Metric
Matching |

**In General** > s.a. types of metrics.

* __Idea__:
Establish junction / matching conditions that a metric and its derivatives must satisfy
across a hypersurface in order for conditions such as field equations to
be satisfied at least in a distributional sense, e.g., distributional
sources corresponding to thin matter shells in general relativity.

* __Lichnerowicz conditions__:
In general
relativity, the Lorentzian metric *g*_{ab} and
its first derivatives ∂_{a} *g*_{bc} must
be continuous across a discontinuity surface; Higher derivatives need not be.

* __Note on validity__: Metrics
are known with thin shell matter for which the metric is not continuous across
the corresponding hypersurface; Marolf and Yaida have conjectured that in general relativity,
in all positive-energy spacetimes, the metric is continuous across hypersurfaces.

**References** > s.a. action for
general relativity [singular hypersurfaces]; gravitating
matter; models in canonical general relativity.

@ __Spacelike / timelike hypersurface__: Israel NCB(66), NCB(67);
in Misner et al 73, #21.13; Ipser & Sikivie
PRD(84)
[domain walls]; Fayos et al PRD(96)
[spherical symmetry].

@ __Spacelike / timelike, beyond thin wall__: Garfinkle & Gregory PRD(90).

@ __Null hypersurface__: Penrose in(72) [spinors];
Redmount ["contranormal" coordinates];
Dray & 't Hooft CMP(85)
[two Schwarzschild metrics separated by null shell]; Clarke & Dray
CQG(87);
Gemelli GRG(02)
[rev, timelike/null]; Poisson gq/02.

@ __General hypersurface__: Barrabès CQG(89);
Mars & Senovilla CQG(93)gq/02;
Ferraris et al in(96); Nozari & Mansouri JMP(02);
Vera CQG(02)gq [and
symmetries]; Raju a0804-MG5
[distributional matter, shocks].

@ __Perturbations__:
Mukohyama CQG(00)ht;
Mars et al CQG(07);
Copeland & Wands JCAP(07) [and cosmology].

@ Lemaître-Tolman-Bondi solutions:
Khakshournia & Mansouri G&C(08)
[and FLRW spacetimes]; Khakshournia GRG(10)-a0907 [and Vaidya exterior solution].

@ __Other special types__: Israel PRS(58)
[spherically symmetric]; Grøn & Rippis GRG(03)gq [Schwarzschild-FLRW spacetimes];
Kirchner CQG(04)
[spherically symmetric]; Copeland & Wands JCAP(07)ht/06 [cosmological];
Mena & Natário JGP(09)
[stationary].

@ __And energy conditions__: Goldwirth & Katz CQG(95)gq/94;
Marolf & Yaida PRD(05)gq.

@ __At spacetime singularities__: Khakshournia & Mansouri ht/99 [spherically
symmetric, with singular hypersurface]; Rosenthal a1011.

@ __Other topics__: Schmidt GRG(84)gq/01 [and
surface tension]; Taylor CQG(04)
[at a corner].

> __Related topics__: see boundaries in field theory; constraints and solutions
in general relativity [gluing of solutions].

**In Modified Gravity Theories**

@ __Higher-order gravity__: Deruelle et al PTP(08)-a0711 [*f*(*R*)
gravity]; Senovilla PRD(13)-a1303, CQG(14)-a1402 [for *f*(*R*)-gravity, and consequences]; Reina et al CQG(16)-a1510 [junction conditions in quadratic gravity].

@ __Other theories__: Bressange CQG(00)gq [shells
in Einstein-Cartan theory]; Macías
et al PRD(02)
[metric-affine gravity]; Giacomini et al PRD(06)gq [with
spinning sources]; Padilla & Sivanesan JHEP(12)-a1206 [generalized scalar-tensor theories]; de la Cruz-Dombriz et al JCAP(14)-a1406 [extended teleparallel gravity].

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jul
2016