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

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value of a certain nondimensional quantity was

likely to exist beyond which a laminar flow gives

rise to a ‘‘sinuous’’ motion. This was followed by

observations of the flow for tubes with different

diameter L, different mean velocities U across the

tube section, and with the kinematic viscosity

 = = being altered through changes in tempera-

ture. The experime nts confirmed the existence of

such a critical value for what is now called the

Reynolds number:

Re ¼



The dimensional analysis argument can be repro-

duced in the following form: the physical dimension

for the inertial term in [1a] is U

=L, while that for

the viscous term is U=L

. The ratio between them

is precisely Re = LU=. For small values of Re

viscosity dominates and the flow is laminar, whereas

for large values of Re the inertial term dominates,

and the flow becomes more complicated and

eventually turbulent. In applications, different types

of Reynolds number can be used depending on the

choice of the characteristic velocity and length, but

in an y case, the larger the Reynolds number, the

more complicated the flow.

The Reynolds Equations

Another advance put forward by Reynolds in a

subsequent article was to decompose the flow into a

mean component and the remaining fluctuations. In

terms of the velocity and pressure fields this can be

written as

u ¼



u þ u

; p ¼



p þ p

½2

with



u and



p representing the mean components and

and p

, the fluctuations. By substituting [2] into

[1], one finds the Reynolds-averaged Navier– Stokes

(RANS) equations for the mean flow:







u þð



u rÞ



u þr



p ¼f þr

r



u ¼ 0

It differs from [1] only by the addition of the

Reynolds stress tensor:

 ¼

 u

¼ u



i;j¼1

In a laminar flow, the fluctuations are negligible,

otherwise this decomposition shows how they

influence the mean flow through this additional

turbulent stresses.

The Closure Problem and Turbulence

Models

The RANS equations cannot be solved directly for the

mean flow since the Reynolds stresses are unknown.

Equations for these stress terms can be derived but they

involve further unknown moments. This continues

with equations for moments of a given order depend-

ing on new moments up to a higher order, leading to

an infinite system of equations known as the Fried-

man–Keller system. For practical applications,

approximations closing the system at some finite

order are needed, in what is called the closure problem.

Several ad hoc approximations exist, the most famous

being the Boussinesq eddy-viscosity approximation, in

which the turbulent fluctuations are regarded as

increasing the viscosity of the flow. Prandtl’s mixing-

length hypothesis yields a prescription for the compu-

tation of this eddy viscosity, and together they form the

basis of the algebraic models of turbulence. Other

models involve additional equations, such as the k-

and k-! models. Most of the practical computations of

industrial flows are based on such lower-order models,

and a large amount of research is done to determine

appropriate values for the various ad hoc parameters

which appear in these models and which are highly

dependent on the geometry of the flow. This depen-

dency can be explained by the fact that the RANS is

supposed to model the mean flow even at the large

scales of motion, which are highly affected by the

geometry.

Computational fluid dynamics (CFD) is indeed a

fundamental tool in turbulence, both for research and

engineering applications. From the theoretical side,

direct numerical simulations (DNS), which attempt to

resolve all the active scales of the flow, reveal some

fundamental mechanisms involved in the transition to

turbulence and in vortex stretching. As for applica-

tions, DNS applies to flows up to low-Reynolds

turbulence, with the current computational power

not allowing for a full resolution of all the scales

involved in high-Reynolds flows. And the current rate

of evolution of computational power predicts that

this will continue so for several decades.

An intermediate CFD method between RANS and

DNS is the large-eddy simulation (LES), which

attempts to fully resolve the large scales while

modeling the turbulent motion at the smaller scales.

Several models have been proposed which have their

own advantages and limitations as compared to

RANS and DNS. It is currently a subject of intense

research, particularly for the development of suitable

models for the structure functions near the boundary.

Theoretical results on fully developed turbulence play

a fundamental role in the modeling process.

298 Turbulence Theories

LESs are a promising tool and they have been

successfully applied to a number of situations. The

choice of the best method for a given application,

however, depends very much on the Reynolds

number of the flow and the prior knowledge of

similar situations for adjusting the parameters.

Elements of the Statistical Theory

Several types of averages can be used. The ensemble

average is taken with respect to a number of experi-

ments at nearly identical conditions. Despite the

irregular motion of, say, the velocity vector u

(n)

(x, t)

of each experiment n = 1, ..., N, the average value



uðx; tÞ¼

n¼1

ðnÞ

ðx; tÞ

is expected to behave in a more regular way. This

type of averaging is usually denoted with the symbol

hi. This notion can be cast into the context of a

probability space (M, , P), where M is a set,  is a

-algebra of subsets of M, and P is a probability

measure on . The velocity field is a random

variable in the sense that it is a density function

! 7!u(x, t, !) from M into the space of time-

dependent divergence-free velocity fields. The mean

velocity field in this context is regarded as

huðx; tÞi ¼

uðx; t;!ÞdPð!Þ

Other flow quantities such as energy and correla-

tions in space and time can be expressed by means

of a function ’ = ’(u( , )) of the velocity field,

with their mean value given by

h’ðuð; ÞÞi ¼

’ðuð; ;!ÞdPð!Þ

In general, the statistics of the flow are allowed to

change with time. A particular situation is when

statistical equilibrium is reached, so that hu(x, t)i

and, more generally, h’(u( , þt))i are independent

of t . In this case, an ergodic assumption is usually

invoked, which means that for ‘‘most’’ individual

flows u( , , !

