 We can model a black hole, neglecting the effect of the Pauli 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, exactly as for a single gravitating particle. 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, since the surface and the point are disjoint, the properties of the one don’t have an immediate bearing on the other. 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. If the argument were valid, it would show a discontinuity of the metric at r = 0, not that r = 0 cannot be a point. 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 ""Pauli 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, exactly as for a [[OriginOfCurvature single gravitating particle]].. 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, since the surface and the point are disjoint, the properties of the one don’t have an immediate bearing on the other. 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. If the argument were valid, it would show a discontinuity of the metric at ""r = 0"", not that ""r = 0"" cannot be a point. 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:
""""We can model a black hole, neglecting the effect of the ""Pauli 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, exactly as for a [[OriginOfCurvature single gravitating particle]].. 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, since the surface and the point are disjoint, the properties of the one don’t have an immediate bearing on the other. 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. If the argument were valid, it would show a discontinuity of the metric at ""r = 0"", not that ""r = 0"" cannot be a point. 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.
"" ""Fundamental particles are Fermions and obey the ""exclusion principle"" which prohibits placing more than one of each type of Fermion at ""r = 0"". More realistically, we consider large numbers of particles in a region surrounding ""r = 0"". This does not alter the qualitative features of the description, although there may be a region in which the metric is not defined (not just a point). In practice, black holes are believed to form from the collapse of neutron stars. 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”//.
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.
Deletions:
We can model a black hole, neglecting the effect of the ""Pauli 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, since the surface and the point are disjoint, the properties of the one don’t have an immediate bearing on the other. 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. If the argument were valid, it would show a discontinuity of the metric at ""r = 0"", not that ""r = 0"" cannot be a point. 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.
Fundamental particles are Fermions and obey the ""exclusion principle"" which prohibits placing more than one of each type of Fermion at ""r = 0"". More realistically, we consider large numbers of particles in a region surrounding ""r = 0"". This does not alter the qualitative features of the description, although there may be a region in which the metric is not defined (not just a point). In practice, black holes are believed to form from the collapse of neutron stars. 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”//.
""""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.
Additions:
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).
We can model a black hole, neglecting the effect of the ""Pauli 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, since the surface and the point are disjoint, the properties of the one don’t have an immediate bearing on the other. 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. If the argument were valid, it would show a discontinuity of the metric at ""r = 0"", not that ""r = 0"" cannot be a point. 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.
Fundamental particles are Fermions and obey the ""exclusion principle"" which prohibits placing more than one of each type of Fermion at ""r = 0"". More realistically, we consider large numbers of particles in a region surrounding ""r = 0"". This does not alter the qualitative features of the description, although there may be a region in which the metric is not defined (not just a point). In practice, black holes are believed to form from the collapse of neutron stars. 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”//.
In relational quantum gravity, a neutron star will collapse as described by Oppenheimer and Volkoff. Rotation will also slow down the collapse. However, a black hole is not strictly a “hole” but is described on a continuous chart containing ""r = 0"". It would perhaps be more realististic to describe a black hole as a compact neutron star. However, as “black hole” is now used to describe a number of astronomical objects, it seems better to stick with the usual terminology.
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, or has irregularities due to infalling matter, relativistic matter (potentially containing all particle types) 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 compacting the hole, and hence the greater the amplitudes of states of sufficent energy to be radiated to infinity, and the greater the consequent radiation.
Deletions:
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).
We can model a black hole, neglecting the effect of the ""Pauli 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.
Fundamental particles are Fermions and obey the ""exclusion principle"" which prohibits placing more than one of each type of Fermion at ""r = 0"". More realistically, we consider large numbers of particles in a region surrounding ""r = 0"". This does not alter the qualitative features of the description, although there may be a region in which the metric is not defined (not just a point). In practice, black holes are believed to be formed from the collapse of neutron stars. 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”//.
