Deviation of Geodesics in the Gravitational Field of Finslerian Space-Time

(1)

Shonan Institute of Technology

Shonan 工nstitute of _Teohnology

　　M 跏 0 血 S　or 　SHON ▲N

L別■letTVTR 　OF 　T 圏喝日冒OLOOY

　　　Vo1．27幽　No．1，1993

Deviation

　of　

Geodesics

in

the

Gra

_▼

itational

Fie

_璽

d

　　

of　

Finslerian

Space

−

Time

P

．

C

．　

STAvRINos

＊ _and 　

H

．　

KAwAGucHI

＊＊

　　　一．般相対論によれば，重力の影響

F

にある質点の運動は 4 次元り一マァ空間の測地線の微分方程式によって記述される。このとき，リーr ソ計．量は重力ポテソシ．1・ルを表わしている。さらに，重力場のなかで質点の代わりに _{有限}のひろがりをもった物体を自由落下させるとぎ潮汐力が生ずる。これは，測地的偏差によって表現され，その加速度はリーマン・クリストッ 7r 一ルの曲率テンソル _で_{記述される} n 　　　この論文では，上記のり一マン_空間を n 次元のフィ γ スラー空間ヘー般化した場合の測地的偏．差を与える微分方程式について研究し，その成果を拡張してフ _ィソスラー空間の_{接触} リーマン _{空間}における微分方程式を求める。さらlc，一般化された 7 _イソスラー空間の場合にも測地的偏差の微分方程式を検証する。最後に，定曲率の接触リーマ _ン_{空間}における測地的偏差の_{微分}_方程式を導く。

1

．　

1ntroduction

　 The 　profound 　role of_theequation of

Riemannian

　geodesic　

deviation

has

been

　re−

cognized 　

by

　the　general　relativity 　

for

　a　

long

time （cf．　

Ref

．11_）_）．　 It　is　known 　that　if　there 　are

deviations

　in　

geodesically

　moving 　

free

parti

−

cles，　they 　will 　

be

　caused 　

by

　the　curvature 　of

the　_space _，　_which 　physically　

is

interpreted

by

the　_existence 　 of　tidal　

forces

．　

The

　relative

accelerations 　of　nearby 　time ・

like

　geodesics　are

caused 　

by

_thecurvature 　of _thespace −time ．

In

Riemannian

　spaces ，　 the　curvature 　tensor

l

〜jikt　enters 　

fully

in

　the　equation 　of　geodesic

deviation

　and 　it　produces 　the　relative 　accelera 。

tlons．

In

　the　Finslerian 　approach ，　 the　curvature

of　a　

Finsler

　space ・time　

is

　characterized 　not only 　by　the　tensor　

1

ヒ丿

ikt

but

　also 　

by

　the　tensors

S

_ん，

1

已 and 　

K

？ t（

Ref．

2

），

6

），

8

））。　

Thus

，　

the

question　arises 　when 　

it

is

possible

　to丘nd 　a　

full

interpretation

ofthe　curvature 　of　a　

Finsler

space 　in　terms 　of　geodesic　

deviations

．　

These

presented

　lnanother 　　　　　　Department 　of　 Mathe ・ considerations 　 will　

be

　　＊　Associate　Professor，　　　matiCS _，　 UniVerSity 　　　　（1992 年 5 月 11H 　　　　演した。）　＊＊情報工学科　助教授　　　　平尹戊 4t・t” 9 _月 28 of　Athens ，15771，　Greece．本学情報工学科を訪問し講 N受付 work ．