) (i.e, for almost all !

with respect to

the probability measure P), the time averages along

this flow converge to the mean ensemble value as

the period of the average increases to the mean value

obtained by the ensemble average:

lim

T!1

’ðuð; þs;!

ÞÞds

’ðuð; ;!ÞÞdPð!Þ

Based on this assumption, the averages may in

practice be calculated as time averages over a

sufficiently large period T. There is a related

argument for substituting space averages by time

averages and based on the mechanics of turbulence

which is called the ‘‘Taylor hypothesis.’’

Another fundamental concept in the statistical

theory is that of homogeneity, which is the spatial

analog of the statistical equilibrium in time.

In homogeneous turbulence, the statistical qua ntities

of a flow are independent of translations in space,

that is,

h’ðuð þ‘; Þi¼h’ðuð; Þi

for all ‘ 2 R

. The concept of isotropic turbulence

assumes further independence with respect to

rotations and reflections in the frame of reference,

that is,

h’ðQ

uðQ ; Þi¼h’ðuð; Þi

for all orthogonal transformations Q in R

, with

adjoint Q

Under the homogeneity assumption, mean quan-

tities can be defined independently of position in

space, such as the mean kinetic energy per unit mass

e ¼

hjuðxÞj

i¼

j¼1

hju

ðxÞj

and the mean rate of viscous energy dissipation per

unit mass and unit time

 ¼ 

i¼1

hjru

ðxÞj

i¼

i;j¼1



ðxÞ





The mean kinetic energy can be written as

e = trR(0)=2, where

trRð‘Þ¼R

ð‘ÞþR

ð‘Þ;‘2 R

;

is the trace of the correlation tensor

Rð‘Þ¼huðxÞuðx þ ‘Þi ¼ ðR

ð‘ÞÞ

i;j¼1

¼ðhu

ðxÞu

ðx þ ‘ÞiÞ

i;j¼1

which measures the correlation between the velocity

components at different positions in space. From the

homogeneity assumption, this tensor is a function

only of the relative position ‘. Then, assuming that

the Fourier transform of trR(‘) exists, and denoting

it by Q(), for  2 R

, we have

trRð‘Þ¼

ð2Þ

3=2

QðÞe

i‘ 

d

¼ 2

SðÞe

i‘ 

d

Turbulence Theories 299

where S() is the energy spectrum defined by

SðÞ¼

2ð2Þ

3=2

jj¼

QðÞdðÞ

8>0

with d() denoting the area element of the

2-sphere of radius jj. Then we can write

e ¼

hjuðxÞj

i¼

trRð0Þ

SðÞd

By expanding the velocity coordinates into Four-

ier modes exp (‘ ), with  jj þ d and

interpreting them as ‘‘eddies’’ with characteristic

wave number jj, the quantity S()d can be

interpreted as the energy of the component of the

flow formed by the ‘‘eddies’’ with characteristic

wave number between  and  þ d.

Similarly,

 ¼ 2



SðÞd

and we obtain the dissipation spectrum 2

S(),

which can be interpreted as the density of energy

dissipation occurring at wave number .

In the previous arguments it is assumed that the

flow extends to all the space R

. This avoids the

presence of boundaries, addressing the idealized case

of fully developed turbulence. It is sometimes

customary to assume as well that the flow is

periodic in space to avoid problems with unbounded

domains such as infinite kinetic energy.

The random nature of turbulent flows was greatly

explored by Taylor in the early twentieth century,

who introduced most of the concepts described

above. Another important concept he introduced

was the Taylor microlength ‘

, which is a char-

acteristic length for the small scales based on the

correlation tensor. A microscale Reynolds number

based on the Taylor microlength is very often used

in applications.

Kolmogorov Theory

An inspiring concept in the theory of turbulence is

Richardson’s ‘‘energy cascade’’ process. For large

Reynolds numbers the nonlinear term dominates the

viscosity according to the dimensional analysis, but

this is valid only for the large-scale structures. The

small scales have their own characteristic length and

velocity. In the cascade process, the inertial term is

responsible for the transfer of energy to smaller and

smaller scales until small enough scales are reached

for which viscosity becom es important (Figure 3). At

those smallest scales kinetic energy is finally dis-

sipated into heat. It should be emphasized that

turbulence is a dissipative process; no matter how

large the Reynolds number is, viscosity plays a role

in the smallest scales.

The Kolmogorov theory of locally isotropic

turbulence allows for inhomogeneity and anisotropy

in the large scales, which contain most of the energy,

assuming that with the cascade transfer of energy to

smaller scales, the orienting effects generated in the

large scales become weaker and weaker so that for

sufficiently small eddies the motion becomes statis-

tically homogeneous, isotropic, and independent of

the particular energy-productive mechanisms. He

proposed that the statistical regime of the small-

scale eddies is then universal and depends only on 

and . The equilibrium range is defined as the range

of scales in which this universality holds.

