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Missing Mass, Galaxy Ageing, Supernova Redshift, MOND and Pioneer
I might describe this post as a pre-preprint. I have not yet sent it
anywhere, and would like to get comments and criticisms before I do. Title Does a Teleconnection between Quantum States account for Missing Mass, Galaxy Ageing, Supernova Redshift, MOND and Pioneer? Author Charles Francis MA (Cantab) PhD (Lond) Abstract: There have been previous suggestions, notably by Einstein, that the affine connection in general relativity might be replaced with a teleparallel one. This paper investigates the empirical implications of using a teleparallel displacement of momentum between initial and final states quantum theory. An exact formulation is possible in a closed FRW cosmology in which cosmological redshift is given by 1+z =3D (a_o/a(t))^2. This is consistent with observation for a universe expanding at half the currently accepted rate, twice as old, and requiring a quarter of the critical density for closure. To first order, supernova redshifts indicating Omega_M=3D0.3, Omega_Lambda=3D0.7 in standard cosmology are consistent with a closed universe close to critical density and with zero cosmological constant. The effect of expansion is found in the anomalous Pioneer redshift and in the flattening of galaxies' rotation curves. Milgrom's phenomenological law (MOND) is precisely obeyed, and appears in the model as an optical effect, affecting observation but not dynamics. 1 Introduction 1.1 Teleparallelism There are well known and substantial difficulties in formulating quantum theories in curved spacetime (Dirac, 1964; Fulling, 1989; Wald, 1994). Einstein (1930) found problems with electrodynamics in curved space time, and suggested that the affine connection of general relativity might be replaced with a teleparallel connection. Such a replacement can be motivated in the orthodox interpretation of quantum mechanics in which it does not make sense to talk of position between measurements and hence it does not make sense to talk of transport along a path. This is not a theory of teleparallel gravity in the usual sense, as described by e.g. Arcos and Pereira (2004). The connection is only defined for measured states and will be called a teleconnection. We require only longitudinal teleparallel displacement of momentum. Teleparallel displacement requires a preferred coordinate system. This will be determined from the requirements of wave mechanics and exhibited in the instance of a homogeneous isotropic cosmos. Torsion will be removed as part of wave function collapse and in the classical correspondence gravity is described by curvature, as is normal in general relativity. It will be shown that this prescription reduces to the affine connection and that geodesic motion is preserved for classical particles and of a beam of light. 1.2 Comparison with the Standard Model As described in section 3, the model predicts that the cosmological redshift factor, 1+z, varies with the square of the expansion parameter, not linearly. The square law applies to cosmological redshift; gravitational redshift is as in general relativity, as required by the principle of equivalence. It follows that the rate of expansion of the universe is half that predicted by the standard model, the universe is twice as old as has been thought, and critical density for closure is a quarter of the standard value, dispensing with at least the bulk of missing mass and resolving any ageing issues arising from recent observations of mature galaxies at z=3D1.4 and greater (e.g. Mullis et al., 2005). The existence of mature galaxies at this red shift may genuinely be described as prediction rather than retrodiction, since the implications of the square red shift law were discussed in public forum with Farmer Oz before the observations had been announced. The model predicts that cosmological redshift is present in all measurements using Doppler. This has been observed. For some years the Pioneer spacecraft have been sending back Doppler information appearing to indicate an anomalous acceleration toward the sun (Anderson et al., 2002). The anomalous data can now be understood as an optical effect due to cosmological redshift not evidence of actual acceleration. No other explanation has yet been given for the Pioneer redshift. The effect is also present in the observation of distant galaxies, and precisely accounts for flattening of galaxies' rotation curves consistent with MOND, the phenomenological law found by Milgrom (1994). Again this appears as an optical effect arising from the treatment of redshift, not a real change to Newtonian dynamics. All cosmological tests using redshift measurement will have to be reinterpreted to demonstrate the consistency of this model with data. In standard cosmology a best fit with supernova data is found for the concordance model, Omega=3DOmega_M=3D0.3, Omega_Lambda=3D0.7 (Reiss et al= ..., 2004; Filippenko, 2004, and references cited therein). To first order in z, for a closed cosmos with zero cosmological constant the redshift magnitude relation found here is close to that of the standard model with Omega=3D0.41 and Omega_Lambda=3D0.59. The fit improves when terms O(z^2) are taken into account, but it remains to find the best fit values of Omega and Omega_Lambda using a computer solution. Further consistency tests are possible. The standard concordance model is consistent with evidence from the Two-Degree Field Galaxy Redshift Survey (2dFGRS; Peacock et al. 2001; Percival et al., 2001; Efstathiou, 2002), and from the Wilkinson Microwave Anisotropy Probe (WMAP; Spergal, 2003, and references cited therein). Evidence for dark energy and accelerated expansion also comes from the integrated Sachs-Wolfe effect (Afshordi, Loh & Strauss; 2004; Boughn & Crittendon, 2004; Fosalba et al., 2003; Nolta et al., 2004; Scranton et al., 2004). It will be necessary to analyse these tests and any of them could potentially falsify the model. It is not unreasonable to suspect that the optical distortion due to the square redshift law will, in each case, affect Omega and Omega_Lambda in the same way, but rigorous testing requires resources and data not immediately available to the author. 