Most recent edit on 2009-04-23 09:05:41 by CharlesFrancis
Additions:
.
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,
Deletions:
.
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 <span class=math><i>a</i><sub>P. is subject to a factor of <span class=math>¼, where one factor of <span class=math>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 <span class=math>P has an orbital velocity <span class=math><i>v</i><sub>P"",
Edited on 2009-04-23 09:02:00 by CharlesFrancis
Additions:
These coordinates are defined such that the speed of light is unity, but, due to the variability of the 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).
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 <span class=math><i>a</i><sub>P. is subject to a factor of <span class=math>¼, where one factor of <span class=math>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 <span class=math>P has an orbital velocity <span class=math><i>v</i><sub>P,
<img class=right title="quantum coordinates in orbital motion about a massive body in an expanding cosmology" alt="
OriginOfCurvature-14" src="images/quantumcoordinates/
QuantumCoordinates-14N.gif">The matter distribution does not alter <span class=math>τ-ρ</span> coordinates or the non-physical metric. So, it does not alter the rotational velocity, <span class=math><i>v</i><sub>P, required to define quantum coordinates. If the true orbital velocity of Piers in the gravitational field of a massive body at <span class=math>O is <span class=math><i>v<sub>g</i> the orbital velocity in quantum coordinates is
where the signs of each term are the same. The net acceleration toward <span class=math>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 <span class=math><i>H</i><sub>0</sub><i>c</i> ⁄ 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 <span class=math>O. The second yields an illusory velocity corresponding to an illusory inverse acceleration law. This term agrees with the phenomenological law used in <a href=
http://www.teleconnection.info/rqg/GalaxyRotationCurves#MOND>MOND</a>"»" and accounts for the flattening of
galaxy rotation curves.
Deletions:
These coordinates are defined such that the speed of light is unity, but, due to the variability of the of k : ρ → k(ρ) , lines of constant ρ are not geodesic (as is the case for a standard Friedmann cosmology). In relational quantum gravity, Hilbert space is defined locally in an inertial reference frame.
Definition: Quantum coordinates are inertial, locally Minkowski coordinates.
In quantum coordinates, the speed of light is unity. In order to analyse redshift effects between remote initial and final quantum states, we need to study the relationship between quantum coordinates, defined locally by an inertial observer, and global
τ−ρ coordinates.
For the study of
galaxy rotation curves we are interested in circular geodesic motion. First consider an inertial observer, Spike, at a distance,
r, from
O, where
r is constant over the timescales of observation and sufficiently large that Spike’s clock cannot be calibrated to a clocks at
O . Neither Spike nor
O is assumed to be on a line of constant
ρ but we will assume that their
peculiar velocities» are small, and that the time coordinate is
cosmic time. Spike wishes to make
O the centre of coordinates, while retaining the constraint of quantum coordinates that the speed of light is unity. Quantum coordinates require constant momentum for inertial objects. In other words, quantum coordinates are locally defined with respect to a local inertial frame of reference. Inertial frames at constant distance from
O with uniform acceleration toward
O are in circular motion, so quantum coordinates with an origin at
O are in circular motion about
O relative to
τ−ρ coordinates.
The Pioneer drift is equivalent to an acceleration of Spike toward
O,
where there is a factor of two due to the non-physical metric. To take account of this acceleration, Spike introduces rotating coordinates with orbital velocity,
vP. 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, Spike introduces an orbital velocity
vP,
Applying this argument in the gravitational field of a massive body at
O, this shift is added to the kinematic shift due to the true orbital velocity,
vg, required by Newtonian gravity. The net shifted orbital velocity in quantum coordinates is then
and the net apparent acceleration toward
O in quantum coordinates is
When Spike transforms back to coordinates at his own origin, the apparent acceleration
H0c ⁄ 32 due to circular motion is removed, leaving a shift equivalent to total acceleration
in
τ−ρ coordinates. This shift appears in the wave function of a photon with an initial state in Spike’s reference frame and detected by a remote observer, Piers. The first term on the right hand side is simply the acceleration due to gravity and corresponds to Spike’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.
Edited on 2008-09-20 01:42:11 by CharlesFrancis
Additions:
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.
Deletions:
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 and the interpretation of redshifts by standard formulae leads to an illusory component in the velocities of astronomical objects.
