- Email: [email protected]
A criterion for local instability of a geodesic flow from the Jacobi equation in Fermi basis
A Criterion for Local Instability of a Geodesic Flow from the Jacobi Equation in Fermi Basis MAREK Astronomical
Ohsenator];.
SZYDtOtVSKI
Jagellonian
University.
JERZY Department
of PhyGcs.
Ptdagq$cnl
Orb
171. 30-244
Cracow.
Poland
SZCZFSM
University.
Podchorqiych
2. ?O-C4h Crcrcow.
Poland
and
hIAREli Copernicus
Astrowmical
I. GWI)I
WC shall metric
investigate
tensor
Center.
Bnrtycka
IXVIA’I‘ION
the gcodcsic
g. The ccluntion
tIIESL\DA
llow
IS. 00-716
IX)U,\lION
IN FERXII
on Ricmannian
of gcodcsics
Warsaw.
(or
Poland
B>\\sIS
pscudoricmannian)
spaces with
rends: V”U
=
0,
(1)
whcrc u is the vector field tangent to the geodesic and V denotes the covariant derivative. Therefore, Ict LIS consider ;I gcodcsic congruence on the manifold M. Then let us take a curve C orthogonal to the consrucncc, i.e. a curve which intersects each geodesic just one
time. We shall assume that the affinc zero at the point of intersection with curve
C along
gcoclcsics
parameter the curve
to a new position
s measured along the geodesic is equal to C. Next by displacing all the points of the
labcllcd
by s WC construct
the family
of curves
C,. In this way wt’ obtain a certain two-dimensional submanifold in the region where the congrucncc is defined. Let us dcnotc by u~,.~) the vector field tangent to geodesic at the point for which the parameters mcasurcd along gcodcsic and along C, curves arc (s, A). Let ub also dcnotc by z(T,i) the vector field tangent to C, curve at the same point. Obviously [u. $1 = 0. Because
the connection
I- is torsion-free, T(u,
5) = c,<
(2)
WC have: -
V$l 232
-
[u, $1 = 0.
3-t
and hence: Y “Y= = r,u. The vector
< informs
us about
all start from some arbitrarily We shall now calculate geodesic.
the relative
position
of points
chosen curve C. the accelleration with
which
on neighbour the
vector
geodesics
E varies
which
along
the
We have: R(u.
where
(3)
R(u,
g) is the
;I,u = Y’,T’,u
curvature
tensor.
-
By
r’,r’,u
-
Y,“,<,U, the
combining
(J)
formulae
(1).
(2)
and
(4)
u-e
obtain: Y”Y$l and since (3) is valid
\ve finally
arrive
= R(u,
c;“)u.
(-5)
$,u.
(6)
at the formula: Y”Y,$
= R(u,
As
it has just been mentioned we are interest4 in relative motion of geodesics not of about the position of point from uhich points on them. The vector f contains information to u and is the gcodcsic starts. This information is carried by the component of < parallel unimportant
for our
purpose.
Therefore
we shall
consider
the component
of c orthogonal
to II. Let us write: 5 = whcrc
n is the vector
orthogonal
I1 +
Al
,
to u, 1. E IR. WC have: g(<. u) = JJ(n, 11) +
Lkxxusc
(7)
g(n, u) = 0. then by normalizing
Q(ll,
the tangent g(f,
u).
vector
(8)
g(u, u) = 1 wc obtain:
u) = EL,
(9)
2lld n=;l-
g(&
U)U.
(10)
vector I’iclcl II is collcd ;I gcodcsic deviation field and carries information about position of ncighbour gcotlcsics. WC shall now find an equation for the acccllcration vector n along the gcoclcsic. From the equations (9) and (IO) wc have:
The
~“~“(11 ht:
-r”[ll
+
g(f,
u)]
=
T,,Il
+
+
g(T,,;17
g(f,
U)UJ
u)u
=
= -i”Il
‘c .f 011
the
other
obtain
bvc’
hLld:
f?(U,
I1 i
~(2.
U)U)U
=
+
II
g(T$l,
+
g(f, u)u
U)U), =
T,,Il.
It
Illc;llls
that:
T,,n.
= f<(U,
(Ill
fl)ll
f
g(:,
U)f?(U,
tJ)U
=
f<(U,
1l)U.
fjcllcc
that r,,YUn
The
R(u.
rclativc of the
equation
(13)
callccl
;I
= R(u,
deviation
gcodcsic
n)u.
equation
(12, plays
;I
crucial
role
in our
iri/
the
n \2xtor
furthcr-
clixussion. WC arc
gcodcsic.
now
able
to
tell
b\hat
Since /~ij = [Sq(n, n)]’
ti -cl !
Ill:
i5 fhc
;1cCcIICr;ltic)n
of
the
lcngtli
‘. wc have: =
-L IIli
y(T,,n. ’
11)
=
4 1”
g(r1,
-c’,,Il)
of
;tlong
Local stability
Thus we arrive at the following -$
a geodesic
of
23.5
flow
equation:
/nj’ = -2R(
u, n, u, n) + 2g(V,n.
V,n).
(13)
of the Riemann tensor. where R(u. n, u. n) = R,,krdn’uknl and Rllkl are components It is suitable to write the geodesic deviation equation in terms of a linear operator K,:T,M - TqM. defined as VlA E T,M:K(A) = - R(A, u)u. One can easily check that this operator is symmetric in the sense of metric 5: g(A, The geodesic
deviation
equation
K(B))
= g(B.
in terms of the K-operator C,V,n
In local coordinate
system
K(A)).
(14) reads:
= K(n).
(15)
(x’}, we have:
I\; = (dx’,
R(u, a,)~) = (dx’, R5,,k,~‘“~ki3;) = R:,,k,d”Ilk,
(16)
or h’; =
--R;,,,kll”‘llk.
