Francoise J.-P., Naber G.L., Tsun T.S. (editors) Encyclopedia of Mathematical Physics

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Bogomolny (1997)) suggest that the X

are asymp-

totically Poisson distributed:

Conjecture 1 For any bounded function g : Z

0

we have

g Nðx; LÞðÞdx !

k¼0

gðkÞ

L

½14

as T !1.

One may also consider larger intervals, where

L !1 as X !1. In this case, the assumption on

the independence of the X

predicts a central-limit

theorem. Weyl’s law [3] implies that the expectation

value is asymptotically, for T !1,

Nðx; LÞdx  L ½15

This asymptotics holds for any sequence of L

bounded away from zero (e.g., L constant, or

L !1).

Define the variance by



ðX; LÞ¼

Nðx; LÞLðÞ

dx ½16

In view of the above conjecture, one expects



(X, L)  L in the limit X !1, L=

ﬃﬃﬃﬃ

!0 (the

variance exhibits a less universal behavior in the

range L 

ﬃﬃﬃﬃ

(the notation A  B means there is a

constant c > 0 such that A  cB), cf. Sarnak (1995),

and a central-limit theorem for the fluctuations

around the mean:

Conjecture 2 For any bounded function g : R !C

we have

Nðx; LÞL

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ



ðx; LÞ

ﬃﬃﬃﬃﬃﬃ

2

1

gðtÞe

ð1=2Þt

dt ½17

as X, L !1, L  X.

The main tool in the attempts to prove the above

conjectures has been the Selberg trace formula. It

relates sums over eigenvalues of the Laplacians to

sums over lengths of closed geodesics on the

hyperbolic surface. The trace formula is in its

simplest form in the case of compact hyperbolic

surfaces; we have

j¼0

hð

Þ¼

AreaðMÞ

4

1

hðÞtanhðÞ d

2H



n¼1

‘



gðn‘



2 sinhðn‘



=2Þ

½18

where H



is the set of all primitive oriented closed

geodesics ,and‘



their lengths. The quantity 

related to the eigenvalue 

by the equation 

= 

1=4. The trace formula [18] holds for a large class of

even test functions h. For example, it is sufficient to

assume that h is infinitely differentiable, and that the

Fourier transform of h,

gðtÞ¼

2

hðÞe

it

d ½19

has compact support. The trace form ula for non-

compact surfaces has additional terms from the

parabolic elements in the corresponding group, and

includes also sums over the resonances of the

continuous part of the spectrum. The noncompact

modular surface behaves in many ways like a

compact surface. In particular, Selberg showed that

the number of eigenvalues embe dded in the con-

tinuous spectrum satisfies the same Weyl law as in

the compact case (Sarnak 2003).

Setting

hðÞ¼

½X;XþL

AreaðMÞ

4





½20

where 

[X, XþL]

is the characteristic function of the

interval [X, X þ L], we may thus view N(X, L)as

the left-hand side of the trace formula. The above

test function h is, however, not admissible, and

requires appropriate smoothing. Luo and Sarnak (cf.

Sarnak (2003)) developed an argument of this type

to obtain a lower bound on the average number

variance,



ðX; L

ÞdL



ﬃﬃﬃﬃ

ðlog XÞ

½21

in the regime

ﬃﬃﬃﬃ

= log X  L 

ﬃﬃﬃﬃ

, which is

consistent with the Poisson conjecture 

(X, L)  L.

Bogomolny, Levyraz, and Schmit suggested a remark-

able limiting formula for the two-point correlation

function for the modular surface (cf. Bogomolny

et al. (1997) and Bogomolny (2006)), based on an

analysis of the correlations between multiplicities of

lengths of closed geodesics. A rigorous analysis of the

fluctuations of multiplicities is given by Peter (cf.

Bogomolny (2006)). Rudnick (2005) has recently

established a smoothed version of Conjecture 2 in the

regime

ﬃﬃﬃﬃ

!1;

ﬃﬃﬃﬃ

L log X

!0 ½22

where the charac teristic function in [20] is replaced

by a certain class of smooth test functions.

All of the above approaches use the Selberg trace

formula, exploiting the particular properties of the

Arithmetic Quantum Chaos 215

distribution of lengths of closed geodesics in

arithmetic hyperbolic surfaces. These will be dis-

cussed in more detail in the next section, following

the work of Bogomolny, Georgeot, Giannoni and

Schmit, Bolte, and Luo and Sarnak (see Bogomolny

et al. (1997) and Sarnak (1995) for references).

Distribution of Lengths of Closed

Geodesics

The classical prime geodesic theorem asserts that the

number N(‘) of primitive closed geodesics of length

less than ‘ is asymptotically

Nð‘Þ

‘

½23

One of the significant geometrical characteristics of

arithmetic hyperbolic surfaces is that the number of

closed geodesics with the same length ‘ grows

exponentially with ‘. This phenomenon is most

easily explained in the case of the modular surface,

where the set of lengths ‘ appearing in the lengths

spectrum is characterized by the condition

2 coshð‘=2Þ¼jtr j½24

where  runs over all elements in SL(2, Z) with

jtrj >2. It is not hard to see that any integer n > 2

appears in the set {jtr j:  2 SL(2, Z)}, and hence

the set of distinct lengths of closed geodesics is

L¼f2 arcoshðn=2Þ: n ¼ 3; 4; 5; ...g½25

Therefore, the number of distinct lengths less than ‘

is asymptotically (for large ‘)

ð‘Þ¼#ðL \ ½0;‘Þ  e

‘=2

½26

Equations [26] and [23] say that on average the

number of geodesics with the same lengths is at least

}

‘=2

=‘.

