Coordinate-free Solutions for Cosmological Superspace

11 Pages·0.16 MB·English

by  D. S. Salopek

COORDINATE-FREE SOLUTIONS FOR COSMOLOGICAL SUPERSPACE
D. S. Salopek
Department of Physics and Astronomy, University of British Columbia, Vancouver, Canada V6T 1Z1
(March 5, 1997)
Hamilton-Jacobi theory for general relativity provides an elegant covariant formulation of the
gravitationalfield. Ageneral‘coordinate-free’methodofintegratingthefunctionalHamilton-Jacobi
equation for gravity and matter is described. This series approximation method represents a large
generalizationofthespatialgradientexpansionthathadbeenemployedearlier. Additionalsolutions
may be constructed using a nonlinear superposition principle. This formalism may be applied to
problems in cosmology.
8 I. INTRODUCTION
9
9
General relativity was formulated to describe the gravitational field in a manner which was independent of any
1
particular choice of reference frame. It is invariant under ‘general coordinate transformations’. However, when one

n
actually solves the Einstein field equations, say in a cosmological setting, one typically invokes a particular choice of
a
coordinates. Asanimprovementtothissituation,onewouldprefertoutilize a‘coordinate-free’methodofcomputing
J
solutions to Einstein gravity. In fact, Hamilton-Jacobi (HJ) theory for general relativity provides such a description
6
of the gravitationalfield [1], [2], [3]. In solving the HJ equation, one need not specify the choice of time hypersurface
1
nor the spatial coordinates.

1 Coordinate-free descriptions have provenuseful in many fields of theoreticalphysics. Vector analysis for Euclidean
v space is a good example. Although one normally associates coordinates with vectors, it is possible to interpret a
1 vector geometrically in a coordinate free way using a magnitude and a direction. In addition, by using his bra and
6
ket notation, Dirac [4] was able to formulate quantum mechanics in a form which was independent of the choice of
1
basis states. The bra could be interpreted as an abstract vector without referring to a particular basis.
1
It has been known since the early 1960’s that the HJ equation for general relativity does not refer to the lapse
0
8 and shift parameters which characterize the gauge freedom of gravity [5]. Hence the HJ equation is the natural
9 startingpointforacoordinate-freeanalysisofgravity. However,theHJequationisanonlinear,functionaldifferential
/ equation which characterizes an ensemble of universes, superspace. It was generally believed that superspace was
h
too complicated to solve in its entirety. Beginning with Misner [6], researchers were content to solve homogeneous
p
- minisuperspace models where one considered only a finite number of degrees of freedom for the gravitational field.
o These investigations have been proven to be quite fruitful in quantum cosmology (see, e.g., Louko and Ruback [7]).
r
t (For a discussion of future trends in quantum cosmology, consult Hartle [8] as well as Barvinsky and Kamenshchik
s
[9].) However,there are some questions that these models cannot address since they do not include inhomogeneities.
a
: For such models, the choice of time hypersurface is degenerate: a hypersurface of uniform φ is the same as that of
v
uniformscalarfactora. Inordertoobtainadeeperunderstandingforthenatureoftimeinasemiclassicalsetting[10],
i
X [11],itisessentialto includethe roleofinhomogeneities. (However,there aremanyadditionalobstaclesinaddressing
the question of time in a quantum context [12], [13], [14].)
r
a It is somewhat surprising that one can actually obtain solutions to semiclassicalsuperspace using a series approxi-
mationmethod. TheprototypesolutionfortheHamilton-Jacobi(HJ)equationforgeneralrelativityutilizedaspatial
gradient approximation. It was developed by Salopek and Stewart [1] and advanced further by Parry, Salopek and
Stewart [2]. In effect, they demonstratedthat one can decompose superspace into a discrete sum of minisuperspaces.
Here,ageneralmethodwillbegivenforconstructing(HJ)solutionswhichutilizevariousotherseriesapproximations.
These techniques may be applied profitably to cosmology and other areas of astrophysics.
