arXiv subjects
Publications and source records attributed to Scott Funkhouser.
We survey some past conditional results on the distribution of large differences between consecutive primes and examine how the Hardy-Littlewood prime k-tuples conjecture can be applied to this question.
By manipulating the integral expression for the proper radius $R_e$ of the cosmological event horizon (CEH) in a Friedmann-Robertson-Walker (FRW) universe, we obtain an analytical expression for the change $\dd R_e$ in response to a uniform fluctuation $\ddρ$ in the average cosmic background density $ρ$. We stipulate that the fluctuation arises within a vanishing interval of proper time, during which the CEH is approximately stationary, and evolves subsequently such that $\ddρ/ρ$ is constant. The respective variations $2πR_e \dd R_e$ and $\dd E_e$ in the horizon entropy $S_e$ and enclosed energy $E_e$ should be therefore related through the cosmological Clausius relation. In that manner we find that the temperature $T_e$ of the CEH at an arbitrary time in a flat FRW universe is $E_e/S_e$, which recovers asymptotically the usual static de Sitter temperature. Furthermore, it is proven that during radiation-dominance and in late times the CEH conforms to the fully dynamical First Law $T_e \drv S_e = P\drv V_e - \drv E_e$, where $V_e$ is the enclosed volume and $P$ is the average cosmic pressure.
The change in entropy, /DeltaS, associated with the quasi-static absorption of a particle of energy u by a Schwarzschild black hole (ScBH) is approximately (u/T)-s, where T is the Hawking temperature of the black hole and s is the entropy of the particle. Motivated by the statistical interpretation of entropy, it is proposed here that absorption should be suppressed, but not forbidden, when /DeltaS<0, which requires the absorption cross-section to be sensitive to /DeltaS. A purely thermodynamic formulation of the probability for absorption is obtained from the standard relationship between microstates and entropy. If /DeltaS>>1 and s<<u/T then the probability for the particle not to be absorbed is approximately exp[-u/T], which is identical to the probability for quantum mechanical reflection by the horizon of a ScBH. The manifestation of quantum behaviors in the new probability function may intimate a fundamental physical unity between thermodynamics and quantum mechanics.
The holographic indeterminacy resulting from the quantization of spacetime leads to an inherent uncertainty (lpL)1/2 in the relative positions of two events, separated by a distance L, in a direction transverse to a null ray connecting the events, where lP is the Planck length. The new indeterminacy principle leads to a critical condition in which the holographic uncertainty in the relative transverse positions of two diametrically opposed particles on the surface a body becomes greater than the average distance between particles in the body. The Chandrasekhar mass and the characteristic nuclear density emerge as the minimum mass and density of a baryonic body that could meet the critical criteria. Neutron stars are therefore identified as a class of bodies in which holographic indeterminacy may have physical consequences.
The holographic upper bound on entropy is applied to the gravitational action associated with the non-relativistic contraction of a nebula. A critical radius is identified, as a function of the initial radius and mass, for which the number of bits associated with the action would equal the maximum number of bits allowed to the body. The gravitational action of a typical star approximately saturates the holographic bound, perhaps suggesting a physical link between holographic principles and astrophysical processes.
The existence of a positive cosmological constant leads naturally to two fundamental scales of length, being the De Sitter horizon and the radius of the cell associated with a holographic degree of freedom. Associated with each of those scales of length are a macroscopic gravitational mass and a microscopic quantum mechanical mass. Three of those four fundamental masses have been discussed in the literature, and this present work identifies the physical significance of the remaining mass, being the gravitational mass associated with the holographic length. That mass, which is of the order 10^{12}kg and inversely proportional to the sixth root of the cosmological constant, represents the mass of the black hole whose evaporation time is equal to the fundamental cosmic time, which is of the order the current age of the universe. It also represents the minimum mass of a black hole that is capable of accreting a particle whose Compton wavelength is equal to the fundamental holographic length, which is of the order the Compton wavelength of the nucleon.
An empirically validated, phenomenological model relating the parameters of an astronomical body to the stochastic fluctuations of its granular components is generalized in terms of fractal scaling laws. The mass of the particle constituting the preponderance of the mass of a typical galaxy is determined from the generalized model as a function of the fractal dimension. For a fractal dimension between 1 and 3 the mass of the dominant particle in galaxies is, roughly, between the Planck mass and 1eV. If the dimension is near 2 then the fractal model is identical to the original stochastic model, and the mass of the dominant particle must be of order near the nucleon mass. Two additional expressions for the mass of the dominant particle in the universe are obtained from basic quantum considerations and from the existence of a cosmological constant. It follows that the fractal dimension 2 is favored and that the mass of the dominant particle is proportional to sixth root of the cosmological constant and of order near the nucleon mass.