The prime geodesic theorem [23] holds equally for

all hyperbolic surfaces with finite area, while [26] is

specific to the modular surface. For general arith-

metic surfaces, we have the upper bound

ð‘Þce

‘=2

½27

for some constant c > 0 that may depend on the

surface. Although one expects N

(‘) to be asympto-

tic to (1=2)N(‘) for generic surfaces (since most

geodesics have a time-reversal partner which thus

has the same length, and otherwise all lengths are

distinct), there are examples of nonarithmetic Hecke

triangles where numerical and heuristic arguments

suggest N

(‘)  c

‘

=‘ for suitable constants c

> 0

and 0 < c

< 1=2 (cf. Bogomolny (2006)). Hence

exponential degeneracy in the length spectrum seems

to occur in a weaker form also for nonarithmetic

surfaces.

A further useful property of the length spectrum

of arithmetic surfaces is the bounded clustering

property: there is a constant C (again surface

dependent) such that

#ðL \ ½‘; ‘ þ 1Þ  C ½28

for all ‘. This fact is evident in the case of the

modular surface; the general case is proved by Luo

and Sarnak (cf. Sarnak (1995)).

Quantum Unique Ergodicity

The unit tangent bundle of a hyperbolic surface nH

describes the physical phase space on which the

classical dynamics takes place. A convenient para-

metrization of the unit tangent bundle is given by

the quotie nt nPSL(2, R – this may be seen be means

of the Iwasawa decomposition for an element

g 2 PSL(2, R),

g ¼

1 x

1=2

0 y

1=2



cos =2 sin =2

sin =2 cos =2

½29

where x þ iy 2 H represents the position of the

particle in nH in half-plane coordinates, and  2

[0, 2) the direction of its velocity. Multiplying the

matrix [29] from the left by





and writing the

result again in the Iwasawa form [29], one obtains

the action

ðz;Þ7!

az þ b

cz þ d

; 2 argðcz þ dÞ



½30

which represents precisely the geome tric action of

isometries on the unit tangent bundle.

The geodesic flow 

on nPSL(2, R) is repre-

sented by the right translation



:g 7!g

t=2

t=2



½31

The Haar measure  on PSL(2, R) is thus trivially

invariant under the geodesic flow. It is well known

that  is not the only invariant measure, that is, 

not uniquely ergodic, and that there is in fact an

abundance of invariant measures. The simplest

examples are those with uniform mass on one, or a

countable collection of, clos ed geodesics.

To test the distribution of an eigenfunction

’

in phase space, one associates with a function

216 Arithmetic Quantum Chaos

a 2 C

(nPSL(2, R)) the quantum observable

Op(a), a zeroth order pseudodifferential operator

with principal symbol a. Using semiclassical tech-

niques based on Friedrich’s symmetrization, one

can show that the matrix element



ðaÞ¼hOpðaÞ’

;’

i½32

is asymptotic (as j !1) to a positive functional

that defines a probability measure on

nPSL(2, R). Therefore, if M is compact, any

weak limit of 

represents a probability measure

on nPSL(2, R). Egoro v’s theorem (see Quantum

Ergodicity and Mixing of Eigenfunctions) in turn

implies that any such limit must be invariant

under the geodesic flow, and the main challenge

in proving QUE is to rule out all invariant

measures apart from Haar.

Conjecture 3 (Rudnick and Sarnak (1994); see

Sarnak (1995, 2003)). For every compact hyperbolic

surface nH, the sequence 

converges weakly to .

Lindenstrauss has proved this conjecture for

compact arithmetic hyperbolic surfaces of congru-

ence type (such as the second example in the section

‘‘Hyperbolic surfaces’’) for special bases of eigen-

functions, using ergodic-theoretic methods. These

will be disc ussed in more detail in the next section.

His results extend to the noncompact case, that is, to

the modular surface where =PSL(2, Z). Here he

shows that any weak limit of subseque nces of 

of the form c, where c is a constant with values in

[0, 1]. One believes that c = 1, but with present

techniques it cannot be ruled out that a proportion

of the mass of the eigenfunction escapes into the

noncompact cusp of the surface. For the modular

surface, c = 1 can be proved under the assumption of

the genera lized Riemann hypothesis (see the section

‘‘Eigenfunctions and L-functions’’ and Sarnak

(2003)). QUE also holds for the continuous part of

the spectrum, which is furnished by the Eisenstein

series E(z, s), where s = 1=2 þ ir is the spectral

parameter. Note that the measures associated with

the matrix elements



ðaÞ¼hOpðaÞEð; 1=2 þ irÞ; Eð; 1=2 þ irÞi ½33

are not probability measures but only Radon

measures, since E(z, s) is not square-integrable. Luo

and Sarnak, and Jakobson have shown that

lim

r !1



ðaÞ



ðbÞ

ðaÞ

ðbÞ

½34

for suitable test functions a, b 2 C

(nPSL(2, R))

(cf. Sarnak (2003)).