2 Teleparallel Quantum Theory 2.1 Coordinate Space Each local region, O, of a continuous manifold, M, can be considered as a subset, U is a subset of R^n, together with a map, psi:U-O. Let U be denoted by axes a=3D0,...,n. Definition: A coordinate space is any such subset, U is a subset of R^n, together with metric yta_a_b. Here yta is not the physical metric, but is an abstract metric used for mapping. Curvature is naturally conceived in terms of the scaling distortions of maps. In general straight lines in coordinate space are not geometrically straight, but for a sufficiently short line segment the deviation from straightness is not detectable, and, to first order, a short rod placed at x will appear as a small displacement vector, A^a, defined, as usual, as the difference in the coordinates of one end of the rod from the other. A coordinate space vector can be defined by inverting the scaling distortions of the map. This is done by choosing primed locally Minkowski coordinates with an origin at x. We define the matrix kappa^a_b(x)=3Dx^aprime_,b(x). 2.1.1 2.1.1 applies pointwise; kappa is defined using different local Minkowski coordinates for each origin x. 2.1.1 is meaningless as a differential equation and gives paradoxical results when treated as one. For the vector, A^a, at position x, the corresponding coordinate space vector, barred to distinguish it from an ordinary, or physical, vector, is defined by Abar^a(x) =3D kappa^a_b(x) A^b(x). 2.1.2 This ensures that the coefficients of a coordinate space vector are equal to the coefficients of the corresponding vector in the primed, Minkowski, coordinates and preserves the inner product: yta_m_n Abar^m Bbar^n =3D yta_m_n kappa^m_a(x) kappa^n_b(x) A^a B^b =3D yta_m_n x^m_,a(x) x^n_,b(x) A^a B^b =3D g_a_b(x) A^a B^b 2.1.3 2.1.3 is true for any vectors A, . So g_a_b(x) =3D yta_m_n k^m_a(x) k^n_b(x) 2.1.4 =2E 2.1.4 gives the metric in terms of the variable scale coefficients, k^m_a(x), of coordinate space vectors compared to physical vectors. 2.2 Teleparallel Displacement A short rod placed at x is described by vector, A(x). An identical short rod is placed at y, so that its coordinate space vector is parallel to . It is described by a vector, A(y), whose length is unchanged A^2(x)=3DA^2(y). Definition: A(y) is teleparallel to A(x) if and only if the coordinate space components are proportional kappa^a_b(y) A^b(y) =3D Abar^a(y) is proportional to Abar^a(x) =3D kappa^a_b(x) A^b(x) 2,2,1 =2E In general this is not equality because of the different scaling distortions at x and y. 2.2.1 generalises Einstein's definition (Einstein, 1930), in which coordinate space is tangent to local Minkowski space at x. In this case axes can be chosen such that kappa^a_b(x)=3Ddelta(a_b) and 2.2.1 reduces to kappa^a_b(y) A^b(y) =3D Abar^a(y) is proportional to Abar^a(x) =3D A^a(x) 2,2,2 Multiply both sides of 2.2.2 by yta_ag k^g_m(y) and use 2.1.4: A_m(y) is proportional to A^a(x)yta_ag k^g_m(y) =3D A_g(x)k^g_m(x) 2.2.3. The magnitudes of the left and right hand sides of 2.2.3 are the same and we have equality A_m(y) =3D A_g(x)kappa^g_m(x) 2.2.4 Parallel transport is a sequence of infinitesimal parallel displacements in the form 2.2.4. 2.3 Quantum Coordinates It is natural to define quantum coordinates such that momentum is always teleparallel to momentum observed at the origin. Definition: In quantum coordinates the plane wave state, |pbar is defined at any location by replacing 3-vectors with coordinate space 4-vectors, pbar=3D(Ebar,pbar) and xbar=3D(tbar,xbar). Then, at time x^0=3D= t, x|pbar =3D t,x|pbar =3D (1/2pi)^3/2 e^-ixbar.pbar 2.3.1 This definition replaces the affine connection of classical general relativity in the quantum domain. It is justified empirically because observation yields precise values of cosmological redshift. It follows that in quantum coordinates inertial objects move uniformly, and massless particles travel at constant speed, 1. In general teleparallel displacement requires flat coordinate space, but, for light emitted from a distant object at time, t, and detected at time, t0, we merely require flatness in the time-radial plane. If we require spherical symmetry then, for some function, g:R-R, and real b, the coordinate space metric yta is ds^2=3D b^2(dtau^2-dro^2) - g^2(r0)(dtheta^2_sin^2(theta)dphi^2). 2.3.2 b and g will be determined in section 3.1. An FRW cosmology can be written in conformally flat coordinates, so that for a=3Da(t) with a0=3Da(t0) the metric is ds^2=3D (a/a0)^2(dtau^2-dro^2 - f^2(r0)(dtheta^2+sin^2(theta)dphi^2). 2.3.3 where the radial coordinate is r=3Dro, sin ro, sinh ro for a space of positive, zero, or negative curvature respectively, and coordinate time, tau, is related to cosmic time, t, by dtau =3D (a0/a)^2 dt. 2.3.4 Then, for an FRW cosmology mapped onto coordinate space 2.4.1, near the origin, the coordinate space vectors are, to first order, dtbar =3D 1/b dt, drbar =3D a/ba0 dr, dthetabar =3D f/g dtheta, dphibar =3D f/g dphi 2.3.5 2.4 Geodesic Motion In local coordinates with origin at time t=3D0 and, with metric ds^2=3D dt^2-dr^2 - r^2(dtheta^2+sin^2(theta)dphi^2). 