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

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gases, and thermodynamic insights underpin much

of our qualitative understanding.

Few-Body Problems

The Two-Body Problem

For N = 2, the relative motion of the two bodies can

be reduced to the force-free motion of the center of

mass and the problem of the relative motion. If

r = r

 r

, then

€

r ¼Gðm

þ m

jrj

½2

often called the Kepler problem. It represents motion

of a particle of unit mass under a central inverse-square

force of attraction. Energy and angular momentum are

constant, and the motion takes place in a plane passing

through the origin. Using plane polar coordinates (r, )

in this plane, the equations for the energy and angular

momentum reduce to

E ¼





Gðm

þ m

½3

L ¼ r

 ½4

(Note that these are not the energy and angular

momentum of the two-body problem, even in the

barycentric frame of the center of mass; E and L must

be multiplied by the reduced mass m

=(m

þ m

).)

Using eqns [3] and [4], the problem is reduced to

quadratures. The solution shows that the motion is on

a conic section (ellipse, circle, straight line, parabola,

or hyperbola), with the origin at one focus.

This reduction depends on the existence of integrals

of the equations of motion, and these in turn depend

on symmetries of the underlying Lagrangian or

Hamiltonian. Indeed, eqns [1] yield ten first integrals:

six yield the rectilinear motion of the center of mass,

three the total angular momentum, and one the energy.

Furthermore, eqn [2] may be transformed, via the

Kustaanheimo–Stiefel (KS) transformation, to a four-

dimensional simple harmonic oscillator. This reveals

further symmetries, corresponding to further invar-

iants: the three components of the Lenz vector.

Another manifestation of the abundance of symme-

tries of the Kepler problem is the fact that there exist

action-angle variables in which the Hamiltonian

depends only on one action, that is, H = H(L).

Another application of the KS transformation is one

that has practical importance: it removes the singular-

ity of (i.e., regularizes) the Kepler problem at r = 0,

which is troublesome numerically.

To illustrate the character of the KS transforma-

tion, we consi der briefly the planar case, which can

be handled with a complex variable obeying the

equation of motion

€

z = z=jzj

(after scaling

eqn (2)). By introducing the Levi-Civita transforma-

tion z = Z

and Sundman’s transformation of the

time, that is, dt=d = jzj, the equation of motion

transforms to Z

= hZ=2, where h = j

=2  1=jzj is

the constant of energy. The KS transformation is a

very similar exercise using quaternions.

The Restricted Three-Body Problem

The simplest three-body problem is given by the

motion of a test particle in the gravitational field of

two particles, of positive mass m

, m

, in circular

Keplerian motion. This is called the circular

restricted three-body problem, and the two massive

particles are referred to as primaries. In a rotating

frame of reference, with origin at the center of mass

of these two particles, which are at rest at positions

, r

, the equation of motion is

€

r þ 2 

r þ  ð  rÞ

¼ Gr

jr  r



½5

where r is the position of the massless particle and 

is the angular velocity of the frame.

This problem has three degrees of freedom but only

one known integral: it is the Hamiltonian in the

rotating frame, and equivalent to the Jacobi integral, J.

One consequence is that Liouville’s theorem is not

applicable, and more elaborate arguments are required

to decide its integrability. Certainly, no general

analytical solution is known.

There are five equilibrium solutions, discovered

by Euler and Lagrange (see Figure 1). They lie at

critical points of the effective potential in the

Figure 1 The equilibrium solutions of the circular restricted

three-body problem. A rotating frame of reference is chosen in

which two particles are at rest on the x-axis. The massless

particle is at equilibrium at each of the five points shown. Five

similar configurations exist for the general three-body problem;

these are the ‘‘central’’ configurations.

576 Gravitational N-Body Problem (Classical)

rotating frame, and demarcate possible regions of

motion.

Throughout the twentieth century, much numerical

effort was used in finding and classifying periodic

orbits, and in determining their stability and bifurca-

tions. For example, there are families of periodic orbits

close to each primary; these are perturbed Kepler

orbits, and are referred to as satellite motions. Other

important families are the series of Liapounov orbits

starting at the equilibrium points.

Some variants of the restricted three-body pro-

blem include the following:

1. The elliptic restricted three-body problem, in

which the primaries move on an elliptic Kepler-

ian orbit; in suitable coordinates the equation of

motion closely resembles eqn [5], except for a

factor on the right side which depends explicitly

on the independent variable (transformed time);

this system has no first integral.

2. Sitnikov’s pr oblem, which is a special case of the

elliptic problem, in which m

= m

, and the

motion of the massless particle is confined to

the axis of symmetry of the Keplerian motion;

this is still nonintegrable, but simple enough to

allow extensive analysis of such fundamental

issues as integrability and stochasticity.

3. Hill’s problem, which is a scaled version suitable

for examining motions close to one primary; its

importance in applications began with studies of

the motion of the moon, and it remains vital for

understanding the motion of asteroids.

