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 mass-energy present in the spacetime is due to some kind of massless radiation.
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 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.
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 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 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 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.
Examples
Null dust solutions include two large and important families of exact solutions:
- pp-wave spacetimes (which model generalizations of the plane waves familiar from electromagnetism),
- Robinson–Trautman null dusts (which model radiation expanding from a radiating object).
The pp-waves 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 0-521-46136-7.. This standard monograph gives many examples of null dust solutions.