(17)
The next suitable procedure is to introduce the so called Fermi basis [I]. The construction vectors on the hypersurface T,H of this basis is following. We choose N - 1 orthonormal orthogonal to u at point q of the geodesic. Then we identify the Nth basis vector with the vector u at point (1 i.e. E,v = u. As a result we obtain the basis E,, . . ., E, in TqM such that g(E,,, E,,) = ii,,,, for all cr$ = 1, . . ., N. Then WC move this basis parallelly along the basis gcodcsic (i.e. G,E, = 0, for N = 1, . . . , N). In this way we define an orthonormnl along the gcodcsic. WC can write in this basis: n = rl“E,,
(Y = 1, . . ., N
V,,n = E, F,
V,V,n From the equations K(n)
:
dZllrn = E, ds2 ’
(18)
(18) we obtain: = n”K(E,,)
Finally
the geodesic
where
h’z = -R:vtrv.
deviation
= II”R(E,~, E,,)E, equation
We can associate
in Fermi
= r~“E,R:b,v,~ = -E,
R$,,,v,l”.
basis takes the following
form:
a dual basis {E”} with the basis {Et,}, so that: (E”,
EP) = L$.
(21)
Bccausc: K;; = -(E”,
R(E,,, u)u),
then $
K;; = -‘?,[(E”.
R(E,,,
u)u>]
(10)
= - (V,E”,
R(E,,, U)U> - (E“,
V,( R(E,,. u)u)).
is equal
The first term E;‘[T’,(R(E,.
u)u)]’
to zero.
whereas
the second
= E:‘~c~~~R;,,,,,fl’E;;rl” = E:[11’~~R;,,,,f1’E;l’rl”
Hence.
one reads:
+ 1l*~~R;,,,,,11’E;l’rl”j
= E;‘~,Ri,,,,,~1’~1’E;;‘rl”.
we obtain:
__ (“1 The
equation
(22)
implies
that the
T-iR;,,,,, = 0 (the so called equation (20) can he easily
2. A CKITEKION
It is generally there
esists
informs
ii
to rckvritc
solved
FOK LOCAL
;i comples
;I simple us about
the local
has constant spaces).
In
components
these
caws
in spaces
the geodesic
problcrii
to solve
of averaging instability
deviation
for
brhich
deviation
[2]
INSTAI~ILIT~ FKOhI THE A\‘F:KAGEI) GEOI)ESIC EQUATIOS
procedure
the gcodcsic
K-operator symmetric
locally
=
deviation
implie
of a gwdcsic
equation
T’,,I-,,I1
the geodesic which
that
flui\..
D’n
=
-grad,,
Severthclc45
of the Ricci
It is convenient
in the following
-
equation.
the sign
DE~~I:\TION
for
this
scaI;~r
purpose
form:
(2.3)
L’,,(n),
tls’
3
(grad,,)’
= g” --. 311’
and l,',,(u)=
convcnicncc
?‘hc
V,,(n)
25
;I
-4(II.
ot the equatiorl
certain
potential
(23)
whose
R(l1,
n)u)
consists
gradient
=
;f\‘(Il.
u.
in fact that dctcrniincs
Il.
Ll).
bc can look
the
rh5
up011 the qu:intit>.
ot’ the gcocicsic
clcviation
equation.
/‘i-00/~ Consists
ll”dlli
iI1 simple
nlgcbra:
+ $“~~~,f~;,,,ll”fl’ll’)= i( R;,~,rl”ll’rlA + ~“/i,,,,,l,“ll’i =
Hcncc
-i[(R(u.
1l)U)’ +
(li(l1,
)
Il)ll)J
-=
-R(u.
I
1l)ll.
WC hnvc V,(n)
as :I quantity For- arbitrary
playing vector
the role ficlcls
=
of ;1 potential
!f<(u,
II. C. I))
(Z-1)
in the _ccodcsic d~viatioll
A and 1) wc can tlcfinq
K(A.
II. u. n)
ttlc Ricrrlann
=
I))Ii).
t’clu;ltion.
cur\‘aturc
tcnwr
ah:
(‘5)
Local stability
We can also define
the two-dimensional
of a geodesic
curvature
flow
237
KAiB along
the direction
determined
by
these fields: R(A, From
the equations
B, A. B) = K,,,,[g(A,
A)g(B,
B) -
= iK,:,g(n.
n),
because g(u. u) = 1. and g(n. u) = 0. While investigating the geodesic deviation neither
B)g(A.
B)].
(26)
(23) and (26) we obtain: v,(n)
precisely
g(A,
the tangent
vector
we
u, nor
the
are
(27)
not
deviation
in
general
vector
able
n for
to
determine
the specific
initial
value of the parameter S. Therefore it seems reasonable to discuss the geodesic deviation equation o_riginating from a certain averaged potential P(n). We shall define this averaged potential V(n) by choosing the vectors u and n at random, i.e. we assume that every direction determined by the bivector u A n is equally probable. In this way we associate the averaging mean which
procedure
the set of dimension
bivectors
with
the space of two-directions.
have the following
,, ,)QQ = ;(A”
We shall denote the pair of indices tensor in the space of two-directions:
(iliz)
that this tensor
B’: _ Al: &I).
by an ovcrnll
Q:ll = sI,,,&:,: chcckcd
we
form: (A
It can he easily
By the space of two-directions
A A B = !(A” B” - A’: B”)a,, A 3,:. This is a linear space M = i N(N - l), where rV = dim M. The components of
all bivectors is equal to
-
index
I. Let us introduce
the metric
S#,,&,~
(28)
is symmetric: Q=,I = Q=,,*
The mclric
C,, satisfies C(A
the following A 11, c
(29)
identity:
A I)) = $(A.
C)g(IS,
I)) -
g(A,
I))@,
C)
(30)
I3y virtue of the Schwartz inequality the quadratic form defined by the Cl,-matrix is positively dcfinitc, hcncc C defines an cuclidcnn metric in the space of two-directions. In order to dcfinc a malrix invcrsc to C,, wc have to clcfinc a unit operator in the space of two-directions. We shall denote its components by 0: and dcfinc as ii;(A
A
1%)’
=
(f\
A
Ii)‘.
(31)
Hence,
b’h. = ;(cb;‘,q Let us dcnotc
by C”
the inverse
of C,,.
