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

Подождите немного. Документ загружается.

toward a distant guide star (IM Pegasi). Superconduct-

ing current loops encircling each rotor were designed to

measure the change in direction of the rotors by

detecting the change in magnetic flux through the loop

generated by the London magnetic moment of the

spinning superconducting film. The spacecraft was in a

polar orbit at 650 km altitude. The primary science goal

of GPB was a 1% measurement of the 41 marcsec yr

1

frame dragging or Lense–Thirring effect caused by the

rotation of the Earth; its secondary goal was to measure

to six parts in 10

the larger 6.6 arcsec yr

1

geodetic

precession caused by space curvature.

The Binary Pulsar

The binary pulsar PSR 1913 þ 16, discovered in 1974,

provided important new tests of GR. The pulsar, with

a pulse period of 59 ms, was observed to be in orbit

about an unseen companion (now generally thought to

be a dead pulsar), with a period of 8h. Through

precise timing of apparent variations in the pulsar

‘‘clock’’ caused by the Doppler effect, the important

orbital parameters of the system could be measured

with exquisite precision. These included nonrelativistic

‘‘Keplerian’’ parameters, such as the eccentricity e,and

the orbital period (at a chosen epoch) P

,aswellasa

set of relativistic ‘‘post-Keplerian’’ (PK) parameters.

The first PK parameter, h

!i, is the mean rate of

advance of periastron, the analog of Mercury’s

perihelion shift. The second, denoted 

, is the effect

of special relativistic time dilation and the gravita-

tional redshift on the observed phase or arrival time of

pulses, resulting from the pulsar’s orbital motion and

the gravitational potential of its companion. The third,

, is the rate of decrease of the orbital period; this is

taken to be the result of gravitational radiation

damping (apart from a small correction due to the

acceleration of the system in our rotating galaxy). Two

other parameters, s and r, are related to the Shapiro

time delay of the pulsar signal if the orbital inclination

is such that the signal passes in the vicinity of the

companion; s is a direct measure of the orbital

inclination sin i. According to GR, the first three PK

effects depend only on e and P

, which are known, and

on the two stellar masses, which are unknown. By

combining the observations of PSR 1913 þ 16 (see

Table 2) with the GR predictions, one obtains both a

measurement of the two masses and a test of GR, since

the system is overdetermined. The results are

¼ 1:4414  0:0002M



; m

¼ 1:3867  0:0002M



OBS

¼ 1:0013  0:0021 ½9

Other relativistic binary pulsars may provide even

more stringent tests. These include the relativistic

neutron star/white dwarf binary pulsar J1141-6545,

with a 0.19 day orbital period, which may ultimately

lead to a very strong bound on the phenomenon of

dipole gravitational radiation, predicte d by many

alternative theories of gravity, but not by GR; and

the remarkable ‘‘double pulsar’’ J0737-3039, a

binary system with two detected pulsars, in a

0.10 day orbit seen almost edge on and a periastr on

advance of 178 per year. For further discussion of

binary pulsar tests, see Stairs (2003 ).

Gravitational-Wave Tests

The detection of gravitational radiation by either

laser interferometers or resonant cryogenic bars will

usher in a new era of gravitational-wave astronomy

(Barish and Weiss 1999). Furthermore, it will yield

new and interesting tests of GR in its radiative

regime (Will 1999).

GR predicts that gravitational waves possess only two

polarization modes independently of the source; they are

transverse to the direction of propagation and quad-

rupolar in their effect on a detector. Other theories of

gravity may predict up to four additional modes of

polarization. A suitable array of gravitational antennas

could delineate or limit the number of modes present in a

given wave. If distinct evidence were found of any mode

other than the two transverse quadrupolar modes of

GR, the result would be disastrous for the theory.

Table 2 Parameters of the binary pulsars PSR 1913 þ 16 and J0737-3039

Parameter Symbol Value

in PSR1913 þ 16 Value

in J0737-3039

Keplerian parameters

Eccentricity e 0.6171338(4) 0.087779(5)

Orbital period P

(day) 0.322997448930(4) 0.102251563(1)

Post-Keplerian parameters

Periastron advance h

!i(



1

) 4.226595(5) 16.90(1)

Redshift/time dilation 

(ms) 4.2919(8) 0.382(5)

Orbital period derivative

(10

12

) 2.4184(9)

Shapiro delay ( sin i ) s 0.9995(4)

Numbers in parentheses denote errors in last digit.

