Most recent edit on 2008-09-01 00:31:41 by CharlesFrancis
Additions:
Before one can study physics, one first has to set up a coordinate system. This must be done relative to matter, i.e. relative to a reference frame. Setting up a coordinate system from reference matter requires the use of Einstein’s synchronisation procedure, or an equivalent procedure. This necessitates that reference matter is chosen within a neighbourhood of the observer. If we want to compare a coordinate system here to one on a spacecraft, we need a rule to determine the meaning of parallel for vectors defined in different coordinate systems.
In non-Euclidean geometry, an affine connection is a rule defining parallel vectors, when the vectors are defined in tangent space at points separated by a small displacement. An affine connection allows us to parallel transport a vector from one place to another along a path in such a way that we may say that it remains parallel to itself in each part of the motion. If we slide a ruler, for example on a spacecraft like Pioneer, even a short way into space, we cease to be in direct contact with it and have to rely on electromagnetic transmissions to read the ruler. Sliding a ruler using parallel transport is clearly sound for coordinate systems local to the Earth, but we cannot physically transport rulers indefinitely far into space, and certainly not from distant stars. Instead we must rely on the transmission of light. In practice, we deduce the properties of distant coordinate systems from properties we find in light from distant stellar objects. We can do this only if we know the laws governing the transmission of light over large distances.
In the early part of the 20th century, attempts were made, notably by Einstein himself, and by Herman Weyl, to unify general relativity with classical electromagnetism. Those attempts failed and it seems that theoretical physicists have lost interest. The Levi-Civita connection is almost universally accepted, and its inconsistency is largely ignored. This is dangerous. Our knowledge of stellar kinematics and of cosmology is almost entirely dependent on spectroscopy» (doppler shift»). There is no direct empirical evidence that the standard Doppler» law holds for the transmission of light across astronomical distances. At the same time, observations appear to reveal a universe with bizarre and paradoxical properties. There is still no sound theoretical basis for the cosmological constant, for exotic cold dark matter», for cosmic inflation» or for modifying Newton’s law of gravity (MOND»). A significant minority of cosmologists recognise that there are serious problems in standard cosmology, but, in the main, empirical evidence which plainly contradicts the standard model, such as galaxy lensing profiles, and the rotation curves of globular clusters», is ignored.
A number of Einstein’s attempts at unification with classical electromagnetism were centred on the investigation of distant parallelism». Relational quantum gravity uses as similar, but not identical, notion, and seeks unification between general relativity and quantum electrodynamics. In relational quantum gravity, general relativity is regarded as correct in the classical correspondence, but must be extended for the treatment of quantum phenomena. Since light consists of photons, light is treated quantum mechanically, even in the description of its transmission over astronomical distances. This is done using a form of distant parallelism which I have called the teleconnection to distinguish it from other teleparallel theories of gravity. I will show that the teleconnection reduces to the Levi-Civita connection in the classical correspondence, and that, when cosmological expansion can be neglected, predicted redshifts are identical to those of general relativity, but it turns out that cosmological expansion affects the calculation of redshift even, in the case of the Pioneer anomaly», within the environs of the solar system.
Deletions:
In non-Euclidean geometry, an affine connection is a rule defining parallel vectors, when the vectors are defined in tangent space at points separated by a small displacement. An affine connection allows us to parallel transport a vector from one place to another along a path in such a way that we may say that it remains parallel to itself in each part of the motion. Sliding a ruler using parallel transport is clearly sound for coordinate systems local to the Earth, but we cannot physically transport rulers indefinitely far into space, and certainly not from distant stars. Instead we must rely on the transmission of light. If we do slide a ruler, for example on a spacecraft like Pioneer, even a short way into space, we cease to be in direct contact with it and have to rely on electromagnetic transmissions to read the ruler. In practice, we deduce the properties of distant coordinate systems from properties we find in light from distant stellar objects. We can do this only if we know the laws governing the transmission of light over large distances.
In the early part of the 20th century, attempts were made, notably by Einstein himself, and by Herman Weyl, to unify general relativity with classical electromagnetism. Those attempts failed and it seems that theoretical physicists have lost interest. The Levi-Civita connection is almost universally accepted, and its inconsistency is largely ignored. This is dangerous. Our knowledge of stellar kinematics and of cosmology is almost entirely dependent on spectroscopy» (doppler shift»). There is no direct empirical evidence that the standard Doppler» law holds for the transmission of light across astronomical distances. At the same time, observations appear to reveal a universe with bizarre and paradoxical properties. There is still no sound theoretical basis for the cosmological constant», for exotic cold dark matter», for cosmic inflation» or for modifying Newton’s law of gravity (MOND»). A significant minority of cosmologists recognise that there are serious problems in standard cosmology, but, in the main, empirical evidence which plainly contradicts the standard model, such as galaxy lensing profiles, and the rotation curves of globular clusters», is ignored.
A number of Einstein’s attempts at unification with classical electromagnetism were centred on the investigation of distant parallelism». Relational quantum gravity uses as similar, but not identical, notion, and seeks unification between general relativity and quantum electrodynamics. In relational quantum gravity, general relativity is regarded as correct in the classical correspondence, but must be extended for the treatment of quantum phenomena. Since light consists of photons, light is treated quantum mechanically, even in the description of its transmission over astronomical distances. This is done using a form of distant parallelism which I have called the teleconnection to distinguish it from other teleparallel theories of gravity. I will show that the teleconnection reduces to the Levi-Civitaconnection in the classical correspondence, and that, when cosmological expansion can be neglected, predicted redshifts are identical to those of general relativity, but it turns out that cosmological expansion affects the calculation of redshift even, in the case of the Pioneer anomaly», within the environs of the solar system.
Edited on 2008-08-13 03:57:55 by CharlesFrancis
Additions:
← The Teleconnection in a Friedmann Cosmology ↑ →
Deletions:
← The Teleconnection in a Friedmann Cosmology →
Edited on 2008-05-04 11:28:39 by ErikAnderson [Fixed first instance of link to Levi-Civita]
Additions:
To introduce the idea of the teleconnection I will here describe it diagramatically for a Friedmann Cosmology. In this case the analysis is particularly simple. The result is that the cosmological redshift of light from a distant galaxy is proportional to the square of the expansion parameter, not linear with it as predicted by the Levi-Civita connection. A more general, and more mathematical, treatment will be given on the next page.
Deletions:
To introduce the idea of the teleconnection I will here describe it diagramatically for a Friedmann Cosmology. In this case the analysis is particularly simple. The result is that the cosmological redshift of light from a distant galaxy is proportional to the square of the expansion parameter, not linear with it as predicted by the Levi-Civita connection. A more general, and more mathematical, treatment will be given on the next page.
Edited on 2008-05-02 10:41:15 by CharlesFrancis
Additions:
To introduce the idea of the teleconnection I will here describe it diagramatically for a Friedmann Cosmology. In this case the analysis is particularly simple. The result is that the cosmological redshift of light from a distant galaxy is proportional to the square of the expansion parameter, not linear with it as predicted by the Levi-Civita connection. A more general, and more mathematical, treatment will be given on the next page.
