Additions:
Additions:
An unjustified term in an equation should be considered as a “fiddle factor”, and best regarded as meaning that the theory behind the equation contains an unknown fault, rather than that the term necessarily models reality. While a non-zero cosmological constant is indicated in the analysis of observation using standard general relativity, we will see from [[Supernova Cosmological Supernovae Redshifts]] that the teleconnection model gives at least as good a match with data while setting ""Λ = 0"".
Deletions:
An unjustified term in an equation should be considered as a “fiddle factor”, and best regarded as meaning that the theory behind the equation contains an unknown fault, rather than that the term necessarily models reality. While a non-zero cosmological constant is indicated in the analysis of observation using standard general relativity, we will see in [[CosmologicalImplications Cosmological Implications]] that the teleconnection model gives at least as good a match with data while setting ""Λ = 0"".
Additions:
[[SpacetimeStructure The Emergence of Spacetime Structure ↑]] [[SingularitiesRevisited Singularities Revisited →]]
Deletions:
[[SpacetimeStructure Einstein’s Field Equation ↑]] [[SingularitiesRevisited Singularities Revisited →]]
Additions:
======[[OriginOfCurvature ←]] The Emergence of Spacetime Structure [[RelationalQuantumGravity ↑]] [[SingularitiesRevisited →]]======
Deletions:
======[[OriginOfCurvature ←]] The Emergence of Spacetime Structure [[RelationalQuantumGravity ↑]] [[CosmologicalImplications →]]======
Additions:
[[SpacetimeStructure Einstein’s Field Equation ↑]] [[SingularitiesRevisited Singularities Revisited →]]
Deletions:
====""""Pre-expansion as an Ametric Phase====
The description of matter using states in Hilbert space requires at least that position can be measured in principle. But in the initial phase after the big bang, measurement of position is impossible in principle; it is not possible to abstract Hilbert space from properties of measurement. Since Hilbert space no longer applies, some other mathematical structure is required to describe evolution from the big bang. Research will be required to identify the precise properties of such a structure, which would describe particle interactions without using the concept of spacetime in any form. Spin networks appear to have some of the requisite properties. Here I merely a few general remarks regarding behaviour near the big bang.
In a discrete manifold it is not possible to divide the early universe into indefinitely small regions which did not communicate. At an initial singularity all particles are at the same place, and relative position has no meaning. Rather than rapid inflation from a small size, there was an initial phase during which we cannot talk of spatial dimension or size and when horizons did not exist. There is a minimum interaction time and several interactions are required to establish a distance between elementary particles. It might have taken thousands, or many thousands, of discrete intervals of proper time to establish the properties of a Riemannian manifold. Prior to that the image is one of perfect chaos, in which any photon may interact with any charged particle, so that the entire is causally connected. Because positions cannot be distinguished during the ametric phase, this phase can only lead to an isotropic initial condition for normal expansion.
It does not appear necessary to postulate that all the matter initially contained in the universe participates in the creation of spacetime. Indeed, if some matter remains disconnected from the observable universe it could account for the observed matter/antimatter imbalance without the need to postulate an exotic and unobserved process in particle physics, viz. the decay of the proton.
There must be a first time at which sufficient interactions had taken place that relative position between particles became possible. A lower bound for the duration of the initial period can be estimated by applying a Doppler shift to one interval of discrete time as appropriate to the high energies of particles near the big bang. Typical quoted energies for particles near the big bang are in the order of a factor ""1030"" greater than rest mass. In this case the discrete interval of proper time ""10−65 ""s for an electron is redshifted to ""10−35 ""s, within range of the time scales normally postulated for the end of inflation and the beginning of normal expansion.
====""""Black holes====
General relativity is known to be valid on large scales and describes matter fields, not pointlike particles. However, on small scales we observe that matter consists of pointlike particles (up to quantum effects). The treatment of [[OriginOfCurvature A Gravitating Particle]] placed an elementary particle in a position eigenstate at ""r = 0"", in a continuous manifold and found that the event horizon of the Schwarzschild geometry was also at the point, ""r = 0"". Although ""r"" is related to the Schwarzschild radial coordinate ""ρ"" by ""ρ = r + 2Gm"", the region ""ρ < 2Gm"" does not map to these coordinates (the manifold with ""r"" as radial coordinate is not a ""chart"" on the maximally extended Schwarzschild geometry, because ""r = 0"" is a single point in a continuous chart).
The argument describes a pointlike particle at ""r = 0"", surrounded by the exterior region of a Schwarzschild geometry. For a pointlike particle, it makes no physical sense to extend the coordinate system interior to the particle. The extension exists mathematically, but has no physical meaning. By considering a classical body, such at the earth, as a composition of these pointlike particles, and by replacing the pointlike structure with a density, we restore the field equations, as an excellent large scale approximation.
