General Relativity Today
General Relativity Today∗†
Thibault Damour
Institut des Hautes Etudes Scientifiques
35 route de Chartres, 91440 Bures-sur-Yvette, France
Abstract: After recalling the conceptual foundations and the basic struc-
ture of general relativity, we review some of its main modern developments
(apart from cosmology) : (i) the post-Newtonian limit and weak-field tests in
the solar system, (ii) strong gravitational fields and black holes, (iii) strong-field
and radiative tests in binary pulsar observations, (iv) gravitational waves, (v)
general relativity and quantum theory.
1 Introduction
The theory of general relativity was developed by Einstein in work that extended
from 1907 to 1915. The starting point for Einstein’s thinking was the compo-
sition of a review article in 1907 on what we today call the theory of special
relativity. Recall that the latter theory sprang from a new kinematics governing
length and time measurements that was proposed by Einstein in June of 1905
[1], [2], following important pioneering work by Lorentz and Poincaré. The
theory of special relativity essentially poses a new fundamental framework (in
place of the one posed by Galileo, Descartes, and Newton) for the formulation of
physical laws: this framework being the chrono-geometric space-time structure
of Poincaré and Minkowski. After 1905, it therefore seemed a natural task to
formulate, reformulate, or modify the then known physical laws so that they fit
within the framework of special relativity. For Newton’s law of gravitation, this
task was begun (before Einstein had even supplied his conceptual crystallization
in 1905) by Lorentz (1900) and Poincaré (1905), and was pursued in the period
from 1910 to 1915 by Max Abraham, Gunnar Nordström and Gustav Mie (with
these latter researchers developing scalar relativistic theories of gravitation).
Meanwhile, in 1907, Einstein became aware that gravitational interactions
possessed particular characteristics that suggested the necessity of generalizing
the framework and structure of the 1905 theory of relativity. After many years
of intense intellectual effort, Einstein succeeded in constructing a generalized
∗Talk given at the Poincaré Seminar “Gravitation et Expérience” (28 October 2006, Paris);
to appear in the proceedings to be published by Birkhäuser.
†Translated from the French by Eric Novak.
http://arxiv.org/abs/0704.0754v1
theory of relativity (or general relativity) that proposed a profound modification
of the chrono-geometric structure of the space-time of special relativity. In 1915,
in place of a simple, neutral arena, given a priori, independently of all material
content, space-time became a physical “field” (identified with the gravitational
field). In other words, it was now a dynamical entity, both influencing and
influenced by the distribution of mass-energy that it contains.
This radically new conception of the structure of space-time remained for
a long while on the margins of the development of physics. Twentieth century
physics discovered a great number of new physical laws and phenomena while
working with the space-time of special relativity as its fundamental framework,
as well as imposing the respect of its symmetries (namely the Lorentz-Poincaré
group). On the other hand, the theory of general relativity seemed for a long
time to be a theory that was both poorly confirmed by experiment and without
connection to the extraordinary progress springing from application of quantum
theory (along with special relativity) to high-energy physics. This marginaliza-
tion of general relativity no longer obtains. Today, general relativity has become
one of the essential players in cutting-edge science. Numerous high-precision ex-
perimental tests have confirmed, in detail, the pertinence of this theory. General
relativity has become the favored tool for the description of the macroscopic uni-
verse, covering everything from the big bang to black holes, including the solar
system, neutron stars, pulsars, and gravitational waves. Moreover, the search
for a consistent description of fundamental physics in its entirety has led to the
exploration of theories that unify, within a general quantum framework, the
description of matter and all its interactions (including gravity). These theo-
ries, which are still under construction and are provisionally known as string
theories, contain general relativity in a central way but suggest that the funda-
mental structure of space-time-matter is even richer than is suggested separately
by quantum theory and general relativity.
2 Special Relativity
We begin our exposition of the theory of general relativity by recalling the
chrono-geometric structure of space-time in the theory of special relativity. The
structure of Poincaré-Minkowski space-time is given by a generalization of the
Euclidean geometric structure of ordinary space. The latter structure is sum-
marized by the formula L2 = (∆x)2 + (∆y)2 + (∆z)2 (a consequence of the
Pythagorean theorem), expressing the square of the distance L between two
points in space as a sum of the squares of the differences of the (orthonormal)
coordinates x, y, z that label the points. The symmetry group of Euclidean ge-
ometry is the group of coordinate transformations (x, y, z) → (x′, y′, z′) that
leave the quadratic form L2 = (∆x)2 + (∆y)2 +(∆z)2 invariant. (This group is
generated by translations, rotations, and “reversals” such as the transformation
given by reflection in a mirror, for example: x′ = −x, y′ = y, z′ = z.)
The Poincaré-Minkowski space-time is defined as the ensemble of events (ide-
alizations of what happens at a particular point in space, at a particular moment
in time), together with the notion of a (squared) interval S2 defined between
any two events. An event is fixed by four coordinates, x, y, z, and t, where
(x, y, z) are the spatial coordinates of the point in space where the event in
question “occurs,” and where t fixes the instant when this event “occurs.” An-
other event will be described (within the same reference frame) by four different
coordinates, let us say x+∆x, y+∆y, z+∆z, and t+∆t. The points in space
where these two events occur are separated by a distance L given by the for-
mula above, L2 = (∆x)2+(∆y)2+(∆z)2. The moments in time when these two
events occur are separated by a time interval T given by T = ∆t. The squared
interval S2 between these two events is given as a function of these quantities,
by definition, through the following generalization of the Pythagorean theorem:
S2 = L2 − c2 T 2 = (∆x)2 + (∆y)2 + (∆z)2 − c2(∆t)2 , (1)
where c denotes the speed of light (or, more precisely, the maximum speed of
signal propagation).
Equation (1) defines the chrono-geometry of Poincaré-Minkowski space-time.
The symmetry group of this chrono-geometry is the group of coordinate trans-
formations (x, y, z, t) → (x′, y′, z′, t′) that leave the quadratic form (1) of the
interval S invariant. We will show that this group is made up of linear trans-
formations and that it is generated by translations in space and time, spatial
rotations, “boosts” (meaning special Lorentz transformations), and reversals of
space and time.
It is useful to replace the time coordinate t by the “light-time” x0 ≡ ct, and
to collectively denote the coordinates as xµ ≡ (x0, xi) where the Greek indices
µ, ν, . . . = 0, 1, 2, 3, and the Roman indices i, j, . . . = 1, 2, 3 (with x1 = x, x2 = y,
and x3 = z). Equation (1) is then written
S2 = ηµν ∆x
µ ∆xν , (2)
where we have used the Einstein summation convention1 and where ηµν is a
diagonal matrix whose only non-zero elements are η00 = −1 and η11 = η22 =
η33 = +1. The symmetry group of Poincaré-Minkowski space-time is therefore
the ensemble of Lorentz-Poincaré transformations,
x′µ = Λµν x
ν + aµ , (3)
where ηαβ Λ
ν = ηµν .
The chrono-geometry of Poincaré-Minkowski space-time can be visualized
by representing, around each point x in space-time, the locus of points that
are separated from the point x by a unit (squared) interval, in other words the
ensemble of points x′ such that S2xx′ = ηµν(x
′µ−xµ)(x′ν −xν) = +1. This locus
is a one-sheeted (unit) hyperboloid.
If we were within an ordinary Euclidean space, the ensemble of points x′
would trace out a (unit) sphere centered on x, and the “field” of these spheres
1Every repeated index is supposed to be summed over all of its possible values.
centered on each point x would allow one to completely characterize the Eu-
clidean geometry of the space. Similarly, in the case of Poincaré-Minkowski
space-time, the “field” of unit hyperboloids centered on each point x is a visual
characterization of the geometry of this space-time. See Figure 1. This figure
gives an idea of the symmetry group of Poincaré-Minkowski space-time, and
renders the rigid and homogeneous nature of its geometry particularly clear.
Figure 1: Geometry of the “rigid” space-time of the theory of special relativity.
This geometry is visualized by representing, around each point x in space-time,
the locus of points separated from the point x by a unit (squared) interval. The
space-time shown here has only three dimensions: one time dimension (repre-
sented vertically), x0 = ct, and two spatial dimensions (represented horizon-
tally), x, y. We have also shown the ‘space-time line’, or ‘world-line’, (moving
from the bottom to the top of the “space-time block,” or from the past towards
the future) representing the history of a particle’s motion.
The essential idea in Einstein’s article of June 1905 was to impose the group
of transformations (3) as a symmetry group of the fundamental laws of physics
(“the principle of relativity”). This point of view proved to be extraordinarily
fruitful, since it led to the discovery of new laws and the prediction of new phe-
nomena. Let us mention some of these for the record: the relativistic dynamics
of classical particles, the dilation of lifetimes for relativistic particles, the re-
lation E = mc2 between energy and inertial mass, Dirac’s relativistic theory
of quantum spin 1
particles, the prediction of antimatter, the classification of
particles by rest mass and spin, the relation between spin and statistics, and
the CPT theorem.
After these recollections on special relativity, let us discuss the special fea-
ture of gravity which, in 1907, suggested to Einstein the need for a profound
generalization of the chrono-geometric structure of space-time.
3 The Principle of Equivalence
Einstein’s point of departure was a striking experimental fact: all bodies in an
external gravitational field fall with the same acceleration. This fact was pointed
out by Galileo in 1638. Through a remarkable combination of logical reason-
ing, thought experiments, and real experiments performed on inclined planes,2
Galileo was in fact the first to conceive of what we today call the “universality of
free-fall” or the “weak principle of equivalence.” Let us cite the conclusion that
Galileo drew from a hypothetical argument where he varied the ratio between
the densities of the freely falling bodies under consideration and the resistance of
the medium through which they fall: “Having observed this I came to the con-
clusion that in a medium totally devoid of resistance all bodies would fall with
the same speed” [3]. This universality of free-fall was verified with more pre-
cision by Newton’s experiments with pendulums, and was incorporated by him
into his theory of gravitation (1687) in the form of the identification of the iner-
tial massmi (appearing in the fundamental law of dynamics F = mi a) with the
gravitational mass mg (appearing in the gravitational force, Fg = Gmgm
mi = mg . (4)
At the end of the nineteenth century, Baron Roland von Eötvös verified
the equivalence (4) between mi and mg with a precision on the order of 10
and Einstein was aware of this high-precision verification. (At present, the
equivalence between mi and mg has been verified at the level of 10
−12 [4].) The
point that struck Einstein was that, given the precision with whichmi = mg was
verified, and given the equivalence between inertial mass and energy discovered
by Einstein in September of 1905 [2] (E = mi c
2), one must conclude that all of
the various forms of energy that contribute to the inertial mass of a body (rest
mass of the elementary constituents, various binding energies, internal kinetic
energy, etc.) do contribute in a strictly identical way to the gravitational mass of
this body, meaning both to its capacity for reacting to an external gravitational
field and to its capacity to create a gravitational field.
In 1907, Einstein realized that the equivalence betweenmi andmg implicitly
contained a deeper equivalence between inertia and gravitation that had impor-
tant consequences for the notion of an inertial reference frame (which was a fun-
damental concept in the theory of special relativity). In an ingenious thought
experiment, Einstein imagined the behavior of rigid bodies and reference clocks
within a freely falling elevator. Because of the universality of free-fall, all of the
objects in such a “freely falling local reference frame” would appear not to be
accelerating with respect to it. Thus, with respect to such a reference frame,
the exterior gravitational field is “erased” (or “effaced”). Einstein therefore pos-
tulated what he called the “principle of equivalence” between gravitation and
inertia. This principle has two parts, that Einstein used in turns. The first
part says that, for any external gravitational field whatsoever, it is possible to
2The experiment with falling bodies said to be performed from atop the Leaning Tower of
Pisa is a myth, although it aptly summarizes the essence of Galilean innovation.
locally “erase” the gravitational field by using an appropriate freely falling local
reference frame and that, because of this, the non-gravitational physical laws
apply within this local reference frame just as they would in an inertial reference
frame (free of gravity) in special relativity. The second part of Einstein’s equiv-
alence principle says that, by starting from an inertial reference frame in special
relativity (in the absence of any “true” gravitational field), one can create an
apparent gravitational field in a local reference frame, if this reference frame is
accelerated (be it in a straight line or through a rotation).
4 Gravitation and Space-Time Chrono-Geometry
Einstein was able (through an extraordinary intellectual journey that lasted
eight years) to construct a new theory of gravitation, based on a rich general-
ization of the 1905 theory of relativity, starting just from the equivalence prin-
ciple described above. The first step in this journey consisted in understanding
that the principle of equivalence would suggest a profound modification of the
chrono-geometric structure of Poincaré-Minkowski space-time recalled in Equa-
tion (1) above.
To illustrate, let Xα, α = 0, 1, 2, 3, be the space-time coordinates in a lo-
cal, freely-falling reference frame (or locally inertial reference frame). In such a
reference frame, the laws of special relativity apply. In particular, the infinites-
imal space-time interval ds2 = dL2 − c2 dT 2 between two neighboring events
within such a reference frame Xα, X ′α = Xα + dXα (close to the center of this
reference frame) takes the form
ds2 = dL2 − c2 dT 2 = ηαβ dXα dXβ , (5)
where we recall that the repeated indices α and β are summed over all of their
values (α, β = 0, 1, 2, 3). We also know that in special relativity the local energy
and momentum densities and fluxes are collected into the ten components of
the energy-momentum tensor Tαβ. (For example, the energy density per unit
volume is equal to T 00, in the reference frame described by coordinates Xα =
(X0, X i), i = 1, 2, 3.) The conservation of energy and momentum translates
into the equation ∂β T
αβ = 0, where ∂β = ∂/∂ X
The theory of special relativity tells us that we can change our locally in-
ertial reference frame (while remaining in the neighborhood of a space-time
point where one has “erased” gravity) through a Lorentz transformation, X ′α =
Λαβ X
β. Under such a transformation, the infinitesimal interval ds2, Equation
(5), remains invariant and the ten components of the (symmetric) tensor Tαβ
are transformed according to T ′αβ = Λαγ Λ
γδ. On the other hand, when
we pass from a locally inertial reference frame (with coordinates Xα) to an
extended non-inertial reference frame (with coordinates xµ; µ = 0, 1, 2, 3), the
transformation connecting the Xα to the xµ is no longer a linear transforma-
tion (like the Lorentz transformation) but becomes a non-linear transformation
Xα = Xα(xµ) that can take any form whatsoever. Because of this, the value of
the infinitesimal interval ds2, when expressed in a general, extended reference
frame, will take a more complicated form than the very simple one given by
Equation (5) that it had in a reference frame that was locally in free-fall. In
fact, by differentiating the non-linear functions Xα = Xα(xµ) we obtain the
relation dXα = ∂Xα/∂xµ dxµ. By substituting this relation into (5) we then
obtain
ds2 = gµν(x
λ) dxµ dxν , (6)
where the indices µ, ν are summed over 0, 1, 2, 3 and where the ten functions
gµν(x) (symmetric over the indices µ and ν) of the four variables x
λ are de-
fined, point by point (meaning that for each point xλ we consider a refer-
ence frame that is locally freely falling at x, with local coordinates Xαx ) by
gµν(x) = ηαβ ∂X
x (x)/∂x
µ ∂Xβx (x)/∂x
ν . Because of the nonlinearity of the
functions Xα(x), the functions gµν(x) generally depend in a nontrivial way on
the coordinates xλ.