The General Three-Body Problem

Exact solutions When all three particles have

nonzero masses, the equations of motion become

€

¼r

where the potential energy is

W ¼G

1 i < j 3

 r

Then the exact solutions of Euler and Lagrange

survive in the form of homographic solutions. In

these solutions, the configuration remains geometri-

cally similar, but may rotate and/or pulsate in the

same way as in the two-body problem.

Let us repre sent the position vector r

in the

planar three-body problem by the complex number

. Then, it is easy to see that we have a solution of

the form z

(t) = z(t)z

, provided that

€

z ¼C

jzj

and

¼r

Wðz

; z

for some constant C. Thus, z(t) may take the form of

any solution of the Kepler problem, while the

complex numbers z

must correspond to what is

called a central configuration. These are in fact

critical points of the scale-free function W

ﬃﬃ

, where

I (the ‘‘moment of inertia of the system’’) is given by

I =

; and C = W=I.

The existence of other important classes of

periodic solutions can be proved analytically, even

though it is not possible to express the solution in

closed form. Examples include hierarchical three-

body systems, in which two masses m

, m

exhibit

nearly elliptic relative motion, while a third mass

orbits the barycenter of m

and m

in another nearly

elliptic orbit. In the mathematical literature, this is

referred to as motion of elliptic–elliptic type. More

surprisingly, the existence of a periodic solution in

which the three bodies travel in succession along the

same path, shaped like a figure 8 (cf. Figure 2), was

established by Chenciner and Montgomery (2000),

following its independent discovery by Moore using

numerical methods. Another interesting periodic

motion that was discovered numerically, by

Schubart, is a solution of the collinear three-body

problem, and so collisions are inevitable. In this

motion, the body in the middle alternately encoun-

ters the other two bodies.

–2

–4

Figure 2 A rare example of a scattering encounter between

two binaries (which approach from upper right and lower left)

which leads to a permanently bound triple system describing the

‘‘figure-8’’ periodic orbit. A fourth body escapes at the bottom.

Note the differing scales on the two axes. (Reproduced with

permission from Heggie DC (2000) A new outcome of binary-

binary scattering. Monthly Notices of the Royal Astronomical

Society 318(4): L61–L63; ª Blackwell Publishing Ltd.)

Gravitational N-Body Problem (Classical) 577

Singularities As Schubart’s solution illustrates,

two-body encounters can occur in the three -body

problem. Such singularities can be regularized just as

in the pure two-body problem. Triple collisions

cannot be regularized in general, and this singularity

has been studied by the technique of ‘‘blowup.’’ This

has been worked out most thoroughly in the collinear

three-body problem, which has only two degrees of

freedom. The general idea is to transform to two

variables, of which one (denoted by r, say) deter-

mines the scale of the system, while the other (s)

determines the configuration (e.g., the ratio of

separations of the three masses). By scaling the

corresponding velocities and the time, one obtains a

system of three equations of motion for s and the

two velocities which are perfectly regular in the limit

r ! 0. In this limit, the energy integral restricts the

solutions of the system to a manifold (called the

collision manifold). Exactly the same manifold

results for zero-energy solutions, which permits a

simple visualization. Equilibria on the collision

manifold correspond to the Lagrangian collinear

solutions in which the system either expands to

infinity or contracts to a three-body collision.

Qualitative ideas Reference has already been made

to motion of elliptic –elliptic type. In a motion of

elliptic–hyperbolic type, there is again an ‘‘inner’’

pair of bodies describing nearly Keplerian motion,

while the relative motion of the third body is nearly

hyperbolic. In applications, this is referred to as a

kind of scattering encounter between a binary and a

third body. When the encounter is sufficiently close,

it is possible for one member of the binary to be

exchanged with the third body. One of the major

historical themes of the general three-body problem

is the classification of connections between these

different types of asymptotic motion. It is possible to

show, for instance, that the measure of initial

conditions of hyperbolic–elliptic type leading asymp-

totically to elliptic–elliptic motion (or any other type

of permanently bound motion) is zero. Much of the

study of such problems has been carried out

numerically.

There are many ways in which the stability of

three-body motions may be approached. One exam-

ple is furnished by the central configurations already

referred to. They can be used to establish sufficient

conditions for ensuring that exchange is impossible,

and similar conclusions.

A powerful tool for qualitative study of three-

body motions is Lagrange’s identity, which is now

thought of as the reduction to three bodies of the

virial theorem. Let the size of the system be

characterized by the ‘‘moment of inertia’’ I. Then it

is easy to show that

¼ 4T þ 2W

where T, W are, respectively, the kinetic an d

potential energies of the system. Usually, the bary-

centric frame is adopted. Since E = T þ V is con-

stant and T  0, it follows that the system is not

bounded for all t > 0 unless E < 0.

Perturbation theory The question of the integr-

ability of the general three-body problem has

stimulated much research, including the famous

study by Poincare´ which established the nonexis-

tence of integrals beyond the ten classical ones.

