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formed by instantons made using Taubes’ gluing

construction, described previously. The complement

of this collar is compact. In the interior of the moduli

space, there are a finite number of special points

corresponding to U(1)-reductions of the bundle P.

This is the way in which the moduli space ‘‘sees’’ the

integral structure of the intersection form since such

reductions correspond to integral homology classes

with self-intersection 1. Neighborhoods of these

special points are modeled on quotients C

=U(1); that

is, cones on copies of CP

. The upshot is that (for

generic Riemannian metrics on M) the moduli space

gives a cobordism from the manifold M to a set of

copies of CP

which can be counted in terms of the

intersection form, and the result follows easily from

standard topology. More sophisticated versions of the

argument extended the results to rule out some

indefinite intersection forms.

On the other hand, the original proofs of item (2)

used ‘‘invariants’’ defined by instanton moduli spaces

(Donaldson 1990). The general scheme exploits the

same construction outlined in the previous section.

We suppose that M is a simply connected 4-manifold

with b

(M) = 1 þ 2p,wherep > 0 is an integer.

(Here b

(M) is, as usual, the number of positive

eigenvalues of the intersection matrix.) Ignoring some

technical restrictions, there is a map

 : R

! H



ðM

where R

is a graded ring freely generated by

the homology (below the top dimension) of the

4-manifold M and M

is the moduli space of

anti-self-dual SU(2)-instantons on a bundle with

= k > 0. For an element W in R

of the

appropriate degree, one obtains a number by

evaluating, or integrating, (W)onM

. The main

technical difficulty here is that the moduli space M

is rarely compact, so one needs to make sense of this

‘‘evaluation.’’ With all the appropriate technicalities

in place, these invariants could be shown to

distinguish various homotopy-equivalent, homeo-

morphic 4-manifolds. All these early developments

are described in detail in the book by Donaldson

and Kronheimer (1990).

Basic Classes

Until the early 1990s, these instanton invariants

couldonlybecalculatedinisolatedfavorable

cases (although the calculations which were

made, through the work of many mathematicians,

led to a large number of further results about

4-manifold topology). Deeper understanding of

their structure came with the work of Kronhe imer

and M rowka. This work was, in large part,

motivated by a natural question in geometric

topology. Any homology class  2 H

(M; Z)can

be represented by an embedded, connected,

smooth surface. One can define an inte ger

g()

to be the minimal genus of such a representative.

The problem is to find

g(), or at least bounds on

it. A well-known conjecture, ascribed to Thom,

was that when M is the complex projective plane

the minimal genus is realized by a complex curve;

that is,

gðdHÞ¼

ðd  1Þðd  2Þ

where H is the standard generator of H

(CP

) and

d  1.

The new geometrical idea introduced by

Kronheimer and Mrowka was to study instantons

over a 4-manifold M with singularities along a

surface   M. F or such connections, there is a real

parameter: the limit of the trace of the holonomy

around small circles linking the surface. By

varying this parameter, they were able to inter-

polate between moduli spaces of nonsingular

instantons on different bundles over M and obtain

relations between the different invariants. T hey

alsofoundthatifthegenusof is suitably small

then some of the invariants are forced to vanish,

thus, conversely, getting information about

g for

4-manifolds with nontrivial invariants. For exam-

ple, they showed that if M is a K3 surface then

g() = (1/2)(   þ 2).

The structural results of Kronheimer and Mrowka

(1995) introduced the notion of a 4-manifold of

‘‘simple type.’’ Write the invariant defined above by

the moduli space M

as I

: R

!Q. Then I

vanishes except on terms of degree 2d(k), where

d(k ) = 4k  3(1 þ p). We can put all these together

to define

I =

: R

!Q. The ring R

is a

polynomial ring generated by classes  2 H

(M),

which have degree 2 in R

, and a class X of degree

4inR

, corresponding to the generator of H

(M).

The 4-manifold is of simple type if

IðX

WÞ¼4Ið WÞ

for all W 2 R

. Under this condition, Kronheimer

and Mrowka showed that all the invariants are

determined by a finite set of ‘‘basic’’ classes

, ..., K

2 H

(M) and rational numbers 

, ..., 

To express the relation, they form a generating

function

ðÞ¼I ðe



ÞþI





This is a priori a formal power series in H

(M) but a

posteriori the series converges and can be regarded

476 Gauge Theory: Mathematical Applications

as a function on H

(M). Kronheimer and Mrowka’s

result is that

ðÞ¼exp

  



r¼1





It is not known whether all simply connected

4-manifolds are of simple type, but Kronheimer and

Mrowka were able to show that this is the case for a

multitude of examples. They also introduced a

weaker notion of ‘‘finite type,’’ and this condition

was shown to hold in general by Munoz and

Froyshov. The overall result of this work of

Kronheimer and Mrowka was to make the calcula-

tion of the instanton invariants for many familiar

4-manifolds a comparatively straightforward matter.

