 > Q

. + + < ik > Q

.

where the indices < i1 > = 0, 1, 2, 3, denote possible types of namaps.

Definition 4. A locally anisotropic spacetime V (n+ m) is nearly conformally projective to the locally anisotropic spacetime V (n+ m) , nc : V (n+ m) V (n+ m) , if there is a nite chain of namaps from V (n+ m) to V (n+ m) .

For nearly conformal maps we formulate:

Theorem 4. For every xed triples ( N a j ,. , U V

(n+ m)) and ( N a j ,. , U

V (n+ m)) and given components of nonlinear connection, dconnection and dmetric being of class C r (U ), C r (U ) , r > 3, there is a nite chain of namaps

nc : U U .The proof is to performed by introducing a nite number of na-maps with

corresponding components of deformation parameters and deformation tensors inorder to transform step by step the coefcients of d-connection into the

).

Now we introduce the concept of the Category of locally anisotropic space-times, C(V (n+ m)). The elements of C(V (n+ m)) consist from objects

Ob C V (n+ m) = V (n+ m) , V (n+ m)< i1 > , V (n+ m)

< i2> , . . .

being locally anisotropic spacetimes, for simplicity in this work, hav-ing common Nconnection structures, and morphisms Mor C(V (n+ m)) ={nc (V (n+ m)< i1> , V

(n+ m)< i2> )} being chains of namaps interrelating locally

anisotropic spacetimes. We point out that we can consider equivalent models of


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physical theories on every object of C(V (n+ m)). One of the main purposesof the nextsection is to develop a dtensor and dvariational formalism on C(V (n+ m)), i.e. onlocally anisotropic multispaces, interrelated with ncmaps. Taking into accountthe distinguished character of geometrical objects on locally anisotropic space-times we call tensors on C(V (n+ m)) as distinguished tensors on locally anisotropicspacetime Category, or dctensors.

Finally, we emphasize that the presented in this Section denitions and theo-rems can be generalized for (super) vector bundles with arbitrary given structuresof nonlinear connection, linear dconnection and metric structures [15, 12].

5. THE NEARLY AUTOPARALLEL TENSORINTEGRAL

The aim of this Section is to dene the tensor integration not only for bitensors,objects dened on the same curved space, but for dctensors, dened on twospaces, V (n+ m) and V (n+ m), even it is necessary on locally anisotropic multispaces.A. Mo or tensorintegral formalism [16] having a lot of applications in classicaland quantum gravity [23, 24, 14] was extended for locally isotropic multispacesin [22]. The unispacial locally anisotropic version is given in [15, 12, 13].

Let T u V (n+ m) and T u V (n+ m) be tangent spaces in corresponding points uU V (n+ m) and uU V (n+ m) and, respectively, T u V

(n+ m) and T u V (n+ m) be their

duals (in general, in this Section we shall not consider that a common coordinati-zation is introduced for open regions U and U ). We call as the dctensors on thepair of spaces ( V (n+ m) , V (n+ m)) the elements of distinguished tensor algebra

T u V (n+ m) T u V (n+ m)

T u V (n+ m) T u V (n+ m)

dened over the space V (n+ m) V (n+ m) , for a given nc : V (n+ m) V (n+ m) .We admit the convention that underlined and nonunderlined indices refer,

respectively, to the points u and u. Thus Q . , for instance, are the components of

dctensor QT u V (n+ m)

T u V (n+ m)

.Now, we dene the transport dctensors. Let open regions U and U be homeo-

morphic to sphere R 2(n+ m) and introduce isomorphism u , u between T u V (n+ m)

and T u V (n+ m) (given by map nc : U U ). We consider that for every dvectorvT u V (n+ m) corresponds the vector u ,u (v ) = vT u V (n+ m) , with componentsv being linear functions of v :

v = h (u , u)v , v = h (u , u)v ,

where h

(u , u) are the components of dctensor associated with 1u ,u

. In a similarmanner we have

v = h (u , u)v , v = h (u , u)v .


25/44


In order to reconcile just presented denitions and to assure the identityfor trivial maps V (n+ m) V (n+ m) , u = u , the transport dc-tensors must satisfyconditions:

h (u , u)h (u , u) =

, h

(u , u)h

(u , u) =

and lim (u u)h (u , u) =

, lim (u u)h (u , u) =

.

Let S pU V (n+ m) is a homeomorphic to p-dimensional sphere and suggestthat chains of namaps are used to connect the regions,

Unc (1) S p nc (2) U .

