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*, \(\mathbb R\)).

* __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].

@ __Global aspects__: Nawarajan & Visser IJMPD(16)-a1601
[physical considerations on orientability, the use of tetrads as variables, and other issues].

@ __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 EPJwc(14)-a1401;
Cirilo-Lombardo & Prudêncio IJGMP(14) [from supergeometries];
> s.a. emergent gravity [analog models of spacetime metrics, including acoustic].

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

> __Related topics__:
see affine connection; fluids; jacobi
metric; lines [space of timelike lines]; types of distances
[Lorentzian length spaces].

> __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];
Hintz & Uhlmann a1705
[reconstruction of Lorentzian manifolds from boundary light observation sets].

@ __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* → \(\mathbb R\) 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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