Most recent edit on 2008-08-25 02:03:31 by CharlesFrancis
Additions:
← A Gravitating Particle ↑ →
A Gravitating Particle ↑ The Emergence of Spacetime Structure →
Deletions:
← A Gravitating Particle ↑ →
A Gravitating Particle ↑ Einstein’s Field Equation →
Edited on 2008-08-25 00:51:12 by CharlesFrancis
Additions:
← A Gravitating Particle ↑ →
Relational quantum gravity modifies the standard, continuum, form of qed and introduces an effective minimal interval of proper time between interactions of an elementary particle. The result of this modification is that instead of Minkowski spacetime, we find a curved spacetime obeying Einstein’s field equation.
The k-Calculus
Special relativity was described by Einstein as a principle theory, as distinct from a fundamental theory, because it derives its results from principles of measurement, and does not explain fundamental physical mechanisms. It enables us to derive the mathematical properties of Minkowski spacetime, but does not explain the underlying structure, or give the reason why light behaves as it does. In Foundations Of Special Relativity, the structure of Minkowski spacetime was found from the radar method, using two way transmission of light. As used in the radar method, light is the sensor for the properties of spacetime, not the cause of them. The radar method assumes a greater importance in relational quantum gravity. In qed, two-way transmission of photons is responsible for the electromagnetic force, and hence for the all the structures of matter we observe in our immediate environment. The radar method can use light as a sensor for the properties of spacetime because it uses the same physical process, two-way photon exchange, which is also the cause of the structure of spacetime.
The k-calculus» was introduced by Hermann Bondi» as a simple means of introducing special relativity. In the k-calculus, the metric is determined by the reflection of light (the radar method). Any other method of measurement may be used provided that it is calibrated to, and hence equivalent to, radar. In special relativity k may be identified with Doppler shift, k = 1 + z . The treatment of stationary observers generalises the k-calculus to the case in which k may be identified with gravitational redshift. This approach is directly based on the equivalence principle, has advantages in (relative) mathematical and conceptual simplicity, and leads to a formulation of general relativity which is mathematically equivalent to other formulations.
Here I generalise the radar method to allow that the minimum time for the return of a photon may depend not only on the speed of light, c, but also on an effective minimum proper time interval between the absorption and re-emission of a reflected photon. In the idealised case of a single elementary particle, neglecting spin, the geometrical effect is calculated. I will show that an effective minimal time interval of reflection proportional to the mass of the reflecting elementary particle leads the replacement of Minkowski geometry by Schwarzschild.
A Modification to Radar
Consistent with Einstein’s 1905 paper and the internationally agreed empirical definition of the metre, Bondi's k-calculus for special relativity postulates instantaneous reflection of radar at the event whose position is to be determined. Although reflection clearly takes place on a very small timescale, there is no empirical basis on which we can say it is actually instantaneous. A natural generalisation is to hypothesise a small time delay between absorption and emission in proper time of a fundamental charged particle (electron or quark) reflecting electromagnetic radiation.
An intrinsic delay between the interactions of elementary particles affects the empirical definition of spacetime measurement (e.g. SI units»). We seek to analyse the geometric implications. The metric is determined as in the k-calculus for special relativity, from the minimum time for the return of information reflected at an event. But now this minimum net time depends not only on the maximum theoretical speed of information, c, but also on an effective least proper time between absorption and emission in the reflection of a photon. Let the effective time delay be 4GM, where M is the mass of the reflecting particle and 4G is a constant of proportionality. It will be seen that G may be identified with the gravitational constant. Special relativity can be recovered in the limit in which G goes to zero (allowing G to go to zero introduces the Landau pole, so this limit may not be strictly valid).
There are several reasons for introducing such a delay. Firstly, as shown here, the delay perturbs the metric, resulting in a curved spacetime obeying Einstein’s field equation. If the reflection of a photon were instantaneous, the physical metric would be Minkowski. The intrinsic time delay in reflection, 4GM, causes a small amount of curvature, in accordance with Einstein's field equation. Secondly, it is well known that some small scale modification is needed to qed in order to remove the ultraviolet divergence and resolve the Landau pole. The delay introduced here is an effective cut-off and allows the construction of a consistent qed. Finally, a minimum time between interactions proportional to mass may be related to the concept of inertia; if the interactions of muons and electrons with photons are discrete and identical, then it is natural that the acceleration due to the electromagnetic field will be proportional to the frequency of interaction. An intrinsic delay between interactions proportional to mass will result in accelerations inversely proportional to mass.
The calculation performed here applies to fundamental stable particles which can emit and absorb photons. In the real world these are charged particles with spin, electrons or quarks. It does not directly apply to macroscopic bodies. If radar is used to measure the position of the moon, for example, then an individual reflected photon can be said only to measure the position of a single electron in the surface of the moon. A classical radar pulse contains many such photons, the time delay at each reflection being dependant on the mass of an electron, not the mass of the moon. A more realistic calculation should also take into account charge and spin, and would be expected to yield Kerr-Newman geometry. It is not known how to carry out such a calculation within the k-calculus, but it appears reasonable to separate off the contributions due to charge and spin, and to regard the calculation below of a Schwarzschild geometry surrounding a particle in a position eigenstate as a genuine indicator of an effective time interval between the interactions of an elementary particles.
A Particle in a Position Eigenstate
Since I am discussing measurement of position, I will describe eigenstates of position. There is no such thing as a perfect eigenstate of position of a massive point particle. Nonetheless such states span Hilbert space and will be sufficient for a description of geometry. For the purpose of analysis, I will consider a static system and calculate the metric at particular time. I consider only a system with a single gravitating particle, at O. I will assume distance and time scales such that cosmological expansion is negligible.
 An isolated elementary particle in an eigenstate of position has spherical symmetry and spacetime diagrams may be used to show a radial coordinate in n dimensions without loss of generality. A spacetime diagram is drawn, showing a tangent space with an origin at O, so that light is shown at 45°, lines of equal time are horizontal and time is proper time for the gravitating particle. In tangent space at O, the coordinate distance between Beth and the particle is r. |
 As shown in tangent space at O, Beth uses the radar method to determine a distance coordinate for the particle. Beth cannot resolve the points A and E where a photon is absorbed and emitted and places the particle at apparent position P. If the effective delay in the reflection is 4GM, then the coordinate coordinate distance of P from Beth is ρ = r + 2GM. |
 As calculated for stationary observers, the physical metric (with angular directions suppressed) is
 
So, Beth’s clock runs fast by a factor k compared to proper time, t, for the particle at O. Minkowski coordinates (t, r) for tangent space at O are stretched by a factor k−1 in the time direction and by k in the radial direction compared to Minkowski coordinates, (T, R) for tangent space at B, as determined by Beth using the radar method. Since the radar method determines that R = T, we have
So,
Using the apparent position ρ as the radial coordinate, and substituting ρ, we find the Schwarzschild metric:
 |
It will be observed that the event horizon, ρ = 2GM, maps to r = 0. Only space outside the event horizon is mapped in these coordinates. The space inside the event horizon, ρ < 2GM, has no physical meaning. A particle which is point-like in a tangent space at the position of the particle, is mapped to the event horizon in a tangent space at the position of a distant observer. The effect of the discrete interval of proper time between interactions is a singularity at ρ = 0 which “magnifies” a point-like particle to the size of the event horizon.
The Path of Light
 With r as the radial coordinate, the metric is
Superficially, it appears that the speed of light is less than unity in these coordinates. Light should then be drawn at a greater than 45° to the horizontal. The inconsistency is resolved because this refers to the mean velocity of light, as determined from measurement using the radar method. The true path of light is plotted at 45°. The coordinate singularity, or event horizon in the Schwarzschild geometry, is actually at the position of the particle itself, while the singularity at ρ = 0 is an illusory point, which does not correspond to a position in space. |
By considering the coordinate transformation r → r + δr, we replace the delta function with a uniform density over a small region. We now take the limit δr → 0 to see that Einstein’s field equation holds for the single particle case
The derivation here applies only to the case where the gravitating source is a single particle with known position. The next section considers the issues in the general case.
A Gravitating Particle ↑ Einstein’s Field Equation →
Deletions:
The Physical Origin of Curvature
In relational quantum gravity, the most fundamental description of matter is illustrated by
Feynman diagrams. In Feynman diagrams, space, and hence curvature, have no meaning, and emerge only in the classical correspondence from the mean behaviour of large populations of particles. The aim of this section is to illustrate that the curvature of space is a consequence of a requirement that there is an effective minimal interval of proper time between interactions of an elementary particle.
