Weyl
Tensor |

**In General** > s.a. bianchi
models; curvature; FLRW
geometry; riemann
tensor.

$ __Def__: The "trace-free part" of
the Riemann tensor, defined on a manifold of dimension *n* ≥ 3 by

\[ \def\_#1{_{#1}^{\,}} \def\ii{{\rm i}}

C\_{abcd}:= R\_{abcd} - {2\over n-2}\,(g\_{a[c}\,R\_{d]b} - g\_{b[c}\,R\_{d]a})

+ {2\over(n-1)(n-2)}\,R\,g\_{a[c}\,g\_{d]b} \;;\]

The definition can also be written in terms of the Weyl-Schouten tensor *S*_{ab}, as

\[ C\_{abcd}:= R\_{abcd} - {2\over n-2}\,(g\_{a[c}\,R\_{d]b} - g\_{b[c}\,R\_{d]a})\;,

\quad{\rm with}\quad S\_{ab}:= R\_{ab}
- {1\over2(n-1)}\,R\,g\_{ab}\;. \]

* __Properties__: It is
conformally invariant, if expressed with indices *C*_{abc}^{d},
and its trace over any two indices vanishes; The number of independent
components in *n* dimensions is \({1\over12}\)*n* (*n*+1)
(*n*+2) (*n*–3) [@ e.g., in Gursky & Viaclovsky AM(07)].

* __Use__: In general relativity
it contains the information on gravitational radiation, since the "trace part" of the Riemann tensor is determined by the
matter.

@ __General references__: Weyl MZ(18);
in Weinberg 72; in Wald 84; Ehlers & Buchert GRG(09)-a0907 [Newtonian limit]; Dewar & Weatherall a1707-conf [in geometrized Newtonian gravitation].

@ __Related topics and uses__: Schmidt GRG(03)gq
[square]; Hussain et al IJMPD(05)-a0812 [collineations];
Danehkar MPLA(09)
[significance]; Hofmann et al al PRD(13)-a1308 [limitations of the interpretation in terms of incoming and outgoing waves];
> s.a. gravitational entropy; phenomenology of gravity.

**Electric Part**

$ __Def__: The symmetric
trace-free tensor defined, for a hypersurface Σ with
unit normal *n ^{a}* in spacetime, by

*E*_{ab}:= *C*_{ambn} *n*^{m} *n*^{n} .

* __And physics__: Corresponds
to tidal forces; Near spatial infinity, using an appropriately rescaled curvature
on the hyperboloid \({\cal D}\) [> see asymptotic
flatness at spi], it
represents the way in which nearby geodesics tear apart from each other.

* __Potential__: It admits a potential *E*,
such that *E*_{ab}
= –\({1\over4}\)(*D*_{a }*D*_{b}* E* + *E* *h*_{ab});
This is used to define 4-momentum.

@ __References__: Ashtekar in(80); Bonnor CQG(95);
Maartens et
al CQG(97)gq/96 [and
gravitational degrees of freedom].

**Magnetic Part**

$ __Def__: Given a spacelike
hypersurface Σ with unit normal *n*^{a}, it is the
symmetric tracefree tensor defined by

(*H*_{ab} or) *B*_{ab}:= \({1\over2}\)**C*_{ambn}
*n*^{m} *n*^{n}, with **C*_{ambn}:=
*ε*_{ampq}* C*^{pq}_{bn} .

* __And physics__: It has
no direct physical significance, and all the conserved quantities constructed from it vanish identically;
The tensor itself vanishes for stationary or axisymmetric spacetimes that are asymptotically flat at spatial
infinity; However, it has been shown to be responsible for destroying the pure Kasner-like approach
to the singularity in BKL evolution.

* __Potential__: The one
constructed from the appropriate curvature on the hyperboloid \({\cal D}\) at
spatial infinity admits a potential *K*_{ab},
such that *B*_{ab} = –\({1\over4}\)*ε*_{mnb}* D*^{m}*K*^{ n}_{a}.

