CORRECTION: An earlier version of this post improperly converted radians to arcseconds, and some table entries were errors, among other errors. [July 28, 2022]

Calculations below provide a demonstration on how using the Rayleigh criterion (R = λ/D) may be used to distinguish between two sources of radiation emitted from the surface of the Moon.  The two sources, which are emitted are: ● 0.11 mm apart ● viewed from a distance of 1.2 AU, and ● to be detected by a 16,000-meter detector.

According to the calculation, the radiation required for proper resolution would be gamma rays of wavelength 9.9 x 10^-6 micrometers (3.0 x 10¹⁹ Hz) , or 124 keV.

The average distance to the Moon from Earth is 385,000 km (385,000,000 m). However, detectors must be farther than b _Moon  =  9 x 10 ¹⁰ m (0.6 AU). 

The distance b _Moon  is described in previous posts as a hypothetical spherical boundary (b) between which negative and positive gravitational accelerations are in the neighborhood of 10^-10 m/sec² and depend on the gravitational object’s Schwarzschild radius (a) and the radius of the observable universe (c), such that b² = ac.

If detectors were placed 16,000 meters apart, located 1.8 x 10¹¹ m (2 · b _Moon = 1.2AU) from the gamma emission source on the Moon, and the image to resolve is 0.11 mm (1.1 x 10^-4), then:

Tan^-1 (1.1 x 10^-4 m/1.8 x 10¹¹ m) = 6.1 x 10^-16 radians = 1.3 x 10^-10 arcseconds

The Rayleigh criterion R (in arcseconds) = 0.21 λ/D, where λ is the wavelength in micrometres and D is the size of the telescope in metres, or the smallest distance between two gamma ray detectors.^[1]

D = 16,000 meters *

R = 1.3 x 10^-10 arcseconds

(1.3 x 10^-10 x 16,000)/0.21  = 9.9 x 10^-6 μm (gamma rays)

                                                  = 9.9 x 10^-6 μm = 3.0 x 10¹⁹ Hz  = 124 keV

*A 16,000 meter detector would be a very large. However, using a large number of smaller detectors spread across a 16,000-meter area and detecting a small fraction of the emissions may be sufficient to form a conclusion.

The resolution can be improved and the size D may be reduced, when used in conjunction with higher energy gamma rays, such as the decay of Technicium-99, which is used in nuclear medicine, has a half-life of six hours and produces 140 keV gamma rays, or Cobalt-60, which emits two gamma rays with energies of 1.17 and 1.33 MeV and has a half-life of 5.27 years.

Similar calculations can be made with respect to the Earth. Here are some preliminary Python scripts used to calculate the experimental parameters relative to the Moon and the Earth. The purpose of this experiment is to test orientability using gamma rays under extraordinary spatial constraints relative to a gravitational field.

Using a Box with Two Small Holes

For example, a box with two very small apertures containing Cobalt-60 emitting 1.1 – 1.3 MeV gamma rays is placed on the surface of the Moon (see image below). The center between the apertures lies on a plane perpendicular to the gravitational field.

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The entire speculation is, given high energy photons across large spatial distances, can orientability under the influence of a gravitational field be tested using the Rayleigh Criterion?

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 Parameters for Three Experimental Apparatus

Location	Distance between Apertures	Distance to Detectors	Required Energy of Gamma Rays
Moon	~ 0.11 mm	~ 1.2 AU	~ 124 keV
Mars	~ 0.9 mm	~ 4 AU	~ 54 keV
Earth	~ 9 mm	~ 13 AU	~ 18 keV

Click here to visit a site on gamma ray detection, NASA’s GLAST Burst Monitor. For a list of comparable scales, see this NASA planet distance chart.

 

 Mass and Schwarzschild Radii of Various Celestial Objects:*

Object	Mass	RSchwarzschild
Neutron Star	2.8 × 1030 kg	4.2 × 103 m	= 4.2 km
Sun	2.0 × 1030 kg	3.0 × 103 m	= 3 km
Jupiter	1.9 × 1027 kg	2.2 m	= 2.2 m
Earth	6.0 × 1024 kg	8.7 × 10-3 m	= 8.7 mm
Moon	7.3 × 1022 kg	1.1 × 10-4 m	= 0.11 mm !

* https://astronomy.swin.edu.au/cosmos/S/Schwarzschild+Radius

“When formal observations begin in July, DKIST, with its 13-foot mirror, will be the most powerful solar telescope in the world. Located on Haleakalā (the tallest summit on Maui), the telescope will be able to observe structures on the surface of the sun as small as 18.5 miles (30 kilometers). This resolution is over five times better than that of DKIST’s predecessor, the Richard B. Dunn Solar Telescope in New Mexico.”

— Neel V. Patel, MIT Technology Review

 

There are some equations I’m working on.^[1]  They are intended to describe a projection from the outside surface of a sphere to the inside surface of a concentric spherical shell. This paradigm corresponds to a hypothetical gravitational field that has a net zero gravitational force in a universe with a single black hole.^[2]

In two dimensions, the equations are scaled and rotated logarithmic spirals, which preserve the angles between its tangent and the intersecting circle — the polar slope angle is constant, which is consistent with flat planar projections. The equations in three dimensions will be published in a later post.

