General relativity (GR) is a
theory of
gravitation that
was developed by
Albert Einstein
between 1907 and 1915. According to general relativity, the
observed gravitational attraction between
masses results from the warping of
space and time by those masses.
Before the advent of general relativity,
Newton's law of universal
gravitation had been accepted for more than two hundred years
as a valid description of the gravitational force between masses.
Under Newton's model, gravity was the result of an attractive force
between massive objects. Although even Newton was bothered by the
unknown nature of that force, the basic framework was extremely
successful at describing motion.
However, experiments and observations show that Einstein's
description accounts for several effects that are unexplained by
Newton's law, such as minute anomalies in the
orbits of
Mercury and
other
planets. General relativity also
predicts novel effects of gravity, such as
gravitational waves,
gravitational lensing and an effect of
gravity on
time known as
gravitational time dilation.
Many of these predictions have been confirmed by experiment, while
others are the subject of ongoing research.
For example, although
there is indirect evidence for gravitational waves, direct evidence
of their existence is still being sought by several teams of
scientists in experiments such as the LIGO and GEO 600
projects.
General relativity has developed into an essential tool in modern
astrophysics. It provides the
foundation for the current understanding of
black holes, regions of space where
gravitational attraction is so strong that not even light can
escape. Their strong gravity is thought to be responsible for the
intense
radiation emitted by certain types
of astronomical objects (such as
active galactic nuclei or
microquasars). General relativity is also part
of the framework of the standard
Big Bang
model of
cosmology.
Although general relativity is not the only relativistic theory of
gravity, it is the simplest such theory that is consistent with the
experimental data. Nevertheless, a number of open questions remain:
the most fundamental is how general relativity can be reconciled
with the laws of
quantum physics to produce
a complete and self-consistent theory of
quantum gravity.
From special to general relativity
In September 1905, Albert Einstein published his theory of
special relativity, which reconciles
Newton's laws of motion with
electrodynamics (the interaction
between objects with
electric
charge). Special relativity introduced a new framework for all
of physics by proposing new concepts of
space
and
time. Some then-accepted physical theories
were inconsistent with that framework; a key example was Newton's
theory of
gravity, which describes the
mutual attraction experienced by bodies due to their
mass.
Several physicists, including Einstein, searched for a theory that
would reconcile Newton's law of gravity and special relativity.
Only Einstein's theory proved to be consistent with experiments and
observations. To understand the theory's basic ideas, it is
instructive to follow Einstein's thinking between 1907 and 1915,
from his simple thought experiment involving an observer in free
fall to his fully geometric theory of gravity.
Equivalence principle
A person in a
free-fall elevator
experiences weightlessness during their fall, and objects either
float alongside them or drift at constant speed. Since everything
in the elevator is falling together, no gravitational effect can be
observed. In this way, the experiences of an observer in free fall
are indistinguishable from those of an observer in deep space, far
from any sufficient source of gravity. Such observers are the
privileged ("inertial") observers Einstein described in his theory
of
special relativity: observers
for whom
light travels along straight lines at
constant speed.
Einstein hypothesized that the similar experiences of weightless
observers and inertial observers in special relativity represented
a fundamental property of gravity, and he made this the cornerstone
of his theory of general relativity, formalized in his
equivalence principle. Roughly
speaking, the principle states that a person in a free-falling
elevator cannot tell that they are in free fall. Every experiment
in such a free-falling environment has the same results as it would
for an observer at rest or moving uniformly in deep space, far from
all sources of gravity.
Gravity and acceleration
Just as most effects of gravity can be made to vanish by observing
them in free fall, the same effects can be
produced by
observing objects in an
accelerated
frame of reference. An observer in a closed room cannot tell which
of the following is true:
- Objects are falling to the floor because the room is resting on
the surface of the Earth and the objects are being pulled down by
gravity.
- Objects are falling to the floor because the room is aboard a
rocket in space, which is accelerating at 9.81 m/s^{2} and is far from any
source of gravity. The objects are being pulled towards the floor
by the same "inertial force" that presses the driver of an
accelerating car into his seat.
Conversely, any effect observed in an accelerated reference frame
should also be observed in a gravitational field of corresponding
strength. This principle allowed Einstein to predict several novel
effects of gravity in 1907, as explained in the
next section.
An observer in an accelerated reference frame must introduce what
physicists call
fictitious forces
to account for the acceleration experienced by himself and objects
around him. One example, the force pressing the driver of an
accelerating car into his or her seat, has already been mentioned;
another is the force you can feel pulling your arms up and out if
you attempt to spin around like a top. Einstein's key insight was
that the constant, familiar pull of the Earth's gravitational field
is fundamentally the same as these fictitious forces. Since
fictitious forces are always proportional to the
mass of the object on which they act, an object in a
gravitational field should feel a gravitational force proportional
to its mass, as embodied in
Newton's law of
gravitation.