In non-Euclidean geometry, an affine connection is a rule defining parallel vectors, when the vectors are defined in tangent space at points separated by a small displacement. An affine connection allows us to parallel transport a vector from one place to another along a path in such a way that we may say that it remains parallel to itself in each part of the motion. Sliding a ruler using parallel transport is clearly sound for coordinate systems local to the Earth, but we cannot physically transport rulers indefinitely far into space, and certainly not from distant stars. Instead we must rely on the transmission of light. If we do slide a ruler, for example on a spacecraft like Pioneer, even a short way into space, we cease to be in direct contact with it and have to rely on electromagnetic transmissions to read the ruler. In practice, we deduce the properties of distant coordinate systems from properties we find in light from distant stellar objects. We can do this only if we know the laws governing the transmission of light over large distances.
In standard general relativity it is assumed that photon momentum is parallel transported through large distances, using the Levi-Civita connection in a manner analogous to sliding a ruler across a curved surface while maintaining parallellism in each part of the motion. It is sometimes argued that, since the the laws of physics are locally the same in all coordinates systems, Maxwell’s equations, which govern the propagation of light, must hold locally in empty space, and hence that the Levi-Civita connection holds in empty space. The argument relies on induction», which cannot be rigorously justified, and it ignores the fact that we cannot define local coordinates in empty space, since empty space contains neither reference matter nor matter whose position can be measured. More seriously still, it ignores the known inconsistency between classical electromagnetism and general relativity. It was pointed out by Einstein that the Levi-Civita connection appears to be wrong. In particular, it leads to an inconsistency in the paths of light transmitted by a distant object, seen in the mismatch of Alf’s and Beth’s coordinates. Moreover, the application of the Levi-Civita connection takes no account of the propagation of a photon wave function in a curved spacetime, which would imply that a photon created with precise momentum in a distant galaxy would not have precise momentum when detected on Earth.
In the early part of the 20th century, attempts were made, notably by Einstein himself, and by Herman Weyl, to unify general relativity with classical electromagnetism. Those attempts failed and it seems that theoretical physicists have lost interest. The Levi-Civita connection is almost universally accepted, and its inconsistency is largely ignored. This is dangerous. Our knowledge of stellar kinematics and of cosmology is almost entirely dependent on spectroscopy» (doppler shift»). There is no direct empirical evidence that the standard Doppler» law holds for the transmission of light across astronomical distances. At the same time, observations appear to reveal a universe with bizarre and paradoxical properties. There is still no sound theoretical basis for the cosmological constant», for exotic cold dark matter», for cosmic inflation» or for modifying Newton’s law of gravity (MOND»). A significant minority of cosmologists recognise that there are serious problems in standard cosmology, but, in the main, empirical evidence which plainly contradicts the standard model, such as galaxy lensing profiles, and the rotation curves of globular clusters», is ignored.
A number of Einstein’s attempts at unification with classical electromagnetism were centred on the investigation of distant parallelism». Relational quantum gravity uses as similar, but not identical, notion, and seeks unification between general relativity and quantum electrodynamics. In relational quantum gravity, general relativity is regarded as correct in the classical correspondence, but must be extended for the treatment of quantum phenomena. Since light consists of photons, light is treated quantum mechanically, even in the description of its transmission over astronomical distances. This is done using a form of distant parallelism which I have called the teleconnection to distinguish it from other teleparallel theories of gravity. I will show that the teleconnection reduces to the Levi-Civitaconnection in the classical correspondence, and that, when cosmological expansion can be neglected, predicted redshifts are identical to those of general relativity, but it turns out that cosmological expansion affects the calculation of redshift even, in the case of the Pioneer anomaly», within the environs of the solar system.
The teleconnection defines the way in which Beth converts the value of momentum calculated by Alf to the value found in her own coordinate system, preserving parallelism. Wave evolution (red) takes place in a tangent space. Discontinuity is associated with wave function collapse, at which point the teleconnection describes the projection of the evolved initial state to the final state (blue). The teleconnection does not apply when there is an intermediate measured (or classical) state. The same recipe applies whether Alf and Beth are separated by astronomical distances, or are in the same laboratory, and reduces to the Levi-Civita connection when the initial and final states are separated by small times. The classical correspondence is conceived as a limit of many quantum motions in which curved spacetime (green) emerges as the envelope of flat spaces (pale blue), and in which parallel transport is found in the limit of small teleparallel displacements.</td></table>
<img alt="TeleconnectionIntro-33" title="redshift as a function of expansion under the Levi-Civita connection" src="images/teleconnectionintro/TeleconnectionIntro-33.gif" vspace=3>""
Deletions:
To introduce the idea of the teleconnection I will here describe it diagramatically for a Friedmann Cosmology. In this case the analysis is particularly simple. The result is that the cosmological redshift of light from a distant galaxy is proportional to the square of the expansion parameter, not linear with it as predicted by the affine connection. A more general, and more mathematical, treatment will be given on the next page.
In non-Euclidean geometry, an affine connection is a rule defining parallel vectors, when the vectors are defined at points separated by a small displacement. An affine connection allows us to parallel transport a vector from one place to another along a path in such a way that we may say that it remains parallel to itself in each part of the motion. Sliding a ruler using parallel transport is clearly sound for coordinate systems local to the Earth, but we cannot physically transport rulers indefinitely far into space, and certainly not from distant stars. Instead we must rely on the transmission of light. If we do slide a ruler, for example on a spacecraft like Pioneer, even a short way into space, we cease to be in direct contact with it and have to rely on electromagnetic transmissions to read the ruler. In practice, we deduce the properties of distant coordinate systems from properties we find in light from distant stellar objects. We can do this only if we know the laws governing the transmission of light over large distances.
In standard general relativity it is assumed that photon momentum is parallel transported through large distances, using the Levi-Civita connection in a manner analogous to sliding a ruler across a curved surface while maintaining parallellism in each part of the motion. It is sometimes argued that, since the the laws of physics are locally the same in all coordinates systems, Maxwell’s equations, which govern the propagation of light, must hold locally in empty space, and hence that the Levi-Civita connection holds in empty space. The argument relies on induction», which cannot be rigorously justified, and it ignores the fact that we cannot define local coordinates in empty space, since empty space contains neither reference matter nor matter whose position can be measured. More seriously still, it ignores the known inconsistency between classical electromagnetism and general relativity. It was pointed out by Einstein that the Levi-Civita connection appears to be wrong. In particular, it leads to an inconsistency in the paths of light transmitted by a distant object, seen in the mismatch of Alf’s and Beth’s coordinates. Moreover, the application of the affine connection takes no account of the propagation of a photon wave function in a curved spacetime, which would imply that a photon created with precise momentum in a distant galaxy would not have precise momentum when detected on Earth.