A Hamilton-Jacobi description of general relativity is attractive for severalother reasons:
(1) Primitive formulation of quantum theory for the gravitational field. In the semiclassical approximation, the
wavefunctionalisgivenbyaphasefactor,Ψ ei /h¯,wherePlanck’sconstanth¯ isassumedtobetiny. Thegenerating
S
∼
functional then satisfies the HJ equation. If is real, then one is describing classical phenomena. If is complex,
S S S
one may describe quantum phenomena such as tunnelling or the initial wavefunction of the universe. It is generally
believed that the fluctuations for structure formation as well as microwave background fluctuations are generated
during an inflationary epoch. For such models, it is absolutely essential to quantize the gravitationalfield, including
both scalar and tensor modes (see, for example, [21], [10]). The current status of inflation after the detection of the
cosmic microwave backgroundanisotropy is discussed in refs. [15], [16].
(2) Systematic solution of constraint equations. In superspace, the momentum constraint may be given a simple
geometric interpretation and it is easy to solve. The energy constraint fully characterizes the dynamics of the gravi-
1
tational system, and it is difficult to integrate directly. As will be demonstratedin the present work,it will be solved
using a series approximation.
(3)Nonlinearanalysisofgravity. Inthequasi-nonlinearregime,onemayemployHJtheorytoanalyzetheformation
of cosmologicalpancakes [17] during the matter-dominated epoch of the Universe [18], [19].
Spatial gradient expansions in general relativity have a long history [20]. Within a HJ context, a spatial gradient
expansion has been applied successfully to cosmology including a detailed computation of microwave background
fluctuationsandgalaxy-galaxycorrelationsfunctionarisingfrominflation[21],[22]. Sodaetal[23]havegeneralizedthe
expansionto encompassBrans-Dickegravity,andChiba[24]hasformulatedthegradientexpansionforn-dimensional
Einstein gravity. The HJ equation for long-wavelength fields has often been invoked in an attempt to recover the
inflation potential from cosmologicalobservations [25], [26], [27].
InSec. II,theHJequationforascalarfieldinteractingwithgravityispresentedalongwiththeanalogousequation
for a dust field interacting with gravity. I quickly review the spatial gradient expansion of the generating functional
in a way which is easy to generalize to other situations. In Sec. III, the HJ equation including a scalar field is solved
usingaTaylorseriesinthescalarfieldφ. InSec. IV,theanalogousmethodisappliedforadustfield,whichdescribes
collisionless, pressureless matter. In Sec. V, the Superposition Principle for Hamilton-Jacobi theory allows one to
construct complicated solutions of the HJ from known solutions which depend on a parameter. Conclusions and a
summary follow in Sec. VI.
(Units are chosen so that c = 8πG = 8π/m2 = 1. The sign conventions of Misner, Thorne and Wheeler [28] will
P
be adopted throughout.)
II. HAMILTON-JACOBI EQUATION FOR GENERAL RELATIVITY WITH MATTER
Inthe presentwork,twosituationswillbe considered: (1)ascalarfieldinteractingwithgravityand(2)adustfield
interacting with gravity.
A. Scalar field interacting with gravity
The action for a single scalar field interacting with Einstein gravity is
1 1
= d4x√ g (4)R gµν∂ φ∂ φ V(φ) ; (2.1)
µ ν
I − 2 − 2 −
Z (cid:18) (cid:19)
(4)R is the Ricci scalar of the space-time metric g . For simplicity, the scalar field potential is assumed to be
µν
1
V(φ)= m2φ2+V (2.2)
0
2
which describes a massive scalar field with a cosmologicalconstant term. (In general the solution methods described
in this paper will work for all potentials which are regular at φ=0.)
