Metri Tangent Bundles
P.C. Stavrinos and P. Manouselis
Abstrat
TherelationbetweenspinorofSL(2;C)groupandtensorsintheframework
of lagrange spaes is studied. A geometrial extension to generalized metri
tangent bundles is developed by meansofspinor. Also, thespinorial equation
ofausalityfor theunique solutionofthe null-oneintheFinsleror Lagrange
spaeisgivenexpliitely.
MathematisSubjet Classiation:53C60,53B50,81R25
Key words: Finslerspae-time, Lagrangespae, spinor,generalizedtangentmetri
bundle,spinor-onnetion,null-one.
1 Introdution
The theory of spinors on pseudo{Riemannian spaes has been reognized by many
authors, e.g. [1℄, [2℄, [3℄ fortheimportant role ithas playedfrom the mathematial
andphysialpointofview.
Thespinorsthatwearedealingwithhere,areassoiatedwiththegroupSL(2;C).
InpartiularSL(2;C) atsonC 2
.Eah elementof C 2
representsatwo{omponent
spinor.ThisgroupistheoveringgroupoftheLorentzgroupinwhihthetensorsare
desribed[2℄.Theorrespondenebetweenspinorsandtensorsis ahievedbymeans
ofmixedquantitiesinitiallyintroduedbyInfeldandVanderWaerden.
Theorrespondeneof tensorsandspinors establishesahomomoerhism between
theLorentzgroupand theoveringgroupSL(2;C).
Inthefollowing,wegivesomeimportantrelationsbetweenspinorsandtensorson
ageneralmanifoldofmetrig
.
Let :SS !V 4
be ahomomorphism between spinorspaes S;S and four{
vetors belonging to theV 4
spae, then the omponentsof , whih are alled the
Pauli{spinor matries, aregivenby
0
AB 0 =
1
p
2
1 0
0 1
; 1
AB 0 =
1
p
2
0 1
1 0
Balkan Journal of Geometry and Its Appliations, Vol.2, No.2, 1997, pp. 119-130
(1) 2
AB 0
= 1
p
2
0 i
i 0
; 3
AB 0
= 1
p
2
1 0
0 1
:
Thehermitian spinorialequivalentnotationof
AB 0
is givenby
AB 0
=
BA 0
=
B 0
A
.Greekletters; ;:::representtheusual spae{timeindies takingthevalues
0;1;2;3andtheRomanapitalindiesA;B;A 0
;B 0
arethespinorindiestakingthe
values0;1.Thetensorindiesareraisedand loweredbymeansofthemetritensor,
whereastheraisingandloweringofspinorindiesisgivenbythespinormetritensors
"
AC
;"
B 0
C 0
whihareofskew{symmetriform.Thus,fortwospinors A
;n A
0
wehave
therelations,
(2)
A
="
AB
B
; n A
0
="
A 0
B 0
n
B 0
A
= B
"
BA
; n
A 0
=n B
0
"
B 0
A 0
;
moreoverwehave,
A
n
A
= A
n B
"
BA
=
A
"
AB n
B
=
B n
B
:
ForarealvetorV
itsspinorequivalentis
(3a) V
AB 0
=V
AB 0
;
where
AB 0
are givenbytherelation(1).Also,thefollowingformulasaresatised,
(3b)
AB 0
AB
0
= g
;
AB 0
AB
0
= Æ
:
ThespinorequivalentofatensorT
isgivenby
(4) T
= AB
0
CD 0
T
AB 0
CD 0
andthetensororrespondingto thespinorT
AB 0
CD 0
is,
(5) T
AB 0
CD 0
=
AB 0
CD 0
T
:
The relationshipbetweenthe matries
and thegeometri tensor g
, as well
asitsspinorequivalentare
(6)
g
AB 0
CD 0
="
AC
"
B 0
D 0
g
AB 0
CD 0 =
AB 0
CD 0
g
="
AC
"
B 0
D 0
g AB
0
CD 0
= AB
0
CD 0
g
="
AC
"
B 0
D 0
:
TheomplexonjugationofthespinorS
AB 0
is
(7) S
AB 0
=S
A 0
B :
Furthermore, for a real vetor V
the spinor hermitian equivalene yields V
B 0
A
=
V 0
. Ifavetory k
isanull{vetor,
(8) y k
y
k
=g
k y
k
y
=0;
thenitsspinorequivalentwilltaketheform
(9) y
k
= k
AB 0
A
B
0
;
where, A
; B
0
representsthetwo{omponentspinorsof SL(2;C)group.