2.4.1 for light emitted at time x^0=3Dt and detected at the origin at time t0, momentum is given by Pbar_0(x) =3D -i(a0/a)^2 dbar_0, Pbar_1(x) =3D -i(a0/a)^2 dbar_0, Pbar_2 =3D Pbar_3 =3D 0 2.4.2 Light from a distant object is not observable unless Pbar_2 =3D Pbar_3 =3D 0. Using 2.1.2 and 2.3.2, in quantum coordinates, momentum at the time of emission is Pbar_0(x) =3D -i(a0/a)^2 dbar_0, Pbar_1(x) =3D -i(a0/a)^2 dbar_0, 2.4.3 2.4.2 follows immediately on teleparallel displacement to the origin. 2.4.2 obtains in quantum coordinates, defined from a reference frame using a clock at the time of detection. In particular, 2.4.2 applies to cosmological red shift (section 3.1). In this instance the initial state is determined from the final measurement, so that there is no collapse. More usually in quantum theory, two measurements are required; one for the initial state at time t and one for the final state at t0. Collapse signifies a change from one reference frame to the other. The effect of expansion is that unit distance is rescaled as compared to the radius of the universe. Rescaling removes torsion and there is a corresponding renormalisation of energy momentum such that 2.4.2 is replaced by: Pbar_0(x) =3D -i(a0/a) dbar_0, Pbar_1(x) =3D -i(a0/a) dbar_0, Pbar_2 =3D Pbar_3 =3D 0 2.4.4 Let k^g_m be the scale factors between Minkowski coordinate spaces at x and y, so that k^g_m =3D a/a0 kappa^g_m. 2.4.5 Then, by 2.2.4, parallel displacement from the point x at time, x^0=3Dt , to y at time, y^0=3Dt_0, yields A_m(y) =3D a/a0 A_g(x)k^g_m(x) 2.4.6 =2E After renormalisation of energy-momentum, 2.4.6 gives P_m(y) =3D P_g(x)k^g_m(x) When y =3D x+dx 2.4.7 is the standard formula for infinitesimal parallel displacement in a tangent space. In the classical correspondence the motion may be described as a sequence of states |f(t^i) at instances t_i with 0 t_(i+1) - t_i chi in the limit as chi tends to zero. The state at any instant may be regarded as an initial state using coordinates with metric 2.3.3. The state at the next instant may be regarded as a final state, which after rescaling, becomes the initial state for the next part of the motion. Momentum is parallel displaced in each coordinate space using 2.4.7. The cumulative effect of such infinitesimal parallel displacements is parallel transport. So teleparallelism of momentum between initial and final states in quantum theory, together with renormalisation of momentum, gives rise to parallel transport in the classical domain. Geodesic motion of a classical particle follows as the cumulative effect of displacements in the direction of momentum over small time increments. The same argument follows for a classical beam of light, in which each photon wave function is localised within the beam at any time. 3 Comparison with the Standard Model 3.1 Cosmological Redshift Typical experiments in quantum theory require two measurements, one for the initial state and one for the final state, and are such that reference matter used for the initial measurement is rigidly related to that used for the final one; either the same coordinate axes and clock are used in both measurements or the coordinate axes and clocks are calibrated to each other. As seen in section 2.4 this requires a renormalisation of energy momentum, such that geodesic motion and the principle of equivalence are restored. But in measurements on light from a distant object it is not possible to define a prior relationship between the reference matter used for the final measurement and the matter from which the photon is emitted. Light received at the origin has been transmitted from an event on the light cone, so that the only information we have about the initial state comes from measurement of the final state. Then there is no renormalisation of energy-momentum, and cosmological redshift is read from 2.4.2: 1+z =3D (a_0/a)^2 3.1.1 For small r 1+z =3D 2 r adot / a 3.1.2 [I am using adot for da/dt] Thus coordinates in which radial distance from Earth is calculated from redshift exhibit a stretch of factor two in the radial direction, so that b=3D1/2. The time taken for a pulse of light to traverse a small angular distance, dtheta is rdtheta, so g=3D2f and the metric, 2.3.3, in quantum coordinates is: ds^2=3D a^2((dtau^2-dro^2)/4 - 4f^2(r0)(dtheta^2+sin^2(theta)dphi^2). 3.1.3 Hubble's constant, H=3D2adot/a , is read from 3.1.2. This is consistent with observation for a universe expanding at half the currently accepted rate, twice as old, and requiring a quarter of the critical density for closure. Let Omega=3D32pi G ro /3H^2, Omega_R=3D-4k/H0^2a0^2 Omega_Lambda=3D4Lambda/3H0^2 where k=3D-1,0,1 (quadruple the standard values, so that Omega=3D1 is critical density). Then, from Friedmansn's equation, adot/a=3D H0/2( Omega(1+z)^3/2 + Omega_R(1+z) + Omega_Lambda )^1/2 3.1.4 requiring that Omega + Omega_R + Omega_Lambda =3D 1. To calculate angular distance to an object emitting light at time te, integrate the speed of light after applying redshift l =3D Int dt (a0/a)^2 dt =3D horrible formula too nasty for ASCII 3.1.5 From which the redshift magnitude relation can be calculated. To first order in z it is m ~ 5log z + 10.86(1.5 - 0.125Omega)z 3.1.6 =2E For the critical value of density, Omeg=3D1, 3.1.6 gives the same plot as the standard magnitude-redshift relation with Omega_Lambda=3D0.59 and Omega=3D0.41, corresponding to an apparently accelerated expansion in the standard model. This appears consistent with values from recent measurements (Reiss et al., 2004; Filippenko, 2004, and references cited therein), such that Omega_Lambda=3D0.7 and Omega=3D0.3. The fit improves when terms O(z^2) are taken into account. 