In the early part of the 20th century, attempts were made, notably by Einstein himself, and by Herman Weyl, to unify general relativity with classical electromagnetism. Those attempts failed and it seems that theoretical physicists have lost interest. The affine connection is almost universally accepted, and its inconsistency is largely ignored. This is dangerous. Our knowledge of stellar kinematics and of cosmology is almost entirely dependent on spectroscopy» (doppler shift»). There is no direct empirical evidence that the standard Doppler» law holds for the transmission of light across astronomical distances. At the same time, observations appear to reveal a universe with bizarre and paradoxical properties. There is still no sound theoretical basis for the cosmological constant», for exotic cold dark matter», for cosmic inflation» or for modifying Newton’s law of gravity (MOND»). A significant minority of cosmologists recognise that there are serious problems in standard cosmology, but, in the main, empirical evidence which plainly contradicts the standard model, such as galaxy lensing profiles, and the rotation curves of globular clusters», is ignored.
A number of Einstein’s attempts at unification with classical electromagnetism were centred on the investigation of distant parallelism». Relational quantum gravity uses as similar, but not identical, notion, and seeks unification between general relativity and quantum electrodynamics. In relational quantum gravity, general relativity is regarded as correct in the classical correspondence, but must be extended for the treatment of quantum phenomena. Since light consists of photons, light is treated quantum mechanically, even in the description of its transmission over astronomical distances. This is done using a form of distant parallelism which I have called the teleconnection to distinguish it from other teleparallel theories of gravity. I will show that the teleconnection reduces to the standard affine connection in the classical correspondence, and that, when cosmological expansion can be neglected, predicted redshifts are identical to those of general relativity, but it turns out that cosmological expansion affects the calculation of redshift even, in the case of the Pioneer anomaly», within the environs of the solar system.
The teleconnection defines the way in which Beth converts the value of momentum calculated by Alf to the value found in her own coordinate system, preserving parallelism. Wave evolution (red) takes place in a tangent space. Discontinuity is associated with wave function collapse, at which point the teleconnection describes the projection of the evolved initial state to the final state (blue). The teleconnection does not apply when there is an intermediate measured (or classical) state. The same recipe applies whether Alf and Beth are separated by astronomical distances, or are in the same laboratory, and reduces to the affine connection when the initial and final states are separated by small times. The classical correspondence is conceived as a limit of many quantum motions in which curved spacetime (green) emerges as the envelope of flat spaces (pale blue), and in which parallel transport is found in the limit of small teleparallel displacements.</td></table>
<img alt="TeleconnectionIntro-33" title="redshift as a function of expansion under the affine connection" src="images/teleconnectionintro/TeleconnectionIntro-33.gif" vspace=3>""
Edited on 2008-04-14 00:46:29 by CharlesFrancis
Additions:
<table width=100% cellspacing=0 cellpadding=0><td align=centre><img class="centre" alt="TeleconnectionIntro-6" title="Alf’s and Beth’s maps in a closed expanding universe" src="images/teleconnectionintro/TeleconnectionIntro-6N.gif"></td></table>
<table width=100% cellspacing=0 cellpadding=0><td align=centre><img class="centre" alt="TeleconnectionIntro-10" title="Beth’s tangent chart" src="images/teleconnectionintro/TeleconnectionIntro-10N.gif"></td></table> Beth defines tangent space vectors in these coordinates (red), denoted by a bar, and which can be translated radially in tangent space and are equal to physical vectors at her own origin. She defines <a href=http://www.teleconnection.info/rqg/FoundationsOfQuantumTheory#PlaneWaveStates>plane» wave states</a> obeying <a href=http://www.teleconnection.info/rqg/Evolution#Newton’sFirstLaw>Newton’s» first law</a> using barred vectors,
<table width=100% cellspacing=0 cellpadding=0><td><img class="right" alt="TeleconnectionIntro-13" title="Alf’s tangent space is half the size of Beth’s" src="images/teleconnectionintro/TeleconnectionIntro-13N.gif">Similarly Alf defines a tangent chart at <span class=math>A</span> at cosmic time <span class=math><i>t</i><sub>1</sub></span>,
Deletions:
<table width=100% cellspacing=0 cellpadding=0><td align=centre><img class="centre" alt="TeleconnectionIntro-6" title="Alf’s and Beth’s maps in a closed expanding universe" src="images/teleconnectionintro/TeleconnectionIntro-6.gif"></td></table>
<table width=100% cellspacing=0 cellpadding=0><td align=centre><img class="centre" alt="TeleconnectionIntro-10" title="Beth’s tangent chart" src="images/teleconnectionintro/TeleconnectionIntro-10.gif"></td></table> Beth defines tangent space vectors in these coordinates (red), denoted by a bar, and which can be translated radially in tangent space and are equal to physical vectors at her own origin. She defines <a href=http://www.teleconnection.info/rqg/FoundationsOfQuantumTheory#PlaneWaveStates>plane» wave states</a> obeying <a href=http://www.teleconnection.info/rqg/Evolution#Newton’sFirstLaw>Newton’s» first law</a> using barred vectors,
<table width=100% cellspacing=0 cellpadding=0><td><img class="right" alt="TeleconnectionIntro-13" title="Alf’s tangent space is half the size of Beth’s" src="images/teleconnectionintro/TeleconnectionIntro-13.gif">Similarly Alf defines a tangent chart at <span class=math>A</span> at cosmic time <span class=math><i>t</i><sub>1</sub></span>,
Edited on 2008-04-14 00:43:07 by CharlesFrancis
Additions:
Since the calculation of wave evolution is carried out by an observer at a particular place and time, it is natural that the calculation should be done using the metric at that place and time. In other words, wave evolution is determined in a tangent space. More strictly this generalises the notion of tangent space (it can be regarded as a fibre»), but for our purpose the important properties are those of a tangent space, and I will refer to it as such. As for the wave function itself, no ontology is attached to tangent space. 3-momentum has been defined from the Fourier transform of the probability amplitude, and found to be a well defined, locally conserved, property. Now, if momentum is a well defined property for Alf on a space craft or another planet, then it is also a well defined property for Beth on Earth, because Alf can communicate to Beth his value of momentum. Alf's value of momentum does not have to be the same as Beth's, but we seek a way of converting his value to the value Beth will use.
<table width=100% cellspacing=0 cellpadding=0><td align=centre><img class="centre" alt="TeleconnectionIntro-6" title="Alf’s and Beth’s maps in a closed expanding universe" src="images/teleconnectionintro/TeleconnectionIntro-6.gif"></td></table>
<table width=100% cellspacing=0 cellpadding=0><td align=centre><img class="centre" alt="TeleconnectionIntro-10" title="Beth’s tangent chart" src="images/teleconnectionintro/TeleconnectionIntro-10.gif"></td></table> Beth defines tangent space vectors in these coordinates (red), denoted by a bar, and which can be translated radially in tangent space and are equal to physical vectors at her own origin. She defines <a href=http://www.teleconnection.info/rqg/FoundationsOfQuantumTheory#PlaneWaveStates>plane» wave states</a> obeying <a href=http://www.teleconnection.info/rqg/Evolution#Newton’sFirstLaw>Newton’s» first law</a> using barred vectors,
<table width=100% cellspacing=0 cellpadding=0><td><img class="right" alt="TeleconnectionIntro-13" title="Alf’s tangent space is half the size of Beth’s" src="images/teleconnectionintro/TeleconnectionIntro-13.gif">Similarly Alf defines a tangent chart at <span class=math>A</span> at cosmic time <span class=math><i>t</i><sub>1</sub></span>,
Deletions:
Since the calculation of wave evolution is carried out by an observer at a particular place and time, it is natural that the calculation should be done using the metric at that place and time. In other words, wave evolution is determined in a tangent space. As for the wave function itself, no ontology is attached to tangent space. 3-momentum has been defined from the Fourier transform of the probability amplitude, and found to be a well defined, locally conserved, property. Now, if momentum is a well defined property for Alf on a space craft or another planet, then it is also a well defined property for Beth on Earth, because Alf can communicate to Beth his value of momentum. Alf's value of momentum does not have to be the same as Beth's, but we seek a way of converting his value to the value Beth will use.
