General relativity (GR, also known as the
general theory of relativity or GTR) is the
geometric theory of gravitation published
by Albert Einstein in 1915 and the current
description of gravitation in modern physics.
General relativity generalizes special relativity
and Newton's law of universal gravitation,
providing a unified description of gravity
as a geometric property of space and time,
or spacetime. In particular, the curvature
of spacetime is directly related to the energy
and momentum of whatever matter and radiation
are present. The relation is specified by
the Einstein field equations, a system of
partial differential equations.
Some 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, gravitational lensing, the
gravitational redshift of light, and the gravitational
time delay. The predictions of general relativity
in relation to classical physics 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. For example, it implies 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 ample
evidence that the intense radiation emitted
by certain kinds of astronomical objects is
due to black holes; for example, microquasars
and active galactic nuclei result from the
presence of stellar black holes and supermassive
black holes, respectively. The bending of
light by gravity can lead to the phenomenon
of gravitational lensing, in which 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 observed directly
by the physics collaboration LIGO. In addition,
general relativity is the basis of current
cosmological models of a consistently expanding
universe.
Widely acknowledged as a theory of extraordinary
beauty, general relativity has often been
described as the most beautiful of all existing
physical theories.
== History ==
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
presentation to the Prussian Academy of Science
in November 1915 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 and radiation
are 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 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 electrically 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
match that observational presumption. 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 during the total solar eclipse
of May 29, 1919, 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 quasars as one of these objects'
astrophysical manifestations. Ever more precise
solar system tests confirmed the theory's
predictive power, and relativistic cosmology,
too, became amenable to direct observational
tests.Over the years, general relativity has
acquired a reputation as a theory of extraordinary
beauty. Subrahmanyan Chandrasekhar has noted
that at multiple levels, general relativity
exhibits what Francis Bacon has termed, a
"strangeness in the proportion" (i.e. elements
that excite wonderment and surprise). It juxtaposes
fundamental concepts (space and time versus
matter and motion) which had previously been
considered as entirely independent. Chandrasekhar
also noted that Einstein's only guides in
his search for an exact theory were the principle
of equivalence and his sense that a proper
description of gravity should be geometrical
at its basis, so that there was an "element
of revelation" in the manner in which Einstein
arrived at his theory. Other elements of beauty
associated with the general theory of relativity
are its simplicity, symmetry, the manner in
which it incorporates invariance and unification,
and its perfect logical consistency.
== From classical mechanics to general relativity
==
General relativity can be 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 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 curved
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 a rocket that
is accelerating at a rate equal to that of
the gravitational field.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,
spacetime 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 resulting Newton–Cartan
theory 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 (special) 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 includes translations,
rotations and boosts.) The differences between
the two become significant when 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
image). 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 or conformal geometry.
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: pressure and shear. Using
the equivalence principle, this tensor is
readily generalized to curved spacetime. 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:
On the left-hand side is the Einstein tensor,
a specific divergence-free combination of
the Ricci tensor
R
μ
ν
{\displaystyle R_{\mu \nu }}
and the metric. Where
G
μ
ν
{\displaystyle G_{\mu \nu }}
is symmetric. In particular,
R
=
g
μ
ν
R
μ
ν
{\displaystyle R=g^{\mu \nu }R_{\mu \nu }\,}
is the curvature scalar. The Ricci tensor
itself is related to the more general Riemann
curvature tensor as
R
μ
ν
=
R
α
μ
α
ν
.
{\displaystyle R_{\mu \nu }={R^{\alpha }}_{\mu
\alpha \nu }.\,}
On the right-hand side,
T
μ
ν
{\displaystyle T_{\mu \nu }}
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 = 8πG/c4, 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 results are the vacuum
Einstein equations,
R
μ
ν
=
0.
{\displaystyle R_{\mu \nu }=0.\,}
=== Alternatives to general relativity ===
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
Whitehead's theory, Brans–Dicke theory,
teleparallelism, f(R) gravity 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, pseudo-Riemannian manifold
representing spacetime, and the energy–momentum
contained in that spacetime. 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, i.e. it is
background independent. 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 solutions
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 many years of
research that followed Einstein's initial
publication.
=== Gravitational time dilation and frequency
shift ===
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 will follow the curvature of spacetime
as it passes near a star. This effect was
initially confirmed by observing the light
of stars or distant quasars being deflected
as it passes the Sun.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 delay), 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 γ, which encodes the influence
of gravity on the geometry of space.
