Conformal Transformations and Hubble Parameter

  1. Here is a simple demonstration that the Hubble effect is compatible
    with both Hubble's law and the large value of the Hubble time.

    A special conformal transformation is

    Xm' = 1/S (Xm + XnXn Am)

    where Am is a four-vector and

    S = (1 + 2 Am Xm + AnAn XmXm)

    If we assume Am is timelike and go to its rest frame where Am =
    [0,0,0,a] then

    S = (1 + 2 at + (t^2 - r^2) a^2) = (1 + at)^2 - (ar)^2

    It turns out that in general

    dXm' dXm' = 1/S^2 dXm dXm

    This becomes singular when S=0. That happens when

    t = r - 1/a

    The implication is that a is very small and this t represents the
    Hubble time.

    For small a,

    dXi' = 1/S^2 ( dXi + 2a (t dxi - xi dt) )
    dt' = 1/S^2 ( dt + 2a (t dt - xi dxi) )

    so

    v' = [ v' + 2a (t v - x) ] / [ 1 + 2a(t - x.v) ] = [ v' + 2a
    (t v - x) ] * [ 1 - 2a(t - x.v) ]

    as long as t - x.v does not become large. To first order

    v' = v - 2a x + 2a (x.v) v

    = v - 2a (1-v^2) x - 2a v ^ (x ^ v)

    If v << 1 then simply

    v' = v - 2ax

    that is, there is an apparent velocity proportional to the distance.
    An object at rest in the distant frame, v' = 0, appears to us to have
    an apparent velocity radially outward

    v = 2ax

    which is just Hubble's law.

    -drl
     
  2. jcsd
  3. On May 30, 12:31 pm, DRLunsford <antimatte...@yahoo.com> wrote:
    > Here is a simple demonstration that the Hubble effect is compatible
    > with both Hubble's law and the large value of the Hubble time.
    >
    > A special conformal transformation is
    >
    > Xm' = 1/S (Xm + XnXn Am)
    >
    > where Am is a four-vector and
    >
    > S = (1 + 2 Am Xm + AnAn XmXm)
    >
    > If we assume Am is timelike and go to its rest frame where Am =
    > [0,0,0,a] then
    >
    > S = (1 + 2 at + (t^2 - r^2) a^2) = (1 + at)^2 - (ar)^2
    >
    > It turns out that in general
    >
    > dXm' dXm' = 1/S^2 dXm dXm
    >
    > This becomes singular when S=0. That happens when
    >
    > t = r - 1/a
    >
    > The implication is that a is very small and this t represents the
    > Hubble time.
    >
    > For small a,
    >
    > dXi' = 1/S^2 ( dXi + 2a (t dxi - xi dt) )
    > dt' = 1/S^2 ( dt + 2a (t dt - xi dxi) )
    >
    > so
    >
    > v' = [ v' + 2a (t v - x) ] / [ 1 + 2a(t - x.v) ] = [ v' + 2a
    > (t v - x) ] * [ 1 - 2a(t - x.v) ]
    >
    > as long as t - x.v does not become large. To first order
    >
    > v' = v - 2a x + 2a (x.v) v
    >
    > = v - 2a (1-v^2) x - 2a v ^ (x ^ v)
    >
    > If v << 1 then simply
    >
    > v' = v - 2ax
    >
    > that is, there is an apparent velocity proportional to the distance.
    > An object at rest in the distant frame, v' = 0, appears to us to have
    > an apparent velocity radially outward
    >
    > v = 2ax
    >
    > which is just Hubble's law.
    >
    > -drl


    Sorry this was worded strangely. I meant was that conformal geometry
    over spacetime was compatible both with Hubble's law and the large
    value of the Hubble time. These ideas have direct application to the
    Pioneer anomaly.

    -drl
     
  4. On 2007-05-31, DRLunsford <antimatter33@yahoo.com> wrote:
    > On May 30, 12:31 pm, DRLunsford <antimatte...@yahoo.com> wrote:
    >> Here is a simple demonstration that the Hubble effect is compatible
    >> with both Hubble's law and the large value of the Hubble time.
    >> [...]

    > Sorry this was worded strangely. I meant was that conformal geometry
    > over spacetime was compatible both with Hubble's law and the large
    > value of the Hubble time. These ideas have direct application to the
    > Pioneer anomaly.


    You might be interested in the following papers. They respectively rule
    out FRW expansion and the cosmological constant as potential sources of
    the Pioneer anomaly.

    Marc Lachieze-Rey
    Cosmology in the Solar System: Pioneer effect is not cosmological
    http://arxiv.org/abs/gr-qc/0701021

    Valeria Kagramanova, Jutta Kunz, Claus L=C3=A4mmerzahl
    Solar system effects in Schwarzschild--de Sitter spacetime
    http://arxiv.org/abs/gr-qc/0602002

    Igor
    =2E
     
  5. On Jun 3, 2:16 pm, Igor Khavkine <igor...@gmail.com> wrote:
    > On 2007-05-31, DRLunsford <antimatte...@yahoo.com> wrote:
    >
    > > On May 30, 12:31 pm, DRLunsford <antimatte...@yahoo.com> wrote:
    > >> Here is a simple demonstration that the Hubble effect is compatible
    > >> with both Hubble's law and the large value of the Hubble time.
    > >> [...]