Simple dimensional analysis shows that the only

algebraic combination of  and  with dimension of

length is ‘



= (

=)

1=4

, which is then interpreted as

that near which the viscous effect becomes impor-

tant and hence most of the energy dissipation takes

place. The scale ‘



is known as Kolmogorov

dissipation length.

Kolmogorov theory gives particular attention to

moments involving differences of velocities, such as

the pth-order structure funct ion

ð‘Þ¼

def

hðuðx þ ‘eÞe  uðxÞeÞ

where e may be taken as an arbitrary unit vector,

thanks to the isotropy assumption. By restricting the

search for universal laws for the structure functions

only for small value s of ‘ anisotropy and inhomo-

geneity are allowed in the large scales.

Figure 3 Illustration of the eddy breakdown process in which

energy is transferred to smaller eddies and so on until the smallest

scales are reached and the energy is dissipated by viscosity.

300 Turbulence Theories

The theory assumes a wide separation between

the energy-containing scales, of order say ‘

, and the

energy-dissipative scales, of order ‘



, so that the

cascade process occurs within a wide range of scales

‘ such that ‘

 ‘  ‘



. In this range, termed the

inertial range, the viscous effects are still negligible

and the statistical regime should depend only on .

Then, the Kolmogorov ‘‘two-thirds law’’ asserts that

within the inertial range the second-order correla-

tions must be proportional to (‘)

2=3

, that is,

ð‘Þ¼C

ð‘Þ

2=3

for some constant C

known as the Kolmogorov

constant in physical space (there is a related constant

in spectral space). The argument extends to higher-

order structure functions, yielding

ð‘Þ¼C

ð‘Þ

p=3

Kolmogorov’s derivation of these results was not by

dimensional analysis, it was in fact a more convincing

self-similarity argument based on the universality

assumed for the equilibrium range. A different argu-

ment without resorting to universality assumptions,

however, was applied to the third-order structure

function, yielding the more precise ‘‘four-fifths law’’:

ð‘Þ¼

‘

The ‘‘Kolmogorov five-thirds law’’ concerns the

energy spectrum S() and is the spectral version of

the two-thirds law, given by Obukhoff:

SðÞ¼C



2=3



5=3

The constant C

is the Kolmogorov constant

in spectral space. The spectral version of the

dissipation length is the Kolmogorov wave number





= (=

)

1=4

A typical distribution of energy in a turbulent

flow is depicted in Figure 4. The energy is

concentrated on the large scales, while the dissipa-

tion is concentrated near the Kolmogorov scale ‘



The four-fifths law becomes visible as a straight line

in the logarithmic scale.

A more precise mechanism for the energy cascade

assumes that in the inertial range, eddies with length

scale ‘ transfer kinetic energy to smaller eddies during

their characteristic timescale, also known as circula-

tion time. If u

‘

is their characteristic velocity, then



‘

= ‘=u

‘

is their circulation time, so that the kinetic

energy transferred from these eddies during this time is



‘



‘



‘

In statistical equilibrium, the energy lost to the

smaller scales equals the energy gained from the

larger scales, and that should also equal the total

kinetic energy dissipated by viscous effects. Hence,



‘

 , and we find

 

‘

It also follows that 

‘

= ‘=u

‘

= ‘(‘ )

1=3

= 

1=3

‘

2=3

that the circulation time decreases with the length

scale and becomes of the order of the viscous

dissipation time (=)

1=2

precisely when ‘  ‘



A similar relation between  and the large scales

can also be obtained with heuristic arguments: let e

be the mean kinetic energy and ‘

, a characteristic

length for the large scales. Then u

given by e = u

is a characteristic velocity for the large scales, and



= ‘

is the large-scale circulation time. In

statistical equilibrium, the rate  of kinetic energy

dissipated per unit time and unit mass is expected to

be of the order of e=

, hence

 

‘

which is called the ‘‘energy dissipation law.’’

(κ)

κ κ

ln κ

Inertial range

ln κ

Equilibrium ran

Inertial range

Equilibrium ran

(κ)

ln κ

2νκ

(κ)

Figure 4 A typical distribution for the energy spectrum S() and the dissipation spectrum 2

S() in spectral space in

nonlogarithmic and logarithmic scales. The energy is mostly concentrated on the large scales while the dissipation is concentrated

near the dissipation scale. In the logarithmic scale, the four-fifths law for the energy spectrum stands out as a straight line with

slope 4=5 over the inertial range.

Turbulence Theories 301

From the energy dissipation law, several relations

between characteristic quantities of turbulent flows can

be obtained, such as ‘

=‘



 Re

3=4

,forRe = ‘

=.

Now, assuming the active scales in a turbulent

flow exist down to the Kolmogorov scale ‘



, one

needs a three-dimensional grid with mesh spacing ‘



to resolve all the scales, which means that the

number N of degrees of freedom of the system is of

the order of N  (‘

=‘



)

(see Figure 5). This

number can be estimated in terms of the Reynolds

number by N  Re

9=4

. This relation is important in

predicting the computational power needed to

simulate all the active scales in turbul ent flows.

Several such universal laws can be deduced and

extended to other situations such as turbulent

boundary layers, with the famous logarithmic law

of the wall. They play a fundamental role in

turbulence modeling and closure, for the calculation

of the mean flow and other quantities.

