The Hubble law beyond the General Relativity

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The Hubble law beyond the General Relativity

• L.M.Tomilchik
• B.I.Stepanov Institute of Physics of NAS of
  Belarus
• N.G.Kembrovskaya
• Belarusian State University, Minsk
• Gomel, July 2009

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Gedanken experiment in the history of Physics

• The experimental discovery of the Universe
  expansion was happened not in 1929 but in
  1910-1911 (before the creation of GR)
• The expansion law
• R(t) is the time-dependent distance between the
  light signal source and detector.
• is their relative radial velocity

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Creation of the Special Relativity

• (A) Einstein
• Relativity Principle
• Speed of light independence on the source
  velocity
• Starting point location clock synchronization
  procedure (gedanken experiment) for derivation of
  the Lorentz transformation
• (B) Poincare
• Relativity Principle
• Group of the isometric symmetry transformations
  of the four-dimensional pseudo Euclidian
  space-time (Lorentz group). Consistently rigorous
  algebraic consideration

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Einstein and Poincare response (most likely
version)

• Einstein modified his initial synchronization
  procedure for the pair of mutually motionless
  clocks.
• Poincare extended the group of the space-time
  transformations taking into account the
  nonisometric (changing the scales) ones i.e.,
  the group of the special conformal
  transformations.

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Location clock synchronization procedure
A
B
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Location clock synchronization procedure I
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Location clock synchronization procedure II
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Location clock synchronization procedure in the
expanding space
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Two possibilities
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Version (I)
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Comments for abovementioned

• The situation looks as existence of the uniform
  acceleration of the clock B with
  respect to clock A directed to the observation
  point.

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The outcomes

• It is noninertiality FR effect
• It could be treated as an existence of the
  homogeneous gravitational field (equivalence
  principle)
• The Universe expansion leads to the (local)
  quadratic time inhomogeneity and (possibly)
  conformal time inhomogeneity

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The outcomes (continue)

• 4. The experimental prediction there exist the
  universal uniform blue-shifted Doppler drift in
  the location-type experiments.

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Linear transformation of light cone generatrixs
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Longitudinal Doppler effect
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Longitudinal Doppler effect (continue)
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Nonisometric Subgroup of SO(4,2) (besides
Poincare group)
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Standard interpretation of SCT transformations
from the comoving inertial RF to the uniformly
accelerated RF
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Conformal deformationof the light cone
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The dependence of distance on red shift I
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The dependence of distance on red shift II
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The Hubble law I
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The Hubble law II

• The function f(z) The zero position
  of f(z)

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Accelerating Universe(Riess et al., 1998
Perlmutter et al. 1999)

• The observed phenomenon
• Deflection from linearity of the conventional
  Hubble law VH0R with maximum at zexp0.460.13
• The interpretation
• zexp is the point of transition from the Universe
  expansion deceleration to acceleration. Effect
  follows from the existence of the Dark Energy,
  amount of which is 70 of the full gravitating
  energy in the Metagalaxy.

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Location-type experiment
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Mapping of the world lines
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Observable consequences

• Under condition we have
• time
  distance
• the forward
• the backward
• Acceleration
• Frequency

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Pioneer Anomaly (PA) (1998)

• The observable phenomenon
• Unexpected uniform blue-shifted drift of the
  frequency ?obs
• The interpretation
• An anomalous sunward directed acceleration
• ap (8,74?1,33)?10-10 m/s2

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Conclusion I

• The conformal transformation of time
• leads to the following consequences
• The cosmological location distance can be
  determined as an explicit function of red shift
  R(z).
• The combination of this function with the SR
  expression for the longitudinal Doppler-effect
  V(z) gives the explicit analytic expression for
  the ratio V(z) / H0 R(z).
• The ratio coincides with the Hubble law in the
  limit of ,and possesses a maximum at
• The appearance of this maximum is a pure
  kinematic manifestation of the time inhomogeneity
  and does not need any special gravitational
  sources (like the dark energy).

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Conclusion II

• 2. In the location-type experiments uniform
  blue-shifted frequency drift appears, which
  mimics constant acceleration towards the observer
  
• The Pioneer Anomaly is the first really observed
  effect of that kind. The observed drift can be
  used for local experimental determination of
  Hubble constant.
• This effect can be observed in principle at any
  frequency even between mutually moveless emitter
  and receiver in the absence of any gravitating
  centers.

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Thank You for Your attention.