Dynamics of Totally Constrained Systems. I: Classical Theory
  Classical Theory  
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Yukawa Institute for Theoretical Physics, Kyoto University
, Uji 611
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Dynamics of Totally Constrained Systems. I
This is the first of a series of papers in which a new formulation of quantum theory is developed for totally constrained systems, that is, canonical systems in which the hamiltonian is written as a linear combination of constraints ha with arbitrary coefficients. The main purpose of the present paper is to clarify that classical dynamics of a totally constrained system is nothing but the foliation of the constraint submanifold in phase space by the involutive system of infinitesimal canonical transformations Ya generated by the constraint functions. From this point of view it is shown that statistical dynamics for an ensemble of a totally constrained system can be formulated in terms of a relative distribution function without gauge fixing or reduction. There the key role is played by the fact that the canonical measure in phase space and the vector fields Ya induce natural conservative measures on acausal submanifolds, which are submanifolds transversal to the dynamical foliation. Further it is shown that the structure coefficients c,?;p defined by {h., hp}=2:.,c~ph7 should weakly commute with ha, 2:,,{h 7, c,?;p} ;,;Q, in order that the description in terms of the relative distribution function is consistent. The overall picture on the classical dynamics given in this paper provides the basic motivation for the quantum formulation developed in the subsequent papers.

1. Introduction
In the canonical approach to
Dirac
prescription is widely adopted, in which the classical constraints ha;::;;;;O are formulat
ed as constraints on physical states of the form
evolution equation is lost.
This problem is closely related with the fact that opera
tors corresponding to the physical quantities which play the role of time are excluded
from observables in this formulation. 2> Further the Dirac quantization of general
relativity does not give a mathematically welldefined formulation apart from the
regularization and the operator ordering problems because Eq. (11) does not have
normalizable solutions in general even if the spatial diffeomorphism freedom is
eliminated before quantization.
As discussed in Ref. 3), this difficulty comes from the too formal application of the
Dirac procedure.
Since the hamiltonian is written as a linear combination of the
constraint functions in general relativity, the hamiltonian constraints carry all the
information on dynamics.
Hence if we formulate the hamiltonian constraint as the
condition on state vectors as above, each state vector becomes a dynamical object.
This should be compared with the ordinary quantum mechanics.
There state vectors
are used to describe dynamics of a system, but each state is not a dynamical object.
Dynamics is described by a oneparameter family of states qJ(f) satisfying the
Each state vector in this family merely carries information by
possible measurements at each instant t.
This observation indicates that we should impose the hamiltonian constraints on
the object which picks up all the possible state vectors allowed by dynamics. The
most natural object of this nature will be the probability amplitude lJf( (])) which
assigns the probability to each state vector, taking account of the probabilistic nature
of quantum mechanics.
Here one may notice that a similar phenomenon occurs in the classical dynamics
of a system with hamiltonian constraints. For simplicity let us consider a system
with a single hamiltonian constraint h ~ 0 on a phase space r. If we reduce this
system to a canonical system without constraint, dynamics is described by a curve
ro(t) in a reduced phase space 10 which is a solution to a canonical equation of motion.
This curve corresponds to the family of states (J)(t) in quantum mechanics. On the
other hand in the original phase space this curve corresponds to a curve r contained
in the constraint hypersurface l:H, which corresponds to lJf above. This analogy
becomes better if we consider an ensemble of systems instead of a single system, for
which ro(t) is replaced by a family of distribution functions po(t) on 10 and r by a
distribution function p on r. Clearly p does not represent a state but is a dynamical
object which picks up possible states allowed by dynamics. Hence p should be
constant along each curve in r corresponding to a solution to the equation of motion
in 10. This implies that it is not normalizable on r. This may be regarded as the
essential reason why the solutions to Eq. (11) are unnormalizable. It is obviously
absurd to postulate dynamics so that it picks up p which is normalizable in r.
Applying the quantum hamiltonian constraints on states just corresponds to such an
approach.
This observation suggests that the investigation of the structure of classical
statistical dynamics of totally constrained systems will shed a good light on how to
find a consistent quantum formulation of them. This is the motivation of the present
paper. Since a variety of forms exist for the canonical formulation of gravity and
since the structure of the problem is common in all the theories with general covarian
ce, we consider a generic totally constrained system in most part of the paper.
The organization of the paper is as follows. First in 2 we consider a simple
totally constrained system obtained by embedding a canonical system without con
straint to a larger phase space in order to find how to interpret the unnormalizable
distribution function on the extended phase space. Then in 3 on the basis of the
result obtained there we describe how to formulate the statistical dynamics of a
generic totally constrained system with a single constraint without reduction and not
referring to special time variables. Further the general structure of reduction and its
freedom is examined because it is relevant to the time variable problem. In particu
lar by applying it to the totally constrained system describing a relativistic particle in
curved spacetime, it is shown that the background spacetime should have a Killing
vector in order that there is a natural reduction of this system. In the subsequent two
sections the formulation obtained for a single constraint system is extended to a
multiple constraint system. First in 4 an overview on the canonical structure of
general relativity in terms of the ADM variables is given in order to clarify that
dynamics of a totally constrained system with multiple constraints is completely
determined by the foliation of the constraint submanifold by the involutive system of
the infinitesimal canonical transformations generated by the constraint functions.
Then in 5 based on this viewpoint, statistical dynamics for multiple constraint
systems is formulated in terms of the relative distribution function by proving the
existence of natural conservative induced measures on acausal submanifolds. Sec
tion 6 is devoted to discussion.
2. Embedding of an unconstrained system into a constrained system
In this section we study the dynamics of a simple constrained system obtained by
embedding an unconstrained canonical system into a larger phase space. The main
purpose is to find the way how to formulate the dynamics of a constrained system and
its ensemble without reducing it into an unconstrained one.
2.1. Canonical system
A canonical dynamical system with no constraint is specified by a triplet (r, w, h)
of a phase space, a symplectic form and a hamiltonian.4> The phase spacer is a 2n
dimensional smooth manifold, the symplectic form w is a closed nondegenerate
2form on r,
and the hamiltonian h is a smooth function on R x r.
Let :I (F) and X(r) be the sets of all smooth functions and all smooth vector
fields on r, respectively. Then for any /E :f(r), w uniquely determines a vector field
XfEX(r), which is called the infinitesimal canonical transformation generated by /,
through the equation
From the identity
x=fxod+dofx
the infinitesimal transformation xf preserves (i) and Q,
xfw=O, xf2=0.