The local chrono-geometry of space-time thus appears to be given, not by
the simple Minkowskian metric (2), with constant coefficients ηµν , but by a
quadratic metric of a much more general type, Equation (6), with coefficients
gµν(x) that vary from point to point. Such general metric spaces had been
introduced and studied by Gauss and Riemann in the nineteenth century (in
the case where the quadratic form (6) is positive definite). They carry the
name Riemannian spaces or curved spaces. (In the case of interest for Einstein’s
theory, where the quadratic form (6) is not positive definite, one speaks of a
pseudo-Riemannian metric.)
We do not have the space here to explain in detail the various geometric
structures in a Riemannian space that are derivable from the data of the in-
finitesimal interval (6). Let us note simply that given Equation (6), which
gives the distance ds between two infinitesimally separated points, we are able,
through integration along a curve, to define the length of an arbitrary curve
connecting two widely separated points A and B: LAB =
ds. One can then
define the “straightest possible line” between two given points A and B to be
the shortest line, in other words the curve that minimizes (or, more generally,
extremizes) the integrated distance LAB. These straightest possible lines are
called geodesic curves. To give a simple example, the geodesics of a spherical
surface (like the surface of the Earth) are the great circles (with radius equal
to the radius of the sphere). If one mathematically writes the condition for
a curve, as given by its parametric representation xµ = xµ(s), where s is the
length along the curve, to extremize the total length LAB one finds that x
must satisfy the following second-order differential equation:
d2 xλ
+ Γλµν(x)
= 0 , (7)
where the quantities Γλµν , known as the Christoffel coefficients or connection
coefficients, are calculated, at each point x, from the metric components gµν(x)
by the equation
Γλµν ≡
gλσ(∂µ gνσ + ∂ν gµσ − ∂σ gµν) , (8)
where gµν denotes the matrix inverse to gµν (g
µσ gσν = δ
ν where the Kronecker
symbol δµν is equal to 1 when µ = ν and 0 otherwise) and where ∂µ ≡ ∂/∂xµ
denotes the partial derivative with respect to the coordinate xµ. To give a
very simple example: in the Poincaré-Minkowski space-time the components of
the metric are constant, gµν = ηµν (when we use an inertial reference frame).
Because of this, the connection coefficients (8) vanish in an inertial reference
frame, and the differential equation for geodesics reduces to d2 xλ/ds2 = 0,
whose solutions are ordinary straight lines: xλ(s) = aλ s + bλ. On the other
hand, in a general “curved” space-time (meaning one with components gµν that
depend in an arbitrary way on the point x) the geodesics cannot be globally
represented by straight lines. One can nevertheless show that it always remains
possible, for any gµν(x) whatsoever, to change coordinates x
µ → Xα(x) in such
a way that the connection coefficients Γαβγ , in the new system of coordinates
Xα, vanish locally, at a given point Xα0 (or even along an arbitrary curve).
Such locally geodesic coordinate systems realize Einstein’s equivalence principle
mathematically: up to terms of second order, the components gαβ(X) of a
“curved” metric in locally geodesic coordinates Xα (ds2 = gαβ(X) dX
α dXβ)
can be identified with the components of a “flat” Poincaré-Minkowski metric:
gαβ(X) = ηαβ +O((X−X0)2), where X0 is the point around which we expand.
5 Einstein’s Equations: Elastic Space-Time
Having postulated that a consistent relativistic theory of the gravitational field
should include the consideration of a far-reaching generalization of the Poincaré-
Minkowski space-time, Equation (6), Einstein concluded that the same ten
functions gµν(x) should describe both the geometry of space-time as well as
gravitation. He therefore got down to the task of finding which equations must
be satisfied by the “geometric-gravitational field” gµν(x). He was guided in this
search by three principles. The first was the principle of general relativity, which
asserts that in the presence of a gravitational field one should be able to write
the fundamental laws of physics (including those governing the gravitational
field itself) in the same way in any coordinate system whatsoever. The second
was that the “source” of the gravitational field should be the energy-momentum
tensor T µν . The third was a principle of correspondence with earlier physics:
in the limit where one neglects gravitational effects, gµν(x) = ηµν should be
a solution of the equations being sought, and there should also be a so-called
Newtonian limit where the new theory reduces to Newton’s theory of gravity.
Note that the principle of general relativity (contrary to appearances and
contrary to what Einstein believed for several years) has a different physical
status than the principle of special relativity. The principle of special relativity
was a symmetry principle for the structure of space-time that asserted that
physics is the same in a particular class of reference frames, and therefore that
certain “corresponding” phenomena occur in exactly the same way in different
reference frames (“active” transformations). On the other hand, the principle
of general relativity is a principle of indifference: the phenomena do not (in
general) take place in the same way in different coordinate systems. However,
none of these (extended) coordinate systems enjoys any privileged status with
respect to the others.
The principle asserting that the energy-momentum tensor T µν should be the
source of the gravitational field is founded on two ideas: the relations E = mi c
and the weak principle of equivalence mi = mg show that, in the Newtonian
limit, the source of gravitation, the gravitational mass mg, is equal to the total
energy of the body considered, or in other words the integral over space of
the energy density T 00, up to the factor c−2. Therefore at least one of the
components of the tensor T µν must play the role of source for the gravitational
field. However, since the gravitational field is encoded, according to Einstein,
by the ten components of the metric gµν , it is natural to suppose that the
source for gµν must also have ten components, which is precisely the case for
the (symmetric) tensor T µν .
In November of 1915, after many years of conceptually arduous work, Ein-
stein wrote the final form of the theory of general relativity [6]. Einstein’s equa-
tions are non-linear, second-order partial differential equations for the geometric-
gravitational field gµν , containing the energy-momentum tensor Tµν ≡ gµκ gνλ T κλ
on the right-hand side. They are written as follows:
Rµν −
Rgµν =
Tµν (9)
where G is the (Newtonian) gravitational constant, c is the speed of light, and
R ≡ gµν Rµν and the Ricci tensor Rµν are calculated as a function of the
connection coefficients Γλµν (8) in the following way:
Rµν ≡ ∂α Γαµν − ∂ν Γαµα + Γαβα Γβµν − Γαβν Γβµα . (10)
One can show that, in a four-dimensional space-time, the three principles
we have described previously uniquely determine Einstein’s equations (9). It
is nevertheless remarkable that these equations may also be developed from
points of view that are completely different from the one taken by Einstein. For
example, in the 1960s various authors (in particular Feynman, Weinberg and
Deser; see references in [4]) showed that Einstein’s equations could be obtained
from a purely dynamical approach, founded on the consistency of interactions
of a long-range spin 2 field, without making any appeal, as Einstein had, to
the geometric notions coming from mathematical work on Riemannian spaces.
Let us also note that if we relax one of the principles described previously (as
Einstein did in 1917) we can find a generalization of Equation (9) in which one
adds the term +Λ gµν to the left-hand side, where Λ is the so-called cosmological
constant. Such a modification was proposed by Einstein in 1917 in order to be
able to write down a globally homogeneous and stationary cosmological solution.
Einstein rejected this additional term after work by Friedmann (1922) showed
the existence of expanding cosmological solutions of general relativity and after
the observational discovery (by Hubble in 1929) of the expanding motion of
galaxies within the universe. However, recent cosmological data have once again
made this possibility fashionable, although in the fundamental physics of today
one tends to believe that a term of the type Λ gµν should be considered as a
particular physical contribution to the right-hand side of Einstein’s equations
(more precisely, as the stress-energy tensor of the vacuum, T Vµν = − c
Λ gµν),
rather than as a universal geometric modification of the left-hand side.
Let us now comment on the physical meaning of Einstein’s equations (9).
The essential new idea is that the chrono-geometric structure of space-time,
Equation (6), in other words the structure that underlies all of the measurements
that one could locally make of duration, dT , and of distance, dL, (we recall
that, locally, ds2 = dL2 − c2 dT 2) is no longer a rigid structure that is given a
priori, once and for all (as was the case for the structure of Poincaré-Minkowski
space-time), but instead has become a field, a dynamical or elastic structure,
which is created and/or deformed by the presence of an energy-momentum
distribution. See Figure 2, which visualizes the “elastic” geometry of space-
time in the theory of general relativity by representing, around each point x,
the locus of points (assumed to be infinitesimally close to x) separated from x by
a constant (squared) interval: ds2 = ε2. As in the case of Poincaré-Minkowski
space-time (Figure 1), one arrives at a “field” of hyperboloids. However, this
field of hyperboloids no longer has a “rigid” and homogeneous structure.
Figure 2: “Elastic” space-time geometry in the theory of general relativity. This
geometry is visualized by representing, around each space-time point x, the locus
of points separated from x by a given small positive (squared) interval.
The space-time field gµν(x) describes the variation from point to point of
the chrono-geometry as well as all gravitational effects. The simplest example
of space-time chrono-geometric elasticity is the effect that the proximity of a
mass has on the “local rate of flow for time.” In concrete terms, if you separate
two twins at birth, with one staying on the surface of the Earth and the other
going to live on the peak of a very tall mountain (in other words farther from
the Earth’s center), and then reunite them after 100 years, the “highlander”
will be older (will have lived longer) than the twin who stayed on the valley
floor. Everything takes place as if time flows more slowly the closer one is to
a given distribution of mass-energy. In mathematical terms this effect is due
to the fact that the coefficient g00(x) of (dx
0)2 in Equation (6) is deformed
with respect to its value in special relativity, gMinkowski00 = η00 = −1, to become
gEinstein00 (x) ≃ −1 + 2GM/c2r, where M is the Earth’s mass (in our example)
and r the distance to the center of the Earth. In the example considered above
of terrestrial twins the effect is extremely small (a difference in the amount of
time lived of about one second over 100 years), but the effect is real and has
been verified many times using atomic clocks (see the references in [4]). Let us
mention that today this “Einstein effect” has important practical repercussions,
for example in aerial or maritime navigation, for the piloting of automobiles, or
even farm machinery, etc. In fact, the GPS (Global Positioning System), which
uses the data transmitted by a constellation of atomic clocks on board satellites,
incorporates the Einsteinian deformation of space-time chrono-geometry into its
software. The effect is only on the order of one part in a billion, but if it were not
taken into account, it would introduce an unacceptably large error into the GPS,
which would continually grow over time. Indeed, GPS performance relies on the
high stability of the orbiting atomic clocks, a stability better than 10−13, or in
other words 10,000 times greater than the apparent change in frequency(∼ 10−9)
due to the Einsteinian deformation of the chrono-geometry.
6 TheWeak-Field Limit and the Newtonian Limit
To understand the physical consequences of Einstein’s equations (9), it is useful
to begin by considering the limiting case of weak geometric-gravitational fields,
namely the case where gµν(x) = ηµν + hµν(x), with perturbations hµν(x) that
are very small with respect to unity: |hµν(x)| ≪ 1. In this case, a simple
calculation (that we encourage the reader to perform) starting from Definitions
(8) and (10) above, leads to the following explicit form of Einstein’s equations
(where we ignore terms of order h2 and hT ):
� hµν − ∂µ ∂α hαν − ∂ν ∂α hαµ + ∂µν hαα = −
16 πG
T̃µν , (11)
where � = ηµν ∂µν = ∆−∂20 = ∂2/∂x2+∂2/∂y2+∂2/∂z2− c−2 ∂2/∂t2 denotes
the “flat” d’Alembertian (or wave operator; xµ = (ct, x, y, z)), and where indices
in the upper position have been raised by the inverse ηµν of the flat metric ηµν
(numerically ηµν = ηµν , meaning that −η00 = η11 = η22 = η33 = +1). For
example ∂α hαν denotes η
αβ ∂α hβν and h
α ≡ ηαβ hαβ = −h00 + h11 + h22 +
h33. The “source” T̃µν appearing on the right-hand side of (11) denotes the
combination T̃µν ≡ Tµν − 12 T
α ηµν (when space-time is four-dimensional).
The “linearized” approximation (11) of Einstein’s equations is analogous to
Maxwell’s equations
�Aµ − ∂µ ∂αAα = −4π Jµ , (12)
connecting the electromagnetic four-potential Aµ ≡ ηµν Aν (where A0 = V ,
Ai = A, i = 1, 2, 3) to the four-current density Jµ ≡ ηµν Jν (where J0 = ρ is the
charge density and J i = J is the current density). Another analogy is that the
structure of the left-hand side of Maxwell’s equations implies that the “source”
Jµ appearing on the right-hand side must satisfy ∂
µ Jµ = 0 (∂
µ ≡ ηµν ∂ν),
which expresses the conservation of electric charge. Likewise, the structure of
the left-hand side of the linearized form of Einstein’s equations (11) implies that
the “source” Tµν = T̃µν − 12 T̃
α ηµν must satisfy ∂
µ Tµν = 0, which expresses the
conservation of energy and momentum of matter. (The structure of the left-
hand side of the exact form of Einstein’s equations (9) implies that the source
Tµν must satisfy the more complicated equation ∂µ T
µν+Γµσµ T
σν+Γνσµ T
µσ = 0,
where the terms in ΓT can be interpreted as describing an exchange of energy
and momentum between matter and the gravitational field.) The major dif-
ference is that, in the case of electromagnetism, the field Aµ and its source Jµ
have a single space-time index, while in the gravitational case the field hµν and
its source T̃µν have two space-time indices. We shall return later to this anal-
ogy/difference between Aµ and hµν which suggests the existence of a certain
relation between gravitation and electromagnetism.
We recover the Newtonian theory of gravitation as the limiting case of Ein-
stein’s theory by assuming not only that the gravitational field is a weak defor-
mation of the flat Minkowski space-time (hµν ≪ 1), but also that the field hµν
is slowly varying (∂0 hµν ≪ ∂i hµν) and that its source Tµν is non-relativistic
(Tij ≪ T0i ≪ T00). Under these conditions Equation (11) leads to a Poisson-
type equation for the purely temporal component, h00, of the space-time field,
∆h00 = −
16 πG
T̃00 = −
(T00 + Tii) ≃ −
T00 , (13)
where ∆ = ∂2x + ∂
y + ∂
z is the Laplacian. Recall that, according to Laplace
and Poisson, Newton’s theory of gravity is summarized by saying that the grav-
itational field is described by a single potential U(x), produced by the mass
density ρ(x) according to the Poisson equation ∆U = −4 πGρ, that deter-
mines the acceleration of a test particle placed in the exterior field U(x) ac-
cording to the equation d2 xi/dt2 = ∂i U(x) ≡ ∂U/∂xi. Because of the relation
mi = mg = E/c
2 one can identify ρ = T 00/c2. We therefore find that (13)
reproduces the Poisson equation if h00 = +2U/c
2. It therefore remains to ver-
ify that Einstein’s theory indeed predicts that a non-relativistic test particle is
accelerated by a space-time field according to d2 xi/dt2 ≃ 1
c2 ∂i h00. Einstein
understood that this was a consequence of the equivalence principle. In fact,
as we discussed in Section 4 above, the principle of equivalence states that the
gravitational field is (locally) erased in a locally inertial reference frame Xα
(such that gαβ(X) = ηαβ +O((X −X0)2)). In such a reference frame, the laws
of special relativity apply at the point X0. In particular an isolated (and elec-
trically neutral) body must satisfy a principle of inertia in this frame: its center
of mass moves in a straight line at constant speed. In other words it satisfies
the equation of motion d2Xα/ds2 = 0. By passing back to an arbitrary (ex-
tended) coordinate system xµ, one verifies that this equation for inertial motion
transforms into the geodesic equation (7). Therefore (7) describes falling bodies,
such as they are observed in arbitrary extended reference frames (for example a
reference frame at rest with respect to the Earth or at rest with respect to the
center of mass of the solar system). From this one concludes that the relativis-
tic analog of the Newtonian field of gravitational acceleration, g(x) = ∇U(x),
is gλ(x) ≡ −c2 Γλµν dxµ/ds dxν/ds. By considering a particle whose motion is
slow with respect to the speed of light (dxi/ds ≪ dx0/ds ≃ 1) one can easily
verify that gi(x) ≃ −c2 Γi00. Finally, by using the definition (8) of Γαµν , and the
hypothesis of weak fields, one indeed verifies that gi(x) ≃ 1
c2 ∂i h00, in perfect
agreement with the identification h00 = 2U/c
2 anticipated above. We encour-
age the reader to personally verify this result, which contains the very essence
of Einstein’s theory: gravitational motion is no longer described as being due
to a force, but is identified with motion that is “as inertial as possible” within a
space-time whose chrono-geometry is deformed in the presence of a mass-energy
distribution.
