Properties of the Spatial Sections of the Space-Time of a Rotating System

0.12 MB·English

by  Paschalis G. Paschali

Properties of the Spatial Sections of the
Space-Time of a Rotating System.
2
1 Paschalis G. Paschali∗
0
Department of Mathematics and Statistics
2
University of Cyprus
n
a Nicosia, P.O.Box 537
J Cyprus

9 Georgios C. Chrysostomou†
2
Frederick University, Pallouriotissa, 1036 Nicosia, Cyprus

]
c
January 31, 2012
q
-
r
g
[
Abstract

1
v Westudythesymmetrygroupofthegeodesicequationsofthespatial
7 solutions of the space-time generated by a noninertial rotating system of
8 reference. ItisasevendimensionalLiegroup,whichisneithersolvablenor
0 nilpotent. The variational symmetries form a five dimensional solvable
6 subgroup. Using the symplectic structure on the cotangent bundle we
.
1 study the resulting Hamiltonian system, which is closely related to the
0 geodesic flowon thespatial sections. Wehavealso studiedsomeintrinsic
2 and extrinsic geometrical properies of the spatial sections.
1
:
v Keywords
i
X
1 Introduction
r
a
InClassicalMechanicsandprerelativisticscience,space andtime areseparated
and independent. The time is absolute but the space is not. In the theory
of Relativity both space and time are relative, yet there is a combination of
them which is absolute. In this sense this theory introduces the concept of
space-time. This invariant combination can be expressed by the metric. In the
Special Theory of Relativity the space-time manifold is flat and its metric with
respect to an inertial system of reference takes the well known Lorentzianform
ds2 = c2dt2+dx2+dy2+dz2 (1)
−
∗email:[e-mail]
†email: eng.cg@fit.ac.cy
1
In the General Theory of Relativity the presence of a gravitationalfield affects
the geometry of space-time. Here the space-time manifold is curved. Actually
thegravitationalfieldisthecurvatureofthespace-timeandEinstein’sequations
relate the distribution of mass and energy with the curvature of space-time.
The cornerstone of the general theory is the principle of equivalence, which
states that every gravitationalfield is locally (but not globally) equivalent to a
noninertialsystemofreference. Ifthereisnogravitationalfieldbutwedecideto
useanoninertialsystemofreferencetheexpressionofthemetricwillnotbethe
aboveLorentzianmetricofEq. (1),butstillallthecomponentsoftheRiemann
tensorwillbezero,sincethereisaglobaltransformationfromtheinertialsystem
to the noninertial that we are using. So with respect to a noninertial system
of coordinates the space-time is still flat, but on the other hand the space itself
may be curved. What we will study here are some geometrical properties of
the spatial part of the space-time with respect to a rotating coordinate system
with constant angular velocity ω around the z-axis. This is one of the simplest
possiblecases. Usingsimilartechniqueswecanstudymorecomplicateproblems
resulting either from a more complicate system of reference, or because of the
presence of a gravitational field. These problems can help us to understand
better the principle of equivalence as well as the relation between gravity and
the noninertial systems. We can also study the possibility of changes in the
topology of the spatial part of the space-time.
Substituting x and y by xcosωt ysinωt and xsinωt+ycosωt in Eq. (1)
−
we obtain:
ds2 = ω2(x2+y2) c2 dt2+dx2+dy2+dz2 2ωydxdt+2ωxdydt (2)
− −
(cid:2) (cid:3)
ThegeometryofthespacewithrespecttothisrotatingobserverisnotEuclidean
since the relation between the circumference and the diameter of a circle is not
any more π. Due to the Lorentz contraction along the circumference this is
larger than π by a factor of (1 ω2r2/c2)−1/2. In cylindrical coordinates the
−
above metric becomes:
ds2 =(ω2r2 c2)dt2+2ωr2dφdt+dz2+r2dφ2+dr2 (3)
−
In this paper we have found the intrinsic and extrinsic curvatures of the
spatial sections of the above space-time. They are both negative and blow up
where the coordinate systemwe are using ceases to have physical meaning. We
havealsofoundthesymmetrygroupofthegeodesicequationsanditsvariational
subgroup. The first one is neither solvable nor nilpotent. On the other hand
the variational subgroup is solvable. Using the Hamilton-Jacobi theory for the
Hamiltonian system which is induced on the cotangent bundle we have found
the expression of the geodesic flow in terms of the affine time t.