Hecke Operators, Entropy

and Measure Rigidity

For compact surfaces, the sequence of probability

measures approaching the matrix elements 

relatively compact. That is, every infinite sequence

contains a convergent subsequence. Lindenstrauss’

central idea in the proof of QUE is to exploit the

presence of Hecke operators to understand the

invariance properties of possible quantum limits.

We will sketch his argument in the case of the

modular surface (ignoring issues related to the non-

compactness of the surface), where it is most

transparent.

For every positive integer n, the Hecke operator

acting on continuous functions on nH with

=SL(2, Z) is defined by

f ðzÞ¼

ﬃﬃﬃ

a;d¼1

ad¼n

d1

b¼0

az þ b



½35

The set M

of matrices with integer coefficients and

determinant n can be expressed as the disjoint union

[

a;d¼1

ad¼n

[

d1

b¼0



0 d



½36

and hence the sum in [35] can be viewed as a sum

over the cosets in this decomposition. We note the

product formula

djgcdðm;nÞ

mn=d

½37

The Hecke operators are normal, form a com-

muting family, and in addition they commute with

the Laplacian . In the following, we consider an

orthonormal basis of eigenfunctions ’

of  that

are simultaneously eigenfunctions of all Hecke

operators. We will refer to such eigenfunctions as

Hecke eigenfunctions. The above assumption is

automatically satisfied, if the spectrum of  is

simple (i.e., no eigenvalues coincide), a property

conjectured by Cartier and supported by numerical

computations. Lindenstrauss’ work is based on the

following two observations. Firstly, all quantum

limits of Hecke eigenfunctions are geodesic-flow

invariant measures of positive entropy, and sec-

ondly, the only s uch m easure of positive entropy

that is recurrent under Hecke correspondences is

the Lebesgue measure.

The first property is proved by Bourgain and

Lindenstrauss (2003) and refines arguments of

Rudnick and Sarnak (1994) and Wolpert (2001) on

the distribution of Hecke points (see Sarnak (2003) for

Arithmetic Quantum Chaos 217

references to these papers). For a given point z 2 H

the set of Hecke points is defined as

ðzÞ :¼ M

z ½38

For most primes, the set T

(z) comprises (p þ 1)

k1

distinct points on nH. For each z, the Hecke

operator T

may now be interpreted as the

adjacency matrix for a finite graph embedded in

nH, whose vertices are the Hecke points T

(z).

Hecke eigenfunctions ’

with

’

¼ 

ðnÞ’

½39

give rise to eigenfunctions of the adjacency matrix.

Exploiting this fact, Bourgain and Lindenstrauss

show that for a large set of integers n

j’

ðzÞj



w2T

ðzÞ

j’

ðwÞj

½40

that is, pointwise values of j’

cannot be substan-

tially larger than its sum over Hecke points. This,

and the observation that Hecke points for a large set

of integers n are sufficiently uniformly distributed

on nH as n !1, yields the estimate of positive

entropy with a quantitative lower bound.

Lindenstrauss’ proof of the second property,

which shows that Lebesgue measure is the only

quantum limit of Hecke eigenfunctions, is a result of

a currently very active branch of ergodic theory:

measure rigidity. Invariance under the geodesic flow

alone is not sufficient to rule out other possible limit

measures. In fact, there are uncountably many

measures with this property. As limits of Hecke

eigenfunctions, all quantum limits possess an addi-

tional property, namely recurrence under Hecke

correspondences. Since the explanation of these is

rather involved, let us recall an analogous result in a

simpler setup. The map 2:x 7!2x mod 1 defines a

hyperbolic dynamical system on the unit circle with

a wea lth of invariant measures, similar to the case of

the geodesic flow on a surface of negative curvature.

Furstenberg conjectured that, up to trivial invariant

measures that are localized on finitely many rational

points, Lebesgue measure is the only 2-invariant

measure that is also invariant unde r action of

3:x 7!3x mod 1. This fundamental problem is

still unsolved and one of the central conjectures in

measure rigidity. Rudolph, however, showed that

Furstenberg’s conjecture is true if one restricts the

statement to 2-invariant measures of positive

entropy (cf. Lindenstrauss (to appear)). In Linden-

strauss’ work, 2 plays the role of the geodesic

flow, and 3 the role of the Hecke correspondences.

Although here it might also be interesting to ask

whether an analog of Furstenberg’s conjecture

holds, it is ines sential for the proof of QUE due to

the positive entropy of quantum limits discussed in

the previous paragraph.