In

　the　present　paper ，　we 　study 　the　

devia

−

tion　equation 　in　Finsler　spaces ．

In

paragraph

§2we 　give　an 　

interpretation

　of　the　equation

of　geodesic　

deviation

　as　

it

has

　given　

by

　H ．

Rund

（

Ref

．8）．　

Also

_，　in §3we 　extend 　the　

form

of　this　equation 　for　the　tangent　Riemannian

space 　of　a　

Finsler

　space ，　and 　we 　examine 　the

deviation

　of　geodesics 　

for

　a　　generalized Finsler　 _metric 　 gi_ゴ＝α‘ゴ十β馬　（see　

Ref

．

1

））．

Finally

in

_§4，　considering 　a　tangent　

Rieman

−

nian 　space 　of 　constant 　curvature ，　we 　

derive

the　equation 　of　

geodesic

deviations

．

2 ． Tidal　Forces　in　a　Finsler　

Space

・

Time

　’

We

　 consider 　 a　

Finsler

fundamental

function

F

giゴ which 　

is

　given　

by

，　　　　　　

1

∂2F2 （x，y＞

　

9ij＝

2 　

∂y‘

み

砂』．』・

space ・time，　 with

and 　 _metric 　 tensor

　，

dxi

y

’

　＝

ii

（2 ．1）

and 　

dse

＝

gii

（x，_ア）

d

ガ

d

_κゴ

，　 then 　the　movement

of　a　

free

　particle　on 　a　geodesic　

in

　the　gravita − tional丘eld ，　

is

derived

by

　the　variational

princip夏e

・

1

・・一・（

2

．

2

）

which 　

implies

　the　equation 　of　the　geodesics

in　the　

Finsler

　space −time ，

(2)

Shonan Institute of Technology ShonanInstitute ofTechnology

vaMZ

flk\re er xM+rn`k(x,y)xthxtk=O

(2.3)

with

.･i=!liig, ."i=glil'

and the

Christoffel

symbols,

rinic=-;-[Oag.h,k+aog.i,n-gi:t].

(2.4)

From a more general viewpoint, a

Finsler

space

is

_a

fibered

spaoe, so thatthe movernent of a

_particle

in a

Finsler

_tangent

bundle

is represented

by

the curve

(xt(t),yi(t)),

which

is

distinguished

in

the

horizontal

h-path

on the base manifold and the vertical v-path on

the tangent space.

Therefore,

introducing

the

_generalized

element of

length

da2==gt,(x,y)dx`dr,+glei(x,y)byilly'

(2･5)

where

Llyi==:

dy`+IV]･(x,y)dx'

, the _geodesics will

be

derived

by

the

varia-tional _principle,

6jda=o

d2xt

drJ

dxk

da2

+

E`'(X'

X')

da

rEIT.'

dxJ

d2xk

+Cil(x,x')

da

do2

=O '

Deviations of such curves

be

examined

in

a separate

geodesic deviation in the

fibered

Finslerian

gauge

studied in

Ref.

3)･

In

the

following,

we shall _get of _geodesic

deviation

of a

respect to

the

third curvature

the space. An interpretation of

deviation

of two neighbouring moving

in

the

Finslerian

can be _given

by

their relative

So

it

is

possible to

be

forces

with relation to the

In this case we

follow

H. Letx`(2,s)cF`

be

a

(2.6)

(x`(t),y`(t))

will

paper.

The

framework

of the

approach

has

been

the equation

Finsler

space, with

tensor

K)`nk

of

the

geodesic

free

particles, gravitational

field,

accelerations.

revealed the tidal

curvature

Kiihs.

Rund

(see

Ref.

8)). two-parameter

family

es

27

#

em

1e

of

_geodesics

in a Finsler space-time where 2

is

the aMne parameter or the proper time, and the s

denotes

the

family

of

geodesics,

then we assume the

equation

of

geodesic

deviation,

02zi

+K)1,(x,e)ed2hEk=:O

<2,7)

6jlE

where z means _the

deviations

vector, which rneasures the relative acceleration between

two

neighbouring

_particles.

e`=

OoX:

,

66Z2i

==2ii,elt =

[oOx21

+E,i,(x,y)zt]e"

K}`,,(x,y)=[OoF.';za-Ollll,`haoG.l]

-[O,E.i;t.OSIIkOaG,i] +Flt`kEiMh-Fm`nEfMk

1 G`(x,y)=-2-rh`k(x,yiyey'

OGn

OGh

+Cwn

FZk"=rkiiNCnh -Ctin

byi

' a)ld

a)tt

i2zi

is

different

from

Hence,

if

the vector

6at

zero, then itwill irnply

K)1,"O.

