Internat. J. Math. & Math. Sci.
VOL. 13 NO. 3 (1990) 545-554
545
SPACE TIME MANIFOLDS AND CONTACT STRUCTURES
K.L. DUGGAL
Department of Mathematics and Statistics
University of Windsor, Windsor
Ontario, Canada NgB 3P4
(Received February 27, 1989)
ABSTRACT.
A new class of contact manifolds (carrlng a global non-vanlshlng tlmellke
vector field) is introduced to establish a relation between spacetlme manifolds and
contact
structures.
We show that odd dimensional strongly causal (in particular,
As examples,
globally hyperbolic) spacetimes can carry a regular contact structure.
we
present
a causal
spacetlme with a non regular contact structure and a physical
model [Gdel Universe] of Homogeneous contact manifold.
Finally, we construct a model
of 4-dimenslonal spacetlme of general relativity as a contact CR-submanlfold.
KEY WORDS AND PHRASES.
Contact manifold, global tfmellke vector field, spacetlme
manifold, Lorentzlan geometry, submanlfold.
1980 AMS SUBJECT CLASSIFICATION CODES. 53C25, 83C50.
I.
INTRODUCTION.
A smooth (2n+l)-dimensional manifold N is called contact manifold if it carries a
global 1-form n such that
everywhere on N, where d is the exterior derivative of the contact from n.
N is
then orlentable. The name contact is due to S. Lie [I]. Classlcally there have been
two large classes of contact manifolds, namely, the principle clrcle bundle of the
boothby-wang flbratlons includlng the odd-dlmenslonal spheres and the tangent sphere
A thorough dlsusslon may be found in [2]. The theory of contact manifolds
bundles.
has been wldely used in mathematlcal physics. For example, in thermodynamics [3], a
contact manifold is named as thermodynamic phase space (TPS) with contact form
du
Tds + pdv +**., where u, s, v, p and T are the internal energy, the entropy, the
volume, the pressure and the temperature respectlvely.
Contact manifolds have an interplay with Cauchy Riemann (CR) manifolds [4,5] in
The 2n-dlmenslonal contact distribution D
{X TN/n(X)
0} has
a complex structure.
Thus, the complexlflcation of D has a holomorphlc sub-bundle H
such that (N,H) is a CR-manlfold if the contact structure is normal [6]. Recently,
the present author introduced a new area of research, namely, Lorentzlan Geometry of
CR submaniflds with applications to relativity [7,8]; cf. also [9]. As normal contact
manlfold is an example of CR manifolds (see Blalr [2] p. 62), a systematic study on
the Lorentzan geometry (mathematical theory for relativity) of contact manifolds is
the following way.
needed.
K. L. DUGGAL
546
Motivated by above, as a first step, the objective of this paper is to establish
a relation between spacetlme manifolds and contact structures.
Our study
is
llne
in
with
the
latest
trend of
relating
the Riemannlan and
Lorentzlan geometry (see, for example [13, 14, 17]).
2.
PRELIMINARIES.
A spacetlme (M,g) is a connected smooth Hausdorff manifold of dimension
a Lorentz metric g of
global
tlmellke
signature
vector
field.
(-,+,...,+) and
is
All
manifolds
non-compact
time oriented,
admit
2 with
that is, has a
Lorentz
metrics,
Any
whereas, a compact manifold is Lorentz iff its Euler characteristic vanishes.
compact spacetlme contains a closed timellke curve. For example, on the cylinder
2
2
2
const, are closed tlmelIke curves.
R
S with ds
-dt + de
the circles t
relativity,
each point of spacetlme corresponds to an event relative to time.
In
Thus,
physically, closed tlmellke curves are forbidden as this raises the possibility that a
person might meet himself in the past.
non-compact.
poknt
[a,b]
Let C:
of C for t
Therefore, all physical spacetlmes are assumed
M be a curve in M.
b if llm C(t)
t
b-
The point p
M is called the end
A nonspacellke curve is future (resp., past)
p.
Inextendible if it has no future (resp., past) endpolnt.
past) inextendlble curve C does one of the following:
(I) Traps within a compact
A nonspacellke future (resp.
set.
set.
(2) Does not trap in any compact set but continually re-enter a compact
(3) Does not trap in any compact set and does not re-enter in any such set more than a
flnlte number of times.
If (3) holds then C goes off to the edge of spacetlme to infinity or a singularity
point.
For
respectlvely.
