Recent interrelated progress in Lorentzian, Finslerian and ......Lorentzian, Finslerian and...
Transcript of Recent interrelated progress in Lorentzian, Finslerian and ......Lorentzian, Finslerian and...
1 Causal structure2 Visibility and lensing
3 Causal boundary
Recent interrelated progress inLorentzian, Finslerian and Riemannian Geometry
Miguel Sanchez Caja
Universidad de Granada
Workshop on Lorentzian GeometryLaboratoire Jacques-Louis Lions, Paris 6
November 14th, 2012
M. Sanchez Lorentz, Finsler and Riemann Geometry
1 Causal structure2 Visibility and lensing
3 Causal boundary
Introduction
Stationary to Randers correspondence. Equivalence:
Conformal structure of stationary spacetimes←→ Geometry of Randers spaces
Applicability:
→ Precise description of spacetime elements in terms ofFinsler counterparts
← New geometric elements and results in Randers spaces,extensible to general Finsler manifolds
Broad relation
Lorentzian Geometry ←→ Finsler Geometry
(including the Riemannian one!)
M. Sanchez Lorentz, Finsler and Riemann Geometry
1 Causal structure2 Visibility and lensing
3 Causal boundary
Introduction
Stationary to Randers correspondence. Equivalence:
Conformal structure of stationary spacetimes←→ Geometry of Randers spaces
Applicability:
→ Precise description of spacetime elements in terms ofFinsler counterparts
← New geometric elements and results in Randers spaces,extensible to general Finsler manifolds
Broad relation
Lorentzian Geometry ←→ Finsler Geometry
(including the Riemannian one!)
M. Sanchez Lorentz, Finsler and Riemann Geometry
1 Causal structure2 Visibility and lensing
3 Causal boundary
Introduction
Stationary to Randers correspondence. Equivalence:
Conformal structure of stationary spacetimes←→ Geometry of Randers spaces
Applicability:
→ Precise description of spacetime elements in terms ofFinsler counterparts
← New geometric elements and results in Randers spaces,extensible to general Finsler manifolds
Broad relation
Lorentzian Geometry ←→ Finsler Geometry
(including the Riemannian one!)
M. Sanchez Lorentz, Finsler and Riemann Geometry
1 Causal structure2 Visibility and lensing
3 Causal boundary
Starting point
Normalized standard stationary spacetime:V = (R×M, gL = −1dt2 + π∗ω ⊗ dt + dt ⊗ π∗ω + π∗g)ω 1-form, g Riemannian metric on M, π : R×M → M projection
∂t timelike (future-directed) Killing vector fieldgL ≡ −dt2 + 2ωdt + g
Normalized: −1dt2. Useful for conformal elements such aslightlike vectors/geodesics (otherwise: −βdt2)
Global “standard” (but not unique) splitting not toorestrictive: it always hold locally and [Javaloyes & — ’08]:
A spacetime is (globally) conformal to a standard stationaryone iff it admits a complete timelike conformal vector fieldand it is distinguishing (and, so, strongly causal and causallycontinuous).
M. Sanchez Lorentz, Finsler and Riemann Geometry
1 Causal structure2 Visibility and lensing
3 Causal boundary
Starting point
Normalized standard stationary spacetime:V = (R×M, gL = −1dt2 + π∗ω ⊗ dt + dt ⊗ π∗ω + π∗g)ω 1-form, g Riemannian metric on M, π : R×M → M projection
∂t timelike (future-directed) Killing vector fieldgL ≡ −dt2 + 2ωdt + g
Normalized: −1dt2. Useful for conformal elements such aslightlike vectors/geodesics (otherwise: −βdt2)
Global “standard” (but not unique) splitting not toorestrictive: it always hold locally and [Javaloyes & — ’08]:A spacetime is (globally) conformal to a standard stationaryone iff it admits a complete timelike conformal vector fieldand it is distinguishing (and, so, strongly causal and causallycontinuous).
M. Sanchez Lorentz, Finsler and Riemann Geometry
1 Causal structure2 Visibility and lensing
3 Causal boundary
Starting point
Appearance of Finsler GeometryWith these elements ω, g construct the functions F± : TM → R
F±(v) =√
g(v , v) + ω(v)2 ± ω(v)
Finsler metrics of Randers type on M, “Fermat metrics”
Connection with the spacetime geometry (Cap, Jav. Mas. ’11):A curve γ(t) = (±t, c(t)), t ∈ [a, b] is lightlike and future/pastdirected iff F±(c) = 1. In this case:
the arrival time b− a is equal to the F±-length of c ,∫ ba F±(c)
(Fermat principle) γ is a pregeodesic iff c is a geodesic for F±
(i.e., a critical point of the arrival time/length functionalc 7→
∫F±(c) parametrized with F±-length.
M. Sanchez Lorentz, Finsler and Riemann Geometry
1 Causal structure2 Visibility and lensing
3 Causal boundary
Starting point
Appearance of Finsler GeometryWith these elements ω, g construct the functions F± : TM → R
F±(v) =√
g(v , v) + ω(v)2 ± ω(v)
Finsler metrics of Randers type on M, “Fermat metrics”Connection with the spacetime geometry (Cap, Jav. Mas. ’11):A curve γ(t) = (±t, c(t)), t ∈ [a, b] is lightlike and future/pastdirected iff F±(c) = 1. In this case:
the arrival time b− a is equal to the F±-length of c ,∫ ba F±(c)
(Fermat principle) γ is a pregeodesic iff c is a geodesic for F±
(i.e., a critical point of the arrival time/length functionalc 7→
∫F±(c) parametrized with F±-length.
M. Sanchez Lorentz, Finsler and Riemann Geometry
1 Causal structure2 Visibility and lensing
3 Causal boundary
Starting point
These elementary considerations suggest the possibility to relatethe conformal geometry of standard stationary spacetimes and thegeometry of the corresponding class of Finsler manifolds, i.e.Randers spaces.
Aims:
1 Causal structure ←→ Finslerian distances(Caponio, Javaloyes, — arxiv ’09)
2 Visibility and gravitational lensing ←→convexity of Finsler hypersurfaces(Caponio, Germinario, — arxiv ’11).
3 Causal boundaries ←→ Cauchy, Gromov and Busemannboundaries in Finslerian (and Riemannian) settings(Flores, Herrera, — ’10, ’12).
M. Sanchez Lorentz, Finsler and Riemann Geometry
1 Causal structure2 Visibility and lensing
3 Causal boundary
Starting point
These elementary considerations suggest the possibility to relatethe conformal geometry of standard stationary spacetimes and thegeometry of the corresponding class of Finsler manifolds, i.e.Randers spaces. Aims:
1 Causal structure ←→ Finslerian distances(Caponio, Javaloyes, — arxiv ’09)
2 Visibility and gravitational lensing ←→convexity of Finsler hypersurfaces(Caponio, Germinario, — arxiv ’11).
3 Causal boundaries ←→ Cauchy, Gromov and Busemannboundaries in Finslerian (and Riemannian) settings(Flores, Herrera, — ’10, ’12).
M. Sanchez Lorentz, Finsler and Riemann Geometry
1 Causal structure2 Visibility and lensing
3 Causal boundary
Starting point
These elementary considerations suggest the possibility to relatethe conformal geometry of standard stationary spacetimes and thegeometry of the corresponding class of Finsler manifolds, i.e.Randers spaces. Aims:
1 Causal structure ←→ Finslerian distances(Caponio, Javaloyes, — arxiv ’09)
2 Visibility and gravitational lensing ←→convexity of Finsler hypersurfaces(Caponio, Germinario, — arxiv ’11).
3 Causal boundaries ←→ Cauchy, Gromov and Busemannboundaries in Finslerian (and Riemannian) settings(Flores, Herrera, — ’10, ’12).
M. Sanchez Lorentz, Finsler and Riemann Geometry
1 Causal structure2 Visibility and lensing
3 Causal boundary
Starting point
These elementary considerations suggest the possibility to relatethe conformal geometry of standard stationary spacetimes and thegeometry of the corresponding class of Finsler manifolds, i.e.Randers spaces. Aims:
1 Causal structure ←→ Finslerian distances(Caponio, Javaloyes, — arxiv ’09)
2 Visibility and gravitational lensing ←→convexity of Finsler hypersurfaces(Caponio, Germinario, — arxiv ’11).
3 Causal boundaries ←→ Cauchy, Gromov and Busemannboundaries in Finslerian (and Riemannian) settings(Flores, Herrera, — ’10, ’12).