Poincare´’s work was an important landmark in the

application to the three-body problem of perturba-

tion methods. If one mass dominates, that is, m



and m

 m

, then the motion of m

and m

relative to m

is a mildly perturbed two-body

motion, unless m

and m

are close together. Then

it is beneficial to describe the motion of m

relative

to m

by the parameters of Keplerian motion. These

would be constant in the absence of m

, and vary

slowly because of the perturbation by m

. This was

the idea be hind Lagrange’s very general method of

variation of parameters for solving systems of

differential equations. Numerous methods were

developed for the iterative solution of the resulting

equations. In this way, the solution of such a three-

body problem could be represented as a type of

trigonometric series in which the arguments are the

angle variab les describing the two approximate

Keplerian motions. These were of immense value in

solving problems of celestial mechanics, that is, the

study of the motions of planets, their satellites,

comets, and asteroids.

A major step forward was the introduction of

Hamiltonian methods. A three-body problem of the

type considered here has a Hamiltonian of the form

H ¼ H

ðL

ÞþH

ðL

ÞþR

where H

, i = 1, 2, are the Hamiltonians describing

the interaction between m

and m

,andR is the

‘‘disturbing function.’’ It depends on all the vari-

ables, but is small compared with the H

. Now

perturbation theory reduces to the task of perform-

ing canonical transformations which simplify R as

much as possible.

Poincare´’s major contribution in this area was to

show that the series solutions produced by perturba-

tion methods are not, in general, convergent, but

578 Gravitational N-Body Problem (Classical)

asymptotic. Thus, they were of practical rather than

theoretical value. For example, nothing could be

proved about the stability of the solar system using

perturbation methods. It took the further analytic

development of KAM theory to rescue this aspect of

perturbation theory. This theory can be used to

show that, provide d that two of the three masses are

sufficiently small, then for almost all initial condi-

tions the motions remain close to Kepl erian for all

time. Unfortunately, now it is the practical aspect of

the theory which is missing; though we have

introduced this topic in the context of the three-

body problem, it is extensible to any N-body system

with N 1 small masses in nearly Keplerian motion

about m

, but to be applicable to the solar system

the masses of the planets would have to be

ridiculously small.

Numerical methods Numerical integrations of the

three-body problem were first carried out near the

beginning of the twentieth century, and are now

commonplace. For typical scattering events, or other

short-lived solutions, there is usually little need to go

beyond common Runge–Kutta methods, provided

that automatic step-size control is adopted. When

close two-body approaches occur, some regulariza-

tion based on the KS transformation is often

exploited. In cases of prolonged elliptic–elliptic

motion, an analytic approximation based on Kepler-

ian motion may be adequate. Otherwise (as in

problems of planetary motion, where the evolution

takes place on an extremely long timescale), meth-

ods of very high ord er are often used. Symplect ic

methods, which have been developed in the context

of Hamiltonian problems, are increasingly adopted

for problems of this kind, as their long-term error

behavior is generally much superior to that of

methods which ignore the geometrical properties of

the equations of motion.

Four- and Five-Body Problems

Many of the foregoing remarks, on central config-

urations, numerical methods, KAM theory, etc.,

apply equally to few-body problems with N > 3.

Of special interest from a theoretical point of view is

the occurr ence of a new kind of singularity, in which

motions become unbounded in finite time. For

N = 4, the known examples also require two-body

collisions, but noncollision orbits exhibiting finite-

time blowup are known for N = 5.

One of the practical (or, at least, astronomical)

applications is again to scattering encounters, this

time involving the approach of two binaries on a

hyperbolic relative orbit. Numerical results show

that a wide variety of outcomes is possible, includ-

ing even the creation of the figure-8 periodic orbit of

the three-body problem, while a fourth body escapes

(Figure 2).

Many-Body Problems

Many of the concepts already introduced, such as

the virial theorem, apply equally well to the many-

body classical gravitational problem. This section

refers mainly to the new features which arise when

N is not small. In particular, statistical descriptions

become central. The applications also have a

different emphasis, moving from problems of plane-

tary dynamics (celestial mechanics) to those of

stellar dynamics. Typically, N lies in the range

–10

Evolution of the Distribution Function

The most useful statistical description is obtained if

the correlations we neglect and focus on the one-

particle distribution funct ion f (r, v, t), which can be

interpreted as the number density at time t at the

point in phase space corresponding to position r and

velocity v. Several processes contribute to the

evolution of f.

Collective effects When the effects of near neigh-

bors are neglected, the dynamics is described by the

Vlasov–Poisson system

þ v 



@ðr; tÞ



¼ 0 ½ 6

 ¼ 4Gm

f ðr; v; tÞd

v ½7

where  is the gravitational potential and m is the

mass of each body. Obvious ex tensions are neces-

sary if not all bodies have the same mass.

Solutions of eqn [6] may be found by the method

of characteristics, which is most useful in cases

where the equation of motion

€

r = r is integr-

able, for exa mple, in stationary, spherical potentials.

An example is the solution

f ¼jEj

7=2

½8

where E is the specific energy of a body, that is,

E = v

=2 þ . This satisfies eqn [6] provided that 

is static. Equation [7] is satisfied provided that 

satisfies a case of the Lane–Emden equation, which

is easy to solve in this case.