Definition 5. The tensor integral in u S p

of a dctensor N .... 1 p

(u , u), com-

pletely antisymmetric on the indices 1 , . . . , p , over domain S p , is dened as

N ... (u , u) = I

U (S p)

N .... 1 ... p (u , u)d S

1 ... p

(56)

= (S p )h (u , u)h (u , u) N .... 1 p (u , u)d S 1 p ,where dS 1 p = u 1 u p .

Let suppose that transport dctensors h and h admit covariant derivationsof order two and postulateexistence of deformation dctensor B .. (u , u) satisfyingrelations

D h (u , u) = B

.. (u , u)h

(u , u) (57)

and, taking into account that D = 0,

D h (u , u) = B

.. (u , u)h

(u , u).

By using theformulas for torsion and, respectively, curvature of connection wecan calculate next commutators:

D[ D ]h = R.. + T

. B

.. h

. (58)

On the other hand from (57) one follows that

D[ D ]h = D[ B.. ] + B

..[ | | . B

.. ] . h

, (59)

where | | denotes that index is excluded from the action of antisymmetrization[ ]. From (58) and (59) we obtain

D[ B.. ] . + B[ | | B.. ] = R

.. + T

. B

.. . (60)


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Let S p be the boundary of S p 1 . The Stokes type formula for the tensorintegral (56) is dened as

I S p N .... 1 ... p d S 1 ... p = I S p+ 1 ( p) D[ | N

.... | 1 ... p]d S 1 ... p ,

where( p) D[ | N

.... | 1 ... p ] = D[ | N

.... | 1 ... p ]

+ pT .[ 1 | N .... | 2 ... p ] B

..[ | N

... .| 1 ... p ] + B

..[ | N

.... | 1 ... p ] .

We dene the dual element of the hypersurfaces element d S j1 ... j p as

d S 1...

q p=

1

p!

1...

k p

1...

pd S1 ... p , (61)

where 1 ... q is completely antisymmetric on its indices and

12 ... (n+ m) = | g | , g = det | g | ,

g is taken as the dmetric (10). The dual of dctensor N . .. 1 ... p is dened as

the dctensor N .. 1 ... n+ m p. satisfying

N .... 1 ... p =

1

p!

N .. 1 ... n+ m p. 1 ... n+ m p 1 ... p . (62)

Using (61) and (62) we can write

I S p N .... 1 ... p d S

1 ... p = S p+ 1 p D N .. 1 ... n+ m p 1 . d S 1 ... n+ m p 1 , (63)where

p D N .. 1 ... n+ m p 1 . = D N

.. 1 ... n+ m p 1 . B

.. N

.. 1 ... n+ m p 1 .

+ B..

N .. 1 ... n+ m p 1

. + ( 1) (n+ m p)(n + m p + 1)T [

. N .| . | 1 ... n+ m p 1 ]

. .

To verify the equivalence of (62) and (63) we must take in consideration that

D 1 ... k = 0 and 1 ... n+ m p 1 ... p 1 ... n+ m p 1 ... p = p!(n + m p)![ 11

p ] p .

The developed tensor integration formalism will be used in the next section fordenition of conservation laws on spaces with local anisotropy.

6. TENSOR INTEGRALS AND CONSERVATION LAWS

The denition of conservation laws on curved and/or locally anisotropicspaces is a challenging task because of absence of global and local groups of automorphisms of such spaces. Our main idea is to use chains of namaps from


27/44


a given, called hereafter as the fundamental, locally anisotropic spacetime to anauxiliary one with trivial curvatures and torsions admitting a global group of au-tomorphisms. The aim of this section is to formulate conservation laws for locallyanisotropic gravitational elds by using dcobjects and tensorintegral values, namaps and variational calculus on the Category of locally anisotropic spacetimes. R.Miron and M. Anastasiei [7] calculated the divergence of the energymomentumdtensor on vector bundles provided with Nconnection structure (the same for-mulas hold for (pseudo) Riemannian locally anisotropic spacetimes)

D E =11

U , (64)

where

E = R

12

R

is the Einstein dtensor, and concluded that the dvector

U =12

G R T

G R T

+ R

T

vanishes if and only if the dconnection D is without torsion. On V (n+ m) the dtorsion T could be effectively induced with respect to an anholonomic frame

and became trivial after transition to a holonomic frame.No wonder that conservation laws, in usual physical theories being a con-

sequence of global (for usual gravity of local) automorphisms of the fundamen-tal spacetime, are more sophisticate on the spaces with local anisotropy. Hereit is important to emphasize the multiconnection character of locally anisotropicspacetimes. For example, for a dmetric (10) on V (n+ m) we can equivalently denean auxiliar linear connection D constructed from by using the usual formulas forChristoffel symobls with the operators of partial differential equations (5) chaingedrespectively into the locally adapted to a Nconnection ones (1). We conclude that

by using auxiliary symmetric dconnections, we can also use the symmetric dconnection from (22) we construct a model of locally anisotropic gravitywhich looks like locally isotropic on the spacetime V (n+ m) . More general gravi-tational models with local anisotropy can be obtained by using deformations of connection ,