M. Sanchez Lorentz, Finsler and Riemann Geometry
1 Causal structure2 Visibility and lensing
3 Causal boundary
Preliminaries on Finsler distancesCausal HierarchyFurther results
FIRST PART
1. CAUSAL STRUCTURE
M. Sanchez Lorentz, Finsler and Riemann Geometry
1 Causal structure2 Visibility and lensing
3 Causal boundary
Preliminaries on Finsler distancesCausal HierarchyFurther results
Notion of Finsler and Randers metric
Let F : TM → R continuous and smooth away 0:
F Finsler metric: essentially, F a positively homogeneousnorm at each p ∈ M
Positive homogeneity F (λv) = λF (v) for λ > 0
The unit sphere is assumed strongly convex
Randers metric: R =√
g + ω2 + ω for some Riemanniang (and h := g + ω2) and 1-form ω.
In particular, Fermat metrics F± are Randers (and viceversa)
Reversed Finsler metric: F rev(v) := F (−v)
In particular,for Fermat metrics (F +)rev(v) = F−(v)
if ω 6= 0, Randers metrics are non-reversible (R 6= Rrev)
M. Sanchez Lorentz, Finsler and Riemann Geometry
1 Causal structure2 Visibility and lensing
3 Causal boundary
Preliminaries on Finsler distancesCausal HierarchyFurther results
Notion of Finsler and Randers metric
Let F : TM → R continuous and smooth away 0:
F Finsler metric: essentially, F a positively homogeneousnorm at each p ∈ M
Positive homogeneity F (λv) = λF (v) for λ > 0
The unit sphere is assumed strongly convex
Randers metric: R =√
g + ω2 + ω for some Riemanniang (and h := g + ω2) and 1-form ω.
In particular, Fermat metrics F± are Randers (and viceversa)
Reversed Finsler metric: F rev(v) := F (−v)
In particular,for Fermat metrics (F +)rev(v) = F−(v)
if ω 6= 0, Randers metrics are non-reversible (R 6= Rrev)
M. Sanchez Lorentz, Finsler and Riemann Geometry
1 Causal structure2 Visibility and lensing
3 Causal boundary
Preliminaries on Finsler distancesCausal HierarchyFurther results
Notion of Finsler and Randers metric
Let F : TM → R continuous and smooth away 0:
F Finsler metric: essentially, F a positively homogeneousnorm at each p ∈ M
Positive homogeneity F (λv) = λF (v) for λ > 0
The unit sphere is assumed strongly convex
Randers metric: R =√
g + ω2 + ω for some Riemanniang (and h := g + ω2) and 1-form ω.
In particular, Fermat metrics F± are Randers (and viceversa)
Reversed Finsler metric: F rev(v) := F (−v)
In particular,for Fermat metrics (F +)rev(v) = F−(v)
if ω 6= 0, Randers metrics are non-reversible (R 6= Rrev)
M. Sanchez Lorentz, Finsler and Riemann Geometry
1 Causal structure2 Visibility and lensing
3 Causal boundary
Preliminaries on Finsler distancesCausal HierarchyFurther results
Notion of generalized distance
Taking infimum of lengths connecting two points, each Finslermetric induces a generalized distance d . This means:
1 all the axioms of a distance hold but symmetry
2 for sequences xn: d(x , xn)→ 0 ⇐⇒ d(xn, x)→ 0
Centered at any point x0, there are forward balls (d(x0, x) < r)and backward balls (d(x , x0) < r) that may differ but generate thesame topology (in the Finslerian case, the manifold topology)
Symmetrized distance: ds(x , y) = (d(x , y) + d(y , x))/2
Remark Even in the Finslerian case, ds does not come from alength space.
M. Sanchez Lorentz, Finsler and Riemann Geometry
1 Causal structure2 Visibility and lensing
3 Causal boundary
Preliminaries on Finsler distancesCausal HierarchyFurther results
Notion of generalized distance
Taking infimum of lengths connecting two points, each Finslermetric induces a generalized distance d . This means:
1 all the axioms of a distance hold but symmetry
2 for sequences xn: d(x , xn)→ 0 ⇐⇒ d(xn, x)→ 0
Centered at any point x0, there are forward balls (d(x0, x) < r)and backward balls (d(x , x0) < r) that may differ but generate thesame topology (in the Finslerian case, the manifold topology)
Symmetrized distance: ds(x , y) = (d(x , y) + d(y , x))/2
Remark Even in the Finslerian case, ds does not come from alength space.
M. Sanchez Lorentz, Finsler and Riemann Geometry
1 Causal structure2 Visibility and lensing
3 Causal boundary
Preliminaries on Finsler distancesCausal HierarchyFurther results
Notion of generalized distance
Taking infimum of lengths connecting two points, each Finslermetric induces a generalized distance d . This means:
1 all the axioms of a distance hold but symmetry
2 for sequences xn: d(x , xn)→ 0 ⇐⇒ d(xn, x)→ 0
Centered at any point x0, there are forward balls (d(x0, x) < r)and backward balls (d(x , x0) < r) that may differ but generate thesame topology (in the Finslerian case, the manifold topology)
Symmetrized distance: ds(x , y) = (d(x , y) + d(y , x))/2
Remark Even in the Finslerian case, ds does not come from alength space.
M. Sanchez Lorentz, Finsler and Riemann Geometry
1 Causal structure2 Visibility and lensing
3 Causal boundary
Preliminaries on Finsler distancesCausal HierarchyFurther results
Notion of generalized distance
Taking infimum of lengths connecting two points, each Finslermetric induces a generalized distance d . This means:
1 all the axioms of a distance hold but symmetry
2 for sequences xn: d(x , xn)→ 0 ⇐⇒ d(xn, x)→ 0
Centered at any point x0, there are forward balls (d(x0, x) < r)and backward balls (d(x , x0) < r) that may differ but generate thesame topology (in the Finslerian case, the manifold topology)
Symmetrized distance: ds(x , y) = (d(x , y) + d(y , x))/2
Remark Even in the Finslerian case, ds does not come from alength space.
M. Sanchez Lorentz, Finsler and Riemann Geometry
1 Causal structure2 Visibility and lensing
3 Causal boundary
Preliminaries on Finsler distancesCausal HierarchyFurther results
Finslerian Hopf-Rinow
Theorem
Let (M,F ) be a Finsler manifold with generalized distance d.They are equivalent:
(a) d is forward (resp. backward) complete (Cauchy sequences).
(b) The Finsler manifold (M,F ) is forward (resp. backward)geodesically complete.
(c) At some (and then all) point p ∈ M, expp (resp. ˜expp) isdefined on all of TpM.
(d) Heine-Borel property: every closed and forward (resp.backward) bounded subset of (M, d) is compact.
Moreover, in this case (M,F ) is convex, i.e., every pair of pointsp, q ∈ M can be joined by a minimizing geodesic from p to q.
M. Sanchez Lorentz, Finsler and Riemann Geometry
1 Causal structure2 Visibility and lensing
3 Causal boundary
Preliminaries on Finsler distancesCausal HierarchyFurther results
Finslerian Hopf-Rinow
Remark. Relation with ds :
1 d is either forward or backward complete =⇒2 ds satisfies Heine-Borel ⇐⇒ all Bs(x , r) compact =⇒3 ds complete
M. Sanchez Lorentz, Finsler and Riemann Geometry
1 Causal structure2 Visibility and lensing
3 Causal boundary
Preliminaries on Finsler distancesCausal HierarchyFurther results
Basic idea
For p ∈M, d+ ≡ d distance of F+ ≡ F:
I +(0, p) determined by the graph of d+(p, ·):t0 × B+(p, t0) = I +(0, p) ∩ (t0 ×M)
M. Sanchez Lorentz, Finsler and Riemann Geometry
1 Causal structure2 Visibility and lensing
3 Causal boundary
Preliminaries on Finsler distancesCausal HierarchyFurther results
Results
Theorem
The slices t0 ×M are Cauchy hypersurfaces of (R×M, gL)iff d+ is forward and backward complete.
(R×M, g) is globally hyperbolic (causal + J+(z) ∩ J−(z ′)compact) iff B+
s (p, r) are compact ∀p ∈ M, r > 0(Heine-Borel property)
(R×M, gL) is causally simple (causal + J±(z) closed)iff (M,F ) is convex
(Full characterization of Causality, as standard stationary s-t arealways causally continuous.)
M. Sanchez Lorentz, Finsler and Riemann Geometry
1 Causal structure2 Visibility and lensing
3 Causal boundary
Preliminaries on Finsler distancesCausal HierarchyFurther results
Results
Theorem
The slices t0 ×M are Cauchy hypersurfaces of (R×M, gL)iff d+ is forward and backward complete.