Conversely any vector field which satisfies this equation is an infinitesimal canonical
transformation generated by some function on r, at least locally.
In terms of this infinitesimal canonical transformation the Poisson bracket of two
functions I and g is defined by
It follows from this definition that
X<f.ul= [Xf, Xu].
Thus the correspondence f't+Xf gives a homomorphism from :t(r) into X(r) as Lie
algebras, whose kernel consists of constant functions.
In appropriate local coordinates (qi, pJ w can be always written as
On the other hand the behavior of an ensemble of the system is described by a
distribution function pE : (R X F), which satisfies the two conditions: i) JrpQ = 1 for
each t, and ii) pQ is preserved by the equation of motion. This second condition
yields the equation of motion for p,
which implies that p is constant along each integration curve in R X r. When we
introduce Pt defined by Pt(u)=p(t, u)(uET), the expectation value of /E:f(Rxr)
at a time t is given by
In this local coordinate system Xf is expressed as
which leads to the familiar expression
and its timederivative by
Finally the dynamics in the phase space is determined by the hamiltonian h in the
following way. Let the canonical coordinate of R in R X r be t, and for each value
oft let Y=Xh be the infinitesimal canonical transformation generated by h regarded
as a function on r. Then for a single system, its possible histories are given by the
integration curves of the vector field at+ Y on R x r when t is regarded as the time
variable, and the value of /E :I(T) along each curve, when regarded as a function of
t, satisfies the equation
In particular in a local coordinate system (t, ua) of R X reach integral curve follows
the canonical equation of motion
2.2. Embedding into a totally constrained system
The above canonical dynamical system can be embedded into totally constrained
canonical systems with larger phase spaces by various ways. Here we consider the
simplest one.
Let (f, w, h) be a canonical system defined by
Then the infinitesimal canonical transformation Y generated by his expressed as
iiJ:=dPol\dq0 +m,
h(q0,Po, u):=h(q0, u)+Po.
Hence by the projection 1r define<;} by
1r: t
w
(q', Po, u)
it is mapped to 1r* Y= Y + Ot, and each integral curve of Y to a solution to the
equation of motion in R X r. Therefore, noting that his conserved and 1r is injective
on each h =const surface, one sees that the original canonical system is equivalent to
the extended canonical system with a constraint h =const.
In this embedding only the integral curves of Y in the extended phase space,
which we call the hamiltonian flow, have a physical significance, while the canonical
time variable for the extended system, which is denoted by r, just plays the role of a
parameter of these curves. Hence for an arbitrary function N(r) the system with h
replaced by H = Nh is also equivalent to the original system under the constraint h
=const. In particular for the special choice of the constraint, h=0, this equivalence
holds for an arbitrary function NE <;I(Rx f) since
where A:::::B means that A=B under the constraint. We express this last situation
by saying that the original canonical system is embedded into the totally constraint
system (f, w, h). In this expression fi is understood to play the double roles, one as
the generating function of the hamiltonian flow and the other giving the constraint fi
=0.
2.3. Distribution function on the extended phase space
The distribution function p for the unconstrained system, if it is regarded as a
function on f, is constant along the hamiltonian flow from Eqs. (2 14) and (2 21).
This is quite natural since each pure dynamical state of the constrained system is not
represented by a point but by a hamiltonian flow line in the extended phase space.
Hence, taking account of the constraint, it is natural to consider the distribution
function j5 on the extended phase space defined by
where o( *) is the delta function. From this definition it follows that i5 is character
ized as a distribution on f satisfying the two equations,
i5= po(ii),
Here by a distribution on f we mean a functional f on the space of smooth
functions with compact supports in f, which is expressed as
the above condition implies that fhfl.I, = fl.I2 where 8 is the diffeomorphism from .E1
onto .E2 determined by the hamiltonian flow. Hence the requirement that fi.I, coin
cides with that given by Eq. (228) completely determines fl.x for any maximal acausal
hypersurface.
The explicit form of the measure is given by
when it can be identified with a function on f.
Since the original phase space r at time t can be identified with the intersection
of the q0= t hypersurface .Et and the constraint hypersurface J;H in f, the expectation
value of /E :E(R x r) at a time t is expressed in terms of j5 and Et:=o(q0  t) as
This fixes the interpretation and the normalization of the distribution function j5 in
the extended phase space.
From the dynamical point of view these q0 =const surfaces have no special
significance in the extended phase space, apart from the fact that they are 'Cauchy
surfaces' for (5. In fact we can easily extend the expression (228) to that for the
expectation value on an arbitrary hypersurface J; which is transversal to all the
hamiltonian flow lines. Let us call such a surface a maximal acausal hypersuiface and
denote the corresponding expectation value by <!>I. Then for two maximal acausal
hypersurfaces .E1 and .E2, <!>I, and <!>I2 should coincide for any constant of motion
/. If we express <!>I in terms of a measure fl.I as
Hence p cannot give a finite measure on t by simple renormalization. Nevertheless
it can be interpreted as giving a relative probability density under some limited
situations.
To see this, let us define the conditional probability for physical quantities A
to take value in Lll, ... under the condition c/>=r, by
one can rewrite the above expression for <!>.E as
E.E=I{Jz, c/>}l8(c/> r)'
if the maximal acausal surface J: is specified by the condition c/>= r( =const). It is
easy to see that the righthand side of this equation coincides with Eq. (228) for cf>=q0
and r=t.
2.4. Probability interpretation of the relative distribution function
where C is a normalization constant and EALl) is the characteristic function of the
region where the value of I is contained in Ll. However, this naive interpretation
does not work by itself because the integration of p over the whole extended phase
space diverges:
d(f[yQ)= f(/Q) [yd(!Q)=O.
where lxi.E implies that the differential form xis regarded as a positive measure on J:.
In order to see this, first note that, for any constant of motion/, flfQ is a closed form
from
Hence by applying Stokes' theorem to a region bounded by J:1 and J:z one gets J.EJdJL.E,
= J.EJdJL.E.. On the other hand from the identity
I" ) ._ .
Pr (n+=,...,'.1A1 ' "'r .I1L1Im.H Pr(/1EL11, , cf>ELlP)
Pr( c/>ELlP)
Then the expectation value of any /E c;J(f) determined from this probability coin
cides with the righthand side of Eq. (233), if and only if {h, c/>} is given by some
function of cf> on J:H. This condition requires that c/> is expressed in terms of a
function g and some constant of motion k as
Let us call a function satisfying this condition a good time variable.
