1. 1 Mathematical Theory Of Cosmological Redshift in Static Lobachevskian Universe. Mistake of Edwin Hubble. J. Georg von Brzeski Vadim von Brzeski www.helioslabs.com CCC2 Port Angeles, Washington, USA, September 2008

2. 2 Goals Present a mathematical model of cosmological redshift in static space –Based on our previously published papers [2,3,4,5] Explain cosmological redshift as a physical realization of abstract Lobachevskian geometry [1,6,8] Present an alternative, logically and mathematically coherent, explanation to the “expansion” driven by the Big Bang Analyze Edwin Hubble’s mistake and its legacy

3. 3 Scientifically Required Properties of a Formula for Cosmological Redshift Explain existing observations & extend to new areas Expressed by conceptually coherent, clear & acceptable mathematical formula It should uniformly shift entire spectrum & preserve wavelength ratios It should also be –Scale invariant –Source independent –Linear fn of distance for “small” distances (already experimentally observed)

4. 4 Key Concepts Geodesics –Paths of shortest distances –In physics, commonly identified with rays of light Horospheres in Lobachevskian space. –Spheres of infinite radius (limit spheres) orthogonal to equivalence classes of geodesics having common point at infinity and tangent at that point to the boundary at infinity –Can be interpreted as surfaces of constant phase of EM wave (wavefronts) Mapping of hyperbolic distances onto Euclidean distances

5. 5 Behavior of Parallel Geodesics in Lobachevskian Space Horospheres: Surfaces of constant phase (horespherical wavefronts); orthogonal to geodesics Geodesics – Diverge Exponentially. Volume ~ exp(R) Foliation of L 3 by horospherical waves Ω. Illusion of space “expansion” in astronomy based on E 3. l0l0 l

6. 6 Parallel Geodesics in Euclidean Space Horospheres: Surfaces of constant phase (plane waves) orthogonal to geodesics Parallel Geodesics – Equally spaced l on entire Euclidean space. Volume ~ R n l l l l d d Foliation of E 3 by plane waves Ω.

7. 7 Key Theorem (Lobachevsky): Rate of divergence of geodesics in LG l l0l0 exp(δ) = l0l0 l Theorem gives a novel approach to measuring distance in space without involving the notion of time. Clocks: NO, diffraction gratings: YES. Reference horosphere Parallel geodesics Parallel horospheres Unit radius, R = 1. Poincare ball model of LG.

8. 8 Mapping of Distances in Lobachevskian Space into Euclidean Space d = tanh(δ) –d : Euclidean distance in E 3 –δ : Hyperbolic distance in L 3 Similar to S 2  E 2 Mercator projection via tan() function

9. 9 Formula for Cosmological Redshift Distance Measured by Diffraction Gratings From distance mapping and Lobachevsky’s theorem ln l l0l0 δ = d = tanh(δ) We get the Formula for Cosmological Redshift = tanh(δ) = tanh( ln ( ) ) = tanh( ln (1 + z) ), R = 1. λ λ0λ0 Geodesics separated by λ 0 at source will be separated by λ > λ 0 at detector. and d 1 d R = tanh (ln (1 + z)) Arbitrary R.

10. 10 Properties of Our Model Physical realization of geometrical theorem of abstract LG Uniformly shifts entire spectrum –Preserves wavelength ratios Scale invariant Monotonically increasing fn of distance Linear fn of distance for “small” distances Source independent Easy to compute

11. 11 Relationship of Our Formula to Actual Hubble Observations Our formula : Recalling that for x << 1 ln (1 + x) ~ x tanh(x) ~ x Thus : d R = tanh (ln (1 + z)) d R = z or z = Kd, where K = 1/R This is exactly what Hubble found - redshift is a linear function of distance. Hubble experimentally discovered evidence for Lobachevskian geometry of the Universe and failed to recognize properly what he observed [7].