(R×M, g) is globally hyperbolic (causal + J+(z) ∩ J−(z ′)compact) iff B+
s (p, r) are compact ∀p ∈ M, r > 0(Heine-Borel property)
(R×M, gL) is causally simple (causal + J±(z) closed)iff (M,F ) is convex
(Full characterization of Causality, as standard stationary s-t arealways causally continuous.)
M. Sanchez Lorentz, Finsler and Riemann Geometry
1 Causal structure2 Visibility and lensing
3 Causal boundary
Preliminaries on Finsler distancesCausal HierarchyFurther results
Results
Theorem
The slices t0 ×M are Cauchy hypersurfaces of (R×M, gL)iff d+ is forward and backward complete.
(R×M, g) is globally hyperbolic (causal + J+(z) ∩ J−(z ′)compact) iff B+
s (p, r) are compact ∀p ∈ M, r > 0(Heine-Borel property)
(R×M, gL) is causally simple (causal + J±(z) closed)iff (M,F ) is convex
(Full characterization of Causality, as standard stationary s-t arealways causally continuous.)
M. Sanchez Lorentz, Finsler and Riemann Geometry
1 Causal structure2 Visibility and lensing
3 Causal boundary
Preliminaries on Finsler distancesCausal HierarchyFurther results
Consequences for Finsler manifolds
Remark. Compactness of B+s (p, r) for a Randers metric
=⇒ (R×M, gL) is glob. hyp
=⇒ (R×M, gL) admits a spacelike Cauchy hypersurfaceMf = (f (x), x) : x ∈ M=⇒ The spacetime admits a splitting R×Mf with Cauchy slicesand Fermat metric Ff ≡ F − df=⇒ for some f , the F − df is forward and backward completeRanders metric! this property is extensible to any Finsler metric(Matveev’12).
In general, the compactness of B+s (p, r) (Heine-Borel) is
the optimal assumption for classical Finsler theorems(Myers, sphere...)
M. Sanchez Lorentz, Finsler and Riemann Geometry
1 Causal structure2 Visibility and lensing
3 Causal boundary
Preliminaries on Finsler distancesCausal HierarchyFurther results
Consequences for Finsler manifolds
Remark. Compactness of B+s (p, r) for a Randers metric
=⇒ (R×M, gL) is glob. hyp=⇒ (R×M, gL) admits a spacelike Cauchy hypersurfaceMf = (f (x), x) : x ∈ M
=⇒ The spacetime admits a splitting R×Mf with Cauchy slicesand Fermat metric Ff ≡ F − df=⇒ for some f , the F − df is forward and backward completeRanders metric! this property is extensible to any Finsler metric(Matveev’12).
In general, the compactness of B+s (p, r) (Heine-Borel) is
the optimal assumption for classical Finsler theorems(Myers, sphere...)
M. Sanchez Lorentz, Finsler and Riemann Geometry
1 Causal structure2 Visibility and lensing
3 Causal boundary
Preliminaries on Finsler distancesCausal HierarchyFurther results
Consequences for Finsler manifolds
Remark. Compactness of B+s (p, r) for a Randers metric
=⇒ (R×M, gL) is glob. hyp=⇒ (R×M, gL) admits a spacelike Cauchy hypersurfaceMf = (f (x), x) : x ∈ M=⇒ The spacetime admits a splitting R×Mf with Cauchy slicesand Fermat metric Ff ≡ F − df
=⇒ for some f , the F − df is forward and backward completeRanders metric! this property is extensible to any Finsler metric(Matveev’12).
In general, the compactness of B+s (p, r) (Heine-Borel) is
the optimal assumption for classical Finsler theorems(Myers, sphere...)
M. Sanchez Lorentz, Finsler and Riemann Geometry
1 Causal structure2 Visibility and lensing
3 Causal boundary
Preliminaries on Finsler distancesCausal HierarchyFurther results
Consequences for Finsler manifolds
Remark. Compactness of B+s (p, r) for a Randers metric
=⇒ (R×M, gL) is glob. hyp=⇒ (R×M, gL) admits a spacelike Cauchy hypersurfaceMf = (f (x), x) : x ∈ M=⇒ The spacetime admits a splitting R×Mf with Cauchy slicesand Fermat metric Ff ≡ F − df=⇒ for some f , the F − df is forward and backward completeRanders metric! this property is extensible to any Finsler metric(Matveev’12).
In general, the compactness of B+s (p, r) (Heine-Borel) is
the optimal assumption for classical Finsler theorems(Myers, sphere...)
M. Sanchez Lorentz, Finsler and Riemann Geometry
1 Causal structure2 Visibility and lensing
3 Causal boundary
Preliminaries on Finsler distancesCausal HierarchyFurther results
Consequences for Finsler manifolds
Remark. Compactness of B+s (p, r) for a Randers metric
=⇒ (R×M, gL) is glob. hyp=⇒ (R×M, gL) admits a spacelike Cauchy hypersurfaceMf = (f (x), x) : x ∈ M=⇒ The spacetime admits a splitting R×Mf with Cauchy slicesand Fermat metric Ff ≡ F − df=⇒ for some f , the F − df is forward and backward completeRanders metric! this property is extensible to any Finsler metric(Matveev’12).
In general, the compactness of B+s (p, r) (Heine-Borel) is
the optimal assumption for classical Finsler theorems(Myers, sphere...)
M. Sanchez Lorentz, Finsler and Riemann Geometry
1 Causal structure2 Visibility and lensing
3 Causal boundary
Preliminaries on Finsler distancesCausal HierarchyFurther results
Cauchy developments
A ⊂ V achronal setD+(A) = z ∈ V : γ ∩ A 6= ∅ for all γ past-inextensible causalcurve starting at pH+(A) = z ∈ D
+(A) : I +(z) ∩ D+(A) = ∅
Proposition
For A ⊂ M ≡ 0 ×M:D+(A) = (t, y) : 0 ≤ t < d+(x , y) ∀x 6∈ AH+(A) = (t, y) : t = infx 6∈Ad+(x , y)(= d+(M\A, y)
H+(A) is constructed from the level sets of d+(M\A, · )
M. Sanchez Lorentz, Finsler and Riemann Geometry
1 Causal structure2 Visibility and lensing
3 Causal boundary
Preliminaries on Finsler distancesCausal HierarchyFurther results
Cauchy developments
A ⊂ V achronal setD+(A) = z ∈ V : γ ∩ A 6= ∅ for all γ past-inextensible causalcurve starting at pH+(A) = z ∈ D
+(A) : I +(z) ∩ D+(A) = ∅
Proposition
For A ⊂ M ≡ 0 ×M:D+(A) = (t, y) : 0 ≤ t < d+(x , y) ∀x 6∈ AH+(A) = (t, y) : t = infx 6∈Ad+(x , y)(= d+(M\A, y)
H+(A) is constructed from the level sets of d+(M\A, · )
M. Sanchez Lorentz, Finsler and Riemann Geometry
1 Causal structure2 Visibility and lensing
3 Causal boundary
Preliminaries on Finsler distancesCausal HierarchyFurther results
Applications to Finsler
Remark. The results on horizons yield results on the distancefunction to a set in a Randers manifold, and viceversa.For example:
Theorem
Let C ⊂ M a closed subset, p ∈ M\C is a differentiable point ofthe distance from C iff p is crossed by exactly one minimizingsegment
generalizable to any Finsler manifold(Sabau-Tanaka’12)
M. Sanchez Lorentz, Finsler and Riemann Geometry
1 Causal structure2 Visibility and lensing
3 Causal boundary
Preliminaries on Finsler distancesCausal HierarchyFurther results
Applications to Finsler
Remark. The results on horizons yield results on the distancefunction to a set in a Randers manifold, and viceversa.For example:
Theorem
Let C ⊂ M a closed subset, p ∈ M\C is a differentiable point ofthe distance from C iff p is crossed by exactly one minimizingsegment generalizable to any Finsler manifold(Sabau-Tanaka’12)
M. Sanchez Lorentz, Finsler and Riemann Geometry
1 Causal structure2 Visibility and lensing
3 Causal boundary
The physical problemPreliminaries on convex hypersurfacesMain ideas and results
SECOND PART
2. VISIBILITY AND LENSING
M. Sanchez Lorentz, Finsler and Riemann Geometry
1 Causal structure2 Visibility and lensing
3 Causal boundary
The physical problemPreliminaries on convex hypersurfacesMain ideas and results
Visibility of particles
Fermat principle (Kovner’90, Perlick’90): if a first arriving causalcurve connecting a point and a line exists then it is a lightlikegeodesic. BUT:
Does such a geodesic exist?
