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

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ones. In the vacuum region, these are all given by the

Schwarzschild family. A theorem of Birkhoff shows

that in the vacuum region any spherically symmetric

metric, even without assuming stationarity, belongs to

the family of Schwarzschild metrics, parametrized by a

positive mass parameter m. Thus, regardless of

possible motions of the matter, as long as they remain

spherically symmetric, the exterior metric is the

Schwarzschild one for some constant m. This has the

following consequence for stellar dynamics: imagine

following the collapse of a cloud of pressureless fluid

(‘‘dust’’). Within Newtonian gravity, this dust cloud

will, after finite time, contract to a point at which the

density and the gravitational potential diverge. How-

ever, this result cannot be trusted as a sensible physical

prediction because, even if one supposes that New-

tonian gravity is still valid at very high densities, a

matter model based on noninteracting point particles

is certainly not. Consider, next, the same situation in

the Einstein theory of gravity: here a new question

arises, related to the form of the Schwarzschild metric

outside of the spherically symmetric body:

g ¼V

þ V

2

þ r

d

;

¼ 1 

2Gm

;

t 2 R; r 2

2Gm

; 1



½2

Here d

is the line element of the standard

2-sphere. Since the metric [2] seems to be singular as

r = 2m is approached (from now on, we use units in

which G = c = 1), there arises the need to understand

what happens at the surface of the star when the

radius r = 2m is reached. One thus faces the need of

a careful study of the geometry of the metric [2]

when r = 2m is approached, and crossed.

The first key feature of the metric [2] is its

stationarity, of course, with Killing vector field X

given by X = @

. A Killing field, by definition, is a

vector field the local flow of which generates isome-

tries. A spacetime (the term spacetime denotes a

smooth, paracompact, connected, orientable, and

time-orientable Lorentzian manifold) is called station-

ary if there exists a Killing vector field X which

approaches @

in the asymptotically flat region (where r

goes to 1; see below for precise definitions) and

generates a one-parameter group of isometries. A

spacetime is called static if it is stationary and if the

stationary Killing vector X is hypersurface orthogonal ,

that is, X

[

^ dX

[

= 0, where X

[

= X



= g







A spacetime is called axisymmetric if there exists a

Killing vector field Y, which generates a one-parameter

group of isometries and which behaves like a rotation

in the asymptotically flat region, with all orbits

2-periodic. In asymptotically flat spacetimes, this

implies that there exists an axis of symmetry, that is, a

set on which the Killing vector vanishes. Killing vector

fields which are a nontrivial linear combination of a

time translation and of a rotation in the asymptotically

flat region are called stationary rotating, or helical.

There exists a technique, due independently to

Kruskal and Szekeres, of attaching together two

regions r > 2m and two regions r < 2m of the

Schwarzschild metric, as in Figure 1, to obtain a

manifold with a metric which is smooth at r = 2m.

In the extended spacetime, the hypersurface {r = 2 m}

is a null hypersurface e, the Schwarzschild event

horizon. The stationary Killing vector X = @

extends to a Killing vector in the extended spacetime

which becomes tangent to and null on e. The global

properties of the Kruskal–Szekeres extension of the

exterior Schwarzschild spacetime make this spacetime

a natural model for a nonrotating black hole. It is

worth noting here that the exterior Schwarzschild

spacetime [2] admits an infinite number of noniso-

metric vacuum extensions, even in the class of

maximal, analytic, simply connected ones. The

Kruskal–Szekeres extension is singled out by the

properties that it is maximal, vacuum, analytic, simply

connected, with all maximally extended geodesics

either complete, or with the area r of the orbits of the

isometry groups tending to zero along them.

We can now come back to the problem of the

contracting dust cloud according to the Einstein

theory. For simplicity, we take the density of the

dust to be uniform – the so-called Oppenheimer–

Snyder solution. It then turns out that, in the course

of collapse , the surface of the dust will eventually

cross the Schwarzschild radius, leav ing behind a

Schwarzschild black hole. If one follows the dust

cloud further, a singularity will eventually form, but

will not be visible from the ‘‘outside regi on’’ where

r > 2m. For a collapsing body of the mass of the

Sun, say, one has 2m = 3 km. Thus, standard

phenomenological matter models such as that for

dust can still be trusted, so that the previous

objection to the Newtonian scenario does not apply.

There is a rotating generalization of the Schwarz-

schild metric, namely the two-parameter family of

exterior Kerr metrics, whi ch in Boyer–Lindquist

coordinates takes the form

g ¼

 a

sin







2asin

ðr

þa

Þ



dt d’

ðr

þa

a

sin





sin

d’





þ d

½3

Stationary Black Holes 39

with 0 a < m. Here =r

þa

cos

,=r

þa



2mr and r

< r < 1 where r

= m þ(m

a

)

1=2

When a= 0, the Kerr metric reduces to the

Schwarzschild metric. The Kerr metric is again a

vacuum solution, and it is stationary with X= @

the

asymptotic time translation, as well as axisymmetr ic

with Y = @

’

the generator of rotations. Similarly to

the Schwarzschild case, it turns out that the metric

can be smoothly extended across r= r

, with {r= r

}

being a smooth null hypersurface e in the extension.