= + P

+ Q ,

were, for simplicity, is chosen to be also metric and satisfy the Einstein equa-tions (15). The dvector U is interpreted as an effective source of local anisotropyon V (n+ m) satisfying the generalized conservation laws (64). The deformation dtensor P is could be generated (or not) by deformations of type (29)(33) fornamaps.


28/44


From (56) we obtain a tensor integral on C(V (n+ m)) of a dtensor:

N . (u) = I S p N .... 1 ... p (u)h

(u , u)h

(u , u)d S

1 ... p .

We point out that tensorintegrals can be dened not only for dctensors butand for dtensors on V (n+ m). Really, suppressing indices and in (62) and(63), considering instead of a deformation dctensor a deformation tensor

B.. (u , u) = B.. (u) = P

. (u) (65)

(we consider deformations induced by a nctransform) and integration I S p . . . d S

1 ... p in locally anisotropic spacetime V (n+ m) we obtain from (56) atensorintegral on C(V (n+ m)) of a dtensor:

N . (u) = I S p N .. 1 ... p (u)h (u , u)h (u , u)d S1 ... p .

Taking into account (59) we can calculate that curvature

R.. = D[ B.. ] + B

.. [ | | B

.. ] + T

... B

..

of connection . (u) = . (u) + B

.. . (u), with B

.. (u) taken from (65), van-

ishes, R .. = 0. So, we can conclude that a locally anisotropic spacetime V (n+ m)

admits a tensor integral structure on C(V (n+ m)) for dtensors associated to the de-formation tensor B ..

(u) if the ncimage V (n+ m) is locally parallelizable. That way

we generalize the one space tensor integral constructions from [14, 13], were thepossibility to introduce tensor integral structure on a curved space was restricted bythe condition that this space is locally parallelizable. For q = n + m the relations(63), written for dtensor N

. (we change indices , , . . . into , , . . . ) extend

the Gauss formula on C(V (n+ m)):

I Sq 1 N . d S = I Sq

q 1 D N . dV , (66)

where dV = | g

|du 1 . . . du q and

q 1 D N . = D N

. T . N

B.. N

. + B

.. N

. . (67)

Let consider physical values N . on V (n+ m) dened on its density N

. , i. e.

N . = I Sq 1 N

. d S (68)

with this conservation law (due to (66)):

q 1

D N

.

= 0. (69)We note that these conservation laws differ from covariant conservation lawsfor well known physical values such as density of electric current or of


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energymomentum tensor. For example, taking the density E . , with correspond-ing to (67) and (69) conservation law,

q 1 D E

= D E

T

. E

. B

.. E

= 0, (70)

we can dene values (see (66) and (68))

P = I Sq 1 E . d S .

The dened conservation laws (70) for E . are not related with those for energy

momentum tensor E . from the Einstein equations for thealmost Hermitian gravity[7] or with a E with vanishing divergence D E

. = 0. So E

. = E

. . A similar

conclusion was made in [14] for the unispacial locally isotropic tensor integral.

In the case of multispatial tensor integration we have another possibility (rstlypointed in [22] for EinsteinCartan spaces), namely, to identify E

. from (70) with

the naimage of E . on locally anisotropic spacetime V (n+ m) . We shall consider

this construction in the next Section.

7. CONSERVATION LAWS FOR ANISOTROPIC BACKROUNDS

Let us consider a xed background locally anisotropic spacetime V (n+ m) withgiven metric g

= (g

i j, h ab ) and dconnection

. For simplicity, we suppose

that the metricity conditions are satised and that the connection is torsionlessand with vanishing curvature. Considering a nctransform from the fundamentallocally anisotropic spacetime V (n+ m) to an auxiliary one V (n+ m) we are interestedin the equivalents of the Einstein equations on V (n+ m) .

We suppose that a part of gravitational degrees of freedom is pumped outinto the dynamics of deformation dtensors for dconnection, P , and metric, B = (bi j , bab ). The remained part of degrees of freedom is coded into the metricg

and dconnection

.