In the Riemannian spae, the ovariant derivative of x{ dependent spinors will
taketheform
(10)
D
A
=
A
x
+L A
B
B
;
D
A 0
=
A 0
x
+L A
0
B 0
B 0
;
D
A
=
A
x
L B
A
B
;
D
A 0
=
A 0
x
L B
0
A 0
B 0
;
where A
;
A
; A
0
;
A 0
represents two{omponent spinors and L A
B
;L A
0
B 0
the spinor
aÆne onnetions.In thease that wehave spinorswith twoindies, the ovariant
derivativewill beintheform
(11) D
AB 0
=
AB 0
x
+L A
C
CB 0
+L B
0
C 0
AC 0
:
Applyingthis formulatothespinormetritensors"
AC
;"
B 0
C 0
weget
(12) D
"
AB
=
"
AB
x
L C
A
"
CB L
C
B
"
AC :
If
D
"
AB
=0;
we shall say that the spinor onnetion oeÆients L A
B
are metrial together with
therelations
(13) D
AB
0 =0; D
"
AB
=0; D
"
A 0
B 0
=0;D
"
A 0
B 0
=0:
Fromtherelation(12)weimmediatelyobtain
L
BA
=L
AB
;
whereweusedtherelation
L
AB
=L C
B
"
CA :
Alsofromtherelation13a) wehave
(14) D
AB 0
=
AB 0
+L
0 L
C
A
CB 0
L D
0
B 0
AD 0
=0:
2 Generalization of the Equivalent of Two
Component{Spinors with Tensors
Theabovementionedwell{knownproedureforSL(2;C)groupbetweenspinorsand
tensorsinapseudo{Riemannianspae{timeanbeappliedtomoregeneralizedmetri
spaesor bundles.For exampleG.Asanov[6℄appliedthismethod forFinslerspaes
(FS),wherethetwo{omponentspinorsn(x;y)dependonthepositionanddiretion
variablesorn(x i
;z
),withz
asalarforagaugeapproah.Conerningthisapproah
someresults were given relatively to the gaugeovariant derivative of spinors and
the Finsleriantetrad. In ourpresent study we give the relation between spinors of
SL(2;C)groupandtensorsin theframeworkofLagrangespaes(LS).
Theexpansionfor theovariantderivatives,onnetions non{linearonnetions,
torsionsandurvaturesarethemainpurposeof ourapproah.
Inthe following, weshall study theasethat thevetorsof LS are null{vetors
andonsequentlyfullltherelation(9).InFinslertypespae{timethemetritensor
g
ij
(x;y) depends on the position and diretional variables,where thevetory may
beidentiedwiththeframeveloity([6℄h.t).So,avetorv i
willbealled nullif
g
ij (x;v)v
i
v j
=0:
In this ase there is no unique solution for the light{one [7℄, [8℄. The problem of
ausalityissolvedonsideringtheveloityasaparameterandthemotionofapartile
inFinslerspae isdesribedbyapair(x;y).Themetriform in suh aasewill be
givenby
(15a) ds
2
=g
ij
(x;v)dx i
dx j
:
Whenapartileismovingin thetangentbundleof aFinsler(Lagrange)spae{time
itsline{elementwill begivenby
(16) d 2
=G
ab dx
a
dx b
=g (0)
ij
(x;y)dx i
dx j
+g (1)
(x;y)Æy
Æy
;
y
= dx
dt
;
wheretheindiesi;j and; takingthevalues1;2;3;4and
Æy
=dy
+N
j dx
j
:
Thuswehave
Theorem 2.1. The null{geodesi ondition (15) is satised for a partile whih is
moving in the tangent bundle of Finslerspae{time of metri d 2
