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].

@ __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"].

**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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send feedback and suggestions to bombelli at olemiss.edu – modified 4
feb
2017