Physical consequences
In 1907, Einstein was still eight years away from completing the
general theory of relativity. Nonetheless, he was able to make a
number of novel, testable predictions that were based on his
starting point for developing his new theory: the equivalence
principle.
The gravitational redshift of a light
wave as it moves upwards against a gravitational field (caused by
the yellow star below).
The first new effect is the
gravitational frequency shift
of light. Consider two observers aboard an accelerating
rocket-ship. Aboard such a ship, there is a natural concept of "up"
and "down": the direction in which the ship accelerates is "up",
and unattached objects accelerate in the opposite direction,
falling "downward". Assume that one of the observers is "higher up"
than the other. When the lower observer sends a light signal to the
higher observer, the acceleration causes the light to be
red-shifted, as may be calculated from
special relativity; the second observer
will measure a lower
frequency for the
light than the first. Conversely, light sent from the higher
observer to the lower is
blue-shifted,
that is, shifted towards higher frequencies. Einstein argued that
such frequency shifts must be also observed in a gravitational
field. This is illustrated in the figure at left, which shows a
light wave that is gradually red-shifted as it works its way
upwards against the gravitational acceleration. This effect has
been confirmed experimentally, as described
below.
This gravitational frequency shift corresponds to a
gravitational time dilation:
Since the "higher" observer measures the same light wave to have a
lower frequency than the "lower" observer, time must be passing
faster for the higher observer. Thus, time runs slower for
observers who are lower in a gravitational field.
It is important to stress that, for each observer, there are no
observable changes of the flow of time for events or processes that
are at rest in his or her reference frame. Five-minute-eggs as
timed by each observer's clock have the same consistency; as one
year passes on each clock, each observer ages by that amount; each
clock, in short, is in perfect agreement with all processes
happening in its immediate vicinity. It is only when the clocks are
compared between separate observers that one can notice that time
runs more slowly for the lower observer than for the higher. This
effect is minute, but it too has been confirmed experimentally in
multiple experiments, as described
below.
In a similar way, Einstein predicted the
gravitational
deflection of light: in a gravitational field, light is
deflected downward. Quantitatively, his results were off by a
factor of two; the correct derivation requires a more complete
formulation of the theory of general relativity, not just the
equivalence principle.
Tidal effects
The equivalence between gravitational and inertial effects does not
constitute a complete theory of gravity. Notably, it does not
answer the following simple question: what keeps people on the
other side of the world from falling off? When it comes to
explaining gravity near our own location on the Earth's surface,
noting that our reference frame is not in free fall, so that
fictitious forces are to be
expected, provides a suitable explanation. But a freely falling
reference frame on one side of the Earth cannot explain why the
people on the opposite side of the Earth experience a gravitational
pull in the opposite direction.
A more basic manifestation of the same effect involves two bodies
that are falling side by side towards the Earth. In a reference
frame that is in free fall alongside these bodies, they appear to
hover weightlessly – but not exactly so. These bodies are not
falling in precisely the same direction, but towards a single point
in space: namely, the Earth's
center of
gravity. Consequently, there is a component of each body's
motion towards the other (see the figure). In a small environment
such as a freely falling lift, this relative acceleration is
minuscule, while for
skydivers on opposite
sides of the Earth, the effect is large. Such differences in force
are also responsible for the
tides in the
Earth's oceans, so the term "
tidal
effect" is used for this phenomenon.
The equivalence between inertia and gravity cannot explain tidal
effects – it cannot explain variations in the gravitational field.
For that, a theory is needed which describes the way that matter
(such as the large mass of the Earth) affects the inertial
environment around it.
From acceleration to geometry
In exploring the equivalence of gravity and acceleration as well as
the role of tidal forces, Einstein discovered several analogies
with the
geometry of
surfaces. An example is the transition from an
inertial reference frame (in which free particles coast along
straight paths at constant speeds) to a rotating reference frame
(in which extra terms corresponding to
fictitious forces have to be introduced in
order to explain particle motion): this is analogous to the
transition from a
Cartesian coordinate system (in which the coordinate
lines are straight lines) to a
curved coordinate system (where
coordinate lines need not be straight).
A deeper analogy relates tidal forces with a property of surfaces
called
curvature. For
gravitational fields, the absence or presence of tidal forces
determines whether or not the influence of gravity can be
eliminated by choosing a freely falling reference frame. Similarly,
the absence or presence of curvature determines whether or not a
surface is
equivalent to a
plane. In the summer of 1912, inspired
by these analogies, Einstein searched for a geometric formulation
of gravity.