(rel. 16) with the
assumption,the veloityv istakenasaparameter ofthe absoluteparallelism
(17) Æy
=0:
Theprevious treatment of null-vetorsin Finsler spaes an also be onsidered for
LagrangespaesinvolvingLagrangianswhiharenothomogeneous[9℄,[8℄.Theintro-
dutionof spinors; of theoveringgroupSL(2;C) in themetritensorg(x;;)
undertheorrespondenebetweenspinorsandtensorsin LS,
(x;y)!(x;V 0
)!(x;
A
; A
0
)
preservestheanisotropyofspaewithtorsions.Inthisaseallobjetsdependonthe
position and spinors,e.g. the Paulimatriese i
AA 0
(x;;).Suh anapproahan be
developedforaseond{orderspinorbundleapplyingthemethodanalogousto[4℄.In
virtueofrelation(8), anullvetorin spinorformanbeharaterizedby
(18) g
AA 0
BB 0
A
A
0
B
B
0
=e i
AA 0e
j
BB 0
g
ij
A
A
0
B
B
0
=0:
Proposition 2.2.In atangentbundleof metri(Finsler, Lagrange)
G=g
ij
(x;y)dx i
dx j
+h
ab (x;y)Æy
a
Æy b
;
if the vetor y is a null, then the orresponding spinor metri of the bundle will be
given inthe form
(19) G=g
AA 0
BB 0
d A
d A
0
d B
d B
0
+h
AA 0
BB 0
Æ(
B
B
0
)Æ(
A
A
0
)
orequivalently
G=g
AA 0
BB 0d
A
d A
0
d B
d B
0
+h
AA 0
BB 0Æy
AA 0
Æy BB
0
;
wherey AA
0
= A
A
0
, wheny isnullvetor(f.[2 ℄).
Proof.Therelation(19)isobviousbyvirtueof(6)and(9).
Remark. A generalized spinor an be onsidered as the square root of a Finsler
(Lagrange)nullvetor.
3 Adapted Frames and Linear Connetions
InthegeneralaseofaLS,thespinorequivalenttothemetritensor
g
ij
=
2
L
y i
y j
; L= 1
2 F
2
isgivenby
(20) g
ij
=e AA
0
i e
BB 0
j g
AA 0
BB 0
:
The orresponding Lagrangian will be L : MC 2
C 2
! R, with the property
L(x;;)=L(x;y),where LrepresentstheLagrangianin aLagrangespae.Wean
adoptthespinorequivalentformoftheadaptedframesandtheirdualsinaLS,
Æ
Æx
;
y i
! Æ
Æx
;
A
;
A
0
!
; (dx
;Æy i
)!(dx
;Æ A
;Æ A
0
)
aswellas thespinorounterpartofthenon{linearonnetionN i
ofaLS,
N i
!(N A
;N A
0
):
ThegeometrialobjetsÆ A
;Æ A
0
aregivenby
(21) Æ
A
=d A
+N A
dx
; Æ A
0
=d A
0
+N A
0
dx
:
Invirtueof(3),thebases
;
AA 0
arerelatedasfollows
(22)
=e AA
0
AA 0
;
where
=
x
and
AA 0
=
A
A
0 .
Theorem 3.1. In a Lagrange spae the spinor equivalent of the adapted basis
(Æ=Æx
;=y
)anditsdual (dx
;Æy
)aregiven by
(23)
a) Æ
Æx
=e AA
0
A
A 0
N A
A N
A 0
A 0
b)
P e
AA 0
P
=
AA 0
; P =fi;g
) dx
=e
AA 0
d A
d A
0
d) Æy
=( A
0
d A
+ A
d A
0
)e
AA 0
+( A
0
N A
+
A
N A
0
)e
AA 0
e
AA 0
d B
d B
0
:
Proof.Therelations(23)arederivedfrom(21)and(22).
Proposition 3.2.If y
;N
j
represent anull vetorand anon{linear onnetion in
aLagrangespae, thenitsorresponding spinor representationsaregiven by
(24) dy
=e
AA 0(
A 0
d A
+ A
d A
0
) N
j
=e
AA 0(
A 0
N A
j +
A
N A
0
j ):
Proof.Therelation(24)isobviousbeauseof(23d).
