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| In [[differential geometry]] and [[theoretical physics]], the '''Petrov classification''' describes the possible algebraic [[symmetry|symmetries]] of the [[Weyl tensor]] at each [[Spacetime#Basic concepts|event]] in a [[Lorentzian manifold]].
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| It is most often applied in studying [[exact solutions]] of [[Einstein's field equations]], but strictly speaking the classification is a theorem in pure mathematics applying to any Lorentzian manifold, independent of any physical interpretation. The classification was found in 1954 by [[A. Z. Petrov]].
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| ==The classification theorem==
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| We can think of a fourth [[Tensor#Tensor rank|rank]] [[tensor]] such as the [[Weyl tensor]], ''evaluated at some event'', as acting on the space of [[bivector]]s at that event like a [[linear operator]] acting on a vector space:
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| :<math> X^{ab} \rightarrow \frac{1}{2} \, {C^{ab}}_{mn} X^{mn} </math>
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| Then, it is natural to consider the problem of finding [[eigenvalues]] <math>\lambda</math> and [[eigenvectors]] (which are now referred to as eigenbivectors) <math>X^{ab}</math> such that
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| :<math>\frac{1}{2} \, {C^{ab}}_{mn} \, X^{mn} = \lambda \, X^{ab} </math>
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| In (four dimensional) Lorentzian spacetimes, there is a six dimensional space of antisymmetric bivectors at each event. However, the symmetries of the Weyl tensor imply that any eigenbivectors must belong to a four dimensional subset.
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| Thus, the Weyl tensor (at a given event) can in fact have ''at most four'' linearly independent eigenbivectors.
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| Just as in the theory of the eigenvectors of an ordinary linear operator, the eigenbivectors of the Weyl tensor can occur with various [[Multiplicity (mathematics)|multiplicities]]. Just as in the case of ordinary linear operators, any multiplicities among the eigenbivectors indicates a kind of ''algebraic symmetry'' of the Weyl tensor at the given event. Just as you would expect from the theory of the eigenvalues of an ordinary linear operator on a four dimensional vector space, the different types of Weyl tensor (at a given event) can be determined by solving a [[Characteristic polynomial#Characteristic equation|characteristic equation]], in this case a [[quartic equation]].
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| These eigenbivectors are associated with certain null vectors in the original spacetime, which are called the '''principal null directions''' (at a given event).
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| The relevant [[multilinear algebra]] is somewhat involved (see the citations below), but the resulting classification theorem states that there are precisely six possible types of algebraic symmetry. These are known as the '''Petrov types''':
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| [[Image:Petrov.png|frame|right|The ''Penrose diagram'' showing the possible degenerations of the Petrov type of the Weyl tensor]]
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| *'''Type I''' : four simple principal null directions,
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| *'''Type II''' : one double and two simple principal null directions,
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| *'''Type D''' : two double principal null directions,
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| *'''Type III''' : one triple and one simple principal null direction,
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| *'''Type N''' : one quadruple principal null direction,
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| *'''Type O''' : the Weyl tensor vanishes.
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| The possible transitions between Petrov types are shown in the figure, which can also be interpreted as stating that some of the Petrov types are "more special" than others. For example, type '''I''', the most general type, can ''degenerate'' to types '''II''' or '''D''', while type '''II''' can degenerate to types '''III''', '''N''', or '''D'''.
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| Different events in a given spacetime can have different Petrov types. A Weyl tensor that has type '''I''' (at some event) is called '''algebraically general'''; otherwise, it is called '''algebraically special''' (at that event). Type '''O''' spacetimes are said to be '''[[conformally flat]]'''.