Edited on 2008-09-20 01:41:28 by CharlesFrancis
Additions:
← 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 and the interpretation of redshifts by standard formulae leads to an illusory component in the velocities of astronomical objects.
Deletions:
← Quantum Coordinates ↑ →
For the study of galaxy rotation curves we are interested in circular geodesic motion. First consider an inertial observer, Spike, at a distance, r, from O, where r is constant over the timescales of observation and sufficiently large that Spike’s clock cannot be calibrated to a clocks at O . Neither Spike nor O is assumed to be on a line of constant ρ but we will assume that their peculiar velocities» are small, and that the time coordinate is cosmic time. Spike wishes to make O the centre of coordinates, while retaining the constraint of quantum coordinates that the speed of light is unity. Quantum coordinates require constant momentum for inertial objects. In other words, quantum coordinates are locally defined with respect to a local inertial frame of reference. Inertial frames at constant distance from O with uniform acceleration toward O are in circular motion, so quantum coordinates with an origin at O are in circular motion about O relative to τ−ρ coordinates.
Applying this argument in the gravitational field of a massive body at O, this shift is added to the kinematic shift due to the true orbital velocity, vg, required by Newtonian gravity. The net shifted orbital velocity in quantum coordinates is then
Deletions:
The teleconnection has been described using τ−ρ coordinates for a Penrose diagram. The general form of the metric is
For the study of galaxy rotation curves we are interested in circular geodesic motion. First consider an inertial observer, Spike, at a distance, r, from O, where r is constant over the timescales of observation and sufficiently large that Spike’s clock cannot be calibrated to a clocks at O . Neither Spike nor O is assumed to be on a line of constant ρ but we will assume that their peculiar velocities» are small, and that the time coordinate is cosmic time. Spike wishes to make O the centre of coordinates, while retaining the constraint of quantum coordinates that the speed of light is unity. Quantum coordinates require constant momentum for inertial objects. In other words, quantum coordinates are locally defined with respect to a local inertial frame of reference. Inertial frames at constant distance from O with uniform acceleration toward O are in circular motion, so quantum coordinates with an origin at O are in circular motion about O relative to τ−ρ coordinates.
Applying this argument in the gravitational field of a massive body at O, this shift is added to the kinematic shift due to the true orbital velocity, vg, required by Newtonian gravity. The net shifted orbital velocity in quantum coordinates is then
Edited on 2008-09-19 11:16:58 by CharlesFrancis
Additions:
The teleconnection has been described using τ−ρ coordinates for a Penrose diagram. The general form of the metric is
Deletions:
The teleconnection has been described using τ−ρ coordinates for a Penrose diagram. The general form of the metric is
Edited on 2008-09-19 11:16:34 by CharlesFrancis
Additions:
The teleconnection has been described using τ−ρ coordinates for a Penrose diagram. The general form of the metric is
Deletions:
The teleconnection has been described using τ−ρ coordinates for a Penrose diagram. The general form of the metric is
Edited on 2008-09-19 11:16:13 by CharlesFrancis
Additions:
The teleconnection has been described using τ−ρ coordinates for a Penrose diagram. The general form of the metric is
Deletions:
The teleconnection has been described using τ−ρ coordinates for a Penrose diagram. The general form of the metric is
Edited on 2008-09-19 11:14:32 by CharlesFrancis
Additions:
The teleconnection has been described using τ−ρ coordinates for a Penrose diagram. The general form of the metric is
Deletions:
The teleconnection has been described using τ−ρ coordinates for a Penrose diagram. The general form of the metric is
Edited on 2008-09-19 10:12:29 by CharlesFrancis
Additions:
Applying this argument in the gravitational field of a massive body at O, this shift is added to the kinematic shift due to the true orbital velocity, vg, required by Newtonian gravity. The net shifted orbital velocity in quantum coordinates is then
in τ−ρ coordinates. This shift appears in the wave function of a photon with an initial state in Spike’s reference frame and detected by a remote observer, Piers. The first term on the right hand side is simply the acceleration due to gravity and corresponds to Spike’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.
