General relativity or the
general theory
of relativity is the
geometric
theory of
gravitation published by
Albert Einstein in 1915. It is the current
description of gravitation in modern
physics. It unifies
special relativity and
Newton's law of universal
gravitation, and describes gravity as a geometric property of
space and
time,
or
spacetime. In particular, the
curvature of spacetime is directly related to the
four-momentum (
mass-energy and linear
momentum) of whatever
matter
and
radiation are present. The relation is
specified by the
Einstein field
equations, a system of
partial differential
equations.
Many predictions of general relativity differ significantly from
those of classical physics, especially concerning the passage of
time, the geometry of space, the motion of bodies in
free fall, and the propagation of
light. Examples of such differences include
gravitational time dilation, the
gravitational redshift of
light, and the
gravitational time
delay. General relativity's predictions have been confirmed in
all
observations and
experiments to date. Although general relativity is
not the only relativistic
theory of gravity, it is the simplest theory that is consistent
with experimental data. However, unanswered questions remain, the
most fundamental being how general relativity can be reconciled
with the laws of
quantum physics
to produce a complete and self-consistent theory of
quantum gravity.
Einstein's theory has important astrophysical implications. It
points towards the existence of
black
holes—regions of space in which space and time are distorted in
such a way that nothing, not even light, can escape—as an end-state
for massive
stars. There is evidence that such
stellar black holes as well as
more massive varieties of black hole are responsible for the
intense
radiation emitted by certain types
of astronomical objects such as
active galactic nuclei or
microquasars. The bending of light by gravity
can lead to the phenomenon of
gravitational lensing, where multiple
images of the same distant astronomical object are visible in the
sky.
General relativity also predicts the
existence of gravitational waves,
which have since been measured indirectly; a direct measurement is
the aim of projects such as LIGO. In
addition, general relativity is the basis of current
cosmological models of a consistently
expanding universe.
History
First page from Einstein's manuscript explaining general
relativity
Soon after publishing the
special
theory of relativity in 1905, Einstein started thinking about
how to incorporate
gravity into his new
relativistic framework. In 1907, beginning with a simple
thought experiment involving an observer
in free fall, he embarked on what would be an eight-year search for
a relativistic theory of gravity. After numerous detours and false
starts, his work culminated in the November, 1915 presentation to
the
Prussian Academy of
Science of what are now known as the
Einstein field equations. These
equations specify how the geometry of space and time is influenced
by whatever matter is present, and form the core of Einstein's
general theory of relativity.
The Einstein field equations are
nonlinear
and very difficult to solve. Einstein used approximation methods in
working out initial predictions of the theory. But as early as
1916, the astrophysicist
Karl
Schwarzschild found the first non-trivial exact solution to the
Einstein field equations, the so-called
Schwarzschild metric. This solution
laid the groundwork for the description of the final stages of
gravitational collapse, and the objects known today as
black holes. In the same year, the first steps
towards generalizing Schwarzschild's solution to
electrically charged objects were taken,
which eventually resulted in the
Reissner-Nordström solution, now
associated with charged black holes. In 1917, Einstein applied his
theory to the
universe as a whole,
initiating the field of relativistic
cosmology. In line with contemporary
thinking, he assumed a static universe, adding a new parameter to
his original field equations—the
cosmological constant—to reproduce
that "observation". By 1929, however, the work of
Hubble and others had shown that our
universe is expanding. This is
readily described by the expanding cosmological solutions found by
Friedmann in 1922, which do not
require a cosmological constant.
Lemaître used these solutions to
formulate the earliest version of the
big
bang models, in which our universe has evolved from an
extremely hot and dense earlier state. Einstein later declared the
cosmological constant the biggest blunder of his life.
During that period, general relativity remained something of a
curiosity among physical theories. It was clearly superior to
Newtonian gravity, being
consistent with
special
relativity and accounting for several effects unexplained by
the Newtonian theory. Einstein himself had shown in 1915 how his
theory explained the
anomalous
perihelion advance of the planet
Mercury without any arbitrary parameters
("fudge factors"). Similarly, a 1919 expedition led by
Eddington confirmed general relativity's
prediction for the deflection of starlight by the Sun, making
Einstein instantly famous. Yet the theory entered the mainstream of
theoretical physics and
astrophysics only with the developments between
approximately 1960 and 1975, now known as the
Golden age of general
relativity. Physicists began to understand the concept of a
black hole, and to identify these
objects' astrophysical manifestation as
quasars. Ever more precise solar system tests
confirmed the theory's predictive power, and relativistic
cosmology, too, became amenable to direct observational
tests.
From classical mechanics to general relativity
General relativity is best understood by examining its similarities
with and departures from
classical
physics. The first step is the realization that classical
mechanics and Newton's law of gravity admit of a geometric
description. The combination of this description with the laws of
special relativity results in a heuristic derivation of general
relativity.
Geometry of Newtonian gravity
At the base of
classical
mechanics is the notion that a
body's motion can be described as a
combination of free (or
inertial) motion,
and deviations from this free motion. Such deviations are caused by
external forces acting on a body in accordance with Newton's second
law of motion, which states
that the net
force acting on a body is equal
to that body's (inertial)
mass multiplied by
its
acceleration. The preferred
inertial motions are related to the geometry of
space and
time: in the standard
reference frames of classical
mechanics, objects in free motion move along straight lines at
constant speed. In modern parlance, their paths are
geodesics, straight
world
lines in
spacetime.
Conversely, one might expect that inertial motions, once identified
by observing the actual motions of bodies and making allowances for
the external forces (such as
electromagnetism or
friction), can be used to define the geometry of
space, as well as a time
coordinate.
However, there is an ambiguity once
gravity
comes into play. According to
Newton's law of gravity, and
independently verified by experiments such as that of
Eötvös and its successors (see
Eötvös experiment),
there is a
universality of
free fall (also known as the weak
equivalence principle, or the
universal equality of inertial and passive-gravitational mass): the
trajectory of a
test body in
free fall depends only on its position and initial
speed, but not on any of its material properties. A simplified
version of this is embodied in Einstein's elevator experiment,
illustrated in the figure on the right: for an observer in a small
enclosed room, it is impossible to decide, by mapping the
trajectory of bodies such as a dropped ball, whether the room is at
rest in a gravitational field, or in free space aboard an
accelerated rocket.
Given the universality of free fall, there is no observable
distinction between inertial motion and motion under the influence
of the gravitational force. This suggests the definition of a new
class of inertial motion, namely that of objects in free fall under
the influence of gravity. This new class of preferred motions, too,
defines a geometry of space and time—in mathematical terms, it is
the
geodesic motion associated with a
specific
connection which
depends on the
gradient of the
gravitational potential. Space, in
this construction, still has the ordinary
Euclidean geometry. However,
space
time as a whole is more complicated. As can be shown
using simple
thought experiments
following the free-fall trajectories of different test particles,
the result of transporting spacetime vectors that can denote a
particle's velocity (time-like vectors) will vary with the
particle's trajectory; mathematically speaking, the Newtonian
connection is not
integrable.