The null generator K of e is the limit of the

stationary-rotating Killing field X þ!Y, where

!= a=(2mr

). On the other hand, the Killing vector

X is timelike only outside the hypersurface {r = m þ

a

cos

)

1=2

}, on which X becomes null. In the

region between r

and r= m þ(m

a

cos

)

1=2

which is called the ergore gion, X is spacelike. It is

also spacelike on and tangent to e, except where the

axis of rotation meets e, where X is null. Based on

the above properties, the Kerr family provides

natural models for rotating black holes.

Unfortunately, as opposed to the spherically

symmetric case, there are no known explicit collap-

sing solutions with rotating matter, in particular no

known solutions having the Kerr metric as final

state.

The aim of the theory outlined below is to

understand the general geometrical features of

stationary black holes, and to give a classification

of models satisfying the field equations.

Model-Independent Concepts

Some of the notions used informally in the

introductory section will now be made more

precise. The mathematical notion of black hole is

meant to capture the idea of a region of spacetime

which cannot be seen by ‘‘outside observers.’’ Thus,

at the outset, one assumes that there exists a family

of physically preferred observers in the spacetime

under consideration. When considering isolated

physical systems, it is na tural to define the ‘‘exterior

observers’’ as observers which are ‘‘very far’’ away

from the system under consideration. The standard

way of making this mathematically precise is by

using conformal completions, discussed in more

detail in the article about asymptotic structure in

this encyclopedia: a pair (

g) is called a con-

formal completion at infinity, or simply conformal

completion, of (m, g)if

m is a manifold with

boundary such that:

1. m is the interior of

2. there exists a function , with the property that

the metric

g, defined as 

g on m, extends by

continuity to the boundary of

m, with the

r = constant > 2m

r = constant > 2

r = constant < 2m

= constant < 2m

= constant

Singularity (r

= 0)

Singularity (r

= 0)

= 2m

III

Figure 1 The Kruskal–Szekeres extension of the Schwarzschild solution. (Adapted with permission from Nicolas J-P (2002) Dirac fields

on asymptotically flat space-times. Dissertationes Mathematicae 408: 1–85.)

40 Stationary Black Holes

extended metric remaining of Lorentzian signa-

ture; and

3.  is positive on m, differentiable on

m, vanishes

on the boundary

i :¼

m n m

with d nowhere vanishing on i.

The bounda ry i of

m is called Scri, a phonic

shortcut for ‘‘script I.’’ The idea here is the

following: forcing  to vanish on i ensures that i

lies infinitely far away from any physical object – a

mathematical way of capturing the notion ‘‘very far

away.’’ The condition that d does not vanish is a

convenient technical condition which ensures that i

is a smooth three-dimensional hypersurface, instead

of some, say, one- or two-dimensional object, or of a

set with singularities here and there. Thus, i is an

idealized description of a family of observers at

infinity.

To distinguish between various points of i, one

sets

¼fpoints in i which are to the futur e of the

physical spacetimeg



¼fpoints in i which are to the past of the

physical spacetimeg

(Recall that a point q is to the future, respectively to

the past, of p if there exists a future directed,

respectively past directed, cau sal curve from p to q.

Causal curves are curves  such that their tangent

vector

 is causal everywhere, g(

,

)  0.) One

then defines the black hole region b as

b :¼fpoints in m which are

not in the past of i

½4

By definition, points in the black hole region cannot

thus send information to i

; equivalently, observers

on i

cannot see points in b. The white-hole region

w is defined by changing the time orientation in [4].

A key notion related to the concept of a black hole is

that of future (e

) and past (e



) event horizons,

:¼ @b; e



:¼ @w ½5

Under mild assumptions, event horizons in station-

ary spacetimes with matter satisfying the null-energy

condition,



‘



‘



 0 for all null vectors ‘



½6

are smooth null hypersurfaces, analytic if the metric

is analytic.

In order to develop a reasonable theory, one

also needs a regularity condition for the interior of

spacetime. This has to be a condition which does not

exclude singularities (otherwise the Schwarzschild

and Kerr black holes would be excluded), but which

nevertheless guarantees a well-behaved exterior

region. One such condition, assumed in all the

results described below, is the existence in m of an

asymptotically flat spacelike hypersurface s with

compact interior. Further, either s has no boundary

or the boundary of s lies on

[ e



. To make

things precise, for any spacelike hypersurface let g

be the induced metric, and let K

denote its extrinsic

curvature. A spacelike hypersurface s

ext

diffeo-

morphic to R

minus a ball will be called asympto-

tically flat if the fields (g

, K

) satisfy the fall-off

conditions

 

jþrj@

‘

jþþr

‘

‘

þ rjK

jþþr

‘

‘

k1

jCr

1

½7

for some constants C, k  1. A hypersurface s (with

or without boundary) will be said to be asymptotically

flat with compact interior if s is of the form s

int

[

ext

,withs

int

compact and s

ext

asymptotically flat.

There exists a canonical way of constructing a

conformal completion with good global properties

for stationary spacetimes which are asymptotically

flat in the sense of [7], and which are vacuum

sufficiently far out in the asymptotic region. This

conformal completion is referred to a s the standard

completion and will be assumed from now on.

