‚t

It is in this hyperbolic case that conservation laws have been most e¬ectively

used. Everything developed previously in this chapter for the Laplace operator

and Poisson equations on Riemannian manifolds has an analog for the wave

operator and wave equations on Lorentzian manifolds, which by de¬nition carry

a metric of signature (n, 1). Indeed, even the coordinate formulae for conserva-

tion laws that we derived in the preceding section are easily altered by a sign

change to give corresponding conservation laws for the wave equation. Our goal

in this section is to see how certain analytic conclusions can be drawn from

these conservation laws.

Before doing this, we will illustrate the usefulness of understanding the wave

operator geometrically, by presenting a result of Christodoulou asserting that

the Cauchy problem for the non-linear hyperbolic equation

n+3

z = z n’1 (3.68)

3.5. CONSERVATION LAWS FOR WAVE EQUATIONS 119

has solutions for all time, given su¬ciently small initial data.6 The proof exploits

conformal invariance of the equation in an interesting way, and this is what we

want to explain. Note that (3.68) is the hyperbolic analog of the maximally

n+2

symmetric non-linear Poisson equation ∆z = z n’2 considered previously; the

change in exponent re¬‚ects the fact that the number of independent variables is

now n+1, instead of n. This equation will be of special interest in our discussion

of conservation laws, as well.

The idea for proving the long-time existence result is to map Minkowski

space Ln+1, which is the domain for the unknown z in (3.68), to a bounded

domain, in such a way that the equation (3.68) corresponds to an equation

for which short-time existence of the Cauchy problem is already known. With

su¬ciently small initial data, the “short-time” will cover the bounded domain,

and back on Ln+1 we will have a global solution.

The domain to which we will map Ln+1 is actually part of a conformal com-

pacti¬cation of Minkowski space, analogous to the conformal compacti¬cation

of Euclidean space constructed in §3.1.1. This compacti¬cation is di¬eomorphic

to a product S 1 — S n , and topologically may be thought of as the result of

adding a point at spatial-in¬nity for each time, and a time-at-in¬nity for each

spatial point. Formally, one can begin with a vector space with inner-product

of signature (n + 1, 2), and consider the projectivized null-cone; it is a smooth,

real quadric hypersurface in Pn+2, which in certain homogeneous coordinates

is given by

2 2 2 2

ξ1 + ξ 2 = · 1 + · · · + · n ,

evidently di¬eomorphic to S 1 — S n . The (n + 1, 2)-inner-product induces a

Lorentz metric on this hypersurface, well-de¬ned up to scaling, and its conformal

isometry group has identity component SO o (n + 1, 2), which we will revisit in

considering conservation laws. What is important for us now is that among the

representative Lorentz metrics for this conformal structure one ¬nds

g = ’dT 2 + dS 2 ,

where T is a coordinate on S 1 , and dS 2 is the standard metric on S n . In

certain spherical coordinates (“usual” spherical coordinates applied to Rn, after

stereographic projection, this may be written

g = ’dT 2 + dR2 + (sin2 R) dZ 2,

where R ∈ [0, π], and dZ 2 is the standard metric on the unit (n ’ 1)-sphere.

Now we will conformally embed Minkowski space Ln+1 as a bounded domain

in the ¬nite part R — Rn of S 1 — S n , the latter having coordinates (T, R, Z).

The map •(t, r, z) = (T, R, Z) is given by

« «

arctan(t + r) + arctan(t ’ r)

T

R = arctan(t + r) ’ arctan(t ’ r) ,

Z z

6 See [Chr86]; what is proved there is somewhat more general.

120 CHAPTER 3. CONFORMALLY INVARIANT SYSTEMS

and one can easily check that

•— (’dT 2 + dR2 + (sin2 R) dZ 2) = „¦2 (’dt2 + dr2 + r2 dz 2),

where dZ 2 and dz 2 are both the standard metric on the unit (n ’ 1)-sphere; the

conformal factor is

1 1

„¦ = 2(1 + (t + r)2 )’ 2 (1 + (t ’ r)2 )’ 2 ,

and the right-hand side is a multiple of the ¬‚at Minkowski metric. The image

of • is the “diamond”

D = {(T, R, Z) : R ’ π < T < π ’ R, R ≥ 0}.

Note that the initial hyperplane {t = 0} corresponds to {T = 0}, and that with

¬xed (R, Z), as T ’ π ’ R, t ’ ∞. Consequently, the long-time Cauchy prob-

lem for the invariant wave equation (3.68) corresponds to a short-time Cauchy

problem on the bounded domain D for some other equation.

We can see what this other equation is without carrying out tedious calcula-

tions by considering the conformally invariant wave operator, an analog of the

conformal Laplacian discussed in §3.1.3. This is a di¬erential operator

n’1 n+3

: “(D 2(n+1) ) ’ “(D 2(n+1) )

c

between certain density line bundles over a manifold with Lorentz metric. With

n’1

a choice of Lorentz metric g representing the conformal class, u ∈ “(D 2(n+1) ) is

represented a function ug , and the density c u is represented by the function

n’1

( c u)g = g (ug ) + 4n Rg ug ,

where Rg is the scalar curvature and g is the wave operator associated to g.

We interpret our wave equation (3.68) as a condition on a density represented in

the ¬‚at (Minkowski) metric g0 by the function u0 , and the equation transformed

by the map • introduced above should express the same condition represented

in the new metric g. The representative functions are related by

n’1 n+3

ug = „¦ ’ = „¦’

u0 , ( c u)g ( c u)0 ,

2 2

so the condition (3.68) becomes

n’1

g (ug ) + R g ug = ( c u)g

4n

n+3

„¦’

= ( c u)0

2

n+3

n+3

„¦’ n’1

= u0

2

n+3

n’1

= ug .