Account of the theory of Relativity can be found for example in the books
of Misner, Thorne and Wheeler [11], Weinberg [16], O’ Neill [13], Hawking and
Ellis [7] and Landau and Lifschitz [8].
2
2 The geometry of spatial sections.
Let the metric of a space-time be given by:
ds2 =g dxαdxβ (4)
αβ
TheGreekindicestakethevalues0,1,2,3andtheLatinindicesthevalues1,2,3.
Theindexzerocorrespondstothetime ofthe eventandx1,x2,x3 determineits
location in space. As a result we have:
ds2 =g dx0dx0+2g dx0dxi+g dxidxj (5)
00 0i ij
tostudythespatialsectionsofthespace-timemanifoldwemustconsidertheset
ofalleventsthataresimultaneoustoagivenevent. Usingthe nullgeodesicswe
can prove that two nearby events are simultaneous if they satisfy the condition
(dx) = 0. As a result the metric on the spatial sections in contravariant form
0
is
dl2 =gijdx dx
i j
and in covariant form it becomes
g g
dl2 = g 0i 0j dxidxj (6)
ij
(cid:16) − g00 (cid:17)
Applying this to Eqs. (2) and (3) we get the metric on the spatial sections of a
noninertial rotating system in cartesian coordinates in the form:
dl2 =(1+Ay2)dx2 2Axydxdy+(1+Ax2)dy2+dz2 (7)
−
In cylindrical coordinates we have
r2
dl2 =dr2+ dφ2+dz2 (8)
1 ω2r2/c2
−
In Eq. (7) the factor A is given by:
ω2
A=
c2 ω2(x2+y2)
−
The metric in Eq. (8) is static so it can be used to define and to calculate
finite distances and the result will not depend on the cosmic curve connecting
the two points. On the other hand we must bear in mind that the concept
of simultaneous events which is the foundation of the positive definite spatial
metric of Eq. (6) is only a local one and it may not be possible to define
simultaneous events in a consistent way over the wlole manifold since g = 0
0i
6
These arguments indicate that the concept of a spatial slice is maybe more
complicatethantheconceptofahypersurfaceforexampleandheremayappear
some questions of topological character.
Weturnnowtosomegeometricalpropertiesofthemetric(8). Thez-axisisa
geodesicparametrizedbyarclengththatisthesamewiththecoordinatez. This
3
isnottrueforthe x,y,r orφcoordinates. Thisisaspaceofnegativecurvature.
The Riemann tensor has only 6 functionally independent components that are
all zero except R , which is given by:
1212
3c4ω2r2
R = (9)
1212 −(c2 ω2r2)3
−
Here we use the notation x1 = r,x2 = φ and x3 = z. We can easily calculate
the scalar curvature and get the following result:
6c2ω2
R=gkigljR = (10)
klij −(c2 ω2r2)2
−
This is negative and blows up when r =c/ω. This is exactly where the coordi-
nates of the rotating system cease to have meaning since beyond that distance
we have speeds higher than the speed of light in vacuum. Using the metric
(8) we can get for the element of the arclength along the circle with both z,r
constant the expression:
cr
dl = dφ
√c2 ω2r2
−
fromwherewegetthatthe ratioofthecircumferenceto thediameter isgreater
than π by the Lorentz factor.
The intrinsic geometry does not give any informationabouthow the spatial
slices are impended in the four dimensional space-time. Geometrically the im-
pendingcanbefoundbystudyingtheunitnormalvectoronthespatialsections
and is somehow represented by the extrinsic curvature, which is a symmetric
tensor parallel to the spatial slices. The extrinsic curvature is closely related
to the Weingarten map, which is the Jacobian of the Gauss map and can be
calculatedasthecovariantderivativeoftheunitnormalalongthespatialslices.
