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OCR for page 61
8
Gravitation Theory:
Highlights
NEUTRON STARS
The discovery of radio pulsars in 1967, and their subsequent identi-
fication as rotating neutron stars, exhibited for the first time objects in
the universe with gravity so strong that the effects of general relativity
must be important in their structure. Subsequently, neutron stars were
also discovered in x-ray binary systems, in which gas from a normal
star is accreting onto the neutron star and releasing gravitational
binding energy as x rays. The discovery of the binary radio pulsar PSR
1913 + 16 has provided an impressive example of relativistic ejects in
its orbital motion (see earlier sections on Perihelion Advance,
Einstein's Only Handle in Chapter 2 and on Binary Pulsar in Chapter
5~. However, it has been difficult to discover direct evidence for
general relativistic effects in observations of neutron stars themselves.
One likely case is the observation of a spectral line in hard x-ray
observations of gamma-ray bursters, which can be understood as the
positron annihilation line emitted at the surface of a neutron star,
redshifted about 10 percent by the general relativistic gravitational
redshift.
The theory of stellar structure and stellar pulsation, which was
originally developed for normal stars, has now been successfully
extended to neutron stars, taking full account of general relativity. The
most important new effects are the emission of gravitational waves and
61
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62 GRA VITA TION
the existence of relativistic instabilities (not present in Newtonian
theory), which limit the physical range of stellar possibilities. Detailed
results are available for the frequencies and damping rates for nonra-
dial oscillations of neutron stars. These oscillations generate gravita-
tional-wave emission and might be excited during the birth of a neutron
star in stellar collapse. Surprising new effects have been uncovered for
gravitational-wave emission by rotating stars. A general theorem has
been proved that says roughly that all perfect fluid, rotating stars are
unstable to the emission of gravitational radiation via a secular
instability. The inclusion of viscosity, always present to some degree in
nature, allows the instability to exist only above some critical threshold
in stellar rotation rate. The recent discovery of a pulsar with a rotational
period of only 1.5 ms has shown the relevance of these results, and it
now seems quite possible that there exists a class of fast pulsars with
rotational rates at or near the instability threshold, which could be
sources of periodic gravitational radiation.
GRAVITATIONAL COLLAPSE AND BLACK HOLES
There is a maximum mass limit for neutron stars. The exact limit
depends on the equation of state for nuclear matter, which is not well
known. Nevertheless, the upper mass limit is certainly less than about
5 solar masses and seems likely to be in the range of 1.5-2.5 solar
masses. Any stellar core with a mass exceeding the upper limit that
undergoes gravitational collapse must collapse to indefinitely high
central density to form a singularity. It is generally believed among
theorists that in such circumstances a black hole will always form so
that the final singularity will be hidden from external observers. A
black hole is a region of space-time where the gravitational field is so
strong that not even light can escape. Inside a black hole, there exists
a space-time singularity, a place where the space-time curvature
becomes infinite and all the known laws of physics may break down.
The hypothesis that space-time singularities always remain hidden
from observers is called the cosmic censorship hypothesis. It remains
unproved. A singularity that is visible to external observers, in vio-
lation of the hypothesis, would be naked singularity. Naked singulari-
ties have been the object of a considerable amount of study and
speculation, but most relativists believe that they do not exist, with the
exception of the big bang itself.
The evidence for existence of black holes is impressive but not
conclusive. At least one binary stellar x-ray source, Cygnus X-1, has a
mass greater than the upper mass limit for neutron stars and must be of
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GRA VITA TION THEOR Y.: HIGHLIGHTS 63
compact size as evidenced by millisecond variability of its emission.
Alternative models are possible but implausible. Other x-ray sources
may well contain black holes; a strong possibility is LMC-X3 in the
Large Magellanic Cloud.
Supermassive black holes of a million to a billion solar masses might
be able to form in the nuclei of galaxies by stellar coalescence and
accretion. Direct evidence for such black holes remains weak. A1-
though some galactic nuclei are found to possess a large accumulation
of dark matter at the core, this mass has not been shown to be so
compact that it must be a black hole. A popular model of quasars and
active galactic nuclei postulates the existence of a supermassive black
hole undergoing accretion of surrounding matter, with enormous
amounts of gravitational energy released in thermal and nonthermal
radiation coming from an accretion disk or a chaotic accretion region
around the black hole. Accretion may produce electrodynamic effects
or jets of outflowing matter.
A black hole may be born in a more or less excited state, depending
on the degree of disorder in the gravitational collapse of its progenitor
star, but it quickly relaxes to a stationary state by the emission of
gravitational waves. The stationary states of black holes are remark-
ably simple, according to the uniqueness theorems, which state that a
stationary black hole must belong to the three-parameter family of
black-hole solutions of the Einstein equations, the Schwarzschild-
Kerr-Newman solutions. The three parameters are the mass, the total
angular momentum, and the electric charge. The mass, angular mo-
mentum, and charge of a black hole cannot disappear because these
quantities are conserved charges coupled to long-range fields.
In an astrophysical environment, any electric charge on a black hole
will be quickly and almost completely neutralized through the conduc-
tivity of the surrounding plasma. Since the electromagnetic interaction
is so much stronger than gravity, it only takes a tiny amount of charged
plasma to neutralize even a maximally charged black hole. On the other
hand, the angular momentum of a black hole will persist, and a black
hole may remain rotating for hundreds of millions of years or more.
A rotating black hole has free energy that can be tapped externally.
The energy can be tapped by immersing the black hole in a suitable
configuration of conductors and magnetic fields, in which case it acts as
a kind of electrical generator, or it can be tapped mechanically by
suitable arrangements of particles traversing a certain region near the
black hole, called the ergosphere. When all the free energy is removed
from a rotating black hole, it is reduced to a nonrotating state.
The energetics of black holes are governed by remarkably simple
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64 GRA VI TA TI ON
laws, the four laws of black-hole dynamics, which in essence are the
four laws of thermodynamics as applied to black holes. The role of
entropy is played by a purely geometric quantity, the surface area of
the black hole. The area theorem states that, in classical physics, the
surface area of a black hole never decreases; this theorem is known as
the second law of black-hole dynamics from its parallelism to the
Second Law of Thermodynamics, which asserts that entropy never
decreases for an isolated system.
Small disturbances of stationary black holes, for instance due to
small particles falling in or to impinging electromagnetic or gravita-
tional waves, can be worked out in linear perturbation theory. This
theory reduces to the solution of certain remarkably simple wave
equations and is in essence complete.
QUANTUM PARTICLE CREATION BY BLACK HOLES
A major advance in fundamental understanding of the laws of
physics was achieved in the discovery that, when quantum effects are
considered, a black hole emits quanta of radiation just as if it were a
blackbody at a finite temperature. The temperature is inversely pro-
portional to the mass of the black hole. The temperature of a black hole
is its key feature that makes possible the identification of the laws of
black-hole dynamics with the laws of thermodynamics. The emission
of quanta by black holes, known as the Hawking process, causes black
holes to become gradually smaller and finally to decay away entirely.
This effect seems unobservable for stellar mass black holes, for which
the temperature is less than a microkelvin and whose lifetime is greater
than 1070 years. On the other hand, if small black holes, with a mass of
about 10'5 g, were created in the big bang, they could be observable
today. Their lifetime would be roughly 20 billion years, the age of the
universe, and their temperatures would become high just before their
final decay, so that they would emit a burst of hard electromagnetic
radiation. Such bursts have not been found to date.
QUANTUM EFFECTS IN THE EARLY UNIVERSE
Gravity becomes comparable in strength with the other fundamental
forces of nature only at the Planck energy, about 10'9 GeV. The only
known places in the present universe where energies reach this level
are at space-time singularities. Those inside black holes are thought to
be invisible to us, according to the cosmic censorship hypothesis. The
initial singularity of the big bang is in principle observable to us,
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GRA VI TA TION THEOR Y.: HIGHLIGHTS 65
although it is shrouded in the hot dense matter of the primeval fireball.
The ejects of quantum gravity, imprinted on the universe at times so
early that the temperature exceeded the Planck temperature, could
have affected the present universe in important ways. An important
effect could have been the damping of initial anisotropies, by quantum
particle creation, to leave the almost perfectly isotropic universe that
we see today. On the other hand, residual anisotropies in the cosmic
background radiation could have been created by quantum effects at
somewhat later times, for instance during an inflationary era in
cosmology (see the section on The Inflationary Universe in Chapter
121.
ALTERNATIVE THEORIES
The tremendous advances in experimental tests of relativity have
changed the theoretical scene greatly in the last decade. Theories that
were viable and indeed admirable have now been stringently con-
strained or even ruled out by solar-system tests and by observations of
the binary pulsar PSR 1913+16. This progress has increased the
confidence of most gravitation theorists that general relativity is indeed
the correct classical theory of gravity, at least in the long-distance,
low-energy domain, despite the fact that many of its most important
effects, such as detection of gravitational radiation and magnetic
gravity, remain to be demonstrated. Thus, although some work con-
tinues on alternative theories, most ongoing theoretical work is based
on general relativity.
EXACT SOLUTIONS OF THE EINSTEIN EQUATIONS
The Einstein equations are a nonlinear set of coupled partial differ-
ential equations, and their complete solution is unknown. The discov-
ery of exact particular solutions has played an important role in the
progress of relativity; for instance, the Kerr solution, which is now
known to be the unique solution for a rotating, uncharged, stationary
black hole, was first found in a systematic search for certain exact
solutions known as algebraically special.
Great progress has been made on solution of the Einstein equations
in the more general case of a stationary, axisymmetric, vacuum
space-time, which is now known to be completely soluble in principle.
Soliton methods from mathematical physics have also been applied to
this problem.
There has even been reason to hope for the complete and general
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66 GRAVITATION
solution of the Einstein equations. A set of ideas called twister theory
has been developed in a new approach to the issues both of classical
and of quantum general relativity. Twistor theory has close connec-
tions to modern mathematics, specifically to algebraic topology and
algebraic geometry. Twistor theory has already produced new exact
solutions for non-Abelian gauge theories in field theory (some of the
instanton solutions) and has also produced large new classes of
complex valued solutions to the Einstein equations. There has been
progress toward a general solution by twister techniques, though as Yet
~ _ _ ~ _ ~ _ ~ _ ,
it has not been achieved.
The initial-value problem for the Einstein evolution equations is
itself a deep problem, on which good progress has been made. Known
exact solutions for the initial-value constraint equations are few, but
constructive methods are now available that give, in principle, the
general solution of the Cauchy (spacelike) initial-value problem from
freely specifiable initial data for the gravitational field and matter fields.
Characteristic (lightlike) initial-value surfaces are likewise often useful,
especially in the study of gravitational radiation.
ASYMPTOTIC PROPERTIES OF SPACE-TIME
An isolated system in general relativity is represented by a space-
time that becomes asymptotically flat (Minkowskian) at infinity. Phys-
ics should become simple at infinity; and mass, angular momentum,
and gravitational waves should become easily measurable there. How-
ever, the nonlinearities in the Einstein equations make the study of
— 7
infinity a subtle one.
In a general, asymptotically flat space-time in which gravitational
waves are propagating toward infinity, the Riemann curvature tensor
falls off only as 1/r in the directions (called null infinity) in which both
t and r become large. This slow falloff of the curvature causes many
difficulties in principle for the measurement of the properties of isolated
systems by distant observers, for instance in the definition of angular
momentum. It has been found that if one generalizes the space-time
manifold into a four-complex-dimensional manifold by allowing the
four space-time coordinates to take complex values instead of just real
ones, then a reference system of remarkable simplicity exists at
infinity. In it the most troublesome asymptotic terms in the geometry
vanish. The new complex space that arises at null infinity is called
H-space or the nonlinear graviton. Asymptotic properties of space-
time at spacelike infinity (t fixed, r large) also reveal subtleties. The
correct definition for the angular momentum of an isolated system as
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GRA VI TA TION THEOR Y.: HIGHLIGHTS 67
measured at spacelike infinity is a problem that has only recently been
resolved.
NUMERICAL RELATIVITY
In the absence of analytic techniques for the general solution of the
Einstein equations, relativists have turned to large-scale numerical
techniques to solve important problems, such as the collapse of stellar
cores or the collision of black holes. The inclusion of general relativity
in spherically symmetric computations of stellar collapse is now
routinely done when necessary. Nonspherical systems, which unlike
spherical ones admit gravitational radiation, are much more difficult to
simulate numerically and require both state-of-the-art numerical tech-
niques and the largest computers. These computations also require
state-of-the-art theoretical analyses of the Cauchy and characteristic
initial-value problems of general relativity. Finally, great care is needed
for the numerical treatment of hydrodynamics in these simulations.