    > > Sorry this was worded strangely. I meant was that conformal geometry
    > > over spacetime was compatible both with Hubble's law and the large
    > > value of the Hubble time. These ideas have direct application to the
    > > Pioneer anomaly.

    >
    > You might be interested in the following papers. They respectively rule
    > out FRW expansion and the cosmological constant as potential sources of
    > the Pioneer anomaly.
    >
    > Marc Lachieze-Rey
    > Cosmology in the Solar System: Pioneer effect is not cosmologicalhttp://arxiv.org/abs/gr-qc/0701021
    >
    > Valeria Kagramanova, Jutta Kunz, Claus L=C3=A4mmerzahl
    > Solar system effects in Schwarzschild--de Sitter spacetimehttp://arxiv.org/abs/gr-qc/0602002
    >
    > Igor
    > =2E


    Thanks for that.

    Here is the exact statement of the "conformal Hubble effect" as I'm
    calling it. I will just state the results - I am going to write up a
    short paper on this fascinating subject soon with the details.

    Again, the SCTs are of the form

    X'm = 1/S ( Xm + XaXa Am) where S = ( 1 + 2AmXm + XaXa AbAb)

    for a given four-vector Am. We assume that Am is time-like and go to
    its rest frame, so it takes the form (0,0,0,a). We calculate dx' and
    dt', divide them, and get the following formula for the transformation
    of the velocity under SCT:

    Let

    V = (Vx,Vy,Vz)
    R = (x,y,z)
    r = |R|
    s = t + 1/a
    L = 2(s - V.R)/(s^2 - r^2)

    Then

    V' = (V - LR) / (Ls - 1)

    and this is exact, not approximate. So without any approximation we
    get a modified Hubble effect - if we assume the distant object is
    stationary in its frame of reference, that is, V' = 0, then

    V = LR

    Now since V is proportional to R, the magnitude of V is (V.R)/r.
    Dotting both sides with R we get

    V.R = L r^2

    Solving for V.R we get

    (V.R) = 2 s r^2 / (s^2 + r^2)

    so

    |V| = 2 s r / ( s^2 + r^2 )

    A beautiful formula! Expressing it in terms of a and t rather than the
    "translated time" s, we have

    |V| = 2 a r (1 + at) / (1 + 2at + a^2(t^2 - r^2))

    For a << 1 this reduces to the previous result

    |V| = 2 a r

    There are some very strange interpretive issues here - the distant
    object is fixed in its frame, but appears to have a non-zero speed to
    us! Does r therefore change? No! We must constantly re-scale the units
    so that r remains fixed in order to have a consistent interpretation.
    Therefore Hubble's law is NOT dynamic evidence of an original cosmic
    process, but simple kinematical evidence of a geometry in which local
    length is not fixed. This rescaling can already be seen inside the
    solar system, and can be used to account for the Pioneer anomaly -
    something FRW cosmology and a non-zero CC cannot do!
     
  6. sr

    sr 0

    DRLunsford wrote:

    > Here is the exact statement of the "conformal Hubble effect"
    > as I'm calling it. I will just state the results - I am going
    > to write up a short paper on this fascinating subject soon with
    > the details.


    I'm puzzled about some aspects of what you've written in this
    thread. Maybe it will be clearer in your more expansive paper,
    so for now I'll just ask some dumb questions and hope I don't
    try your patience too much...


    > Again, the SCTs are of the form
    >
    > X'm = 1/S ( Xm + XaXa Am) where S = ( 1 + 2AmXm + XaXa AbAb)
    >
    > for a given four-vector Am. We assume that Am is time-like and go
    > to its rest frame, so it takes the form (0,0,0,a).


    OK... so... you're effectively starting from some frame, choosing
    a time-like vector Am=(0,0,0,a) and performing the SCT.

    > We calculate dx'and dt', divide them, and get the following
    > formula for the transformation of the velocity under SCT:


    SCTs correspond to uniform accelerations, right? So you're
    transforming to a uniformly-accelerated frame?

    > Let
    >
    > V = (Vx,Vy,Vz)
    > R = (x,y,z)
    > r = |R|
    > s = t + 1/a
    > L = 2(s - V.R)/(s^2 - r^2)
    >
    > Then
    >
    > V' = (V - LR) / (Ls - 1)
    >
    > and this is exact, not approximate. So without any approximation we
    > get a modified Hubble effect - if we assume the distant object is
    > stationary in its frame of reference, that is, V' = 0, [...]


    So... the distant object is non-accelerating, and the dashed frame
    corresponds to that inertial frame where the distant object is
    at rest (right?)

    If that's correct, it must mean the original undashed frame
    is accelerating - since the two frames are related by an SCT.(?)


    > then
    >
    > V = LR
    >
    > Now since V is proportional to R, [...]


    Sorry, I failed to follow that step from what you've written.
    Your L above appears to contain an "R", so V=LR isn't simply
    proportional to R. What am I missing?

