Constraint Equations in General Relativity| Constraint Equations in General Relativity |
In General
> s.a. canonical formulation [constraint algebra]; initial-value
formulation; ADM and connection formulation.
* Idea: They are the
Einstein equation with one or more indices projected arthogonally
to a spacelike hypersurface or, in differential geometry terms,
just the Gauss-Codazzi equations with the Ricci tensor substituted
for in terms of the matter stress-energy.
* In terms of first and
second fundamental form: For real, Lorentzian gravity
Gab qma nb = Da Kam − Dm K = 8πG Tab qma nb = 8πG jm
2 Gab na nb = 3R − Kab Kab + K2 = 16πG Tab na nb = 16πG ρ ,
where N:= [−(∇a t)
(∇a t)]−1/2
is the lapse, na:=
−N∂a t is the unit normal to Σ,
and K:= Kaa
= qab Kab.
* For complex / euclidean general relativity:
Flip the signs of the two KK terms in the second constraint equation.
@ General references:
Moncrief PRD(72) [redundancy],
& Teitelboim PRD(72) [Hamiltonian and diffeomorphism];
Dittrich CQG(06)gq/05 [diff-invariant Hamiltonian constraints];
Balasin & Wieland a0912
[simple Hamiltonian constraint for specific value of the Barbero-Immirzi parameter];
Mars GRG(13)-a1303 [for general hypersurfaces, and applications to shells];
Rácz CQG(16)-a1508 [as evolutionary systems].
@ On closed / compact manifolds: Isenberg CQG(95) [constant mean curvature],
& Moncrief CQG(96) [non-constant mean curvature];
Choquet-Bruhat CQG(04)gq/03-in [compact nΣ];
Maxwell JHDE(05)gq [rough/low-regularity];
Holst et al PRL(08) [far-from-constant mean curvature],
CMP(09)-a0712 [without near-CMC assumption];
Dilts CQG(14)-a1310 [with boundary];
Canepa et al AHP-a2010 [with null boundary].
Variables and Solution Methods
* Lichnerowicz-York solution method:
A conformal technique initiated by Lichnerowicz and perfected by York, the only
efficient and robust method of generating consistent initial data; In the spatially
compact case, the complete scheme consists of the ADM Hamiltonian and momentum
constraints, the ADM Euler-Lagrange equations, York's constant-mean-curvature (CMC)
condition, and a lapse-fixing equation (LFE) that ensures propagation of the CMC
condition by the Euler-Lagrange equations; The variables are a conformal factor
ψ, a spatial metric gij,
a symmetric tensor Aij and a scalar
τ, in terms of which the physical metric and extrinsic curvature are
hij = ψ4 gij, Kij = ψ−2 Aij + \(1\over3\)ψ4 gij τ ;
The Hamiltonian constraint is rewritten as the Lichnerowicz-York
equation for the conformal factor ψ of the physical metric
ψ4 gij,
given an initial unphysical 3-metric gij;
The CMC condition and LFE introduce a distinguished foliation (definition of simultaneity)
on spacetime, and separate scaling laws for the canonical momenta and their trace are used.
@ General references: Isenberg & Marsden JGP(84) [York map];
Maxwell CMP(05)gq/03 [constant mean curvature conformal method];
Anderson et al CQG(05)gq/04 [physical degrees of freedom];
Pfeiffer et al PRD(05) [stationary + gravitational wave];
O'Murchadha APPB(05)gq [uses of Lichnerowicz-York equation];
Tiemblo & Tresguerres GRG(06)gq/05 [and Poincaré gauge theory, single condition];
Martín et al IJGMP(09)-a0709 [framework for solutions];
Maxwell a0804 [freely-specified mean curvature];
Corvino & Pollack a1102-fs;
Maxwell CQG(14) [conformal method and conformal thin-sandwich method];
Tafel GRG(15)-a1405 [(2+1)-decomposition approach];
Cang AHP(15)-a1405 [application of fixed-point theorems];
Alves a1603-wd
[Hamiltonian constraint, reduction to a first-order equation];
Anderson ATMP-a1812
[Lichnerowicz-Choquet-Bruhat-York conformal method, in the far from CMC-regime];
Canepa et al a2001 [tetrad variables].