The elementary objects of
geometry –
points,
lines,
triangles
– are traditionally defined in three-dimensional
space or on two-dimensional
surfaces. In 1907, however, the mathematician
Hermann Minkowski introduced a
geometric formulation of Einstein's
special theory of relativity in
which the geometry included not only
space,
but also
time. The basic entity of this new
geometry is four-
dimensional spacetime. The orbits of moving bodies are
lines in spacetime; the orbits of bodies
moving at constant speed without changing direction correspond to
straight lines.
For surfaces, the generalization from the geometry of a plane – a
flat surface – to that of a general curved surface had been
described in the early nineteenth century by
Carl Friedrich Gauss. This description
had in turn been generalized to higher-dimensional spaces in a
mathematical formalism introduced by
Bernhard Riemann in the 1850s. With the
help of
Riemannian geometry,
Einstein formulated a geometric description of gravity in which
Minkowski's spacetime is replaced by distorted, curved spacetime,
just as curved surfaces are a generalization of ordinary plane
surfaces.
After he had realized the validity of this geometric analogy, it
took Einstein a further three years to find the missing cornerstone
of his theory: the equations describing how
matter influences spacetime's curvature. Having
formulated what are now known as
Einstein's equations (or, more
precisely, his field equations of gravity), he presented his new
theory of gravity at several sessions of the
Prussian Academy of Sciences in
late 1915.
Geometry and gravitation
Paraphrasing the doyen of American relativity research,
John Wheeler, Einstein's geometric
theory of gravity can be summarized thus: spacetime tells matter
how to move; matter tells spacetime how to curve. What this means
is addressed in the following three sections, which explore the
motion of so-called test particles, examine which properties of
matter serve as a source for gravity, and, finally, introduce
Einstein's equations, which relate these matter properties to the
curvature of spacetime.
Probing the gravitational field
Converging geodesics: two lines of
longitude (green) that start out in parallel at the equator (red)
but converge to meet at the pole
In order to map a body's gravitational influence, it is useful to
think about what physicists call probe or
test particles: particles that are influenced
by gravity, but are so small and light that we can neglect their
own gravitational effect. In the absence of gravity and other
external forces, a test particle moves along a straight line at a
constant speed. In the language of
spacetime, this is equivalent to saying that such
test particles move along straight
world
lines in spacetime. In the presence of gravity, however,
spacetime is
non-Euclidean, or
curved. In such a spacetime, straight
world lines may not exist. Instead, test particles move along lines
called
geodesics, which are "as straight as
possible".
A simple analogy is the following: In
geodesy, the science of measuring Earth's size and
shape, a geodesic (from Greek "geo", Earth, and "daiein", to
divide) is the shortest route between two points on the Earth's
surface. Approximately, such a route is a
segment of a
great
circle, such as a
line of
longitude or the
equator. These paths
are certainly not straight, simply because they must follow the
curvature of the Earth's surface. But they are as straight as is
possible subject to this constraint.
The properties of geodesics differ from those of straight lines.
For example, in a plane, parallel lines never meet, but this is not
so for geodesics on the surface of the Earth: for example, lines of
longitude are parallel at the equator, but intersect at the poles.
Analogously, the world lines of test particles in free fall are
spacetime geodesics,
the straightest possible lines in spacetime. But still there are
crucial differences between them and the truly straight lines that
can be traced out in the gravity-free spacetime of special
relativity. In special relativity, parallel geodesics remain
parallel. In a gravitational field with tidal effects, this will
not, in general, be the case. If, for example, two bodies are
initially at rest relative to each other, but are then dropped in
the Earth's gravitational field, they will move towards each other
as they fall towards the Earth's center.
Compared with planets and other astronomical bodies, the objects of
everyday life (people, cars, houses, even mountains) have little
mass. Where such objects are concerned, the laws governing the
behavior of test particles are sufficient to describe what happens.
Notably, in order to deflect a test particle from its geodesic
path, an external force must be applied. A person sitting on a
chair is trying to follow a geodesic, that is, to
fall freely towards the center of the Earth. But
the chair applies an external upwards force preventing the person
from falling. In this way, general relativity explains the daily
experience of gravity on the surface of the Earth
not as
the downwards pull of a gravitational force, but as the upwards
push of external forces. These forces deflect all bodies resting on
the Earth's surface from the geodesics they would otherwise follow.
For matter objects whose own gravitational influence cannot be
neglected, the laws of motion are somewhat more complicated than
for test particles, although it remains true that spacetime tells
matter how to move.
Sources of gravity
In
Newton's description of
gravity, the gravitational force is caused by matter. More
precisely, it is caused by a specific property of material objects:
their
mass. In Einstein's theory and related
theories of gravitation,
curvature at every point in spacetime is also caused by whatever
matter is present. Here, too, mass is a key property in determining
the gravitational influence of matter. But in a relativistic theory
of gravity, mass cannot be the only source of gravity. Relativity
links mass with energy, and energy with momentum.