In relational quantum gravity, a neutron star will collapse as described by Oppenheimer and Volkoff. Rotation will also slow down the collapse. However, a black hole is not strictly a “hole” but is described on a continuous chart containing ""r = 0"". It would perhaps be more realististic to describe a black hole as a compactified neutron star. However, as “black hole” is now used to describe a number of astronomical objects, it seems better to stick with the usual terminology.
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, or has irregularities due to infalling matter, relativistic matter (potentially containing all particle types) 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.
Additions:
Fundamental particles are Fermions and obey the ""exclusion principle"" which prohibits placing more than one of each type of Fermion at ""r = 0"". More realistically, we consider large numbers of particles in a region surrounding ""r = 0"". This does not alter the qualitative features of the description, although there may be a region in which the metric is not defined (not just a point). In practice, black holes are believed to be formed from the collapse of neutron stars. 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”//.
In relational quantum gravity, a neutron star will collapse as described by Oppenheimer and Volkoff. Rotation will also slow down the collapse. However, a black hole is not strictly a “hole” but is described on a continuous chart containing ""r = 0"". It would perhaps be more realististic to describe a black hole as a compactified neutron star. However, as “black hole” is now used to describe a number of astronomical objects, it seems better to stick with the usual terminology.
Deletions:
Fundamental particles are Fermions and obey the ""exclusion principle"" which prohibits placing more than one of each type of Fermion at ""r = 0"". More realistically, we consider large numbers of particles in a region surrounding ""r = 0"". This does not alter the qualitative features of the description. In practice, black holes are believed to be formed from the collapse of neutron stars. 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”//.
In relational quantum gravity, a neutron star will collapse as described by Oppenheimer and Volkoff. Rotation will also slow down the collapse. However, a black hole is not strictly a “hole”, but is described on a continuous chart containing ""r = 0"". It would perhaps be more realististic to describe a black hole as a compactified neutron star. However, as “black hole” is now used to describe a number of astronomical objects, it seems better to stick with the usual terminology.
Additions:
""Inflation""
""Pre-expansion as an Ametric Phase""
""Black Holes""
Fundamental particles are Fermions and obey the ""exclusion principle"" which prohibits placing more than one of each type of Fermion at ""r = 0"". More realistically, we consider large numbers of particles in a region surrounding ""r = 0"". This does not alter the qualitative features of the description. In practice, black holes are believed to be formed from the collapse of neutron stars. 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”//.
In relational quantum gravity, a neutron star will collapse as described by Oppenheimer and Volkoff. Rotation will also slow down the collapse. However, a black hole is not strictly a “hole”, but is described on a continuous chart containing ""r = 0"". It would perhaps be more realististic to describe a black hole as a compactified neutron star. However, as “black hole” is now used to describe a number of astronomical objects, it seems better to stick with the usual terminology.
Deletions:
""Inflation""
""Pre-expansion as an Ametric Phase""
""Black Holes""
Fundamental particles are Fermions and obey the ""exclusion principle"" which prohibits placing more than one of each type of Fermion at ""r = 0"". More realistically, we consider large numbers of particles in a region surrounding ""r = 0"". This does not alter the qualitative features of the description. In practice, black holes are believed to be formed from the collapse of neutron stars. 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 probable structure of a real black hole is as described by Oppenheimer and Volkoff. Rotation will also slow down the collapse. It is seen that a black hole is not strictly a “hole”, but is described on a continuous chart containing ""r = 0"". It would perhaps be more realististic to describe a black hole as a compactified neutron star. However, as “black hole” is now used to describe a number of astronomical objects, it seems better to stick with the usual terminology.
Additions:
Fundamental particles are Fermions and obey the ""exclusion principle"" which prohibits placing more than one of each type of Fermion at ""r = 0"". More realistically, we consider large numbers of particles in a region surrounding ""r = 0"". This does not alter the qualitative features of the description. In practice, black holes are believed to be formed from the collapse of neutron stars. 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 probable structure of a real black hole is as described by Oppenheimer and Volkoff. Rotation will also slow down the collapse. It is seen that a black hole is not strictly a “hole”, but is described on a continuous chart containing ""r = 0"". It would perhaps be more realististic to describe a black hole as a compactified neutron star. However, as “black hole” is now used to describe a number of astronomical objects, it seems better to stick with the usual terminology.
