Cosmological Time in (2+1) - Gravity

The initial singularity of U(M0) is the segment J0 = {x1=x3=0, 0≤x2≤r}.

“The initial singularity of U ( M 0 ) is the segment J 0 = { x 1 = x 3 = 0 0 ≤ x 2 ≤ r } .”

For any maximal globally hyperbolic flat (2+1)-spacetime M with compact space S of genus g>1, the cosmological time CT is a global time on M.

“For any spacetime M of this type, the length of time that the events have been in existence is M defines a global time, called the cosmological time CT of M , which reveals deep intrinsic properties of spacetime.”

• Also stated in: § 4 The cosmological time of (2+1)-spacetimes · p. 9 · plain

  “The cosmological time function, τ : M → (0 , ∞ [ , is surjective and regular, so that it defines a global time on M .”

• AssumptionThe authors state this as an assumption: The space is future expanding.
• AssumptionThe authors state this as an assumption: The maximal hyperbolic flat spacetimes considered are time-oriented and their orientations locally agree with the usual ones on M2+1.

In the elementary deformation M0, the CT level surfaces S~_a of U(M0) are only C^1-embedded into U(M0).

“The surfaces ˜ S a are only C 1 -embedded into U ( M 0 ).”

When M is of simplicial type, the corresponding real tree T is actually a simplicial real tree.

“When M is of simplicial type, the corresponding real tree T is actually a simplicial real tree.”

The asymptotic state for t→-π/2, i.e. the initial singularity of D(M), is equal to the initial singularity of M.

“The asymptotic state for t →-π/ 2 (i.e. the initial singularity) is equal to the initial singularity of M .”

ST(S) is dense in GT(S) in the induced topology by (R≥0)^C.

“Proposition 4.23 ST ( S ) is dense in GT ( S ) in the induced topology by ( R ≥ 0 ) C .”

The future asymptotic states of M and M^- glue at the level surface {t=0} of the CT where expansion ends and collapse begins.

“The future asymptotic states of M and M -'glue' at the level surface { t = 0 } of the CT where the expansion ends and the collapse begins.”

There is a natural isometric action of the fundamental group π1(S) on T.

“There is a natural isometric action of the fundamental group π 1 ( S ) on T .”

The other end-point i(p) of a(p) belongs to the boundary ∂U(M) of U(M) in M2+1.

“The other end-point of a ( p ) , denoted by i ( p ) , belongs to the boundary ∂U ( M ) of U ( M ) in M 2+1 .”

Each S~_a (S_a) is a future Cauchy surface.

“Each ˜ S a ( S a ) is a future Cauchy surface.”

F is the disjoint union of two sublaminations, where L is the maximal multi-curve sublamination of F; either L or F' may be empty.

“2) F is the disjoint union of two sublaminations”

• Also stated in: § Proposition 4.10 For every a ∈ (0 , ∞ [ : · p. 13 · hedged

  “where L is the maximal multi-curve sublamination of F . Note that either L or F ′ may be empty.”

Each segment of T is a rectifiable space-like curve in ∂U(M) with its own length.

“More precisely, each segment of T is a rectifiable space-like curve in ∂U ( M ) with its own length.”

Each level surface S~_a of U(M) maps onto S_a of M and is its universal covering.

“Each level surface ˜ S a of U ( M ) maps onto S a of M and is its universal covering.”

If M is the spacetime of simplicial type corresponding to the weighted multi-curve L on F, then L_a is made by the boundary components of the flat annular components embedded into S_a associated to the components of L.

“If M is the spacetime of simplicial type which corresponds to the weighted multi-curve L on the surface F , then L a is just made by the boundary components of the flat annular components embedded into S a , which are associated to the components of L .”

If L is non empty, extending L_a to L'_a by foliating the flat annular regions of S_a by closed geodesics parallel to the boundary components, the map p_a sends L'_a onto a F.

“The above map p a sends L ′ a onto a F .”