3.2 Anomalous Pioneer Redshift It is known that the anomalous Pioneer redshift may be accounted for by an "acceleration in time" equal to the Hubble constant (Anderson et al., 2002). Acceleration in time was rejected by JPL, because, using conventional physics, it is incompatible with ranging data, and they elected to express their result in the form of an equivalent classical acceleration. A laboratory moving with respect to the cosmic fluid uses locally Minkowski coordinates, which can be transformed locally to comoving coordinates and extended globally to coordinates with metric 3.1.3. Then the time coordinate obeys 2.3.4, and exhibits acceleration with respect to proper time. The value of the acceleration in time is H0/2 but redshift is doubled in 3.1.2 so the resultant shift is the same. Consistent with NASA's findings, expansion affects Doppler measurements but not geodesic motion of classical bodies, as described in section 2.4. Thus Pioneer redshift simulates constant acceleration toward the origin of coordinates. This is purely an optical effect. There is no corresponding classical acceleration and planetary motions are unaffected. Further experiment is required, because the data available to NASA came from space craft for which either only ranging or only Doppler determination of position was available. If both methods were used simultaneously on the same object, it would rigorously establish whether Pioneer exhibits an anomalous acceleration, or merely an anomalous redshift as predicted here. Undoubtedly more accurate data could be obtained with an experiment designed for the purpose. Nonetheless, in the absence of an alternative explanation, the value of 8x10^-8 cm s^-2 given by Pioneer can be regarded as experimental confirmation of a property of the model and is possibly the most accurate current empirical determination of Hubble's constant. 3.3 Flattening of Galaxies' Rotation Curves A star, S, in a galaxy, centre G, is subject to an inward acceleration due to gravity toward G. Then the red shift due to orbital velocity is maximised when S is on a diameter perpendicular to the Earth. In addition there is an observed redshift equivalent to the Pioneer acceleration g_r=3D-Hc toward the Earth. For simplicity of calculation assume that the galaxy lies in a plane with the Earth. Set up locally Minkowski coordinates with an origin at G and with the y-axis directed toward the Earth and S on the x-axis. The Pioneer redshift is interpreted as an acceleration which can be resolved into a part, gy, equal to the acceleration of G toward the Earth and a part equal to an acceleration gx toward G. Quantum coordinates with metric 3.1.3 are stretched in time by a factor of 2 and in the transverse direction by a factor of 1/2, so the redshift corresponding to expansion from G is subject to a factor of 1/4. Acceleration is the second time derivative of the transverse distance, and requires a factor of 8 in these coordinates, so we find a net factor of 32, giving a perceived inward acceleration g_x =3D -Hc/32 3.3.1 The redshift is interpreted as being due to the motion of a body in orbit about G with orbital velocity v_p. Then v_p^2 / x =3D Hc/32 or v_p =3D root(Hcx/32). 3.3.2 This simulated velocity, v_p, is independent of galactic mass and would appear in Minkowski coordinates with an origin at any point in space. If the true orbital velocity of the star S due to gravity is vg then the observed orbital velocity is v =3D v_g + v_p =3D root(GM/x) + root(Hcx/32) 3.3.3 =2E 3.3.3 recognises that, since the alteration to redshift is an optical effect, it is correct to add velocities, not accelerations as would be the case for a dynamical law. Then the apparent acceleration toward G is v^2/x =3D GM/x^2 + root(GMHc/8)/x + Hc/32 3.3.4 =2E The first term in 3.4.4 is the acceleration due to gravity. The last is simply the component of Pioneer acceleration toward G, and appears also in the absence of a source of gravity. This leaves an unmodelled acceleration equivalent to a redshift due to velocity, v^2 =3D root(GMHc/8) 3.3.5 , in precise agreement with MOND, the phenomenological law proposed by Milgrom (1994) which retains Newton's square law for accelerations xdoubledot a_M for some constant a_M, but replaces it with xdoubledot =3D - (GMa_M)^1/2 /x for xdoubledot a_M 3.3.6 and gives a good match with data. The best fit value of a_M from observations on thousands of stars is 1x10^-8 cm s-2 in precise agreement with a_M =3D Hc/8 using the value 8x10^-8 cm s-2 found from the observations on Pioneer. References Afshordi N., Loh Y.