- CqAp,).
Obviously
(31)
WC have CK,C”
= Oi.. Thus
q=/, = ~(g/1~ls/:12 _ s/l~:s,:li)~ We shall not rcwritc V,(n)
the potential = iR(n.
V,(n)
(32)
in terms of bivcctors:
II, n, u) = ~R,,klll’flirl’ll’ = iR,,(n
A
u)‘(n
A
u)’
(33)
WC have used the known symmetry of the Ricmann tensor IFl,, = !R,,. In general quantity R(I\, It. it. I)) dcfincs a certain quadratic form in the space of two-directions: whcrc
R(A. As
it
is
well
known,
I%, r\, 15) = R(i\ there
exists
A
1%. t\
a basis
A
ii)
{E,,},
=
R,,(/\
n = I,
A
I{)‘(,\
. ., iN(
A
N -
the
It)’
1) in the
space
of
M.
139
er al.
SZY~OWSKI
two-directions for which lR(E,. Eb) = Adbob and C(E,. Eb) = b,,. R(A A B. A A B) = 1.,(x’) + . _ . + AM(rM)‘. where A A B = Y’E,. We can now define the averaged potential P(n): R(A where assume fix the sphere
A B. A
In such
basis
we have:
B)dC’,
A
(34)
IS the volume of M - 1 dimensional sphere Sl’-‘. In the other words we volS,!‘-’ that none of the geodesics is privileged and neither is the deviation vector. We only length of bivectors by assuming that C(A A B. A A B) = g(n. n). This defines the S;‘-’ of the radius r = [g(n. n)]‘,’ In the space of two-directions. Hence.
I,~~,_,
P(n)= zvoIk ,,-, ,
[A,(x~)~ + .
+ A,,(.r”)‘]dL’.
and because: .M - I ,
~‘e arrive
(.rU):dV
= ‘O’i;
r’ = +
volS:‘-‘g(n.
n).
at the formula: I
‘(n)= 2MI
IAl + .
+
A,,]Lq(Il.
n)
However A, + whcrc
. + J’,,, =
IR,,C” = ;R ,,,:,,,:[g”“$J’:
R is the Ricci scalar.
Thcrcforc
the avcragcd
clircction
dctcrmincs
oC gcoclcsic
tangent
vector
the: gcodcsic
N(N
to unity).
fQ(n.
I)
deviation
and the dircctir)n
is normalized
-
J = ;[ R,:,$=
potential
I
c3 (II) = This potential
- pg”‘:
-grad,,
has the form: (35)
in which
vector
The corresponding
D?n^ -= ds?
= R.
II).
equation
of dcvintion
+ R ,,,, g”“]
(with
WC randomly
the only
equation
choose
the
that
the
constraint
reads:
P(n).
!A)
II2 7
ds-
If u’c introduce
;I Fermi
basis (E,}
2R
fl’ = -
along
,V(;\’
-
I)
the randomly
2X 7d? ff” = iV(iV dsOne see from equation is containccl in the Ricci scalar for locnI
instability
spaces implies
of geodesics.
chaos in a gcodcsic
A’ is the
dimension
(36)
chosen
gcoclcsic.
WC obtain: (37)
I) Ti”
that the full information cunccrning the local instability R. The negativity of the Ricci scalar is a sufficient condition The
property
rz-
of local
of the
space
2R N(N
-
1)
011 tvhich
instability of geodesics on compact the mean timescale for mixins as
-12 1
flow. We can estimate
[
where
ii’,
the
gcodtzsic
(‘S) flow
is defined.
The
Local stability
of a geodesic flow
239
relaxation time r is a time-scale in which the length (nl of the deviation vector increases e-times in average. The inverse of t corresponds to the positive Lyapunov exponent [2]. It is worth noticing that the equation (37) has the same form as in the case of maximally symmetric spaces for which the Ricci scalar is proportional to the Gauss curvature. As it is known, a geodesic flow on the compact space with negative Gauss curvature becomes an Anosov flow [3]. It implies that after averaging the Ricci scalar contains the full information about the local instability and plays the role analogous to that of Gauss curvature in the case of maximally symmetric spaces. Of course, if M is a compact Riemannian manifold such that its curvature in an arbitrary two-direction is negative (such manifolds do exist [-I]), then the geodesic flow is an Anosov flow. The geodesic flows on compact Riemannian manifolds will obviously have the same property. In this sense we can say that the criterion of negativity of the Ricci scalar is a generalization of classical works of Anosov.
3. APPLICATIONS 3.1.
Locnl iustabiliry of r~llllritIir~lcrlsio,lal~~on~oge~~cowcosmological models
The Maupertuis principle of classical mechanics [4] allows for representing the trajectories of a Hamiltonian system as a geodesic flow in certain region of the configurational space (for details see [2]). The system with the Hamiltonian: H = +“h(cl)l“,~~b cl”, p,, arc coordinntcs and momenta cncrgy, is equivalent pscudoricrnannian) manifold with the metric:
whcrc
T = $d’p,,p,, is the kinetic
+ V(q). respectively, V(q) is the potential to,a gcodcsic flow on a Ricmannian
and (or
ds’ = 2 Wcl,,,,dq” dclh, ds = 2Wdt, whcrc 2W = l/r - VI, and II is the mcasurcd along geodesics which formula of (30). This means that Lagrange function L = ~a,,Q’cj’ -
total energy. Note that there is a new affinc parnmctcr s is conncctcd with the dynamical time I by the second the equations of motion of the system dcscribcd by the V(q) arc reduced to the geodesic equation:
d’ ds’.- (1’ + ds = dr where
2W = I/z - V(q)l.
Because
the subspace I-,, of the phase-space
(39)
r;k
drl’ d# ds ds
-
= 0,
2W(q(s(O)),
the total energy is conserved
(40) the motion
takes place in
r such that
rh = {(p,
(I):
~u"~J,/J,
+
V(q) = /I}
(41)
or in the subspace Q of the tangent fibre bundle TM:
Q = ((4, 0): Cl,,(C/)lj’Cj’= 2W). Hence the problem is reduced to determining the geodesics on a Riemannian where M is a configurational space and g is the metric: S,, = WQ,,.