Will it remain in a (reasonably big) realistic region R× Dsuch that our conformally stationary model R×M?
Will massive particles (timelike geodesics) arrive in a finiteproper time (length) prior to their desintegration?
Lensing: can we receive more than once particles from thesame source emitted at the same or different times?
Problems related to convexity.
M. Sanchez Lorentz, Finsler and Riemann Geometry
1 Causal structure2 Visibility and lensing
3 Causal boundary
The physical problemPreliminaries on convex hypersurfacesMain ideas and results
Visibility of particles
Fermat principle (Kovner’90, Perlick’90): if a first arriving causalcurve connecting a point and a line exists then it is a lightlikegeodesic. BUT:
Does such a geodesic exist?
Will it remain in a (reasonably big) realistic region R× Dsuch that our conformally stationary model R×M?
Will massive particles (timelike geodesics) arrive in a finiteproper time (length) prior to their desintegration?
Lensing: can we receive more than once particles from thesame source emitted at the same or different times?
Problems related to convexity.
M. Sanchez Lorentz, Finsler and Riemann Geometry
1 Causal structure2 Visibility and lensing
3 Causal boundary
The physical problemPreliminaries on convex hypersurfacesMain ideas and results
Visibility of particles
Fermat principle (Kovner’90, Perlick’90): if a first arriving causalcurve connecting a point and a line exists then it is a lightlikegeodesic. BUT:
Does such a geodesic exist?
Will it remain in a (reasonably big) realistic region R× Dsuch that our conformally stationary model R×M?
Will massive particles (timelike geodesics) arrive in a finiteproper time (length) prior to their desintegration?
Lensing: can we receive more than once particles from thesame source emitted at the same or different times?
Problems related to convexity.
M. Sanchez Lorentz, Finsler and Riemann Geometry
1 Causal structure2 Visibility and lensing
3 Causal boundary
The physical problemPreliminaries on convex hypersurfacesMain ideas and results
Visibility of particles
Fermat principle (Kovner’90, Perlick’90): if a first arriving causalcurve connecting a point and a line exists then it is a lightlikegeodesic. BUT:
Does such a geodesic exist?
Will it remain in a (reasonably big) realistic region R× Dsuch that our conformally stationary model R×M?
Will massive particles (timelike geodesics) arrive in a finiteproper time (length) prior to their desintegration?
Lensing: can we receive more than once particles from thesame source emitted at the same or different times?
Problems related to convexity.
M. Sanchez Lorentz, Finsler and Riemann Geometry
1 Causal structure2 Visibility and lensing
3 Causal boundary
The physical problemPreliminaries on convex hypersurfacesMain ideas and results
Visibility of particles
Fermat principle (Kovner’90, Perlick’90): if a first arriving causalcurve connecting a point and a line exists then it is a lightlikegeodesic. BUT:
Does such a geodesic exist?
Will it remain in a (reasonably big) realistic region R× Dsuch that our conformally stationary model R×M?
Will massive particles (timelike geodesics) arrive in a finiteproper time (length) prior to their desintegration?
Lensing: can we receive more than once particles from thesame source emitted at the same or different times?
Problems related to convexity.
M. Sanchez Lorentz, Finsler and Riemann Geometry
1 Causal structure2 Visibility and lensing
3 Causal boundary
The physical problemPreliminaries on convex hypersurfacesMain ideas and results
Visibility of particles
Fermat principle (Kovner’90, Perlick’90): if a first arriving causalcurve connecting a point and a line exists then it is a lightlikegeodesic. BUT:
Does such a geodesic exist?
Will it remain in a (reasonably big) realistic region R× Dsuch that our conformally stationary model R×M?
Will massive particles (timelike geodesics) arrive in a finiteproper time (length) prior to their desintegration?
Lensing: can we receive more than once particles from thesame source emitted at the same or different times?
Problems related to convexity.
M. Sanchez Lorentz, Finsler and Riemann Geometry
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The physical problemPreliminaries on convex hypersurfacesMain ideas and results
Previous: Riemannian convexity
(M, gR) Riemannian, D ⊂ M open domain with smooth ∂DNotions of convexity for ∂D:
1 Infinitesimal convexity: second fundamental form positivesemi-definite with respect to the inner normal⇐⇒ Hessφ negative semidefinite on T (∂D) for any smoothφ : D → [0,∞) with ∂D = φ−1(0) and φ regular on ∂D.
2 Local convexity:locally each exp(Tp(∂D) does not touch D.
Local =⇒ infinitesimal trivially (and pointwise)
Non-trivial converse (Do Carmo, Warner ’70)Bishop ’74 proved the converse but
1 his proof required smoothness C 4 (also for g)2 It cannot be extended to the Finslerian setting (Borisenko,
Olin ’10)
M. Sanchez Lorentz, Finsler and Riemann Geometry
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The physical problemPreliminaries on convex hypersurfacesMain ideas and results
Previous: Riemannian convexity
(M, gR) Riemannian, D ⊂ M open domain with smooth ∂DNotions of convexity for ∂D:
1 Infinitesimal convexity: second fundamental form positivesemi-definite with respect to the inner normal⇐⇒ Hessφ negative semidefinite on T (∂D) for any smoothφ : D → [0,∞) with ∂D = φ−1(0) and φ regular on ∂D.
2 Local convexity:locally each exp(Tp(∂D) does not touch D.
Local =⇒ infinitesimal trivially (and pointwise)
Non-trivial converse (Do Carmo, Warner ’70)
Bishop ’74 proved the converse but1 his proof required smoothness C 4 (also for g)2 It cannot be extended to the Finslerian setting (Borisenko,
Olin ’10)
M. Sanchez Lorentz, Finsler and Riemann Geometry
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The physical problemPreliminaries on convex hypersurfacesMain ideas and results
Previous: Riemannian convexity
(M, gR) Riemannian, D ⊂ M open domain with smooth ∂DNotions of convexity for ∂D:
1 Infinitesimal convexity: second fundamental form positivesemi-definite with respect to the inner normal⇐⇒ Hessφ negative semidefinite on T (∂D) for any smoothφ : D → [0,∞) with ∂D = φ−1(0) and φ regular on ∂D.
2 Local convexity:locally each exp(Tp(∂D) does not touch D.
Local =⇒ infinitesimal trivially (and pointwise)
Non-trivial converse (Do Carmo, Warner ’70)Bishop ’74 proved the converse but
1 his proof required smoothness C 4 (also for g)2 It cannot be extended to the Finslerian setting (Borisenko,
Olin ’10)
M. Sanchez Lorentz, Finsler and Riemann Geometry
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The physical problemPreliminaries on convex hypersurfacesMain ideas and results
General Finslerian results
Approach by Bartolo, Caponio, Germinario, — ’10:
previous notions extensible to Finslerian manifolds
intermediate notion: ∂D is geometrically convex when: nogeodesic in D connecting some p, q ∈ D touches ∂D.
Theorem
For any Finsler manifold, and domain D with C2,1
locboundary ∂D:
1 The infinitesimal, geometric and local notions of convexity areequivalent.
M. Sanchez Lorentz, Finsler and Riemann Geometry
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The physical problemPreliminaries on convex hypersurfacesMain ideas and results
General Finslerian results
Approach by Bartolo, Caponio, Germinario, — ’10:
previous notions extensible to Finslerian manifolds
intermediate notion: ∂D is geometrically convex when: nogeodesic in D connecting some p, q ∈ D touches ∂D.
Theorem
For any Finsler manifold, and domain D with C2,1
locboundary ∂D:
1 The infinitesimal, geometric and local notions of convexity areequivalent.
M. Sanchez Lorentz, Finsler and Riemann Geometry
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The physical problemPreliminaries on convex hypersurfacesMain ideas and results
General Finslerian results
Theorem (Bartolo, Caponio, Germinario, — ’10)
For any Finsler manifold, and domain D with C2,1
locboundary ∂D:
1 The infinitesimal, geometric and local notions of convexity areequivalent.
2 If all BDs (p, r) (in particular, if Bs(p, r)∩D) are compact: ∂D
is convex iff D is convex.
In the last case, if D is not contractible then any p, q ∈ D can beconnected by infinitely many geodesics contained in D.