Returning to the event horizon e = e

[ e



it is not very difficult to show that every Killing

vector field X is necessarily tangent to e. Since

the latter set is a null Lipschitz hypersurface, it

follows that X is either null or spacelike on e. This

leads to a preferred class of event horizons, called

Killing horizons. By definition, a Killing horizon

associated with a Killing vector K is a null hypersur-

face which coincides with a connected component of

the set

HðKÞ :¼fp 2 m: gðK; KÞðpÞ¼0; KðpÞ 6¼ 0g½8

A simple example is provided by the ‘‘boost Killing

vector field’’ K = z@

þ t@

in Minkowski spacetime:

H(K) has four connected components,



:¼ft ¼ z;t > 0g;;2f1g

The closure



H of H is the set {jtj= jzj}, which is not

a manifold, because of the crossing of the null

hyperplanes {t =  z}att = z = 0. Horizons of this

type are referred to as bifurcate Ki lling horizons,

with the set {K(p) = 0} being called the bifurcation

surface of H(K). The bifurcate horizon structure in

the Kruszkal–Szekeres–Schwarzschild spacetime can

be clearly seen in Figures 1 and 2.

Stationary Black Holes 41

The Vishveshwara–Carter lemma shows that if a

Killing vector K is hypersurface orthogonal, K

[

= 0, then the set H(K ) defined in [8] is a union

of smooth null hypersurfaces, with K being

tangent to the null geodesics threading H (‘‘H is

generated by K’’), and so is indeed a Kill ing

horizon. It has been shown by Carter t hat the

same conclusion can be reached if the hypothesis

of hypersurface orthogonality is replaced by that

of existence of two linearly independent Killing

vector fields.

In stationary-axisymmetric spacetimes, a Killing

vector K tangent to the generators of a Killing

horizon H can be normalized so that K = X þ !Y,

where X is the Killing vector field which asymptotes

to a time translation in the asymptotic region, and Y

is the Killing vector field which generates rotations

in the asymptotic region. The constant ! is called

the angular velocity of the Killing horizon H.

On a Killing horizon H(K), one necessarily has



ðK



Þ¼2K



½9

Assuming the so-called dominant-energy condition

on T



(see Positive Energy Theorem and Other

Inequalities in GR), it can be shown that  is constant

(recall that Killing horizons are always connected in

the terminology used in this article); it is called the

surface gravity of H. A Killing horizon is called

degenerate when  = 0, and nondegenerate other-

wise; by an abuse of terminology, one similarly talks

of degenerate black holes, etc. In Kerr spacetimes we

have  = 0 if and only if m = a. A fundamental

theorem of Boyer shows that degenerate horizons

are closed. This implies that a horizon H(K) such that

K has zeros in



H is nondegenerate, and is of bifurcate

type, as described above. Further, a nondegenerate

Killing horizo n with complete geodesic generators

always contains zeros of K in its closure. However, it

is not true that existence of a nondegenerate horizon

implies that of zeros of K : take the Killing vector field

þ t@

in Minkowski spacetime from which the

2-plane {z = t = 0} has been removed. The universal

cover of that last spacetime provides a spacetime in

which one cannot restore the points which have been

artificially removed, without violating the manifold

property.

The domain of outer communications (DOC) of a

black hole spacetime is defined as

hhmii :¼ m nfb [ wg½10

Thus, hhmii is the region lying outside of the white-

hole region and outside of the black hole region ; it is

the region which can both be seen by the outside

observers and influenced by those.

The subset of hhmii where X is spacelike is called

the ergoregion. In the Schwarzschild spacetime,

! = 0 and the ergoregion is empty, but neither of

these is true in Kerr with a 6¼ 0.

A very convenient method for visualizing the

global structure of spacetimes is provided by the

Carter–Penrose diagrams. An example of such a

diagram is given in Figure 2.

–

Singularity (r = 0)

Singularity (r

= 0)

III

r = constant < 2M

= constant > 2M

= constant < 2M

= constant > 2

r = 2M

= 2M

r = 2M

= 2M

= infinity

r = infinity

t = constant

= constant

Figure 2 The Carter–Penrose diagram for the Kruskal–Szekeres spacetime. There are actually two asymptotically flat regions, with

corresponding i



and e



defined with respect to the second region, but not indicated on this diagram. Each point in this diagram represents

a two-dimensional sphere, and coordinates are chosen so that light cones have slopes 1. Regions are numbered as in Figure 1.(Adapted

with permission from Nicolas J-P (2002) Dirac fields on asymptotically flat space-times. Dissertationes Mathematicae 408: 1–85.)

42 Stationary Black Holes

A corollary of the topological censorship theorem

of Friedman, Schleich, and Witt is that DOCs of

regular black hole spacetimes satisfying the domi-

nant-energy condition are simply connected. This

implies that connected components of event hor-

izons in stationary spacetimes have R  S

topology.

The discussion of t he conc epts as soci ated with

stationary-black hole spacetimes can be concluded

by summarizing the properties of the Schwarzs-

child and Kerr geometries: the extended

Kerr spacetime with m > a is a black hole space-

time with the hypersurface {r = r

} forming a

nondegenerate, bifurcate Killing horizon generated

by the vector field X þ !Y and surface gravity

given by

 ¼

ðm

 a

1=2

2m½m þðm

 a

1=2



In the case a = 0, where the angular velocity !

vanishes, X is hypersurface orthogonal and becomes

the generator of H. The bifurcation surface in this

case is the totally geodesic 2-sphere, along which the

four regions in Figure 1 are joined.