Intermittency

The universality hypothesis based on a constant mean

energy dissipation rate throughout the flow received

some criticisms and was later modified by Kolmo-

gorov in an attempt to account for observed large

deviations on the mean rate of energy dissipation. Such

phenomenon of intermittency is related to the vortex

stretching and thinning mechanism, which leads to the

formation of coherent structures of vortex filaments of

high vorticity and low dissipation (Figure 6). These

filaments have diameter as small as the Kolmogorov

scale and longitudinal length extending from the

Taylor scale up to the large scales and with a lifetime

of the order of the large-scale circulation time.

It has been argued based on experimental evidence

that intermittency leads to modified power laws

(‘) / ‘

(p)

, (p) < p=3, for high-order (p > 3) struc-

ture functions. The issues of intermittency and

coherent structures and whether and how they could

affect the deductions of the universality theory such as

the power laws for the structure functions are far from

settled and are currently one of the major and most

fascinating issues being addressed in turbulence

theory. Several phenomenological theories attempt to

adjust the universality theory to the existence of such

coherent structures. Multifractal models, for instance,

suppose that the eddies generated in the cascade

process do not fill up the space and form multifractal

structures. Field-theoretic renormalization group

develops techni ques based on quantum field renor-

malization theory. Intermediate asymptotics also

exploits self-similar analysis and renormalization

theory but wi th a somewhat different flavor. Detailed

mathematical analysis of the vorticity equatio ns is

also playing a major role in the understanding of the

dynamics of the vorticity field.

Mathematical Aspects

of Turbulence Theory

From a mathematical perspective, it is fundamental to

develop a rigorous background upon which to study

the physical quantities of a turbulent flow. The first

problem in the mathematical theory is related to the

deterministic nature of chaotic systems assumed in

dynamical system theory and believed to hold in

turbulence. This has actually not been proved for the

Navier–Stokes equations. It is in fact one of the most

outstanding open problems in mathematics to deter-

mine whether given an initial condition for the velocity

field there exists, in some sense, a unique solution of

the Navier–Stokes equations starting with this initial

condition and valid for all later times. It has been

proved that a global solution (i.e., valid for all later

ᐉ



Figure 5 A schematic representation of a flow structure

displaying a range of active scales and a three-dimensional

grid with linear dimension ‘

and mesh length ‘



, sufficient to

represent all the active scales in a turbulent flow. The number of

degrees of freedom is the number of blocks: (‘

=‘



)

Figure 6 A portion of rotating fluid gets stretched and thinned

as the flow speeds up, generating one of many coherent

structures of high vorticity and low dissipation.

302 Turbulence Theories

times) exists but which may not be unique, and it has

been proved that unique solutions exist which may not

be global (i.e., they are guaranteed to exist as unique

solutions only for a finite time).

The difficulty here is the possible existence of

singularities in the vorticity field (vorticity becoming

infinite at some points in space and time). Depending

on how large the singularity set is, uniqueness may fail

in strictly mathematical terms. The existence of

singularities may not be a purely mathematical

curiosity, it may in fact be related with the inter-

mittency phenomenon. Rigorous studies of the vorti-

city equation may continue to reveal more fundamental

aspects on vortex dynamics and coherent structures.

The statistical theory has also been put into a firm

foundation with the notion of statistical solution of the

Navier–Stokes equations. It addresses the existence

and regularity of the probability distribution assumed

for turbulent flows and of the fundamental elements of

the statistical theory such as correlation functions and

spectra. Based on that, a number of relations between

physical quantities of turbulent flows may be derived

in a mathematically sound and definitive way. This

does not replace other theories, it is mostly a

mathematical framework upon which other techni-

ques can be applied to yield rigorous results.

Despite the difficulties in the mathematical theory

of the NSE some successes have been collected such

as estimates for the number of degrees of freedom in

terms of frac tal dimensions of suitable sets asso-

ciated with the solutions of the Navier–Stokes

equations, and partial estimates of a number of

relations derived in the statistical theory of fully

developed turbulence.

See also: Bifurcations in Fluid Dynamics; Geophysical

Dynamics; Incompressible Euler Equations:

Mathematical Theory; Intermittency in Turbulence;

Inviscid Flows; Lagrangian Dispersion (Passive Scalar);

Stochastic Hydrodynamics; Variational Methods in

Turbulence; Viscous Incompressible Fluids:

Mathematical Theory; Vortex Dynamics; Wavelets:

Application to Turbulence.

Further Reading

Adzhemyan LT, Antonov NV, and Vasiliev AN (1999) The Field

Theoretic Renormalization Group in Fully Developed Turbu-

lence. Amsterdam: Gordon and Breach.

Barenblatt GI and Chorin AJ (1998) New perspectives in

turbulence: scaling laws, asymptotics, and intermittency.

SIAM Review 40(2): 265–291.

Batchelor GK (1953) The Theory of Homogeneous Turbulence.

Cambridge Monographs on Mechanics and Applied Mathe-

matics. New York: Cambridge University Press.

Chorin AJ (1994) Vorticity and Turbulence. Applied Mathema-

tical Sciences vol. 103, New York: Springer.

Constantin P (1994) Geometric statistics in turbulence. SIAM

Review 36: 73–98.