Finding the Newtonian theory as a limiting case of Einstein’s theory is ob-
viously a necessity for seriously considering this new theory. But of course,
from the very beginning Einstein explored the observational consequences of
general relativity that go beyond the Newtonian description of gravitation. We
have already mentioned one of these above: the fact that g00 = η00 + h00 ≃
−1 + 2U(x)/c2 implies a distortion in the relative measurement of time in the
neighborhood of massive bodies. In 1907 (as soon as he had developed the
principle of equivalence, and long before he had obtained the field equations
of general relativity) Einstein had predicted the existence of such a distortion
for measurements of time and frequency in the presence of an external gravita-
tional field. He realized that this should have observable consequences for the
frequency, as observed on Earth, of the spectral rays emitted from the surface of
the Sun. Specifically, a spectral ray of (proper local) frequency ν0 emitted from
a point x0 where the (stationary) gravitational potential takes the value U(x0)
and observed (via electromagnetic signals) at a point x where the potential is
U(x) should appear to have a frequency ν such that
g00(x0)
g00(x)
≃ 1 + 1
[U(x)− U(x0)] . (14)
In the case where the point of emission x0 is in a gravitational potential well
deeper than the point of observation x (meaning that U(x0) > U(x)) one has
ν < ν0, in other words a reddening effect on frequencies. This effect, which was
predicted by Einstein in 1907, was unambiguously verified only in the 1960s,
in experiments by Pound and collaborators over a height of about twenty me-
ters. The most precise verification (at the level of ∼ 10−4) is due to Vessot
and collaborators, who compared a hydrogen maser, launched aboard a rocket
that reached about 10,000 km in altitude, to a clock of similar construction
on the ground. Other experiments compared the times shown on clocks placed
aboard airplanes to clocks remaining on the ground. (For references to these
experiments see [4].) As we have already mentioned, the “Einstein effect” (14)
must be incorporated in a crucial way into the software of satellite positioning
systems such as the GPS.
In 1907, Einstein also pointed out that the equivalence principle would sug-
gest that light rays should be deflected by a gravitational field. Indeed, a gener-
alization of the reasoning given above for the motion of particles in an external
gravitational field, based on the principle of equivalence, shows that light must
itself follow a trajectory that is “as inertial as possible,” meaning a geodesic
of the curved space-time. Light rays must therefore satisfy the geodesic equa-
tion (7). (The only difference from the geodesics followed by material particles
is that the parameter s in Equation (7) can no longer be taken equal to the
“length” along the geodesic, since a “light” geodesic must also satisfy the con-
straint gµν(x) dx
µ dxν = 0, ensuring that its speed is equal to c, when it is
measured in a locally inertial reference frame.) Starting from Equation (7) one
can therefore calculate to what extent light is deflected when it passes through
the neighborhood of a large mass (such as the Sun). One nevertheless soon
realizes that in order to perform this calculation one must know more than the
component h00 of the gravitational field. The other components of hµν , and in
particular the spatial components hij , come into play in a crucial way in this
calculation. This is why it was only in November of 1915, after having obtained
the (essentially) final form of his theory, that Einstein could predict the total
value of the deflection of light by the Sun. Starting from the linearized form of
Einstein’s equations (11) and continuing by making the “non-relativistic” sim-
plifications indicated above (Tij ≪ T0i ≪ T00, ∂0 h≪ ∂i h) it is easy to see that
the spatial component hij , like h00, can be written (after a helpful choice of
coordinates) in terms of the Newtonian potential U as hij(x) ≃ +2U(x) δij/c2,
where δij takes the value 1 if i = j and 0 otherwise (i, j = 1, 2, 3). By inserting
this result, as well as the preceding result h00 = +2U/c
2, into the geodesic
equation (7) for the motion of light, one finds (as Einstein did in 1915) that
general relativity predicts that the Sun should deflect a ray of light by an angle
θ = 4GM/(c2b) where b is the impact parameter of the ray (meaning its mini-
mum distance from the Sun). As is well known, the confirmation of this effect
in 1919 (with rather weak precision) made the theory of general relativity and
its creator famous.
7 The Post-Newtonian Approximation and Ex-
perimental Confirmations in the Regime of
Weak and Quasi-Stationary Gravitational Fields
We have already pointed out some of the experimental confirmations of the
theory of general relativity. At present, the extreme precision of certain mea-
surements of time or frequency in the solar system necessitates a very careful
account of the modifications brought by general relativity to the Newtonian de-
scription of space-time. As a consequence, general relativity is used in a great
number of situations, from astronomical or geophysical research (such as very
long range radio interferometry, radar tracking of the planets, and laser tracking
of the Moon or artificial satellites) to metrological, geodesic or other applica-
tions (such as the definition of international atomic time, precision cartography,
and the G.P.S.). To do this, the so-called post-Newtonian approximation has
been developed. This method involves working in the Newtonian limit sketched
above while keeping the terms of higher order in the small parameter
ε ∼ v
∼ |hµν | ∼ |∂0 h/∂i h|2 ∼ |T 0i/T 00|2 ∼ |T ij/T 00| ,
where v denotes a characteristic speed for the elements in the system considered.
For all present applications of general relativity to the solar system it suffices
to include the first post-Newtonian approximation, in other words to keep the
relative corrections of order ε to the Newtonian predictions. Since the theory of
general relativity was poorly verified for a long time, one found it useful (as in
the pioneering work of A. Eddington, generalized in the 1960s by K. Nordtvedt
and C.M. Will) to study not only the precise predictions of the equations (9)
defining Einstein’s theory, but to also consider possible deviations from these
predictions. These possible deviations were parameterized by means of several
non-dimensional “post-Newtonian” parameters. Among these parameters, two
play a key role: γ and β. The parameter γ describes a possible deviation from
general relativity that comes into play starting at the linearized level, in other
words one that modifies the linearized approximation given above. More pre-
cisely, it is defined by writing that the difference hij ≡ gij − δij between the
spatial metric and the Euclidean metric can take the value hij = 2γ U δij/c
2 (in a
suitable coordinate system), rather than the value hGRij = 2U δij/c
2 that it takes
in general relativity, thus differing by a factor γ. Therefore, by definition γ takes
the value 1 in general relativity, and γ− 1 measures the possible deviation with
respect to this theory. As for the parameter β (or rather β−1), it measures a pos-
sible deviation (with respect to general relativity) in the value of h00 ≡ g00−η00.
The value of h00 in general relativity is h
00 = 2U/c
2 − 2U2/c4, where the first
term (discussed above) reproduces the Newtonian approximation (and cannot
therefore be modified, as the idea is to parameterize gravitational physics be-
yond Newtonian predictions) and where the second term is obtained by solving
Einstein’s equations (9) at the second order of approximation. One then writes
an h00 of a more general parameterized type, h00 = 2U/c
2 − 2 β U2/c4, where,
by definition, β takes the value 1 in general relativity. Let us finally point out
that the parameters γ−1 and β−1 completely parameterize the post-Newtonian
regime of the simplest theoretical alternatives to general relativity, namely the
tensor-scalar theories of gravitation. In these theories, the gravitational inter-
action is carried by two fields at the same time: a massless tensor (spin 2) field
coupled to T µν , and a massless scalar (spin 0) field ϕ coupled to the trace Tαα .
In this case the parameter −(γ − 1) plays the key role of measuring the ratio
between the scalar coupling and the tensor coupling.
All of the experiments performed to date within the solar system are com-
patible with the predictions of general relativity. When they are interpreted
in terms of the post-Newtonian (and “post-Einsteinian”) parameters γ − 1 and
β−1, they lead to strong constraints on possible deviations from Einstein’s the-
ory. We make note of the following among tests performed in the solar system:
the deflection of electromagnetic waves in the neighborhood of the Sun, the grav-
itational delay (‘Shapiro effect’) of radar signals bounced from the Viking lander
on Mars, the global analysis of solar system dynamics (including the advance of
planetary perihelia), the sub-centimeter measurement of the Earth-Moon dis-
tance obtained from laser signals bounced off of reflectors on the Moon’s surface,
etc. At present (October of 2006) the most precise test (that has been published)
of general relativity was obtained in 2003 by measuring the ratio 1 + y ≡ f/f0
between the frequency f0 of radio waves sent from Earth to the Cassini space
probe and the frequency f of coherent radio waves sent back (with the same
local frequency) from Cassini to Earth and compared (on Earth) to the emitted
frequency f0. The main contribution to the small quantity y is an effect equal, in
general relativity, to yGR = 8(GM/c
3 b) db/dt (where b is, as before, the impact
parameter) due to the propagation of radio waves in the geometry of a space-time
deformed by the Sun: ds2 ≃ −(1−2U/c2) c2 dt2+(1+2U/c2)(dx2+dy2+dz2),
where U = GM/r. The maximum value of the frequency change predicted
by general relativity was only |yGR| . 2 × 10−10 for the best observations,
but thanks to an excellent frequency stability ∼ 10−14 (after correction for the
perturbations caused by the solar corona) and to a relatively large number of
individual measurements spread over 18 days, this experiment was able to verify
Einstein’s theory at the remarkable level of ∼ 10−5 [7]. More precisely, when
this experiment is interpreted in terms of the post-Newtonian parameters γ− 1
and β − 1, it gives the following limit for the parameter γ − 1 [7]
γ − 1 = (2.1± 2.3)× 10−5 . (15)
As for the best present-day limit on the parameter β−1, it is smaller than 10−3
and comes from the non-observation, in the data from lasers bounced off of the
Moon, of any eventual polarization of the Moon’s orbit in the direction of the
Sun (‘Nordtvedt effect’; see [4] for references)
4(β − 1)− (γ − 1) = −0.0007± 0.0010 . (16)
Although the theory of general relativity is one of the best verified theories in
physics, scientists continue to design and plan new or increasingly precise tests of
the theory. This is the case in particular for the space mission Gravity Probe B
(launched by NASA in April of 2004) whose principal aim is to directly observe a
prediction of general relativity that states (intuitively speaking) that space is not
only “elastic,” but also “fluid.” In the nineteenth century Foucalt invented both
the gyroscope and his famous pendulum in order to render Newton’s absolute
(and rigid) space directly observable. His experiments in fact showed that,
for example, a gyroscope on the surface of the Earth continued, despite the
Earth’s rotation, to align itself in a direction that is “fixed” with respect to the
distant stars. However, in 1918, when Lense and Thirring analyzed some of the
consequences of the (linearized) Einstein equations (11), they found that general
relativity predicts, among other things, the following phenomenon: the rotation
of the Earth (or any other ball of matter) creates a particular deformation of the
chrono-geometry of space-time. This deformation is described by the “gravito-
magnetic” components h0i of the metric, and induces an effect analogous to
the “rotation drag” effect caused by a ball of matter turning in a fluid: the
rotation of the Earth (minimally) drags all of the space around it, causing it to
continually “turn,” as a fluid would.3 This “rotation of space” translates, in an
observable way, into a violation of the effects predicted by Newton and confirmed
by Foucault’s experiments: in particular, a gyroscope no longer aligns itself in a
direction that is “fixed in absolute space,” rather its axis of rotation is “dragged”
by the rotating motion of the local space where it is located. This effect is much
too small to be visible in Foucalt’s experiments. Its observation by Gravity
Probe B (see [8] and the contribution of John Mester to this Poincaré seminar)
is important for making Einstein’s revolutionary notion of a fluid space-time
tangible to the general public.
Up till now we have only discussed the regime of weak and slowly varying
gravitational fields. The theory of general relativity predicts the appearance
of new phenomena when the gravitational field becomes strong and/or rapidly
varying. (We shall not here discuss the cosmological aspects of relativistic grav-
itation.)
8 Strong Gravitational Fields and Black Holes
The regime of strong gravitational fields is encountered in the physics of grav-
itationally condensed bodies. This term designates the final states of stellar
evolution, and in particular neutron stars and black holes. Recall that most of
the life of a star is spent slowly burning its nuclear fuel. This process causes
the star to be structured as a series of layers of differentiated nuclear structure,
surrounding a progressively denser core (an “onion-like” structure). When the
initial mass of the star is sufficiently large, this process ends into a catastrophic
phenomenon: the core, already much denser than ordinary matter, collapses in
on itself under the influence of its gravitational self-attraction. (This implosion
of the central part of the star is, in many cases, accompanied by an explosion
of the outer layers of the star—a supernova.) Depending on the quantity of
mass that collapses with the core of a star, this collapse can give rise to either
a neutron star or a black hole.
A neutron star condenses a mass on the order of the mass of the Sun inside
a radius on the order of 10 km. The density in the interior of a neutron star
(named thus because neutrons dominate its nuclear composition) is more than
100 million tons per cubic centimeter (1014 g/cm3)! It is about the same as the
density in the interior of atomic nuclei. What is important for our discussion is
that the deformation away from the Minkowski metric in the immediate neigh-
borhood of a neutron star, measured by h00 ∼ hii ∼ 2GM/c2R, where R is the
radius of the star, is no longer a small quantity, as it was in the solar system.
In fact, while h ∼ 2GM/c2R is on the order of 10−9 for the Earth and 10−6 for
the Sun, one finds that h ∼ 0.4 for a typical neutron star (M ≃ 1.4M⊙, R ∼ 10
3Recent historical work (by Herbert Pfister) has in fact shown that this effect had already
been derived by Einstein within the framework of the provisory relativistic theory of gravity
that he started to develop in 1912 in collaboration with Marcel Grossmann.
km). One thus concludes that it is no longer possible, as was the case in the
solar system, to study the structure and physics of neutron stars by using the
post-Newtonian approximation outlined above. One must consider the exact
form of Einstein’s equations (9), with all of their non-linear structure. Because
of this, we expect that observations concerning neutron stars will allow us to
confirm (or refute) the theory of general relativity in its strongly non-linear
regime. We shall discuss such tests below in relation to observations of binary
pulsars.
A black hole is the result of a continued collapse, meaning that it does
not stop with the formation of an ultra-dense star (such as a neutron star).
(The physical concept of a black hole was introduced by J.R. Oppenheimer and
H. Snyder in 1939. The global geometric structure of black holes was not un-
derstood until some years later, thanks notably to the work of R. Penrose. For
a historical review of the idea of black holes see [9].) It is a particular structure
of curved space-time characterized by the existence of a boundary (called the
“black hole surface” or “horizon”) between an exterior region, from which it is
possible to emit signals to infinity, and an interior region (of space-time), within
which any emitted signal remains trapped. See Figure 3.
r = 0 SINGULARITY
r = 2M
HORIZON
FLASH
OF LIGHT
EMITTED
FROM CENTER
COLLAPSING
space
Figure 3: Schematic representation of the space-time for a black hole created
from the collapse of a spherical star. Each cone represents the space-time history
of a flash of light emitted from a point at a particular instant. (Such a “cone
field” is obtained by taking the limit ε2 = 0 from Figure 2, and keeping only
the upper part, in other words the part directed towards the future, of the
double cones obtained as limits of the hyperboloids of Figure 2.) The interior
of the black hole is shaded, its outer boundary being the “black hole surface”
or “horizon.” The “inner boundary” (shown in dark grey) of the interior region
of the black hole is a space-time singularity of the big-crunch type.
