Acta Mathematica Academiae Paedagogicae Nyı́regyháziensis
31 (2015), 139–151
www.emis.de/journals
ISSN 1786-0091
EINSTEIN EQUATIONS OF G-NATURAL COMPLEX
FINSLER METRICS
ANNAMÁRIA SZÁSZ
Dedicated to Professor Lajos Tamássy on the occasion of his 90th birthday
Abstract. In this paper, we endow the holomorphic tangent bundle with a
generalized Sasaki type lift G of the fundamental metric tensor of a complex
Finsler space. In order to build the Einstein equations on the holomorphic
tangent bundle, we determine the Levi-Civita complex linear connection
corresponding to this metric. As an application, we give some solutions of
the complex Einstein equations in a weakly gravitational space.
1. Preliminaries
In the papers [3, 4] are studied complex Einstein equations for the weakly
gravitational field and for complex version of Schwartzschild metric. This
study is based on the idea to write the complex Einstein equations for these
metrics relative to the Chern-Finsler connection, which is metrical but with
torsion. For such theory to be consistent some restrictions are required, called
conservation laws, because the connection used is with torsion.
An alternative to this theory is the one expressed in the following. We
extend the metric structure of the weakly gravitational field to one on the
holomorphic tangent bundle T 0 M of a complex manifold M , and we then
consider the Levi-Civita connection of this lifted metric, which is metrical and
torsion-free. Therefore the complex Einstein equations with respect to the
Levi-Civita connection have the classical from. In particular, if the space is
vacuum the complex Einstein equations reduce to the vanishing of the complex
Ricci tensor. Basically, the idea seems to be simple, but the first problem is
how to get such a lift and how they can be general. Then is the issue of writing
2010 Mathematics Subject Classification. 53B40, 53C60.
Key words and phrases. Holomorphic tangent bundle, Chern-Finsler connection, complex
Einstein equations.
This paper is supported by the Sectoral Operational Programme Human Resources Development (SOP HRD), ID134378 financed from the European Social Fund and by the
Romanian Government.
139
140
ANNAMÁRIA SZÁSZ
the curvature tensors and Ricci tensors on T 0 M . For this we turn again to the
well-known adapted frames of Chern-Finsler connection, and express all in this
complex adapted frames. A similar idea has been applied to the real case by
M. Anastasiei and H. Shimada, [5]. Finally, we propose to solve these complex
Einstein equations, at least in same particular cases of weakly gravitational
metric.
Let M be a complex manifold of complex dimension n. We consider z ∈
M , and so √
z = (z 1 , . . . , z n ) are complex coordinate in a local chart. Since
k
k
z = x + −1xk+n , k = 1, . . . , n, the complex coordinates induce the real
coordinates {x1 , x2 , . . . , x2n } on M . Let TR M be the real tangent bundle. Its
complexified tangent bundle TC M splits into the sum of holomorphic tangent
bundle T 0 M and its conjugate T 00 M , under the action of the natural complex
structure J on M. The holomorphic tangent bundle T 0 M is itself a complex
manifold, and the coordinates in
chart will be denoted by u = (z k , η a ),
√ a local
a
a
a+n
k, a = 1, . . . , n. with η = y + −1y , a = 1, . . . , n. Trough this paper the
indices i, j, k, . . . and a, b, c, . . . run over {1, . . . , n}, where the second denotes
geometric objects in local fibers of the holomorphic tangent bundle. This
notation of the indices is important for the clarity of the notions in geometry
of T 0 M manifold.
0
Consider the sections of the complexified tangent bundle
n ToC M . Let V T M ⊂
T 0 (T 0 M ) be the vertical bundle, locally spanned by ∂η∂ a
, and V T 00 M
a=1,n
its conjugate. The idea of the complex non-linear connection, briefly c.n.c. is
an instrument in ‘linearization’ of the geometry of T 0 M manifold. A c.n.c. is
a supplementary subbundle to V T 0 M in T 0 (T 0 M ), i.e. T 0 (T 0 M ) = HT 0 M ⊕
V T 0 M . The horizontal distribution Hu T 0 M is locally spanned by
(1)
∂
δ
∂
= k − Nka a ,
k
δz
∂z
∂η
where Nka (z, η) are the coefficients of the c.n.c. The pair {δk := δzδk , ∂˙a := ∂η∂ a }
will be called the adapted frame of the c.n.c., which obey the transformation
0j
0j
laws δk = ∂z
δ 0 and ∂˙a = δak δjb ∂z
∂˙ 0 , where δak is the Kronecker symbol. By
∂z k j
∂z k b
conjugation everywhere we obtain an adapted frame {δk̄ , ∂˙ā } on Tu00 (T 0 M ). The
dual adapted frame are {dz k , dη a } and {dz̄ k , dη̄ a } its conjugate, where
(2)
δη a = dη a + Nka (z, η)dz k .
