Additions:
where ""f(ρ) = sinρ"", ""ρ"", or ""sinhρ"" for space with positive, zero or negative curvature respectively. Coordinate time, ""τ"", is related to cosmic time, ""t"", by ""a(τ)dτ = dt"". An observer, Alf, at ""A"" at cosmic time ""t1"" and coordinate time ""τ1"", defines radial unprimed locally ""Minkowski Coordinates"", ""(t, r, θ, φ)"" based on cosmic time, ""t"", such that ""dt = a(τ)dτ"" and ""r = a(τ)ρ"". For ""
"" cross terms in the metric can be neglected,
It follows that for a vector (including a displacement vector) ""x = (xt, xr, xθ, xφ)"", in locally ""Minkowski Coordinates"" at ""(t1, A)"",
This definition preserves Newton’s first law and the constancy of the speed of light in ""τ−ρ"" coordinates with non-physical metric ""
"". It is required to retain the formal structure of [[RelativisticQuantumTheory relativistic quantum theory]], and preserves the momentum space wave function as a constant of the motion. A plane wave is sometimes regarded as a physical field in a substantive spacetime. Here, in accordance with the orthodox interpretation, it is seen as a mathematical construction in a non-physical space defined by an observer, Alf, at ""A"". The evolution of the wave function is determined in ""τ−ρ"" coordinates, using non-physical metric ""
"", and is such that it corresponds to the standard wave function in locally ""Minkowski Coordinates"".
""
""**Proof:** One period of light in locally ""Minkowski Coordinates"", ""(t, r, θ, φ)"", with an origin at ""A"" at cosmic time ""t1"" is represented by a timelike vector of magnitude ""λ1"". In ""τ−ρ"" coordinates, its timelike component is ""λ1 ⁄ a1"". The corresponding barred quantity is ""λ1 ⁄ Aa1"". The barred vector is translated to ""(τ, B)"". The corresponding unbarred quantity is found from the physical metric, ""g"", and has magnitude ""a0λ1 ⁄ a12"". On transforming to primed locally ""Minkowski Coordinates"" ""(t', r', θ', φ')"", with an origin at ""B"", we find
where the factor ""k"" describes gravitational redshift. While the calculation of parallel displacement in ""τ−ρ"" coordinates is straightforward, the determination of the relationship between ""τ−ρ"" coordinates and inertial locally ""Minkowski Coordinates"" depends on both the motion of the observer and on local variations in geometry, summarised in the factor ""k"". The calculation is complicated by the fact that, in the general case, motion with ""ρ = const"" is not inertial. The consequence is that an an inertial observer will need to modify wave functions in such a way that they are subjected to apparent accelerations, or Doppler shifts which do not reflect acceleration in the classical domain.
in agreement with the calculation of the form of the ""metric for stationary observers"". Locally ""Minkowski Coordinates"" at the point ""x"" with ""r = 0"" are found by substituting ""dt = k−1dt'"" and ""dr = kdr'"", ""rdα = r'dθ'"", ""r sinα dβ = r' sinθ' dφ'"". For small ""r"",
In locally ""Minkowski Coordinates"" at ""A"", gravitational redshift is given by a factor ""(g00)−½"", in agreement with standard general relativity.
The classical correspondence is found when there is sufficient information that states may be treated as being continuously measured. The quantum state is defined on a synchronous slice using quantum time, ""t"". In order to describe a continuous classical motion, we must take a collection of synchronous slices, and treat each slice as the final state of one quantum step and as the initial state of the next quantum step, then allow the step size to go to zero, to obtain a [[http://en.wikipedia.org/wiki/Foliation foliation]]. At each stage of the motion, quantum theory is formulated in a space with a non-physical metric ""
"". Classical motion is determinate and may be described as an ordered sequence, ""
"", of ""effectively measured"" states at instances ""ti"" such that ""0 < ti+1 - ti < δ"" where ""δ"" is sufficiently small that there is negligible alteration in predictions in the limit ""δ → 0"". Each state, ""
"", is a multiparticle state in [[MultiparticleStates Fock space]], which has been relabelled using mean properties determined by an effective classical measurement. For the motion between times ""ti"" and ""ti+1"", ""
"" may be regarded as the initial state and ""
"" may be regarded as the final state. Since the time evolution of mean properties is determinate, there is no collapse and ""
"" is the initial state for the motion to ""ti+2"". Classical formulae are recovered by considering a sequence of initial and final effectively measured states in the limit as the maximum time interval between them goes to zero. In each stage of the motion momentum is parallel displaced using the teleconnection, which gives the same result as parallel displacement in tangent space. The cumulative effect of such infinitesimal parallel displacements is ""parallel transport"". So the teleconnection between initial and final states in quantum theory leads to parallel transport in the classical domain. The same argument holds for a classical beam of light, in which each photon wave function is localised within the beam at any time, and for a classical field which has a measurable value at each point where it is defined (up to the possible addition of an arbitrary gauge function).
Deletions:
where ""f(ρ) = sinρ"", ""ρ"", or ""sinhρ"" for space with positive, zero or negative curvature respectively. Coordinate time, ""τ"", is related to cosmic time, ""t"", by ""a(τ)dτ = dt"". An observer, Alf, at ""A"" at cosmic time ""t1"" and coordinate time ""τ1"", defines radial unprimed locally Minkowski coordinates, ""(t, r, θ, φ)"" based on cosmic time, ""t"", such that ""dt = a(τ)dτ"" and ""r = a(τ)ρ"". For """" cross terms in the metric can be neglected,
It follows that for a vector (including a displacement vector) ""x = (xt, xr, xθ, xφ)"", in locally Minkowski coordinates at ""(t1, A)"",
This definition preserves Newton’s first law and the constancy of the speed of light in ""τ−ρ"" coordinates with non-physical metric """". It is required to retain the formal structure of [[RelativisticQuantumTheory relativistic quantum theory]], and preserves the momentum space wave function as a constant of the motion. A plane wave is sometimes regarded as a physical field in a substantive spacetime. Here, in accordance with the orthodox interpretation, it is seen as a mathematical construction in a non-physical space defined by an observer, Alf, at ""A"". The evolution of the wave function is determined in ""τ−ρ"" coordinates, using non-physical metric """", and is such that it corresponds to the standard wave function in locally Minkowski coordinates.
""""**Proof:** One period of light in locally Minkowski coordinates, ""(t, r, θ, φ)"", with an origin at ""A"" at cosmic time ""t1"" is represented by a timelike vector of magnitude ""λ1"". In ""τ−ρ"" coordinates, its timelike component is ""λ1 ⁄ a1"". The corresponding barred quantity is ""λ1 ⁄ Aa1"". The barred vector is translated to ""(τ, B)"". The corresponding unbarred quantity is found from the physical metric, ""g"", and has magnitude ""a0λ1 ⁄ a12"". On transforming to primed locally Minkowski coordinates ""(t', r', θ', φ')"", with an origin at ""B"", we find
where the factor ""k"" describes gravitational redshift. While the calculation of parallel displacement in ""τ−ρ"" coordinates is straightforward, the determination of the relationship between ""τ−ρ"" coordinates and inertial locally Minkowski coordinates depends on both the motion of the observer and on local variations in geometry, summarised in the factor ""k"". The calculation is complicated by the fact that, in the general case, motion with ""ρ = const"" is not inertial. The consequence is that an an inertial observer will need to modify wave functions in such a way that they are subjected to apparent accelerations, or Doppler shifts which do not reflect acceleration in the classical domain.
in agreement with the calculation of the form of the ""metric for stationary observers"". Locally Minkowski coordinates at the point ""x"" with ""r = 0"" are found by substituting ""dt = k−1dt'"" and ""dr = kdr'"", ""rdα = r'dθ'"", ""r sinα dβ = r' sinθ' dφ'"". For small ""r"",
In locally Minkowski coordinates at ""A"", gravitational redshift is given by a factor ""(g00)−½"", in agreement with standard general relativity.