In the ADM formalism the line element is written as
ds2 = N2+γijN N dt2+2N dtdxi+γ dxidxj , (2.3)
i j i ij
−
where N and N are the lapse and (cid:0)shift functions res(cid:1)pectively, and γ is the 3-metric. (For a elegant review, consult
i ij
D’Eath [29]). One can then rewrite the action in Hamiltonian form,
= d4x πφφ˙+πijγ˙ N Ni . (2.4)
ij i
I − H− H
Z (cid:16) (cid:17)
where and aretheenergyandmomentumdensities,respectively. IntheHamilton-Jacobiformalism,onereplaces
i
H H
the momenta by functional derivatives of the generating functional, [γ (x),φ(x)],
ab
S
δ δ
πij(x)= S , πφ(x)= S ; (2.5)
δγ (x) δφ(x)
ij
the generating functional associates a complex number for each field configuration φ(x) on a space-like hypersurface
whose geometry is described by the 3-metric γ (x). The energy and momentum constraints are now given by:
ij
2
δ δ
0= (x)=γ−1/2[2γik(x)γjl(x) γij(x)γkl(x)] S S +
H − δγ (x)δγ (x)
ij kl
2
1 δ 1 1
γ−1/2 S +γ1/2V[φ(x)] γ1/2R+ γ1/2γijφiφj, (2.6a)
2 δφ(x) − 2 2
(cid:18) (cid:19)
and
δ δ δ
0= (x)= 2 γ S + S γ + S φ . (2.6b)
i ik kl,i ,i
H − δγ (x) δγ (x) δφ(x)
(cid:18) kj (cid:19),j kl
The Hamilton-Jacobi equation (2.6a) governs the evolution of the generating functional, [γ (x),φ(x)], in
ab
S ≡ S
superspace. It is quadratic in the momenta of fields. The momentum constraint is linear in momenta. Higgs [5]
showedthatitlegislatesthatthegeneratingfunctionalisinvariantunderreparametrizationsofthespatialcoordinates
(‘spatial gauge-invariance’). It may be solved, for example, by assuming that the generating functional is an integral
of some function of the curvature, say d3xγ1/2f(R), or some other combination of spatial invariants.
R
B. Dust field interacting with gravity
The case of dust interacting with gravity is of high interest in cosmology. It is generally believed that most of
the Universe consists of dark-matter whose dynamics may be described by a dust field which is pressureless and
collisionless. Secondly,hypersurfacesofuniformχ ingeneralrelativitycomeclosestto describingLorentzframesthat
provedsousefulinflatspacetime. Infact,aLorentzframemaybeconsideredtobeaproperlysynchronizedcollection
of dust particles. It may be necessary to introduce dust to interpret a quantum theory of the gravitationalfield [31],
[13].
The action
1 1
= d4x√ g (4)R n(gµν∂ χ∂ χ+1) , (2.7)
µ ν
I − 2 − 2
Z (cid:20) (cid:21)
for a dust field, χ, interacting with gravity, is similar to that of a scalar field. The new ingredient is the rest number
density n n(t,x) which is a Lagrange multiplier that ensures that the 4-velocity
≡
Uµ = gµνχ (2.8)
,ν
−
satisfies UµU = 1. Hence χ may be interpreted as a velocity potential. In Hamiltonian form, the action is
µ
−
= d4x πχχ˙ +πijγ˙ N Ni . (2.9)
ij i
I − H− H
Z
(cid:0) (cid:1)
where πχ is the canonical momentum of the dust field. Replacing the momenta by functional derivatives of ,
S
δ δ
πij(x)= S , πχ(x)= S , (2.10)
δγ (x) δχ(x)
ij
the energy and momentum constraint equations become:
δ δ
0= (x)=γ−1/2 S S [2γil(x)γjk(x) γij(x)γkl(x)]
H δγ (x)δγ (x) −
ij kl
δ 1
+ 1+γijχ χ S γ1/2R, (2.11a)
,i ,j
δχ(x) − 2
q
δ δ δ
0= (x)= 2 γ S + S γ + S χ . (2.11b)
i ik kl,i ,i
H − δγ (x) δγ (x) δχ(x)
(cid:18) kj (cid:19),j kl
In contrast to the Hamiltonian for a scalar field (2.6a), the energy constraint (2.11a) is linear in the canonical
momentum of the dust field. In a quantum context, the Hamiltonian constraint for dust and gravity is thus very
similar to the Schrodinger equation (or its covariantgeneralization, the Tomonoga-Schwingerequation [30]) that has
been so successful in flat space-time.
3
C. Review of Spatial Gradient Expansion
The spatial gradient approximation for the HJ equation (2.6a) with a scalar field will be quickly reviewed. The
essential aspects will highlighted in order to illustrate what must be done in other situations.