(42) space (M,
g)
(43)
In constructing the multidimensional sional Einstein-Maxwell action:
s=
cosmological
&
dDx(-detjj,,)“‘(fi
I
ii
models
we start
from
a multidimen-
- 2i),
where D = 1 + n + ii is the dimension of total space-time. n and fi are dimensions of the physical and internal spaces respectively. i is the D-dimensional cosmological constant, 6 is the D-dimensional gravitational constant. According to the philosophy of Kaluza and Klein we can represent the metric of a D-dimensional spacetime in the form:
0. 1. ., II, a.6 = I.. ., if, h is a function of time. The metric S;,(,, is associated with that of the physical space g!,,, by the relation g,,V = I\~‘~,,,.. where N.’ = /T-‘” (“-I’. We shall assume that the metric ,c,,~ describes a ii-dimensional space of constant curvature, i.e. R,,,, = k?g<,,,. The above assumptions allow for representing the action (44) as arising from the Lagrangian:
il.\‘=
L = (-det
&JZ
&
R -
&3pj~” -
V(Q)
[ whet-c G = 6/(detCqGrh)“‘,
$,= h--‘{C[l
V(d))=
(46) I
+ fi/(rl -
l)]}‘,” In h, h’= (SnG)‘,‘.
and
5 {4-2I-((,,_,)(;+,I _ ,) j’s]
- gLsp
[
(g,“I,’j’ y)
-2K
(47)
‘.
Aflcr the climcnsional rccluction this milltidirncnsionnl systcrn bcconics equivalent gravity and scalar ficlrl couplccl niinim~~lly. By choosing such ;I coorclin;ltc systcrn that:
to
(-IS) Lvhcrc R,, is the metric of a constant wc’ finally arrive at the formula:
ciIrvnturc
[_ = (dct J,,)‘F{hq-+(,I lvhcre the clot clcnotcs the system rcacls: I/ and the constant
pscudoricmannian
-
the diffcrcntiation
=
condition
-+(,c
spxc.
+ $?c-‘-]
with rcspcct
- 1,;> +
is: I/ = 0. The
I,:’
and by introducing
$
I- &4 -
to time.
cj’ - g&-‘;
~lamiltonian
sonic
Hence
ntx
variable
V(C/I)} the Hamiltonian
+ h-)V(Q), (50) induce5
(50)
;I gcoclcsic
!lo\\
011 ;I
space with metric:
cl.sJ = 2Cl’(d~j~ - ‘I:?) ;. bvticre 5 = of a gcoclcsic
for
2r1( r1 - 1) :. c/l = KqJ 211cl ?\I/ = h-’ v(&K) flow is detcrminccl by ttic Iiicci scalar:
(51) -
J r?e-’
’ “I’‘-‘I
‘,
Local
illstal)ili[\
Local stability of a geodesic flow
case when V = 0 or A = 0 and ii = 0 the Ricci scalar (52) is equal the fact that the corresponding systems are integrable. If the internal
In particular This reflects Ricci
flat (R
to zero. space is
= 0) we have: sgn R = sgn(Ri).
This means that the system
is locally
unstable sgn R =
Therefore
the
system
sgn R = sgn (RA)
3.2.
Choric
In general Hausdorff defines
is unstable
and the system
hellclriolrr relativity
differential
locally is unstable
(53)
if RA < 0. In the case of (? = 0 and n = 3: -sgn(Ri?). if
(54)
Rt? > 0.
when
Analogously
for
R and A are of opposite
fi = 0.
we
have
signs.
of gt~0dcsic.s in spficctirncs the spacetime
manifold
the causal structure
is modelled
and g is a Lorenzian
of k! imposes
certain
by a pair metric. restrictions
(AI,
g) where
The Lorenzian on topology
bf
is a C”-class
structure
which
of the spacetime
[I]. Namely the spacetime is non-compact or if it is compact its Euler characteristic must vanish. Timclike or null geodesics in spncctimc have a deep physical meaning since they rcprcscnt historics 01 obscrvcrs or photons. Thcrcforc it is interesting to ask whcthcr gcodcsic flows in spacctimcs can bc locally unstable or can possess a mising property. It has been cstnblishcd that the so called Mixmastcr models exhibit a chaotic behaviour of scale factors. Howcvcr the scale factors alons have no physical meaning. Therefore our clucstion about chaos in gcodcsics is a physical one and can bc answcrcd within the frames of above dcscribcd formalism. In the cast when the spacctime is compact and its Ricci scalar is ncgativc the gcodcsic llows are Anosov tlows and consequently are crgodic. It should bc strcsscd, however, that the compact spacctimcs can bc pathological due to cxistcncc of closed timclikc trajcctorics [I]. When
the spacetime
is not compact
and has a negative
Ricci
scalar,
then
the gcodcsic
flows have the property of local instability. Projections of these gcodcsics on hypcrsurfaces of constant time represent the trajectories of observers or photons and can become Anosov flows provided the Ricci scalar is negative. As an example we can take the bchaviour of geodesics
in the so called
Ellis small
and Prigogine [5]. They show time, when suitably projected
that into
world,
which
has been investigated
by Lockhart,
non-spacelike geodesics in Robertson-Walker a three-dimensional compactified hypersurface
Misra spaceof a
constant negative curvature, change into geodesics paramctrised by a new affinc parameter. It is interesting that in Mixmaster models the criterion of ergodicity of geodesic flows on hypcrsurfaces of constant time is equivalent to the corresponding condition (R < 0) for a whole moclcl. In the other words chaos in geodesics is a consequence of chaotic nonlinear dynamics of the spacetime which is an arena for all events. When the spacctime is compact and its Ricci scalar is negative, then the geodesic flows have mixing property. The negativity of the Ricci scalar has a simple physical meaning. Namely the condition R < 0 implies E + 311 < 0. where F is the energy density and 1~ is the pressure. It means that the strong energy condition is violated. On the other hand the violation of the strong, energy condition solves the horizon problem in Friedman world models [6].