<table width=100% cellspacing=0 cellpadding=0><td align=centre><img class="centre" alt="TeleconnectionIntro-6" title="Alf’s and Beth’s maps in a closed expanding universe" src="images/teleconnectionintro/TeleconnectionIntro-6N.gif"></td></table>
<table width=100% cellspacing=0 cellpadding=0><td align=centre><img class="centre" alt="TeleconnectionIntro-10" title="Beth’s tangent chart" src="images/teleconnectionintro/TeleconnectionIntro-10N.gif"></td></table> Beth defines tangent space vectors in these coordinates (red), denoted by a bar, and which can be translated radially in tangent space and are equal to physical vectors at her own origin. She defines <a href=http://www.teleconnection.info/rqg/FoundationsOfQuantumTheory#PlaneWaveStates>plane» wave states</a> obeying <a href=http://www.teleconnection.info/rqg/Evolution#Newton’sFirstLaw>Newton’s» first law</a> using barred vectors,
<table width=100% cellspacing=0 cellpadding=0><td><img class="right" alt="TeleconnectionIntro-13" title="Alf’s tangent space is half the size of Beth’s" src="images/teleconnectionintro/TeleconnectionIntro-13N.gif">Similarly Alf defines a tangent chart at <span class=math>A</span> at cosmic time <span class=math><i>t</i><sub>1</sub></span>,
Edited on 2008-04-07 17:17:50 by ErikAnderson [New Diagrams]
Additions:
<table width=100% cellspacing=0 cellpadding=0><td align=centre><img class="centre" alt="TeleconnectionIntro-6" title="Alf’s and Beth’s maps in a closed expanding universe" src="images/teleconnectionintro/TeleconnectionIntro-6N.gif"></td></table>
<table width=100% cellspacing=0 cellpadding=0><td align=centre><img class="centre" alt="TeleconnectionIntro-10" title="Beth’s tangent chart" src="images/teleconnectionintro/TeleconnectionIntro-10N.gif"></td></table> Beth defines tangent space vectors in these coordinates (red), denoted by a bar, and which can be translated radially in tangent space and are equal to physical vectors at her own origin. She defines <a href=http://www.teleconnection.info/rqg/FoundationsOfQuantumTheory#PlaneWaveStates>plane» wave states</a> obeying <a href=http://www.teleconnection.info/rqg/Evolution#Newton’sFirstLaw>Newton’s» first law</a> using barred vectors,
<table width=100% cellspacing=0 cellpadding=0><td><img class="right" alt="TeleconnectionIntro-13" title="Alf’s tangent space is half the size of Beth’s" src="images/teleconnectionintro/TeleconnectionIntro-13N.gif">Similarly Alf defines a tangent chart at <span class=math>A</span> at cosmic time <span class=math><i>t</i><sub>1</sub></span>,
Deletions:
<table width=100% cellspacing=0 cellpadding=0><td align=centre><img class="centre" alt="TeleconnectionIntro-6" title="Alf’s and Beth’s maps in a closed expanding universe" src="images/teleconnectionintro/TeleconnectionIntro-6.gif"></td></table>
<table width=100% cellspacing=0 cellpadding=0><td align=centre><img class="centre" alt="TeleconnectionIntro-10" title="Beth’s tangent chart" src="images/teleconnectionintro/TeleconnectionIntro-10.gif"></td></table> Beth defines tangent space vectors in these coordinates (red), denoted by a bar, and which can be translated radially in tangent space and are equal to physical vectors at her own origin. She defines <a href=http://www.teleconnection.info/rqg/FoundationsOfQuantumTheory#PlaneWaveStates>plane» wave states</a> obeying <a href=http://www.teleconnection.info/rqg/Evolution#Newton’sFirstLaw>Newton’s» first law</a> using barred vectors,
<table width=100% cellspacing=0 cellpadding=0><td><img class="right" alt="TeleconnectionIntro-13" title="Alf’s tangent space is half the size of Beth’s" src="images/teleconnectionintro/TeleconnectionIntro-13.gif">Similarly Alf defines a tangent chart at <span class=math>A</span> at cosmic time <span class=math><i>t</i><sub>1</sub></span>,
Edited on 2008-04-03 04:08:02 by CharlesFrancis
Additions:
 Similarly Alf defines a tangent chart at A at cosmic time t1,
The square redshift law applies to [[http://en.wikipedia.org/wiki/Spectroscopy spectroscopic]] measurements of the wavelength of light, and takes into account that, as viewed by Beth, Alf’s coordinate axes are enlarged by a factor |
<span class=math><i>a</i><sub>0</sub> ⁄ <i>a</i><sub>1</sub>. Alf’s measurement of energy-momentum is relative to his own coordinate axes. This factor must be removed in the calculation of energy/momentum of light from a distant source. Thus, the energy transferred by a photon passing from Alf to Beth is inverse to expansion, not inverse to the square of expansion, giving the same relation as is given by parallel transport under the <a href=http://www.teleconnection.info/rqg/GeneralRelativity#TheLevi-CivitaConnection>Levi-Civita» connection</a>"". The square redshift law affects spectroscopic measurements of light from distant sources, but does not affect energy transfer.
is recovered for a classical ray of light which is observable in each part of the motion. For example, the cosmic background radiation» is continuously observable and obeys the linear redshift law, not the square law applicable to the observation of a photon from a distant source, and in which the detection of the photon takes place after diffraction or refraction of the quantum wave function.
Deletions:
|
Similarly Alf defines a tangent chart at A at cosmic time t1,
The square redshift law applies to [[http://en.wikipedia.org/wiki/Spectroscopy spectroscopic]] measurements of the wavelength of light, and takes into account that, as viewed by Beth, Alf’s coordinate axes are enlarged by a factor |
<span class=math><i>a</i><sub>0</sub> ⁄ <i>a</i><sub>1</sub>. Alf’s measurement of energy-momentum is relative to his own coordinate axes. This factor must be removed in the calculation of energy/momentum of light from a distant source. Thus, the energy transferred by a photon passing from Alf to Beth is inverse to expansion, not inverse to the square of expansion, giving the same relation as is given by parallel transport under the <a href=http://www.teleconnection.info/rqg/GeneralRelativity#TheLevi-CivitaConnection>Levi-Civita» connection</a>"". The square redshift law affects spectroscopic measurements of light from distant sources, but does not affect energy transfer.
is recovered for a classical ray of light which is observable in each part of the motion. For example, the cosmic background radiation» is continuously observable and obeys the linear redshift law, not the square law applicable to the observation of a photon from a distant source.