From this, one can deduce that spacetime is
curved. The result is a geometric formulation of
Newtonian gravity using only
covariant concepts, i.e. a description which is valid in any
desired coordinate system. In this geometric description,
tidal effects—the relative acceleration of
bodies in free fall—are related to the derivative of the
connection, showing how the modified geometry is caused by the
presence of mass.
Relativistic generalization
As intriguing as geometric Newtonian gravity may be, its basis,
classical mechanics, is merely a limiting case of
relativistic mechanics. In the language
of
symmetry: where gravity can be
neglected, physics is
Lorentz
invariant as in special relativity rather than
Galilei invariant as in classical
mechanics. (The defining symmetry of special relativity is the
Poincaré group which also
includes translations and rotations.) The differences between the
two become significant when we are dealing with speeds approaching
the
speed of light, and with
high-energy phenomena.
With Lorentz symmetry, additional structures come into play. They
are defined by the set of light cones (see the image on the left).
The light-cones define a causal structure: for each
event A, there is a set of events that
can, in principle, either influence or be influenced by A via
signals or interactions that do not need to travel faster than
light (such as event B in the image), and a set of events for which
such an influence is impossible (such as event C in the image).
These sets are observer-independent. In conjunction with the
world-lines of freely falling particles, the light-cones can be
used to reconstruct the space-time's semi-Riemannian metric, at
least up to a positive scalar factor. In mathematical terms, this
defines a
conformal
structure.
Special relativity is defined in the absence of gravity, so for
practical applications, it is a suitable model whenever gravity can
be neglected. Bringing gravity into play, and assuming the
universality of free fall, an analogous reasoning as in the
previous section applies: there are no global
inertial frames. Instead there are
approximate inertial frames moving alongside freely falling
particles. Translated into the language of spacetime: the straight
time-like lines that define a gravity-free
inertial frame are deformed to lines that are curved relative to
each other, suggesting that the inclusion of gravity necessitates a
change in spacetime geometry.
A priori, it is not clear whether the new local frames in free fall
coincide with the reference frames in which the laws of special
relativity hold—that theory is based on the propagation of light,
and thus on
electromagnetism, which
could have a different set of preferred frames. But using different
assumptions about the special-relativistic frames (such as their
being earth-fixed, or in free fall), one can derive different
predictions for the
gravitational
redshift, that is, the way in which the frequency of light
shifts as the light propagates through a gravitational field (cf.
below). The actual measurements show that free-falling frames
are the ones in which light propagates as it does in special
relativity. The generalization of this statement, namely that the
laws of special relativity hold to good approximation in freely
falling (and non-rotating) reference frames, is known as the
Einstein
equivalence
principle, a crucial guiding principle for generalizing
special-relativistic physics to include gravity.
The same experimental data shows that time as measured by clocks in
a gravitational field—
proper time, to
give the technical term—does not follow the rules of special
relativity. In the language of spacetime geometry, it is not
measured by the
Minkowski metric.
As in the Newtonian case, this is suggestive of a more general
geometry. At small scales, all reference frames that are in free
fall are equivalent, and approximately Minkowskian. Consequently,
we are now dealing with a curved generalization of Minkowski space.
The
metric tensor
that defines the geometry—in particular, how lengths and angles are
measured—is not the Minkowski metric of special relativity, it is a
generalization known as a semi- or
pseudo-Riemannian metric. Furthermore,
each Riemannian metric is naturally associated with one particular
kind of connection, the
Levi-Civita connection, and this is,
in fact, the connection that satisfies the equivalence principle
and makes space locally Minkowskian (that is, in suitable
locally inertial coordinates, the
metric is Minkowskian, and its first partial derivatives and the
connection coefficients vanish).
Einstein's equations
Having formulated the relativistic, geometric version of the
effects of gravity, the question of gravity's source remains. In
Newtonian gravity, the source is mass. In special relativity, mass
turns out to be part of a more general quantity called the
energy-momentum tensor, which
includes both
energy and
momentum densities as well
as
stress (that is,
pressure and shear). Using the equivalence
principle, this tensor is readily generalized to curved space-time.
Drawing further upon the analogy with geometric Newtonian gravity,
it is natural to assume that the
field
equation for gravity relates this tensor and the
Ricci tensor, which describes a particular
class of tidal effects: the change in volume for a small cloud of
test particles that are initially at rest, and then fall freely. In
special relativity,
conservation
of energy-momentum corresponds to the statement that the
energy-momentum tensor is
divergence-free. This formula, too, is readily
generalized to curved spacetime by replacing partial derivatives
with their curved-
manifold counterparts,
covariant derivatives studied
in
differential geometry. With
this additional condition—the covariant divergence of the
energy-momentum tensor, and hence of whatever is on the other side
of the equation, is zero— the simplest set of equations are what
are called Einstein's (field) equations:
- R_{ab} - {\textstyle 1 \over 2}R\,g_{ab} = \kappa
T_{ab}.\,
On the left-hand side is the
Einstein
tensor, a specific divergence-free combination of the
Ricci tensor R_{ab} and the metric. In
particular,
- R=R_{cd}g^{cd}\,
is the curvature scalar. The Ricci tensor itself is related to the
more general Riemann curvature tensor as
- \quad R_{ab}={R^d}_{adb}.\,
On the right-hand side,
T_{ab} is the
energy-momentum tensor. All tensors are written in
abstract index notation. Matching
the theory's prediction to observational results for
planetary orbits (or,
equivalently, assuring that the weak-gravity, low-speed limit is
Newtonian mechanics), the proportionality constant can be fixed as
κ = 8π
G/
c^{4}, with
G the
gravitational constant and
c the
speed of light. When
there is no matter present, so that the energy-momentum tensor
vanishes, the result are the
vacuum Einstein
equations,
- R_{ab}=0.\,
There are
alternatives to general
relativity built upon the same premises, which include
additional rules and/or constraints, leading to different field
equations. Examples are
Brans-Dicke
theory,
teleparallelism, and
Einstein-Cartan theory.
Definition and basic applications
The derivation outlined in the previous section contains all the
information needed to define general relativity, describe its key
properties, and address a question of crucial importance in
physics, namely how the theory can be used for
model-building.
Definition and basic properties
General relativity is a
metric theory of
gravitation. At its core are
Einstein's equations, which describe
the relation between the
geometry of a
four-dimensional, semi-
Riemannian
manifold representing
spacetime on the one hand, and the
energy-momentum contained in that spacetime
on the other. Phenomena that in
classical mechanics are ascribed to the
action of the force of gravity (such as
free-fall,
orbital motion,
and
spacecraft trajectories), correspond to inertial motion
within a
curved geometry of spacetime in
general relativity; there is no gravitational force deflecting
objects from their natural, straight paths. Instead, gravity
corresponds to changes in the properties of space and time, which
in turn changes the straightest-possible paths that objects will
naturally follow. The curvature is, in turn, caused by the
energy-momentum of matter. Paraphrasing the relativist
John Archibald Wheeler, spacetime
tells matter how to move; matter tells spacetime how to
curve.