REFERESCES
Ohsenator];.
SZYDtOtVSKI
Jagellonian
University.
JERZY Department
of PhyGcs.
Ptdagq$cnl
Orb
171. 30-244
Cracow.
Poland
SZCZFSM
University.
Podchorqiych
2. ?O-C4h Crcrcow.
Poland
and
hIAREli Copernicus
Astrowmical
I. GWI)I
WC shall metric
investigate
tensor
Center.
Bnrtycka
IXVIA’I‘ION
the gcodcsic
g. The ccluntion
tIIESL\DA
llow
IS. 00-716
IX)U,\lION
IN FERXII
on Ricmannian
of gcodcsics
Warsaw.
(or
Poland
B>\\sIS
pscudoricmannian)
spaces with
rends: V”U
=
0,
(1)
whcrc u is the vector field tangent to the geodesic and V denotes the covariant derivative. Therefore, Ict LIS consider ;I gcodcsic congruence on the manifold M. Then let us take a curve C orthogonal to the consrucncc, i.e. a curve which intersects each geodesic just one
time. We shall assume that the affinc zero at the point of intersection with curve
C along
gcoclcsics
parameter the curve
to a new position
s measured along the geodesic is equal to C. Next by displacing all the points of the
labcllcd
by s WC construct
the family
of curves
C,. In this way wt’ obtain a certain two-dimensional submanifold in the region where the congrucncc is defined. Let us dcnotc by u~,.~) the vector field tangent to geodesic at the point for which the parameters mcasurcd along gcodcsic and along C, curves arc (s, A). Let ub also dcnotc by z(T,i) the vector field tangent to C, curve at the same point. Obviously [u. $1 = 0. Because
the connection
I- is torsion-free, T(u,
5) = c,<
(2)
WC have: -
V$l 232
-
[u, $1 = 0.
3-t
and hence: Y “Y= = r,u. The vector
< informs
us about
all start from some arbitrarily We shall now calculate geodesic.
the relative
position
of points
chosen curve C. the accelleration with
which
on neighbour the
vector
geodesics
E varies
which
along
the
We have: R(u.
where
(3)
R(u,
g) is the
;I,u = Y’,T’,u
curvature
tensor.
-
By
r’,r’,u
-
Y,“,<,U, the
combining
(J)
formulae
(1).
(2)
and
(4)
u-e
obtain: Y”Y$l and since (3) is valid
\ve finally
arrive
= R(u,
c;“)u.
(-5)
$,u.
(6)
at the formula: Y”Y,$
= R(u,
As
it has just been mentioned we are interest4 in relative motion of geodesics not of about the position of point from uhich points on them. The vector f contains information to u and is the gcodcsic starts. This information is carried by the component of < parallel unimportant
for our
purpose.
Therefore
we shall
consider
the component
of c orthogonal
to II. Let us write: 5 = whcrc
n is the vector
orthogonal
I1 +
Al
,
to u, 1. E IR. WC have: g(<. u) = JJ(n, 11) +
Lkxxusc
(7)
g(n, u) = 0. then by normalizing
Q(ll,
the tangent g(f,
u).
vector
(8)
g(u, u) = 1 wc obtain:
u) = EL,
(9)
2lld n=;l-
g(&
U)U.
(10)
vector I’iclcl II is collcd ;I gcodcsic deviation field and carries information about position of ncighbour gcotlcsics. WC shall now find an equation for the acccllcration vector n along the gcoclcsic. From the equations (9) and (IO) wc have:
The
~“~“(11 ht:
-r”[ll
+
g(f,
u)]
=
T,,Il
+
+
g(T,,;17
g(f,
U)UJ
u)u
=
= -i”Il
‘c .f 011
the
other
obtain
bvc’
hLld:
f?(U,
I1 i
~(2.
U)U)U
=
+
II
g(T$l,
+
g(f, u)u
U)U), =
T,,Il.
It
Illc;llls
that:
T,,n.
= f<(U,
(Ill
fl)ll
f
g(:,
U)f?(U,
tJ)U
=
f<(U,
1l)U.
fjcllcc
that r,,YUn
The
R(u.
rclativc of the
equation
(13)
callccl
;I
= R(u,
deviation
gcodcsic
n)u.
equation
(12, plays
;I
crucial
role
in our
iri/
the
n \2xtor
furthcr-
clixussion. WC arc
gcodcsic.
now
able
to
tell
b\hat
Since /~ij = [Sq(n, n)]’
ti -cl !
Ill:
i5 fhc
;1cCcIICr;ltic)n
of
the
lcngtli
‘. wc have: =
-L IIli
y(T,,n. ’
11)
=
4 1”
g(r1,
-c’,,Il)
of
;tlong
Local stability
Thus we arrive at the following -$
a geodesic
of
23.5
flow
equation:
/nj’ = -2R(
u, n, u, n) + 2g(V,n.
V,n).
(13)
of the Riemann tensor. where R(u. n, u. n) = R,,krdn’uknl and Rllkl are components It is suitable to write the geodesic deviation equation in terms of a linear operator K,:T,M - TqM. defined as VlA E T,M:K(A) = - R(A, u)u. One can easily check that this operator is symmetric in the sense of metric 5: g(A, The geodesic
deviation
equation
K(B))
= g(B.
in terms of the K-operator C,V,n
In local coordinate
system
K(A)).
(14) reads:
= K(n).
(15)
(x’}, we have:
I\; = (dx’,
R(u, a,)~) = (dx’, R5,,k,~‘“~ki3;) = R:,,k,d”Ilk,
(16)
or h’; =
--R;,,,kll”‘llk.
