General relativity
General relativity |
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General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the
Reconciliation of general relativity with the laws of quantum physics remains a problem, however, as there is a lack of a self-consistent theory of quantum gravity. It is not yet known how gravity can be unified with the three non-gravitational forces: strong, weak and electromagnetic.
Einstein's theory has
Widely acknowledged as a theory of extraordinary beauty, general relativity has often been described as the most beautiful of all existing physical theories.[2]
History
The Einstein field equations are
During that period, general relativity remained something of a curiosity among physical theories. It was clearly superior to
General relativity has acquired a reputation as a theory of extraordinary beauty.[2][19][20] 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.[21] Other elements of beauty associated with the general theory of relativity are its simplicity and symmetry, the manner in which it incorporates invariance and unification, and its perfect logical consistency.[22]
In the preface to Relativity: The Special and the General Theory, Einstein said "The present book is intended, as far as possible, to give an exact insight into the theory of Relativity to those readers who, from a general scientific and philosophical point of view, are interested in the theory, but who are not conversant with the mathematical apparatus of theoretical physics. The work presumes a standard of education corresponding to that of a university matriculation examination, and, despite the shortness of the book, a fair amount of patience and force of will on the part of the reader. The author has spared himself no pains in his endeavour to present the main ideas in the simplest and most intelligible form, and on the whole, in the sequence and connection in which they actually originated."[23]
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.[24][25]
Geometry of Newtonian gravity
At the base of
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
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
Relativistic generalization
As intriguing as geometric Newtonian gravity may be, its basis, classical mechanics, is merely a
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
Special relativity is defined in the absence of gravity. For practical applications, it is a suitable model whenever gravity can be neglected. Bringing gravity into play, and assuming the universality of free fall motion, an analogous reasoning as in the previous section applies: there are no global
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
The same experimental data shows that time as measured by clocks in a gravitational field—
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
On the left-hand side is the Einstein tensor, , which is symmetric and a specific divergence-free combination of the Ricci tensor and the metric. In particular,
is the curvature scalar. The Ricci tensor itself is related to the more general Riemann curvature tensor as
On the right-hand side, is a constant and is the energy–momentum tensor. All tensors are written in abstract index notation.[41] 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 is found to be , where is the
In general relativity, the world line of a particle free from all external, non-gravitational force is a particular type of geodesic in curved spacetime. In other words, a freely moving or falling particle always moves along a geodesic.
The geodesic equation is:
where is a scalar parameter of motion (e.g. the proper time), and are
Total force in general relativity
In general relativity, the effective
A conservative total force can then be obtained as[citation needed]
where L is the angular momentum. The first term represents the force of Newtonian gravity, which is described by the inverse-square law. The second term represents the centrifugal force in the circular motion. The third term represents the relativistic effect.
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.[45]
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
While general relativity replaces the
As it is constructed using tensors, general relativity exhibits
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.[53]
Einstein's equations are nonlinear partial differential equations and, as such, difficult to solve exactly.
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.[59] 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[60] 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.[61] 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.[62]
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,
Gravitational redshift has been measured in the laboratory
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.[72]
This and related predictions follow from the fact that light follows what is called a light-like or
Closely related to light deflection is the Shapiro Time 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.[77] 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.[78]
Gravitational waves
Predicted in 1916
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).
Some exact solutions describe gravitational waves without any approximation, e.g., a wave train traveling through empty space
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.[89]
The effect can also be derived by using either the exact Schwarzschild metric (describing spacetime around a spherical mass)
In general relativity the perihelion shift , expressed in radians per revolution, is approximately given by[95]
where:
- is the semi-major axis
- is the orbital period
- is the speed of light in vacuum
- is the orbital eccentricity
Orbital decay
According to general relativity, a
The first observation of a decrease in orbital period due to the emission of gravitational waves was made by
Geodetic precession and frame-dragging
Several relativistic effects are directly related to the relativity of direction.
Near a rotating mass, there are gravitomagnetic or
Interpretations
Neo-Lorentzian Interpretation
This section needs expansion with: a description of the interpretation. You can help by adding to it. (April 2024) |
Examples of physicists who support neo-Lorentzian explanations of general relativity are Franco Selleri and Antony Valentini.[109]
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.[110] 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.[111] The
Gravitational lensing has developed into a tool of
Gravitational-wave astronomy
Observations of binary pulsars provide strong indirect evidence for the existence of gravitational waves (see
Observations of gravitational waves promise to complement observations in the electromagnetic spectrum.[121] 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.[122] In February 2016, the Advanced LIGO team announced that they had detected gravitational waves from a black hole merger.[81][82][83]
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
Astronomically, the most important property of compact objects is that they provide a supremely efficient mechanism for converting gravitational energy into electromagnetic radiation.
