A Fundamental Conservation as a Uniﬁcation of Quantum Theory and Relativity

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Introduction
Special relativity (SR) eloquently conforms to M v 2 2 (total kinetic energy) in Noether's theorem as, [3] and energy is thus conserved (time-transitionally invariant).However, in GR, energy evolves as spacetime changes.Einstein has shown us that when the space through which particles move is dynamic, the total energy of those particles is not conserved.Moreover, the energy stored in the cosmological constant must expand at a rate of k 3 , in proportion to the volume of expanding space.An additional challenge to vacuum energy is the unstable nature of the uneven distribution of matter throughout the universe.The pervading justification for redshift photon energy loss is the lack of an associated symmetry.Conservation laws conventionally define invariance with respect to time.For example, the Euler-Lagrange equations (in general coordinates), [4] Then conservation is shown by the first order derivative of some quantity, with respect to time, being equal to zero, However, this article proposes a fundamental conservation of total Hamiltonian energy within the entire scope of cosmology.

The Supernova Cosmology Project with Einstein-de Sitter Model
The 1998 supernova data [5] have concluded that the observed magnitude of nearby and distant type LA supernovae, as compared with cosmological predictions of models with zero vacuum energy and mass densities (ranging from the critical density ρ c down to zero), has formally ruled out the Einstein-de Sitter model of closed ordinary matter (i.e.Ω M = 1) at the 7σ to 8σ confidence level for two different fitting methods.Moreover, the best fit to this divergence implies that, in the present epoch, the vacuum energy density ρ Λ is larger than the energy density attributable to mass (ρ m C 2 ).Therefore, cosmic expansion is now accelerating.However, an alternate interpretation of this data is presented, in defiance of a requirement for any dark component of energy density: Theorem 1 Time interval ∆t contracts (decreases) inversely proportional to the metric expansion of space ar , independent of relative motion (Note that this is distinct from γ time dilation).
Normalizing ∆t n from D, Where ∆t n is an interval of time at distance D n , and K is an undetermined minute constant (≈ 4.000E −24) that becomes significant in a matter-dominated universe.
Thus, accelerating expansion is alternatively explained as being generally constant, such that ä = 0 (excluding local variation) with a decrease in time interval ∆t n , which has an equivalent effect as an increase in velocity v. Thus,

Corollary 1.1 Universal expansion, with decreasing time intervals, appears as accelerated expansion.
Note that this offers an alternative to dark components, as functions with decreasing time intervals are equivalent to functions with an anti-derivative.See figure 1 Figure 1: Velocity ȧ(t ) with decreasing time intervals appears as acceleration ä(t ) Table 1 lists eleven hypothetical sla points as predicted in the Einstein-de Slitter model with uniform time intervals, compared with contracted time intervals (∆t n , per theorem 1).The scatter plot in figure 2 (with logarithmic horizontal axis) shows three trend lines with corresponding values of K ≈ (Ω M , Ω Λ ) = (0, 1), (0.5, 0.5), (1, 0).Note: ä(t ) = 0.

Per theorem 1, velocity ∆d ∆t n increases with distance ar . With this proportionate increase in velocity, energy density ρ proportionally increases, due to increased velocities in particle kinetic and internal energies (compression, energy of nuclear binding, etc.). To the observer at ar 0 , energy at ar n [mpc] density measures ρ n with greater energy per unit of time.
∆ρ n ∆ρ = ∆t n ∆t

Conservation of Energy Density Over Flat Space
Einstein had contemplated that his original static model of GR was unstable, and might require the cosmological constant to offset gravity from collapsing.However, this alternate model is inherently more stable: Corollary 1.3 For galactic scales, at distance ar n , the average force of energy density ρ, approaching from below ar n , is counterbalanced by the average force of increasing energy density ρ n approaching from above ar n , Thus, a fundamental conservation and coordinate symmetry of energy density, with respect to spacetime, is established.See figure 3,

Galaxy Rotation Curve with Increased Density
The discrepancies between theoretical and observed galaxy rotation curves involve both density and velocity.Conventionally, the dependence of circular velocity V ci r c on radial distance R assumes M , m and velocity to be fixed over large scales in Kepler's law, [6] Moreover, gravitational lensing demonstrates the existence of a much greater Mass (density) than the sum of the stars within the galaxy.However, this alternate model specifically addresses these two issues and provides an explanation, Corollary 1. 4 Per theorem 1 and corollary 1.2, velocity ∆d ∆t n and density ρ n are measured with increased magnitude per distance ar n .This directly extends to energy density within galaxies and the effects on rotational velocity, such that: As R increases, centripetal force is perfectly balanced by increases in v ( ∆d ∆t n ) and, subsequently, ρ n , Note: total mass M inside the circle of the radius r can be obtained by doing an integration of mass density in a volume.M = ρ n d t Note:  • Along with time dilation γ, time contraction ∆t n is a distinct and necessary factor in deriving proper time • Ω = 1 (flat space) • The expanding universe is homogeneous, isotropic, and asymptotically flat.

