1

• t |K = (3.72)

22 0 r „

’ pr )2 + 1 p2 ) + F (z))dK.

√ ( 1 (p0

1

= (3.73)

2„

22

In the region of J 1 (Ln+1, R) where F (z) ≥ 0 this integrand is positive, and

from (3.71) we obtain the bound

e dy ¤ e dy.

T B

This says that “energy travels with at most unit speed””no more energy can

end up in T than was already present in B. If F (z) ≥ 0 everywhere, then one

can obtain another consequence of the expression (3.73) by writing

•t ’ •t ¤ •t ¤ E(t0 ) = E.

•t =

K B T B

This gives an upper bound for

√

||(dz|K )||2 2 = ((p0 ’ pr )2 + p2 )dK ¤ 2 2E

„

L

K

which holds for the entire backward null cone; that is, our bound is independent

of t0. Here the L2 -norm is with respect to Euclidean measure.

We should also mention that the spatial-translation and Lorentz rotations

give rise to conserved quantities that may be thought of as linear and angular

momenta, respectively. The uses of these are similar to, though not as extensive

as, the uses of the conserved energy.

3.5.2 The Conformally Invariant Wave Equation

We now determine some additional conservation laws for the conformally in-

variant wave equation (3.68). Again, we could duplicate the process used for

Poisson equations by calculating restrictions of the right-invariant vector ¬elds

of SOo (n + 1, 2) to the image of an embedding J 1 (Ln+1, R+ ) ’ SOo (n + 1, 2),

and contracting with the left-invariant Lagrangian. Instead, we will illustrate

the more concrete, coordinate-based approach, though we will still make some

use of the geometry.

The Dilation Conservation Law

To ¬nd the conservation law corresponding to dilation symmetry of (3.68), we

have to ¬rst determine a formula for this symmetry on J 1 (Ln+1, R+ ), and then

124 CHAPTER 3. CONFORMALLY INVARIANT SYSTEMS

apply the Noether prescription. For this, we will ¬rst determine the vector ¬eld™s

action on J 0 (Ln+1, R+ ); the lift of this action to J 1(Ln+1 , R+) is determined

by the requirement that it preserve the contact line bundle.

By analogy with (3.30), we have an embedding of J 0(Ln+1 , R+ ) into the

null-cone of Ln+1,2 given by

«

1

¬ ·

t

2

(t, yi , z) ’ z n’1 ¬ ·,

i

y

2 2

1

2 (’t + |y| )

and the dilation matrix (in blocks of size 1, 1, n, 1) acts projectively on this slice

of the null-cone by

® « « ’1 ® «

1 1

r 000

n’1 ¬ · def ¬ 0 1 0 0 · n’1 ¬ ·

tr t

2

· · z 2 ¬

z r ¬ · = ¬ · .

° » 0 0 In 0 ° »

i i

yr y

1 2 1 2

2 ||(tr , yr )|| ) 2 ||(t, y)||

0 00r

Taking the derivative with respect to r and setting r = 1 gives the vector ¬eld

¯

Vdil = ’ n’1 z ‚z + t ‚t + yi ‚yi .

‚ ‚ ‚

2

The scaling in the z-coordinate re¬‚ects an interpretation of the unknown z(t, y)

¯

as a section of a certain density line bundle. We then ¬nd the lift from Vdil ∈

V(J 0 (Ln+1 , R+)) to Vdil ∈ V(J 1 (Ln+1, R+ )) by the requirement that the con-

tact form θ = dz ’ p0 dt ’ pi dyi be preserved up to scaling; that is,

¯

Vdil = Vdil + v0 p0 + vi pi

must satisfy

LVdil θ ≡ 0 (mod {θ}),

where v0 and vi are the unknown coe¬cients of the lift. This simple calculation

yields

Vdil = ’ n’1 z ‚z + t ‚t + yi ‚yi ’ n+1 pa ‚pa .

‚ ‚ ‚ ‚

2 2

Then one can compute even for the general wave equation (3.67) that

n’1

LVdil (L(t, y, z, p)dt § dy) = ((n + 1)F (z) ’ § dy.

2 zf(z))dt

Tentatively following the Noether prescription for the general wave equation,

we set

•dil = Vdil Λ,

where Λ is given by (3.70), and because Λ ≡ L dt § dy modulo {I}, we can

calculate

LVdil Λ ’ Vdil Π

d•dil =

≡ LVdil (L dt § dy) ’ 0 (mod EΛ )

((n + 1)F (z) ’ n’1 zf(z))dt § dy (mod EΛ ).

= 2

3.5. CONSERVATION LAWS FOR WAVE EQUATIONS 125

The condition on the equation (3.67) that •dil be a conservation law is therefore

2(n+1)

F (z) = Cz ,

n’1

so the PDE is n+3

z = C z n’1 ,

n’1

as expected (cf. (3.68)); we work with C = 1, C = 2(n+1) . Now, one can

calculate that restricted to any Legendre submanifold the conserved density is

•dil = L(t, y, z, p)(t dy ’ y i dt § dy(i)) + ( n’1 z + tp0 + yi pi )(p0 dy + pj dt § dy(j) ).

