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 Sab, 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 Cabcd,
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 ua
— for example the unit normal to a hypersurface Σ or a (possibly non-hypersurface-orthogonal)
matter 4-velocity vector — by
Eab:= Cambn um 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 Eab
= −\({1\over4}\)(Da
Db E
+ E hab);
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
ua as in the definition of the
electric part, it is the symmetric traceless tensor defined by
(Hab or) Bab:= \({1\over2}\)*Cambn um un, with *Cambn:= εampq C pqbn .
* 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 Kab, such that
Bab
= −\({1\over4}\)εmnb
Dm
K na.
* Purely magnetic spacetimes:
Spacetimes in which the electric part of the Weyl tensor, Cabcd
ub ud
= 0, for some timelike unit vector field ua, 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:= \({1\over2}\)MabMab
= \({1\over16}\)(Cabcd
Ccdab −
\({\ii\over2}\)Cabcd
εcdmn
Cmnab)
J:= \({1\over6}\)MabMcb
Mac [??]
= \({1\over96}\) (Cabcd
Ccdmn
Cmnab
− \({\ii\over2}\)Cabcd
Ccdmn
εmnpq
Cpqab) ,
where Mab:= Eab
+ i Bab 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
Cabcd
and Tab.
@ Potential: Edgar & Senovilla CQG(04)gq [for all dimensions];
> s.a. lanczos tensor.
@ Other related topics: Hall & Sharif NCB(03)gq/04 [metric from Cabcd and Tab];
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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