Proposition 3.3. The null{geodesi equation of spinor equivalene in a LS or FS
isgivenby
(25)
A 0
d A
(e
AA 0
N A
d
A 0
+1)+ A
d A
0
(e
AA 0
N A
0
d
A
+1)=0:
Proof.Invirtueofrelations(18)and(23,d)weobtaintherelation(25).
AÆneonnetionsand aÆnespinor onnetionsare dened in theframes ofLS
bythefollowingformulas
D
Æ=Æx
Æ
Æx
=L k
Æ
Æx k
; D
Æ=Æx
A
=L B
A
B
;
(26) D
Æ=Æx
A
0
!
=L B
0
A 0
B
0
; D
= A
Æ
Æx
=C
A Æ
Æx
;
D
= A
B
0
!
=C C
0
B 0
A
C
0
; D
= A
0
B
=C C
BA 0
C
;
D
= A
0
B
0
=C C
0
B 0
A 0
C
; D
= A
B
=C C
BA
C
;
D
= A
0
Æ
Æx
=C
A 0
Æ
Æx
:
Weangivetheovariantderivativesofthehigherordergeneralizedspinors AB
0
BA 0
(x;;),
(27)
r
AB 0
BA 0
= Æ
AB 0
:::
BA 0
:::
Æx
+L A
C
CB 0
:::
BA 0
:::
+L B
0
C 0
AC 0
:::
BA 0
:::
L C
B
AB 0
:::
CA 0
:::
L C
0
A 0
AB
0
:::
BC 0
:::
r
E
AB 0
:::
BA 0
:::
=
AB 0
:::
BA 0
:::
E
+C A
CE
CB 0
:::
BA 0
:::
+C B
0
C 0
E
AC 0
:::
BA 0
:::
C C
BE
AB 0
CA 0 C
C 0
EA 0
AB
0
BC 0
r
Z 0
AB
0
:::
BA 0
:::
=
AB 0
:::
BA 0
:::
Z
0 +C
A
CZ 0
CB
0
:::
BA 0
:::
+C B
0
C 0
Z 0
AC
0
:::
BA 0
:::
C C
0
Z 0
A 0
AB
0
:::
BC 0
:::
:
Proposition3.4.Ifthe onnetionsdenedbythe relations(26) areof theCartan{
type, thenthe spinor equivalentrelationsare givenby
(28)
A
0
Æ A
Æx k
+L A
Ck
C
A
0
+ A
Æ A
0
Æx k
+L A
0
C 0
k
C 0
A
= 0;
(e AA
0
)
1
( A
0
r
E
A
+ A
r
E
A 0
) = 1;
(e AA
0
)
1
( A
0
r
Z
A
+ A
r
Z 0
A
0
) = 1:
Proof.Applyingtherelations(27)toanullvetorywiththeCartan{typeproperties
y
jk
= 0 and y
j
= Æ
[5℄ [6℄, and taking into aount the (3a), (9) we obtain
therelations(30). (Aswehavementionedpreviouslythey{ovariantderivativehas
orrespondedtothespinorovariantderivatives).
4 Torsions and Curvatures
The spinortorsions orresponding to the torsionsof LS are given by an analogous
methodto thatone wederivedin [4℄foradeformedbundle.Thetorsiontensoreld
T ofaD{onnetionisgivenbyT(X;Y)=D
X
Y D
Y
X [X;Y℄.
Relativelytoanadaptedframewehavetherelations
(29)
a) T
Æ
Æx k
; Æ
Æx
=T
k Æ
Æx
+T A
k
A
+T A
0
k
A
0
b) T
A
; Æ
Æx
=T
A Æ
Æx
+T B
A
B
+T B
0
A
B
0
) T
A
0
; Æ
Æx
!
=T
A 0
Æ
Æx
+T B
A 0
B
+T B
0
A 0
B
0
d) T
A
;
B
=T
BA Æ
Æx
+T C
BA
C
+T C
0
BA
C
0
e) T
A
;
B
0
!
=T
B 0
A Æ
Æx
+T C
B 0
A
C
+T C
0
B 0
A
C
0
f) T
A
0
;
B
!
=T
BA 0
Æ
Æx
+T C
BA 0
C
+T C
0
BA 0
C
0
g) T
A
0
;
B
0
!