While general relativity replaces the
scalar gravitational potential of classical
physics by a symmetric
rank-two
tensor, the latter reduces to the former in
certain
limiting
cases. For
weak
gravitational fields and
slow speed relative to the speed
of light, the theory's predictions converge on those of
Newton's law of universal
gravitation.
As it is constructed using
tensors, general
relativity exhibits
general
covariance: its laws—and further laws formulated within the
general relativistic framework—take on the same form in all
coordinate systems. Furthermore,
the theory does not contain any invariant geometric background
structures. It thus satisfies a more stringent
general principle of
relativity, namely that the
laws of
physics are the same for all observers.
Locally, as expressed in the
equivalence principle,
spacetime is
Minkowskian, and the
laws of physics exhibit
local
Lorentz invariance.
Model-building
The core concept of general-relativistic model-building is that of
a
solution of
Einstein's equations. Given both Einstein's equations and
suitable equations for the properties of matter, such a solution
consists of a specific semi-Riemannian manifold (usually defined by
giving the metric in specific coordinates), and specific matter
fields defined on that manifold. Matter and geometry must satisfy
Einstein's equations, so in particular, the matter's
energy-momentum tensor must be divergence-free. The matter must, of
course, also satisfy whatever additional equations were imposed on
its properties. In short, such a solution is a model universe that
satisfies the laws of general relativity, and possibly additional
laws governing whatever matter might be present.
Einstein's equations are nonlinear
partial differential
equations and, as such, difficult to solve exactly.
Nevertheless, a number of
exact solution are
known, although only a few have direct physical applications. The
best-known exact solutions, and also those most interesting from a
physics point of view, are the
Schwarzschild solution, the
Reissner-Nordström
solution and the
Kerr metric, each
corresponding to a certain type of
black
hole in an otherwise empty universe, and the
Friedmann-Lemaître-Robertson-Walker
and
de Sitter universes, each
describing an expanding cosmos. Exact solutions of great
theoretical interest include the
Gödel
universe (which opens up the intriguing possibility of
time travel in curved spacetimes), the
Taub-NUT solution (a model universe that
is
homogeneous, but
anisotropic), and
Anti-de Sitter space (which has
recently come to prominence in the context of what is called the
Maldacena conjecture).
Given the difficulty of finding exact solutions, Einstein's field
equations are also solved frequently by
numerical integration on a computer,
or by considering small perturbations of exact solutions. In the
field of
numerical relativity,
powerful computers are employed to simulate the geometry of
spacetime and to solve Einstein's equations for interesting
situations such as two colliding
black
holes. In principle, such methods may be applied to any system,
given sufficient computer resources, and may address fundamental
questions such as
naked
singularities. Approximate solutions may also be found by
perturbation theories such as
linearized gravity and its
generalization, the
post-Newtonian expansion, both of
which were developed by Einstein. The latter provides a systematic
approach to solving for the geometry of a spacetime that contains a
distribution of matter that moves slowly compared with the speed of
light. The expansion involves a series of terms; the first terms
represent Newtonian gravity, whereas the later terms represent ever
smaller corrections to Newton's theory due to general relativity.
An extension of this expansion is the
parametrized
post-Newtonian (PPN) formalism, which allows quantitative
comparisons between the predictions of general relativity and
alternative
theories.
Consequences of Einstein's theory
General relativity has a number of physical consequences. Some
follow directly from the theory's axioms, whereas others have
become clear only in the course of the ninety years of research
that followed Einstein's initial publication.
Gravitational time dilation and frequency shift
Schematic representation of the
gravitational redshift of a light wave escaping from the surface of
a massive body
Assuming that the
equivalence
principle holds, gravity influences the passage of time. Light
sent down into a
gravity well is
blueshifted, whereas light sent in the
opposite direction (i.e., climbing out of the gravity well) is
redshifted; collectively, these two effects
are known as the gravitational frequency shift. More generally,
processes close to a massive body run more slowly when compared
with processes taking place farther away; this effect is known as
gravitational time dilation.
Gravitational redshift has been measured in the laboratory and
using astronomical observations. Gravitational time dilation in the
Earth's gravitational field has been measured numerous times using
atomic clocks, while ongoing
validation is provided as a side-effect of the operation of the
Global Positioning System
(GPS). Tests in stronger gravitational fields are provided by the
observation of
binary pulsars. All
results are in agreement with general relativity. However, at the
current level of accuracy, these observations cannot distinguish
between general relativity and other theories in which the
equivalence principle is valid.
Light deflection and gravitational time delay
General relativity predicts that the path of light is bent in a
gravitational field; light passing a massive body is deflected
towards that body. This effect has been confirmed by observing the
light of stars or distant
quasars being
deflected as it passes the
Sun.
Deflection of light (sent out from the
location shown in blue) near a compact body (shown in gray)
This and related predictions follow from the fact that light
follows what is called a light-like or null
geodesic—a generalization of the straight lines
along which light travels in
classical
physics. Such geodesics are the generalization of the
invariance of
lightspeed in
special relativity. As one examines
suitable model spacetimes (either the exterior
Schwarzschild solution or, for more
than a single mass, the
post-Newtonian expansion), several
effects of gravity on light propagation emerge. Although the
bending of light can also be derived by extending the
universality of free fall to
light, the angle of deflection resulting from
such calculations is only half the value given by general
relativity.
Closely related to light deflection is the gravitational time delay
(or Shapiro effect), the phenomenon that light signals take longer
to move through a gravitational field than they would in the
absence of that field. There have been numerous successful tests of
this prediction. In the
parameterized
post-Newtonian formalism (PPN), measurements of both the
deflection of light and the gravitational time delay determine a
parameter called \gamma, which encodes the influence of gravity on
the geometry of space.
Gravitational waves
Ring of test particles floating in
space
Ring of test particles influenced by
gravitational wave
One of several analogies between weak-field gravity and
electromagnetism is that, analogous to
electromagnetic waves, there
are
gravitational waves: ripples
in the metric of spacetime that propagate at the
speed of light. The simplest type of such a
wave can be visualized by its action on a ring of freely floating
particles (upper image to the right). A sine wave propagating
through such a ring towards the reader distorts the ring in a
characteristic, rhythmic fashion (lower, animated image to the
right). Since Einstein's equations are
non-linear, arbitrarily strong gravitational
waves do not obey
linear
superposition, making their description difficult. However, for
weak fields, a linear approximation can be made. Such linearized
gravitational waves are sufficiently accurate to describe the
exceedingly weak waves that are expected to arrive here on Earth
from far-off cosmic events, which typically result in relative
distances increasing and decreasing by 10^{-21} or less.
Data-analysis methods routinely make use of the fact that these
linearized waves can be
Fourier
decomposed.
Some
exact solutions describe
gravitational waves without any approximation, e.g., a wave train
traveling through empty space or so-called
Gowdy universes, varieties of an expanding
cosmos filled with gravitational waves. But for gravitational waves
produced in astrophysically relevant situations, such as the merger
of two black holes,
numerical
methods are presently the only way to construct appropriate
models.
Orbital effects and the relativity of direction
General relativity differs from classical mechanics in a number of
predictions concerning orbiting bodies. It predicts an overall
rotation (
precession) of planetary
orbits, as well as orbital decay caused by the emission of
gravitational waves and effects related to the relativity of
direction.