(17)
The next suitable procedure is to introduce the so called Fermi basis [I]. The construction vectors on the hypersurface T,H of this basis is following. We choose N - 1 orthonormal orthogonal to u at point q of the geodesic. Then we identify the Nth basis vector with the vector u at point (1 i.e. E,v = u. As a result we obtain the basis E,, . . ., E, in TqM such that g(E,,, E,,) = ii,,,, for all cr$ = 1, . . ., N. Then WC move this basis parallelly along the basis gcodcsic (i.e. G,E, = 0, for N = 1, . . . , N). In this way we define an orthonormnl along the gcodcsic. WC can write in this basis: n = rl“E,,
(Y = 1, . . ., N
V,,n = E, F,
V,V,n From the equations K(n)
:
dZllrn = E, ds2 ’
(18)
(18) we obtain: = n”K(E,,)
Finally
the geodesic
where
h’z = -R:vtrv.
deviation
= II”R(E,~, E,,)E, equation
We can associate
in Fermi
= r~“E,R:b,v,~ = -E,
R$,,,v,l”.
basis takes the following
form:
a dual basis {E”} with the basis {Et,}, so that: (E”,
EP) = L$.
(21)
Bccausc: K;; = -(E”,
R(E,,, u)u),
then $
K;; = -‘?,[(E”.
R(E,,,
u)u>]
(10)
= - (V,E”,
R(E,,, U)U> - (E“,
V,( R(E,,. u)u)).
is equal
The first term E;‘[T’,(R(E,.
u)u)]’
to zero.
whereas
the second
= E:‘~c~~~R;,,,,,fl’E;;rl” = E:[11’~~R;,,,,f1’E;l’rl”
Hence.
one reads:
+ 1l*~~R;,,,,,11’E;l’rl”j
= E;‘~,Ri,,,,,~1’~1’E;;‘rl”.
we obtain:
__ (“1 The
equation
(22)
implies
that the
T-iR;,,,,, = 0 (the so called equation (20) can he easily
2. A CKITEKION
It is generally there
esists
informs
ii
to rckvritc
solved
FOK LOCAL
;i comples
;I simple us about
the local
has constant spaces).
In
components
these
caws
in spaces
the geodesic
problcrii
to solve
of averaging instability
deviation
for
brhich
deviation
[2]
INSTAI~ILIT~ FKOhI THE A\‘F:KAGEI) GEOI)ESIC EQUATIOS
procedure
the gcodcsic
K-operator symmetric
locally
=
deviation
implie
of a gwdcsic
equation
T’,,I-,,I1
the geodesic which
that
flui\..
D’n
=
-grad,,
Severthclc45
of the Ricci
It is convenient
in the following
-
equation.
the sign
DE~~I:\TION
for
this
scaI;~r
purpose
form:
(2.3)
L’,,(n),
tls’
3
(grad,,)’
= g” --. 311’
and l,',,(u)=
convcnicncc
?‘hc
V,,(n)
25
;I
-4(II.
ot the equatiorl
certain
potential
(23)
whose
R(l1,
n)u)
consists
gradient
=
;f\‘(Il.
u.
in fact that dctcrniincs
Il.
Ll).
bc can look
the
rh5
up011 the qu:intit>.
ot’ the gcocicsic
clcviation
equation.
/‘i-00/~ Consists
ll”dlli
iI1 simple
nlgcbra:
+ $“~~~,f~;,,,ll”fl’ll’)= i( R;,~,rl”ll’rlA + ~“/i,,,,,l,“ll’i =
Hcncc
-i[(R(u.
1l)U)’ +
(li(l1,
)
Il)ll)J
-=
-R(u.
I
1l)ll.
WC hnvc V,(n)
as :I quantity For- arbitrary
playing vector
the role ficlcls
=
of ;1 potential
!f<(u,
II. C. I))
(Z-1)
in the _ccodcsic d~viatioll
A and 1) wc can tlcfinq
K(A.
II. u. n)
ttlc Ricrrlann
=
I))Ii).
t’clu;ltion.
cur\‘aturc
tcnwr
ah:
(‘5)
Local stability
We can also define
the two-dimensional
of a geodesic
curvature
flow
237
KAiB along
the direction
determined
by
these fields: R(A, From
the equations
B, A. B) = K,,,,[g(A,
A)g(B,
B) -
= iK,:,g(n.
n),
because g(u. u) = 1. and g(n. u) = 0. While investigating the geodesic deviation neither
B)g(A.
B)].
(26)
(23) and (26) we obtain: v,(n)
precisely
g(A,
the tangent
vector
we
u, nor
the
are
(27)
not
deviation
in
general
vector
able
n for
to
determine
the specific
initial
value of the parameter S. Therefore it seems reasonable to discuss the geodesic deviation equation o_riginating from a certain averaged potential P(n). We shall define this averaged potential V(n) by choosing the vectors u and n at random, i.e. we assume that every direction determined by the bivector u A n is equally probable. In this way we associate the averaging mean which
procedure
the set of dimension
bivectors
with
the space of two-directions.
have the following
,, ,)QQ = ;(A”
We shall denote the pair of indices tensor in the space of two-directions:
(iliz)
that this tensor
B’: _ Al: &I).
by an ovcrnll
Q:ll = sI,,,&:,: chcckcd
we
form: (A
It can he easily
By the space of two-directions
A A B = !(A” B” - A’: B”)a,, A 3,:. This is a linear space M = i N(N - l), where rV = dim M. The components of
all bivectors is equal to
-
index
I. Let us introduce
the metric
S#,,&,~
(28)
is symmetric: Q=,I = Q=,,*
The mclric
C,, satisfies C(A
the following A 11, c
(29)
identity:
A I)) = $(A.
C)g(IS,
I)) -
g(A,
I))@,
C)
(30)
I3y virtue of the Schwartz inequality the quadratic form defined by the Cl,-matrix is positively dcfinitc, hcncc C defines an cuclidcnn metric in the space of two-directions. In order to dcfinc a malrix invcrsc to C,, wc have to clcfinc a unit operator in the space of two-directions. We shall denote its components by 0: and dcfinc as ii;(A
A
1%)’
=
(f\
A
Ii)‘.
(31)
Hence,
b’h. = ;(cb;‘,q Let us dcnotc
by C”
the inverse
of C,,.
- CqAp,).