The Physical Content of Einstein’s Field Equation
“Matter tells space how to curve. Space tells matter how to move.” — John Archibald Wheeler.
The physical content of Einstein’s Field equation has been famously summarised by [[http://en.wikipedia.org/wiki/John_Archibald_Wheeler» John Archibald Wheeler. The first part is straightforward: in words, Einstein’s field equation,
simply says the Einstein curvature tensor is proportional to the stress energy tensor, i.e. “matter tells space how to curve”. The second part, “space tells matter how to move”, is given by the
from which follows the law of local ,
Conservation of energy and momentum have been shown as a requirement of a relativistic theory of particle interactions. In special relativity the structure of Minkowski spacetime was found from the radar method, using two way transmission of light.
Special relativity was described by Einstein as a principle theory, as distinct from a fundamental theory, because it derives its results from principles of measurement, and does not explain fundamental physical mechanisms. It enables us to derive the mathematical properties of Minkowski spacetime, but does not explain the underlying structure, or give the reason why light behaves as it does. As used in the radar method, light is the sensor for the properties of spacetime, not the cause of them. The radar method assumes a greater importance in relational quantum gravity. In [[QuantumElectrodynamics qed] two-way transmission of photons is responsible for the electromagnetic force, and hence for the all the structures of matter we observe in our immediate environment. Space has no meaning in Feynman diagrams and emerges in the macroscopic properties of structures described mathematically as graphs. The radar method can use light as a sensor for the properties of spacetime because it uses the same physical process, two-way photon exchange, which is also the cause of the structure of spacetime. Relational quantum gravity modifies the standard, continuum, form of qed and introduces an effective minimal interval of proper time between interactions of an elementary particle. The result of this modification is that instead of the emergence of Minkowski spacetime, we will find in the classical correspondence a curved spacetime obeying Einstein’s field equation.
The k-calculus
The <a href=http://en.wikipedia.org/wiki/Bondi_k-calculus><span» class=math><i>k</i>-</span>calculus</a><sup>»</sup> was introduced by Hermann Bondi» as a simple means of introducing special relativity. In the <span class=math><i>k</i>-</span>calculus the metric is determined by the reflection of light (the radar method). Any other method of measurement may be used provided that it is calibrated to, and hence equivalent to, radar. In special relativity <span class=math><i>k</i> may be identified with Doppler shift, . The treatment of general relativity given in relational quantum gravity generalises the <span class=math><i>k</i>-</span>calculus to the case of <a href=http://www.teleconnection.info/rqg/GeneralRelativity#StationaryObservers>stationary» observers</a>, in which <span class=math><i>k</i>-</span> may be identified with gravitational redshift. The approach is directly based on the equivalence principle, has advantages in (relative) mathematical and conceptual simplicity, and leads to a formulation of general relativity which is mathematically equivalent to other formulations.
Relational quantum gravity applies the same approach to extend classical general relativity with the introduction of the teleconnection. The teleconnection enables the formulation of a relativistic quantum theory in which the initial and final states are described in reference frames separated by astronomical distances. Later on this site, I show that the teleconnection has cosmological implications and describe empirical evidence which supports a model in which the transmission of light (photons) over astronomical distances is described by the teleconnection.
Here I generalise the radar method to allow that the minimum time for the return of a photon may depend not only on the speed of light, c, but also on a minimum proper time interval between the absorption and re-emission of a reflected photon. In the idealised case of a single elementary particle, neglecting spin, is calculated and show that of a minimal time interval of reflection proportional to the mass of the reflecting elementary particle leads the replacement of Minkowski geometry by Schwarzschild. This result will be used in the next section to show Einstein’s field equation.
 I do not infer that radar is the fundamental definition of distance. In relational quantum gravity, the fundamental structures of matter is aplenum consisting of arrangements of particles described by Feynman diagrams. The background in a Feynman diagram has no mathematical meaning. Distance scales are emergent quantities arising from the structure of the plenum. Radar provides a direct measurement of distance because it uses the same physical process, namely two way photon exchange, as is seen in diagrams for stable configurations of matter. |
| This notional diagram restores a time ordering between the interactions, showing photon exchange as a two way process. Such a diagram is at best a rough approximation to the truth. There is no fixed time interval between absorption and emission, and we can only describe the radius of the orbit of an electron using a probability amplitude, not a numerical value. |
Systems containing larger numbers of particles also contain greater structure. Spacetime structure emerges only as a macroscopic property of systems containing many particles. |
The calculation performed here applies to fundamental stable particles which can emit and absorb photons. In the real world these are charged particles with spin, electrons or quarks. It does not directly apply to macroscopic bodies. If radar is used to measure the position of the moon, for example, then an individual reflected photon can be said only to measure the position of a single electron in the surface of the moon. A classical radar pulse contains many such photons, the time delay at each reflection being dependant on the mass of an electron, not the mass of the moon. A more realistic calculation should also take into account charge and spin, and would be expected to yield Kerr-Newman geometry. It is not known how to carry out such a calculation within the k-calculus, but it appears reasonable to separate off the contributions due to charge and spin, and to regard the calculation of Schwarzschild as a genuine indicator of an effective time interval between the interactions of an elementary particles.
A Modification to Radar
Consistent with Einstein’s 1905 paper and the internationally agreed empirical definition of the metre, Bondi's k-calculus for special relativity postulates instantaneous reflection of radar at the event whose position is to be determined. Although reflection clearly takes place on a very small timescale, there is no empirical basis on which we can say it is actually instantaneous. A natural generalisation is to hypothesise a small time delay between absorption and emission in proper time of a fundamental charged particle (electron or quark) reflecting electromagnetic radiation.
There are several reasons for introducing such a delay. Firstly, as shown here, the delay perturbs the metric, resulting in a curved spacetime obeying Einstein’s field equation. As Bondi’s k-calculus shows, if the reflection of a photon were instantaneous, the physical metric would be radially Minkowski. The intrinsic time delay in reflection, 4GM, causes a small amount of curvature, in accordance with Einstein's field equation. Secondly, it is well known that some small scale modification is needed to qed in order to remove the ultraviolet divergence and resolve the Landau pole. The delay introduced here is an effective cut-off and allows the construction of a consistent qed. Finally, a minimum time between interactions proportional to mass may be related to the concept of inertia; if the interactions of muons and electrons with photons are discrete and identical, then it is natural that the acceleration due to the electromagnetic field will be proportional to the frequency of interaction.
An intrinsic delay between the interactions of elementary particles affects the empirical definition of spacetime measurement (e.g. SI units»). We seek to analyse the geometric implications. The metric is determined as in the k-calculus for special relativity, from the minimum time for the return of information reflected at an event. But now this minimum net time depends not only on the maximum theoretical speed of information, c, but also on the least proper time between absorption and emission in the reflection of a photon. Let the time delay be , where M is the mass of the reflecting particle and 4G is a constant of proportionality. It will be seen that G may be identified with the gravitational constant. Special relativity can be recovered in the limit in which G goes to zero (allowing G to go to zero introduces the Landau pole, so this limit may not be valid).
A Particle in a Position Eigenstate
Since I am discussing measurement of position, I will describe <a href=http://www.teleconnection.info/rqg/Observables#ObservableOperators>eigenstates</a»> of position in a quantum theory. There is no such thing as a perfect eigenstate of position of a massive point particle. Nonetheless such states span Hilbert space and will be sufficient for a full description of geometry. For the purpose of analysis, I will consider a static system and calculate the metric at particular time. I consider only a system with a single gravitating particle, at at O. I will assume distance and time scales such that expansion is negligible.
An isolated elementary particle in an eigenstate of position has spherical symmetry and spacetime diagrams may be used to show a radial coordinate in n dimensions without loss of generality. A spacetime diagram is drawn, showing a tangent space with an origin at O, so that light is shown at 45o, lines of equal time are horizontal and time is proper time for the gravitating particle. In tangent space at O, the coordinate distance between Beth and the particle is r.