Classification of Stationary Solutions

(‘‘No-Hair Theorems’’)

We confine attention to the ‘‘outside region’’ of

black holes, the DOC. (Except f or the degenerate

case discussed later, the ‘‘inside’’(black hole)

region is not stationary, so that this r estriction

already follows from the requirement of stationar-

ity.) For reasons of space, we only consider

vacuum solutions; there exists a similar t heory

for electro-vacuum black holes. (There is a some-

what less developed theory f or black hole space-

times in the presence of nonabelian gauge fields.)

In connection with a collapse scenario, the vacuum

condition begs the question: collapse of what? The

answer is twofol d: firs t, the re a re large classes of

solutions of Einstein equations describing pure

gravitational waves. It is believed that sufficiently

strong such solutions will form black holes.

(Whether or not they will do that is related to the

cosmic censorship conjecture, see Spacetime Toplogy,

Casual Structure and Singularities.) Consider, next, a

dynamical situation in which matter is initially present.

The conditions imposed in this section correspond

then to a final state in which matter has either been

radiated away to infinity, or has been swallowed by

the black hole (as in the spherically symmetric

Oppenheimer–Snyder collapse described above).

Based on the facts below, it is expected that the

DOCs of appropriately regular, stationary, vacuum

black holes are isometrically diffeomorphic to those

of Kerr black holes:

1. The rigidity theorem (Hawking). Event horizons in

regular, nondegenerate, stationary, analytic

vacuum black holes are either Killing horizons for

X, or there exists a second Killing vector in hhmii.

2. The Killing horizons theorem (Sudarsky–Wald).

Nondegenerate stationary vacuum black holes

such that the event horizon is the union of Killing

horizons of X are static.

3. The Schwarzschild black holes exhaust the family

of static regular vacuum black holes (Israel,

Bunting – Masood-ul-Alam, Chrus

ciel).

4. The Kerr black holes satisfying

> a

½11

exhaust the family of nondegenerate, stationary-

axisymmetric, vacuum, connected black holes.

Here m is the total Arnowitt–Deser–Misner

(ADM) mass, while the product am is the total

ADM angular momentum. (Of course, these

quantities generalize the constants a and m

appearing in the Kerr metric.) The framework

for the proof has been set up by Carter, and the

statement above is due to Robinson.

The above results are collectively known under

the name of no-hair theorems, and they have not

provided the final answer to the problem so far.

There are no a priori reasons known for the

analyticity hypothesis in the rigidity theorem.

Further, degener ate horizons have been completely

understood in the static case only.

Yet another key open question is that of the

existence of nonconnected regular stationary-

axisymmetric vacuum black holes. The following

result is due to Weinstein: let @s

, a = 1, ..., N,be

the connected components of @s.LetX

[

= g







where X



is the Killing vector field which asymptoti-

cally approaches the unit normal to s

ext

. Similarly, set

[

= g







, Y



being the Killing vector field

associated with rotations. On each @s

, there exists

aconstant!

such that the vector X þ !

Y is tangent

to the generators of the Killing horizon intersecting

.Theconstant!

is called the angular velocity of

the associated Killing horizon. Define

¼

8

dX

[

½12

¼

4

dY

[

½13

Stationary Black Holes 43

Such integrals are called Komar integrals. One

usually thinks of L

as the angular momentum of

each connected component of the black hole. Set



¼ m

 2!

½14

Weinstein shows that one necessarily has 

> 0.

The problem at hand can be reduced to a harmonic-

map equation, also known as the Ernst equation,

involving a singular map from R

with Euclidean

metric  to the two-dimensional hyperbol ic space.

Let r

> 0, a = 1, ..., N  1, be the distance in R

along the axis between neighboring black holes as

measured with respect to the (unphysical) metric .

Weinstein proved that for nondegenerate regular

black holes the inequality [11] holds, and that the

metric on hhmii is determined up to isometry by the

3N  1 parameters

ð

; ...;

; L

; ...; L

; r

; ...; r

N1

Þ½15

just described, with r

, 

> 0. These results by

Weinstein contain the no-hair theorem of Carter

and Robinson as a special case. Weinstein also

shows that, for every N  2 and for every set of

parameters [15] with 

, r

> 0, there exists a

solution of the problem at hand. It is known that

for some sets of parameters [15] the solutions will

have ‘‘strut singularities’’ between some pairs of

neighboring black holes, but the existence of the

‘‘struts’’ for all sets of parameters as above is not

known, and is one of the main open problems in our

understanding of stationary-axisymmetric electro-

vacuum black holes. The existence and uniqueness

results of Weinstein remain valid when strut

singularities are allowed in the metric at the outset,

although such solutions do not fall into the category

of regular black holes discussed here.

Acknowledgments

This work was partially supported by a Polish

Research Committee grant 2 P03B 073 24 and by

the Erwin Schroedinger Institute, Vienna.