3-Manifolds: Casson’s Invariant

Gauge theory has also entered into 3-manifold

topology. In 1985, Casson introduced a new

integer-valued invariant of oriented homology

3-spheres which ‘‘counts’’ the set Z of equivalence

classes of irreducible flat SU(2)-connections, or

equivalently irreducible representations 

(Y) !

SU(2). Casson’s approach (Akbulut and McCarthy

1990) was to use a Heegard splitting of a

3-manifold Y into two handle bodies Y

, Y



with

asurface as common boundary. Then 

()maps

onto 

(Y) and a flat SU(2) connection on Y is

determined by its restric tion to .LetM



be the

moduli space of irreducible flat connections over 

(as discussed in the last section) and let L



 M



the subsets which extend over Y



.ThenL



are

submanifolds of half the dimension of M



and the

set Z can be identified with the intersection L

\ L



The Casson invariant is one-half the algebraic

intersection number of L

and L



. Casson showed

that this is independent of the Heegard splitting

(and is also, in fact, an integer, although this is not

obvious). He showed that when Y is changed by

Dehn surgery along a knot, the invariant changes

by a term computed from the Alexander polynomial

of the knot. This makes the Casson invariant

computable in examples. (For a discussion of

Casson’s formula see Donaldson (1999).) Taube s

showed that the Casson invariant could also be

obtained in a more differential-geometric fashion,

analogous to the instanton invariants of 4-manifolds

(Taubes 1990).

3-Manifolds: Floer Theory

Independently, at about the same time, Floer

(1989) introduced more sophisticated invariants –

the Floer homology groups – of homology

3-spheres, using gauge theory. This development

ran parallel to his introduction of similar ideas in

symplectic geometry. Suppose, for simplicity, that

the se t Z of equivalence classes of irreducible flat

connections is finite. For pairs 



, 

in Z,Floer

considered the instantons on the tube Y  R

asymptotic to 



at 1. T here is an infinite set

of moduli s paces of such instantons, labeled by a

relative Chern class, but the dimensions of these

moduli spaces agree modulo 8 . This gives a relative

index (



, 

) 2 Z=8. If (



, 

) = 1thereisa

moduli space of dimension 1 (possibly empty), but

the translations of the tube act on this moduli space

and, dividing by translations, we get a finite set.

The number of points in this set, counted with

suitable signs, gives a n integer n(



, 

). Then,

Floer considers the f ree abelian groups



2Z

Zhi

generated by the set Z and a map @ : C



defined by

@ðh



iÞ ¼

nð



;

Þh

Here the sum runs over the 

with (



, 

) = 1.

Floer showed that @

= 0 and the homology



(Y) = ker @=Im @ is independent of the metric

on Y (and various other choices made in implement-

ing the construction in detail). The chain complex



and hence the Floer homology can be graded by

Z=8, using the relative index, so the upshot is to

define 8 abelian groups HF

(Y): invariants of the

3-manifold Y. The Casson invariant appears now as

the Euler characteristic of the Floer homology.

There has been extensive work on extending these

ideas to other 3-manifolds (not homology spheres)

and gauge groups, but this line of research does not

yet seem to have reached a clear-cut conclusion.

Part of the motivation for Floer’s work came from

Morse theory, and particul arly the approach to

this theory expounded by Witten (1982). The

Chern–Simons functional is a map

CS : A=G!R=Z

from the space of SU(2)-connections over Y.

Explicitly, in a trivialization of the bundle

CSðAÞ¼

A ^ dA þ

A ^ A ^ A

It appears as a boundary term in the Chern–Weil

theory for the second Chern class, in a similar way

as holonomy appears as a boundary term in the

Gauss–Bonnet theorem. The set Z can be identified

with the critical points of CS and the instantons on

the tube as integral curves of the gradient vector

Gauge Theory: Mathematical Applications 477

field of CS. Floer’s definition mimics the definition

of homology in ordinary Morse theory, taking

Witten’s point of view. It can be regarded formally

as the ‘‘middle-dimensional’’ homology of the

infinite-dimensional space A=G. See Atiyah (1988)

and Cohen et al. (1995) for discussions of these

ideas.

The Floer theory interacts with 4-manifold invar-

iants, making up a structure approximating to

a(3þ 1)-dimensional topological field theory

(Atiyah 1988). Roughly, the numerical invariants

of closed 4-manifolds generalize to invariants for a

4-manifold M with bounda ry Y taking values in the

Floer homology of Y. If two such manifolds are

glued along a common boundary, the invariants of

the result are obtained by a pairing in the Floer

groups. There are, however, at the moment, some

substantial technical restrictions on this picture. This

theory, as well as Floer’s original construction, is

developed in detail by Donaldson (2002). At the time

of writing, the Floer homology groups are still difficult

to compute in examples. One important tool is a

surgery-exact sequence found by Floer (Braam and

Donaldson 1995), related to Casson’s surgery

formula.