The union of the initial points T = {i(p); p∈U(M)} is an R-tree.

“The union of the initial points T = { i ( p ); p ∈ U ( M ) } is an R -tree.”

Each non static toric spacetime determines a curve γ: (0,∞) → T*_1, γ(a)=(w(a),ω(a)), where T*_1 is the cotangent bundle of the Teichmüller space T_1 of conformal structures on the torus up to isomorphism isotopic to the identity.

“Each non static spacetime determines a curve γ : (0 , ∞ [ → T ∗ 1 , γ ( a ) = ( w ( a ) , ω ( a )), where T ∗ 1 is the cotangent bundle of the Teichm¨ uller space T 1 of conformal structures on the torus up to isomorphism isotopic to the identity.”

• cites [B-G]Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

The same conclusion holds if F is of finite area.

“The same conclusion holds if F is of finite area.”

In the simplicial case, the set of edges of T consists of the union of the space-like segments which form the initial singularity of the different tiles of the form B(σ,vσ,r)+t(Rσ).

“In this case, the set of edges of T consists of the union of the space-like segments which form the initial singularity of the different tiles of the form B ( σ, v σ , r ) + t ( R σ ) (see section 3).”

One can invert the above construction and associate to each measured geodesic lamination (F,μ) of the hyperbolic surface F a suitable geometric R-tree.

“One can invert the above construction and associate to each measured geodesic lamination ( F , µ ) of the hyperbolic surface F a suitable geometric R -tree.”

Each component of W_a tilde is either isometric to an open set of I(a) or is a flat band which embeds into I(1,3,a), and projects onto an annulus of S_a.

“Each component of ˜ W a is either isometric to an open set of I ( a ) or is a flat band which embeds into I (1 , 3 , a ) , and projects onto an annulus of S a .”

S_a can be obtained by cutting F0 along c and by inserting an annulus of depth r.

“As shown in figure 4, S a can be obtained by cutting F 0 along c and by inserting a annulus of depth r .”

The projection of γ onto T_1 is a complete geodesic connecting two boundary points.

“In particular the projection of γ onto T 1 is a complete geodesic connecting two boundary points.”

A homeomorphic (not isometric) copy of T can easily be embedded into I(1).

“A homeomorphic (not isometric) copy of T can easily be embedded into I (1).”

For every a in (0,∞), l_{S_a}/a belongs to T_g ⊂ (R≥0)^C.

“Proposition 4.27 For every a ∈ (0 , ∞ [ , l S a /a belongs to T g ∈ ( R ≥ 0 ) C .”

When F is compact, every region R is bounded by infinitely many lines of L~.

“When F is compact, every region R (defined above) is bounded by infinitely many lines of ˜ L .”

For every a in (0,∞), S_a tilde is only C1-embedded into U(M), so it carries an induced C1-Riemannian metric.

“2) ˜ S a is only C 1 -embedded into U ( M ) , so that it carries an induced C 1 -Riemannian metric.”

There exists a geodesic lamination F on F=I(1)/Γ, lifting to F tilde on I(1), such that for every a, p_a(L_a tilde)=a F tilde and any leaf of L_a tilde is isometrically mapped onto a leaf of a F tilde.

“1) There exists a geodesic lamination F on F = I (1) / Γ , which lifts to a geodesic lamination ˜ F on I (1) , such that, for every a , one has p a ( ˜ L a ) = a ˜ F and any leaf of ˜ L a is isometrically mapped onto a leaf of a ˜ F .”

All the vertices of T are of infinite valence.

“Note that, as demonstrated in remark 3.4, all the vertices of T are of infinite valence.”

For every a in (0,∞), the locus L_a tilde at which the embedding of S_a tilde into U(M) is no longer C2 is a geodesic lamination of S_a tilde.

“3) The locus ˜ L a at which the embedding of ˜ S a into U ( M ) is no longer C 2 is a geodesic lamination of ˜ S a .”