The metric (4) in the canonical 1+3 language takes the form:
ds2 =g (dxi+Nidt)(dxj +Njdt) N2dt2 =(g NiNj N2)dt2
ij ij
− − (11)
+g Njdxidt+g Nidxjdt+g dxidxj
ij ij ij
where Ni are the shift functions and N is the lapse function. Comparing this
expression with Eq. (8) we get the shift and lapse functions of our space-time:
Nr =Nz =0 (12)
Nφ =ω (13)
N =c (14)
So the contravariant normal vector connecting the spatial sections that corre-
spond to time t and t+dt is
(dt,0, ωdt,0)
−
4
and its proper length is cdt as we should expect. So the unit contravariant
vector is
1 ω
ηα = ,0, ,0 (15)
(cid:16)c −c (cid:17)
In covariantform is
η =(c,0,0,0) (16)
α
Since the metric g is static, the tensor of the extrinsic curvature canbe calcu-
ij
lated using the following relation:
1
K = N +N (17)
ij j;i i;j
2N(cid:16) (cid:17)
wherethesemicolonrepresentscovariantdifferentiationwithrespecttothemet-
ricofthespatialsections. WecaneasilycalculatethequantitiesN ,N andN
r φ z
and we obtain:
N =N =0 (18)
r z
ωc2r2
N = (19)
φ c2 ω2r2
−
Using these relations in (17) we can prove that all the components of the ex-
trinsic curvature are zero except the component K , which is given by:
rφ
ωc3r
K = (20)
rφ −(c2 ω2r2)3
−
andit has againa singularity atthe positions r =c/ω. The Riemannian tensor
R andtheaboveextrinsiccurvaturetensorarerelatedbytheGauss-Codazzi
ijkl
equations.
3 The symmetry group of the geodesics.
Hereweshallstudy the Liepointsymmetriesandthe variationalsymmetriesof
the geodesic equations. A symmetry groupof a system of differential equations
isagroupactingonthespaceofindependentanddepandentvariablesinsucha
way that solutions are mapped into other solutions. A local group of transfor-
mationsisavariationalsymmetryifitnecessarilyleavesthevariationalintegral
invariant. Every variational group is a symmetry group of the corresponding
Euler-Lagrangeequationsbuttheoppositeisnottrue. Thesymmetryapproach
to differential equations can be found, for example, in the books of Olver [12],
Bluman and Cole [3], Bluman and Kumei [4], Fushchich and Nikitin [6] and
Obsiannikov [14]. In our problem the geodesics satisfy the following equations:
d2xl dxj dxk
+Γl =0 (21)
ds2 jk ds ds
whereΓl aretheChristoffelsymbolsandsisthelengthofthegeodesics,which
jk
weusetoparametrizethem. Herewedonothavetoworryaboutnullgeodesics
5
since the metric (8) on the spatial sections is positive definite. Because of the
form of this metric the Christoffel symbols are given by the following relations:
Γl =0 (22)
jk
1 ∂g
Γj = ii (23)
ii −2g ∂xj
jj
1∂lng
Γi = ii (24)
ij 2 ∂xj
1∂lng
Γi = ii (25)
ii 2 ∂xi
where the indices i,j,k are all different. Using these in the metric (8) we get
the differential equations of the geodesics in the form:
d2r c4r dφ 2
=0 (26)
ds2 − (c2 ω2r2)2(cid:16)ds(cid:17)
−
d2φ 2c2 dφdr
+ =0 (27)
ds2 r(c2 ω2r2) ds ds
−
d2z
=0 (28)
dr2
We can also find the geodesics by considering the partial differential equation
∂w ∂w ∂w 2 c2 ω2r2 ∂w 2 ∂w
gij = + − + =1 (29)
∂xi∂xj (cid:16)∂r(cid:17) c2r2 (cid:16)∂φ(cid:17) (cid:16)∂z(cid:17)
A complete solution of this equation has the form
w(r,φ,z,a ,a )=k
1 2
where k is a constant. The equations of the geodesics in parametric form take
the form:
∂w
=b
1
∂a
1
∂w
=b
2
∂a
2
where b and b are arbitrary constants.
