Questions & Problems

Minkowskian Spacetime

1a.     All mass shares an origin with the Big Bang, yet the Universe expands causing objects to move away from each other.  For one second after the Big Bang, what is the formula for the length of the worldline?  The Minkowskian grid spaces events with “real” distances between intervals.  Where does the worldline point one second after the Big Bang event, i.e., does the worldline point in a particular direction?  Towards what quadrant of the heavens does it extend from the one second local event?  For static movement through time, is the Big Bang moving away from the local mass or is the local mass moving away from the BB?

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1b.     Does the passage of time alone increase the distance from the Big Bang?  Does the Big Bang become the center of the Universe?  For static movement through time, does the passage of time actually result in the expansion of space?  Is the worldline for such static mass really equal the radius of the Universe?  From where you are, point to the Big Bang.  Is the Big Bang today just an infinitely thin membrane moving away from here at the velocity of light?

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Hubble’s Constant

1c.     Edwin Hubble while working with his assistant Milton Humason determined that the Universe was expanding.  Today, the universal rate of the Universe’s expansion is referred to as Hubble’s Constant or simply H0.  One of the most recent estimates of (H0) taken from the surveys utilizing the Hubble telescope is 71 ±4 km per second per megaparsec.[1]  (A megaparsec is 3.085678 × 1022 meters.)  Therefore, the mean measurement of the velocity of separation between two points in space that are located one megaparsec apart is 71 km per sec.  At what distance of separation, measured in megaparsecs, do two points recede from each other at the velocity of light?  What is the calculation in meters?  Can two points recede from each other faster than c?  Would this be a violation of Special Relativity?

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1d.     If two points, residing at an initial position of one megaparsec apart, move toward each other at 71 kilometers per second, as in a time-reversed movie, how long in seconds will it take for them to come together?  A sidereal year (the year used in science) is 31,558,150 seconds.  How long in years?  Is there anything significant about this time?

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1e.     If two points that lay 5,000 megaparsecs from each other were to move together at 71 km/s/mpc, what would be their initial velocity of approach?  What would be their final velocity of approach, assuming no acceleration?  Would not all points in the Universe come together simultaneously?  Could this coming together be interpreted as the Big Bang?  Is this approach velocity greater or less than the velocity of light, i.e., <c or >c?  Would this be a violation of Special Relativity?  Given this answer, is there any room for inflation in the early Universe, i.e., an initial expansion at many times the velocity of light?

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[1] Hubble’s constant has been estimated using different techniques, but most estimates lie within the four  kilometer per second error window.

Kepler’s 3rd Law of Planetary Motion

2a.     Johannes Kepler 400 years ago established his three laws of planetary motion.  The third law states that the period of the planets’ orbits around the sun are proportional to the 3/2s power of their semi-major radii.  The radius of Earth’s orbit is one astronomical unit (AU).  What is the period of a satellite expressed in years that is orbiting the Sun with at a radius of 2AU?

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2b.     The earliest directly observable event in the Universe is when hydrogen ions cooled sufficiently to capture free electrons in the primordial plasma, resulting in neutral and transparent hydrogen.  Today we observe this event as cosmic background radiation (CBR).  Hydrogen has an ionization temperature of 1,100 K.  Assume that the expansion of the Universe since the CBR event caused the initial wavelengths of the released radiation to stretch to their current observable length.  Since the black-body mean temperature of the observed radiation from this event is 2.726 K, what is the ratio between the original temperature of the radiation when released and today’s temperature?

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2c.     If it is assumed that the wavelengths of the radiation stretched with the expansion of the Universe, what was the radius of the Universe at the time of the CBR event?

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2d.     If the age of the Universe changed according to Kepler’s third law, i.e., to the 3/2 power of change in radius, what was the age of the Universe when it first became transparent?  Is this in line with current estimates?

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The Velocity of Light

3a.     What is the velocity of the satellite orbiting at a radius of 2AU in question 2a relative to the orbiting velocity of Earth?  Stated another way, what is the relative orbital circumference of the satellite divided by the period for its orbit?