See also: Asymptotic Structure and Conformal Infinity;

Black Hole Mechanics; Critical Phenomena in

Gravitational Collapse; Einstein Equations: Exact

Solutions; Einstein Equations: Initial Value Formulation;

Geometric Flows and the Penrose Inequality; Spacetime

Topology, Causal Structure and Singularities.

Further Reading

Carter B (1973) Black hole equilibrium states. In: de Witt C and

de Witt B (eds.) Black Holes, Proceedings of the Les Houches

Summer School. New York: Gordon and Breach.

Chrus

ciel PT (2002) Black holes. In: Friedrich H and

Frauendiener J (eds.) Proceedings of the Tu¨ bingen Workshop

on the Conformal Structure of Space-Times, Springer Lecture

Notes in Physics, vol. 604, pp. 61–102 (gr-qc/0201053).

Springer: Heidelberg.

Hawking SW and Ellis GFR (1973) The Large Scale Structure of

Space–Time. Cambridge: Cambridge University Press.

Heusler M (1996) Black Hole Uniqueness Theorems. Cambridge:

Cambridge University Press.

Heusler M (1998) Stationary black holes: uniqueness and beyond.

Living Reviews Relativity 1 (http://www.livingreviews.org/

lrr-1988-6).

Nicolas J-P (2002) Dirac fields on asymptotically flat space-times.

Dissertationes Mathematicae 408: 1–85.

O’Neill B (1995) The Geometry of Kerr Black Holes. Wellesley,

MA: A.K. Peters.

Volkov MS and Gal’tsov DV (1999) Gravitating non-abelian

solitons and black holes with Yang–Mills fields. Physics

Reports 319: 1–83 (hep-th/9810070).

Wald RM (1984) General Relativity. Chicago: University of

Chicago Press.

Stationary Phase Approximation

J J Duistermaat, Universiteit Utrecht, Utrecht,

The Netherlands

Introduction

An oscillatory integral is an integral of the form

Ið!Þ¼

i!’ðÞ

aðÞd ½1

Here the integration is over a smooth k-dimensional

manifold  which is provided with a smooth density

d. The real variable ! plays the role of a frequency

variable, whereas the real-valued smooth function ’ on

 is called the phase function. The amplitude function a

is assumed to be a compactly supported complex

(vector-) valued smooth function on .Thetopicof

this article is the asymptotic behavior of the oscillatory

integral I(!) as the frequency ! tends to infinity.

When the manifold  is not compact and the

amplitude function is not compactly supported, then a

smooth cutoff function may be used to write the

integral as the sum of an integral with a compactly

supported amplitude and one with an amplitude which

is equal to zero in a large compact subset of .The

44 Stationary Phase Approximation

latter integral can be studied if suitable assumptions

are made about the asymptotic behavior of the phase

function and the amplitude at infinity, but this is not

the subject of this article. The use of the exponential

function with purely imaginary argument instead of

the sine and the cosine is just a matter of convenience.

The first observation abou t oscillatory integrals in

the next section is the principle of stationary phase,

which states that the contributions to the integral

which are not rapidly decreasing as ! !1 only

come from the stationary points of ’,thepoints 2 

where the total derivative d’()of’ is equal to zero.

This principle is closely related to the observation that

a superposition of waves is maximal at points where

the waves are in phase, an observation which goes

back to Huygens (1690).

Assume that 

is a nondegenerate stationary point of

’.Thatis,d’(

) = 0andtheHessianD

’(

)of’ at



is nondegenerate. Then 

is an isolated stationary

point of ’, and the contribution to I(!) of a neighbor-

hood of 

has an asymptotic expansion of the form

Ið!Þe

i!’ð

r¼0

k=2r

; r !1

Here the leading coefficient c

is the product of a(

)

with a nonzero constant which only depends on

’(

) and the density d at 

. For increasing r the

coefficients c

depend on the derivatives of ’ and a

at 

of increasi ng ord er (see the sect ion ‘‘The

metho d of stationar y pha se’’).

Usually, even if all the objects are analytic in a

neighborhood of 

, the asymptotic power series

does not converge. However, there are exceptional

cases where the stationary phase approximation is

exact. Assume, for instance, that  is a compact

manifold provided with a symplectic form , ’ is the

Hamiltonian function of a Hamiltonian circle action

on  with isolated fixed points, and a()d = 

=k!.

Then the stationary points of ’ are the fixed points

of the circle action, each stationar y point of ’ is

nondegenerate and I(!) is equal to the sum over the

finitely many stationary points of only the leading

terms of the asymptotic expansions at the stationary

points. This Duistermaat–Heckman formula is a

consequence of a more general localization formula

in equivaria nt cohomolo gy (see the sect ion ‘‘Exact

stationar y phase’’).

For the purpose of applications, but also in the

analysis of oscillatory integrals, it is worthwhile to

allow complex-valued phase functions, but with a

local minimum for the imaginary part at the

stationary point 

of the real parts. That is, the

real part of the exponent i!’() has a local

maximum at 

. An extreme case occurs when

’() = i () for a real-valued function which has a

nondegenerate local minimum at 

, in which case

the integrand is a sharply peaking Gaussian density

at 

. When ’ and a are analytic near 

, then the

method of steepest descent consists of deforming the

path of integration in the complex domain in such a

way that the integrand becomes such a sharply

peaking Gaussian density. During the deformation,

the integral does not change because of Cauchy’s

integral theorem.

