Petrov Classification - Physical Interpretation

Physical Interpretation

According to general relativity, the various algebraically special Petrov types have some interesting physical interpretations, the classification then sometimes being called the classification of gravitational fields.

Type D regions are associated with the gravitational fields of isolated massive objects, such as stars. More precisely, type D fields occur as the field of a gravitating object which is completely characterized by its mass and angular momentum. (A more general object might have nonzero higher multipole moments.) The two double principal null directions define "radially" ingoing and outgoing null congruences near the object which is the source of the field.

The electrogravitic tensor (or tidal tensor) in a type D region is very closely analogous to the gravitational fields which are described in Newtonian gravity by a Coulomb type gravitational potential. Such a tidal field is characterized by tension in one direction and compression in the orthogonal directions; the eigenvalues have the pattern (-2,1,1). For example, a spacecraft orbiting the Earth experiences a tiny tension along a radius from the center of the Earth, and a tiny compression in the orthogonal directions. Just as in Newtonian gravitation, this tidal field typically decays like, where is the distance from the object.

If the object is rotating about some axis, in addition to the tidal effects, there will be various gravitomagnetic effects, such as spin-spin forces on gyroscopes carried by an observer. In the Kerr vacuum, which is the best known example of type D vacuum solution, this part of the field decays like .

Type III regions are associated with a kind of longitudinal gravitational radiation. In such regions, the tidal forces have a shearing effect. This possibility is often neglected, in part because the gravitational radiation which arises in weak-field theory is type N, and in part because type III radiation decays like, which is faster than type N radiation.

Type N regions are associated with transverse gravitational radiation, which is the type astronomers are trying to detect with LIGO. The quadruple principal null direction corresponds to the wave vector describing the direction of propagation of this radiation. It typically decays like, so the long-range radiation field is type N.

Type II regions combine the effects noted above for types D, III, and N, in a rather complicated nonlinear way.

Type O regions, or conformally flat regions, are associated with places where the Weyl tensor vanishes identically. In this case, the curvature is said to be pure Ricci. In a conformally flat region, any gravitational effects must be due to the immediate presence of matter or the field energy of some nongravitational field (such as an electromagnetic field). In a sense, this means that any distant objects are not exerting any long range influence on events in our region. More precisely, if there are any time varying gravitational fields in distant regions, the news has not yet reached our conformally flat region.

Gravitational radiation emitted from an isolated system will usually not be algebraically special. The peeling theorem describes the way in which, as one moves farther way from the source of the radiation, the various components of the radiation field "peel" off, until finally only type N radiation is noticeable at large distances. This is similar to the electromagnetic peeling theorem.