Let N be a c.n.c. on T 0 M . An h−metric on T 0 M is a d−tensor field hG =
gj k̄ (z, η)dz j ⊗ dz̄ k , with gj k̄ (z, η) = gkj̄ (z, η), det kgj k̄ (z, η)k 6= 0. A v−metric
on T 0 M is a d−tensor field vG = hab̄ (z, η)δη a ⊗ δ η̄ b , with hab̄ (z, η) = hbā (z, η),
det khab̄ (z, η)k 6= 0. From here we obtain that an (h, v)−metric on T 0 M is
tensor field G = hG + vG. So, this metric will be written in the next form:
(3)
G(z, η) = gj k̄ (z, η)dz j ⊗ dz̄ k + hab̄ (z, η)δη a ⊗ δ η̄ b .
EINSTEIN EQUATIONS OF COMPLEX FINSLER METRICS
141
A distinguished complex linear connection, briefly d-c.l.c., D on T 0 M is
called compatible with the metric G if DG = 0.
Definition 1. An n−dimensional complex Finsler space is a pair (M, F ),
where F : T 0 M → R+ is a continuous function satisfying the conditions:
a)
b)
c)
d)
0 M := T 0 M \ {0};
]
L := F 2 is smooth on T
F (z, η) ≥ 0, the equality holds if and only if η = 0;
F (z, λη) = |λ|F (z, η) for ∀λ ∈ C;
the Hermitian matrix
gj k̄ = δja δkb
(4)
∂2L
,
∂η a ∂ η̄ b
0M .
]
is positive-definite on T
Then, gj k̄ is called the fundamental metric tensor of the complex Finsler
∂gj k̄ a
∂gj k̄ a
∂L a
∂L a
space. Consequently, from c) we have ∂η
a η = ∂ η̄ a η̄ = L, ∂η a η = ∂ η̄ a η̄ = 0,
and L = δaj δbk gj k̄ η a η̄ b .
From [1, 7] we know there exists a unique metric Hermitian connection D,
of type (1, 0)−type, which satisfies in addition DJX Y = JDX Y , for all X
horizontal vectors, called the Chern-Finsler connection, in brief C-F, which
have a special meaning in complex Finsler geometry. The C-F connection
a
DΓ = (Nja , Lijk , Cbc
, 0, 0) is locally given by the following coefficients:
(5) Nja = δka δbl g m̄k
∂glm̄ b
η = δka δbl Lklj η b ;
∂z j
Lhjk = g l̄h δk gj l̄ ;
a
Cbc
= δha δbj g l̄h ∂˙c gj l̄ ,
where the non-vanishing expressions of the C-F connection are Dδk δj = Lhjk δh ,
d ˙
D∂˙b ∂˙a = Cab
∂d and its conjugates.
A particular situation of the d−tensor gj k̄ from (4) is:
Definition 2. If gj k̄ depends only on the variable z, then we say that the space
(M, F ) is purely Hermitian.
The metric tensor gj k̄ from (4) determines a metric structure GS on TC (T 0 M ),
called the Sasaki lift of gj k̄ , [7], p.96:
(6)
GS = gj k̄ dz j ⊗ dz̄ k + δaj δbk gj k̄ (z, η)δη a ⊗ δ η̄ b .
We introduce a generalization of the lift (6), which defines also an (h, v)−metric
on TC (T 0 M ):
(7)
G(z, η) = gj k̄ (z, η)dz j ⊗ dz̄ k + hab̄ (z, η)δη a ⊗ δ η̄ b ,
where gj k̄ is the fundamental metric tensor of the Finsler space (M, F ), and
hab̄ is an arbitrary d−tensor of (0, 2)−type.
142
ANNAMÁRIA SZÁSZ
2. The Levi-Civita connection on T 0 M
From the standard definition of a complex linear connection on the manifold T 0 M , extended on the complexified tangent bundle TC (T 0 M ), a complex linear connection ∇ can be decomposed in the sum ∇ = ∇0 + ∇00 ,
where ∇0 : Γ(TC (T 0 M )) → Γ(TC (T 0 M ) ⊗ T 0 (T 0∗ M )) and ∇00 : Γ(TC (T 0 M )) →
Γ(TC (T 0 M ) ⊗ T 00 (T 0∗ M )), which can be decomposed in
∇0 = ∇0h + ∇0v and ∇00 = ∇00h + ∇00v .
