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Einstein's Equations
In the general theory of relativity, the Einstein field equations (EFE; also known as Einstein's equations) relate the geometry of spacetime to the distribution of matter within it. The equations were published by Einstein in 1915 in the form of a tensor equation which related the local ' (expressed by the Einstein tensor) with the local energy, momentum and stress within that spacetime (expressed by the stress–energy tensor). Analogously to the way that electromagnetic fields are related to the distribution of charges and currents via Maxwell's equations, the EFE relate the spacetime geometry to the distribution of mass–energy, momentum and stress, that is, they determine the metric tensor of spacetime for a given arrangement of stress–energy–momentum in the spacetime. The relationship between the metric tensor and the Einstein tensor allows the EFE to be written as a set of nonlinear partial differential equations when used in this way. The solutions of the EFE ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
General Relativity
General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics. General relativity generalizes special relativity and refines Newton's law of universal gravitation, providing a unified description of gravity as a geometric property of space and time or four-dimensional spacetime. In particular, the ' 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 second order partial differential equations. Newton's law of universal gravitation, which describes classical gravity, can be seen as a prediction of general relativity for the almost flat spacetime geometry around stationary mass distributions. Some predictions of general relativity, however, are beyond Newton's law of universal gra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geodesics In General Relativity
In general relativity, a geodesic generalizes the notion of a "straight line" to curved spacetime. Importantly, the world line of a particle free from all external, non-gravitational forces is a particular type of geodesic. In other words, a freely moving or falling particle always moves along a geodesic. In general relativity, gravity can be regarded as not a force but a consequence of a curved spacetime geometry where the source of curvature is the stress–energy tensor (representing matter, for instance). Thus, for example, the path of a planet orbiting a star is the projection of a geodesic of the curved four-dimensional (4-D) spacetime geometry around the star onto three-dimensional (3-D) space. Mathematical expression The full geodesic equation is : +\Gamma^\mu _=0\ where ''s'' is a scalar parameter of motion (e.g. the proper time), and \Gamma^\mu _ are Christoffel symbols (sometimes called the affine connection coefficients or Levi-Civita connection coefficients) s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cosmological Constant
In cosmology, the cosmological constant (usually denoted by the Greek capital letter lambda: ), alternatively called Einstein's cosmological constant, is the constant coefficient of a term that Albert Einstein temporarily added to his field equations of general relativity. He later removed it. Much later it was revived and reinterpreted as the energy density of space, or vacuum energy, that arises in quantum mechanics. It is closely associated with the concept of dark energy. Einstein originally introduced the constant in 1917 to counterbalance the effect of gravity and achieve a static universe, a notion that was the accepted view at the time. Einstein's cosmological constant was abandoned after Edwin Hubble's confirmation that the universe was expanding. From the 1930s until the late 1990s, most physicists agreed with Einstein's choice of setting the cosmological constant to zero. That changed with the discovery in 1998 that the expansion of the universe is accelerating, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Metric Tensor (general Relativity)
In general relativity, the metric tensor (in this context often abbreviated to simply the metric) is the fundamental object of study. It may loosely be thought of as a generalization of the gravitational potential of Newtonian gravitation. The metric captures all the geometric and causal structure of spacetime, being used to define notions such as time, distance, volume, curvature, angle, and separation of the future and the past. Notation and conventions Throughout this article we work with a metric signature that is mostly positive (); see sign convention. The gravitation constant G will be kept explicit. This article employs the Einstein summation convention, where repeated indices are automatically summed over. Definition Mathematically, spacetime is represented by a four-dimensional differentiable manifold M and the metric tensor is given as a covariant, second- degree, symmetric tensor on M, conventionally denoted by g. Moreover, the metric is required to be nondeg ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gravitational Waves
Gravitational waves are waves of the intensity of gravity generated by the accelerated masses of an orbital binary system that propagate as waves outward from their source at the speed of light. They were first proposed by Oliver Heaviside in 1893 and then later by Henri Poincaré in 1905 as waves similar to electromagnetic waves but the gravitational equivalent. Gravitational waves were later predicted in 1916 by Albert Einstein on the basis of his general theory of relativity as ripples in spacetime. Later he refused to accept gravitational waves. Gravitational waves transport energy as gravitational radiation, a form of radiant energy similar to electromagnetic radiation. Newton's law of universal gravitation, part of classical mechanics, does not provide for their existence, since that law is predicated on the assumption that physical interactions propagate instantaneously (at infinite speed)showing one of the ways the methods of Newtonian physics are unable to explain ph ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linearized Gravity