One expands the generating functional
∞
[γ (x),φ(x)] = (2n) (spatialgradientexpansion) (2.12)
ab
S S
n=0
X
in a series of terms according to the number of spatial gradient terms that they contain. It is quite important that
each term in the series expansion satisfy the momentum constraint,
δ (2n) δ (2n) δ (2n)
0= (2n)(x) 2 γ S + S γ + S φ . (2.13)
Hi ≡− ikδγ (x) δγ (x) kl,i δφ(x) ,i
(cid:18) kj (cid:19),j kl
The zeroth order term
(0) = 2 d3xγ1/2H(φ) , (2.14)
S −
Z
is the simplest such term that one can imagine; it contains no spatial gradients where the function H(φ) satisfies
2
2 ∂H 1
H2 = + V(φ) . (2.15)
3 ∂φ 3
(cid:18) (cid:19)
The volume element d3xγ1/2 appearing in eq.(2.14) is obviously invariant under spatial coordinate transformations.
The second order term contains two spatialgradients and is an integralover the 3-curvature and a term quadratic in
spatial derivatives of φ:
(2) = γ1/2d3x J(φ)R+K(φ)γabφ φ ; (2.16)
,a ,b
S
Z
(cid:0) (cid:1)
J and K are arbitrary functions of φ which are chosen to satisfy the second order HJ equation. The higher order
terms proceed along similar lines: e.g., (4) consists of all invariant terms with four spatial derivatives.
S
What preciselyis the expansionparameterinthe gradientexpansion? To whatdoes the index 2nrefer? The index
2n is related to the conformal weight of the functional (2n). To clarify this point, it is useful to introduce a scaling
S
factor, s. If one rescales the 3-metric using the homogeneous conformal factor, s,
γ (x) s2γ (x) (2.17)
ab ab
→
one finds that
(2n)[s2γ ,φ] = s(3 2n) (2n)[γ ,φ]. (2.18)
ab − ab
S S
Hence the gradient expansion is an expansion of the generating functional in powers of the scaling factor s:
∞
[s2γ ,φ]= s(3 2n) (2n)[γ ,φ], (2.19)
ab − ab
S S
n=0
X
where (2n)[γ ,φ] are simply the coefficients of s(3 2n). At the end of the calculation, one sets the scaling factor to
ab −
S
unity because it is just a counting parameter.
The interpretation of the spatial gradient expansion as an expansion in powers of the conformal weight may be
trivially extended to other situations. For example, instead of rescaling the metric, one may choose to rescale the
matter field, φ(x) sφ(x) (or χ(x) sχ(x)) and then expand in powers of s. Such a simple adjustment leads to
→ →
radically different forms of the generating functional as will be demonstrated in Sec. III and Sec. IV. (In order to
reduce the introduction of extraneous notation, one in practice expands in powers of the desired field, and typically
foregoes any mention of the scaling factor s.)
4
III. TAYLOR SERIES EXPANSION IN THE SCALAR FIELD
For the case of a scalar field interacting with gravity, we will now consider a solution for the generating functional
in the form of a power series of the scalar field:
∞
[γ (x),φ(x)] = (m) (series in φ). (3.1)
ab
S S
m=0
X
The zeroth order term will be assumed to be independent of φ, but otherwise, it is an arbitrary functional of the
3-metric,
(0) (0)[γ (x)], (3.2a)
ab
S ≡S
which is invariant under reparametrizations of the spatial coordinates:
δ (0) δ (0)
0= 2 γ S + S γ . (3.2b)
ik kl,i
− δγ (x) δγ (x)
(cid:18) kj (cid:19),j kl
A. Equations for Scalar Field
One substitutes the series into the HJ equation 2.6a, and collects terms of like order, to find:
δ (1) 2 δ (0) δ (0)
S =γR 2γV + 2[2γ (x)γ (x) γ (x)γ (x)] S S , (zerothorderterms), (3.3a)
0 il jk ij kl
δφ(x) − − − δγ (x)δγ (x)
(cid:18) (cid:19) ij kl
δ (1) δ (2) δ (0) δ (1)
S S = 2[2γ (x)γ (x) γ (x)γ (x)] S S , (firstorderterms) (3.3b)
il jk ij kl
δφ(x) δφ(x) − − δγ (x)δγ (x)
ij kl
δ (1) δ (3) 1 δ (2) 2 γ γ
S S = S m2φ2 γabφ φ (secondorderterms) (3.3c)
,a ,b
δφ(x) δφ(x) −2 δφ(x) − 2 − 2
(cid:18) (cid:19)
δ (1) δ (1) δ (0) δ (2)
[2γ (x)γ (x) γ (x)γ (x)] S S +2 S S ,
il jk ij kl
− − δγ (x)δγ (x) δγ (x)δγ (x)
(cid:18) ij kl ij kl (cid:19)
It is straightforwardto derive a general expression for higher order terms in analogy with ref. [2].