The equivalence between mass and
energy, as
expressed by the formula
E = mc^{2},
is perhaps the most famous consequence of special relativity. In
relativity, mass and energy are two different ways of describing
one physical quantity. If a physical system has energy, it also has
the corresponding mass, and vice versa. In particular, all
properties of a body that are associated with energy, such as its
temperature or the
binding energy of systems such as
nuclei or
molecules,
contribute to that body's mass, and hence act as sources of
gravity.
In special relativity, energy is closely connected to
momentum. Just as space and time are, in that
theory, different aspects of a more comprehensive entity called
spacetime, energy and momentum are merely different aspects of a
unified, four-dimensional quantity that physicists call
four-momentum. In consequence, if energy is a
source of gravity, momentum must be a source as well. The same is
true for quantities that are directly related to energy and
momentum, namely internal
pressure and
tension. Taken together, in
general relativity it is mass, energy, momentum, pressure and
tension that serve as sources of gravity: they are how matter tells
spacetime how to curve. In the theory's mathematical formulation,
all these quantities are but aspects of a more general physical
quantity called the
energy-momentum
tensor.
Einstein's equations
Einstein's equations are the
centerpiece of general relativity. They provide a precise
formulation of the relationship between spacetime geometry and the
properties of matter, using the language of mathematics. More
concretely, they are formulated using the concepts of
Riemannian geometry, in which the
geometric properties of a space (or a spacetime) are described by a
quantity called a
metric. The
metric encodes the information needed to compute the fundamental
geometric notions of distance and angle in a curved space (or
spacetime).
Distances corresponding to 30 degrees
difference in longitude, at different latitudes.
A spherical surface like that of the Earth provides a simple
example. The location of any point on the surface can be described
by two coordinates: the geographic
latitude
and
longitude. Unlike the Cartesian
coordinates of the plane, coordinate differences are not the same
as distances on the surface, as shown in the diagram on the right:
for someone at the equator, moving 30 degrees of longitude westward
(magenta line) corresponds to a distance of roughly . On the other
hand, someone at a latitude of 55 degrees, moving 30 degrees of
longitude westward (blue line) covers a distance of merely .
Coordinates therefore do not provide enough information to describe
the geometry of a spherical surface, or indeed the geometry of any
more complicated space or spacetime. That information is precisely
what is encoded in the metric, which is a function defined at each
point of the surface (or space, or spacetime) and relates
coordinate differences to differences in distance. All other
quantities that are of interest in geometry, such as the length of
any given curve, or the angle at which two curves meet, can be
computed from this metric function.
The metric function and its rate of change from point to point can
be used to define a geometrical quantity called the
Riemann curvature tensor, which
describes exactly how the space (or spacetime) is curved at each
point. In general relativity, the metric and the Riemann curvature
tensor are quantities defined at each point in spacetime. As has
already been mentioned, the matter content of the spacetime defines
another quantity, the
Energy-momentum tensor
T, and the principle that "spacetime tells matter
how to move, and matter tells spacetime how to curve" means that
these quantities must be related to each other. Einstein formulated
this relation by using the Riemann curvature tensor and the metric
to define another geometrical quantity
G, now
called the
Einstein tensor, which
describes some aspects of the way spacetime is curved.
Einstein's equation then states that
- \mathbf{G}=\frac{8\pi G}{c^4}\mathbf{T},
i.e., up to a constant multiple, the quantity
G
(which measures curvature) is equated with the quantity
T (which measures matter content). The constants
involved in this equation reflect the different theories that went
into its making:
G is the
gravitational constant that is
already present in Newtonian gravity;
c is the
speed of light, the key constant in special
relativity; and
π is one of the basic constants
of geometry.
This equation is often referred to in the plural as
Einstein's
equations, since the quantities
G and
T are each determined by several functions of the
coordinates of spacetime, and the equations equate each of these
component functions.
A solution of
these equations describes a particular geometry of
space and time; for example, the
Schwarzschild solution describes the
geometry around a spherical, non-rotating mass such as a
star or a
black hole, whereas
the
Kerr solution describes a rotating
black hole. Still other solutions can describe a
gravitational wave or, in the case of the
Friedmann-Lemaître-Robertson-Walker
solution, an expanding universe. The simplest solution is the
uncurved
Minkowski spacetime,
the spacetime described by special relativity.
Experimental tests
No scientific theory is
apodictically
true; each is a model that must be checked by experiment. A
theory is falsified if it unambiguously fails even a single
experiment.
Newton's law of
gravity was accepted because it accounted for the motion of
planets and moons in the solar system with exquisite accuracy.