A

physical

6:zi

explanation of can

be

given

by the tidal

622

forces

in

the gravitational

field

of the

Finsler

space.

The

condition

K)`bklO

is

equivalent

to the existence of

(Rpv-gpvR),,=:.U'geO

(2.8)

q=S-gU(ah.R.",+R=caR.a2+Rx..Raan

and we

have

considered 7p"e"=-ILuv

(cf.

Ref.

x

12)}.

The

vector

U'

acts as a source or sink well or as

force

density

of the energy-mo･

mentum tensor

T),..

When

the relative acceleration

between

the paths

is

zero,

(tidal

foroes

are zero) in all

families

of geodesics of the

Finsler

time we

have

&i,,=O.

(2.9)

(3)

ShonanInstitute ofTechnology

lleviatien

of

Geodesicsin the GravitationalFVeldof FVnslerian

SPace-7Vme

If

in

a

_point

P

of the space, the relative accelerations of the nearby time-like _geodesics

become

infinite,

it

means thatthere

is

a

`"physi-cal" singularity

in

that point and also the

curvature

becomes

infinite.

In

spaces, where the

dimension

is

1arger

than two, the

deviation

vector when

it

rnoves

along the fundamental _geodesic, remains vertical on this

_path,

but

it

also turns round this one

(Ref.

7),_p.29>.

3.

Deviation

of

Vertical

Geodesics

in a

"Tangent

Riemannian"

Space-Time

We

shall restrict our study in a tangent

Riemannian

space-time

(x;

constant), with .metnc,

ds.2=gti(x,

y)dJ,idyi

.

The

_{gravitational}

field in a Finsler

space-time

F.C`)

is

_governed

by

the

Riemannian

curvature tensor

_S,ijk

which is derived

from

the

Cartan

connection

Cd`k(y).

Let

F.{2)

be

a two-dimensional _geodesic

surface of the

Finsler

space-time F.(`', and suppose that "Fl.(2) rnay

be

represented

para-metrically

by

the equations

y`(u,

U)

/i=1,

2,

3,

4

where u,

U

are the

Gaussian

_parameters of

the surface.

The

functions

_yi<u,

U)

being

of

class

C`,

we denote the tangent vectors of

the _parameters

lines,

u and

U,

by

nt,gt, respectively

e,.,ay`(:hU!,

.i=ay`(ouiU)

(3.1)

so that we

_get

Oe` 02y Oni

liTU=

oii/o

U-=

b-u'

'

(3･2)

Now, we shall

derive

the equation of the

vertical _geodesic

deviation

on ]Fl.C`)

(x:

const).

Two nearby

geodesics

C,

C' on F.C2'as above, will

be

satisfied

by

the equation,

dd2,y,`+c.i.(x,y)`lilg-i-ddY;=o

with

ds=F(x,

dy),

(the

_parameter

s=u plays the role of afine parameter or _proper time)

and,

ck

=

:.

liik

d.

+

g.

IVdu

(3.3)

is

the

deviation

vector

between

C

and

C'･

The

parameter

U

will symbolize the

devia-tion

from

a

base

_geodesic to

its

infinitesimally

nearby _geodesic.

The

covariant

derivative

and the

commuta-tion relation of a vector

field

X`(yk)

on a

tangent

Riemannian

space,

have

the

follow-ing

form,

OIF

Xil.

