1. Introduction
The geometry of Lagrange spaces is applied to the description of classical general relativity and electrodynamics. First, the Einstein equations are given in a new form, where the geometrical objects related to the internal variables are separated from those related to the external variables. After this, several special Lagrange spaces are analyzed. The almost Riemannian Lagrange spaces are rather simple for explicit calculations and they recover all classical results of general relativity and electrodynamics.
The theory of Finsler spaces with \((\alpha, \beta)\)-metrics was
introduced by Matsumoto [1]. The natural extension of this
theory is based on the canonial Cartan nonlinear connection \(N\)
[2].
In [3], Bucataru studied the Finsler
space with \((\alpha, \beta)\)-metrics have nonholonomic frames which
are useful for unifying theories in theoretical physics.
The notion of Lorentz nonlinear connection \(N\) was introduced
by Hassan, which depends only on the metric \(L(\alpha,
\beta)\), so the spaces \(FL^{n}=(M, L(\alpha, \beta), N)\) are
called the Finsler-Lagrange spaces with \((\alpha, \beta)\)-metrics.
This theory has been applied in the study of gravitational and
electromagnetic [4,5].
The present paper organized the Euler-Lagrange spaces with \((\alpha,
\beta)\)-metrics and Lorentz equations. Also, Einstein equations
for Lagrange space with \((\alpha, \beta)\)-metrics, in particularly
Randers metric by means of canonical \(N\)-metrical connection is presented.
2. Preliminaries
The present section deals with some fundamental concepts and facts of
Finsler-Lagrange geometry [
6,
7,
8].
2.1.Finsler-Lagrange space with \((\alpha,\beta)\)-metrics
Let \(F^{n}=(M, F(x,y))\) be a Finsler space with \((\alpha,
\beta)\)-metric and \(F(x,y)\) be a fundamental function of the form
$$ F(x, y)=\hat{F}(\alpha(x, y), \beta(x, y)),$$
where, \(\hat{F}\) is a differentiable function of two variables:
$$\alpha^{2}(x, y)=a_{ij}(x)y^{i}y^{j},$$
$$\beta(x, y)=b_{i}(x)y^{i}.$$
The notion \(\alpha\) in the above equation represent the pseudo-Riemannian metric on the base manifold
\(M\) which gives the gravitational part of \(F(x, y)\) whereas \(\beta\) is
the eletromagnetic \(1\)-form on \(M\).
Denoting \(L(\alpha(x, y),\beta(x,y))=\hat{F}(\alpha(x,
y),\beta(x,y))\), gives \(L^{n}=(M, L)\) Lagrange
space with fundamental metric tensor \(g_{ij}(x, y)\) of the form
$$g_{ij}=\frac{1}{2}\frac{\partial^{2}L}{\partial y^{i}\partial
y^{j}}.$$
Which further modified as
\begin{equation}
g_{ij}=\rho
a_{ij}+\rho_{0}b_{i}b_{j}+\rho_{-1}(b_{i}l_{j}+b_{j}l_{i})+\rho_{-2}l_{i}l_{j},
\end{equation}
(1)
where \(b_{i}=\frac{\partial \beta}{\partial y^{i}}\),
\(l_{i}=a_{ij}y^{j}=\alpha \frac{\partial \alpha}{\partial y^{j}}\),
\(\rho\), \(\rho_{0}\), \(\rho_{-1}\) and \(\rho_{-2}\) are invariants of
the space \(L^{n}\):
\begin{equation}
\rho=\frac{1}{2\alpha}L_{\alpha},\,\;\;\;\;\;\;\;\;\,\rho_{0}=\frac{1}{2}L_{\beta\beta},\,\;\;\;\;\;\;\;\;\,\rho_{-1}=\frac{1}{2}L_{\alpha\beta},
\,\;\;\;\;\;\;\;\;\,\rho_{-2}=\frac{1}{2\alpha^{2}}(L_{\alpha\alpha}-\frac{1}{\alpha}L_{\alpha})
\end{equation}
(2)
with
\begin{equation*}
L_{\alpha}=\frac{\partial L}{\partial
\alpha},\,\,\;\;\;\;\;\;\;\;\,\,L_{\beta}=\frac{\partial L}{\partial
\beta},\,\;\;\;\;\;\;\;\;\,\,L_{\alpha\alpha}=\frac{\partial^{2} L}{\partial
\alpha^{2}}\,\;\;\;\;\;\;\;\;
L_{\beta\beta}=\frac{\partial^{2} L}{\partial
\beta^{2}},\,\,\;\;\;\;\;\;\;\;\,L_{\alpha\beta}=\frac{\partial^{2} L}{\partial
\alpha \partial \beta}.
