There are two types of fundamental quantum gravitational mass amplitude states that
are denoted by the subscripts D and P. The D amplitudes lead to Einstein's usual
general relativity mass density functions. The P amplitudes lead to Einstein's
additional pressure mass densities, 3P/c2. Both of these densities
appear in the stress energy momentum tensor of general relativity.
Here they appear as solutions to a non-linear Schrödinger equation and carry
three quantising parameters (lD,m) and (lP,m), The
lD,lP values are subsets of the usual electronic quantum
variable l which is here denoted by l' to avoid confusion. The m parameter is
exactly the same as the electronic quantum theory m, there the z component of
angular momentum. In this paper, these parametric relations are briefly displayed
followed by an account of the connection to the spherical harmonic functions
symmetry system that is necessarily involved. Taken together, the two types of mass
density can be integrated over configuration space to give quantised general
relativity galactic masses in the form of cosmological mass spectra as was shown in
previous papers. Here this aspect has been extended to ensure that every galaxy
component of the spectra has a quantised black hole core with a consequent
quantised surface area. This is achieved by replacing the original free core radius
parameter rε with the appropriate Schwarzschild radius associated
with the core mass. Explanations are given for the choices of two further,
originally free, parameters, tb, θ0. The main result from
this paper is a quantum classification scheme for galaxies determined by the form
of their dark matter spherical geometry.
Using a new isothermal gravity equilibrium theory, the dust universe model together with a cosmological Schroedinger equation are applied to solving the problem of generating mass spectra. The masses generated can range from sub fundamental particle rest masses to masses greater than that of the universe. The ranges all depend on a quantum integer number l, related to the isotropic index n, which can lie between unity and infinity. One such mass obtained is given by l=8 and can represent a small galaxy. The rotation curves for stars, in motion, within this galaxy are examined for flatness and found to have gradients of approximately, -10^(-23). Examination of the Newtonian gravitation potential associated with these mass quanta reveals that it is, consistent with the dust universe model, based on Einstein's cosmological constant, Lambda, rather than on Newton's gravitational constant, G, as this last constant disappears by fractional cancellation within the theory structure. Thus this quantization of gravity is based on the cosmological constant. There is found within this theory structure a simulation of negative mass from suitably geometrically orientated positive mass. It is suggested that this feature could supply an explanation for the character of "dark energy" mass as being due to suitably orientated positive mass. However, this last point needs further study.
Much of the introductory section of this paper is devoted to displaying some previously obtained formulae, incorporating a change of notation and variables and giving some explanation of the relation of the work to Newtonian gravitation theory. This section all refers to a quantisation of gravity concentrated on and limited to galaxies with totally spherically symmetric cores and halos. Only the radial variable r is involved and the emphasis is on the dark matter concept. All the following sections are devoted to generalising the theory to additionally incorporate a dependence of galactic structure on the theta and phi spherical angular coordinates. The theory is derived using Schroedinger quantum theory in much the same way as it was used in developing the theory of atomic structure. The theoretical structure to be developed in this papers is a hybrid formulation involving three fundamental theoretical facets, general relativity, Schroedinger quantum mechanics and a new theoretical version of isothermal gravity self equilibrium. The combined structure has only become possible because of the discovery of an infinite discrete set of equilibrium states associated with this later theory, the l parameter states. The configuration space structure of these states has been found to be available in Schr\"odinger theory from a special inverse square law potential which appears to supply an inverse cube self attraction to the origin that maintains galaxies in an isolated steady state self gravity quantum condition. The arbitrary numerical coefficients of these Schroedinger states can also depend on l and are appropriately imported from the isothermal equilibrium theory. The work discussed here is much about how these l states can be interleaved with with the usual Schroedinger parameter for angular momentun which I call l-prime to avoid confusion. The l values have been found to be two possible cases of infinite subsets of the l-prime values, a D set for the usual mass density distributions in galaxies and an P set for Einstein's extra pressure term density 3P/c^2. However these identifications are just a working hypothesis. The usual atomic electron theory approach of separation of variables is used to solve the general gravitational Schr\"odinger equation and it turns out to be rather simpler than the atomic electronic situation. Two version of adapting the Schr\"odinger equation to hold the isothermal l states are given. The first I call a transplant operation that in fact is a replacement of appropriate Schroedinger l-prime angular momentum state representations with isothermal l state representations. The second version is in the conclusions section and involves simply displaying restricted Schroedinger representations that describe various gravitational situations. Also in this section, it is made clear that each of the one component Schroedinger representations can be replaced with an equivalent two component representation consisting of a Laplace equation together with a quantised energy equation. Finally, I display the mapping of the angular symmetry defining letters from atomic theory into the quantum theory structure of the isothermal l states. The main products of the theory are a quantisation of the gravitational field with explicitly a refined collections of mass accumulation spectra and a generalisation of Newtonian gravitation theory based on general relativity.