B. Solutions for a Scalar Field
The generating functional (n) for any order may be written in terms of a recursion relation. The first few terms
S
are:
(0) (0)[γ (x)], (3.4a)
ab
S ≡S
δ (0) δ (0) 1/2
(1) = d3xγ1/2φ(x) R 2V0 2γ−1[2γil(x)γjk(x) γij(x)γkl(x)] S S (3.4b)
S − − − δγ (x)δγ (x)
Z (cid:20) ij kl (cid:21)
δ (0) δ (1) δ (1)
(2) = d3xφ(x)[2γ (x)γ (x) γ (x)γ (x)] S S S , (3.4c)
il jk ij kl
S −Z − δγij(x)δγkl(x),δφ(x)
1 1 δ (2) 2 γ γ
(3) = d3xφ(x) S m2φ2 γabφ φ (3.4d)
,a ,b
S 3Z (− 2 (cid:18)δφ(x)(cid:19) − 2 − 2
δ (1) δ (1) δ (0) δ (2) δ (1)
[2γ (x)γ (x) γ (x)γ (x)] S S +2 S S S .
il jk ij kl
− − (cid:18)δγij(x)δγkl(x) δγij(x)δγkl(x)(cid:19)),δφ(x)
The validity ofthesethese formulaewillbe demonstratedbyderiving (2). The functionalequation(3.3b)defining
(2) may be rewritten in the form S
S
5
δ (2) δ (0) δ (1) δ (1)
S = 2[2γ (x)γ (x) γ (x)γ (x)] S S / S . (3.5)
il jk ij kl
δφ(x) − − δγ (x)δγ (x) δφ(x)
ij kl
Given (1), the right hand side is known. Hence eq.(3.5) has the form of an infinite dimensional gradient which may
S
be integrated using a contour integral [2] in φ field-space (the 3-metric is held fixed during such an integration). For
the same reasons given in earlier work [10], one may choose an arbitrary contour of integration, and I will use a
straight line parameterized by
φ(x)=tφ(x), 0 t 1, (3.6)
≤ ≤
to connect the origin, φ (x)=0 to the ‘final point’ φ (x)=φ(x):
0 f
1 δ (2)
(2) = d3x dtφ(x) S (3.7)
S δφ(x)
Z Z0
After counting powers of t, the integral over t gives 1/2, and hence (2) is given by eq.(3.4c). Integrability of (2) in
eq.(3.5) is guaranteedif the previous order terms, (0) and (1), areSgauge-invariant[10]. S
S S
1. First Example for a Scalar Field
Wewillnowgivesomeexplicitexamples. Since (0)[γ (x)]isarbitrary,thepossibilitiesarelimitless,butasampling
ab
S
is instructive.
(0) = 2H d3xγ1/2, H = V /3, (3.8a)
0 0 0
S −
Z
p
(1) = d3xγ1/2φ(x)R1/2, (3.8b)
S
Z
(2) = H d3xγ1/2 φ2+ φ |k φ . (3.8c)
0
S − Z " (cid:18)√R(cid:19) (cid:18)√R(cid:19)k#
|
Nospatialderivativesofφappearin (1). However,theydoappearin (2),andtheycannotberemovedbyintegration
S S
by parts.