An important extension of the theory occurs if the

real-valued phase function and the amplitude are

allowed to depend smoothly on additional para-

meters x, which vary in an n-dimensional smooth

manifold M. The amplitude is also allowed to

depend on !, with an asymptotic expansion of

the form

aðx;;!Þ

r¼0

ðx;Þ!

mþðk=2Þr

as ! !1 ½2

The expansion is supposed to be locally uniformly in

(x, ) and to allow termwise differentiations of any

order with respect to the variables (x, ). Then the

integral

Iðx;!Þ¼

i!’ðx;Þ

aðx;;!Þd

is called an oscillatory integral of order m. Here the

function x 7!I(x, !) is viewed as a continuous

superposition of the -dependent family of oscilla-

tory functions x 7!e

i!’(x, )

a(x, ).

The example which formed the point of departure

of Airy (1838) is that e

i!’(x, )

a(x, , !) is the wave

which arrives at the points x in spacetime which is

sent out by a point  on a reflecting mirror. That is,

at x one collects (= integrates over ) all the waves

sent out by the vari ous points  of the mirror . The

main point of the theory, however, is that in great

generality the solutions of linear partial differential

equations, such as classical wave equations or

quantum mechanical Schro¨ dinger equations, can be

represented, as functions of x, as oscillatory inte-

grals. This construction has led to decisive progress

in the general theory of linear partial differential

equations with smoothly varying coefficients.

According to the principle of stationary phase, the

main asymptotic contributions to the integral come

from the points  such that @’(x, )=@ = 0. The

phase function ’ 2 C

(M  ) is called nondegene-

rate if the (n þ k)  k-matrix

’ðx;Þ

@ðx;Þ@

has rank k when

@’ðx;Þ

@

¼ 0 ½3

Stationary Phase Approximation 45

This is the natural condition to ensure that the set

’

:¼ðx;Þ2M  

@’ðx;Þ

@

¼ 0





is a smooth n-dimensional submanifold of M  .

The condition [3], moreover, implies that the mapping



’

: S

’

3ðx;Þ7! x;

@’ðx;Þ





is a smooth immersion from S

’

into the cotangent

bundle T



M of M. Note that  = @’(x, )=@x is

coordinate invariantly defined as a linear form on

the tangent space T

M of M at the point x. That

is,  2 (T



= the dual space of T

M, and (T



is the fiber of T



M over x. In classical mechanics,



M is the phase space of the position space M,

and a linear form  on T

M is called a momen-

tum vector at the position x.If denotes the cano-

nical symplectic form on T



M, then 



’

 = 0. The

immersion 

’

locally embeds S

’

onto a smooth

n-dimensional submanifold 

’

of M, which is a

Lagrangian manifold in T



M, which by definition

means that 



’

 = 0.

Oscillatory integrals with very different phase

functions and amplitudes can define the same

!-dependent functions on M. The theory of

Ho¨ rmander (1971, section 3.1) says that the germs

of the Lagrangian manifolds 

’

and 

are the same

if and only if ’ and define the same class of

oscillatory integrals. Moreover, every Lagrangian

submanifold  of T



M is locally of the form 

’

for some nondegenerate phase function ’. In this

way, the mapping ’ 7!

’

defines a bij ection

between the set of equivalence classes of germs of

nondegenerate phase functions and the set of germs

of Lagrangian submanifolds of T



M.Let be an

immersed Lagrangian submani fold of T



M.A

global oscillatory integral of order m on M, defined

by , is a locally finite sum u(x, !) of oscillatory

integrals of order m with nondegenerate phase

functions ’ such that 

’

 . The leading terms of

the amplitudes correspond to a section s of a

canonically defined complex line bundle  over ,

which is called the principal symbol of u (see the

section ‘‘The princ ipal symbol on the Lagrangi an

manifol d’’).

If P is a linear partial differential operator, such as

the wave operators, in which the coefficients may

depend in a smooth way on x and in a polynomial

way on !, then the co ndition that Pu is asymptoti-

cally small implies that p = 0on, in which p is a

smooth function on T



M, called the principal

symbol of P. Because  is a Lagrangi an manifold,

the equation p = 0 implies that  is invariant under

the flow of the Hamiltonian system with Hamilton

function equal to p. Furthermore, the principal

symbol s of u satisfies a homogeneous first-order

ordinary differential equation along the solution

curves of the Hamiltonian system. Conversely, these

properties can be used to construct global oscillatory

integrals u which asymptotically satisfy Pu = 0 and

have prescribed initial values. This theory, due to

Maslov (1972), may be viewed as a far reaching

generalization of the WKB method.

Let  :T



M !M :(x, ) 7!x denote the canonical

projection from T



M onto M. The projections into

M of the solution curves in a Lagrangian submani-

fold  of T



M, of a Hamiltonian system which

leaves  invariant, are the ray bundles of geome-

trical optics. If  is not transversal to the fiber of



M at (x, ), then the ray bundle exhibits a caustic

at the point x 2 M, and the oscillatory integral is

asymptotically of larger order than !

near x.