The classical correspondence is found when there is sufficient information that states may be treated as being continuously measured. The quantum state is defined on a synchronous slice using quantum time, ""t"". In order to describe a continuous classical motion, we must take a collection of synchronous slices, and treat each slice as the final state of one quantum step and as the initial state of the next quantum step, then allow the step size to go to zero, to obtain a [[http://en.wikipedia.org/wiki/Foliation foliation]]. At each stage of the motion, quantum theory is formulated in a space with a non-physical metric """". Classical motion is determinate and may be described as an ordered sequence, """", of ""effectively measured"" states at instances ""ti"" such that ""0 < ti+1 - ti < δ"" where ""δ"" is sufficiently small that there is negligible alteration in predictions in the limit ""δ → 0"". Each state, """", is a multiparticle state in [[Multiparticle Fock space]], which has been relabelled using mean properties determined by an effective classical measurement. For the motion between times ""ti"" and ""ti+1"", """" may be regarded as the initial state and """" may be regarded as the final state. Since the time evolution of mean properties is determinate, there is no collapse and """" is the initial state for the motion to ""ti+2"". Classical formulae are recovered by considering a sequence of initial and final effectively measured states in the limit as the maximum time interval between them goes to zero. In each stage of the motion momentum is parallel displaced using the teleconnection, which gives the same result as parallel displacement in tangent space. The cumulative effect of such infinitesimal parallel displacements is ""parallel transport"". So the teleconnection between initial and final states in quantum theory leads to parallel transport in the classical domain. The same argument holds for a classical beam of light, in which each photon wave function is localised within the beam at any time, and for a classical field which has a measurable value at each point where it is defined (up to the possible addition of an arbitrary gauge function).
Additions:
""
""====""""====
======[[TeleconnectionIntro ←]] The Teleconnection [[RelationalQuantumGravity ↑]] [[QuantumCoordinates →]]======
A ""connection"" defines the notion of parallel in vector spaces defined at nearby points of a manifold. The teleconnection defines the parallel displacement of momentum in quantum mechanics from an initial state to a final state when the reference matter used to describe the initial state is remote from that used to describe the final state. When the initial and final states are determined with respect to nearby reference matter, the teleconnection is equivalent to the ""Levi-Civita connection"".
""Absolute Space and Time""
""Suppression of Expansion in a Neighbourhood""
""The Teleconnection in a Friedmann Cosmology""
""Cosmological Redshift""
""Energy Transfer""
""Geometries with Expansion""
""Gravitational Redshift""
""The Levi-Civita Connection""
====""""Absolute Space and Time====
The purpose of the teleconnection is to retain the fundamental arguments leading from special relativity to quantum electrodynamics and to general relativity, and to use them to show how apparent fundamental incompatibilities can be reconciled. The teleconnection adapts the probabilistic structure described in quantum theory to the situation studied in general relativity, in which a clock at the position of the emission of a quantum particle does not measure time at the same rate as an identical clock at the position at which the particle is detected.
In a ""Friedmann cosmology"" the existence of ""cosmic time"", or time on a geodesic from the big bang conflicts with Einstein's precepts in which only local law, and local time as measured on a clock, are physically important or meaningful. The teleconnection assumes that since cosmic time makes sense on the large scale in a Friedmann cosmology, it should also make sense on the small scale, even after taking local variations in geometry into account. Cosmic time gains a greater importance than in standard general relativity, and plays a role very much like Newton's absolute time. To describe the transmission of photons over large distances, the ""inner product"" in quantum theory is defined using an integration over all space, and, in an expanding cosmology, only makes complete sense on a synchronous slice. Synchronous slices for given cosmic time yield a "preferred" frame, in which the teleconnection is defined, and play the role of Newton's absolute space. Nonetheless the fundamental elements of the model are particles of matter, and in contrast to Newton’s ideas, time and space appear as emergent organisational principles, not as physical background.
====""""Suppression of Expansion in a Neighbourhood====
""
""The inner product is defined to generate probabilities. To specify a probability, it is necessary to specify the known conditions to which the probability applies. We may interpret the collapse of the wave function as the change in probability when the known condition changes. The quantum state describes knowledge of the particle. “Knowledge” in this context refers to information which is available in principle from the physical situation, whether or not it is known in practice by a particular observer. The knowledge about a particle which is possible in principle depends on the physical relationship of that particle with other matter and, in the context of an expanding universe, will have empirical consequences which cannot be reconciled with a view of the wave function as a physical field.
The teleconnection will be applied to two types of physical situation, depending on whether the clock used to determine the final state can be calibrated to the clock used to determine the initial state, for example by Einstein’s calibration procedure.
1. When there is a physical calibration between the clocks used to measure initial and final states quantum theory is formulated without expansion.
2. When no calibration is possible, as is the case for light from a distant stellar object, the quantum theory must be formulated taking into account expansion between coordinates used for initial and final states.
This should be not be understood as a change in physical behaviour, but as a change in the mathematical formulation of quantum theory as a theory of probabilities, reflecting the difference in information available to the observer depending on whether synchronisation is possible between the clocks used to determine the initial and final states. Because a factor of the expansion parameter is removed in the calculation of ""energy/momentum"", both formulations will give the same classical quantities, but there will be a difference in predictions for spectral shifts, determined, for example, by diffraction. This is legitimate for interpretations of quantum theory in which the wave function is a device for the calculation of probabilities, but excludes interpretations which attribute any form of ontological existence to the wave function.
Thus, the rule regarding the evolution of the wave function from an initial state at ""A"" to a remote final state at ""B"" depends on whether the coordinates at ""B"" can be physically calibrated to those at ""A"", for example by Einstein’s synchronisation procedure. When there is a physical calibration, quantum theory is formulated without expansion. When no calibration is possible, as is the case for light from a distant stellar object, the quantum theory must be formulated taking into account expansion between coordinates used for initial and final states. Wave evolution takes place in coordinates defined from the initial state, so that interference effects depend on wavelengths modified by a factor of the expansion parameter. This factor is removed in the calculation of energy-momentum from the wavelength, because physical quantities are defined with respect to current coordinates. When a synchronisation procedure exists between clocks used to describe the initial and final states, expansion is suppressed. This is not a change in the physical behaviour of light, but a change in the mathematical formulation of quantum theory as a theory of probabilities, and reflects the different information available to the observer.
At the present time, it is not possible to write down conditions under which Einstein’s synchronisation procedure is possible. It is known that the [[http://en.wikipedia.org/wiki/Pioneer_anomaly anomalous Pioneer blue shift]] appeared after radar lock was lost. After radar lock was lost with Pioneer, it was no longer possible to calibrate processes on Pioneer to processes on Earth. Synchronisation became impossible in practice, creating the conditions under which the anomalous shift was observed. The shift is seen not as an indication of a change in motion, but as a consequence of a change in the conditions under which quantum theory is formulated. There are indications of an anomalous shift during [[http://en.wikipedia.org/wiki/Flyby_anomaly planetary flybys]], when radar lock is also lost, but no indication of a shift for planets in the outer solar system, whose distance and orbital periods are known to good accuracy and together give information equivalent to clock synchronisation. It remains to be established whether the anomaly appears in consequence of some deep reason dependent on local geometry and motion according to which maintenance of radar lock is impossible in principle, or whether it is sufficient that a synchronisation procedure cannot be carried out in practice. Hopefully future missions will cast greater light on the onset of the anomaly.