When a maximal acausal hypersurface J; is given, we can always find a good time
variable which is constant on J:. However, its freedom is just a rescaling of the
variable so that it is in general impossible to find a good time variable which is
constant on each of more than two given acausal hypersurfaces. Though this fact
has no significance in the classical framework, it seems to have a deep implication in
the quantum framework in connection with the unitarity problem as will be discussed
in a subsequent paper.5>
3. Nontrivial system with a single hamiltonian constraint
In this section we show that by a simple generalization of the formulae in the
previous section we can discuss the dynamics of a totally constrained system with a
single hamiltonian constraint without reducing it to an unconstrained system.
3.1. Dynamics in the extended phase space
Let (T, w, h) be a totally constrained system with a single hamiltonian constraint
h=O, and YEX(T) be the infinitesimal canonical transformation Xh. Then from the
consideration in the previous section, it is natural to interpret that each integration
curve of Y on the constraint hypersurface J:H yields a possible time evolution of the
system. Hence if Y has a zeropoint on J:H, it represents a solution for which any
physical quantity takes a fixed value. Since such a solution is quite unphysical, we
assume that Y does not vanish on J:H. On the other hand, if there is a closed orbit
in the hamiltonian flow, it represents a completely periodic world like the antide
Sitter spacetime. We do not consider such causality violating cases in this paper
either. We further assume that the hamiltonian flow is not ergodic. This is equiva
lent to requiring that the foliation of J:H by the hamiltonian flow has a locally trivial
bundle structure whose fiber is homeomorphic to R. Thus it has a global section.
We call an extension of such a global section off J:H hypersurface, a maximal acausal
hypersurface.
When we consider an ensemble of totally constrained systems with the same
structure, they are represented by a set of hamiltonian flow lines in r, each of which
intersects with a maximal acausal hypersurface at a single point. In the limit that
the ensemble consists of very large number of members, these interesection points
determine a measure fl.E on each maximal acausal hypersurface J;. From its
definition this measure is preserved by the mapping among maximal acausal hypersur
faces determined by the hamiltonian flow. Let c be a positivedefinite constant of
motion. Then, since cfvQ is a closed form and yields a measure with the same
property as fl.E on each maximal acausal hypersurface as shown in the previous
section, the RadonNykodim derivative of fl.E by ciJyQI yields a function which is
constant along the hamiltonian flow on the constraint surface J:H. Thus if we regard
this function as a distribution whose support is contained in J:H, we are naturally led
to the distribution function p on r which satisfies
in the distribution sense. From its definition the expectation value of a physical
quantity /E :F(r) on a maximal acausal hypersurface S for the ensemble is given by
We will show later that we must allow for a nontrivial choice of c for a natural time
variable to satisfy this condition even in simple cases.
3.2. Reduction
As clarified in the previous section, the dynamics of a totally constrained system
(r, w, h) can be described with no reference to a special time variable. Now let us
study the relation of this description with that in terms of a time variable in a
canonical system with no constraint which is obtained by reduction.
In practical situations each maximal acausal hypersurface is specified by the
condition =const in terms of a physical quantity on the phase spacer. Here the
constant may have to be a special value and the other values may not give maximal
acausal hypersurface in general. We call such a function an instant function.
In terms of the instant function the dynamics of the totally constrained system is
formulated in the following way. First one selects an appropriate instant function t/11
from measured quantities, and specifies a maximal acausal hypersurface, say, by t/11
=0. Then measurements of various physical quantities determine a measure f.i.,,
which in turn determines the value of the distribution function p on the maximal
acausal hypersurface S1. p is uniquely extended over the phase space by Eqs. (31)
and (32), at least around the constraint hypersurface SH. Once the distribution
function is determined, one can calculate the expectation value of any physical
quantity at an instant specified by any instant function. Here, though the distribution
function depends on the choice of the constant of motion c in the above procedure, this
freedom does not affect the predictions on the expectation values since c and p come
into the theory always in the combination cp.
The reason why we have introduced the apparently superfluous freedom of c in
the definition of p is to widen the concept of good time variables introduced in the
previous section. Let be an instant function such that = r( =const) gives a
maximal acausal hypersurface for any r in some open interval of R. If we require
that the probability measures on these hypersurfaces derived from the natural
measure pQ as in Eq. (236) coincide with the conserved measure f.l., we obtain the
condition
where /( ) is an appropriate function of . This condition is equivalent to the
condition that is a function of t/lo which is a solution to the equation
In general a canonical dynamical system without constraint (To, cvo, ho) is a
reduction of the totally constrained system (r, cv, h) if there exists a diffeomorphic
embedding
which satisfies the following two conditions:
<J>*(at+ Yo)=kY (kE :I(r)),
ii) <J>*w(Z1, Z2)=wo(Z1, Z2) Z1, Z2E.X({t} X To),
where Yo is the infinitesimal canonical transformation on To generated by ho. A
convenient characterization of (}) is given by the following wellknown result.
Proposition 3.1. The necessary and sufficient condition for the mapping (}) to give a
reduction is that the following equation holds:
1) Necessity
From condition ii) for the reduction mapping there exists a 1form ~ such that
<J>*cv=cvo~1\dt. Since cv and Wo are both closed, we obtain d~l\dt=O. From this
it follows that we can choose~ so that it is closed. Hence <J>*cv can be written as <J>*cv
= cvo dpl\ dt in terms of some function p. Applying Ia,+Yo on this expression leads to
This is equivalent to condition i).
which are related by
t=<J>*r/J, <J>*at=V,
The function rp, when extended off the constraint surface ~H, yields a time variable on
(})* cv = Wo dho 1\ dt .
The reduction mapping (}) induces a function r/J and a vector field V on ~H defined
On the other hand from condition i)
Ia.+Yo(})*cv=d(p ho)(dtP+ Yop)dt.
2) Sufficiency
If the equation in the proposition holds, condition ii) is obvious. Further it is
easy to check that(})*J.~.(at+Yo)cv=la,+Yo(})*cv=O. Hence there is a function k on r such
that
r, and foliates r into a family of acausal hypersurfaces .l't={uEFI(u)=t}. On the
order hand the vector field V, when extended off .l'H, generates a oneparameter
family of transformations v, on r such that v,(.l't)=.l't+r, and :F(I'o) can be identified
with the restriction to .l'H of the set of functions on r that are invariant by these
transformations. Hence the reduced phase space I'o can be naturally identified with
.l'o n.l'H. Under this identification the reduction mapping fP can be written as
Hence, when an acausal hypersurface .l'o is given, the vector field V completely
determines the reduction. For this reason we call V the reduction field.