\end{equation*}
The totally symmetric Cartan tensor defined by
\[C_{ijk}=\frac{1}{4}\frac{\partial^{3}L}{\partial y^{i}\partial
y^{j}\partial y^{k}}=\frac{1}{2}\frac{\partial g_{ij}}{\partial
y^{k}}.\]
By means of \(g_{ij}\) from (1) taking into account the formulae from [
9], one obtains
\begin{equation}
2C_{ijk}=\sigma_{(i,j,k)}(\rho_{-1}a_{ij}b_{k}+\rho_{-2}a_{ij}l_{k}+\frac{1}{3}r_{-1}b_{i}b_{j}b_{k}+r_{-2}b_{i}b_{j}l_{k}+r_{-3}b_{i}l_{j}l_{k}+\frac{1}{3}r_{-4}l_{i}l_{j}l_{k}),
\end{equation}
(3)
where \(\sigma_{(i,j,k)}\) means the cyclic sum in the indices \(i\),
\(j\), \(k\).
2.2.Variational problem and Lorentz non-linear Connection
Let \(L:TM\longrightarrow\mathbb{R}\) be a regular Lagrangian and
\(c:t\in[0,1]\longrightarrow (x^{i}(t))\in U\subset M\) be a regular
curve. The functional defined as follows:
\[
I(c)=\int^{1}_{0}L(\alpha(x, y), \beta(x, y))dt,
\]
gives Euler-Lagrange equations as:
\begin{equation}
E_{i}(L)=\frac{\partial L}{\partial
x^{i}}-\frac{d}{dt}(\frac{\partial L}{\partial
y^{i}})=0,\,\,\,\,\,y^{i}=\frac{dx^{i}}{dt}.
\end{equation}
(4)
The co-vector \(E_{i}(L)\) can also be expressed as:
\begin{equation}
E_{i}(L)=E_{i}(\alpha^{2})+2\frac{\rho_{-1}}{\rho}E_{i}(\beta)+2\frac{d\alpha}{dt}\frac{\partial
\alpha}{\partial y^{i}}.
\end{equation}
(5)
If \(c\) is an extremal curve,
i.e., \(c\) is a solution of
Euler-Lagrange equation (4), then along \(c\) the energy
of a Lagrangian \(L\) is:
\[
E_{L}=y^{i}\frac{\partial L}{\partial y^{i}}-L.
\]
Now, let us fix the parametrization of the curve \(c\) by a natural
parameter \(t=s\), with respect to the Riemannian metric \(\alpha^{2}(x,
dx/dt)\) given by:
\begin{equation}
ds^{2}=\alpha^{2}(x, \frac{dx}{dt})dt^{2}.
\end{equation}
(6)
Thus, along the extremal curve \(c\) parameterized by arc lengths
\(t=s\), we have \(\alpha^{2}(x, dx/ds)=1\) and \(d\alpha/ds=0\),
\(dL/ds=0\), which implies that \(d\beta/ds=0\),
\(dL_{\alpha}/ds=0\),\(dL_{\beta}/ds=0\).
Since \(E_{i}(\beta)\) is given by:
\begin{equation}
\label{e:r7}E_{i}(\beta)=F_{ij}(x)\frac{dx^{j}}{ds},\,\,\,\,F_{ij}=\frac{\partial
b_{j}}{\partial x^{i}}-\frac{\partial
b_{i}}{\partial x^{j}}.
\end{equation}
Then, Miron and Hassan [
5] obtained the following theorems:
Theorem 1.
Consider the natural parametrization \(t=s\), the Euler-Lagrange equations
of the Lagrangian \(L(\alpha, \beta)\) are given by
\begin{equation}
E_{i}(\alpha^{2})+2\frac{\rho_{-1}}{\rho}F_{ij}(x)y^{j}=0,\,\,\,y^{i}=\frac{\partial
x^{i}}{\partial s}.