====""""The Teleconnection in a Friedmann Cosmology====
Using ""τ−ρ"" coordinates for a ""Penrose diagram"", in a Friedmann cosmology the metric is
""
"".
where ""f(ρ) = sinρ"", ""ρ"", or ""sinhρ"" for space with positive, zero or negative curvature respectively. Coordinate time, ""τ"", is related to cosmic time, ""t"", by ""a(τ)dτ = dt"". An observer, Alf, at ""A"" at cosmic time ""t1"" and coordinate time ""τ1"", defines radial unprimed locally Minkowski coordinates, ""(t, r, θ, φ)"" based on cosmic time, ""t"", such that ""dt = a(τ)dτ"" and ""r = a(τ)ρ"". For """" cross terms in the metric can be neglected,
""
""
For ""
"",
""
"".
Alf defines a non-physical metric, ""
"", using ""τ−ρ"" coordinates, such that,
""
"".
where ""A"" and ""B"" are real numbers, whose values are to be determined. Alf defines barred vectors in ""τ−ρ"" coordinates.
<<""Definition: For a vector x = (xτ, xρ, xθ, xφ) at (τ1, A), the corresponding barred vector is""
""
""
<<
This definition will apply to classical energy-momentum, but not to quantum energy-momentum which appears in the wave function. Barred quantum energy-momentum will be defined separately.
<<""Definition: For the displacement, x = (xτ, xρ, xθ, xφ), from (τ1, A), the corresponding barred displacement vector is""
""
""
It follows that for a vector (including a displacement vector) ""x = (xt, xr, xθ, xφ)"", in locally Minkowski coordinates at ""(t1, A)"",
""
""
and that, for barred vectors ""
"" and ""
"" at ""(τ1, A)"", the barred dot product, evaluated with non-physical metric ""
"", satisfies
""
""
Alf formulates quantum states locally in Hilbert space at time ""t1"", and defines ""plane wave states"" at ""t1"" using
""
""
Quantum theory is then reformulated in terms of barred quantities, under the requirement that the inner product is preserved.
""
""
This requires that ""
"", and we have the definition:
<<""Definition: Barred momentum in quantum theory is""
""
""
It follows that the momentum space wave function,
""
""
is preserved when it is given that ""
"" vanishes outside a sphere of radius ""πa"", and when the wave function is normalised such that ""
"".
The teleconnection defines the inner product between the initial state of a particle emitted at ""(τ1, A)"" and the final state of that particle detected some later time, by translating barred momentum from in ""τ−ρ"" coordinates with constant non-physical metric ""
"".
<<""The Teleconnection: For the barred momentum 
, the plane wave state at time τ is given in τ−ρ coordinates by""
""
""
""where the barred dot product uses the non-physical metric, 
.""
This definition preserves Newton’s first law and the constancy of the speed of light in ""τ−ρ"" coordinates with non-physical metric """". It is required to retain the formal structure of [[RelativisticQuantumTheory relativistic quantum theory]], and preserves the momentum space wave function as a constant of the motion. A plane wave is sometimes regarded as a physical field in a substantive spacetime. Here, in accordance with the orthodox interpretation, it is seen as a mathematical construction in a non-physical space defined by an observer, Alf, at ""A"". The evolution of the wave function is determined in ""τ−ρ"" coordinates, using non-physical metric """", and is such that it corresponds to the standard wave function in locally Minkowski coordinates.
====""""Cosmological Redshift====
A second observer, Beth, at ""(τ0, B)"", remote from ""(τ1, A)"" and with ""τ0 > τ1"", also defines quantum states in Hilbert space.
<<""Theorem: Let a1 = a(t1) and a0 = a(t0) . Light emitted at time t1 with wavelength λ1 from a distant point A, and detected at B, at time t0 with wavelength λ0 is redshifted according to""
""
""
""""**Proof:** One period of light in locally Minkowski coordinates, ""(t, r, θ, φ)"", with an origin at ""A"" at cosmic time ""t1"" is represented by a timelike vector of magnitude ""λ1"". In ""τ−ρ"" coordinates, its timelike component is ""λ1 ⁄ a1"". The corresponding barred quantity is ""λ1 ⁄ Aa1"". The barred vector is translated to ""(τ, B)"". The corresponding unbarred quantity is found from the physical metric, ""g"", and has magnitude ""a0λ1 ⁄ a12"". On transforming to primed locally Minkowski coordinates ""(t', r', θ', φ')"", with an origin at ""B"", we find
""
""
It follows that, for small ""r"",
""
"".
Thus recession velocity due to expansion is half the value calculated from Doppler, and coordinates in which radial distance from Earth is calculated from redshift exhibit a stretch of factor half in the radial direction. So ""A = 2"". The time taken for a pulse of light at this distance to traverse a small angular distance ""dθ"" is ""
"". So ""B = ½"" (a stretch of factor two in angular directions gives ""4π"" in a circle, which may be related to the spin states of Fermions). Thus, the non-physical metric, ""
"",is
""
""
====""""Energy Transfer====
The squared redshift law appears at first to be at odds with the claim that parallel displacement under the teleconnection reduces to parallel transport in the classical correspondence. This is due to the manner in which expansion is treated in the quantum theory and is resolved at the point of the collapse of the wave function. The square redshift law applies to [[http://en.wikipedia.org/wiki/Spectroscopy spectroscopic]] measurements of light from distant sources, when the detection of the photon takes place //after// diffraction or refraction of the quantum wave function. It does not affect energy transfer because, in order to formulate the inner product between an initial state at time ""t0"" and a final state at time ""t1"", Beth enlarges the coordinate axes at time ""t0"" by a factor ""a0 ⁄ a1"". This affects spectroscopic measurements, but, since the initial measurement of energy-momentum is relative to the coordinate axes at time ""t0"". this factor does not appear in the calculation of energy/momentum of light from a distant source. The energy ""
"" measured locally by Beth at time ""t0"" of a photon emitted at time ""t1"" with energy ""
"" from a from a distant source at time ""t1"" is given by
""
""
Thus we have the same relation for energy transferred as is given by parallel transport under the ""Levi-Civita connection"".
====""""Geometries with Expansion====
In general relativity, time is determined from a clock locally. Although cosmic time is defined globally, its definition, based on ""Weyl’s postulate"" depends on the local time of many galaxies on geodesics from the Big Bang. The Friedmann models are based on the particular assumptions that the distribution of matter is homogeneous and isotropic. This is observably true in reasonable approximation when the matter distribution is averaged over large enough distances. So, we expect these models to give a reasonable description of the universe at large scales. But these models can only be approximate because they take no account of local mass distributions or of peculiar motions of galaxies and the orbits of stars within a galaxy. On their own, the Friedmann models say nothing about what happens at smaller scales, at which matter is clearly not homogeneous. However, it is natural to think that local fluctuations in geometry due to the inhomogeneous local matter distribution can be treated as perturbations to a Friedmann model (at least for points where the gravitational field is not large).
Since it does not make sense to talk of expansion locally, ""a(τ)"" is a global parameter. We may define space-like hypersurfaces with ""a(τ) = const"" and define ""τ"" to be a global time parameter with ""τ = const"" on any surface with scale factor ""a(τ) = const"". The teleconnection postulates that we may define a non-physical metric, ""
"", as for a Friedmann cosmology in which the speed of light is constant. In ""τ−ρ"" coordinates,
""
""
This seems reasonable for points which are not hidden by an [[http://en.wikipedia.org/wiki/Event_horizon event horizon]]. Points inside an event horizon will be considered again later.