So, in the adapted frame of the C-F c.n.c. {δk , ∂˙a , δk̄ , ∂˙ā }, ∇ is defined by
the following coefficients:
(8)
1
2
3
4
¯
1
2
3
4
¯
1
2
3
4
1
2
3
4
2
Fjbd ∂˙d
3
Fjbı̄ δı̄
4
¯
Fjbd ∂˙d¯;
∇δk δj = Lijk δi + Adjk ∂˙d + Aı̄jk δı̄ + Adjk ∂˙d¯;
ı̄
d ˙
i
δı̄ + Bak
∂d¯;
δi + Ldak ∂˙i + Bak
∇δk ∂˙a = Bak
¯
d ˙
i
d ˙
∂d¯;
∂d + Lı̄j̄k δı̄ + Dj̄k
∇δk δj̄ = Dj̄k
δi + Dj̄k
¯
ı̄
i
d ˙
∂d + Eāk
δı̄ + Ldāk ∂˙d¯;
∇δk ∂˙ā = Eāk
δi + Eāk
∇∂˙b δj =
1
i
Cjb
δi
+
+
2
1
+
4
3
¯
d ˙
∇∂˙b ∂˙a = Giab δi + Cab
∂d + Gı̄ab δı̄ + Gdab ∂˙d¯;
∇∂˙b δj̄ =
1
i
Hj̄b
δi
1
+
2
d ˙
Hj̄b
∂d
2
+
3
ı̄
Cj̄b
δı̄
3
+
4
d¯ ˙
Hj̄b
∂d¯;
4
i
d ˙
ı̄
d¯ ˙
∇∂˙b ∂˙ā = Māb
δi + Māb
∂d + Māb
δı̄ + Cāb
∂d¯
and its conjugates, by ∇X̄ Ȳ = ∇X Y .
Since ∇G = 0 and ∇ is a symmetric connection, direct calculus leads to
Theorem 1. The Hermitian manifold (T 0 M, GH ) admits a unique complex
linear connection, which is symmetric and metrical in respect to G, defined by
(3). This is called the Levi-Civita connection on T 0 M , and its local coefficients
are represented in the local adapted frame {δk , ∂˙a , δk̄ , ∂˙ā } by the following nonzero expressions:
(9)
1
1
Lijk = g l̄i (δk gj l̄ + δj gkl̄ );
2
2
1
¯
Lcak = [hdc (δk had¯) + ∂˙a Nkc ];
2
3
1
1
i
Lij k̄ = Dk̄j
= g l̄i (δk̄ gj l̄ − δl̄ gj k̄ );
2
2
4
1
¯
c
Dj̄k
= −Dkc j̄ = [δj̄ Nkc − hdc (∂˙d¯gkj̄ )];
2
2
1
¯
c
Eāk
= hdc [(∂˙ā Nke )hed¯ − (∂˙d¯Nke )heā ];
2
2
1 ¯
Fjbc = [hdc (δj had¯) − ∂˙a Njc ];
2
EINSTEIN EQUATIONS OF COMPLEX FINSLER METRICS
4
Lcak̄
1
=
2
c
Hk̄a
1
i
i
Cka
= Bak
1 ¯
= hdc [δk̄ had¯ − (∂˙d¯Nk̄ē )haē ];
2
1 l̄i ˙
¯
= g [∂a gkl̄ + (δk Nl̄d )had¯];
2
2
1 ¯
c
= hdc (∂˙b had¯ + ∂˙a hbd¯);
Cab
2
3
1
1
i
Cjib̄ = Eāj
= g l̄i [∂˙b̄ gj l̄ − (δl̄ Njd )hdb̄ ];
2
and its conjugates.
143
1
Giab
1
¯
¯
= g l̄i [(∂˙b Nl̄d )had¯ + (∂˙a Nl̄d )hbd¯];
2
4
1 ¯
Hjcb̄ = − hdc [(∂˙d¯Nje )heb̄ + (∂˙b̄ Nje )hed¯];
2
3
1
1
¯
i
i
= g l̄i [(∂˙ā Nl̄d )hbd¯ − δl̄ hbā ];
= Mbā
Māb
2
4
2
1 ¯
c
Cacb̄ = Mb̄a
= hdc (∂˙b̄ had¯ − ∂˙d¯hab̄ ),
2
This connection is not h- or v-metrical.
To study the Levi-Civita connection, we may consider a similar connection,
which help us to express easier the different properties of the Levi-Civita cone be a d-c.l.c. on TC (T 0 M ):
nection. Indeed let D
1
2
1
2
3
e δ δj̄ = Li δı̄ ;
e δ δj = Li δi ; D
e δ ∂˙a = Ld ∂˙d ; D
(10) D
ak
jk
k
k
k
j̄k
e ˙ δj = C i δi ; D
e ˙ ∂˙a = C d ∂˙d ;
D
ja
ab
∂a
∂b
4
e δ ∂˙a = Ld ∂˙ ;
D
ak d
k
4
3
e ˙ δj̄ = C ı̄ δi ; D
e ˙ ∂˙ā = C d¯ ∂˙
D
āb d
∂a
∂b
j̄a
and its conjugates, where the local coefficients are expressed in (9).