:::

dy,

+C.i,X}

,

(3.4)

X`lnlkmX`lklh=SAnM

-

(3-5)

Using

the notation

Pi,

it

is

known

that

l,--

Fa

<3.6)

so that we can write for the vector

field

e`,

p,gi="[g;l

+c,

i,eic

]

,

(3.7)

Taking

account of the relations

(3.1)

and

(3.2),

we _get,

ei4.n"=[

£

_{tllilb+C.ikehnk]=n`r.e"}

(3.8)

or

by

virture of

(3･7)

we have

(D,e`)nk

==

(Dlen`)gk

.

(3.9)

If

we consider the commutation relation

(Ref･

2>,

p.

30),

we get,

D,4et-LL.D,ei==F-tS,t.,ek

(3.lo)

(DLD.ei-qD,ei)Etnm=F-2S.iikeketnm.

(3.11)

The

geodesics

C

and

C'

can

be

denoted

by

U,

U+e, where e=dU is constant.

Along

these paths

aef･=O

and

(efeU)(D.ei)

are valid so that we

have

DtrD.e`=O .

If

we substitute the above relations to

(3.11),

by

virture of

(3.3)

and

(3.9),

we find

(4)

ShonanInstitute of Technology

uaptzptJc\tept

(QD.e`)etnm=Fr

2S,i.,nZ6kgm .

(3.12)

By

the relation

(3.9),

the

equation

(3.12)

become

D2ni

e2-d-s-,

+S.'tk6knM6`F-!==O,

<3.13)

where u*-u=f(u)=ds

is

of the order of

magnitude of e, and we have chosen the

function

_f(u)

so that

f"(u)==O.

Hence,

the

relation

(3.13)

will take the

form,

D2ci

+Sii,,ejCketF-2=O .

(3.14>

ds2

Beeause

the vector

ei

is

in

the

direction

of the tangential

line

element _y`,

(3.14)

can

be

wrltten,

DisCl+F-2s,i,,:itl'ck::t!=o.

(3.ls)

If

we _put

S"kt8je`=Se`ko

in

(3.14),

we _get

-Dd-

/-,-C,`-

₊

_S,

i,SCk _=-o

,

(3.16)

Dei

+Soikegk=:O,

(3.17)

ds

where 0`=DC`/ds: the components of the rota-tion of

C`,

(cf.S2).

The

relation

(3･17)

is

analogous _with the _equation

(47)

_of

E･

Cartan

(Ref.

5))with respect to the curvature tensor

Sj`kt.

The

relation

(3.14)

is

the equation of the

vertical _geodesic

deviation

in which the first term shows the relative acceleration

between

two test_particlesand thesecond one represents

the tide-producing

_{gravitational}

forces which are expressed in terms of Riemannian

curva-ture tensor

Sj`kt

of a tangent

Riemannian

space-time.

It

will

be

noted that the equation

(3.14)

can

be

related totheso-called

Einstein's

equations in the tangent space, i.e.relation

(3.18).

D2ci

The

nullification of -entails

from

(3.14)

dsa

that

Sif..=O

and this occurs only when

Ricci

tensor

Siv=O

(cf.

Ref.

6),9)).

In

this case,

we-

have

an "ernpty" region and the space

in

a

Riernannian

space.

In

the equation of the

gravitational

field

of a tangent

Riemannian

space, as was proposed

by

Y. Takano

(cf.

Ref.

12)).

ee

27 #

eg

1

_g

1 SwL/2-Sgi,{x,y)=NTij(x,y),

(3.ls)

where x=:internal gravitational constant, when TLd==O,

(Ttf=internal

energy-momentum

tensor) we

have

Sij=O,

which

is

equivalent

to

_SJikt=:o

_(cf･

Ref. 12)).Hence, the velocity

lilE`-between

the nearby _particles

is

constant, the v-paths are zero and

the

follow

_particles

h-paths are _parallel with each other, on the

base

Riemannian

manifold.

The

tidal

forces

are

dependent

on the

geometry

of the space.

Namely

they are

produced

by

the nature of the space

itself.