In the theory of general relativity, linearized gravity is the application of perturbation theory to the metric tensor that describes the geometry of spacetime. As a consequence, linearized gravity is an effective method for modeling the effects of gravity when the gravitational field is weak. The usage of linearized gravity is integral to the study of gravitational waves and weak-field gravitational lensing. Weak-field approximation The Einstein field equation (EFE) describing the geometry of spacetime is given as (using natural units) :R_ - \fracRg_ = 8\pi GT_ where R_ is the Ricci tensor, R is the Ricci scalar, T_ is the energy–momentum tensor, and g_ is the spacetime metric tensor that represent the solutions of the equation. Although succinct when written out using Einstein notation, hidden within the Ricci tensor and Ricci scalar are exceptionally nonlinear dependencies on the metric which render the prospect of finding exact solutions impractical in most systems. H ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Minkowski Space
In mathematical physics, Minkowski space (or Minkowski spacetime) () is a combination of three-dimensional Euclidean space and time into a four-dimensional manifold where the spacetime interval between any two events is independent of the inertial frame of reference in which they are recorded. Although initially developed by mathematician Hermann Minkowski for Maxwell's equations of electromagnetism, the mathematical structure of Minkowski spacetime was shown to be implied by the postulates of special relativity. Minkowski space is closely associated with Einstein's theories of special relativity and general relativity and is the most common mathematical structure on which special relativity is formulated. While the individual components in Euclidean space and time may differ due to length contraction and time dilation, in Minkowski spacetime, all frames of reference will agree on the total distance in spacetime between events.This makes spacetime distance an invariant. Beca ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Metric Expansion Of Space
The expansion of the universe is the increase in distance between any two given gravitationally unbound parts of the observable universe with time. It is an intrinsic expansion whereby the scale of space itself changes. The universe does not expand "into" anything and does not require space to exist "outside" it. This expansion involves neither space nor objects in space "moving" in a traditional sense, but rather it is the metric (which governs the size and geometry of spacetime itself) that changes in scale. As the spatial part of the universe's spacetime metric increases in scale, objects become more distant from one another at ever-increasing speeds. To any observer in the universe, it appears that all of space is expanding, and that all but the nearest galaxies (which are bound by gravity) recede at speeds that are proportional to their distance from the observer. While objects within space cannot travel faster than light, this limitation does not apply to the effects of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rotating Black Hole
A rotating black hole is a black hole that possesses angular momentum. In particular, it rotates about one of its axes of symmetry. All celestial objects – planets, stars (Sun), galaxies, black holes – spin. Types of black holes There are four known, exact, black hole solutions to the Einstein field equations, which describe gravity in general relativity. Two of those rotate: the Kerr and Kerr–Newman black holes. It is generally believed that every black hole decays rapidly to a stable black hole; and, by the no-hair theorem, that (except for quantum fluctuations) stable black holes can be completely described at any moment in time by these 11 numbers: * mass-energy ''M'', * linear momentum ''P'' (three components), * angular momentum ''J'' (three components), * position ''X'' (three components), * electric charge ''Q''. These numbers represent the conserved attributes of an object which can be determined from a distance by examining its electromagnetic and grav ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Exact Solutions In General Relativity
In general relativity, an exact solution is a solution of the Einstein field equations whose derivation does not invoke simplifying assumptions, though the starting point for that derivation may be an idealized case like a perfectly spherical shape of matter. Mathematically, finding an exact solution means finding a Lorentzian manifold equipped with tensor fields modeling states of ordinary matter, such as a fluid, or classical non-gravitational fields such as the electromagnetic field. Background and definition These tensor fields should obey any relevant physical laws (for example, any electromagnetic field must satisfy Maxwell's equations). Following a standard recipe which is widely used in mathematical physics, these tensor fields should also give rise to specific contributions to the stress–energy tensor T^. (A field is described by a Lagrangian, varying with respect to the field should give the field equations and varying with respect to the metric should give the s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spacetime Symmetries
Spacetime symmetries are features of spacetime that can be described as exhibiting some form of symmetry. The role of symmetry in physics is important in simplifying solutions to many problems. Spacetime symmetries are used in the study of exact solutions of Einstein's field equations of general relativity. Spacetime symmetries are distinguished from internal symmetries. Physical motivation Physical problems are often investigated and solved by noticing features which have some form of symmetry. For example, in the Schwarzschild solution, the role of spherical symmetry is important in deriving the Schwarzschild solution and deducing the physical consequences of this symmetry (such as the nonexistence of gravitational radiation in a spherically pulsating star). In cosmological problems, symmetry plays a role in the cosmological principle, which restricts the type of universes that are consistent with large-scale observations (e.g. the Friedmann–Lemaître–Robertson–Walker (F ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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