As shown in tangent space at O, Beth uses the radar method to determine a distance coordinate for the particle. Beth cannot resolve the points A and E where a photon is absorbed and emitted and places the particle at apparent position P. If the effective delay in the reflection is 4GM, then the coordinate coordinate distance of P from Beth is
As calculated for <a href=http://www.teleconnection.info/rqg/GeneralRelativity#StationaryObservers>stationary» observers</a>, the physical metric (with angular directions suppressed) is
So, Beth’s clock runs fast by a factor k compared to proper time t for the particle at O. Minkowski coordinates (t, r') for tangent space at O are stretched by a factor k-1 in the time direction and by k in the radial direction compared to Minkowski coordinates (T,R) for tangent.space at B, as determined by Beth using the radar method. Since the radar method determines that we have
So,
Using the apparent position r as the radial coordinate, and substituting k, we find the Schwarzschild metric:
It will be observed that the event horizon, , maps to . Thus, a particle which is point-like in a tangent space at the position of the particle, is mapped to the event horizon in a tangent space at the position of a distant observer. The effect of the discrete interval of proper time between interactions is to “magnify” a point-like particle to the size of the event horizon. The space inside the event horizon, , has no physical meaning.
The Path of Light
With r as the radial coordinate, the metric is
Only space outside the event horizon is mapped in these coordinates. Superficially, it appears that the speed of light is less than unity in these coordinates. Light should then be drawn at a greater than 45° to the horizontal. The inconsistency is resolved because this refers to the mean velocity of light, as determined from measurement using the radar method. The true path of light is plotted at 45°. The coordinate singularity, or event horizon in the Schwarzschild geometry, is actually at the position of the particle itself, while the singularity at r = 0 is an illusory point, which does not correspond to a position in space.
By considering the coordinate transformation , we replace the delta function with a uniform density over a small region. We now take the limit to see that Einstein’s field equation holds for the single particle case
The derivation here applies only to the case where the gravitating source is a single particle with known position.
A general consequence of Einstein Field equation is that changes in the mass distribution change the manifold. This creates difficulties which are resolved by using coordinates defined from lightspeed = c. These coordinates are independent of the mass distribution. The last section applied the radar method to a single, static, elementary particle in an eigenstate of position, neglecting spin and assuming an intrinsic time delay in reflection at the particle. Schwarzschild geometry was found in coordinates in which light is drawn at 45°, and the event horizon is a singular point at r=0, showing that Einstein’s field equation holds for the single particle case
Embedding these coordinates into a <a href=http://www.teleconnection.info/rqg/LargeScaleStructure>Penrose» diagram</a> for the universe leads intuitively to the conjecture that the underlying cause of curvature is intrinsic time delay in reflection by a charged particle, and that consistency, together with conservation momentum, require the general application of Einstein’s field equation.
Quantum theory is defined using plane wave motions on a Penrose diagram. The factor k is only invoked to determine the redshift between the initial and final states. Clearly k, (and hence Gab) is a parameter, not an operator. The parameter G depends on the actual, but intrinsically unknown, distribution of matter, not the quantum state.
The next section considers the issues in the general case.
Edited on 2008-08-25 00:50:22 by CharlesFrancis
Additions:
The Physical Origin of Curvature
In relational quantum gravity, the most fundamental description of matter is illustrated by
Feynman diagrams. In Feynman diagrams, space, and hence curvature, have no meaning, and emerge only in the classical correspondence from the mean behaviour of large populations of particles. The aim of this section is to illustrate that the curvature of space is a consequence of a requirement that there is an effective minimal interval of proper time between interactions of an elementary particle.
“Matter tells space how to curve. Space tells matter how to move.” — John Archibald Wheeler.
The physical content of Einstein’s Field equation has been famously summarised by [[http://en.wikipedia.org/wiki/John_Archibald_Wheeler» John Archibald Wheeler. The first part is straightforward: in words, Einstein’s field equation,
simply says the Einstein curvature tensor is proportional to the stress energy tensor, i.e. “matter tells space how to curve”. The second part, “space tells matter how to move”, is given by the
from which follows the law of local ,
Conservation of energy and momentum have been shown as a requirement of a relativistic theory of particle interactions. In special relativity the structure of Minkowski spacetime was found from the radar method, using two way transmission of light.
Special relativity was described by Einstein as a principle theory, as distinct from a fundamental theory, because it derives its results from principles of measurement, and does not explain fundamental physical mechanisms. It enables us to derive the mathematical properties of Minkowski spacetime, but does not explain the underlying structure, or give the reason why light behaves as it does. As used in the radar method, light is the sensor for the properties of spacetime, not the cause of them. The radar method assumes a greater importance in relational quantum gravity. In [[QuantumElectrodynamics qed] two-way transmission of photons is responsible for the electromagnetic force, and hence for the all the structures of matter we observe in our immediate environment. Space has no meaning in Feynman diagrams and emerges in the macroscopic properties of structures described mathematically as graphs. The radar method can use light as a sensor for the properties of spacetime because it uses the same physical process, two-way photon exchange, which is also the cause of the structure of spacetime. Relational quantum gravity modifies the standard, continuum, form of qed and introduces an effective minimal interval of proper time between interactions of an elementary particle. The result of this modification is that instead of the emergence of Minkowski spacetime, we will find in the classical correspondence a curved spacetime obeying Einstein’s field equation.
The k-calculus
The <a href=http://en.wikipedia.org/wiki/Bondi_k-calculus><span» class=math><i>k</i>-</span>calculus</a><sup>»</sup> was introduced by Hermann Bondi» as a simple means of introducing special relativity. In the <span class=math><i>k</i>-</span>calculus the metric is determined by the reflection of light (the radar method). Any other method of measurement may be used provided that it is calibrated to, and hence equivalent to, radar. In special relativity <span class=math><i>k</i> may be identified with Doppler shift, . The treatment of general relativity given in relational quantum gravity generalises the <span class=math><i>k</i>-</span>calculus to the case of <a href=http://www.teleconnection.info/rqg/GeneralRelativity#StationaryObservers>stationary» observers</a>, in which <span class=math><i>k</i>-</span> may be identified with gravitational redshift. The approach is directly based on the equivalence principle, has advantages in (relative) mathematical and conceptual simplicity, and leads to a formulation of general relativity which is mathematically equivalent to other formulations.
Relational quantum gravity applies the same approach to extend classical general relativity with the introduction of the teleconnection. The teleconnection enables the formulation of a relativistic quantum theory in which the initial and final states are described in reference frames separated by astronomical distances. Later on this site, I show that the teleconnection has cosmological implications and describe empirical evidence which supports a model in which the transmission of light (photons) over astronomical distances is described by the teleconnection.
Here I generalise the radar method to allow that the minimum time for the return of a photon may depend not only on the speed of light, c, but also on a minimum proper time interval between the absorption and re-emission of a reflected photon. In the idealised case of a single elementary particle, neglecting spin, is calculated and show that of a minimal time interval of reflection proportional to the mass of the reflecting elementary particle leads the replacement of Minkowski geometry by Schwarzschild. This result will be used in the next section to show Einstein’s field equation.
| I do not infer that radar is the fundamental definition of distance. In relational quantum gravity, the fundamental structures of matter is aplenum consisting of arrangements of particles described by Feynman diagrams. The background in a Feynman diagram has no mathematical meaning. Distance scales are emergent quantities arising from the structure of the plenum. Radar provides a direct measurement of distance because it uses the same physical process, namely two way photon exchange, as is seen in diagrams for stable configurations of matter. |
| This notional diagram restores a time ordering between the interactions, showing photon exchange as a two way process. Such a diagram is at best a rough approximation to the truth. There is no fixed time interval between absorption and emission, and we can only describe the radius of the orbit of an electron using a probability amplitude, not a numerical value. |
Systems containing larger numbers of particles also contain greater structure. Spacetime structure emerges only as a macroscopic property of systems containing many particles. |
The calculation performed here applies to fundamental stable particles which can emit and absorb photons. In the real world these are charged particles with spin, electrons or quarks. It does not directly apply to macroscopic bodies. If radar is used to measure the position of the moon, for example, then an individual reflected photon can be said only to measure the position of a single electron in the surface of the moon. A classical radar pulse contains many such photons, the time delay at each reflection being dependant on the mass of an electron, not the mass of the moon. A more realistic calculation should also take into account charge and spin, and would be expected to yield Kerr-Newman geometry. It is not known how to carry out such a calculation within the k-calculus, but it appears reasonable to separate off the contributions due to charge and spin, and to regard the calculation of Schwarzschild as a genuine indicator of an effective time interval between the interactions of an elementary particles.