We can model a black hole, neglecting the effect of the exclusion principle, by placing large numbers of elementary pointlike particles at ""r = 0"". We will then have a large mass, ""M"", at ""r = 0"" surrounded by the exterior region of a Schwarzschild geometry. There is again no physical meaning to the interior region. A curious feature is that ""r = 0"" cannot be enclosed in a surface of arbitrarily small surface area. However, this is not inconsistent and is no more counter intuitive than, for example, that in a closed homogeneous isotropic universe a circle of sufficiently large radius will have zero circumference. Since the surface and the point are disjoint, the properties of the one don't have an immediate bearing on the other. If the argument were valid, it would show a discontinuity of the metric at ""r = 0"", not that ""r = 0"" cannot be a point. However in relational quantum gravity this argument has no meaning, because the very notion of a surface breaks down on small distance scales. The manifold is not conceived as some kind of metaphysical entity generalising the properties of Newton’s absolute space, but rather as a collection of potential measurement results, arising from the operational definition of time and space coordinates.
More realistically, allowing that fundamental particles are fermions, we may consider large numbers of particles in a region surrounding ""r = 0"". This does not alter the qualitative features of the description. In either case, the black hole is not strictly a “hole”, but is described on a continuous chart containing ""r = 0"".
In practice black holes are believed to be formed from the collapse of neutron stars. Neutrons are Fermions. In their [[http://prola.aps.org/abstract/PR/v55/i4/p374_1 seminal paper]] of 1939, Oppenheimer and Volkoff say //“A discussion of the probable effect of deviations from the Fermi equation of state suggests that actual stellar matter after the exhaustion of thermonuclear sources of energy will, if massive enough, contract indefinitely, although more and more slowly, never reaching true equilibrium”//. The Pauli exclusion principle prohibits placing more than one fermion at ""r = 0"", so the probable structure of a real black hole is as described by Oppenheimer and Volkoff. Rotation will also slow down the collapse.
""
""Hawking radiation is not possible, since this depends on the classical structure of spacetime in the vicinity of the event horizon. Nonetheless a black hole can be expected to radiate. In Penrose coordinates, wave functions for particles are plane waves and can be emitted to infinity provided that there is sufficient energy in the initial state. There is always sufficient energy to emit zero mass particles, which can have arbitrarily low energies at infinity. Matter in the hole will have high energy from gravitational collapse, and in addition, as the hole becomes more compact and particles approach ""r = 0"", wave functions have components with ever increasing energies. For a hole of ""1,000,000"" solar masses, the energy required of an electron to escape to infinity is ""952"" kg, eleven orders of magnitude less than the maximal energy ""pmax = 4.08 × 1014"" kg corresponding to the lattice spacing. We may conclude that localisation of matter near ""r = 0"" creates energy states from which electrons (and other particles) are radiated with relativistic velocities. Since angular momentum of matter falling into the hole will generate a disc, the direction of radiation is in the axis of rotation, suggesting that this is the mechanism for relativistic jets. In a case where a disc is poorly defined, relativistic matter relativistic matter will be radiated from the hole in all directions, and will interact with surrounding matter in the host galaxy, creating a quasar. It is to be expected that the greater the mass of the black hole, the greater the gravitational force compactifying the hole, and hence the greater the amplitudes of states of sufficent energy to be radiated to infinity, and the greater the consequent radiation. One may think that gamma ray bursts result from a star falling into the hole with a resulting sudden increase in radiated energy.
[[SpacetimeStructure Einstein’s Field Equation ↑]] [[CosmologicalImplications Cosmological Implications →]]
Additions:
""""Hawking radiation is not possible, since this depends on the classical structure of spacetime in the vicinity of the event horizon. Nonetheless a black hole can be expected to radiate. In Penrose coordinates, wave functions for particles are plane waves and can be emitted to infinity provided that there is sufficient energy in the initial state. There is always sufficient energy to emit zero mass particles, which can have arbitrarily low energies at infinity. Matter in the hole will have high energy from gravitational collapse, and in addition, as the hole becomes more compact and particles approach ""r = 0"", wave functions have components with ever increasing energies. For a hole of ""1,000,000"" solar masses, the energy required of an electron to escape to infinity is ""952"" kg, eleven orders of magnitude less than the maximal energy ""pmax = 4.08 × 1014"" kg corresponding to the lattice spacing. We may conclude that localisation of matter near ""r = 0"" creates energy states from which electrons (and other particles) are radiated with relativistic velocities. Since angular momentum of matter falling into the hole will generate a disc, the direction of radiation is in the axis of rotation, suggesting that this is the mechanism for relativistic jets. In a case where a disc is poorly defined, relativistic matter relativistic matter will be radiated from the hole in all directions, and will interact with surrounding matter in the host galaxy, creating a quasar. It is to be expected that the greater the mass of the black hole, the greater the gravitational force compactifying the hole, and hence the greater the amplitudes of states of sufficent energy to be radiated to infinity, and the greater the consequent radiation. One may think that gamma ray bursts result from a star falling into the hole with a resulting sudden increase in radiated energy.