486 General Relativity: Experimental Tests

According to GR, gravitational waves propagate with

thesamespeed,c, as light. In other theories, the speed

could differ from c because of coupling of gravitation to

‘ ‘b ackground’’ gravitational fields, or propagation of the

waves into additional spatial dimensions. Another way

in which the speed of gravitational waves could differ

from c is if gravitation were propagated by a massive

field (a massive graviton), in which case v

would be

given by, in a local inertial frame,

¼ 1 

 1 



½10

where m

, E, and f are the graviton rest mass,

energy, and frequency, respectively, and 

= h=m

is the graviton Compton wavelength (it is assumed

that 

 c=f ).

The most obvious way to measure the speed of

gravitational waves is to compare the arrival times of a

gravitational wave and an electromagnetic wave from

the same event (e.g., a supernova). For a source at a

distance of 600 million light years (a typical distance

for the currently operational detectors), and a differ-

ence in times on the order of seconds, the bound on the

difference j1  v

=cj could be as small as a part in

. It is worth noting that a 2002 report that the

speed of gravity had been measured by studying light

from a quasar as it propagated past Jupiter was

fundamentally flawed. That particular measurement

was not sensitive to the speed of gravity.

Conclusions

The past four decades have witnessed a systematic,

high-precision experimental verification of Einstein’s

theories. Relativity has passed every test with flying

colors. A central theme of future work will be to test

strong-field gravity in the vicinity of black holes and

neutron stars, and to see how well GR works on

cosmological scales. Gamma-ray, X-ray, microwave,

infrared, neutrino, and gravitational-wave astronomy

will all play a critical role in probing these largely

unexplored aspects of GR.

GR is now the ‘‘standard model’’ of gravity. But,

as in particle physics, there may be a world beyond

the standard model. Quantum gravity, strings, and

branes may lead to testabl e effects beyond Einstein’s

GR. Searches for such effects using laboratory

experiments, particle accelerators, space instrumen-

tation, and cosmological observations are likely to

continue for some time to come.

See also: Cosmology: Mathematical Aspects; Einstein

Equations: Exact Solutions; General Relativity: Overview;

Geometric Flows and the Penrose Inequality;

Gravitational Lensing; Gravitational Waves; Standard

Model of Particle Physics.

Further Reading

Barish BC and Weiss R (1999) LIGO and the detection of

gravitational waves. Physics Today 52: 44.

Haugan MP and Will CM (1987) Modern tests of special

relativity. Physics Today 40: 69.

Mattingly D (2005) Modern tests of Lorentz invariance. Living

Reviews in Relativity 8: 5 (online article cited on 1 April 2005,

http://www.livingreviews.org/lrr-2005-5).

Stairs IH (2003) Testing general relativity with pulsar timing.

Living Reviews in Relativity 6: 5 (online article cited on 1

April 2005, http://www.livingreviews.org/lrr-2003-5).

Will CM (1993) Theory and Experiment in Gravitational Physics.

Cambridge: Cambridge University Press.

Will CM (1999) Gravitational radiation and the validity of

general relativity. Physics Today 52: 38.

Will CM (2001) The confrontation between general relativity

and experiment. Living Reviews in Relativity 4: 4 (online

article cited on 1 April 2005, http://www.livingreviews.org/

lrr-2001-4).

General Relativity: Overview

R Penrose, University of Oxford, Oxford, UK

The Principle of Equivalence

The special theory of relativity is founded on two

basic principles: that the laws of physics should be

independent of the uniform motion of an inertial

frame of reference, and that the speed of light

should have the same constant value in any such

frame. In the years between 1905 and 1915,

Einstein pondered deeply on what was , to him, a

profound enigma, which was the issue of why these

laws retain their proper form only in the case of an

inertial frame. In special relativity, as had been the

case in the earlier dynamics of Galilei–Newton, the

laws indeed retain their basic form only when the

reference frame is unaccelerated (which includes it

being nonrotating). It demonstrated a particular

prescience on the part of Einstein that he should

have demanded the seemingly impossible require-

ment that the very same dynamical laws should

hold also in an accelerating (or even rotating)

reference frame. The key realization came to him

late in 1907, when sitting in his chair in the Bern

General Relativity: Overview 487

patent office he had the ‘‘happiest thought’’ in his life,

namely that if a person were to fall freely in a

gravitational field, then he would not notice that field

at all while falling. The physical point at issue is

Galileo’s early insight (itself having roots even earlier

from Simon Stevin in 1586 or Ioannes Philiponos in

the fifth or sixth century) that the acceleration

induced by gravity is independent of the body upon

which it acts. Accordingly, if two neighboring bodies

are accelerated together in the same gravitational

field, then the motion of one body, in the (nonrotat-

ing) reference frame of the other, will be as though

there were no gravitational field a t all. To put this

another way, the effect of a gravitational force is just

like that of an accelerating reference system, and can

be eliminated by free fall. This is now known as the

‘‘principle of equivalence.’’