(I) and
(2),
C
is
called
totally and
partially
imprisoned
in
Carter [I0] has given following example of a causal spacetlme M having
imprisoned nonspacellke inextendlble curves.
2
2
ds
(cosh t
l)2(dy
The metric is:
dt
2)
dtdy + dz
(1)
Identify
Identi fy ef ter shifting
en irr(]tionel number
(Fig. i)
M--RxSIxS
{(t,y,z)
R3: (t,y,z)
(t,y,z+l)and
.
(t,y,z)- (y,y+l,z+a),
where
a
A Cauchy hypersurface S in M is a subset that meets exactly
M is globally hyperbolic Iff it
once every inextendible nonspacellke curve in
is an irrational number.
SPACE TIME MANIFOLDS AND CONTACT STRUCTURES
hypersurface.
Cauchy
a
admits
spacetlme
product
a
is
According
manifold
of
[11]
Geroch
to
form (M
the
547
globally
a
-dt 2
RxS: g
hyperbolic
(S,G) a
G) with
Minkowskl spacetlme and Einstein static universe are
compact Riemannlan manifold.
Given any two points p,q of M, q is in the chronological future
simple examples.
<<
(resp. past) of p, denoted by p
q (resp. q
<<
p), if there
is a future
(resp. past)
The chronological future (resp. past) of p are
directed tlmellke curve from p to q.
l+(p)
The causal future
{q c M: q << p}.
{q c M: p << q} and l-(p)
{q E M: p < q} for
q} and J-(p)
(resp. past) of p are the sets J+(p)
{q E M: p
nonspacellke curves. This means that M, with no closed nonspacellke curve is causal
space. Also, as any compact spacetlme contains a closed tlmellke curve, it fails to
the sets
The sets
be causal.
J+(p)
and
every
J-(p)
has
point
l+(p)
are always open in any space time, but the sets
M is strongly causal if its
small nelghbourhoods which no nonspacellke curve
A strongly causal M is globally hyperbolic, if for each p,
arbitrarily
intersects more than once.
J+(p) N J-(q)
q of M,
l-(p)
and
are neither open nor closed in general.
is compact.
PROPOSITION I.
If the strong causality condition holds on a compact set of a
then there can be non imprisoned inextendlble curves
spacetlme,
(Hawking and Ellis
For details on above (with examples) we refer [13].
[12]).
CONTACT SPACE TIMES.
3.
(2n+1)-dlmenslonal
A
manifold
form
there
if
exists
smooth
a
manifold
tensor
N
fleld
is
of
called
type
an
(l,l),
almost
a
contact
vector
,
metric
field
l-
and a semi-Riemannlan metric g such that:
(2)
g(,)
g(X,Y)
e, g(,bY)
erl(X) rl(Y)
2n.
-I according as
is
spacellke or tlmellke and rank(@)
g(X,Y) for every X,Y of N, then N is called a contact metric manifold.
+I and g definite, N is the usual contact metric structure [2].
Here, we
For
-I,
study a larger class to allow indefinite metric. For example, if
then N contains a tlmellke vector field. However, the signature of g is restricted by
where c
If
+I or
dn(X,Y)
the following result.
PROPOSITION
manifold (n
PROOF.
n
> 2)
2.
The
2n-dlmenslonal
cannot carry a
Lorentz
contact
distribution D(
O) of a contact
metric.
D admits a Hermltlan structure which cannot carry a Lorentz metric unless
(for details see Flaherty [14]). Based on above proposition the followlng holds.
THEOREM I.
For a contact metric manifold N, the following are equivalent:
(a) N is a spacetlme manifold
(b) the contact vector field
is
tlmellke and the contact distribution D (defined
spacellke and D tlmellke is
0) is spacellke [except when dlm(N)
3, then
by n
posslble].
In this paper, a contact manifold with a Lorentz metric will be called a contact
An almost contact manifold is said to be normal if the NiJenhuls tensor
spacetlme.
of
satisfies:
[, ] + 2dn
(R)
0
(3)
K. L. DUGGAL
548
EXAMPLE.
coordinate
Consider
system
a
(Xi; yl,
tlmellke vector fleld, say
2t.
(2n+l)-dlmenslonal spacetlme manifold N with a local
t) i
1,...,n. N being tlme
admits a global
Define a l-form
yidxl)
so that
=I/2(dt E
.
norlented
.