Deletions:
Applying this argument in the gravitational field of a massive body at O, this shift is added to the kinematic shift due to the true orbital velocity, vg, required by Newtonian gravity. The net shifted orbital velocity in quantum coordinatesis then
in τ−ρ coordinates. This shift appears in the wave function of a photon with an initial state in Spike’s reference frame and detected by a remote observer, Piers. The first term on the right hand side is simply the acceleration due to gravity and corresponds to Spike’s orbital velocity about O. The second yields and illusory velocity obeying an illusory inverse acceleration law. This term agrees with the phenomenological law used in MOND and accounts for the flattening of galaxy rotation curves.
Edited on 2008-09-19 10:05:09 by CharlesFrancis
Additions:
For the study of galaxy rotation curves we are interested in circular geodesic motion. First consider an inertial observer, Spike, at a distance, r, from O, where r is constant over the timescales of observation and sufficiently large that Spike’s clock cannot be calibrated to a clocks at O . Neither Spike nor O is assumed to be on a line of constant ρ but we will assume that their peculiar velocities» are small, and that the time coordinate is cosmic time. Spike wishes to make O the centre of coordinates, while retaining the constraint of quantum coordinates that the speed of light is unity. Quantum coordinates require constant momentum for inertial objects. In other words, quantum coordinates are locally defined with respect to a local inertial frame of reference. Inertial frames at constant distance from O with uniform acceleration toward O are in circular motion, so quantum coordinates with an origin at O are in circular motion about O relative to τ−ρ coordinates.
Applying this argument in the gravitational field of a massive body at O, this shift is added to the kinematic shift due to the true orbital velocity, vg, required by Newtonian gravity. The net shifted orbital velocity in quantum coordinatesis then
Deletions:
For the study of galaxy rotation curves we are interested in circular geodesic motion. First consider an inertial observer, Spike, at a distance, r, from O, where r is constant over the timescales of observation and sufficiently large that Spike’s clock cannot be calibrated to a clocks at O . Neither Spike nor O is assumed to be on a line of constant ρ but we will assume that their peculiar velocities» are small, and that the time coordinate is cosmic time. Spike wishes to make O the centre of coordinates, while retaining the constraint of quantum coordinates that light is unity. Quantum coordinates require constant momentum for inertial objects. In other words, quantum coordinates are locally defined with respect to a local inertial frame of reference. Inertial frames at constant distance from O with uniform acceleration toward O are in circular motion, so quantum coordinates with an origin at O are in circular motion about O relative to τ−ρ coordinates.
Applying this argument in the gravitational field of a massive body at <span class=math>O, this shift is added to the kinematic shift due to the true orbital velocity, <span class=math><i>v<sub>g"", required by Newtonian gravity. The net shifted orbital velocity in quantum coordinatesis then
Edited on 2008-09-19 09:58:56 by CharlesFrancis
Additions:
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 the of k : ρ → k(ρ) , lines of constant ρ are not geodesic (as is the case for a standard Friedmann cosmology). In relational quantum gravity, Hilbert space is defined locally in an inertial reference frame.
Definition: Quantum coordinates are inertial, locally Minkowski coordinates.
In quantum coordinates, the speed of light is unity. In order to analyse redshift effects between remote initial and final quantum states, we need to study the relationship between quantum coordinates, defined locally by an inertial observer, and global τ−ρ coordinates.
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
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.
For the study of galaxy rotation curves we are interested in circular geodesic motion. First consider an inertial observer, Spike, at a distance, r, from O, where r is constant over the timescales of observation and sufficiently large that Spike’s clock cannot be calibrated to a clocks at O . Neither Spike nor O is assumed to be on a line of constant ρ but we will assume that their peculiar velocities» are small, and that the time coordinate is cosmic time. Spike wishes to make O the centre of coordinates, while retaining the constraint of quantum coordinates that light is unity. Quantum coordinates require constant momentum for inertial objects. In other words, quantum coordinates are locally defined with respect to a local inertial frame of reference. Inertial frames at constant distance from O with uniform acceleration toward O are in circular motion, so quantum coordinates with an origin at O are in circular motion about O relative to τ−ρ coordinates.
Deletions:
/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 the of k : ρ → k(ρ) , lines of constant ρ are not geodesic (as is the case for a standard Friedmann cosmology. In relational quantum gravity, Hilbert space is defined locally in an inertial reference frame.