The reduction field yields a reference dynamics in describing the dynamics of the
totally constrained system as time evolution, and the hamiltonian in the reduced
system is essentially the generator of the deviation of the hamiltonian flow from this
reference dynamics. To see this, let Ot be the mapping from .l'o to .Et determined by
the hamiltonian flow. Then 1Jt=IJt8t yields a family of transformations on I'o=.l'o
n .I;H deSCribing the deviation Of the hamiltonian flOW from the reference dynamics.
For a given point uEF the tangent vector z~,<ult) of the curve 7Jt(u) is given by
where k is a function determined by the condition k Y ~ 1. Hence from the condition
i) for the reduction mapping we see that Z(t) is nothing but the infinitesimal canonical
transformation Yo generated by ho.
An arbitrary vector cannot be a reduction field. It must be approximately a
canonical transformation as the following proposition shows.
Proposition 3.2. A vector field V on r is a reduction field if and only if it satisfies
the following conditions:
ii) there exists a function on r such that V~1 and vw/\d1xH=O.
In particular any infinitesimal canonical transformation satisfying i) is a reduction
field. For this case the reduced hamiltonian ho can be chosen to be independent of t.
1) Necessity
From the definition of the reduction field i) is obvious, and for the time variable
in the reduction the former half of condition ii) is satisfied by definition. Further
since fP*w=wodho/\dt from Proposition 3.1,
fP*( dfy(JJ 1\ d) = dla, rp* (1) 1\ dt = 0 .
This is equivalent to the latter half of ii).
2) Sufficiency By taking the function in condition ii) as a time variable, let us construct the
mapping ~:R X 10+ In from and V exactly in the same way as described below
Eq. (39). Then it obviously follows that ~*at= V and ~*=t. Hence from ii) we
obtain O=~*(dlvwi\d)=d(la,~*w)/\dt. This implies that there exist ho, aE':E(R
X10) such that Ia,~*w=dho+adt. Since lx2 =0, a is expressed as a=iJoho. Hence
~*w is written in terms of 2form wo such that la,wo=O as
~* (l) = Wo dho 1\ dt .
Since w is closed, wo is also closed. Further from
a,wo/\ dt =a,~*w 1\ dt = ~*( vW 1\ d)=0,
we obtain a,wo=O. Hence from Proposition 3.1 ~ yields a reduction mapping.
Though the reduction fields are rather restricted, we can always find a reduction
~ for which ho coincides with an arbitrarily given function on R X 10. To see this, let
be a function on r such that Y={h, }=1=0 on In, and put V=Xp+aX, where p
and a are functions to be determined so that V is a reduction field. First from
condition i) in Proposition 3.2 a is uniquely determined as a= {h, p}/{h, }. Since
vw= da 1\ d, condition ii) simply reduces to V= X,p~ 1. As X, is transversal
to In, we can always find a solution p to this equation such that P coincides with an
arbitrarily given function on In. Since ho= ~*PIIH and Vp~ Yp/Y, this implies that
ho can be an arbitrary function on R X To.
This observation shows that reduction and the corresponding reduced
hamiltonian have a physical significance only when the original system has some kind
of timetranslation symmetry which induces a reduction field in the phase space. In
generic cases for which no such symmetry exists it is more natural to discuss the
dynamics in the original phase space with a constraint in the way described in the
previous section.
3.3. Example: a relativistic particle in curved background
We illustrate the argument so far by a simple totally constrained system describ
ing a relativistic particle moving on a curved background (M, g) with a mass which
may be positiondependent.
The action for this system is given by
This action is equivalent to
By eliminating v~' in St with the help of this equation we get
where N is regarded as an independent variable. Hence the original system is
equivalent to the totally constrained canonical system
<r. w, h): r= T*M,
where T* M is the cotangent bundle of M and 8 is its canonical 1form.
This totally constrained system is not simply reducible as the hamiltonian con
straint is quadratic in the momentum unlike the system considered in 2. However,
one can discuss its dynamics in the sense discussed in this section except for special
cases. In fact, the generating vector Y of the hamiltonian flow of this system is given
by
y_ p,. axap z1( aagx''"" p,p<J+ aaxu'" ) apa,. '
which vanishes at points where p,.=O and au;ax'"=O. Hence if there is no point such
that U =0 and aU/ax~'=O, the hamiltonian flow has global acausal hypersurfaces.
As shown in the previous section, we can always find a reduction of this system
into an unconstrained system. However, in order for the corresponding reduction
field to be associated with some time translation of the system, the system must have
a symmetry. To see this, let K be a vector field on M. Then it induces a vector field
K on the phase space T* M which is expressed in the local coordinate system (x'", Pu)
as
It is easy to see that this is an infinitesimal canonical transformation generated by the
function p,.K'". The condition for this field to be a reduction field is given by the
following proposition.
Proposition 3.3. The vector field K yields a reduction field of a totally constrained
system (T* M, w, h) if and only if Kh ~ 0. In particular for the constraint h given by
Eq. (3 16) this is equivalent to the condition that K is a Killing vector of the metric
g=Ug.
Since K is an infinitesimal canonical transformation, the former half of the
proposition is obvious from Proposition 3.2. For h given by Eq. (316) the condition
is expressed as
o~ K( u1h)= tfp( fJuaKa) ff~'' ffu"p,p" ,
{7,.,( ifvaKa) + Pv( ifpaKa) = 0 ,
where c7,., is a covariant derivative with respect to if. This equation is equivalent to
which implies that K is a Killing vector of the metric if.
This restriction on the system is closely related with the condition for the system
to have a good time variable which is independent of p,.,. To see this, let r/J be a
function on M. Then, since Yr/J is now given by
the condition for r/J to be a good time variable, Eq. (3 5), is written as
which is equivalent to
p~'pv{7,.,{7vr/J ~ {7~'r/J{l,.,U~o,
UP,.,Pvr/J= ~ g,.,vP).r/J{l;.U.
This last equation is written in terms of the covariant derivative with respect to the
metric ifw= Ug,.,v as
Hence the metric if,.,v must have a Killing vector K such that
In particular if,.,v must be static.