The solution just referred to is an example of an

equilibrium solution. In an equilibrium solution, the

virial theorem takes the form 4T þ 2W = 0, where

Gravitational N-Body Problem (Classical) 579

T, W are appropriate mean-field approximations for

the kinetic and potential energy, respectively. It

follows that E = T, where E = T þ V is the total

energy. An increase in E causes a decrease in T,

which implies that a self-gravitating N-body system

exhibits a negative specific heat.

There is little to choose between one equilibrium

solution and another, except for their stability. In

such an equilibrium, the bodies orbit within the

potential on a timescale of the crossing time, which

is conventionally defined to be

5=2

ð2jEjÞ

3=2

The most important evolutionary phenomenon of

collisionless dynamics is violent relaxation. If f is not

time indepe ndent then  is time dependent in

general. Also, from the equation of motion of one

body, E varies according to dE=dt = @=@t, and so

energy is exchanged between bodies, which leads to

an evolution of the distribution of energies. This

process is known as violent relaxati on.

Two other relaxation processes are of importance:

1. Relaxation is possible on each energy hypersur-

face, even in a static potential, if the potential is

nonintegrable.

2. The range of collective phenomena becomes

remarkably rich if the system exhibits ordered

motions, as in rotating systems. Then an impor-

tant role is played by resonant motions, espe-

cially resonances of low order. The

corresponding theory lies at the basis of the

theory of spiral structure in galaxies, for instance.

Collisional effects The approximations of colli-

sionless stellar dynamics suppress two important

processes:

1. The exponential divergence of stellar orbits,

which takes place on a timescale of order t

Even in an integrable potential, therefore, f

evolves on each energy hypersur face.

2. Two-body relaxation. It operates on a timescale of

order (N= ln N)t

, where N is the number of

particles. Although this two-body relaxation time-

scale, t

, is much longer than any other timescale

we have considered, this process leads to evolution

of f (E), and it dominates the long-term evolution

of large N-body systems . It is usually modeled by

adding a collision term of Fokker–Planck type on

the right-hand side of eqn [6].

In this case, the only equilibrium solutions

in a steady potential are those in which f (E) /

exp (E), where  is a constant. Then eqn [7]

becomes Liouville’s equation, and for the case of

spherical symmetry the relevant solutions are those

corresponding to the isothermal sphere.

Collisional Equilibrium

We consider the collisional evolution of an N-body

system further in a later subsection and here develop

fundamental ideas about the isothermal model . The

isothermal model has infinite mass, and much has

been learned by considering a model confined within

an adiabatic boundary or enclosure. There is a series

of such models, characterized by a single dimension-

less parameter, which can be taken to be the ratio

between the central density and the density at the

boundary, 

=

(Figure 3).

These models are extrema of the Boltzmann

entropy S = k

f ln f d

,wherek is the Boltzmann

constant, and the integration is taken over all

available phase space. Their stability may be

determined by evaluating the second variation of S.

It is found that it is negative definite, so that S is a

local maximum and the configuration is stable, only

if 

=

< 709 approximately. A physical explana-

tion for this is the following. In the limit when



=

’ 1, the self-gravity (which causes the spatial

inhomogeneity) is weak, and the system behaves like

an ordinary perfect gas. When 

=

 1, however,

the system is highly inhomogeneous, consisting of a

core of low mass and high density surrounded by an

extensive halo of high mass and low density.

Consider a transfer of energy from the deep interior

to the envelope. In the envelope, which is restrained

by the enclosure, the additional energy causes a rise

in temperature, but this is small, because of the very

large mass of the halo. Extraction of energy from

around the core, however, causes the bodies there to

sink and accelerate, and, because of the negative

–6

–5

–4

–3

–2

–1

–1 –0.5 0 1 2 30.5 1.5 2.5

Log density

radius

Figure 3 The density profile of the nonsingular isothermal

model, with conventional scalings.

580 Gravitational N-Body Problem (Classical)

specific heat of a self-gravitating system, they gain

more kinetic energy than they lost in the original

transfer. Now the system is hotter in the core than in

the halo, and the transfer of energy from the interior

to the exterior is self-sustaining, in a gravothermal

runaway. The isothermal model with large density

contrast is therefore unstable.

The negative specific heat, and the lack of an

equilibrium which maximizes the entropy, are two

examples of the anomalous thermo dynamic beha-

vior of the self-gravitating N-body problem. They

are related to the long-range nature of the gravita-

tional interaction, the importance of boundary

terms, and the nonextensivity of the energy. Another

consequence is the inequivalence of canonical and

microcanonical ensembles.

Numerical Methods

The foregoing considerations are difficult to extend

to systems without a boundary, although they are a

vital guide to the behavior even in this case. Our

knowledge of such systems is due largely to

numerical experiments, which fall into several

classes:

1. Direct N-body calculations. These minimize the

number of simplifying assumptions, but are

expensive. Special-purpose hardware is readily

available, which greatly accelerates the necessary

calculations. Great care has to be taken in the

treatment of few-body configurations, which

otherwise consume almost all resour ces.