A theoretical value for the total positively gravitating mass of the universe is implied by the mathematical structure of the dust universe model. A simple formula is obtained that gives the value of this mass quantity in terms of Newton's gravitational constant, G, the Cosmological constant or Einstein's Lambda and the velocity of light, c. This result depends on taking a fundamental view of an epoch time conditioned relation, obtained earlier, between the universe's content of positively gravitating mass density and the universe's content of negatively gravitating mass density, The value obtained is approximately 2.00789x1053 kg, The approximation aspect depends on the currently measured or assumed values for G and Lambda.
A descriptive account is given for the dust universe Friedman lambda model of the universe developed earlier from general relativity by the present author. This description is wrapped around a new and very simple derivation of the model from first principles. The mathematics of this derivation rests on two classical physical equations, the formula for black body radiation and Newton's inverse square law of gravitation, so that without its descriptive wrapping the new derivation which does not involve general relativity directly would occupy about one page of this paper. The descriptive aspect is devoted to showing how the dust universe model can be decomposed into a many subunit form where each galaxy is seen as being a thermal cavity subunit. The time evolution of the whole universe can consequently be seen as being a bundling together of the thermal cavity elements to make up the time evolution structure of the whole universe. Finally, a cosmological Schr?dinger equation derived earlier by the present author is significantly generalized to make possible individual quantum state descriptions of the separate galactic thermal cavity elements. Some possible future generalizations of the structure are discussed.
The need for the cosmological constant, Lambda, in Einstein's field equations to be an absolute mathematical constant over all the time that they are used to describe some astrophysics process is demonstrated. Only if that condition holds will the conservation laws of mass and momentum hold as in classical physics. The Friedman equations that can be deduced rigorously from general relativity are consequently equally restricted to a constant valued Lambda and for the same reasons. However, the standard cosmological model, is not constructed from one solution of the Friedman equations but rather from at least three different but rigorous solutions patched together at times where they are physically thought to join. This is because the known solutions are thought to represent different conditions of mass movement, highly erratic or thermal at time near the big bang or more particle like and organized into systems at time near now, just to mention two types of activity when there obviously could be a continuous range of activities of mass types. Clearly this idea of how things have evolved after the big bang is very plausible, if the big bang idea is accepted as fact. The apparent need to patch solutions together over time creates great mathematical difficulties for cosmology because the three functions selected have to join smoothly which is the same as saying that they have to be differentiable not once but twice if accelerations are taken into account as they must be if the Friedman equations are to hold through the join. It is not clear whether or not this patching process can be rigorously achieved. However it is clear that the big bang concept does violate Einstein's field equations at $t=0$ because this concept implies that mass and momentum comes from nowhere. It is shown that all of these problems can be removed by introducing a continuously variable over time structure into the definition of temperature for the dust universe model. This only affects the value of the temperature that is associated with a given time and make no difference to the validity of the dust universe model with regard to it being a rigorous solution to the Einstein Field equations for all time from minus infinity to plus infinity.
In this paper, the general relativistic replacement for the Newtonian inverse square law of gravitation is obtained from the Friedman Cosmology equations. This version of the inverse square law is shown to contain information about the amount of dark energy mass contained in a specific region through a mass term $M_\Lambda^-$ dependent on Einstein's Lambda and, importantly for this paper, it also contains information about the amount of dark matter mass in the same region through a term $M_P^+$. This work derives from the Dust Universe Model which gives a complete cosmological description of the movement and evolution of the astrophysical space substratum which as usual is represented by a spatially uniform or constant mass density distribution at zero pressure. Thus definite spatial regions of the substratum can only be regarded as holding regions for un clumped mass, as primitive galaxies might be described. Consequently, to describe actual galaxies that have condensed from such a region, the more general solution of Einstien's Field eqtions involving the pressure term is needed to explain clumping and the resultant galactic form. The general relativist version of the inverse square law is written in a form applicable to the case of bound circular orbiting about a spherically symmetric central gravitational spatially distributed source force. Thus the behaviour of masses cycling within or outside the source region can be analysed. The formula for the galactic rotation curves for stars rotating within or outside the source region is obtained. A very simple galactic model is used consisting of just two components, the halo and the bulge with all visible orbiting stars, The conclusion is that the pressure term from general relativity and in the consequent Friedman equations is adequate to explain the constancy of the function of rotational velocity as a function of orbital distance from the centre of gravity starting at the massive core of the galaxy. A simple and parameter adaptable computer program using Mathematica has been constructed to display diagrams of galactic rotation curves. This program is available for downloading.