1 2
Using the method of Lie groups we shall study the symmetry group of the
geodesics. A vector field G acting on the space of independent and dependent
variables is a symmetry of the equations (26)-(28) if and only if it satisfies the
relations
pr(2)G(Eq26)=pr(2)G(Eq27)=pr(2)G(Eq28)=0
where pr(2)G is the second prolongationor extension of the vector field G. If
∂ ∂ ∂ ∂
G=Σ +R +Φ +Z (30)
∂s ∂r ∂φ ∂z
6
then
∂ ∂ ∂
pr(2)G=G+(R˙ Σ˙r˙) +(Φ˙ Σ˙φ˙) +(Z˙ Σ˙z˙)
− ∂r˙ − ∂φ˙ − ∂z˙
∂ ∂
+(R¨ Σ¨r˙ 2Σ˙r¨) +(Φ¨ Σ¨φ˙ 2Σ˙φ¨) (31)
− − ∂r¨ − − ∂φ¨
∂
+(Z¨ Σ¨z˙ 2Σ˙z¨)
− − ∂z¨
If we apply this to the geodesic equations (26)-(27) we get the following condi-
tions:
2c4r
R¨ Σ¨r˙ 2Σ˙r¨ φ˙(Φ˙ Σ˙φ˙)
− − − (c2 ω2r2)2 −
− (32)
c4(c2+3ω2r2)
Rφ˙2 =0
− (c2 ω2r2)3
−
2c2
Φ¨ Σ¨φ˙ 2Σ˙φ¨+ φ˙(R˙ Σ˙r˙)
− − r(c2 ω2r2) −
− (33)
2c2 2c2(c2 3ω2r2)
+ (Φ˙ Σ˙φ˙)r˙ − Rφ˙r˙ =0
r(c2 ω2r2) − − r2(c2 ω2r2)2
− −
Z¨ Σ¨z˙ 2Σ˙z¨=0 (34)
− −
Expanding Eq. (34) and using Eqs. (26)-(28) we obtain:
Z +2Z r˙+2Z φ˙+2Z z˙+2Z r˙φ˙+2Z r˙z˙+2Z φ˙z˙
ss sr sφ sz rφ rz φz
+Z a(r)φ˙2 Z b(r)φ˙r˙+Z r˙2+Z φ˙2+Z z˙2 Σ z˙
r φ rr φφ zz ss
− − (35)
2Σ z˙r˙ 2Σ z˙φ˙ 2Σ z˙2 2Σ z˙r˙φ˙ 2Σ z˙2r˙ 2Σ φ˙z˙2
rs sφ sz rφ rz φz
− − − − − −
Σ a(r)z˙φ˙2+Σ b(r)z˙φ˙r˙ Σ z˙r˙2 Σ z˙φ˙2 Σ z˙3 =0
r φ rr φφ zz
− − − −
Themethodweusedissimilarwiththeonein[9]and[15]. Condition(35)after
a long calculation gives the following for Σ and Z:
1
Σ= c s2+c s+c +z(c s+c ) (36)
1 2 3 5 6
2
1
Z =c z2+c z+c +s c z+c (37)
5 8 9 1 7
(cid:16)2 (cid:17)
where all the c are arbitrary constants. Using the results above in Eqs. (32)
i
and (33) after a long but straightforward calculation we obtain the following
final result:
Σ=c s+c z+c (38)
2 3 4
R=0 (39)
Φ=c (40)
1
7
Z =c s+c z+c (41)
5 6 7
So the symmetries of the geodesic equations form a seven dimensional group
with generators
x~ =∂ ,x~ =s∂ ,x~ =z∂ ,x~ =∂ ,x~ =s∂ ,x~ =z∂ ,x~ =∂
1 φ 2 s 3 s 4 s 5 z 6 z 7 z
The multiplication table for this Lie group is given by the relations:
[x~ ,x~ ]= x~ ,[x~ ,x~ ]= x~ ,[x~ ,x~ ]= x~ ,
2 3 3 2 4 4 5 2 5
− − −
[x~ ,x~ ]=x~ x~ ,[x~ ,x~ ]= x~ ,[x~ ,x~ ]= x~ ,
3 5 6 2 3 6 3 3 7 4
− − −
[x~ ,x~ ]=x~ ,[x~ ,x~ ]= x~ ,[x~ ,x~ ]= x~
4 5 7 5 6 5 6 7 7
− −
The subroup g = [G,G] is five dimensional and is generated by the vectors
x~ x~ , x~ , x~ , x~ and x~ .
2 6 3 4 5 7
−
If we calculate the group [g,g] we will see that this is the same as g. So the
derived series of ideals terminates. So the symmetry group of the geodesics is
neither nilpotent, nor solvable,in contraryto systems like the Mawell-Bloch[5]
or the Hilbert-Einstein equations in Cosmology [15].