Foias C, Manley OP, Rosa R, and Temam R (2001) Navier–Stokes

Equations and Turbulence. Encyclopedia of Mathematics and its

Applications, vol. 83. Cambridge: Cambridge University Press.

Friedlander S and Topper L (1961) Turbulence. Classic Papers on

Statistical Theory. New York: Interscience Publisher.

Frisch U (1995) Turbulence. The Legacy of A. N. Kolmogorov.

Cambridge: Cambridge University Press.

Hinze JO (1975) Turbulence. McGraw-Hill Series in Mechanical

Engineering. New York: McGraw-Hill.

Holmes P, Lumley JL, and Berkooz G (1996) Turbulence,

Coherent Structures, Dynamical Systems, and Symmetry.

Cambridge: Cambridge University Press.

Lesieur M (1997) Turbulence in Fluids. Fluid Mechanics and its

Applications, 3rd edn. vol. 40. Dordrecht: Kluwer Academic.

Monin AS and Yaglom AM (1975) Statistical Fluid Mechanics:

Mechanics of Turbulence 2. Cambridge: MIT Press.

Sagaut P (2001) Large Eddy Simulation for Incompressible Flows.

Berlin: Springer.

Schlichting H and Gersten K (2000) Boundary Layer Theory, 8th

edn. Berlin: Springer.

Tennekes H and Lumley JL (1972) A First Course in Turbulence.

Cambridge, MA: MIT Press.

Vishik MI and Fursikov AV (1988) Mathematical Problems of

Statistical Hydrodynamics. Dordrecht: Kluwer.

Wilcox DC (2000) Turbulence Modeling for CFD, 2nd edn.

Anaheim, CA: DCW Industries Inc.

Twistor Theory: Some Applications

L Mason, University of Oxford, Oxford, UK

Introduction

Roger Penrose introduced twistor theory as a geome-

trical framework for basic physics in order to unify

quantum theory and gravity. This program has had

many successes along the way, but the long-term goals

of reformulating and superceding the established

theories of basic physics are still a long way from

being fulfilled. Nevertheless, the successes have had

many important applications across mathematics and

mathematical physics. This article will concentrate on

three areas of application: integrable systems, geome-

try, and perturbative gauge theory (via twistor-string

theory). It is intended to be self-contained as far as

possible, but the reader may well find it easier to first

read the article Twistors.

Twistor Theory: Some Applications 303

Twistor Theory

A basic motivation of twistor theory is to bring out

the complex (holomorphic) geometry that underlies

real spacetime. In genera l relativity, a spacetime is a

4-manifold with metric g of signature (1, 3), and

when it is flat, that is, g = dt

 dx

 dy

 dz

where (t, x, y, z) are coordinates on R

, it is called

Minkowski space. The first appearance of a com-

plex structure arises from the fact that, at a given

event, the celestial sphere of light rays (directions of

zero length with respect to g) naturally has the

structure of the Riemann sphere, CP

, in such a way

that Lorentz transformations (linear transformations

of the tangent space preserving the metric) act on

this sphere by Mo¨ bius transformations. These are

the maximal group of complex analytic transforma-

tions of CP

Twistor space extends this idea to the whole of

Minkowski space. Denoted PT, the twistor space for

Minkowski space is complex projective 3-space, CP

the space of one-dimensional subspaces of C

;itisa

three-dimensional complex manifold obtained by add-

ing a ‘ ‘plane at infinity’ ’ to C

. Explicitly, we can

introduce homogeneous coordinates Z



2 C

 {0}

with  =0,1, 2,3 but where Z



Z



for  2C {0}.

Affine coordinates on a C

chart Z

6¼0can

be obtained by setting (z

,)= (Z

). Physically, points of twistor space corre-

spond to spinning massless particles in Minkowski

space. Mathematically, the correspondence can be

understood as the Klein correspondence.

The Klein Correspondence

The correspondence between PT and Minkowski

space can be extended first to complexified Minkowski

space so that the coordinates are allowed to take on

values in C, and then to its conformal compactification

by including the ‘‘light cone at infinity.’’ It then

coincides with the classical complex Klein correspon-

dence. The Klein correspondence is the one-to-one

correspondence between lines in CP

and points of a

four complex-dimensional quadric, CM,inCP

.The

4-quadric CM can be understood as conformally

compactified complexified Minkowski space. Introdu-

cing affine coordinates (z

, z

, )onPT and (t, x, y, z)

on CM,wefindthatapoint(t, x, y, z)inCM

corresponds to a line in PT according to



t  zxþ iy

x  iytþ z







Alternatively, fixing (, z

, z

) in these equations

gives a 2-plane in complex Minkowski space

corresponding to all the lines in PT through

(, z

, z

). Such 2-planes are called ‘‘-planes.’’

They are totally null (i.e., the tangent vectors not

only have zero length but are also mutually

orthogonal) and also self-dual (under the differential

geometer’s notion of Hodge duality).

This complex correspondence can also be

restricted to give correspondences for R

with

metrics of positive-definite, Euclidean, signature or

ultrahyperbolic, (2, 2), signature. A particular sim-

plification in Euclidean signature is that the complex

-planes intersect the real slice in a point. The

conformal compactification of Euclidean R

is the

4-sphere S

given by adding a single point at infinity,

and so we have a projection p : PT !S

whose

fibers are holomorphically embedded CP

s. These

fibers can be characterized as the lines in PT that

are invariant under a quaternionic complex con-

jugation which is an antiholomorpic map^: PT !

