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
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
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
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!
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.
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
[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