Deletions:
""""Hawking radiation is not possible, since this depends on the classical structure of spacetime in the vicinity of the event horizon. Nonetheless a black hole can be expected to radiate. Wave functions for particles are plane waves and can be emitted to infinity provided that there is sufficient energy in the initial state. There is always sufficient energy to emit zero mass particles, which can have arbitrarily low energies at infinity. Matter in the hole will have high energy from gravitational collapse, and in addition, as the hole becomes more compact and particles approach ""r = 0"", wave functions have components with ever increasing energies. For a hole of ""1,000,000"" solar masses, the energy required of an electron to escape to infinity is ""952"" kg, eleven orders of magnitude less than the maximal energy ""pmax = 4.08 × 1014"" kg corresponding to the lattice spacing. We may conclude that localisation of matter near ""r = 0"" creates energy states from which electrons (and other particles) are radiated with relativistic velocities. Since angular momentum of matter falling into the hole will generate a disc, the direction of radiation is in the axis of rotation, suggesting that this is the mechanism for relativistic jets. In a case where a disc is poorly defined, relativistic matter relativistic matter will be radiated from the hole in all directions, and will interact with surrounding matter in the host galaxy, creating a quasar. It is to be expected that the greater the mass of the black hole, the greater the gravitational force compactifying the hole, and hence the greater the amplitudes of states of sufficent energy to be radiated to infinity, and the greater the consequent radiation. One may think that gamma ray bursts result from a star falling into the hole with a resulting sudden increase in radiated energy.
Additions:
""""Hawking radiation is not possible, since this depends on the classical structure of spacetime in the vicinity of the event horizon. Nonetheless a black hole can be expected to radiate. Wave functions for particles are plane waves and can be emitted to infinity provided that there is sufficient energy in the initial state. There is always sufficient energy to emit zero mass particles, which can have arbitrarily low energies at infinity. Matter in the hole will have high energy from gravitational collapse, and in addition, as the hole becomes more compact and particles approach ""r = 0"", wave functions have components with ever increasing energies. For a hole of ""1,000,000"" solar masses, the energy required of an electron to escape to infinity is ""952"" kg, eleven orders of magnitude less than the maximal energy ""pmax = 4.08 × 1014"" kg corresponding to the lattice spacing. We may conclude that localisation of matter near ""r = 0"" creates energy states from which electrons (and other particles) are radiated with relativistic velocities. Since angular momentum of matter falling into the hole will generate a disc, the direction of radiation is in the axis of rotation, suggesting that this is the mechanism for relativistic jets. In a case where a disc is poorly defined, relativistic matter relativistic matter will be radiated from the hole in all directions, and will interact with surrounding matter in the host galaxy, creating a quasar. It is to be expected that the greater the mass of the black hole, the greater the gravitational force compactifying the hole, and hence the greater the amplitudes of states of sufficent energy to be radiated to infinity, and the greater the consequent radiation. One may think that gamma ray bursts result from a star falling into the hole with a resulting sudden increase in radiated energy.
Deletions:
""""Hawking radiation is not possible, since this depends on the classical structure of spacetime in the vicinity of the event horizon. Nonetheless a black hole can be expected to radiate. in which a photon is reflected from an elementary particle after a delay ""2GM"". Wave functions for particles are plane waves in this diagram and can be emitted to infinity provided that there is sufficient energy in the initial state. There is always sufficient energy to emit zero mass particles, which can have arbitrarily low energies at infinity. Matter in the hole will have high energy from gravitational collapse, and in addition, as the hole becomes more compact and particles approach ""r = 0"", wave functions have components with ever increasing energies. For a hole of ""1,000,000"" solar masses, the energy required of an electron to escape to infinity is ""952"" kg, eleven orders of magnitude less than the maximal energy ""pmax = 4.08 × 1014"" kg corresponding to the lattice spacing. We may conclude that localisation of matter near ""r = 0"" creates energy states from which electrons (and other particles) are radiated with relativistic velocities. Since angular momentum of matter falling into the hole will generate a disc, the direction of radiation is in the axis of rotation, suggesting that this is the mechanism for relativistic jets. In a case where a disc is poorly defined, relativistic matter relativistic matter will be radiated from the hole in all directions, and will interact with surrounding matter in the host galaxy, creating a quasar. It is to be expected that the greater the mass of the black hole, the greater the gravitational force compactifying the hole, and hence the greater the amplitudes of states of sufficent energy to be radiated to infinity, and the greater the consequent radiation. One may think that gamma ray bursts result from a star falling into the hole with a resulting sudden increase in radiated energy.