-S., Strauss M. S. (2004) Phys. Rev. D69, 083524 Arcos H. I., Pereira J. G. (2004) Int.J.Mod.Phys. D13 2193-2240 Anderson J. D., Laing P. A., Lau E. L., Liu A. S., Nieto M. M., Turyshev S. G. (2002) Study of the Anomalous Acceleration of Pioneer 10 and 11, Phys.Rev. D65 082004. Boughn S., & Crittendon (2004) Nature 427,45 Dirac P. A. M. (1964) Lectures on Quantum Mechanics, Belfer Grad School of Science, New York. Efstathiou G., et al. (2002) MNRAS, 330, L29. Einstein A. (1930) Auf die Riemann-metric und den Fern-Parallelismus gegr=FCndete einheitliche Field-Theorie, Math Ann., 102, 658-697. Filippenko A. V. (2004) Type Ia Supernova and Cosmology, White Dwarfs: Probes of Galactic Structure and Cosmology, ed. E. M. Sion, H. L. Shipman, and S. Vennes (Kluwer: Dordrecht) astro-ph/0410609. Fosalba P., et al, (2003) ApJ., 597, L89. Francis C., (2004) http://groups.google.co.uk/group/uk....g/3c1784a594b0 dd02?hl=3Den& Fulling S. A. (1989) Aspects of Quantum Field Theory in Curved Space-Time, Cambridge University Press . C. R. Mullis, P. Rosati, G. Lamer, H. Boehringer, A. Schwope, P. Schuecker, R. Fassbender (2005) Discovery of an X-ray-Luminous Galaxy Cluster at z=3D1.4, ApJ Letters, 623, L85-L88, astro-ph/0503004 Milgrom M. (1994) Ann. Phys., (NY) 229, 384 Also see astro-ph/0112069 Nolta M. R. et al., (2004) ApJ., 608, 10. Peacock J. A., et al. 2001 Nature, 410, 169 Percival W. et al. 2001, MNRAS, 327, 1297 Reiss A. G. et al (2004) Astrophys.J., 607, 665-687. Scranton et al (2004) Physics Review Letters astro-ph/0307335 Spergal D. L. et al (2003) ApJS, 613, 200 Wald R. M. (1994) Quantum Field Theory in Curved Spacetime and Black Hole Thermodynamics, University of Chicago Press. Regards --=20 Charles Francis |
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Charles Francis writes
I might describe this post as a pre-preprint. I have not yet sent it anywhere, and would like to get comments and criticisms before I do. This is long, and has a nasty chunk of maths to start off with. I guess few people will want to follow it (and I am unable to anyway). A somewhat edited 'highlights' might be found with some astute snipping: Title Does a Teleconnection between Quantum States account for Missing Mass, Galaxy Ageing, Supernova Redshift, MOND and Pioneer? Answer: yes. Author Charles Francis MA (Cantab) PhD (Lond) A mathematician (hence the arcane maths in places) Note that cut&paste glitch has displayed "=" as "=3D". 1.1 Teleparallelism Light from a distant object is not observable unless Pbar_2 =3D Pbar_3 =3D 0. Using 2.1.2 and 2.3.2, in quantum coordinates, momentum at the time of emission is Pbar_0(x) = -i(a0/a)^2 dbar_0, Pbar_1(x) = -i(a0/a)^2 dbar_0, 2.4.3 When y = x+dx 2.4.7 is the standard formula for infinitesimal parallel displacement in a tangent space. 3 Comparison with the Standard Model 3.1 Cosmological Redshift Typical experiments in quantum theory require two measurements, one for the initial state and one for the final state, Light received at the origin has been transmitted from an event on the light cone, so that the only information we have about the initial state comes from measurement of the final state. Then there is no renormalisation of energy-momentum, and cosmological redshift is read from 2.4.2: 1+z = (a_0/a)^2 3.1.1 For small r 1+z = 2 r adot / a 3.1.2 [I am using adot for da/dt] Hubble's constant, H=2adot/a , is read from 3.1.2. This is consistent with observation for a universe expanding at half the currently accepted rate, twice as old, and requiring a quarter of the critical density for closure. [Oz: solving a whole bunch of problems at a stroke?] Let Omega=32pi G ro /3H^2, Omega_R=-4k/H0^2a0^2 Omega_Lambda=4Lambda/3H0^2 where k=-1,0,1 (quadruple the standard values, so that Omega=3D1 is critical density). Then, from Friedmansn's equation, adot/a= H0/2( Omega(1+z)^3/2 + Omega_R(1+z) + Omega_Lambda )^1/2 3.1.4 requiring that Omega + Omega_R + Omega_Lambda = 1. [After this it should be understandable allowing for = being =3D etc.] From which the redshift magnitude relation can be calculated. To first order in z it is m ~ 5log z + 10.86(1.5 - 0.125Omega)z 3.1.6 =2E For the critical value of density, Omeg=3D1, 3.1.6 gives the same plot as the standard magnitude-redshift relation with Omega_Lambda=3D0.59 and Omega=3D0.41, corresponding to an apparently accelerated expansion in the standard model. This appears consistent with values from recent measurements (Reiss et al., 2004; Filippenko, 2004, and references cited therein), such that Omega_Lambda=3D0.7 and Omega=3D0.3. The fit improves when terms O(z^2) are taken into account. 3.2 Anomalous Pioneer Redshift It is known that the anomalous Pioneer redshift may be accounted for by an "acceleration in time" equal to the Hubble constant (Anderson et al., 2002). Acceleration in time was rejected by JPL, because, using conventional physics, it is incompatible with ranging data, and they elected to express their result in the form of an equivalent classical acceleration. A laboratory moving with respect to the cosmic fluid uses locally Minkowski coordinates, which can be transformed locally to comoving coordinates and extended globally to coordinates with metric 3.1.3. Then the time coordinate obeys 2.3.4, and exhibits acceleration with respect to proper time. The value of the acceleration in time is H0/2 but redshift is doubled in 3.1.2 so the resultant shift is the same. Consistent with NASA's findings, expansion affects Doppler measurements but not geodesic motion of classical bodies, as described in section 2.4. Thus Pioneer redshift simulates constant acceleration toward the origin of coordinates. This is purely an optical effect. There is no corresponding classical acceleration and planetary motions are unaffected. Further experiment is required, because the data available to NASA came from space craft for which either only ranging or only Doppler determination of position was available. If both methods were used simultaneously on the same object, it would rigorously establish whether Pioneer exhibits an anomalous acceleration, or merely an anomalous redshift as predicted here. Undoubtedly more accurate data could be obtained with an experiment designed for the purpose. Nonetheless, in the absence of an alternative explanation, the value of 8x10^-8 cm s^-2 given by Pioneer can be regarded as experimental confirmation of a property of the model and is possibly the most accurate current empirical determination of Hubble's constant. 