Definition: ; <i>Quantum coordinates</i> are inertial, locally Minkowski coordinates
In quantum coordinates, the speed of light is unity. In order to analyse redshift effects between remote initial and final quantum states, we need to study the relationship between quantum coordinates, defined locally by an inertial observer, and global <span class=math>τ−ρ</span> coordinates.
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 lens, quantum wave effects are transmitted using parallel displacement of momentum in τ−ρ coordinates using the non-physical metric, , given by
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.
For the study of [[GalaxyRotationCurves galaxy rotation curves]] we are interested in circular geodesic motion. First consider an inertial observer, Spike, at a distance, <span class=math><i>r</i>, from <span class=math>O, where <span class=math><i>r</i> is constant over the timescales of observation and sufficiently large that Spike’s clock cannot be calibrated to a clocks at <span class=math>O . Neither Spike nor <span class=math>O is assumed to be on a line of constant <span class=math>ρ but we will assume that their [[http://en.wikipedia.org/wiki/Peculiar_velocity peculiar velocities]] are small, and that the time coordinate is cosmic time. Spike wishes to make <span class=math>O the centre of coordinates, while retaining the constraint of quantum coordinates that light is unity. Quantum coordinates require constant momentum for inertial objects. In other words, quantum coordinates are locally defined with respect to a local inertial frame of reference. Inertial frames at constant distance from <span class=math>O with uniform acceleration toward <span class=math>O are in circular motion, so quantum coordinates with an origin at <span class=math>O are in circular motion about <span class=math>O relative to <span class=math>τ−ρ"" coordinates.
Edited on 2008-09-19 09:49:24 by CharlesFrancis
Additions:
/The teleconnection has been described using τ−ρ coordinates for a Penrose diagram. The general form of the metric is
Deletions:
The [[Teleconnection teleconnection]] has been described using <a href="http://www.teleconnection.info/rqg/Teleconnection#TheTeleconnectionInAFriedmannCosmology"><span» class=math>τ−ρ</span> coordinates</a> for a <a href=http://www.teleconnection.info/rqg/LargeScaleStructure#CosmologicalExpansion>Penrose» diagram</a>. The <a href="http://www.teleconnection.info/rqg/Teleconnection#GeometriesWithExpansion">general» form of the metric</a>"" is
Oldest known version of this page was edited on 2008-09-19 09:45:11 by CharlesFrancis []
Page view:
← Quantum Coordinates ↑ →
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. The calculations yield some surprising results.
Quantum Coordinates
The [[Teleconnection teleconnection]] has been described using <a href="
http://www.teleconnection.info/rqg/Teleconnection#TheTeleconnectionInAFriedmannCosmology"><span» class=math>τ−ρ</span> coordinates</a> for a <a href=
http://www.teleconnection.info/rqg/LargeScaleStructure#CosmologicalExpansion>Penrose» diagram</a>. The <a href="
http://www.teleconnection.info/rqg/Teleconnection#GeometriesWithExpansion">general» form of the metric</a> is
<img alt="
QuantumCoordinates-2" title="The metric in a Friedmann cosmology" src="images/quantumcoordinates/
QuantumCoordinates-2.gif" vspace=3>
These coordinates are defined such that the speed of light is unity, but, due to the variability of the of <span class=math><i>k</i> : ρ → <i>k</i>(ρ) , lines of constant ρ are not geodesic (as is the case for a standard Friedmann cosmology. In relational quantum gravity, [[RelativisticQuantumTheory Hilbert space]] is defined locally in an inertial reference frame.
<Quantum coordinates are inertial, locally Minkowski coordinates<<
In quantum coordinates, the speed of light is unity. In order to analyse redshift effects between remote initial and final quantum states, we need to study the relationship between quantum coordinates, defined locally by an inertial observer, and global
τ−ρ coordinates.
====<a name="
BendingOfLight"></a>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 lens, quantum wave effects are transmitted using parallel displacement of momentum in <span class=math>τ−ρ</span> coordinates using the non-physical metric, <img alt="
QuantumCoordinates-4" title="The non-physical metric" src="images/quantumcoordinates/
QuantumCoordinates-4.gif" align=top vspace=0>, given by
<img alt="
QuantumCoordinates-5" title="The non-physical line element" src="images/quantumcoordinates/
QuantumCoordinates-5.gif" align=top vspace=3>
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.