It should be made clear that this is a particular

feature of only the gravitational field. From the

perspective of Newtonian dynamics, it is a conse-

quence of the seemingly accidental fact that the

concept of (passive) ‘‘mass’’ m that features in

Newton’s law of gravitational attraction, where the

attractive force due to the gravitational field of

another body, of mass M, has the form

GmM

is the same as – or, at least, proportional to – the

inertial mass m of the body which is being acted

upon. Thus, the impedance to acceleration of a body

and the strength of the attractive force on that body

are, in the case of gravity (and only in the case of

gravity), in proportion to one another, so that the

acceleration of a body in a gravitational field is

independent of its mass (or, indeed, of any other

localized magnitude) possessed by it. (The fact that

the active gravitational mass, here given by the

quantity M, is also in proportion to its own passive

gravitational mass – from Newton’s third law – may

be regarded as a feature of the general Lagrangian/

Hamiltonian framework of physics. But see Bondi

(1957).) Other forces of nature do not have this

property. For example, the electrostatic force on a

charged body, by an electric field, acts in proportion to

the electric charge on that body, whereas, the

impedance to acceleration is still the inertial mass of

that body, so the acceleration induced depends on the

charge-to-mass ratio. Accordingly, it is the gravita-

tional field alone which is equivalent to an acceleration.

Einstein’s fundam ental idea, therefore, was to

take the view that the ‘‘relativity principle’’ could

as well be applied to accelerating reference frames as

to inertial ones, where the same physical laws would

apply in each, but where now the perceived

gravitational field would be different in the two

frames. In accordance with this perspective, Einstein

found it necessary to adopt a different viewpoint

from the Newtonian one, both with regard to the

notion of ‘‘gravitational force’’ and to the very

notion of an ‘‘inertial frame.’’ According to the

Newtonian perspective, it would be appropriate to

describe the action of the Earth’s gravitational field,

near some specific place on the Earth’s surface, in

terms of a ‘‘Newtonian inertial frame’’ in which the

Earth is ‘‘fixed’’ (here we ignore the Earth’s rotation

and the Earth’s motion about the Sun ), and we

consider that there is a constant gravitational field

of force (directed towards the Earth’s center). But

the Einsteinian perspective is to regard that frame as

noninertial where, instead, it would be a frame

which falls freely in the Earth’s (Newtonian)

gravitational field that would be regarded as a

suitable ‘‘Einsteinian inertial frame.’’ Generally, to

be inertial in Einstein’s sense, the frame would refer

to free fall under gravity, so that the Newtonian

field of gravitational force would appear to have

disappeared – in accordance with his ‘‘happiest

thought’’ that Einstein had had in the Bern patent

office. We see that the concept of a gravitational

field must also be changed in the passage from

Newton’s to Einstein’s viewpoint. For in Newton’s

picture we indeed have a ‘‘gravitational force’’

directed towards the ground with a magnitude of

gm, where m is the mass of the body being acted

upon and g is the ‘‘acceleration due to gravity’’ at

the Earth’s surface, whereas in Einstein’s picture we

have specifically eliminated this ‘‘gravitational

force’’ by the choice of ‘‘Einsteinian inertial frame.’’

It might at first seem puzzling that the gravitational

field has appeared to have been removed altogether by

this device, and it is natural to wonder how gravita-

tional effects can have any physical role to play at all

from this point of view! However, this would be to go

too far, as the Newtonian gravitational field may vary

from place to place – as it does, indeed, in the case of

the Earth’s field, since it is directed towards the Earth’s

center, which is a different spatial direction at different

places on the Earth’s surface. Our considerations up to

this point really refer only to a small neighborhood of a

point. One might well take the view that a ‘‘frame’’

ought really to describe things also at widely separated

places at once, and the considerations of the para-

graphs above do not really take this into consideration.

The Tidal Effect

To proceed further, it will be helpful to consider an

astronaut A in free fall, high above the Earth’s

surface. Let us first adopt a Newtonian perspective.

488 General Relativity: Overview

We shall be concer ned only with the instantaneous

accelerations due to gravity in the neighborhood of

A, so it will be immaterial whether we regard the

astronaut as falling to the ground or – more

comfortably! – in orbit about the Earth.

Let us imagine that the astronaut is initially

surrounded, nearby, by a sphere of particles, with A

at the centre, which are taken to be initially at rest

with respect to A (see Figure 1). To a first

approximation, all the particles will share the same

acceleration as the astronaut, so they will seem to the

astronaut to hover motionless all around. But now let

us be a little more precise about the accelerations.

