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Gravitational Plane Wave
In general relativity, a gravitational plane wave is a special class of a vacuum pp-wave spacetime, and may be defined in terms of Brinkmann coordinates by ds^2= (u)(x^2-y^2)+2b(u)xyu^2+2dudv+dx^2+dy^2 Here, a(u), b(u) can be any smooth functions; they control the waveform of the two possible polarization modes of gravitational radiation. In this context, these two modes are usually called the plus mode and cross mode, respectively. See also *vacuum solution (general relativity) In general relativity, a vacuum solution is a Lorentzian manifold whose Einstein tensor vanishes identically. According to the Einstein field equation, this means that the stress–energy tensor also vanishes identically, so that no matter or non ... {{relativity-stub Exact solutions in general relativity ... [...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 gravitat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pp-wave Spacetime
In general relativity, the pp-wave spacetimes, or pp-waves for short, are an important family of exact solutions of Einstein's field equation. The term ''pp'' stands for ''plane-fronted waves with parallel propagation'', and was introduced in 1962 by Jürgen Ehlers and Wolfgang Kundt. Overview The pp-waves solutions model radiation moving at the speed of light. This radiation may consist of: * electromagnetic radiation, * gravitational radiation, * massless radiation associated with Weyl fermions, * ''massless'' radiation associated with some hypothetical distinct type relativistic classical field, or any combination of these, so long as the radiation is all moving in the ''same'' direction. A special type of pp-wave spacetime, the plane wave spacetimes, provide the most general analogue in general relativity of the plane waves familiar to students of electromagnetism. In particular, in general relativity, we must take into account the gravitational effects of the energy ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Brinkmann Coordinates
Brinkmann coordinates are a particular coordinate system for a spacetime belonging to the family of pp-wave metrics. They are named for Hans Brinkmann. In terms of these coordinates, the metric tensor can be written as :ds^2 = H(u,x,y) du^2 + 2 du dv + dx^2 + dy^2 where \partial_, the coordinate vector field dual to the covector field dv, is a null vector field. Indeed, geometrically speaking, it is a null geodesic congruence with vanishing optical scalars. Physically speaking, it serves as the wave vector defining the direction of propagation for the pp-wave. The coordinate vector field \partial_ can be spacelike, null, or timelike at a given event in the spacetime, depending upon the sign of H(u,x,y) at that event. The coordinate vector fields \partial_, \partial_ are both spacelike vector fields. Each surface u=u_, v=v_ can be thought of as a wavefront. In discussions of exact solutions to the Einstein field equation, many authors fail to specify the intended ran ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Smooth Function
In mathematical analysis, the smoothness of a function (mathematics), function is a property measured by the number of Continuous function, continuous Derivative (mathematics), derivatives it has over some domain, called ''differentiability class''. At the very minimum, a function could be considered smooth if it is differentiable everywhere (hence continuous). At the other end, it might also possess derivatives of all Order of derivation, orders in its Domain of a function, domain, in which case it is said to be infinitely differentiable and referred to as a C-infinity function (or C^ function). Differentiability classes Differentiability class is a classification of functions according to the properties of their derivatives. It is a measure of the highest order of derivative that exists and is continuous for a function. Consider an open set U on the real line and a function f defined on U with real values. Let ''k'' be a non-negative integer. The function f is said to be of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Waveform
In electronics, acoustics, and related fields, the waveform of a signal is the shape of its graph as a function of time, independent of its time and magnitude scales and of any displacement in time.David Crecraft, David Gorham, ''Electronics'', 2nd ed., , CRC Press, 2002, p. 62 In electronics, the term is usually applied to periodically varying voltages, currents, or electromagnetic fields. In acoustics, it is usually applied to steady periodic sounds—variations of pressure in air or other media. In these cases, the waveform is an attribute that is independent of the frequency, amplitude, or phase shift of the signal. The term can also be used for non-periodic signals, like chirps and pulses. The waveform of an electrical signal can be visualized in an oscilloscope or any other device that can capture and plot its value at various times, with a suitable scales in the time and value axes. The electrocardiograph is a medical device to record the waveform of the ele ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polarization (waves)
Polarization (also polarisation) is a property applying to transverse waves that specifies the geometrical orientation of the oscillations. In a transverse wave, the direction of the oscillation is perpendicular to the direction of motion of the wave. A simple example of a polarized transverse wave is vibrations traveling along a taut string ''(see image)''; for example, in a musical instrument like a guitar string. Depending on how the string is plucked, the vibrations can be in a vertical direction, horizontal direction, or at any angle perpendicular to the string. In contrast, in longitudinal waves, such as sound waves in a liquid or gas, the displacement of the particles in the oscillation is always in the direction of propagation, so these waves do not exhibit polarization. Transverse waves that exhibit polarization include electromagnetic waves such as light and radio waves, gravitational waves, and transverse sound waves (shear waves) in solids. An electromagnetic wa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gravitational Radiation
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 phe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vacuum Solution (general Relativity)
In general relativity, a vacuum solution is a Lorentzian manifold whose Einstein tensor vanishes identically. According to the Einstein field equation, this means that the stress–energy tensor also vanishes identically, so that no matter or non-gravitational fields are present. These are distinct from the electrovacuum solutions, which take into account the electromagnetic field in addition to the gravitational field. Vacuum solutions are also distinct from the lambdavacuum solutions, where the only term in the stress–energy tensor is the cosmological constant term (and thus, the lambdavacuums can be taken as cosmological models). More generally, a vacuum region in a Lorentzian manifold is a region in which the Einstein tensor vanishes. Vacuum solutions are a special case of the more general exact solutions in general relativity. Equivalent conditions It is a mathematical fact that the Einstein tensor vanishes if and only if the Ricci tensor vanishes. This follows from th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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