Eigenfunctions and L-Functions

An even eigen function ’

(z) for =SL(2, Z) has the

Fourier expansion

’

ðzÞ¼

n¼1

ðnÞy

1=2

i

ð2nyÞcosð2nxÞ½41

We associate with ’

(z) the Dirichlet series

Lðs;’

Þ¼

n¼1

ðnÞn

s

½42

which converges for Re s large enough. These series

have an analytic continuation to the entire complex

plane C and satisfy a functional equation,

ðs;’

Þ¼ð1  s;’

Þ½43

where

ðs;’

Þ¼

s



s þ i





s  i



Lðs;’

Þ½44

If ’

(z) is in addition an eigenfunction of all Hecke

operators, then the Fourier coefficients in fact

coincide (up to a normalization constant) with the

eigenvalues of the Hecke operators

ðmÞ¼

ðmÞa

ð1Þ½45

If we normalize a

(1) = 1, the Hecke relations [37]

result in an Euler product formula for the

L-function,

Lðs;’

Þ¼

p prime

1  a

ðpÞp

s

þ p

12s



1

½46

These L-functions behave in many other ways like

the Riemann zeta or classical Dirichlet L-functions.

In particular, they are expected to satisfy a Riemann

hypothesis, that is, all nontrivial zeros are con-

strained to the critical line Ims = 1=2.

Questions on the distribution of Hecke eigenfunc-

tions, such as QUE or value distribution properties,

can now be translated to analytic properties of

L-functions. We will discuss two examples.

The asymptotics in [6] can be esta blished

by proving [6] for the choices g = ’

, k = 1, 2, ...,

that is,

j’

’

dA !0 ½ 47

218 Arithmetic Quantum Chaos

Watson discovered the remarkable relation (Sarnak

2003)

’





ð

;’

 ’

ð1; sym

’

Þð1; sym

’

Þð1; sym

’

½48

The L-functions (s, g) in Watson’s formula are

more advanced cousins of those introduced earlier

(see Sarnak (2003) for details). The Riemann

hypothesis for such L-functions then implies, via

[48], a preci se rate of convergence to QUE for the

modular surface,

j’

g dA ¼

g dA þ Oð

1=4þ

Þ½49

for any >0, where the implied constant depends

on  and g.

A second example on the connection between

statistical properties of the matrix elements



(a) = hOp(a)’

, ’

i (for fixed a and ran dom j) and

values L-functions has appeared in the work of Luo

and Sarnak (cf. Sarnak (2003)). Define the variance



ðaÞ¼

NðÞ







ðaÞðaÞ



½50

with N() =#{j: 

 }; cf. [3]. Following a conjec-

ture by Feingold–Peres and Eckhardt et al. (see Sarnak

(2003) for references) for ‘‘generic’’ quantum chaotic

systems, one expects a central-limit theorem for the

statistical fluctuations of the 

(a), where the normal-

ized variance N()

1=2



(a) is asymptotic to the

classical autocorrelation function C(a), see eqn [54].

Conjecture 4 For any bounded function g : R !C

we have

NðÞ







ðaÞðaÞ

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ



ðaÞ

ﬃﬃﬃﬃﬃﬃ

2

1

gðtÞe

ð1=2Þt

dt ½51

as  !1.

Luo and Sarnak prove that in the case of the

modular surface the variance ha s the asymptotics

lim

!1

NðÞ

1=2



ðaÞ¼hBa; ai½52

where B is a non-negative self-adjoint operator

which commutes with the Laplacian  and all

Hecke operators T

. In particular, we have

B’

;’



Cð’

Þ’

½53

where

CðaÞ:¼

nPSLð2;RÞ

a 

ðgÞðÞaðgÞdðgÞdt ½54

is the classical autocor relation function for the

geodesic flow with respect to the observable a

(Sarnak 2003). Up to the arithmetic factor

(1=2)L(1=2, ’

), eqn [53] is consistent with the

Feingold–Peres prediction for the variance of generic

chaotic systems. Furthermore, recent estimates of

moments by Rudnick and Soundararajan (2005)

indicate that Conjecture 4 is not valid in the case of

the modular surface.

Quantum Eigenstates of Cat Maps

Cat maps are probably the simplest area-preserving

maps on a compact surface that are highly chaotic.

They are defined as linear automorphisms on the

torus T

= R



: T

½55

where a point  2 R

(mod Z

) is mapped to

A(mod Z

); A is a fixed matrix in GL(2, Z) with

eigenvalues off the unit circle (this guarantees

hyperbolicity). We view the torus T

as a symplectic

manifold, the phase space of the dynamical system.

Since T

is compact, the Hilbert space of quantum

states is an N-dimensional vector space H

, N

integer. The semiclassical limit, or limit of small

wavelengths, corresponds here to N !1.

It is convenient to identify H

with L

(Z=NZ),

with the inner product

;

i¼

Q mod N

ðQÞ

ðQÞ½56

For any smooth function f 2 C

), define a

quantum observable

ðf Þ¼

n2Z

f ðnÞT

ðnÞ

where

f (n) are the Fourier coefficients of f, and

(n) are translation operators

ðnÞ¼e

in

½57

½t

ðQÞ¼ ðQ þ 1Þ

½t

ðQÞ¼e

2iQ=N

ðQÞ

½58

The operators Op

(a) are the analogs of the

pseudodifferential operators discussed in the section

‘‘Quantum unique ergodicity.’’

