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

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Mathematical Theory Of Cosmological Redshift in Static Lobachevskian Universe.Mistake of Edwin Hubble.

J. Georg von Brzeski

Vadim von Brzeski


Port Angeles, Washington, USA, September 2008


  • 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

Scientifically required properties of a formula for cosmological redshift
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)

Key concepts
Key Concepts Cosmological Redshift

  • 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

Behavior of parallel geodesics in lobachevskian space
Behavior of Parallel Geodesics Cosmological Redshiftin Lobachevskian Space


Surfaces of

constant phase




to geodesics

Geodesics –



Volume ~ exp(R)



Foliation of L3 by horospherical waves Ω.

Illusion of space “expansion” in astronomy based on E3.

Parallel geodesics in euclidean space
Parallel Geodesics in Euclidean Space Cosmological Redshift


Surfaces of constant

phase (plane waves)

orthogonal to geodesics



Parallel Geodesics –

Equally spaced l

on entire Euclidean


Volume ~ Rn





Foliation of E3 by plane waves Ω.

Key theorem lobachevsky rate of divergence of geodesics in lg
Key Theorem (Lobachevsky): Cosmological RedshiftRate of divergence of geodesics in LG

Reference horosphere









Unit radius, R = 1.

Poincare ball model of LG.



Theorem gives a novel approach to measuring distance in space without

involving the notion of time. Clocks: NO, diffraction gratings: YES.

Mapping of distances in lobachevskian space into euclidean space
Mapping of Distances in Lobachevskian Space into Euclidean Space

  • d = tanh(δ)

    • d : Euclidean distance in E3

    • δ : Hyperbolic distance in L3

  • Similar to S2 E2 Mercator projection via tan() function

Formula for cosmological redshift distance measured by diffraction gratings

l Space





Formula for Cosmological RedshiftDistance Measured by Diffraction Gratings

  • From distance mapping and Lobachevsky’s theorem

δ =

d = tanh(δ)


  • We get the Formula for Cosmological Redshift


= tanh(δ) = tanh( ln ( ) ) = tanh( ln (1 + z) ), R = 1.



= tanh (ln (1 + z))

Arbitrary R.


Geodesics separated by λ0at source will be

separated by λ > λ0at detector.

Properties of our model
Properties of Our Model Space

  • 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

Relationship of our formula to actual hubble observations
Relationship of Our Formula to Actual Hubble Observations Space


  • Our formula :

  • Recalling that for x << 1

    ln (1 + x) ~ x

    tanh(x) ~ x

  • Thus :

= tanh (ln (1 + z))



= 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].

Graphical representation of tanh ln 1 z
Graphical Representation of tanh(ln(1+z) Space


tanh( ln(1+z) )

(von Brzeski et. al.)

Valid for all z,

0 ≤ z < ∞

z ≈ KD (Hubble observations)

Linear behavior, valid only for small z.

Test of our formula for redshift
Test of Our Formula for Redshift Space


= tanh (ln (1 + z))

  • 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:



β = = tanh (ln (1 + z))




Relativistic Doppler

effect as shown in

all references.

1 + β

1 + z =



1 - β

Hubble s mistake and it s legacy
Hubble’s Mistake and It’s Legacy Space

  • 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

Application of our model
Application of Our Model Space

  • 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

  • vrel = 0.81c

    • if QSO is located in the galaxy

Faint galaxy count
Faint Galaxy Count Space

  • Data shows that there are more faint galaxies than would follow from Euclidean universe

    • Euclidean volume ~ Rn

  • 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

Conclusions Space

  • 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 :


= tanh (ln (1 + z))


References Space

  • Bonola, R., Non-Euclidean Geometry, Dover,NY 1955. This book has an original paper by N.I. Lobachevsky

  • von Brzeski, J.G., von Brzeski,V., Topological Frequency Shifts, Electromagnetic Field in Lobachevskian Geometry, PIER 39,p.289, 2003.

  • von Brzeski,J.G., von Brzeski,V., Topological Intensity Shifts, Electromagnetic Field in Lobachevskian Geometry, PIER 43, p.161,2003.

  • 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.

  • 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.

  • Buseman,H., Kelly,P.J., Projective Geometry and Projective Metrics, Academic Press, NY, 1953.

  • 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.

  • Iversen, B., Hyperbolic Geometry, Cambridge Univ.Press, 1993.

  • 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.