Then, from a classlcal theorem of Darboux [2], N is a contact manlfold with
and tlmellke contact vector field
The Lorentz metric g:
contact from
n
=I/4
g
((dxl) 2 + (dyi) 2)
Z
]
-I.
gives a contact metric structure to the spacetlme N for
With respect to an orthonormal
contact spacetlme.
U
2(
2yl, Un+i
i
i
+
yit)’
@-basis
Un+i’ SUn+i
$Ui
Therefore, N is a
(Ui; Un+i,
), where
Ui
the metrix of the components of g is given by:
0
yl
o
lj
o
-yJ
0
yiyj
lj
THEOREM 2.
An odd dimensional globally hyperbolic spacetime can carry a contact
structure.
PROOF.
[M2n,G,J 2
-I] and G(JX,JY)
Construct a globally hyperbolic spacetlme
d
Denote a vector field on N by X
((X) -,
X)where X
t is the coordinate of R and () is a smooth
on N
d
O) is timelike global vector field. Then with
Consider
an
almost
-
Hermltlan
manifold
G(X,Y) for every X,Y of M 2n.
2
N
g =-dt + G}.
"tRM
2n,
tangent to M 2n
is
Set
dt so that
functlon
d
-,
((X)
g[(n(x) -f, x),
(O,JX)
X)
(u(Y) -f, Y)]
G(X,Y)
n(x)n(Y),
-I.
we recover a contact metric structure on N for
n
In particular, let Rq be n-dlmensional pseudo-Euclldean space of signature (,...,-,+,..,+) with q negative and n-q positive eigenvalues. Hence, for a local
coordinate system
(Xi),
ds
Define for r
i
. .
l,...,n, its metric
q
2
dx +
n
dx
q+l
is given by:
2
i
0 (see Wolf [15 section 4.2])
S
n
{X e R
n
{X e
n+
X
n+
X
2
2
+ X2 +
2
X
2
n+l
r
+ X
2
n+l
-r
and
HI
R1
2
+
2}
+ X
2}
SPACE TIME MANIFOLDS AND CONTACT STRUCTURES
Topologically
of
the
S
n
of
sphere
covering
curvature-r
of
-2
is R
n
HI
n-I and H
radius
r
topologically
is
S
S
n
(ni)
In
is S
with
n
RI,
R
n-I
positive
n
Also, S is a Lorentzlan analogue
curvature
thus
and,
549
a
r
-2.
The
space
hyperbolic
are called de-Sltter (anti de-sitter) spaces
Universal
of
[12].
negative
n
S is
globally hyperbolic but H is not. Thus, we have
COROLLARY. Odd dimensional de-sltter spaces can carry a contact structure.
4.
REGULAR CONTACT SPACE TIMES.
A contact structure on N is regular if every point p of
coordinate nelghbourhood
this
of
such that the integral curves of
N has
a
cubical
passing through
pass
neighborhood only once.
If N is compact, then the maximal integral curves
are homeomorphlc to circles and therefore Boothby-Wang theorem [16] will hold.
For noncompact spacetlmes, comparing the definitions of strongly causal spacetlmes and
regular contact spaces, we get
THEOREM 2.
An odd dimensional strongly causal spacetlme can carry a regular
contact structure.
COROLLARY.
Odd dimensional globally hyperbolic spacetlmes can carry a regular
contact structure.
Well-known examples are Minkowski spacetime, Lorentz spheres and Robertson-Walker
spacetimes.
On the other hand, there exist causal spacetimes which can not carry a regular
To illustrate this point consider a 3-dimensional contact manifold
contact structure.
M 3 defined by n
cosh z dt
slnh z dy and
system (t,y,z). The Lorentz metric
g
cosh 2z dy
2
+ dz
cosh2z[dy 2
dt
2
2]
cosh
zt
+ sinh
(I + slnh 2z)dtdy
+ dz
2
zy
for a coordinate
n
(4)
dtdy.
cosh2z=cosht-1) -l
gives a contact metric structure on spacetlme M
3. In particular, for
and t > 0, M3 is Carter’s example (see section 2) of a causal
inextendlble
nonspacellke
imprisoned
curves.
Note
that
the
through (y,y+1,a) induce an irratlonal flow as a is irratlonal.
not regular.
of
spacetlme with
integral
curves
Therefore
is
The discussion, so far, opens the way to address the following basic problem.
"Characterlse regular
contact spacetlmes"
For the characterlsatlon of regular contact Riemannlan manifolds, we refer [16]. The
followlng results may be useful.