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3b.     If the radius of the Universe (R) is cT (See question 1b.), then what is the formula for the velocity of light?

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3c.     If the formulation for the velocity of light is R/T today, can it be assumed to have been the same at the time of the CBR event?  What would the velocity of light be at the time of the CBR event?  What proportion is this relative to today?

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3d.     Assume that the change in radius of the Universe since the CBR event to have been approximately 210, (Actually the expansion proportion would be 210.1045.) what was the proportionate change in the age of the Universe as a power of 2?  What is the formula for the change in the velocity of light relative to the proportionate change in the radius of the Universe?

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3e.     If the velocity of light was different in the past than it is today, would this be directly observable?  Why or why not?

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Radius and Mass

4a.     If the mass of the moon is doubled by the addition of another equal mass moon and the density of the matter is not increased, what would be the increase in the moon’s radius?

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4b.     The radius of a black hole’s event horizon is a spacetime barrier and not a surface of matter.  Assuming that Earth underwent infinite compression, what would be the radius of a black hole with Earth’s mass (5.9722 • 1024 kg)?

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4c.     What would the radius of a black hole’s horizon be if the mass of the Sun were to be compressed to the critical density?  (Assume Sun’s mass of 1.9890 • 1030 kg.)

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4d.     What is the ratio of the critical density necessary to create a black hole with the Earth’s mass relative to that necessary for the Sun?

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4e.     If a large volume of space has a certain mass density, what is the formula that determines the critical radius where that mass density would curve into a black hole?

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4f.      Based on this formula, what is the critical radius that would cause the mass and density calculated here to form a black hole?  Is there any mass density given a sufficient quantity of space that would not generate a Schwarzschild critical radius of curvature?  No matter how small the mass density, given sufficient volume of space with that mass density there is a critical radius that would provide a closed Schwarzschild curvature of space.

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Gravitational Curvature Constant

5a.     The following table shows the orbital radius and periods of the various planets.  What is the Earth’s radius cubed divided by the square of the Earth’s year?

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5b.     What are the radii cubed divided by the square of the orbital periods for each of the other planet’s?

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5c.     Assuming that the Universe’s radius is equal to the velocity of light times the age of the Universe, i.e., 2.99792458 × ms-1 • 4.32346 × 1017 s or 1.29614 × 1026 m, what is the cube of its radius divided by the square of its age?

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5d.     What was the cube of the Universe’s radius divided by the square of its age at the time of the CBR event?  [See question 2.]  Is this a constant for all ages of the Universe?

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5e.     What is the calculated mass of the Universe, i.e.,

times the gravitational constant G?  [Make sure to carry all units.]

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5f.      If we declare that the Kepler proportion for the Universe to be K, then it can be stated:

Has this equation held true throughout the history of the Universe?  Has K always been its current calculation?

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Big Bang Radius

6a.     Build a grid that lists masses from 109 to 10-15 kg.  In the column to the right show the corresponding Compton’s wavelength.  To the right of that show the Schwarzschild radius for the indicated mass.  Since the formulas for radius have inverse relationship relative to each other, determine by trial and error where the wavelength radius and Schwarzschild radius are the same.

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6b.     Using the mass from 6b, what is the corresponding unification radius for the Universe, i.e., at what radius does the Universe exhibit both complete gravitational curvature and an equal length quantum waveform?

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6c.     What is the most logical radius for the Universe at the Big Bang?  Why?

Reversing the Universe’s Expansion

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6d.     If the present radius of the Universe is 1.29614266 • 1026 meters and its age is 4.32346655 • 1017 seconds, what was the age of the Universe when the radius was ½ the present radius?  What was the velocity of light?

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6e.     What is the ratio between the Big Bang radius and the radius now?  What is the ratio of the age of the Universe to now?  What is the ratio of the velocity of light to now?

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6f.      Using the spreadsheet identify how many iterations must the Universe be cut in half to reduce the current radius to that of the Big Bang?

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Mass and G

7a.     If at the Big Bang c was its current measurement (~3 • 108 ms-1), G was its current calculated value, and if at that time the

Universe had the same mass as it does today, i.e., ~1053 kg, what would have been the mandatory radius of curvature?