Edited on 2008-04-03 03:55:07 by CharlesFrancis
Additions:
It follows that, for a classical motion, momentum is parallel transported from ti to ti−1 in the normal way. The result is parallel transport. An overall factor,
is recovered for a classical ray of light which is observable in each part of the motion. For example, the cosmic background radiation» is continuously observable and obeys the linear redshift law, not the square law applicable to the observation of a photon from a distant source.
Deletions:
It follows that momentum is parallel transported from ti to ti−1 in the normal way, and that the result is parallel transport. An overall factor,
is recovered.
Edited on 2008-04-03 03:38:53 by CharlesFrancis
Additions:
The teleconnection applies to a quantum motion between an initial state and a final state when it is not meaningful to describe the intervening state in terms of measured physical quantities. In the classical correspondence, time is taken as a continuous variable and motion may be described as a sequence of measured states 
at instances t = tn < tn−1 < … < t1 < t0 in the limit in which max(ti−1 − ti) → 0. The state at any instant, ti, may be regarded as an initial state, and the state at the next instant, ti−1, may be regarded as a final state for that part of the motion. Quantum theory is redefined and the final state then becomes the initial state for the next part of the motion. At each instant, ti, when quantum theory is redefined, the coordinate system is rescaled, removing a factor a(ti−1) ⁄ a(ti). The paths of light and of the galaxies become continuous in the limit, and the classical notion of continuous expansion is restored.
It follows that momentum is parallel transported from ti to ti−1 in the normal way, and that the result is parallel transport. An overall factor,
Deletions:
The teleconnection applies to a quantum motion between an initial state and a final state when it is not meaningful to describe the intervening state in terms of measured physical quantities. In the classical correspondence, time is taken as a continuous variable and motion may be described as a sequence of measured states at instances t = tn < tn-1 < … < t1 < t0 in the limit in which max(ti − tn+1) → 0. The state at any instant, ti, may be regarded as an initial state, and the state at the next instant, ti−1, may be regarded as a final state for that part of the motion. Quantum theory is redefined and the final state then becomes the initial state for the next part of the motion. At each instant, ti, when quantum theory is redefined, the coordinate system is rescaled, removing a factor a(ti) ⁄ a(ti+1). The paths of light and of the galaxies become continuous in the limit, and the classical notion of continuous expansion is restored.
It follows that momentum is parallel transported from ti+1 to ti in the normal way, and that the result is parallel transport. An overall factor,
Edited on 2008-04-03 03:23:45 by CharlesFrancis
Additions:
The square redshift law applies to spectroscopic» measurements of the wavelength of light, and takes into account that, as viewed by Beth, Alf’s coordinate axes are enlarged by a factor a0 ⁄ a1. Alf’s measurement of energy-momentum is relative to his own coordinate axes. This factor must be removed in the calculation of energy/momentum of light from a distant source. Thus, the energy transferred by a photon passing from Alf to Beth is inverse to expansion, not inverse to the square of expansion, giving the same relation as is given by parallel transport under the Levi-Civita connection. The square redshift law affects spectroscopic measurements of light from distant sources, but does not affect energy transfer.
When Beth performs a measurement, the wave function collapses. Her measurement determines both the final state of a quantum motion of a particle travelling from Alf to Beth, and the initial state for the next part of the motion. At the point of the measurement quantum theory is reformulated with coordinate axes in scale by a factor a1 ⁄ a0.
Deletions:
The square redshift law applies to spectroscopic» measurements of the wavelength of light, and takes into account that, on the space in which wave functions are defined, Alf’s coordinate axes are enlarged by a factor a0 ⁄ a1 relative to Beth’s. Alf’s measurement of energy-momentum is relative to his own coordinate axes. This factor must be removed in the calculation of energy/momentum of light from a distant source. Thus, the energy transferred by a photon passing from Alf to Beth is inverse to expansion, not inverse to the square of expansion, giving the same relation as is given by parallel transport under the Levi-Civita connection. The square redshift law affects spectroscopic measurements of light from distant sources, but does not affect energy transfer.
When Beth performs a measurement, the wave function collapses. Her measurement determines both the final state of a quantum motion of a particle travelling from Alf to Beth, and the initial state for the next part of the motion. At the point of the measurement quantum theory is reformulated with coordinate axes according to the current scale factor, removing a factor of the expansion from redshift.
Edited on 2008-04-02 07:30:57 by CharlesFrancis
Additions:
When Beth performs a measurement, the wave function collapses. Her measurement determines both the final state of a quantum motion of a particle travelling from Alf to Beth, and the initial state for the next part of the motion. At the point of the measurement quantum theory is reformulated with coordinate axes according to the current scale factor, removing a factor of the expansion from redshift.
Deletions:
When Beth performs a measurement, the wave function collapses. Her measurement determines both the final state of a quantum motion of a particle travelling from Alf to Beth, and the initial state for the next part of the motion. At the point of the measurement quantum theory is reformulated with coordinate axes according to the current scale factor. This removing a factor a(ti) ⁄ a(ti+1) from redshift.
Edited on 2008-04-02 07:10:09 by CharlesFrancis
Additions:
The square redshift law applies to spectroscopic» measurements of the wavelength of light, and takes into account that, on the space in which wave functions are defined, Alf’s coordinate axes are enlarged by a factor a0 ⁄ a1 relative to Beth’s. Alf’s measurement of energy-momentum is relative to his own coordinate axes. This factor must be removed in the calculation of energy/momentum of light from a distant source. Thus, the energy transferred by a photon passing from Alf to Beth is inverse to expansion, not inverse to the square of expansion, giving the same relation as is given by parallel transport under the Levi-Civita connection. The square redshift law affects spectroscopic measurements of light from distant sources, but does not affect energy transfer.
When Beth performs a measurement, the wave function collapses. Her measurement determines both the final state of a quantum motion of a particle travelling from Alf to Beth, and the initial state for the next part of the motion. At the point of the measurement quantum theory is reformulated with coordinate axes according to the current scale factor. This removing a factor a(ti) ⁄ a(ti+1) from redshift.
Deletions:
Wave Function Collapse
The rule regarding collapse of the wave function in the quantum theory depends on whether the scaling of coordinates used to describe the final state can be physically calibrated to that of the initial state, for example by parallel transport of classical matter (e.g. a clock) or by Einstein’s synchronisation procedure. When there is a physical calibration quantum theory is formulated without expansion. In this case the predictions of the teleconnection are identical to those of classical general relativity, as shown in the mathematical treatment. When no calibration is possible, as is the case for light from a distant stellar object, expansion is possible and the quantum theory must be formulated taking into account the change in scale between coordinates used for initial and final states. This leads to predictions concerning cosmological redshift distinct from those of classical general relativity.
