Weyl Tensor |

**In General** > s.a. bianchi models;
curvature; FLRW geometry;
riemann tensor; Weyl
Curvature Hypothesis [Penrose].

$ __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}\,S\_{d]b} - g\_{b[c}\,S\_{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),
re GRG(22);
in Weinberg 72;
in Wald 84;
Ehlers & Buchert GRG(09)-a0907 [Newtonian limit];
Dewar & Weatherall FP(18)-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 traceless tensor defined
with respect to a unit timelike vector *u*^{a}
— for example the unit normal to a hypersurface Σ or a (possibly non-hypersurface-orthogonal)
matter 4-velocity vector — by

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

* __And physics__: It 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];
Munoz & Bruni CQG(23)-a2211,
PRD(23)-a2302 [numerical code].

**Magnetic Part**

$ __Def__: Given a unit timelike vector
*u*^{a} as in the definition of the
electric part, it is the symmetric traceless tensor defined by

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

* __And physics__: In a weak-field approximation
to the gravitational field, its effects are similar to those arising from the Lorentz force
in electromagnetism, and it is responsible for gravitomagnetic effects like frame dragging;
In cosmological perturbation theory a non-zero magnetic Weyl tensor is associated with the
vector modes of the first post-Newtonian contribution, and 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];
Danehkar IJMPD(20)-a2006-GRF.

@ __And physics__: Ellis & Dunsby ApJ(97)ap/94 [evolution in general relativity and "Newtonian gravity"];
Bruni & Sopuerta CQG(03)gq/03 [approach to the singularity]; 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;
Poynting Vector; riemann tensor [symmetries];
spin coefficients [NP formalism]; self-dual
solutions.

* __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]; general relativity actions.

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