Applying the theory of unfoldings of singularities

to the phase function, one can determine the

structurally stable caustics and obtain normal

forms of the oscillatory integrals in the structurally

stable cases (see the section ‘‘Caustics ’’).

If we also integrate over the frequency variable !,

then we obtain the Fourier integral distributions u of

Ho¨ rmander (1971, sections 1.2 and 3.2). In this case

the corresponding Lagrangian manifold is conic in

the sense that if (x, ) 2 , then (x, ) 2  for every

>0. The wave front set of u, which is the

microlocal singular locus of the distribution u,is

contained in  , with equality if the principal symbol

of u is not equal to zero at the corresponding

stationary points of the phase function. Fourier

integral operators are defined as the linear operators

acting on distributions, of which the distribution

kernels are Fourier integral distributions. Under a

suitable transversality con dition for the Lagrangian

manifolds of the distribution kernels, the composi-

tion of two Fourier integral operators is again a

Fourier integral operator, and the principal symbol

of the composition is a product of the principal

symbols. The proof is an application of the method

of stationary phase. Fourier integral operators are a

very powerful tool in the analysis of linear partial

differential operators with smooth ly varying co effi-

cients (see Ho¨ rmander (1985)).

The Principle of Stationary Phase

The principle of stationary phase says that if the

phase function ’ has no stationary points in

the support of the amplitude function a, then the

46 Stationary Phase Approximation

oscillatory integral [1] is rapidly decreasing, in the

sense that for every N we have I(!) = O(!

N

)as

! !1. For the proof, one introduces a vect or field v

on  such that v’ = 1 on a neighborhood of the

support of a. Then e

i!’

= (i!)

1

v(e

i!’

), and an

integration by parts in [1] yields that

Ið!Þ¼

i!’ðÞ

vaÞðÞd

where

v denotes the transposed of the linear partial

differential operator v. Iterating this, the rapid

decrease of I(!) follows.

Using cutoff functions, I(!) is, modulo a rapidly

decreasing function, equal to an oscillatory integral

with phase function ’ and an amplitude which

has support in an arbitrarily small neighborhood of

the set of stationary points of ’. In this sense,

the contributions to the integral which are not

rapidly decreasing come only from the stationary

points of ’.

The Method of Stationary Phase

Assume that 

is a nondegenerate stationary point

of ’. Then 

is an isolated stationary point of ’.

Using local coordinates near 

, the contribution to

[1] from the neighborhood of 

can be written as an

oscillatory integral with =R

and a pase function

’ which has a nondegenerate stationary point at 0.

Write Q = D

’(0). According to the Morse lemma,

there is smooth substitution of variables  = T(y)

such that T(0) = 0, DT(0) = I, and ’(T()) = ’(0) þ

hQy, yi=2 for all y in a neighborhood of 0 in R

Applying this substitution of variables to [1] we

obtain

Ið!Þ¼e

i!’ð0Þ

i!hQy;yi=2

bðyÞdy

where b is a compactly supported smooth function

on R

with b(0) = a(0). Now the Fourier transform

of the funct ion y 7!e

i!hQy, yi=2

is equ al to the function

 7! det

2i



1=2

1

;i=2

½4

Both in the definition of the square root of the

determinant and in the proof one uses the analytic

continuation to the domain of complex-valued

symmetric bilinear forms Q for which the imaginary

part of Q is positive definite. For purely imaginary

Q we have the familiar formula for the Fourier

transform of a Gaussian density (see Ho¨ rmander

(1990, theore m 7.6.1)). The Taylor expansion of the

exponential factor in [4] then yields that

i!hQy;yi=2

bðyÞdy

 det

2i



1=2

r¼0

ð2i!Þ

r

 Q

1

;



bðyÞ



y¼0

as ! !1(see Ho¨ rmander (1990, lemma 7.7.3)).

It is important for the applications that, if the

phase function and amplitude depend smoothly on

parameters, all the constructions can be made to

depend smoothly on the parameters.

Exact Stationary Phase

Suppose that we have given an action of a Lie group

G on the manifold . Let g denote the Lie algebra of

G. For any g 2 G and X 2 g the corresponding

diffeomorphism of  and vector field on  is

denoted by g



and X



, respectively. If () denotes

the algebra of smooth differential forms on , then

we consider the algebra Sg



 () of all ()-

valued polynomials on g, where Sg



denotes the

algebra of all polynomial functions on g.OnSg





() we have the action of g 2 G which sends  to

X 7!g





((Ad gX)). Let A = (Sg



 ())

denote

the subalgebra of all G-invariant elements of Sg





(). The equivariant exterior derivative D is

defined by

ðDÞðXÞ¼dððXÞÞ  i



ððXÞÞ

If  is homogeneous as a differential form of degree p

and homogeneous as a polynomial on g of degree q,

then r = p þ 2q is called the total degree of .LetA

denote the space of sums of such  2 A of total degree r.

Then D

= D: A

rþ1

and D

 D

r1

= 0. The space

():= ker D

=Im D

r1

is called the equivariant

cohomology in degree r,inthemodelofCartan (1950).