Following [25, 22] we apply the rst order formalism and consider B

andP as independent variables on V (n+ m) . Using notations

P = P , = ,

B = | g | B , g = | g | g , g = | g | g

and making identications

B + g = g , P

= ,

we take the action of locally anisotropic gravitational eld on V (n+ m)

in this form:

S (g) = (2c1) 1 q uL (g) , (71)


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where

L (g) = B D P D P + (g + B ) P P P P

and the interaction constant is taken 1 = 4c4 k , (c is the light constant and k isNewton constant) in order to obtain concordance with the Einstein theory in thelocally isotropic limit.

We construct on V (n+ m) a locally anisotropic gravitational theory with matterelds (denoted as A with A being a general index) interactions by postulating thisLagrangian density for matter elds

L (m) = L (m) g + B ;

u (g + B ); A;

Au

. (72)

Starting from (71) and (72) the total action of locally anisotropic gravity onV (n+ m) is written as

S = (2c1) 1 q uL (g) + c 1 (m)L (m) . (73)Applying variational procedure on V (n+ m) , similar to that presented in [25] butin our case adapted to Nconnection by using derivations (5) instead of partialderivations (1), we derive from (73) the locally anisotropic gravitational eldequations

= 1(t + T ) (74)and matter eld equations

L (m)

A= 0, (75)

where / A denotes the variational derivation.In (74) we have introduced these values: the energymomentum dtensor for

locally anisotropic gravitational eld

1 t = ( | g | ) 1 L(g)

g= K + P P P P

+12

g

g ( P P P P ), (76)

where

K = D K ,

2K = B P ( g ) B

P ( g ) + g h ( P )

+ g g P G ( B ) + g B P B P ,

2 = D D B + g D D B g D D ( B )


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and the energymomentum dtensor of matter

T = 2L (m)

g g

g

L (m)

g . (77)

As a consequence of (75)(77) we obtain the dcovariant on V (n+ m) conservationlaws

D (t + T ) = 0. (78)

We have postulated the Lagrangian density of matter elds (72) in a form as totreat t + T as the source in (74).

Now we formulate the main results of this Section:

Proposition 1. The dynamics of the locally anisotropic gravitational elds, mod-eled as solutions of the Einstein equations (15) and of matter eld equations on lo-cally anisotropic spacetime V (n+ m) , can be locally equivalently modeled on a back-ground locally anisotropic spacetime V (n+ m) provided with a trivial d-connectionand metric structure (with vanishing dtensors of torsion and curvature) by equa-tions (74) and (75) on condition that the deformation tensor P is a solution of the Cauchy problem posed for the basic equations for a chain of namaps fromV (n+ m) to V (n+ m) .

Proposition 2. The local dtensor conservation laws for Einstein locallyanisotropic gravitational elds can be written in the form (78) for both locallyanisotropic gravitational (76) and matter (77) energymomentum dtensors. Theselaws are dcovariant on the background space V (n+ m) and must be completed with invariant conditions of type (34)((37)) for every deformation parameters of a chain of namaps from V (n+ m) to V (n+ m) .

The above presented considerations consist proofs of both propositions.We emphasize that the nonlocalization of both locally anisotropic and isotrop-

ic gravitational energymomentum values on the fundamental (locally anisotropicor isotropic) spacetime V (n+ m) is a consequence of the absence of global group au-tomorphisms for generic curved spaces. Considering gravitational theories fromthe view of multispaces and their mutual maps (directed by the basic geomet-ric structures on V (n+ m) such as Nconnection, dconnection, dtorsion and dcurvature components, see the coefcients for basic naequations (29)(33)), wecan formulate local dtensor conservation laws on auxiliary globally automorphicspaces being related with some covering regions of the spacetime V (n+ m) by meansof chains of namaps. Finally, we remark that as a matter of principle we can alsouse dconnection deformations in order to modelate the locally anisotropic grav-itational interactions with nonvanishing torsion and nonmetricity. In this case wemust introduce a corresponding source in (78) and dene generalized conservationlaws as in (64).


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8. EINSTEIN SPACES GENERATED BY FINSLER METRICS

In this Section we analyze the conditions when four dimensional (4D) vacuumand nonvacuum solutions of the Einstein equations can be induced by Finsler likemetrics depending on three variables; we construct such solutions in explicit form.

8.1. Two Dimensional Finsler Metrics

There is a class of 2D Finsler metrics

h ab ( xi , yc ) =12

2 F 2 xi , yc

ya yb(79)

generated by the socalled Finsler metric function F = F xi , yc , where theindices i , j, k , . . . run values 1 and 2 on a 2D base manifold V (2) and ycoordinateindices a , b, c, . . . = 3, 4 are used for 2D bers Y x of the tangent bundle T V (2) .Because for Finsler spaces the function F is homogeneous on yvariables we canexpress

F ( x1 , x2 , y3 , y4) = y3 f ( x1 , x2 , z), (80)

where z = y4 / y3 and

f = f ( x, z) = f ( x1 , x2 , z) .= F x1 , x2 , 1, z .