=== Gravitational waves ===
Predicted in 1916 by Albert Einstein, there
are gravitational waves: ripples in the metric
of spacetime that propagate at the speed of
light. These are one of several analogies
between weak-field gravity and electromagnetism
in that, they are analogous to electromagnetic
waves. On February 11, 2016, the Advanced
LIGO team announced that they had directly
detected gravitational waves from a pair of
black holes merging.The simplest type of such
a wave can be visualized by its action on
a ring of freely floating particles. A sine
wave propagating through such a ring towards
the reader distorts the ring in a characteristic,
rhythmic fashion (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
{\displaystyle 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 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 ====
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 Mercury's anomalous perihelion shift, 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).
Relativistic precession has been observed
for all planets that allow for accurate precession
measurements (Mercury, Venus, and Earth),
as well as in binary pulsar systems, where
it is larger by five orders of magnitude.In
general relativity the perihelion shift σ,
expressed in radians per revolution, is approximately
given by:
σ
=
24
π
3
L
2
T
2
c
2
(
1
−
e
2
)
,
{\displaystyle \sigma ={\frac {24\pi ^{3}L^{2}}{T^{2}c^{2}(1-e^{2})}}\
,}
where:
L
{\displaystyle L}
is the semi-major axis
T
{\displaystyle T}
is the orbital period
c
{\displaystyle c}
is the speed of light
e
{\displaystyle e}
is the orbital eccentricity
==== Orbital decay ====
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.
This is not the case 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. Because neutron stars
are immensely 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 0.3%.Near a rotating mass,
there are 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.
== 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). Detection
of these waves is a major goal of current
relativity-related research. Several land-based
gravitational wave detectors are currently
in operation, most notably the interferometric
detectors GEO 600, LIGO (two detectors), TAMA
300 and VIRGO. Various pulsar timing arrays
are using millisecond pulsars to detect gravitational
waves in the 10−9 to 10−6 Hertz frequency
range, which originate from binary supermassive
blackholes. A European space-based detector,
eLISA / NGO, is currently under development,
with a precursor mission (LISA Pathfinder)
having launched in December 2015.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. In February 2016, the Advanced
LIGO team announced that they had detected
gravitational waves from a black hole merger.
=== Black holes and other compact objects
===
Whenever the ratio of an object's mass to
its radius becomes sufficiently large, 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 of around 1.4 solar masses, and 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. Usually
a galaxy has one supermassive black hole with
a few million to a few billion solar masses
in its center, and its presence is thought
to have played an important role in the formation
of the galaxy and larger cosmic structures.
Astronomically, the most important property
of compact objects is that they provide a
supremely 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 the
supermassive black hole's geometry.
=== Cosmology ===
The current models of cosmology are based
on Einstein's field equations, which include
the cosmological constant Λ since it has
important influence on the large-scale dynamics
of the cosmos,
R
μ
ν
−
1
2
R
g
μ
ν
+
Λ
g
μ
ν
=
8
π
G
c
4
T
μ
ν
{\displaystyle R_{\mu \nu }-{\textstyle 1
\over 2}R\,g_{\mu \nu }+\Lambda \ g_{\mu \nu
}={\frac {8\pi G}{c^{4}}}\,T_{\mu \nu }}
where
g
μ
ν
{\displaystyle g_{\mu \nu }}
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% of all matter appears to be 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.An 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).
=== Time travel ===
Kurt Gödel showed that solutions to Einstein's
equations exist that contain closed timelike
curves (CTCs), which allow for loops in time.
The solutions require extreme physical conditions
unlikely ever to occur in practice, and it
remains an open question whether further laws
of physics will eliminate them completely.
Since then, other—similarly impractical—GR
solutions containing CTCs have been found,
such as the Tipler cylinder and traversable
wormholes.
== 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 geodesics) 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 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 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
solutions 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 artifact 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 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 "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 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 were considered to be
one of the two pillars of modern physics,
then quantum theory, the basis of understanding
matter from elementary particles to solid
state physics, would be the other. However,
how to reconcile quantum theory with general
relativity is still an open question.
=== 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 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.
Some have argued that at low energies, this
approach proves successful, in that it results
in an acceptable effective (quantum) field
theory of gravity. At very high energies,
however, the perturbative results are badly
divergent and lead to models devoid of predictive
power ("perturbative 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. evolution equations
above), the result is the Wheeler–deWitt
equation (an analogue of the Schrödinger
equation) which, regrettably, turns out to
be ill-defined without a proper ultraviolet
(lattice) cutoff. 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 the lattice theory of gravity based
on the Feynman Path Integral approach and
Regge Calculus, 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 many unambiguous observational
and experimental tests. However, 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. 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, while numerical relativists
run increasingly powerful computer simulations
(such as those describing merging black holes).