PT with no fixed points. (Here quaternionic means

that on the nonprojective twistor space, T = C

, the

conjugation has the propert y

Z = Z so that it

defines a second complex structure anticommuting

with the standard one; this is sufficient to express

T = Q

, where Q denotes the quaternions. The

complex structures i, j, and k of the quaternions

are given by identifying i with

ﬃﬃﬃﬃﬃﬃﬃ

1

on C

and j

with^and k = ij.)

The Penrose Transform

A basic task of twistor theory is to transform

solutions to the field equations of mathematical

physics into objects on twistor space. This works

well for linear massless fields such as the Weyl

neutrino equation, Maxwell’s equations for electro-

magnetism and linearized gravity. In its general

form, this transform has become known as the

Penrose transform. Such fields correspond to freely

prescribable holomorphic functions f (, z

, z

) (or,

more precisely, analytic cohomology classes) on

regions of twistor space. The field can be obtained

from this function by means of a contour integral.

The simplest of these integral form ulas is

ðx

Þ¼

f ; t  z þ ðx þ iyÞ; x  iyð

þðt þ zÞÞd

and differentiation under the integral sign leads

easily to the fact that  satisfies the wave equation















¼ 0

This formula was originally discovered by Bateman.

Note that f must have singularities on twistor space

to yield a nontrivial  and even then, there are many

choices of f that yield zero. For a solution  defined

304 Twistor Theory: Some Applications

over a region U in spacetime, the function f is

correctly understood as a representative of a Cech

cohomology class defined on the region U

in twistor

space swept out by the lines corre sponding to points

of U. Furthermore, the function f should be taken

globally to be a function of homogeneity 2,

f (Z



) = 

2

f (Z



). This formula has generalizations

to massless fields of all helicities in which a field of

helicity s corresponds to a function (Cech cocycle) of

homogeneity degree 2s  2.

The Penrose transform has found important

applications in representation theory and integral

geometry. For a review, the reader is referred to

Baston and Eastwood (1989), the relevant survey

articles in Bailey and Baston (1990),orMason and

Hughston (1990, chapter 1).

Twistor Theory and Nonlinear Equations

The Penrose transform for the Maxwell equations

and linearized gravity turns out to be linearizations

of correspondences for the nonlinear analogs of

these equations: the Einstein vacuum equations and

the Yang–Mills equations. However, the construc-

tions only work when these fields are anti-self-dual.

This is the condition that the curvature 2-forms

satisfy F



= iF, where  denotes the Hodge dual

(which, up to certain factors of i, has the effect of

interchanging electric and magnetic fields); it is a

nonlinear generalization of the right-handed circular

polarization condition. Explicitly, in terms of space-

time indices a, b, ... =0, 1, 2, 3, F



= (1=2)"

abcd

where "

0123

= 1 and "

abcd

= "

[abcd]

. In Minkowski

signature, the i factor in the anti-self-duality condi-

tion implies that real fields cannot be anti-self-dual.

Thus, these extensions are not sufficient to fulfill the

ambitions of twistor theory to incorporate real

classical nonlinear physics in Minkowski space.

However, the factor of i is not present in Euclidean

and ultrahyperbolic signature, so the anti-self-

duality condition is consistent with real fields in

these signatures and this is where the main applica-

tions of these constructions have been.

The Nonlinear Graviton Construction

and Its Generalizations

The first nonlinear twistor construction was due to

Penrose (1976), and was inspired by Newman’s

(1976) construction of ‘‘heavens’’ from the infinities

of asymptotically flat spacetimes in general

relativity.

The nonlinear graviton construction proceeds

from the definition of twisto rs in flat spacetime as

-planes in complexified Minkowski space. It is

natural to ask which complexified metrics admit a

full family of -surfaces, that is, 2-surfaces that are

totally null and self-dual. The answer is that a full

family of -surfaces exists iff the conformally

invariant part of the curvature tensor, the Weyl

tensor, is anti-self-dual. If this is the case, twistor

space can be defined to be the (necessarily three-

dimensional) space of such -surfaces.

The remarkable fact is that the twistor space,

together with its complex structure, is sufficient to

determine the original spacetime. Twistor space is

again a three-dimensional complex manifold, and

contains holomorphically embedded rational curves,

s, at least one for each point of the spacetime.

However, holomorphic rigidity implies that the

family of rational curves is precisely four-

dimensional over the complex numbers. Further-

more, incidence of a pair of curves can be taken to

imply that the corresponding points in spacetime lie

on a null geodesic and this yields a conformal

structure on spacetime. Further structures on twistor

space can be imposed to give the complex spacetime

a metric that is vacuum, perhaps with a cosmologi-

cal constant. The correspondence is stable under

small deformations and so the data defining the

twistor space is effectively freely prescribable, see

Penrose (1976).