In this paper, the general relativistic replacement for the inverse square law of gravitation is obtained from the Friedman Cosmology Equation.
The stress energy momentum tensor with its invariant contracted scalar from Einstein's field equations are used to show that the pressure term they involve is responsible for inducing additional mass density, above that which is historically thought to be present. This has significant consequences for both the dark matter and dark energy problems. The additional material in the case of normally gravitating material might reasonably be taken to be the missing dark matter required for the stability of galaxies. The negative pressure of the dark energy is shown to induce just the right amount of extra negatively gravitating material to account for its usually assumed equation of state and also confirming the physical dark energy density I have deduced as being present in astrospace in earlier papers.
It is shown that negatively gravitating particles can consistently be considered to exist and interact with normal positively gravitating particles in the contexts of general relativity and classical Newtonian gravitational theory. This issue arises from the discovery of dark energy which is considered to be causing an acceleration of the expansion of the universe. The issue is, can this dark energy occur in particulate form? A related issue was studied in the fifties by Herman Bondi. He investigated the possible existence of negative mass in the general relativity context, long before dark energy appeared on the scene. He came to a paradoxical conclusion that seemed to rule out the actual physical existence of negatively gravitating particles. This paradox does not occur in this work because only positive mass particle are involved whatever their gravitational character may be. The structure of the differential equations that would apply in the case of a binary pair of opposite gravitational character components are used to show and explain how they can become consistent in general relativity or classical gravitation theory. This involves explaining a non-obvious relation between the principle of equivalence and Newton's ?Action Equals Reaction? principle. A path structure for a mixed mass binary pair is set up which satisfies the equations of motion and does not have paradoxical properties. The force structure of the system is checked with a known classical dynamical test for the force per unit mass involved in the component particles motions. This test is used to demonstrate that the basic assumptions of this theory are incorporated into the consequential orbital structure. An alternative to the ?Action Equals Reaction? principle more appropriate to the astronomical situation is suggested. An animation using Mathematica has been derived, and is available, and shows how a mixed gravity binary pair move under their mutual gravitational action.
Astronomical measurements of the Omegas for mass density, cosmological constant lambda and curvature k are shown to be sufficient to produce a unique and detailed cosmological model describing dark energy influences based on the Friedman equations. The equation of state Pressure turns out to be identically zero at all epochs as a result of the theory. The partial omega, for dark energy, has the exact value, minus unity, as a result of the theory and is in exact agreement with the astronomer?s measured value. Thus this measurement is redundant as it does not contribute to the construction of the theory for this model. Rather, the value of omega is predicted from the theory. The model has the characteristic of changing from deceleration to acceleration at exactly half the epoch time at which the input measurements are taken. This is a mysterious feature of the model for which no explanation has so far been found. An attractive feature of the model is that the acceleration change time occurs at a red shift of approximately 0.8 as predicted by the dark energy workers. Using a new definition of dark energy density it is shown that the contribution of this density to the acceleration process is via a negative value for the gravitational constant, -G, exactly on a par with gravitational mass which occurs via the usual positive value for G.
In an earlier paper, it was shown that the cosmological model that was introduced in a sequence of three earlier papers under the title A Dust Universe Solution to the Dark Energy Problem, originally described by the Friedman equations, can be expressed as a solution to a non-linear Schr?dinger equation. In this paper, a large collection of solutions to this Schr?dinger equation are found and discussed in the context of relaxing the uniform mass density condition usually employed in cosmology theory. The surprising result is obtained that this non-linear equation can have its many solutions linearly superposed to obtain solution of the cosmology theory problem of great generality and applicability.
Sommerfeld?s numerical quantum-electronic structure for the energies of the states of the hydrogen-like atoms is shown to present a scheme that is able to express a unique observer point of view. The perspective of this observer is essentially how he, if fixed on a trapped electron, would see his and his electron's state of containment within the full atomic quantum state. This particular internal view of a quantum state is then shown to have strong analytic powers in the numerically very different scale of gravitation theory. This unexpected analogy becomes possible when it is recognized that a basic part of gravitation theory can be expressed in terms completely analogous to the quantum relativistic electromagnetic structure involved in Sommerfeld's formula for quantum-state energy.