Unlike the spatial gradient expansion, the above series is not analytic since (1) contains a square root of R. If
S
R < 0, is imaginary, which is classically forbidden. If the sign of the square root is chosen accordingly, these
S
solutions would be exponentially suppressed in a quantum analysis:
Ψ eiS, Ψ2 =exp 2 d3xγ1/2φ√ R , (forφ 0). (3.9)
∼ | | − − ≥
(cid:20) Z (cid:21)
2. Second Example for a Scalar Field
Another simple example arises if we take (0) =0:
S
(0) =0, (3.10a)
S
(1) = d3xγ1/2φ(R 2V )1/2 , (3.10b)
0
S −
Z
(2) =0, (3.10c)
S
1 φ 1 1 δ (1) δ (1)
(3) = d3xγ1/2 m2φ2+ γabφ,aφ,b+γ−1[2γil(x)γjk(x) γij(x)γkl(x)] S S , (3.10d)
S −3 √R 2V 2 2 − δγ (x)δγ (x)
Z − 0 (cid:20) ij kl (cid:21)
where
δ (1) γ1/2
S = (R 2V0)γab Rab+D|ab γabD2 φ(R 2V0)−1/2 . (3.10e)
δγ (x) 2 − − − −
ab
h ih i
If R>2V , then (1) describes the classically forbidden regime.
0
S
6
IV. TAYLOR SERIES EXPANSION IN DUST FIELD
A. Equations for Dust Field
I will now consider an expansion of in powers of χ:
S
∞
[γ (x),χ(x)]= (m) (series in χ). (4.1)
ab
S S
m=0
X
This expansion is quite general, and it will be applicable in most instances except when one encounters a singular
point (which will require a separate treatment). The zeroth order term will be assumed to be independent of χ:
(0) (0)[γ (x)]. (4.2)
ab
S ≡S
Expanding all terms in powers of χ, one obtains the following equations:
δ (1) 1 δ (0) δ (0)
S = γ1/2R γ−1/2[2γil(x)γjk(x) γij(x)γkl(x)] S S , (4.3a)
δχ(x) 2 − − δγ (x)δγ (x)
ij kl
δ (2) δ (0) δ (1)
S = 2γ−1/2[2γil(x)γjk(x) γij(x)γkl(x)] S S , (4.3b)
δχ(x) − − δγ (x)δγ (x)
ij kl
δ (3) 1 δ (1) δ (1) δ (1) δ (0) δ (2)
δχS(x) =−2(cid:16)χ|aχ|a(cid:17)δχS(x) −γ−1/2[2γil(x)γjk(x)−γij(x)γkl(x)] (cid:20)δγSij(x)δγSkl(x) +2δγSij(x)δγSkl(x)(cid:21) . (4.3c)
B. Solutions for Dust Field
The above equations (4.3) may be integrated immediately:
(0) (0)[γ (x)], (4.4a)
ab
S ≡S
1 δ (0) δ (0)
(1) = d3xχ(x) γ1/2R γ−1/2[2γil(x)γjk(x) γij(x)γkl(x)] S S , (4.4b)
S 2 − − δγ (x)δγ (x)
Z (cid:20) ij kl (cid:21)
δ (0) δ (1)
(2) = d3xχ(x)γ−1/2[2γil(x)γjk(x) γij(x)γkl(x)] S S , (4.4c)
S − − δγ (x)δγ (x)
Z ij kl
1 1 δ (1)
S(3) = 3 d3xχ(x)(− 2 χ|aχ|a δχS(x) (4.4d)
Z (cid:16) (cid:17)
δ (1) δ (1) δ (0) δ (2)
γ−1/2[2γil(x)γjk(x) γij(x)γkl(x)] S S +2 S S .