* __Purely magnetic spacetimes__:
Spacetimes in which the electric part of the Weyl tensor, *C*_{abcd}* u*^{b}* u*^{d} =
0, for some timelike unit vector field *u*^{a}, vanishes;
2004, Examples of purely magnetic spacetimes are known and are relatively easy
to construct, if no restrictions are placed on the energy-momentum tensor; However,
it has long been conjectured that purely magnetic vacuum spacetimes (with or
without a cosmological constant) do not exist; For irrotational dust, the only solutions are FLRW spacetimes.

@ __Purely magnetic__: Haddow JMP(95)gq;
Van den Bergh CQG(03)gq/02,
CQG(03)gq,
Zakhary & Carminati GRG(05)
[vacuum no-go results]; Lozanovski CQG(02),
& Carminati CQG(03) [locally rotationally
symmetric]; Barnes gq/04-proc;
Wylleman CQG(06)gq [irrotational
dust, any cosmological constant]; Wylleman & Van den Bergh PRD(06)gq [classification]; Hervik et al SPP(14)-a1301 [and purely electric, in higher dimensions].

@ __And physics__: Ellis & Dunsby ApJ(97)ap/94 [evolution
in general relativity and "Newtonian gravity"]; Clifton et al GRG(17) [effect on universal expansion, with regularly arranged discrete masses].

**Invariants** > s.a. petrov-pirani
classification; riemann tensor.

* __Vacuum 4D spacetime__: There are only 4 independent
algebraic curvature invariants, and they can be expressed in terms of the two complex invariants

*I*:= \({1\over2}\)*M*^{ab}*M*_{ab}
= \({1\over16}\)(*C*_{ab}^{cd}* C*_{cd}^{ab} –
\({\ii\over2}\)*C*_{ab}^{cd} *ε*_{cd}^{mn}* C*_{mn}^{ab})

*J*:= \({1\over6}\)*M*^{ab}*M*_{cb }*M*_{ac}
[??] = \({1\over96}\) (*C*_{ab}^{cd}* C*_{cd}^{mn}* C*_{mn}^{ab} – \({\ii\over2}\)*C*_{ab}^{cd}* C*_{cd}^{mn} *ε*_{mn}^{pq}* C*_{pq}^{ab})
,

where *M*_{ab}:= *E*_{ab} + i *B*_{ab} is the sum of the electric and
magnetic parts of the Weyl tensor.

@ __General references__: Nita & Robinson gq/01 =
Nita GRG(03);
Beetle & Burko PRL(02)gq [radiation
scalars].

@ __Classification, in higher dimensions__: Boulanger & Erdmenger CQG(04)ht [8D];
Ortaggio CQG(09)-a0906 [Bel-Debever
characterization]; Coley & Hervik CQG(10)-a0909 [higher-dimensional
Lorentzian manifolds]; Senovilla CQG(10)-a1008 [based
on its superenergy tensor]; Godazgar CQG(10)-a1008 [spinor classification]; Coley et al CQG(12)-a1203 [5D, refinement]; Batista GRG(13)-a1301; Batista & da Cunha JMP(13)-a1212 [6D]; > s.a. spin coefficients [Newman-Penrose and GHP formalisms].

**Related Concepts** > s.a. Peeling; riemann tensor [symmetries];
spin coefficients [NP formalism]; self-dual
solutions; Weyl Curvature Hypothesis [Penrose].

* __Determining the metric__:
The spacetime metric is generically determined up to a constant factor by *C*_{abc}^{d}
and *T*_{ab}.

@ __Potential__: Edgar & Senovilla CQG(04)gq [for all dimensions]; > s.a.
lanczos tensor.

@ __Other related topics__: Hall & Sharif NCB(03)gq/04 [metric from *C*_{abc}^{d} and *T*_{ab}]; Mantica & Molinari IJGMP(14)-a1212 [Weyl-compatible tensors]; Ortaggio & Pravdová PRD(14)-a1403 [in higher dimensions, asymptotic behavior at null infinity]; > s.a. curvature [Bianchi identities].

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jul 2017