Putting these two requirements together, we find that there should exist a function
r/J such that {7~'r/J is a Killing vector of if and 17,..r/J{l~'r/J=const in order that there is a
good time variable which is a function on M and whose gradient field generates a
reduction field. It is easy to see that those conditions are satisfied if and only if g has
a static Killing vector along which U is constant (cf. Kuchai''s article in Ref. 2)). For
such cases the reduced hamiltonian is given by p,.,/7~'r/J.
For example for a relativistic free particle in Minkowski spacetime for which U
=m2 =const, the translation Killing vector a~'iJ,., satisfies these conditions when a~' is
a constant timelike vector. The time function and the reduced hamiltonian are
given by r/J=a,..x~' and ho=a~'p,.,. On the other hand the boost and rotation Killing
vectors do not correspond to good time variables on M because they are not gradient
vectors.
4. General relativity as a totally constrained system
Before extending the argument on the totally constrained system with a single
constraint to a more generic case, we give an overview on the structure of the totally
constrained system obtained from general relativity.6l The main purpose is to clarify
that the classical dynamics of general relativity as a totally constrained system is
nothing but a foliation of the constraint submanifold such that each leaf is onetoone
correspondence with a 4dimensional diffeomorphism class of solutions to the Einstein
equations. This fact will be used to establish an interpretation of generic systems in
the next section.
4.1. ADM canonical formulation on the 3 metric space
For simplicity we only consider globally hyperbolic vacuum spacetime (M, g),
and assume that it is spatially compact. Hence M is diffeomorphic to R X S where S
is a compact space. Let q(t) be the induced 3metric on {t}x S, K(t) the extrinsic
curvature tensor of {t} x S, and n(t)=(l/N)(ot v) the unit normal to {t} x S where v
EX({t} x S). Then by regarding q(t), K(t), N(t) and v(t) as quantities on S with the
time parameter t, K(t) is written as
where n is the covariant derivative with respect to q. Further the EinsteinHilbert
action for the spacetime (M, g),
is expressed in terms of these quantities as
By introducing the momentum variable tJk conjugate to qik by
this action is put into the canonical form,
S= jdt[<p, ti>(<h.L, N>+<hn, v>)],
where h.L, hn and p are linear functionals of functions, vector fields and 2ndrank
covariant tensor fields on S, respectively, defined by
Let 'I /(S) be the set of all smooth (p, q)type tensor fields on S, and if be a
reference Riemannian metric on S. Then by taking the completion with respect to
the inner product
..5lf(S):={qEHR2(r:Iz0(S))jq is positive definite on S},
we obtain the Sobolev space HRn( r:I /(S)) which is a real Hilbert space. In particular,
if we define the space of 3metrics on S by
..5lf(S) becomes an open subset of HR 2( r:I z0(S)). Hence it has a natural Hilbert
manifold structure, and its tangent space and cotangent space are both isomorphic to
the original Hilbert space:
<hD, X>=O '\1 XEX(S),
<hj_, J>=O '\1 jE :F(S),
H=<hj_, N>+<hD, y).
where the total phase space r is given by the cotangent bundle T*..5lf(S), the
symplectic form formally by w=<Bpl\ Bq>, and the hamiltonian H by
Though we can give an exact expression for w by introducing the basis of HR2( r:I z0
(S)), we will not do it because the argument in this section is formal. Since the
hamiltonian is written as a linear combination of the constraint functionals, this
canonical system is a totally constrained system.
4.2. Dynamical foliation of the Phase space and Di.ffo(M)classes
Let l:D and l:H be the submanifolds of r defined by
Further, since the operator L defined by
(a, (3)=<a, L/J>,
where /Jik= ijilijkm/3tm, is given by L=~~=o( 1)n.Jn and elliptical, it defines an injec
tion from Tq*(..5lf(S)) into the space of linear functionals on HR0(r:Iz0(S)). Under the
identification by this mapping the momentum p can be regarded as an element of
Tq*(..5lf(S)).
From this observation the action Eq. (4 5) defines a canonical system (r, w, H)
with an infinite number of constraints:
J:D:={uEFj<hD, X>(u)=0'\1 XEX(S)},
};H:={uEFj<hj_, J>(u)=0'\1 jE :F(S)}.
{<hD, X1>, <hD, X2>}=<hD, [XI, Xz]>,
{<hD, X>, <hj_, J>}=<hj_, xf>'
{<hj_, fi), <hj_, fz)}=<hD, f1Dfz fzDf!),
Then from the Poisson bracket structure among the constraints,
the infinitesimal canonical transformation XH generated by His tangential both to l:D
and l:H, and each integration curve of XH on J:Dnl:H yields a solution to the Einstein
equations. However, this correspondence is not onetoone because the same
spacetime allows an infinite number of different slicings with the same N and v.
Further a different choice of the lapse function N and the shift vector v yields a
different curve in r for the same spacetime.
This ill correspondence between the spacetime solutions to the Einstein equations
and the curves in the phase space, which arises due to the general covariance of
general relativity, can be made welldefined by considering the subspace spanned by
the integration curves instead of each curve. To see this, let us denote the set of
constraints symbolically by ha and the corresponding infinitesimal canonical transfor
mations by Ya=Xh.. Then from the first class nature of ha shown above, {Ya} yields
an inVOlUtiVe System On };D n };H:
where cir is a set of functions on r. Hence we obtain a foliation of the constraint
submanifold l:D n l:H = U .~C.~ where each leaf C.~ is a connected component of the
integration submanifolds.
For an arbitrary nondegenerate curve r contained in a leaf C.~, its tangent vector
X is written in terms of some set of functions Na as X= ":E. aNa Ya because {Ya} spans
the tangent space of C at any point. Hence it is an integration curve of the
hamiltonian flow for the hamiltonian H =":E. aNaha, and corresponds to some spacetime
solution (M, g) to the Einstein equations. Further, if two curves /'I and 1'2 are
contained in the same leaf and intersect with each other at a point u, they correspond
to solutions to the Einstein equations with the same initial data for different lapse
functions and shift vectors. Hence from the uniqueness of the initial value problem
for the Einstein equations the corresponding spacetime solutions (M, g1) and (M, g2)
are isometric if they are maximally extended. The same conclusion holds even if
these two curves do not intersect. For one can find another curve /'3 in the same leaf
which intersects both with /'I and /'2, which implies that (M, g3)~(M, gt) and (M, g3)
~(M, !Jz), hence (M, g~)~(M, g2). Therefore all the curves contained in the same leaf
corresponds to a unique 4dimensional diffeomorphism class of the spacetime solu
tions to the Einstein equations.