Arithmetic Quantum Chaos 219

A quantization of 

is a unitary operator U

(A)

on L

(Z=NZ) satisfying the equation

ðAÞ

1

ðf ÞU

ðAÞ¼Op

ðf  

Þ½59

for all f 2 C

). There are explicit formulas for

(A) when A is in the group

 ¼



2 SLð2; ZÞ: ab  cd  0 mod 2



½60

These may be viewed as analogs of the Shale–Weil

or metaplectic representation for SL(2). for example,

the quantization of

A ¼



½61

yields

ðAÞ ðQÞ¼N

1=2

mod N

exp



2i

ðQ

 QQ

þ Q



ðQ

Þ½62

In analogy with [1], we are interested in the

statistical features of the eigenvalues and eigenfunc-

tions of U

(A), that is, the solutions to

ðAÞ’ ¼ ’; k’k

ðZ=NZÞ

¼ 1 ½63

Unlike typical quantum-chaotic maps, the statistics

of the N eigenvalues



;

; ...;

2 S

½64

do not follow the distributions of unitary random

matrices in the limit N !1, but are rather singular

(Keating 1991). In analogy with the Selberg trace

formula for hyperbolic surfaces [18], there is an

exact trace form ula relating sums over eigenvalues

of U

(A) with sums over fixed points of the classical

map (Keating 1991).

As in the case of arithmetic surfaces, the eigenfunc-

tions of cat maps appear to behave more generically.

The analog of the Schnirelman–Zelditch–Colin de

Verdie` re theorem states that, for any orthonormal

basis of eigenfunctions {’

}

j = 1

we have, for all

f 2 C

hOpðf Þ’

;’

f ðÞd ½65

as N !1, for all j in an index set J

of full density,

that is, #J

 N. Kurlberg and Rudnick (see

Rudnick (2001)) have characterized special bases of

eigenfunctions {’

}

j = 1

(termed Hecke eigenbases,

in analogy with arithmetic surfaces) for which QUE

holds, generalizing earlier work of Degli Esposti,

Graffi, and Isola (1995). That is, [65] holds for all

j = 1, ..., N. Rudnick and Kurlberg, and more

recently Gurevich and Hadani, have established

results on the rate of convergence analogous to

[49]. These results are unconditional. Gurevich and

Hadani use methods from algebraic geometry based

on those developed by Deligne in his proof of the

Weil conjectures (an analog of the Riemann hypoth-

esis for finite fields).

In the case of quantum-cat maps, there are values

of N for whi ch the number of coinciding eigenvalues

can be large, a major difference to what is expected

for the modular surface. Linear combinations of

eigenstates with the same eigenvalue are as well

eigenstates, and may lead to different quantum

limits. Indeed, Faure, Nonnenmacher, and De Bie` vre

(see De Bie` vre (to appear)) have shown that there

are subsequences of values of N, so that, for all

f 2 C

hOpðf Þ’

;’

f ðÞd þ

f ð0Þ½66

that is, half of the mass of the quantum limit

localizes on the hyperbolic fixed point of the map.

This is the first, and to date the only, rigorous result

concerning the existence of scarred eigenfunctions in

systems with chaotic classical limit.

Acknowledgment

The author is supported by an EPSRC Advanced

Research Fellowship.

See also: Quantum Ergodicity and Mixing of

Eigenfunctions; Random Matrix Theory in Physics.

Further Reading

Aurich R and Steiner F (1993) Statistical properties of highly

excited quantum eigenstates of a strongly chaotic system.

Physica D 64(1–3): 185–214.

Berry MV and Keating JP (1999) The Riemann zeros and

eigenvalue asymptotics. SIAM Review 41(2): 236–266.

Bogomolny EB (2006) Quantum and arithmetical chaos. In:

Cartier PE, Julia B, Moussa P, and Vanhove P (eds.) Frontiers

in Number Theory, Physics and Geometry on Random

Matrices, Zeta Functions, and Dynamical Systems, Springer

Lecture Notes. Les Houches.

Bogomolny EB, Georgeot B, Giannoni M-J, and Schmit C (1997)

Arithmetical chaos. Physics Reports 291(5–6): 219–324.

De Bie` vre S Recent Results on Quantum Map Eigenstates,

Proceedings of QMATH9, Giens 2004 (to appear).

Hejhal DA and Rackner BN (1992) On the topography of Maass

waveforms for PSL(2, Z). Experiment. Math. 1(4): 275–305.

Keating JP (1991) The cat maps: quantum mechanics and classical

motion. Nonlinearity 4(2): 309–341.

220 Arithmetic Quantum Chaos

Lindenstrauss E Rigidity of multi-parameter actions. Israel

Journal of Mathematics (Furstenberg Special Volume) (to

appear).

Marklof J (2006) Energy level statistics, lattice point problems and

almost modular functions. In: Cartier PE, Julia B, Moussa P,

and Vanhove P (eds.) Frontiers in Number Theory, Physics and

Geometry on Random Matrices, Zeta Functions, and Dynami-

cal Systems, Springer Lecture Notes. Les Houches.

Rudnick Z (2001) On quantum unique ergodicity for linear maps

of the torus. In: European Congress of Mathematics,

(Barcelona, 2000), Progr. Math., vol. 202, pp. 429–437.