Additions:
""""Hawking radiation is not possible, since this depends on the classical structure of spacetime in the vicinity of the event horizon. Nonetheless a black hole can be expected to radiate. in which a photon is reflected from an elementary particle after a delay ""2GM"". Wave functions for particles are plane waves in this diagram and can be emitted to infinity provided that there is sufficient energy in the initial state. There is always sufficient energy to emit zero mass particles, which can have arbitrarily low energies at infinity. Matter in the hole will have high energy from gravitational collapse, and in addition, as the hole becomes more compact and particles approach ""r = 0"", wave functions have components with ever increasing energies. For a hole of ""1,000,000"" solar masses, the energy required of an electron to escape to infinity is ""952"" kg, eleven orders of magnitude less than the maximal energy ""pmax = 4.08 × 1014"" kg corresponding to the lattice spacing. We may conclude that localisation of matter near ""r = 0"" creates energy states from which electrons (and other particles) are radiated with relativistic velocities. Since angular momentum of matter falling into the hole will generate a disc, the direction of radiation is in the axis of rotation, suggesting that this is the mechanism for relativistic jets. In a case where a disc is poorly defined, relativistic matter relativistic matter will be radiated from the hole in all directions, and will interact with surrounding matter in the host galaxy, creating a quasar. It is to be expected that the greater the mass of the black hole, the greater the gravitational force compactifying the hole, and hence the greater the amplitudes of states of sufficent energy to be radiated to infinity, and the greater the consequent radiation. One may think that gamma ray bursts result from a star falling into the hole with a resulting sudden increase in radiated energy.
Deletions:
""Hawking radiation is not possible, since this depends on the classical structure of spacetime in the vicinity of the event horizon. Nonetheless a black hole can be expected to radiate. in which a photon is reflected from an elementary particle after a delay ""2GM"". Wave functions for particles are plane waves in this diagram and can be emitted to infinity provided that there is sufficient energy in the initial state. There is always sufficient energy to emit zero mass particles, which can have arbitrarily low energies at infinity. Matter in the hole will have high energy from gravitational collapse, and in addition, as the hole becomes more compact and particles approach ""r = 0"", wave functions have components with ever increasing energies. For a hole of ""1,000,000"" solar masses, the energy required of an electron to escape to infinity is ""952"" kg, eleven orders of magnitude less than the maximal energy ""pmax = 4.08 × 1014"" kg corresponding to the lattice spacing. We may conclude that localisation of matter near ""r = 0"" creates energy states from which electrons (and other particles) are radiated with relativistic velocities. Since angular momentum of matter falling into the hole will generate a disc, the direction of radiation is in the axis of rotation, suggesting that this is the mechanism for relativistic jets. In a case where a disc is poorly defined, relativistic matter relativistic matter will be radiated from the hole in all directions, and will interact with surrounding matter in the host galaxy, creating a quasar. It is to be expected that the greater the mass of the black hole, the greater the gravitational force compactifying the hole, and hence the greater the amplitudes of states of sufficent energy to be radiated to infinity, and the greater the consequent radiation. One may think that gamma ray bursts result from a star falling into the hole with a resulting sudden increase in radiated energy.
Additions:
""Hawking radiation is not possible, since this depends on the classical structure of spacetime in the vicinity of the event horizon. Nonetheless a black hole can be expected to radiate. in which a photon is reflected from an elementary particle after a delay ""2GM"". Wave functions for particles are plane waves in this diagram and can be emitted to infinity provided that there is sufficient energy in the initial state. There is always sufficient energy to emit zero mass particles, which can have arbitrarily low energies at infinity. Matter in the hole will have high energy from gravitational collapse, and in addition, as the hole becomes more compact and particles approach ""r = 0"", wave functions have components with ever increasing energies. For a hole of ""1,000,000"" solar masses, the energy required of an electron to escape to infinity is ""952"" kg, eleven orders of magnitude less than the maximal energy ""pmax = 4.08 × 1014"" kg corresponding to the lattice spacing. We may conclude that localisation of matter near ""r = 0"" creates energy states from which electrons (and other particles) are radiated with relativistic velocities. Since angular momentum of matter falling into the hole will generate a disc, the direction of radiation is in the axis of rotation, suggesting that this is the mechanism for relativistic jets. In a case where a disc is poorly defined, relativistic matter relativistic matter will be radiated from the hole in all directions, and will interact with surrounding matter in the host galaxy, creating a quasar. It is to be expected that the greater the mass of the black hole, the greater the gravitational force compactifying the hole, and hence the greater the amplitudes of states of sufficent energy to be radiated to infinity, and the greater the consequent radiation. One may think that gamma ray bursts result from a star falling into the hole with a resulting sudden increase in radiated energy.