    > [...]
    >
    > |V| = 2 s r / ( s^2 + r^2 )
    >
    > A beautiful formula! Expressing it in terms of a and t
    > rather than the "translated time" s, we have
    >
    > |V| = 2 a r (1 + at) / (1 + 2at + a^2(t^2 - r^2))
    >
    > For a << 1 this reduces to the previous result
    >
    > |V| = 2 a r
    >
    > There are some very strange interpretive issues here - the
    > distant object is fixed in its frame, but appears to have a
    > non-zero speed to us! Does r therefore change? No! We must
    > constantly re-scale the units so that r remains fixed in order
    > to have a consistent interpretation. [...]


    But since you're using an SCT between the two frames, doesn't
    that mean one frame or the other is in uniform acceleration?
    (If so, I'm not sure what's been gained here.)

    (OK, that's enough dumb questions for now. :-)

    - strangerep.
     
  7. On 3 juin, 19:16, Igor Khavkine <igor...@gmail.com> wrote:
    > On 2007-05-31, DRLunsford <antimatte...@yahoo.com> wrote:


    >
    > You might be interested in the following papers. They respectively rule
    > out FRW expansion and the cosmological constant as potential sources of
    > the Pioneer anomaly.
    >
    > Marc Lachieze-Rey
    > Cosmology in the Solar System: Pioneer effect is not cosmologicalhttp://arxiv.org/abs/gr-qc/0701021
    >
    > Valeria Kagramanova, Jutta Kunz, Claus L=C3=A4mmerzahl
    > Solar system effects in Schwarzschild--de Sitter spacetimehttp://arxiv.org/abs/gr-qc/0602002
    >
    > Igor
    > =2E

    +++
    I am surprised that in these papers, dealing with a Schwarzschild
    solution with a non vanishing cosmological constant, there no
    references to the Lemaitre 1933 paper "l"expansion de
    L'univers" (translated in 1987 "The expanding universe" in GRG).

    In this article, he derived in chapter 11 (directly from the Einstein
    equation, using a generic spherically symmetric metric ) an analytic
    solution with cosmological constant for both Schwarzschild and
    Painleve forms and derived the analytic radial geodesic equation (with
    cosmological constant):

    r = 2B [Sh ((3A/2 )(t- R))]^2/3

    A^2 = Lambda/3
    B^3= Gm/(4A^2)

    R is the radial coordinate attached to the radially free falling
    observer.

    Let's notice the parity (in power 2/3) of the function describing both
    incoming (black hole) and outgoing geodesics (the white hole).
    He did not noticed explicitly the last point (Finkelstein will
    rediscover it 25 years later!)

    In fact the the main purpose of this chapter was to demonstrate that
    there was no singularity on the horizon of the Schwarzschild space
    time and to derive the "Lemaitre form" of the Schwarzschild solution.

    As a "byproduct" he derived some forms of the metric and the geodesic
    radial equation with cosmological constant.!

    Jacques
     
  8. [SOLVED] Conformal Transformations and Hubble Parameter

    On 3 juin, 19:16, Igor Khavkine <igor...@gmail.com> wrote:
    > On 2007-05-31, DRLunsford <antimatte...@yahoo.com> wrote:


    >
    > You might be interested in the following papers. They respectively rule
    > out FRW expansion and the cosmological constant as potential sources of
    > the Pioneer anomaly.
    >
    > Marc Lachieze-Rey
    > Cosmology in the Solar System: Pioneer effect is not cosmologicalhttp://arxiv.org/abs/gr-qc/0701021
    >
    > Valeria Kagramanova, Jutta Kunz, Claus L=C3=A4mmerzahl
    > Solar system effects in Schwarzschild--de Sitter spacetimehttp://arxiv.org/abs/gr-qc/0602002
    >
    > Igor
    > =2E

    +++
    I am surprised that in these papers, dealing with a Schwarzschild
    solution with a non vanishing cosmological constant, there no
    references to the Lemaitre 1933 paper "l"expansion de
    L'univers" (translated in 1987 "The expanding universe" in GRG).

    In this article, he derived in chapter 11 (directly from the Einstein
    equation, using a generic spherically symmetric metric ) an analytic
    solution with cosmological constant for both Schwarzschild and
    Painleve forms and derived the analytic radial geodesic equation (with
    cosmological constant):

    r = 2B [Sh ((3A/2 )(t- R))]^2/3

    A^2 = Lambda/3
    B^3= Gm/(4A^2)

    R is the radial coordinate attached to the radially free falling
    observer.

    Let's notice the parity (in power 2/3) of the function describing both
    incoming (black hole) and outgoing geodesics (the white hole).
    He did not noticed explicitly the last point (Finkelstein will
    rediscover it 25 years later!)

    In fact the the main purpose of this chapter was to demonstrate that
    there was no singularity on the horizon of the Schwarzschild space
    time and to derive the "Lemaitre form" of the Schwarzschild solution.

    As a "byproduct" he derived some forms of the metric and the geodesic
    radial equation with cosmological constant.!

    Jacques
     
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