The calculation of the general form of the metric for observers at constant ""ρ"" goes through as for ""stationary observers"", but now we have an additional factor of the expansion parameter. Thus the physical metric has the form
""
""
where the factor ""k"" describes gravitational redshift. While the calculation of parallel displacement in ""τ−ρ"" coordinates is straightforward, the determination of the relationship between ""τ−ρ"" coordinates and inertial locally Minkowski coordinates depends on both the motion of the observer and on local variations in geometry, summarised in the factor ""k"". The calculation is complicated by the fact that, in the general case, motion with ""ρ = const"" is not inertial. The consequence is that an an inertial observer will need to modify wave functions in such a way that they are subjected to apparent accelerations, or Doppler shifts which do not reflect acceleration in the classical domain.
With the substitutions ""dt' = a(τ)dτ"" ""r' = a(τ)ρ"", ""θ' = θ"", and ""φ' = φ"", and for ""
""
""
""
For ""
"",
""
""
in agreement with the calculation of the form of the ""metric for stationary observers"". Locally Minkowski coordinates at the point ""x"" with ""r = 0"" are found by substituting ""dt = k−1dt'"" and ""dr = kdr'"", ""rdα = r'dθ'"", ""r sinα dβ = r' sinθ' dφ'"". For small ""r"",
""
""
Barred vectors are defined in ""τ−ρ"" coordinates as for a ""Friedmann cosmology"".
<<""Definition: For a vector x = (xτ, xρ, xθ, xφ), at (τ1, A), the corresponding barred vector is""
""
""
For a vector ""x = (xt, xr, xα, xβ)"" in locally Minkowski, ""t-r"" , coordinates at ""(t1, A)"" where ""t1 ↔ τ1""
""
""
Alf formulates quantum states locally in Hilbert space at time ""t1"", and defines ""plane wave states"" at ""t1"" using
""
""
Quantum theory is reformulated globally using barred quantities, under the requirement that the inner product is preserved.
""
""
This requires that ""
"", and we have the definition:
<<""Definition: Barred momentum is""
""
""
We will be interested in weak fields and the transmission of photons over astronomical distances, for which momentum may be taken as radial up to the bending of light by a lens. It will be sufficient to consider the approximation
""
""
====""""Gravitational Redshift====
Let ""A"" and ""B"" be sufficiently close that gravitational lensing effects may be ignored. Then, in coordinates with an origin at ""A"", the momentum of a photon passing from ""A"" at ""t1"" to ""B"" at ""t2"" is radial,
""
""
Let the redshift factor at ""A"" at time ""t1"" be ""kA"", and let the redshift factor at ""B "" at ""t2"" be ""kB"", and let ""a1 = a(t1)"" and ""a2 = a(t2)"". Then, barred momentum is
""
""
Barred momentum is translated to ""(τ2, B)"" in ""τ−ρ"" coordinates. Beth converts to locally Minkowski ""t'-r'"" coordinates, and finds, with the removal of a factor of the expansion,
""
""
which is the same as is found by ""parallel displacement"" of momentum through a small distance in tangent space when the metric is
""
""
In locally Minkowski coordinates at ""A"", gravitational redshift is given by a factor ""(g00)−½"", in agreement with standard general relativity.
====""""The Levi-Civita Connection====
The classical correspondence is found when there is sufficient information that states may be treated as being continuously measured. The quantum state is defined on a synchronous slice using quantum time, ""t"". In order to describe a continuous classical motion, we must take a collection of synchronous slices, and treat each slice as the final state of one quantum step and as the initial state of the next quantum step, then allow the step size to go to zero, to obtain a [[http://en.wikipedia.org/wiki/Foliation foliation]]. At each stage of the motion, quantum theory is formulated in a space with a non-physical metric """". Classical motion is determinate and may be described as an ordered sequence, """", of ""effectively measured"" states at instances ""ti"" such that ""0 < ti+1 - ti < δ"" where ""δ"" is sufficiently small that there is negligible alteration in predictions in the limit ""δ → 0"". Each state, """", is a multiparticle state in [[Multiparticle Fock space]], which has been relabelled using mean properties determined by an effective classical measurement. For the motion between times ""ti"" and ""ti+1"", """" may be regarded as the initial state and """" may be regarded as the final state. Since the time evolution of mean properties is determinate, there is no collapse and """" is the initial state for the motion to ""ti+2"". Classical formulae are recovered by considering a sequence of initial and final effectively measured states in the limit as the maximum time interval between them goes to zero. In each stage of the motion momentum is parallel displaced using the teleconnection, which gives the same result as parallel displacement in tangent space. The cumulative effect of such infinitesimal parallel displacements is ""parallel transport"". So the teleconnection between initial and final states in quantum theory leads to parallel transport in the classical domain. The same argument holds for a classical beam of light, in which each photon wave function is localised within the beam at any time, and for a classical field which has a measurable value at each point where it is defined (up to the possible addition of an arbitrary gauge function).
[[Teleconnection The Teleconnection ↑]] [[QuantumCoordinates Illusory Velocity →]]
Deletions:
""""====""""====
======[[Teleconnection ←]] Illusory Velocity [[RelationalQuantumGravity ↑]] [[ParticlesOrFields →]]======
In order to analyse redshift effects between remote initial and final quantum states, we need to study the relationship between inertial reference frames defined locally with speed of light equal to unity and ""τ−ρ"" coordinates used in a Penrose diagram of the universe with a non-physical metric, ""
"". The factors of two in the non-physical metric are equivalent to a two way stretch of the wave function, and yields surprising results in conjunction with local fluctuations in geometry. Gravitational lenses in deep space have four times greater effect for given mass. The interpretation of redshifts by standard formulae leads to an illusory component in the velocities of astronomical objects.
""Quantum Coordinates""
""Bending of Light""
""Pioneer Blueshift""
""Circular Orbits""
====""""Quantum Coordinates====
""
""The [[Teleconnection teleconnection]] has been described using ""τ−ρ coordinates"" for a ""Penrose diagram"". The ""general form of the metric"" is
""
""
These coordinates are defined such that the speed of light is unity, but, due to the variability of ""k : ρ → k(ρ)"" , lines of constant ""ρ"" are not geodesic (as is the case for a standard Friedmann cosmology). Quantum theory was formulated with the non-physical metric, ""
"", given by
""
""
If we wish to formulate quantum theory coordinates with an origin which is not on a line of constant ""ρ"" then we must retain the condition that the speed of light is unity.
<<""Definition: Quantum coordinates are defined such that the speed of light is unity in each direction from the origin, and the time coordinate is τ, and the non-physical metric is given locally by ""
""
""
""τ-ρ"" coordinates are a special case of quantum coordinates, but it is not immediately clear how to generalise the angular coefficents to give a global description of the non-physical metric in the general case. Quantum coordinates have the property that they are not affected by the mass distribution. For small distances, parallel displacement of momentum in quantum coordinates is identical to parallel displacement in tangent space (locally Minkowski coordinates).
====""""Bending of Light====
The teleconnection model predicts geodesic motion for a classical ray of light for which classical time and space coordinates can be determined at each point within a local reference frame. Thus, there is no change in the prediction of bending of light around the Sun. For bending by a distant [[http://en.wikipedia.org/wiki/Gravitational_lensing gravitational lens]], quantum wave effects are transmitted using parallel displacement of momentum in ""τ−ρ"" coordinates using the non-physical metric, """", given by
""""
So, in the calculation of the deflection by a lens, we must halve the radial distance and double the angular distance, increasing the angle of deflection by a factor of four. Thus the mass required for a given strength of gravitational lens is a quarter of that which would be required in a standard no-CDM theory.