This d-c.l.c. is metrical with respect to G, i.e.
gj k̄|m = gj k̄ |d = gj k̄|m̄ = gj k̄ |d¯ = hab̄|m = hab̄ |d = hab̄|m̄ = hab̄ |d¯ = 0,
where with ”| ”, ”|”, ”¯| ”, ”¯|” are notated the h−, v−, h̄− and v̄−covariant
e
derivatives with respect to D.
(11)
e are
Proposition 1. The non-zero components of the torsion of the d-c.l.c. D
(12)
e ∂˙ā , δj ) = Υ
e ∂˙a , δj ) = Q
e k̄ , δj ) = τei δi ; v T(δ
e k̄ , δj ) = Θ
e i δi ; hT(
ei δi ;
e d ∂˙d ; hT(
hT(δ
jā
ja
j k̄
j k̄
e ∂˙b̄ , ∂˙a ) = χ
e ∂˙ā , δj ) = ρed ∂˙d ; v T(δ
e k̄ , ∂˙a ) = Σ
e ∂˙a , δj ) = Ped ∂˙d ;
e d ∂˙d ; v T(
v T(
edab̄ ∂˙d ; v T(
jā
ja
ak̄
and their conjugates.
After a straightforward computation we obtain the expressions for (12)
3
3
(13)
1
i
i
eija = Cja
e d = δk̄ Njd ; Υ
e ijā = Cjā
;
; Q
τejik̄ = Lij k̄ ; Θ
j k̄
4
4
2
e d = Ld ; Ped = ∂˙b N d − Ld .
χ
edab̄ = Cadb̄ ; ρedjā = ∂˙ā Njd ; Σ
ja
j
jb
bj̄
bj̄
e has twenty components in the form (see p. 44 of [7]):
The curvature of D
1
1
1
3
3
3
ei = Ahk {δh Li + Lm Li };
(14) R
jk mh
jk
jkh
eı̄ = Ahk {δh Lı̄ + Lm̄ Lı̄ };
R
j̄k
j̄k m̄h
j̄kh
144
ANNAMÁRIA SZÁSZ
3
1
1
3
3
3
1
1
ei = δh Li − δk̄ Li + Lm Li − Li Lm + (δh N ē )C i − (δk̄ N e )C i ;
R
jh
jē
h
je
j k̄h
j k̄
j k̄ mh
mk̄ jh
k̄
2
2
2
4
4
4
e dakh = Ahk {δh Liak + Leak Ldeh };
Ω
e d¯ = Ahk {δh Ld¯ + Lē Ld¯ };
Ω
ākh
āk
āk ēh
4
2
1
1
3
2
4
1
1
1
1
1
3
3
3
3
3
3
3
1
3
1
3
1
1
2
2
2
2
2
2
2
4
4
4
4
4
4
4
4
2
4
2
4
2
4
3
1
3
1
4
2
4
2
4
2
4
2
e d = δh Ld − δk̄ Ld + Le Ld − Ld Le + (δh N ē )C d − (δk̄ N e )C d ;
Ω
ah
aē
h
ae
ak̄h
ak̄
ak̄ eh
ek̄ ah
k̄
e i = ∂˙c Li − δk C i + Lm C i − Li C i + (∂˙c N e )C i ;
Π
ej
k
mk jc
jk mc
jk
jc
jkc
ı̄
ı̄
e ı̄ = ∂˙c Lı̄ − δk C ı̄ + Lm Cmc
Π
− Lı̄mk Cj̄c
+ (∂˙c Nke )Ceij̄ ;
j̄kc
j̄k
j̄c
j̄k
e i = ∂˙c Li − δ C i + Lm C i − Li C m + (∂˙c N e )C i ;
Π
je
k jc
jkc
jk
jk mc
mk jc
k
d
d
d
e
d
Peakc
= ∂˙c Ldak − δk Cac
+ Leak Cec
− Ldek Cac
+ (∂˙c Nke )Cae
;
d
d
d
e
d
Peakc
= ∂˙c Ldak − δk Cac
+ Leak Cec
− Ldek Cac
+ (∂˙c Nke )Cae
;
d
d
e
d
d
Peakc
= ∂˙c Ldak − δk Cac
− Ldek Cac
+ (∂˙c Nke )Cae
;
+ Leak Cec
3
1
1
i
e i = δh C i − ∂˙ Lijh + C m Limh − C i Lm
− (∂˙b Nhe )Cje
Θ
;
b
jbh
jb
jb
mb jh
4
2
2
d
ed = δh C d − ∂˙ Ldah + C e Ldeh − C d Leah − (∂˙ Nhe )Cje
Q
;
b
b
abh
ab
eb
ab
1
1
1
3
3
3
e i = Acb {∂˙c C i + C m C i };
Ξ
jbc
jb
jb mc
e ı̄ = Acb {∂˙c C ı̄ + C m̄ C ı̄ };
Ξ
j̄bc
j̄b
j̄b m̄c
1
3
3
1
3
1
e i = ∂˙c C i − ∂˙ C i + C m C i − C i C m ;
Ξ
b jc
jbc
jb
jb mc
mb jc
2
2
2
4
4
4
d
d
e
d
Seabc
= Acb {∂˙c Cab
+ Cab
Cec
};
d¯
ē
d¯
d¯
};
Cēc
+ Cāb
= Acb {∂˙c Cāb
Seābc
4
2
4
2
4
2
d
d
e
Seadb̄c = ∂˙c Cadb̄ − ∂˙b̄ Cac
+ Caeb̄ Cec
− Cedb̄ Cac
,
where Ahk means the difference between the terms in the brackets and the
terms obtained by replacing k with h.