The cones shown in this figure are called “light cones.” They are defined as
the locus of points (infinitesimally close to x) such that ds2 = 0, with dx0 =
cdt ≥ 0. Each represents the beginning of the space-time history of a flash of
light emitted from a certain point in space-time. The cones whose vertices are
located outside of the horizon (the shaded zone) will evolve by spreading out
to infinity, thus representing the possibility for electromagnetic signals to reach
infinity.
On the other hand, the cones whose vertices are located inside the horizon
(the grey zone) will evolve without ever succeeding in escaping the grey zone.
It is therefore impossible to emit an electromagnetic signal that reaches infinity
from the grey zone. The horizon, namely the boundary between the shaded
zone and the unshaded zone, is itself the history of a particular flash of light,
emitted from the center of the star over the course of its collapse, such that
it asymptotically stabilizes as a space-time cylinder. This space-time cylinder
(the asymptotic horizon) therefore represents the space-time history of a bubble
of light that, viewed locally, moves outward at the speed c, but which globally
“runs in place.” This remarkable behavior is a striking illustration of the “fluid”
character of space-time in Einstein’s theory. Indeed, one can compare the pre-
ceding situation with what may take place around the open drain of an emptying
sink: a wave may move along the water, away from the hole, all the while run-
ning in place with respect to the sink because of the falling motion of the water
in the direction of the drain.
Note that the temporal development of the interior region is limited, ter-
minating in a singularity (the dark gray surface) where the curvature becomes
infinite and where the classical description of space and time loses its meaning.
This singularity is locally similar to the temporal inverse of a cosmological sin-
gularity of the big bang type. This is called a big crunch. It is a space-time
frontier, beyond which space-time ceases to exist. The appearance of singulari-
ties associated with regions of strong gravitational fields is a generic phenomenon
in general relativity, as shown by theorems of R. Penrose and S.W. Hawking.
Black holes have some remarkable properties. First, a uniqueness theorem
(due to W. Israel, B. Carter, D.C. Robinson, G. Bunting, and P.O. Mazur)
asserts that an isolated, stationary black hole (in Einstein-Maxwell theory) is
completely described by three parameters: its mass M , its angular momen-
tum J , and its electric charge Q. The exact solution (called the Kerr-Newman
solution) of Einstein’s equations (11) describing a black hole with parameters
M,J,Q is explicitly known. We shall here content ourselves with writing the
space-time geometry in the simplest case of a black hole: the one in which
J = Q = 0 and the black hole is described only by its mass (a solution discov-
ered by K. Schwarzschild in January of 1916):
ds2 = −
1− 2GM
c2 dt2 +
1− 2GM
+ r2(dθ2 + sin2 θ dϕ2) . (17)
We see that the purely temporal component of the metric, g00 = −(1−2GM/c2r),
vanishes when the radial coordinate r takes the value r = rH ≡ 2GM/c2. Ac-
cording to the earlier equation (14), it would therefore seem that light emitted
from an arbitrary point on the sphere r0 = rH , when it is viewed by an observer
located anywhere in the exterior (in r > rH), would experience an infinite red-
dening of its emission frequency (ν/ν0 = 0). In fact, the sphere rH = 2GM/c
is the horizon of the Schwarzschild black hole, and no particle (that is capable of
emitting light) can remain at rest when r = rH (nor, a fortiori, when r < rH).
To study what happens at the horizon (r = rH) or in the interior (r < rH) of
a Schwarzschild black hole, one must use other space-time coordinates than the
coordinates (t, r, θ, ϕ) used in Equation (17). The “big crunch” singularity in
the interior of a Schwarzschild black hole, in the coordinates of (17), is located
at r = 0 (which does not describe, as one might believe, a point in space, but
rather an instant in time).
The space-time metric of a black hole space-time, such as Equation (17)
in the simple case J = Q = 0, allows one to study the influence of a black
hole on particles and fields in its neighborhood. One finds that a black hole
is a gravitational potential well that is so deep that any particle or wave that
penetrates the interior of the black hole (the region r < rH) will never be able
to come out again, and that the total energy of the particle or wave that “falls”
into the black hole ends up augmenting the total mass-energy M of the black
hole. By studying such black hole “accretion” processes with falling particles
(following R. Penrose), D. Christodoulou and R. Ruffini showed that a black hole
is not only a potential well, but also a physical object possessing a significant
free energy that it is possible, in principle, to extract. Such black hole energetics
is encapsulated in the “mass formula” of Christodoulou and Ruffini (in units
where c = 1)
Mirr +
4GMirr
4G2M2irr
, (18)
where Mirr denotes the irreducible mass of the black hole, a quantity that can
only grow, irreversibly. One deduces from (18) that a rotating (J 6= 0) and/or
charged (Q 6= 0) black hole possesses a free energy M − Mirr > 0 that can,
in principle, be extracted through processes that reduce its angular momentum
and/or its electric charge. Such black hole energy-extraction processes may lie
at the origin of certain ultra-energetic astrophysical phenomena (such as quasars
or gamma ray bursts). Let us note that, according to Equation (18), (rotating or
charged) black holes are the largest reservoirs of free energy in the Universe: in
fact, 29% of their mass energy can be stored in the form of rotational energy, and
up to 50% can be stored in the form of electric energy. These percentages are
much higher than the few percent of nuclear binding energy that is at the origin
of all the light emitted by stars over their lifetimes. Even though there is not, at
present, irrefutable proof of the existence of black holes in the universe, an entire
range of very strong presumptive evidence lends credence to their existence. In
particular, more than a dozen X-ray emitting binary systems in our galaxy
are most likely made up of a black hole and an ordinary star. Moreover, the
center of our galaxy seems to contain a very compact concentration of mass
∼ 3 × 106M⊙ that is probably a black hole. (For a review of the observational
data leading to these conclusions see, for example, Section 7.6 of the recent book
by N. Straumann [6].)
The fact that a quantity associated with a black hole, here the irreducible
mass Mirr, or, according to a more general result due to S.W. Hawking, the
total area A of the surface of a black hole (A = 16 πG2M2irr), can evolve only by
irreversibly growing is reminiscent of the second law of thermodynamics. This
result led J.D. Bekenstein to interpret the horizon area, A, as being propor-
tional to the entropy of the black hole. Such a thermodynamic interpretation
is reinforced by the study of the growth of A under the influence of external
perturbations, a growth that one can in fact attribute to some local dissipative
properties of the black hole surface, notably a surface viscosity and an electrical
resistivity equal to 377 ohm (as shown in work by T. Damour and R.L. Zna-
jek). These “thermodynamic” interpretations of black hole properties are based
on simple analogies at the level of classical physics, but a remarkable result by
Hawking showed that they have real content at the level of quantum physics.
In 1974, Hawking discovered that the presence of a horizon in a black hole
space-time affected the definition of a quantum particle, and caused a black
hole to continuously emit a flux of particles having the characteristic spectrum
(Planck spectrum) of thermal emission at the temperature T = 4 ~G∂M/∂A,
where ~ is the reduced Planck constant. By using the general thermodynamic
relation connecting the temperature to the energy E = M and the entropy S,
T = ∂M/∂S, we see from Hawking’s result (in conformity with Bekenstein’s
ideas) that a black hole possesses an entropy S equal (again with c = 1) to
. (19)
The Bekenstein-Hawking formula (19) suggests an unexpected, and perhaps pro-
found, connection between gravitation, thermodynamics, and quantum theory.
See Section 11 below.
9 Binary Pulsars and Experimental Confirma-
tions in the Regime of Strong and Radiating
Gravitational Fields
Binary pulsars are binary systems made up of a pulsar (a rapidly spinning
neutron star) and a very dense companion star (either a neutron star or a white
dwarf). The first system of this type (called PSR B1913+16) was discovered by
R.A. Hulse and J.H. Taylor in 1974 [10]. Today, a dozen are known. Some of
these (including the first-discovered PSR B1913+16) have revealed themselves
to be remarkable probes of relativistic gravitation and, in particular, of the
regime of strong and/or radiating gravitational fields. The reason for which a
binary pulsar allows for the probing of strong gravitational fields is that, as we
have already indicated above, the deformation of the space-time geometry in the
neighborhood of a neutron star is no longer a small quantity, as it is in the solar
system. Rather, it is on the order of unity: hµν ≡ gµν − ηµν ∼ 2GM/c2R ∼ 0.4.
(We note that this value is only 2.5 times smaller than in the extreme case of a
black hole, for which 2GM/c2R = 1.) Moreover, the fact that the gravitational
interaction propagates at the speed of light (as indicated by the presence of
the wave operator, � = ∆ − c−2∂2/∂t2 in (11)) between the pulsar and its
companion is found to play an observationally significant role for certain binary
pulsars.
Let us outline how the observational data from binary pulsars are used to
probe the regime of strong (hµν on the order of unity) and/or radiative (effects
propagating at the speed c) gravitational fields. (For more details on the obser-
vational data from binary pulsars and their use in probing relativistic gravita-
tion, see Michael Kramer’s contribution to this Poincaré seminar.) Essentially,
a pulsar plays the role of an extremely stable clock. Indeed, the “pulsar phe-
nomenon” is due to the rotation of a bundle of electromagnetic waves, created
in the neighborhood of the two magnetic poles of a strongly magnetized neutron
star (with a magnetic field on the order of 1012 Gauss, 1012 times the size of
the terrestrial magnetic field). Since the magnetic axis of a pulsar is not aligned
with its axis of rotation, the rapid rotation of the pulsar causes the (inner)
magnetosphere as a whole to rotate, and likewise the bundle of electromagnetic
waves created near the magnetic poles. The pulsar is therefore analogous to a
lighthouse that sweeps out space with two bundles (one per pole) of electromag-
netic waves. Just as for a lighthouse, one does not see the pulsar from Earth
except when the bundle sweeps the Earth, thus causing a flash of electromag-
netic noise with each turn of the pulsar around itself (in some cases, one even
sees a secondary flash, due to emission from the second pole, after each half-
turn). One can then measure the time of arrival at Earth of (the center of) each
flash of electromagnetic noise. The basic observational data of a pulsar are thus
made up of a regular, discrete sequence of the arrival times at Earth of these
flashes or “pulses.” This sequence is analogous to the signal from a clock: tick,
tick, tick, . . .. Observationally, one finds that some pulsars (and in particular
those that belong to binary systems) thus define clocks of a stability comparable
to the best atomic clocks [11]. In the case of a solitary pulsar, the sequence of
its arrival times is (in essence) a regular “arithmetic sequence,” TN = aN + b,
where N is an integer labelling the pulse considered, and where a is equal to the
period of rotation of the pulsar around itself. In the case of a binary pulsar, the
sequence of arrival times is a much richer signal, say TN = aN + b+∆N , where
∆N measures the deviation with respect to a regular arithmetic sequence. This
deviation (after the subtraction of effects not connected to the orbital period
of the pulsar) is due to a whole ensemble of physical effects connected to the
orbital motion of the pulsar around its companion or, more precisely, around
the center of mass of the binary system. Some of these effects could be pre-
dicted by a purely Keplerian description of the motion of the pulsar in space,
and are analogous to the “Rœmer effect” that allowed Rœmer to determine,
for the first time, the speed of light from the arrival times at Earth of light
signals coming from Jupiter’s satellites (the light signals coming from a body
moving in orbit are “delayed” by the time taken by light to cross this orbit and
arrive at Earth). Other effects can only be predicted and calculated by using a
relativistic description, either of the orbital motion of the pulsar, or of the prop-
agation of electromagnetic signals between the pulsar and Earth. For example,
the following facts must be accounted for: (i) the “pulsar clock” moves at a
large speed (on the order of 300 km/s ∼ 10−3c) and is embedded in the varying
gravitational potential of the companion; (ii) the orbit of the pulsar is not a
simple Keplerian ellipse, but (in general relativity) a more complicated orbit
that traces out a “rosette” around the center of mass; (iii) the propagation of
electromagnetic signals between the pulsar and Earth takes place in a space-time
that is curved by both the pulsar and its companion, which leads to particular
effects of relativistic delay; etc. Taking relativistic effects in the theoretical de-
scription of arrival times for signals emitted by binary pulsars into account thus
leads one to write what is called a timing formula. This timing formula (due to
T. Damour and N. Deruelle) in essence allows one to parameterize the sequence
of arrival times, TN = aN + b +∆N , in other words to parameterize ∆N , as a
function of a set of “phenomenological parameters” that include not only the
so-called “Keplerian” parameters (such as the orbital period P , the projection
of the semi-major axis of the pulsar’s orbit along the line of sight xA = aA sin i,
and the eccentricity e), but also the post-Keplerian parameters associated with
the relativistic effects mentioned above. For example, effect (i) discussed above
is parameterized by a quantity denoted γT ; effect (ii) by (among others) the
quantities ω̇, Ṗ ; effect (iii) by the quantities r, s; etc.
The way in which observations of binary pulsars allow one to test rela-
tivistic theories of gravity is therefore the following. A (least-squares) fit be-
tween the observational timing data, ∆obsN , and the parameterized theoreti-
cal timing formula, ∆thN (P, xA, e; γT , ω̇, Ṗ , r, s), allows for the determination of
the observational values of the Keplerian (P obs, xobsA , e
obs) and post-Keplerian
(γobsT , ω̇
obs, Ṗ obs, robs, sobs) parameters. The theory of general relativity pre-
dicts the value of each post-Keplerian parameter as a function of the Keple-
rian parameters and the two masses of the binary system (the mass mA of
the pulsar and the mass mB of the companion). For example, the theoretical
value predicted by general relativity for the parameter γT is γ
T (mA,mB) =
en−1(GMn/c3)2/3mB(mA + 2mB)/M
2, where e is the eccentricity, n = 2π/P
the orbital frequency, andM ≡ mA+mB. We thus see that, if one assumes that
general relativity is correct, the observational measurement of a post-Keplerian
parameter, for example γobsT , determines a curve in the plane (mA,mB) of the
two masses: γGRT (mA,mB) = γ
T , in our example. The measurement of two
post-Keplerian parameters thus gives two curves in the (mA,mB) plane and
generically allows one to determine the values of the two masses mA and mB,
by considering the intersection of the two curves. We obtain a test of general
relativity as soon as one observationally measures three or more post-Keplerian
parameters: if the three (or more) curves all intersect at one point in the plane
of the two masses, the theory of general relativity is confirmed, but if this is
not the case the theory is refuted. At present, four distinct binary pulsars have
allowed one to test general relativity. These four “relativistic” binary pulsars
are: the first binary pulsar PSR B1913+16, the pulsar PSR B1534+12 (dis-
covered by A. Wolszczan in 1991), and two recently discovered pulsars: PSR
J1141−6545 (discovered in 1999 by V.M. Kaspi et al., whose first timing results
are due to M. Bailes et al. in 2003), and PSR J0737−3039 (discovered in 2003
by M. Burgay et al., whose first timing results are due to A.G. Lyne et al. and
M. Kramer et al.). With the exception of PSR J1141−6545, whose companion
is a white dwarf, the companions of the pulsars are neutron stars. In the case
of PSR J0737−3039 the companion turns out to also be a pulsar that is visible
from Earth.
