Null dust solution

By definition, the Einstein tensor of a null dust solution has the form

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.

specifies the direction in which the radiation is moving; the scalar multiplier

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 solutions, while an incoherent mixture can be modeled by a null dust.

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,

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

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.