By introducing the function K ( x, z) = f f , where the prime denotes thepartial derivation on z, the metric coefcients (79) are computed

h 3 = h33 = K z2 2 f f z + f 2 , h = h 34 = K z + f f , h4 = h 44 = K .

(81)

We note that if the 2D Finsler metric coefcients formally depended on fourvariables, by introducing the function f ( x1 , x2 , z) one has obtained an explicitdependence only on three coordinates.

Consider a vertical 2D d-metric

h ab ( xi , z) =h3( xi , z) h( xi , z)h( xi , z) h4( xi , z)

(82)

which, by applying a matrix transform (21) can be diagonalized

h a b ( xi , z) = 3( xi , z) 0

0 4( xi , z). (83)


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We can generate by a Finsler metric function a 2D diagonal (pseudo) Riemannianmetric

h a b (t , r ) = 3( xi

) 00 4( xi )(84)

depending only on coordinates x = { xi } if we choose the square of f functions( xi , z) = f 2( xi , z) from (80) to be (see formulas (81))

s( xi , z) = z2 2( xi ) + 1( xi ). (85)

This is the simplest case when a 2D diagonal metric (84) is dened by a trivialFinsler squared f function (85) depending on z2 and two functions 1,2( xi ).

The problem of denition of a corresponding Finsler metric function becamesmore difcult if we try to generate not a diagonal 2D metric (84) depending onlyon two variables ( xi ), but a nondiagonal one depending on three variables ( xi , z)(see (82)). There are three classes of such type Finsler generated 2D metrics.

8.1.1. Euler Nonhomogeneous Equations and Finsler Metrics

The rst class of 2D Finsler metric is dened by the condition when the func-tion s( xi , z) is chosen as to solve the rst equation in (81) when the coefcient

a1( xi , z) of a nondiagonal 2D dmetric (82) are prescribed. The rest of compo-nents of the vertical dmetric, b1( xi , z) and h( xi , z), are not arbitrary ones butthey must be found by using partial derivatives s = s / z and s = 2s / z2 , incorrespondence with the formulas (81).

The basic equation is

z2s 2 zs + 2s = 2a1 (86)

which for a1 = 0 and variables x i treated as some parameters is the socalled Eulerequation [30] having solutions of type

C 1( xi ) z2 + C 2( xi ) z.

By integrating on the zvariable we can construct the solution s(a 1) of (86) for anonvanishing right part,

s(a 1) ( xi , z) = zC (0) ( xi ) + z2C (1) ( xi ) + z

z

const 2

d

const 1

d a1( xi , )

3, (87)

where C (0) ( xi )and C (1) ( xi ) are some arbitrary functions, the index ( a 1) emphasizesthat the Finsler metric is associated to the value a1( xi , )andthe const 1and const 2in the integrals should be chosen from some boundary conditions.


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The 2D d-metric coefcients h( xi , ) and b1( xi , ) are computed

b1

=1

2s and h = zb

1+

s

2,

where s = s(a 1 ) .

8.1.2. The Simplest Case

If thecoefcient b1( xi , ) is given, the squared 2D metric Finsler function is tobe found from the last formula in (81), b1 = s / 2. By considering the coordinates(r , t ) as parameters, and integrating on z we obtain

s(b) = 2 z

const 1 d

const 2 d b1( xi , ) + zS(0) ( xi ) + S(1) ( xi ),where S(0) ( xi ) and S(1) ( xi ) are some functions on variables x i .

The corresponding 2D vertical dmetric coefcients h( xi , ) and a1( xi , )are computed

a1 = s z2b1 2 zh and h = zb1 +s2

,

where s = s(b).

8.1.3. Prescribed Nondiagonal Coefcients

In this case one choose the coefcient h( xi , ) for denition of the squaredFinsler metric function s( xi , ). As the basic equation we consider the equation

zs s = h

which has the solution

s(h) = 1( xi ) + z22( xi ) 2

z

const 1

d

const 2

d h( xi , )

2

depending on two arbitrary functions 1,2 (t , r ) .The explicit formulas for the rest of 2D vertical dmetric coefcients b1( xi , )

and a1( xi , ) follows from

h = zb1 + s2and a1 = s z2b1 2 zh

where s = s(h) .