3-Manifolds: Jones–Witten Theory

There is another, quite different, way in which ideas

from gauge theory have entered 3-manifold topo-

logy. This is the Jones–Witten theory of knot and

3-manifold invariants. This theory falls outside the

main line of this article, but we will say a little about

it since it draws on many of the ideas we have

discussed. The goal of the theory is to construct a

family of (2 þ 1)-dimensional topological field the-

ories indexed by an integer k, assigning complex

vector space H

() to a surface  and an invariant

in H

(@Y) to a 3-manifold-with-boundary Y.If@Y is

empty, the vector space H

(@Y) is taken to be C,so

one seeks numerical invariants of closed 3-manifolds.

Witten’s (1989) idea is that these invariants of closed

3-manifolds are Feynmann integrals

A=G

i2kCSðAÞ

This functional integral is probably a schematic

rather than a rigorous notion. The data associated

with surfaces can, however, be defined rigorously. If

we fix a complex structure I on , we can define a

vector space H

(, I)tobe

ð; IÞ¼H

ðMðÞ; L

where M() is the moduli space of stable holo-

morphic bundles/flat unitary connections over 

and L is a certain holomorphic line bundle over

M(). These are the spaces of ‘‘conformal blocks’’

whose dimension is given by the Verlinde formulas.

Recall that M(), as a symplectic manifold, is

canonically associated with the surface , without

any choice of complex structure. The Hilbert

spaces H

(, I) can be regarded as the quantization

of this symplectic manifold, in the general frame-

work of geometric quantization: the inverse of k

plays t he role of Planck’s constant. What is not

obvious is that this quantization is independent of

the complex structure chosen on the Riemann

surface: that is, that there is a natural identification

of the vector spaces (or at least the associated

projective spaces) formed by using different com-

plex structures. This was established rigorously by

Hitchin (1990) and Axelrod et al. (1991),who

constructed a projectively flat connection on the

bundle of spaces H

(, I) over the space of complex

structures I on .Ataformallevel,these

constructions are derived from the construction of

the metaplectic representation of a linear symplec-

tic group, since the M



are symplectic quotients of

an affine symplectic space.

The Jones–Witten invariants have been rigorously

established by indirect means, but it seems that there

is still work to be done in developing Witten’s point

of view. If Y

is a 3-manifold with boundary, one

would like to have a geometric definition of a vector

in H

(@Y

). This should be the quantized version of

the submanifold L

(which is Lagrangian in M



)

entering into the Casson theory.

Seiberg–Witten Invariants

The instanton invariants of a 4-manifold can be

regarded as the integrals of certain natural differ-

ential forms over the moduli spaces of instantons.

Witten (1988) showed that these invariants could be

obtained as functional integrals, involving a variant

of the Feynman integral, over the space of connec-

tions and certain auxiliary fields (insofar as this

latter integral is defined at all). A geometric

explanation of Witten’s construction was given by

Atiyah and Jeffrey (199 0). Developing this point of

view, Witten made a series of predictions about the

instanton invariants, many of which were subse-

quently verified by other means. This line of work

culminated in 1994 where, applying developments

in supersymmetric Yang–Mills QFT, Seiberg and

Witten introduced a new system of invariants and a

precise prediction as to how these should be related

to the earlier ones.

The Seiberg–Witten invariants (Witten 1994) are

associated with a Spin

structure on a 4-ma nifold M.

If M is simply connected this is specified by a class

K 2 H

(M; Z ) lifting w

(M). One has spin bundles

478 Gauge Theory: Mathematical Applications

, S



!M with c



) = K. The Seiberg–Witten

equation is for a spinor field  – a section of S

and a connection A on the complex line bundle



. This gives a connection on S

and hence a

Dirac operator

:ðS

Þ!ðS



The Seiberg–Witten equations are

 ¼ 0; F

¼ ðÞ

where  : S

!

is a certain natural quadratic

map. The crucial differential-geometric feature of

these equations arises from the Weitzenbock

formula



 ¼r



 þ

 þ ðF

Þ

where R is the scalar curvature and  is a natural

map from 

to the endomorphisms of S

. Then  is

adjoint to  and

hððÞÞ; i¼jj

It follows easily from this that the moduli space of

solutions to the Seiberg–Witten equation is compact.

The most important invariants arise when K is

chosen so that

K  K ¼ 2ðMÞþ3 signðMÞ

where (M) is the Euler characteristic and sign(M)is

the signature. (This is just the condition for K to

correspond to an almost-complex structure on M.) In

this case, the moduli space of solutions is zero

dimensional (after generic perturbation) and the

Seiberg–Witten invariant SW(K) is the number of

points in the moduli space, counted with suitable signs.