However, as the precision of experimental measurements gradually
improved, some discrepancies with Newton's predictions were
observed. These discrepancies were accounted for in the general
theory of relativity, but the predictions of that theory must also
be checked with experiment. Three experimental tests were devised
by Einstein himself and are now known as the classical tests of the
theory:
Newtonian (red) vs. Einsteinian orbit
(blue) of a single planet orbiting a spherical star.
- Newtonian gravity predicts that the orbit
which a single planet traces around a
perfectly spherical star should be an ellipse. Einstein's theory predicts a more
complicated curve: the planet behaves as if it were travelling
around an ellipse, but at the same time, the ellipse as a whole is
rotating slowly around the star. In the diagram on the right, the
ellipse predicted by Newtonian gravity is shown in red, and part of
the orbit predicted by Einstein in blue. For a planet orbiting the
Sun, this deviation from Newton's orbits is
known as the anomalous
perihelion shift. The first measurement of this effect, for the
planet Mercury, dates back to 1859.
The most accurate results for Mercury and for other planets to date
are based on measurements which were undertaken between 1966 and
1990, using radio telescopes.
General relativity predicts the correct anomalous perihelion shift
for all planets where this can be measured accurately (Mercury, Venus and the
Earth).
- According to general relativity, light does not travel along
straight lines when it propagates in a gravitational field.
Instead, it is deflected
in the presence of massive bodies. In particular, starlight is
deflected as it passes near the Sun, leading to
apparent shifts of up 1.75 arc seconds in
the stars' positions in the night sky (an arc second is equal to
1/3600 of a degree). In the framework of Newtonian gravity, a
heuristic argument can be made that leads to light deflection by
half that amount. The different predictions can be tested by
observing stars that are close to the Sun during a solar eclipse. In this way, a British
expedition to Brazil and West Africa in 1919, directed by Arthur Eddington, confirmed that Einstein's
prediction was correct, and the Newtonian predictions wrong.
Eddington's results were not very accurate; subsequent observations
of the deflection of the light of distant quasars by the Sun, which utilize highly accurate
techniques of radio astronomy, have
confirmed Eddington's results with significantly better precision
(the first such measurements date from 1967, the most recent
comprehensive analysis from 2004).
- Gravitational redshift
was first measured in a laboratory setting in 1959 by Pound and Rebka. It is also seen in
astrophysical measurements, notably for light escaping the White Dwarf Sirius B.
The related gravitational
time dilation effect has been measured by transporting atomic clocks to altitudes of between tens and
tens of thousands of kilometers (first by Hafele and Keating in 1971; most
accurately to date by Gravity Probe
A launched in 1976).
Of these tests, only the perihelion advance of Mercury was known
prior to Einstein's final publication of general relativity in
1916. The subsequent experimental confirmation of his other
predictions, especially the first measurements of the deflection of
light by the sun in 1919, catapulted Einstein to international
stardom. These three experimental tests justified adopting general
relativity over Newton's theory and, incidentally, over a number of
alternatives to
general relativity that had been proposed.
Further tests of general relativity include precision measurements
of the
Shapiro effect or
gravitational time delay for light, most recently in 2002 by the
Cassini space probe. One set of
tests focuses on effects predicted by general relativity for the
behavior of
gyroscopes travelling through
space. One of these effects,
geodetic
precession, has been tested with
Lunar laser ranging
experiments (high precision measurements of the orbit of the
Moon). Another, which is related to rotating
masses, is called
frame-dragging. It
is due to be tested by the
Gravity Probe
B satellite experiment launched in 2004, with results expected
in late 2008.
By cosmic standards, gravity throughout the solar system is weak.
Since the differences between the predictions of Einstein's and
Newton's theories are most pronounced when gravity is strong,
physicists have long been interested in testing various
relativistic effects in a setting with comparatively strong
gravitational fields. This has become possible thanks to precision
observations of
binary pulsars. In
such a star system, two highly compact
neutron stars orbit each other. At least one of
them is a
pulsar – an astronomical object
that emits a tight beam of radiowaves. Similar to the way that the
rotating beam of a lighthouse means that an observer sees the
lighthouse blink, these beams strike the Earth at very regular
intervals, and can be observed as a highly regular series of
pulses. General relativity predicts specific deviations from the
regularity of these radio pulses. For instance, at times when the
radio waves pass close to the other neutron star, they should be
deflected by the star's gravitational field. The observed pulse
patterns are impressively close to those predicted by general
relativity.