====""""Pioneer Blueshift====
""
""In the absence of calibration between clocks on a distant body and clocks on the earth, the energy-momentum of a distant body is constant in ""τ-ρ"" coordinates with non-physical metric """". Meanwhile vectors on Earth are contracted because of the expansion of the Universe. Let the classical momentum of Pioneer be ""p"" at the time, ""t0"", of loss of radar lock. Ignoring radiation pressure and local gravitational effects such as the gravity of the Sun (set ""k = 1"" in the metric), for a signal emitted at time ""t"" and detected on Earth, classical energy is given by (note that ""t0"" is now the earlier time)
""
""
Classical energy is proportional to the rate of clocks on the distant body, so signals show a frequency drift, ""H0"", toward the blue. This effect appears to have been observed in the [[Pioneer anomalous Pioneer blueshift]].
====""""Circular Orbits====
""
""For the study of [[GalaxyRotationCurves galaxy rotation curves]] we are interested in circular geodesic motion. Consider an inertial observer, Piers, at a point ""P"" at constant distance, ""r"", from some arbitrary point ""O"", on a line of constant ""ρ"" ""r"" is sufficiently large that Piers’ clock cannot be calibrated to a clock at ""O"". Piers wishes to make ""O"" the centre of coordinates, while retaining the constraints that the speed of light is unity and time is ""τ"". The Pioneer drift is equivalent to an acceleration of ""P"" toward ""O"",
""
""
where there is a factor of two due to the non-physical metric. To take account of this acceleration, Piers introduces rotating coordinates with orbital velocity, ""vP"" (""P"" is not at rest with respect to Piers). Because of the factors of ""22"" in the non-physical metric, the orbital velocity corresponding to ""aP"". is subject to a factor of ""¼"", where one factor of ""½"" is from the time coordinate in the denominator of velocity, and another is from the transverse space coordinate. Thus, Piers defines coordinates in which ""P"" has an orbital velocity ""vP"",
""
""
Hence,
""
""
""
""The matter distribution does not alter ""τ-ρ"" coordinates or the non-physical metric. So, it does not alter the rotational velocity, ""vP"", required to define quantum coordinates. If the true orbital velocity of Piers in the gravitational field of a massive body at ""O"" is ""vg"" the orbital velocity in quantum coordinates is
""
""
where the signs of each term are the same. The net acceleration toward ""O"" of a point at rest in quantum coordinates is
""
""
We use an interpretation in which the wave function describes a state of knowledge used for the calculation of probabilities. When a distant observer, Spike, observes photons from Piers, he applies the classical constraint that there is no expansion in gravitationally bound systems. This removes the apparent acceleration ""H0c ⁄ 32"", leaving a spectral shift equivalent to total inward acceleration
""
""
The first term on the right hand side is simply the acceleration due to gravity and corresponds to Piers’s orbital velocity about ""O"". The second yields an illusory velocity corresponding to an illusory inverse acceleration law. This term agrees with the phenomenological law used in ""MOND"" and accounts for the flattening of ""galaxy rotation curves"".
[[QuantumCoordinates Quantum Coordinates ↑]] [[ParticlesOrFields Particles or Fields? →]]
Additions:
""""====""""====
======[[Teleconnection ←]] Illusory Velocity [[RelationalQuantumGravity ↑]] [[ParticlesOrFields →]]======
In order to analyse redshift effects between remote initial and final quantum states, we need to study the relationship between inertial reference frames defined locally with speed of light equal to unity and ""τ−ρ"" coordinates used in a Penrose diagram of the universe with a non-physical metric, """". The factors of two in the non-physical metric are equivalent to a two way stretch of the wave function, and yields surprising results in conjunction with local fluctuations in geometry. Gravitational lenses in deep space have four times greater effect for given mass. The interpretation of redshifts by standard formulae leads to an illusory component in the velocities of astronomical objects.
""Quantum Coordinates""
""Bending of Light""
""Pioneer Blueshift""
""Circular Orbits""
====""""Quantum Coordinates====
""""The [[Teleconnection teleconnection]] has been described using ""τ−ρ coordinates"" for a ""Penrose diagram"". The ""general form of the metric"" is
""""
These coordinates are defined such that the speed of light is unity, but, due to the variability of ""k : ρ → k(ρ)"" , lines of constant ""ρ"" are not geodesic (as is the case for a standard Friedmann cosmology). Quantum theory was formulated with the non-physical metric, """", given by
""""
If we wish to formulate quantum theory coordinates with an origin which is not on a line of constant ""ρ"" then we must retain the condition that the speed of light is unity.
<<""Definition: Quantum coordinates are defined such that the speed of light is unity in each direction from the origin, and the time coordinate is τ, and the non-physical metric is given locally by ""
""""
""τ-ρ"" coordinates are a special case of quantum coordinates, but it is not immediately clear how to generalise the angular coefficents to give a global description of the non-physical metric in the general case. Quantum coordinates have the property that they are not affected by the mass distribution. For small distances, parallel displacement of momentum in quantum coordinates is identical to parallel displacement in tangent space (locally Minkowski coordinates).
====""""Bending of Light====
The teleconnection model predicts geodesic motion for a classical ray of light for which classical time and space coordinates can be determined at each point within a local reference frame. Thus, there is no change in the prediction of bending of light around the Sun. For bending by a distant [[http://en.wikipedia.org/wiki/Gravitational_lensing gravitational lens]], quantum wave effects are transmitted using parallel displacement of momentum in ""τ−ρ"" coordinates using the non-physical metric, """", given by
""""
So, in the calculation of the deflection by a lens, we must halve the radial distance and double the angular distance, increasing the angle of deflection by a factor of four. Thus the mass required for a given strength of gravitational lens is a quarter of that which would be required in a standard no-CDM theory.
====""""Pioneer Blueshift====
""""In the absence of calibration between clocks on a distant body and clocks on the earth, the energy-momentum of a distant body is constant in ""τ-ρ"" coordinates with non-physical metric """". Meanwhile vectors on Earth are contracted because of the expansion of the Universe. Let the classical momentum of Pioneer be ""p"" at the time, ""t0"", of loss of radar lock. Ignoring radiation pressure and local gravitational effects such as the gravity of the Sun (set ""k = 1"" in the metric), for a signal emitted at time ""t"" and detected on Earth, classical energy is given by (note that ""t0"" is now the earlier time)
""""
Classical energy is proportional to the rate of clocks on the distant body, so signals show a frequency drift, ""H0"", toward the blue. This effect appears to have been observed in the [[Pioneer anomalous Pioneer blueshift]].