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There exists at least one r₁(φ₁) and a scaled inverse r₂(φ₁ + φ₂), where every product of r₁(φ) with r₂ (φ) is constant. See the proof at the bottom of the page. [Click the image to enlarge.]

See equations (1-5) below, which include:
  (1) a logarithmic spiral in polar form,  
  (2) the inverse of equation 1,  
  (3) equation 2 scaled and rotated, and  
  (4) domain, scaling and rotational factors.

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(1)  

(2)  

(3)  

(4)  

(5)  

The variable P is the subtended projection angle in units 1=π (or 1=1/2 turn), and the variable b is the radius of the neutral surface, where all projection lines meet.

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Below is an example drawn to represent the projection in three dimensions (not to scale) in a universe containing only a single black hole — a very small universe with a very large black hole!
[Click the image to enlarge.]

[1] For a more dynamic view of these equations, visit desmos.com.
[2] Here is where I am on all this speculation. Click the link to see a recently reported-measurable experiment inside (see page 18). This link has a graphic on page 24 that might be helpful.

Happy Pi Day, everyone!

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PROOF:

   

Fig 1:  dx=2r²/R  [proof]

Without hypothesizing on how such a classical state exists, a possible “confinement argument” is represented by this drawing that shows that for particles P, if they repel, work has to be performed across distance x to exist inside surface (S).

Resistance to compression is a quantum mechanical reaction that violates classical mechanics.

Because there are limits to the energy of “stationary” massive particles, as there are limits to the capacity for a particle to be stationary, defined by Plank’s Constant, I imagine a circumstance at the core of a collapsing neutron star, under such enormous compression at the core, that the capacity of a particle to remain confined at the very center is lost, and propelled outward, and unable to be replaced by another similar particle, for the very same reason, and as the remaining innermost shell of the core continues to be compressed by the increasing pressure of the collapse, a chain reaction ensues, and the star collapses simultaneously from both the outside inward and the inside outward.

How can I construct a plausible mathematical argument that would prove such a collapse takes place, if indeed it does?

Time delay in quantum coherence and tunneling?

Well, maybe in my dreams.  The counter-arguments to hollow black holes would include statements to the effect that the particles would not all necessarily be fundamental particles,  nor that are all of the same type; another counter-argument made might be that certain properties of the bodies to reach such a heightened state of entropy contradicts the possibility of such implicit order in its behavior (coherence).  Or that the observed diameter of existing “black holes” and neutron stars, and the periods of their orbiting celestial objects, rule out their “hollowness.”

It would be possible that the theoretical fundamental particles in question, ie. those that comprise the entirety of the hypothetical surface, may be a particle that changes state with a period. For example, changing from having mass to not having mass, and these flavor changes corresponding to the period, may also correlate its position and direction relative to the surface and any coherence, or tunneling, across the interior of the sphere (or it’s surface): whether on the inside and moving toward the outside, or vice-versa.

Whether the shell of a “hollow black hole” could be high enough in its density to equal in mass a solid sphere of a similar diameter, ie. to within a few orders of magnitude— to within the resolving power of observations required to discriminate black hole from neutron star diameters, and how these diameters correspond to the periods of surrounding orbiting bodies.

Random notes’ quotes:

Einstein: from the point of view in General Relativity that one finds it “appropriate to introduce a tensor T_uv,  ”^[1]  as “It is only the circumstance that we have no sufficient knowledge of the electromagnetic field of concentrated charges that compels us, provisionally, to leave undetermined in presenting the theory [GR], the true form of this tensor.”^[2] And simultaneously,  to “introduce a negative pressure for which there exists no physical justification.” ^[3] the “cosmological constant”.

Fig: The Rotational Inertia of a Point Mass in a Galaxy as Constant

The diagram above describes a pair of conditions that will manifest a constant rotational inertia about the center of a galaxy: 1) the total mass, and 2) the cross-sectional area of the galactic disk are proportional to the radius. A larger version of the diagram is located here.

Net Gravitational Energy = 0

In Standard Cosmology, the motion of matter—such as stars, and planets about stars, and objects about planets—are all moving about the centers of galaxies. These paths are presumed elliptical and are thoroughly examined in literature. The orbital motions of galaxies about other galaxies, however, are less understood; less observed, less measured.

I hypothesize that the orbital motions of galaxies about other galaxies include non-planar orbital periods that are very, very large. And if this chaotic dance is to exist over hundreds of billions of Earth years, this chaotic mixing of galaxies in the closed space of the Universe, albeit across the vast spaces would include paths that are not elliptical about a single stationary point, or foci. As an alternative, the paths may resemble more the motion of the turbulent mixing of oceans, such as the Atlantic Conveyor Belt Circulation here on Earth. [2]

I attempt to explain this turbulent mixing of galaxies as subject to a law that is distinct in its character.  Perhaps, with the simple modification that the total gravitational energy driving the turbulent mixing is gravitationally positive in composition, like that of Einstein’s Cosmological Constant; and opposed, by the total gravitational energy bound in collections of matter contained in the negative fields, as those described by classical gravitational theory.

