Lorentzian
Geometry |

**In General** > s.a. differential
geometry; riemann
tensor / metric types and matching;
types of lorentzian geometries [including scalar invariants, generalizations].

* __Idea__: A Lorentzian
metric is one with signature (–, +, ..., +).

* __Remark__: This signature convention gives
normal signs to spatial components, while the opposite ones gives *p*^{m}* p*_{m}
= *m*^{2} for a relativistic particle.

$ __Lorentzian structure__:
A reduction of the bundle of frames *F*(*M*) to the
Lorentz group, as a subgroup of GL(*n*, **R**).

* __Conditions__: The necessary
and sufficient condition for the existence of a Lorentzian structure on a manifold *M* is
that *M* be
non-compact, or that the Euler number *χ*(*M*) = 0.

@ __Books__: Steenrod 51; O'Neill 83; Beem et al 96; Hall 04.

@ __General references__: Iliev JGP(00)gq/98 [relation with Riemannian geometry]; Müller & Sánchez JDMV-math/06; Chen 11 [submanifolds, *δ*-invariants and applications]; news ea(11)apr [visualization through tendex lines and vortex lines]; Gilkey et al IJGMP(13) [with boundary, universal curvature identities].

@ __Emergent geometry__: Wilczek PRL(98)ht [metrics from volumes and gauge symmetries]; Brown a0911 [metric
as spacetime property or emergent field]; Mukohyama & Uzan PRD(13)-a1301, Kehayias et al PRD(14)-a1403 [Lorentzian signature as emergent from Riemannian one and classical fields]; Majid a1401-proc; Cirilo-Lombardo & Prudêncio IJGMP(14) [from supergeometries]; > s.a. emergent gravity [analog models of spacetime metrics].

@ __Related topics__: Bugajska JMP(89)
[open spin 4-manifolds].

> __Related topics__:
see affine connection; fluids; jacobi metric; lines [space of timelike lines].

> __Online resources__:
see Technische Universität Berlin group [2012].

**Specific Concepts and Results** > s.a. causality; holonomy;
minkowski space; spectral geometry; simplex (Lorentzian geometry case).

* __Flat deformation theorem__:
Given a semi-Riemannian analytic metric *g* on a manifold, there always
exists a two-form *F*, a scalar function *c*, and an arbitrarily
prescribed scalar constraint depending on the point *x* of the manifold
and on *F* and *c*, say Ψ(*c*, *F*, *x*)
= 0, such that the deformed metric *η*
= *c g* – *ε* *F*^{2} is
semi-Riemannian and flat; It implies that every (Lorentzian analytic) metric
*g* can be written in the extended Kerr-Schild form *η*_{ab}:=
*a* *g*_{ab} – 2 *b k*_{(a} *l*_{b)}, where
*η* is flat and *k*_{a}, *l*_{a}
are two null covectors such that *k*_{a} *l*^{ a} =
–1.

* __Result__: Given a metric *g* with
scalar curvature *R*,
there is another *g'* with *R*' = 0 iff for all *φ* ∈ C_{0}^{∞},

–(1/8) ∫ *Rφ* d*μ* < ∫ |∇*φ*|^{2} d*μ* (sufficient
condition: ||* R *||_{L3/2} < some known *c*) .

@ __General references__: Kim BAusMS(90); Tod CQG(92)
[diagonalizability]; Pezzaglia & Adams
gq/97-conf
[(–,+,+,+) vs (+,–,–,–)]; Gerhardt GRG(03)m.DG/02 [volume
estimates]; Milson et al IJGMP(05)gq/04 [alignment];
Sánchez DG&A(06)
[compact, causality]; Llosa & Carot CQG(09)-a0809 [flat
deformation theorem and symmetries]; Kim JGP(09), JGP(11)
[volume comparison between hypersurfaces]; Pugliese et al a0910-proc
[deformations]; de Siqueira a1006 [every *n*-dimensional
pseudo-Riemannian manifold is conformal to one of constant curvature?]; Kim JMP(11) [covering spaces and homotopy classes of causal curves]; Impera JGP(12) [Hessian and Laplacian comparison theorems].

@ __Extensions__: Chruściel JDG(10)
[conformal boundary extensions]; Low CQG(12) [maximal extensions and Zorn's lemma]; > s.a. spacetime
boundaries and completions.

> __Related topics__:
see distance; Extremal Surface; geodesics; Hypersurface; Osserman Manifolds; Pythagorean
Theorem; Splitting Theorem; world
function.

**Space of Lorentzian Geometries** > s.a. distance between geometries;
spacetime singularities; solutions
of general relativity.

* C^{k} __open
topology__: (a.k.a. Whitney fine or uniform convergence topology)
A neighborhood basis for *g* is

*B*_{f}(*g*):=
{*g'* | for all *p* ∈ *M*
||*g *– *g'*||(*p*) < *f*(*p*),
..., ||∂^{k}*g* – ∂^{k}*g'*||(*p*) < *f*(*p*)}
,

where *f* : *M* → **R** is continuous and
strictly positive and the norms are calculated using some positive-definite (inverse) metric; Intuitively, for C^{0} the
light cones are close, for C^{1} the geodesic
systems are close, for C^{2}
the curvature tensors are also close; This is an extremely fine topology; For example, on a non-compact *M* a sequence *g*_{(n)} of metrics cannot converge unless there exists a compact subset in *M* such that for sufficiently large *n* all metrics coincide outside it [@ in Golubitsky & Guillemin 73].

* W^{k} __compact-open topology__:
A neighborhood subbasis is

*B*_{U, δ}(*g*):=
{*g'* | ||(*g* –
*g'*)|_{U} ||_{Wk} < *δ*}
,

where *U* is an open set of compact closure in *M* and *δ* a positive constant.

* __Partial order__: For
each *A* ⊂ *M*, *g* <_{A} *g*'
iff for all *p* ∈ *A*,
all non-spacelike vectors with respect to *g* are non-spacelike with
respect to *g*' (the light cones of *g* are narrower).

@ __General references__: Geroch JMP(70),
in(70); Hawking GRG(71);
in Hawking & Ellis 73; Lerner CMP(73);
in Beem et al 96.

@ __Structures__: Beem 81 [Lorentzian distance function]; Bombelli JMP(00)gq [pseudodistance];
Noldus CQG(02)-a1104 [topology]; García-Parrado & Senovilla
mp/02-proc,
CQG(03)gq/02 [causal
equivalence];
Noldus
CQG(04)gq/03,
CQG(04)gq/03 [distance];
Bombelli & Noldus CQG(04)gq.

@ __Limits of spacetimes__: Geroch CMP(69);
Bampi & Cianci IJTP(80); Paiva et al CQG(93)gq; Sormani a1006-fs [convergence].

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send feedback and suggestions to bombelli at olemiss.edu – modified 7
sep
2014