A Modification to Radar
Consistent with Einstein’s 1905 paper and the internationally agreed empirical definition of the metre, Bondi's k-calculus for special relativity postulates instantaneous reflection of radar at the event whose position is to be determined. Although reflection clearly takes place on a very small timescale, there is no empirical basis on which we can say it is actually instantaneous. A natural generalisation is to hypothesise a small time delay between absorption and emission in proper time of a fundamental charged particle (electron or quark) reflecting electromagnetic radiation.
There are several reasons for introducing such a delay. Firstly, as shown here, the delay perturbs the metric, resulting in a curved spacetime obeying Einstein’s field equation. As Bondi’s k-calculus shows, if the reflection of a photon were instantaneous, the physical metric would be radially Minkowski. The intrinsic time delay in reflection, 4GM, causes a small amount of curvature, in accordance with Einstein's field equation. Secondly, it is well known that some small scale modification is needed to qed in order to remove the ultraviolet divergence and resolve the Landau pole. The delay introduced here is an effective cut-off and allows the construction of a consistent qed. Finally, a minimum time between interactions proportional to mass may be related to the concept of inertia; if the interactions of muons and electrons with photons are discrete and identical, then it is natural that the acceleration due to the electromagnetic field will be proportional to the frequency of interaction.
An intrinsic delay between the interactions of elementary particles affects the empirical definition of spacetime measurement (e.g. SI units»). We seek to analyse the geometric implications. The metric is determined as in the k-calculus for special relativity, from the minimum time for the return of information reflected at an event. But now this minimum net time depends not only on the maximum theoretical speed of information, c, but also on the least proper time between absorption and emission in the reflection of a photon. Let the time delay be , where M is the mass of the reflecting particle and 4G is a constant of proportionality. It will be seen that G may be identified with the gravitational constant. Special relativity can be recovered in the limit in which G goes to zero (allowing G to go to zero introduces the Landau pole, so this limit may not be valid).
A Particle in a Position Eigenstate
Since I am discussing measurement of position, I will describe <a href=http://www.teleconnection.info/rqg/Observables#ObservableOperators>eigenstates</a»> of position in a quantum theory. There is no such thing as a perfect eigenstate of position of a massive point particle. Nonetheless such states span Hilbert space and will be sufficient for a full description of geometry. For the purpose of analysis, I will consider a static system and calculate the metric at particular time. I consider only a system with a single gravitating particle, at at O. I will assume distance and time scales such that expansion is negligible.
An isolated elementary particle in an eigenstate of position has spherical symmetry and spacetime diagrams may be used to show a radial coordinate in n dimensions without loss of generality. A spacetime diagram is drawn, showing a tangent space with an origin at O, so that light is shown at 45o, lines of equal time are horizontal and time is proper time for the gravitating particle. In tangent space at O, the coordinate distance between Beth and the particle is r.
As shown in tangent space at O, Beth uses the radar method to determine a distance coordinate for the particle. Beth cannot resolve the points A and E where a photon is absorbed and emitted and places the particle at apparent position P. If the effective delay in the reflection is 4GM, then the coordinate coordinate distance of P from Beth is
As calculated for <a href=http://www.teleconnection.info/rqg/GeneralRelativity#StationaryObservers>stationary» observers</a>, the physical metric (with angular directions suppressed) is
So, Beth’s clock runs fast by a factor k compared to proper time t for the particle at O. Minkowski coordinates (t, r') for tangent space at O are stretched by a factor k-1 in the time direction and by k in the radial direction compared to Minkowski coordinates (T,R) for tangent.space at B, as determined by Beth using the radar method. Since the radar method determines that we have
So,
Using the apparent position r as the radial coordinate, and substituting k, we find the Schwarzschild metric:
It will be observed that the event horizon, , maps to . Thus, a particle which is point-like in a tangent space at the position of the particle, is mapped to the event horizon in a tangent space at the position of a distant observer. The effect of the discrete interval of proper time between interactions is to “magnify” a point-like particle to the size of the event horizon. The space inside the event horizon, , has no physical meaning.
The Path of Light
With r as the radial coordinate, the metric is
Only space outside the event horizon is mapped in these coordinates. Superficially, it appears that the speed of light is less than unity in these coordinates. Light should then be drawn at a greater than 45° to the horizontal. The inconsistency is resolved because this refers to the mean velocity of light, as determined from measurement using the radar method. The true path of light is plotted at 45°. The coordinate singularity, or event horizon in the Schwarzschild geometry, is actually at the position of the particle itself, while the singularity at r = 0 is an illusory point, which does not correspond to a position in space.
Einstein’s Field Equation
For a single particle in an eigenstate of position, stress energy is given by
By considering the coordinate transformation , we replace the delta function with a uniform density over a small region. We now take the limit to see that Einstein’s field equation holds for the single particle case
The derivation here applies only to the case where the gravitating source is a single particle with known position.
A general consequence of Einstein Field equation is that changes in the mass distribution change the manifold. This creates difficulties which are resolved by using coordinates defined from lightspeed = c. These coordinates are independent of the mass distribution. The last section applied the radar method to a single, static, elementary particle in an eigenstate of position, neglecting spin and assuming an intrinsic time delay in reflection at the particle. Schwarzschild geometry was found in coordinates in which light is drawn at 45°, and the event horizon is a singular point at r=0, showing that Einstein’s field equation holds for the single particle case
Embedding these coordinates into a <a href=http://www.teleconnection.info/rqg/LargeScaleStructure>Penrose» diagram</a> for the universe leads intuitively to the conjecture that the underlying cause of curvature is intrinsic time delay in reflection by a charged particle, and that consistency, together with conservation momentum, require the general application of Einstein’s field equation.
Quantum theory is defined using plane wave motions on a Penrose diagram. The factor k is only invoked to determine the redshift between the initial and final states. Clearly k, (and hence Gab) is a parameter, not an operator. The parameter G depends on the actual, but intrinsically unknown, distribution of matter, not the quantum state.
The next section considers the issues in the general case.
Deletions:
← The Physical Origin of Curvature ↑ →
In relational quantum gravity, the most fundamental description of matter is illustrated by
Feynman diagrams. In Feynman diagrams, space, and hence curvature, have no meaning, and emerge only in the classical correspondence from the mean behaviour of large populations of particles. Relational quantum gravity modifies the standard, continuum, form of qed and introduces an effective minimal interval of proper time between interactions of an elementary particle. The result of this modification is that instead of the emergence of Minkowski spacetime, we will find a curved spacetime obeying Einstein’s field equation.
“Matter tells space how to curve. Space tells matter how to move.” — John Archibald Wheeler.
The physical content of Einstein’s Field equation has been famously summarised by John Archibald Wheeler». The first part is straightforward: in words, Einstein’s field equation,
simply says the Einstein curvature tensor is proportional to the stress energy tensor, i.e. “matter tells space how to curve”. The second part, “space tells matter how to move”, is given by the contracted Bianchi identity
from which follows the law of local energy-momentum conservation,
Space has no meaning in Feynman diagrams and emerges in the macroscopic properties of structures described mathematically as graphs». Conservation of energy and momentum was derived from the integral formulae for probability amplitudes associated with these graphs, and follows from the principle that the interactions of elementary particles are always and everywhere the same. The view that space is an emergent property of the interactions of particles casts a deeper light on Einstein’s field equation. Instead of the duality between matter and space described by Wheeler, at a fundamental level we have only matter. Conservation of energy-momentum constrains the structure of spacetime, not the other way around.
Special relativity was described by Einstein as a principle theory, as distinct from a fundamental theory, because it derives its results from principles of measurement, and does not explain fundamental physical mechanisms. It enables us to derive the mathematical properties of Minkowski spacetime, but does not explain the underlying structure, or give the reason why light behaves as it does. In Foundations Of Special Relativity, the structure of Minkowski spacetime was found from the radar method, using two way transmission of light. As used in the radar method, light is the sensor for the properties of spacetime, not the cause of them. The radar method assumes a greater importance in relational quantum gravity. In qed, two-way transmission of photons is responsible for the electromagnetic force, and hence for the all the structures of matter we observe in our immediate environment. The radar method can use light as a sensor for the properties of spacetime because it uses the same physical process, two-way photon exchange, which is also the cause of the structure of spacetime.