Black holes are also sought-after targets in the search for gravitational waves (cf.
Cosmology
The current models of cosmology are based on
where is the spacetime metric.
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.[141] There is no generally accepted description of this new kind of matter, within the framework of known particle physics[142] or otherwise.[143] 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.[144]
An
Exotic solutions: time travel, warp drives
Kurt Gödel showed[150] 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. Stephen Hawking introduced chronology protection conjecture, which is an assumption beyond those of standard general relativity to prevent time travel.
Some exact solutions in general relativity such as Alcubierre drive present examples of warp drive but these solutions requires exotic matter distribution, and generally suffers from semiclassical instability. [151]
Advanced concepts
Asymptotic symmetries
The spacetime symmetry group for special relativity is the Poincaré group, which is a ten-dimensional group of three Lorentz boosts, three rotations, and four spacetime translations. It is logical to ask what symmetries if any might apply in General Relativity. A tractable case might be to consider the symmetries of spacetime as seen by observers located far away from all sources of the gravitational field. The naive expectation for asymptotically flat spacetime symmetries might be simply to extend and reproduce the symmetries of flat spacetime of special relativity, viz., the Poincaré group.
In 1962 Hermann Bondi, M. G. van der Burg, A. W. Metzner[152] and Rainer K. Sachs[153] addressed this asymptotic symmetry problem in order to investigate the flow of energy at infinity due to propagating gravitational waves. Their first step was to decide on some physically sensible boundary conditions to place on the gravitational field at light-like infinity to characterize what it means to say a metric is asymptotically flat, making no a priori assumptions about the nature of the asymptotic symmetry group—not even the assumption that such a group exists. Then after designing what they considered to be the most sensible boundary conditions, they investigated the nature of the resulting asymptotic symmetry transformations that leave invariant the form of the boundary conditions appropriate for asymptotically flat gravitational fields. What they found was that the asymptotic symmetry transformations actually do form a group and the structure of this group does not depend on the particular gravitational field that happens to be present. This means that, as expected, one can separate the kinematics of spacetime from the dynamics of the gravitational field at least at spatial infinity. The puzzling surprise in 1962 was their discovery of a rich infinite-dimensional group (the so-called BMS group) as the asymptotic symmetry group, instead of the finite-dimensional Poincaré group, which is a subgroup of the BMS group. Not only are the Lorentz transformations asymptotic symmetry transformations, there are also additional transformations that are not Lorentz transformations but are asymptotic symmetry transformations. In fact, they found an additional infinity of transformation generators known as supertranslations. This implies the conclusion that General Relativity (GR) does not reduce to special relativity in the case of weak fields at long distances. It turns out that the BMS symmetry, suitably modified, could be seen as a restatement of the universal soft graviton theorem in quantum field theory (QFT), which relates universal infrared (soft) QFT with GR asymptotic spacetime symmetries.[154]
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.[155]
Aware of the importance of causal structure,
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[157]), 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.[158]
Early studies of black holes relied on explicit solutions of Einstein's equations, notably the spherically symmetric Schwarzschild solution (used to describe a
Even more remarkably, there is a general set of laws known as
There are many other types of horizons. In an expanding universe, an observer may find that some regions of the past cannot be observed ("
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
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.
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
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.[175] 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.[176] Such formulations of Einstein's field equations are the basis of numerical relativity.[177]
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.[178]
Nevertheless, there are possibilities to define a system's total mass, either using a hypothetical "infinitely distant observer" (
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.[185] However, how to reconcile quantum theory with general relativity is still an open question.
Quantum field theory in curved spacetime
Ordinary
Quantum gravity
The demand for consistency between a quantum description of matter and a geometric description of spacetime,[190] 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.[191] Despite major efforts, no complete and consistent theory of quantum gravity is currently known, even though a number of promising candidates exist.[192][193]
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.
