# Null dust solution > Mediated Wiki article. Canonical URL: https://mediated.wiki/source/Null_dust_solution > Markdown URL: https://mediated.wiki/source/Null_dust_solution.md > Source: https://en.wikipedia.org/wiki/Null_dust_solution > Source revision: 1284539644 > License: Creative Commons Attribution-ShareAlike 4.0 International (https://creativecommons.org/licenses/by-sa/4.0/) In [mathematical physics](/source/Mathematical_physics), a **null dust solution** (sometimes called a **null fluid**) is a [Lorentzian manifold](/source/Lorentzian_manifold) in which the [Einstein tensor](/source/Einstein_tensor) is [null](/source/Causal_structure#Tangent_vectors). Such a spacetime can be interpreted as an [exact solution](/source/Exact_solutions_in_general_relativity) of [Einstein's field equation](/source/Einstein's_field_equation), in which the only [mass–energy](/source/Mass%E2%80%93energy) present in the [spacetime](/source/Spacetime) is due to some kind of massless [radiation](/source/Radiation). ## Mathematical definition By definition, the Einstein tensor of a null dust solution has the form G^{ab} = 8 \pi \Phi \, k^{a} \, k^{b} where \vec{k} is a [null vector](/source/Causal_structure#Tangent_vectors) field. This definition makes sense purely geometrically, but if we place a [stress–energy tensor](/source/Stress%E2%80%93energy_tensor) on our spacetime of the form 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 \vec k specifies the direction in which the radiation is moving; the scalar multiplier \Phi specifies its intensity. ## Physical interpretation Physically speaking, a null dust describes either [gravitational radiation](/source/Gravitational_radiation), or some kind of nongravitational radiation which is described by a [relativistic](/source/Theory_of_relativity) [classical field theory](/source/Classical_field_theory) (such as [electromagnetic radiation](/source/Electromagnetic_radiation)), or a combination of these two. Null dusts include [vacuum solutions](/source/Vacuum_solution_(general_relativity)) as a special case. Phenomena which can be modeled by null dust solutions include: - a beam of [neutrinos](/source/Neutrino) assumed for simplicity [to be massless](/source/Neutrino#Mass) (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](/source/Einstein_field_equations) 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](/source/Electrovacuum_solution), 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](/source/Frame_fields_in_general_relativity) 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 - \vec{e}_0, \; \vec{e}_1, \; \vec{e}_2, \; \vec{e}_3 (a [timelike](/source/Timelike) unit [vector field](/source/Vector_field) and three [spacelike](/source/Spacelike) [unit vector](/source/Unit_vector) fields, respectively) can always be found in which the Einstein tensor has a particularly simple appearance: - G^{\hat{a}\hat{b}} = 8 \pi \epsilon \, \left[ \begin{matrix} 1 & 0 & 0 & \pm 1 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ \pm1 & 0 & 0 & 1 \end{matrix} \right] Here, \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 \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](/source/Isotropy_group) as the null vector field \vec{k}. It is generated by two parabolic Lorentz transformations (pointing in the \vec{e}_3 direction) and one rotation (about the \vec{e}_3 axis), and it is isometric to the three-dimensional Lie group E(2), the [isometry group](/source/Isometry_group) of the [euclidean plane](/source/Euclidean_plane). ## Examples Null dust solutions include two large and important families of exact solutions: - [pp-wave spacetimes](/source/Pp-wave_spacetime) (which model generalizations of the [plane waves](/source/Plane_wave) familiar from [electromagnetism](/source/Electromagnetism)), - [Robinson–Trautman null dusts](/source/Robinson%E2%80%93Trautman_null_dusts) (which model radiation expanding from a radiating object). The pp-waves include the [gravitational plane waves](/source/Gravitational_plane_wave) and the [monochromatic electromagnetic plane wave](/source/Monochromatic_electromagnetic_plane_wave). A specific example of considerable interest is - the [Bonnor beam](/source/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](/source/Kinnersley%E2%80%93Walker_photon_rocket) solutions, which include the [Vaidya null dust](/source/Vaidya_metric), which includes the [Schwarzschild vacuum](/source/Schwarzschild_metric). ## See also - [Vaidya metric](/source/Vaidya_metric) - [Lorentz group](/source/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](/source/Cambridge_University_Press). ISBN 0-521-46136-7.. This standard monograph gives many examples of null dust solutions. --- Adapted from the Wikipedia article [Null dust solution](https://en.wikipedia.org/wiki/Null_dust_solution) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Null_dust_solution?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.