====""""Circular Orbits====
""""For the study of [[GalaxyRotationCurves galaxy rotation curves]] we are interested in circular geodesic motion. Consider an inertial observer, Piers, at a point ""P"" at constant distance, ""r"", from some arbitrary point ""O"", on a line of constant ""ρ"" ""r"" is sufficiently large that Piers’ clock cannot be calibrated to a clock at ""O"". Piers wishes to make ""O"" the centre of coordinates, while retaining the constraints that the speed of light is unity and time is ""τ"". The Pioneer drift is equivalent to an acceleration of ""P"" toward ""O"",
""""
where there is a factor of two due to the non-physical metric. To take account of this acceleration, Piers introduces rotating coordinates with orbital velocity, ""vP"" (""P"" is not at rest with respect to Piers). Because of the factors of ""22"" in the non-physical metric, the orbital velocity corresponding to ""aP"". is subject to a factor of ""¼"", where one factor of ""½"" is from the time coordinate in the denominator of velocity, and another is from the transverse space coordinate. Thus, Piers defines coordinates in which ""P"" has an orbital velocity ""vP"",
""""
Hence,
""""
""""The matter distribution does not alter ""τ-ρ"" coordinates or the non-physical metric. So, it does not alter the rotational velocity, ""vP"", required to define quantum coordinates. If the true orbital velocity of Piers in the gravitational field of a massive body at ""O"" is ""vg"" the orbital velocity in quantum coordinates is
""""
where the signs of each term are the same. The net acceleration toward ""O"" of a point at rest in quantum coordinates is
""""
We use an interpretation in which the wave function describes a state of knowledge used for the calculation of probabilities. When a distant observer, Spike, observes photons from Piers, he applies the classical constraint that there is no expansion in gravitationally bound systems. This removes the apparent acceleration ""H0c ⁄ 32"", leaving a spectral shift equivalent to total inward acceleration
""""
The first term on the right hand side is simply the acceleration due to gravity and corresponds to Piers’s orbital velocity about ""O"". The second yields an illusory velocity corresponding to an illusory inverse acceleration law. This term agrees with the phenomenological law used in ""MOND"" and accounts for the flattening of ""galaxy rotation curves"".
[[QuantumCoordinates Quantum Coordinates ↑]] [[ParticlesOrFields Particles or Fields? →]]
Deletions:
""""====""""====
======[[TeleconnectionIntro ←]] The Teleconnection [[RelationalQuantumGravity ↑]] [[QuantumCoordinates →]]======
A ""connection"" defines the notion of parallel in vector spaces defined at nearby points of a manifold. The teleconnection defines the parallel displacement of momentum in quantum mechanics from an initial state to a final state when the reference matter used to describe the initial state is remote from that used to describe the final state. When the initial and final states are determined with respect to nearby reference matter, the teleconnection is equivalent to the ""Levi-Civita connection"".
""Absolute Space and Time""
""Suppression of Expansion in a Neighbourhood""
""The Teleconnection in a Friedmann Cosmology""
""Cosmological Redshift""
""Energy Transfer""
""Geometries with Expansion""
""Gravitational Redshift""
""The Levi-Civita Connection""
====""""Absolute Space and Time====
The purpose of the teleconnection is to retain the fundamental arguments leading from special relativity to quantum electrodynamics and to general relativity, and to use them to show how apparent fundamental incompatibilities can be reconciled. The teleconnection adapts the probabilistic structure described in quantum theory to the situation studied in general relativity, in which a clock at the position of the emission of a quantum particle does not measure time at the same rate as an identical clock at the position at which the particle is detected.
In a ""Friedmann cosmology"" the existence of ""cosmic time"", or time on a geodesic from the big bang conflicts with Einstein's precepts in which only local law, and local time as measured on a clock, are physically important or meaningful. The teleconnection assumes that since cosmic time makes sense on the large scale in a Friedmann cosmology, it should also make sense on the small scale, even after taking local variations in geometry into account. Cosmic time gains a greater importance than in standard general relativity, and plays a role very much like Newton's absolute time. To describe the transmission of photons over large distances, the ""inner product"" in quantum theory is defined using an integration over all space, and, in an expanding cosmology, only makes complete sense on a synchronous slice. Synchronous slices for given cosmic time yield a "preferred" frame, in which the teleconnection is defined, and play the role of Newton's absolute space. Nonetheless the fundamental elements of the model are particles of matter, and in contrast to Newton’s ideas, time and space appear as emergent organisational principles, not as physical background.
====""""Suppression of Expansion in a Neighbourhood====
""""The inner product is defined to generate probabilities. To specify a probability, it is necessary to specify the known conditions to which the probability applies. We may interpret the collapse of the wave function as the change in probability when the known condition changes. The quantum state describes knowledge of the particle. “Knowledge” in this context refers to information which is available in principle from the physical situation, whether or not it is known in practice by a particular observer. The knowledge about a particle which is possible in principle depends on the physical relationship of that particle with other matter and, in the context of an expanding universe, will have empirical consequences which cannot be reconciled with a view of the wave function as a physical field.
The teleconnection will be applied to two types of physical situation, depending on whether the clock used to determine the final state can be calibrated to the clock used to determine the initial state, for example by Einstein’s calibration procedure.
1. When there is a physical calibration between the clocks used to measure initial and final states quantum theory is formulated without expansion.
2. When no calibration is possible, as is the case for light from a distant stellar object, the quantum theory must be formulated taking into account expansion between coordinates used for initial and final states.
This should be not be understood as a change in physical behaviour, but as a change in the mathematical formulation of quantum theory as a theory of probabilities, reflecting the difference in information available to the observer depending on whether synchronisation is possible between the clocks used to determine the initial and final states. Because a factor of the expansion parameter is removed in the calculation of ""energy/momentum"", both formulations will give the same classical quantities, but there will be a difference in predictions for spectral shifts, determined, for example, by diffraction. This is legitimate for interpretations of quantum theory in which the wave function is a device for the calculation of probabilities, but excludes interpretations which attribute any form of ontological existence to the wave function.
Thus, the rule regarding the evolution of the wave function from an initial state at ""A"" to a remote final state at ""B"" depends on whether the coordinates at ""B"" can be physically calibrated to those at ""A"", for example by Einstein’s synchronisation procedure. When there is a physical calibration, quantum theory is formulated without expansion. When no calibration is possible, as is the case for light from a distant stellar object, the quantum theory must be formulated taking into account expansion between coordinates used for initial and final states. Wave evolution takes place in coordinates defined from the initial state, so that interference effects depend on wavelengths modified by a factor of the expansion parameter. This factor is removed in the calculation of energy-momentum from the wavelength, because physical quantities are defined with respect to current coordinates. When a synchronisation procedure exists between clocks used to describe the initial and final states, expansion is suppressed. This is not a change in the physical behaviour of light, but a change in the mathematical formulation of quantum theory as a theory of probabilities, and reflects the different information available to the observer.
At the present time, it is not possible to write down conditions under which Einstein’s synchronisation procedure is possible. It is known that the [[http://en.wikipedia.org/wiki/Pioneer_anomaly anomalous Pioneer blue shift]] appeared after radar lock was lost. After radar lock was lost with Pioneer, it was no longer possible to calibrate processes on Pioneer to processes on Earth. Synchronisation became impossible in practice, creating the conditions under which the anomalous shift was observed. The shift is seen not as an indication of a change in motion, but as a consequence of a change in the conditions under which quantum theory is formulated. There are indications of an anomalous shift during [[http://en.wikipedia.org/wiki/Flyby_anomaly planetary flybys]], when radar lock is also lost, but no indication of a shift for planets in the outer solar system, whose distance and orbital periods are known to good accuracy and together give information equivalent to clock synchronisation. It remains to be established whether the anomaly appears in consequence of some deep reason dependent on local geometry and motion according to which maintenance of radar lock is impossible in principle, or whether it is sufficient that a synchronisation procedure cannot be carried out in practice. Hopefully future missions will cast greater light on the onset of the anomaly.