Those particles which are initially located in a vertical

line from A, that is, either directly below A, at B, or

directlyaboveA,atT,willhave,likeA,an

acceleration which is in the direction AO, where O

is the Earth’s center. But for the bottom point B, the

acceleration will be slightly greater than that at A, and

for the top point T, the acceleration will be slightly

less than the acceleration at A, because of the slightly

differing distances from O. Thus, relative to A, both

will initially accelerate away from A. With regard to

particles in the sphere which are initially in a circle in

the horizontal plane through A, the direction to O

will now be somewhat inwards, so that the particles

at these points H

will accelerate, relative to A,

slightly inwards. Accordingly, the entire sphere of

particles will begin to get distorted into a prolate

spheroid (elongated ellipsoid of revolution). This is

referred to as the tidal distortion, for the good reason

that it is precisely the same physical effect which is

responsible for the tides in the Earth’s oceans, where

for this illustration we are to think of the Earth’s

center as being at A, the Moon (or Sun) to be situated

at O, and the sphere of particles to represent the

surface of the water of the Earth’s oceans.

It is not hard to calculate (reverting, now, to our

original picture) that, as a reflection of Newton’s

inverse-square law of gravitational attraction, the

amount of (small) outward vertical displacement

from A (at B and T) will be twice the inward

horizontal displacement (over the circle of points

); accordingly, the sphere will ini tially be distorted

into an ellipsoid of the same volume. This depends

upon there being no gravitating matter insi de the

sphere. The presence of such matter would con-

tribute a volume-reducing effect in proportion to the

total mass surrounded. (An extreme case illustrating

this would occur if we take our sphere of particles to

surround the entire Earth, where the volume-

reducing effect would be manifest in the accelera-

tions towards the ground at all points of the

surrounding sphere.)

Gravity as Curved Spacetime

It is appropriate to take a spacetime view of these

phenomena (Figure 2). The distortions that we have

been considering are, in fact, direct manifestations

(a) (b)

Figure 1 (a) Tidal effect. The astronaut A surrounded by a

sphere of nearby particles initially at rest with respect to A. In

Newtonian terms, they have an acceleration towards the Earth’s

center E, varying slightly in direction and magnitude (single-

shafted arrows). By subtracting A’s acceleration from each, we

obtain the accelerations relative to A (double-shafted arrows);

this relative acceleration is slightly inward for those particles

displaced horizontally from A, but slightly outward for those

displaced vertically from A. Accordingly, the sphere becomes

distorted into a (prolate) ellipsoid of revolution, with symmetry

axis in the direction AE. The initial distortion preserves volume.

(b) Now move A to the Earth’s center E and the sphere of

particles to surround E just above the atmosphere. The

acceleration (relative to A = E) is inward all around the sphere,

with an initial volume reduction acceleration 4GM, where M is

the total mass surrounded. Reproduced with permission from

Penrose R (2004) The Road to Reality: A Complete Guide to the

Laws of the Universe. London: Jonathan Cape.

(a)

(b)

Figure 2 Spacetime versions of Figure 1 in terms of the

relative distortion of neighboring geodesics. (a) Geodesic

deviation in empty space (basically Weyl curvature) as seen in

the world lines of A and surrounding particles (one spatial

dimension suppressed), as might be induced from the gravita-

tional field of a nearby body E. (b) The corresponding inward

acceleration (basically Ricci curvature) due to the mass density

within the bundle of geodesics. Reproduced with permission

from Penrose R (2004) The Road to Reality: A Complete Guide

to the Laws of the Universe. London: Jonathan Cape.

General Relativity: Overview 489

of spacetime curvature, according to Einstein’s

viewpoint. We are to think of the world line of a

particle, falling freely under gravity (Eins teinian

inertial motion), as described as some kind of

geodesic in spacetime. We shall be coming to this

more completely shortly, but for the moment it will

be helpful to picture the behavior of geodesics

within an ordinary curved 2-surface S (Figure 3).

If S has positive (Gaussian) curvature, then there

will be a tendency for geodesics on S to bend

towards each other, so that a pair of infinitesimally

separated geodesics which are initially parallel will

begin to get closer together as we move along them;

if S has negative (Gaussian) curvature, then there

will be a corresponding tendency for geodesics on S

to bend away from each other. This is what happens

in two dimensions, where the intr insic curvature at a

point is given by a single number. However, we are

now concerned with a four-dimensional space,

where the notion of curvature requires many more

components. We see in Figure 2 that we are indeed

to expect mixtures of convergence and divergence of

geodesics, which suggests that there are both

positive and negative curvature components

involved, the positive curvature being in the hor-

izontally displaced directions from A and the

negative curvature in the vertically displaced direc-

tions. In a curved space of dimension 4, as is the

case for a curved spacetime, we can expect 20

independent components of curvature at each point

altogether. In the present situation, the others would

be called into play when differing velocities of A are

considered.