One particular set of observations is related to eminently useful
practical applications, namely to
satellite navigation systems
such as the
Global Positioning
System that are used both for precise
positioning and
timekeeping. Such systems rely on two sets of
atomic clocks: clocks aboard
satellites orbiting the Earth, and reference clocks stationed on
the Earth's surface. General relativity predicts that these two
sets of clocks should tick at slightly different rates, due to
their different motions (an effect already predicted by special
relativity) and their different positions within the Earth's
gravitational field. In order to ensure the system's accuracy, the
satellite clocks are either slowed down by a relativistic factor,
or that same factor is made part of the evaluation algorithm. In
turn, tests of the system's accuracy (especially the very thorough
measurements that are part of the definition of
universal coordinated time) are testament to the
validity of the relativistic predictions.
A number of other tests have probed the validity of various
versions of the
equivalence
principle; strictly speaking, all measurements of gravitational
time dilation are tests of the
weak version of that principle,
not of general relativity itself. So far, general relativity has
passed all observational tests.
Astrophysical applications
Models based on general relativity play an important role in
astrophysics, and the success of these
models is further testament to the theory's validity.
Gravitational lensing
Since light is deflected in a gravitational field, it is possible
for the light of a distant object to reach an observer along two or
more paths. For instance, light of a very distant object such as a
quasar can pass along one side of a massive
galaxy and be deflected slightly so as to
reach an observer on Earth, while lightpassing along the opposite
side of that same galaxy is deflected as well, reaching the same
observer from a slightly different direction. As a result, that
particular observer will see one astronomical object in two
different places in the night sky. This kind of focussing is
well-known when it comes to
optical
lenses, and hence the corresponding gravitational effect is
called
gravitational
lensing.
Observational astronomy uses
lensing effects as an important tool to infer properties of the
lensing object. Even in cases where that object is not directly
visible, the shape of a lensed image provides information about the
mass distribution responsible for the light
deflection. In particular, gravitational lensing provides one way
to measure the distribution of
dark
matter, which does not give off light and can be observed only
by its gravitational effects. One particularly interesting
application are large-scale observations, where the lensing masses
are spread out over a significant fraction of the observable
universe, and can be used to obtain information about the
large-scale properties and evolution of our cosmos.
Gravitational waves
Gravitational waves, a direct
consequence of Einstein's theory are ripples in space-time,
distortions of geometry which propagate at the speed of light.
(They should not be confused with the
gravity waves of
fluid dynamics, which are a different
concept.)
Indirectly, the effect of gravitational waves has been detected in
observations of specific binary stars. Such pairs of stars
orbit each other and, as they do so, gradually lose
energy by emitting gravitational waves. For ordinary stars like our
sun, this energy loss would be too small to be detectable. However,
in 1974, this energy loss was observed in a
binary pulsar called
PSR1913+16. In such a system, one of the orbiting
stars is a pulsar. This has two consequences: a pulsar is an
extremely dense object known as
neutron
star, for which gravitational wave emission is much stronger
than for ordinary stars. Also, a pulsar emits a narrow beam of
electromagnetic radiation
from its magnetic poles. As the pulsar rotates, its beam sweeps
over the Earth, where it is seen as a regular series of radio
pulses, just as a ship at sea observes regular flashes of light
from the rotating light in a lighthouse. This regular pattern of
radio pulses functions as a highly accurate "clock". It can be used
to time the double star's orbital period, and it reacts sensitively
to distortions of space-time in its immediate neighborhood.
The discoverers of
PSR1913+16,
Russell Hulse and
Joseph Taylor, were awarded the
Nobel prize in physics in 1993. Since then, several other binary
pulsars have been found. The most useful are those in which both
stars are pulsars, since they provide the most accurate tests of
general relativity.
Currently, one major goal of research in relativity is the direct
detection of gravitational waves. To this end, a number of
land-based
gravitational
wave detectors are in operation, and a mission to launch a
space-based detector,
LISA, is
currently under development, with a precursor mission (
LISA Pathfinder) due for launch in late
2009. If gravitational waves are detected, they could be used to
obtain information about compact objects such as
neutron stars and
black
holes, and also to probe the state of the early
universe fractions of a
second after the
Big
Bang.
Black holes
When mass is concentrated into a sufficiently
compact region of space, general relativity
predicts the formation of a
black hole –
a region of space with a gravitational attraction so strong that
not even light can escape. Certain types of black holes are thought
to be the final state in the
evolution of massive
stars. On the other hand,
supermassive black holes with the
mass of
millions or
billion of
Suns are
assumed to reside in the cores of most
galaxies, and they play a key role in current models
of how galaxies have formed over the past billions of years.
Matter falling onto a compact object is one of the most efficient
mechanisms for releasing
energy in the form
of
radiation, and matter falling onto
black holes is thought to be responsible for some of the brightest
astronomical phenomena imaginable. Notable examples of great
interest to astronomers are
quasars and
other types of
active galactic
nuclei. Under the right conditions, falling matter accumulating
around a black hole can lead to the formation of
jet, in which focused beams of matter are
flung away into space at speeds near
that of
light.