Basel: Birkha¨user.

Rudnick Z (2005) A central limit theorem for the spectrum of the

modular group, Park city lectures. Annales Henri Poincare´ 6:

863–883.

Sarnak P Arithmetic quantum chaos. The Schur lectures (1992)

(Tel Aviv), Israel Math. Conf. Proc., 8, pp. 183–236. Bar-Ilan

Univ., Ramat Gan, 1995.

Sarnak P (2003) Spectra of hyperbolic surfaces. Bulletin of the

American Mathematical Society (N.S.) 40(4): 441–478.

Asymptotic Structure and Conformal Infinity

J Frauendiener, Universita

tTu

bingen, Tu

bingen,

Germany

Introduction

A major motivation for studying the asymptotic

structure of spacetimes has been the need for a

rigorous description of what should be understood by

an ‘‘isolated system’’ in Einstein’s theory of gravity.

As an example, consider a gravitating system some-

where in our universe (e.g., a galaxy, a cluster of

galaxies, a binary system, or a star) evolving accord-

ing to its own gravitational interaction, and possibly

reacting to gravitational radiation impinging on it

from the outside. Thereby it will emit gravitational

radiation. We are interested in describing these waves

because they provide us with important information

about the physics governing the system.

To adequately describe this situation, we need to

idealize the real situation in an appropriate way, since

it is hopeless to try to analyze the behavior of the

system in its interaction with the rest of the universe.

We are mainly interested in the behavior of the

system, an d not so much in other processes taking

place at large distances from the system. Since we

would like to ignore those regions, we need a way to

isolate the system from their influence.

The notion of an isolated system allows us to

select individual subsystems of the universe and

describe their properties regardless of the rest of the

universe so that we can assign to each subsystem

such physical attributes as its energy–momentum,

angular momentum, or its emitted radiation field.

Without this notion, we would always have to take

into account the interaction of the system with its

environment in full detail.

In general relativity (GR) it turns out to be a rather

difficult task to describe an isolated system and the

reason is – as always in Einstein’s theory – the fact

that the metric acts both as the physical field and as

the background. In other theories, like electrody-

namics, the physical field, such as the Maxwell field,

is very different from the background field, the flat

metric of Minkowski space. The fact that the metric

in GR plays a dual role makes it difficult to extract

physical meaning from the metric because there is no

nondynamical reference point.

Imagine a system alone in the universe. As we

recede from the system we would expect its influence

to decrease. So we expect that the spacetime which

models this situation mathematically will resemble

the flat Minkowski spacetime and it will approximate

it even better the farther away we go. This implies

that one needs to impose fall-off conditions for the

curvature and that the manifold will be asymptoti-

cally flat in an appropriate sense. However, there is

the problem that fall-off conditions necessarily imply

the use of coordinates and it is awkward to decide

which coordinates should be ‘‘good ones.’’ Thus, it is

not clear whether the notion of an asymptotically flat

spacetime is an invariant concept.

What is needed, therefore, is an invariant defini-

tion of asymptotically flat spacetimes. The key

observation in this context is that ‘‘infinity’’ is far

away with respect to the spacetime metric. This

means that geodesics heading away from the system

should be able to ‘‘run forever,’’ that is, be defined

for arbitrary values of their affine parameter s.

‘‘Infinity’’ will be reached for s !1. However,

suppose we do not use the spacetime metric g but a

metric

g which is scaled down with respe ct to g, that

is, in such a way that

g =

g for some function .

Then it might be possible to arrange  in such a way

that geodesics for the metric

g cover the same events

(strictly speaking, this holds only for null geodesics,

but this is irrelevant for the present plausibility

argument) as those for the metric g yet that their

affine parameter

s (which is also scaled down with

respect to s) approaches a finite value

for s !1.

Then we could attach a boundary to the spacetime

manifold consisting of all the limit points corre-

sponding to the events with

s =

on the

g-geodesics.

Asymptotic Structure and Conformal Infinity 221

This boundary would have to be interpreted as

‘‘infinity’’ for the spacetime because it takes infinitely

long for the g-geodesics to get there.

We arrived at this idea of attaching a boundary by

considering the metric structure only ‘‘up to arbi-

trary scaling,’’ that is, by looking at metrics which

differ only by a factor. This is the conformal

structure of the spacetime manifold in question. By

considering the spacetime only from the point of

view of its conformal structure we obtain a picture

of the spacetime which is essentially finite but which

leaves its causal properties unchanged, and hence in

particular the properties of wave propagation. This

is exactly what is needed for a rigorous treatment of

radiation emitted by the system.

Infinity for Minkowski Spacetime

The above discussion suggests that we should consider

the spacetime metric only up to scale, that is,

to focus on the conformal structure of the spacetime

in question. Since we are interested in systems which

approach Minkowski spacetime at large distances

from the source, it is illuminating to study Minkowski

spacetime as a preliminary example. So consider the

manifold M = R

equipped with the flat metric

g ¼ dt

 dr

 r

d

½1

where r is the standar d radial coordinate defined by

= x

þ y

þ z

and

d

¼ d

þ sin

d

is the standard metric on the unit sphere S

. We now

introduce retarded and advanced time coordinates,

which are adapted to the null cone and hence to the

conformal structure of g by the definition

u ¼ t  r; v ¼ t þ r

and obtain the metric in the form

g ¼ du dv 

ðv  uÞ

d

The coordinates u and v both take arbitrary real values

but they are restricted by the relation v  u = 2r  0.