Deletions:
Hawking radiation is not possible, since this depends on the classical structure of spacetime in the vicinity of the event horizon. Nonetheless a black hole can be expected to radiate. in which a photon is reflected from an elementary particle after a delay ""2GM"". Wave functions for particles are plane waves in this diagram and can be emitted to infinity provided that there is sufficient energy in the initial state. There is always sufficient energy to emit zero mass particles, which can have arbitrarily low energies at infinity. Matter in the hole will have high energy from gravitational collapse, and in addition, as the hole becomes more compact and particles approach ""r = 0"", wave functions have components with ever increasing energies. For a hole of ""1,000,000"" solar masses, the energy required of an electron to escape to infinity is ""952"" kg, eleven orders of magnitude less than the maximal energy ""pmax = 4.08 × 1014"" kg corresponding to the lattice spacing. We may conclude that localisation of matter near ""r = 0"" creates energy states from which electrons (and other particles) are radiated with relativistic velocities. Since angular momentum of matter falling into the hole will generate a disc, the direction of radiation is in the axis of rotation, suggesting that this is the mechanism for relativistic jets. In a case where a disc is poorly defined, relativistic matter relativistic matter will be radiated from the hole in all directions, and will interact with surrounding matter in the host galaxy, creating a quasar. It is to be expected that the greater the mass of the black hole, the greater the gravitational force compactifying the hole, and hence the greater the amplitudes of states of sufficent energy to be radiated to infinity, and the greater the consequent radiation. One may think that gamma ray bursts result from a star falling into the hole with a resulting sudden increase in radiated energy.
Additions:
Hawking radiation is not possible, since this depends on the classical structure of spacetime in the vicinity of the event horizon. Nonetheless a black hole can be expected to radiate. in which a photon is reflected from an elementary particle after a delay ""2GM"". Wave functions for particles are plane waves in this diagram and can be emitted to infinity provided that there is sufficient energy in the initial state. There is always sufficient energy to emit zero mass particles, which can have arbitrarily low energies at infinity. Matter in the hole will have high energy from gravitational collapse, and in addition, as the hole becomes more compact and particles approach ""r = 0"", wave functions have components with ever increasing energies. For a hole of ""1,000,000"" solar masses, the energy required of an electron to escape to infinity is ""952"" kg, eleven orders of magnitude less than the maximal energy ""pmax = 4.08 × 1014"" kg corresponding to the lattice spacing. We may conclude that localisation of matter near ""r = 0"" creates energy states from which electrons (and other particles) are radiated with relativistic velocities. Since angular momentum of matter falling into the hole will generate a disc, the direction of radiation is in the axis of rotation, suggesting that this is the mechanism for relativistic jets. In a case where a disc is poorly defined, relativistic matter relativistic matter will be radiated from the hole in all directions, and will interact with surrounding matter in the host galaxy, creating a quasar. It is to be expected that the greater the mass of the black hole, the greater the gravitational force compactifying the hole, and hence the greater the amplitudes of states of sufficent energy to be radiated to infinity, and the greater the consequent radiation. One may think that gamma ray bursts result from a star falling into the hole with a resulting sudden increase in radiated energy.
Deletions:
Hawking radiation is not possible, since this depends on the classical structure of spacetime in the vicinity of the event horizon. Nonetheless a black hole can be expected to radiate. in which a photon is reflected from an elementary particle after a delay ""2GM"", as described in RQG III. Wave functions for particles are plane waves in this diagram and can be emitted to infinity provided that there is sufficient energy in the initial state. There is always sufficient energy to emit zero mass particles, which can have arbitrarily low energies at infinity. Matter in the hole will have high energy from gravitational collapse, and in addition, as the hole becomes more compact and particles approach ""r = 0"", wave functions have components with ever increasing energies. For a hole of 1,000,000 solar masses, the energy required of an electron to escape to infinity is ""952"" kg, eleven orders of magnitude less than the maximal energy ""pmax = 4.08 ¥ 1014"" kg corresponding to the lattice spacing. We may conclude that localisation of matter near ""r = 0"" creates energy states from which electrons (and other particles) are radiated with relativistic velocities. Since angular momentum of matter falling into the hole will generate a disc, the direction of radiation is in the axis of rotation, suggesting that this is the mechanism for relativistic jets. In a case where a disc is poorly defined, relativistic matter relativistic matter will be radiated from the hole in all directions, and will interact with surrounding matter in the host galaxy, creating a quasar. It is to be expected that the greater the mass of the black hole, the greater the gravitational force compactifying the hole, and hence the greater the amplitudes of states of sufficent energy to be radiated to infinity, and the greater the consequent radiation. One may think that gamma ray bursts result from a star falling into the hole with a resulting sudden increase in radiated energy.