One attempt to overcome these limitations is
Another approach starts with the
Depending on which features of general relativity and quantum theory are accepted unchanged, and on what level changes are introduced,[205] 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,[192] dynamical triangulations,[206] causal sets,[207] twistor models[208] or the path integral based models of quantum cosmology.[209]
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.[210]
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 that the theory is incomplete.[211] The problem of quantum gravity and the question of the reality of spacetime singularities remain open.[212] Observational data that is taken as evidence for dark energy and dark matter could indicate the need for new physics.[213]
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,[214] while numerical relativists run increasingly powerful computer simulations (such as those describing merging black holes).[215] In February 2016, it was announced that the existence of gravitational waves was directly detected by the Advanced LIGO team on 14 September 2015.[83][216][217] A century after its introduction, general relativity remains a highly active area of research.[218]
See also
- Alcubierre drive – Hypothetical FTL transportation by warping space (warp drive)
- Alternatives to general relativity – Proposed theories of gravity
- Contributors to general relativity
- Derivations of the Lorentz transformations
- Ehrenfest paradox – Paradox in special relativity
- Einstein–Hilbert action – Concept in general relativity
- Einstein's thought experiments – Albert Einstein's hypothetical situations to argue scientific points
- General relativity priority dispute – Debate about credit for general relativity
- Introduction to the mathematics of general relativity – non-technical introduction to the mathematics of general relativity
- Nordström's theory of gravitation – Predecessor to the theory of relativity
- Ricci calculus – Tensor index notation for tensor-based calculations
- Timeline of gravitational physics and relativity
- Weak Gravity Conjecture– Conjecture that gravity must be the weakest force
References
- ^ "GW150914: LIGO Detects Gravitational Waves". Black-holes.org. Retrieved 18 April 2016.
- ^ a b Landau & Lifshitz 1975, p. 228 "...the general theory of relativity...was established by Einstein, and represents probably the most beautiful of all existing physical theories."
- ^ Poincaré 1905
- University of St. Andrews, archived from the originalon 4 February 2015, retrieved 4 February 2015
- ^ Pais 1982, ch. 9 to 15, Janssen 2005; an up-to-date collection of current research, including reprints of many of the original articles, is Renn 2007; an accessible overview can be found in Renn 2005, pp. 110ff. Einstein's original papers are found in Digital Einstein, volumes 4 and 6. An early key article is Einstein 1907, cf. Pais 1982, ch. 9. The publication featuring the field equations is Einstein 1915, cf. Pais 1982, ch. 11–15
- ^ Moshe Carmeli (2008).Relativity: Modern Large-Scale Structures of the Cosmos. pp.92, 93.World Scientific Publishing
- ^ Grossmann for the mathematical part and Einstein for the physical part (1913). Entwurf einer verallgemeinerten Relativitätstheorie und einer Theorie der Gravitation (Outline of a Generalized Theory of Relativity and of a Theory of Gravitation), Zeitschrift für Mathematik und Physik, 62, 225–261. English translate
- ^ Schwarzschild 1916a, Schwarzschild 1916b and Reissner 1916 (later complemented in Nordström 1918)
- ^ Einstein 1917, cf. Pais 1982, ch. 15e
- ^ Hubble's original article is Hubble 1929; an accessible overview is given in Singh 2004, ch. 2–4
- ^ As reported in Gamow 1970. Einstein's condemnation would prove to be premature, cf. the section Cosmology, below
- ^ Pais 1982, pp. 253–254
- ^ Kennefick 2005, Kennefick 2007
- ^ Pais 1982, ch. 16
- ^ Thorne 2003, p. 74
- ^ Israel 1987, ch. 7.8–7.10, Thorne 1994, ch. 3–9
- ^ Sections Orbital effects and the relativity of direction, Gravitational time dilation and frequency shift and Light deflection and gravitational time delay, and references therein
- ^ Section Cosmology and references therein; the historical development is in Overbye 1999
- ^ Wald 1984, p. 3
- ^ Rovelli 2015, pp. 1–6 "General relativity is not just an extraordinarily beautiful physical theory providing the best description of the gravitational interaction we have so far. It is more."
- ^ Chandrasekhar 1984, p. 6
- ^ Engler 2002
- ^ The following exposition re-traces that of Ehlers 1973, sec. 1
- ^ Al-Khalili, Jim (26 March 2021). "Gravity and Me: The force that shapes our lives". www.bbc.co.uk. Retrieved 9 April 2021.
- ^ Arnold 1989, ch. 1
- ^ Ehlers 1973, pp. 5f
- ^ Will 1993, sec. 2.4, Will 2006, sec. 2
- ^ Wheeler 1990, ch. 2
- ^ Ehlers 1973, sec. 1.2, Havas 1964, Künzle 1972. The simple thought experiment in question was first described in Heckmann & Schücking 1959
- ^ Ehlers 1973, pp. 10f
- ^ Good introductions are, in order of increasing presupposed knowledge of mathematics, Giulini 2005, Mermin 2005, and Rindler 1991; for accounts of precision experiments, cf. part IV of Ehlers & Lämmerzahl 2006
- ^ An in-depth comparison between the two symmetry groups can be found in Giulini 2006
- ^ Rindler 1991, sec. 22, Synge 1972, ch. 1 and 2
- ^ Ehlers 1973, sec. 2.3
- ^ Ehlers 1973, sec. 1.4, Schutz 1985, sec. 5.1
- ^ Ehlers 1973, pp. 17ff; a derivation can be found in Mermin 2005, ch. 12. For the experimental evidence, cf. the section Gravitational time dilation and frequency shift, below
- ^ Rindler 2001, sec. 1.13; for an elementary account, see Wheeler 1990, ch. 2; there are, however, some differences between the modern version and Einstein's original concept used in the historical derivation of general relativity, cf. Norton 1985
- ^ Ehlers 1973, sec. 1.4 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
- ^ Ehlers 1973, p. 16, Kenyon 1990, sec. 7.2, Weinberg 1972, sec. 2.8
- Bianchi identities, the Einstein tensor satisfies a further four identities reduces these to six independent equations, e.g. Schutz 1985, sec. 8.3
- ^ Kenyon 1990, sec. 7.4
- ISBN 978-0-471-92567-5.
