Tomaschitz, R. (1996). Tachyonic chaos and causality in the open universe, Chaos, Solitons & Fractals 7, 753-768, DOI: 10.1016/0960-0779(95)00106-9

 

 

Abstract (ScienceDirect, CDS, SAO/NASA ADS)

The chaoticity of classical world lines in extended Robertson-Walker cosmologies is pointed out and related to the fractal limit set of the covering group of the spacelike slices. We investigate the possible existence of tachyons (faster-than-light particles) in this context. The cosmic time and the 3-space co-ordinates comoving with the galactic background provide a distinguished frame of reference. With respect to this comoving frame the causality of tachyonic events can be unambiguously defined, contrary to Minkowski space. We study the dynamics of tachyons and show that tachyonic world-lines are chaotic in the center of the spacelike slices.

 

 

 

Title:

 

Tachyonic chaos and causality in the open universe.

Authors:

 

Tomaschitz, R.

Publication:

 

Chaos Solitons Fractals, Vol. 7, No. 5, p. 753 - 768

Publication Date:

 

05/1996

Origin:

 

ARI

ARI Keywords:

 

Cosmology: Robertson-Walker Metric, Cosmology: Elementary Particles, Cosmology: Chaos

Bibliographic Code:

 

1996CSF.....7..753T

 

 

 

Fig. 2a. The horizon at infinity of the Poincaré half-space H3. A spacelike slice (F,Γ) is realized in H3 as a polyhedron F with face-identification. The face-pairing transformations generate a discrete group Γ which gives, if applied to the polyhedron, a tessellation Γ(F) of H3 with polyhedral images. This tessellation induces by continuity a tiling on the boundary of H3 which is depicted here. The qualitative structure of the fractal limit set Λ(Γ) depends on the topology of the 3-slices, which is in turn determined by the covering group. For quasi-Fuchsian groups [26] like here, the limit set is a Jordan curve (not self-similar). In all three examples, the 3-space fibers over an open interval, with Riemann surfaces (g = 19) as fibers. The depicted tilings correspond to 3-slices which are globally non-isometric, but have the same topology and curvature. They make deformations as schematized in Fig. 1(a–c) quantitative. For example, the polyhedral tiling of H3 is obtained by placing hemispheres onto the circular arcs. The colors label generations in the tiling procedure [34]. The chaotic trajectories in the 3-space have covering trajectories with initial and terminal points in Λ(Γ). If the end points are not in Λ(Γ) but close to it, then the trajectory is regular, but it can shadow a chaotic trajectory over a long time. The convex hull C(Λ) of Λ(Γ) is the intersection of all hyperbolic half-spaces which contain Λ(Γ). Projected into (F,Γ), it constitutes the center C(Λ)\Γ of the 3-space, see Ref. [29] for an explicit example of this projection.

Fig. 2a  full size image

 

 

 

Fig. 2b. The horizon at infinity of the Poincaré half-space H3. A spacelike slice (F,Γ) is realized in H3 as a polyhedron F with face-identification. The face-pairing transformations generate a discrete group Γ which gives, if applied to the polyhedron, a tessellation Γ(F) of H3 with polyhedral images. This tessellation induces by continuity a tiling on the boundary of H3 which is depicted here. The qualitative structure of the fractal limit set Λ(Γ) depends on the topology of the 3-slices, which is in turn determined by the covering group. For quasi-Fuchsian groups [26] like here, the limit set is a Jordan curve (not self-similar). In all three examples, the 3-space fibers over an open interval, with Riemann surfaces (g = 19) as fibers. The depicted tilings correspond to 3-slices which are globally non-isometric, but have the same topology and curvature. They make deformations as schematized in Fig. 1(a–c) quantitative. For example, the polyhedral tiling of H3 is obtained by placing hemispheres onto the circular arcs. The colors label generations in the tiling procedure [34]. The chaotic trajectories in the 3-space have covering trajectories with initial and terminal points in Λ(Γ). If the end points are not in Λ(Γ) but close to it, then the trajectory is regular, but it can shadow a chaotic trajectory over a long time. The convex hull C(Λ) of Λ(Γ) is the intersection of all hyperbolic half-spaces which contain Λ(Γ). Projected into (F,Γ), it constitutes the center C(Λ)\Γ of the 3-space, see Ref. [29] for an explicit example of this projection.