We can further show that this correspondence is oneto one. Take two curves /'I
c C1 and rzC C2 and suppose that the corresponding spacetime solutions (M, gt) and
(M, gz) are isometric. Then there exist isometric diffeomorphisms to a spacetime (M,
g), (])1: (M,gt)>(M, g) and (/)2: (M,g2)>(M, g). Let St=(])I({ti}xS) and S2=(/)2({t2}
x S) be two spacelike constanttime hypersurfaces in M, and choose two families of
time slicings in M such that they contain a common time slice, one of them cont<}ins
5 1 and the other S2. Further let the two curves in the phase space determined by
these slicings be /'3 and /'4 Then from the construction /'3 n 1'1 =t= 0, /'4 n 1'2 =t= 0 and /'3
n /'4=t=0. This implies that there is a curve which connects a point in C1 and a point C2.
Hence from the connectedness of each leaf C1 and C2 must coincide with each other.
Thus we have found that the connected components of the integration manifolds
of the involutive system {Ya} are in onetoone correspondence with the 4dimensional
diffeomorphism classes of the spacetime solutions to the Einstein equations. In other
words the classical dynamics of general relativity is completely determined by the
foliation of the constraint submanifold in terms of the infinitesimal canonical transfor
mations generated by the constraints. We can discard the lapse function and the shift
vector, or the corresponding hamiltonian. For this reason we can simply say that the
canonical theory of general relativity is given by a totally constrained system ( T*.5li
(S), w, {hv, h..L}). We will call each leaf of the foliation a causal submanifold.
4.3. Elimination of the diffeomorphism constraints
As the structure constants of the Poisson brackets among hv are genuinely
constant from Eq. (4 18), the corresponding infinitesimal canonical transformations
are involutive on the whole phase space and generate the action of the diffeomorphism
group of S, Diffo(S) where the subscript 0 denotes the connected component containing
the unit element. Since all the measurable quantities are invariant under these
transformations, it is desirable to eliminate this kinematical gauge symmetry from the
canonical theory, especially when one considers the quantization of the theory. Now
we will show that the classical dynamics has the same structure as above even after
the elimination of this gauge freedom.
First of all note that for an q(S)valued functional N on r which transforms
covariantly under Diffo(S) as
N(a*u)=a*(N(u))
<h..L, N> is invariant under Diffo(S) as a functional on r from
Similarly for a .X(S)valued functional i7 on r which is Diffo(S)covariant, <hv, i7) is
invariant under Diffo(S). Further by inspecting the argument on the correspondence
between a curve in a causal submanifold and the hamiltonian flow generated by the
hamiltonian H=<h..L, N>+<hv, v> one easily sees that Nand v can be replaced by
some appropriate Diffo(S)covariant functionals N and i7. Hence the connected
integration surfaces of the involutive system generated by the Diffo(S)invariant
functionals (h..L, N> and <hv, i7) give the same foliation as that given by <hv, v> and
<h..L,N>.
Further if <hv, ,;)=frO at a point uEF for some ~E.X(S), there exists a functional
l: F+.X(S) such that (hv, l>=frO at the same point u. Thus I:v can be redefined as
I:v={uEFI<hv, i/)(u)=OV i7: F+.X(S); Diffo(S)covariant}.
Similarly In can be expressed as
In={uErl<h..L, N>(u)=OV N: r q(s); Diffo(S)covariant}.
These arguments indicate that the original canonical system can be naturally
projected on r/Diffo(S). To confirm this, let us denote all the functions on r which
are invariant under Diffo(S) by qtnv:
Then it is easily shown that qtnv is closed with respect to the Poisson algebra and
qtnvq o=q D where
Further since{!, <ho, ~>}=0 implies that Xf is tangential to Io, {!, g}= Xfg vanishes
on Iv for /Eqtnv and gEq o. Hence {qtnv, q v}=q v. This implies that q v forms
an ideal of qtnv and the Poisson bracket in qtnv naturally induces a Poisson bracket
in Jltnv:=qtnv/ q v. Each element of Jltnv is just a set of functions in qtnv which
coincide with each other on Iv.
Let 1r: F> F/Diffo(S) be the natural projection and put nnv:=;r(Io). Then from
the arguments above nnv is characterized as
nnv={uEF/Diffo(S)i<hv, J7)(u)=OV J7: F>X(S); Diffo(S)covariant},
(4. 28)
and Jltnv is naturally identified with q(nnv). Further the constraint hv is trivialized
on nnv and the causal submanifolds in Inn Iv is bijectively mapped to the causal
submanifolds in ;r(.En) n nnv generated by <h.l_,
N>l.~:oEnnv
Let haEJltnv be a generating set of all the functions of the form <h.l, N>b such
For any Diffo(S)covariant functional N: F> q(S) there exists a set of
elements tlaEJltnv such that <h.l, N>="i:.atlaha,
ii) ;r(.En) n nnv={uEnnvlha(u)=OV a},
iii) {ha, hp} = "i:.rC~phr.
Further let us denote ;r(In) n nnv by the same symbol In. Then the arguments so far
show that the canonical dynamics of general relativity is described by the totally
constrained system (nnv, Wtnv, {ha}) and the causal submanifolds in In is onetoone
correspondence with the Diffo(M)class of the spacetime solutions to the Einstein
equations.
4.4. Cotangentbundle structure of nnv
In the last statement of the previous subsection Wtnv is understood to be the
symplectic form corresponding to the Poisson brackets in Jltnv. Hence in order to
make the statement rigorous it should be shown that nnv has a manifold structure and
the required symplectic form exists. Now we prove these facts by showing that nnv
can be identified with T*(.511(S)/Diffo(S)) and Wtnv coincides with the canonical
symplectic form corresponding to the cotangent bundle.