If this curvature radius is the minimum that is possible under General Relativity, what would the radius of the Universe have been at the Big Bang, assuming that the mass of the Universe has been unchanged?  What would the radius of curvature have been if instead c was 1038 ms-1?  Under “the black hole curvature limitation” of General Relativity, could the mass of the Universe have been the same as today if the velocity of light was 1038 ms-1?

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7b.     What was K at the Big Bang?  What was GM at the Big Bang?

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7c.     Since GM has always been a constant, were G and the mass of the Universe (M) always what they are now?  Could G and M have changed in inverse proportion to one another thus maintaining the GM equality as the Universe aged?  Does it make sense that the entire present mass of the Universe (~1053 kilograms) and all of its stuff, i.e., its stars, dark matter, black holes and galaxies, could be compressed into a space with a radius of 10-35 meters?

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7d.     The radius of a black hole is directly proportional to its mass.  Thus, if black hole A has double the mass of black hole B, A will have double B’s radius.  If it is assumed that the Universe is a black hole, does it make sense that the Universe’s mass would have to double to double the corresponding radius of the Universal Black Hole?

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7e.     If the mass of the Universe were to double every time that the Universe’s radius doubles, then what would happen to the gravitational curvature factor G?

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7f.      Does it make sense that G, also known as the gravitational curvature factor, changes with the Universe’s curvature during its expansion?  If G is inversely proportional to the Universe’s mass and if the radius of the Universe is directly proportional to mass what was the gravitational curvature constant relative to today, assuming that the Universe has increased 8 • 1060 times since the Big Bang?

What would G have been at the Big Bang?

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The Universal Black Hole

8a.     If neutral-energy freefall velocity in a gravitational field is equal to

,

what is the escape velocity for that same gravitational system?

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8b.     So if

,

then what is the freefall velocity of a mass when r = RS. (Iignore the negative sign in the equation.)?  If this fraction is the proportion of the velocity of light, then what is that mass’ velocity when the radius of the freefall mass equals that of the black hole’s event horizon?  Under the current paradigm for black holes the event horizon does not present a barrier to movement.  Should a mass pass the horizon on its way to the central singularity, it would continue to accelerate.  What would the velocity of freefall be when the mass reaches ½ of the Schwarzschild radius?

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8c.     If the black hole were to have the radius of the Universe, i.e., 1.29614 • 1026 meters, what would the velocity of freefall be in meters per second when that mass reaches half way to the center, i.e., 6.4807 • 1025 m?  What would it be when it reached one Planck radius from the central singularity (1.61624281 • 10-35 m)?

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8d.     There are four components to the Schwarzschild metric:

is the rate of change in the time of the mass within curved spacetime, dt is the rate of change in distant time, dr is the change in radius, and is the change in the mass’ angle relative to the gravitational center.  The signs of component indicates the direction of movement, i.e., both local time and distant time carry a positive sign, thus times movement is positive.  The negative sign for two components indicating the directions of movement in space represent the inward motion of curved spacetime.  What happens to the curvature factor

when r equals RS?  This is the event horizon. What is the equation solution at r = RS?  Does the equation hold up at r = RS?  Do infinity solutions occur?  Is the horizon penetrable?

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8e.     What happens to the radius of a black hole if its mass increases?  Do black holes tend to expand or contract over time?  As time passes, does our Universe’s radius increase or decrease?  Is the mass of the Universal Black Hole increasing or decreasing as it ages?  What is the spacetime vector for an expanding black hole?

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8f.      Would the interior of the Universal Black Hole be the present or be in the past?

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8g.     What would be the surface (event horizon) velocity of this black hole be?  If that velocity is the velocity of light or 3 • 108 ms-1, what would that velocity be relative to today when the Universal Black Hole was ¼ its current radius according to the Schwarzschild metric?

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Gravity’s Acceleration

9a.     If the neutral-energy freefall velocity for a mass is vfreefall = RS/r, then the change in this velocity would be the acceleration.  If acceleration is the first derivative of velocity, what is the formula for the acceleration experienced by a mass in freefall?  [Calculate the derivative with respect to r.]