M. Sanchez Lorentz, Finsler and Riemann Geometry
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The physical problemPreliminaries on convex hypersurfacesMain ideas and results
General Finslerian results
Theorem (Bartolo, Caponio, Germinario, — ’10)
For any Finsler manifold, and domain D with C2,1
locboundary ∂D:
1 The infinitesimal, geometric and local notions of convexity areequivalent.
2 If all BDs (p, r) (in particular, if Bs(p, r)∩D) are compact: ∂D
is convex iff D is convex.
In the last case, if D is not contractible then any p, q ∈ D can beconnected by infinitely many geodesics contained in D.
M. Sanchez Lorentz, Finsler and Riemann Geometry
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The physical problemPreliminaries on convex hypersurfacesMain ideas and results
Basic ideas: existence
Existence of connecting causal geodesics in a prescribed R× D
Lightlike geodesics. Optimal conditions:
1 (geometric) light-convexity of R× ∂D in R×M⇐⇒ Convexity of ∂D in (M,F ).
2 + completeness of the space of connecting curves:⇐⇒ Compactness of B D
s (p, r) (⇐ Bs(p, r) ∩ D)
Timelike geodesics. Reduction to the lightlike case:γ timelike geodesic in R×M with gL(γ′, γ′) = −c2
⇐⇒ γ(s) = (cs, γ(s)) is a lightlike geodesic in R× (R×M)with the product metric (≡ du2 + gL)
M. Sanchez Lorentz, Finsler and Riemann Geometry
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The physical problemPreliminaries on convex hypersurfacesMain ideas and results
Basic ideas: existence
Existence of connecting causal geodesics in a prescribed R× D
Lightlike geodesics. Optimal conditions:
1 (geometric) light-convexity of R× ∂D in R×M⇐⇒ Convexity of ∂D in (M,F ).
2 + completeness of the space of connecting curves:⇐⇒ Compactness of B D
s (p, r) (⇐ Bs(p, r) ∩ D)
Timelike geodesics. Reduction to the lightlike case:γ timelike geodesic in R×M with gL(γ′, γ′) = −c2
⇐⇒ γ(s) = (cs, γ(s)) is a lightlike geodesic in R× (R×M)with the product metric (≡ du2 + gL)
M. Sanchez Lorentz, Finsler and Riemann Geometry
1 Causal structure2 Visibility and lensing
3 Causal boundary
The physical problemPreliminaries on convex hypersurfacesMain ideas and results
Basic ideas: existence
Existence of connecting causal geodesics in a prescribed R× D
Lightlike geodesics. Optimal conditions:
1 (geometric) light-convexity of R× ∂D in R×M⇐⇒ Convexity of ∂D in (M,F ).
2 + completeness of the space of connecting curves:⇐⇒ Compactness of B D
s (p, r) (⇐ Bs(p, r) ∩ D)
Timelike geodesics. Reduction to the lightlike case:γ timelike geodesic in R×M with gL(γ′, γ′) = −c2
⇐⇒ γ(s) = (cs, γ(s)) is a lightlike geodesic in R× (R×M)with the product metric (≡ du2 + gL)
M. Sanchez Lorentz, Finsler and Riemann Geometry
1 Causal structure2 Visibility and lensing
3 Causal boundary
The physical problemPreliminaries on convex hypersurfacesMain ideas and results
Basic ideas: existence
Existence of connecting causal geodesics in a prescribed R× D
Lightlike geodesics. Optimal conditions:
1 (geometric) light-convexity of R× ∂D in R×M⇐⇒ Convexity of ∂D in (M,F ).
2 + completeness of the space of connecting curves:⇐⇒ Compactness of B D
s (p, r) (⇐ Bs(p, r) ∩ D)
Timelike geodesics. Reduction to the lightlike case:γ timelike geodesic in R×M with gL(γ′, γ′) = −c2
⇐⇒ γ(s) = (cs, γ(s)) is a lightlike geodesic in R× (R×M)with the product metric (≡ du2 + gL)
M. Sanchez Lorentz, Finsler and Riemann Geometry
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The physical problemPreliminaries on convex hypersurfacesMain ideas and results
Basic ideas: multiplicity
Multiplicity (lensing)
Local, around a given lightlike pregeodesic γ(t) = (t, c(t)):existence of conjugate points for γ c admits conjugate point(gravitational lensing)
Global:non trivial topology of R× D D non-contractible(topological lensing)
M. Sanchez Lorentz, Finsler and Riemann Geometry
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3 Causal boundary
The physical problemPreliminaries on convex hypersurfacesMain ideas and results
Basic ideas: multiplicity
Multiplicity (lensing)
Local, around a given lightlike pregeodesic γ(t) = (t, c(t)):existence of conjugate points for γ c admits conjugate point(gravitational lensing)
Global:non trivial topology of R× D D non-contractible(topological lensing)
M. Sanchez Lorentz, Finsler and Riemann Geometry
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3 Causal boundary
The physical problemPreliminaries on convex hypersurfacesMain ideas and results
Results: lightlike geodesics
M ×R standard stationary, D ⊂ M a C 2 domain
Theorem
Assume that all BDs (p, r) are compact (which happens, in
particular, when Bs(p, r) ∩ D are compact ). They are equivalent:
1 (R× D, gL) is causally simple (⇔ (D,F ) is convex)
2 (∂D; F ) is convex
3 (R× ∂D; gL) is light-convex.
4 Any point w ∈ R× D and any line lq, q ∈ D connected inR× D by a future–pointing lightlike geodesic minimizing the(future) arrival time T (⇔ idem for past)
In this case, if D is not contractible, infinitely many connectinglightlike geodesics with diverging arrival times exist
M. Sanchez Lorentz, Finsler and Riemann Geometry
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The physical problemPreliminaries on convex hypersurfacesMain ideas and results
Results: lightlike geodesics
M ×R standard stationary, D ⊂ M a C 2 domain
Theorem
Assume that all BDs (p, r) are compact (which happens, in
particular, when Bs(p, r) ∩ D are compact ). They are equivalent:
1 (R× D, gL) is causally simple (⇔ (D,F ) is convex)
2 (∂D; F ) is convex
3 (R× ∂D; gL) is light-convex.
4 Any point w ∈ R× D and any line lq, q ∈ D connected inR× D by a future–pointing lightlike geodesic minimizing the(future) arrival time T (⇔ idem for past)
In this case, if D is not contractible, infinitely many connectinglightlike geodesics with diverging arrival times exist
M. Sanchez Lorentz, Finsler and Riemann Geometry
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The physical problemPreliminaries on convex hypersurfacesMain ideas and results
Results: lightlike geodesics
M ×R standard stationary, D ⊂ M a C 2 domain
Theorem
Assume that all BDs (p, r) are compact (which happens, in
particular, when Bs(p, r) ∩ D are compact ). They are equivalent:
1 (R× D, gL) is causally simple (⇔ (D,F ) is convex)
2 (∂D; F ) is convex
3 (R× ∂D; gL) is light-convex.
4 Any point w ∈ R× D and any line lq, q ∈ D connected inR× D by a future–pointing lightlike geodesic minimizing the(future) arrival time T (⇔ idem for past)
In this case, if D is not contractible, infinitely many connectinglightlike geodesics with diverging arrival times exist
M. Sanchez Lorentz, Finsler and Riemann Geometry
1 Causal structure2 Visibility and lensing
3 Causal boundary
The physical problemPreliminaries on convex hypersurfacesMain ideas and results
Results: lightlike geodesics
M ×R standard stationary, D ⊂ M a C 2 domain
Theorem
Assume that all BDs (p, r) are compact (which happens, in
particular, when Bs(p, r) ∩ D are compact ). They are equivalent:
1 (R× D, gL) is causally simple (⇔ (D,F ) is convex)
2 (∂D; F ) is convex
3 (R× ∂D; gL) is light-convex.
4 Any point w ∈ R× D and any line lq, q ∈ D connected inR× D by a future–pointing lightlike geodesic minimizing the(future) arrival time T (⇔ idem for past)
In this case, if D is not contractible, infinitely many connectinglightlike geodesics with diverging arrival times exist
M. Sanchez Lorentz, Finsler and Riemann Geometry
1 Causal structure2 Visibility and lensing
3 Causal boundary
The physical problemPreliminaries on convex hypersurfacesMain ideas and results
Results: lightlike geodesics
M ×R standard stationary, D ⊂ M a C 2 domain
Theorem
Assume that all BDs (p, r) are compact (which happens, in
particular, when Bs(p, r) ∩ D are compact ). They are equivalent:
1 (R× D, gL) is causally simple (⇔ (D,F ) is convex)
2 (∂D; F ) is convex
3 (R× ∂D; gL) is light-convex.