The square redshift law applies to the measurement of a final state at time t0 of a particle produced at an earlier time t1, such that the momentum space wave function is given by
The integral stands for a sum over the basis states of a Hilbert space with dimension N. Use a spherical lattice with radial lattice spacing χ. If Beth now sets ups quantum theory to measure an initial state at time t0 in the same way she will define the momentum space wave function by
If the lattice spacing is unchanged, the integral stands for a sum over basis states on a Hilbert space with dimension (a0 ⁄ a1)N. Thus the Hilbert space used by Beth for local measurements of energy has a different dimension and lattice spacing from the one used to measure a particle with an initial state at t0. Because quantum theory is redefined at time t0 on a different lattice, the energy measured locally by Beth is given by
A factor a0 ⁄ a1 is thus removed, and the energy transfered by a particle is identical to that given by parallel transport under the Levi-Civita connection. Thus, the square redshift law affects spectroscopic measurements, but does not affect energy transfer.
Edited on 2008-04-01 22:52:00 by CharlesFrancis
Additions:
Alf and Beth’s maps are each defined at constant cosmic time with radial coordinate equal to proper distance. The metric is as described in Large Scale Structure of the Universe. Alf and Beth are here drawn in different galaxies. For convenience Beth’s galaxy is taken as the origin of both maps. The maps are spherical in three dimensional space with one dimension suppressed. Identical galaxies are shown on geodesics emanating from the big bang in the x-t plane. As drawn, each galaxy has the same radial dimension but the galaxies acquire an increasing angular stretch towards the edge of the map. The scaling distortions are physically meaningless and can be removed by embedding the x-y plane onto the surface of a sphere with B at the North pole. The circles on the maps correspond to lines of latitude at 15° and 30° intervals for Beth’s and Alf’s respective maps. The region corresponding to the Southern hemisphere is shaded blue. The outermost circumference is infinitely stretched and represents a point, the South pole. The galaxy at the South pole is so stretched that it surrounds the map. The radial path of light is shown in yellow, with the signal from Alf to Beth shown in red. The region outside Beth’s past light cone is shaded grey. A space-like radial vector has the same apparent and proper length at any point (green).
Deletions:
Alf and Beth’s maps are each defined at constant cosmic time with radial coordinate equal to proper distance. The metric is as described in Large Scale Structure of the Universe. Alf and Beth are here drawn in different galaxies. For convenience Beth’s galaxy is taken as the origin of both maps. The maps are spherical in three dimensional space with one dimension suppressed. Identical galaxies are shown on geodesics emanating from the big bang in the x-t plane. As drawn, each galaxy has the same radial dimension but the galaxies acquire an increasing angular stretch towards the edge of the map. The scaling distortions are physically meaningless and can be removed by embedding the x-y plane onto the surface of a sphere with B at the North pole. The circles on the maps correspond to lines of latitude at 15° and 30° intervals for Beth’s and Alf’s respective maps. The region corresponding to the Southern hemisphere is shaded blue. The outermost circumference is infinitely stretched and represents a point, the South pole. The galaxy at the South pole is so stretched that it surrounds the map. The radial path of light is shown in yellow, with the signal from Alf to Beth shown in red. The region outside Beth’s past light cone is shaded grey. A space-like radial vector has the same apparent and proper length at any point (cyan).
Edited on 2008-04-01 22:25:06 by CharlesFrancis
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Deletions:
Edited on 2008-03-30 14:03:26 by CharlesFrancis
Additions:
is recovered.
Deletions:
is recovered. If Beth’s ancestor, Charles, had set up a trapped signal (green) in a laboratory on Beth’s planet, such that its wavelength could be measured at any time, then the classical linear redshift law found from the affine connection would be found.
Edited on 2008-03-29 01:51:13 by CharlesFrancis
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Deletions:
Edited on 2008-03-28 02:18:39 by CharlesFrancis
Additions:
Edited on 2008-03-28 02:16:24 by CharlesFrancis
Additions:
In standard general relativity it is assumed that photon momentum is parallel transported through large distances, using the Levi-Civita connection in a manner analogous to sliding a ruler across a curved surface while maintaining parallellism in each part of the motion. It is sometimes argued that, since the the laws of physics are locally the same in all coordinates systems, Maxwell’s equations, which govern the propagation of light, must hold locally in empty space, and hence that the Levi-Civita connection holds in empty space. The argument relies on induction», which cannot be rigorously justified, and it ignores the fact that we cannot define local coordinates in empty space, since empty space contains neither reference matter nor matter whose position can be measured. More seriously still, it ignores the known inconsistency between classical electromagnetism and general relativity. It was pointed out by Einstein that the Levi-Civita connection appears to be wrong. In particular, it leads to an inconsistency in the paths of light transmitted by a distant object, seen in the mismatch of Alf’s and Beth’s coordinates. Moreover, the application of the affine connection takes no account of the propagation of a photon wave function in a curved spacetime, which would imply that a photon created with precise momentum in a distant galaxy would not have precise momentum when detected on Earth.
The teleconnection is an extension of classical general relativity and applies to the treatment of quantum mechanics. It was proposed to reconcile the propagation of the wave function in standard quantum theory using Minkowski metric with curvature in classical spacetime. Instead of defining a connection between vectors at nearby points, the teleconnection is defined remotely, between vectors used to describe the initial and final states when the reference matter used to describe the initial state is remote from that used to describe the final state.
Energy Transfer
The square redshift law applies to the measurement of a final state at time t0 of a particle produced at an earlier time t1, such that the momentum space wave function is given by
The integral stands for a sum over the basis states of a Hilbert space with dimension N. Use a spherical lattice with radial lattice spacing χ. If Beth now sets ups quantum theory to measure an initial state at time t0 in the same way she will define the momentum space wave function by
If the lattice spacing is unchanged, the integral stands for a sum over basis states on a Hilbert space with dimension (a0 ⁄ a1)N. Thus the Hilbert space used by Beth for local measurements of energy has a different dimension and lattice spacing from the one used to measure a particle with an initial state at t0. Because quantum theory is redefined at time t0 on a different lattice, the energy measured locally by Beth is given by
A factor a0 ⁄ a1 is thus removed, and the energy transfered by a particle is identical to that given by parallel transport under the Levi-Civita connection. Thus, the square redshift law affects spectroscopic measurements, but does not affect energy transfer.
The teleconnection applies to a quantum motion between an initial state and a final state when it is not meaningful to describe the intervening state in terms of measured physical quantities. In the classical correspondence, time is taken as a continuous variable and motion may be described as a sequence of measured states at instances t = tn < tn-1 < … < t1 < t0 in the limit in which max(ti − tn+1) → 0. The state at any instant, ti, may be regarded as an initial state, and the state at the next instant, ti−1, may be regarded as a final state for that part of the motion. Quantum theory is redefined and the final state then becomes the initial state for the next part of the motion. At each instant, ti, when quantum theory is redefined, the coordinate system is rescaled, removing a factor a(ti) ⁄ a(ti+1). The paths of light and of the galaxies become continuous in the limit, and the classical notion of continuous expansion is restored.
