← Illusory Velocity ↑ →
In order to analyse redshift effects between remote initial and final quantum states, we need to study the relationship between inertial reference frames defined locally with speed of light equal to unity and
τ−ρ coordinates used in a Penrose diagram of the universe with a non-physical metric,
. The factors of two in the non-physical metric are equivalent to a two way stretch of the wave function, and yields surprising results in conjunction with local fluctuations in geometry. Gravitational lenses in deep space have four times greater effect for given mass. The interpretation of redshifts by standard formulae leads to an illusory component in the velocities of astronomical objects.
Quantum Coordinates
The
teleconnection has been described using
τ−ρ coordinates for a
Penrose diagram. The
general form of the metric is
These coordinates are defined such that the speed of light is unity, but, due to the variability of
k : ρ → k(ρ) , lines of constant
ρ are not geodesic (as is the case for a standard Friedmann cosmology). Quantum theory was formulated with the non-physical metric,
, given by
If we wish to formulate quantum theory coordinates with an origin which is not on a line of constant
ρ then we must retain the condition that the speed of light is unity.
Definition: Quantum coordinates are defined such that the speed of light is unity in each direction from the origin, and the time coordinate is τ, and the non-physical metric is given locally by
τ-ρ coordinates are a special case of quantum coordinates, but it is not immediately clear how to generalise the angular coefficents to give a global description of the non-physical metric in the general case. Quantum coordinates have the property that they are not affected by the mass distribution. For small distances, parallel displacement of momentum in quantum coordinates is identical to parallel displacement in tangent space (locally Minkowski coordinates).
Bending of Light
The teleconnection model predicts geodesic motion for a classical ray of light for which classical time and space coordinates can be determined at each point within a local reference frame. Thus, there is no change in the prediction of bending of light around the Sun. For bending by a distant
gravitational lens, quantum wave effects are transmitted using parallel displacement of momentum in
τ−ρ coordinates using the non-physical metric,
, given by
So, in the calculation of the deflection by a lens, we must halve the radial distance and double the angular distance, increasing the angle of deflection by a factor of four. Thus the mass required for a given strength of gravitational lens is a quarter of that which would be required in a standard no-CDM theory.
Pioneer Blueshift
In the absence of calibration between clocks on a distant body and clocks on the earth, the energy-momentum of a distant body is constant in
τ-ρ coordinates with non-physical metric
. Meanwhile vectors on Earth are contracted because of the expansion of the Universe. Let the classical momentum of Pioneer be
p at the time,
t0, of loss of radar lock. Ignoring radiation pressure and local gravitational effects such as the gravity of the Sun (set
k = 1 in the metric), for a signal emitted at time
t and detected on Earth, classical energy is given by (note that
t0 is now the earlier time)
Classical energy is proportional to the rate of clocks on the distant body, so signals show a frequency drift,
H0, toward the blue. This effect appears to have been observed in the
anomalous Pioneer blueshift.
Circular Orbits
For the study of
galaxy rotation curves we are interested in circular geodesic motion. Consider an inertial observer, Piers, at a point
P at constant distance,
r, from some arbitrary point
O, on a line of constant
ρ r is sufficiently large that Piers’ clock cannot be calibrated to a clock at
O. Piers wishes to make
O the centre of coordinates, while retaining the constraints that the speed of light is unity and time is
τ. The Pioneer drift is equivalent to an acceleration of
P toward
O,
where there is a factor of two due to the non-physical metric. To take account of this acceleration, Piers introduces rotating coordinates with orbital velocity,
vP (
P is not at rest with respect to Piers). Because of the factors of
22 in the non-physical metric, the orbital velocity corresponding to
aP. is subject to a factor of
¼, where one factor of
½ is from the time coordinate in the denominator of velocity, and another is from the transverse space coordinate. Thus, Piers defines coordinates in which
P has an orbital velocity
vP,
Hence,
The matter distribution does not alter
τ-ρ coordinates or the non-physical metric. So, it does not alter the rotational velocity,
vP, required to define quantum coordinates. If the true orbital velocity of Piers in the gravitational field of a massive body at
O is
vg the orbital velocity in quantum coordinates is
where the signs of each term are the same. The net acceleration toward
O of a point at rest in quantum coordinates is
We use an interpretation in which the wave function describes a state of knowledge used for the calculation of probabilities. When a distant observer, Spike, observes photons from Piers, he applies the classical constraint that there is no expansion in gravitationally bound systems. This removes the apparent acceleration
H0c ⁄ 32, leaving a spectral shift equivalent to total inward acceleration
The first term on the right hand side is simply the acceleration due to gravity and corresponds to Piers’s orbital velocity about
O. The second yields an illusory velocity corresponding to an illusory inverse acceleration law. This term agrees with the phenomenological law used in
MOND and accounts for the flattening of
galaxy rotation curves.
Quantum Coordinates ↑ Particles or Fields? →
The teleconnection has been described using τ−ρ coordinates for a Penrose diagram. The general form of the metric is 
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