3.3 Flattening of Galaxies' Rotation Curves A star, S, in a galaxy, centre G, is subject to an inward acceleration due to gravity toward G. Then the red shift due to orbital velocity is maximised when S is on a diameter perpendicular to the Earth. In addition there is an observed redshift equivalent to the Pioneer acceleration g_r=3D-Hc toward the Earth. For simplicity of calculation assume that the galaxy lies in a plane with the Earth. Set up locally Minkowski coordinates with an origin at G and with the y-axis directed toward the Earth and S on the x-axis. The Pioneer redshift is interpreted as an acceleration which can be resolved into a part, gy, equal to the acceleration of G toward the Earth and a part equal to an acceleration gx toward G. Quantum coordinates with metric 3.1.3 are stretched in time by a factor of 2 and in the transverse direction by a factor of 1/2, so the redshift corresponding to expansion from G is subject to a factor of 1/4. Acceleration is the second time derivative of the transverse distance, and requires a factor of 8 in these coordinates, so we find a net factor of 32, giving a perceived inward acceleration g_x =3D -Hc/32 3.3.1 The redshift is interpreted as being due to the motion of a body in orbit about G with orbital velocity v_p. Then v_p^2 / x =3D Hc/32 or v_p =3D root(Hcx/32). 3.3.2 This simulated velocity, v_p, is independent of galactic mass and would appear in Minkowski coordinates with an origin at any point in space. If the true orbital velocity of the star S due to gravity is vg then the observed orbital velocity is v =3D v_g + v_p =3D root(GM/x) + root(Hcx/32) 3.3.3 =2E 3.3.3 recognises that, since the alteration to redshift is an optical effect, it is correct to add velocities, not accelerations as would be the case for a dynamical law. Then the apparent acceleration toward G is v^2/x =3D GM/x^2 + root(GMHc/8)/x + Hc/32 3.3.4 =2E The first term in 3.4.4 is the acceleration due to gravity. The last is simply the component of Pioneer acceleration toward G, and appears also in the absence of a source of gravity. This leaves an unmodelled acceleration equivalent to a redshift due to velocity, v^2 =3D root(GMHc/8) 3.3.5 , in precise agreement with MOND, the phenomenological law proposed by Milgrom (1994) which retains Newton's square law for accelerations xdoubledot a_M for some constant a_M, but replaces it with xdoubledot =3D - (GMa_M)^1/2 /x for xdoubledot a_M 3.3.6 and gives a good match with data. The best fit value of a_M from observations on thousands of stars is 1x10^-8 cm s-2 in precise agreement with a_M =3D Hc/8 using the value 8x10^-8 cm s-2 found from the observations on Pioneer. -- Oz This post is worth absolutely nothing and is probably fallacious. Use functions]. BTOPENWORLD address has ceased. DEMON address has ceased. |
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[Mod. note: this is off-topic but I'll post it anyway because it may
act as a warning to users of the group. If I have the time and energy, I fix MIME damage before posting, but much the best solution is not to post in such formats in the first place -- mjh] In article , Oz writes: Note that cut&paste glitch has displayed "=" as "=3D". Actually, this was probably caused by the software sending "quoted-printable" instead of "plain text" to the newsgroup. Newsgroups are for plain text, not for encoded stuff or for attachments. While it is possible that a quoted-printable text might display correctly with some newsreaders if it includes an appropriate MIME header, many newsreaders don't support this. The solution is not "get a better newsreader" since there is no standard which specifies MIME for newsgroups. Quoted-printable stuff is usally thrust upon the world via Microsoft software; often the user is unaware that he is posting plain text. (Again, the fact that it looks right to him is not reliable, since it might be converted back to plain text.) Note that while there is some justification for encoding stuff which contains 8-bit characters such as ä Å ø ß etc, since while 8-bit transmission is generally OK these days there is not a standard 8-bit character set, there is no justification for replacing = with =3D since = is part of the good old 7-bit printable US-ASCII character set; certainly not if it is the ONLY character which is encoded. What happens is that the two characters after the = refer to the hexadecimal encoding of the character (see the tables below). While replacing = with =3D is the most common (as well as replacing special characters), sometimes one sees =20 for a space (often at the end of a line) and =2E for ".". The solution is for the moderators to reject posts which are not in 7-bit printable US-ASCII (except for examples like this post!) or correct them before posting to the newsgroup (perhaps with editor macros making use of the tables below). Best, of course, is for all users to make sure they are posting only plain text to the newsgroup, and also to avoid using special characters. +------------------------------------------+ | 0 1 2 3 4 5 6 7 | +---+--------------------------------------+ | 0 | NUL DLE SP 0 @ P ` p | | 1 | SOH DC1 ! 