Deletions:
More realistically, allowing that fundamental particles are Fermions and obey the ""exclusion principle"" which prohibits placing more than one of each type of Fermion at ""r = 0"". Instead we consider large numbers of particles in a region surrounding ""r = 0"". This does not alter the qualitative features of the description. 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 probable structure of a real black hole is as described by Oppenheimer and Volkoff. Rotation will also slow down the collapse. It is seen that a black hole is not strictly a “hole”, but is described on a continuous chart containing ""r = 0"". It would perhaps be more realististic to describe a black hole as a compactified neutron star. However, as “black hole” is now used to describe a number of astronomical objects, it seems better to stick with the usual terminology.
Additions:
More realistically, allowing that fundamental particles are Fermions and obey the ""exclusion principle"" which prohibits placing more than one of each type of Fermion at ""r = 0"". Instead we consider large numbers of particles in a region surrounding ""r = 0"". This does not alter the qualitative features of the description. 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 probable structure of a real black hole is as described by Oppenheimer and Volkoff. Rotation will also slow down the collapse. It is seen that a black hole is not strictly a “hole”, but is described on a continuous chart containing ""r = 0"". It would perhaps be more realististic to describe a black hole as a compactified neutron star. However, as “black hole” is now used to describe a number of astronomical objects, it seems better to stick with the usual terminology.
Deletions:
More realistically, allowing that fundamental particles are Fermions and obey the ""exclusion principle"" which prohibits placing more than one of each type of Fermion at ""r = 0"". Instead we consider large numbers of particles in a region surrounding ""r = 0"". This does not alter the qualitative features of the description. 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”//.
Thus, the probable structure of a real black hole is as described by Oppenheimer and Volkoff. Rotation will also slow down the collapse. It is seen that a black hole is not strictly a “hole”, but is described on a continuous chart containing ""r = 0"". It would perhaps be more realististic to describe a black hole as a compactified neutron star. However, as “black hole” is now used to describe a number of astronomical objects, it seems better to stick with the usual terminology.
Additions:
Thus, the probable structure of a real black hole is as described by Oppenheimer and Volkoff. Rotation will also slow down the collapse. It is seen that a black hole is not strictly a “hole”, but is described on a continuous chart containing ""r = 0"". It would perhaps be more realististic to describe a black hole as a compactified neutron star. However, as “black hole” is now used to describe a number of astronomical objects, it seems better to stick with the usual terminology.
Deletions:
Thus, the probable structure of a real black hole is as described by Oppenheimer and Volkoff. Rotation will also slow down the collapse. It is seen that a black hole is not strictly a “hole”, but is described on a continuous chart containing ""r = 0"". It would perhaps be more realististic to describe a black hole as a compactified neutron star. However as “black hole” is now used to describe many astronomical it seems better to stick with the usual terminology.
Additions:
More realistically, allowing that fundamental particles are Fermions and obey the ""exclusion principle"" which prohibits placing more than one of each type of Fermion at ""r = 0"". Instead we consider large numbers of particles in a region surrounding ""r = 0"". This does not alter the qualitative features of the description. 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”//.
Thus, the probable structure of a real black hole is as described by Oppenheimer and Volkoff. Rotation will also slow down the collapse. It is seen that a black hole is not strictly a “hole”, but is described on a continuous chart containing ""r = 0"". It would perhaps be more realististic to describe a black hole as a compactified neutron star. However as “black hole” is now used to describe many astronomical it seems better to stick with the usual terminology.
Deletions:
More realistically, allowing that fundamental particles are Fermions and obey the ""exclusion principle"", we may consider large numbers of particles in a region surrounding ""r = 0"". This does not alter the qualitative features of the description. 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.