In Euclidean signature, again the complex

-planes intersect the real spacetime in a point, so

the tw istor space again fibers over spacetime. The

twistor fibration can be constr ucted as the projecti-

vized bundle of self-dual spinors or more commonly

as the unit sphere bundle in the space of self-dual

2-forms (Atiyah et al. 1978). In the latter formula-

tion, the complex structure on the twistor space

arises from the direct sum of the naturall y defined

complex structures on the horizontal and vertical

tangent spaces to the bundle; that on the vertical

subspace is the standard one on the sphere, and that

on the horizontal subspace is a multiple of the self-

dual 2-form at the given point of the fiber.

There are now large families of extensions,

generalizations, and reductions of this construction.

They are all based on the idea of realizing a space

with a given complexified geometric structure as the

parameter space of a family of holom orphically

embedded submanifol ds inside a twistor space. In

general, the most useful of these constructions are

those in which the ‘‘spacetime’’ is obtained as the

space of rational curves in a twistor space. This is

because the equations that are solved on the

corresponding spacetime can be though t of as a

completely integrable system in which the integr-

ability condition for the generalized -surfaces is

interpreted as the consistency condition of a Lax

Twistor Theory: Some Applications 305

pair or more general linear system. For a more

detailed discussion from this point of view, see

Mason and Woodhouse (1996, chapter 13).

The Anti-Self-Dual Yang–Mills Equation

and Its Twistor Correspondence

The anti-self-dual Yang–Mills equations extend

Maxwell’s equations for electr omagnetism in the

right-circularly polarized case. They are a family of

equations that depend on a choice of Lie group G,

usually taken to be a group of complex matrices;

Maxwell’s equations arise when G = U(1).

Introduce coordinates x

, a = 0, 1, 2, 3, on R

with

metric ds

= dx

 dx

 dx

 dx

(this is a metric of

ultrahyperbolic signature – Euclidean signature can

be obtained by choosing the coo rdinates to be

complex, but with (x

, x

) the complex conjugates

of (x

, x

)). The dependent variables are the compo-

nents A

of a connection D

= @

 A

, where

= @=@x

and A

= A

) 2 Lie G, the Lie algebra

of G. This connection defines a method of differ-

entiating vector-valued functions s in some repre-

sentation of G. The freedom in changing bases for

the vector bundle induce the gauge transformations

1

g  g

1

g, g(x) 2 G on A

; two connec-

tions that are related by a gauge transformation are

deemed to be the same. The self-dual Yang–Mills

equations are the condition

½D

; D

¼½D

; D

¼½D

; D

½D

; D

¼0

They are the compatibility conditions

½D

þ D

; D

þ D

¼0

for the linear system of equations

ðD

 D

Þs ¼ðD

 D

Þs ¼ 0 ½1

where  2 C and s is an n-component column

vector. The se latter equations form a ‘‘Lax pair’’

for the system.

The Ward (1977) construction provides a one–one

correspondence between gauge equivalence classes

of solut ions of the self-dual Yang–Mills equations

and holomorphic vector bundles on regions in

twistor space. The key point here is that eqn [1]

defines parallel propagation along -planes. To each

point Z in twistor space, we can associate the vector

space E

of solutions to eqn [1] along the

corresponding -plane. These vector spaces vary

holomorphically with Z and that is what one means

by a holomorphic vector bundle E !PT. The

remarkable fact is that the anti-self-dual Yang–

Mills field can be reconstructed up to gauge from E,

and, in effect, for local analytic solutions, E can be

represented by freely prescribable ‘‘patching’’ data

consisting of local holomorphic matrix-valued func-

tions on twistor space. To construct the solution on

spacetime, one must first find a Birkhoff factoriza-

tion of the patching data on each Riemann sphere in

twistor space corresponding to points of the appro-

priate region in spacetime. On each Riemann sphere,

the Birkhoff factorization starts with the given

patching function with values in GL(n, C) on the

real axis in the complex plane, and expresses it as a

product of functions with values in GL(n, C) one of

which extends over the upper-half plane, and the

other over the lower-half complex plane. The anti-

self-dual conn ection can be obtained by differentiat-

ing the resulting matrices. See Penrose (1984, 1986),

Ward and Wells (1990),orMason and Woodhouse

(1996) for a full discussion, and Atiyah (1979) for

the formulation appropriate to Euclidean signature.

Completely Integrable Systems

In effect, the twistor constructions amount to

providing a geometric general local solution to the

anti-self-duality equations; the twistor data is, for a

local solution, freely prescribable. In this sense, they

demonstrate complete integrability of the anti-self-

duality equations. The reconstruction of a solution

on spacetime from twistor data is not a quadrature –

it involves, in the anti-self-dual Yang–Mills case, a

Birkhoff fact orization (also sometimes referred to as

the solution to a Riemann–Hilbert problem), and in

the case of the anti-self-dual Einstei n equations, the

construction of a family of rational curves inside a

complex manifold. Nevertheless, such constructions

are a familiar part of the apparatus of the theory of

integrable systems.

In Ward (1985), this connection with integrable

systems was developed further, and the anti-self-

dual Yang–Mills equations were shown to yield

many important integrable systems under symmetry

reduction. Ward’s list has been extended and now

includes many of the most famous examples of

integrable systems such as the Painleve´ equation s,

the Korteweg–de Vries (KdV) equation, the non-

linear Schro¨ dinger equation, the n-wave equations,

and so on, see Mason and Woodhouse (1996) for a

review. There are some notable omissions from the

list such as the Kadomtsev–Petviashvili (KP) and

Davey–Stewartson equations (at least if one restricts

oneself to finite-dimensional gauge groups; reduc-

tions using infinite dimensional gauge groups have

been obtained).