The k-Calculus
The k-calculus» was introduced by Hermann Bondi» as a simple means of introducing special relativity. In the k-calculus, the metric is determined by the reflection of light (the radar method). Any other method of measurement may be used provided that it is calibrated to, and hence equivalent to, radar. In special relativity k may be identified with Doppler shift, k = 1 + z . The treatment of stationary observers generalises the k-calculus to the case in which k may be identified with gravitational redshift. This approach is directly based on the equivalence principle, has advantages in (relative) mathematical and conceptual simplicity, and leads to a formulation of general relativity which is mathematically equivalent to other formulations.
Here I generalise the radar method to allow that the minimum time for the return of a photon may depend not only on the speed of light, c, but also on an effective minimum proper time interval between the absorption and re-emission of a reflected photon. In the idealised case of a single elementary particle, neglecting spin, the geometrical effect is calculated. I will show that an effective minimal time interval of reflection proportional to the mass of the reflecting elementary particle leads the replacement of Minkowski geometry by Schwarzschild. This result will be used in the next section to show Einstein’s field equation.
Primitive Structures of Matter
 The use of the radar method should not be taken to imply that radar is fundamental definition of distance. In relational quantum gravity, the fundamental structures of matter is a plenum consisting of arrangements of particles described by Feynman diagrams. The background in a Feynman diagram has no mathematical meaning, so that spacetime background has no physical meaning. Distances are emergent quantities arising from the structure of the plenum. Radar provides a direct measurement of distance because it uses the same physical process, two way photon exchange, as is seen in diagrams for stable configurations of matter. |
 These notional diagrams restore time ordering between the interactions, showing photon exchange as a two way process. Such a diagram is at best a rough approximation to the truth. There is no fixed time interval between absorption and emission, and we can only describe the radius of the orbit of an electron with a probability amplitude, not a numerical value.
Systems containing larger numbers of particles also contain greater structure. For large structures the uncertainties in position are small compared to the scale of the structure. Spacetime structure emerges as a macroscopic property of systems containing many particles. |
A Modification to Radar
Consistent with Einstein’s 1905 paper and the internationally agreed empirical definition of the metre, Bondi's k-calculus for special relativity postulates instantaneous reflection of radar at the event whose position is to be determined. Although reflection clearly takes place on a very small timescale, there is no empirical basis on which we can say it is actually instantaneous. A natural generalisation is to hypothesise a small time delay between absorption and emission in proper time of a fundamental charged particle (electron or quark) reflecting electromagnetic radiation.
An intrinsic delay between the interactions of elementary particles affects the empirical definition of spacetime measurement (e.g. SI units»). We seek to analyse the geometric implications. The metric is determined as in the k-calculus for special relativity, from the minimum time for the return of information reflected at an event. But now this minimum net time depends not only on the maximum theoretical speed of information, c, but also on an effective least proper time between absorption and emission in the reflection of a photon. Let the effective time delay be 4GM, where M is the mass of the reflecting particle and 4G is a constant of proportionality. It will be seen that G may be identified with the gravitational constant. Special relativity can be recovered in the limit in which G goes to zero (allowing G to go to zero introduces the Landau pole, so this limit may not be strictly valid).
There are several reasons for introducing such a delay. Firstly, as shown here, the delay perturbs the metric, resulting in a curved spacetime obeying Einstein’s field equation. If the reflection of a photon were instantaneous, the physical metric would be Minkowski. The intrinsic time delay in reflection, 4GM, causes a small amount of curvature, in accordance with Einstein's field equation. Secondly, it is well known that some small scale modification is needed to qed in order to remove the ultraviolet divergence and resolve the Landau pole. The delay introduced here is an effective cut-off and allows the construction of a consistent qed. Finally, a minimum time between interactions proportional to mass may be related to the concept of inertia; if the interactions of muons and electrons with photons are discrete and identical, then it is natural that the acceleration due to the electromagnetic field will be proportional to the frequency of interaction. An intrinsic delay between interactions proportional to mass will result in accelerations inversely proportional to mass.
The calculation performed here applies to fundamental stable particles which can emit and absorb photons. In the real world these are charged particles with spin, electrons or quarks. It does not directly apply to macroscopic bodies. If radar is used to measure the position of the moon, for example, then an individual reflected photon can be said only to measure the position of a single electron in the surface of the moon. A classical radar pulse contains many such photons, the time delay at each reflection being dependant on the mass of an electron, not the mass of the moon. A more realistic calculation should also take into account charge and spin, and would be expected to yield Kerr-Newman geometry. It is not known how to carry out such a calculation within the k-calculus, but it appears reasonable to separate off the contributions due to charge and spin, and to regard the calculation below of a Schwarzschild geometry surrounding a particle in a position eigenstate as a genuine indicator of an effective time interval between the interactions of an elementary particles.
A Particle in a Position Eigenstate
Since I am discussing measurement of position, I will describe eigenstates of position. There is no such thing as a perfect eigenstate of position of a massive point particle. Nonetheless such states span Hilbert space and will be sufficient for a description of geometry. For the purpose of analysis, I will consider a static system and calculate the metric at particular time. I consider only a system with a single gravitating particle, at O. I will assume distance and time scales such that cosmological expansion is negligible.
| An isolated elementary particle in an eigenstate of position has spherical symmetry and spacetime diagrams may be used to show a radial coordinate in n dimensions without loss of generality. A spacetime diagram is drawn, showing a tangent space with an origin at O, so that light is shown at 45°, lines of equal time are horizontal and time is proper time for the gravitating particle. In tangent space at O, the coordinate distance between Beth and the particle is r. |
| As shown in tangent space at O, Beth uses the radar method to determine a distance coordinate for the particle. Beth cannot resolve the points A and E where a photon is absorbed and emitted and places the particle at apparent position P. If the effective delay in the reflection is 4GM, then the coordinate coordinate distance of P from Beth is ρ = r + 2GM. |
As calculated for stationary observers, the physical metric (with angular directions suppressed) is
So, Beth’s clock runs fast by a factor k compared to proper time, t, for the particle at O. Minkowski coordinates (t, r) for tangent space at O are stretched by a factor k−1 in the time direction and by k in the radial direction compared to Minkowski coordinates, (T, R) for tangent space at B, as determined by Beth using the radar method. Since the radar method determines that R = T, we have
So,
Using the apparent position ρ as the radial coordinate, and substituting ρ, we find the Schwarzschild metric:
|
It will be observed that the event horizon, ρ = 2GM, maps to r = 0. Only space outside the event horizon is mapped in these coordinates. The space inside the event horizon, ρ < 2GM, has no physical meaning. A particle which is point-like in a tangent space at the position of the particle, is mapped to the event horizon in a tangent space at the position of a distant observer. The effect of the discrete interval of proper time between interactions is a singularity at ρ = 0 which “magnifies” a point-like particle to the size of the event horizon.