4 Any point w ∈ R× D and any line lq, q ∈ D connected inR× D by a future–pointing lightlike geodesic minimizing the(future) arrival time T (⇔ idem for past)
In this case, if D is not contractible, infinitely many connectinglightlike geodesics with diverging arrival times exist
M. Sanchez Lorentz, Finsler and Riemann Geometry
1 Causal structure2 Visibility and lensing
3 Causal boundary
The physical problemPreliminaries on convex hypersurfacesMain ideas and results
Results: lightlike geodesics
M ×R standard stationary, D ⊂ M a C 2 domain
Theorem
Assume that all BDs (p, r) are compact (which happens, in
particular, when Bs(p, r) ∩ D are compact ). They are equivalent:
1 (R× D, gL) is causally simple (⇔ (D,F ) is convex)
2 (∂D; F ) is convex
3 (R× ∂D; gL) is light-convex.
4 Any point w ∈ R× D and any line lq, q ∈ D connected inR× D by a future–pointing lightlike geodesic minimizing the(future) arrival time T (⇔ idem for past)
In this case, if D is not contractible, infinitely many connectinglightlike geodesics with diverging arrival times exist
M. Sanchez Lorentz, Finsler and Riemann Geometry
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The physical problemPreliminaries on convex hypersurfacesMain ideas and results
Results: timelike geodesics
R×M standard stationary, non-necessarily normalized (−βdt2),product metric for Ru ≡ (R, du2) and (R×M, gL), Πu projectionon the first factor, in addition to ΠM . Fermat Fβ on R×M:
Fβ =
√Π∗Mh +
Π∗udu2
β ΠM+ Π∗Mω =
√hβ + ω1.
(extra dimension plus non-conformal invariance)
M. Sanchez Lorentz, Finsler and Riemann Geometry
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Results: timelike geodesics
Theorem
Assume that all BDs (p, r) are compact (which happens, in
particular, when Bs(p, r) ∩ D are compact ). Then:
(Ru × ∂D; Fβ) is convex ⇔for any length l > 0, each point w ∈ R× D and line lq arejoined by a future (and a past) pointing timelike geodesic inR× D,with length l minimizing the arrival time (amongcausal curves of length l).
In this case, if D is not contractible, a sequence of such connectinggeodesics with diverging lengths exists.
M. Sanchez Lorentz, Finsler and Riemann Geometry
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Further results: asympt. flat spacetimes
Asymptotically flat stationary spacetime:
(M, g) complete, outside a compact subset C , isdiffeomorphic to Rn \ B(0,R0), elements g , ω, β turningEuclidean with large radial coordinate r .
Model isolated systems -gravity outside a star.
Makes sense to speak on (stationary) large ballsB(0,R)×R ⊂ R×M and spheres.
Application: Large spheres in asympt. flat spacetimes are alwayslight-convex (and all the previous results are applicable) but,typically, (including reasonable matter, when gravity attracts) theyare not time-convex
M. Sanchez Lorentz, Finsler and Riemann Geometry
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Further results: asympt. flat spacetimes
Asymptotically flat stationary spacetime:
(M, g) complete, outside a compact subset C , isdiffeomorphic to Rn \ B(0,R0), elements g , ω, β turningEuclidean with large radial coordinate r .
Model isolated systems -gravity outside a star.
Makes sense to speak on (stationary) large ballsB(0,R)×R ⊂ R×M and spheres.
Application: Large spheres in asympt. flat spacetimes are alwayslight-convex (and all the previous results are applicable) but,typically, (including reasonable matter, when gravity attracts) theyare not time-convex
M. Sanchez Lorentz, Finsler and Riemann Geometry
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THIRD PART
3. CAUSAL BOUNDARY
M. Sanchez Lorentz, Finsler and Riemann Geometry
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Introduction
Causal boundary ∂V of a spacetime V :
Involved structure: conformal structure (Causality).Intrinsic alternative to common Penrose conformal boundaryapplicable to any strongly causal spacetime
Purpose: attach a boundary endpoint P ∈ ∂V to anyinextensible future or past directed timelike curve γ
M. Sanchez Lorentz, Finsler and Riemann Geometry
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Introduction
Basic idea: the boundary point would be represented byP = I−(γ) or F = I +(γ) or, more precisely, a pair (P,F ).
M. Sanchez Lorentz, Finsler and Riemann Geometry
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Introduction
Long story from Geroch, Kronheimer & Penrose ’72 until its recentredefinition Flores, Herrera & — ’11):
As a point set, the completion V is composed by pairs (P,F )(“(IP,IF)”, for example V 3 p ≡ (I +(p), I−(p)) ∈ V )
Chronological relation: (P,F ) (P ′,F ′)⇔ F ∩ P ′ 6= ∅
M. Sanchez Lorentz, Finsler and Riemann Geometry
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Introduction
Long story until its recent redefinition:
As a point set, the completion V is composed by pairs (P,F )(“(IP,IF)”, for example V 3 p ≡ (I +(p), I−(p)) ∈ V )Chronological relation: (P,F ) (P ′,F ′)⇔ F ∩ P ′ 6= ∅Subtle topology... non always Hausdorff
M. Sanchez Lorentz, Finsler and Riemann Geometry
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Introduction
When computed for standard stationary spacetimes, relations with
Cauchy boundary
Gromov boundary
Busemann-type boundary
for Randers manifolds:
previous study for any Riemannian and Finslerianmanifold with interest by itself
M. Sanchez Lorentz, Finsler and Riemann Geometry
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Introduction
When computed for standard stationary spacetimes, relations with
Cauchy boundary
Gromov boundary
Busemann-type boundary
for Randers manifolds:
previous study for any Riemannian and Finslerianmanifold with interest by itself
M. Sanchez Lorentz, Finsler and Riemann Geometry
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Cauchy boundary for Riemannian manifolds
Remark: it may be non-locally compact
M. Sanchez Lorentz, Finsler and Riemann Geometry
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Gromov boundary for Riemannian manifolds
Classical Gromov’s compactification for complete Riemannianmanifolds
L1(M, g) 1-Lipschitz functions (pointwise topology)
x ∈ M can be seen in L1(M, g) as dx : y 7→ d(x , y) and alsodx + C for any C ∈ Rf ∼ f ′ ⇐⇒ f − f ′ =constant (quotient topology)
each x ∈ M is represented in L1(M, g)/ ∼ as the class of −dx
MG = closure of M in L1(M, g)/ ∼
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Gromov boundary for Riemannian manifolds
What about if (M, g) is not complete? Repeat construction:
Cauchy MC : completion, no compactification (MC may benon-locally compact)
Gromov MG : compactification even in the incomplete case.
1 MC → MG in a natural way and continuous but:the inclusion is an embedding ⇔ MC is locally compact
2 MG = M ∪ ∂CGM ∪ ∂GM
∂CGM: limits of bounded sequences (∂CM ⊂ ∂CGM)∂GM: limits of unbounded sequences
M. Sanchez Lorentz, Finsler and Riemann Geometry
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Gromov boundary for Riemannian manifolds
What about if (M, g) is not complete? Repeat construction:
Cauchy MC : completion, no compactification (MC may benon-locally compact)
Gromov MG : compactification even in the incomplete case.
1 MC → MG in a natural way and continuous but:the inclusion is an embedding ⇔ MC is locally compact
2 MG = M ∪ ∂CGM ∪ ∂GM
∂CGM: limits of bounded sequences (∂CM ⊂ ∂CGM)∂GM: limits of unbounded sequences
M. Sanchez Lorentz, Finsler and Riemann Geometry
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Gromov boundary for Riemannian manifolds
What about if (M, g) is not complete? Repeat construction:
Cauchy MC : completion, no compactification (MC may benon-locally compact)
Gromov MG : compactification even in the incomplete case.
