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];
Lapiedra & Morales-Lladosa PRD(19)-a1910 [discontinuous source].

@ __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];
Huber a1908.

@ 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];
Chatterjee & Anand a1810 [at fractal hypersurfaces].

> __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].

@ __Scalar-tensor theories__:
Padilla & Sivanesan JHEP(12)-a1206 [generalized];
Avilés et al a1910 [null or non-null, arbitrary dimensionality].

@ __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];
de la Cruz-Dombriz et al JCAP(14)-a1406 [extended teleparallel gravity].

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