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| ==Newman–Penrose formalism==
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| The [[Newman–Penrose formalism]] is often used in practice for the classification. Consider the following set of bivectors:
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| :<math>U_{ab}=-l_{[a}m_{b]}</math>
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| :<math>V_{ab}=n_{[a}m_{b]}</math>
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| :<math>W_{ab}=m_{[a}\bar{m}_{b]}-n_{[a}l_{b]}.</math>
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| The Weyl tensor can be expressed as a combination of these bivectors through
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| :<math>\begin{align}C_{abcd}&= \Psi_0U_{ab}U_{cd} \\
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| &\, \, \, +\Psi_1(U_{ab}W_{cd}+W_{ab}U_{cd}) \\
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| &\, \, \, +\Psi_2(V_{ab}U_{cd}+U_{ab}V_{cd}+W_{ab}W_{cd}) \\
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| &\, \, \, +\Psi_3(V_{ab}W_{cd}+W_{ab}V_{cd}) \\
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| &\, \, \, +\Psi_4V_{ab}V_{cd}\end{align}</math>
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| where the <math>\{\Psi_j\}</math> are the [[Weyl scalar]]s. The six different Petrov types are distinguished by which of the Weyl scalars vanish. The conditions are
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| *'''Type I''' : <math>\Psi_0=0</math>,
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| *'''Type II''' : <math>\Psi_0=\Psi_1=0</math>,
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| *'''Type D''' : <math>\Psi_0=\Psi_1=\Psi_3=\Psi_4=0</math>,
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| *'''Type III''' : <math>\Psi_0=\Psi_1=\Psi_2=0</math>,
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| *'''Type N''' : <math>\Psi_0=\Psi_1=\Psi_2=\Psi_3=0</math>,
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| *'''Type O''' : <math>\Psi_0=\Psi_1=\Psi_2=\Psi_3=\Psi_4=0</math>.
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| ==Bel criteria==
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| Given a [[Metric (mathematics)|metric]] on a Lorentzian manifold <math>M</math>, the Weyl tensor <math>C</math> for this metric may be computed. If the Weyl tensor is ''algebraically special'' at some <math>p \in M</math>, there is a useful set of conditions, found by Lluis<!--sic--> (or Louis) Bel and Robert Debever,<ref>[http://arxiv.org/find/gr-qc/1/au:+Ortaggio_M/0/1/0/all/0/1 Marcello Ortaggio (2009), ''Bel-Debever criteria for the classification of the Weyl tensors in higher dimensions.'']</ref> for determining precisely the Petrov type at <math>p</math>. Denoting the Weyl tensor components at <math>p</math> by <math>C_{abcd}</math> (assumed non-zero, i.e., not of type '''O'''), the '''Bel criteria''' may be stated as:
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| * <math>C_{abcd}</math> is type '''N''' if and only if there exists a vector <math>k(p)</math> satisfying
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| :<math>C_{abcd} \, k^d =0</math>
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| where <math>k</math> is necessarily null and unique (up to scaling).
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| * If <math>C_{abcd}</math> is '''not type N''', then <math>C_{abcd}</math> is of type '''III''' if and only if there exists a vector <math>k(p)</math> satisfying
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| :<math>C_{abcd}\, k^bk^d=0= {^*C}_{abcd}\, k^bk^d</math>
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| where <math>k</math> is necessarily null and unique (up to scaling).
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| * <math>C_{abcd}</math> is of type '''II''' if and only if there exists a vector <math>k</math> satisfying
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| :<math>C_{abcd}\, k^bk^d=\alpha k_ak_c</math> and <math>{}^*C_{abcd}\, k^bk^d=\beta k_ak_c</math> (<math>\alpha \beta \neq 0</math>) | |
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| where <math>k</math> is necessarily null and unique (up to scaling).
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| * <math>C_{abcd}</math> is of type '''D''' if and only if there exists ''two linearly independent vectors'' <math>k</math>, <math>k'</math> satisfying the conditions
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| :<math>C_{abcd}\, k^bk^d=\alpha k_ak_c</math>, <math>{}^*C_{abcd}\, k^bk^d=\beta k_ak_c</math> (<math>\alpha \beta \neq 0</math>)
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| and
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| :<math>C_{abcd}\, k'^bk'^d=\gamma k'_ak'_c</math>, <math>{}^*C_{abcd}\, k'^bk'^d=\delta k'_ak'_c</math> (<math>\gamma \delta \neq 0</math>).
