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

By definition, the Einstein tensor of a null dust solution has the form ${\displaystyle G^{ab}=8\pi \Phi \,k^{a}\,k^{b}}$ where ${\displaystyle {\vec {k}}}$ 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 ${\displaystyle T^{ab}=\Phi \,k^{a}\,k^{b}}$, 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 (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

${\displaystyle {\vec {e}}_{0},\;{\vec {e}}_{1},\;{\vec {e}}_{2},\;{\vec {e}}_{3}}$

(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:

${\displaystyle G^{{\hat {a}}{\hat {b}}}=8\pi \epsilon \,\left[{\begin{matrix}1&0&0&\pm 1\\0&0&0&0\\0&0&0&0\\\pm 1&0&0&1\end{matrix}}\right]}$

Here, ${\displaystyle {\vec {e}}_{0}}$ is everywhere tangent to the world lines of our adapted observers, and these observers measure the energy density of the incoherent radiation to be ${\displaystyle \epsilon }$.

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 ${\displaystyle {\vec {k}}}$. It is generated by two parabolic Lorentz transformations (pointing in the ${\displaystyle {\vec {e}}_{3}}$ direction) and one rotation (about the ${\displaystyle {\vec {e}}_{3}}$ axis), and it is isometric to the three-dimensional Lie group ${\displaystyle E(2)}$, 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

• Vaidya metric
• Lorentz group

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.