The basic scaling laws for structures in a fractal universe require that the characteristic quantity of action associated with astronomical bodies should be of order near the maximum possible action allowed by the holographic upper bound. That conclusion is consistent with the observed parameters of galaxies and clusters.
Five fundamental scales of mass follow from holographic limitations, a self-similar law for angular momentum and the basic scaling laws for a fractal universe with dimension 2. The five scales correspond to the observable universe, clusters, galaxies, stars and the nucleon. The fundamental scales form naturally a self-similar hierarchy, generating new relationships among the parameters of the nucleon,the cosmological constant and the Planck scale. There is implied a sixth fundamental scale thatcorresponds to the electrostatic force within an atom. Identifying the implied scale as such leads to new relationships among the fundamental charge, the mass of the electron and cosmological parameters. Theseconsiderations also suggest that structures on the scale of galaxies and larger must be bound by non-Newtonian forces.
According to a model of quantum cosmology the maximum number of degrees of freedom allowed in our three dimensions was determined by the size of seven extra dimensions in an initial excited state before inflation. The size of the extra dimensions can be inferred from a simple scheme for unifying the strong force and gravity. Coupled with the Bekenstein-Hawking entropy bound, these considerations lead to a scaling law for the cosmological constant that has been proposed independently by several authors.
An ensemble of pure numbers of order near 10^122 is produced naturally from the fundamental parameters of modern cosmology. This new large-number coincidence problem is resolved by demonstrating implicit physical connections that follow from the standard cosmological model. However, the occurrence of the new large-number coincidence combined with the known coincidence among pure numbers of order near 10^40 poses a distinct problem that is resolved with a scaling law for the cosmological constant that was originally proposed by Zel'dovich.
The coincidence problem among the pure numbers of order near 10^{40} is resolved with the Raychaudhuri and Friedmann-Robertson-Lemaitre-Walker equations and a trivial relationship involving the fine structure constant. The fact that the large number coincidence occurs only in the same epoch in which other coincidences among cosmic parameters occur could be considered a distinct coincidence problem suggesting an underlying physical connection. A natural set of scaling laws for the cosmological constant and the critical acceleration are identified that would resolve the coincidence among cosmic coincidences.
The 'dark galaxy' VIRGOHI21 seems to be composed of an unusually high proportion of darkmatter and is situated in a strong external gravitational field. As such it offers a rare test for theories of modified dynamics. If the system is bound then its dynamics are inconsistent with those predicted by theMOND theory.
From the Eddington-Weinberg relationship, which may be explained by the holographic principle and the cosmic coincidence in a flat Universe, it follows that the characteristic gravitational acceleration aN associated with the nucleon and its Compton wavelength is of order the Hubble acceleration H0c in this epoch. A natural scaling for the cosmological constant is obtained from this acceleration term. It also happens that the critical acceleration a0 associated with the MOND theory is of order aN.
Mach's "fixed stars" are actually not fixed at all. The distant clusters of galaxies are not only receding from each observer but they are also accelerating since the rate of cosmological expansion is not constant. If the distant cosmic masses in someway constitute the frame of inertial reference then an additional force should be generated among local bodies in reaction to the apparent cosmological accelerations of the distant galaxies.
If Mach's Principle explains the Newtonian inertial reaction to acceleration then the role of the 'fixed stars' should also be manifest through Hamilton's formulation of mechanics. This consistency may be achieved if the expression for relativistic energy contains a cosmological coefficient that is (currently) equal to one. The presence of the required cosmological term exactly identifies the rest energy of a body as its gravitational potential energy due to the mass of the Universe.
If a physical significance should be attributed to the cosmological large number relationship obtained from Sciama's formulation of Mach's Principle, then a number of interesting physical conclusions may be drawn. The Planck length is naturally obtained as the amplitude of waves in a medium whose properties are implied by the relationship. The relativistic internal energy associated with a rest mass is explicitly related to the gravitational potential energy of the Universe, and consistency with the Einstein photon energy is demonstrated. Broader cosmological consequences of this formulation are addressed.