Null dust solution


In mathematical physics, a null dust solution 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.

Mathematical definition

By definition, the Einstein tensor of a null dust solution has the form
where is a null vector field. This definition makes sense purely geometrically, but if we place a stress–energy tensor on our spacetime of the form
then Einstein's field equation is satisfied, and 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 relativistic classical field theory, 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.
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 be measured by an observer.
In the case of a null dust solution, an adapted frame
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 and one rotation, and it is isometric to the three-dimensional Lie group, the isometry group of the euclidean plane.

Examples

Null dust solutions include two large and important families of exact solutions:
The pp-waves include the gravitational plane waves 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.