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| where <math>{{}^*C}_{abcd}</math> is the dual of the Weyl tensor at <math>p</math>.
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| In fact, for each criterion above, there are equivalent conditions for the Weyl tensor to have that type. These equivalent conditions are stated in terms of the dual and self-dual of the Weyl tensor and certain bivectors and are collected together in Hall (2004).
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| The Bel criteria find application in general relativity where determining the Petrov type of algebraically special Weyl tensors is accomplished by searching for null vectors.
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| ==Physical Interpretation==
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| 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'''.
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| '''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 congruence]]s near the object which is the source of the field.
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| 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 <math>O(r^{-3})</math>, where <math>r</math> is the distance from the object.
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| If the object is rotating about some [[axis of rotation|axis]], in addition to the tidal effects, there will be various [[gravitomagnetism|gravitomagnetic]] effects, such as [[spin-spin forces]] on [[gyroscopes]] carried by an observer. In the [[Kerr metric|Kerr vacuum]], which is the best known example of type '''D''' vacuum solution, this part of the field decays like <math>O(r^{-4})</math>.
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| '''Type III''' regions are associated with a kind of [[Longitudinal wave|longitudinal]] gravitational radiation. In such regions, the tidal forces have a [[Shear (fluid)|shearing]] effect. This possibility is often neglected, in part because the gravitational radiation which arises in [[Weak-field approximation|weak-field theory]] is type '''N''', and in part because type '''III''' radiation decays like <math>O(r^{-2})</math>, which is faster than type '''N''' radiation.
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| '''Type N''' regions are associated with [[Transversality (mathematics)|transverse]] gravitational radiation, which is the type astronomers are trying to detect with [[LIGO]].
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| The quadruple principal null direction corresponds to the [[wave vector]] describing the direction of propagation of this radiation. It typically decays like <math>O(r^{-1})</math>, so the long-range radiation field is type '''N'''.
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| '''Type II''' regions combine the effects noted above for types '''D''', '''III''', and '''N''', in a rather complicated nonlinear way.
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| '''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 tensor|Ricci]]''. In a conformally flat region, any gravitational effects must be due to the immediate presence of matter or the [[classical field theory|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 function|news]] has not yet reached our conformally flat region.
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| [[Gravitational radiation]] emitted from an isolated system will usually not be algebraically special.
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| 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]].
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| ==Examples==
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| In some (more or less) familiar solutions, the Weyl tensor has the same Petrov type at each event:
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| *the [[Kerr metric|Kerr vacuum]] is everywhere type '''D''',
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| *certain [[Robinson/Trautman spacetimes|Robinson/Trautman vacuums]] are everywhere type '''III''',
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| *the [[pp-wave spacetimes]] are everywhere type '''N''',
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| *the [[Robertson-Walker metric|FRW models]] are everywhere type '''O'''.
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| More generally, any [[spherically symmetric spacetime]] must be of type '''D''' (or '''O'''). All algebraically special spacetimes having various types of [[stress-energy tensor]] are known, for example, all the type '''D''' vacuum solutions.
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| Some classes of solutions can be invariantly characterized using algebraic symmetries of the Weyl tensor: for example, the class of non-conformally flat null [[electrovacuum solution|electrovacuum]] or [[null dust solution|null dust]] solutions admitting an expanding but nontwisting null congruence is precisely the class of ''Robinson/Trautmann spacetimes''. These are usually type '''II''', but include type '''III''' and type '''N''' examples.
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| ==Generalization to higher dimensions== | |
| A. Coley, R. Milson, V. Pravda and A. Pravdová (2004) developed a generalization of algebraic classification to arbitrary spacetime dimension <math>d</math>. Their approach uses a null [[vielbein|frame basis]] approach, that is a frame basis containing two null vectors <math>l</math> and <math>n</math>, along with <math>d-2</math> spacelike vectors. Frame basis components of the [[Weyl tensor]] are classified by their transformation properties under local [[Lorentz transformations|Lorentz boosts]]. If particular Weyl components vanish, then <math>l</math> and/or <math>n</math> are said to be '''Weyl-Aligned Null Directions''' (WANDs). In four dimensions, <math>l</math> is a WAND if and only if it is a principal null direction in the sense defined above. This approach gives a natural higher-dimensional extension of each of the various algebraic types '''II''','''D''' etc. defined above.