− − (cid:20)δγij(x)δγkl(x) δγij(x)δγkl(x)(cid:21))
One may interpret the higher order terms, (1), (2), ..., as describing the evolution in time χ(x) of the the initial
state (0)[γ (x)]. S S
ab
S
1. First Example for a Dust Field
If (0) is proportional to the volume of a given 3-geometry,the first few terms are:
S
(0) =C d3xγ1/2, (4.5a)
S
Z
R 3C2
(1) = d3xγ1/2χ + , (4.5b)
S 2 4
Z (cid:20) (cid:21)
R 9C2 1
S(2) =C d3xγ1/2 χ2 8 + 16 + 2χ|kχ|k . (4.5c)
Z (cid:20) (cid:18) (cid:19) (cid:21)
7
2. Second Example for a Dust Field
As a second example, consider a series whose first term is proportional to an integral of the spatial curvature:
(0) =E d3xγ1/2R, (4.6a)
S
Z
R 3
(1) = d3xγ1/2χ 2E2 RabR R2 . (4.6b)
ab
S 2 − − 8
Z (cid:20) (cid:18) (cid:19)(cid:21)
V. SUPERPOSITION OF HAMILTON-JACOBI SOLUTIONS
One of the very attractive features of quantum mechanics is the principle of linear superposition: if ψ and ψ are
1 2
solutions of the quantum theory, then ψ(y) = ψ +ψ is a solution as well. This principle has no direct classical
1 2
interpretation. Nonetheless, there are situations where one can indeed construct additional solutions to the HJ
equation from other known solutions. One can enunciate a Principle of Superposition for Hamilton-Jacobi theory
which is motivated by a semiclassical treatment of the quantum theory.
For a quantum system with configuration variable, y, suppose that one is fortunate enough to find a solution to
the Schrodingerequation,ψ(y a)whichdepends onacontinuousparametera. Thenanylinearsuperpositionofthese
|
solutions is also a solution:
ψ(y)= daw(a)ψ(y a), (5.1)
|
Z
where the weighting function w(a) is arbitrary. If we work in the semiclassical limit, h¯ 0, then ψ(y a) and w(a)
→ |
may be approximated by phase factors,
ψ(y a) eiS(ya)/h¯, w(a)=eig(a)/h¯; (5.2)
|
| ∼
S(y a) is then a solution of the HJ equation which depends on a parameter a. The resulting integral (5.1) may
|
approximated using the stationary phase approximation,
ψ(y)=exp[i(S(y a)+g(a))/¯h] (5.3a)
|
where a a(y) is now chosen so that the phase of the integrand has a maximum or minimum,
≡
∂
0= [S(y a)+g(a)] . (5.3b)
∂a |
Hence the principle of linear superposition in quantum mechanics and the stationary phase approximation lead to
the Principle of Superposition for Hamilton-Jacobi theory:
If S(y a) is a solution of the HJ equation which depends on a continuous parameter a, then
|
T(y)=S(y a)+g(a) (5.4a)
|
is also a solution provided that a a(y) is determined by the stationary phase condition,
≡
∂S(y a) ∂g(a)
0= | + . (5.4b)
∂a ∂a
The proof given above was motivated by the quantum theory. In a gravitational context, it is not clear that a
consistentquantumtheoryexists (atpresent). However,one canverify the principle totally within a classicalcontext
by noting that
∂T(y) ∂S(y a) ∂S(y a) ∂a ∂g(a) ∂a
= | + | + . (5.5)
∂y ∂y (cid:12) ∂a ∂y ∂a ∂y
(cid:12)a
(cid:12)
The last two terms vanish by virtue of the statio(cid:12)nary phase condition (5.4b), and thus a derivative of S(y) with
(cid:12)
respect to y coincides with a derivative of S(y a) with respect to y (holding a fixed):
|
∂T(y) ∂S(y a)
= | . (5.6)
∂y ∂y (cid:12)
(cid:12)a
(cid:12)
Since S appears in the HJ equation only in terms of its deriva(cid:12)tives with respect to the configuration variables, the
(cid:12)
principle is justified.