\end{equation}
(8)
Choosing \(\gamma^{i}_{jk}(x)\) as the Christoffel symbols of
pseudo-Riemannian metric \(\alpha^{2}\) and
\[
\sigma(x,
y)=\frac{\rho_{-1}}{\rho},\,\,\,\,F^{i}_{j}(x)=a^{ih}(x)F_{hj}(x),
\]
we have the following result.
Theorem 2.
The Euler-Lagrange equation (8) are equivalent to the
Lorentz equations as:
\begin{equation}
\frac{d^{2}x^{i}}{ds^{}2}+\gamma^{i}_{jk}(x)\frac{dx^{j}}{ds}\frac{dx^{k}}{ds}=\sigma
(x, \frac{dx}{ds})F^{i}_{j}(x)\frac{dx^{j}}{ds}.
\end{equation}
(9)
If Euler-Lagrange equations \(E_{i}(L)\)=0, then we determine a
canonical semispray \(S\) as:
\[
S=y^{i}\frac{\partial}{\partial x^{i}}-2
G^{i}\frac{\partial}{\partial y^{i}},
\]
where \(2G^{i}(x, y)=\gamma^{i}_{jk}(x)-\sigma(x,
y)F^{i}_{j}(x)y^{j}\). Then, the integral curve of \(S\) are given by
the Lorentz equation (9).
Now, let us consider the non-linear connection \(N\) with the
coefficients as:
\[
N^{i}_{j}=\gamma^{i}_{jk}(x)y^{k}-\sigma(x)F^{i}_{j}(x)
\]
Thus, the variation of autoparallel curves of a non-linear
connections work should be progressed in 2003.
Since the autoparallel curves of \(N\) are given by the Lorentz
equation (8), we call it as the Lorentz non-linear
connection of the metric \(L\) and so \(FL^{n}\) is the Finsler-
Lagrange \((\alpha, \beta)\)-metric \(L(\alpha, \beta)\) and the
Lorentz non-linear connection \(N\). The semispray \(S\) associated to
\(N\) has the coefficients as:
\begin{equation}
2G^{i}=N^{i}_{j}y^{j}.
\end{equation}
(10)
2.3.Properties of the Lorentz non-linear connection
- (i) The Berwald connection \(B\Gamma(N)=(B^{i}_{jk}(x,y),
0)\) of \(N\) has the coefficients
\[
B^{i}_{jk}(x,
y)=\gamma^{i}_{jk}(x)-\dot{\sigma}_{k}F^{i}_{j}(x),
\]
where \(\dot{\sigma}_{k}=\frac{\partial \sigma}{\partial y^{k}}\)
- (ii) The weak torsion of \(N\) is
\[
L^{i}_{jk}=\dot{\sigma}_{j}F^{i}_{k}(x)-\dot{\sigma}_{k}F^{i}_{j}(x).
\]
Clearly, if \(b_{i}=grad_{i}\varphi(x)\), then \(L^{i}_{jk}=0\).
- (iii) The adapted bases are
\[
\frac{\delta}{\delta x^{i}}=\frac{\partial}{\partial
x^{i}}-N^{j}_{i}\frac{\partial}{\partial y^{j}}.
\]
- (iv) The integrability tensor
\[
R^{i}_{jk}=\frac{\delta N^{i}_{j}}{\delta x^{k}}-\frac{\delta N^{i}_{k}}{\delta
x^{j}},
\]
of \(N\) is
\[
R^{i}_{jk}=y^{h}\rho^{i}_{hjk}(x)+\sigma_{j}F^{i}_{k}-\sigma_{k}F^{i}_{j}-\sigma(F^{i}_{j|k}-F^{i}_{k|j}),
\]
where \(\sigma_{j}=\frac{\partial \sigma}{\partial x^{j}}\) and
`\(|\)’ if the covariant derivative with respect to the Levi-Civita
connection of \(\alpha^{2}\) and \(\rho^{i}_{hjk}(x)\) is the
curvature tensor of the Levi-Civita connection.
- (v) The Lorentz non-linear connection \(N\) is integrable if
and only if the d-tensor of integrability \(R^{i}_{jk}\) vanishes.