In the system PSR B1913+16, three post-Keplerian parameters have been
measured (ω̇, γT , Ṗ ), which gives one test of the theory. In the system PSR
J1141−65, three post-Keplerian parameters have been measured (ω̇, γT , Ṗ ), which
gives one test of the theory. (The parameter s is also measured through scin-
tillation phenomena, but the use of this measurement for testing gravitation is
more problematic.) In the system PSR B1534+12, five post-Keplerian param-
eters have been measured, which gives three tests of the theory. In the system
PSR J0737−3039,six post-Keplerian parameters,4 which gives four tests of the
theory. It is remarkable that all of these tests have confirmed general relativ-
ity. See Figure 4 and, for references and details, [4, 11, 12, 13], as well as the
contribution by Michael Kramer.
Note that, in Figure 4, some post-Keplerian parameters are measured with
such great precision that they in fact define very thin curves in the mA,mB
plane. On the other hand, some of them are only measured with a rough
fractional precision and thus define “thick curves,” or “strips” in the plane of
the masses (see, for example, the strips associated with Ṗ , r and s in the case
of PSR B1534+12). In any case, the theory is confirmed when all of the strips
(thick or thin) have a non-empty common intersection. (One should also note
that the strips represented in Figure 4 only use the “one sigma” error bars, in
other words a 68% level of confidence. Therefore, the fact that the Ṗ strip for
PSR B1534+12 is a little bit disjoint from the intersection of the other strips is
not significant: a “two sigma” figure would show excellent agreement between
observation and general relativity.)
In view of the arguments presented above, all of the tests shown in Figure 4
confirm the validity of general relativity in the regime of strong gravitational
fields (hµν ∼ 1). Moreover, the four tests that use measurements of the pa-
rameter Ṗ (in the four corresponding systems) are direct experimental confir-
mations of the fact that the gravitational interaction propagates at the speed
c between the companion and the pulsar. In fact, Ṗ denotes the long-term
variation 〈dP/dt〉 of the orbital period. Detailed theoretical calculations of the
motion of two gravitationally condensed objects in general relativity, that take
into account the effects connected to the propagation of the gravitational inter-
action at finite speed[14], have shown that one of the observable effects of this
propagation is a long-term decrease in the orbital period given by the formula
ṖGR(mA,mB) = −
192 π
1 + 73
e2 + 37
(1− e2)7/2
4In the case of PSR J0737−3039, one of the six measured parameters is the ratio xA/xB
between a Keplerian parameter of the pulsar and its analog for the companion, which turns
out to also be a pulsar.
s ≤ 1
0 0.5 1 1.5 2 2.5
PSR J1141−6545
intersection
0 0.5 1 1.5 2 2.5
PSR B1534+12
intersection
0 0.5 1 1.5 2 2.5
PSR J0737−3039
intersection
0 0.5 1 1.5 2 2.5
2.5 ω
s ≤ 1
PSR B1913+16
intersection
Figure 4: Tests of general relativity obtained from observations of four binary
pulsars. For each binary pulsar one has traced the “curves,” in the plane of
the two masses (mA = mass of the pulsar, mB = mass of the companion),
defined by equating the theoretical expressions for the various post-Keplerian
parameters, as predicted by general relativity, to their observational value, de-
termined through a least-squares fit to the parameterized theoretical timing
formula. Each “curve” is in fact a “strip,” whose thickness is given by the
(one sigma) precision with which the corresponding post-Keplerian parameter
is measured. For some parameters, these strips are too thin to be visible. The
grey zones would correspond to a sine for the angle of inclination of the or-
bital plane with respect to the plane of the sky that is greater than 1, and are
therefore physically excluded.
The direct physical origin of this decrease in the orbital period lies in the mod-
ification, produced by general relativity, of the usual Newtonian law of gravi-
tational attraction between two bodies, FNewton = GmAmB/r
AB. In place of
such a simple law, general relativity predicts a more complicated force law that
can be expanded in the symbolic form
FEinstein =
GmAmB
+ · · ·
, (20)
where, for example, “v2/c2” represents a whole set of terms of order v2A/c
v2B/c
2, vA vB/c
2, or even GmA/c
2 r or GmB/c
2 r. Here vA denotes the speed
of body A, vB that of body B, and rAB the distance between the two bod-
ies. The term of order v5/c5 in Equation (20) is particularly important. This
term is a direct consequence of the finite-speed propagation of the gravitational
interaction between A and B, and its calculation shows that it contains a com-
ponent that is opposed to the relative speed vA − vB of the two bodies and
that, consequently, slows down the orbital motion of each body, causing it to
evolve towards an orbit that lies closer to its companion (and therefore has a
shorter orbital period). This “braking” term (which is correlated with the emis-
sion of gravitational waves), δFEinstein ∼ v5/c5 FNewton, leads to a long-term
decrease in the orbital period ṖGR ∼ −(v/c)5 ∼ −10−12 that is very small, but
whose reality has been verified with a fractional precision of order 10−3 in PSR
B1913+16 and of order 20% in PSR B1534+12 and PSR J1141−6545 [4, 11, 13].
To conclude this brief outline of the tests of relativistic gravitation by binary
pulsars, let us note that there is an analog, for the regime of strong gravitational
fields, of the formalism of parametrization for possible deviations from general
relativity mentioned in Section 6 in the framework of weak gravitational fields
(using the post-Newtonian parameters γ−1 and β−1). This analog is obtained
by considering a two-parameter family of relativistic theories of gravitation,
assuming that the gravitational interaction is propagated not only by a tensor
field gµν but also by a scalar field ϕ. Such a class of tensor-scalar theories
of gravitation allows for a description of possible deviations in both the solar
system and in binary pulsars. It also allows one to explicitly demonstrate that
binary pulsars indeed test the effects of strong fields that go beyond the tests
of the weak fields of the solar system by exhibiting classes of theories that
are compatible with all of the observations in the solar system but that are
incompatible with the observations of binary pulsars, see [4, 13].
10 Gravitational Waves: Propagation, Genera-
tion, and Detection
As soon as he had finished constructing the theory of general relativity, Ein-
stein realized that it implied the existence of waves of geometric deformations
of space-time, or “gravitational waves” [15, 2]. Mathematically, these waves are
analogs (with the replacement Aµ → hµν) of electromagnetic waves, but concep-
tually they signify something remarkable: they exemplify, in the purest possible
way, the “elastic” nature of space-time in general relativity. Before Einstein
space-time was a rigid structure, given a priori, which was not influenced by
the material content of the Universe. After Einstein, a distribution of matter
(or more generally of mass-energy) that changes over the course of time, let us
say for concreteness a binary system of two neutron stars or two black holes,
will not only deform the chrono-geometry of the space-time in its immediate
neighborhood, but this deformation will propagate in every possible direction
away from the system considered, and will travel out to infinity in the form of a
wave whose oscillations will reflect the temporal variations of the matter distri-
bution. We therefore see that the study of these gravitational waves poses three
separate problems: that of generation, that of propagation, and, finally, that of
detection of such gravitational radiation. These three problems are at present
being actively studied, since it is hoped that we will soon detect gravitational
waves, and thus will be able to obtain new information about the Universe [16].
We shall here content ourselves with an elementary introduction to this field
of research. For a more detailed introduction to the detection of gravitational
waves see the contribution by Jean-Yves Vinet to this Poincaré seminar.
Let us first consider the simplest case of very weak gravitational waves,
outside of their material sources. The geometry of such a space-time can be
written, as in Section 6, as gµν(x) = ηµν+hµν(x), where hµν ≪ 1. At first order
in h, and outside of the source (namely in the domain where Tµν(x) = 0), the
perturbation of the geometry, hµν(x), satisfies a homogeneous equation obtained
by replacing the right-hand side of Equation (11) with zero. It can be shown that
one can simplify this equation through a suitable choice of coordinate system.
In a transverse traceless (TT) coordinate system the only non-zero components
of a general gravitational wave are the spatial components hTTij , i, j = 1, 2, 3 (in
other words hTT00 = 0 = h
0i ), and these components satisfy
� hTTij = 0 , ∂j h
ij = 0 , h
jj = 0 . (21)
The first equation in (21), where the wave operator � = ∆ − c−2 ∂2t appears,
shows that gravitational waves (like electromagnetic waves) propagate at the
speed c. If we consider for simplicity a monochromatic plane wave (hTTij =
ζij exp(ik ·x− i ω t)+ complex conjugate, with ω = c |k|), the second equation
in (21) shows that the (complex) tensor ζij measuring the polarization of a
gravitational wave only has non-zero components in the plane orthogonal to
the wave’s direction of propagation: ζij k
j = 0. Finally, the third equation
in (21) shows that the polarization tensor ζij has vanishing trace: ζjj = 0.
More concretely, this means that if a gravitational wave propagates in the z-
direction, its polarization is described by a 2 × 2 matrix,
ζxx ζxy
ζyx ζyy
, which
is symmetric and traceless. Such a polarization matrix therefore only contains
two independent (complex) components: ζ+ ≡ ζxx = −ζyy, and ζ× ≡ ζxy =
ζyx. This is the same number of independent (complex) components that an
electromagnetic wave has. Indeed, in a transverse gauge, an electromagnetic
wave only has spatial components ATi that satisfy
�ATi = 0 , ∂j A
j = 0 . (22)
As in the case above, the first equation (22) means that an electromagnetic wave
propagates at the speed c, and the second equation shows that a monochromatic
plane electromagnetic wave (ATi = ζi exp(ik · x− i ω t)+ c.c., ω = c |k|) is de-
scribed by a (complex) polarization vector ζi that is orthogonal to the direction
of propagation: ζj k
j = 0. For a wave propagating in the z-direction such a
vector only has two independent (complex) components, ζx and ζy. It is in-
deed the same number of components that a gravitational wave has, but we
see that the two quantities measuring the polarization of a gravitational wave,
ζ+ = ζxx = −ζyy, ζ× = ζxy = ζyx are mathematically quite different from
the two quantities ζx, ζy measuring the polarization of an electromagnetic wave.
However, see Section 11 below.
We have here discussed the propagation of a gravitational wave in a back-
ground space-time described by the Minkowski metric ηµν . One can also con-
sider the propagation of a wave in a curved background space-time, namely by
studying solutions of Einstein’s equations (9) of the form gµν(x) = g
µν(x) +
hµν(x) where hµν is not only small, but varies on temporal and spatial scales
much shorter than those of the background metric gBµν(x). Such a study is nec-
essary, for example, for understanding the propagation of gravitational waves
in the cosmological Universe.
The problem of generation consists in searching for the connection between
the tensorial amplitude hTTij of the gravitational radiation in the radiation zone
and the motion and structure of the source. If one considers the simplest case of
a source that is sufficiently diffuse that it only creates waves that are everywhere
weak (gµν − ηµν = hµν ≪ 1), one can use the linearized approximation to Ein-
stein’s equations (9), namely Equations (11). One can solve Equations (11) by
the same technique that is used to solve Maxwell’s equations (12): one fixes the
coordinate system by imposing ∂α hαµ − 12 ∂µ h
α = 0 (analogous to the Lorentz
gauge condition ∂αAα = 0), then one inverts the wave operator by using re-
tarded potentials. Finally, one must study the asymptotic form, at infinity, of
the emitted wave, and write it in the reduced form of a transverse and traceless
amplitude hTTij satisfying Equations (21) (analogous to a transverse electromag-
netic wave ATi satisfying (22)). One then finds that, just as charge conservation
implies that there is no monopole type electro-magnetic radiation, but only
dipole or higher orders of polarity, the conservation of energy-momentum im-
plies the absence of monopole and dipole gravitational radiation. For a slowly
varying source (v/c≪ 1), the dominant gravitational radiation is of quadrupole
type. It is given, in the radiation zone, by an expression of the form
hTTij (t, r,n) ≃
[Iij(t− r/c)]TT . (23)
Here r denotes the distance to the center of mass of the source, Iij(t) ≡
d3x c−2
T 00(t,x)
xixj − 1
x2δij
is the quadrupole moment of the mass-energy distri-
bution, and the upper index TT denotes an algebraic projection operation for
the quadrupole tensor Iij (which is a 3 × 3 matrix) that only retains the part
orthogonal to the local direction of wave propagation ni ≡ xi/r with vanish-
ing trace (ITTij is therefore locally a (real) 2× 2 symmetric, traceless matrix of
the same type as ζij above). Formula (23) (which was in essence obtained by
Einstein in 1918 [15]) is only the first approximation to an expansion in powers
of v/c, where v designates an internal speed characteristic of the source. The
prospect of soon being able to detect gravitational waves has motivated theo-
rists to improve Formula (23): (i) by describing the terms of higher order in
v/c, up to a very high order, and (ii) by using new approximation methods that
allow one to treat sources containing regions of strong gravitational fields (such
as, for example, a binary system of two black holes or two neutron stars). See
below for the most recent results.
Finally, the problem of detection, of which the pioneer was Joseph Weber in
the 1960s, is at present giving rise to very active experimental research. The
principle behind any detector is that a gravitational wave of amplitude hTTij
induces a change in the distance L between two bodies on the order of δL ∼ hL
during its passage. One way of seeing this is to consider the action of a wave
hTTij on two free particles, at rest before the arrival of the wave at the positions
xi1 and x
2 respectively. As we have seen, each particle, in the presence of the
wave, will follow a geodesic motion in the geometry gµν = ηµν + hµν (with
h00 = h0i = 0 and hij = h
ij ). By writing out the geodesic equation, Equation
(7), one finds that it simply reduces (at first order in h) to d2xi/ds2 = 0.
Therefore, particles that are initially at rest (xi = const.) remain at rest in a
transverse and traceless system of coordinates! This does not however mean
that the gravitational wave has no observable effect. In fact, since the spatial
geometry is perturbed by the passage of the wave, gij(t,x) = δij + h
ij (t,x),
one finds that the physical distance between the two particles xi1, x
2 (which is
observable, for example, by measuring the time taken for light to make a round
trip between the two particles) varies, during the passage of the wave, according
to L2 = (δij + h
ij )(x
2 − xi1)(x
2 − x
The problem of detecting a gravitational wave thus leads to the problem of
detecting a small relative displacement δL/L ∼ h. By using Formula (23), one
finds that the order of magnitude of h, for known or hoped for astrophysical
sources (for example,a very close system of two neutron stars or two black holes),
situated at distances such that one may hope to see several events per year
(r & 600 million light-years), is in fact extremely small: h . 10−22 for signals
whose characteristic frequency is around 100 Hertz. Several types of detectors
have been developed since the pioneering work of J. Weber [16]. At present,
the detectors that should succeed in the near future at detecting amplitudes
h ∼ δL/L ∼ 10−22 are large interferometers, of the Michelson or Fabry-Pérot
type, having arms that are many kilometers in length into which a very powerful
monochromatic laser beam is injected. Such terrestrial interferometric detectors
presently exist in the U.S.A. (the LIGO detectors [17]), in Europe (the VIRGO
[18] and GEO 600 [19] detectors) and elsewhere (such as the TAMA detector
in Japan). Moreover, the international space project LISA [20], made up of
an interferometer between satellites that are several million kilometers apart,
should allow one to detect low frequency (∼ one hundredth or one thousandth
of a Hertz) gravitational waves in a dozen years or so. This collection of gravi-
tational wave detectors promises to bring invaluable information for astronomy
by opening a new “window” on the Universe that is much more transparent
than the various electromagnetic (or neutrino) windows that have so greatly
expanded our knowledge of the Universe in the twentieth century.