- ISBN 978-0-19-852957-6.
- ^ Brans & Dicke 1961, Weinberg 1972, sec. 3 in ch. 7, Goenner 2004, sec. 7.2, and Trautman 2006, respectively
- ^ Wald 1984, ch. 4, Weinberg 1972, ch. 7 or, in fact, any other textbook on general relativity
- ^ At least approximately, cf. Poisson 2004a
- ^ Wheeler 1990, p. xi
- ^ Wald 1984, sec. 4.4
- ^ Wald 1984, sec. 4.1
- ^ For the (conceptual and historical) difficulties in defining a general principle of relativity and separating it from the notion of general covariance, see Giulini 2007
- ^ section 5 in ch. 12 of Weinberg 1972
- ^ Introductory chapters of Stephani et al. 2003
- ^ A review showing Einstein's equation in the broader context of other PDEs with physical significance is Geroch 1996
- ^ For background information and a list of solutions, cf. Stephani et al. 2003; a more recent review can be found in MacCallum 2006
- ^ Chandrasekhar 1983, ch. 3,5,6
- ^ Narlikar 1993, ch. 4, sec. 3.3
- ^ Brief descriptions of these and further interesting solutions can be found in Hawking & Ellis 1973, ch. 5
- ^ Lehner 2002
- ^ For instance Wald 1984, sec. 4.4
- ^ Will 1993, sec. 4.1 and 4.2
- ^ Will 2006, sec. 3.2, Will 1993, ch. 4
- ^ Rindler 2001, pp. 24–26 vs. pp. 236–237 and Ohanian & Ruffini 1994, pp. 164–172. Einstein derived these effects using the equivalence principle as early as 1907, cf. Einstein 1907 and the description in Pais 1982, pp. 196–198
- ^ Rindler 2001, pp. 24–26; Misner, Thorne & Wheeler 1973, § 38.5
- ^ Pound–Rebka experiment, see Pound & Rebka 1959, Pound & Rebka 1960; Pound & Snider 1964; a list of further experiments is given in Ohanian & Ruffini 1994, table 4.1 on p. 186
- ^ Greenstein, Oke & Shipman 1971; the most recent and most accurate Sirius B measurements are published in Barstow, Bond et al. 2005.
- ^ Starting with the Hafele–Keating experiment, Hafele & Keating 1972a and Hafele & Keating 1972b, and culminating in the Gravity Probe A experiment; an overview of experiments can be found in Ohanian & Ruffini 1994, table 4.1 on p. 186
- ^ GPS is continually tested by comparing atomic clocks on the ground and aboard orbiting satellites; for an account of relativistic effects, see Ashby 2002 and Ashby 2003
- ^ Stairs 2003 and Kramer 2004
- ^ General overviews can be found in section 2.1. of Will 2006; Will 2003, pp. 32–36; Ohanian & Ruffini 1994, sec. 4.2
- ^ Ohanian & Ruffini 1994, pp. 164–172
- ^ Cf. Kennefick 2005 for the classic early measurements by Arthur Eddington's expeditions. For an overview of more recent measurements, see Ohanian & Ruffini 1994, ch. 4.3. For the most precise direct modern observations using quasars, cf. Shapiro et al. 2004
- ^ This is not an independent axiom; it can be derived from Einstein's equations and the Maxwell Lagrangian using a WKB approximation, cf. Ehlers 1973, sec. 5
- ^ Blanchet 2006, sec. 1.3
- ^ Rindler 2001, sec. 1.16; for the historical examples, Israel 1987, pp. 202–204; in fact, Einstein published one such derivation as Einstein 1907. Such calculations tacitly assume that the geometry of space is Euclidean, cf. Ehlers & Rindler 1997
- ^ 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. Rindler 2001, sec. 11.11
- ^ For the Sun's gravitational field using radar signals reflected from planets such as Venus and Mercury, cf. Shapiro 1964, Weinberg 1972, ch. 8, sec. 7; for signals actively sent back by space probes (transponder measurements), cf. Bertotti, Iess & Tortora 2003; for an overview, see Ohanian & Ruffini 1994, table 4.4 on p. 200; 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. Stairs 2003, sec. 4.4
- ^ Will 1993, sec. 7.1 and 7.2
- Bibcode:1916SPAW.......688E. Archived from the originalon 21 March 2019. Retrieved 12 February 2016.