As

a special case, we can restrict our study

in

the

_geodesic

deviation

between

the

geo-desically

moving particles,to the indicatrix

4,

narnely

for

a

Riemannian

hypersurface

of

F.C`),

with equation

F(x,y)==1,

x=const.

We

consider the angle

dip,

between

the .x'X

tarytangent vectors of

F.(4),

dp==(yk,

yk+d)ik),

then

dg

is

defined

by,

dq2=egp.

dyP

dy'

e= ±

1 (3･19)

where _g",

is

the angular metric,

gpv="aiO-;S-,

(3.2o)

so that a vertical _path on the indicatrix

4

with _g as _a _parameter, will

be

_governed

by

the equation

d2yi

dyk

dyt

drp2

+Ck`i(Y) -d-g

ags'

=O ･

(3.21)

On

the

indicatrix

dp:=ds

(Ref.

8),p.

209).

If

Sapra

is

the curvature tensor of

4

then the equation of _geodesic

deviation

can be written

Di.Z,i+F'2Sp",,be,Eg'td-p'zkgr/l

¢

-=O

_(3･22)

be,2i-yey,rygy:

[yh=:ill/li-,

==ii,2i,33'4]

where 2k _presents the "angular

deviation"

vector _and

Shwk=SapraYZY{

Yi

Yt ･

In

the above equation we can substitute the

(5)

-38-Shonan Institute of Technology

Deviation of Geodesicsin the _{Gravitational}flieldof FVnslerian Eipace-Time

curvature tensor

Sp"re

with Rfi"re,where Rp"re

is

the Riernannian curvature on the

in-dicatrix

(cf.

Ref.

6)),

Sapro=Rapre'(IIhrgpfiugaogpr)･

(3･23)

Hence,

the equation

(3.22)

becomes

by

virture

of

(3･23)

D2zi

+F'2(Rpara-g7gpe-gggpr)

dgz

xbes2igtlll'2k

[IrLJ`-=O.

(3.24)

Furthermore,

we consider an

interesting

example of

generalized

metric giy=atJ(x)+

PhiJ(x,y)

(cf-

Ref-

1)),

with

hij==ao-oiatoaio-a",

aio=airvi, croo=aioyi,

P

is

a

_parameter

and

F2=goo==aoo=gljYlyj.

We

shall examine the

form

of equation

(3.14)

in

the case of generalized tangent space

M(D. The Cartan-like symbols

have

been

calculated

(cf.

Ref.

1),

10))

cr,=Pcr"(aioig!ocrjkNexo-o2ftioiaprocrko)+P2(crNo-eihftyD.

(3.25)

If we substitute

(3.25)

to

F"2SiJmn=CkjmCikn-CkjnCikm

(3.26)

and after some calculations, we get the

deviation

equation,

(1-P)F-2StjmA=(1'2cv6o2)X;nj{n

+((rifo3-cro-oi)T;ijiano'P(cxeoEmiin-cre"D'BiJmn) -P2croTo2(akoaiea".'6:+P2Aij.n)

+P3aero'MJma

-cro-o2

T:

in ,

where we

have

_put,

Aijmn

== HLjmlLknm

H;tjk

H;nici

_,

HLkj

=::

hivYk

H}dmn=arinH6Int+6SakmH}kn-aiohJn')rmT6YnHhOt

BYmn=

TLkmH]}Jk

+

TLjnHL-ki

11i`ic=Xeako

, 7;,ii.==TZj.'7}ic.

Emjin=ILejkHLkn

Xa=i32cro-e2figa.o,

XLeiin=XAain

･

4. Deviation of

Vertical

Geodesics

in

a

Tangent

Riemannian

Space

of

Constant

Curyature

Finsler

spaces with constant curvature are

important

for

geometry as well

for

Physics

(cf･

Ref.

4),

6)).

In

our case, we assume the

deviation

equations

for

a tangent

Riemannian

space

M:

(n24)

of constant curvature.

The

following

is

valid:

{cf.

Ref.

6)

th.

31.6p.