Quantum Mechanics Time Problem
The logical contrapositive of theorem 1 is that the unit of time increases in microspace: Theorem 2 As scales approach Planck length, time intervals dilate (independent of their relative motion in SR) to a range, represented as an integral from −t n past to +t n future.Subsequently, corresponding values of position, energy, density, and charge become superimposed within this dilated range.
Figure 5 shows how both GR and QM are unified by this single basic premise.The entire range of scales, from 10 26 to 10 −35 , is illustrated with corresponding time units contracting in macrospace and expanding in microspace.Notice how a familiar projectile in classic-space appears normally orthogonal to the observer, yet a rotating body in macrospace appears skewed along the line of sight, as well as accelerated, as a result of decreased units of time.Also notice that a particle in macrospace appears to be in a wider range of positions, as a result of increased units of time (similar to a photograph with a delayed shutter).
Using p = ħk for momentum, the dominate wave function P si ⃗ k 0 includes wave vector ⃗ k 0 : So a particular orbit might appear as a torus.If the "range" is subatomic (< the orbit diameter) a projectile might appear as a partial torus.See figure 6 6 Vacuum Energy Density (ρ hep ) in QM Theorem 2 ("As scales approach Planck length, time intervals dilate to a range, represented as an integral from -t n past to +t n future.As well as corresponding values of position, energy, density, and charge become superimposed within this range".),and corollary 2.1 ("wavelength is inversely proportional to distance") provides a reasonable alternative to the unreasonable sum of vacuum energy (even within a restricted cutoff of photon energy being equal to Planck energy): Corollary 2. 3 As measured from the classic scale, the Casimir force (U) between plates a distance x micrometers apart represents a much greater range (n) of expanded time interval, along with associated values of position, energy, momentum, and charge.This range (n) increases as x decreases.
Where (q) is general positional coordinates.Thus, the assumed force measured in a unit of volume is instead a much greater integral over, both time (-t n past to +t n future) and position (-q n to q n ).Note that as λ decreases U increases (See figure 7, Theorem 1, is supported by the following correlation study: "On Possible Systematic Redshifts Across the Disks of Galaxies" [9].This study shows a deviation from Kepler's orbital laws, specifically on the subject of increased velocity on the far sides of multiple galaxies.Although not conclusive, it does justify the consideration of this article.Note that multiple galaxy surveys with increased velocities across their minor axis.Thus, velocity within the same body appears to increase per distance."Velocity observations in 25 galaxies have been examined for possible systematic redshifts across their disks: a possible origin for the redshifts could be the radiation fields.Velocities increase towards the far sides in most cases.This is so for the ionized gas, for neutral hydrogen, and in some cases for the stars.The effect is seen as velocity gradients along the minor axes, as well as in velocity fields of neutral hydrogen in other parts of the galaxies.Deviation of the kinematic major axis from the optical axis is found for 10 galaxies, and, in 9 of these, the largest velocities occur on the far side.In the central regions of four galaxies are found large velocity gradients in the same direction.While expanding motions provide an explanation for some of these features, it remains difficult to thereby explain all the peculiarities found.The faintness of the data available in this preliminary study should be noticed.Observations specially programmed for this subject would be necessary." Figure 8 shows 'table 1', on page 258 which lists 25 galaxies, correlation coefficients, and relevant columns (including sources of data):

Prediction as Supportive Evidence
One prediction of decreasing time intervals would be: Galaxies with a negative z value (approaching instead of receding, in our local group) would also correlate with distance, such that the furthest galaxies would appear to approach with the fastest velocity.

Conclusions
In order to define the fundamental conservation and symmetry of spacetime, within the broad scope of cosmology, it is necessary to consider some independent parameters representing constant energy.Once this conservation is established, a simple and parsimonious resolution to applications in General Relativity, Quantum Mechanics, and the Cosmological constant becomes both plausible and reasonable.

Figure 2 :
Figure 2: Hypothetical sla points as predicted in the Einstein-de Slitter model with uniform time intervals, compared with contracted time intervals

Figure 3 :
Figure 3: Fundamental conservation and coordinate symmetry of energy density, with respect to spacetime

Figure 4 :
Figure 4: Flat galaxy rotation curve explained with fundamental conservation

Figure 5 :
Figure 5: Gr and QM are unified

Figure 7 :
Figure 7: Casimir force energy U represents an integral (range) of both time and position

Table 1 :
Predicted Einstein-de Slitter model with uniform time intervals, compared with contracted time intervals.In successive columns: