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Ideas and Phenomena of
General Relativity
Not everyone who reads this report will be familiar with the beautiful and
simple ideas that underlie Einstein's general relativity or with the vast range of
phenomena for which gravitational physics is important. In this chapter the
Committee on Gravitational Physics briefly describes some key ideas in general
relativity and some key phenomena of gravitational physics that are discussed in
Chapter 3.
A. .~.~Y .~.~./hS IN ~ EN [~ ~./h.~. ~ EL.~,4.~.! \71 r.~\r
Gravity Is Geometry. Gravity is the geometry of four-dimensional spacetime.
That is the central idea of Einstein's 1915 general theory of relativity the clas-
sical theory of relativistic gravitation. It is not difficult to imagine a curved
space. The curved surface of a sphere or a car fender are two-dimensional
examples. But gravitational effects arise from the curvature offour-dimensional
spacetime with three space dimensions and one time dimension. It is more
difficult to imagine a notion of curvature involving time, but the Global Position-
ing System (described in Box 2.1) provides an everyday practical example of its
implications.
In Newtonian physics two identically constructed clocks run at the same rate
no matter what their positions in space. But in relativity a stationary clock above
Earth's surface runs fast compared to a clock at the surface by 1 part in ten
thousand billion for each kilometer in height. That tiny difference is the result of
the curvature of spacetime produced by the mass of Earth a small effect indeed,
but large enough that the Global Positioning System would fail in a few minutes
24
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IDEAS AND PHENOMENA OF GENERAL RELATIVITY
25
BOX 2.1 General Relativity and Daily Life
There is no better illustration of the unpredictable application of fundamental
science in daily life than the story of general relativity and the Global Positioning
System (GPS). Built at a cost of more than $10 billion mainly for military naviga-
tion, the GPS has been rapidly transformed into a thriving, multibillion-dollar com-
mercial industry. GPS is based on an array of 24 Earth-orbiting satellites, each
carrying a precise atomic clock. With a hand-held GPS receiver that detects radio
emissions from any of the satellites that happen to be overhead, a user can deter-
mine latitude, longitude, and altitude to an accuracy that currently can reach 50
feet, and local time to 50 billionths of a second. Apart from the obvious military
uses, the GPS is finding applications in airplane navigation, wilderness recreation,
sailing, and interstate trucking. Even Hollywood has met the GPS, pitting James
Bond in "Tomorrow Never Dies" against an evil genius able to insert deliberate
errors into the system and send British ships into harm's way.
Because the satellite clocks are moving in high-speed orbits and are far from
Earth, they tick at different rates than clocks on the ground. Gravity and speed
contribute comparable amounts to the total discrepancy. The offset is so large
that, if left uncompensated, it would lead to navigational errors that would accumu-
late at a rate greater than 6 miles per day. In GPS, the relativity is accounted for
by electronic adjustments to the rates of the satellite clocks, and by mathematical
corrections built into the computer chips that solve for the user's location.
Schematic illustration of segments used in operation of the Global Positioning System. (Adapted
from a figure courtesy of the Aerospace Corporation.)
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26 GRAVITATIONS PHYSICS: E~LOHNG THE STRUT OF SPACE ^D TIME
BOX 2.2 Newtonian and Einstein Gravity Compared
In Newton's 300-year-old theory of gravity, a mass attracts other masses with a
force of gravity that decreases as the inverse of the square of the distance be-
tween them. Masses move in response to the forces acting on them, including
gravitational forces, according to Newton's laws of motion.
In Einstein's 1915 general theory of relativity, a mass curves the one time di-
mension and three space dimensions of spacetime according to Einstein's equa-
tion. The spacetime curvature is greatest near the mass and vanishes at a dis-
tance. Other masses move along the straightest possible paths in this curved
spacetime. Einstein's theory thus expresses both the gravitational effect of mass
and the response of mass to that effect in terms of the geometry ofspacetime. The
Newtonian idea of a gravitational force acting at a distance between bodies was
replaced by the idea of a body moving in response to the curvature of spacetime.
In relativity, mass and energy are the same thing according to Einstein's fa-
mous E= mc2 relation. Not only mass but also any form of energy will curve
spacetime. Gravity itself carries energy, and even small propagating ripples in
spacetime cause further curvature. The equations of Einstein's theory keep track
of this complex feedback interrelationship between energy and curvature.
Newton's theory of gravity is not wrong. It is a correct approximation to Ein-
stein's theory when spacetime curvature is small and the velocities of masses are
much smaller than the velocity of light. The first general relativistic corrections
beyond Newtonian theory (called "post-Newtonian") are responsible for small devi-
ations to the motion of light and to the orbits of the planets from those predicted by
Newton. Measurements of these deviations are among the most precise tests of
general relativity.
The founders of gravitational physics Isaac Newton (1642-1727) and Albert Einstein (1879-
1955). (Courtesy of the American Institute of Physics Emilio Segre Visual Archives.)
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IDEAS AND PHENOMENA OF GENERAL RELATIVITY
27
if this effect of the spacetime curvature implied by general relativity were not
taken into account.
Mass Produces Spacetime Curvature, and Spacetime Curvature Determines
the Motion of Mass. Einstein's equation makes a quantitative connection be-
tween mass and the amount of curvature of spacetime it produces. (See Box 2.2.)
Just as Earth curves spacetime near its surface, so too does the Sun produce a
slight curvature of spacetime in its vicinity. The curvatures produced near the
surface of a black hole or a neutron star, or at the beginning of the universe, are
much greater. These are realms of strong gravitational physics. According to
general relativity, Earth follows an elliptical orbit about the Sun, not because it is
attracted to the Sun by a gravitational force, but because it is following the
straightest possible path through the spacetime that has been curved by the Sun.
The Principle of Equivalence. General relativity predicts that a tiny asteroid,
or indeed any other body, could follow the same path around the Sun as Earth
does. Each body is following a path determined by the geometry of spacetime,
not by its mass. This universality of free fall called the principle of equiva-
lence is one of the foundations of general relativity. It is one of the most
accurately tested predictions in all of physics. The equality of accelerations of
different bodies in the curved spacetime of the Sun has been verified to a few
parts in a thousand billion. Were a violation of this equality ever detected it
would signal either new physical interactions or a revision in our ideas about the
nature of space, time, and gravity.
Described below are some important phenomena in gravitational physics.
Strong gravitational physics plays a central role in all these examples. The
essential features of general relativity are present, and the Newtonian approxima-
tion is inadequate.
Gravitational Waves. Einstein's theory predicts that ripples in spacetime
curvature can propagate with the speed of light through otherwise empty space-
a gravitational wave. Mass in motion is the source of a gravitational wave. In
turn, gravitational waves can be detected through the motion of masses produced
as the ripple in spacetime curvature passes by. The weak coupling of mass to
spacetime curvature means that an extraordinarily energetic, strong-gravity event,
such as the coalescence of two massive stars, is required to produce gravitational
waves copious enough to be detected by gravitational wave receivers now under
construction. By contrast, the indirect detection of gravitational waves from the
Hulse-Taylor binary pulsar system resulted from the observation of the minus-
cule shortening of the period of a pair of neutron stars orbiting about each other.
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28 GRAVITATIONS PHYSICS: E~LOHNG THE STRUT OF SPACE ^D TIME
This weak coupling of gravity to matter is the reason that gravitational waves
have not yet been detected directly. But this weak coupling also means that the
universe is largely transparent to gravitational waves. Once produced, little is
absorbed. A gravitational wave receiver could therefore enable researchers to see
phenomena in the universe that are visible in no other way.
Black Holes. Perhaps no other concept in physics has made as deep an
impact on public consciousness as has the black hole. General relativity predicts
that a black hole is created whenever mass is compressed into a volume small
enough that the gravitational pull at the surface is too large for anything to
escape, no matter how fast it accelerates. The surface of a black hole called its
event horizon is like a one-way membrane. Mass, information, and observers
can fall into it, but nothing can emerge from it. Although black holes in nature
are typically produced by complex gravitational collapse, such as gave rise to
binary x-ray sources or as occurred at the centers of galaxies, general relativity
predicts that they are remarkably simple objects completely characterized by just
a few parameters.
Black holes exhibit many properties of ordinary objects: they have mass and
spin and can have electric charge; they can oscillate, change shape, show tides,
and emit gravitational radiation; they can exhibit electric polarizability, resistiv-
ity, eddy currents, and threaded magnetic fields; they can act as generators and
engines for the most energetic phenomena in the universe. Yet all this richness of
physics is described cleanly by the Einstein equation coupled to ordinary matter.
As S. Chandrasekhar put it: "The black holes of nature are the most perfect
macroscopic objects there are in the universe: the only elements in their con-
struction are our concepts of space and time. And since the general theory of
relativity provides only a single unique family of solutions for their descriptions,
they are the simplest objects as well." (The Mathematical Theory of Black Holes,
Oxford University Press, New York, 1992, p. 1.)
According to quantum mechanics, black holes exhibit yet more remarkable
properties as participants in the second law of thermodynamics. They possess an
entropy proportional to their area whose statistical mechanical origin is begin-
ning to be understood. They radiate like blackbodies with a temperature in-
versely proportional to their mass. Thus, as they radiate they heat up and radiate
even more. They may radiate completely away, producing for a brief explosive
moment the strongest spacetime curvatures since the big bang.
The Universe and the Big Bang. Gravity governs the structure and evolution
of the universe on the largest scales of space and time. This is true even though
gravity is the weakest of the four fundamental forces. Gravitation is universal,
acts at long range, and cannot be canceled since it has no negative "charges."
Cosmology and gravitational physics are thus inextricably linked. From cosmo
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IDEAS AND PHENOMENA OF GENERAL RELATIVITY
29
logical observations and Einstein's theory, the future fate of the universe can be
extrapolated and its origins reconstructed.
Galaxies the basic building blocks of the present universe are distributed
uniformly on the largest distance scales. They recede from one another in a way
that shows that the universe is expanding. Observations show that the universe
was even simpler earlier that it is now. The origin of the universe was an initial
state of extremely high density, pressure, and spacetime curvature about 13 bil-
lion years ago the big bang. Although extreme in these measures, the big bang
was remarkably regular. It was an explosive event that happened everywhere in
space at the same time, producing matter in nearly perfect thermal equilibrium.
Such a uniform, expanding universe is described by solutions of Einstein's equa-
tion known as the Friedmann-Robertson-Walker (FRW) cosmological models.
They are characterized by a few cosmological parameters whose values are the
subject of ever refined observational searches. The simplest FRW models come
in two varieties: models in which space is closed like the surface of a sphere, and
models in which space is unlimited or open. The closed models end in a finite
time in a "big crunch," whereas the open models expand forever. The closed
models have a higher density than the open ones, and the dividing density be-
tween them is called the critical density to close the universe. The real universe
cannot exhibit exactly the perfect uniformity of the FRW models, since small
quantum fluctuations away from uniformity must have occurred. These tiny
"seeds" grew by the action of gravitational attraction to form the galaxies and
stars we see today.
Cosmic Backgrounds. Matter cooled as the universe expanded from its
initial hot beginning. Protons and neutrons formed in the first microsecond after
the start of the big bang; during the first few minutes they combined to form
primordial nuclei, chiefly hydrogen and helium. About 300,000 years after the
big bang, nuclei combined with electrons to make atoms. Once most of the
electrons combined into atoms, matter was cool enough to be transparent to light.
This light from the big bang has been propagating to us ever since. The subse-
quent expansion has cooled it to a temperature of only 2.73 degrees above abso-
lute zero, but it still comes toward us from every direction, forming the cosmic
background radiation. This light from the early universe is detectable by sensi-
tive instruments on the ground and in space, giving the most compelling evidence
for the big bang. Small variations of a few microdegrees that are observed in the
temperature of the cosmic background radiation are evidence for the slight initial
concentrations of density that grew to be the galaxies today. From the details of
these fluctuations their amplitude, angular distribution, and spectrum we can
learn a great deal about the universe. If we could observe the similar background
of gravitational waves, we could see back to the earliest moments of the big bang.
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30 GRAVITATIONS PHYSICS: E~LOHNG THE STRUT OF SPACE ED TIME
Binary Pulsars. Gravitational physics is central to some of the most dra-
matic and large-scale phenomena in nature. The big bang, black holes, explosive
gravitational collapse, quasars, pulsars, and x-ray sources are all examples. Yet
because gravity couples universally to all matter, its effects are in principle ob-
servable in any physical system. Just as remarkable, just as beautiful, and just as
confirming as the dramatic phenomena mentioned above are the minute, pre-
cisely observable predictions of relativistic gravity for the deviations of the paths
of orbiting bodies from the laws of Newton. These effects have been observed
with impressive accuracy in the solar system. They are observed even more
cleanly in binary neutron stars pairs of neutron stars orbiting about each other.