Let us see how we are to accommodate the above

considerations within the standard framework of

differential geometry. So far, we have not really

deviated from Newtonian theory, even though we

have been considering ‘‘geodesics’’ in a four-dimen-

sional spacetime. In fact, it is perfectly legitimate to

view Newtonian theory in this way (see Newtonian

Limit of General Relativity), although the 4-geometry

description is somewhat more complicated than one

might wish. This is due to the fact that the infinite

speed at which gravitation is taken to act in

Newtonian theory demands that the ‘‘metric’’ of

Newtonian spacetime is degenerate. (In effect, one

would have a degenerate ‘‘dual metric’’ G

,of

matrix rank 3, which plays a role in defining spatial

displacements and a very degenerate ‘‘metric’ ’ G

,of

matrix rank 1, which defines temporal differences,

where G

= 0; see Newtonian Limit of General

Relativity.) Accordingly, there is no unique notion of

‘‘geodesic’’ defined by the metric in Newtonian

theory.

It is strik ing that althoug h the insights provided

by the principle of equiva lence are to some

considerable extent independent of special relativity

(since we see from the paragraphs prior to the

preceding one that a curved-spacetime-geometry

view of gravity is natural in the ligh t of the

equivalence principle alone), it is the nondegenerate

metric g

, (and its inverse g

) that special relativity

gives us locally, which leads to an elegant space-

time theory of gravity. Although the metric g

is Lorentzian (with prefe rred choice of signature

þ here) rather than positive definite, so that

the spacetime is not strictly a Riemannian one, the

change of signature makes little difference to

the local formalism. In particular, the fact that the

metric defines a unique (torsion-free) connection

preserving it is unaffected by the signature. This

connection is the one defined by Christoffel’s symbols



ð@

þ @

 @

where @

stands for coordinate derivative @=@x

so that the covariant derivative of a vector V

given by

¼ @

þ V



(Here the standard ‘‘physicist’s conventions’’ are

being used, whereby notation such as ‘‘g

’’ and

‘‘V

’’ can be used interchangeably either for the sets

of components of the metric tensor g and the vector

V, respectively, or alternatively for the entire

geometrical metric tensor g or vector V, in each

case; moreover, the summation convention is being

assumed, or this can alternatively be understood in

terms of abstract indices. (For the abstract-index

notation for tensors, see Penrose and Rindler (1984),

especially Chapters 2 and 4. Sign and index-ordering

conventions used here follow those given in that

book. Many other authors use conventions which

differ from these in various, usually minor

respects.))

(a) (b)

γ ′

′

Figure 3 Geodesic deviation when M is a 2-surface (a) of

positive (Gaussian) curvature, when the geodesics , 



bend

towards each other, and (b) of negative curvature, when they

bend apart. Reproduced with permission from Penrose R (2004)

The Road to Reality: A Complete Guide to the Laws of the

Universe. London: Jonathan Cape.

490 General Relativity: Overview

Physical Interpretation of the Metric

Some words of clarification are needed, as to the

meaning of the metric tensor g

in relativity theory.

In the early discussions by Einstein and others, the

spacetime metric tended to be interpreted in terms of

little ‘‘rulers’’ placed on a curved manifold.

Although this is natural in the Riemannian

(positive-definite) case, it is not quite so appropriate

for the Lorentzian geometry of spacetime manifolds.

An ordinary physical ruler has a spacetime descrip-

tion as a timelike strip, and it does not naturally

express the spatial separation between two

spacelike-separated events. In order for a ruler to

measure such a spacelike separation, it would be

necessary for the two events to be simultaneous in

the ruler’s rest frame, and for this to be assured,

some further mechanism would be needed, such as

Einstein’s procedure for ensuring simultaneity by the

use of light signals from the two events to be

received simultaneously at their midpoint on the

ruler. Clearly this complicate s the issue, and it turns

out to be much preferable to concentrate on

temporal displacements rather than spatial ones.

The idea that spacetime geometry should really be

regarded as ‘‘chronometry,’’ in this way, has been

stressed by a number of distinguished expositors of

relativity theory, most notably John L Synge (1956,

1960) and Hermann Bondi (1961, 1964, 1967).