In February 2016, it was announced that the
existence of gravitational waves was directly
detected by the Advanced LIGO team on September
14, 2015. A century after its introduction,
general relativity remains a highly active
area of research.
== See also ==
== Notes ==
== References ==
== Further reading ==
=== Popular books ===
Geroch, R. (1981), General Relativity from
A to B, Chicago: University of Chicago Press,
ISBN 0-226-28864-1
Lieber, Lillian (2008), The Einstein Theory
of Relativity: A Trip to the Fourth Dimension,
Philadelphia: Paul Dry Books, Inc., ISBN 978-1-58988-044-3
Wald, Robert M. (1992), Space, Time, and Gravity:
the Theory of the Big Bang and Black Holes,
Chicago: University of Chicago Press, ISBN
0-226-87029-4
Wheeler, John; Ford, Kenneth (1998), Geons,
Black Holes, & Quantum Foam: a life in physics,
New York: W. W. Norton, ISBN 0-393-31991-1
=== Beginning undergraduate textbooks ===
Callahan, James J. (2000), The Geometry of
Spacetime: an Introduction to Special and
General Relativity, New York: Springer, ISBN
0-387-98641-3
Taylor, Edwin F.; Wheeler, John Archibald
(2000), Exploring Black Holes: Introduction
to General Relativity, Addison Wesley, ISBN
0-201-38423-XCS1 maint: Multiple names: authors
list (link)
=== Advanced undergraduate textbooks ===
B. F. Schutz (2009), A First Course in General
Relativity (Second Edition), Cambridge University
Press, ISBN 978-0-521-88705-2
Cheng, Ta-Pei (2005), Relativity, Gravitation
and Cosmology: a Basic Introduction, Oxford
and New York: Oxford University Press, ISBN
0-19-852957-0
Gron, O.; Hervik, S. (2007), Einstein's General
theory of Relativity, Springer, ISBN 978-0-387-69199-2
Hartle, James B. (2003), Gravity: an Introduction
to Einstein's General Relativity, San Francisco:
Addison-Wesley, ISBN 0-8053-8662-9
Hughston, L. P. & Tod, K. P. (1991), Introduction
to General Relativity, Cambridge: Cambridge
University Press, ISBN 0-521-33943-XCS1 maint:
Multiple names: authors list (link)
d'Inverno, Ray (1992), Introducing Einstein's
Relativity, Oxford: Oxford University Press,
ISBN 0-19-859686-3
Ludyk, Günter (2013). Einstein in Matrix
Form (1st ed.). Berlin: Springer. ISBN 978-3-642-35797-8.
=== Graduate-level textbooks ===
Carroll, Sean M. (2004), Spacetime and Geometry:
An Introduction to General Relativity, San
Francisco: Addison-Wesley, ISBN 0-8053-8732-3
Grøn, Øyvind; Hervik, Sigbjørn (2007),
Einstein's General Theory of Relativity, New
York: Springer, ISBN 978-0-387-69199-2
Landau, Lev D.; Lifshitz, Evgeny F. (1980),
The Classical Theory of Fields (4th ed.),
London: Butterworth-Heinemann, ISBN 0-7506-2768-9
Misner, Charles W.; Thorne, Kip. S.; Wheeler,
John A. (1973), Gravitation, W. H. Freeman,
ISBN 0-7167-0344-0
Stephani, Hans (1990), General Relativity:
An Introduction to the Theory of the Gravitational
Field, Cambridge: Cambridge University Press,
ISBN 0-521-37941-5
Wald, Robert M. (1984), General Relativity,
University of Chicago Press, ISBN 0-226-87033-2
== External links ==
Einstein Online – Articles on a variety
of aspects of relativistic physics for a general
audience; hosted by the Max Planck Institute
for Gravitational Physics
NCSA Spacetime Wrinkles – produced by the
numerical relativity group at the NCSA, with
an elementary introduction to general relativity
Einstein's General Theory of Relativity on
YouTube (lecture by Leonard Susskind recorded
September 22, 2008 at Stanford University).
Series of lectures on General Relativity given
in 2006 at the Institut Henri Poincaré (introductory/advanced).
General Relativity Tutorials by John Baez.
Brown, Kevin. "Reflections on relativity".
Mathpages.com. Retrieved May 29, 2005.
Carroll, Sean M. "Lecture Notes on General
Relativity". arXiv:gr-qc/9712019.
Moor, Rafi. "Understanding General Relativity".
Retrieved July 11, 2006.
Waner, Stefan. "Introduction to Differential
Geometry and General Relativity" (PDF). Retrieved
2015-04-05.
