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

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.

[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 |
+---+--------------------------------------+

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.

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

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.

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

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

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.

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.