2. Hierarchical methods, including tree methods,

which shorten the calculation of forces by

grouping distant masses. They are most ly used

for collisionless problems.

3. Grid-based methods, which are used for colli-

sionless problems.

4. Fokker–Planck methods, which usually require a

theoretical knowledge of the statistical effects of

two-, three- and four-body interactions. Other-

wise they can be very flexible, especially in the

form of Monte Carlo codes.

5. Gas codes. The behavior of a self-gravitating

system is simulated surprisingly well by modeling

it as a self-gravitating perfect gas, rather like a

star.

Collisional Evolution

Consider an isolate d N-body system, which is

supposed initially to be given by a spherically

symmetric equilibrium solution of eqns [6] and [7],

such as eqn [8]. The temperature decreases with

increasing radius, and a gravothermal runaway

causes the ‘‘collapse’’ of the core, which reaches

extremely high density in finite time. (This collapse

takes place on the long two-body relaxation time-

scale, and so it is not the rapid collapse, on a free-

fall timescale, which the name rather suggests.)

At sufficiently high densities, the timescale of

three-body reactions becomes competitive. These

create bound pairs, the excess energy being removed

by a third body. From the point of view of the one-

particle distribution function, f, these reactions are

exothermic, causing an expansion and cooling of the

high-density central regions. This temperat ure inver-

sion drives the gravothermal runaway in reverse,

and the core expands, until contact with the cool

envelope of the system restores a normal tempera-

ture profile. Core collapse resumes once more, and

leads to a chaotic sequ ence of expansi ons and

contractions, called gravothermal oscillations

(Figure 4).

The monotonic addition of energy during the

collapsed phases causes a secular expansion of the

system, and a general increase in all timescales. In

each relaxation time, a small fraction of the masses

escape, and eventually (it is thought) the system

consists of a dispersing collection of mutually

unbound single masses, binaries, and (presumably)

stable higher-order systems.

It is very remarkable that the long-term fate of

the largest self-gravitating N-body system appears

to be intimately linked with the three-body

problem.

N = 64 K stars

100

1000

Central density, ρ

0 4000 800060002000

T [N-body ]

Figure 4 Gravothermal oscillations in an N-body system with

N = 65 536. The central density is plotted as a function of time in

units such that t

= 2

ﬃﬃﬃ

. (Source: Baumgardt H, Hut P,

and Makino J, with permission.)

Gravitational N-Body Problem (Classical) 581

See also: Boltzmann Equation (Classical and Quantum);

Chaos and Attractors; Dynamical Systems and

Thermodynamics; KAM Theory and Celestial Mechanics;

Lyapunov Exponents and Strange Attractors;

Nonequilibrium Statistical Mechanics: Interaction

between Theory and Numerical Simulations; Quantum

N-Body Problem; Stability Problems in Celestial

Mechanics; Stability Theory and KAM.

Further Reading

Arnol’d VI, Kozlov VV, and Neishtadt AI (1993) Mathematical

Aspects of Classical and Celestial Mechanics, 2nd edn. Berlin:

Springer.

Barrow-Green J (1997) Poincare´ and the Three Body Problem.

Providence: American Mathematical Society.

Binney J and Tremaine S (1988) Galactic Dynamics. Princeton:

Princeton University Press.

Chenciner A and Montgomery R (2000) A remarkable periodic

solution of a three-body problem in the case of equal masses.

Annals of Mathematics 152: 881–901.

Dauxois T et al. (eds.) (2003) Dynamics and Thermodynamics of

Systems with Long Range Interactions. Berlin: Springer.

Diacu F (2000) A century-long loop. Mathematical Intelligencer

22(2): 19–25.

Heggie DC and Hut P (2003) The Gravitational Million-Body

Problem. Cambridge: Cambridge University Press.

Marchal C (1990) The Three-Body Problem. Amsterdam: Elsevier.

Murray CD and Dermott SF (2000) Solar System Dynamics.

Cambridge: Cambridge University Press.

Padmanabhan T (1990) Statistical mechanics of gravitating

systems. Physics Reports 188: 285–362.

Siegel CL, Moser JK, and Kalme CI (1995) Lectures on Celestial

Mechanics. Berlin: Springer.

Steves BJ and Maciejewski AJ (eds.) (2001) The Restless Universe.

Applications of Gravitational N-Body Dynamics to Planetary,

Stellar and Galactic Systems. Bristol: IoP and SUSSP.

Szebehely V (1967) Theory of Orbits: The Restricted Problem of

Three Bodies. New York: Academic Press.

Gravitational Waves

G Gonza´ lez and J Pullin, Louisiana State University,

Baton Rouge, LA, USA

In elementary physics presentations, one learns about

electricity and magnetism, and also about gravity.

