Null dust solution
Encyclopedia
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 mass-energy present in the spacetime
is due to some kind of massless radiation
.
where
is a null vector field. This definition makes sense in the absence of any physical interpretation, but if we place a stress-energy tensor
on our spacetime which happens to have the form
then Einstein's field equation is trivially satisfied, and in addition, such a stress-energy 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.
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:
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 solution
s, while an incoherent mixture can be modeled by a null dust.
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 stress-energy 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 
, the isometry group
of the euclidean plane.
The pp-waves include the gravitational plane wave
s and the monochromatic electromagnetic plane wave
. A specific example of considerable interest is
Robinson–Trautman null dusts include the Kinnersley–Walker photon rocket solutions, which include the Vaidya null dust, which includes the Schwarzschild vacuum
.
Einstein tensor
In differential geometry, the Einstein tensor , named after Albert Einstein, is used to express the curvature of a Riemannian manifold...
is null. Such a spacetime can be interpreted as an exact solution
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....
of Einstein's field equation, in which the only mass-energy present in the spacetime
Spacetime
In physics, spacetime is any mathematical model that combines space and time into a single continuum. Spacetime is usually interpreted with space as being three-dimensional and time playing the role of a fourth dimension that is of a different sort from the spatial dimensions...
is due to some kind of massless radiation
Radiation
In physics, radiation is a process in which energetic particles or energetic waves travel through a medium or space. There are two distinct types of radiation; ionizing and non-ionizing...
.
Mathematical definition
The Einstein tensor of a null dust must have the formwhere
is a null vector field. This definition makes sense in the absence of any physical interpretation, but if we place a stress-energy tensorStress-energy tensor
The stress–energy tensor is a tensor quantity in physics that describes the density and flux of energy and momentum in spacetime, generalizing the stress tensor of Newtonian physics. It is an attribute of matter, radiation, and non-gravitational force fields...
on our spacetime which happens to have the form
then Einstein's field equation is trivially satisfied, and in addition, such a stress-energy 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 relativisticTheory of relativity
The theory of relativity, or simply relativity, encompasses two theories of Albert Einstein: special relativity and general relativity. However, the word relativity is sometimes used in reference to Galilean invariance....
classical field theory
Classical field theory
A classical field theory is a physical theory that describes the study of how one or more physical fields interact with matter. The word 'classical' is used in contrast to those field theories that incorporate quantum mechanics ....
(such as electromagnetic radiation
Electromagnetic radiation
Electromagnetic radiation is a form of energy that exhibits wave-like behavior as it travels through space...
), or a combination of these two. Null dusts include vacuum solutions
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.More generally, a...
as a special case.
Phenomena which can be modeled by null dust solutions include:
- a beam of neutrinoNeutrinoA neutrino is an electrically neutral, weakly interacting elementary subatomic particle with a half-integer spin, chirality and a disputed but small non-zero mass. It is able to pass through ordinary matter almost unaffected...
s assumed for simplicity to be massless (treated according to classical physics), - a very high-frequency 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
Einstein field equations
The Einstein field equations or Einstein's equations are a set of ten equations in Albert Einstein's general theory of relativity which describe the fundamental interaction of gravitation as a result of spacetime being curved by matter and energy...
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 solution
Electrovacuum solution
In general relativity, an electrovacuum solution is an exact solution of the Einstein field equation in which the only nongravitational mass-energy present is the field energy of an electromagnetic field, which must satisfy the source-free Maxwell equations appropriate to the given geometry...
s, while an incoherent mixture can be modeled by a null dust.
Einstein tensor
The components of a tensor computed with respect to a frame fieldFrame fields in general relativity
In general relativity, a frame field is a set of four orthonormal vector fields, one timelike and three spacelike, defined on a Lorentzian manifold that is physically interpreted as a model of spacetime...
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
Vector field
In vector calculus, a vector field is an assignmentof a vector to each point in a subset of Euclidean space. A vector field in the plane for instance can be visualized as an arrow, with a given magnitude and direction, attached to each point in the plane...
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 stress-energy 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 , the isometry groupIsometry group
In mathematics, the isometry group of a metric space is the set of all isometries from the metric space onto itself, with the function composition as group operation...
of the euclidean plane.
Examples
Null dust solutions include two large and important families of exact solutions:- pp-wave spacetimePp-wave spacetimeIn 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...
s (which model generalizations of the plane wavePlane waveIn the physics of wave propagation, a plane wave is a constant-frequency wave whose wavefronts are infinite parallel planes of constant peak-to-peak amplitude normal to the phase velocity vector....
s familiar from electromagnetismElectromagnetismElectromagnetism is one of the four fundamental interactions in nature. The other three are the strong interaction, the weak interaction and gravitation...
), - Robinson–Trautman null dusts (which model radiation expanding from a radiating object).
The pp-waves include the gravitational plane wave
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 byds^2=[a+2bxy]du^2+2dudv+dx^2+dy^2...
s and the monochromatic electromagnetic plane wave
Monochromatic electromagnetic plane wave
In general relativity, the monochromatic electromagnetic plane wave spacetime is the analog of the monochromatic plane waves known from Maxwell's theory...
. A specific example of considerable interest is
- the Bonnor beamBonnor beamIn 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. It is named after William B...
, 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
Schwarzschild metric
In Einstein's theory of general relativity, the Schwarzschild solution describes the gravitational field outside a spherical, uncharged, non-rotating mass such as a star, planet, or black hole. It is also a good approximation to the gravitational field of a slowly rotating body like the Earth or...
.