225,

Ref.

9))

F2Snidk=S(hnjhtk-hltkhiD

(4.1)

or

F2Si`Jk==S(htk6S-hty6k)

,

S=const.

If

we substitute this equation to

(3.14)

we

_get

D2nt

dy,

+S(htk6S･-hiffiDvivicn,=o

(4.2)

where we symbolize the tangent vectors with

v`, vk and nd is the

deviation

vector.

From

(4.2)

we

have,

I-

]-d-?-sn-,`

+

sv,vk

ni-

svdnJv`

=o.

(4.3)

If

we apply the relation

(4.3)

for

the

gravitational

field

to the

Finsler

M'.

with respect to nearby vertica1 _geodesies of

M`.

as

we

have

mentioned above, and considering the relations viv`=1 and vknk=O then the

equation

(4.3)

is

reduced to the form,

d2nt

+Sni=O

(4.4)

ish-which

has

the obvious solution

zi=sinA(NVS)

(4･5)

where the vector 2i is_parallel to n`.

In

consequence of the equation

(4.1)

and

Ref.

6)

th.

31.6

p.

225,

the indicatrix

4

of

M(.`'

isa

Riemannian

space of constant curvature,

Rapre=(S+1)(gargp"-gltsgrr)

(4-6)

where Raprb is the curvature of the

indicatrix

and

g.fi=h" yE'

y'p

[Yt

--

911ia]

'

(6)

vautXptJi(\resu

Hence,

repeating the above _procedure, we take a similar result

for

the

deviation

equation of the

indicatrix

4,

d2ei

+(s+1)ei=o

(4.7)

ds2

where we _put e`the

deviation

vector

between

two

infinitesimally

nearby

_geodesics

on the

indicatrix

l}.

5.

Conelusion

So

in

Riemannian

space-time, _as

in

_the

Finslerian approach of space-time, the

ex-istence

of the _{gravitational}

field

is

manifested

by

the

deviation

of two-test _particles,which

are

located

to _nearby _geodesics.

The

_study

of geodesic

deviation

is

closely related to the existence of tidal

forces

which are

intrinsically

produced

by

the geometry

(curvature

tensor)

of the space.

The

kind

of equation _of _geodesic

deviation

reveals _physically the nature of

these

forces.

In

the

tangent

Riemannian

space

the

devia-tions of yer'tical geodesics are

important

be-cause they

imply

the existence of the vertical

component of the energy-momentum tensor

Tl".t

of

the

Einstein's

equations, so

that

in

case of "the empty vertical space", the vertical

ca 27

#

ee1e

geodesics are

free

of any

deviations･

Referenceg

1) A.K. Aringazing and G.S. Asanov: Rep.

Math. Phys. 25 (1988),183.

2) G.S. Asanoy: Finsler Geometry Relativity

and Gauge Theories, D. Reidel, Dordrecht 1985.

3) G.S. Asanov and P.C. Stavrinos: Rep. Math.

Phys.

4) G.A. Asanov: Fortschr. Phys. 39 {1991)3,

185-210.

5) E.Cartan: Les espaces deFinsler, Actual.79,

Paris1934.

6) M. Matsumoto: Foundations of Finsler

ometry and SpecialFinslerSpaces,Kaiseisha

Pres. Kaiseisha 1986.

7) C.W.Misner-K.S.Thorne-I.A.Wheeler: tation Publ. Freeman. Freeman, San Francisco

1973.

8) H.Rund: TheDifferentialGeometryofFinsler Spaces, Springer, Berlin1959.

9) H. Shimada:

_J.

Korean Math. Soc. Vol. 14,

No. 1,1977.

10) H.Shimada: Symp. en Finsler Geometry at

Awara Sept.29,1989.

11>

J.

L.Synge: Relativity: The _generalTheory,

North Holland. Amsterdam 1960.

12) Y. Takano: Proc. Int. Symp. Rel. Un. Field

Theory p. 17, 1978.