∆: MGL(F) → GT(S) is a bijection, that is it can be naturally inverted.

“Proposition 4.19 ∆ : MGL ( F ) → GT ( S ) is a bijection, that is it can be naturally inverted.”

• cites [O]Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

The final singularity of D(M), t→π/2, coincides with the initial singularity of M^-.

“The final singularity ( t → π/ 2) coincides with the initial singularity of M -.”

T_g and the image Im(I) are disjoint subsets of (R≥0)^C.

“Moreover T g and the image Im(I) are disjoint subsets of ( R ≥ 0 ) C”

• cites [O]Not matched

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For the toric case, γ is the complete orbit of the Teichmüller flow with initial data (w(1),ω(1)).

“It is not hard to verify that γ is just the complete orbit of the Teichm¨ uller flow with initial data ( w (1) , ω (1)) (see [Ab],[B-G]).”

• cites [Ab]Not matched

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• cites [B-G]Not matched

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The Fenchel-Nielsen flow generalizes to the earthquake flow with initial data (F,F), which has real analytic orbits.

“Concerning proposition 4.27, the Fenchel-Nielsen flow generalizes to the earthquake flow (one uses again the density 4.23) with initial data ( F, F ) which has real analytic orbits [Ke2].”

• cites [Ke2]Not matched

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L_a tilde is the pull-back of a geodesic lamination L_a of S_a.

“˜ L a is in fact the pull-back of a geodesic lamination L a of S a .”

For a spacetime of simplicial type, the curve in T_g is given by the Fenchel-Nielsen flow obtained by twisting the hyperbolic surface F along the closed geodesic of the multi-curve.

“Concerning proposition 4.27, in the special case of a spacetime of simplicial type, the curve in T g is just given by the Fenchel-Nielsen flow obtained by 'twisting' the hyperbolic surface F along the closed geodesic of the multi-curve (see [T3] and also [B-P]).”

• cites [T3]Not matched

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• cites [B-P]Not matched

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The cosmological time lifts to a regular cosmological time on the covering, τ~ : U(M) → (0,∞[.

“It lifts to a regular cosmological time on the covering, ˜ τ : U ( M ) → (0 , ∞ [ .”

Starting from any locally finite family of weighted geodesic lines in I(1), the simplicial deformation produces a globally hyperbolic spacetime structure on R^2×R with a regular cosmological time.

“Similarly, by starting from any locally finite family of weighted geodesic lines in I (1), the simplicial deformation that we have just described produces a globally hyperbolic spacetime structure on R 2 × R with a regular cosmological time.”

(p')^{-1}(L) ∩ S_a is the disjoint union of flat annular components of S_a, like in the case of a spacetime of simplicial type.

“so that ( p ′ ) -1 ( L ) ∩ S a is the disjoint union of flat annular components of S a , like in the case of a spacetime of simplicial type.”

Flat annuli occur only if L is non empty.

“Flat annuli do occur only if L is non empty.”

For any p∈U(M) there is a unique time-like geodesic arc a(p) contained in U(M), which starts at p and is directed in the past of p, such that the Lorentzian length of a(p) equals τ~(p).

“For any p ∈ U ( M ) ⊂ M 2+1 there is a unique time-like geodesic arc a ( p ) contained in U ( M ) , which starts at p and is directed in the past of p , such that the Lorentzian length of a ( p ) equals ˜ τ ( p ) .”

If p and q are identified in M by the action of π1(S), so are a(p) and a(q).

“If p and q are identified in M by the action of π 1 ( S ) , so are a ( p ) and a ( q ) .”

The set p^{-1}(L hat) is the union of components of the type B(σ, v_σ, r)+w.

“3) The set p -1 ( ̂ L ) is the union of components of the type B ( σ, v σ , r ) + w ,”

The cosmological time determines a curve γ_M: (0,∞) → T_g.

“Hence, the cosmological time determines a curve γ M : (0 , ∞ [ → T g .”

