Petrov's classification

The Petrov classification (sometimes the Petrov – Pirani classification , rarely the Petrov – Pirani – Penrose classification ) describes possible algebraic symmetries of the Weil tensor for each event on a pseudo-Riemannian manifold .

This classification is most actively used in the study of exact solutions of Einstein's equations , although generally speaking it is an abstract mathematical result that does not depend on any physical interpretation. The classification was first proposed in 1954 by A.Z. Petrov and in 1957 independently by Felix Pirani .

Content

Classification Theorem

A rank 4 tensor with antisymmetry in the first and second pair of indices, for example, the Weyl tensor , at each point of the manifold can be represented as a linear operator $C$ ${\ displaystyle C}$ : $T_{p}\times T_{p}\rightarrow T_{p}\times T_{p}$ ${\ displaystyle T_ {p} \ times T_ {p} \ rightarrow T_ {p} \ times T_ {p}}$ acting in the vector space of bivectors :

X^{ab}\rightarrow {\frac {1}{2}}\,{C^{ab}}_{mn}X^{mn}

{\ displaystyle X ^ {ab} \ rightarrow {\ frac {1} {2}} \, {C ^ {ab}} _ {mn} X ^ {mn}}

In this case, it is natural to pose the problem of finding the eigenvalues $\lambda$ ${\ displaystyle \ lambda}$ and eigenvectors (or eigenvectors) $X^{ab}$ ${\ displaystyle X ^ {ab}}$ such that

{\frac {1}{2}}\,{C^{ab}}_{mn}\,X^{mn}=\lambda \,X^{ab}

{\ displaystyle {\ frac {1} {2}} \, {C ^ {ab}} _ {mn} \, X ^ {mn} = \ lambda \, X ^ {ab}}

In four-dimensional pseudo-Riemannian manifolds, at each point the space of bivectors is six-dimensional. However, the Weil tensor symmetries limit the dimension of the space of proper bivectors to four. Thus, the Weyl tensor at a given point can have a maximum of four linearly independent eigenvectors.

Just as in the usual theory of eigenvectors of a linear operator , the eigenvectors of the Weyl tensor can be multiple. The multiplicity of proper bivectors indicates some additional algebraic symmetry of the Weyl tensor at a given point; this means that the type of symmetry of the Weil tensor can be determined by solving the 4th order equation for its eigenvalues.

The proper bivectors of the Weil tensor are associated with certain isotropic vectors on the manifold, which are called the main isotropic directions (at a given point). The classification theorem states that there are exactly six possible types of algebraic symmetry, which are known as Petrov types :

Penrose diagram showing the hierarchy of Petrov types for the Weil tensor

Type I : four major isotropic directions,
Type II : one double and two single main isotropic directions,
Type D : two double isotropic directions,
Type III : one triple and one single direction,
Type N : one isotropic direction with a multiplicity of 4,
Type O : Weil tensor is zero.

The Weyl tensor of type I (at a point) is called algebraically general ; tensors of other types are called algebraically special . Different points in space-time can have a different type of Petrov. Possible transitions between Petrov types are shown in the figure, which can also be interpreted so that some types of Petrov are more special than others. For example, type I , the most common type, can degenerate to types II or D , while type II can go over to types III , N , or D.

Bela Criteria

For a pseudo-Riemannian (Lorentzian) manifold $M$ ${\ displaystyle M}$ , the Weyl tensor can be calculated from the metric tensor . If at some point $p\in M$ ${\ displaystyle p \ in M}$ The Weyl tensor is algebraically special ; there is an effective set of rules (which was discovered by Louis Bel) for determining the Petrov type at a point $p$ ${\ displaystyle p}$ . We denote the components of the Weyl tensor at the point $p$ ${\ displaystyle p}$ across $C_{abcd}$ ${\ displaystyle C_ {abcd}}$ (and suppose that they are not equal to zero, that is, it is not type O ), then Bel's criteria can be expressed as follows:

$C_{abcd}$ ${\ displaystyle C_ {abcd}}$ has type N if and only if there is a single (up to a factor) isotropic vector $k(p)$ ${\ displaystyle k (p)}$ satisfying

C_{abcd}\,k^{d}=0

{\ displaystyle C_ {abcd} \, k ^ {d} = 0}

If $C_{abcd}$ ${\ displaystyle C_ {abcd}}$ does not belong to type N then $C_{abcd}$ ${\ displaystyle C_ {abcd}}$ belongs to type III if and only if there is a single (up to a factor) isotropic vector $k(p)$ ${\ displaystyle k (p)}$ satisfying

C_{abcd}\,k^{b}k^{d}=0

{\ displaystyle C_ {abcd} \, k ^ {b} k ^ {d} = 0}

$C_{abcd}$ ${\ displaystyle C_ {abcd}}$ has type II if and only if there is a single (up to a factor) isotropic vector $k$ ${\ displaystyle k}$ satisfying

C_{abcd}\,k^{b}k^{d}=\alpha k_{a}k_{c}

{\ displaystyle C_ {abcd} \, k ^ {b} k ^ {d} = \ alpha k_ {a} k_ {c}}

and

{}^{*}C_{abcd}\,k^{b}k^{d}=\beta k_{a}k_{c}

{\ displaystyle {} ^ {*} C_ {abcd} \, k ^ {b} k ^ {d} = \ beta k_ {a} k_ {c}}

(

\alpha \beta \neq 0

{\ displaystyle \ alpha \ beta \ neq 0}

)