There are several properties that make black holes most promising
sources of gravitational waves. One reason is that black holes are
the most compact objects that can orbit each other as part of a
binary system; as a result, the gravitational waves emitted by such
a system are especially strong. Another reason follows from what
are called
black hole uniqueness
theorems: over time, black holes retain only a minimal set of
distinguishing features (since different hair styles are a crucial
part of what gives different people their different appearances,
these theorems have become known as "no hair" theorems). For
instance, in the long term, the collapse of a hypothetical matter
cube will not result in a cube-shaped black hole. Instead, the
resulting black hole will be indistinguishable from a black hole
formed by the collapse of a spherical mass, but with one important
difference: in its transition to a spherical shape, the black hole
formed by the collapse of a cube will emit gravitational
waves.
Cosmology
One of the most important aspects of general relativity is that it
can be applied to the
universe as a whole.
A key point is that, on large scales, our universe appears to be
constructed along very simple lines: All current observations
suggest that, on average, the structure of the cosmos should be
approximately the same, regardless of an observer's location or
direction of observation: the universe is approximately
homogeneous and
isotropic. Such comparatively simple universes can
be described by simple solutions of Einstein's equations. The
current
cosmological models of
the universe are obtained by combining these simple solutions to
general relativity with theories describing the properties of the
universe's
matter content, namely
thermodynamics,
nuclear- and
particle physics. According to these
models, our present universe emerged from an extremely dense
high-temperature state (the
Big
Bang)roughly 14
billion
years ago, and has been
expanding ever since.
Einstein's equations can be generalized by adding a term called the
cosmological constant. When
this term is present,
empty space itself acts
as a source of attractive or, unusually, repulsive gravity.
Einstein originally introduced this term in his pioneering 1917
paper on cosmology, with a very specific motivation: contemporary
cosmological thought held the universe to be static, and the
additional term was required for constructing static model
universes within the framework of general relativity. When it
became apparent that the universe is not static, but expanding,
Einstein was quick to discard this additional term; prematurely, as
we know today: From about 1998 on, a steadily accumulating body of
astronomical evidence has shown that the expansion of the universe
is
accelerating in a way that suggests
the presence of a cosmological constant or, equivalently, of a
dark energy with specific properties
that pervades all of space.
Modern research: general relativity and beyond
General relativity is very successful in providing a framework for
accurate models which describe an impressive array of physical
phenomena. On the other hand, there are many interesting open
questions, and in particular, the theory as a whole is almost
certainly incomplete.
In contrast to all other modern theories of
fundamental interactions, general
relativity is a
classical theory:
it does not include the effects of
quantum physics. The quest
for a quantum version of general relativity addresses one of the
most fundamental open questions in physics. While there are
promising candidates for such a theory of
quantum gravity, notably
string theory and
loop quantum gravity, there is at
present no consistent and complete theory. It has long been hoped
that a theory of quantum gravity would also eliminate another
problematic feature of general relativity: the presence of
spacetime singularities. These
singularities are boundaries ("sharp edges") of spacetime at which
geometry becomes ill-defined, with the consequence that general
relativity itself loses its predictive power. Furthermore, there
are so-called
singularity theorems
which predict that such singularities
must exist within
the universe if the laws of general relativity were to hold without
any quantum modifications. The best-known examples are the
singularities associated with the model universes that describe
black holes and the
beginning of the
universe.
Other attempts to modify general relativity have been made in the
context of
cosmology. In the modern
cosmological models, most energy in the universe is in forms that
have never been detected directly, namely
dark energy and
dark
matter. There have been several controversial proposals to
obviate the need for these enigmatic forms of matter and energy, by
modifying the laws governing gravity and the dynamics of
cosmic expansion, for example
modified Newtonian
dynamics.
It is possible that another reason to modify Einstein's theory can
be found much closer to home, in the shape of what is called the
Pioneer anomaly, after the
Pioneer 10 and
Pioneer 11 space
probes. Taking into account all known effects, gravitational or
otherwise, it is possible to make very specific predictions for
these probes' trajectories. Yet observations show ever-so-slight
divergences between these predictions and the actual positions. The
possibility of new physics has not been ruled out, despite thorough
attempts to find more conventional explanations.
Beyond the challenges of quantum effects and cosmology, research on
general relativity is rich with possibilities for further
exploration: mathematical relativists explore the nature of
singularities and the fundamental properties of Einstein's
equations, ever more comprehensive computer simulations of specific
spacetimes (such as those describing merging black holes) are run,
and the race for the first direct detection of gravitational waves
continues apace.More than ninety years after the theory was first
published, research is more active than ever.