It is seen that a black hole is not strictly a “hole”, but is described on a continuous chart containing ""r = 0"". It would perhaps be more realististic to describe a black hole as a compactified neutron star. However as “black hole” is now used to describe many astronomical it seems better to stick with the usual terminology.
Additions:
We can model a black hole, neglecting the effect of the ""Pauli 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 and obey the ""exclusion principle"", we may consider large numbers of particles in a region surrounding ""r = 0"". This does not alter the qualitative features of the description. 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.
It is seen that a black hole is not strictly a “hole”, but is described on a continuous chart containing ""r = 0"". It would perhaps be more realististic to describe a black hole as a compactified neutron star. However as “black hole” is now used to describe many astronomical it seems better to stick with the usual terminology.
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.
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 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.
It is seen that a black hole is not strictly a “hole”, but is described on a continuous chart containing ""r = 0"". It would perhaps be more realististic to describe a black hole as a compactified neutron star. However as “black hole” is now used to describe many astronomical it seems better to stick with the terminology.
Additions:
In classical general relativity, a ""singularity"" may be described as a point at which the known laws of physics necessarily break down. The singularities of interest to physics are the big bang (and the big crunch) and black holes. We may expect that known laws of physics break down not just at a singularity, but also close to it.
""Inflation""
""Pre-expansion as an Ametric Phase""
""Black Holes""
This explanation confuses the coordinate speed of light, which depends on the chosen coordinate system, with the local speed of light ""c"", which is necessarily constant according to Einstein’s original arguments for [[FoundationsOfSpecialRelativity special relativity]]. It amounts to saying that the universe must have been expanding faster than itself. No matter what the initial rate of expansion, by choosing the varying scale on the time axis, it is //always// possible to define Penrose coordinates, in which the coordinate speed of light is constant (the first two figures on this page). It is then seen that, irrespective of inflation, the early universe consisted of causally disconnected regions. The horizon problem cannot be solved by inflation.
"" ""The description of a particle by the state "" "" implies that the particle's position has been measured relative to an apparatus. 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, even 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. [[http://en.wikipedia.org/wiki/Spin_network Spin networks]] appear to have some of the requisite properties. Here I merely a few general remarks regarding behaviour near the big bang.
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 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.
It is seen that a black hole is not strictly a “hole”, but is described on a continuous chart containing ""r = 0"". It would perhaps be more realististic to describe a black hole as a compactified neutron star. However as “black hole” is now used to describe many astronomical it seems better to stick with the terminology.
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In classical general relativity, a ""singularity"" may be described as a point at which the known laws of physics necessarily break down. The singularities of interest to physics are the big bang (and the big crunch) and black holes. We may expect that known laws of physics break down not just at a singularity, but also close to it.
""Inflation""
""Pre-expansion as an Ametric Phase""
""Black Holes""
This explanation confuses the coordinate speed of light, which depends on the chosen coordinate system, with the local speed of light ""c"", which is necessarily constant according to Einstein’s original arguments for [[FoundationsOfSpecialRelativity special relativity]]. It amounts to saying that the universe must have been expanding faster than itself. No matter what the initial rate of expansion, by choosing the varying scale on the time axis it is always possible to define Penrose coordinates, in which the coordinate speed of light is constant (the first two figures on this page). It is then seen that, irrespective of inflation, the early universe consisted of causally disconnected regions such that the horizon problem remains.
""""The description of a particle by the state """" implies that the particle's position has been measured relative to an apparatus. 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, even 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. [[http://en.wikipedia.org/wiki/Spin_network Spin networks]], appear to have some of the requisite properties. Here I merely a few general remarks regarding behaviour near the big bang.
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.
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Deletions:
Additions:
In classical general relativity, a ""singularity"" may be described as a point at which the known laws of physics necessarily break down. The singularities of interest to physics are the big bang (and the big crunch) and black holes. We may expect that known laws of physics break down not just at a singularity, but also close to it.
