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J. Georg von Brzeski Vadim von Brzeski helioslabs CCC2

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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

- 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

- 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)

- 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

Horospheres:

Surfaces of

constant phase

(horespherical

wavefronts);

orthogonal

to geodesics

Geodesics –

Diverge

Exponentially.

Volume ~ exp(R)

l0

l

Foliation of L3 by horospherical waves Ω.

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

Horospheres:

Surfaces of constant

phase (plane waves)

orthogonal to geodesics

d

d

Parallel Geodesics –

Equally spaced l

on entire Euclidean

space.

Volume ~ Rn

l

l

l

l

Foliation of E3 by plane waves Ω.

Reference horosphere

Parallel

horospheres

l

=

exp(δ)

l0

l0

l

Unit radius, R = 1.

Poincare ball model of LG.

Parallel

geodesics

Theorem gives a novel approach to measuring distance in space without

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

- d = tanh(δ)
- d : Euclidean distance in E3
- δ : Hyperbolic distance in L3

- Similar to S2 E2 Mercator projection via tan() function

l

ln

l0

λ

λ0

- From distance mapping and Lobachevsky’s theorem

δ =

d = tanh(δ)

and

- We get the Formula for Cosmological Redshift

d

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

1

d

= tanh (ln (1 + z))

Arbitrary R.

R

Geodesics separated by λ0at source will be

separated by λ > λ0at detector.

- 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

d

- Our formula :
- Recalling that for x << 1
ln (1 + x) ~ x

tanh(x) ~ x

- Thus :

= tanh (ln (1 + z))

R

d

= z

or z = Kd, where K = 1/R

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].

ln(1+z)

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.

d

= 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:

R

v

β = = tanh (ln (1 + z))

c

1/2

λ

Relativistic Doppler

effect as shown in

all references.

1 + β

1 + z =

=

λ0

1 - β

- 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

- 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

- 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

- 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

= tanh (ln (1 + z))

R

- 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.