Additions:
We can model a black hole, neglecting the effect of the exclusion principle, by placing large numbers of elementary pointlike particles at ""r = 0"". We will then have a large mass, ""M"", at ""r = 0"" surrounded by the exterior region of a Schwarzschild geometry. There is again no physical meaning to the interior region. A curious feature is that ""r = 0"" cannot be enclosed in a surface of arbitrarily small surface area. However, this is not inconsistent and is no more counter intuitive than, for example, that in a closed homogeneous isotropic universe a circle of sufficiently large radius will have zero circumference. Since the surface and the point are disjoint, the properties of the one don't have an immediate bearing on the other. If the argument were valid, it would show a discontinuity of the metric at ""r = 0"", not that ""r = 0"" cannot be a point. However in relational quantum gravity this argument has no meaning, because the very notion of a surface breaks down on small distance scales. The manifold is not conceived as some kind of metaphysical entity generalising the properties of Newton’s absolute space, but rather as a collection of potential measurement results, arising from the operational definition of time and space coordinates.
Deletions:
We can model a black hole, neglecting the effect of the exclusion principle, by placing large numbers of elementary pointlike particles at ""r = 0"". We will then have a large mass, "M"", at ""r = 0"" surrounded by the exterior region of a Schwarzschild geometry. There is again no physical meaning to the interior region. A curious feature is that ""r = 0"" cannot be enclosed in a surface of arbitrarily small surface area. However, this is not inconsistent and is no more counter intuitive than, for example, that in a closed homogeneous isotropic universe a circle of sufficiently large radius will have zero circumference. Since the surface and the point are disjoint, the properties of the one don't have an immediate bearing on the other. If the argument were valid, it would show a discontinuity of the metric at ""r = 0"", not that ""r = 0"" cannot be a point. However in relational quantum gravity this argument has no meaning, because the very notion of a surface breaks down on small distance scales. The manifold is not conceived as some kind of metaphysical entity generalising the properties of Newton’s absolute space, but rather as a collection of potential measurement results, arising from the operational definition of time and space coordinates.
Additions:
There must be a first time at which sufficient interactions had taken place that relative position between particles became possible. A lower bound for the duration of the initial period can be estimated by applying a Doppler shift to one interval of discrete time as appropriate to the high energies of particles near the big bang. Typical quoted energies for particles near the big bang are in the order of a factor ""1030"" greater than rest mass. In this case the discrete interval of proper time ""10−65 ""s for an electron is redshifted to ""10−35 ""s, within range of the time scales normally postulated for the end of inflation and the beginning of normal expansion.
General relativity is known to be valid on large scales and describes matter fields, not pointlike particles. However, on small scales we observe that matter consists of pointlike particles (up to quantum effects). The treatment of [[OriginOfCurvature A Gravitating Particle]] placed an elementary particle in a position eigenstate at ""r = 0"", in a continuous manifold and found that the event horizon of the Schwarzschild geometry was also at the point, ""r = 0"". Although ""r"" is related to the Schwarzschild radial coordinate ""ρ"" by ""ρ = r + 2Gm"", the region ""ρ < 2Gm"" does not map to these coordinates (the manifold with ""r"" as radial coordinate is not a ""chart"" on the maximally extended Schwarzschild geometry, because ""r = 0"" is a single point in a continuous chart).
Deletions:
The theory of inflation is used in standard cosmology to
There must be a first time at which sufficient interactions had taken place that relative position between particles became possible. A lower bound for the duration of the initial period can be estimated by applying a Doppler shift to one interval of discrete time as appropriate to the high energies of particles near the big bang. Typical quoted energies for particles near the big bang are in the order of a factor ""1030"" greater than rest mass. In this case the discrete interval of proper time ""10−65""s for an electron is redshifted to ""10−-35""s, within range of the time scales normally postulated for the end of inflation and the beginning of normal expansion.