The Path of Light
With r as the radial coordinate, the metric is
Superficially, it appears that the speed of light is less than unity in these coordinates. Light should then be drawn at a greater than 45° to the horizontal. The inconsistency is resolved because this refers to the mean velocity of light, as determined from measurement using the radar method. The true path of light is plotted at 45°. The coordinate singularity, or event horizon in the Schwarzschild geometry, is actually at the position of the particle itself, while the singularity at ρ = 0 is an illusory point, which does not correspond to a position in space. |
The Physical Origin of Curvature ↑ Einstein’s Field Equation →
Edited on 2008-08-13 04:03:28 by CharlesFrancis
Additions:
← The Physical Origin of Curvature ↑ →
Deletions:
← The Physical Origin of Curvature →
Edited on 2008-08-08 00:46:26 by CharlesFrancis
Additions:
| The use of the radar method should not be taken to imply that radar is fundamental definition of distance. In relational quantum gravity, the fundamental structures of matter is a plenum consisting of arrangements of particles described by Feynman diagrams. The background in a Feynman diagram has no mathematical meaning, so that spacetime background has no physical meaning. Distances are emergent quantities arising from the structure of the plenum. Radar provides a direct measurement of distance because it uses the same physical process, two way photon exchange, as is seen in diagrams for stable configurations of matter. |
Superficially, it appears that the speed of light is less than unity in these coordinates. Light should then be drawn at a greater than <span class=math>45°</span> to the horizontal. The inconsistency is resolved because this refers to the mean velocity of light, as determined from measurement using the radar method. The true path of light is plotted at <span class=math>45°</span>. The coordinate singularity, or event horizon in the Schwarzschild geometry, is actually at the position of the particle itself, while the singularity at <span class=math>ρ = 0</span> is an illusory point, which does not correspond to a position in space. </td></table>""
Deletions:
| The use of the radar method should not be taken to imply that radar is fundamental definition of distance. In relational quantum gravity, the fundamental structures of matter is a plenum consisting of arrangements of particles described by Feynman diagrams. The background in a Feynman diagram has no mathematical meaning, so that spacetime background has no physical meaning. Distances are emergent quantities arising from the structure of the plenum. Radar provides a direct measurement of distance because it uses the same physical process, two way photon exchange, as is seen in diagrams for stable configurations of matter. |
Superficially, it appears that the speed of light is less than unity in these coordinates. Light should then be drawn at a greater than <span class=math>45°</span> to the horizontal. The inconsistency is resolved because this refers to the mean velocity of light, as determined from measurement using the radar method. The true path of light is plotted at <span class=math>45°</span>. The coordinate singularity, or event horizon in the Schwarzschild geometry, is actually at the position of the particle itself, while the singularity at <span class=math>ρ = 0</span> is an illusory point, which does not correspond to a position in space. </td></table>""
Edited on 2008-08-07 23:57:27 by CharlesFrancis
Additions:
Superficially, it appears that the speed of light is less than unity in these coordinates. Light should then be drawn at a greater than <span class=math>45°</span> to the horizontal. The inconsistency is resolved because this refers to the mean velocity of light, as determined from measurement using the radar method. The true path of light is plotted at <span class=math>45°</span>. The coordinate singularity, or event horizon in the Schwarzschild geometry, is actually at the position of the particle itself, while the singularity at <span class=math>ρ = 0</span> is an illusory point, which does not correspond to a position in space. </td></table>""
Deletions:
Superficially, it appears that the speed of light is less than unity in these coordinates. Light should then be drawn at a greater than <span class=math>45°</span> to the horizontal. The inconsistency is resolved because this refers to the mean velocity of light, as determined from measurement using the radar method. The true path of light is plotted at <span class=math>45°</span>. The coordinate singularity, or event horizon in the Schwarzschild geometry, is actually at the position of the particle itself, while the singularity at <span class=math>&rho = 0</span> is an illusory point, which does not correspond to a position in space. </td></table>""
Edited on 2008-06-30 09:38:32 by CharlesFrancis
Additions:
These notional diagrams restore time ordering between the interactions, showing photon exchange as a two way process. Such a diagram is at best a rough approximation to the truth. There is no fixed time interval between absorption and emission, and we can only describe the radius of the orbit of an electron with a probability amplitude, not a numerical value.
Systems containing larger numbers of particles also contain greater structure. For large structures the uncertainties in position are small compared to the scale of the structure. Spacetime structure emerges as a macroscopic property of systems containing many particles. |
| An isolated elementary particle in an eigenstate of position has spherical symmetry and spacetime diagrams may be used to show a radial coordinate in n dimensions without loss of generality. A spacetime diagram is drawn, showing a tangent space with an origin at O, so that light is shown at 45°, lines of equal time are horizontal and time is proper time for the gravitating particle. In tangent space at O, the coordinate distance between Beth and the particle is r. |
| As shown in tangent space at O, Beth uses the radar method to determine a distance coordinate for the particle. Beth cannot resolve the points A and E where a photon is absorbed and emitted and places the particle at apparent position P. If the effective delay in the reflection is 4GM, then the coordinate coordinate distance of P from Beth is ρ = r + 2GM. |
As calculated for stationary observers, the physical metric (with angular directions suppressed) is
So, Beth’s clock runs fast by a factor k compared to proper time, t, for the particle at O. Minkowski coordinates (t, r) for tangent space at O are stretched by a factor k−1 in the time direction and by k in the radial direction compared to Minkowski coordinates, (T, R) for tangent space at B, as determined by Beth using the radar method. Since the radar method determines that R = T, we have
|
<table width=100% cellspacing=0 cellpadding=0><td><img class=right title="The true and apparent paths of light" alt="OriginOfCurvature-22" src="images/originofcurvature/OriginOfCurvature-22N.gif">With <span class=math><i>r</span> as the radial coordinate, the metric is<br>
Deletions:
 These notional diagrams restore time ordering between the interactions, showing photon exchange as a two way process. Such a diagram is at best a rough approximation to the truth. There is no fixed time interval between absorption and emission, and we can only describe the radius of the orbit of an electron with a probability amplitude, not a numerical value.
Systems containing larger numbers of particles also contain greater structure. For large structures the uncertainties in position are small compared to the scale of the structure. Spacetime structure emerges as a macroscopic property of systems containing many particles. |
 An isolated elementary particle in an eigenstate of position has spherical symmetry and spacetime diagrams may be used to show a radial coordinate in n dimensions without loss of generality. A spacetime diagram is drawn, showing a tangent space with an origin at O, so that light is shown at 45°, lines of equal time are horizontal and time is proper time for the gravitating particle. In tangent space at O, the coordinate distance between Beth and the particle is r. |
 As shown in tangent space at O, Beth uses the radar method to determine a distance coordinate for the particle. Beth cannot resolve the points A and E where a photon is absorbed and emitted and places the particle at apparent position P. If the effective delay in the reflection is 4GM, then the coordinate coordinate distance of P from Beth is ρ = r + 2GM. |
 As calculated for stationary observers, the physical metric (with angular directions suppressed) is
So, Beth’s clock runs fast by a factor k compared to proper time, t, for the particle at O. Minkowski coordinates (t, r') for tangent space at O are stretched by a factor k−1 in the time direction and by k in the radial direction compared to Minkowski coordinates, (T, R) for tangent space at B, as determined by Beth using the radar method. Since the radar method determines that R = T, we have
|
<table width=100% cellspacing=0 cellpadding=0><td><img class=right title="The true and apparent paths of light" alt="OriginOfCurvature-22" src="images/originofcurvature/OriginOfCurvature-22.gif">With <span class=math><i>r</span> as the radial coordinate, the metric is<br>
Edited on 2008-06-30 04:37:27 by CharlesFrancis
Additions:
<img alt="tabspace" src="images/tabspace.gif" vspace=3 ><img title="Substitute r = ρ +2GM" alt="OriginOfCurvature-21" src="images/originofcurvature/OriginOfCurvature-21.gif" vspace=3><br>
Deletions:
<img alt="tabspace" src="images/tabspace.gif" vspace=3 ><img title="Substitute <i>r</i> = ρ +2<i>GM</i>" alt="OriginOfCurvature-21" src="images/originofcurvature/OriginOfCurvature-21.gif" vspace=3><br>
Edited on 2008-06-30 04:36:31 by CharlesFrancis
Additions:
<img alt="tabspace" src="images/tabspace.gif" vspace=3 ><img title="Substitute <i>r</i> = ρ +2<i>GM</i>" alt="OriginOfCurvature-21" src="images/originofcurvature/OriginOfCurvature-21.gif" vspace=3><br>
Deletions:
<img alt="tabspace" src="images/tabspace.gif" vspace=3 ><img title="Substitute r" alt="OriginOfCurvature-21" src="images/originofcurvature/OriginOfCurvature-21.gif" vspace=3><br>
Edited on 2008-06-30 04:33:15 by CharlesFrancis
Additions:
Superficially, it appears that the speed of light is less than unity in these coordinates. Light should then be drawn at a greater than <span class=math>45°</span> to the horizontal. The inconsistency is resolved because this refers to the mean velocity of light, as determined from measurement using the radar method. The true path of light is plotted at <span class=math>45°</span>. The coordinate singularity, or event horizon in the Schwarzschild geometry, is actually at the position of the particle itself, while the singularity at <span class=math>&rho = 0</span> is an illusory point, which does not correspond to a position in space. </td></table>""
Deletions:
Superficially, it appears that the speed of light is less than unity in these coordinates. Light should then be drawn at a greater than <span class=math>45°</span> to the horizontal. The inconsistency is resolved because this refers to the mean velocity of light, as determined from measurement using the radar method. The true path of light is plotted at <span class=math>45°</span>. </td></table>"" The coordinate singularity, or event horizon in the Schwarzschild geometry, is actually at the position of the particle itself, while the singularity at <span class=math>&rho = 0</span> is an illusory point, which does not correspond to a position in space.
