Spacetime Boundaries and Completions |

**In General** > s.a. lorentzian geometry [extensions].

* __Idea__: Boundaries are
attached to spacetime manifolds in certain directions in one of two contexts,

(i) The spacetime is non-compact and "infinite" (an affine parameter along
a geodesic has an infinite range of values) along those directions, and is
asymptotic to a spacetime of a highly symmetric, well-known form (mainly Minkowski
or AdS spacetime); Compactifying spacetime along those directions by a conformal
transformation allows one to define fields on the boundary that encode information
on the rate of approach of various metric and curvature components to those of the
asymptotic spacetime, and this information can be related to observable quantities;

(ii) The spacetime has a singularity a finite affine parameter distance from points
in its interior, and compactifying it by defining a boundary there allows to
study the structure of the singularity better.

@ __Approaches__: Dodson IJTP(78);
Rácz JMP(93),
CQG(10)-a0803;
Marolf & Ross CQG(03)gq [future/past set pairs];
Flores et al MAMS-a1011
[Gromov, Cauchy and causal boundaries for Riemannian, Finslerian and Lorentzian manifolds].

@ __Conformal compactification__: Penrose PRS(65) [at null infinity];
Herranz & Santander JPA(02);
Dussan & Magid JGP(07);
Hořava & Melby-Thompson GRG(11) [with anisotropic scaling];
Antoniadis & Cotsakis MPLA(15)-a1505 [Zeeman fine topology, and consequences];
> s.a. asymptotic flatness.

@ __With a cosmological constant__:
Ashtekar & Magnon CQG(84);
Henneaux & Teitelboim CMP(85).

@ __And phenomenology__: Nakao et al PRD(10) [visible borders of spacetime generated in high-energy collisions].

**Specific Types of Boundaries**

* __b-boundary__: A way
of attaching endpoints to inextendible endless curves in a spacetime;
The b-boundary of a manifold *M* with connection is constructed
by forming the Cauchy completion of the frame bundle L*M* equipped
with a certain Riemannian metric, the b-metric *G*.

@ __Causal boundary and completion__:
Geroch et al PRS(72) [strongly causal spacetime];
Budic & Sachs JMP(74) [causally continuous spacetime];
Beem GRG(77) [metric topology];
Rácz GRG(88);
Flores & Harris CQG(07)gq/06 [topology, static];
Flores CMP(07)gq/06;
Alana & Flores GRG(07)-a0704 [for product manifolds];
Flores et al ATMP-a1001 [final definition];
Flores et al CQG(11)-a1103 [isocausal spacetimes with non-equivalent causal boundaries];
Flores et al CQG(13)-a1209 [computability];
Costa e Silva et al a1811 [globally hyperbolic spacetimes with timelike boundary];
Gaztañaga a1911-proc [new condition].

@ __General references__: Geroch et al JMP(82);
Szabados CQG(88),
CQG(89);
Szekeres in(94);
Harris gq/03-in [outline];
Bailleul JMP-a1009 [probabilistic view].

@ __b-boundary__: Schmidt GRG(71);
Dodson GRG(79) [modification];
Ståhl CMP(99)gq,
gq/00/JMP,
gq/00/JMP;
Sánchez Sánchez et al a1702.

@ __Properties__: Ashley GRG(02)gq [stability of essential singularities].

@ __For Friedmann solution__: Bosshard CMP(76);
Johnson JMP(77);
Amores & Gutiérrez JGP(99).

@ __Abstract boundary__: Scott & Szekeres JGP(94)gq;
Ashley & Scott CM-gq/03 [singularity theorems];
Whale & Scott JGP(11) [and edges, singularities];
Barry & Scott CQG(11)-a1409 [attached-point topology],
CQG(14)-a1408 [strongly-attached-point topology];
Whale et al CQG(15)-a1508 [singularity theorem].

@ __Other boundaries__: Meyer JMP(86);
Harris JMP(98)gq/97,
CQG(00) [future chronological boundary];
Harada & Nakao PRD(04)gq ["border"];
Sherif & Amery a1810 [*g* and *a* boundary].

**Specific Types of Spacetimes** > s.a. Kaluza-Klein theory;
reissner-nordström; types of spacetimes.

@ __Schwarzschild__: Lynden-Bell & Katz MNRAS(90);
Ho gq/93/PLA;
Peeters et al CQG(95)gq/94;
Heinzle & Steinbauer JMP(02)gq/01.

@ __Other types__: Flores & Sánchez JHEP(08)-a0712 [wave-type spacetimes].

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