====""""The Teleconnection in a Friedmann Cosmology====
Using ""τ−ρ"" coordinates for a ""Penrose diagram"", in a Friedmann cosmology the metric is
"""".
where ""f(ρ) = sinρ"", ""ρ"", or ""sinhρ"" for space with positive, zero or negative curvature respectively. Coordinate time, ""τ"", is related to cosmic time, ""t"", by ""a(τ)dτ = dt"". An observer, Alf, at ""A"" at cosmic time ""t1"" and coordinate time ""τ1"", defines radial unprimed locally Minkowski coordinates, ""(t, r, θ, φ)"" based on cosmic time, ""t"", such that ""dt = a(τ)dτ"" and ""r = a(τ)ρ"". For """" cross terms in the metric can be neglected,
""""
For """",
"""".
Alf defines a non-physical metric, """", using ""τ−ρ"" coordinates, such that,
"""".
where ""A"" and ""B"" are real numbers, whose values are to be determined. Alf defines barred vectors in ""τ−ρ"" coordinates.
<<""Definition: For a vector x = (xτ, xρ, xθ, xφ) at (τ1, A), the corresponding barred vector is""
""""
<<
This definition will apply to classical energy-momentum, but not to quantum energy-momentum which appears in the wave function. Barred quantum energy-momentum will be defined separately.
<<""Definition: For the displacement, x = (xτ, xρ, xθ, xφ), from (τ1, A), the corresponding barred displacement vector is""
""""
It follows that for a vector (including a displacement vector) ""x = (xt, xr, xθ, xφ)"", in locally Minkowski coordinates at ""(t1, A)"",
""""
and that, for barred vectors """" and """" at ""(τ1, A)"", the barred dot product, evaluated with non-physical metric """", satisfies
""""
Alf formulates quantum states locally in Hilbert space at time ""t1"", and defines ""plane wave states"" at ""t1"" using
""""
Quantum theory is then reformulated in terms of barred quantities, under the requirement that the inner product is preserved.
""""
This requires that """", and we have the definition:
<<""Definition: Barred momentum in quantum theory is""
""""
It follows that the momentum space wave function,
""""
is preserved when it is given that """" vanishes outside a sphere of radius ""πa"", and when the wave function is normalised such that """".
The teleconnection defines the inner product between the initial state of a particle emitted at ""(τ1, A)"" and the final state of that particle detected some later time, by translating barred momentum from in ""τ−ρ"" coordinates with constant non-physical metric """".
<<""The Teleconnection: For the barred momentum , the plane wave state at time τ is given in τ−ρ coordinates by""
""""
""where the barred dot product uses the non-physical metric, .""
This definition preserves Newton’s first law and the constancy of the speed of light in ""τ−ρ"" coordinates with non-physical metric """". It is required to retain the formal structure of [[RelativisticQuantumTheory relativistic quantum theory]], and preserves the momentum space wave function as a constant of the motion. A plane wave is sometimes regarded as a physical field in a substantive spacetime. Here, in accordance with the orthodox interpretation, it is seen as a mathematical construction in a non-physical space defined by an observer, Alf, at ""A"". The evolution of the wave function is determined in ""τ−ρ"" coordinates, using non-physical metric """", and is such that it corresponds to the standard wave function in locally Minkowski coordinates.
====""""Cosmological Redshift====
A second observer, Beth, at ""(τ0, B)"", remote from ""(τ1, A)"" and with ""τ0 > τ1"", also defines quantum states in Hilbert space.
<<""Theorem: Let a1 = a(t1) and a0 = a(t0) . Light emitted at time t1 with wavelength λ1 from a distant point A, and detected at B, at time t0 with wavelength λ0 is redshifted according to""
""""
""""**Proof:** One period of light in locally Minkowski coordinates, ""(t, r, θ, φ)"", with an origin at ""A"" at cosmic time ""t1"" is represented by a timelike vector of magnitude ""λ1"". In ""τ−ρ"" coordinates, its timelike component is ""λ1 ⁄ a1"". The corresponding barred quantity is ""λ1 ⁄ Aa1"". The barred vector is translated to ""(τ, B)"". The corresponding unbarred quantity is found from the physical metric, ""g"", and has magnitude ""a0λ1 ⁄ a12"". On transforming to primed locally Minkowski coordinates ""(t', r', θ', φ')"", with an origin at ""B"", we find
""""
It follows that, for small ""r"",
"""".
Thus recession velocity due to expansion is half the value calculated from Doppler, and coordinates in which radial distance from Earth is calculated from redshift exhibit a stretch of factor half in the radial direction. So ""A = 2"". The time taken for a pulse of light at this distance to traverse a small angular distance ""dθ"" is """". So ""B = ½"" (a stretch of factor two in angular directions gives ""4π"" in a circle, which may be related to the spin states of Fermions). Thus, the non-physical metric, """",is
""""
====""""Energy Transfer====
The squared redshift law appears at first to be at odds with the claim that parallel displacement under the teleconnection reduces to parallel transport in the classical correspondence. This is due to the manner in which expansion is treated in the quantum theory and is resolved at the point of the collapse of the wave function. The square redshift law applies to [[http://en.wikipedia.org/wiki/Spectroscopy spectroscopic]] measurements of light from distant sources, when the detection of the photon takes place //after// diffraction or refraction of the quantum wave function. It does not affect energy transfer because, in order to formulate the inner product between an initial state at time ""t0"" and a final state at time ""t1"", Beth enlarges the coordinate axes at time ""t0"" by a factor ""a0 ⁄ a1"". This affects spectroscopic measurements, but, since the initial measurement of energy-momentum is relative to the coordinate axes at time ""t0"". this factor does not appear in the calculation of energy/momentum of light from a distant source. The energy """" measured locally by Beth at time ""t0"" of a photon emitted at time ""t1"" with energy """" from a from a distant source at time ""t1"" is given by
""""
Thus we have the same relation for energy transferred as is given by parallel transport under the ""Levi-Civita connection"".
====""""Geometries with Expansion====
In general relativity, time is determined from a clock locally. Although cosmic time is defined globally, its definition, based on ""Weyl’s postulate"" depends on the local time of many galaxies on geodesics from the Big Bang. The Friedmann models are based on the particular assumptions that the distribution of matter is homogeneous and isotropic. This is observably true in reasonable approximation when the matter distribution is averaged over large enough distances. So, we expect these models to give a reasonable description of the universe at large scales. But these models can only be approximate because they take no account of local mass distributions or of peculiar motions of galaxies and the orbits of stars within a galaxy. On their own, the Friedmann models say nothing about what happens at smaller scales, at which matter is clearly not homogeneous. However, it is natural to think that local fluctuations in geometry due to the inhomogeneous local matter distribution can be treated as perturbations to a Friedmann model (at least for points where the gravitational field is not large).
Since it does not make sense to talk of expansion locally, ""a(τ)"" is a global parameter. We may define space-like hypersurfaces with ""a(τ) = const"" and define ""τ"" to be a global time parameter with ""τ = const"" on any surface with scale factor ""a(τ) = const"". The teleconnection postulates that we may define a non-physical metric, """", as for a Friedmann cosmology in which the speed of light is constant. In ""τ−ρ"" coordinates,
""""
This seems reasonable for points which are not hidden by an [[http://en.wikipedia.org/wiki/Event_horizon event horizon]]. Points inside an event horizon will be considered again later.
The calculation of the general form of the metric for observers at constant ""ρ"" goes through as for ""stationary observers"", but now we have an additional factor of the expansion parameter. Thus the physical metric has the form
""""
where the factor ""k"" describes gravitational redshift. While the calculation of parallel displacement in ""τ−ρ"" coordinates is straightforward, the determination of the relationship between ""τ−ρ"" coordinates and inertial locally Minkowski coordinates depends on both the motion of the observer and on local variations in geometry, summarised in the factor ""k"". The calculation is complicated by the fact that, in the general case, motion with ""ρ = const"" is not inertial. The consequence is that an an inertial observer will need to modify wave functions in such a way that they are subjected to apparent accelerations, or Doppler shifts which do not reflect acceleration in the classical domain.