3. Pure Hermitian metric
The geometrical objects associated with G are generally complicated. Some
simplifications appear for particular choices for gj k̄ and hab̄ . We studied in
a previous paper, [9], the case δja δk̄b̄ hab̄ = a(F1 2 ) gj k̄ , and G. Munteanu and N.
EINSTEIN EQUATIONS OF COMPLEX FINSLER METRICS
145
Aldea studied the case δja δk̄b̄ hab̄ = gj k̄ , [7, 2]. Here we resume for a detailed
analysis the following particular case of the Sasaki type metric:
GH (z, η) = gj k̄ (z)dz j ⊗ dz̄ k + hab̄ (z)δη a ⊗ δ η̄ b .
(15)
The Chern-Finsler c.l.c. of the complex Finsler space (M, F ) is reduced to
CF
lm̄ b
DΓ = Nja = δia δbl g m̄i ∂g
η , Lijk = g m̄i ∂g∂zkjm̄ , 0, 0, 0 . The v− and v̄−covariant
∂z j
derivatives coincides with the partial derivatives with respect to η a and η̄ a ,
respectively. By direct calculation we prove:
Proposition 2. The Levi-Civita connection of the metric (15) is given by the
following non-zero coefficients
1
1
Lijk = g l̄i (∂k gj l̄ + ∂j gkl̄ );
2
2
1 ¯
Lcak = [hdc (∂k had¯) + ∂˙a Nkc ];
2
3
1
1
i
Lij k̄ = Dk̄j
= g l̄i (∂k̄ gj l̄ − ∂l̄ gj k̄ );
2
2
4
1 ¯
c
= hdc [∂k̄ had¯ − (∂˙d¯Nk̄ē )haē ];
Lcak̄ = Hk̄a
2
2
1
¯
Fjbc = [hdc (∂j had¯) − ∂˙a Njc ];
2
1
3
1
¯
i
i
Māb
= Mbā
= g l̄i [(∂˙ā Nl̄d )hbd¯ − ∂l̄ hbā ],
2
(16)
where ∂k =
∂
.
∂z k
e from (15) is reduced to
The curvature of the d-c.l.c. D
(17)
1
1
1
3
3
3
ei = Ahk {∂h Li + Lm Li };
R
jkh
jk
jk mh
eı̄ = Ahk {∂h Lı̄ + Lm̄ Lı̄ };
R
j̄kh
j̄k
j̄k m̄h
3
1
3
1
3
1
ei = ∂h Li − ∂k̄ Li + Lm Li − Li Lm ;
R
jh
mk̄ jh
j k̄h
j k̄
j k̄ mh
2
2
2
4
4
4
e dakh = Ahk {∂h Liak + Leak Ldeh };
Ω
e d¯ = Ahk {∂h Ld¯ + Lē Ld¯ };
Ω
ākh
āk
āk ēh
4
2
4
2
4
2
e d = ∂h Ld − ∂k̄ Ld + Le Ld − Ld Le .