General relativity is known to be valid on large scales and describes matter fields, not pointlike particles. However, on small scales we observe that matter consists of pointlike particles (up to quantum effects). The treatment of [[OriginOfCurvature A Gravitating Particle]] placed an elementary particle in a position eigenstate at ""r = 0"", in a continuous manifold and found that the event horizon of the Schwarzschild geometry was also at the point, ""r = 0"". Although ""r is related to the Schwarzschild radial coordinate ""ρ"" by ""ρ = r + 2Gm"", the region ""ρ < 2Gm"" does not map to these coordinates (the manifold with ""r"" as radial coordinate is not a ""chart"" on the maximally extended Schwarzschild geometry, because ""r = 0"" is a single point in a continuous chart).
Additions:
====""""Pre-expansion as an Ametric Phase====
The theory of inflation is used in standard cosmology to
The description of matter using states in Hilbert space requires at least that position can be measured in principle. But in the initial phase after the big bang, measurement of position is impossible in principle; it is not possible to abstract Hilbert space from properties of measurement. Since Hilbert space no longer applies, some other mathematical structure is required to describe evolution from the big bang. Research will be required to identify the precise properties of such a structure, which would describe particle interactions without using the concept of spacetime in any form. Spin networks appear to have some of the requisite properties. Here I merely a few general remarks regarding behaviour near the big bang.
In a discrete manifold it is not possible to divide the early universe into indefinitely small regions which did not communicate. At an initial singularity all particles are at the same place, and relative position has no meaning. Rather than rapid inflation from a small size, there was an initial phase during which we cannot talk of spatial dimension or size and when horizons did not exist. There is a minimum interaction time and several interactions are required to establish a distance between elementary particles. It might have taken thousands, or many thousands, of discrete intervals of proper time to establish the properties of a Riemannian manifold. Prior to that the image is one of perfect chaos, in which any photon may interact with any charged particle, so that the entire is causally connected. Because positions cannot be distinguished during the ametric phase, this phase can only lead to an isotropic initial condition for normal expansion.
It does not appear necessary to postulate that all the matter initially contained in the universe participates in the creation of spacetime. Indeed, if some matter remains disconnected from the observable universe it could account for the observed matter/antimatter imbalance without the need to postulate an exotic and unobserved process in particle physics, viz. the decay of the proton.
There must be a first time at which sufficient interactions had taken place that relative position between particles became possible. A lower bound for the duration of the initial period can be estimated by applying a Doppler shift to one interval of discrete time as appropriate to the high energies of particles near the big bang. Typical quoted energies for particles near the big bang are in the order of a factor ""1030"" greater than rest mass. In this case the discrete interval of proper time ""10−65""s for an electron is redshifted to ""10−-35""s, within range of the time scales normally postulated for the end of inflation and the beginning of normal expansion.
====""""Black holes====
General relativity is known to be valid on large scales and describes matter fields, not pointlike particles. However, on small scales we observe that matter consists of pointlike particles (up to quantum effects). The treatment of [[OriginOfCurvature A Gravitating Particle]] placed an elementary particle in a position eigenstate at ""r = 0"", in a continuous manifold and found that the event horizon of the Schwarzschild geometry was also at the point, ""r = 0"". Although ""r is related to the Schwarzschild radial coordinate ""ρ"" by ""ρ = r + 2Gm"", the region ""ρ < 2Gm"" does not map to these coordinates (the manifold with ""r"" as radial coordinate is not a ""chart"" on the maximally extended Schwarzschild geometry, because ""r = 0"" is a single point in a continuous chart).
The argument describes a pointlike particle at ""r = 0"", surrounded by the exterior region of a Schwarzschild geometry. For a pointlike particle, it makes no physical sense to extend the coordinate system interior to the particle. The extension exists mathematically, but has no physical meaning. By considering a classical body, such at the earth, as a composition of these pointlike particles, and by replacing the pointlike structure with a density, we restore the field equations, as an excellent large scale approximation.
We can model a black hole, neglecting the effect of the exclusion principle, by placing large numbers of elementary pointlike particles at ""r = 0"". We will then have a large mass, "M"", at ""r = 0"" surrounded by the exterior region of a Schwarzschild geometry. There is again no physical meaning to the interior region. A curious feature is that ""r = 0"" cannot be enclosed in a surface of arbitrarily small surface area. However, this is not inconsistent and is no more counter intuitive than, for example, that in a closed homogeneous isotropic universe a circle of sufficiently large radius will have zero circumference. Since the surface and the point are disjoint, the properties of the one don't have an immediate bearing on the other. If the argument were valid, it would show a discontinuity of the metric at ""r = 0"", not that ""r = 0"" cannot be a point. However in relational quantum gravity this argument has no meaning, because the very notion of a surface breaks down on small distance scales. The manifold is not conceived as some kind of metaphysical entity generalising the properties of Newton’s absolute space, but rather as a collection of potential measurement results, arising from the operational definition of time and space coordinates.
