 Null dust solution

In mathematical physics, a null dust solution (sometimes called a null fluid) is a Lorentzian manifold in which the Einstein tensor is null. Such a spacetime can be interpreted as an exact solution of Einstein's field equation, in which the only massenergy present in the spacetime is due to some kind of massless radiation.
Contents
Mathematical definition
The Einstein tensor of a null dust must have the form where is a null vector field. This definition makes sense in the absence of any physical interpretation, but if we place a stressenergy tensor on our spacetime which happens to have the form then Einstein's field equation is trivially satisfied, and in addition, such a stressenergy tensor has a clear physical interpretation in terms of massless radiation. The vector field specifies the direction in which the radiation is moving; the scalar multiplier specifies its intensity.
Physical interpretation
Physically speaking, a null dust describes either gravitational radiation, or some kind of nongravitational radiation which is described by a relativistic classical field theory (such as electromagnetic radiation), or a combination of these two. Null dusts include vacuum solutions as a special case.
Phenomena which can be modeled by null dust solutions include:
 a beam of neutrinos assumed for simplicity to be massless (treated according to classical physics),
 a very highfrequency electromagnetic wave,
 a beam of incoherent electromagnetic radiation.
In particular, a plane wave of incoherent electromagnetic radiation is a linear superposition of plane waves, all moving in the same direction but having randomly chosen phases and frequencies. (Even though the Einstein field equation is nonlinear, a linear superposition of comoving plane waves is possible.) Here, each electromagnetic plane wave has a well defined frequency and phase, but the superposition does not. Individual electromagnetic plane waves are modeled by null electrovacuum solutions, while an incoherent mixture can be modeled by a null dust.
Einstein tensor
The components of a tensor computed with respect to a frame field rather than the coordinate basis are often called physical components, because these are the components which can (in principle) be measured by an observer.
In the case of a null dust solution, an adapted frame
(a timelike unit vector field and three spacelike unit vector fields, respectively) can always be found in which the Einstein tensor has a particularly simple appearance:
Here, is everywhere tangent to the world lines of our adapted observers, and these observers measure the energy density of the incoherent radiation to be .
From the form of the general coordinate basis expression given above, it is apparent that the stressenergy tensor has precisely the same isotropy group as the null vector field . It is generated by two parabolic Lorentz transformations (pointing in the direction) and one rotation (about the axis), and it is isometric to the three dimensional Lie group E(2), the isometry group of the euclidean plane.
Examples
Null dust solutions include two large and important families of exact solutions:
 ppwave spacetimes (which model generalizations of the plane waves familiar from electromagnetism),
 Robinson–Trautman null dusts (which model radiation expanding from a radiating object).
The ppwaves include the gravitational plane waves and the monochromatic electromagnetic plane wave. A specific example of considerable interest is
 the Bonnor beam, an exact solution modeling an infinitely long beam of light surrounded by a vacuum region.
Robinson–Trautman null dusts include the Kinnersley–Walker photon rocket solutions, which include the Vaidya null dust, which includes the Schwarzschild vacuum.
See also
References
 Stephani, Hans; Kramer, Dietrich; Maccallum, Malcolm; Hoenselaers, Cornelius; & Herlt, Eduard (2003). Exact Solutions of Einstein's Field Equations. Cambridge: Cambridge University Press. ISBN 0521461367.. This standard monograph gives many examples of null dust solutions.
Categories:
Wikimedia Foundation. 2010.
Look at other dictionaries:
Van Stockum dust — In general relativity, the van Stockum dust is an exact solution of the Einstein field equation in which the gravitational field is generated by dust rotating about an axis of cylindrical symmetry. Since the density of the dust is increasing with … Wikipedia
Exact solutions in general relativity — In general relativity, an exact solution is a Lorentzian manifold equipped with certain tensor fields which are taken to model states of ordinary matter, such as a fluid, or classical nongravitational fields such as the electromagnetic field.… … Wikipedia
Golden age of general relativity — The Golden Age of General Relativity is the period roughly from 1960 to 1975 during which the study of general relativity, which had previously been regarded as something of a curiosity, entered the mainstream of theoretical physics. During this… … Wikipedia
Ppwave 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. These solutions model radiation moving at the speed of light. This radiation may consist of:*… … Wikipedia
Bonnor beam — In general relativity, the Bonnor beam is an exact solution which models an infinitely long, straight beam of light. It is an explicit example of a pp wave spacetime.The Bonnor beam is obtained by matching together two regions:* a uniform plane… … Wikipedia
William Morris Kinnersley — is an American physicist who is well known for his contributions to general relativity.Kinnersley earned his Ph.D. from Cal Tech in 1968, under the direction of Kip Thorne. In 1969 he published the famous Kinnersley photon rocket, an exact null… … Wikipedia
CartanKarlhede algorithm — One of the most fundamental problems of Riemannian geometry is this: given two Riemannian manifolds of the same dimension, how can one tell if they are locally isometric? This question was addressed by Elwin Christoffel, and completely solved by… … Wikipedia
BransDicke theory — In theoretical physics, the Brans Dicke theory of gravitation (sometimes called the Jordan Brans Dicke theory) is a theoretical framework to explain gravitation. It is a well known competitor of Einstein s more popular theory of general… … Wikipedia
Gödel metric — The Gödel metric is an exact solution of the Einstein field equations in which the stress energy tensor contains two terms, the first representing the matter density of a homogeneous distribution of swirling dust particles, and the second… … Wikipedia
Известные учёныерелятивисты — Служебный список статей, созданный для координации работ по развитию темы. Данное предупреждение не устанавл … Википедия