- Bibcode:1918SPAW.......154E. Archived from the originalon 21 March 2019. Retrieved 12 February 2016.
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- ^ S2CID 124959784.
- ^ a b c "Gravitational waves detected 100 years after Einstein's prediction". NSF – National Science Foundation. 11 February 2016.
- ^ Most advanced textbooks on general relativity contain a description of these properties, e.g. Schutz 1985, ch. 9
- ^ For example Jaranowski & Królak 2005
- ^ Rindler 2001, ch. 13
- ^ Gowdy 1971, Gowdy 1974
- ^ See Lehner 2002 for a brief introduction to the methods of numerical relativity, and Seidel 1998 for the connection with gravitational wave astronomy
- ^ Schutz 2003, pp. 48–49, Pais 1982, pp. 253–254
- ^ Rindler 2001, sec. 11.9
- ^ Will 1993, pp. 177–181
- ^ In consequence, in the parameterized post-Newtonian formalism (PPN), measurements of this effect determine a linear combination of the terms β and γ, cf. Will 2006, sec. 3.5 and Will 1993, sec. 7.3
- VLBI measurements of planetary positions; see Will 1993, ch. 5, Will 2006, sec. 3.5, Anderson et al. 1992; for an overview, Ohanian & Ruffini 1994, pp. 406–407
- ^ Kramer et al. 2006
- ^ Dediu, Magdalena & Martín-Vide 2015, p. 141.
- ^ S2CID 245124502.
- ^ Stairs 2003, Schutz 2003, pp. 317–321, Bartusiak 2000, pp. 70–86
- ^ Weisberg & Taylor 2003; for the pulsar discovery, see Hulse & Taylor 1975; for the initial evidence for gravitational radiation, see Taylor 1994
- ^ Kramer 2004
- ^ Penrose 2004, § 14.5, Misner, Thorne & Wheeler 1973, § 11.4
- ^ Weinberg 1972, sec. 9.6, Ohanian & Ruffini 1994, sec. 7.8
- ^ Bertotti, Ciufolini & Bender 1987, Nordtvedt 2003
- ^ Kahn 2007
- ^ A mission description can be found in Everitt et al. 2001; a first post-flight evaluation is given in Everitt, Parkinson & Kahn 2007; further updates will be available on the mission website Kahn 1996–2012.
- ^ Townsend 1997, sec. 4.2.1, Ohanian & Ruffini 1994, pp. 469–471
- ^ Ohanian & Ruffini 1994, sec. 4.7, Weinberg 1972, sec. 9.7; for a more recent review, see Schäfer 2004
- ^ Ciufolini & Pavlis 2004, Ciufolini, Pavlis & Peron 2006, Iorio 2009
- ^ Iorio 2006, Iorio 2010
- ISBN 978-1-134-00389-1.
- ^ For overviews of gravitational lensing and its applications, see Ehlers, Falco & Schneider 1992 and Wambsganss 1998
- ^ For a simple derivation, see Schutz 2003, ch. 23; cf. Narayan & Bartelmann 1997, sec. 3
- ^ Walsh, Carswell & Weymann 1979
- ^ Images of all the known lenses can be found on the pages of the CASTLES project, Kochanek et al. 2007
- ^ Roulet & Mollerach 1997
- ^ Narayan & Bartelmann 1997, sec. 3.7
- ^ Barish 2005, Bartusiak 2000, Blair & McNamara 1997
- ^ Hough & Rowan 2000
- S2CID 56073764
- ^ Danzmann & Rüdiger 2003
- ^ "LISA pathfinder overview". ESA. Retrieved 23 April 2012.