Ω
ah
ek̄ ah
ak̄h
ak̄
ak̄ eh
Let K be the curvature tensor field of the Levi-Civita connection ∇. We
shall denote its components by the same letters as N. Aldea in [2], indexed
with two types of indices with the understanding that different indices means
146
ANNAMÁRIA SZÁSZ
different components. From the sixty different curvature components, where
will be 21 non-zero:
e be a d-c.l.c. on TC (T 0 M ) which local coefficients are
Proposition 3. Let D
expressed in (17). With respect to the local adapted frame relative to the ChernFinsler c.n.c., the curvature local coefficients of the Levi-Civita d-c.l.c. on
(T 0 M, GH ) are
(18)
i
ei ;
Rjkh
=R
jkh
3
3
1
i
i
Rj̄kh
= Ahk {Likj̄|h + Lm
kh Lmj̄ };
ı̄
eı̄ ;
Rj̄kh
=R
j̄kh
1
3
3
1
3
3
ei + Lm Li − Lm Li + Lm̄ Li ;
Rji k̄h = R
jh mk̄
k̄j hm̄
j k̄h
j k̄ mh
3
3
1
Rjı̄ k̄h = Lı̄k̄j|h + Lı̄mk̄ Lm
jh ;
(19)
ed ;
Ωdakh = Ω
akh
d
ed ;
Ω =Ω
ak̄h
(20)
Πdjkc =
ak̄h
2
d
Fjc|h
2
2
2
d
;
− Fjcd Leck + (∂˙c Nke )Fje
4
3
2
2
2
2
4
4
d
Πdj̄kc = −Ljc|k
+ Lm
F d − Lecj Ldek + (∂˙c Nke )Ldj̄e ;
kj mc
4
3
4
m̄ d
d e
d
Πdjk̄c = −Fjc|
k̄ − Fje Lck̄ + Lk̄j Lcm̄ ;
1
(21)
1
2
3
1
ī
Pākc
=
1
ī
−Mcā|k
=
2
4
ē
d
−Lb̄j Eēh
−
1
2
e
ī
Lck Meā
1
ī
+ (∂˙c Nke )Meā
;
2
d
− (∂˙b̄ Nhe )Fje
(22)
Θdjb̄h
(23)
i
i
Qiab̄h = Mb̄a|h
+ Lēb̄h Mēa
+ Mam̄b̄ Lihm̄ ;
1
1
4
1
4
1
3
1
ī
Qīab̄h = Maī b̄|h + Lēb̄h Maē
;
(24)
Ξij̄bc
=
1
4
ē
i
};
Acb {Lbj̄ Mēc
4
1
4
1
2
1
i
i
Ξij b̄c = Lēbj̄ Mēc
− Fjce Mb̄e
;
d
Ξdjb̄c = Lēb̄j Mcē
;
1
(25)
1
i
i
i
m̄ i
i
Pākc
= −Māc|k
− Leck Māe
− Mcā
Lkm̄ + (∂˙c Nke )Māe
;
2
1
4
m̄ d
m d
d
Lcm̄ };
Fmc + Mbā
= Acb {Māb
Sābc
EINSTEIN EQUATIONS OF COMPLEX FINSLER METRICS
Sadb̄c
2
m d
Mb̄a Fmc
4
m̄ d
Mab̄ Lcm̄ ;
1
1
=
147
+
and the rest are zero.
H
H
H̄
V
i
i
; Rjk := Rjī īk ; Πj̄k :=
The Ricci curvature tensors are Rjk := Rjki
; Rj̄k := Rj̄ki
V
H
d
i
. From which we obtain the following Ricci
; S āb := Sābd
Πdj̄kd ; P āb := Pāib
H
V
H
V
scalars r := g j̄k Rj̄k ; π := g j̄k Πj̄k ; p := hāb P āb ; s := hāb S āb .
Using some idea from the real case ([6]), a generalization of the classical
Einstein equations for an n−dimensional complex Finsler space is
1
(26)
Rᾱβ − ρ · Gβ ᾱ = χTᾱβ ,
2
where to standardize the notation we use the Greek letters α, β = 1, . . . , n for
the two types of indices; Rᾱβ denotes the components of the Ricci tensors; ρ denotes the the Ricci scalar curvature; Gβ ᾱ represents the metric tensors gj k̄ and
hab̄ , respectively; χ is the universal constant; Tᾱβ are the energy-momentum
tensors ([3]). Since the Levi-Civita connection ∇ is without torsion, the conservation laws to the Einstein equations (26) are satisfied, i.e. ∇α (Rαβ − 12 ρδβα ) = 0,
or equivalently ∇α Tαβ = 0.
By this, as in classical theory, in vacuum we have
Proposition 4. In vacuum the Ricci tensors of the Levi-Civita connection on
T 0 M are vanishing.
4. Solutions for the Complex Einstein Equation in Vacuum for
a weakly gravitational metric
In this section our goal is to solve the complex Einstein equations for the
particular case of a 2−dimensional complex Finsler space in vacuum, when the
fundamental metric tensor has a form of a weakly gravitational metric [3]:
(27)
where (ηj k̄ ) :=
gj k̄ (z, η) = ηj k̄ + pj k̄ ,
1 −i
, is the Minkowski metric and
i −1
2Φ 2Φ i c2
c2
(pj k̄ ) :=
2Φ
−i 2Φ
c2 c2
is a small perturbation of ηj k̄ , and Φ has the meaning of a gravitational poten2
tial. Here Φ is a real valued smooth function in T 0 M , Φ 6= c2 , and c ∈ R, c 6= 0.