The most ambitious numerical calculation carried out to date in pure
general relativity, without any matter present, is the head-on collision
of two identical nonrotating black holes. The numerical results show
that the two holes coalesce to form a single one, and gravitational
waves amounting to about a part in 103 of the total rest mass of the
system are radiated to infinity.
EMISSION OF GRAVITATIONAL RADIATION
Inspired by experiments to detect gravitational radiation, investiga-
tors have studied many source models. The calculations carried out
include perturbation studies of gravitational collapse and black holes,
approximate models of collapsing cores and of colliding neutron stars,
and full-scale numerical calculations of gravitational collapse, colliding
neutron stars, and colliding black holes (see Figure 8.11. As noted in
Chapter 2, the results of such calculations are essential for making the
important estimates of the strengths and frequencies of gravitational
waves near the Earth (Figures 6.2 and 6.3 in Chapter 61.
Doubts were raised about the validity of the quadrupole formula for
gravitational-wave luminosity and radiation reaction of weak-field,
slow-motion systems. Careful investigation of this formula by tech-
niques of applied mathematics have strongly reinforced the belief in its
validity. The experimental confirmation of the prediction of this
formula for the binary pulsar PSR 1913+16 has emphasized its
importance.
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68 GRAVITATION
A__ ~ _.:
- ~~) - ,~., - ~:.~
FIGURE 8.1 Numerical simulation of gravitational radiation from two colliding black
holes of equal mass. The right axis is the axis of symmetry for the collision, and the left
axis lies in the equator. Wave amplitude is plotted upward. This is the outgoing wave at
a time of about t = 37 (mass) after the collision.
THE POSITIVE ENERGY THEOREM
Gravitational binding energy is negative, because gravity is an
attractive force. When a body of given mass becomes so compact that
the effects of general relativity become significant for its structure, the
binding energy becomes comparable with the total rest energy of the
matter making up the body. The possibility thus arises that the total
energy of the body could become negative, should the binding energy
actually dominate. It was conjectured 20 years ago that the total energy
of a body could never become negative in the General Theory of
Relativity. Heuristically one expects that any body attempting to
violate this condition would lose stability and collapse to form a black
hole before its total energy could become negative. A general form of
this conjecture was finally proved in 1979 by two mathematicians using
sophisticated arguments from differential geometry, and several gen-
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GRA VITA TION THEOR Y.: HIGHLIGHTS 69
eral forms of this positive energy theorem have now been proved.
Mathematicians were attracted to the problem after relativists publi-
cized the importance and apparent difficulty of the conjecture.
A quite different and more direct proof of the positive energy
theorem was given in 1981 by a particle physicist, using an argument
motivated by supergravity theories (see section below on quantum
gravity). Two relativists had earlier shown that the Hamiltonian of
supergravity the expectation value of which is the total energy is
formally nonnegative because it is a sum of perfect squares of certain
fermionic charges. When this formal argument is made concrete, it
indeed yields a rigorous proof of positive energy in general relativity.
QUANTUM FIELD THEORY IN CURVED SPACE-TIME
The discovery of the Hawking process by which black holes radiate
particles quantum mechanically led to extensive development of the
theory of quantum-matter fields in curved background space-time. A
deeper understanding of the Hawking process was achieved together
with a compelling and suggestive unification of the laws of black-hole
mechanics and the laws of thermodynamics. The theory served as a
laboratory in which ideas eventually to be important in a quantum
theory of gravity could be tested in a simpler situation. Many concep-
tually interesting and unanticipated ideas emerged. The reaction of a
moving particle detector to a curved space-time, the possibility of CPT
nonconservation in quantum gravity, and the possibility of quantum-
mechanical evolution from pure to mixed states are three examples.
QUANTUM GRAVITY
The last decade has seen a remarkable growth in the theoretical
effort devoted to the construction of a quantum theory of gravity. The
unification of gravity and quantum physics had always been under-
stood to be a fundamental question. The activity of the past decade was
much stimulated by new techniques arising from gauge theories that
could be applied to answer new questions in quantum gravity and to the
ever more active search in particle physics for a unified theory of all
interactions, which must at the end include gravity.