Witten’s conjecture relating the invariants, in its

simplest form, is that when M has simple type the

classes K for which SW(K) is nonzero are exactly the

basic classes K

of Kronheimer and Mrowka and that



¼ 2

CðMÞ

SWðK

where C(M) = 2 þ (1/4)(7(M) þ 11 sign(M)). This

assertsthatthetwosetsofinvariantscontainexactly

the same information about the 4-manifold.

The evidence for this conjecture, via calculations of

examples, is very strong. A somewhat weaker

statement has been proved rigorously by Feehan and

Leness (2003). They use an approach suggested by

Pidstragatch and Tyurin, studying moduli spaces of

solutions to a nonabelian version of the Seiberg–

Witten equations. These contain both the instanton

and abelian Seiberg–Witten moduli spaces, and the

strategy is to relate the topology of these two sets by

standard localization arguments. (This approach is

related to ideas introduced by Thaddeus (1994) in the

case of bundles over Riemann surfaces.) The serious

technical difficulty in this approach stems from the

lack of compactness of the nonabelian moduli spaces.

The more general versions of Witten’s conjecture

(Moore and Witten 1997) (e.g., when b

(M) = 1)

contain very complicated formulas, involving mod-

ular forms, which presumably arise as contributions

from the compactification of the moduli spaces.

Applications

Regardless of the connection with the instanton

theory, one can go ahead directly to apply the

Seiberg–Witten invar iants to 4-manifold topology,

and this has been the main direction of research

since the 1990s. The features of the S eiberg–Witten

theory which have led to the most prominent

developments are the following.

1. The reduction of the equations to two dimensions

is very easy to understand. This has led to proofs

of the Thom conjecture and wide-ranging gen-

eralizations (Ozsvath and Szabo 2000).

2. The Weitzenboch formula implies that, if M has

positive scalar curvature, then solutions to the

Seiberg–Witten equations must have  = 0. This

has led to important interactions with four-

dimensional Riemannian geometry (Lebrun 1996).

3. In the case when M is a symplectic manifold,

there is a natural deformation of the Seiberg–

Witten equations, discovered by Taubes (1996),

who used it to show that the Seiberg–Witten

invariants of M are nontrivial. More generally,

Taubes showed that for large values of the

deformation parameter the solutions of the

deformed equation localize around surfaces in

the 4-manifold and used this to relate the

Seiberg–Witten invariants to the Gromov theory

of pseudoholomorphic curves. These results of

Taubes have completely transformed the subject

of four-dimensional symplectic geometry.

Bauer and Furuta (2004) have combined the

Seiberg–Witten theory with more sophisticated

algebraic topology to obtain further results about

4-manifolds. They consider the map from the space of

connections and spinor fields defined by the formulas

on the left-hand side of the equations. The general

idea is to obtain invariants from the homotopy class

of this map, under a suitable notion of homotopy.

A technical complication arises from the gauge group

action, but this can be reduced to the action of a single

U(1). Ignoring this issue, Bauer and Furuta have

obtained invariants in the stable homotopy groups

lim

N !1



Nþr

), which reduce to the ordinary

numerical invariants when r = 1. Using these invar-

iants, they obtain results about connected sums of

Gauge Theory: Mathematical Applications 479

4-manifolds, for which the ordinary invariants are

trivial. Using refined cobordism invariants ideas,

Furuta made great progress towards resolving the

question of which intersection forms arise from

smooth, simply connected 4-manifolds. A well-

known conjecture is that, if such a manifold is spin,

then the second Betti number satisfies

ðMÞ

jsignðMÞj

Furuta (2001) proved that b

(M) (10/8)jsign(M)jþ2.

An important and very recent achievement, bringing

together many different lines of work, is the proof of

‘‘Property P’’ in 3-manifold topology by Kronheimer

and Mrowka (2004). This asserts that one cannot

obtain a homotopy sphere (counter-example to the

Poincare´ conjecture) by þ1-surgery along a nontrivial

knot in S

. The proof uses work of Gabai and

Eliashberg to show that the manifold obtained by

0-framed surgery is embedded in a symplectic

4-manifold; Taubes’ results to show that the Seiberg–

Witten invariants of this 4-manifold are nontrivial;

Feehan and Leness’ partial proof of Witten’s con-

jecture to show that the same is true for the instanton

invariants; and the gluing rule and Floer’s exact

sequence to show that the Floer homology of the

þ1-surgered manifold is nontrivial. It follows then

from the definition of Floer homology that the funda-

mental group of this manifold is not trivial; in fact,

it must have an irreducible representation in SU(2).

See also: Cotangent Bundle Reduction; Floer Homology;

Gauge Theories from Strings; Gauge Theoretic Invariants

of 4-Manifolds; Instantons: Topological Aspects;

Knot Homologies; Moduli Spaces: An Introduction;

Nonperturbative and Topological Aspects of Gauge

Theory; Seiberg–Witten Theory; Topological Quantum

Field Theory: Overview; Variational Techniques for

Ginzburg–Landau Energies.