PROPOSITION 3 (O’Neill [17]). Maxlmal Integral curves of a vector fleld V of a
spacetlme are inextendlble.
This
means
that
the
maximal integral curves of the regular contact vector
On the topology of such spacetlmes, we know that Open sets of
the form l+(p) N l-(q) for any p,q of any spacetlme N form a basis of a topological
structure on N, called Alexandrov topology.
This topology is related with the
strongly causal spacetlmes (and, therefore, regular contact spacetlmes) by the
field
are extendlble.
followlng result:
550
K.L. DUGGAL
(Penrose [18])
PROPOSITION 4.
The Alexandrov topology for a spacetlme N agrees
with the given manifold topology iff N is strongly causal.
A contact manifold N is said to be homogeneous if there is a connected Lie group
G acting transitively and effective as a group of dlfferentlable homomorphlsm on N
which leave
of
are
invarlant.
Boothby and Wang [16] have proved that the integral curves
regular for homogeneous
necessarily
contact
manifolds
but
N need not be
of
the
contact
Also, following holds:
compact.
THEOREM
4
(Boothby-Wang)
[16]).
The
integral
curves
vector
of a homogeneous contact manifold are either closed curves or open arcs.
field
Relating
above
result
with
homogeneous
spacetlmes,
we
the
following
Gd’del[
Universe,
present
example:
Kurt G6"dell
[19] discovered a homogeneous spacetlme, called
whose metric g is a direct sum of the metric:
+
exp(2
ax)dy 2
gl
-dt2
()
dx2 1/2
2exp(()
(5)
ax)dtdy,
on the 3-dlmenslonal manifold M 3 defined by the coordinates (t,x,y), where a > 0 is a
on the manifold R defined by the coordinate z. For
constant, and the metric g2
dz2
9olutlons
(M3,gl).
of
the
Einstein’s
field
equations,
Transforming into new coordinates
exp((2)ax)
cosh 2r +
ay exp((/2)ax)
tan
I/2 (
+ at
sin
(/2)t’)
the metric gl takes the new form
2
(s
2 + dr
gl
2a-2[-dt
Inh4r-slnh2r)d
it
is
sufficient
to
(t’,r,) by
cos
slnh 2r,
slnh 2r,
exp(-2r)tan
I/,
2 + 2() slnh2r d#dt].
r =0 coordinate axis
Matter
world line
(r, ) constant}
r (log (1,VJ
(closed spocelike
curve
r) log (1,)
(closed timelike
curve
r: log(1,vr}
(closed null
curve}
(Fig. 2)
consider
only
SPACE TIME MANIFOLDS AND CONTACT STRUCTURES
The
flow
of
vector
the
fluid
(a/J2)
is u
2).
0 (see fig.
symmetric about the axis r
t"
The
551
space
is
rotatlonally
0
The light cones on the axis r
The light cone opens out and tips over as r increases
contain the direction of u.
At a radius r
is a null
in closed tlmellke curves.
log (l+J2),
is a tlmellke vector and circles of constant r,
For a greater value of r,
t’ are closed tlmellke curves. (M3,g I) is a contact spacetlme define by
(a/)
(/a)(dt’ + Ad),
+ A
where A2
slnh2r. Thus,
resulting
@
vector.
I@],
[t
slnh4r
Gdel universe is an example of a homogeneous contact spacetlme such that the integral
>
[for r
curves of
log (I + 2) and t’ constant] are closed tlmellke curves.
Further research in this direction is needed to find the topology of homogeneous
For results on these topics with respet to Riemanlan metric see
contact spacetlmes.
[16].
CONTACT STRUCTURE AND PHYSICAL SPACE TIME.
5.
By a physical
spacetlme
we
4-dimenslonal
mean
spacetime
the
in
context
of
relativity (special or general).
As contact manifolds are odd dimensional (there do
exist even dimensional contact manifolds [20] which we are not discussing here), to
relate them with the physical spacetlme, we embed physical spacetlme as submanifold of
contact
There
manifold.
manifolds.
the ambient manifold.
[4,5]
manifolds
are
three
popular
classes
of
submanlfolds
of
contact
First, Invarlant submanifolds [21] which inherit almost all properties of
(See
Second, antl-lnvarlant submanlfolds [22]. Third, contact CR
Genner [23] for details on the isometric embedding of
In particular, every 4-dlmenslonal spacetlme M can be seen (locally) as a
n
46 or 87 according as M
I0.
submanlfold of a Mlnkowskl space R of n
Globally, n
is compact or noncompact).
