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Inflation

The past light cone can be described as a causal horizon, meaning that an event outside the past light cone cannot be a cause of anything inside the past light cone. The causal horizon, or past light cone, of an event in our past is necessarily smaller than our own causal horizon. If the past light cones of two events do not intersect, then we can say that those events are causally disconnected, meaning that they cannot share a common cause.

The further an event is in the past, the smaller is its past light cone, or causal horizon. As we look at the universe further and further away, we should observe increasing numbers of ever smaller causally disconnected regions. In the absence of common cause, random fluctuations should create differences between these regions. However, we observe extreme isotropy in the Cosmic microwave background. This is the horizon problem.

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.

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 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.

Pre-expansion as an Ametric Phase

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. 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. Perfect chaos in physical conditions gives rise to perfect order in a probabilistic description as required by quantum theory. 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 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 Pauli exclusion principle, by placing large numbers of elementary pointlike particles at r = 0. We will then have a large mass, M, 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.

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 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.

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 p_max = 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, 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.

Since matter is freely radiated from states with high energies, in the absence of further infalling matter, the black hole will rapidly cool and reach a state in which there is little radiation. Black holes in the early universe can be expected to have an irregular structure and large amounts of infalling matter, which will generate quasars. As matter ceases to fall into the hole, radiation takes the form of jets, perpendicular to the hole, and finally the hole becomes quiet. Infalling matter will trigger further radiation. A gamma ray burst may result from a star falling into the hole and causing a sudden increase in radiated energy. Similarly, galaxy collisions, or near collisions, will disturb orbits and increase the amount of matter entering the hole, causing the galaxy to light up.


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