In [3], Proposition 4.1 is proved, that the metric (27) is Finsler if the following statements are satisfied:
2
i) Φ > c2 ;
ii) Φ is homogeneous function respect to η;
∂Φ
iii) iΦ·2 = Φ·1 , where Φ·a = ∂η
a , a = 1, 2,
148
ANNAMÁRIA SZÁSZ
and it can be used in the study of the weakly gravitational fields in the complex
Finsler space (M, F ), with
(28) L =
2Φ
1+ 2
c
2Φ
|η | − i 1 − 2
c
1 2
η 1 η̄ 2
2Φ
2Φ
2 1
+ i 1 − 2 η η̄ − 1 − 2 |η 2 |2 .
c
c
Remark 1. In the following, we will use the letters j, k, l, . . . = 1, . . . , n for
both the horizontal and vertical indices.
Using some elementary calculus, it is obtained the local expressions of the
Chern-Finsler c.l.c.:
(29)
−2i
(η 1 − iη 2 )Φk ;
c2 1 − 2Φ
c2
2i
Φk = iL22k ;
=− 2
2Φ
c 1 − c2
Nk1 = 0;
Nk2 =
L1jk = 0;
L21k
1
Cjk
= 0;
2
Cjk
= 0, for j, k = 1, 2,
∂Φ
where Φk := ∂z
k , k = 1, 2.
The above requirements ii) and iii) imply Φ·1 (η 1 − iη 2 ) = iΦ·2 (η 1 − iη 2 ) = 0,
and their conjugates, which implies the following
Theorem 2 ([4, Theorem 3.1]). Let (M, F ) be a complex Finsler space, where
L = F 2 is the metric (28). Then (M, F ) is either a purely Hermitian space or
a locally Minkowski space with η 1 = iη 2 .
In the following we assume that (M, F ) is a purely Hermitian space, i.e.
gj k̄ (z, η) = gj k̄ (z), which implies that form (27) the function Φ = Φ(z). We
consider the corresponding Hermitian metric GH given by (15) customized for
the weakly gravitational metric (27)
Gwg (z, η) = ηj k̄ dz j ⊗ dz̄ k + gj k̄ δη j ⊗ δ η̄ k ,
1 i 1 −i
−
k̄j
where (ηj k̄ ) :=
, with the inverse matrix (η ) := 2i 21 , and
i −1
−2
2
2Φ
√
1 + 2Φ
2 −i 1 − c2 c
(31)
(gj k̄ (z, η))j k̄=1,2 =
, where i := −1;
2Φ
2Φ
i 1 − c2 − 1 − c2
(30)
with its inverse
(32)
g k̄j (z, η))j k̄=1,2 =
1
2
i
2
−i
!
2
1+ 2Φ
2
− 2 1−c2Φ
( c2 )
.
EINSTEIN EQUATIONS OF COMPLEX FINSLER METRICS
149
In order to solve the Einstein equations for this particular case of Finsler
space, we can express the coefficients of the Levi-Civita connection corresponding to the metric (30). The non-vanishing local expressions of the above mentioned connection are:
2
−ik
(Φ + (−1)k Φk );
(33)
L2jk = 2
c 1 − 2Φ
c2
2
2
Fjk
=
c2
−ik
(Φ + Φk );
1 − 2Φ
c2
−ij
(Φ1̄ + iΦ2̄ );
c2
1
ij
2
Mj̄2
= 2 (iΦ1̄ + Φ2̄ ), j, k = 1, 2.
c
The non-zero local expression of the Ricci tensors with respect to the LeviCivita connection (33) are as follows:
1
1
Mj̄2
=
H
(34)
Pj̄1 =
H
Pj̄2 =
c4
2Φ
c2
ij (Φ1 Φ1̄ + iΦ1 Φ2̄ − iΦ2 Φ1̄ − Φ2 Φ2̄ ) ;
ij
(Φ11̄ + iΦ12̄ − iΦ21̄ − Φ22̄ ) +
c2
2i
(1 − i2−j ) (Φ1 Φ1̄ + iΦ1 Φ2̄ − iΦ2 Φ1̄ − Φ2 Φ2̄ ) ;
+ 4
2Φ
c 1 − c2
V
Sj̄2 =
2i
1−
c4
ij
1−
2Φ
c2
(1 − i)(Φ + Φ1 ) (Φ1̄ − Φ2̄ ) ,
j = 1, 2.
Now we are able to write the complex Einstein equations of this space.