For every a in (0,∞), the surface S_a tilde is the graph of a positive convex function defined on the plane {x3=0} in M^{2+1}.

“1) ˜ S a is the graph of a positive convex function defined on the plane { x 3 = 0 } in M 2+1 .”

For each r > 0, ∆(r F) = r ∆(F), taking either the r-multiple of the measure or the r-multiple of the distance; ∆ respects the positive rays.

“For each r > 0 , ∆( r F ) = r ∆( F ; here we take either the r -multiple of the measure or the r -multiple of the distance. We can shortly say that ' ∆ respects the positive rays'.”

• cites [O]Not matched

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The orbit of D(M) in T_g is the union of two earthquake rays associated to M, pointing to the future, and to M^-, towards the past.

“The orbit of D ( M ) in T g is given by the union of two earthquake rays associated to M (pointing to the future) and to M -(towards the past); note that the qualitative behaviour is similar to what we have remarked for g = 1.”

The union of p_a(L_a tilde) over a covers the suspension F hat of F tilde.

“That is, the union of p a ( ˜ L a ) 's covers the suspension ̂ F of ˜ F .”

The (2+1)-dimensional Einstein equation with vanishing cosmological constant implies that M is Riemann flat.

“The (2+1)-dimensional Einstein equation with vanishing cosmological constant actually implies that M is (Riemann) flat.”

• cites [D-J-'t H]Not matched

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• cites [W]Not matched

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The initial singularity can be interpreted as an isometric action of the fundamental group of S on a suitable real tree.

“The initial singularity can be interpreted as an isometric action of the fundamental group of S on a suitable real tree.”

• Also stated in: § 1 Introduction. · p. 2 · plain

  “The initial singularity can be interpreted as the isometric action of the fundamental group of S on a suitable 'real tree'.”

The initial singularity admits two complementary descriptions: one in terms of real trees and the second in terms of geodesic laminations.

“Thus the initial singularity admits two complementary descriptions: one, in terms of real trees and, the second, in terms of geodesic laminations.”

The lack of smoothness of the CT level surfaces takes place on a geodesic lamination on S_a and is an observable large scale manifestation of the intrinsic geometry of the initial singularity.

“This lack of smoothness takes place on a 'geodesic lamination' on S a and is a observable large scale manifestation of the intrinsic geometry of the initial singularity.”

When the cosmological time is regular, the τ-constant level surfaces and their properties have a direct physical meaning as they are observables.

“Thus, when the cosmological time is regular, the τ -constant level surfaces and their properties have a direct physical meaning as they are observables.”

The initial singularity faithfully manifests itself as a lack of smoothness of the embedding of the CT level surfaces into the spacetime M.

“The initial singularity faithfully manifests itself as a lack of smoothness of the embedding of the CT level surfaces into the spacetime M .”

The past/future asymptotic states of the cosmological time recover and decouple the linear and translational parts of the ISO(2,1)-valued holonomy of the flat spacetime.

“In particular, the past/future asymptotic states of the cosmological time recover and decouple the linear and the translational parts of the ISO (2 , 1)-valued holonomy of the flat spacetime.”

• Also stated in: § 1 Introduction. · p. 2 · boosted

  “Remarkably, they recover and decouple the linear and the translational parts of the holonomy.”

Each element γ'∈Γ' is of the form γ' = γ + t(γ), where γ∈Γ and t(γ)∈R^3 is a translation; t:Γ→R^3 is a cocycle representing an element of H^1(Γ,R^3).

“Each element γ ′ ∈ Γ ′ is of the form γ ′ = γ + t ( γ ), where γ ∈ Γ and t ( γ ) ∈ R 3 is a translation. t : Γ → R 3 is a cocycle representing an element of H 1 (Γ , R 3 ).”

The cosmological time determines a real analytic curve in the Teichmüller space of Riemann surfaces of genus g, which connects an interior point (associated to the linear part of the holonomy) with a point on Thurston's natural boundary (associated to the initial singularity).