- ""Inflation""
- ""Pre-expansion as an Ametric Phase""
- ""Black Holes""
Deletions:
In classical general relativity, a ""singularity"" may be described as a point at which the known laws of physics necessarily break down. The singularities of interest to physics are the big bang (and the big crunch) and black holes. We may expect that known laws of physics break down not just at a singularity, but also close to it.
- ""Inflation""
- ""Pre-expansion as an Ametric Phase""
- ""Black Holes""
Additions:
Deletions:
Additions:
Deletions:
Additions:
""Inflation""
""Pre-expansion as an Ametric Phase""
""Black Holes""
"" ""[[http://en.wikipedia.org/wiki/Inflation_(cosmology) Inflation]] hypothesises that the horizon problem can be resolved by a period of ultra-rapid expansion in the early universe. Although inflation is described in a number of textbooks, I have not found a coherent explanation of inflation, consistent with the principles of general relativity. As it is usually described, the basic idea is to observe that in general relativity the coordinate speed of light is not necessarily constant. For example, a Friedmann cosmology can be shown expanding from a point singularity at the big bang. In this description the coordinate speed of light is greater nearer to the big bang. It is then stated that if expansion were rapid enough near the initial singularity the speed of light could be so great that light could cross horizons and causally disconnected regions could become causally connected.
This explanation confuses the coordinate speed of light, which depends on the chosen coordinate system, with the local speed of light ""c"", which is necessarily constant according to Einstein’s original arguments for [[FoundationsOfSpecialRelativity special relativity]]. It amounts to saying that the universe must have been expanding faster than itself. No matter what the initial rate of expansion, by choosing the varying scale on the time axis it is always possible to define Penrose coordinates, in which the coordinate speed of light is constant (the first two figures on this page). It is then seen that, irrespective of inflation, the early universe consisted of causally disconnected regions such that the horizon problem remains.
Deletions:
""""[[http://en.wikipedia.org/wiki/Inflation_(cosmology) Inflation]] hypothesises that the horizon problem can be resolved by a period of ultra-rapid expansion in the early universe. Although inflation is described in a number of textbooks, I have not found a coherent explanation of inflation, consistent with the principles of general relativity. As it is usually described, the basic idea is to observe that in general relativity the coordinate speed of light is not necessarily constant. For example, a Friedmann cosmology can be shown expanding from a point singularity at the big bang. In this description the coordinate speed of light is greater nearer to the big bang. It is then stated that if expansion were rapid enough near the initial singularity the speed of light could be so great that light could cross horizons and causally disconnected regions could become causally connected.
This explanation confuses the coordinate speed of light, which depends on the chosen coordinate system, with the local speed of light ""c"", which is necessarily constant according to Einstein’s original arguments for [[FoundationsOfSpecialRelativity special relativity]]. It amounts to saying that the universe must have been expanding faster than itself. No matter what the initial rate of expansion, it is always possible to define Penrose coordinates, in which the coordinate speed of light is constant, showing that the early universe consisted of causally disconnected regions such that the horizon problem remains.
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""[[http://en.wikipedia.org/wiki/Inflation_(cosmology) Inflation]] hypothesises that the horizon problem can be resolved by a period of ultra-rapid expansion in the early universe. Although inflation is described in a number of textbooks, I have not found a coherent explanation, consistent with the principles of general relativity. As it is usually described, the basic idea is to observe that in general relativity the coordinate speed of light is not necessarily constant. For example, a Friedmann cosmology can be shown expanding from a point singularity at the big bang. In this description the coordinate speed of light is greater nearer to the big bang. It is then stated that if expansion were rapid enough near the initial singularity the speed of light could be so great that light could cross horizons and causally disconnected regions could become causally connected.
""Fundamental particles are Fermions and obey the ""exclusion principle"" which prohibits placing more than one of each type of Fermion at ""r = 0"". More realistically, we consider large numbers of particles in a region surrounding ""r = 0"". This does not alter the qualitative features of the description, although there may be a region in which the metric is not defined (not just a point). In practice, black holes are believed to form from the collapse of neutron stars. 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”//.