There appear striking similarities between Newton’s

law of gravitational attraction and Coulomb’s law of

attraction between charges. There are also obvious

differences, the most immediate one being that in

gravitation all masses are positive and always attract

each other, whereas in electromagnetism charges may

attract or repel, depending on their signs. We also

know today that Newton’s theory of gravity is not

considered an entirely correct description of the

gravitational field, particularly when fields are time

dependent and intense. The currently accepted theory

of gravity is Einstein’s theory of general relativity.

The similarity between electromagnetism and

gravitation also holds to a certain extent when the

fields depend on time. This is usually not discussed

in elementary treatments since a full description of

time-dependent gravitational fields requi res the use

of general relativity. It is true, however, that if the

fields are weak, there exist several similarities

between gravitation and electromagnetism. In parti-

cular, one can have waves in the gravitational field

that are able to carry energy from a source to a

receptor.

If one assumes that the metric of spacetime is

close to the flat Minkow ski metric 



, that is,



= 



þ h



with jh



j << 1 in Cartesian coordi-

nates, the Einstein equations of general relativity,

expanding to linear order in h



, become

0 ¼R



















þ @









 @





 &h



 









þ 





½1

These do not look like wave equations. However,

if one chooses ‘‘harmonic coordinates,’’



= 0,

where

is the d’Alembertian constructed with the

full metric and then linearized, the vacuum Einstein

equations become



¼ 0 ½2

where

is the d’Alembertian computed in the flat

Minkowski metric.

Just as in electromagnetism the motion of charges

produces waves, the motion of masses produces

waves in the gravitational field. In the above wave

equations, one would have nonzero right-hand sides

if masses were present. In electromagnetism, the

conservation of charge implies that the lowest order

of ‘‘structure’’ a source must have to produce

electromagnetic waves is that of a time-dependent

dipole. In the gravitational field, the conservation of

momentum implies that the lowest multipolar order

of a source of gravitational waves must be a

quadrupole. Moreover, gravity is a weaker force

than electromagnetism when one considers usually

available situations. One can exert forces of the

orders of fractions of Newton with electromagnetic

582 Gravitational Waves

charges easily collected in tabletop experiments. To

produce similar amou nts of gravitational force, one

needs large quantities of mass. This last fact,

coupled with the quadrupolar nature of the sources

of gravi tational waves, makes their production quite

challenging in experimental terms. The luminosity of

a gravitational wave source is given by the cele-

brated Einstein quadrupole formula,

L ¼



j¼1;k¼1

...



½3

where G is Newton’s gravitational constant, c is the

speed of light, and I

...

is the third-order time

derivative of the traceless part of the qua drupole

mass moment of the source.

Gravity is, however, a domina nt force if one

considers the universe at large (say, at least

planetary) scales. There one would expect gravita-

tional waves to play some role in the dynamics of

the systems. In such systems, the pr esence of

gravitational waves has indeed been experimentally

confirmed. We know of a system of two pulsars in

mutual orbit, PSR1913 þ16, whose orbit has been

tracked with enough accuracy via radioastronomy

to make the influence of gravitational waves

observable. The motion of the pulsars makes the

system an emitter of gravitational waves. Since the

waves carry away energy, the orbit of the system

decreases in radius and the period of oscillation

increases. The system has now been tracked for over

20 years, and the predi ction of the emitted amount

of energy in gravitational waves due to general

relativity has been confirmed with a very significant

degree of accuracy. Penrose was the first to notice

that if one considers how accurately Newton’s

theory plus the corrections due to general relativity

predict the positions of the pulsars in their orbit, this

is in fact the most accurately verified physical

prediction ever.

Technically, even the existence of gravitational

waves at a conceptual mathematical level, was an

open problem for many years. Since the correct

description of the waves is through the general

theory of relativity, a ‘‘gravitational wave’’ should

really be viewed as a ‘‘ripple in spacetime.’’ Disen-

tangling if such ripples are a true physical effect or a

time-dependent coordinate transformation that pro-

pagates – to use the words of Eddington – ‘‘with the

speed of thought’’ took quite a bit of technical

development within the general theory of relativity.

It was only in the 1960s that a clear enou gh

conceptual picture was developed to determine that

gravitational waves were indeed a true physical

phenomenon akin to electromagnetic waves. And in

particular that one can unambiguously characterize

them as transporting energy, momentum, and

angular momentum from a source to an observer.

Gravitational waves are as difficult to detect as

they are to produce. Since all masses fall in the same

way in a gravitational field, one needs to couple to

the gradients of the field to detect gravitational

waves, which diminishes the efficiency. Attempting

to produce gravitational waves via mechanical

means in the lab (e.g., by rotating a bar of metal)

produces too little luminosity, and in addition, the

relatively low frequency implies that the wave zone

is far away, which further decreases the chances for

detection. Up to date, no one has succeeded in

producing a Hertz-like experiment for gravitational

waves and the jury is still out on the issue if future

technologies (e.g., the use of superconductors to

produce waves of gigahertz frequency) will ever

allow such an experiment.