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9b.     If it is assumed that the Universe is a black hole and that all mass exists on the horizon of this black hole, what is the gravitational acceleration formula under General Relativity for the Universe?

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9c.     Assume when the Universe is viewed from its own event horizon, that this formula is reduced to its self-observed equivalent of RS/r2.  This formula is a geometric equivalent of acceleration.  To convert to conventional units multiply by c2.  Since r = RS at the event horizon, what is the gravitational acceleration for the Universe?

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Apparent Acceleration of the Universe’s Expansion

9d.     What is the vector direction of the Universe’s acceleration, inward or outward?  What is the implication for the rate of the Universe’s expansion?

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9e.     If the Universe’s rate of expansion is slowing and if the Universe expands at the velocity of light, then how does the velocity of light change as the Universe expands?

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9f.      What is the measuring stick for large distances?  How does this change as the Universe decelerates?  If the measuring stick changes in this way, does it make the Universe more accessible or less accessible over time?  Does a shortening measuring stick make the apparent distances between masses increase or decrease?

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9g.     Does a Universe whose expansion is slowed by gravity appear like a decelerating Universe or an accelerating Universe if the measuring stick continues to shorten?

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Universe’s Energy Content

10a.   It is assumed that mass changes to the first power with radius and that the velocity of light changes to the -½  power with radius, then how has the energy content of the Universe changed since the Big Bang?  Express this as a power of R.

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10b.   Is this in keeping with the law of the conservation of energy?  In gravitational systems “energy is a constant of motion” within curved spacetime.  Is this principal observed by the expanding Universal Black Hole?

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10c.   What is the Universe’s energy content?  Explain why this has been unchanged since the Big Bang given that the mass has increased by 60 orders of magnitude over the Universe’s history.

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Planck-Wheeler’s Energy

10d.      If the energy equivalent of Planck-Wheeler’s mass is

,

what is Planck-Wheeler’s energy today?  At the Big Bang?  At the time of the release of the cosmic background radiation (CBR event)?  Hint:  First calculate c at the CBR event.  See chapter 2.

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 Mass and Energy Extrema

11a.   Are Planck’s angular momentum, Planck-Wheeler’s radius and time extrema quantities, i.e., either the maximum or minimum quantity possible?  Are Planck-Wheeler’s mass and energy extrema quantities?

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11b.   Light’s energy is equal to Planck’s angular momentum (ħ) times light’s frequency (ω) or

.

Since frequency is equal to the velocity of light divided by light’s wavelength (λ), what is light’s energy formula expressed as a function of wavelength radius ?

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11c.   What is the longest wavelength possible for the Universe?  What is the energy of radiation with this wavelength?  Is this the extrema energy?  We will distinguish this energy quantity, for lack of a better term, as Planck-Otto energy.

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11d.   What is the Planck-Otto extrema equivalent for mass?

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11e.   What was the Planck-Otto energy at the time of the Big Bang?  What was the Planck-Otto mass at the Big Bang?  How do these compare to the Planck-Wheeler’s mass and energy at the Big Bang?  Compare these to the mass and energy of the Universe at the Big Bang?

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Planck’s Angular Momentum

12a.   Max Planck determined that energy was carried in discrete quanta, of which the lowest unit is termed Planck’s constant.  Planck’s constant is actually a unit of angular momentum.  So many units of angular momentum times a frequency constitutes energy content.  Planck’s constant is illustrated by h for the angular momentum of a complete sine wave or ħ for the radial equivalent.  ħ is equal to h/2π.   What is the energy content for radiation with a radial frequency of 5 × 1013 s-1?

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12b.   The units for angular momentum are kg m2 s-1.  If these units are replaced by the Planck-Wheeler equivalents for mass length and time, what would be the result, i.e.,?

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12c.   Planck-Wheeler mass and radius were the same at the Big Bang, but Planck-Wheeler’s time was different by 30 orders of magnitude.  What was Planck’s angular momentum at the Big Bang?