Precession of apsides
Newtonian (red) vs. Einsteinian orbit
(blue) of a lone planet orbiting a star
In general relativity, the
apsides of any
orbit (the point of the orbiting body's
closest approach to the system's
center
of mass) will
precess—the orbit is not
an
ellipse, but akin to an ellipse that
rotates on its focus, resulting in a
rose curve-like shape (see image).
Einstein first derived this result by using an approximate metric
representing the Newtonian limit and treating the orbiting body as
a
test particle. For him, the fact
that his theory gave a straightforward explanation of the
anomalous
perihelion shift of the planet
Mercury, discovered earlier by
Urbain Le Verrier in 1859, was important
evidence that he had at last identified the correct form of the
gravitational field
equations.
The effect can also be derived by using either the exact
Schwarzschild metric (describing
spacetime around a spherical mass) or the much more general
post-Newtonian formalism.
It is due to the influence of gravity on the geometry of space and
to the contribution of
self-energy to a
body's gravity (encoded in the
nonlinearity of Einstein's equations).In
consequence, in the
parameterized
post-Newtonian formalism (PPN), measurements of this effect
determine a linear combination of the terms \beta and \gamma, cf.
and . Relativistic precession has been observed for all planets
that allow for accurate precession measurements (Mercury,
Venus and the
Earth), as well as
in
binary pulsar systems, where it is
larger by five
orders of
magnitude.
Orbital decay
Orbital decay for PSR1913+16: time
shift in seconds, tracked over three decades.
According to general relativity, a
binary system will emit
gravitational waves, thereby losing
energy. Due to this loss, the distance
between the two orbiting bodies decreases, and so does their
orbital period. Within the
solar system
or for ordinary
double stars, the effect
is too small to be observable. Not so for a close
binary pulsar, a system of two orbiting
neutron stars, one of which is a
pulsar: from the pulsar, observers on Earth
receive a regular series of radio pulses that can serve as a highly
accurate clock, which allows precise measurements of the orbital
period. Since the neutron stars are very compact, significant
amounts of energy are emitted in the form of gravitational
radiation.
The first observation of a decrease in orbital period due to the
emission of gravitational waves was made by
Hulse and
Taylor, using the binary pulsar
PSR1913+16 they had discovered in 1974.
This was the first detection of gravitational waves, albeit
indirect, for which they were awarded the 1993
Nobel Prize in physics. Since then, several
other binary pulsars have been found, in particular the double
pulsar
PSR J0737-3039, in which both
stars are pulsars.
Geodetic precession and frame-dragging
Several relativistic effects are directly related to the relativity
of direction. One is
geodetic
precession: the axis direction of a
gyroscope in free fall in curved spacetime will
change when compared, for instance, with the direction of light
received from distant stars—even though such a gyroscope represents
the way of keeping a direction as stable as possible ("
parallel transport"). For the
Moon-
Earth-system, this effect has
been measured with the help of
lunar
laser ranging. More recently, it has been measured for test
masses aboard the satellite
Gravity
Probe B to a precision of better than 1 percent.
Near a rotating mass, there are so-called gravitomagnetic or
frame-dragging effects. A distant
observer will determine that objects close to the mass get "dragged
around". This is most extreme for
rotating
black holes where, for any object entering a zone known as the
ergosphere, rotation is inevitable. Such
effects can again be tested through their influence on the
orientation of gyroscopes in free fall. Somewhat controversial
tests have been performed using the
LAGEOS
satellites, confirming the relativistic prediction.. Also the
Mars Global Surveyor probe
around Mars has been used
- see the entry frame-dragging for
an account of the debate.
A precision measurement is the main aim of the
Gravity Probe B mission, with the results
expected in September 2008.
Astrophysical applications
Gravitational lensing
The deflection of light by gravity is responsible for a new class
of astronomical phenomena. If a massive object is situated between
the astronomer and a distant target object with appropriate mass
and relative distances, the astronomer will see multiple distorted
images of the target. Such effects are known as
gravitational lensing. Depending on the
configuration, scale, and mass distribution, there can be two or
more images, a bright ring known as an
Einstein ring, or partial rings called
arcs.The
earliest example was discovered
in 1979; since then, more than a hundred gravitational lenses have
been observed. Even if the multiple images are too close to each
other to be resolved, the effect can still be measured, e.g., as an
overall brightening of the target object; a number of such
"
microlensing events" have been
observed.
Gravitational lensing has developed into a tool of
observational astronomy. It is used
to detect the presence and distribution of
dark matter, provide a "natural telescope" for
observing distant galaxies, and to obtain an independent estimate
of the
Hubble constant. Statistical
evaluations of lensing data provide valuable insight into the
structural evolution of
galaxies.
Gravitational wave astronomy
Observations of binary pulsars provide strong indirect evidence for
the existence of gravitational waves (see
Orbital decay, above).
However, gravitational waves reaching us from the depths of the
cosmos have not been detected directly, which is a major goal of
current relativity-related research.
Several land-based
gravitational wave
detectors are currently in operation, most notably the interferometric
detector GEO
600, LIGO (three
detectors), TAMA 300 and VIRGO. A
joint US-European space-based detector,
LISA, is currently under development, with
a precursor mission (
LISA
Pathfinder) due for launch in late 2009.
Observations of gravitational waves promise to complement
observations in the
electromagnetic spectrum. They are
expected to yield information about black holes and other dense
objects such as neutron stars and white dwarfs, about certain kinds
of
supernova implosions, and about
processes in the very early universe, including the signature of
certain types of hypothetical
cosmic
string.
Black holes and other compact objects
Whenever an object becomes sufficiently compact, general relativity
predicts the formation of a
black hole, a
region of space from which nothing, not even light, can escape. In
the currently accepted models of
stellar evolution,
neutron stars with around 1.4
solar mass and so-called
stellar black holes with a few to a few
dozen solar masses are thought to be the final state for the
evolution of massive stars.
Supermassive black holes with a few
million to a few
billion solar
masses are considered the rule rather than the exception in the
centers of galaxies, and their presence is thought to have played
an important role in the formation of galaxies and larger cosmic
structures.
Simulation based on the equations of
general relativity: a star collapsing to form a black hole while
emitting gravitational waves
Astronomically, the most important property of compact objects is
that they provide a superbly efficient mechanism for converting
gravitational energy into electromagnetic radiation.
Accretion, the falling of
dust or gaseous matter onto
stellar or
supermassive black holes, is thought
to be responsible for some spectacularly luminous astronomical
objects, notably diverse kinds of
active galactic nuclei on galactic
scales and stellar-size objects such as
microquasars. In particular, accretion can lead
to
relativistic jets, focused beams
of highly energetic particles that are being flung into space at
almost
light speed.General relativity
plays a central role in modelling all these phenomena, and
observations provide strong evidence for the existence of black
holes with the properties predicted by the theory.
Black holes are also sought-after targets in the search for
gravitational waves (cf.