Efforts to attempt to detect gravitational waves

produced by astrophysical phenomena started in the

1960s with pioneering work by Weber. The initially

proposed technology for detection was the construc-

tion of large (1 ton) resonant bars. The idea was

to use sensitive technology to measure the resonance

of the bar as gravitational waves of astrophysical

origin impact on it. Gravitational waves manifest

themselves as a stretching and contraction of

lengths. The contraction or stretchi ng is propor-

tional to the length of the object considered and is

therefore characterized by a dimensionless number,

the ‘‘strain’’ L=L usually called ‘‘h.’’ Conservative

current estimates of possible astrophysical sources

state that on Earth one should not expect strains

larger than 10

22

for events that repeat more

frequently than a few times every year. Detectors

with bar technology are approaching their funda-

mental quantum limits with strains that appear to be

too large for detection to be ensured. This led to the

proposal of a new technology, the use of Michelson-

type interferometers to detect the waves. Currently,

several interferometric detectors are being built in

the US, Europe, Japan, and Australia that expect to

achieve enough sensitivity for detection within a few

years. Contrary to the bars, which are quintessen-

tially narrow-band detectors (most bars operate

900 Hz with a bandwidth of 10 Hz), interfero-

metric detectors are broadband. Current detectors

have a sensitivity curve limited by various sources of

noise that make them suitable for detection within

the 10 Hz–1 kHz band. The broadband nature of the

detectors opens several oppor tunities for the use of

data analysis techniques that can allow the detection

of gravitational waves that have strains even lower

than the noise of the detectors. Moreover, several of

Gravitational Waves 583

the candidate events ‘‘evolve’’ in frequency as they

emit gravitational waves (in the case of the binary

pulsar, for instance, the frequency ‘‘sweeps up’’ as the

system loses energy), and such evolution could be

monitored with interferometric detectors. This

would allow several insights into the physics of the

observed systems.

An important limitation of any type of detector

based on Earth is that the seismic noise increases

quite significantly below 10 Hz. Even if seismic

isolation allo wed sensitivities below 10 Hz, gravity

gradients due to Earth’s seismic motion and due to

clouds would limit gro und-based detectors to 1 Hz

and above. The frequency at which a system emits

gravitational waves is inversely proportional to the

system’s mass (a simple way to see this is to realize

that larger systems move proportionally slower to

their size). However, larger systems generically have

more mass and therefore consequently emit larger

amounts of energy in gravitational waves. This

suggests that setting up detectors in space, free of

the constraints of seismic noise, would offer sig-

nificant promise in detec ting gravitational waves.

Currently, there is a proposal for a space-borne

gravitational wave detector consisting of three

satellites in a solar orbit that trails that of Earth.

Lasers would be sent between the satellites to track

their relative pos itions, which will be separated by

5 mi llion kilometers. Such a detector would be

sensitive in frequencies of 10

4

–10

2

Hz. In such a

frequency band, one expects that compact objects

plunging into supermassive black holes and other

sources will be readily available. Detection of

gravitational waves on Earth is considered marginal,

in the sense that conservative current estimates

cannot guarantee that there will be enough events

to make the detection successful at significant event

rates. Conversely, for the detectors in space, detec-

tion should be guaranteed at high event rates.

Possible sources of gravitational waves to be

detected by the Earth-based interferometric detec-

tors are:

1. Binary systems of compact objects. As the system

orbits, it emits gravitational waves, which makes

the orbit shrink in size and the orbiting period

shorter with the objects eventually merging

together. Potential systems include black hole

binaries, neutron star binaries and mixed black

hole/neutron star binaries. As the system sweeps

up in orbital speed towards the merger, so does

the frequency of the gravitational waves emitted.

For binaries of neutron stars, which usually have

masses slightly larger than the mass of the Sun,

the last few minutes of the binary inspiral will be

detectable by the current generation of gravita-

tional wave detectors, up to a distance of several

mega-parsecs for the initial detectors, increasing

to a few hundreds of mega-parsecs for improve-

ments planned for the next few years. For black

hole binaries, since the masses can be larger, one

expects larger signal-to-noise ratio for the same

distance or to be able to detect at larger

distances.

2. Spinning neutron stars that develop ‘‘mountains’’

or other irregularities in the surface would

produce gravitational wave s of small amplitude

but of a very regular periodic nature. This makes

them prime candidates for data analysis techni-

ques that could exhibit the presence of the wave

even though it is weaker than the background

noise of the interferometers. Integration periods

of several months may be needed for detection,

depending on the size of the asymmetries in the

neutron stars.

3. Supernovas or other violent events are obviously

possible sources of gravitational waves. How-

ever, the quadrupole nature of the waves requires

the events to be asymmetric in order to produce

gravitational waves. Current numerical models of

supernovas are not accurate enough to predict in

a clear way the level of asymmetry to make

reliable predictions of how frequently and at

what intensity could these types of sources be

detected.

4. The primordial background of gravitational

waves produced in the big bang is not expected

to be detectable by the Earth-based detectors.

The precise amplitude of the background is

unknown, depending on details of cosmological

models. The detectors are likely to be able to

constrain some of the models that predict large

amplitudes for the gravitational wave

background.

For the space-based detectors, the situation is

more favorable, since there exist sources of gravita-

tional waves that are guaranteed to be detected.