See also
Notes
- This development is traced e.g. in , in chapters 9 through 15
of , and in . A precis of Newtonian gravity can be found in . It is
impossible to say whether the problem of Newtonian gravity crossed
Einstein's mind before 1907, but by his own admission, his first
serious attempts to reconcile that theory with special relativity
date to that year, cf. .
- This is described in detail in chapter 2 of .
- While the equivalence principle is still part of modern
expositions of general relativity, there are some differences
between the modern version and Einstein's original concept, cf.
.
- E. g. . Einstein himself also explains this in section XX of
his non-technical book Einstein 1961. Following earlier ideas by
Ernst Mach,
Einstein also explored centrifugal force and their
gravitational analogue, cf. .
- More specifically, Einstein's calculations, which are described
in chapter 11b of , use the equivalence principle, the equivalence
of gravity and inertial forces, and the results of special
relativity for the propagation of light and for accelerated
observers (the latter by considering, at each moment, the
instantaneous inertial frame of reference
associated with such an accelerated observer).
- This effect can be derived directly within special relativity,
either by looking at the equivalent situation of two observers in
an accelerated rocket-ship or by looking at a falling elevator; in
both situations, the frequency shift has an equivalent description
as a Doppler
shift between certain inertial frames. For simple derivations
of this, see .
- See chapter 12 of .
- Cf. ; for a non-technical presentation, see .
- These and other tidal effects are described in .
- Tides and their geometric interpretation are explained in
chapter 5 of . This part of the historical development is traced in
.
- For elementary presentations of the concept of spacetime, see
the first section in chapter 2 of , and . More complete treatments
on a fairly elementary level can be found e.g. in and in .
- See for vivid illustrations of curved spacetime.
- Einstein's struggle to find the correct field equations is
traced in chapters 13–15 of .
- E.g. p. xi in .
- A thorough, yet accessible account of basic differential
geometry and its application in general relativity can be found in
.
- See chapter 10 of .
- In fact, when starting from the complete theory, Einstein's
equation can be used to derive these more complicated laws of
motion for matter as a consequence of geometry; however, deriving
from this the motion of idealized test particles is a highly
non-trivial task, cf. .
- A simple explanation of mass-energy-equivalence can be found in
sections 3.8 and 3.9 of .
- See chapter 6 of .
- For a more detailed definition of the metric, but one that is
more informal than a textbook presentation, see chapter 14.4 of
.
- The geometrical meaning of Einstein's equations is explored in
chapters 7 and 8 of ; cf. box 2.6 in . An introduction using only
very simple mathematics is given in chapter 19 of .
- The most important solutions are listed in every textbook on general
relativity; for a (technical) summary of our current
understanding, see .
- More precisely, these are VLBI measurements of planetary positions; see chapter 5
of and section 3.5 of .
- For the historical measurements, see , , and ; Soldner's
original derivation in the framework of Newton's theory is . For
the most precise measurements to date, see .
- See and chapter 3 of . For the Sirius B measurements, see
.
- , Mercury on pp. 253–254, Einstein's rise to fame in sections
16b and 16c.
- For the Cassini measurements of the Shapiro effect, see . For
more information about Gravity Probe B, see the
- .
- An accessible account of relativistic effects in the global
positioning system can be found in ; details are given in .
- An accessible introduction to tests of general relativity is ;
a more technical, up-to-date account is .
- The geometry of such situations is explored in chapter 23 of
.
- Introductions to gravitational lensing and its applications can
be found on the webpages and .
- ; .
- The ongoing search for gravitational waves is described vividly
in and in .
- For an overview of the history of black hole physics from its
beginnings in the early twentieth century to modern times, see the
very readable account by . For an up-to-date account of the role of
black holes in structure formation, see ; a brief summary can be
found in the related article .
- See chapter 8 of and . A treatment that is more thorough, yet
involves only comparatively little mathematics can be found in
.
- An elementary introduction to the black hole uniqueness
theorems can be found in and in .
- Detailed information can be found in Ned Wright's Cosmology
Tutorial and FAQ, ; a very readable introduction is . Using
undergraduate mathematics but avoiding the advanced mathematical
tools of general relativity, provides a more thorough
presentation.
- Einstein's original paper is ; good descriptions of more modern
developments can be found in and .
- Cf. ; .
- With a focus on string theory, the search for quantum gravity
is described in ; for an account from the point of view of loop
quantum gravity, see .
- For dark matter, see ; for dark energy, .
- See .
- See .
- A review of the various problems and the techniques being
developed to overcome them, see .
- See for an account up to that year; up-to-date news can be
found on the websites of major detector collaborations such as
GEO 600 and
LIGO.
- A good starting point for a snapshot of present-day research in
relativity is the electronic review journal Living
Reviews in Relativity.
References
External links
Additional resources, including more advanced material, can be
found in General relativity
resources.