1 A Q a q | | 2 | STX DC2 " 2 B R b r | | 3 | ETX DC3 # 3 C S c s | | 4 | EOT DC4 $ 4 D T d t | | 5 | ENQ NAK % 5 E U e u | | 6 | ACK SYN & 6 F V f v | | 7 | BEL ETB ' 7 G W g w | | 8 | BS CAN ( 8 H X h x | | 9 | HT EM ) 9 I Y i y | | A | LF SUB * : J Z j z | | B | VT ESC + ; K [ k { | | C | FF FS , L \ l | | | D | CR GS - = M ] m } | | E | SO RS . N ^ n ~ | | F | SI US / ? O _ o DEL | +---+--------------------------------------+ +------------------------------------------+ | 8 9 A B C D E F | +---+--------------------------------------+ | 0 | DCS ° À à | | 1 | PU1 ¡ ± Á Ñ á ñ | | 2 | PU2 ¢ ² Â Ò â ò | | 3 | STS £ ³ Ã Ó ã ó | | 4 | IND CCH Ä Ô ä ô | | 5 | NEL MW ¥ µ Å Õ å õ | | 6 | SSA SPA ¶ Æ Ö æ ö | | 7 | ESA EPA § · Ç × ç ÷ | | 8 | HTS ¨ È Ø è ø | | 9 | HTJ © ¹ É Ù é ù | | A | VTS ª º Ê Ú ê ú | | B | PLD CSI « » Ë Û ë û | | C | PLU ST ¼ Ì Ü ì ü | | D | RI OSC ½ Í Ý í ý | | E | SS2 PM Î î | | F | SS3 APC ¿ Ï ß ï | +---+--------------------------------------+ |
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Phillip Helbig---remove CLOTHES to reply
e writes What I would like to know is how reasonable Francis' ideas are. To slightly alter doppler in the way he proposes and fix this range of problems seems to me to be surprising even if you don't agree with the theoretical basis. His theory is a very esoteric and mathematical. He hasn't even posted the full theory developed from first principles because most would find it too scary. But the results are surely worthy of some comment, if just disagreement. -- Oz This post is worth absolutely nothing and is probably fallacious. Use functions]. BTOPENWORLD address has ceased. DEMON address has ceased. |
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In message , Oz
writes His theory is a very esoteric and mathematical. He hasn't even posted the full theory developed from first principles because most would find it too scary. It's not so much that it is scary. It just takes you into an area, interpretation of quantum theory, that has been in the doldrums for many years, leading only to dead ends and arguments. However, if one thinks of a photon emitted from a distant star and being detected here as a quantum event which should be described in braket formulation as photon detected here|photon emitted from distant star it becomes obvious that really we need to unify qm and gtr in order to analyse cosmological red shift in an expanding universe. I got to the square red shift law on theoretical grounds, by looking at how I could used teleparallel displacement to get a consistent mathematical model. But from the point of view of the paper discussed here I am happy to treat it as phenomenological. It appears to me that it actually does give a better match with data than the linear law derived from parallel transport of light. Regards -- Charles Francis |
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Charles Francis writes
In message , Oz writes His theory is a very esoteric and mathematical. He hasn't even posted the full theory developed from first principles because most would find it too scary. It's not so much that it is scary. It just takes you into an area, interpretation of quantum theory, that has been in the doldrums for many years, leading only to dead ends and arguments. However, if one thinks of a photon emitted from a distant star and being detected here as a quantum event which should be described in braket formulation as photon detected here|photon emitted from distant star it becomes obvious that really we need to unify qm and gtr in order to analyse cosmological red shift in an expanding universe. Yes, but how? I got to the square red shift law on theoretical grounds, by looking at how I could used teleparallel displacement to get a consistent mathematical model. Einstein tried this and failed. What are you doing that's different? But from the point of view of the paper discussed here I am happy to treat it as phenomenological. It appears to me that it actually does give a better match with data than the linear law derived from parallel transport of light. It does, but there must be a little more to it than that judging by the refutation of segal's ideas in the quoted website. He seems to have got somewhat different results from the same basic concept, which suggests one of you have made a mistake. -- Oz This post is worth absolutely nothing and is probably fallacious. Use functions]. BTOPENWORLD address has ceased. DEMON address has ceased. |
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In message , Oz
writes Charles Francis writes In message , Oz writes His theory is a very esoteric and mathematical. He hasn't even posted the full theory developed from first principles because most would find it too scary. It's not so much that it is scary. It just takes you into an area, interpretation of quantum theory, that has been in the doldrums for many years, leading only to dead ends and arguments. However, if one thinks of a photon emitted from a distant star and being detected here as a quantum event which should be described in braket formulation as photon detected here|photon emitted from distant star it becomes obvious that really we need to unify qm and gtr in order to analyse cosmological red shift in an expanding universe. Yes, but how? Well that part I did try to explain. Since quantum wave theory only works on a flat space, I am doing it in a teleparallel theory. But then I find that making qm work actually determines the space. I got to the square red shift law on theoretical grounds, by looking at how I could used