Theorem 3. The complex Einstein equations in vacuum corresponding to a
two-dimensional complex Finlser space (M, F ), with the complex Finsler metric
(28) and with the Levi-Civita connection (33) are
2Φ
2
(Φ1 Φ1̄ + iΦ1 Φ2̄ − iΦ2 Φ1̄ − Φ2 Φ2̄ ) + 1 − 2 ρ = 0;
(35)
− 4
c
c 1 − 2Φ
2
c
2i
2Φ
(Φ1 Φ1̄ + iΦ1 Φ2̄ − iΦ2 Φ1̄ − Φ2 Φ2̄ ) + i 1 − 2 ρ = 0;
c
c4 1 − 2Φ
2
c
i2−j
2i
(Φ11̄ + iΦ12̄ − iΦ21̄ − Φ22̄ ) + 4
2
c
c 1−
2Φ
c2
(1 − i2−j ) (Φ1 Φ1̄ +
2Φ
+iΦ1 Φ2̄ − iΦ2 Φ1̄ − Φ2 Φ2̄ ) + (−i) 1 − 2 ρ = 0;
c
2−j
i
2Φ
j
(1 − i)(Φ + Φ1 ) (Φ1̄ − Φ2̄ ) + (−i) 1 − 2 ρ = 0,
c
1 − 2Φ
c2
j
c4
(j = 1, 2), and their conjugates, where ρ represents the scalar curvature.
150
ANNAMÁRIA SZÁSZ
Instead to solve the system of PDE’s (35) we can use the Proposition 3.3
to simplify this. According to the mentioned Proposition, the Ricci tensors in
vacuum are zero, which implies that the Ricci scalars are vanishing too in the
same conditions. Then the system (35) is equivalent with
(36)
Φ1 Φ1̄ + iΦ1 Φ2̄ − iΦ2 Φ1̄ − Φ2 Φ2̄ = 0;
Φ11̄ + iΦ12̄ − iΦ21̄ − Φ22̄ = 0;
(Φ + Φ1 ) (Φ1̄ − Φ2̄ ) = 0.
Proposition 5. The real valued smooth function on T 0 M , Φ(z) = Φ(z 1 , z̄ 1 , z 2 , z̄ 2 ),
is a solution for the system of equations (36), when it satisfies the following
conditions:
a) Φ1 = Φ2 ;
b) Φ1̄ = Φ2̄ .
Example 1. We consider the function Φ(z) = c2 ei(z −z̄ )+i(z −z̄ ) on C2 . Requir2
ing Φ > c2 , by (28), it induce on D := {z ∈ C2 | Im(z 1 + z 2 ) > 0} the purely
Hermitian complex Finsler metric
(37) L = 1 + eZ |η 1 |2 − i 1 − eZ η 1 η̄ 2 + i 1 − eZ η 2 η̄ 1 − 1 − eZ |η 2 |2 ,
2
1
1
2
2
where Z = i(z 1 − z̄ 1 ) + i(z 2 − z̄ 2 ). Since Φ satisfies the conditions a) and b)
from Proposition 4.1, the Sasaki type metric defined by (30) is a solution for
the complex Einstein equations in vacuum.
We notice that besides the solutions provided by Proposition 4.1, there exists
also different type of solutions:
Example 2. Let Φ(z) = c2 e−(z +z̄ )+z +z̄ a real valued function on T 0 M . Re2
quiring Φ > c2 , by (28), the purely Hermitian complex Finsler metric induced
on D := {z ∈ C2 | Re(z 2 − z 1 ) > 0} is
(38) L = 1 + eZ |η 1 |2 − i 1 − eZ η 1 η̄ 2 + i 1 − eZ η 2 η̄ 1 − 1 − eZ |η 2 |2 ,
2
1
1
2
2
where Z = −(z 1 + z̄ 1 ) + z 2 + z̄ 2 . Because Φ is a solution for the system of
equations (36), a solution for the complex Einstein equations in vacuum is the
Sasaki type metric defined by (30).
References
[1] M. Abate and G. Patrizio. Finsler metrics—a global approach, volume 1591 of Lecture
Notes in Mathematics. Springer-Verlag, Berlin, 1994.
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space. Bull. Transilv. Univ. Braşov Ser. B (N.S.), 9(44):39–46, 2002.
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[4] N. Aldea and G. Munteanu. New candidates for a Hermitian approach of gravity. Int. J.
Geom. Methods Mod. Phys., 10(9):1350041, 22, 2013.
EINSTEIN EQUATIONS OF COMPLEX FINSLER METRICS
151
[5] M. Anastasiei and H. Shimada. Deformations of Finsler metrics. In Finslerian geometries
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Received December 13, 2013.
Faculty of Mathematics and Computer Science,
Transilvania University of Braşov,
Romania
E-mail address: am.szasz@yahoo.com