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Einstein Tensor in Scalar Curvature Finsler Spaces.

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~~
Annalen der Physik. 7. Folge, Band 37, Heft 2, 1980, S. 151-154
~
J.A. B'irth, Leipzig
Einstein Tensor in Scalar Curvature Finsler Spaces
By N. ISIIIKAWA
Department of Physics, Nagoya University, Xagoye (Japan)
A b s t r a c t . Einstein tensor of Finsler space which has internal coordinate dependent scalar
curvatiire is presented.
Einstein Tensor in Finsler-Ranmen init skalarer Iiriimmuiig
Inlial t s i i b e r s i c h t . I m folgenden wird der Einstein-Tensor in Finder-Ra~nnenniit von internen
Koorclinaten abhiingiger skalarer Krumniung dargestellt.
1. Introduction
It is known that the usual Einstein tensor defined by Gli = Rii - ( I / 2 ) g i i B does
not satisfy the divergence equation G"Ji = 0 in Finsler spacetiiiie. This is because the
torsion tensor in Finsler space does not vanish in general and these terms behave
like sources of energy-iiioiiientuin. However in the case of constant curvature space
these ternis are also divergent,. But by suitable niodification of the Einstein tensor we
can obtain a divergence-free tensor Gii. I n this paper we show that there also exists
such a divergence-free tensor in Finsler space with scalar curvature spacet.inie independent from external coordinates (explained below). Before discussing this possibility, we briefly review the concept of Finsler spaces.
A Finsler space [l]is a differentiable manifold with a metric function ds = L(x,d.c)
positively hotnogeneous and of degree one in dx, i.e. L(x, k dx) = U ( x ,dx)for k > 0.
Analogous t o the Rieliiannian space, a metric tensor qij is defined by:
Here, by the homogeneity of L, we can m e a vector y in Lsinstead of dx. The covariant
differential of a vector X iis defined by
DXi G axi XkF@zi
XkCi
k i Dti
J ,
+
+
where F& and C i j are the Christoffel synibols which are calculated once L(x, y) is
given:
152
and tlie subscript 0 means
h'1 = y ' ~ i , etc.
0
Covariant, derivatives are defined by
where
I n contrast t o the Riemannian space there are three curvature tensors:
gikAlk
P riik
.'
= A , - A,,
(=
g i l q i k ) = 0,i (C{jklr
+
C,j&},
Srijk(z
gj,s:jk)= .ih-{Cr&i>,
and three torsion tensors:
Ri7h- - niojh-7
PiJ = C&,
cg .
Bianehi's identity wliicli is needed in the following discussion is :
+ +
A?,,, A,/j
Aljk,
ojtr{RRjkjl
P;jrRi,> = 0 I n Firisler spaces the spaces of scalar curvature are defined as follows: The invariant
constructed from Rijkland two vectors ( X i , Yi)
oih-zAjtl
+
is called tlie Riemannian curvature with respect to ( X , Y ) . Setting Y i= yi, one
obtains
x)
BERWALD[S] showed that the space has scalar curvature if R ( z .y,
i s X-independent.
Further ifK(x, y) = constant tlie space is called the constant curvature space. The condition that the space has scalar curvature is (Riik=_ g&.)
3R,,h-= L2(K 1
&J
- h' 1 k k i j )
+ 3K(ytl,ha- M h i ? )
where
h,,
G
gi, -
YiYl
L2 -
The condition of constant curvature space is
K;i = Kli = 0.
Einstein Tensor in Scalar Curvature Finsler Spaces
153
2. Einstein Tensor
Starting point of the argument is Bianchi’s identity
(1)
ajki(Rijkll
PijrRL] = 0.
Contracting the equation (1)by (i,k) and (h, j) we obtain the following equation
(2Rf- SiR)li = gh’ (PiirRli PiCJZ;. Pil$!i)Now we use the condition of scalar curvature space. I n performing the calculation
we notice the following relations :
giils == 0,
yyj = 0 ,
1
pi.- q,,,
+
+
21
+
-
Pfi, - Pfi, = -silo,
where
8.71 =fJ4
- Ilk
s
7
-
gils.
71
By using these relations we obtain
For scalar function K , by Ricci’s identity,
Kl.1.
2 7 - KIili = -KIrC& - Kl r P‘.
21
Since g i j ( x ,y) is homogeneous of degree zero in y,
C&yi= P&yi= 0 ,
we obtain
Kli,+Ji= KIiliyi.
Applying this relation to the second terni in r.h.s. we obtain the following relation
[2R$- S$,R - KSy’yl
L2
+(28; - SjS) K[i$]lj
3
L2
= - (28; - (3;s)
Kljliyj - 8yiyzK,j.
3
We thus obtain the following theorem:
T h e o r e m . If Kli = 0 then the following tensor Gii defined by
..
1 g21R
..
1
._
( # i t - 1 91~
Rz?
-- - Ksyiyvi +
@if
Kl,!Ji,
2
2
~
s)
satisfies the divergence-free equation
Gii - 0.
Ii Since the covariant derivative fi/Sxi inay be considered as the derivative with respect
to the external coordinates (xi),Kji might be considered as the space whose scalar
curvature only depends on the internal coordinates (yi). B’ollowing TAKANO
[3] if we
use the Einstein equations
1 gifR = Ti7
Rii - 2
1
.. = Tiii
sii - - gvx
2
154
H. ISHIKAWA
our Gii can be written as follows
The r.h.s. of this equation might be considered as total energy-inoiiientuni. It goes
without saying that our Gii reduces to the Einstein tensor of constant curvature space
obtained by RUND[4] for Kli = 0. Further if we impose K = 0 then it reduces
to HORVBTH'S
[ 5 ] Einstein tensor if we choose an absolute parallelism as yi.
Finally we mention the following point. For applications to physical processes especially t o particle physics as Yukawa's bilocal theory, we may suitably assume that the scalar
curvature depends only on internal freedom SO we iiiay assuine Kli = 0.
References
[l] H. RUND,The differential geometry of Finder spares. Springer-Verlag, Berlin-Heicielberg
New York 1969.
M. MATSUMOTO,
The theory of Finder connections. Pub]. Study Group of Geometry Yo. 3 (1970).
r2l L. BERWALD,
Ann. Math. Stat. 4S, 7% (1947).
131 Y. TAKANO,
Nuovo Cimento Lett. 10, 747 (1975).
[4] H. RUND,Monatshefto Math. 66, 241 (1962).
[5] J. I. HORVATH,
Nuovo Ciniento Stippl. 9, 444 (1968); see also Y. TAKANO,
in Proc. 1ntelnatio:ul
Symposium on Relativity PC Unified Field Theory (1975-76) P. 1 7 (S. S. Bose Inst. of Phys.
SCS., Calcutta-9).
Bei der Redaktion eingegangen itin 14. August 1979.
Snschr. d. Verf.: Dr. HIROHISA
ISHIKAWA
Department of Physics
Nagoya University
Nagoys (Japan)
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