Classification used in differential geometry and general relativity
In
differential geometry
and
theoretical physics
, the
Petrov classification
(also known as Petrov?Pirani?Penrose classification) describes the possible algebraic
symmetries
of the
Weyl tensor
at each
event
in a
Lorentzian manifold
.
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
and independently by
Felix Pirani
in 1957.
Classification theorem
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We can think of a fourth
rank
tensor
such as the
Weyl tensor
,
evaluated at some event
, as acting on the space of
bivectors
at that event like a
linear operator
acting on a vector space:
Then, it is natural to consider the problem of finding
eigenvalues
and
eigenvectors
(which are now referred to as eigenbivectors)
such that
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.
Thus, the Weyl tensor (at a given event) can in fact have
at most four
linearly independent eigenbivectors.
The eigenbivectors of the Weyl tensor can occur with various
multiplicities
and any multiplicities among the eigenbivectors indicates a kind of
algebraic symmetry
of the Weyl tensor at the given event. The different types of Weyl tensor (at a given event) can be determined by solving a
characteristic equation
, in this case a
quartic equation
. All the above happens similarly to the theory of the eigenvectors of an ordinary linear operator.
These eigenbivectors are associated with certain
null vectors
in the original spacetime, which are called the
principal null directions
(at a given event).
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
:
- Type I
: four simple principal null directions,
- Type II
: one double and two simple principal null directions,
- Type D
: two double principal null directions,
- Type III
: one triple and one simple principal null direction,
- Type N
: one quadruple principal null direction,
- Type O
: the Weyl tensor vanishes.
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
.
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). In General Relativity, type
O
spacetimes are
conformally flat
.
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, constructed out of
tetrads
of
null vectors
(note that in some notations, symbols l and n are interchanged):
The Weyl tensor can be expressed as a combination of these bivectors through
where the
are the
Weyl scalars
and c.c. is the complex conjugate. The six different Petrov types are distinguished by which of the Weyl scalars vanish. The conditions are
- Type I
:
,
- Type II
:
,
- Type D
:
,
- Type III
:
,
- Type N
:
,
- Type O
:
.
Bel criteria
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Given a
metric
on a Lorentzian manifold
, the Weyl tensor
for this metric may be computed. If the Weyl tensor is
algebraically special
at some
, there is a useful set of conditions, found by Lluis (or Louis) Bel and Robert Debever,
[1]
for determining precisely the Petrov type at
. Denoting the Weyl tensor components at
by
(assumed non-zero, i.e., not of type
O
), the
Bel criteria
may be stated as:
- is type
N
if and only if there exists a vector
satisfying
where
is necessarily null and unique (up to scaling).
- If
is
not type N
, then
is of type
III
if and only if there exists a vector
satisfying
where
is necessarily null and unique (up to scaling).
- is of type
II
if and only if there exists a vector
satisfying
- and
(
)
where
is necessarily null and unique (up to scaling).
- is of type
D
if and only if there exists
two linearly independent vectors
,
satisfying the conditions
- ,
(
)
and
- ,
(
).
where
is the dual of the Weyl tensor at
.
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).
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.
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
.
Type D
regions are associated with the gravitational fields of isolated massive objects, such as stars. More precisely, type
D
fields occur as the exterior 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 have detected 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
.
Examples
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In some (more or less) familiar solutions, the Weyl tensor has the same Petrov type at each event:
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.
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
or
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.
Generalization to higher dimensions
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A. Coley, R. Milson, V. Pravda and A. Pravdova (2004) developed a generalization of algebraic classification to arbitrary spacetime dimension
. Their approach uses a null
frame basis
approach, that is a frame basis containing two null vectors
and
, along with
spacelike vectors. Frame basis components of the
Weyl tensor
are classified by their transformation properties under local
Lorentz boosts
. If particular Weyl components vanish, then
and/or
are said to be
Weyl-Aligned Null Directions
(WANDs). In four dimensions,
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.
An alternative, but inequivalent, generalization was previously defined by de Smet (2002), based on a
spinorial approach
. However, de Smet's approach is restricted to 5 dimensions only.
See also
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References
[
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]
- Coley, A.; et al. (2004). "Classification of the Weyl tensor in higher dimensions".
Classical and Quantum Gravity
.
21
(7): L35?L42.
arXiv
:
gr-qc/0401008
.
Bibcode
:
2004CQGra..21L..35C
.
doi
:
10.1088/0264-9381/21/7/L01
.
S2CID
31859828
.
- de Smet, P. (2002). "Black holes on cylinders are not algebraically special".
Classical and Quantum Gravity
.
19
(19): 4877?4896.
arXiv
:
hep-th/0206106
.
Bibcode
:
2002CQGra..19.4877D
.
doi
:
10.1088/0264-9381/19/19/307
.
S2CID
15772816
.
- d'Inverno, Ray (1992).
Introducing Einstein's Relativity
. Oxford:
Oxford University Press
.
ISBN
0-19-859686-3
.
See sections 21.7, 21.8
- Hall, Graham (2004).
Symmetries and Curvature Structure in General Relativity (World Scientific Lecture Notes in Physics)
. Singapore: World Scientific Pub. Co.
ISBN
981-02-1051-5
.
See sections 7.3, 7.4 for a comprehensive discussion of the Petrov classification
.
- MacCallum, M.A.H. (2000). "Editor's note: Classification of spaces defining gravitational fields".
General Relativity and Gravitation
.
32
(8): 1661?1663.
Bibcode
:
2000GReGr..32.1661P
.
doi
:
10.1023/A:1001958823984
.
S2CID
116370483
.
- Penrose, Roger (1960). "A spinor approach to general relativity".
Annals of Physics
.
10
(2): 171?201.
Bibcode
:
1960AnPhy..10..171P
.
doi
:
10.1016/0003-4916(60)90021-X
.
- Petrov, A.Z. (1954). "Klassifikacya prostranstv opredelyayushchikh polya tyagoteniya".
Uch. Zapiski Kazan. Gos. Univ
.
114
(8): 55?69.
English translation
Petrov, A.Z. (2000). "Classification of spaces defined by gravitational fields".
General Relativity and Gravitation
.
32
(8): 1665?1685.
Bibcode
:
2000GReGr..32.1665P
.
doi
:
10.1023/A:1001910908054
.
S2CID
73540912
.
- Petrov, A.Z. (1969).
Einstein Spaces
. Oxford: Pergamon.
ISBN
0080123155
.
, translated by R. F. Kelleher & J. Woodrow.
- Stephani, H.; Kramer, D.; MacCallum, M.; Hoenselaers, C. & Herlt, E. (2003).
Exact Solutions of Einstein's Field Equations (2nd edn.)
. Cambridge:
Cambridge University Press
.
ISBN
0-521-46136-7
.
See chapters 4, 26