- (vi) The dual basis \((dx^{i}, \delta y^{i})\) of
\((\delta/\delta x^{i}, \partial/\partial y^{i})\) is determined by
\begin{eqnarray*}
\delta y^{i}&=&dy^{i}+N^{i}_{j}dx^{j},\\
&=&dy^{i}+\gamma^{i}_{jk}y^{k}dx^{j}-\sigma F^{i}_{j}dx^{j},\\
&=&\delta y^{i}-\sigma F^{i}_{j}dx^{j}.
\end{eqnarray*}
- (vii) The autoparallel curves of Lorentz non-linear
connection are given by the system of differential equations:
\[
\frac{\delta y^{i}}{dt},\,\,\,\,\,y^{i}=\frac{dx^{i}}{dt}.
\]
- (viii) In the parametrization \(S\) with \(\alpha^{2}(x,
dx/ds)=1\), the property (vii) are the Lorentz equation
(8).
- (ix) The exterior differential of \(1\)-forms \(\delta y^{i}\)
of the form
\[
d\delta y^{i}=\frac{1}{2}R^{i}_{jk}dx^{k}\wedge
dx^{j}+B^{i}_{jk}\delta y^{k}\wedge dx^{j}.
\]
2.4.Canonical N-metrical connection
The metric \(N\)-linear connection is called the canonical N-linear
connection or the Cartan connection of the Lagrange space. The space
\(FL^{n}=(M, L(\alpha, \beta), N)\) has a canonical \(N\)-linear
connection \(C\Gamma(N)\) with the coefficients \((L^{i}_{jk},
C^{i}_{k})\) given by the generalized Christoffel symbols are:
\begin{eqnarray}
\label{e:r11}\left.\begin{array}{c}
L^{i}_{jk}=\frac{1}{2}g^{is}\left(\frac{\delta g_{sk}}{\delta x^{j}}+\frac{\delta g_{js}}{\delta x^{k}}-\frac{\delta g_{jk}}{\delta x^{s}}\right)
\\\\
C^{i}_{jk}=\frac{1}{2}g^{is}\left(\frac{\partial g_{sk}}{\partial y^{j}}+\frac{\partial g_{js}}{\partial y^{k}}-\frac{\partial g_{jk}}{\partial y^{s}}\right) \\
\end{array}\right\}
\end{eqnarray}
(11)
The \(1\)-form connection \(C\Gamma(N)\) is:
\[
\omega^{i}_{j}=L^{i}_{jk}dx^{k}+C^{i}_{jk}\delta y^{k}.
\]
By the property (ix), the structure equations of \(C\Gamma(N)\)
expressed in the following theorem [
10]:
Theorem 3.
The structure equations of the canonical \(N\)-linear metrical
connection \(C\Gamma(N)\) of the space \(FL^{n}\) are as follows:
\[
d(dx^{i})-dx^{k}\wedge \omega^{i}_{k}=-^{1}\Omega^{i},\\
d(\delta y^{i})-\delta y^{k}\wedge \omega^{i}_{k}=-^{2}\Omega^{i},\\
d\omega^{i}_{k}-\omega^{k}_{j}\wedge \omega^{i}_{k}=-\Omega^{i}_{j},
\]
where \(^{1}\Omega\), \(^{2}\Omega \) are the \(2\)-forms of torsion.
Here
\begin{eqnarray*}
^{1}\Omega^{i}&=&C^{i}_{jk}dx^{j}\wedge \delta y^{k},\\
^{2}\Omega^{i}&=&\frac{1}{2}R^{i}_{jk}dx^{j}\wedge
dx^{k}+P^{i}_{jk}dx^{j}\wedge\delta y^{k}
\end{eqnarray*}
and \(\Omega\) is the \(2\)-form of curvature.
\[
\Omega^{i}_{j}=\frac{1}{2}R^{i}_{jkh}dx^{k}\wedge
dx^{h}+P^{i}_{jkh}dx^{k}\wedge \delta
y^{h}+\frac{1}{2}S^{i}_{jkh}\delta y^{k}\wedge \delta y^{h},
\]
where \(R^{i}_{jk}\) is tensor given in property (iv),
\(P^{i}_{jk}=B^{i}_{jk}-L^{i}_{jk}\) and \(R^{i}_{jkh}\), \(P^{i}_{jkh}\),
\(S^{i}_{jkh}\) are the curvature tensor of \(C\Gamma(N)\).