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| An alternative, but inequivalent, generalization was previously defined by de Smet (2002), based on a [[Spinors|spinorial approach]]. However, the de Smet is restricted to 5 dimensions only.
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| == See also ==
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| *[[Classification of electromagnetic fields]]
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| *[[Exact solutions in general relativity]]
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| *[[Segre classification]]
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| *[[Peeling theorem]]
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| *[[Plebanski tensor]]
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| *[[Penrose limit]]
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| == References ==
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| <references/>
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| *{{cite journal | author=Coley, A. et al.. | title=Classification of the Weyl tensor in higher dimensions | year=2004 | doi=10.1088/0264-9381/21/7/L01 | journal=Classical and Quantum Gravity | volume=21 | issue=7 | pages=L35–L42 | arxiv=gr-qc/0401008|bibcode = 2004CQGra..21L..35C }}
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| *{{cite journal | author=de Smet, P. | title=Black holes on cylinders are not algebraically special | year=2002 | doi=10.1088/0264-9381/19/19/307 | journal=Classical and Quantum Gravity | volume=19 | issue=19 | pages=4877–4896 |arxiv=hep-th/0206106|bibcode = 2002CQGra..19.4877D }}
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| *{{cite book | author=d'Inverno, Ray | title=Introducing Einstein's Relativity | location=Oxford | publisher=Oxford University Press | year=1992 | isbn=0-19-859686-3}} ''See sections 21.7, 21.8''
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| * {{cite book | author = Hall, Graham | title=Symmetries and Curvature Structure in General Relativity (World Scientific Lecture Notes in Physics) | location= Singapore | publisher=World Scientific Pub. Co | year=2004 | isbn=981-02-1051-5}} ''See sections 7.3, 7.4 for a comprehensive discussion of the Petrov classification''.
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| *{{cite journal | author=MacCallum, M.A.H. | title=Editor's note: Classification of spaces defining gravitational fields | journal=General Relativity and Gravitation | year=2000 | volume=32 | issue=8 | pages=1661–1663|bibcode = 2000GReGr..32.1661P }}
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| *{{cite journal | author=Penrose, Roger | title=A spinor approach to general relativity | journal=Annals of Physics | year=1960 | volume=10 | pages=171–201|bibcode = 1960AnPhy..10..171P |doi = 10.1016/0003-4916(60)90021-X }}
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| *{{cite journal | author=Petrov, A.Z. | title=Klassifikacya prostranstv opredelyayushchikh polya tyagoteniya | journal=Uch. Zapiski Kazan. Gos. Univ. | year=1954 | volume=114 | pages=55–69 | issue=8}} English translation {{cite journal | author=Petrov, A.Z. | title=Classification of spaces defined by gravitational fields | journal=General Relativity and Gravitation | year=2000 | volume=32 | pages=1665–1685 | doi=10.1023/A:1001910908054 | issue=8|bibcode = 2000GReGr..32.1665P }}
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| *{{cite book | author = Petrov, A.Z. | title=Einstein Spaces | location=Oxford | publisher=Pergamon | year=1969 | isbn= 0080123155}}, translated by R. F. Kelleher & J. Woodrow.
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| *{{cite book | author=Stephani, H.; Kramer, D.; MacCallum, M.; Hoenselaers, C.; & Herlt, E. | title=Exact Solutions of Einstein's Field Equations (2nd edn.) | location=Cambridge | publisher=Cambridge University Press | year=2003 | isbn=0-521-46136-7}} ''See chapters 4, 26''
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| [[Category:Tensors in general relativity]]
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| [[Category:Exact solutions in general relativity]]
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| [[Category:Differential geometry]]
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