Gravitational waves,
above). Merging black hole binaries should lead to some of the
strongest gravitational wave signals reaching detectors here on
Earth, and the phase directly before the merger ("chirp") could be
used as a "
standard candle" to
deduce the distance to the merger events–and hence serve as a probe
of cosmic expansion at large distances. The gravitational waves
produced as a stellar black hole plunges into a supermassive one
should provide direct information about supermassive black hole's
geometry.
Cosmology
The current models of cosmology are based on Einstein's equations
including
cosmological
constant Λ, which has important influence on the large-scale
dynamics of the cosmos,
- R_{ab} - {\textstyle 1 \over 2}R\,g_{ab} + \Lambda\ g_{ab} =
\kappa\, T_{ab}
where
g_{ab} is the
spacetime metric.
Isotropic and
homogeneous
solutions of these enhanced equations, the
Friedmann-Lemaître-Robertson-Walker
solutions, allow physicists to model a universe that has
evolved over the past 14
billion years from a hot, early
Big Bang phase. Once a small number of
parameters (for example the universe's mean
matter density) have been
fixed by astronomical observation, further observational data can
be used to put the models to the test. Predictions, all successful,
include the initial abundance of chemical elements formed in a
period of
primordial
nucleosynthesis, the large-scale structure of the universe, and
the existence and properties of a "
thermal echo" from the early cosmos, the
cosmic background
radiation.
Astronomical observations of the cosmological expansion rate allow
the total amount of matter in the universe to be estimated,
although the nature of that matter remains mysterious in part.
About 90 percent of all matter appears to be so-called
dark matter, which has mass (or,
equivalently, gravitational influence), but does not interact
electromagnetically and, hence, cannot be observed directly. There
is no generally accepted description of this new kind of matter,
within the framework of known
particle
physics or otherwise. Observational evidence from redshift
surveys of distant
supernovae and
measurements of the cosmic background radiation also show that the
evolution of our universe is significantly influenced by a
cosmological constant resulting in an
acceleration of cosmic expansion or, equivalently, by a form of
energy with an unusual
equation of
state, known as
dark energy, the
nature of which remains unclear.
A so-called
inflationary phase, an
additional phase of strongly accelerated expansion at cosmic times
of around 10^{-33} seconds, was hypothesized in 1980 to account for
several puzzling observations that were unexplained by classical
cosmological models, such as the nearly perfect homogeneity of the
cosmic background radiation. Recent measurements of the cosmic
background radiation have resulted in the first evidence for this
scenario. However, there is a bewildering variety of possible
inflationary scenarios, which cannot be restricted by current
observations. An even larger question is the physics of the
earliest universe, prior to the inflationary phase and close to
where the classical models predict the big bang
singularity. An authoritative
answer would require a complete theory of
quantum gravity, which has not yet been
developed (cf. the section on
quantum gravity,
below).
Advanced concepts
Causal structure and global geometry
In general relativity, no material body can catch up with or
overtake a light pulse. No influence from an event A can reach any
other location X before light sent out at A to X. In consequence,
an exploration of all light worldlines (
null geodesic) yields key
information about the spacetime's causal structure. This structure
can be displayed using
Penrose-Carter
diagrams in which infinitely large regions of space and
infinite time intervals are shrunk ("
compactified") so as to fit
onto a finite map, while light still travels along diagonals as in
standard
spacetime diagrams.
Aware of the importance of causal structure,
Roger Penrose and others developed what is
known as
global geometry. In global
geometry, the object of study is not one particular
solution (or
family of solutions) to Einstein's equations. Rather, relations
that hold true for all geodesics, such as the
Raychaudhuri equation, and additional
non-specific assumptions about the nature of
matter (usually in the form of so-called
energy conditions) are used to derive
general results.
Horizons
Using global geometry, some spacetimes can be shown to contain
boundaries called
horizons, which
demarcate one region from the rest of
spacetime. The best-known examples are
black holes: if mass is compressed into a
sufficiently compact region of space (as specified in the
hoop conjecture, the relevant length scale
is the
Schwarzschild radius),
no light from inside can escape to the outside. Since no object can
overtake a light pulse, all interior matter is imprisoned as well.
Passage from the exterior to the interior is still possible,
showing that the boundary, the black hole's
horizon, is
not a physical barrier.
Early studies of black holes relied on
explicit solutions of
Einstein's equations,
notably the spherically symmetric
Schwarzschild solution (used to
describe a
static black hole) and
the axisymmetric
Kerr solution (used
to describe a rotating,
stationary black hole, and introducing
interesting features such as the
ergosphere). Using global geometry, later studies
have revealed more general properties of black holes. In the long
run, they are rather simple objects characterized by eleven
parameters specifying
energy,
linear momentum,
angular momentum, location at a specified
time and
electric charge. This is
stated by the
black hole uniqueness
theorems: "black holes have no hair", that is, no
distinguishing marks like the hairstyles of humans. Irrespective of
the complexity of a gravitating object collapsing to form a black
hole, the object that results (having emitted
gravitational waves) is very
simple.
Even more remarkably, there is a general set of laws known as
black hole mechanics, which is
analogous to the
laws of
thermodynamics. For instance, by the second law of black hole
mechanics, the area of the event horizon of a general black hole
will never decrease with time, analogous to the
entropy of a thermodynamic system. This limits the
energy that can be extracted by classical means from a rotating
black hole (e.g. by the
Penrose
process). There is strong evidence that the laws of black hole
mechanics are, in fact, a subset of the laws of thermodynamics, and
that the black hole area is proportional to its entropy. This leads
to a modification of the original laws of black hole mechanics: for
instance, as the second law of black hole mechanics becomes part of
the second law of thermodynamics, it is possible for black hole
area to decrease—as long as other processes ensure that, overall,
entropy increases. As thermodynamical objects with non-zero
temperature, black holes should emit
thermal radiation. Semi-classical
calculations indicate that indeed they do, with the surface gravity
playing the role of temperature in
Planck's
law. This radiation is known as
Hawking radiation (cf. the
quantum
theory section, below).
There are other types of horizons. In an expanding universe, an
observer may find that some regions of the past cannot be observed
("
particle horizon"), and some
regions of the future cannot be influenced (event horizon). Even in
flat Minkowski space, when described by an accelerated observer
(
Rindler space), there will be
horizons associated with a semi-classical radiation known as
Unruh radiation.
Singularities
Another general—and quite disturbing—feature of general relativity
is the appearance of spacetime boundaries known as singularities.
Spacetime can be explored by following up on timelike and lightlike
geodesics—all possible ways that light and particles in free fall
can travel. But some solutions of Einstein's equations have "ragged
edges"—regions known as
spacetime
singularities, where the paths of light and falling particles
come to an abrupt end, and geometry becomes ill-defined. In the
more interesting cases, these are "curvature singularities", where
geometrical quantities characterizing spacetime curvature, such as
the
Ricci scalar, take on infinite
values. Well-known examples of spacetimes with future
singularities—where
worldlines end—are the
Schwarzschild solution, which
describes a singularity inside an eternal static black hole, or the
Kerr solution with its ring-shaped
singularity inside an eternal rotating black hole. The
Friedmann-Lemaître-Robertson-Walker
solution, and other spacetimes describing universes, have past
singularities on which worldlines begin, namely
big bang singularities, and some have future
singularities (
big crunch) as well.