The extreme smallness of the expected gravitational signals has led a num-
ber of experimentalists to contribute, over many years, a wealth of ingenuity
and know-how in order to develop technology that is sufficiently precise and
trustworthy (see [17, 18, 19, 20]). To conclude, let us also mention how much
concerted theoretical effort has been made, both in calculating the general rel-
ativistic predictions for gravitational waves emitted by certain sources, and in
developing methods adapted to the extraction of the gravitational signal from
the background noise in the detectors. For example, one of the most promising
sources for terrestrial detectors is the wave train for gravitational waves emitted
by a system of two black holes, and in particular the final (most intense) portion
of this wave train, which is emitted during the last few orbits of the system and
the final coalescence of the two black holes into a single, more massive black
hole. We have seen above (see Section 9) that the finite speed of propagation
of the gravitational interaction between the two bodies of a binary system gives
rise to a progressive acceleration of the orbital frequency, connected to the pro-
gressive approach of the two bodies towards each other. Here we are speaking
of the final stages in such a process, where the two bodies are so close that they
orbit around each other in a spiral pattern that accelerates until they attain
(for the final “stable” orbits) speeds that become comparable to the speed of
light, all the while remaining slightly slower. In order to be able to determine,
with a precision that is acceptable for the needs of detection, the dynamics of
such a binary black hole system in such a situation, as well as the gravitational
amplitude hTTij that it emits, it was necessary to develop a whole ensemble of
analytic techniques to a very high level of precision. For example, it was neces-
sary to calculate the expansion (20) of the force determining the motion of the
two bodies to a very high order and also to calculate the amplitude hTTij of the
gravitational radiation emitted to infinity with a precision going well beyond the
quadrupole approximation (23). These calculations are comparable in complex-
ity to high-order calculations in quantum field theory. Some of the techniques
developed for quantum field theory indeed proved to be extremely useful for
these calculations in the (classical) theory of general relativity (such as certain
resummation methods and the mathematical use of analytic continuation in the
number of space-time dimensions). For an entryway into the literature of these
modern analytic methods, see [21], and for an early example of a result obtained
by such methods of direct interest for the physics of detection see Figure 5 [22],
which shows a component of the gravitational amplitude hTTij (t) emitted during
the final stages of evolution of a system of two black holes of equal mass. The
first oscillations shown in Figure 5 are emitted during the last quasi-circular
orbits (accelerated motion in a spiral of decreasing radius). The middle part
of the signal corresponds to a phase where, having moved past the last stable
orbit, the two black holes “fall” toward each other while spiraling rapidly. In
fact, contrarily to Newton’s theory, which predicts that two condensed bodies
would be able to orbit around each other with an orbit of arbitrarily small ra-
dius (basically up until the point that the two bodies touch), Einstein’s theory
predicts a modified law for the force between the two bodies, Equation (20),
whose analysis shows that it is so attractive that it no longer allows for sta-
ble circular orbits when the distance between the two bodies becomes smaller
than around 6G(mA +mB)/c
2. In the case of two black holes, this distance is
sufficiently larger than the black hole “radii” (2GmA/c
2 and 2GmB/c
2) that
one is still able to analytically treat the beginning of the “spiralling plunge” of
the two black holes towards each other. The final oscillations in Figure 5 are
emitted by the rotating (and initially highly deformed) black hole formed from
the merger of the two initial, separate black holes.
−200 −100 0 100
−0.48
−0.38
−0.28
−0.18
−0.08
inspiral + plunge
merger + ring−down
Figure 5: The gravitational amplitude h(t) emitted during the final stages of
evolution of a system of two equal-mass black holes. The beginning of the signal
(the left side of the figure), which is sinusoidal, corresponds to an inspiral motion
of two separate black holes (with decreasing distance); the middle corresponds
to a rapid “inspiralling plunge” of the two black holes towards each other; the
end (at right) corresponds to the oscillations of the final, rotating black hole
formed from the merger of the two initial black holes.
Up until quite recently the analytic predictions illustrated in Figure 5 con-
cerning the gravitational signal h(t) emitted by the spiralling plunge and merger
of two black holes remained conjectural, since they could be compared to neither
other theoretical predictions nor to observational data. Recently, worldwide ef-
forts made over three decades to attack the problem of the coalescence of two
black holes by numerically solving Einstein’s equations (9) have spectacularly
begun to bear fruit. Several groups have been able to numerically calculate
the signal h(t) emitted during the final orbits and merger of two black holes
[23]. In essence, there is good agreement between the analytical and numerical
predictions. In order to be able to detect the gravitational waves emitted by
the coalescence of two black holes, it will most likely be necessary to properly
combine the information on the structure of the signal h(t) obtained by the two
types of methods, which are in fact complementary.
11 General Relativity and Quantum Theory: From
Supergravity to String Theory
Up until now, we have discussed the classical theory of general relativity, ne-
glecting any quantum effects. What becomes of the theory in the quantum
regime? This apparently innocent question in fact opens up vast new prospects
that are still under construction. We will do nothing more here than to touch
upon the subject, by pointing out to the reader some of the paths along which
contemporary physics has been led by the challenge of unifying general relativity
and quantum theory. For a more complete introduction to the various possi-
bilities “beyond” general relativity suggested within the framework of string
theory (which is still under construction) one should consult the contribution of
Ignatios Antoniadis to this Poincaré Seminar.
Let us recall that, from the very beginning of the quasi-definitive formula-
tion of quantum theory (1925–1930), the creators of quantum mechanics (Born,
Heisenberg, Jordan; Dirac; Pauli; etc.) showed how to “quantize” not only
systems with several particles (such as an atom), but also fields, continuous dy-
namical systems whose classical description implies a continuous distribution of
energy and momentum in space. In particular, they showed how to quantize (or
in other words how to formulate within a framework compatible with quantum
theory) the electromagnetic field Aµ, which, as we have recalled above, satisfies
the Maxwell equations (12) at the classical level. They nevertheless ran into dif-
ficulty due to the following fact. In quantum theory, the physics of a system’s
evolution is essentially contained in the transition amplitudes A(f, i) between
an initial state labelled by i and a final state labelled by f . These amplitudes
A(f, i) are complex numbers. They satisfy a “transitivity” property of the type
A(f, i) =
A(f, n)A(n, i) , (24)
which contains a sum over all possible intermediate states, labelled by n (with
this sum becoming an integral when there is a continuum of intermediate pos-
sible states). R. Feynman used Equation (24) as a point of departure for a
new formulation of quantum theory, by interpreting it as an analog of Huy-
gens’ Principle: if one thinks of A(f, i) as the amplitude, “at the point f ,” of
a “wave” emitted “from the point i,” Equation (24) states that this amplitude
can be calculated by considering the “wave” emitted from i as passing through
all possible intermediate “points” n (A(n, i)), while reemitting “wavelets” start-
ing from these intermediate points (A(f, n)), which then superpose to form the
total wave arriving at the “final point f .”
Property (24) does not pose any problem in the quantum mechanics of dis-
crete systems (particle systems). It simply shows that the amplitude A(f, i)
behaves like a wave, and therefore must satisfy a “wave equation” (which is in-
deed the case for the Schrödinger equation describing the dependence of A(f, i)
on the parameters determining the final configuration f). On the other hand,
Property (24) poses formidable problems when one applies it to the quantiza-
tion of continuous dynamical systems (fields). In fact, for such systems the
“space” of intermediate possible states is infinitely larger than in the case of
the mechanics of discrete systems. Roughly speaking, the intermediate possible
states for a field can be described as containing ℓ = 1, 2, 3, . . . quantum excita-
tions of the field, with each quantum excitation (or pair of “virtual particles”)
being described essentially by a plane wave, ζ exp(i kµ x
µ), where ζ measures
the polarization of these virtual particles and kµ = ηµν kν , with k
0 = ω and
ki = k, their angular frequency and wave vector, or (using the Planck-Einstein-
de Broglie relations E = ~ω, p = ~k) their energy-momentum pµ = ~ kµ.
The quantum theory shows (basically because of the uncertainty principle) that
the four-frequencies (and four-momenta) pµ = ~ kµ of the intermediate states
cannot be constrained to satisfy the classical equation ηµν p
µ pν = −m2 (or in
other words E2 = p2 +m2 ; we use c = 1 in this section). As a consequence,
the sum over intermediate states for a quantum field theory has the following
properties (among others): (i) when ℓ = 1 (an intermediate state containing
only one pair of virtual particles, called a one-loop contribution), there is an in-
tegral over a four-momentum pµ,
d4p =
dp; (ii) when ℓ = 2 (two pairs
of virtual particles; a two-loop contribution), there is an integral over two four-
momenta p
1 , p
d4p1 d
4p2; etc. The delicate point comes from the fact that
the energy-momentum of an intermediate state can take arbitrarily high values.
This possibility is directly connected (through a Fourier transform) to the fact
that a field possesses an infinite number of degrees of freedom, corresponding
to configurations that vary over arbitrarily small time and length scales.
The problems posed by the necessity of integrating over the infinite domain
of four-momenta of intermediate virtual particles (or in other words of account-
ing for the fact that field configurations can vary over arbitrarily small scales)
appeared in the 1930s when the quantum theory of the electromagnetic field
Aµ (called quantum electrodynamics, or QED) was studied in detail. These
problems imposed themselves in the following form: when one calculates the
transition amplitude for given initial and final states (for example the collision
of two light quanta, with two photons entering and two photons leaving) by
using (24), one finds a result given in the form of a divergent integral, because
of the integral (in the one-loop approximation, ℓ = 1) over the arbitrarily large
energy-momentum describing virtual electron-positron pairs appearing as pos-
sible intermediate states. Little by little, theoretical physicists understood that
the types of divergent integrals appearing in QED were relatively benign and,
after the second world war, they developed a method (renormalization theory)
that allowed one to unambiguously isolate the infinite part of these integrals,
and to subtract them by expressing the amplitudes A(f, i) solely as a function of
observable quantities [24] (work by J. Schwinger, R. Feynman, F. Dyson etc.).
The preceding work led to the development of consistent quantum theories
not only for the electromagnetic field Aµ (QED), but also for generalizations of
electromagnetism (Yang-Mills theory or non-abelian gauge theory) that turned
out to provide excellent descriptions of the new interactions between elementary
particles discovered in the twentieth century (the electroweak theory, partially
unifying electromagnetism and weak nuclear interactions, and quantum chro-
modynamics, describing the strong nuclear interactions). All of these theories
give rise to only relatively benign divergences that can be “renormalized” and
thus allowed one to compute amplitudes A(f, i) corresponding to observable
physical processes [24] (notably, work by G. ’t Hooft and M. Veltman).
What happens when we use (24) to construct a “perturbative” quantum
theory of general relativity (namely one obtained by expanding in the number
ℓ of virtual particle pairs appearing in the intermediate states)? The answer is
that the integrals over the four-momenta of intermediate virtual particles are
not at all of the benign type that allowed them to be renormalized in the simpler
case of electromagnetism. The source of this difference is not accidental, but is
rather connected with the basic physics of relativistic gravitation. Indeed, as we
have mentioned, the virtual particles have arbitrarily large energies E. Because
of the basic relations that led Einstein to develop general relativity, namely
E = mi and mi = mg, one deduces that these virtual particles correspond to
arbitrarily large gravitational masses mg. They will therefore end up creating
intense gravitational effects that become more and more intense as the number
ℓ of virtual particle pairs grows. These gravitational interactions that grow
without limit with energy and momentum correspond (by Fourier transform) to
field configurations concentrated in arbitrarily small space and time scales. One
way of seeing why the quantum gravitational field creates much more violent
problems than the quantum electromagnetic field is, quite simply, to go back to
dimensional analysis. Simple considerations in fact show that the relative (non-
dimensional) one-loop amplitude A1 must be proportional to the product ~G
and must contain an integral
d4k. However, in 1900 Planck had noticed that
(in units where c = 1) the dimensions of ~ and G were such that the product
~G had the dimensions of length (or time) squared:
≃ 1.6× 10−33 cm, tP ≡
≃ 5.4× 10−44 s . (25)
One thus deduces that the integral
d4k f(k) must have the dimensions of a
squared frequency, and therefore that A1 must (when k → ∞) be of the type,
A1 ∼ ~G
d4k/k2. Such an integral diverges quadratically with the upper
limit Λ of the integral (the cutoff frequency, such that |k| ≤ Λ), so that A1 ∼
~GΛ2 ∼ t2P Λ2. The extension of this dimensional analysis to the intermediate
states with several loops (ℓ > 1) causes even more severe polynomial divergences
to appear, of a type such that the power of Λ that appears grows without limit
with ℓ.
In summary, the essential physical characteristics of gravitation (E = mi =
mg and the dimension of Newton’s constant G) imply the impossibility of gener-
alizing to the gravitational case the methods that allowed a satisfactory quantum
treatment of the other interactions (electromagnetic, weak, and strong). Several
paths have been explored to get out of this impasse. Some researchers tried to
quantize general relativity non-perturbatively, without using an expansion in
intermediate states (24) (work by A. Ashtekar, L. Smolin, and others). others
have tried to generalize general relativity by adding a fermionic field to Einstein’s
(bosonic) gravitational field gµν(x), the gravitino field ψµ(x). It is indeed re-
markable that it is possible to define a theory, known as supergravity, that gener-
alizes the geometric invariance of general relativity in a profound way. After the
1974 discovery (by J. Wess and B. Zumino) of a possible new global symmetry
for interacting bosonic and fermionic fields, supersymmetry (which is a sort of
global rotation transforming bosons to fermions and vice versa), D.Z. Freedman,
P. van Nieuwenhuizen, and S. Ferrara; and S. Deser and B. Zumino; showed that
one could generalize global supersymmetry to a local supersymmetry, meaning
that it varies from point to point in space-time. Local supersymmetry is a sort
of fermionic generalization (with anti-commuting parameters) of the geometric
invariance at the base of general relativity (the invariance under any change in
coordinates). The generalization of Einstein’s theory of gravitation that admits
such a local supersymmetry is called supergravity theory. As we have mentioned,
in four dimensions this theory contains, in addition to the (commuting) bosonic
field gµν(x), an (anti-commuting) fermionic field ψµ(x) that is both a space-
time vector (with index µ) and a spinor. (It is a massless field of spin 3/2,
intermediate between a massless spin 1 field like Aµ and a massless spin 2 field
like hµν = gµν − ηµν .) Supergravity was extended to richer fermionic struc-
tures (with many gravitinos), and was formulated in space-times having more
than four dimensions. It is nevertheless remarkable that there is a maximal
dimension, equal to D = 11, admitting a theory of supergravity (the maximal
supergravity constructed by E. Cremmer, B. Julia, and J. Scherk). The initial
hope underlying the construction of these supergravity theories was that they
would perhaps allow one to give meaning to the perturbative calculation (24)
of quantum amplitudes. Indeed, one finds for example that at one loop, ℓ = 1,
the contributions coming from intermediate fermionic states have a sign oppo-
site to the bosonic contributions and (because of the supersymmetry, bosons ↔
fermions) exactly cancel them. Unfortunately, although such cancellations exist
for the lowest orders of approximation, it appeared that this was probably not
going to be the case at all orders5. The fact that the gravitational interaction
constant G has “a bad dimension” remains true and creates non-renormalizable
divergences starting at a certain number of loops ℓ.