“The cosmological time determines a real analytic curve in the Teichm¨ uller space of Riemann surfaces of genus g , which connects an interior point (associated to the linear part of the holonomy) with a point on Thurston's natural boundary (associated to the initial singularity).”

Any maximal globally hyperbolic, future expanding flat spacetime M with compact space homeomorphic to S can be regarded as a deformation of some Minkowskian suspension.

“It has been shown in [Me] that any maximal globally hyperbolic, future expanding flat spacetime M with compact space homeomorphic to S , as above, can be regarded as a 'deformation' of some Minkowskian suspension (see also [W]).”

• cites [Me]Not matched

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• cites [W]Not matched

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When τ is regular, for each q∈N there exists a future-directed time-like unit speed geodesic ray γ_q : (0, τ(q)] → N such that its Lorentzian length equals τ(q).

“For each q ∈ N there exists a future-directed time-like unit speed geodesic ray γ q : (0 , τ ( q )] → N such that:”

• cites [A-G-H]Not matched

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The asymptotic states of the cosmological time are intrinsic features of the spacetime.

“On the other hand, we have seen that the asymptotic states of the cosmological time are intrinsic features of the spacetime.”

By [A-M-T], ρ is a global time function with image the interval (0,+∞).

“(1) ρ is a global time function with image the interval (0 , + ∞ ).”

• cites [A-M-T]Not matched

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Given any global time on a spacetime M=U(M)/Γ', the asymptotic behaviour of the geometry of the corresponding level surfaces reflects in general a property of the specific time and not of the spacetime.

“Given any global time on a spacetime M = U ( M ) / Γ ′ , the asymptotic behaviour of the geometry of the corresponding level surfaces reflects in general a property of the specific time and not of the spacetime.”

The natural action on T_g of the mapping class group Mod_g of S extends to an action on the compactification T_g bar.

“The natural action on T g of the mapping class group Mod g of S extends to an action on the compactification ¯ T g .”

• cites [T3]Not matched

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• cites [B-P]Not matched

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• cites [F-L-P]Not matched

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The R-trees and the associated π1(S)-actions which occur in proposition 4.4 are not arbitrary.

“The R -trees and the associated π 1 ( S )-actions which occur in proposition 4.4 are not arbitrary (see [O] pag. 32).”

• cites [O]Not matched

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The quotient space i(M) = T/π1(S) can be thought as the initial singularity of M.

“The quotient space i ( M ) = T /π 1 ( S ) can be thought as the initial singularity of M .”

The pair (T_g ∪ P+(Im(I)), P+(Im(I))) is homeomorphic to the pair (closed B^{6g-6}, S^{6g-7}).

“Proposition 4.22 The pair ( ¯ T g , ∂ ¯ T g ) = ( T g ∪ P + ( Im ( I )) , P + ( Im ( I ))) is homeomorphic to the pair ( ¯ B 6 g -6 , S 6 g -7 ) , where ¯ B 6 g -6 is the closed ball and S 6 g -7 is its boundary sphere.”

• cites [T3]Not matched

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• cites [B-P]Not matched

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• cites [F-L-P]Not matched

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The dual effect of the initial singularity on the embedding of S_a into M is, in principle, observable for any finite value a of the CT.

“Nevertheless this effect is, in principle, observable for any finite value a of the CT.”

By [A-M-T], if γ: (0,∞) → T_g is any ρ-orbit in T_g, then γ is proper, that is it goes out from any compact set of T_g.

“(ii) γ is proper, that is it goes out from any compact set of T g , roughly it 'goes to ∞ '.”

• cites [A-M-T]Not matched

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By [A-M-T], if γ: (0,∞) → T_g is any ρ-orbit in T_g, then lim_{ρ→0} γ exists in T_g.

“(i) The lim ρ → 0 γ exists in T g .”

• cites [A-M-T]Not matched

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