- ^ Thorne 1995
- ^ Cutler & Thorne 2002
- ^ Miller 2002, lectures 19 and 21
- ^ Celotti, Miller & Sciama 1999, sec. 3
- ^ Springel et al. 2005 and the accompanying summary Gnedin 2005
- ^ Blandford 1987, sec. 8.2.4
- ^ For the basic mechanism, see Carroll & Ostlie 1996, sec. 17.2; for more about the different types of astronomical objects associated with this, cf. Robson 1996
- ^ For a review, see Begelman, Blandford & Rees 1984. 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 Rees 1966
- ^ For stellar end states, cf. Oppenheimer & Snyder 1939 or, for more recent numerical work, Font 2003, sec. 4.1; for supernovae, there are still major problems to be solved, cf. Buras et al. 2003; for simulating accretion and the formation of jets, cf. Font 2003, sec. 4.2. Also, relativistic lensing effects are thought to play a role for the signals received from X-ray pulsars, cf. Kraus 1998
- X-ray bursts for which the central compact object is either a neutron star or a black hole; cf. Remillard et al. 2006 for an overview, Narayan 2006, sec. 5. Observations of the "shadow" of the Milky Way galaxy's central black hole horizon are eagerly sought for, cf. Falcke, Melia & Agol 2000
- ^ Dalal et al. 2006
- ^ Barack & Cutler 2004
- ^ Einstein 1917; cf. Pais 1982, pp. 285–288
- ^ Carroll 2001, ch. 2
- ^ Bergström & Goobar 2003, ch. 9–11; 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. Peebles et al. 1991
- WMAP data, see Spergel et al. 2003
- ^ These tests involve the separate observations detailed further on, see, e.g., fig. 2 in Bridle et al. 2003
- ^ Peebles 1966; for a recent account of predictions, see Coc, Vangioni‐Flam et al. 2004; an accessible account can be found in Weiss 2006; compare with the observations in Olive & Skillman 2004, Bania, Rood & Balser 2002, O'Meara et al. 2001, and Charbonnel & Primas 2005
- ^ Lahav & Suto 2004, Bertschinger 1998, Springel et al. 2005
- polarization, cf. Kamionkowski, Kosowsky & Stebbins 1997 and Seljak & Zaldarriaga 1997
- ^ Evidence for this comes from the determination of cosmological parameters and additional observations involving the dynamics of galaxies and galaxy clusters cf. Peebles 1993, ch. 18, evidence from gravitational lensing, cf. Peacock 1999, sec. 4.6, and simulations of large-scale structure formation, see Springel et al. 2005
- ^ Peacock 1999, ch. 12, Peskin 2007; 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. Peacock 1999, ch. 12
- ^ 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 Mannheim 2006, sec. 9
- ^ Carroll 2001; an accessible overview is given in Caldwell 2004. 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. Mannheim 2006, sec. 10; 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. Buchert 2008
- ^ A good introduction is Linde 2005; for a more recent review, see Linde 2006
- monopole problem; a pedagogical introduction can be found in Narlikar 1993, sec. 6.4, see also Börner 1993, sec. 9.1
- ^ Spergel et al. 2007, sec. 5,6
- ^ 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
- ^ Brandenberger 2008, sec. 2
- ^ Gödel 1949
- S2CID 59575856.
- S2CID 120125096.
- .
- arXiv:1703.05448 [hep-th].
...redacted transcript of a course given by the author at Harvard in spring semester 2016. It contains a pedagogical overview of recent developments connecting the subjects of soft theorems, the memory effect and asymptotic symmetries in four-dimensional QED, nonabelian gauge theory and gravity with applications to black holes. To be published Princeton University Press, 158 pages.
- ^ Frauendiener 2004, Wald 1984, sec. 11.1, Hawking & Ellis 1973, sec. 6.8, 6.9
- ^ Wald 1984, sec. 9.2–9.4 and Hawking & Ellis 1973, ch. 6
- ^ Thorne 1972; for more recent numerical studies, see Berger 2002, sec. 2.1
- ^ Israel 1987. A more exact mathematical description distinguishes several kinds of horizon, notably event horizons and apparent horizons cf. Hawking & Ellis 1973, pp. 312–320 or Wald 1984, sec. 12.2; there are also more intuitive definitions for isolated systems that do not require knowledge of spacetime properties at infinity, cf. Ashtekar & Krishnan 2004
- ^ For first steps, cf. Israel 1971; see Hawking & Ellis 1973, sec. 9.3 or Heusler 1996, ch. 9 and 10 for a derivation, and Heusler 1998 as well as Beig & Chruściel 2006 as overviews of more recent results
- ^ The laws of black hole mechanics were first described in Bardeen, Carter & Hawking 1973; a more pedagogical presentation can be found in Carter 1979; for a more recent review, see Wald 2001, ch. 2. A thorough, book-length introduction including an introduction to the necessary mathematics Poisson 2004. For the Penrose process, see Penrose 1969
- ^ Bekenstein 1973, Bekenstein 1974
- ^ The fact that black holes radiate, quantum mechanically, was first derived in Hawking 1975; a more thorough derivation can be found in Wald 1975. A review is given in Wald 2001, ch. 3
- ^ Narlikar 1993, sec. 4.4.4, 4.4.5
- ^ Horizons: cf. Rindler 2001, sec. 12.4. Unruh effect: Unruh 1976, cf. Wald 2001, ch. 3
- ^ Hawking & Ellis 1973, sec. 8.1, Wald 1984, sec. 9.1
- ^ Townsend 1997, ch. 2; a more extensive treatment of this solution can be found in Chandrasekhar 1983, ch. 3
- ^ Townsend 1997, ch. 4; for a more extensive treatment, cf. Chandrasekhar 1983, ch. 6
- ^ Ellis & Van Elst 1999; a closer look at the singularity itself is taken in Börner 1993, sec. 1.2
- ^ Here one should remind to the well-known fact that the important "quasi-optical" singularities of the so-called eikonal approximations of many wave equations, namely the "caustics", are resolved into finite peaks beyond that approximation.
