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 uⁿ.

* 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 uⁿ,   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^abM_ab = \({1\over16}\)(C_ab^cd C_cd^ab − \({\ii\over2}\)C_ab^cd ε_cd^mn C_mn^ab)
J:= \({1\over6}\)M^abM_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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