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Atiyah MF (1988) Topological Quantum Field Theories, vol. 68,

pp. 135–186. Math. Publ. IHES.

Atiyah MF and Bott R (1982) The Yang–Mills equations over

Riemann surfaces. Philosophical Transactions of the Royal

Society of London, Series A 308: 523–615.

Atiyah MF, Drinfeld V, Hitchin NJ, and Manin YuI (1978)

Construction of instantons. Physics Letters A 65: 185–187.

Atiyah MF and Hitchin NJ (1989) The Geometry and Dynamics of

Magnetic Monopoles. Princeton, NJ: Princeton University Press.

Atiyah MF, Hitchin NJ, and Singer IM (1978) Self-duality in

four-dimensional Riemannian geometry. Proceedings of the

Royal Society of London, Series A 362: 425–461.

Atiyah MF and Jeffrey L (1990) Topological Lagrangians and

cohomology. Journal of Geometry and Physics 7: 119–136.

Atiyah MF and Jones JDS (1978) Topological aspects of Yang–Mills

theory. Communications in Mathematical Physics 61: 97–118.

Axelrod S, Della Pietra S, and Witten E (1991) Geometric

quantisation of Chern–Simons gauge theories. Journal of

Differential Geometry 33: 787–902.

Bauer S and Furuta M, A stable cohomotopy refinement of the

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Braam PJ and Donaldson SK (1995) Floer’s work on instanton

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nications in Mathematical Physics 89: 145–190.

480 Gauge Theory: Mathematical Applications

Hitchin NJ (1987) The self-duality equations over Riemann surfaces.

Proceedings of the London Mathematical Society 55: 59–126.

Hitchin NJ (1990) Flat connections and geometric quantisation.

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General Relativity: Experimental Tests

C M Will, Washington University, St. Louis, MO, USA

Introduction

Einstein’s general theory of relativity has become the

foundation for our understanding of the gravita-

tional interaction. Four decades of high-precisi on

experiments have verified the theory with ever-

increasing precision, with no confirmed evidence of

a deviation from its predictions. The theory is now

the standard framework for much of astronomy,

with its searches for black holes, neutron stars,

gravitational waves, and the origin and fate of the

universe.

Yet modern developments in particle theory

suggest that it may not be the entire story, and that

General Relativity: Experimental Tests 481

modification of the basic theory may be required at

some level. String theory generally predicts a

proliferation of gravity-like fields that could result

in alterations of general relativity (GR) reminiscent

of the Brans–Dicke theory of the 1960s. In the

presence of extra dimensions, the gravity of the four-

dimensional ‘‘brane’’ of a higher-dimensional world

could be somewhat different from a pure four-

dimensional GR. However, any theoretical specula-

tion along these lines must still abide by the best

current empirical bounds. This article will review

experimental tests of GR and the theoretical

implications of the results.

The Einstein Equivalence Principle

The Einstein equivalence principle is a modern

generalization of Einstein’s 1907 idea of an equiva-

lence between gravity and acceleration, or between

free fall and an absence of gravity. It states that:

(1) test bodies fall with the same acceleration

independently of their internal structure or composi-

tion (weak equivalence principle, or WEP); (2) the

outcome of any local nongravitational experiment is

independent of the velocity of the freely falling

reference frame in which it is performed (local

Lorentz invariance, or LLI); and (3) the outcome of

any local nongravitational experiment is indepen-

dent of where and when in the universe it is

performed (local position invar iance, or LPI).

This principle is fundamental to gravitational

theory, for it is possible to argue that, if EEP is

valid, then gravitation and geometry are synon-

ymous. In other words, gravity must be described by

a ‘‘metric theory of gravity,’’ in which (1) spacetime

is endowed with a symmetric metric, (2) the

trajectories of freely falling bodies are geodesics of

that metric, and (3) in local freely falling reference

frames, the nongravitational laws of physics are

those written in the language of special relativity

(see Will (1993) for further details).

GR is a metric theory of gravity, but so are many

others, including the scalar–tensor theory of Brans

and Dicke and many of its modern descendents,

some of which are inspired by string theory.

Tests of the Weak Equivalence Principle

To test the WEP, one compares the acceleration of

two laboratory-sized bodies of different composition

in an external gravitational field. Although legend

suggests that Galileo may have demonstrated this

principle to his students at the Leaning Tower of Pisa,

and Newton tested it by means of pendulum

experiments, the first true high-precision experiments

were done at the end of the nineteenth century by the

Hungarian physicist Baron Roland von Eo¨tvo¨ s and

colleagues.