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8.2. An Ansatz for Finsler Like Vacuum Solutions

Let us consider a particular case of metrics (3) are generated as generalizedLagrange metrics (20) by a diagonalization transform (21) of a Finsler inducedvmetric (83)

g1 + q1 2h3 + n12 4 q1q2h3 + n1n2 4 q1 3 n1 4q1q2h3 + n1n2 4 1 + q22 3 + n22 4 q2 3 n2 4

q1 3 q2 3 3 0n1 4 n2 4 0 4

(88)

with coefcients being some functions of necessary smooth class g1 =

g1( x2), g2 = 1, qi = qi ( x

j, z), n i = n i ( x

j, z), 3 = 3( x

j, z) and 4 = 4( x

j).

Latin indices run respectively i, j, k , . . . = 1, 2 and a , b, c, . . . = 3, 4 and thelocal coordinates are denoted u = ( xi , y3 = z, y4), where one from the coor-dinates x1 , z and y4 could be treated as a timelike coordinate. A metric (88) isdiagonalized,

s2 = g1( x2)(dx 1)2 + (dx 2)2 + a ( x j , z)( ya )2 , (89)

with respect to corresponding anholonomic frames (5) and (6), here we write downonly the elongated differentials

z = dz + qi ( x j , z)dx i , y4 = dy4 + n i ( x j , z)dx i .

The system of Einstein eld equations (15) reduces to four nontrivial secondorder partial differential equations on z for functions qi ( x j , z), n i ( x j , z), 3( x j , z)and 4( x j ),

P3i =qi

2 3

1 3

3 z

2

2 3 z2

,

(90)

P4i = 4

4 3n i z

3 z

2 2n i

z2.

There are two possibilities to satisfy the equations (90): 1/ If the function 3( x j , z) is a nonvanishing solution of

1 3

3 z

2

2 3 z2

= 0, (91)

the Nconnection coefcients qi ( x j , z) could take arbitrary values in correspon-dence to a stated Cauchy problem. 2/ The coefcients qi ( x j , z) 0 if the function 3 does not satisfy the condition (91); we have only one anisotropic directiondistinguished by some nontrivial functions n i ( x j , z).


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The general solution of (90) is written

n i ( xj

, z) = pi (0) ( xi

) z

0 exp 3( x j , )

2 d + n i (0) ( xi

), (92)

where the functions pi (0) ( xi ) and n i (0) ( xi ) have to be dened from some boundary(initial) conditions.

We conclude this Section by formulating the rule for generation by Finsler likemetrics of vacuum solutions of the Einstein equations. Firstly, we take a Finslermetric function (80) and following (81) we induce a nondiagonal 2D vmetric(82). Diagonalizing (21), we obtain a hmetric (83). If the induced coefcient 4depends only on horizontal variables x j , theansatz(88) solves thevacuum Einsteinequations under the conditions that the functions qi ( x j , z), n i ( x j , z), 3( x j , z) and 4( x j ) satisfy the conditions (91) and (92). Instead of starting the procedure byxing the Finsler metric function we can x a necessary type coefcient h3 (h orh 4) and then, as was stated in subsection 8.1.1 (8.1.2 or 8.1.3), we must denethe corresponding class of Finsler metrics. Finally, diagonalizing the vmetric, weobtain the coefcients which must be put into the ansatz (88).

We restricted our constructions only for some trivial Finsler like inducedhcomponents of dmetrics, for simplicity, considering h-components of typegi j = diag [a1( x2), 1] . To induce more general Finsler like hmetrics is possibleby a similar to the presented for vsubspaces procedure.

8.3. Non Vacuum Locally Anisotropic Solutions

In this subsection we generalize the ansatz in order to induce nonvacuumsolutions of the Einstein equations (88). We consider a 4D metric parametrized

g1 + q12h 3 + n12 4 q1q2h3 + n1n2 4 q1 3 n1 4q1q2h3 + n1n2 4 g2 + q2 2 3 + n2 2 4 q2 3 n2 4

q1 3 q2 3 3 0n1 4 n2 4 0 4

(93)

with the coefcients being some functions of necessary smooth class g1 = (r ), g2 = 1/ (r ), qi = qi ( x j , z), n i = n i ( x j , z), 3 = 3( x j , z) and 4 = 4( x j , z) where the hcoordinates are denoted x1 = t (the time like coordinate)and x2 = r . Our aim is to dene the function (r ) which gives a solution of theEinstein equations with diagonal energy momentum dtensor

= [ , p2 , p3 , p4]

for a matter state when p2 = and p3 = p4 .