In this study, we use the metric \(N\)-linear connection \({D}\Gamma(N)=(\bar{L}^{i}_{jk},
\bar{C}^{i}_{jk})\) and has a given d-tensor of torsion
\(\bar{\mathrm{T}}^{i}_{jk}\) and \(\bar{S}^{i}_{jk}\) as follows:
\begin{eqnarray}
\label{e:r12}\left.\begin{array}{c}
\bar{L}^{i}_{jk}=L^{i}_{jk}+\frac{1}{2}g^{ih}\left( g_{jr}\bar{\mathrm{T}}^{r}_{kh}+ g_{jh}\bar{\mathrm{T}}^{r}_{jh}-
g_{hr}\bar{\mathrm{T}}^{r}_{kj}\right),
\\\\
\bar{C}^{i}_{jk}=C^{i}_{jk}+\frac{1}{2}g^{ih}\left( g_{jr}S^{r}_{kh}+ g_{jh}S^{r}_{jh}- g_{hr}S^{r}_{kj}\right), \\
\end{array}\right\}
\end{eqnarray}
(12)
where \((L^{i}_{jk}, C^{i}_{jk})\) are the local coefficients of the
canonical metric N-linear connection \(C\Gamma(N)\) and
\(\bar{\mathrm{T}}^{i}_{jk}\), \(\bar{S}^{i}_{jk}\) simply by
\((\mathrm{T}^{i}_{jk}, S^{i}_{jk})\).
2.5.Einstein Equations on \(TM\)
Let \(TM\) be endowed with a non-linear connection \(N\), an h-v
metric structure \(G\) and a metrical \(N\)-connection \(D\Gamma(N)\)
with a priori given torsions \((\mathrm{T}^{i}_{jk}, S^{i}_{jk})\).
Given an h-v metric \(G\) on \(TM\) becomes a
pseudo-Riemannian manifold of dimension \(2n\). The Einstein
equations written for the connection \(D\Gamma(N)\) on \(TM\) as:
\begin{equation}
\label{e:r13}Ric(D)-\frac{1}{2}Sc(D)G=k\mathrm{T},
\end{equation}
(13)
where \(Ric(D)\) is the Ricci tensor field and \(Sc(D)\) is the scalar
curvature of \(D\Gamma(N)\), \(k\) is constant and \(\mathrm{T}\) is the
energy-momentum tensor field.
In local coordinates, Miron and Anastasiei
stated as [11]:
Theorem 4.
The Einstein equations of the Lagrange space \(L^{n}=(M,L)\)
corresponding to the metric N-linear connection
\(D\Gamma(N)=(L^{i}_{jk}, C^{i}_{jk})\) with the coefficients
(12) have the following form:
\begin{eqnarray*}
R_{ij}=\frac{1}{2}(R+S)g_{ij}=k\mathrm{T}_{ij},\\
S_{ij}=\frac{1}{2}(R+S)g_{ij}=k\mathrm{T}_{(i)(j)},\\
^{1}P^{i}_{j}=k\mathrm{T}_{(i)j},\,\,\,^{2}P^{i}_{j}=-k\mathrm{T}_{i(j)},
\end{eqnarray*}
where \(\mathrm{T}_{ij}\), \(\mathrm{T}_{(i)(j)}\),
\(\mathrm{T}_{i(j)}\) are d-tensor fields.
3. The notion of Randers metric
The preliminaries theories has a remarkable particular case, that
is based on the Randers metric.
\begin{equation}
\label{e:r14}F(x, y)=\alpha(x, y)+\beta(x, y).
\end{equation}
(14)
The Lagrange space \(L^{n}=(M, L)\) with
\begin{equation}
\label{e:r15}L(\alpha(x, y), \beta(x, y))=\hat{F}^{2}(\alpha(x,
y), \beta(x, y))=(\alpha+\beta)^{2}.
\end{equation}
(15)
The invariants (2) of Randers metric are given by:
\begin{equation*}
\rho=\frac{\alpha+\beta}{\alpha},\,\,\rho_{0}=1,\,\,\rho_{-1}=\frac{1}{\alpha},\,\,\rho_{-2}=-\frac{\beta}{\alpha^{3}}.
\end{equation*}
Using the formula (1), we obtain the fundamental metric
tensor \(g_{ij}\),
\begin{equation}
\label{e:r16}g_{ij}=\frac{\alpha+\beta}{\alpha}a_{ij}+b_{i}b_{j}+\frac{1}{\alpha}(b_{i}l_{j}+b_{j}l_{i})-\frac{\beta}{\alpha^{3}}l_{i}l_{j}.