Eo¨ tvo¨ s employed a torsion balance, in which

(schematically) two bodies of different composition

are suspended at the ends of a rod that is supported

horizontally by a fine wire or fiber. One then looks

for a difference in the horizontal accelerations of the

two bodies as revealed by a slight rotation of the

rod. The source of the horizontal gravitational force

could be the Sun, a larg e mass in or near the

laboratory, or, as Eo¨ tvo¨ s recognized, the Earth itself.

A measurement or limit on the fractional difference

in acceleration between two bodies yields a quantity

  2ja

 a

j=ja

þ a

j, called the ‘‘Eo¨ tvo¨ s ratio.’’

Eo¨ tvo¨ s’ experiments showed that  was smaller than

a few parts in 10

, and later classic experiments in

the 1960s and 1970s by Dicke and Braginsky

improved the bounds by several orders of magni-

tude. Additional experiments were carried out

during the 1980s as part of a search for a putative

‘‘fifth force,’’ that was motivated in part by a

re-analysis of Eo¨ tvo¨ s’ original data.

The best limit on  currently comes from experi-

ments carried out during the 1985–2000 period at

the University of Washington (called the ‘‘Eo¨t-

Wash’’ experiments), which used a sophisticated

torsion balance tray to compar e the accelerations of

bodies of different composition toward the Earth,

the Sun, and the galaxy. Another strong bound

comes from ongoing laser ranging to reflectors

deposited on the Moon during the Apollo program

in the 1970s (lunar laser ranging, LLR), which

routinely determines the Earth–Moon distance to

millimeter accuracies. The data may be used to

check the equality of acceleration of the Earth and

Moon toward the Sun. The results from laboratory

and LLR experiments are (Will 2001):



€

ot– Wash

< 4  10

13

;

LLR

< 5  10

13

½1

LLR also shows that gravitational binding energy

falls with the same acceleration as ordinary matter to

1.3  10

 3

(test of the Nordtvedt ef fec t – s ee the se ction

‘‘Bounds on the PPN pa rameters’’ a nd Table 1).

Many of the high-precision, low-noise methods that

were developed for tests of WEP have been adapted

to laboratory tests of the inverse-square law of

Newtonian gravitation at millimeter scales and

below. The goal of these experiments is to search for

additional gravitational interactions involving massive

particles or for the presence of large extra dimensions.

The challenge of these experiments is to distinguish

gravitation-like interactions from electromagnetic and

quantum-mechanical effects. No deviations from

482 General Relativity: Experimental Tests

Newton’s inverse-square law have been found to date

at distances between 10 mm and 10 mm.

Tests of Local Lorentz Invariance

Although special relativity itself never benefited

from the kind of ‘‘crucial’’ experiments, such as the

perihelion advance of Mercury and the deflection of

light, that contributed so much to the initial

acceptance of GR and to the fame of Einstein, the

steady accumulation of experimental support,

together with the successful integration of special

relativity into quantum mechanics, led to its being

accepted by mainstream physicists by the late 1920s,

ultimately to become part of the standard toolkit of

every working physicist.

But in recent years new experiments have placed

very tight bounds on any violations of the Lorentz

invariance, which underlies special relativit y. A

simple way of interpreting this new class of

experiments is to suppose that a coupling of some

external gravitation-like field (not the metric) to the

electromagnetic interactions results in an effective

change in the speed of electromagnetic radiation, c,

relative to the limiting speed of material test

particles, c

; in other words, c 6¼ c

. It can be

shown that such a Lorentz-noninvariant electromag-

netic interaction would cause shifts in the energy

levels of atoms and nuclei that depend on the

orientation of the quantization axis of the state

relative to our velocity relative to the rest of the

universe, and on the quantum numbers of the state,

resulting in orientation dependences of the funda-

mental frequencies of such atomic clocks. The

magnitude of these ‘‘clock anisotropies’’ would be

proportional to  j(c

=c)

 1j, which vanishes if

Lorentz invariance holds (see Will (1993) and

Haugan and Will (1987) for details).

The earliest clock anisotropy experiments were

carried out around 1960 independently by Hughes

and Drever, although their original motivation was

somewhat different. Dramatic improvements were

made in the 1980s using laser-cooled trapped atoms

and ions. This technique made it possible to reduce

the broadening of resonance lines caused by colli-

sions, leading to the impressive bound jj > 10

21

(Will 2001).

Other recent tests of Lorentz invariance violation

include comparisons of resonant cavities with

atomic clocks, tests of dispersion and birefringence

in the propagation of high-energy photons from

astrophysical sources, threshold effects in elemen-

tary particle collisions, and anomalies in neutrino

oscillations. Mattingly (2005) gives a thorough and

up-to-date review of both the theore tical frame-

works for studying these effects and the experimen-

tal results.