Deletions:
In standard general relativity it is assumed that photon momentum is parallel transported through large distances, using the Levi-Civita connection in a manner analogous to sliding a ruler across a curved surface, while maintaining parallellism in each part of the motion. It is sometimes argued that, since the the laws of physics are locally the same in all coordinates systems, Maxwell’s equations, which govern the propagation of light, must hold locally in empty space, and hence that the Levi-Civita connection holds in empty space. The argument relies on induction», which cannot be rigorously justified, and it ignores the fact that we cannot define local coordinates in empty space, since empty space contains neither reference matter nor matter whose position can be measured. More seriously still, it ignores the known inconsistency between classical electromagnetism and general relativity. It was pointed out by Einstein that the Levi-Civita connection appears to be wrong. In particular, it leads to an inconsistency in the paths of light transmitted by a distant object, seen in the mismatch of Alf’s and Beth’s coordinates. Moreover, the application of the affine connection takes no account of the propagation of a photon wave function in a curved spacetime, which would imply that a photon created with precise momentum in a distant galaxy would not have precise momentum when detected on Earth.
The teleconnection applies to a quantum motion between an initial state and a final state when it is not meaningful to describe the intervening state in terms of measured physical quantities. In the classical correspondence, time is taken as a continuous variable and motion may be described as a sequence of measured states 
at instances t = tn < tn-1 < … < t1 < t0 in the limit in which max(ti − tn+1) → 0. The state at any instant, ti, may be regarded as an initial state, and the state at the next instant, ti−1, may be regarded as a final state for that part of the motion. Quantum theory is redefined and the final state then becomes the initial state for the next part of the motion. At each instant, ti, when quantum theory is redefined, the coordinate system is rescaled, removing a factor a(ti) ⁄ a(ti+1). The paths of light and of the galaxies become continuous in the limit, and the classical notion of continuous expansion is restored.
← The Teleconnection in a Friedmann Cosmology →
To introduce the idea of the teleconnection I will here describe it diagramatically for a Friedmann Cosmology. In this case the analysis is particularly simple. The result is that the cosmological redshift of light from a distant galaxy is proportional to the square of the expansion parameter, not linear with it as predicted by the affine connection. A more general, and more mathematical, treatment will be given on the
next page.
The Status Quo
In non-Euclidean geometry, an affine connection is a rule defining parallel vectors, when the vectors are defined at points separated by a small displacement. An affine connection allows us to parallel transport a vector from one place to another along a path in such a way that we may say that it remains parallel to itself in each part of the motion. Sliding a ruler using parallel transport is clearly sound for coordinate systems local to the Earth, but we cannot physically transport rulers indefinitely far into space, and certainly not from distant stars. Instead we must rely on the transmission of light. If we do slide a ruler, for example on a spacecraft like Pioneer, even a short way into space, we cease to be in direct contact with it and have to rely on electromagnetic transmissions to read the ruler. In practice, we deduce the properties of distant coordinate systems from properties we find in light from distant stellar objects. We can do this only if we know the laws governing the transmission of light over large distances.
In standard general relativity it is assumed that photon momentum is parallel transported through large distances, using the Levi-Civita connection in a manner analogous to sliding a ruler across a curved surface, while maintaining parallellism in each part of the motion. It is sometimes argued that, since the the laws of physics are locally the same in all coordinates systems, Maxwell’s equations, which govern the propagation of light, must hold locally in empty space, and hence that the Levi-Civita connection holds in empty space. The argument relies on induction», which cannot be rigorously justified, and it ignores the fact that we cannot define local coordinates in empty space, since empty space contains neither reference matter nor matter whose position can be measured. More seriously still, it ignores the known inconsistency between classical electromagnetism and general relativity. It was pointed out by Einstein that the Levi-Civita connection appears to be wrong. In particular, it leads to an inconsistency in the paths of light transmitted by a distant object, seen in the mismatch of Alf’s and Beth’s coordinates. Moreover, the application of the affine connection takes no account of the propagation of a photon wave function in a curved spacetime, which would imply that a photon created with precise momentum in a distant galaxy would not have precise momentum when detected on Earth.
In the early part of the 20th century, attempts were made, notably by Einstein himself, and by Herman Weyl, to unify general relativity with classical electromagnetism. Those attempts failed and it seems that theoretical physicists have lost interest. The affine connection is almost universally accepted, and its inconsistency is largely ignored. This is dangerous. Our knowledge of stellar kinematics and of cosmology is almost entirely dependent on spectroscopy» (doppler shift»). There is no direct empirical evidence that the standard Doppler» law holds for the transmission of light across astronomical distances. At the same time, observations appear to reveal a universe with bizarre and paradoxical properties. There is still no sound theoretical basis for the cosmological constant», for exotic cold dark matter», for cosmic inflation» or for modifying Newton’s law of gravity (MOND»). A significant minority of cosmologists recognise that there are serious problems in standard cosmology, but, in the main, empirical evidence which plainly contradicts the standard model, such as galaxy lensing profiles, and the rotation curves of globular clusters», is ignored.
Distant Parallelism
A number of Einstein’s attempts at unification with classical electromagnetism were centred on the investigation of distant parallelism». Relational quantum gravity uses as similar, but not identical, notion, and seeks unification between general relativity and quantum electrodynamics. In relational quantum gravity, general relativity is regarded as correct in the classical correspondence, but must be extended for the treatment of quantum phenomena. Since light consists of photons, light is treated quantum mechanically, even in the description of its transmission over astronomical distances. This is done using a form of distant parallelism which I have called the teleconnection to distinguish it from other teleparallel theories of gravity. I will show that the teleconnection reduces to the standard affine connection in the classical correspondence, and that, when cosmological expansion can be neglected, predicted redshifts are identical to those of general relativity, but it turns out that cosmological expansion affects the calculation of redshift even, in the case of the Pioneer anomaly», within the environs of the solar system.
The teleconnection is an extension of classical general relativity and applies to the treatment of quantum mechanics. It was proposed to reconcile the propagation of the wave function in standard quantum theory using Minkowski metric with curvature in classical spacetime. Instead of defining a connection between vectors at nearby points, the teleconnection is defined remotely, between vectors used to describe the initial and final states when the reference matter used to describe the initial state is necessarily remote from that used to describe the final state.
To define the teleconnection, it is observed that quantum theory provides rules for calculating the probability of the measurement of a final state, given the measurement of a initial state. According to the rules, imaginary wave motions are used to calculate the probability that a given initial state will lead to a given final state. Only the measured, i.e. initial and final, states are physically meaningful. The intermediate wave motion should not be thought of as physically real, but as a device for calculating probabilities.
Since the calculation of wave evolution is carried out by an observer at a particular place and time, it is natural that the calculation should be done using the metric at that place and time. In other words, wave evolution is determined in a tangent space. As for the wave function itself, no ontology is attached to tangent space. 3-momentum has been defined from the Fourier transform of the probability amplitude, and found to be a well defined, locally conserved, property. Now, if momentum is a well defined property for Alf on a space craft or another planet, then it is also a well defined property for Beth on Earth, because Alf can communicate to Beth his value of momentum. Alf's value of momentum does not have to be the same as Beth's, but we seek a way of converting his value to the value Beth will use.