\end{equation}
(16)
Its contravariant counterpart \(g^{ij}\) as:
\begin{equation}
\label{e:r17}g^{ij}=\frac{1}{\rho}\left(a^{ij}-\frac{(b^{i}l^{j}+b^{j}l^{i})}{\alpha+\beta}+\frac{b^{2}}{(\alpha+\beta)^{2}}y^{i}y^{j}\right).
\end{equation}
(17)
And we know \(g_{ij}\) is positively defined if \(b^{2}<1\). The
Cartan tensor \(C_{ijk}\) (3) given by:
\begin{equation}
\label{e:r18}C_{ijk}=\sigma_{(i,j,k)}\frac{1}{2}\left(\frac{1}{\alpha}
a_{ij}b_{k}-\frac{\beta}{\alpha^{3}}a_{ij}l_{k}-\frac{1}{\alpha^{3}}b_{i}l_{j}l_{k}+\frac{\beta}{\alpha^{4}}l_{i}l_{j}l_{k}\right).
\end{equation}
(18)
Clearly, we see that \(C_{ijk}\neq 0\). Thus, we have
Theorem 5.
The Cartan tensor \(C_{ijk}\) of Randers metric is non zero
(different from zero).
Moreover, the Randers metric is not reducible to a Riemannian
metric. For this metric (15), the Euler-Lagrange equation
in the natural parametrization given by:
\begin{equation}
\label{e:r19} E_{i}(\alpha^{2})+2\sigma
F_{ij}y^{j}=0,\,\,\,y^{i}=\frac{dx^{i}}{ds},
\end{equation}
(19)
where
\(\sigma=\frac{\rho_{-1}}{\rho}=\frac{1}{2}\frac{L_{\beta}}{\rho}=\alpha\).
From Theorem 2, we have the result
Theorem 6.
The Euler-Lagrange equations of Randers metric
\(L=(\alpha+\beta)^{2}\) is the natural parametrization \(\alpha(x,
dx/ds)=1\) and are given by the Lorentz equation
\begin{equation}
\label{e:r20}\frac{d^{2}x^{i}}{ds^{2}}+\gamma^{i}_{jk}(x)\frac{d
x^{j}}{ds}\frac{dx^{k}}{ds}=\alpha F^{i}_{j}(x)\frac{dx^{j}}{ds}
\end{equation}
(20)
Thus, the coefficients of the canonical semispray and non-linear
connection \(N\) are:
\begin{eqnarray*}
2G^{i}(x,y)=\gamma^{i}_{jk}(x)y^{j}y^{k}-\alpha(x,
y)F^{i}_{j}(x)y^{j},\\
N^{i}_{j}=\gamma^{i}_{jk}(x)y^{k}-\alpha F^{i}_{j}(x).
\end{eqnarray*}
The weak torsion of \(N\) is \(L^{i}_{jk}=0\) and the metric
\(N\)-linear connection \(D\Gamma(N)=(\bar{L}^{i}_{jk},
\bar{C}^{i}_{jk})\) is given in (12) coincide with those
of the Cartan connection. Moreover, taking into account that, with
respect to the canonical \(N\)-linear connection \(N\), we have
\(\frac{\delta F}{\delta x^{i}}=0\).
The torsion tensor of \(D\Gamma(N)\) are:
\begin{equation}
\label{e:r21}\mathrm{T}^{i}_{jk}=0,\,\,\,R^{i}_{jk},\,\,\,C^{i}_{jk},\,\,P^{i}_{jk}=N^{i}_{jk}-L^{i}_{kj},\,\,S^{i}_{jk}=0.
\end{equation}
(21)
In the following and using the properties form [
2], we get
\begin{equation}
\label{e:r22}P^{i}_{jk}y^{k}=0,\,\,\,P^{i}_{jk}y^{j}=0.
\end{equation}
(22)
4. Einstein equations of Lagrange space with Randers metric
Now, we express equation (13) in the basis \((\delta/\delta x^{i}, \partial/\partial
y^{i})\), i.e., adapted to the decomposition of \(T_{u}TM\), \(u\in
TM\) into horizontal and vertical subspaces.