=T
B 0
A 0
Æ
Æx
+T C
B 0
A 0
C
+T C
0
B 0
A 0
C
0 :
Thetorsion(29a))an bewrittenin theform
(30)
D
Æ=Æx
Æ
Æx
D
Æ=Æx
Æ
Æx
Æ
Æx
; Æ
Æx
=L
Æ
Æx
L
Æ
Æx
R A
A
V A
0
A
0
;
wherethebraketshavetheform
(31) [Æ=Æx
;Æ=Æx
℄=R A
A
+V A
0
A
0
;
andÆ=Æx
;R A
;V A
0
aregivenby
Æ
Æx k
=
x k
N A
k
A
N A
0
k
A
0
; R A
= ÆN
A
Æx
ÆN A
Æx
V A
0
= ÆN
A 0
Æx
ÆN A
0
Æx
:
ThetermsR A
;V A
0
representsthespinor{urvaturesofnon{linearonnetionsN A
;N A
0
.
Invirtueoftherelations(29.a), (30),(31)weobtain
(32) T
=L
L
; T A
= R A
; T A
0
= V A
0
:
Similarlyfrom the relations(31b{31g), omparing with thetorsion in thefollowing
(33) T Æ
ÆY P
; Æ
ÆY Q
=D
Æ=ÆY P
Æ
ÆY Q
D
Æ=ÆY Q
Æ
ÆY P
Æ
ÆY P
Æ
ÆY Q
weanobtaintherelations
(34)
T
A
=C
A
; T A
B
= N
A
B
L A
B
T A
0
A
= e
Y A
0
A
; T
AB
=T
AA 0
T
`
AB
=C
`
AB C
`
BA
; T A
0
AB
= R
A 0
AB
; T
A 0
= C
A 0
T A
A 0
= N
A
A
0
; T A
0
B 0 =C
A 0
B 0
P
A 0
B
T B
AA 0
= C B
AA 0
; T A
0
AB 0
=C A
0
AB 0
C A
0
A
B
0
;
wherewehaveput
Æ
ÆY P
=
A
;
A
0
; Æ
ÆY Q
=
Æ
Æx
;
; =B;B 0
and C A
0
A
=C A
0
BA
B
:
So,weobtainthefollowing:
Proposition 4.1. In the adapted basis of a generalized metri tangent bundle the
spinor equivalentof oeÆientsof the torsionT ofa D{onnetion,are given bythe
relations (32){(34).
Proposition4.2.D{onnetionhasnotorsionifandonlyifalltermsoftherelation
(34) areequal tozero.
The urvature tensor eld R of a D{onnetion has the form R (X;Y)Z =
[D
X
;D
Y
℄Z D
[X ;Y℄
Z 8X;Y;Z 2 X(TM). The oeÆients of the urvature ten-
(35)
R k
= ÆL
k
Æx
ÆL k
Æx
+L
L
k
L
L
k
R
A
C
k
A V
A 0
C
k
A 0
R B
A
= ÆL
B
A
Æx
ÆL B
A
Æx
+L
A L
B
L
A L
B
R
L
B
A V
A 0
C
B
A 0
A
R B
0
A 0
= ÆL
B 0
A 0
Æx
ÆL B
0
A 0
Æx
+L B
0
A 0
L
L
B 0
A 0
L
R
A
C
B 0
A 0
A V
D 0
C
B 0
D 0
A 0
P k
A
= ÆL
k
Æ A
ÆC k
A
x
+L
C
k
A C
A L
k
+
N E
A
C k
E +
e
Y A
0
A k
A 0
P
`
AB
= L
`
A
B
ÆC
`
AB
Æx
+L k
A C
`
k B C
k
AB L
`
k +
N n
A
C
`
Bn +
e
Y A
0
A C
`
A 0
B
P B
0
A 0
A
= L
B 0
A 0
A
ÆC B
0
A 0
A
Æx
+L B
A 0
C
B 0
BA C
B
A 0
A L
B 0
B +
ÆN E
A
C B
0
A 0
E +
e
Y E
0
A C
B 0
E 0
A 0
S k
AB
= C
k
A
B
C k
B
A
+C
A C
k
B C
B C
k
A R
A 0
AB C
k
A 0
S m
`AB
= C
m
`A
B
C m
`B
A
+C n
`A C
m
nB C
n
`B C
m
nA R
A 0
AB C
m
A 0
`
S B
0
A 0
AB
= C
B 0
A 0
A
B
C B
0
A 0
B
A
+C D
0
A 0
A C
B 0
D 0
B C
D 0
A 0
B C
B 0
D 0
B R
D 0
AB C
B 0
D 0
A 0
I k
A 0
= ÆC
k
A 0
Æx
L k
A
0 +C
A 0
L
k
L
C
k
A 0
N
A
A
0 C
k
A L
A
A 0
C
k
A
I B
AA 0
= ÆC
B
A 0
A
Æx
L B
A
A
0 +C
A 0
A L
B
L
A C
B
A 0
N
A
0 C
B
A L
D 0
A 0
C
B
D 0
A
I B
0
A 0
C 0
= ÆC
B 0
A 0
C 0
Æx
L B
0
A 0
C
0 +C
B 0
A 0
D 0
L D
0
C 0
L
B 0
E 0
C