====<a name="
PioneerBlueshift"></a>Pioneer Blueshift====
In the absence of calibration between clocks on a distant body and clocks on the earth, the momentum of the distant body is constant in <span class=math>τ-ρ coordinates with non-physical metric <img alt="
QuantumCoordinates-6" title="The non-physical metric" src="images/quantumcoordinates/
QuantumCoordinates-6.gif" align=top vspace=0>. Ignoring local gravitational effects (set <span class=math><i>k</i> = 1 in the metric), as determined in signals detected on Earth classical energy is given by (note that <span class=math><i>t</i><sub>0 is now the earlier time)
<img alt="
QuantumCoordinates-7" title="Parallel displacement of energy component of energy momentum" src="images/quantumcoordinates/
QuantumCoordinates-7.gif" align=top vspace=3>
Classical energy is proportional to the rate of clocks on the distant body, so signals show a frequency drift, <span class=math><i>H</i><sub>0 , toward the blue. This effect appears to have been observed in the [[Pioneer anomalous Pioneer blueshift]].
====<a name="
CircularOrbits"></a>Circular Orbits====
<img class=right title="stationary motion in expanding cosmology in the absence of gravitation" alt="
OriginOfCurvature-9" src="images/quantumcoordinates/
QuantumCoordinates-9.gif">For the study of
galaxy rotation curves we are interested in circular geodesic motion. First consider an inertial observer, Spike, at a distance,
r, from
O, where
r is constant over the timescales of observation and sufficiently large that Spike’s clock cannot be calibrated to a clocks at
O . Neither Spike nor
O is assumed to be on a line of constant
ρ but we will assume that their
peculiar velocities» are small, and that the time coordinate is cosmic time. Spike wishes to make
O the centre of coordinates, while retaining the constraint of quantum coordinates that light is unity. Quantum coordinates require constant momentum for inertial objects. In other words, quantum coordinates are locally defined with respect to a local inertial frame of reference. Inertial frames at constant distance from
O with uniform acceleration toward
O are in circular motion, so quantum coordinates with an origin at
O are in circular motion about
O relative to
τ−ρ coordinates.
The Pioneer drift is equivalent to an acceleration of Spike toward
O,
where there is a factor of two due to the non-physical metric. To take account of this acceleration, Spike introduces rotating coordinates with orbital velocity,
vP. 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, Spike introduces an orbital velocity
vP,
Hence,
Applying this argument in the gravitational field of a massive body at <span class=math>O, this shift is added to the kinematic shift due to the true orbital velocity, <span class=math><i>v<sub>g, required by Newtonian gravity. The net shifted orbital velocity in quantum coordinatesis then
<img alt="
QuantumCoordinates-16" title="net apparent orbital velocity in quantum coordinates" src="images/quantumcoordinates/
QuantumCoordinates-16.gif" align=top vspace=3>
and the net apparent acceleration toward <span class=math>O in quantum coordinates is
<img alt="
QuantumCoordinates-17" title="net apparent centripetal acceleration in quantum coordinates" src="images/quantumcoordinates/
QuantumCoordinates-17.gif" align=top vspace=3>
When Spike transforms back to coordinates at his own origin, the apparent acceleration <span class=math><i>H</i><sub>0</sub><i>c</i> ⁄ 32 due to circular motion is removed, leaving a shift equivalent to total acceleration
<img alt="
QuantumCoordinates-19" title="net apparent orbital velocity in τ-ρ coordinates" src="images/quantumcoordinates/
QuantumCoordinates-19.gif" align=top vspace=3>
in <span class=math>τ−ρ coordinates. This shift appears in the wave function of a photon with an initial state in Spike’s reference frame and detected by a remote observer, Piers. The first term on the right hand side is simply the acceleration due to gravity and corresponds to Spike’s orbital velocity about <span class=math>O. The second yields and illusory velocity obeying an illusory inverse acceleration law. This term agrees with the phenomenological law used in <a href=
http://www.teleconnection.info/rqg/GalaxyRotationCurves#GalaxyRotationCurves>MOND</a>"»" 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 