1 MC → MG in a natural way and continuous but:the inclusion is an embedding ⇔ MC is locally compact
2 MG = M ∪ ∂CGM ∪ ∂GM
∂CGM: limits of bounded sequences (∂CM ⊂ ∂CGM)∂GM: limits of unbounded sequences
M. Sanchez Lorentz, Finsler and Riemann Geometry
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Busemann boundary for Riemannian manifolds
M ≡ (M, g) connected Riemannian manfiold
Typically, Busemann functions are defined when c is a ray(half unit geodesic with no cut locus)bc(x0) = limt→∞(t − d(x0, c(t))) for all x0 ∈ M
Eberlein & O’Neill developed a compactification of anyHadamard manifold in terms of Busemann functions (conetopology), which coincides with Gromov’s one
M. Sanchez Lorentz, Finsler and Riemann Geometry
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Busemann boundary for Riemannian manifolds
M ≡ (M, g) connected Riemannian manfiold
Typically, Busemann functions are defined when c is a ray(half unit geodesic with no cut locus)bc(x0) = limt→∞(t − d(x0, c(t))) for all x0 ∈ M
Eberlein & O’Neill developed a compactification of anyHadamard manifold in terms of Busemann functions (conetopology), which coincides with Gromov’s one
M. Sanchez Lorentz, Finsler and Riemann Geometry
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Busemann boundary for Riemannian manifolds
M ≡ (M, g) connected Riemannian manfiold
Typically, Busemann functions are defined when c is a ray(half geodesic with no cut locus)bc(x0) = limt→∞(t − d(x0, c(t))) for all x0 ∈ M
Eberlein & O’Neill developed a compactification of anyHadamard manifold in terms of Busemann functions (conetopology), which coincides with Gromov’s one
We will admit a “Busemann” function for any curvec : [0,Ω)→ M with |c| ≤ 1bc(x0) = limt→Ω(t − d(x0, c(t)))
bc is ∞ at some x0 ∈ M iff bc ≡ ∞.B(M): set of finite Busemann functions
M. Sanchez Lorentz, Finsler and Riemann Geometry
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Busemann boundary for Riemannian manifolds
As a subset B(M) ⊂ L1(M, g)Busemann completion MB = B(M)/ ∼ (quotient by additiveconstant –included as a subset in MG )
B(M), and then MB , are topological spaces with thechronological topology (different to the induced fromL1(M, g)) defined by means of a limit operator Lgiven fn ⊂ B(M), the subset L(fn) ⊂ B(M) is defined by:f ∈ L(fn) iff
(a) f ≤ lim infn fn and(b) ∀g ∈ B(M) with f ≤ g ≤ lim supn fn, it is g = f .
M. Sanchez Lorentz, Finsler and Riemann Geometry
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Busemann boundary for Riemannian manifolds
As a subset B(M) ⊂ L1(M, g)Busemann completion MB = B(M)/ ∼ (quotient by additiveconstant –included as a subset in MG )
B(M), and then MB , are topological spaces with thechronological topology (different to the induced fromL1(M, g))
defined by means of a limit operator Lgiven fn ⊂ B(M), the subset L(fn) ⊂ B(M) is defined by:f ∈ L(fn) iff
(a) f ≤ lim infn fn and(b) ∀g ∈ B(M) with f ≤ g ≤ lim supn fn, it is g = f .
M. Sanchez Lorentz, Finsler and Riemann Geometry
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Busemann boundary for Riemannian manifolds
As a subset B(M) ⊂ L1(M, g)Busemann completion MB = B(M)/ ∼ (quotient by additiveconstant –included as a subset in MG )
B(M), and then MB , are topological spaces with thechronological topology (different to the induced fromL1(M, g)) defined by means of a limit operator Lgiven fn ⊂ B(M), the subset L(fn) ⊂ B(M) is defined by:f ∈ L(fn) iff
(a) f ≤ lim infn fn and(b) ∀g ∈ B(M) with f ≤ g ≤ lim supn fn, it is g = f .
M. Sanchez Lorentz, Finsler and Riemann Geometry
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Busemann boundary for Riemannian manifolds
Properties of the Busemann completion:
1 MB is sequentially compact
2 MB is T1, and points in ∂BM may be non-T2 related
3 MC → MB → MG (naturally) but MB topology is coarser.
4 MB = MG (as pointsets and, then, topologically)⇐⇒ ∂BM is Hausdorff
Discrepancy: MB compactifies directions (finite or asympt)MG may contain non-endpoints of curves in M
5 MB = M ∪ ∂CM (finite directions) ∪∂BM (asymptotic)
MC : Busemann functions for curves c with Ω <∞MB: Busemann for c with Ω =∞
M. Sanchez Lorentz, Finsler and Riemann Geometry
1 Causal structure2 Visibility and lensing
3 Causal boundary
Riemannian boundariesFinslerian boundariesC-boundary on stationary s-t
Cauchy boundary for Finslerian manifolds
Cauchy completions for a Finsler manifold:(M, d), d associated to F
1 Two types of Cauchy sequences (ordering)Cauchy boundaries ∂+
C M, ∂−C M plus the symmetrized one∂sCM = ∂−C M ∩ ∂+
C M
2 The extension dQ of d to, say, M+C = M ∪ ∂+
C M is not ageneralized distance but a quasidistance( dQ(xn, x)→ 0 6=⇒ dQ(x , xn)→ 0)
Topology on M+C generated by the forward balls different to
generated by backward balls∂+C M may be only a T0 space
M. Sanchez Lorentz, Finsler and Riemann Geometry
1 Causal structure2 Visibility and lensing
3 Causal boundary
Riemannian boundariesFinslerian boundariesC-boundary on stationary s-t
Cauchy boundary for Finslerian manifolds
Cauchy completions for a Finsler manifold:(M, d), d associated to F
1 Two types of Cauchy sequences (ordering)Cauchy boundaries ∂+
C M, ∂−C M plus the symmetrized one∂sCM = ∂−C M ∩ ∂+
C M
2 The extension dQ of d to, say, M+C = M ∪ ∂+
C M is not ageneralized distance but a quasidistance( dQ(xn, x)→ 0 6=⇒ dQ(x , xn)→ 0)
Topology on M+C generated by the forward balls different to
generated by backward balls∂+C M may be only a T0 space
M. Sanchez Lorentz, Finsler and Riemann Geometry
1 Causal structure2 Visibility and lensing
3 Causal boundary
Riemannian boundariesFinslerian boundariesC-boundary on stationary s-t
Finslerian Gromov completions
Gromov completions for a Finsler manifold:
1 Non-symmetric notions of LipschitzianL+
1 (M, d): f (y)− f (x) ≤ d(x , y)(L−1 (M, d) : f (x)− f (y) ≤ d(x , y))
2 Two Gromov’s compactifications M±G ,say: M+
G = closure of M in L+1 (M, d)/ ∼
In a natural way, i : M+C → M+
G but:
i continuous iff the backward balls generate a finer topologyon M+
C than the forward balls
i embedding when:
M+C is locally compact [also Riemannian condition] AND
dQ is a generalized distance
M. Sanchez Lorentz, Finsler and Riemann Geometry
1 Causal structure2 Visibility and lensing
3 Causal boundary
Riemannian boundariesFinslerian boundariesC-boundary on stationary s-t
Finslerian Gromov completions
Gromov completions for a Finsler manifold:
1 Non-symmetric notions of LipschitzianL+
1 (M, d): f (y)− f (x) ≤ d(x , y)(L−1 (M, d) : f (x)− f (y) ≤ d(x , y))
2 Two Gromov’s compactifications M±G ,say: M+
G = closure of M in L+1 (M, d)/ ∼
In a natural way, i : M+C → M+
G but:
i continuous iff the backward balls generate a finer topologyon M+
C than the forward balls
i embedding when:
M+C is locally compact [also Riemannian condition] AND
dQ is a generalized distance
M. Sanchez Lorentz, Finsler and Riemann Geometry
1 Causal structure2 Visibility and lensing
3 Causal boundary
Riemannian boundariesFinslerian boundariesC-boundary on stationary s-t
Finslerian Gromov completions
Gromov completions for a Finsler manifold:
1 Non-symmetric notions of LipschitzianL+
1 (M, d): f (y)− f (x) ≤ d(x , y)(L−1 (M, d) : f (x)− f (y) ≤ d(x , y))
2 Two Gromov’s compactifications M±G ,say: M+
G = closure of M in L+1 (M, d)/ ∼
In a natural way, i : M+C → M+
G but:
i continuous iff the backward balls generate a finer topologyon M+
C than the forward balls
i embedding when:
M+C is locally compact [also Riemannian condition] AND
dQ is a generalized distance
M. Sanchez Lorentz, Finsler and Riemann Geometry
1 Causal structure2 Visibility and lensing
3 Causal boundary
Riemannian boundariesFinslerian boundariesC-boundary on stationary s-t
Finslerian Busemann completions
Busemann completions for a Finsler manifold:
1 There are also two completions M±B constructed by usingBusemann functions (which depends on the order of thearguments in d)
2 As in the Riemannian case, each M±B is naturally included inM±G (with a coarser topology) and both boundaries coincideiff ∂±B M is Hausdorff.
3 The spacetime viewpoint suggests further relations betweenM+
B and M−B .