Edited on 2008-06-30 04:32:09 by CharlesFrancis
Additions:
Superficially, it appears that the speed of light is less than unity in these coordinates. Light should then be drawn at a greater than <span class=math>45°</span> to the horizontal. The inconsistency is resolved because this refers to the mean velocity of light, as determined from measurement using the radar method. The true path of light is plotted at <span class=math>45°</span>. </td></table>"" The coordinate singularity, or event horizon in the Schwarzschild geometry, is actually at the position of the particle itself, while the singularity at <span class=math>&rho = 0</span> is an illusory point, which does not correspond to a position in space.
Deletions:
Superficially, it appears that the speed of light is less than unity in these coordinates. Light should then be drawn at a greater than <span class=math>45°</span> to the horizontal. The inconsistency is resolved because this refers to the mean velocity of light, as determined from measurement using the radar method. The true path of light is plotted at <span class=math>45°</span>. </td></table>""
Edited on 2008-06-29 23:48:13 by CharlesFrancis
Additions:
Space has no meaning in Feynman diagrams and emerges in the macroscopic properties of structures described mathematically as graphs». Conservation of energy and momentum was derived from the integral formulae for probability amplitudes associated with these graphs, and follows from the principle that the interactions of elementary particles are always and everywhere the same. The view that space is an emergent property of the interactions of particles casts a deeper light on Einstein’s field equation. Instead of the duality between matter and space described by Wheeler, at a fundamental level we have only matter. Conservation of energy-momentum constrains the structure of spacetime, not the other way around.
Special relativity was described by Einstein as a principle theory, as distinct from a fundamental theory, because it derives its results from principles of measurement, and does not explain fundamental physical mechanisms. It enables us to derive the mathematical properties of Minkowski spacetime, but does not explain the underlying structure, or give the reason why light behaves as it does. In Foundations Of Special Relativity, the structure of Minkowski spacetime was found from the radar method, using two way transmission of light. As used in the radar method, light is the sensor for the properties of spacetime, not the cause of them. The radar method assumes a greater importance in relational quantum gravity. In qed, two-way transmission of photons is responsible for the electromagnetic force, and hence for the all the structures of matter we observe in our immediate environment. The radar method can use light as a sensor for the properties of spacetime because it uses the same physical process, two-way photon exchange, which is also the cause of the structure of spacetime.
| The use of the radar method should not be taken to imply that radar is fundamental definition of distance. In relational quantum gravity, the fundamental structures of matter is a plenum consisting of arrangements of particles described by Feynman diagrams. The background in a Feynman diagram has no mathematical meaning, so that spacetime background has no physical meaning. Distances are emergent quantities arising from the structure of the plenum. Radar provides a direct measurement of distance because it uses the same physical process, two way photon exchange, as is seen in diagrams for stable configurations of matter. |
<img class=left title="Rough diagram of a hydrogen molecule" alt="OriginOfCurvature-7" src="images/originofcurvature/OriginOfCurvature-7.gif"><br><br><br><br><br><br><br><br><br>Systems containing larger numbers of particles also contain greater structure. For large structures the uncertainties in position are small compared to the scale of the structure. Spacetime structure emerges as a macroscopic property of systems containing many particles.</td></table>
There are several reasons for introducing such a delay. Firstly, as shown here, the delay perturbs the metric, resulting in a curved spacetime obeying <a href=http://www.teleconnection.info/rqg/Gravitation#Einstein’sLawOfGravitation>Einstein’s» field equation</a>. If the reflection of a photon were instantaneous, the physical metric would be Minkowski. The intrinsic time delay in reflection, <span class=math>4<i>GM, causes a small amount of curvature, in accordance with Einstein's field equation. Secondly, it is well known that some small scale modification is needed to qed in order to remove the <a href=http://www.teleconnection.info/rqg/Regularisation#TheUltravioletDivergence>ultraviolet» divergence</a> and resolve the <a href=http://www.teleconnection.info/rqg/Regularisation#TheLandauPole>Landau» pole</a>. The delay introduced here is an effective cut-off and allows the construction of a consistent qed. Finally, a minimum time between interactions proportional to mass may be related to the concept of inertia; if the interactions of muons and electrons with photons are discrete and identical, then it is natural that the acceleration due to the electromagnetic field will be proportional to the frequency of interaction. An intrinsic delay between interactions proportional to mass will result in accelerations inversely proportional to mass.
It will be observed that the event horizon, <span class=math>ρ = 2<i>GM</i>, maps to <span class=math><i>r</i> = 0. Only space outside the event horizon is mapped in these coordinates. The space inside the event horizon, <span class=math>ρ < 2<i>GM</i>, has no physical meaning. A particle which is point-like in a tangent space at the position of the particle, is mapped to the event horizon in a tangent space at the position of a distant observer. The effect of the discrete interval of proper time between interactions is a singularity at <span class=math><i>ρ</i> = 0"" which “magnifies” a point-like particle to the size of the event horizon.
Deletions:
Special relativity was described by Einstein as a principle theory, as distinct from a fundamental theory, because it derives its results from principles of measurement, and does not explain fundamental physical mechanisms. It enables us to derive the mathematical properties of Minkowski spacetime, but does not explain the underlying structure, or give the reason why light behaves as it does. In special relativity, the structure of Minkowski spacetime was found from the radar method, using two way transmission of light. As used in the radar method, light is the sensor for the properties of spacetime, not the cause of them. The radar method assumes a greater importance in relational quantum gravity. In qed, two-way transmission of photons is responsible for the electromagnetic force, and hence for the all the structures of matter we observe in our immediate environment. The radar method can use light as a sensor for the properties of spacetime because it uses the same physical process, two-way photon exchange, which is also the cause of the structure of spacetime.
Space has no meaning in Feynman diagrams and emerges in the macroscopic properties of structures described mathematically as graphs». Conservation of energy and momentum was derived from the integral formulae for probability amplitudes associated with these graphs, and follows from the principle that the interactions of elementary particles are always and everywhere the same. The view that space is an emergent property of the interactions of particles casts a deeper light on Einstein’s field equation. Instead of the duality between matter and space described by Wheeler, at a fundamental level we have only matter. The emergent structure of spacetime is constrained by conservation of energy-momentum, not the other way around.
Relational quantum gravity applies the same approach to extend classical general relativity with the introduction of the teleconnection. The teleconnection enables the formulation of a relativistic quantum theory in which the initial and final states are described in reference frames separated by astronomical distances. Later on this site, I show that the teleconnection has cosmological implications and describe empirical evidence which supports a model in which the transmission of light (photons) over astronomical distances is described by the teleconnection.
| The use of the radar method should not be taken to imply that radar is fundamental definition of distance. In relational quantum gravity, the fundamental structures of matter is a plenum consisting of arrangements of particles described by Feynman diagrams. The background in a Feynman diagram has no mathematical meaning, so that spacetime background has no physical meaning. Distances are emergent quantities arising from the structure of the plenum. Radar provides a direct measurement of distance because it uses the same physical process, namely two way photon exchange, as is seen in diagrams for stable configurations of matter. |
<img class=left title="Rough diagram of a hydrogen molecule" alt="OriginOfCurvature-7" src="images/originofcurvature/OriginOfCurvature-7.gif"><br><br><br><br><br><br><br><br><br>Systems containing larger numbers of particles also contain greater structure. Spacetime structure emerges as a macroscopic property of systems containing many particles.</td></table>
There are several reasons for introducing such a delay. Firstly, as shown here, the delay perturbs the metric, resulting in a curved spacetime obeying <a href=http://www.teleconnection.info/rqg/Gravitation#Einstein’sLawOfGravitation>Einstein’s» field equation</a>. As Bondi’s <span class=math><i>k</i>-calculus shows, if the reflection of a photon were instantaneous, the physical metric would be Minkowski. The intrinsic time delay in reflection, <span class=math>4<i>GM, causes a small amount of curvature, in accordance with Einstein's field equation. Secondly, it is well known that some small scale modification is needed to qed in order to remove the <a href=http://www.teleconnection.info/rqg/Regularisation#TheUltravioletDivergence>ultraviolet» divergence</a> and resolve the <a href=http://www.teleconnection.info/rqg/Regularisation#TheLandauPole>Landau» pole</a>. The delay introduced here is an effective cut-off and allows the construction of a consistent qed. Finally, a minimum time between interactions proportional to mass may be related to the concept of inertia; if the interactions of muons and electrons with photons are discrete and identical, then it is natural that the acceleration due to the electromagnetic field will be proportional to the frequency of interaction.