- trapped null surfaces, cf. Penrose 1965
- ^ Hawking 1966
- ^ The conjecture was made in Belinskii, Khalatnikov & Lifschitz 1971; for a more recent review, see Berger 2002. An accessible exposition is given by Garfinkle 2007
- ^ 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 Penrose 1969; a textbook-level account is given in Wald 1984, pp. 302–305. For numerical results, see the review Berger 2002, sec. 2.1
- ^ Hawking & Ellis 1973, sec. 7.1
- ^ Arnowitt, Deser & Misner 1962; for a pedagogical introduction, see Misner, Thorne & Wheeler 1973, § 21.4–§ 21.7
- ^ Fourès-Bruhat 1952 and Bruhat 1962; for a pedagogical introduction, see Wald 1984, ch. 10; an online review can be found in Reula 1998
- ^ Gourgoulhon 2007; for a review of the basics of numerical relativity, including the problems arising from the peculiarities of Einstein's equations, see Lehner 2001
- ^ Misner, Thorne & Wheeler 1973, § 20.4
- ^ Arnowitt, Deser & Misner 1962
- ^ Komar 1959; for a pedagogical introduction, see Wald 1984, sec. 11.2; although defined in a totally different way, it can be shown to be equivalent to the ADM mass for stationary spacetimes, cf. Ashtekar & Magnon-Ashtekar 1979
- ^ For a pedagogical introduction, see Wald 1984, sec. 11.2
- ^ Wald 1984, p. 295 and refs therein; 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
- ^ Townsend 1997, ch. 5
- ^ 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 Szabados 2004
- ^ An overview of quantum theory can be found in standard textbooks such as Messiah 1999; a more elementary account is given in Hey & Walters 2003
- ^ Ramond 1990, Weinberg 1995, Peskin & Schroeder 1995; a more accessible overview is Auyang 1995
- ^ Wald 1994, Birrell & Davies 1984
- ^ For Hawking radiation Hawking 1975, Wald 1975; an accessible introduction to black hole evaporation can be found in Traschen 2000
- ^ Wald 2001, ch. 3
- ^ 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. Carlip 2001, sec. 2
- ^ Schutz 2003, p. 407
- ^ a b Hamber 2009
- ^ A timeline and overview can be found in Rovelli 2000
- ^ 't Hooft & Veltman 1974
- ^ Donoghue 1995
- ^ In particular, a perturbative technique known as renormalization, an integral part of deriving predictions which take into account higher-energy contributions, cf. Weinberg 1996, ch. 17, 18, fails in this case; cf. Veltman 1975, Goroff & Sagnotti 1985; for a recent comprehensive review of the failure of perturbative renormalizability for quantum gravity see Hamber 2009
- ^ An accessible introduction at the undergraduate level can be found in Zwiebach 2004; more complete overviews can be found in Polchinski 1998a and Polchinski 1998b
- messenger particle of gravity, e.g. Green, Schwarz & Witten 1987, sec. 2.3, 5.3
- ^ Green, Schwarz & Witten 1987, sec. 4.2
- ^ Weinberg 2000, ch. 31
- ^ Townsend 1996, Duff 1996
- ^ Kuchař 1973, sec. 3
- ^ These variables represent geometric gravity using mathematical analogues of electric and magnetic fields; cf. Ashtekar 1986, Ashtekar 1987
- ^ For a review, see Thiemann 2007; more extensive accounts can be found in Rovelli 1998, Ashtekar & Lewandowski 2004 as well as in the lecture notes Thiemann 2003
- ^ Isham 1994, Sorkin 1997
- ^ Loll 1998
- ^ Sorkin 2005
- ^ Penrose 2004, ch. 33 and refs therein
- ^ Hawking 1987
- ^ Ashtekar 2007, Schwarz 2007
- ^ Maddox 1998, pp. 52–59, 98–122; Penrose 2004, sec. 34.1, ch. 30
- ^ section Quantum gravity, above
- ^ section Cosmology, above
- ^ Friedrich 2005
- ^ A review of the various problems and the techniques being developed to overcome them, see Lehner 2002
- ^ See Bartusiak 2000 for an account up to that year; up-to-date news can be found on the websites of major detector collaborations such as GEO600 and LIGO
- ^ For the most recent papers on gravitational wave polarizations of inspiralling compact binaries, see Blanchet et al. 2008, and Arun et al. 2008; for a review of work on compact binaries, see Blanchet 2006 and Futamase & Itoh 2006; for a general review of experimental tests of general relativity, see Will 2006
- ^ See, e.g., the Living Reviews in Relativity journal.