Where needed, spatial displacements can then be

defined by the use of temporal ones together with

light signals. This has the additional advantage that

in modern technology, the measurement of (proper)

time far surpasses that of distance in accuracy, to the

extent that the meter is now defined simply by the

requirement that there are exactly 299792458 of

them in a light-second! The proper time interval

between two nearby events is, indeed, measured by a

clock which encounters both events, moving iner-

tially between the two, and very precise atom ic and

nuclear clocks are now a common feature of current

technology. The physical role of the metric g

most clearly seen in the formula

 ¼

ðg

which measures the (proper) time interval  between

an event p and a later event q on its world line, the

integral being taken along this curve, and where now

that curve need not be a geodesic, so that accelerating

(noninertial) motion of the clock is allowed. The

metric (with choice of signature þso that it is

the timelike displacements that are directly provided

as real numbers) is very precisely specified by this

physical requirement, and this tells us that the

pseudo-Riemannian (Lorentzian) structure of space-

time is far from being an arbitrary construction, but

is given to us by Nature with enormous precision.

(Some theorists prefer to use the alternative spacetime

signature þþþ, because this more directly relates

to familiar Newtonian concepts, these being normally

described in spatial terms. The difference is essentially

just a notational one, however. It may be remarked

that the 2-spinor formalism (see Spinors and

Spin Coefficients) fits in much more readily with the

þsignature being used here.) It may be noted,

also that this time measure is ultimately fixed by

quantum principles and the masses of the elementary

ingredients involved (e.g., particle masses) via the

Einstein and Planck relations E = mc

and E = h,so

that there is a natural frequency associated with a

given mass, via  = mc

=h (c being the speed of light

and h being Planck’s constant).

Riemann Curvature and Geodesic

Deviation

The unique torsion-free (Christoffel–Levi-Civita)

connection r

is, via this physically determined

metric, also fixed accordingly by these physical

considerations, as is the notion of a geodesic, and

therefore so also is the curvature. The 20-independent-

component Riemann curvature tensor R

abcd

may be

defined by

ðr

r

ÞV

¼ R

abc

with normal index-raising/lowering conventions, so

that R

abcd

= R

abc

, etc., and we have the standard

classical formula

abc

¼ @



 @



þ 



 



The symmetries R

abcd

cdab

= R

bacd

, R

abcd

þR

bcad

cabd

=0 reduce the number of independent compo-

nents of R

abcd

to 20 (from a potential 4

=64).

Of these, 10 are locally fixed by the kind of

physical requirement indicated above, that in order

to express something that agrees closely with New-

ton’s inverse-square law we require that there should

be a net inward curving of free world lines (the

timelike geodesics that represent local inertial

motions, or ‘ ‘free fall’’ under gravity). Let us see

how this requirement is satisfied in Einstein’s general

relativity.

What we find, from Newton’s theory, is that a

system of test particles which, at some initial time

constitutes a closed 2-surface at rest surrounding

some gravitating matter, will begin to accelerate in

such a way that the volume surrounded is initially

reduced in proportion to the total mass surrounded.

General Relativity: Overview 491

This volume reduction is a direct consequence of

Poisson’s equation r

=4 ( being the gravi-

tational potent ial and  the mass density) and of

Newton’s second law, which tell us that the second

time derivative of the free-fall volume of our initially

stationary closed surface of test particles is indeed

4GM, where M is the total gravitating mass

surrounded (and G is Newton’s constant, as above).

In Einstein’s theory, we can basically carry this over

to our four-dimensional Lorentzian spacetime.

We do, however, find that such a general statement

as this does not exactly hold. Instead of referring to

3-volumes of any size, we must restrict attention to

infinitesimal volumes.

The basic mathematical tool is the equation of

‘‘geodesic deviation,’’ namely the ‘‘Jacobi equation’’:

¼ R

abc

where D describes ‘‘propagation derivative’’

D ¼ t

along a timelike geodesic , where t

is a unit

timelike tangent vector to  (so t

= g

= 1)

which is (consequently) parallel-propagated along ,

¼ 0

(When acting on a scalar quantity defined along ,

we can read ‘‘D’’ as ‘‘d=d,’’ where  measures

proper time along .) The vector u

is what is called

a connecting vector between the geodesic  and

some ‘‘neighboring geodesic’’ 

. We think of the

vector u

as ‘‘connecting’’ a point p on  to some

neighboring point p

on 

, where it is usual to take

to be orthogo nal to t

(i.e., u

= 0). The

derivative Du

measures the rate of change of u

as p and p

move together into the future along .