Given that these examples are all highly symmetric—and thus
simplified—it is tempting to conclude that the occurrence of
singularities is an artefact of idealization. The famous
singularity theorems, proved using the
methods of global geometry, say otherwise: singularities are a
generic feature of general relativity, and unavoidable once the
collapse of an object with realistic matter properties has
proceeded beyond a certain stage and also at the beginning of a
wide class of expanding universes. However, the theorems say little
about the properties of singularities, and much of current research
is devoted to characterizing these entities' generic structure
(hypothesized e.g. by the so-called
BKL
conjecture). The
cosmic
censorship hypothesis states that all realistic future
singularities (no perfect symmetries, matter with realistic
properties) are safely hidden away behind a horizon, and thus
invisible to all distant observers. While no formal proof yet
exists, numerical simulations offer supporting evidence of its
validity.
Evolution equations
Each
solution
of Einstein's equation encompasses the whole history of a
universe — it is not just some snapshot of how things are, but a
whole, possibly matter-filled,
spacetime.
It describes the state of matter and geometry everywhere and at
every moment in that particular universe. Due to its
general covariance, Einstein's theory is
not sufficient by itself to determine the
time evolution of the metric tensor. It must
be combined with a
coordinate
condition, which is analogous to
gauge
fixing in other field theories.
To understand Einstein's equations as
partial differential
equations, it is helpful to formulate them in a way that
describes the evolution of the universe over time. This is done in
so-called "3+1" formulations, where spacetime is split into three
space dimensions and one time dimension. The best-known example is
the
ADM formalism. These
decompositions show that the spacetime evolution equations of
general relativity are well-behaved: solutions always
exist, and are
uniquely defined, once suitable initial
conditions have been specified. Such formulations of Einstein's
field equations are the basis of
numerical relativity.
Global and quasi-local quantities
The notion of evolution equations is intimately tied in with
another aspect of general relativistic physics. In Einstein's
theory, it turns out to be impossible to find a general definition
for a seemingly simple property such as a system's total
mass (or
energy). The main reason
is that the gravitational field—like any physical field—must be
ascribed a certain energy, but that it proves to be fundamentally
impossible to localize that energy.
Nevertheless, there are possibilities to define a system's total
mass, either using a hypothetical "infinitely distant observer"
(
ADM mass) or suitable symmetries (
Komar mass). If one excludes from the system's
total mass the energy being carried away to infinity by
gravitational waves, the result is the so-called
Bondi mass at null infinity. Just as in
classical physics, it can be
shown that these masses are positive. Corresponding global
definitions exist for
momentum and
angular momentum. There have also been a
number of attempts to define
quasi-local quantities, such
as the mass of an isolated system formulated using only quantities
defined within a finite region of space containing that system. The
hope is to obtain a quantity useful for general statements about
isolated systems, such as a more
precise formulation of the
hoop
conjecture.
Relationship with quantum theory
If general relativity is considered one of the two pillars of
modern physics,
quantum theory,
the basis of our understanding of matter from
elementary particles to
solid state physics, is the other.
However, it is still an open question as to how the concepts of
quantum theory will be reconciled with those of general
relativity.
Quantum field theory in curved spacetime
Ordinary
quantum field
theories, which form the basis of modern
elementary particle physics, are
defined in flat
Minkowski space,
which is an excellent approximation when it comes to describing the
behavior of microscopic particles in weak gravitational fields like
those found on Earth. In order to describe situations in which
gravity is strong enough to influence (quantum) matter, yet not
strong enough to require quantization itself, physicists have
formulated quantum field theories in curved spacetime. These
theories rely on
classical general
relativity to describe a curved background spacetime, and define a
generalized quantum field theory to describe the behavior of
quantum matter within that spacetime. Using this formalism, it can
be shown that black holes emit a blackbody spectrum of particles
known as
Hawking radiation,
leading to the possibility that they
evaporate over time. As briefly
mentioned
above, this
radiation plays an important role for the thermodynamics of black
holes.
Quantum gravity
The demand for consistency between a quantum description of matter
and a geometric description of spacetime, as well as the appearance
of
singularities (where
curvature length scales become microscopic), indicate the need for
a full theory of
quantum gravity:
for an adequate description of the interior of black holes, and of
the very early universe, a theory is required in which gravity and
the associated geometry of spacetime are described in the language
of quantum physics. Despite major efforts, no complete and
consistent theory of quantum gravity is currently known, even
though a number of promising candidates exist.
Attempts to generalize ordinary quantum field theories, used in
elementary particle
physics to describe fundamental interactions, so as to include
gravity have led to serious problems. At low energies, this
approach proves successful, in that it results in an acceptable
effective field theory of
gravity. At very high energies, however, the result are models
devoid of all predictive power ("
non-renormalizability").
One attempt to overcome these limitations is
string theory, a quantum theory not of
point particles, but of minute
one-dimensional extended objects. The theory promises to be a
unified description of all
particles and interactions, including gravity; the price to pay is
unusual features such as six
extra
dimensions of space in addition to the usual three. In what is
called the
second
superstring revolution, it was conjectured that both string
theory and a unification of general relativity and
supersymmetry known as
supergravity form part of a hypothesized
eleven-dimensional model known as
M-theory,
which would constitute a uniquely defined and consistent theory of
quantum gravity.
Another approach starts with the
canonical quantization procedures of
quantum theory. Using the initial-value-formulation of general
relativity (cf. the section on evolution equations,
above), the result is
the
Wheeler-deWitt equation
(an analogue of the
Schrödinger equation) which,
regrettably, turns out to be ill-defined. However, with the
introduction of what are now known as
Ashtekar variables, this leads to a
promising model known as
loop
quantum gravity. Space is represented by a web-like structure
called a
spin network, evolving over
time in discrete steps.
Depending on which features of general relativity and quantum
theory are accepted unchanged, and on what level changes are
introduced, there are numerous other attempts to arrive at a viable
theory of quantum gravity, some examples being dynamical
triangulations, causal sets,
twistor models
or the
path-integral based
models of
quantum cosmology.
All candidate theories still have major formal and conceptual
problems to overcome. They also face the common problem that, as
yet, there is no way to put quantum gravity predictions to
experimental tests (and thus to decide between the candidates where
their predictions vary), although there is hope for this to change
as future data from cosmological observations and particle physics
experiments becomes available.
Current status
General relativity has emerged as a highly successful model of
gravitation and cosmology, which has so far passed every
unambiguous observational and experimental test. Even so, there are
strong indications the theory is incomplete. The problem of quantum
gravity and the question of the reality of
spacetime singularities remain open.
Observational data that is taken as evidence for
dark energy and
dark
matter could indicate the need for new physics, and while the
so-called
Pioneer anomaly
might yet admit of a conventional explanation, it, too, could be a
harbinger of new physics. Even taken as is, general relativity is
rich with possibilities for further exploration. Mathematical
relativists seek to understand the nature of singularities and the
fundamental properties of Einstein's equations, and increasingly
powerful computer simulations (such as those describing merging
black holes) are run. The race for the first direct detection of
gravitational waves continues apace, in the hope of creating
opportunities to test the theory's validity for much stronger
gravitational fields than has been possible to date. More than
ninety years after its publication, general relativity remains a
highly active area of research.
