Weyl Tensor

In General > s.a. bianchi models; curvature; riemann tensor.
$ Def: The "trace-free part" of the Riemann tensor, defined for dimension n 3 by

The "remainder" S_ab is sometimes called the Weyl-Schouten tensor.
* 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 (1/12) 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.
@ Related topics and uses: Schmidt GRG(03)gq [square]; > s.a. gravitational thermodynamics [entropy].

Electric Part
$ Def: The symmetric trace-free tensor defined, for a hypersurface with unit normal n^a in spacetime, by

E_ab:= C_ambn n^m nⁿ .

* And physics: Corresponds to tidal forces; Near spatial infinity, using an appropriately rescaled curvature on the hyperboloid [> 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 = – (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 with unit normal n^a, it is the symmetric tracefree tensor defined by

(H_ab or) B_ab:= *C_ambn n^m nⁿ,   with   *C_ambn:= _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 at spatial infinity admits a potential K_ab, such that B_ab = – _mnb D^mK 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 FRW 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-in; Wylleman CQG(06)gq [irrotational dust, any cosmological constant]; Wylleman & Van den Bergh PRD(06)gq [classification].
@ Related topics: Ellis & Dunsby ApJ(97)ap/94 [evolution in general relativity and "Newtonian gravity"].

Invariants > s.a. 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:= M^abM_ab = (1/16) (C_ab^cd C_cd^ab – (i/2) C_ab^cd _cd^mn C_mn^ab)
J:= M^abM_cb M_ac [??] = (1/96) (C_ab^cd C_cd^mn C_mn^ab – (i/2) C_ab^cd C_cd^mn _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.
@ References: Nita & Robinson gq/01 = Nita GRG(03); Beetle & Burko PRL(02)gq [radiation scalars]; Boulanger & Erdmenger CQG(04)ht [8D, classification].

Related Concepts > s.a. Peeling; petrov-pirani classification; 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.
@ Determining the metric: Hall & Sharif NCB(03)gq/04 [from C_abc^d and T_ab].
@ Potential: Edgar & Senovilla CQG(04)gq [all dimensions]; > s.a. lanczos tensor.

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