Mathematically, we express this as the vanishing of

the Lie derivative of u

with respect to t

(with t

extended to a unit vector field which is tangent both

to  and to 

). By taking three independent vectors u

at p, we can form a spatial 3-volume element W and

investigate how this propagates along . We find

W ¼ WR

where the Ricci tensor R

(= R

) is here defined by

¼ R

acb

The Einstein Field Equations

In view of what has been said above, with regard to

the way that the acceleration of volume behaves in

Newtonian theory, it would be natural to ‘‘iden-

tify’’ R

with (4G )the(activegravita-

tional) mass density, with respect to the time

direction t

. In (special) relativity theory we expect

to ident ify mass density with c

2

 energy density

(by E = mc

) and to take energy density as just one

component (the time–time component) of a sym-

metric tensor T

, called the ‘‘energy tensor,’’ and

for s implicity we now take c = 1. The tensor

quantity T

is to incorporate the contributions to

the local mass/energy density of all particles and

fields other than gravity itself. Since we would

require this to work for all choices of time-

direction t

, it would be n atural, accordingly, to

make the identification

¼4GT

Indeed, this was Einstein’s initial choice for a

gravitational field equation. However, this will

actually not do, as Einstein later realized. The

trouble comes from the Bianchi identity

bcde

þr

cade

þr

abde

¼ 0

from which we deduce





¼ 0

where

R ¼ R

This causes trouble in connection with the standard

requirement on the energy tensor, that it satisfy the

local ‘‘conservation law’’

¼ 0

The latter equation is an essential requirement in special

relativity, since it expresses the conservation of energy

and momentum for fields in flat spacetime. In standard

Minkowski coordinates, each of T

, T

satisfies an equation just like the r

= 0ofthe

charge–current vector J

of Maxwell’s theory of

electromagnetism, with now r

= @

= @=@x

,which

expresses global conservation of charge. Similarly to the

way that J

encapsulates density and flux of electric

charge, T

encapsulates density and flux of energy, and

, T

encapsulate the same for the three

components of momentum. So the equation

= 0 is essential in special relativity, for similarly

expressing global conservation of energy and momen-

tum. We find (referring to a local inertial frame) that,

when we pass to general relativity, this equation should

still hold, with r

now standing for covariant

derivative. But the initially proposed field equation

= 4GT

would now give us r

= 0, which

combined with the geometrically necessary r



(

)Rg

) = 0, tells us that R is constant. In turn this

implies the physically unacceptable requirement that

T = T

is constant (since we have R = 4GT).

492 General Relativity: Overview

Einstein eventually became convinced (by 1915)

of the modified field equations



¼8GT

(the ‘‘8’’ rather than ‘‘4’’ being now needed to fit in

with the New tonian limit) and it is these that are

now commonly referred to as ‘‘Einstein’s field

equations.’’ (Some authors prefer to use the singular

form ‘‘field equation,’’ especially if the formula is to

be read as an abstract-index expression rather than a

family of component equations, since the tensors

involved are really single entities.) It may be noted

that the formula can be rewritten as

¼8GT





from which we deduce that in Einstein’s theory the

source of gravity is not simply the mass (or equivalently

energy) density, but there is an additional contribution

from the pressure (momentum flux, i.e., space–space

components of T

). This can have significant implica-

tions for the instability of very large and massive stars in

highly relativistic regimes, where increases in pressure

can, paradoxically, actually increase the tendency for a

star to collapse, owing to its contribution to the

attractive effect of its gravity.

In 1917, Einstein put forward a slight modifica-

tion of his field equations – basically the only

modification that can be made without fundamen-

tally changing the foundations of his theory – by

introducing the very tiny cosmological constant  .

The modified equations are



þ g

¼8GT

and the source of gravity, or active gravitational

mass is now

 þ P

þ P





4G

where (with respect to a local Lorentzian orthonormal

frame, units being chosen so that c = 1)  = T

is the

mass/energy density and P

= T

, P

= T

, P

= T

are the principal pressures. The  -term, for positive ,

provides a repulsive contribution to the gravitational

effect, but it is extremely tiny (and totally ignorable) on

all ordinary scales, beginning to show itself only at the

most vast of observed cosmological distances (since the

effect of  adds up relentlessly at larger and larger

distances). Einstein originally introduced the term in

order to have the possibility of a static universe, where

the attractive gravitational effect of the totality of

ordinary matter would be balanced, overall, by .But

thediscoveryoftheexpansionoftheuniverse(by

Hubbleandothers)ledEinsteintoabandonthe

cosmological term. However, since 1998 (initially

from the supernova observations of Brian Schmidt and

Robert Kirschner, and Saul Perlmutter, see Perlmutter

et al. (1998)), cosmological evidence has mounted in

favor of the presence of a very small positive  -term,

which has resulted in the expansion of the universe

beginning to accelerate. While the presence of Einstein’s

constant -term is consistent with observations, and

remains the simplest explanation of this observed

acceleration, many cosmologists prefer to allow for

what would amount to a ‘‘varying ,’’ and refer to it as

‘‘dark energy.’’