An invarlant submanlfold of a contact manifold inherits
contact struture and, therefore, must be ruled out for our purpose. Out of the other
two classes, we consider CR submanlfolds as they include antl-lnvarlant class.
spacetlmes.
Let M be a real m-dimenslonal submanlfold of a contact metric manifold
We use same symbol g for the
B, g) with semi-Rlemannlan metric metric g.
be tangent to M.
metric of M. Let
(N,
,,
DEFINITION.
M is called a contact
distributions D and D
(I)
(2)
(3)
+/-
CR submanlfold of
N if
there
{}, where D, D I and {} are mutually orthogonal
D is invarlant by
[(D) D];
TM
D
+/-
exist
two
on M satisfying
DOD +/- @
is anti-lnvarlant by
to each other;
[(D i cTM +/-
The same concept was studied under the name
semi-lnvarlant submanlfolds [5].
define a larger class to include indefinite metric.
Dim(D)
0 (resp. dim(D
Here we
+/-
0
implies M is antl-lnvarlant (resp. invarlant), otherwise it is a non-trivlal contact
CR submanlfold.
Based on above definition, we now construct a mathematical model of 4-dlmenslonal
spacetlmes as contact CR submanlfolds.
-
Let (N,g) be a 5-dimensional spacetlme with a local coordinate system
MODEL.
Define a l-form -I/2 [dt- ydx- zds] so that there exists a vector
and
2 on N. N has a contact metric structure (, n,g) with contact form
t
(6)
]
g =I/4 [dx 2 + dy 2 + ds 2 + dz 2
n
(x,s,y,z,t).
field
,
K. L. DUGGAL
552
as the defining equations (2) hold for e
(UI;Vi; )
u
-I.
There exists an orthonormal
2y,
u2
=zZ’
V
#U i
v
2( +yt v2
Vi
i,
U i, i
+.t
2(
S
1,2
Let (M,g) be a 4-dimenslonal spacetlme embedded in N.
orthonormal
for
basis
Tp(M).
Then,
the
complexified
UI+
holomorphic subbundle H generated by (Z,Z) with Z
tangent
iV
submanlfold
of
N
Therefore,
to each other.
space
is an
CT(M) has a
and JZ
-iZ(J is complex
Re(H + H) generated by
{U2} D, D i and are
Thus, there exists a real distribution D
invariant by
and anti-invariant distribution D i
mutually orthogonal
(UI,U2,VI,)
So that
structure on H).
(UI,V I)
-basls
where
by definition, M is a contact CR
and spacellke invariant
field
In general, the dimension of the embedding manifold N may be higher
than 5 (subject to restrictions as stated in [23]).
REMARK.
It is worth mentioning that the embedding manifold has been used as
with
timelike
contact
vector
distribution D.
auxiliary
space
for
deriving physical
of
properties
the
embedded spacetime.
In
particular, several exact solutions of the Einstein’s field equations have been found
by this technique, at least for some cases of low embedding class (the minimum number
of extra dimension is called the embedding class).
For example, the maximal analytic
extension of the Schwarzschild solution was found by embedding technique [24].
However, unfortunately, no systematic research has been done to
embedding class greater than two.
find solutions of
Difficulty is the lack of effective method for
solving embedding equations.
Hopefully,
by
supplying
some
contact structure discussed in this
differential
paper),
geometric
structure
(such as
the
it may help to provide more insight on the
geometry of spacetlme needed for physical problems (including finding exact solutions)
in relativity.
In another direction,
research,
the disussion
Lorentzian
namely,
geome
of
in this
CR
section
submanifolds
leads
to
a new area of
(including
contact
CR
For related
introduced by the present author [7,8].
in this direction and, in general, bridging the gap
progress
references on some
between Differential Geometry and Mathematical Physics of Relativity, see [13, 14, 17,
submanifolds
as
subcase),
25-27].
ACKNOWLEDGEMENT.
This
research was supported by the Natural Sciences and Engineering Research
Council of Canada.
SPACE TIME MANIFOLDS AND CONTACT STRUCTURES
553
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LIE, S., Theorle der Transformatlongruppen, Vol. 2, Leipzig, Tenbner, 1890.
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YANO, K. and KON, M., CR Submanifolds of Kahlerian and Sazakian Manifolds,
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BEJANCU, A., Geometry of CR Submanlfolds, Reidel Publishing Company, Holland,
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IANUS, S., Sulle varieta di Cauchy
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