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Conformal Transformations and Hubble Parameter

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DRLunsford
#1
May31-07, 05:00 AM
P: n/a
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

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DRLunsford
#2
Jun1-07, 05:00 AM
P: n/a
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

Igor Khavkine
#3
Jun4-07, 05:00 AM
P: n/a
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


DRLunsford
#4
Jun6-07, 05:00 AM
P: n/a
Conformal Transformations and Hubble Parameter

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!

sr
#5
Jun7-07, 05:00 AM
P: n/a
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.


jacques
#6
Jun8-07, 05:01 AM
P: n/a
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






jacques
#7
Jun8-07, 05:01 AM
P: n/a
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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