Energy Conservation and Related Matters

One of the features of Einstein’s general relativity

theory that had been deeply puzzling to a good many of

Einstein’s contemporaries, and which may be said to be

still not fully resolved, even today, is ‘ ‘energy conserva-

tion,’ ’ in the presence of a dynamical gravitational field.

We have noted that the energy tensor T

is to

incorporate the contributions of all particles and fields

other than gravity. But what about gravity itself? There

are many physical situations in which energy can be

transferred back and forth from gravitational systems

to nongravitational ones (most strikingly in the example

of the emission of gravitational waves; see Gravitational

Waves). The conservation of energy would make no

sense without an understanding of how energy can be

stored in a gravitational field. At first sight we seem to

see no role for a gravitational contribution to energy in

Einstein’s theory, since the conservation law r

= 0

seems to be a self-contained expression of energy

conservation with no direct contribution from the

gravitational field in the tensor T

.However,thisis

illusory, since the formulation of a global conservation

law from the local covariant expression r

= 0does

notworkincurvedspacetime(basicallybecause,unlike

the charge–current quantity J

of Maxwell’s electro-

dynamical theory, the extra index on T

prevents it

from being regarded as a 1-form). We may take the view

that the energy of gravitation enters nonlocally into the

equation, so that the failure of T

to provide a global

conservation law on its own is an expression of the

gravitational contributions of energy not being taken

into account. This is no doubt a correct attitude to take,

but it is a difficult one to express comprehensively in a

mathematical form. Einstein himself provided a partial

understanding, but at the expense of introducing

concepts known as ‘‘pseudo tensors’’ whose meaning

was too tied up with arbitrary choices of coordinate

systems to provide an overall picture. In modern

approaches, the most clear-cut results come from the

study of asymptotically flat or asymptotically de Sitter

spacetimes (de Sitter space being the empty universe

which takes over the role of Minkowski space when

General Relativity: Overview 493

there is a positive cosmological constant ; see

Cosmology: Mathematical Aspects).

The important role of the ‘‘Weyl conformal tensor’’

abcd

¼R

abcd



ðR

 R

þ R

 R

Rðg

 g

should also be pointed out. This tensor retains all

the symmetries of the full Riemann tensor, but has

the Ricci tensor contribution removed, so that all its

contractions vanish, as is exemplified by

abc

¼ 0

It describes the conformal part of the curvature, that

is, that part that survives under conformal rescalings

of the metric;

7!

where  is a smooth (positive) function of position. The

tensor C

abc

is itself invariant under these conformal

rescalings. This has importance in the asymptotic

analysis of gravitational fields (see Asymptotic Struc-

ture and Conformal Infinity). We may take the view

that C

abcd

describes the degrees of freedom in the free

gravitational field, whereas R

contains the informa-

tion of the sources of gravity. This is analogous to the

Maxwell tensor F

describing the degrees of freedom

in the free electromagnetic field, whereas J

contains

the information of the sources of electromagnetism.

From the observational point of view, general

relativity stands in excellent shape, with full agreement

with all known relevant data, starting with the

anomalous perihelion advance of the planet Mercury

observed by LeVerrier in the mid-nineteenth century,

through clock-slowing, light-bending (lensing) and

time-delay effects, and the necessary corrections to

GPS positioning systems, to the precise orbiting of

double neutron-star systems, with energy loss due to the

emission of gravitational waves. The effects of gravita-

tional lensing now play vital roles in modern cosmology.

To get some idea of the precision in Einstein’s theory, we

may take note of the fact that the double neutron-star

system PSR 1913þ16 has been observed for some

30 years, and the agreement between observation and

theory overall is to about one part in 10

See also: Asymptotic Structure and Conformal Infinity;

Canonical General Relativity; Computational Methods in

General Relativity: the Theory; Cosmology: Mathematical

Aspects; Einstein Equations: Exact Solutions; Einstein

Equations: Initial Value Formulation; Einstein–Cartan

Theory; Einstein’s Equations with Matter; General

Relativity: Experimental Tests; Geometric Flows and the

Penrose Inequality; Gravitational Lensing; Gravitational

Waves; Hamiltonian Reduction of Einstein’s Equations;

Lorentzian Geometry; Newtonian Limit of General

Relativity; Noncommutative geometry and the Standard

Model; Spacetime Topology, Causal Structure and

Singularities; Spinors and Spin Coefficients; Symmetries

and Conservation Laws; Twistor Theory: Some

Applications [in Integrable Systems, Complex Geometry

and String Theory].