Neutron stars are extraordinarily compact somewhat more than the mass of the
Sun in a radius of 10 kilometers. Spacetime in their vicinity is more highly
curved than in any place in the universe other than the big bang and black holes,
and binary neutron stars are therefore among the best laboratories for precision
tests of general relativity. Their orbits can be observed when one of the neutron
stars is a pulsar a magnetized object whose rotation can be observed from the
radio waves it emits, received at Earth as precisely periodic signals. More than
1000 pulsars are known, and 50 are in binary systems with neutron star or white
dwarf companions. Many of them are extraordinarily accurate clocks. The
rotational (spin) period of the Hulse-Taylor binary pulsar PSR1913+16, for ex-
ample, is 0.059029997929613 _ 0.000000000000007 seconds. By noting the
changes in this period induced by the pulsar's orbital motion over decades, the
effects of general relativity can be precisely observed.
Singularities. Einstein's classical theory predicts the formation of a singu-
larity in the interior of a massive body whose gravity collapses it to a sufficiently
compact volume. A singularity is a region of the universe where a classical
description breaks down because it predicts infinite spacetime curvatures or den-
sities of matter. Singularities limit the predictive ability of classical general
relativity and are therefore places where we can expect to find new physics. But
there is considerable evidence that singularities produced in any realistic collapse
are hidden inside the event horizons of black holes where they cannot interfere
with the predictability of physics on the outside. The idea that this always
happens is called the cosmic censorship conjecture. Proving or disproving it is
one of the outstanding challenges of general relativity theory. The singularities
of gravitational collapse may be hidden inside black holes, but Einstein's theory
also predicts that the universe began in a singularity the big bang whose
consequences are all about us.
The Small-Scale Structure of Space and Time. The union of the two most
significant developments of 20th-century physics general relativity and quan-
tum theory is one of the greatest challenges of contemporary theoretical phys-
ics. The result of this union a quantum theory of gravity will have implica
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IDEAS AND PHENOMENA OF GENERAL RELATIVITY
3
lions as profound for our understanding of spacetime on small scales as Einstein's
theory did for that understanding on large scales. The frontier of small scales for
quantum gravitational phenomena is marked by the combination of the quantum
of action it, the velocity of light c, and Newton's gravitational constant G. called
the Planck length:
~(hG / 34~/2 10-33
The corresponding Planck energy is about ten thousand trillion times greater than
the energy reached by the world's largest accelerators. Yet these energies, with
their accompanying enormous curvatures, occurred at the big bang and occur in
the final stages of gravitational collapse. An understanding of spacetime (or
whatever replaces it) on these extreme scales is necessary to understand these
central phenomena of astrophysics. But the small-scale structure of spacetime is
also central to the quest for a unified theory of the fundamental interactions,
because it is only there, where gravity is as strong as any other force, that the full
symmetry between these interactions is likely to emerge.
There are currently two main approaches to constructing this new theory. In
one, Einstein' s picture of gravity as spacetime geometry is fundamental so that a
quantum theory of gravity brings with it a quantum theory of geometry. At the
Planck scale, quantum excitations of geometry are structured like a branched
polymer, and familiar quantities such as lengths and areas can assume only dis-
crete values. It is only because the basic discrete unit of length the Planck
length is so small that space can be approximated by a continuum under ordi-
nary circumstances.
The other approach to quantum gravity unifies gravity with all the other
forces and matter in a natural way. The basic idea is that although elementary
particles appear point-like, they are actually excitations of a one-dimensional
extended object called a string. One mode of oscillation of the string is a gravi-
ton, a quantum of gravity, while other modes are photons, electrons, quarks, and
so on. Furthermore, the interactions between these particles are reproduced by a
splitting and joining interaction between the strings. We thus obtain a compelling
and beautiful unified picture of particles and their interactions, including gravity.
Both of these approaches are still being developed, and the coming decade
promises a much more detailed picture of space and time at small scales.
Representative terms from entire chapter:
black holes