The list of integrable systems obtainable by

symmetry reduction nevertheless remains impressive

and provides a route to the classification of at least

those integrable systems that can be obtained in this

306 Twistor Theory: Some Applications

way. Such systems can be classified by the choice of

ingredients required in the symmetry reduction: the

gauge group, the group of spacetime symmetries to

be reduced by, the choice of Euclidean or ultra-

hyperbolic signature, and the choice of certain

constants of integration that arise in the reduction.

Another implication is that if an integrable system

can be obtained from one of the self-duality

equations by symmetry reduction, then it inherits a

reduced twistor correspondence because the twistor

correspondences share the symmetry groups of the

spacetime field equations. These twistor correspon-

dences can be seen to underlie much of the theory of

these equations; for example, Backlund transforma-

tions of so lutions correspond to elementary alge-

braic operations on the twistor data, similarly the

Kac–Moody Lie algebras of hidden symmetries act

locally on the twistor data by matrix multiplicat ion

of the appropriate loop algebras. Similarly, the

inverse-scattering transform for the KdV and non-

linear Schrodinger equations can be seen to arise as

particular presentation s of the twistor construction.

By and large, although twistor methods have

yielded new insight into the geometry and structure

of systems in dimensions 1 and 2, they have not

necessarily superceded pre-existing techniques for

constructing solutions and analyzing the solution

space. The systems for which twistor method s have

been particularly effective for constructing solutions

and characterizing their properties are in 2 þ 1or

higher dimension. Key examples here are of course

the anti-self-dual Yang–Mills and Einstein equations

themselves, and their single translation reductions.

In the anti-self-dual Yang–Mills case, these reduc-

tions lead either to Ward’s or Manakov and

Zakharov’s chiral model in Lorentzian signature,

2 þ 1, or the Bogomolny equations for monopoles,

the reduction from Euclidean signature. In both

cases, the twistor construction has played a major

role in constructing and studying the solitonic

solutions.

See Ward and Wells (1990), Mason and Wood-

house (1996), Ward’s article in Huggett et al. (1998)

and the first few chapters of Mason et al. (1995),

and Mason et al. (2001) for more examples of

aspects of the theory of integrable systems arising

from twistor correspondences.

Applications to Geometry

These applications are, to a large extent, higher-

dimensional analogs of those discussed above; most

of the problems in geometry to which twistor theory

has been applied are those for which the unde rlying

differential equations are integrable. These start

with the Euclidean signature versions of the original

Ward construction for anti-self-dual Yang– Mills

fields and Penrose’s nonlinear graviton construction

for Ricci-flat anti-self-dual metrics but, as we will

discuss, these constructions have a number of

extensions and generalizations.

The first dramatic application of these construc-

tions was the ADHM construction of Yang–Mills

instantons. These are absolute minima of the Yang–

Mills action, S[A] =

tr(F ^ F



) on the 4-sphere, S

with its round metric. A simple argument shows that

the action is bounde d below by the second Chern

class of the bundle and that this bound is achieved

only for anti-self-dual fields. Thus, the problem was

to characterize all the anti-self-dual Yang–Mills

fields on S

. In this Euclidean context, twistor

space, CP

, fibers over S

and the corresponding

Ward vector bundle is a bundle over all of CP

.It

turns out that all such bundles satisfying a certain

stability condition had been constructed reasonably

explicitly by alge braic geome ters. Since the stability

condition was implied by the context, this could be

turned into an algebraic construction of the general

instanton explicit enough to give some insight into

both the local and global structure of the solution

space. See Atiyah (1979) for a review.

Hitchin used the Euclidean version of the non-

linear graviton to develop the theory of gravitational

instantons that are asymptotically locally Euclidean

(i.e., asymptotically R

=, where  is a finite

subgroup of the rotation group). These were finally

constructed by Kronheimer who again used twistor

theory to identify the appropriate parameter space,

see his article in Mason et al. (2001) and Dancer’s

review of hyper-Ka¨ hler manifolds in LeBrun and

Wang (1999).

Even in four dimensions, there are a number of

variants of the nonlinear graviton construction. The

basic twistor correspondence produces a twistor

space that is a complex 3-manifold PT for

4-manifolds with conformal structures whose Weyl

tensor is anti-self-dual. There are four natural

specializations that have attracted study: (1) the

Ricci-flat case, (2) the Einstein case (with nonzero

cosmological constant), (3) the scalar-flat Ka¨ hler

case, and (4) the hypercomplex case.

The twistor space in the Ricci-flat case admits the

additional structure of a fibration over CP

together

with a holomorphic Poisson structure on the fibers

with values in the pullback of the 1-forms on CP

(alternatively, the bundle of holomorphic 3-forms

should be the pullback of the square of the bundle of

holomorphic 1-forms on CP

). The Einstein case

with nonzero cosmological constant is a variant of

this in which the twistor space admits a

Twistor Theory: Some Applications 307