8
A. Applying the Superposition to the HJ Equation for Gravity
The superposition principle leads to some rather exotic solutions of the HJ equation for general relativity. To
illustrate the basic principles, we will consider a massless scalar field, m = 0 with vanishing cosmological constant,
V =0, which has the following spatial gradient expansion solution for the HJ eq.(2.6a),
0
[γ (x),φ(x)]= (0)+ (2)+..., (5.7a)
ab
S S S
of which the first two terms in a spatial gradient expansion are
(0) = 2C d3xγ1/2e√3/2φ, (5.7b)
S −
Z
1 1 1
(2) = d3xf1/2 e−4φ/√6 Rf + fabφ,aφ,b +ERf . (5.7c)
S C 8 6
Z (cid:20) (cid:18) (cid:19) (cid:21)
C and E are homogeneousbut arbitraryparameters. The new 3-metric f is related to γ by a conformaltransfor-
ab ab
mation:
f =e2φ/√6γ . (5.8)
ab ab
Rf is the Ricci Scalar associated with f .
ab
Consider a new solution of the HJ equation given by the sum
T
1
= C2, (5.9a)
T S− 2
where satisfies the stationary phase condition with respect to C:
T
∂ ∂
0= T = S C. (5.9b)
∂C ∂C −
Using a spatial gradient expansion, one can solve for C:
1 1 1
C =C d3xf1/2e 4φ/√6 Rf + fabφ φ +... (5.10a)
0− C2 − 8 6 ,a ,b
0 Z (cid:18) (cid:19)
where the functional C is given by
0
C = 2 d3xγ1/2e√3/2φ. (5.10b)
0
−
Z
admits the following spatial gradient expansion:
T
= (0)+ (2)+... (5.11a)
T T T
with
2
(0) =2 d3xγ1/2e√3/2φ , (5.11b)
T
(cid:18)Z (cid:19)
(2) =E d3xf1/2Rf + d3xf1/2e 4φ/√6 1 Rf 1 fabφ φ d3xγ1/2e√3/2φ .
− ,a ,b
T Z (cid:20)Z (cid:18)−16 − 12 (cid:19)(cid:21),(cid:18)Z (cid:19)
(5.11c)
One may verify directly that eq.(5.11a) is solution of the HJ eq.(2.6a).
Unlike the original gradient expansion solution (5.7a) for , the new solution is no longer an integral over local
S T
terms, but contains products and quotients of local integrals. Hence very complicated solutions of the HJ equations
may be generated using the nonlinear superposition of local solutions.
9
VI. CONCLUSIONS
Acoordinate-freeanalysisofgeneralrelativitypossessesdistinctadvantagesovertraditionalmethodsofsolvingEin-
stein’s equations. When solving the field equations, one typically makes arbitrary gauge choices. In many situations,
the optimal choice of gauge is not very clear, and a poor choice of gauge can complicate the analysis significantly.
In the coordinate-free method expounded here, one may solve the functional HJ equation without making any gauge
choices. However,applicationsofthe HJ equationtophysicalproblemshavepreviouslybeenhamperedby the lackof
mathematicaltools. Somegeneralandusefultechniqueshavebeendevelopedwhichwillbeappliedlaterincosmology.
However,the functional approachis a radically different way of describing gravity. More work is required in order to
develop a more intuitive understanding of the method.
Although one can obtain exact generalsolutions for gravitationalsuperspace in two spacetime dimensions (see, for
example,[32]),it is doubtful that onemay constructsuchsolutionsin four spacetimedimensions. One must resortto
someapproximationmethod. ThespatialgradientexpansionwastheprototypesolutionoftheHJequationforgravity
and matter. It is basically a Taylor series expansion in the conformal weight factor of the 3-metric. By expanding
the generatingfunctional in terms ofother fields, one may constructnumerous other solutions to the HJ equationfor
general relativity. Explicit expansions in either a scalar field φ or a dust field χ were demonstrated explicitly. These
solutions describe the evolution of some gauge-invariant initial state (0)[γ (x)] as a functional of the matter field.
ab
S
Integrability of these solutions is guaranteed by spatial gauge-invariance.
Many of these solutions depend on continuous parameters. One can construct additional solutions by using the
Superposition Principle for Hamilton-Jacobi Theory which is motivated by the superposition principle in quantum
mechanicsinconjunctionwiththe stationaryphaseapproximation. One canineffectconstructcomplicatedsolutions
by integrating over the continuous parameters.
ACKNOWLEDGMENTS
The author thanks J. M. Stewart, W. Unruh, D. Page, J. Soda and D. Louis-Martinez for useful discussions. This
work was supported by the Natural Sciences and Engineering Research Council of Canada (NSERC).
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