Tests of Local Position Invariance

LPI requires, among other things, that the internal

binding energies of atoms and nuclei be indepen-

dent of location in space and time, when measured

against some standard atom. This means that a

comparison of the rates of two different kinds of

atomic clocks should be independent of location or

epoch, and that the frequency shift between two

identical clocks at di fferent locations is simply a

consequence of the apparent Doppler shift

between a pair of inertial frames momentarily

comoving with the clocks at the moments of

emission and reception, respectively. The relevant

parameter  appears in the formula for the

frequency shift,

f =f ¼ð1 þ Þ=c

½2

Table 1 Current limits on the PPN parameters

Parameter Effect Limit Remarks

  1 (i) Shapiro delay 2.3  10

5

Cassini tracking

(ii) Light deflection 4  10

4

VLBI

  1 (i) Perihelion shift 3  10

3

= 10

7

from helioseismology

(ii) Nordtvedt effect 2.3  10

4

LLR plus bounds on other parameters

 Anisotropy in Newton’s G 10

3

Gravimeter bounds on anomalous Earth tides



Orbit polarization for moving systems 10

4

Lunar laser ranging



Anomalous spin precession for moving bodies 4  10

7

Alignment of solar axis relative to ecliptic



Anomalous self-acceleration for spinning moving bodies 2  10

20

Pulsar spindown timing data



Nordtvedt effect 9  10

4

Lunar laser ranging



2  10

2

Combined PPN bounds



Anomalous self-acceleration for binary systems 4  10

5

Timing data for PSR 1913 þ 16



Violation of Newton’s third law 10

8

Lunar laser ranging



Not independent

Here  = 4    3  10=3  

þ 2

=3  2

=3  

=3.

General Relativity: Experimental Tests 483

where  is the Newtonian gravitational potential. If

LPI holds,  = 0. An early test of this was the

Pound–Rebka experiment of 1960, which measured

the frequency shift of gamma rays from radioactive

iron nuclei in a tower at Harvard University. The

best bounds come from a 1976 experiment in which

a hydrogen maser atomic clock was launched to

10 000 km altitude on a Scout rocket and its

frequency compared via telemetry with an identical

clock on the ground, and a 1993 experiment in

which two different kinds of atomic clocks were

intercompared as a function of the varying solar

gravitational field as seen on Earth (a ‘‘null’’ redshift

experiment). The results are (Will 2001):



Maser

< 2  10

4

;

Null

< 10

3

½3

Recent ‘‘clock comparison’’ tests of LPI include

experiments done at the National Institute of

Standards and Technology (NIST) in Boulder and

at the Observatory of Paris, to look for cosmological

variations in clock rates. The NIST experiment

compared laser-cooled mercury ions with neutral

cesium atoms over a two-year period, while the

Paris experiment compared laser-cooled cesium and

rubidium atomic fountains over five years; the

results sho wed that the fine-structure constant is

constant in time to a part in 10

per year. A better

bound of 6  10

17

1

comes from analysis of

fission yields of the Oklo natural reactor, which

occurred in Africa two billion years ago.

Solar-System Tests

The Parametrized Post-Newtonian Framework

It was once customary to discuss experimental tests of

GR in terms of the ‘‘three classical tests,’’ the

gravitational redshift (which is really a test of the

EEP, not of GR itself; see the section on tests of LPI),

the perihelion advance of Mercury (the first success of

the theory), and the deflection of light (whose

measurement in 1919 made Einstein a celebrity).

However, the proliferation of additional experimental

tests and of well-motivated alternative metric theories

of gravity made it desirable to develop a more general

theoretical framework for analyzing both experiments

and theories. This ‘‘parametrized post-Newtonian

(PPN) framework’’ dates back to Eddington in 1922,

but was fully developed by Nordtvedt and Will in the

period 1968–72 (see Will (1993) for details).

When attention is confined to metric theories of

gravity and, further, the focus is on the slow-motion,

weak-field limit appropriate to the solar system and

similar systems, it turns out that, in a broad class of

metric theories, only the numerical values of a set of

coefficients in the spacetime metric vary from theory to

theory. The resulting PPN framework contains ten

parameters: , related to the amount of spatial curv-

ature generated by mass; , related to the degree of

nonlinearity in the gravitational field; , 

, 

,and



, which determine whether the theory violates LPI

or LLI in gravitational experiments; and 

, 

,and



, which describe whether the theory has appropriate

momentum conservation laws. In GR,  = 1,  = 1,

and the remaining parameters all vanish. In the scalar–

tensor theory of Brans–Dicke,  = (1 þ!

)=(2 þ!

where !

is an adjustable parameter.

A number of well-known relativistic effects can be

expressed in terms of these PPN parameters:

Deflection of light

 ¼

1 þ 



4GM

1 þ 



 1:7505



arcsec ½4

where d is the distance of closest approach of a ray

of light to a body of mass M, and where the second

line is the deflection by the Sun, with radius R



Shapiro time delay

t ¼

1 þ 



4GM

ðr

þ x

 nÞðr

 x

 nÞ



½5

where t is the excess travel time of a round-trip

electromagnetic tracking signal, x

and x

are the

locations relative to the body of mass M of the

emitter and receiver of the round-trip signal (r

and

are the respective distances), and n is the direction

of the outgoing tracking signal.