The scalar curvature is just that of the round metric on the n-sphere, Rg =

n(n ’ 1), so letting u = u0, U = ug , the equation (3.68) is transformed into

(n’1)2 n+3

gU + U = U n’1 . (3.69)

4

3.5. CONSERVATION LAWS FOR WAVE EQUATIONS 121

Finally, suppose given compactly supported initial data u(0, x) = u0 (x) and

ut (0, x) = u1(x) for (3.68). These correspond to initial data U0 (X) and U1(X)

for (3.69), supported in the ball of radius π. The standard result on local

existence implies that the latter Cauchy problem can be solved for all X, in

some time interval T ∈ [0, T0], with a lower-bound for T0 determined by the size

of the initial data. Therefore, with su¬ciently small initial data, we can arrange

T0 ≥ π, and translated back to the original coordinates, this corresponds to a

global solution of (3.68).

We now turn to more general wave equations (3.67), where conservation

laws have been most e¬ectively used.7 Equation (3.67) is the Euler-Lagrange

equation for the action functional

1 2

+ || z||2) + F (z) dy dt,

2 (’zt

Rn

R

where F (z) = f(z), and the gradient z is with respect to the “space” variables

y1 , . . . , yn .

Rather than redevelop the machinery of conformal geometry in the Lorentz

case, we work in the classical setting, on J 1 (Ln+1 , R) with coordinates t, y i , z, pa

(as usual, 1 ¤ i ¤ n, 0 ¤ a ¤ n), contact form θ = dz ’ p0 dt ’ pidyi , Lorentz

inner-product ds2 = ’dt2 + (dyi )2 , and Lagrangian

L(t, y, z, p) = 1 (’p2 + |pi|2) + F (z).

0

2

A normalized representative functional is then

Λ = L dt § dy + θ § (’p0 dy ’ pi dt § dy(i) ), (3.70)

satisfying

dΛ = Π = θ § (dp0 § dy + dpi § dt § dy(i) + f(z)dt § dy).

This is the example discussed at the end of §1.3. As mentioned there, the in-

variance of the equation under time-translation gives an important conservation

law, and its uses will be our ¬rst topic below. In fact, there are conservation laws

associated to space-translations and Lorentz rotations, the latter generated by

ordinary spatial rotations bi yj ‚yj (bi + bj = 0) and Lorentz boosts y i ‚t + t ‚yi ;

‚ ‚ ‚

j j i

however, these seem to have been used less widely in the analysis of (3.67).

Especially interesting is the case of (3.68), which is preserved under a certain

action of the conformal Lorentz group SO 0 (n + 1, 2) on J 1 (Ln+1, R). In par-

ticular, there are extensions to J 1(Ln+1 , R) of the dilation and inversion vector

¬elds on Ln+1 , and these give rise to more conservation laws. We will consider

these after discussing uses of the time-translation conservation law for the more

general wave equations (3.67).

7 This material and much more may be found in [Str89].

122 CHAPTER 3. CONFORMALLY INVARIANT SYSTEMS

3.5.1 Energy Density

‚

The time-translation vector ¬eld ‚t on Ln+1 lifts to J 1 (Ln+1 , R) to a symmetry

‚

of Λ having the same expression, V = ‚t . The Noether prescription gives

12

p2 ) + F (z) dy + p0 pi dt § dy(i) ,

•t = V Λ= (p +

20 i

as calculated in Section 1.3. The coe¬cient of dy here is the energy density

def 1 2

+ |pi|2) + F (z),

e= (p

20

and it appears whenever we integrate •t along a constant-time level surface

Rn = {t} — Rn . The energy function

t

def

e dy ≥ 0,

E(t) =

Rn

t

is constant by virtue of the equation (3.67), assuming su¬cient decay of z and

its derivatives in the space variables for the integral to make sense.

A more substantial application involves a region „¦ ‚ Ln+1 of the form

{||y|| < r0 ’ (t ’ t0 )},

„¦=

t∈(t0 ,t1 )

a union of open balls in space, with initial radius r0 decreasing with speed 1.

The boundary ‚„¦ is T ’ B + K, where

• B = {t0 } — {||y|| ¤ r0} is the initial disc,

• T = {t1} — {||y|| ¤ r0 ’ (t1 ’ t0 )} is the ¬nal disc, and

• K = ∪t∈[t0 ,t1] {||y|| = r0 ’ (t ’ t0)} is part of a null cone.

The conservation of •t on ‚„¦ reads

0= •t = e dy + •t . (3.71)

T ’B

‚„¦ K

The term K •t describes the ¬‚ow of energy across part of the null cone; we will

compute the integrand more explicitly in terms of the area form dK induced

from an ambient Euclidean metric dt2 + (dyi )2 , with the goal of showing

that K •t ≥ 0. This area form is the contraction of the outward unit normal

yi ‚

√ (‚

1

with the ambient Euclidean volume form dt § dy, giving

+ ||y|| ‚y i )

2 ‚t

√

yi

1

’ dt §

dK = √ (dy dy(i) )|K = 2 dy|K .

||y||

2

It is easy to calculate that the restriction to K of •t is

y i pi

1

• t |K = ’

√ (e p )dK.

||y|| 0

2

3.5. CONSERVATION LAWS FOR WAVE EQUATIONS 123

Separating the radial and tangential space derivatives

y i pi

def def

p2 ’ p 2 ,

pr = , p„ = r

i

||y||

we can rewrite this as