12. 12 Graphical Representation of tanh(ln(1+z) ln(1+z) tanh( ln(1+z) ) z ≈ KD (Hubble observations) Linear behavior, valid only for small z. (von Brzeski et. al.) Valid for all z, 0 ≤ z < ∞

13. 13 Test of Our Formula for Redshift Our formula: Represent LG by velocity space, i.e. (signed) distance means relative velocity [2,9] Thus, d  v, R  c From the definition of tanh(x), we get: d R = tanh (ln (1 + z)) v c β = = tanh (ln (1 + z)) 1 + β 1 - β 1/2 = λ λ0λ0 1 + z = Relativistic Doppler effect as shown in all references.

14. 14 Hubble’s Mistake and It’s Legacy Hubble measured redshift z and distance d to some objects He found experimentally z = Kd, linear He erroneously assumed z = Cv : the only cause of redshift was the linear Doppler effect Thus, he equated RHS of the above and obtained relationship: v = Hd, called in all literature the Hubble velocity distance “law” But v = Hd has no experimental basis! Slope, H, called the Hubble constant (parameter), is not a physical quantity –Hubble time, Hubble flow as well

15. 15 Application of Our Model NGC 4319 controversy with binary system –Difference in redshift for 2 component spatially localized system z1 = 0.0225 for NGC 4319 z2 = 2.1100 for QSO If we assume NGC 4319 as a reference, and it’s redshift is due only to distance, then Δz = 2.0875 is due to relative velocity v rel = 0.81c –if QSO is located in the galaxy

16. 16 Faint Galaxy Count Data shows that there are more faint galaxies than would follow from Euclidean universe –Euclidean volume ~ R n Natural explanation of faint galaxy count in Lobachevskian universe –Lobachevskian volume ~ exp(R) From the count of faint galaxies in Lobachevskian universe it might be possible to recover distances to them

17. 17 Conclusions Negative curvature of space causes an illusion of the existence of a global velocity field Illusion was interpreted by Hubble and followers as the effect of “space inflation”, which extrapolated backwards led to a singularity mockingly named by F. Hoyle as the Big Bang Observed cosmological redshift, which increases monotonically with distance, is due to Lobachevskian large scale vacuum given by : d R = tanh (ln (1 + z))

18. 18 References 1.Bonola, R., Non-Euclidean Geometry, Dover,NY 1955. This book has an original paper by N.I. Lobachevsky 2.von Brzeski, J.G., von Brzeski,V., Topological Frequency Shifts, Electromagnetic Field in Lobachevskian Geometry, PIER 39,p.289, 2003. 3.von Brzeski,J.G., von Brzeski,V., Topological Intensity Shifts, Electromagnetic Field in Lobachevskian Geometry, PIER 43, p.161,2003. 4.von Brzeski, J.G., Application of Lobachevsky’s Formula on the Angle of Parallelism to Geometry of Space and to the Cosmological Redshift, Russian Journal of Mathematical Physics, 14,p.366, 2007. 5.von Brzeski,J.G., Expansion of the Universe-Mistake of Edwin Hubble? Cosmological Redshift and Related Electromagnetic Phenomena in Static Lobachevskian (Hyperbolic) Universe, Acta Physica Polonica, 39, No.6, p.1501, 2007. 6.Buseman,H., Kelly,P.J., Projective Geometry and Projective Metrics, Academic Press, NY, 1953. 7.Hubble,E., A Relation Between Distance and Radial Velocity Among Extra Galactic Nebulae, Proc.of National Academy of Sciences, vol.15,No 3, March15, 1929. 8.Iversen, B., Hyperbolic Geometry, Cambridge Univ.Press, 1993. 9.Smorodinsky, Ya. A., Kinematika i Geomietriya Lobachevskogo, ( Kinematics and Lobachevskian Geometry) in Russian, Atomnaya Energiya 1956, Available from Joint Institute for Nuclear Research Library, Dubna, Russian Ferderation.