The teleconnection defines the way in which Beth converts the value of momentum calculated by Alf to the value found in her own coordinate system, preserving parallelism. Wave evolution (red) takes place in a tangent space. Discontinuity is associated with wave function collapse, at which point the teleconnection describes the projection of the evolved initial state to the final state (blue). The teleconnection does not apply when there is an intermediate measured (or classical) state. The same recipe applies whether Alf and Beth are separated by astronomical distances, or are in the same laboratory, and reduces to the affine connection when the initial and final states are separated by small times. The classical correspondence is conceived as a limit of many quantum motions in which curved spacetime (green) emerges as the envelope of flat spaces (pale blue), and in which parallel transport is found in the limit of small teleparallel displacements. |
The Teleconnection in a Friedmann Cosmology
Let Alf be an observer at A, on a space craft or a distant planet, and let Beth be an observer at B, such that Alf can signal to Beth and Alf and Beth are moving with the cosmic fluid, i.e. on geodesics from the big bang as described in Weyl’s postulate. At cosmic time, t1, of emission of a photon passing from Alf to Beth, Alf defines synchronous co-ordinates at cosmic time t1 in 3 dimensions, with the origin at B, using proper distance as the radial coordinate. For definiteness and clarity, I will describe a closed cosmos. The universe can then be mapped onto a 3-dimensional finite space, which will be called Alf’s map. Beth defines Beth’s map in exactly the same way, also with the origin at B, at cosmic time t0 when the photon is detected, and to the same scale, i.e. using units of measurement defined by identical physical processes. For a closed universe Alf’s and Beth’s maps are each contained in a sphere. Let a(t) be the scale factor and let a0 = a(t0) and a1 = a(t1). If the universe expands during the time of travel of the photon from Alf to Beth, then Beth’s map is larger than Alf’s map by a factor a0 ⁄ a1.
Alf and Beth’s maps are each defined at constant cosmic time with radial coordinate equal to proper distance. The metric is as described in Large Scale Structure of the Universe. Alf and Beth are here drawn in different galaxies. For convenience Beth’s galaxy is taken as the origin of both maps. The maps are spherical in three dimensional space with one dimension suppressed. Identical galaxies are shown on geodesics emanating from the big bang in the x-t plane. As drawn, each galaxy has the same radial dimension but the galaxies acquire an increasing angular stretch towards the edge of the map. The scaling distortions are physically meaningless and can be removed by embedding the x-y plane onto the surface of a sphere with B at the North pole. The circles on the maps correspond to lines of latitude at 15° and 30° intervals for Beth’s and Alf’s respective maps. The region corresponding to the Southern hemisphere is shaded blue. The outermost circumference is infinitely stretched and represents a point, the South pole. The galaxy at the South pole is so stretched that it surrounds the map. The radial path of light is shown in yellow, with the signal from Alf to Beth shown in red. The region outside Beth’s past light cone is shaded grey. A space-like radial vector has the same apparent and proper length at any point (cyan).
Beth defines new coordinates, maintaining her own local distance scale, but rescaling Alf’s map so that it is equal in size with Beth’s map, and rescaling the time axis so that light speed is constant in radial directions in the in the new coordinates. The metric is as described in Large Scale Structure of the Universe. Together with non-physical Minkowski metric h, the new coordinates are tangent chart at Beth’s position, B at cosmic time t0.
Beth defines tangent space vectors in these coordinates (red), denoted by a bar, and which can be translated radially in tangent space and are equal to physical vectors at her own origin. She defines plane wave states obeying Newton’s first law using barred vectors,
The wave function,
is strictly a function on an abstract tangent space, and is defined at the origin for each observer. This reflects the orthodox interpretation of quantum theory, as used in relational quantum gravity, in which wave functions do not represent a physical wave property, and are merely used by an observer to calculate probabilities.
Similarly Alf defines a tangent chart at A at cosmic time t1,
so as to maintain his own local distance scale. Alf’s and Beth’s tangent charts are identical up to a scale factor, a1 ⁄ a0.
The teleconnection defines the relationship between Alf’s and Beth’s tangent charts, and makes possible the definition of an inner product beween states in Alf’s and Beth’s formulations of quantum theory. To define the teleconnection, Beth first enlarges Alf’s map by the factor a0 ⁄ a1 (equivalently Alf reduces Beth’s map). |
Definition: The teleconnection is such that photon momentum is represented by a vector (red) of equal magnitude and direction on Beth’s map and on Alf’s enlarged map.
The definition of a teleconnection assumes that if momentum has a precise value at one place and time then it also has a precise value other places and times and is justified empirically in so far as observation yields precise values for cosmological redshift after allowing for dispersion due to dust or other known factors. This is a fundamental assumption in this model, of equal importance to the assumption of the constancy of the speed of light in special relativity. Like that assumption, if it were dropped we would be left, not with a different theory, but with no known consistent theory.
In practice, Beth can compare the scale of her map to that of Alf’s map by studying redshift. There are two scaling effects. First Alf’s map has been enlarged by a factor a0 ⁄ a1. In addition, the scaling on the map changes in time, giving another factor a0 ⁄ a1 (as may be calculated from the physical metric). Thus, the model predicts a net factor for cosmological redshift which varies with the square of the expansion parameter:
Wave Function Collapse
The rule regarding collapse of the wave function in the quantum theory depends on whether the scaling of coordinates used to describe the final state can be physically calibrated to that of the initial state, for example by parallel transport of classical matter (e.g. a clock) or by Einstein’s synchronisation procedure. When there is a physical calibration quantum theory is formulated without expansion. In this case the predictions of the teleconnection are identical to those of classical general relativity, as shown in the mathematical treatment. When no calibration is possible, as is the case for light from a distant stellar object, expansion is possible and the quantum theory must be formulated taking into account the change in scale between coordinates used for initial and final states. This leads to predictions concerning cosmological redshift distinct from those of classical general relativity.
Classical Spacetime
The teleconnection applies to a quantum motion between an initial state and a final state when it is not meaningful to describe the intervening state in terms of measured physical quantities. In the classical correspondence, time is taken as a continuous variable and motion may be described as a sequence of measured states at instances t = tn < tn-1 < … < t1 < t0 in the limit in which max(ti − tn+1) → 0. The state at any instant, ti, may be regarded as an initial state, and the state at the next instant, ti−1, may be regarded as a final state for that part of the motion. Quantum theory is redefined and the final state then becomes the initial state for the next part of the motion. At each instant, ti, when quantum theory is redefined, the coordinate system is rescaled, removing a factor a(ti) ⁄ a(ti+1). The paths of light and of the galaxies become continuous in the limit, and the classical notion of continuous expansion is restored.
It follows that momentum is parallel transported from ti+1 to ti in the normal way, and that the result is parallel transport. An overall factor,
is removed from cosmological redshift, and the usual linear law,
is recovered. If Beth’s ancestor, Charles, had set up a trapped signal (green) in a laboratory on Beth’s planet, such that its wavelength could be measured at any time, then the classical linear redshift law found from the affine connection would be found.
The Teleconnection in a Friedmann Cosmology ↑ The Teleconnection →