E 0
A 0
B 0
N
A
0 L
B 0
C 0
L
D 0
A 0
C
B 0
D 0
C 0
J k
A 0
B
= C
k
A 0
B
C k
B
A
0 +C
A 0
C
k
B C
B C
k
A 0
L
D 0
B
A
0 C
k
D 0
J
AA 0
B
= C
A 0
B
A
C
AB
A
0 +C
k
A 0
A C
k B C
k
AB C
A 0
k L
D 0
A
A
0 C
D 0
B
J B
0
A 0
C 0
A
= C
B 0
A 0
C 0
A
C B
0
A 0
A
A
0 +C
B 0
A 0
D C
D
C 0
A C
B 0
A 0
E C
E
C 0
A L
D 0
A
A
0 C
B 0
C 0
D 0
K
A 0
B 0
=K B
AA 0
B 0
=K B
0
A 0
C 0
D 0
=0:
Sowehave
Theorem4.1.The oeÆientsof theurvaturesofaD-onnetion aregiven by the
Theorem 4.2. In atangent bundlea D{onnetion has no urvature if andonly if
allthe oeÆients(rel.35) of the urvaturesare equaltozero.
5 Disussion
Theadvantageoftheframeworkthatusesaspinorgeometrialrepresentationforthe
generalizedspaesisthatitenablesthedesriptionofpartileswithspins1=2;3=2;:::,
in addition to those with spins0;1;2;:::, et., while the usual tensorsan desribe
onlythelatter kindof partiles.Fromthe mathematialand physial pointof view
spinorsareonsideredto bemorefundamentalthantensors.Theseareassoiatedto
thegroupSL(2;C)whihistheoveringgroupoftheLorentzgroupwithwhihthe
tensorsareassoiated.
Moreover,with thespinorialequivaleneof thenullvetorswean desribepar-
tilessuhasphotons,neutrinoset, inageneralizedmetrispae{time.
Also, thespinorial equationof ausalityfor theuniquesolutionof thenull{one
intheLagrangeorFinslerspaewas givenexpliitlybytheproposition(3.3).
Additionally,thegravitationaleldanbedesribedbyvirtueoftheorresponding
spinorialform ofthemetritensor equivalentto thespinorbundle. Thiswill bethe
objetofourfuturestudy.
Aknowledgements.AversionofthispaperwaspresentedattheFirstConfer-
eneofBalkanSoietyofGeometers,PolitehniaUniversityofBuharest,September
23-27,1996.
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[1℄ R. PenroseandW. Rindler,Spinors andSpae{Time, vol.1,Cambridge,Cam-
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[2℄ M. Carmelli, Classial Fields: General Relativity and Gauge Theory, J. Wiley
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[3℄ R. Wald,General Relativity, TheUniversityofChiagoPress(1984).
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[5℄ R. Miron, P. Watanabe, S. Ikeda, Some Connetions on Tangent Bundle and
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[6℄ G.Asanov,FinslerRelativity andGaugeTheories, (1985),D.ReidelPublishing
Company,Dordreht,Holland.
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DepartmentofMathematis
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andIts Appliations
15784Panepistimiopolis
Athens,Greee.