Obviously
(31)
WC have CK,C”
= Oi.. Thus
q=/, = ~(g/1~ls/:12 _ s/l~:s,:li)~ We shall not rcwritc V,(n)
the potential = iR(n.
V,(n)
(32)
in terms of bivcctors:
II, n, u) = ~R,,klll’flirl’ll’ = iR,,(n
A
u)‘(n
A
u)’
(33)
WC have used the known symmetry of the Ricmann tensor IFl,, = !R,,. In general quantity R(I\, It. it. I)) dcfincs a certain quadratic form in the space of two-directions: whcrc
R(A. As
it
is
well
known,
I%, r\, 15) = R(i\ there
exists
A
1%. t\
a basis
A
ii)
{E,,},
=
R,,(/\
n = I,
A
I{)‘(,\
. ., iN(
A
N -
the
It)’
1) in the
space
of
M.
139
er al.
SZY~OWSKI
two-directions for which lR(E,. Eb) = Adbob and C(E,. Eb) = b,,. R(A A B. A A B) = 1.,(x’) + . _ . + AM(rM)‘. where A A B = Y’E,. We can now define the averaged potential P(n): R(A where assume fix the sphere
A B. A
In such
basis
we have:
B)dC’,
A
(34)
IS the volume of M - 1 dimensional sphere Sl’-‘. In the other words we volS,!‘-’ that none of the geodesics is privileged and neither is the deviation vector. We only length of bivectors by assuming that C(A A B. A A B) = g(n. n). This defines the S;‘-’ of the radius r = [g(n. n)]‘,’ In the space of two-directions. Hence.
I,~~,_,
P(n)= zvoIk ,,-, ,
[A,(x~)~ + .
+ A,,(.r”)‘]dL’.
and because: .M - I ,
~‘e arrive
(.rU):dV
= ‘O’i;
r’ = +
volS:‘-‘g(n.
n).
at the formula: I
‘(n)= 2MI
IAl + .
+
A,,]Lq(Il.
n)
However A, + whcrc
. + J’,,, =
IR,,C” = ;R ,,,:,,,:[g”“$J’:
R is the Ricci scalar.
Thcrcforc
the avcragcd
clircction
dctcrmincs
oC gcoclcsic
tangent
vector
the: gcodcsic
N(N
to unity).
fQ(n.
I)
deviation
and the dircctir)n
is normalized
-
J = ;[ R,:,$=
potential
I
c3 (II) = This potential
- pg”‘:
-grad,,
has the form: (35)
in which
vector
The corresponding
D?n^ -= ds?
= R.
II).
equation
of dcvintion
+ R ,,,, g”“]
(with
WC randomly
the only
equation
choose
the
that
the
constraint
reads:
P(n).
!A)
II2 7
ds-
If u’c introduce
;I Fermi
basis (E,}
2R
fl’ = -
along
,V(;\’
-
I)
the randomly
2X 7d? ff” = iV(iV dsOne see from equation is containccl in the Ricci scalar for locnI
instability
spaces implies
of geodesics.
chaos in a gcodcsic
A’ is the
dimension
(36)
chosen
gcoclcsic.
WC obtain: (37)
I) Ti”
that the full information cunccrning the local instability R. The negativity of the Ricci scalar is a sufficient condition The
property
rz-
of local
of the
space
2R N(N
-
1)
011 tvhich
instability of geodesics on compact the mean timescale for mixins as
-12 1
flow. We can estimate
[
where
ii’,
the
gcodtzsic
(‘S) flow
is defined.
The
Local stability
of a geodesic flow
239
relaxation time r is a time-scale in which the length (nl of the deviation vector increases e-times in average. The inverse of t corresponds to the positive Lyapunov exponent [2]. It is worth noticing that the equation (37) has the same form as in the case of maximally symmetric spaces for which the Ricci scalar is proportional to the Gauss curvature. As it is known, a geodesic flow on the compact space with negative Gauss curvature becomes an Anosov flow [3]. It implies that after averaging the Ricci scalar contains the full information about the local instability and plays the role analogous to that of Gauss curvature in the case of maximally symmetric spaces. Of course, if M is a compact Riemannian manifold such that its curvature in an arbitrary two-direction is negative (such manifolds do exist [-I]), then the geodesic flow is an Anosov flow. The geodesic flows on compact Riemannian manifolds will obviously have the same property. In this sense we can say that the criterion of negativity of the Ricci scalar is a generalization of classical works of Anosov.
3. APPLICATIONS 3.1.
Locnl iustabiliry of r~llllritIir~lcrlsio,lal~~on~oge~~cowcosmological models
The Maupertuis principle of classical mechanics [4] allows for representing the trajectories of a Hamiltonian system as a geodesic flow in certain region of the configurational space (for details see [2]). The system with the Hamiltonian: H = +“h(cl)l“,~~b cl”, p,, arc coordinntcs and momenta cncrgy, is equivalent pscudoricrnannian) manifold with the metric:
whcrc
T = $d’p,,p,, is the kinetic
+ V(q). respectively, V(q) is the potential to,a gcodcsic flow on a Ricmannian
and (or
ds’ = 2 Wcl,,,,dq” dclh, ds = 2Wdt, whcrc 2W = l/r - VI, and II is the mcasurcd along geodesics which formula of (30). This means that Lagrange function L = ~a,,Q’cj’ -
total energy. Note that there is a new affinc parnmctcr s is conncctcd with the dynamical time I by the second the equations of motion of the system dcscribcd by the V(q) arc reduced to the geodesic equation:
d’ ds’.- (1’ + ds = dr where
2W = I/z - V(q)l.
Because
the subspace I-,, of the phase-space
(39)
r;k
drl’ d# ds ds
-
= 0,
2W(q(s(O)),
the total energy is conserved
(40) the motion
takes place in
r such that
rh = {(p,
(I):
~u"~J,/J,
+
V(q) = /I}
(41)
or in the subspace Q of the tangent fibre bundle TM:
Q = ((4, 0): Cl,,(C/)lj’Cj’= 2W). Hence the problem is reduced to determining the geodesics on a Riemannian where M is a configurational space and g is the metric: S,, = WQ,,.