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Further reading
Popular books
- ISBN 978-3-528-06059-6)
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: CS1 maint: location missing publisher (link - ISBN 978-0-226-28864-2
- ISBN 978-1-58988-044-3
- ISBN 978-1-56159-268-5
- ISBN 0-393-03505-0.
- ISBN 978-0-226-87029-8
- ISBN 978-0-393-31991-0
Beginning undergraduate textbooks
- Callahan, James J. (2000), The Geometry of Spacetime: an Introduction to Special and General Relativity, New York: Springer, ISBN 978-0-387-98641-8
- Taylor, Edwin F.; Wheeler, John Archibald (2000), Exploring Black Holes: Introduction to General Relativity, Addison Wesley, ISBN 978-0-201-38423-9
Advanced undergraduate textbooks
- Cheng, Ta-Pei (2005), Relativity, Gravitation and Cosmology: a Basic Introduction, Oxford and New York: Oxford University Press, ISBN 978-0-19-852957-6
- ISBN 978-0-691-01146-2
- Gron, O.; Hervik, S. (2007), Einstein's General theory of Relativity, Springer, ISBN 978-0-387-69199-2
- ISBN 978-0-8053-8662-2
- ISBN 978-0-521-33943-8
- d'Inverno, Ray (1992), Introducing Einstein's Relativity, Oxford: Oxford University Press, ISBN 978-0-19-859686-8
- Ludyk, Günter (2013). Einstein in Matrix Form (1st ed.). Berlin: Springer. ISBN 978-3-642-35797-8.
- Møller, Christian (1955) [1952], The Theory of Relativity, Oxford University Press, OCLC 7644624
- Moore, Thomas A (2012), A General Relativity Workbook, University Science Books, ISBN 978-1-891389-82-5
- ISBN 978-0-521-88705-2
Graduate textbooks
- ISBN 978-0-8053-8732-2
- ISBN 978-0-387-69199-2
- ISBN 978-0-7506-2768-9
- Stephani, Hans (1990), General Relativity: An Introduction to the Theory of the Gravitational Field, Cambridge: Cambridge University Press, ISBN 978-0-521-37941-0
- ISBN 978-1-107-03286-6.
- ISBN 0-7167-0344-0
- ISBN 1461299055
- OCLC 10018614.
Specialists' books
- ISBN 978-0-521-09906-6.
- ISBN 978-0-521-53780-3.
Journal articles
- Flanagan, Éanna É.; Hughes, Scott A. (2005), "The basics of gravitational wave theory", New J. Phys., 7 (1): 204,
- Landgraf, M.; Hechler, M.; Kemble, S. (2005), "Mission design for LISA Pathfinder", Class. Quantum Grav., 22 (10): S487–S492, S2CID 119476595
- Nieto, Michael Martin (2006), "The quest to understand the Pioneer anomaly" (PDF), Europhysics News, 37 (6): 30–34, (PDF) from the original on 24 September 2015
- Valtonen, M. J.; Lehto, H. J.; Nilsson, K.; Heidt, J.; Takalo, L. O.; Sillanpää, A.; Villforth, C.; Kidger, M.; et al. (2008), "A massive binary black-hole system in OJ 287 and a test of general relativity", Nature, 452 (7189): 851–853, S2CID 4412396
External links
- Einstein Online Archived 1 June 2014 at the Wayback Machine – Articles on a variety of aspects of relativistic physics for a general audience; hosted by the Max Planck Institute for Gravitational Physics
- GEO600 home page, the official website of the GEO600 project.
- LIGO Laboratory
- NCSA Spacetime Wrinkles – produced by the numerical relativity group at the NCSA, with an elementary introduction to general relativity
- Courses
- Lectures
- Tutorials
- Einstein's General Theory of Relativity on YouTube (lecture by Leonard Susskind recorded 22 September 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. Archived from the original on 18 December 2015. Retrieved 29 May 2005.
- Carroll, Sean M. (1997). "Lecture Notes on General Relativity". arXiv:gr-qc/9712019.
- Moor, Rafi. "Understanding General Relativity". Retrieved 11 July 2006.
- Waner, Stefan. "Introduction to Differential Geometry and General Relativity". Retrieved 5 April 2015.
- The Feynman Lectures on Physics Vol. II Ch. 42: Curved Space