It will be observed that the event horizon, <span class=math>ρ = 2<i>GM</i>, maps to <span class=math><i>r</i> = 0. Thus, a particle which is point-like in a tangent space at the position of the particle, is mapped to the event horizon in a tangent space at the position of a distant observer. The effect of the discrete interval of proper time between interactions is a singularity at <span class=math><i>ρ</i> = 0 which “magnifies” a point-like particle to the size of the event horizon. Only space outside the event horizon is mapped in these coordinates. The space inside the event horizon, <span class=math>ρ < 2<i>GM</i>"", has no physical meaning.
Edited on 2008-06-29 22:58:48 by CharlesFrancis
Additions:
In relational quantum gravity, the most fundamental description of matter is illustrated by Feynman diagrams. In Feynman diagrams, space, and hence curvature, have no meaning, and emerge only in the classical correspondence from the mean behaviour of large populations of particles. Relational quantum gravity modifies the standard, continuum, form of qed and introduces an effective minimal interval of proper time between interactions of an elementary particle. The result of this modification is that instead of the emergence of Minkowski spacetime, we will find a curved spacetime obeying Einstein’s field equation.
Special relativity was described by Einstein as a principle theory, as distinct from a fundamental theory, because it derives its results from principles of measurement, and does not explain fundamental physical mechanisms. It enables us to derive the mathematical properties of Minkowski spacetime, but does not explain the underlying structure, or give the reason why light behaves as it does. In special relativity, the structure of Minkowski spacetime was found from the radar method, using two way transmission of light. As used in the radar method, light is the sensor for the properties of spacetime, not the cause of them. The radar method assumes a greater importance in relational quantum gravity. In qed, two-way transmission of photons is responsible for the electromagnetic force, and hence for the all the structures of matter we observe in our immediate environment. The radar method can use light as a sensor for the properties of spacetime because it uses the same physical process, two-way photon exchange, which is also the cause of the structure of spacetime.
Deletions:
In relational quantum gravity, the most fundamental description of matter is illustrated by Feynman diagrams. In Feynman diagrams, space, and hence curvature, have no meaning, and emerge only in the classical correspondence from the mean behaviour of large populations of particles. The aim of this section is to illustrate that the curvature of space is a consequence of a requirement that there is an effective minimal interval of proper time between interactions of an elementary particle.
In special relativity, the structure of Minkowski spacetime was found from the radar method, using two way transmission of light. Relational quantum gravity modifies the standard, continuum, form of qed and introduces an effective minimal interval of proper time between interactions of an elementary particle. The result of this modification is that instead of the emergence of Minkowski spacetime, we will find from the radar method a curved spacetime obeying Einstein’s field equation.
Special relativity was described by Einstein as a principle theory, as distinct from a fundamental theory, because it derives its results from principles of measurement, and does not explain fundamental physical mechanisms. It enables us to derive the mathematical properties of Minkowski spacetime, but does not explain the underlying structure, or give the reason why light behaves as it does. As used in the radar method, light is the sensor for the properties of spacetime, not the cause of them. The radar method assumes a greater importance in relational quantum gravity. In qed, two-way transmission of photons is responsible for the electromagnetic force, and hence for the all the structures of matter we observe in our immediate environment. The radar method can use light as a sensor for the properties of spacetime because it uses the same physical process, two-way photon exchange, which is also the cause of the structure of spacetime.
Edited on 2008-06-29 22:54:56 by CharlesFrancis
Additions:
In special relativity, the structure of Minkowski spacetime was found from the radar method, using two way transmission of light. Relational quantum gravity modifies the standard, continuum, form of qed and introduces an effective minimal interval of proper time between interactions of an elementary particle. The result of this modification is that instead of the emergence of Minkowski spacetime, we will find from the radar method a curved spacetime obeying Einstein’s field equation.
Special relativity was described by Einstein as a principle theory, as distinct from a fundamental theory, because it derives its results from principles of measurement, and does not explain fundamental physical mechanisms. It enables us to derive the mathematical properties of Minkowski spacetime, but does not explain the underlying structure, or give the reason why light behaves as it does. As used in the radar method, light is the sensor for the properties of spacetime, not the cause of them. The radar method assumes a greater importance in relational quantum gravity. In qed, two-way transmission of photons is responsible for the electromagnetic force, and hence for the all the structures of matter we observe in our immediate environment. The radar method can use light as a sensor for the properties of spacetime because it uses the same physical process, two-way photon exchange, which is also the cause of the structure of spacetime.
Deletions:
special relativity was described by Einstein as a principle theory, as distinct from a fundamental theory, because it derives its results from principles of measurement, and does not explain fundamental physical mechanisms. It enables us to derive the mathematical properties of Minkowski spacetime, but does not explain the underlying structure, or give the reason why light behaves as it does. As used in the radar method, light is the sensor for the properties of spacetime, not the cause of them. The radar method assumes a greater importance in relational quantum gravity. In qed, two-way transmission of photons is responsible for the electromagnetic force, and hence for the all the structures of matter we observe in our immediate environment.
In special relativity, the structure of Minkowski spacetime was found from the radar method, using two way transmission of light. The radar method can use light as a sensor for the properties of spacetime because it uses the same physical process, two-way photon exchange, which is also the cause of the structure of spacetime. Relational quantum gravity modifies the standard, continuum, form of qed and introduces an effective minimal interval of proper time between interactions of an elementary particle. The result of this modification is that instead of the emergence of Minkowski spacetime, we will find from the radar method a curved spacetime obeying Einstein’s field equation.
Edited on 2008-06-29 22:50:44 by CharlesFrancis
Additions:
special relativity was described by Einstein as a principle theory, as distinct from a fundamental theory, because it derives its results from principles of measurement, and does not explain fundamental physical mechanisms. It enables us to derive the mathematical properties of Minkowski spacetime, but does not explain the underlying structure, or give the reason why light behaves as it does. As used in the radar method, light is the sensor for the properties of spacetime, not the cause of them. The radar method assumes a greater importance in relational quantum gravity. In qed, two-way transmission of photons is responsible for the electromagnetic force, and hence for the all the structures of matter we observe in our immediate environment.
Space has no meaning in Feynman diagrams and emerges in the macroscopic properties of structures described mathematically as graphs». Conservation of energy and momentum was derived from the integral formulae for probability amplitudes associated with these graphs, and follows from the principle that the interactions of elementary particles are always and everywhere the same. The view that space is an emergent property of the interactions of particles casts a deeper light on Einstein’s field equation. Instead of the duality between matter and space described by Wheeler, at a fundamental level we have only matter. The emergent structure of spacetime is constrained by conservation of energy-momentum, not the other way around.
In special relativity, the structure of Minkowski spacetime was found from the radar method, using two way transmission of light. The radar method can use light as a sensor for the properties of spacetime because it uses the same physical process, two-way photon exchange, which is also the cause of the structure of spacetime. Relational quantum gravity modifies the standard, continuum, form of qed and introduces an effective minimal interval of proper time between interactions of an elementary particle. The result of this modification is that instead of the emergence of Minkowski spacetime, we will find from the radar method a curved spacetime obeying Einstein’s field equation.
Deletions:
Conservation of energy and momentum have been shown as a requirement of a relativistic theory of particle interactions. In special relativity, the structure of Minkowski spacetime was found from the radar method, using two way transmission of light.
Special relativity was described by Einstein as a principle theory, as distinct from a fundamental theory, because it derives its results from principles of measurement, and does not explain fundamental physical mechanisms. It enables us to derive the mathematical properties of Minkowski spacetime, but does not explain the underlying structure, or give the reason why light behaves as it does. As used in the radar method, light is the sensor for the properties of spacetime, not the cause of them. The radar method assumes a greater importance in relational quantum gravity. In qed, two-way transmission of photons is responsible for the electromagnetic force, and hence for the all the structures of matter we observe in our immediate environment. Space has no meaning in Feynman diagrams and emerges in the macroscopic properties of structures described mathematically as graphs». The radar method can use light as a sensor for the properties of spacetime because it uses the same physical process, two-way 