Meanwhile, a third way of defining a consistent quantum theory of gravity
was developed, under the name of string theory. Initially formulated as models
for the strong interactions (in particular by G. Veneziano, M. Virasoro, P. Ra-
mond, A. Neveu, and J.H. Schwarz), the string theories were founded upon the
quantization of the relativistic dynamics of an extended object of one spatial di-
mension: a “string.” This string could be closed in on itself, like a small rubber
band (a closed string), or it could have two ends (an open string). Note that
the point of departure of string theory only includes the Poincaré-Minkowski
space-time, in other words the metric ηµν of Equation (2), and quantum theory
(with the constant ~ = h/2π). In particular, the only symmetry manifest in the
classical dynamics of a string is the Poincaré group (3). It is, however, remark-
5Recent work by Z. Bern et al. and M. Green et al., has, however, suggested that such
cancellations take place at all orders for the case of maximal supergravity, dimensionally
reduced to D = 4 dimensions.
able that (as shown by T. Yoneya, and J. Scherk and J.H. Schwarz, in 1974) one
of the quantum excitations of a closed string reproduces, in a certain limit, all
of the non-linear structure of general relativity (see below). Among the other
remarkable properties of string theory [25], let us point out that it is the first
physical theory to determine the space-time dimension D. In fact, this theory
is only consistent if D = 10, for the versions allowing fermionic excitations (the
purely bosonic string theory selects D = 26). The fact that 10 > 4 does not
mean that this theory has no relevance to the real world. Indeed, it has been
known since the 1930s (from work of T. Kaluza and O. Klein) that a space-
time of dimension D > 4 is compatible with experiment if the supplementary
(spatial) dimensions close in on themselves (meaning they are compactified) on
very small distance scales. The low-energy physics of such a theory seems to
take place in a four-dimensional space-time, but it contains new (a priori mass-
less) fields connected to the geometry of the additional compactified dimensions.
Moreover, recent work (due in particular to I. Antoniadis, N. Arkani-Hamed,
S. Dimopoulos, and G. Dvali) has suggested the possibility that the additional
dimensions are compactified on scales that are small with respect to everyday
life, but very large with respect to the Planck length. This possibility opens
up an entire phenomenological field dealing with the eventual observation of
signals coming from string theory (see the contribution of I. Antoniadis to this
Poincaré seminar).
However, string theory’s most remarkable property is that it seems to avoid,
in a radical way, the problems of divergent (non-renormalizable) integrals that
have weighed down every direct attempt at perturbatively quantizing gravity.
In order to explain how string theory arrives at such a result, we must discuss
some elements of its formalism.
Recall that the classical dynamics of any system is obtained by minimizing a
functional of the time evolution of the system’s configuration, called the action
(the principle of least action). For example, the action for a particle of mass
m, moving in a Riemannian space-time (6), is proportional to the length of the
line that it traces in space-time: S = −m
ds. This action is minimized when
the particle follows a geodesic, in other words when its equation of motion is
given by (7). According to Y. Nambu and T. Goto, the action for a string is
S = −T
dA, where the parameter T (analogous to m for the particle) is
called the string tension, and where
dA is the area of the two-dimensional
surface traced out by the evolution of the string in the (D-dimensional) space-
time in which it lives. In quantum theory, the action functional serves (as
shown by R. Feynman) to define the transition amplitude (24). Basically, when
one considers two intermediate configurations m and n (in the sense of the
right-hand side of (24)) that are close to each other, the amplitude A(n,m) is
proportional to exp(i S(n,m)/~), where S(n,m) is the minimal classical action
such that the system considered evolves from the configuration labelled by n to
that labelled by m. Generalizing the decomposition in (24) by introducing an
infinite number of intermediate configurations that lie close to each other, one
ends up (in a generalization of Huygens’ principle) expressing the amplitude
A(f, i) as a multiple sum over all of the “paths” (in the configuration space of
the system studied) connecting the initial state i to the final state f . Each path
contributes a term eiφ where the phase φ = S/~ is proportional to the action
S corresponding to this “path,” or in other words to this possible evolution of
the system. In string theory, φ = −(T/~)
dA. Since the phase is a non-
dimensional quantity, and
dA has the dimension of an area, we see that the
quantum theory of strings brings in the quantity ~/T , having the dimensions
of a length squared, at a fundamental level. More precisely, the fundamental
length of string theory, ℓs, is defined by
ℓ2s ≡ α′ ≡
2 π T
. (26)
This fundamental length plays a central role in string theory. Roughly speak-
ing, it defines the characteristic “size” of the quantum states of a string. If ℓs
is much smaller than the observational resolution with which one studies the
string, the string will look like a point-like particle, and its interactions will be
described by a quantum theory of relativistic particles, which is equivalent to
a theory of relativistic fields. It is precisely in this sense that general relativity
emerges as a limit of string theory. Since this is an important conceptual point
for our story, let us give some details about the emergence of general relativity
from string theory.
The action functional that is used in practice to quantize a string is not
really −T
dA, but rather (as emphasized by A. Polyakov)
= − 1
4 π ℓ2s
−γ γab ∂aXµ ∂bXν ηµν + · · · , (27)
where σa, a = 0, 1 are two coordinates that allow an event to be located on
the space-time surface (or ‘world-sheet’) traced out by the string within the
ambient space-time; γab is an auxiliary metric (dΣ
2 = γab(σ) dσ
a dσb) defined
on this surface (with γab being its inverse, and γ its determinant); and Xµ(σa)
defines the embedding of the string in the ambient (flat) space-time. The dots
indicate additional terms, and in particular terms of fermionic type that were
introduced by P. Ramond, by A. Neveu and J.H. Schwarz, and by others. If
one separates the two coordinates σa = (σ0, σ1) into a temporal coordinate,
τ ≡ σ0, and a spatial coordinate, σ ≡ σ1, the configuration “at time τ” of the
string is described by the functions Xµ(τ, σ), where one can interpret σ as a
curvilinear abscissa describing the spatial extent of the string. If we consider
a closed string, one that is topologically equivalent to a circle, the function
Xµ(τ, σ) must be periodic in σ. One can show that (modulo the imposition of
certain constraints) one can choose the coordinates τ and σ on the string such
that dΣ2 = −dτ2+dσ2. Then, the dynamical equations for the string (obtained
by minimizing the action (27)) reduce to the standard equation for waves on
a string: −∂2Xµ/∂τ2 + ∂2Xµ/∂σ2 = 0. The general solution to this equation
describes a superposition of waves travelling along the string in both possible
directions: Xµ = X
L(τ+σ)+X
R(τ−σ). If we consider a closed string (one that
is topologically equivalent to a circle), these two types of wave are independent
of each other. For an open string (with certain reflection conditions at the
endpoints of the string) these two types of waves are connected to each other.
Moreover, since the string has a finite length in both cases, one can decompose
the left- or right-moving waves X
L(τ + σ) or X
R(τ − σ) as a Fourier series. For
example, for a closed string one may write
Xµ(τ, σ) = X
0 (τ) +
e−2in(τ−σ) +
ãµn√
e−2in(τ+σ)
+ h.c. (28)
Here X
0 (τ) = x
µ + 2 ℓ2s p
µτ describes the motion of the string’s center of mass,
and the remainder describes the decomposition of the motion around the center
of mass into a discrete set of oscillatory modes. Like any vibrating string, a rel-
ativistic string can vibrate in its fundamental mode (n = 1) or in a “harmonic”
of the fundamental mode (for an integer n > 1). In the classical case the com-
plex coefficients aµn, ã
n represent the (complex) amplitudes of vibration for the
modes of oscillation at frequency n times the fundamental frequency. (with aµn
corresponding to a wave travelling to the right, while ãµn corresponds to a wave
travelling to the left.) When one quantizes the string dynamics the position of
the string Xµ(τ, σ) becomes an operator (acting in the space of quantum states
of the system), and because of this the quantities xµ, pµ, aµn and ã
n in (28) be-
come operators. The notation h.c. signifies that one must add the hermitian
conjugates of the oscillation terms, which will contain the operators (aµn)
† and
(ãµn)
†. (The notation † indicates hermitian conjugation, in other words the oper-
ator analog of complex conjugation.) One then finds that the operators xµ and
pµ describing the motion of the center of mass satisfy the usual commutation re-
lations of a relativistic particle, [xµ, pµ] = i ~ ηµν , and that the operators aµn and
ãµn become annihilation operators, like those that appear in the quantum theory
of any vibrating system: [aµn, (a
†] = ~ ηµν δnm, [ã
n, (ã
†] = ~ ηµν δmn. In
the case of an open string, one only has one set of oscillators, let us say aµn.
The discussion up until now has neglected to mention that the oscillation am-
plitudes aµn, ã
n must satisfy an infinite number of constraints (connected with
the equation obtained by minimizing (27) with respect to the auxiliary metric
γab). One can satisfy these by expressing two of the space-time components of
the oscillators aµn, ã
n (for each n) as a function of the other. Because of this, the
physical states of the string are described by oscillators ain, ã
n where the index i
only takes D−2 values in a space-time of dimension D. Forgetting this subtlety
for the moment (which is nevertheless crucial physically), let us conclude this
discussion by summarizing the spectrum of a quantum string, or in other words
the ensemble of quantum states of motion for a string.
For an open string, the ensemble of quantum states describes the states of
motion (the momenta pµ) of an infinite collection of relativistic particles, having
squared massesM2 = −ηµν pµ pν equal to (N−1) m2s, whereN is a non-negative
integer andms ≡ ~/ℓs is the fundamental mass of string theory associated to the
fundamental length ℓs. For a closed string, one finds another “infinite tower”
of more and more massive particles, this time with M2 = 4(N − 1)m2s. In both
cases the integer N is given, as a function of the string’s oscillation amplitudes
(travelling to the right), by
n ηµν(a
† aνn . (29)
In the case of a closed string one must also satisfy the constraint N = Ñ where
Ñ is the operator obtained by replacing aµn by ã
n in (29).
The preceding result essentially states that the (quantized) internal energy
of an oscillating string defines the squared mass of the associated particle. The
presence of the additional term −1 in the formulae given above for M2 means
that the quantum state of minimum internal energy for a string, that is, the
“vacuum” state |0〉 where all oscillators are in their ground state, aµn | 0〉 = 0,
corresponds to a negative squared mass (M2 = −m2s for the open string and
M2 = −4m2s for the closed string). This unusual quantum state (a tachyon) cor-
responds to an instability of the theory of bosonic strings. It is absent from the
more sophisticated versions of string theory (“superstrings”) due to F. Gliozzi,
J. Scherk, and D. Olive, to M. Green and J.H. Schwarz, and to D. Gross and
collaborators. Let us concentrate on the other states (which are the only ones
that have corresponding states in superstring theory). One then finds that the
first possible physical quantum states (such that N = 1) describe some massless
particles. In relativistic quantum theory it is known that any particle is the
quantized excitation of a corresponding field. Therefore the massless particles
that appear in string theory must correspond to long-range fields. To know
which fields appear in this way one must more closely examine which possible
combinations of oscillator excitations a
1 , a
2 , a
3 , . . ., appearing in Formula (29),
can lead to N = 1. Because of the factor n in (29) multiplying the harmonic
contribution of order n to the mass squared, only the oscillators of the fun-
damental mode n = 1 can give N = 1. One then deduces that the internal
quantum states of massless particles appearing in the theory of open strings are
described by a string oscillation state of the form
† | 0〉 . (30)
On the other hand, because of the constraint N = Ñ = 1, the internal quantum
states of the massless particles appearing in the theory of closed strings are
described by a state of excitation containing both a left-moving oscillation and
a right-moving oscillation:
ζµν(a
† (ãν1)
† | 0〉 . (31)
In Equations (30) and (31) the state |0〉 denotes the ground state of all oscillators
(aµn | 0〉 = ãµn | 0〉 = 0).
The state (30) therefore describes a massless particle (with momentum sat-
isfying ηµν p
µ pν = 0), possessing an “internal structure” described by a vector
polarization ζµ. Here we recognize exactly the definition of a photon, the quan-
tum state associated with a wave Aµ(x) = ζµ exp(i kλ x
λ), where pµ = ~ kµ.
The theory of open strings therefore contains Maxwell’s theory. (One can also
show that, because of the constraints briefly mentioned above, the polarization
ζµ must be transverse, k
µ ζµ = 0, and that it is only defined up to a gauge
transformation: ζ′µ = ζµ + a kµ.) As for the state (31), this describes a massless
particle (ηµν p
µ pν = 0), possessing an “internal structure” described by a tensor
polarization ζµν . The plane wave associated with such a particle is therefore of
the form h̄µν(x) = ζµν exp(i kλ x
λ), where pµ = ~ kµ. As in the case of the open
string, one can show that ζµν must be transverse, ζµν k
ν = 0 and that it is only
defined up to a gauge transformation, ζ′µν = ζµν+kµ aν+kν bµ. We here see the
same type of structure appear that we had in general relativity for plane waves.
However, here we have a structure that is richer than that of general relativity.
Indeed, since the state (31) is obtained by combining two independent states of
oscillation, (a
† and (ã
†, the polarization tensor ζµν is not constrained to be
symmetric. Moreover it is not constrained to have vanishing trace. Therefore,
if we decompose ζµν into its possible irreducible parts (a symmetric traceless
part, a symmetric part with trace, and an antisymmetric part) we find that the
field h̄µν(x) associated with the massless states of a closed string decomposes
into: (i) a field hµν(x) (the graviton) representing a weak gravitational wave
in general relativity, (ii) a scalar field Φ(x) (called the dilaton), and (iii) an
antisymmetric tensor field Bµν(x) = −Bνµ(x) subject to the gauge invariance
B′µν(x) = Bµν(x) + ∂µ aν(x) − ∂ν aµ(x). Moreover, when one studies the non-
linear interactions between these various fields, as described by the transition
amplitudes A(f, i) in string theory, one can show that the field hµν(x) truly
represents a deformation of the flat geometry of the background space-time in
which the theory was initially formulated. Let us emphasize this remarkable
result. We started from a theory that studied the quantum dynamics of a string
in a rigid background space-time. This theory predicts that certain quantum
excitations of a string (that propagate at the speed of light) in fact represent
waves of deformation of the space-time geometry. In intuitive terms, the “elas-
ticity” of space-time postulated by the theory of general relativity appears here
as being due to certain internal vibrations of an elastic object extended in one
spatial dimension.
Another suggestive consequence of string theory is the link suggested by
the comparison between (30) and (31). Roughly, Equation (31) states that
the internal state of a closed string corresponding to a graviton is constructed
by taking the (tensor) product of the states corresponding to photons in the
theory of open strings. This unexpected link between Einstein’s gravitation
(gµν) and Maxwell’s theory (Aµ) translates, when we look at interactions in
string theory, into remarkable identities (due to H. Kawai, D.C. Lewellen, and
S.-H.H. Tye) between the transition amplitudes of open strings and those of
closed strings. This affinity between electromagnetism, or rather Yang-Mills
theory, and gravitation has recently given rise to fascinating conjectures (due to
A. Polyakov and J. Maldacena) connecting quantum Yang-Mills theory in flat
space-time to quasi-classical limits of string theory and gravitation in curved
space-time. Einstein would certainly have been interested to see how classical
general relativity is used here to clarify the limit of a quantum Yang-Mills theory.