M. Sanchez Lorentz, Finsler and Riemann Geometry
1 Causal structure2 Visibility and lensing
3 Causal boundary
Riemannian boundariesFinslerian boundariesC-boundary on stationary s-t
Finslerian Busemann completions
Busemann completions for a Finsler manifold:
1 There are also two completions M±B constructed by usingBusemann functions (which depends on the order of thearguments in d)
2 As in the Riemannian case, each M±B is naturally included inM±G (with a coarser topology) and
both boundaries coincideiff ∂±B M is Hausdorff.
3 The spacetime viewpoint suggests further relations betweenM+
B and M−B .
M. Sanchez Lorentz, Finsler and Riemann Geometry
1 Causal structure2 Visibility and lensing
3 Causal boundary
Riemannian boundariesFinslerian boundariesC-boundary on stationary s-t
Finslerian Busemann completions
Busemann completions for a Finsler manifold:
1 There are also two completions M±B constructed by usingBusemann functions (which depends on the order of thearguments in d)
2 As in the Riemannian case, each M±B is naturally included inM±G (with a coarser topology) and both boundaries coincideiff ∂±B M is Hausdorff.
3 The spacetime viewpoint suggests further relations betweenM+
B and M−B .
M. Sanchez Lorentz, Finsler and Riemann Geometry
1 Causal structure2 Visibility and lensing
3 Causal boundary
Riemannian boundariesFinslerian boundariesC-boundary on stationary s-t
Finslerian Busemann completions
Busemann completions for a Finsler manifold:
1 There are also two completions M±B constructed by usingBusemann functions (which depends on the order of thearguments in d)
2 As in the Riemannian case, each M±B is naturally included inM±G (with a coarser topology) and both boundaries coincideiff ∂±B M is Hausdorff.
3 The spacetime viewpoint suggests further relations betweenM+
B and M−B .
M. Sanchez Lorentz, Finsler and Riemann Geometry
1 Causal structure2 Visibility and lensing
3 Causal boundary
Riemannian boundariesFinslerian boundariesC-boundary on stationary s-t
Appearance of the Busemann functions
Standard stationary spacetime:(g Riemannian on M with distance d , π : R×M → M projection)
V = (R×M, gL = −dt2 + 2dt ⊗ π∗ω + π∗g)
Fermat metric(with associated generalized distance d+)
F +(v) =√
g(v , v) + ω(v)2 ∀v ∈ TM
Aim: computation of IP’s (and dual IF’s)P = I−[γ], future-directed timelike curve
γ(t) = (t, c(t)), t ∈ [α,Ω), |c | < 1
M. Sanchez Lorentz, Finsler and Riemann Geometry
1 Causal structure2 Visibility and lensing
3 Causal boundary
Riemannian boundariesFinslerian boundariesC-boundary on stationary s-t
Appearance of the Busemann functions
Standard stationary spacetime:(g Riemannian on M with distance d , π : R×M → M projection)
V = (R×M, gL = −dt2 + 2dt ⊗ π∗ω + π∗g)
Fermat metric(with associated generalized distance d+)
F +(v) =√
g(v , v) + ω(v)2 ∀v ∈ TM
Aim: computation of IP’s (and dual IF’s)P = I−[γ], future-directed timelike curve
γ(t) = (t, c(t)), t ∈ [α,Ω), |c | < 1
M. Sanchez Lorentz, Finsler and Riemann Geometry
1 Causal structure2 Visibility and lensing
3 Causal boundary
Riemannian boundariesFinslerian boundariesC-boundary on stationary s-t
Appearance of the Busemann functions
1 Characterization of (t0, x0) (t1, x1) ⇐⇒ d+(x0, x1) < t1 − t0
2 Application to P = I−[γ], γ(t) = (t, c(t)):P = (t0, x0) ∈ V : (t0, x0) γ(t) for some t ∈ [α,Ω)= (t0, x0) ∈ V : t0 < t − d+(x0, c(t)) for some t ∈ [α,Ω)= (t0, x0) ∈ V : t0 < limt→Ω(t − d+(x0, c(t)))= (t0, x0) ∈ V : t0 < b+
c (x0)
Busemann forward function:b+c (x0) = limt→Ω(t − d+(x0, c(t)))
M. Sanchez Lorentz, Finsler and Riemann Geometry
1 Causal structure2 Visibility and lensing
3 Causal boundary
Riemannian boundariesFinslerian boundariesC-boundary on stationary s-t
Appearance of the Busemann functions
1 Characterization of (t0, x0) (t1, x1) ⇐⇒ d+(x0, x1) < t1 − t0
2 Application to P = I−[γ], γ(t) = (t, c(t)):P = (t0, x0) ∈ V : (t0, x0) γ(t) for some t ∈ [α,Ω)= (t0, x0) ∈ V : t0 < t − d+(x0, c(t)) for some t ∈ [α,Ω)= (t0, x0) ∈ V : t0 < limt→Ω(t − d+(x0, c(t)))= (t0, x0) ∈ V : t0 < b+
c (x0)
Busemann forward function:b+c (x0) = limt→Ω(t − d+(x0, c(t)))
M. Sanchez Lorentz, Finsler and Riemann Geometry
Fundamental correspondence1 Causal structure2 Visibility and lensing
3 Causal boundary
Riemannian boundariesFinslerian boundariesC-boundary on stationary s-t
Fundamental correspondence
B+(M): set of all Busemann functions b+c for (M,F +)
IP‘s on V ≡ B+(M) ∪bc ≡ ∞
1 Past of points (PIP’s): converging c (⇒ Ω <∞)
2 Past of inextensible curves (TIP’s): non-converging c(⇐ Ω =∞)
Remark: for the causal boundary no quotient in the set ofBusemann functions must be carried out
M. Sanchez Lorentz, Finsler and Riemann Geometry
1 Causal structure2 Visibility and lensing
3 Causal boundary
Riemannian boundariesFinslerian boundariesC-boundary on stationary s-t
Computation: c-boundary for static spacetimes
Parts of ∂V
i+, i− apexes (bc ≡ ∞)of a “double cone ”
Horismotic (as lightlikewith no cut points) lineson ∂BM starting at i±
Timelike lines on ∂cM(unique non-trivialS-pairs) connecting i+, i−
Recall: with the chr-topology
M. Sanchez Lorentz, Finsler and Riemann Geometry
1 Causal structure2 Visibility and lensing
3 Causal boundary
Riemannian boundariesFinslerian boundariesC-boundary on stationary s-t
Computation: c-boundary for static spacetimes
Parts of ∂V
i+, i− apexes (bc ≡ ∞)of a “double cone ”
Horismotic (as lightlikewith no cut points) lineson ∂BM starting at i±
Timelike lines on ∂cM(unique non-trivialS-pairs) connecting i+, i−
Recall: with the chr-topology
M. Sanchez Lorentz, Finsler and Riemann Geometry
1 Causal structure2 Visibility and lensing
3 Causal boundary
Riemannian boundariesFinslerian boundariesC-boundary on stationary s-t
Computation: c-boundary for static spacetimes
Parts of ∂V
i+, i− apexes (bc ≡ ∞)of a “double cone ”
Horismotic (as lightlikewith no cut points) lineson ∂BM starting at i±
Timelike lines on ∂cM(unique non-trivialS-pairs) connecting i+, i−
Recall: with the chr-topology
M. Sanchez Lorentz, Finsler and Riemann Geometry
1 Causal structure2 Visibility and lensing
3 Causal boundary
Riemannian boundariesFinslerian boundariesC-boundary on stationary s-t
Computation: c-boundary for stationary spacetimes
Parts of ∂V
“Static part”:i+, i− “apexes” of twodistinct conesHorismotic lines on ∂±BMTimelike lines on ∂sCM(composed of S-pairs)
Locally horismotic lineson ∂±C M\∂sCMThe lines arrive both i+
and i− iff pairings withlines on ∂∓C M\∂sCM
M. Sanchez Lorentz, Finsler and Riemann Geometry
1 Causal structure2 Visibility and lensing
3 Causal boundary
Riemannian boundariesFinslerian boundariesC-boundary on stationary s-t
Computation: c-boundary for stationary spacetimes
Parts of ∂V
“Static part”:i+, i− “apexes” of twodistinct conesHorismotic lines on ∂±BMTimelike lines on ∂sCM(composed of S-pairs)
Locally horismotic lineson ∂±C M\∂sCMThe lines arrive both i+
and i− iff pairings withlines on ∂∓C M\∂sCM
M. Sanchez Lorentz, Finsler and Riemann Geometry