Using Noether’s theorems we can check which of these symmetries are vari-
ational and also we can find the corresponding conserved quantities. A vector
fieldX~ is a variationalsymmetry ofa variationalproblemwith LagrangianL if
and only if it satisfies the following condition:
pr(1)X~ +L~ ξ~=0 (42)
∇·
where
n q
∂ ∂
X~ = λi + ψj (43)
∂ti ∂uj
X X
i=1 j=1
and
n
∂
ξ~= λi (44)
∂ti
X
i=1
EveryvariationalsymmetryX~ generatesaconservationlaw,whichcanbewrit-
ten in the form:
q
~ P~ = Q E (L) (45)
j j
∇· ·
X
j=1
where E (L) are the Euler-Lagrange operators acting on the Lagrangian and
j
Q are the characteristics of X~. In our case L is given by Eq. (8). Using the
j
conditions above we can prove that only the infinitesimal generators X~ with
i
i = 1,2,3,4 and 7 are variational symmetries. They form the Lie subgroup g
v
of the full symmetry group. We can easily check that
[g ,g ]=[g ,[g ,g ]]= X~ ,X~
v v v v v 3 4
{ }
and since [X~ ,X~ ]=0 we infer that the group g is solvable but not nilpotent.
3 4 v
So we have the following theorem:
8
Theorem 1. The Group of symmetries of the geodesic equations of the spatial
sections of the four dimensional space-time with respect to a uniformly rotating
noninertial system of reference is a seven dimensional Lie group, which is nei-
ther nilpotent, nor solvable. on the oher hand the rotational symmetries form a
five dimensional subgroup, which is solvable but not nilpotent.
The equations of Killing, which can be written in the form
ζ +ζ =0 (46)
j;i i;j
determine the infinitesimal generators of the isometry group of a space. Here
the semicolonrepresentscovariantdifferentiationwith respectto the metric(8).
If in Eqs. (38)-(41) we set Σ=c =0 we get the vector
5
∂ ∂
c +(c z+c )
1 6 7
∂φ ∂z
which acts only on the coordinates r,φ and z of the spatial sections. This is of
course a localsymmetry of the geodesic equations so it preserves the geodesics.
If we set c =c =0 we get the vector
6 7
∂
∂φ
which not only preserves the geodesics, but is also a Killing vector. A second
Killing vector can be found if we set c =c =0 and it is given by
6 1
∂
∂z
Finally for c =c =0 we get the vector
1 6
∂
z
∂z
which preserves the geodesics but it is not a Killing vector because it does not
satisfy
∂ξ
z
=0
∂z
which is one of the Killing’s equations. Taking into account the fact that every
Killingvectorpreservesthegeodesicsweinferthattheaboveargumentsindicate
thatthespatialsectionshaveonlytwoKillingvectors,whicharetheonesshown
above.
4 The geodesics from the Hamiltonian point of
view
Here we shall study the geodesics using the canonical symplectic structure on
the cotangent bundle of the spatial sections [1], [2].
9
If M is a Riemannianmanifoldwithmetric g we considerthe Hamiltonian
ij
H with local expression
1
H(xi,p )= gij(xk)p p
i i j
2
where xi,p are canonical coordinates of the contangent bundle T M.
i
∗
We can easily prove that the projections of the inertial curves of H onto M
are the geodesics of g parameterized by their affine parameter t. Hamilton’s
ij
equations can be written in the form:
x˙i =gilp (47)
l
1∂gkl
p˙ = p p (48)
i −2 ∂xi k l
Inverting Eq. (47) we get
p =g x˙i (49)
l il
Differentiating Eq. (47) and using Eqs. (47) and (49) we can find
1 ∂gmn ∂gil
x¨i = gil p p + gkmp p (50)
−2 ∂x m n ∂xk m l
l
Using Eq. (49) and after some algebra we end up with:
1∂g ∂g
x¨i =gil jk ljx˙kx˙j (51)
h2 ∂xl − ∂xk i
which is actually
1 ∂g ∂g ∂g
x¨i = gil jk lj lk x˙kx˙j (52)
2 h ∂xl − ∂xk − ∂xj i
Taking into accountthe definiotion ofChristoffel symbols it becomes clear that
here we have the geodesic Eqs.(21).
Using the metric (8) we get the Hamiltonian in the form:
1 1c2 ω2r2 1
H(xi,p )= p2+ − p2 + p2 (53)
i 2 r 2 c2r2 φ 2 z
WecaneasilyprovethatthisHamiltonianisseparablesinceitsatisfiestheLevi-
Civitaconditions[10]. ThuswecansolvethissystemusingtheHamiltin-Jacobi
method. Hamilton’s characteristic function takes the form
W =W (r)+p φ+p z (54)
1 φ z
and the Hamilton-Jacobi equation becomes
1 dW 2 1c2 ω2r2 1
1 + − p2 + p2 =E (55)
2(cid:16) dr (cid:17) 2 c2r2 φ 2 z
10