The principle that makes this circumstance possible (in hypothesis) is asserted by an argument that is based purely on symmetry and conservation: that the total negative gravitational energy of these bound collections of matter, is equal to the total positive gravitational energy hypothesized to cause the expansion of the Universe, ie. the appearance, by Doppler Shift, as quantified by Hubble’s Law.

The Principle of Universe Expansion is not in dispute; the size and rate of expansion will be argued, however.

DEFINITION: Gravitational Equilibrium: Negative Energy (Orbits of Celestial Bodies) + Positive Energy (Expansion of the Universe) = Zero

This is the principle by which all study here is characterized.

Definitions:

“Internal motions” are those elliptical orbital motions about centers of matter, such as satellites about planets, planets about stars, and stars within the Galactic Bulge; these motions are driven by negative gravitational energy, are closed and bound.

“External motions” are those driven by a net gravitational energy that is positive in sign (and hypothetical), that are particular to the motions of objects outside a zero boundary. These motions not only include those of particles in the Galactic Disk but also the motions of distant Galaxies with respect to one another: closed and unbound.

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[1] With respect to galaxies, I postulate that the boundary between predominately internal motions and external ones is the boundary between the Galalctic Bulge and the Galactic Disk; and that every collection of matter has a similar boundary. This principle and its cause, the central focus of my study, is to be examined continually, and throughout, with increasing detail.

[2] The space of the Universe is postulated to be finite and closed; therefore the Turbulent Mixing of Galaxies across the vast spaces of the Universe are motions along paths that are closed. These paths are closed, but these paths are orbits changing in direction that is not always planar, have average radii similar in scale to the radius of the Universe, and traverse across vast timescales.

Fig. 1: R(t)=size of cosmological space, T(t)=clock-time of cosmological space, dR=change in size of cosmological space, dT=change in the rate of clock-time, dL=change in position of photon. The vector product, dR X dT = -dL

Trial and Error

Given three vectors: Cosmological Expansion, Speed of Light and Time, what field operations can one perform upon them? With respect to General Relativity, the Robertson-Walker Line Element, and the orbital motions of stars about galactic nuclei, I thought I would experiment with multiplication.

The expansion function, R(t) and the time function, T(t) are related here in that the cross product of their derivatives are constant and equal to the translation of light across a cosmological space under the influence of a distribution of gravitational energy that is being defined by the constructions in previous articles. The cross product has been chosen directly as an argument symmetrical to that of an electron moving in a magnetic field.
This article diagrams previously under-emphasized information, and a consequence of the “cosmological cross product”: that the photon moves across a space that contains no net gravitational energy, and so does no work on average. However, photons reaching the Earth from great distances have not average paths, but mostly histories of travel through”empty”, gravitationally positive—though increasingly weak—postive-gravitational space. In other words, excluding gravitationally lensed photons, observed photons from very deep space have not come under any significant gravitational influence of large concentrations of matter (galaxies), ie. about regions of galactic nuclei; though have traveled during a process of Universe expansion.

How the redshift of photons due to expansion, or due to time dilation escaping regions of negative gravitational energy (stars) relates to any “blueshift” that photons may obtain—caused by a path through a positive gravitational expansion—will hopefully be examined subsequently.

Is flat space a coincidence?

Is it possible that a Euclidean space is a requirement for the existence of observers? Is it an accident? And will all observers, past and present, always measure their observable space as Euclidean?

 

What I’ve been playing with is a problem outlined in the sketch “On a Dynamic Equilibrium for the Distribution of the Gravitational Force in Cosmological Space.”  The paper is a casual description of changing function values, and their derivatives, with respect to time. The essence of the paper is to fix the course of these function values to that of a negative gravitational force minimum, a null gravitational force, and a positive gravitational force maximum, over an average collection of bound galactic material in the cosmological space defined by the paper nicknamed Zero.

For no other reason than curiosity, the paper exploits a coincidence in the function under examination and constructs a representation in matrix form. It becomes evident that if the function values are to change as described in the paper—that the derivative of the gravitational force minimum increases as the derivative of the null gravitational force decreases—that the integral between these two points must also increase [image]. Yet it will be an assumption of the theory that this integral does not change, even as the function values and their derivatives do. The brief mention of the problem is at the bottom of page 4, highlighted in yellow.

 

If the integral would be made invariant, forced mathematically, with respect to time, by changing the function in no other way than a the change in the value of the exponent, the consequence would result in the relative flatness of one space-time to another. The interest in another paper, “Analysis and Computation”, will be in how this exponent, and the function values change continuously across time and how a theoretical observer in the present, measures the continually changing gravitational power of galactic nuclei across vast space-time distances.

Is it a coincidence that we exist in a locally flat space? Or does our space only appear locally flat—relatively flat to the past and the future?  Does this hold sway for all observers across all time scales?