teleparallel displacement to get a consistent mathematical model. Einstein tried this and failed. What are you doing that's different? Einstein hated qm, and tried to work with electrodynamics. Remember I have to renormalise momentum at the time of collapse of the wave function. This is necessary on theoretical grounds, but if not my theory would not give geodesic motion. But from the point of view of the paper discussed here I am happy to treat it as phenomenological. It appears to me that it actually does give a better match with data than the linear law derived from parallel transport of light. It does, but there must be a little more to it than that judging by the refutation of segal's ideas in the quoted website. He seems to have got somewhat different results from the same basic concept, which suggests one of you have made a mistake. I haven't yet got to the basis of Segal's idea's, but Ned Wright describes his model as static, which is radically different from mine., He seems to have had a prediction incompatible with the cosmic microwave background. That is likely to be a problem in a static model (consider Olber's paradox). I can't see how I would get that. Regards -- Charles Francis |
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Charles Francis wrote:
In message , Oz writes Charles Francis writes In message , Oz writes His theory is a very esoteric and mathematical. He hasn't even posted the full theory developed from first principles because most would find it too scary. It's not so much that it is scary. It just takes you into an area, interpretation of quantum theory, that has been in the doldrums for many years, leading only to dead ends and arguments. However, if one thinks of a photon emitted from a distant star and being detected here as a quantum event which should be described in braket formulation as photon detected here|photon emitted from distant star it becomes obvious that really we need to unify qm and gtr in order to analyse cosmological red shift in an expanding universe. Yes, but how? Well that part I did try to explain. Since quantum wave theory only works on a flat space, I am doing it in a teleparallel theory. But then I find that making qm work actually determines the space. AFAIK, it has been known how to do QM in curved spacetime for several decades now, so why did you feel the need to come up with this "teleparallel theory"? [snip] Bye, Bjoern |
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Charles Francis writes
Well that part I did try to explain. Since quantum wave theory only works on a flat space, I am doing it in a teleparallel theory. But then I find that making qm work actually determines the space. I think you should expand on this. You say that only a limited number of spaces are consistent. At least one is consistent with the universe we think we like in. That implies to me that the teleconnection in some sense defines the space. Einstein hated qm, and tried to work with electrodynamics. Remember I have to renormalise momentum at the time of collapse of the wave function. This is necessary on theoretical grounds, but if not my theory would not give geodesic motion. Hmmm, I would read it as the reverse. Its a condition required to give geodesic motion, which is undoubtedly an experimentally well-understood requirement. I haven't yet got to the basis of Segal's idea's, but Ned Wright describes his model as static, which is radically different from mine., He seems to have had a prediction incompatible with the cosmic microwave background. That is likely to be a problem in a static model (consider Olber's paradox). I can't see how I would get that. So the criticism of your proposal, based on the criticism of segal's, is entirely misplaced. In that case, doubtless vecchi will come up with another criticism. -- Oz This post is worth absolutely nothing and is probably fallacious. Use functions]. BTOPENWORLD address has ceased. DEMON address has ceased. |
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Charles Francis writes
Well that part I did try to explain. Since quantum wave theory only works on a flat space, I am doing it in a teleparallel theory. But then I find that making qm work actually determines the space. I think you should expand on this. You say that only a limited number of spaces are consistent. At least one is consistent with the universe we think we like in. That implies to me that the teleconnection in some sense defines the space. Einstein hated qm, and tried to work with electrodynamics. Remember I have to renormalise momentum at the time of collapse of the wave function. This is necessary on theoretical grounds, but if not my theory would not give geodesic motion. Hmmm, I would read it as the reverse. Its a condition required to give geodesic motion, which is undoubtedly an experimentally well-understood requirement. I haven't yet got to the basis of Segal's idea's, but Ned Wright describes his model as static, which is radically different from mine., He seems to have had a prediction incompatible with the cosmic microwave background. That is likely to be a problem in a static model (consider Olber's paradox). I can't see how I would get that. So the criticism of your proposal, based on the criticism of segal's, is entirely misplaced. In that case, doubtless vecchi will come up with another criticism. -- Oz This post is worth absolutely nothing and is probably fallacious. Use functions]. BTOPENWORLD address has ceased. DEMON address has ceased. |
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