(42) space (M,
g)
(43)
In constructing the multidimensional sional Einstein-Maxwell action:
s=
cosmological
&
dDx(-detjj,,)“‘(fi
I
ii
models
we start
from
a multidimen-
- 2i),
where D = 1 + n + ii is the dimension of total space-time. n and fi are dimensions of the physical and internal spaces respectively. i is the D-dimensional cosmological constant, 6 is the D-dimensional gravitational constant. According to the philosophy of Kaluza and Klein we can represent the metric of a D-dimensional spacetime in the form:
0. 1. ., II, a.6 = I.. ., if, h is a function of time. The metric S;,(,, is associated with that of the physical space g!,,, by the relation g,,V = I\~‘~,,,.. where N.’ = /T-‘” (“-I’. We shall assume that the metric ,c,,~ describes a ii-dimensional space of constant curvature, i.e. R,,,, = k?g<,,,. The above assumptions allow for representing the action (44) as arising from the Lagrangian:
il.\‘=
L = (-det
&JZ
&
R -
&3pj~” -
V(Q)
[ whet-c G = 6/(detCqGrh)“‘,
$,= h--‘{C[l
V(d))=
(46) I
+ fi/(rl -
l)]}‘,” In h, h’= (SnG)‘,‘.
and
5 {4-2I-((,,_,)(;+,I _ ,) j’s]
- gLsp
[
(g,“I,’j’ y)
-2K
(47)
‘.
Aflcr the climcnsional rccluction this milltidirncnsionnl systcrn bcconics equivalent gravity and scalar ficlrl couplccl niinim~~lly. By choosing such ;I coorclin;ltc systcrn that:
to
(-IS) Lvhcrc R,, is the metric of a constant wc’ finally arrive at the formula:
ciIrvnturc
[_ = (dct J,,)‘F{hq-+(,I lvhcre the clot clcnotcs the system rcacls: I/ and the constant
pscudoricmannian
-
the diffcrcntiation
=
condition
-+(,c
spxc.
+ $?c-‘-]
with rcspcct
- 1,;> +
is: I/ = 0. The
I,:’
and by introducing
$
I- &4 -
to time.
cj’ - g&-‘;
~lamiltonian
sonic
Hence
ntx
variable
V(C/I)} the Hamiltonian
+ h-)V(Q), (50) induce5
(50)
;I gcoclcsic
!lo\\
011 ;I
space with metric:
cl.sJ = 2Cl’(d~j~ - ‘I:?) ;. bvticre 5 = of a gcoclcsic
for
2r1( r1 - 1) :. c/l = KqJ 211cl ?\I/ = h-’ v(&K) flow is detcrminccl by ttic Iiicci scalar:
(51) -
J r?e-’
’ “I’‘-‘I
‘,
Local
illstal)ili[\
Local stability of a geodesic flow
case when V = 0 or A = 0 and ii = 0 the Ricci scalar (52) is equal the fact that the corresponding systems are integrable. If the internal
In particular This reflects Ricci
flat (R
to zero. space is
= 0) we have: sgn R = sgn(Ri).
This means that the system
is locally
unstable sgn R =
Therefore
the
system
sgn R = sgn (RA)
3.2.
Choric
In general Hausdorff defines
is unstable
and the system
hellclriolrr relativity
differential
locally is unstable
(53)
if RA < 0. In the case of (? = 0 and n = 3: -sgn(Ri?). if
(54)
Rt? > 0.
when
Analogously
for
R and A are of opposite
fi = 0.
we
have
signs.
of gt~0dcsic.s in spficctirncs the spacetime
manifold
the causal structure
is modelled
and g is a Lorenzian
of k! imposes
certain
by a pair metric. restrictions
(AI,
g) where
The Lorenzian on topology
bf
is a C”-class
structure
which
of the spacetime
[I]. Namely the spacetime is non-compact or if it is compact its Euler characteristic must vanish. Timclike or null geodesics in spncctimc have a deep physical meaning since they rcprcscnt historics 01 obscrvcrs or photons. Thcrcforc it is interesting to ask whcthcr gcodcsic flows in spacctimcs can bc locally unstable or can possess a mising property. It has been cstnblishcd that the so called Mixmastcr models exhibit a chaotic behaviour of scale factors. Howcvcr the scale factors alons have no physical meaning. Therefore our clucstion about chaos in gcodcsics is a physical one and can bc answcrcd within the frames of above dcscribcd formalism. In the cast when the spacctime is compact and its Ricci scalar is ncgativc the gcodcsic llows are Anosov tlows and consequently are crgodic. It should bc strcsscd, however, that the compact spacctimcs can bc pathological due to cxistcncc of closed timclikc trajcctorics [I]. When
the spacetime
is not compact
and has a negative
Ricci
scalar,
then
the gcodcsic
flows have the property of local instability. Projections of these gcodcsics on hypcrsurfaces of constant time represent the trajectories of observers or photons and can become Anosov flows provided the Ricci scalar is negative. As an example we can take the bchaviour of geodesics
in the so called
Ellis small
and Prigogine [5]. They show time, when suitably projected
that into
world,
which
has been investigated
by Lockhart,
non-spacelike geodesics in Robertson-Walker a three-dimensional compactified hypersurface
Misra spaceof a
constant negative curvature, change into geodesics paramctrised by a new affinc parameter. It is interesting that in Mixmaster models the criterion of ergodicity of geodesic flows on hypcrsurfaces of constant time is equivalent to the corresponding condition (R < 0) for a whole moclcl. In the other words chaos in geodesics is a consequence of chaotic nonlinear dynamics of the spacetime which is an arena for all events. When the spacctime is compact and its Ricci scalar is negative, then the geodesic flows have mixing property. The negativity of the Ricci scalar has a simple physical meaning. Namely the condition R < 0 implies E + 311 < 0. where F is the energy density and 1~ is the pressure. It means that the strong energy condition is violated. On the other hand the violation of the strong, energy condition solves the horizon problem in Friedman world models [6].
REFERESCES