See also
Notes
- This development is traced in chapters 9 through 15 of and in ;
an up-to-date collection of current research, including reprints of
many of the original articles, is ; an accessible overview can be
found in . An early key article is , cf. . The publication
featuring the field equations is , cf. .
- See , and (later complemented in ).
- , cf. .
- Hubble's original article is ; an accessible overview is given
in .
- As reported in . Einstein's condemnation would prove to be
premature, cf. the sectionCosmology, below.
- Cf. .
- Cf. and .
- Cf. .
- Cf. and .
- Cf. the sections
Orbital effects and the relativity of direction,
Gravitational time dilation and frequency shift and
Light deflection and gravitational time delay, and references
therein.
- Cf. the section Cosmology and references
therein; the historical development is traced in .
- The following exposition re-traces that of .
- See, for instance, .
- See .
- See or .
- Cf. ; similar accounts can be found in most other
popular-science books on general relativity.
- See , , and . The simple thought experiment in question was
first described in .
- See .
- Good introductions are, in order of increasing presupposed
knowledge of mathematics, , , and ; for accounts of precision
experiments, cf. part IV of .
- An in-depth comparison between the two symmetry groups can be
found in .
- For instance ; a thorough treatment can be found in .
- E.g.
- Cf. and .
- See ; a derivation can be found e.g. in . For the experimental
evidence, cf. the section
Gravitational time dilation and frequency shift, below.
- Cf. ; for an elementary account, see chapter 2 of ; there are,
however, some differences between the modern version and Einstein's
original concept used in the historical derivation of general
relativity, cf. .
- for the experimental evidence, see once more section
Gravitational time dilation and frequency shift. Choosing a
different connection with non-zero torsion leads to a modified theory known
as Einstein-Cartan theory.
- Cf. ; ; .
- See ; for similar derivations, see sections 1 and 2 of ch. 7 in
. The Einstein tensor is the only divergence-free tensor that is a
function of the metric coefficients, their first and second
derivatives at most, and allows the spacetime of special relativity
as a solution in the absence of sources of gravity, cf. . The
tensors on both side are of second rank, that is, they can each be
thought of as 4×4 matrices, each of which contains ten independent
terms; hence, the above represents ten coupled equations. The fact
that, as a consequence of geometric relations known as
Bianchi identities, the Einstein tensor
satisfies a further four identities reduces these to six
independent equations, e.g. .
- E.g. .
- Cf. and section 3 in ch. 7 of , , and , respectively.
- E.g. , or, in fact, any other text-book on general
relativity.
- At least approximately, cf. .
- E.g. p. xi in .
- E.g. .
- E.g. in .
- For the (conceptual and historical) difficulties in defining a
general principle of relativity and separating it from the notion
of general covariance, see .
- E.g. section 5 in ch. 12 of .
- Cf. the introductory chapters of .
- A review showing Einstein's equation in the broader context of
other PDEs with physical significance is .
- For background information and a list of solutions, cf. ; a
more recent review can be found in .
- E.g. chapters 3, 5, and 6 of .
- E.g. ch. 4 and sec. 3.3. in .
- Brief descriptions of these and further interesting solutions
can be found in .
- See for an overview.
- For instance .
- E.g. .
- Cf. section 3.2 of as well as .
- Cf. and . In fact, Einstein derived these effects using the
equivalence principle as early as 1907, cf. and the description in
.
- ; .
- Pound-Rebka experiment, see , ; ; a
list of further experiments is given in .
- E.g. ; the most recent and most accurate Sirius B measurements
are published in .
- Starting with the Hafele-Keating experiment,
and , and culminating in the Gravity Probe A experiment; an overview of
experiments can be found in .
- GPS is continually tested by comparing atomic clocks on the
ground and aboard orbiting satellites; for an account of
relativistic effects, see and .
- Reviews are given in and .
- General overviews can be found in section 2.1. of Will 2006;
Will 2003, pp. 32–36; .
- Cf. .
- Cf. for the classic early measurements by the Eddington
expeditions; for an overview of more recent measurements, see . For
the most precise direct modern observations using quasars, cf.
.
- This is not an independent axiom; it can be derived from
Einstein's equations and the Maxwell Lagrangian using a WKB
approximation, cf. .
- A brief descriptions and pointers to the literature can be
found in .
- See ; for the historical examples, ; in fact, Einstein
published one such derivation as . Such calculations tacitly assume
that the geometry of space is Euclidean, cf. .
- From the standpoint of Einstein's theory, these derivations
take into account the effect of gravity on time, but not its
consequences for the warping of space, cf. .
- For the Sun's gravitational field using radar signals reflected
from planets such as Venus
and Mercury, cf. , with a pedagogical
introduction to be found in ; for signals actively sent back by
space probes (transponder measurements), cf. ; for an
overview, see ; for more recent measurements using signals received
from a pulsar that is part
of a binary system, the gravitational field causing the time delay
being that of the other pulsar, cf. .
- .
- These have been indirectly observed through the loss of energy
in binary pulsar systems such as the Hulse-Taylor
binary, the subject of the 1993 Nobel Prize in physics. A
number of projects are underway to attempt to observe directly the
effects of gravitational waves. For an overview, see . Unlike
electromagnetic waves, the dominant contribution for gravitational
waves is not the dipole,
but the quadrupole; see .
- Most advanced textbooks on general relativity contain a
description of these properties, e.g. .
- For example .
- .
- See , .
- See for a brief introduction to the methods of numerical
relativity, and for the connection with gravitational wave
astronomy.
- See and .
- See .
- See .
- The most precise measurements are VLBI measurements of planetary positions; see , , ; for
an overview, .
- See .
- A figure that includes error bars is figure 7, in section 5.1,
of .
- See and ; an accessible account can be found in .
- An overview can be found in ; for the pulsar discovery, see ;
for the initial evidence for gravitational radiation, see .
- Cf. .
- See e.g. , .
- See , .
- See and, for a more recent review, .
- See .
- E.g. ,
- E.g. , ; for a more recent review, see .
- E.g. , ,
- A mission description can be found in ; a first post-flight
evaluation is given in ; further updates will be available on the
mission website .
- For overviews of gravitational lensing and its applications,
see and .
- For a simple derivation, see ; cf. .
- See .
- Images of all the known lenses can be found on the pages of the
CASTLES project, .
- For an overview, see .
- See .
- For an overview, ; accessible accounts can be found in and
.
- An overview is given in .
- See .
- See .
- Cf. .
- Cf. .
- See .
- E.g. .
- Cf. and the accompanying summary .
- Cf.
- For the basic mechanism, see ; for more about the different
types of astronomical objects associated with this, cf. .
- For a review, see . Interestingly, to a distant observer, some
of these jets even appear to move faster than
light; this, however, can be explained as an optical illusion
that does not violate the tenets of relativity, see .