Potential sources of gravitational waves are:

1. Merger of the supermassive black holes at the

centers of two galaxies. Given the large amounts

of mass involved, they would be easily detected

and very precise measurements of the system’s

parameters and of various general relativistic

behaviors could be possible. Such systems should

be detectabl e all across the universe, although it

is not expected that such systems form for

redshifts larger than 30.

2. Inspiral of compact objects into the supermassive

black holes at the centers of galaxies (neutron

584 Gravitational Waves

stars, white dwarfs, solar-sized black holes).

These processes will allow the usage of gravita-

tional waves to map precisely the gravitational

field of the supermassive object.

3. White-dwarf binaries and low-mass X-ray bin-

aries. There exist about a dozen such systems

optically observable with gravitational wave

frequencies above 0.1 mHz that the space-based

detectors should be able to detect. There is likely

to be a large population of other systems that are

also detectable and are not optically visible. In

fact, there may be so many of these sources that

time resolution would be impossible, and they

would form a random background.

4. Collapse of supermassive stars. The formation

mechanism for the supermassive black holes in

the centers of galaxies is still uncertain. One

possibility is that they stem from the collapse of

supermassive stars, and in that case a potentially

significant emission of gravitational waves could

take place.

5. Primordial background of gravitat ional waves.

Unfortunately, the abundance of white-dwarf

binaries as a source is expected to cloud the ability

of the space detectors to observe primordial

gravitational waves in an important portion of the

spectrum of the instrument, although it appears

possible at low frequencies, where it could compete

with the bounds set by pulsar timing.

The current Earth-based g ravitational wave pro-

jects include the LIGO project in the US, funded by

the National Science Foundation and jointly oper-

ated by Caltech and MIT and a consortium of

institutions known as the LIGO Science Collabora-

tion. LIGO consists of two 4 km long Fabry–Perot

recycled Michelson interferometers, one in Hanford,

WA, and one in Livingston, LA. In Europe, the

GEO600 project is a 600 m dual-recycled inter-

ferometer near Hanover in Germany and the Virgo

project is a 3 km interferometer near Pisa in Italy

operated by a French–Italian consortium with a

similar optical configuration as LIGO. TAMA300 is

a 300 m interferometer in Japan also with the same

configuration as LIGO. When all these detectors are

in operation, sources seen in coincidence could be

localized by triangulation. TAMA is now operating

close to design sensitivity, GEO600 and LIGO are

likely to operate at design sensitivity in 2006, with

VIRGO following close behind. The space-based

interferometer project is called the LISA project and

is planned as a joint NASA/ESA project. ESA has

approved a launching date for 2015, but it is

plausible that the mission could be launched at an

earlier date.

A direct detection of gravitational waves would be a

breakthrough in experimental science, as well as a

confirmation of the dynamic nature of gravity in

general relativity. Once the detection of gravitational

waves becomes a routine matter, one can imagine a

revolution in astronomy as one uses gravitational

waves to ‘‘see’’ the universe. Since they are so hard to

produce and interfere with, gravitational waves

become an excellent type of ‘‘light’’ to look at the

universe with. Gravitational waves will be produced by

important concentrations of mass, correlating well

with ‘‘interesting’’ astronomical processes, and is not

expected to be affected by the presence of dust or other

interfering objects that could easily obscure electro-

magnetic waves. In addition to this, one has several

‘‘standard candles’’ for gravitational waves (e.g., most

neutron stars have masses that differ by a few percent

from 1.4 solar masses). This could allow, for instance,

to determine with a high degree of accuracy the Hubble

constant. Gravitational waves will also provide insight

into the nuclear equation of state that holds in the

interior of compact objects like neutron stars. Contrary

to ordinary electromagnetic radiation, which

‘‘decoupled’’ from matter only when the universe

became cool enough after the big bang, gravitational

waves could be used to probe the universe further into

the past. The detection could also prove that gravita-

tional waves travel at the speed of light, a prediction of

general relativity and other theories.

An interesting observation is that most astrophysical

objects that are quite visible in the electromagnetic

spectrum are unlikely to be visible in terms of

gravitational waves, and vice versa. This makes the

information we will gather from gravitational wave

astronomy complementary to what we learn from

optical (electromagnetic) astronomy. Moreover, it

should be noted that wavelengths of electromagnetic

waves are typically very small compared to the size of

the astronomical objects they depict. This is due to the

fact that the waves are really not produced by the

objects themselves but by atoms on the surface of the

objects or in regions nearby, usually very hot and in

gaseous form. In contrast, gravitational waves are

produced by the bulk matter of astronomical objects

and their wavelengths are expected to be long as

compared to the objects that produce them. They are

more akin to a sound than to light in this respect,

another reason to suspect that the information we will

get from them is unlike any information obtained

electromagnetically.

Gravitational waves are likely to bring great

surprises. Every time a new window has been

opened on the universe – for instance, the use of

radio waves – our view of the universe has been

revolutionized. Given how differently they operate

Gravitational Waves 585