Assume that  is compact and oriented, and that

the action of G preserves the orientation. If  2 A,

then we denote by (X )

[k]

the volume part of the

differential form (X), and







ðXÞ:¼



ðXÞ

½k

; X 2 g

defines an Ad G-invariant function

 on g.Now

 = D implies that (X)

[k]

id equal to the exterior

derivative of (X )

[k1]

, and therefore

 = 0, in view

of Stokes’ theorem. It follows that integration over 

yields a linear mapping

from H

()to(Sg



)

Ad G

which is called integration in equivariant cohomology.

Stationary Phase Approximation 47

Now assume that also the Lie group G is

compact, and let X 2 g. Then the zero-set Z



in  has finitely many connected components F,

each of which is a smooth and compact submanifold

of . In general, the F’s can have different

dimensions. The linearization LX of the vector

field X



along F acts linearly on the normal bundle

NF of F.If is the curvature form of NF, then

"ðXÞ : ¼ det

2

ðLX  Þ



is called the equivariant Euler form of NF. "(X)isan

invertible element in the algebra 

even

(F ). The

localization formula of Berline–Vergne (1982) and

Atiyah–Bott (1984) now says that if D = 0 then







ðXÞ¼



ðXÞ="ðXÞ



½dim F

Assume that  is a symplectic form on , which

implies that k = 2l is even. Furthermore, assume that

the infinitesimal action of g on  is Hamiltonian,

which means that there exists a G -equivariant

smooth mapping  :  !g



, called the momentum

mapping, such that i



 = d((X)) for every X 2 g.

Here  is viewed as an element of (g



 

())

 A.

Then b(X):=   (X) defines an element b 2 A such

that Db = 0. In turn, this implies that the form

ðXÞ:¼e

i!bðXÞ

¼ e

i!ðXÞ

r¼0

ði!Þ

=r!

is equivariantly closed, and the localization formula

of equivariant cohomology applied to this case

yields the Duistermaat–Heckman (1982, 1983)

formula. Because (X)

[k]

= e

i!(X)

(i!)

=l!, its inte-

gral over  is an oscillatory integral with phase

function (X). The stationary points of (X) are the

zeros of X



and the stationary points of (X) are

nondegenerate if and only if the zeros of X



are

isolated. It follows that in this case the oscillatory

integral is equal to the leading term in the

stationary-phase approximat ion.

The Principal Symbol on the

Lagrangian Manifold

Let u( x, !) be a global oscillatory integral of order m

defined by , and let (x

, 

) 2 . One way to define

the principal symbol of u at (x

, 

) 2  is to test u

with an oscillatory function of the form e

i! (x)

b(x),

in which d (x

) = 

, the support of b is contained

in a small neighborhood of x

, and b(x

) = 1. If u is

locally represented by the phase function ’ and

amplitude a, and (x

, 

) = 

’

, 

), then the phase

function ’(x, )  (x) in the oscillatory integral

hu; e

i

bi¼



ið’ðx;Þ ðxÞ

aðx;ÞbðxÞd dx

has a stationary point at (x

, 

), which means that

 and d intersect at (x

, 

). Her e the 1-form d on

M, which is a section of  :T



M !M, is viewed as a

submanifold of T



M. Locally the Lagrangian sub-

manifolds of T



M which are transversal to the fibers

of  :T



M !M are precisely the manifolds of the

form d . The stationary point of ’  is non-

degenerate if and only if L := T

, 

)

 and

:= T

, 

)

(d ) are transversal. In this case, the

method of stationary phase can be applied in order

to obtain an asymptotic expansion in terms of

powers of !. The coefficient of the leading term of

order !

depends only on the Lagrangian plane L

which is transversal to both L and the tangent space

of the fiber of T



M, and not on the other data of

and b.IfL denotes the set of all Lagrangian planes

in T

, 

)



M) which are transversal to both L and

the fiber, then the complex-valued functions on L

which arise in this way form a one-dimensional

complex vector space L

, 

)

. The L

, 

)

for

, 

) 2  form a complex line bundle  over 

which is canonically isomorphic to the tensor

product of the line bundle of half-densities and the

Maslov line bundle, a line bundle with structure

group Z=4Z (see Duistermaat (1974, section 1.2)).

In this way, the principal symbol s of u can be

viewed as a section of the line bundle  over .

Caustics

Let (x

, 

) be a point in the Lagrangian submanifold

 of T



M. The restriction to  of the projection

 :T



M !M is a diffeomorphism from an open

neighborhood of (x

, 

)in onto an open neigh-

borhood of x

in M, if and only if  is transversal

to the fiber of T



M at (x

, 

). If =

’

for a

nondegenerate phase function ’,(x

, 

) 2 S

’

and

, 

) = 

’

, 

), then this condition is in turn

equivalent to the condition that 

is a nondegenerate

stationary point of  7!’(x

, ). An application of the

method of stationary phase shows that in this case the

oscillatory integral is equal to a progressing wave of

the form e

i! (x)

b(x, !). Here (x) = ’(x, ( x)), where

(x) is the stationary point of  7!’(x, ), and b(x, !)

has an asymptotic expansion as in [2] with k = 0.

If 

is a degenerate stationary point of

 7!’(x

, ) and a

, 

) 6¼ 0, then the oscillatory

integral is not of order O(!

). That is, it is of larger

order than at points where we have a nondegenerate

48 Stationary Phase Approximation