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Putting these values of hmetric into (13) we compute

R11 = R22 =

1

2,

where the dot denote the partial derivative on r . Considering the 2D hsubspaceto be of constant negative scalar curvature,

R = 2 R11 = m2 ,

and that the Einstein equations with anholonomic variables (15) are satised weobtain the relation

= m2 = 33 = 44 . (94)

The solution of (94) is written in the form = (m2r

2 M ) which denes a

2D hmetric

ds 2(h) = (m2r 2 M )dt 2 + (m2r 2 M ) 1dr 2 (95)

being similar to a black hole solution in 2D JackiwTeitelboim gravity [26] anddisplay many of attributes of black holes [27, 28, 29] with that difference that theconstant m is dened by 4D physical values in vsubspace and for deniteness of the theory thehmetric should be supplied with theequationsfor thevcomponentsof the dmetric which in our case is

2 4 z2

12 4

4 z

2 1

2 3 3 z

4 z

+ 2

3 4 = 0.

Prescribing one of the functions 3 , or 4 , the second one is to be dened byintegration on the zvariable (see detalies in [2]). One has a forth order partialdifferential equation for the metric function f ( xi , z), see (80) if we try to inducethe horizontal part in a pure Finsler like manner.

9. NEARLY CONFORMALLY FLAT GRAVITATIONAL FIELDS

We analyze chains of namaps which by corresponding deformation param-eters and deformations of connections induce a vacuum dmetric (89) (for nonvacuum metrics considerations are similar).

The nontrivial canonical dconnection coefcients (11) and dtorsions (12)are respectively computed

L112 = L121 =

12

ln |a1 | x2

, L222 = 12

a1 x2

; C 333 =12

ln | 3 | z

;

L33i

=1

2 3

3

xi qi

h3

z, L3

4i=

4

2 3

n i

z, L4

3i=

1

2

n i

z,

L44i =12

ln | 4 | xi

, (96)


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and

T 33i = T 3i 3 =

qi

z

L33i , T 34i = T

3i 4 = L

34i ,

T 43i = T 4i 3 =

n i z

L43i , T 44i = T

4i 4 = L

44i ,

T 312 = T 321 =

q1 x2

q2 x1

q2q1 z

+ q1q2 z

,

T 412 = T 421 =

n1 x2

n2 x1

q2n1 z

+ q1n2 z

. (97)

The obtained values allow us to dene some namap chains, for instance, from

the Minkowski spacetime V [0] = M 3,1

, where it is pointed the spacetime signature(3,1), to a curved one with local anisotropy, V (2+ 2) , provided by a metric (88)(equivalently, a dmetric (89)).

In this Section we shall consider sets of invertible namaps (the inverse to anatransform is also considered to be a namap) when we could neglect quadraticterms like P P and F F [17] in the basic naequations (29), (31) and (33), forsimplicity, taken in a nonsymmetrized form:

r for na (1) maps

D P. Q

. P

= b P

. + a

; (98)

r for na (2) maps the deformation dtensor is parametrized

P . (u) = ( F ) (99)

and the basic equations are taken

D F Q. F

= F

+

; (100)

r for na (3) maps the deformation dtensor is parametrized

P .

(u) = , (101)

and the basic equations

D = + + Q . . (102)

9.1. Chains of na (1) Maps

We illustrate that the canonical dconnections and dtorsions of the men-tioned vacuum metrics with local anisotropy could be induced by a chain of threena (1) transforms

V [0]na [1]1 V [1]

na [2]1 V [2]na [2]1 V (2+ 2) . (103)


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The rst step in this chain is dened by some deformations of the symmetric(22) and antisymmetric (23) parts of the dconnection

na [1]1 : [1] = [0] + [1] P ,

[1] T =[0] T +

[1] Q ,

where it is considered that on the at background space is chosen a system of coordinates for which [0] = 0 and

[0] T = 0. The values

[1] b = ln L112 ,[1] a = 0, [1] P = L

112 = L

121 ,

[1] Q = 0

solve the system of na (1) equations (98). The resulting auxiliary curved spaceV [1] is provided with a dcovariant differential operator [1] D , dened by thedconnection [1] =

[1] +[1] T .

The second step in the chain (103) is parametrized by the deformations

na [2]1 :[1] =

[1] +[2] P ,

[2] T =[1] T +

[2] Q ,

with associated values

[2] b = L211 1 [1] D L211 ,

[2] a = 0, [2] P = L211 ,

[2] Q = 0

solving the system of na (1) equations (98) for xed initial data on the auxiliaryspace V [1] . The resulting auxiliary curved space V [2] is provided with a dcovariantdifferential operator [2] D , dened by the dconnection [2] =

[2] +[2] T .