2

Typically, one considers the restriction of this form to the constant-time hyper-

planes Rn = {t} — Rn , which is

t

n’1

•dil ≡ (te + rp0 pr + (mod {dt}).

2 zp0 )dy

For example, we ¬nd that for solutions to (3.68) with compact support in y i ,

d n’1

(te + rp0 pr + 2 zp0 )dy = 0. (3.74)

dt Rn

t

For more general wave equations (3.67), an identity like (3.74) holds, but

with a non-zero right-hand side; our conservation law is a special case of this.

The general dilation identity is of considerable use in the analysis of non-linear

wave equations. It is analogous to the “almost-conservation law” derived from

scaling symmetry used to obtain lower bounds on the area growth of minimal

surfaces, as discussed in §1.4.3.

An Inversion Conservation Law

We now consider the inversion symmetry corresponding to the conjugate of

time-translation by inversion in a unit (Minkowski) “sphere”. We will follow

the same procedure as for dilation symmetry, ¬rst determining a vector ¬eld on

J 0 (Ln+1, R+ ) generating this inversion symmetry, then lifting it to a contact-

preserving vector ¬eld on J 1 (Ln+1, R+ ), and then applying the Noether pre-

scription to obtain the conserved density.

The conjugate by sphere-inversion of a time-translation in SO o (n + 1, 2) is

the matrix «

1 b 0 1 b2 2

¬ 0 1 0 ’b ·

¬ ·,

0 0 In 0

000 1

and di¬erentiating its projective linear action on

® «

1

2¬ ·

t

z n’1 ¬ ·

° »

yi

2

1

2 ||(t, y)||

126 CHAPTER 3. CONFORMALLY INVARIANT SYSTEMS

yields the vector ¬eld

¯ ’ 1 (t2 + |y|2) ‚t ’ tyi ‚yi .

n’1 ‚ ‚ ‚

Vinv = tz ‚z

2 2

‚ ‚

Again, the coe¬cients of ‚t and ‚yi describe an in¬nitesimal conformal motion

of Minkowski space”representing an element of the Lie algebra of the Lorentz

‚

conformal group”and the coe¬cient of ‚z gives the induced action on a density

¯

‚ ‚

line bundle. We now look for coe¬cients v0 ‚p0 + vi ‚pi to add to Vinv so that

the new vector ¬eld will preserve θ up to scaling, and the unique solution is

¯ + pi y i + + p0 y i +

n’1 n+1 ‚ n+1 ‚

Vinv = Vinv + 2z 2 tp0 2 tpi ‚pi .

‚p0

In applying the Noether prescription to Λ and Vinv , it will turn out that we

need a compensating term, because LVinv Λ ≡ 0 modulo EΛ . However, instead of

performing this tedious calculation, we can simply test

def

’Vinv Λ

•inv

˜ =

L(t, y, z, p)( 1 (t2 + |y|2 )dy ’ tyi dt § dy(i) )

≡ 2

+( n’1 tz + 1 p0 (t2 + |y|2 ) + tpiyi ) § (p0 dy + pj dt § dy(j) ),

2 2

We ¬nd that on solutions to (3.68),

dt § dy = d( n’1 z 2 dy),

n’1

d•inv =

˜ 2 p0 z 4

and we therefore set

n’1 2

•inv = •inv ’

˜ 4 z dy.

As usual, we consider the restriction of this form to a hyperplane Rn = {t}—Rn,

t

which gives

+ p2 )(t2 + |y|2 ) + + tp0pi yi ’ n’1 2

1 n’1

•inv ≡ 2 (L 2 p0 tz 4z dy,

0

modulo {dt, θ}.

Again, for more general wave equations (3.67), this quantity gives not a

conservation law, but a useful integral identity. The usefulness of the integrand

follows largely from the fact that after adding an exact n-form, the coe¬cient of

dy is positive. One notices this by expanding in terms of radial and tangential

derivatives

1

(p2 + p2 + p2)(t2 + r2) + 2(n ’ 1)p0tz + 4trp0pr ’ (n ’ 1)z 2

•inv = 0 „ r

4

1

+ (t2 + r2)F (z) dy,

2

(3.75)

which suggests completing squares:

1

|p0y + tp|2 + (tp0 + rpr + (n ’ 1)z)2 + (rp„ )2

•inv =

4

(3.76)

n’1

1

+ (t2 + r2)F (z) ’ (nz 2 + 2rzpr ) dy.

2 4

3.5. CONSERVATION LAWS FOR WAVE EQUATIONS 127

The last term is the divergence d( n’1 yi z 2 dy(i) ) modulo {dt, θ}, and the remain-

4

ing terms are positive. The positive expression

def 1