In order to see what happens ‘‘at infinity,’’ we introduce

the coordinates U and V by the relations

u ¼ tan U; v ¼ tan V

Then U and V both take values in the open interval

(=2, =2) with V  U and the metric is trans-

formed to

g ¼

4 cos

U cos

4dU dV sin

ðV  UÞd



½2

Clearly, the metric is undefined at events with

cos U = 0orcosV = 0. These would correspond to

events with u = 1 or v = 1 which do not lie in

M. However, by defining the function

 ¼ 2 cos U cos V

we find that the metric

g =

g with

g ¼ 4dU dV  sin

ðV  UÞd

½3

is conformally equivalent to g and is regular for all

values of U and V (keeping V  U). In fact, by

defining the coordinates

T ¼ V þ U; R ¼ V  U

this metric takes the form

g ¼ dT

 dR

 sin

R d

½4

the metric of the static Einstein universe E. Thus, we

may regard the Minkowski spacetime as the part of

the Einstein cylinder defined by restricting the

coordinates T and R to the region j TjþR < as

illustrated in Figure 1. Although M can be considered

as being diffeomorphic to the shaded part in Figure 1,

these two manifolds are not isometric. This is obvious

from considering the properties of the events lying on

–

Figure 1 The embedding of Minkowski spacetime into the

Einstein cylinder.

222 Asymptotic Structure and Conformal Infinity

the boundary @M of M in E. Fix a point P inside M

and follow a null geodesic with respect to the metric

from P toward the future. It will intersect @M after a

finite amount of its affine parameter has elapsed.

When we follow a null geodesic with respect to g

from P in the same direction, we find that it does not

reach @M for any value of its affine parameter. Thus,

the boundary is at infinity for the metric g but at a

finite location with respect to the metric

g. When we

consider all possible kinds of geodesics for the metric g

we find that @M consists of five qualitatively different

pieces. The future pointing timelike geodesics all

approach the point i

given by (T, R) = (, 0), while

the past-pointing geodesics approach i



with coordi-

nates (, 0). All spacelike geodesics come arbitrarily

close to a point i

with coordinates (0, )(locatedon

the front of the cylinder in Figure 1). Null geodesics,

however, are different. For any point (T,  jTj)with

T 6¼ 0,  on @M there are g-null-geodesics which

come arbitrarily close.

In this sense, we may regard @M as consisting of

limit points obtained by tracing-geodesics for infi-

nite values of their affine parameters. According to

the causal character of the geodesics the set of their

respective limit points is called future/past timelike

infinity i



, spacelike infinity i

or future/past null-

infinity, denoted by I



. These two parts of null-

infinity are three-dimensional regular submanifolds

of the embedding manifold E, while the points i



, i

are regular points in E in the sense that the metric

is regular there. This is not automatic, considering

the fact that infinitely many geodesics converge to a

single point. However, the flatness of Minkowski

spacetime guarantees that the geodesics approach at

just the appropriate rate for the limit points to be

regular.

This example shows that the structure of the

boundary is determined entirely by the metric g of

Minkowski spacet ime. If we had chosen a different

function 

= ! with !>0 then we would not

have obtained the Einstein cylinder but some

different Lorentzian manifold (M

, g

). Yet, the

boundary of M in M

would have had the same

properties.

Asymptotically Flat Spacetimes

The physical idea of an isolated system is captured

mathematically by an asymptotically flat space-

time. Since such a spacetime M is expected to

approach Minkowski space time asymptotically,

the asymptotic structure of M is also expected to

be similar to t hat of M. This expectation is

expressed in

Definition 1 A spacetime (M, g

) is called ‘‘asymp-

totically simple’’ if there exists a manifold-with-

boundary

M with metric

and scalar field  on

M and boundary I = @M such that the following

conditions hold:

1. M is the interior of

M: M= int

=

on M;

3.  and

are smooth on all of



4.  > 0onM ; =0, r

 6¼ 0onI ; and

5. each null geodesic acquires both future and past

endpoints on I .

This definition formalizes the construction which

was explicitly performed above, by which one

attaches a regular (nonempty) boundary to a sp ace-

time after suitably rescaling its metric. Asymptoti-

cally simple spacetimes are exactly those for which

this process of conformal compactification is possi-

ble. The purpose of condition 5 is to exclude

pathological cases. There are spacetimes which do

not satisfy this condition (e.g., the Schwarzschild

spacetime, where some of the null geodesics enter

the event horizon and cannot escape to infinity).

Yet, one would like to include them as being

asymptotically simple in a sense, because they

clearly describe isolated systems. For these cases,

there exists the notion of weakly asymptotically

simple spacetimes.