Let r/JI: .Ev>.511(S) be the restriction of the natural projection from T*(.511(S)) to
.511(S). Then, since the diffeomorphism constraint implies that P vanishes on the
subspace of T*(.511(S)) spanned by tangent vectors to the Diffo(S)orbits from
Eq. (47), r/Jt is surjective and induces a surjective mapping r/J2: nnv>.511(S)/Diffo(S)
such that Jrr/h = r/J2Jr (see the diagram in Fig. 1). Let TJqE Tq(.511(S)) be a vector
tangent to a Diffo(S)orbit passing through qE.51i(S). Then, since it is written in
Further for aEDiffo(S) and vE Tq(5'rt(S))
a*(p, q)(a*v)=<a*p, a*v>=<P. v>=(p, q)(v),
from the diffeomorphism invariance of <P. v>. Hence there is an injection j: nnv
+ T*(5'rt(S)/Diffo(S)) such that for vE Tq(5'rt(S))
It is easily checked that for the natural projection 3: T*(5'rt(S)/Diffo(S))+5'rt(S)
/Diffo(S), z= >3j holds and that j is surjective. Thus nnv can be identified with
T*(5'rt(S) /Diffo(S)).
Next let us show that the symplectic form OJJnv induced from the cotangent bundle
structure of nnv is equivalent to the Poisson brackets in ..Jltnv~ :F(nnv) derived from
the symplectic form w in r. From now on we identify nnv with T*(5'rt(S)/Diffo(S))
and write j;r simply as 7r.
We first show that each element of ..Jltnv uniquely determines a vector field on
T*(5'rt(S)/Diffo(S)), which will turn out to be an infinitesimal canonical transforma
tion on nnv. Let Xtnv be
and for [/]E..Jltnv, let X(fl be
Xtnv={XEX(I.:v)la*X=X\7' aEDiffo(S)},
.X(fJ={XE.Xtnvlw(X, Z)= Zj\7' ZEXtnv}.
Then for each XE.Xtnv 7r*X obviously defines a unique vector field on T*(5'rt(S)/Diffo
(S)). Further let e and Btnv be the canonical 1forms on T*(5'rt(S)) and T*(5'rt(S)
/Diffo(S)), respectively. Then from the commutativity of the diagram in Fig. 1 and
Eq. (431) it follows that (8tnv)"<M>(7r*X)=8<M>(X) for any XE.Xtnv, i.e.,
8(X) = 7r* Btnv(X) \f X E .Xtnv .
Hence for LlX=XXz(XI, XzEXtfJ) and ZE.Xtnv, from
it follows that
As ;r*Z can be any vector field on T*(...'M(S)/Diffo(S)), this equation implies that
;r*L1X=O, i.e., ;r*X1=;r*X2. Hence [/]EJllnv determines a unique vector field on
T*(...'M(S)/Diffo(S)). We denote this vector field by XrfJ.
Next we show that for XEXrfJ and YEXrgJ, w(X, Y)= {!, g}. For X1, X2
EXff1 and Yi, Y2EXrgJ from the definition (433) it follows that
w(X2, 1';)w(XI, YI)=w(X2XI, J';)+w(XI, ; Yi)=O.
{!, g}=w(Xf, Xg)=w(X, Y).
With the help of the equations derived so far for XEXrfJ and YEXrgJ we obtain
[{!, g}]= [w(X, Y)]= [dB(X, Y)]=[ Y(B(X)) Y(B(X)) B([X, Y])]
=;r* Y(Binv(;r*X));r* Y(Binv(;r*X)) fAnv(;r*[X, Y])
This shows that the Poisson brackets induced from w coincide with that defined by
Note here that the arguments so far are not mathematically rigorous because ...'M
(S)/Diffo(S) has conical singularities at metrics with Killing vectors.6> Though these
singularities may have a physical importance in quantization, we will not go into this
problem in this paper.
We can go further and eliminate all the hamiltonian constraints to get the fully
reduced phase space with a symplectic structure which represents the true physical
degrees of freedom as done by Fischer and Marsden. However, we shall not follow
this line because we will then lose the dynamics.
5. General totally constrained systems
Now we discuss the dynamics of a generic totally constrained system. Here a
totally constrained system is defined as a triplet of a phase space, a symplectic form
and a set of constraint functions, (r, w, {ha}). For a technical reason we assume that
the phase space is 2ndimensional smooth manifold with finite n. Further we assume
that the constraints are of first class with the Poisson brackets given by
where dfl are functions on r.
On the basis of the arguments in the previous section we understand that the
physical evolution of the system is onetoone correspondence with each leaf of the
foliation determined by the involutive system of the infinitesimal canonical transfor
mations Ya=Xha on the constraint submanifold .EH={uEFiha(u)=0\7' a}. We call
each leaf a causal submanifold as so far. As is clear from the arguments in the
previous section, this interpretation is equivalent to regard that two solutions to the
canonical equation of motion for the hamiltonian H = ~atlaha with arbitrary functions
tla represent the same physical evolution if they intersect with each other in r.
This is a natural generalization of the argument on the dynamics of a single
totally constrained system with one constraint in 3. Now we extend this generaliza
tion to the statistical dynamics of an ensemble.
5.1. Relative distribution function
From this interpretation of dynamics of a single system and the argument in 3
it is natural to introduce the relative distribution function p on r to describe an
ensemble, which vanishes outside the constraint submanifold and is constant on each
causal submanifold:
Let us define an acausal submanifold as a submanifold of r which intersects with
causal submanifolds transversally. Then for any acausal submanifold I and for any
distribution Pl.~: on I a solution to these equations which coincides with Pl.~: on I is
unique, if it exists, on the causal development of I defined by
where C runs over causal submanifolds. However, such a solution may not exist in
general. In fact the following theorem holds.
THEOREM 5.1. In order that there exists a solution to Eqs. (5 2)(5 3) for arbitrary
initial data on any acausal submanifold, the following condition should be satisfied:
(haP, c/J):=(p, hac/J)=O,
<YaP, <t>>==<p,  Ya<I>>=O,
This condition is satisfied if and only if there exists a function f * 0 such that for c~~ I
corresponding to the constraints h~=!ha
Since pis a distribution, to be exact, Eqs. (52) and (53) are expressed as
where J is an arbitrary smooth function with a compact support on r.
since the commutators among the Ya's are given by
the consistency condition yields
Hence, noting the relation Yrd.s= {hr, c:.s}, we obtain the first condition in the
theorem.
In order to show the latter half of the theorem, first note that the Jacobi identity
for the Poisson brackets among ha yields
Yac.s Y.sca+ ~rd.s~~ {rhr, d.s},
c~=ca(m1) Yaln/,
y aim!~ calm! .
where ca== ~.sc~.s From this it immediately follows that the first condition of the
theorem holds if Ca ~ 0.