Set \((X_{\alpha})=(X_{i}, X_{(i)})\), where
\(X_{i}=\delta/\delta x^{i}\) and \(X_{(i)}=\partial/\partial y^{i}\).
The indices \(i\) will run from \(1\) to \(2n\) and \((i)\) will run from
\(n+1\) to \(2n\). The local vector fields \((X_{a})\) provides a
nonholonomic basis given by
\begin{equation*}
[X_{b}, X_{c}]=W^{a}_{bc}X_{a},
\end{equation*}
which satisfies the following Vranceanu indentities [12]
\begin{equation*}
\sum_{(abc)}[X_{a}(W^{d}_{bc})+W^{e}{ab}W^{d}_{ce}]=0.
\end{equation*}
Let \(D_{X_{c}}X_{b}=\Gamma^{a}_{bc}X_{a}\), then the basis
\((X_{a})\) the torsion \(\mathrm{T}\) of the \(N\)-linear connection \(D\)
has the components:
\begin{equation*}
\mathrm{T}^{a}_{bc}=\Gamma^{a}_{bc}-\Gamma^{a}_{cb}+W^{a}_{bc}.
\end{equation*}
In the basis \((X_{a})\) the curvature \(R\) of the \(N\)-linear
connection \(D\) has the components:
\begin{equation*}
R^{a}_{bcd}=X_{d}\Gamma^{a}_{bc}-X_{c}\Gamma^{a}_{bd}+\Gamma^{e}_{bc}\Gamma^{a}_{ed}-\Gamma^{e}_{bd}\Gamma^{a}_{ec}+\Gamma^{a}_{be}\Gamma^{e}_{cd}.
\end{equation*}
The torsion and curvature components given by:
\begin{equation*}
\mathrm{T}(X_{c}, X_{b})=\mathrm{T}^{a}_{bc}X_{a},\,\,\,R(X_{d},
X_{c})X_{b}=R^{a}_{bcd}X_{a}.
\end{equation*}
In the adapted basis \((X_{a})\) the Bianchi identities of \(D\) of
the form:
\begin{eqnarray*}
\sum_{abc}(D_{a}R^{e}_{dbc}+R^{e}_{dav}\mathrm{T}^{v}_{bc})=0,\\
\sum_{abc}(D_{a}\mathrm{T}^{d}_{bc}+\mathrm{T}^{e}_{ab}\mathrm{T}^{d}_{ec}-R^{d}_{abc})=0,
\end{eqnarray*}
where \(D_{a}=DX_{a}\).
If in these equations the components with respect to
\(X_{i}=\delta/\delta x^{i}\) and \(X_{(i)}=\partial/\partial y^{i}\)
are separated, it comes out that among the coefficients
\(\Gamma^{a}_{bc}\), we have
\begin{equation*}
\Gamma^{i}_{jk}=L^{i}_{jk},\,\,\,\Gamma^{(i)}_{(j)(k)}=C^{i}_{jk}.
\end{equation*}
This is advantage created by the choice of the basis \((X_{a})\) as
well as by the fact that \(D\) is an \(N\)-linear connection.
The set of components \(\mathrm{T}^{a}_{bc}\) of the torsion field
\(\mathrm{T}\) splits into following:
\begin{eqnarray}
\label{e:aa}\left.\begin{array}{c}
\mathbb{T}^{i}_{jk}=\mathrm{T}^{i}_{jk},\,\,\mathrm{T}^{i}_{(j)k}=-C^{i}_{jk},\,\,\,\mathbb{T}^{i}_{j(k)}=-C^{i}_{jk},\,\,\,\mathbb{T}^{(j)(k)}=0,\\
\mathbb{T}^{(i)}_{jk}=R^{i}_{jk},\,\,\,\mathbb{T}^{(i)}_{(j)k}=-P^{i}_{kj},\,\,\,\mathbb{T}^{(i)}_{j(k)}=P^{i}_{jk},\,\,\mathbb{T}^{(i)}_{(j)(k)}=0.
\end{array}\right\}
\end{eqnarray}
(23)
with respect to the basis \((X_{a})\), the Ricci tensor field of the
\(N\)-linear connection \(D\Gamma(N)\) has the components
\begin{equation*}
\mathbb{R}_{ij}=R{ij},\,\,\,\mathbb{R}_{(i)j}=^{1}P_{ij},\,\,\,\mathbb{R}_{i(j)}=-^{2}P_{ij},\,\,\,\mathbb{R}_{(i)(j)}=S_{ij}.
