Contents 1 History 2 From classical mechanics to general relativity 2.1
3 Definition and basic applications 3.1 Definition and basic properties 3.2 Model-building 4 Consequences of Einstein's theory 4.1
4.4.1
5
5.1 Gravitational lensing
5.2
6 Advanced concepts 6.1
7 Relationship with quantum theory 7.1
8 Current status 9 See also 10 Notes 11 References 12 Further reading 12.1 Popular books 12.2 Beginning undergraduate textbooks 12.3 Advanced undergraduate textbooks 12.4 Graduate-level textbooks 13 External links History[edit]
Main articles:
According to general relativity, objects in a gravitational field behave similarly to objects within an accelerating enclosure. For example, an observer will see a ball fall the same way in a rocket (left) as it does on Earth (right), provided that the acceleration of the rocket is equal to 9.8 m/s2 (the acceleration due to gravity at the surface of the Earth). 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.[18] 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.[19]
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.[20] 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.[21]
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
Light cone As intriguing as geometric
Einstein's field equations G μ ν ≡ R μ ν − 1 2 R g μ ν = 8 π G c 4 T μ ν displaystyle G_ mu nu equiv R_ mu nu - textstyle 1 over 2 R,g_ mu nu = 8pi G over c^ 4 T_ mu nu , 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
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.[33] 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.[34] 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[edit]
Main article: 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.[35]
Definition and basic applications[edit]
See also:
Schematic representation of the gravitational redshift of a light wave escaping from the surface of a massive body Assuming that the equivalence principle holds,[53] 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.[54]
Deflection of light (sent out from the location shown in blue) near a compact body (shown in gray)
Gravitational waves[edit] Main article: Gravitational wave Ring of test particles deformed by a passing (linearized, amplified for better visibility) gravitational wave Predicted in 1916[69][70] 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
10 − 21 displaystyle 10^ -21 or less. Data analysis methods routinely make use of the fact that
these linearized waves can be Fourier decomposed.[75]
Some exact solutions describe gravitational waves without any
approximation, e.g., a wave train traveling through empty space[76] or
Gowdy universes, varieties of an expanding cosmos filled with
gravitational waves.[77] 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.[78]
Orbital effects and the relativity of direction[edit]
Main article: Kepler problem in general relativity
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 Mercury's anomalous perihelion shift,
discovered earlier by
σ = 24 π 3 L 2 T 2 c 2 ( 1 − e 2 ) , displaystyle sigma = frac 24pi ^ 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[edit] Orbital decay for PSR1913+16: time shift in seconds, tracked over three decades.[86] 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
Einstein cross: four images of the same astronomical object, produced by a gravitational lens 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.[100]
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.[101] The earliest example was discovered in
1979;[102] since then, more than a hundred gravitational lenses have
been observed.[103] 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.[104]
Artist's impression of the space-borne gravitational wave detector LISA 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.[106] Several land-based gravitational wave detectors are
currently in operation, most notably the interferometric detectors GEO
600,
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 supremely efficient mechanism for converting
gravitational energy into electromagnetic radiation.[117] 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.[118] In particular, accretion can lead to relativistic
jets, focused beams of highly energetic particles that are being flung
into space at almost light speed.[119]
Cosmology[edit] This blue horseshoe is a distant galaxy that has been magnified and warped into a nearly complete ring by the strong gravitational pull of the massive foreground luminous red galaxy. Main article: Physical 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 8pi G c^ 4 ,T_ mu nu where g μ ν displaystyle g_ mu nu is the spacetime metric.[124]
Penrose–Carter diagram of an infinite Minkowski universe 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.[142]
Aware of the importance of causal structure,
The ergosphere of a rotating black hole, which plays a key role when it comes to extracting energy from such a black hole 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
Projection of a Calabi–Yau manifold, one of the ways of compactifying the extra dimensions posited by string theory 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.[181] Some have argued that at low energies, this approach proves successful, in that it results in an acceptable effective (quantum) field theory of gravity.[182] At very high energies, however, the perturbative results are badly divergent and lead to models devoid of predictive power ("perturbative non-renormalizability").[183] Simple spin network of the type used in loop quantum gravity. One attempt to overcome these limitations is string theory, a quantum
theory not of point particles, but of minute one-dimensional extended
objects.[184] The theory promises to be a unified description of all
particles and interactions, including gravity;[185] the price to pay
is unusual features such as six extra dimensions of space in addition
to the usual three.[186] 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[187] form part of a hypothesized eleven-dimensional model
known as M-theory, which would constitute a uniquely defined and
consistent theory of quantum gravity.[188]
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
Observation of gravitational waves from binary black hole merger GW150914.
Science portal
Notes[edit] ^ "GW150914:
References[edit] Alpher, R. A.; Herman, R. C. (1948), "Evolution of the universe",
Nature, 162 (4124): 774–775, Bibcode:1948Natur.162..774A,
doi:10.1038/162774b0
Anderson, J. D.; Campbell, J. K.; Jurgens, R. F.; Lau, E. L. (1992),
"Recent developments in solar-system tests of general relativity", in
Sato, H.; Nakamura, T., Proceedings of the Sixth Marcel Großmann
Meeting on General Relativity, World Scientific, pp. 353–355,
ISBN 981-02-0950-9
Arnold, V. I. (1989), Mathematical Methods of Classical Mechanics,
Springer, ISBN 3-540-96890-3
Arnowitt, Richard; Deser, Stanley; Misner, Charles W. (1962), "The
dynamics of general relativity", in Witten, Louis, Gravitation: An
Introduction to Current Research, Wiley, pp. 227–265
Arun, K.G.; Blanchet, L.; Iyer, B. R.; Qusailah, M. S. S. (2007),
"Inspiralling compact binaries in quasi-elliptical orbits: The
complete 3PN energy flux", Physical Review D, 77 (6),
arXiv:0711.0302 , Bibcode:2008PhRvD..77f4035A,
doi:10.1103/PhysRevD.77.064035
Ashby, Neil (2002), "
Further reading[edit] Popular books[edit] Geroch, R. (1981), General
Beginning undergraduate textbooks[edit] Callahan, James J. (2000), The
Advanced undergraduate textbooks[edit] B. F. Schutz (2009), A First Course in General
Graduate-level textbooks[edit] Carroll, Sean M. (2004),
External links[edit] Wikimedia Commons has media related to General relativity. Wikibooks has more on the topic of: General relativity Wikiversity has learning resources about General relativity
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