On the other hand for h~= !ha, Ca changes as
where m is the number of the constraints. Hence the second condition of the theorem
is satisfied if I is a solution to the equation
However, if the first condition of the theorem is satisfied, we obtain
This is nothing but the consistency condition for the firstorder differential equation
system for r  1 above. Hence the first condition of the theorem implies the second.
Note that for a matrix function A=(Al) on r with det A=FO, the totally con
strained system with the constraints h~=~.sAlh.s is equivalent to the original system.
Hence the precise meaning of the requirement of the theorem is that ~r{hr, d.s} can
be put to zero by such a transformation and that Eqs. (52)~(53) are consistent only
for such choice of the constraints. This result is interesting in relation to the
quantization of the totally constrained system because this condition implies that the
operators corresponding to d.s and ha should commute in a weak sense.
On the basis of this theorem we assume that ca=O from now on. Under this
condition if we put
5.2. Statistical dynamics in terms of conservative measures on acausal submanifolds
where D(SI) and D(S2) are causal developments of S1 and S2, respectively. We
extend this causal mapping to a neighborhood of SH by considering a foliation of the
tubular neighborhood such that the intersection of each leaf with SH coincides with
the foliation of SH by Ya. If a measure f.i<:o on an acausal submanifold So with its
support contained in Son SH is given, this causal mapping uniquely determines a
measure f.i<: on S n D(So) with its support contained in SH such that for any constant
of motion, i.e., a function fE ~(r) which is constant along each leaf,
df.iE=pllv,lvmQIE
dofx+fxod=x,
[x, lv]=l!x,Y(,
In a similar way we obtain
( l)rl[y,dfvrfvm(pQ)
=( 1)7/y, fv,dlvr+I" ..(PQ)
if suppfnSoCdomO. As in the single constraint systems, this measure can be
expressed locally in terms of p, Ya and Q.
is conserved by causal mappings where m is the number of independent constraints.
Let us denote lv,lvm(pQ) simply as x. Then from the identities
Hence for a pair of acausal submanifolds l:1 and l:2 such that D(I:1)=D(I:2), from
Stokes' theorem on (2n m+ I)dimensional submanifold N such that oN=I:1 Ul:2 UI:'
and I:' is parallel to the leaves, we obtain
J(iz lxl J(x1 lxl= J(aNx+ J(x, x=)N(dx=O .
In realistic situations each acausal submanifold is specified by a set of m indepen
dent functions rl>a such that Yar/>p= {rpp, ha} is a regular matrix as, say,
Let us call such a set of functions instant functions. Then the measure given in the
previous theorem is expressed in terms of these instant functions as follows:
THEOREM 5.3. If r/>1, ... , r/>m are instant functions for an acausal submanifold 2:, for
any /E :F(r) the following equality holds:
((ml)1m)! at.._~,m a,.am(IYt d A'P.at )d'A<.pa2 1\ .. I\ dA<.pam 1\ I Y2.. IYm.Q
((ml)1m)! a,..~_a,m a,.am{h 1, 'AP. at}d A'P.a2 1\ ... /\ dA<.pam 1\ I Y2 .. IYmQ
=( l)m(m+1)/2 ~ at.. am{h1, r/>aJ .. {hm, r/>am}.Q,
a1az
By multiplying fp on the both sides of this equation and integrating over r, we obtain
the equation in the theorem.
From these theorems we can formulate the statistical dynamics of an ensemble of
the totally constrained system with multiple constraints in the following way. First,
from the data set obtained by measurements, pick up a set of instant functions (l>a
which take a common set of values in the data set. For simplicity assume that these
values are all zero, and let the corresponding acausal submanifold in r be 1:,, and
define the measure dv by
where c is some fixed positive constant of motion. Then the other data uniquely
determines the distribution p on 1:, through the formula
Extend this distribution p over D(l:,) by the evolution equations Eqs. (5 2) and (5 3).
Then for another set of instant functions <2)a corresponding to an acausal sub
manifold l:2, the expectation value of a function /E q(r) on that submanifold is given
by
if supp/ED(l:,). Of course we do not need the explicit knowledge on the acausal
submanifolds, because from Theorem 53 the expectation values are written as an
integration over r in terms of measures expressed by the constraints and the instant
functions.
Like the case of a single constraint system we can define a set of functions a to
be good time variables if the natural measure II ao( a ra)IQI on a set of acausal
submanifolds
coincides with the conserved measure. This condition is expressed as
cldet{ha, .s}l ~ /( ,, , m) ,
where c is some positive constant of motion and I is some function of m variables.
6. Discussion
In this paper we have shown that the dynamics of a classical totally constrained
system can be consistently formulated without reducing it to an unconstrained system
by solving the constraints or referring to a special time variable. The basic idea has
been to consider the relative distribution function which is constant on each leaf of the
foliation defined by the infinitesimal canonical transformations generated by the
constraint functions, and to normalize it on an acausal submanifold which is transver
sal to the foliation in terms of the conservative measure.
The fact that we can formulate the classical statistical dynamics of a totally
constrained system without referring to a special time variable is very important for
considering a quantum theory of the totally constrained system because a quantum
theory has a similar structure to the classical statistical dynamics in general. In fact
in a subsequent paper we will show that by introducing a similar foliation structure
into a state space of quantum theory and by considering a relative probability
amplitude instead of the relative distribution we can construct a consistent formula
tion of the quantum dynamics of a totally constrained system without referring to a
special time variable under some restrictions.
Though the main purpose of the present paper has been to give a basic motivation
for the quantum formulation developed in the subsequent papers, the results obtained
in the paper may be interesting by themselves. In particular the fact that the
conservative measure can be written only by the canonical volume form and the
constraint functions even for multiconstrained systems seems to be useful in the
arguments of the probability distribution of the initial condition of the universe in the
classical framework and stochastic treatment of general relativity.
Of course the expression for the measure given in this paper cannot be applied to
general relativity directly because we have only considered systems with finite
degrees of freedom. However, it seems possible to extend the formulation to general
relativity by taking an appropriate limit. To examine this limiting procedure
explicitly in some simple situations such as the perturbation theory of general relativ
ity on cosmological background spacetimes and spherical black hole spacetimes with
scalar fields will be interesting.
This work started when the author participated in the program "Geometry and
Gravity" in Newton Institute for Mathematical Science. He would like to thank the
participants of the program for valuable discussions and the staff of the institute for
their hospitality. This work was supported by the GrantinAid for Scientific
Research of the Ministry of Education, Science and Culture of Japan (05640340).