Perihelion advance

2 þ 2  



Pað1  e

Þc

2 þ 2  



 42:98 arcsec=100 yr ½6

where P , a, and e are the period, semimajor axis,

and eccentricity of the planet’s orbit, respectively;

the second line is the value for Mercury.

Nordtvedt effect

 m



4    3 

  













½7

where m

and m

are, respectively, the gravitational

and inertial masses of a body such as the Earth or

484 General Relativity: Experimental Tests

Moon, and E

is its gravitational binding energy. A

nonzero Nordtvedt effect would cause the Earth and

Moon to fall with a different acceleration toward

the Sun. In GR, this effect vanishes.

Precession of a gyroscope

¼ð

þ 

Geo

ÞS



¼

1 þ  þ





ðJ  3nn  JÞ

1 þ  þ





 0:041 arcsec yr

1



Geo

¼

ð1 þ 2Þv 

Gmn

ð1 þ 2Þ6:6 arcsec yr

1

½8

where S is the spin of the gyroscope, and 

and



Geo

are, respectively, the precession angular velo-

cities caused by the dragging of inertial frames

(Lense–Thirring effect) and by the geodetic effect, a

combination of Thomas precession and precession

induced by spatial curvature; J is the angular

momentum of the Earth, and v, n,andr are,

respectively, the velocity, direction, and distance of

the gyroscope. The second line in each case is the

corresponding value for a gyroscope in polar Earth

orbit at about 650 km altitude (Gravity Probe B).

Bounds on the PPN Parameters

Four decades of high-precision experiments, ranging

from the standard light-deflection and perihelion-

shift tests, to LLR, planetary and satellite tracking

tests of the Shapiro time delay, and geophysical and

astronomical observations, have placed bounds on

the PPN parameters that are consistent with GR.

The current bounds are summarized in Table 1 (Will

2001).

To illustrate the dramatic progress of experimen-

tal gravity since the dawn of Einstein’s theory,

Figure 1 shows a history of results for (1 þ )=2,

from the 1919 solar eclipse measurements of

Eddington and his colleagues (which made Einstein

a celebrity), to modern-day measurements using very

long baseline radio interferometry (VLBI), advanced

radar tracking of spacecraft, and the astrometry

satellite Hipparcos. The most recent results include a

2003 measurement of the Shapiro delay, performed

by tracking the ‘‘Cassini’’ spacecraft on its way

to Saturn, and a 2004 measurement of the bending

of light via analysis of VLBI data on 541 quasars

and compact radio galaxies distributed over the

entire sky.

The perihelion advance of Mercury, the first of

Einstein’s successes, is now known to agree with

observationtoafewpartsin10

. During the 1960s

there was controversy about this test when reports of an

excess solar oblateness implied an unacceptably large

Newtonian contribution to the perihelion advance.

However, it is now known from helioseismology, the

study of short-period vibrations of the Sun, that the

oblateness is of the order of a part in 10

,asexpected

from standard solar models, much too small to affect

Mercury’s orbit, within the observational errors.

Gravity Probe B

The NASA Relativity Mission called Gravity Probe B

(GPB) recently completed its mission to measure the

Lense–Thirring and geodetic precessions of gyroscopes

in Earth’s orbit. Launched on 20 April 2004 for a

16-month mission, it consisted of four spherical rotors

coated with a thin layer of superconducting niobium,

spinning at 70–100 Hz, in a spacecraft filled with liquid

helium, containing a telescope continuously pointed

Optical

Radio

VLBI

Hipparcos

Deflection

of light

PSR 1937 + 21

Voyager

Viking

Shapiro

time

delay

1920

1940

1960 1970 1980 1990

2000

1.10

1.05

1.00

0.95

1.05

1.00

0.95

(1 + γ)/2

Year of experiment

Cassini

× 10

–5

)

2 × 10

–4

Figure 1 Measurements of the coefficient (1 þ )=2 from

observations of the deflection of light and of the Shapiro delay

in propagation of radio signals near the Sun. The GR prediction

is unity. ‘‘Optical’’ denotes measurements of stellar deflection

made during solar eclipse, and ‘‘Radio’’ denotes interferometric

measurements of radio-wave deflection. ‘‘Hipparcos’’ denotes the

European optical astrometry satellite. Arrows denote values well

off the chart from one of the 1919 eclipse expeditions and from

others through 1947. Shapiro delay measurements using the

Cassini spacecraft on its way to Saturn yielded tests at the

0.001% level, and light deflection measurements using VLBI

have reached 0.02%.

General Relativity: Experimental Tests 485