@ And conformal structure: Beig & Ó Murchadha CMP(96)gq/94 [and spatial conformal Killing vectors];
Szabados CQG(02)gq/01 [Hamiltonian and Chern-Simons functional];
Butscher CMP(07)gq/02 [conformal constraint equations];
Pfeiffer gq/04-proc [conformal method];
Pfeiffer & York PRL(05),
Walsh CQG(07) [non-uniqueness of solutions];
Gourgoulhon JPCS(07)-a0704 [and 3+1 numerical initial data];
Holst et al a0708;
Nguyen a1507 [non-existence and non-uniqueness results].
@ Black holes: Brandt & Brügmann PRL(97) [multi-black-hole];
Loustó & Price PRD(97),
PRD(98) [data for binary collisions];
Baker & Puzio PRD(99)gq/98 [axisymmetric];
Dain et al PRD(05)gq/04 [multi-black-hole];
Smith GRG(09) [horizon with prescribed geometry];
Baumgarte PRD(12)-a1202.
@ Binary black holes: Baumgarte PRD(00)gq;
Marronetti et al PRD(00)gq,
& Matzner PRL(00)gq [arbitrary P, L];
Dain PRD(01)gq/00 [2 Kerr, head-on].
@ Asymptotically hyperbolic: Isenberg & Park CQG(97)gq/96;
Sakovich CQG(10)-a0910 [Einstein-scalar, constant mean curvature solutions];
Gicquaud & Sakovich CMP(12)-a1012 [non-constant mean curvature].
@ Rough / low-regularity solutions: Maxwell gq/04;
Behzadan & Holst a1504 [asymptotically flat, non-CMC].
@ Other solutions: Maxwell CMP(05)gq/03 [with apparent horizon boundaries];
Choquet-Bruhat et al gq/05,
CQG(07)gq/06,
gq/06 [Einstein-scalar];
Korzyński PRD(06)gq [on dynamical horizons];
Huang CQG(10) [with prescribed asymptotics];
Tafel & Jóźwikowski CQG(14)-a1312;
> s.a. models in numerical relativity.
References
> s.a. numerical relativity; canonical quantum gravity.
@ Reviews: Bartnik & Isenberg gq/04-proc;
Carlotto LRR(21).
@ Gluing solutions: Isenberg et al CMP(02)gq/01 [and wormholes],
AHP(03)gq/02;
Isenberg gq/02-GR16;
Chruściel et al CMP(05)gq/04,
PRL(04)gq [more general];
Isenberg et al ATMP(05)gq [with matter];
Chruściel et al CMP(11)-a1004 [N-body initial-data sets];
> s.a. solution methods.
@ Space of solutions: Ó Murchadha CQG(87) [ADM energy as Morse function];
Chruściel & Delay JGP(04)gq/03 [manifold structure];
Dain gq/04-proc [black holes as boundaries];
Rai & Saraykar a1605 [coupled scalar fields, Hilbert space structure];
Holst et al a1711 [drift method, rev and applications].
@ In other gravity theories: Jacobson CQG(11)-a1108
[generally-covariant theories with tensor matter fields].
@ Constraint propagation:
Frittelli PRD(97) [and numerical evolution];
York gq/98 [and canonical formalism];
York a1512-wd [causal propagation].
@ Related topics: Kuchař & Romano PRD(95)gq [sets that generate true Lie algebras];
Frauendiener & Vogel CQG(05)gq/04 [instability of constraint surface];
Gambini & Pullin GRG(05)gq-GRF [getting rid of constraints in discretization];
Corvino & Schoen JDG(06) [vacuum, asymptotics];
Bojowald et al PRD(06)gq [effective constraints from lqg];
Szabados CQG(08)-a0711 [Hamiltonian constraint for Einstein-Yang-Mills theory as Poisson bracket];
Ita a0904v5 [diffeomorphisms and Gauss' law];
Holst & Kungurtsev PRD(11)-a1107 [bifurcation analysis of conformal formulations].
Various theories: see formulations of general relativity;
linearized general relativity; modified gravity.
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