Having explained the starting point of string theory, we can outline the in-
tuitive reason for which this theory avoids the problems with divergent integrals
that appeared when one tried to directly quantize gravitation. We have seen
that string theory contains an infinite tower of particles whose masses grow
with the degree of excitation of the string’s internal oscillators. The gravita-
tional field appears in the limit that one considers the low energy interactions
(E ≪ ms) between the massless states of the theory. In this limit the gravi-
ton (meaning the particle associated with the gravitational field) is treated as
a “point-like” particle. When we consider more complicated processes (at one
loop, ℓ = 1, see above), virtual elementary gravitons could appear with arbitrar-
ily high energy. It is these virtual high-energy gravitons that are responsible for
the divergences. However, in string theory, when we consider any intermediate
process whatsoever where high energies appear, it must be remembered that
this high intermediate energy can also be used to excite the internal state of
the virtual gravitons, and thus reveal that they are “made” from an extended
string. An analysis of this fact shows that string theory introduces an effective
truncation of the type E . ms on the energies of exchanged virtual particles.
In other words, the fact that there are no truly “point-like” particles in string
theory, but only string excitations having a characteristic length ∼ ℓs, elimi-
nates the problem of infinities connected to arbitrarily small length and time
scales. Because of this, in string theory one can calculate the transition ampli-
tudes corresponding to a collision between two gravitons, and one finds that the
result is given by a finite integral [25].
Up until now we have only considered the starting point of string theory.
This is a complex theory that is still in a stage of rapid development. Let us
briefly sketch some other aspects of this theory that are relevant for this exposé
centered around relativistic gravitation. Let us first state that the more sophis-
ticated versions of string theory (superstrings) require the inclusion of fermionic
oscillators bµn, b̃
n, in addition to the bosonic oscillators a
n, ã
n introduced above.
One then finds that there are no particles of negative mass-squared, and that
the space-time dimension D must be equal to 10. One also finds that the mass-
less states contain more states than those indicated above. In fact, one finds
that the fields corresponding to these states describe the various possible theo-
ries of supergravity in D = 10. Recently (in work by J. Polchinski) it has also
been understood that string theory contains not only the states of excitation of
strings (in other words of objects extended in one spatial direction), but also
the states of excitation of objects extended in p spatial directions, where the
integer p can take other values than 1. For example, p = 2 corresponds to a
membrane. It even seems (according to C. Hull and P. Townsend) that one
should recognize that there is a sort of “democracy” between several different
values for p. An object extended in p spatial directions is called a p-brane. In
general, the masses of the quantum states of these p-branes are very large, be-
ing parametrically higher than the characteristic mass ms. However, one may
also consider a limit where the mass of certain p-branes tends towards zero. In
this limit, the fields associated with these p-branes become long-range fields. A
surprising result (by E. Witten) is that, in this limit, the infinite tower of states
of certain p-branes (in particular for p = 0) corresponds exactly to the infinite
tower of states that appear when one considers the maximal supergravity in
D = 11 dimensions, with the eleventh (spatial) dimension compactified on a
circle (that is to say with a periodicity condition on x11). In other words, in
a certain limit, a theory of superstrings in D = 10 transforms into a theory
that lives in D = 11 dimensions! Because of this, many experts in string theory
believe that the true definition of string theory (which is still to be found) must
start from a theory (to be defined) in 11 dimensions (known as “M -theory”).
We have seen in Section 8 that one point of contact between relativistic grav-
itation and quantum theory is the phenomenon of thermal emission from black
holes discovered by S.W. Hawking. String theory has shed new light upon this
phenomenon, as well as on the concept of black hole “entropy.” The essential
question that the calculation of S.W. Hawking left in the shadows is: what is
the physical meaning of the quantity S defined by Equation (19)? In the ther-
modynamic theory of ordinary bodies, the entropy of a system is interpreted,
since Boltzmann’s work, as the (natural) logarithm of the number of micro-
scopic states N having the same macroscopic characteristics (energy, volume,
etc.) as the state of the system under consideration: S = logN . Bekenstein had
attempted to estimate the number of microscopic internal states of a macroscop-
ically defined black hole, and had argued for a result such that logN was on the
order of magnitude of A/~G, but his arguments remained indirect and did not
allow a clear meaning to be attributed to this counting of microscopic states.
Work by A. Sen and by A. Strominger and C. Vafa, as well as by C.G. Callan
and J.M. Maldacena has, for the first time, given examples of black holes whose
microscopic description in string theory is sufficiently precise to allow for the
calculation (in certain limits) of the number of internal quantum states, N . It
is therefore quite satisfying to find a final result for N whose logarithm is pre-
cisely equal to the expression (19). However, there do remain dark areas in the
understanding of the quantum structure of black holes. In particular, the string
theory calculations allowing one to give a precise statistical meaning to the en-
tropy (19) deal with very special black holes (known as extremal black holes,
which have the maximal electric charge that a black hole with a regular horizon
can support). These black holes have a Hawking temperature equal to zero, and
therefore do not emit thermal radiation. They correspond to stable states in the
quantum theory. One would nevertheless also like to understand the detailed
internal quantum structure of unstable black holes, such as the Schwarzschild
black hole (17), which has a non-zero temperature, and which therefore loses
its mass little by little in the form of thermal radiation. What is the final state
to which this gradual process of black hole “evaporation” leads? Is it the case
that an initial pure quantum state radiates all of its initial mass to transform
itself entirely into incoherent thermal radiation? Or does a Schwarzschild black
hole transform itself, after having obtained a minimum size, into something
else? The answers to these questions remain open to a large extent, although it
has been argued that a Schwarzschild black hole transforms itself into a highly
massive quantum string state when its radius becomes on the order of ℓs [26].
We have seen previously that string theory contains general relativity in
a certain limit. At the same time, string theory is, strictly speaking, infinitely
richer than Einstein’s gravitation, for the graviton is nothing more than a partic-
ular quantum excitation of a string, among an infinite number of others. What
deviations from Einstein’s gravity are predicted by string theory? This question
remains open today because of our lack of comprehension about the connection
between string theory and the reality observed in our everyday environment
(4-dimensional space-time; electromagnetic, weak, and strong interactions; the
spectrum of observed particles; . . .). We shall content ourselves here with out-
lining a few possibilities. (See the contribution by I. Antoniadis for a discussion
of other possibilities.) First, let us state that if one considers collisions between
gravitons with energy-momentum k smaller than, but not negligible with respect
to, the characteristic string mass ms, the calculations of transition amplitudes
in string theory show that the usual Einstein equations (in the absence of mat-
ter) Rµν = 0 must be modified, by including corrections of order (k/ms)
2. One
finds that these modified Einstein equations have the form (for bosonic string
theory)
Rµν +
ℓ2s Rµαβγ R
ν + · · · = 0 , (32)
where
�ναβ ≡ ∂α Γ
νβ + Γ
νβ − ∂β Γµνα − Γ
να , (33)
denotes the “curvature tensor” of the metric gµν . (the quantity Rµν defined in
Section 5 that appears in Einstein’s equations in an essential way is a “trace” of
this tensor: Rµν = R
�µσν .) As indicated by the dots in (32), the terms written
are no more than the two first terms of an infinite series in growing powers of
ℓ2s ≡ α′. Equation (32) shows how the fact that the string is not a point, but
is rather extended over a characteristic length ∼ ℓs, modifies the Einsteinian
description of gravity. The corrections to Einstein’s equation shown in (32) are
nevertheless completely negligible in most applications of general relativity. In
fact, it is expected that ℓs is on the order of the Planck scale ℓp, Equation (25).
More precisely, one expects that ℓs is on the order of magnitude of 10
−32 cm.
(Nevertheless, this question remains open, and it has been recently suggested
that ℓs is much larger, and perhaps on the order of 10
−17 cm.)
If one assumes that ℓs is on the order of magnitude of 10
−32 cm (and that
the extra dimensions are compactified on distances scales on the order of ℓs),
the only area of general relativistic applications where the modifications shown
in (32) should play an important role is in primordial cosmology. Indeed, close
to the initial singularity of the Big Bang (if it exists), the “curvature” Rµναβ
becomes extremely large. When it reaches values comparable to ℓ−2s the infinite
series of corrections in (32) begins to play a role comparable to the first term,
discovered by Einstein. Such a situation is also found in the interior of a black
hole, when one gets very close to the singularity (see Figure 3). Unfortunately, in
such situations, one must take the infinite series of terms in (32) into account,
or in other words replace Einstein’s description of gravitation in terms of a
field (which corresponds to a point-like (quantum) particle) by its exact stringy
description. This is a difficult problem that no one really knows how to attack
today.
However, a priori string theory predicts more drastic low energy (k ≪ ms)
modifications to general relativity than the corrections shown in (32). In fact,
we have seen in Equation (31) above that Einsteinian gravity does not appear
alone in string theory. It is always necessarily accompanied by other long-range
fields, in particular a scalar field Φ(x), the dilaton, and an antisymmetric ten-
sor Bµν(x). What role do these “partners” of the graviton play in observable
reality? This question does not yet have a clear answer. Moreover, if one recalls
that (super)string theory must live in a space-time of dimension D = 10, and
that it includes the D = 10 (and eventually the D = 11) theory of supergravity,
there are many other supplementary fields that add themselves to the ten com-
ponents of the usual metric tensor gµν (in D = 4). It is conceivable that all of
these supplementary fields (which are massless to first approximation in string
theory) acquire masses in our local universe that are large enough that they no
longer propagate observable effects over macroscopic scales. It remains possible,
however, that one or several of these fields remain (essentially) massless, and
therefore can propagate physical effects over distances that are large enough to
be observable. It is therefore of interest to understand what physical effects are
implied, for example, by the dilaton Φ(x) or by Bµν(x). Concerning the latter,
it is interesting to note that (as emphasized by A. Connes, M. Douglas, and
A. Schwartz), in a certain limit, the presence of a background Bµν(x) has the
effect of deforming the space-time geometry in a “non-commutative” way. This
means that, in a certain sense, the space-time coordinates xµ cease to be sim-
ple real (commuting) numbers in order to become non-commuting quantities:
xµxν − xνxµ = εµν where εµν = −ενµ is connected to a (uniform) background
Bµν . To conclude, let us consider the other obligatory partner of the graviton
gµν(x), the dilaton Φ(x). This field plays a central role in string theory. In fact,
the average value of the dilaton (in the vacuum) determines the string theory
coupling constant, gs = e
Φ. The value of gs in turn determines (along with other
fields) the physical coupling constants. For example, the gravitational coupling
constant is given by a formula of the type ~G = ℓ2s(g
s + · · · ) where the dots
denote correction terms (which can become quite important if gs is not very
small). Similarly, the fine structure constant, α = e2/~c ≃ 1/137, which deter-
mines the intensity of electromagnetic interactions is a function of g2s . Because
of these relations between the physical coupling constants and gs (and therefore
the value of the dilaton; gs = e
Φ), we see that if the dilaton is massless (or in
other words is long-range), its value Φ(x) at a space-time point x will depend on
the distribution of matter in the universe. For example, as is the case with the
gravitational field (for example g00(x) ≃ −1 + 2GM/c2r), we expect that the
value of Φ(x) depends on the masses present around the point x, and should
be different at the Earth’s surface than it is at a higher altitude. One may
also expect that Φ(x) would be sensitive to the expansion of the universe and
would vary over a time scale comparable to the age of the universe. However,
if Φ(x) varies over space and/or time, one concludes from the relations shown
above between gs = e
Φ and the physical coupling constants that the latter must
also vary over space and/or time. Therefore, for example, the value, here and
now, of the fine structure constant α could be slightly different from the value
it had, long ago, in a very distant galaxy. Such effects are accessible to detailed
astronomical observations and, in fact, some recent observations have suggested
that the interaction constants were different in distant galaxies. However, other
experimental data (such as the fossil nuclear reactor at Oklo and the isotopic
composition of ancient terrestrial meteorites) put very severe limits on any vari-
ability of the coupling “constants.” Let us finally note that if the fine structure
“constant” α, as well as other coupling “constants,” varies with a massless field
such as the dilaton Φ(x), then this implies a violation of the basic postulate of
general relativity: the principle of equivalence. In particular, one can show that
the universality of free fall is necessarily violated, meaning that bodies with dif-
ferent nuclear composition would fall with different accelerations in an external
gravitational field. This gives an important motivation for testing the principle
of equivalence with greater precision. For example, the MICROSCOPE space
mission [27] (of the CNES) should soon test the universality of free fall to the
level of 10−15, and the STEP space project (Satellite Test of the Equivalence
Principle) [28] could reach the level 10−18.
Another interesting phenomenological possibility is that the dilaton (and/or
other scalar fields of the same type, called moduli) acquires a non-zero mass that
is however very small with respect to the string mass scale ms. One could then
observe a modification of Newtonian gravitation over small distances (smaller
than a tenth of a millimeter). For a discussion of this theoretical possibility and
of its recent experimental tests see, respectively, the contributions by I. Anto-
niadis and J. Mester to this Poincaré seminar.
12 Conclusion
For a long time general relativity was admired as a marvellous intellectual con-
struction, but it only played a marginal role in physics. Typical of the appraisal
of this theory is the comment by Max Born [29] made upon the fiftieth an-
niversary of the annus mirabilis: “The foundations of general relativity seemed
to me then, and they still do today, to be the greatest feat of human thought
concerning Nature, the most astounding association of philosophical penetra-
tion, physical intuition, and mathematical ability. However its connections to
experiment were tenuous. It seduced me like a great work of art that should be
appreciated and admired from a distance.”
Today, one century after the annus mirabilis, the situation is quite different.
General relativity plays a central role in a large domain of physics, including
everything from primordial cosmology and the physics of black holes to the
observation of binary pulsars and the definition of international atomic time.
It even has everyday practical applications, via the satellite positioning sys-
tems (such as the GPS and, soon, its European counterpart Galileo). Many
ambitious (and costly) experimental projects aim to test it (G.P.B., MICRO-
SCOPE, STEP, . . .), or use it as a tool for deciphering the distant universe
(LIGO/VIRGO/GEO, LISA, . . .). The time is therefore long-gone that its con-
nection with experiment was tenuous. Nevertheless, it is worth noting that the
fascination with the structure and physical implications of the theory evoked
by Born remains intact. One of the motivations for thinking that the theory
of strings (and other extended objects) holds the key to the problem of the
unification of physics is its deep affinity with general relativity. Indeed, while
the attempts at “Grand Unification” made in the 1970s completely ignored the
gravitational interaction, string theory necessarily leads to Einstein’s fundamen-
tal concept of a dynamical space-time. At any rate, it seems that one must more
deeply understand the “generalized quantum geometry” created through the in-
teraction of strings and p-branes in order to completely formulate this theory
and to understand its hidden symmetries and physical implications. Einstein
would no doubt appreciate seeing the key role played by symmetry principles
and gravity within modern physics.
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[2] See http://www.einstein.caltech.edu for an entry into the Einstein Col-
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[18] http://www.virgo.infn.it/
[19] http://www/geo600.uni-hanover.de/
[20] http://lisa.jpl.nasa.gov/
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and J.H. Schwarz, String Theory and M-theory: An Introduction,
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http://www.sstd.rl.ac.uk/fundphys/step/
	Introduction
	Special Relativity
	The Principle of Equivalence
	Gravitation and Space-Time Chrono-Geometry
	Einstein's Equations: Elastic Space-Time
	The Weak-Field Limit and the Newtonian Limit
	The Post-Newtonian Approximation and Experimental Confirmations in the Regime of Weak and Quasi-Stationary Gravitational Fields
	Strong Gravitational Fields and Black Holes
	Binary Pulsars and Experimental Confirmations in the Regime of Strong and Radiating Gravitational Fields
	Gravitational Waves: Propagation, Generation, and Detection
	General Relativity and Quantum Theory: From Supergravity to String Theory
	Conclusion