$C_{abcd}$ ${\ displaystyle C_ {abcd}}$ has type D if and only if there are two linearly independent isotropic vectors $k$ ${\ displaystyle k}$ , $k^{'}$ ${\ displaystyle k '}$ satisfying the conditions:

C_{abcd}\,k^{b}k^{d}=\alpha k_{a}k_{c}

{\ displaystyle C_ {abcd} \, k ^ {b} k ^ {d} = \ alpha k_ {a} k_ {c}}

,

{}^{*}C_{abcd}\,k^{b}k^{d}=\beta k_{a}k_{c}

{\ displaystyle {} ^ {*} C_ {abcd} \, k ^ {b} k ^ {d} = \ beta k_ {a} k_ {c}}

(

\alpha \beta \neq 0

{\ displaystyle \ alpha \ beta \ neq 0}

)

and

C_{abcd}\,k'^{b}k'^{d}=\gamma k'_{a}k'_{c}

{\ displaystyle C_ {abcd} \, k '^ {b} k' ^ {d} = \ gamma k '_ {a} k' _ {c}}

,

{}^{*}C_{abcd}\,k'^{b}k'^{d}=\delta k'_{a}k'_{c}

{\ displaystyle {} ^ {*} C_ {abcd} \, k '^ {b} k' ^ {d} = \ delta k '_ {a} k' _ {c}}

(

\gamma \delta \neq 0

{\ displaystyle \ gamma \ delta \ neq 0}

)

Where ${{}^{*}C}_{abcd}$ ${\ displaystyle {{} ^ {*} C} _ {abcd}}$ is the tensor dual to the Weyl tensor at the point $p$ ${\ displaystyle p}$ .

Bel criteria are applied in the general theory of relativity, that is, the Petrov type for the algebraically special Weil tensor is found using zero vectors.

Physical Interpretation

In accordance with the general theory of relativity , the algebraically special types of Petrov have an interesting physical interpretation, so their classification is often called the classification of gravitational fields .

Type D field regions are associated with gravitational fields of isolated massive celestial bodies, such as stars. More precisely, fields of type D arise around stationary objects, which, due to their physical characteristics, have only mass and angular momentum. (A more complex dynamic body has nonzero multipole moments .) Two main isotropic directions define two “radially” converging and diverging isotropic families near the gravitating body.

The electrogravity tensor go tidal tensor ) in areas of type D is similar to gravitational fields, which are described by Newtonian gravity with the Coulomb type of gravitational potential . Such a tidal field is characterized by extension in one direction and compression in orthogonal directions; eigenvalues have a characteristic pattern (-2,1,1). For example, a satellite in orbit around the Earth experiences slight radial tension and weak compression in orthogonal directions. Just like in Newtonian gravity, the tidal field decreases as $O(r^{-3})$ ${\ displaystyle O (r ^ {- 3})}$ where $r$ ${\ displaystyle r}$ - distance from the gravitating body.

If the body rotates around a certain axis, then in addition to tidal effects, various gravitomagnetic effects will appear, such as spin-spin interaction acting on the observer's gyroscopes . In a Kerr vacuum , which is a typical example of a type D vacuum field, this part of the field decreases as $O(r^{-4})$ ${\ displaystyle O (r ^ {- 4})}$ .

Type III regions are associated with the longitudinal portion of a time-varying gravitational field (sometimes called longitudinal gravitational radiation). In these areas, tidal forces have the nature of shifts. This is a rather poorly studied type of field, partly because the gravitational radiation that arises in the approximation of weak fields is of type N , since the field of type III decreases as $O(r^{-2})$ ${\ displaystyle O (r ^ {- 2})}$ , that is, much faster than radiation of type N , and, accordingly, does not detach from the source.

Type N regions are associated with transverse gravitational radiation , which astronomers discovered in 2015 . The fourfold isotropic direction corresponds to the wave vector describing the direction of radiation propagation. The radiation amplitude usually decreases as $O(r^{-1})$ ${\ displaystyle O (r ^ {- 1})}$ so that the gravitational field of a distant source is always radiation and is of type N.

Type II combines the effects of fields of type D , III and N in a rather complex non-linear way.

Regions of type O , or conformally Euclidean domains, are zones in which the Weyl tensor is identically equal to zero. In this case, the curvature tensor is pure Ricci . In conformally Euclidean regions, any gravitational effects arise only due to the instant presence of matter or energy of some non-gravitational field (for example, an electromagnetic field ). In a sense, this means that any remote objects do not affect events in this area; more precisely, if there is some gravitational dynamics in remote areas, the news about it has not yet reached the conformally Euclidean zone under consideration.

The gravitational field and, accordingly, the gravitational radiation emitted by an isolated system, in the general case will not be algebraically special at a finite distance from the source. The splitting theorem describes how different types of fields “split off” as the observer moves away from the radiation source, until only type N radiation is left at long distances. A similar theorem exists in electromagnetism.

Examples

For some exact solutions of the Einstein equations, the Weil tensor is of the same type at each world point :

Kerr metric in vacuum is of type D ,
Robinson-Trautman Space - Type III ,
pp waves are type N ,
the Friedman – Robertson – Walker metric is type O everywhere.

In general, an arbitrary spherically symmetric space-time must be algebraically special, and any static space-time must be of type D.

Literature

From the section of relativity on the site "World of Mathematical Equations" - EqWorld