With the substitutions ""dt' = a(τ)dτ"" ""r' = a(τ)ρ"", ""θ' = θ"", and ""φ' = φ"", and for """"
""""
For """",
""""
in agreement with the calculation of the form of the ""metric for stationary observers"". Locally Minkowski coordinates at the point ""x"" with ""r = 0"" are found by substituting ""dt = k−1dt'"" and ""dr = kdr'"", ""rdα = r'dθ'"", ""r sinα dβ = r' sinθ' dφ'"". For small ""r"",
""""
Barred vectors are defined in ""τ−ρ"" coordinates as for a ""Friedmann cosmology"".
<<""Definition: For a vector x = (xτ, xρ, xθ, xφ), at (τ1, A), the corresponding barred vector is""
""""
For a vector ""x = (xt, xr, xα, xβ)"" in locally Minkowski, ""t-r"" , coordinates at ""(t1, A)"" where ""t1 ↔ τ1""
""""
Alf formulates quantum states locally in Hilbert space at time ""t1"", and defines ""plane wave states"" at ""t1"" using
""""
Quantum theory is reformulated globally using barred quantities, under the requirement that the inner product is preserved.
""""
This requires that """", and we have the definition:
<<""Definition: Barred momentum is""
""""
We will be interested in weak fields and the transmission of photons over astronomical distances, for which momentum may be taken as radial up to the bending of light by a lens. It will be sufficient to consider the approximation
""""
====""""Gravitational Redshift====
Let ""A"" and ""B"" be sufficiently close that gravitational lensing effects may be ignored. Then, in coordinates with an origin at ""A"", the momentum of a photon passing from ""A"" at ""t1"" to ""B"" at ""t2"" is radial,
""""
Let the redshift factor at ""A"" at time ""t1"" be ""kA"", and let the redshift factor at ""B "" at ""t2"" be ""kB"", and let ""a1 = a(t1)"" and ""a2 = a(t2)"". Then, barred momentum is
""""
Barred momentum is translated to ""(τ2, B)"" in ""τ−ρ"" coordinates. Beth converts to locally Minkowski ""t'-r'"" coordinates, and finds, with the removal of a factor of the expansion,
""""
which is the same as is found by ""parallel displacement"" of momentum through a small distance in tangent space when the metric is
""""
In locally Minkowski coordinates at ""A"", gravitational redshift is given by a factor ""(g00)−½"", in agreement with standard general relativity.
====""""The Levi-Civita Connection====
The classical correspondence is found when there is sufficient information that states may be treated as being continuously measured. The quantum state is defined on a synchronous slice using quantum time, ""t"". In order to describe a continuous classical motion, we must take a collection of synchronous slices, and treat each slice as the final state of one quantum step and as the initial state of the next quantum step, then allow the step size to go to zero, to obtain a [[http://en.wikipedia.org/wiki/Foliation foliation]]. At each stage of the motion, quantum theory is formulated in a space with a non-physical metric """". Classical motion is determinate and may be described as an ordered sequence, """", of ""effectively measured"" states at instances ""ti"" such that ""0 < ti+1 - ti < δ"" where ""δ"" is sufficiently small that there is negligible alteration in predictions in the limit ""δ → 0"". Each state, """", is a multiparticle state in [[Multiparticle Fock space]], which has been relabelled using mean properties determined by an effective classical measurement. For the motion between times ""ti"" and ""ti+1"", """" may be regarded as the initial state and """" may be regarded as the final state. Since the time evolution of mean properties is determinate, there is no collapse and """" is the initial state for the motion to ""ti+2"". Classical formulae are recovered by considering a sequence of initial and final effectively measured states in the limit as the maximum time interval between them goes to zero. In each stage of the motion momentum is parallel displaced using the teleconnection, which gives the same result as parallel displacement in tangent space. The cumulative effect of such infinitesimal parallel displacements is ""parallel transport"". So the teleconnection between initial and final states in quantum theory leads to parallel transport in the classical domain. The same argument holds for a classical beam of light, in which each photon wave function is localised within the beam at any time, and for a classical field which has a measurable value at each point where it is defined (up to the possible addition of an arbitrary gauge function).
[[Teleconnection The Teleconnection ↑]] [[QuantumCoordinates Illusory Velocity →]]
Additions:
Additions:
====""""Absolute Space and Time====
====""""Suppression of Expansion in a Neighbourhood====
====""""The Teleconnection in a Friedmann Cosmology====
====""""Cosmological Redshift====
====""""Energy Transfer====
====""""Geometries with Expansion====
====""""Gravitational Redshift====
====""""The Levi-Civita Connection====
Deletions:
====""""Absolute Space and Time====
====""""Suppression of Expansion in a Neighbourhood====
====""""The Teleconnection in a Friedmann Cosmology====
====""""Cosmological Redshift====
====""""Energy Transfer====
====""""Geometries with Expansion====
====""""Gravitational Redshift====
====""""The Levi-Civita Connection====
Additions:
====""""Absolute Space and Time====
====""""Suppression of Expansion in a Neighbourhood====
====""""The Teleconnection in a Friedmann Cosmology====
where ""f(ρ) = sinρ"", ""ρ"", or ""sinhρ"" for space with positive, zero or negative curvature respectively. Coordinate time, ""τ"", is related to cosmic time, ""t"", by ""a(τ)dτ = dt"". An observer, Alf, at ""A"" at cosmic time ""t1"" and coordinate time ""τ1"", defines radial unprimed locally Minkowski coordinates, ""(t, r, θ, φ)"" based on cosmic time, ""t"", such that ""dt = a(τ)dτ"" and ""r = a(τ)ρ"". For """" cross terms in the metric can be neglected,
For """",
====""""Cosmological Redshift====
====""""Energy Transfer====
====""""Geometries with Expansion====
With the substitutions ""dt' = a(τ)dτ"" ""r' = a(τ)ρ"", ""θ' = θ"", and ""φ' = φ"", and for """"
====""""Gravitational Redshift====
====""""The Levi-Civita Connection====
Deletions:
====""""Absolute Space and Time====
====""""Suppression of Expansion in a Neighbourhood====
====""""The Teleconnection in a Friedmann Cosmology====
where ""f(ρ) = sinρ"", ""ρ"", or ""sinhρ"" for space with positive, zero or negative curvature respectively. Coordinate time, ""τ"", is related to cosmic time, ""t"", by ""a(τ)dτ = dt"". An observer, Alf, at ""A"" at cosmic time ""t1"" and coordinate time ""τ1"", defines radial unprimed locally Minkowski coordinates, ""(t, r, θ, φ)"" based on cosmic time, ""t"", such that ""dt = a(τ)dτ"" and ""r = a(τ)ρ"". The metric in ""t-r"" coordinates is
For """",
====""""Cosmological Redshift====
====""""Energy Transfer====
====""""Geometries with Expansion====
With the substitutions ""dt' = a(τ)dτ"" ""r' = a(τ)ρ"", ""θ' = θ"", and ""φ' = φ"",
====""""Gravitational Redshift====
====""""The Levi-Civita Connection====
""The [[Teleconnection teleconnection]] has been described using ""τ−ρ coordinates"" for a ""Penrose diagram"". The ""general form of the metric"" is