Fig. 2b  full size image

 

 

 

Fig. 2c. The horizon at infinity of the Poincaré half-space H3. A spacelike slice (F,Γ) is realized in H3 as a polyhedron F with face-identification. The face-pairing transformations generate a discrete group Γ which gives, if applied to the polyhedron, a tessellation Γ(F) of H3 with polyhedral images. This tessellation induces by continuity a tiling on the boundary of H3 which is depicted here. The qualitative structure of the fractal limit set Λ(Γ) depends on the topology of the 3-slices, which is in turn determined by the covering group. For quasi-Fuchsian groups [26] like here, the limit set is a Jordan curve (not self-similar). In all three examples, the 3-space fibers over an open interval, with Riemann surfaces (g = 19) as fibers. The depicted tilings correspond to 3-slices which are globally non-isometric, but have the same topology and curvature. They make deformations as schematized in Fig. 1(a–c) quantitative. For example, the polyhedral tiling of H3 is obtained by placing hemispheres onto the circular arcs. The colors label generations in the tiling procedure [34]. The chaotic trajectories in the 3-space have covering trajectories with initial and terminal points in Λ(Γ). If the end points are not in Λ(Γ) but close to it, then the trajectory is regular, but it can shadow a chaotic trajectory over a long time. The convex hull C(Λ) of Λ(Γ) is the intersection of all hyperbolic half-spaces which contain Λ(Γ). Projected into (F,Γ), it constitutes the center C(Λ)\Γ of the 3-space, see Ref. [29] for an explicit example of this projection.

Fig. 2c full size image

 

Fig. 2. (a–c) The horizon at infinity of the Poincaré half-space H3. A spacelike slice (F,Γ) is realized in H3 as a polyhedron F with face-identification. The face-pairing transformations generate a discrete group Γ which gives, if applied to the polyhedron, a tessellation Γ(F) of H3 with polyhedral images. This tessellation induces by continuity a tiling on the boundary of H3 which is depicted here. The qualitative structure of the fractal limit set Λ(Γ) depends on the topology of the 3-slices, which is in turn determined by the covering group. For quasi-Fuchsian groups [26] like here, the limit set is a Jordan curve (not self-similar). In all three examples, the 3-space fibers over an open interval, with Riemann surfaces (g = 19) as fibers. The depicted tilings correspond to 3-slices which are globally non-isometric, but have the same topology and curvature. They make deformations as schematized in Fig. 1(a–c) quantitative. For example, the polyhedral tiling of H3 is obtained by placing hemispheres onto the circular arcs. The colors label generations in the tiling procedure [34]. The chaotic trajectories in the 3-space have covering trajectories with initial and terminal points in Λ(Γ). If the end points are not in Λ(Γ) but close to it, then the trajectory is regular, but it can shadow a chaotic trajectory over a long time. The convex hull C(Λ) of Λ(Γ) is the intersection of all hyperbolic half-spaces which contain Λ(Γ). Projected into (F,Γ), it constitutes the center C(Λ)\Γ of the 3-space, see Ref. [29] for an explicit example of this projection.

 

 

description: Roman Tomaschitz (1996) Tachyonic chaos and causality in the open universe, Chaos, Solitons & Fractals 7, 753.

 

Keywords: open Robertson–Walker cosmology with multiply connected hyperbolic 3-space, faster-than-light particles, tachyons, superluminal signal transfer and causality in an absolute spacetime conception, comoving galaxy frame, absolute frame of reference, absolute space and cosmic time, cosmic time order, time inversion in inertial frames, positivity of tachyonic energy, chaoticity of tachyonic world lines, Bernoulli property, chaotic nucleus of the open 3-space, universal covering space of the multiply connected spacelike slices, Poincaré half-space, polyhedral tiling of hyperbolic space, fractal limit sets of Kleinian covering groups, quasi-Fuchsian groups, fibered hyperbolic 3-manifolds, hyperbolic convex hull of a fractal limit set, covering projection, mixing and shadowing in the chaotic center of the universe, cosmic evolution by global metrical deformations, adiabatic deformations of the open 3-space, topological evolution and topology change

 

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