The third, nal, map na [3]1 , which induces a locally anisotropic spacetimeV (2+ 2) with the dconnection (96) and dtorsions (97), could be treated as trivialna (1) map with a simple deformation of the torsion structure

[3] T =[2] T +

[3] Q ,

which is given by the set of values ( [3] b = 0, [2] a = 0, [2] P = 0,[2] Q =

T ), where T

has just the components (97).So, we have proved that the vacuum Einstein elds given by a metric (88)

(equivalently, by a dmetric (89), induced by a Finsler like metric, are nearlyconformally at, being related ba chain of two na (1) maps and a deformationof torsion structure with the Minkowski spacetime. On every spacetime, on theinitial V [0] , two auxiliary, V [1] , and V [2] , and on the nal image, V (2+ 2) one holdsna (1) invariant conditions of type (35).


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The considered vacuum anisotropic spacetimes could be also induced by achain of three na (2) maps from the Minkowski spacetime,

V [0] na[1]2 V [1] na

[2]2 V [2] na

[3]1 V

(2+ 2) . (104)

The rst na 2map from this chain is dened by some deformations of thesymmetric part (22), with the deformation dtensor parametrized as (99), andantisymmetric (23) part of the dconnection

na [1]2 :[1] =

[0] +[1] [1] F ,

[1]

T

=[0]

T

+[1]

Q ,



[1] F = L112 ,

[1] = ln L112 ,[1] = 0, [1] = 1 ,

[1] Q = 0


[1] .

The second na 2map from (104) is parametrized asna [2]2 :

[2] =[1] +

[2] [2] F ,

[2] T =[1] T +

[2] Q ,

with the values[2] F = L

211 ,

[2] = L211 1 [1] D L211 ,

[2] = 0, [2]

= 1 ,[2] Q = 0

solving the system of na (2) equations (100). The second resulting auxiliary curvedspace V [2] is provided with a dcovariant differential operator [2] D , dened bythe dconnection [2] =

[2] .The third step in the chain (104) is a trivial na 2map with pure deformation

of dtorsions given by the values[3] F = 0,

[3] = 0, [3] = 0, [3] = 0, [3] Q = T

,

where the dtorsions are those from (97). We can conclude that a vacuum EinsteinV (2+ 2) spacetime provided with a Finsler like induced metric of type (88) (equiva-lently, by a dmetric (89), could be alternatively induced by a chain of na 2mapsfor which, on every stape, one holds the invariant conditions (36). This is a partic-ular property of this class of dmetrics. We shall prove in the next subsection that


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in a similar fashion we can consider chains of na 3mapa for inducing such typesof vacuum locally anisotropic spacetimes.


Finally, we elucidate the possibility of inducing vacuum Finsler like inducedEinstein spaces by using chains of na (3) maps,

V [0]na [1]3 V [1]

na [2]3 V [2]na [3]3 V (2+ 2) . (105)

Now, the rst na 3map is dened by some deformations of the symmetric

part (22), with the deformation dtensor parametrized as (101), and antisymmetric(23) part of the dconnection

na [1]3 :[1] =

[0] +[1] [1] ,

[1] T =[0] T +

[1] Q ,



[1] P = [1] 12 [1]1 = L112 , [1] = [1]1 [1] = 0, [1]

= 1 ,[1] Q = 0


[1] .The second na 3map from (105) is stated by

na [2]3

: [2]

= [1]

+ [2] [1] ,

[2] T =[1] T +

[2] Q ,

with the values

[2] P =[2] 11 [2]2 = L211 ,

[2] = [2] D [1]2 [2] = 0, [2]

= 2 ,[2] Q = 0

dening a solution of the system of na (3) equations (102).The third step in the chain (105) should be treated asa trivial na 3mapwhenall

parameters and deformations vanishes excepting a pure deformation of dtorsions,[3] Q = T

, where the dtorsions are those from (97).


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10. OUTLINE AND CONCLUSIONS

In this paper, we have studied the problem of formulating the Einstein gen-eral relativity theory with respect to anholonomic frames with associated nonlinearconnection structures, when the dynamics of gravitational and matter eld inter-actions is described by mixed sets of holonomic and anholonomic variables.

We demonstrated that by using anholonomic frames on (pseudo) Riemannianspacetimes we can model locally anisotropic interactions and structures (Finslerlike and more general ones) which are dened in the framework of the generalrelativity theory. The important questions connected with the status of frame sys-tems in Einst

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Nonlinear Connections and Nearly Autoparallel Maps in General Relativity

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