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Universe’s Angular Momentum

12d.   Louis de Broglie determined that electrons and subsequently all matter consists of waveforms.  Since the Universe’s quantum content is a summation of all of the individual quanta of its macro and quantum waveforms, assume that replacement of Planck’s quantities with universal quantities will determine the total angular momentum for the entire Universe.  Using the same technique to calculate Planck’s angular momentum in question 12b what is the total angular momentum (L) for the Universe?  Note:  Replace Planck-Wheeler quantities with universal quantities.

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12e.   Using the same technique, what was the total angular momentum for the Universe at the Big Bang?  How does this compare to Planck’s angular momentum at the Big Bang?

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Frequency in Matter

13a.   What is the formula for the energy of light?

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13b.   What is the formula for the rest energy of matter?  What is the formula for the energy of matter in motion relative to an observer?  What is the formula for Lorentz’s conversion factor γ?

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13c.   Matter’s energy is a compilation of the energy built into the containment forces of the strong,  weak and the electromagnetic forces, and the energy of movement.  We will refer to the containment energy as the potential energy of matter and the energy of movement as the kinetic energy.  What is the formula for the containment energy of matter?

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13d.  If momentum  (p ) is equal to, and momentum energy is , what is the sum of the squares of the contained energy (potential energy) and the kinetic energy?

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13e.  What is the rest frequency of matter?  What is the frequency of movement for matter?  If rest potential energy and momentum energy are perpendicular relationships, what is the formula for the combined frequency of movement and the internal potential energy of the mass?

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Compton’s Frequency

13f.    Compton’s wavelength is the wavelength of light that would result if the mass was converted completely into electromagnetic energy.  Its formula is:

.

The reduced Compton’s wavelength is the radius of the light wave or:

.

What is the frequency of the light for Compton’s wavelength?

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13g.   But matter maintains its integrity and doesn’t spontaneously convert to light.   Does matter at rest generate the waveforms implied by

?  How?

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2nd Law of Thermodynamics

14a.   Entropy is the number of microstates (local conditions) that in combination form the conditions of the macrostate of the overall system.  Is the temperature of the Universe a micro- or macrostate?  Is the local waveform a micro- or macrostate?

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14b.   As the Universe expands and the temperature of the cosmic background radiation declines, is this an increase or decrease in the Universe’s entropy?  Which form of energy exists in a higher form of entropy matter or electromagnetic energy?

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14c.   Does the increase in the wavelength of light traveling through an expanding space change that light’s entropy?  Does the release of each Planck-Otto quantum result in an increase or decrease of the Universe’s entropy?

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14d.   Sometimes entropy is described in terms of the dispersion of heat in a system, while at other times it is described as a statistical state of randomness.  The change in the Universe’s temperature is probably the best example of heat entropy.  Could the release of individual Planck-Otto quanta be a measure of statistical entropy?  Does this version of statistical entropy have advantages to describing the state variables of the system?  What are they?

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14e.   Can the Planck-Otto waveform, which has a radial wavelength equal to the radius of the Universe, cross through the wave’s meridian and form a new wave?  Does this mean that the Planck-Otto waveform can no longer experience increasing entropy?  Why?

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Big Bang Extrema

15a.   Compton determined that if all of a particle of matter were to be converted into electromagnetic energy its frequency would be:

What is the formula for Compton’s radial wavelength D?

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15b.   As matter’s mass increases, does the de Broglie wavelength increase of decrease?

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15c.   If the Schwartzschild radius of a black hole is determined by the formula:

,

does the Schwartzschild curvature increase or decrease as mass increases?

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15d.   Since one radius increases with mass and the other radius decreases with mass, is there a mass that has a Schwartzschild radius that is equal to the deBroglie wave radius?

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15e.   Using the two formulas for radius, can we determine this unique mass?  What is this mass?

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15f.    Is there a point when the mass of the Universe equaled this unique mass?  What’s the implication of having a wavelength for the entire Universe equal to the Schwartzschild radius and a de Broglie radius equal to the identical radius?  Does one push out while the other establishes the spacetime limits?

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