More realistically, allowing that fundamental particles are fermions, we may consider large numbers of particles in a region surrounding ""r = 0"". This does not alter the qualitative features of the description. In either case, the black hole is not strictly a “hole”, but is described on a continuous chart containing ""r = 0"".
In practice black holes are believed to be formed from the collapse of neutron stars. Neutrons are Fermions. In their [[http://prola.aps.org/abstract/PR/v55/i4/p374_1 seminal paper]] of 1939, Oppenheimer and Volkoff say //“A discussion of the probable effect of deviations from the Fermi equation of state suggests that actual stellar matter after the exhaustion of thermonuclear sources of energy will, if massive enough, contract indefinitely, although more and more slowly, never reaching true equilibrium”//. The Pauli exclusion principle prohibits placing more than one fermion at ""r = 0"", so the probable structure of a real black hole is as described by Oppenheimer and Volkoff. Rotation will also slow down the collapse.
Hawking radiation is not possible, since this depends on the classical structure of spacetime in the vicinity of the event horizon. Nonetheless a black hole can be expected to radiate. in which a photon is reflected from an elementary particle after a delay ""2GM"", as described in RQG III. Wave functions for particles are plane waves in this diagram and can be emitted to infinity provided that there is sufficient energy in the initial state. There is always sufficient energy to emit zero mass particles, which can have arbitrarily low energies at infinity. Matter in the hole will have high energy from gravitational collapse, and in addition, as the hole becomes more compact and particles approach ""r = 0"", wave functions have components with ever increasing energies. For a hole of 1,000,000 solar masses, the energy required of an electron to escape to infinity is ""952"" kg, eleven orders of magnitude less than the maximal energy ""pmax = 4.08 ¥ 1014"" kg corresponding to the lattice spacing. We may conclude that localisation of matter near ""r = 0"" creates energy states from which electrons (and other particles) are radiated with relativistic velocities. Since angular momentum of matter falling into the hole will generate a disc, the direction of radiation is in the axis of rotation, suggesting that this is the mechanism for relativistic jets. In a case where a disc is poorly defined, relativistic matter relativistic matter will be radiated from the hole in all directions, and will interact with surrounding matter in the host galaxy, creating a quasar. It is to be expected that the greater the mass of the black hole, the greater the gravitational force compactifying the hole, and hence the greater the amplitudes of states of sufficent energy to be radiated to infinity, and the greater the consequent radiation. One may think that gamma ray bursts result from a star falling into the hole with a resulting sudden increase in radiated energy.
Additions:
"" The use of the radar method of determining distance does not mean that radar is necessarily the fundamental concept 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. Within our immediate environment, the stable structures of matter are bound by the interchange of photons. 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. Thus, we may understand the concept of distance (as defined by the radar method) as a measure of the binding between the particles constituting a stable configuration of matter. |
""
In a famous ""thought experiment"", Eppley and Hannah proposed that gravitational waves are used to determine the position of a particle initially in a state of poor localisation and precise momentum. They argued that, if gravitational waves behave classically then, in principle, waves of indefinitely high frequency and indefinitely low intensity can be used. If the measurement causes collapse, then position is determined to an accuracy dictated by frequency, while the change in momentum is determined by intensity and will be small. Provided that momentum is conserved, then the uncertainty principle will be violated. If, on the other hand, measurement does not cause collapse, then there exists a direct observation of the (poorly localised) wave function. In this case instantaneous collapse could be observed in principle by performing a standard measurement of position, and the speed of light would be exceeded. They concluded that the gravitational field cannot be classical without violating accepted principles of physics, and must therefore satisfy the principles of quantum mechanics.
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. Within our immediate environment, the stable structures of matter are bound by the interchange of photons. 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. Thus, we may understand the concept of distance (as defined by the radar method) as a measure of the binding between the particles constituting a stable configuration of matter. |
""
In a famous ""thought experiment"", Eppley and Hannah proposed that gravitational waves are used to determine the position of a particle initially in a state of poor localisation and precise momentum. They argued that, if gravitational waves behave classically then, in principle, waves of indefinitely high frequency and indefinitely low intensity can be used. If the measurement causes collapse, then position is determined to an accuracy dictated by frequency, while the change in momentum is determined by intensity and will be small. Provided that momentum is conserved, then the uncertainty principle will be violated. If, on the other hand, measurement does not cause collapse, then there exists a direct observation of the (poorly localised) wave function. In this case instantaneous collapse could be observed in principle by performing a standard measurement of position, and the speed of light would be exceeded. They concluded that the gravitational field cannot be classical without violating accepted principles of physics, and must therefore satisfy the principles of quantum mechanics.