In order to arrive at asymptotically flat space-

times, one needs to make certain assumptions about

the behavior of the curvature near the boundary,

thus:

Definition 2 An asymptotically simple spacetime is

called ‘‘asymptotically flat’’ if its Ricci tensor Ric[g]

vanishes in a neighborhood of I .

Note that this definition imposes a rather strong

restriction on the Ricci curvature; less restrictive

assumptions are possible. This condition applies

only near I . Thus, it is possible to consider

spacetimes which contain matter fields as long as

these fields do not extend to infinity.

Other asymptotically simple spacetimes which are

not asymptotically flat are the de Sit ter and anti-de

Sitter spacetimes which are solutions of the Einstein

equations with nonvanishing cosmological constant .

It is a simple consequence of the definition that

the boundary I is a regular three-dimensional

hypersurface of the embedding spacetime

M which

is timelike, spacelike, or null depending on the sign

of . In particular, for the Minkowski spacetime

( = 0) the boundary is necessarily a null hypersur-

face, as noted above.

The requirement that the vacuum Einstein

equations hold near I has several important

Asymptotic Structure and Conformal Infinity 223

consequences. First, I is a null hypersurface with

the special property of being shear-free. This means

that any cross section of a bundle of its null

generators does not suffer any distortions when

moved along the generators. Only exp ansion or

contraction can occur. The global structure of I

is the same as the one from the example above.

Null infinity consists of two connected components,



, each of which is diffeomorphic to S

 R. Thus,

topologically, I



are cylinders. The cone-like

appearance as seen in Figure 1 is artificial. It

depends on the particular conformal factor  chosen

for the conformal compactification. Furthermore, it

is only in very exceptional cases that the metric

g is

regular at i

or i



The most important consequence, however, con-

cerns the conformal Weyl tensor C

bcd

. This is the

part of the full Riemann curvature tensor R

bcd

which

is trace-free. It is invariant under conformal rescal-

ings of the metric. Thus, on M, C

bcd

.When

the vanishing of the Ricci tensor near I is assumed

then it turns out that the Weyl tensor necessarily

vanishes on I . This is the ultimate justification for

calling such manifolds asymptotically flat because the

entire curvature vanishes on I .

Some Consequences

There are several consequences of the existence of

the conformal boundary I . They all can be traced

back to the fact that this boundary can be used to

separate the geometric fields into a universal back-

ground field and dynamical fields which propagate

on it. The background is given by the boundary

points attached to an asymptotically flat spacetime

which always form a three-dimensional null hyper-

surface I with two connected components (in the

sequel, we restrict our attention to I

only; I



treated similarly), each with the topology of a

cylinder. And in each case, I is shear-free.

The BMS Group

Since the structure of null-infinity is universal over

all asymptotically flat spacetimes, it is obvious that

its symmetry group should also possess a universal

meaning. This group, the so-called Bondi–Metzner–

Sachs (BMS) group is in many respects similar to the

Poincare´ group, the symmetry group of M.Itisthe

semidirect product of the Lorentz group with an

abelian group which, however, is not the four-

dimensional translat ion group but an infinite-dimen-

sional group of supertranslations. This group is a

normal subgroup, so the factor group is isomorphic

to the Lorentz group.

In physical terms, the supertranslations arise

because there are infinitely many directions from

which observers at infinity (whose world lines coincide

with the null generators of I in a certain limit) can

observe the system and because each observer is free to

choose its own origin of proper time u. The observers

surrounding the system are not synchronized, because

under the assumptions made there is no natural way to

fix a unique common origin. Hence, a supertranslation

is a shift of the parameter along each null generator of

corresponding to a change of origin for each

individual observer. It can be given as a map S

! R.

A choice of origin on each null generator of I

referred to as a ‘‘cut’’ of I

. It is a two-dimensional

surface of spherical topology which intersects each null

generator exactly once. It is an open question whether

one can always synchronize the observers by imposing

canonical conditions at i

or i



, thereby reducing the

BMS group to the smaller Poincare´group.

The supertranslations contain a unique four-

dimensional normal subgroup. In M these special

supertranslations are the ones which are induced by

the translations of Minkowski spacetime in the

following way. Take the future light cone of some

event P and follow it out to I

, where its intersection

defines an origin for each observer located there.

Now consider the light cone of another event Q

obtained from P by a translation in a spatial

direction. Then the light emitted from Q will arrive

at I

earlier than that from P for observers in the

direction of the translation, while it will be delayed

for observers in the opposite direction. This change

in arrival time defines a specific supertranslation.

Similarly, for a translation in a temporal direction,

the light from Q will arrive later than that from P

for all observers. Thus, every translation in M

defines a particular supertranslation on I

.These

can be characterized in a different way, which is

intrinsic to I

and which can be used in the general

case even though there will be no Killing vectors

present in a general asymptotically flat spacetime. In

an appropriate coordinate system, the asymptotic

translations are given as linear combinations of the

first four spherical harmonics Y

, Y

11

.The

space of asymptotic translations T is in a natural

way isometric to M.

The Peeling Property

Now consider the Weyl tensor C

bcd

M. Since it

vanishes on I where =0 we may form the

quotient

bcd

¼ 

1

bcd

224 Asymptotic Structure and Conformal Infinity