- For stellar end states, cf. or, for more recent numerical work,
; for supernovae, there are still major problems to be solved, cf.
; for simulating accretion and the formation of jets, cf. . Also,
relativistic lensing effects are thought to play a role for the
signals received from X-ray pulsars, cf. .
- The evidence includes limits on compactness from the
observation of accretion-driven phenomena ("Eddington
luminosity"), see , observations of stellar dynamics in the
center of our own Milky
Way galaxy, cf. , and indications that at least some of the
compact objects in question appear to have no solid surface, which
can be deduced from the examination of X-ray bursts for which the central compact
object is either a neutron star or a black hole; cf. for an overview, .
Observations of the "shadow" of the Milky Way galaxy's central
black hole horizon are eagerly sought for, cf. .
- Cf. .
- E.g. .
- Originally ; cf. the description in .
- See .
- See ; use of these models is justified by the fact that, at
large scales of around hundred million light-years and more, our own universe
indeed appears to be isotropic and homogeneous, cf. .
- E.g. with WMAP data, see
.
- These tests involve the separate observations detailed further
on, see, e.g., fig. 2 in .
- See ; for a recent account of predictions, see ; an accessible
account can be found in ; compare with the observations in , , ,
and .
- A review can be found in and ; for more recent results, see
.
- Cf. and, for a pedagogical introduction, see ; for the initial
detection, see and, for precision measurements by satellite
observatories, (COBE) and
(WMAP). Future measurements
could also reveal evidence about gravitational waves in the early
universe; this additional information is contained in the
background radiation's polarization, cf. and .
- Evidence for this comes from the determination of cosmological
parameters and additional observations involving the dynamics of
galaxies and galaxy clusters cf. chapter 18 of , evidence from
gravitational lensing, cf. , and simulations of large-scale
structure formation, see .
- See , and ; in particular, observations indicate that all but a
negligible portion of that matter is not in the form of the usual
elementary particles ("non-baryonic matter"), cf. .
- Namely, some physicists have questioned whether or not the
evidence for dark matter is, in fact, evidence for deviations from
the Einsteinian (and the Newtonian) description of gravity cf. the
overview in .
- See ; an accessible overview is given in . Here, too,
scientists have argued that the evidence indicates not a new form
of energy, but the need for modifications in our cosmological
models, cf. ; aforementioned modifications need not be
modifications of general relativity, they could, for example, be
modifications in the way we treat the inhomogeneities in the
universe, cf. .
- A good introduction is ; for a more recent review, see .
- More precisely, these are the flatness problem, the horizon problem, and
the monopole problem; a pedagogical
introduction can be found in , see also .
- See .
- More concretely, the potential function that is crucial to determining
the dynamics of the inflaton is simply postulated, but not derived from
an underlying physical theory.
- See .
- See , , and
- E.g. and
- See ; for an account of more recent numerical studies, see
.
- For an account of the evolution of this concept, see . A more
exact mathematical description distinguishes several kinds of
horizon, notably event horizons and apparent horizons
cf. or ; there are also more intuitive definitions for isolated
systems that do not require knowledge of spacetime properties at
infinity, cf. .
- For first steps, cf. ; see or for a derivation, and as well as
as overviews of more recent results.
- The laws of black hole mechanics were first described in ; a
more pedagogical presentation can be found in ; for a more recent
review, see chapter 2 of . A thorough, book-length introduction
including an introduction to the necessary mathematics . For the
Penrose process, see .
- See , .
- The fact that black holes radiate, quantum mechanically, was
first derived in ; a more thorough derivation can be found in . A
review is given in chapter 3 of .
- Cf. .
- Horizons: cf. . Unruh effect: , cf. .
- See , .
- See ; a more extensive treatment of this solution can be found
in .
- See ; for a more extensive treatment, cf. .
- See ; a closer look at the singularity itself is taken in
- Namely when there are trapped null surfaces, cf. .
- See .
- The conjecture was made in ; for a more recent review, see . An
accessible exposition is given by .
- The restriction to future singularities naturally excludes
initial singularities such as the big bang singularity, which in
principle be visible to observers at later cosmic time. The cosmic
censorship conjecture was first presented in ; a text-book level
account is given in . For numerical results, see the review .
- Cf. .
- ; for a pedagogical introduction, see .
- and ; for a pedagogical introduction, see ; an online review
can be found in .
- See ; for a review of the basics of numerical relativity,
including the problems arising from the peculiarities of Einstein's
equations, see .
- Cf. .
- .
- Cf. ; for a pedagogical introduction, see ; although defined in
a totally different way, it can be shown to be equivalent to the
ADM mass for stationary spacetimes, cf. .
- For a pedagogical introduction, see .
- See the various references given on p. 295 of ; this is
important for questions of stability—if there were negative mass
states, then flat, empty Minkowski space, which has mass zero, could
evolve into these states.
- E.g. .
- Such quasi-local mass-energy definitions are the Hawking energy,
Geroch
energy, or Penrose's quasi-local energy-momentum based on
twistor methods; cf. the
review article .
- An overview of quantum theory can be found in standard
textbooks such as ; a more elementary account is given in .
- Cf. textbooks such as , , or ; a more accessible overview can
be found in .
- Cf. and .
- For Hawking radiation , ; an accessible introduction to black
hole evaporation can be found in .
- Cf. chapter 3 in .
- Put simply, matter is the source of spacetime curvature, and
once matter has quantum properties, we can expect spacetime to have
them as well. Cf. section 2 in .
- E.g. p. 407ff. in .
- A timeline and overview can be found in .
- See .
- In particular, a technique known as renormalization, an
integral part of deriving predictions which take into account
higher-energy contributions, cf. chapters 17 and 18 of , fails in
this case; cf. .
- An accessible introduction at the undergraduate level can be
found in ; more complete overviews can be found in and .
- At the energies reached in current experiments, these strings
are indistinguishable from point-like particles, but, crucially,
different modes
of oscillation of one and the same type of fundamental string
appear as particles with different (electric and other) charge, e.g. . The theory is
successful in that one mode will always correspond to a
graviton, the
messenger particle of gravity, e.g.
.
- E. g. .
- E.g. .
- E.g. , .
- Cf. section 3 in .
- These variables represent geometric gravity using mathematical
analogues of electric and magnetic fields; cf. , .
- For a review, see ; more extensive accounts can be found in ,
as well as in the lecture notes .
- See e.g. the systematic expositions in and .
- See .
- See .
- See ch. 33 in and references therein.
- Cf. .
- E.g. , .
- Cf. ; .
- Cf. the section Quantum gravity,
above.
- Cf. the section Cosmology, above.
- 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.
- For the most recent papers on gravitational wave polarizations
of inspiralling compact binaries, see , and ; for a review of work
on compact binaries, see and ; for a general review of experimental
tests of general relativity, see .
- A good starting point for a snapshot of present-day research in
relativity is the electronic review journal Living
Reviews in Relativity.
References
- ; original paper in Russian:
Further reading
- Popular books
- Beginning undergraduate textbooks
- Advanced undergraduate textbooks
- Graduate level textbooks
External links
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