\end{equation*}
By the pseudo-Riemannian metric \(G\) has the components \(G_{ab}\)
given by:
\begin{eqnarray*}
G_{ij}=g_{ij},\,\,G_{i(j)}=0,\,\,\,G_{(i)j}=0,\,\,\,G_{(i)(j)}=g_{ij},\\
G^{ij}=g^{ij},\,\,G^{i(j)}=0,\,\,\,G^{(i)j}=0,\,\,\,G^{(i)(j)}=g^{ij},
\end{eqnarray*}
where \(g_{ij}\) and \(g^{ij}\) are given in (16) and
(17) respectively.
Thus, the tensor field \(\mathbb{R}^{a}_{b}=G^{ac}R_{cb}\) and the
scalar curvature \(Sc(D)\) have in the frame \(X_{a}\) the components
are
\begin{equation*}
\mathbb{R}^{i}_{j}=R^{i}_{j},\,\,\mathbb{R}^{(i)}_{j}=^{1}P^{i}_{j},\,\,\,\mathbb{R}^{i}_{(j)}=^{2}P^{i}_{j},\,\,\
\mathbb{R}^{(i)}_{(j)}=S^{i}_{j},\,\,\,Sc(D)=R+S,
\end{equation*}
where \(R=g^{ij}R_{ij}\) and \(S=g^{ij}S_{ij}\).
Theorem 7.
The Einstein equations of the Largrange space with Randers metric
corresponding to the metric \(N\)-linear connection
\(D\Gamma(N)=(L^{i}_{jk}, C^{i}_{k})\) have the following form:
\begin{eqnarray}
\label{e:r23}\left.\begin{array}{c}
R_{ij}=\frac{1}{2}(R+S)g_{ij}, \\
^{1}P^{i}_{j}=0,\,\,\,\,^{2}P^{i}_{j}=0, \\
S_{ij}=\frac{1}{2}(R+S)g_{ij}, \\
\end{array}\right\}
\end{eqnarray}
(24)
where \(g_{ij}\) given in (16).
Proof.
Making use of the formulae in Theorem (4), one can shows that
from theorem and corresponding d-tensor fields in (12),
(21), (22) and (23) are equivalent to get
(24).
In vacuum, which corresponds to the case \(\mathrm{T}_{ij}=0\), if
we multiply this with \(G^{ij}=g^{ij}\) the equation (13)
of Randers metric can be written in the form:
\begin{equation}
\label{e:r24} R_{ij}-\frac{1}{2}Sc(D)G_{ij}=0,\,\,\,
\text{or}\,\,\,\,R_{ij}-\frac{1}{2}Sc(D)g_{ij}=0,
\end{equation}
(25)
which implies that \(Sc(D)-nSc(D)=o\). Hence, \(Sc(D)=0\) for \(n>1\).
Thus, the equation (25) takes the form \(R_{ij}=0\) and
immediately, we obtain the following result:
Lemma 8.
For the vacuum state, the Einstein equations of the Lagrange space
with Randers metric corresponding to the metric connection
\(D\Gamma(N)=(L^{i}_{jk}, C^{i}_{k})\) are as follows:
\begin{equation}
R_{ij}=0,\,\,\,S_{ij}=0,\,\,^{1}P^{i}_{j}=0,\,\,\,^{2}P^{i}_{j}=0.
\end{equation}
(26)
5. Conclusion
The development of the geometry of Lagrange spaces, using the
fundamental concepts from Analytical Mechanics as: the integral of
action, the Euler-Lagrange equations, the law of conservation of
energy and symplectic form etc. The geometry of a Lagrange space
is mostly derived from the Euler-Lagrange equations. This paper
is devoted to derived the Euler Lagrange equations of Randers
metric. Then, by using canonical N-metrical connection,
characterized the Einstein equations of Finsler Lagrange space with
Randers metric.
Acknowledgments
The author wishes to express his profound gratitude to the reviewers for their useful comments on the manuscript.}
Author Contributions
All authors contributed equally in writing of this paper. All authors read and approved the final manuscript.}
Conflicts of Interest:
The authors declare no conflict of interest.