The standard approach to field theory in the 1950s and 1960s was
through the perturbation theory for scattering amplitudes. This is not
always a sufficient tool in non-Abelian gauge theories such as quantum
chromodynamics, nor will it suffice for gravity. On the one hand,
gravitational scattering processes are too weak for observation. On the
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70 GRA VITATION
other, the perturbation theory for these processes has divergences too
strong to be controlled by renormalization. New techniques or new
ideas were necessary, and they emerged through a fruitful exchange
with particle theory ("ghosts" for example, originated in studies of
quantum gravity). Euclidean functional integrals, successful in other
areas of field theory, were applied to formulate a quantum theory of
gravity based on the Lagrangian of general relativity. When a
Euclidean formulation is applied to field theories of flat space-time, it
is just a different technique; however, when it is applied to gravity it
yields a different quantum theory. Further, it yields the theory in a way
in which it can be approximated semiclassically in regimes far from the
domain of validity of perturbation theory. New questions could thus be
asked, and novel results emerged. For example, in this theory pure
states can evolve into mixed states in striking contrast to the usual
situation in quantum mechanics.
There was also progress in the more traditional canonical approach
to quantum gravity. Functional integral techniques clarified some of
this approach's central problems, and promising new formulations of
the canonical framework were worked out. The gravitational measure
in the path integral, the existence of trace anomalies for the stress
tensor, and solutions describing topological nontrivial configurations
are just some examples. More recently, non-Abelian anomalies and the
quantum breaking of coordinate invariance provide other striking
illustrations involving gravity, gauge theories, and recent mathematics.
Relativists have tended to interpret quantum gravity in terms of the
quantum version of Einstein's theory. General relativity works well in
the classical long-range limit. It has also been shown to be the unique
theory of gravity in this limit, on the basis of a few observational facts
taken together with the properties of special relativistic quantum
mechanics. It is not self-evident, however, that it is correct on the scales
of 10-33 cm (10~9 GeV) that characterize strong quantum gravitational
phenomena. The 1 970s and 1980s therefore saw the investigation of new
theories that were generalizations of Einstein's theory and also some
radically different approaches. The twin motivations for these new
initiatives were the hopes that a new theory might be more tractable at
small distances than Einstein's theory seems to be, and the need for a
new theory to realize the goal of the unification of all interactions.
The developments of the past decade have seen a dramatic increase
in the diversity of approaches to a quantum theory of gravity. Clearly,
at present, a variety of approaches offers the best hope for a solution
to this fundamental problem. One cannot help but be excited and
impressed by the beauty and potential of these ideas.
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GRA VITA TION THEOR Y.: HIGHLIGHTS 71
One of the most significant developments of the past decade has been
the emerging close relationship between particle physics and gravita-
tion physics on this fundamental frontier. The search by particle
physics for a unified theory has led to the problems of gravitational
physics, and the search for a quantum gravity has led gravitational
physics to field theory. Goals, techniques, and to some extent people
are now shared between the cutting edges of these two areas.
Supergravity, induced gravity, higher derivative Lagrangians, twis-
tor theory, geometric quantization, discrete gravity, Kaluza-Klein
theories, and string and superstring theories are just some of the
headings under which new theories of quantum gravity might be
grouped. It would be inappropriate to review them all here. Each has
its promise and successes, but none has succeeded. We shall mention
just two approaches that are currently under intense study by particle
and gravitation physicists.
Supergravity
Symmetries between fields of different spins and different statistics
are the basis of supergravity theories, which promote this symmetry
into a local gauge invariance. The gravitational field is symmetrically
related to a larger collection of fields that describe all particles and all
interactions. Supersymmetric theories have a number of remarkable
properties, such as a less rapidly divergent perturbation theory than
ordinary gravity. Despite the absence of immediate direct experimental
tests (a situation that is rapidly improving, however), they have
captured the imagination of many theorists as one of the few viable
avenues leading toward a unification of the forces of nature.
Kaluza-Klein Theories
There appear to be only four dimensions to space-time, but in the
framework of Kaluza-Klein theories appearances are deceiving. These
generalizations of Einstein's theory envisage a world of many (e.g., ten
or eleven) dimensions in which all but four are curled up so as to be
unnoticeable on our macroscopic scales. In such theories the matter
degrees of freedom are space-time degrees of freedom in the extra
dimensions. Kaluza-Klein theories offer the hope of a purely geometric
unification of gravity with other matter interactions and perhaps even
the explanation of the four-dimensional character of our physical
world.
Representative terms from entire chapter:
black holes
