CONTINUED FROM IMMEDIATELY-PREVIOUS BLOG ENTRY, #51
Part II.: Intimations of a faster-than-light Inter-Stellar Drive -- "Side-Stepping" the V = c Relativistic Momentum Equation Singularity -- An F.E.D. Hypothesis.
[¿A "Non-Celeritarian" Pathway from the "Tardyonic Meta-Phase", through the "Luxonic Meta-Phase", to the "Tachyonic Meta-Phase" of "Matter-Energy", or of '"Energy-Matter"', and back again?]
[Note: as usual in this thread, I use visible-light-spectrum color-ordinality to highlight categorial, or system, dialectical ordinalities -- "ROY G. BIV"].
Dear Reader,
. . .
A little further "hyper-numerical" pattern recognition may lead us to some further clues as to what the Musean hpernumber value --
. . .
A little further "hyper-numerical" pattern recognition may lead us to some further clues as to what the Musean hpernumber value --
1/(i + E)
-- might signify in a physical sense, once we place in mind the following physical information. Consider the following "tri-chotomy" of physical categories:
1. Bodies of mass-energy, including "particles" of mass-energy, that characteristically exhibit Velocity magnitudes V < c, are characterized as being composed of "Tardyonic" matter-energy, or of matter-energy in the "Tardyonic" state or 'meta-phase'. Matter-energy which has "non-zero" [e.g., "positive"] "rest mass" [as well as, possibly, matter-energy that has "negative rest-mass"] -- including all of the "ordinary matter" that we experience -- is constituent of this category of "Tardyonic" matter-energy.
2. Bodies of mass-energy, including "particles" of mass-energy, that characteristically exhibit Velocity magnitudes V = c, are characterized as being composed of "Luxonic", or "Light-like", matter-energy, or of matter-energy in a "Luxonic" state or 'meta-phase'. E.g., Matter-energy that has "zero" rest mass, such as the "photons" that are defined to be the <<monads>> of light-"wave" <<arithmoi>>, are constituent of this category of "Luxonic" matter-energy.
3. Bodies of mass-energy, including "particles" of mass-energy, that characteristically exhibit Velocity magnitudes V > c, iff such are ever observed, are to be characterized as being composed of "Tachyonic" matter-energy, or of matter-energy in a "Tachyonic" state or 'meta-phase'.
Matter-energy of this "Tachyonic" category has not, at least not so far, been encountered experientially / experimentally, to my knowledge.
There is a "Tachyon Theory" that predicts the existence of "particles" of "imaginary mass", but that "Tachyon Theory" asserts that such "particles" must travel "backwards" in time, and, therefore, is not the same as the theory described here.
Indeed, the whole idea that, because approaching the speed of light is empirically associated with a relative "slowing down" of clocks/time, that "reaching" the speed of light must mean the "stoppage" of time/change, and that "exceeding" the speed of light must mean a speeding up, again, of time/change, but in a "negative" sense of time/change reversal, bespeaks a mis-apprehension of the very nature of actual "time", and a <<mentalite'>> with profound and ancient -- Parmenidean -- dissonances, which still vitiate "modern" science and mathematics to this very day.
This "Parmenidean hang-over" meme "wants" us to view time as really being like space, only more so, such that experiential, empirical time, change, dynamics, etc., are all merely human, sensuous delusions, and such that true reality is "statical", "changeless" or "immutable", and "eternal".
Goedel's thought was deeply infected with this quasi-Parmenidean meme, leading him, for example, to interpret his "time-travel solution" to the Einstein Equations of General Relativity accordingly, viewing it as a kind of neo-Zenoan, neo-Parmenidean, <<reductio ad absurdum>> denial of the very existence of time, as well as to adopt a kind of "Mathematical Platonism" regarding mathematical [idea-]objects, regarding these objects, and the truths about them, as hailing from an "immutable"/"change-less", "transcendental", "immaterial", "eternal" and "eternally statical", domain, one akin to Plato's <<arithmoi eidetikoi>> "causal heaven".
The initial concept of "space-time", and the [x, y, z, ict] "four-vector" formalism for that '"time-space"', worked out by Hermann Minkowski, one of Einstein's teachers, for the Special Theory of Relativity, re-stimulated that meme in Einstein, and led Einstein to what Einstein himself designated as his greatest scientific blunder -- his rejection of the startling, but empirically correct, prediction, immanent in the tensor equations -- in the system of ten "simultaneous" nonlinear partial differential equations -- that express the system of Einstein's General Theory of Relativity, the prediction of a spatially self-expanding universe of time/energy-matter/space, which led Einstein, in a panic, to superimpose a static equilibrium upon those equations, by the value he chose for the "cosmological constant" that he then added to those equations.
He thus allowed "a mere observationalist", Hubble, to "score the coup" of announcing to the world the world-historic discovery of the self-expanding nature of universal space[-time], a "coup" which Einstein could have been the one to score, years earlier, simply by holding to the manifold integrity of his work, of his vast, General-Relativistic, discovery, by proudly and publicly announcing the prediction of the universal non-Parmenidean non-equilibrium that was immanent in his General-Relativistic equations.
1. Bodies of mass-energy, including "particles" of mass-energy, that characteristically exhibit Velocity magnitudes V < c, are characterized as being composed of "Tardyonic" matter-energy, or of matter-energy in the "Tardyonic" state or 'meta-phase'. Matter-energy which has "non-zero" [e.g., "positive"] "rest mass" [as well as, possibly, matter-energy that has "negative rest-mass"] -- including all of the "ordinary matter" that we experience -- is constituent of this category of "Tardyonic" matter-energy.
2. Bodies of mass-energy, including "particles" of mass-energy, that characteristically exhibit Velocity magnitudes V = c, are characterized as being composed of "Luxonic", or "Light-like", matter-energy, or of matter-energy in a "Luxonic" state or 'meta-phase'. E.g., Matter-energy that has "zero" rest mass, such as the "photons" that are defined to be the <<monads>> of light-"wave" <<arithmoi>>, are constituent of this category of "Luxonic" matter-energy.
3. Bodies of mass-energy, including "particles" of mass-energy, that characteristically exhibit Velocity magnitudes V > c, iff such are ever observed, are to be characterized as being composed of "Tachyonic" matter-energy, or of matter-energy in a "Tachyonic" state or 'meta-phase'.
Matter-energy of this "Tachyonic" category has not, at least not so far, been encountered experientially / experimentally, to my knowledge.
There is a "Tachyon Theory" that predicts the existence of "particles" of "imaginary mass", but that "Tachyon Theory" asserts that such "particles" must travel "backwards" in time, and, therefore, is not the same as the theory described here.
Indeed, the whole idea that, because approaching the speed of light is empirically associated with a relative "slowing down" of clocks/time, that "reaching" the speed of light must mean the "stoppage" of time/change, and that "exceeding" the speed of light must mean a speeding up, again, of time/change, but in a "negative" sense of time/change reversal, bespeaks a mis-apprehension of the very nature of actual "time", and a <<mentalite'>> with profound and ancient -- Parmenidean -- dissonances, which still vitiate "modern" science and mathematics to this very day.
This "Parmenidean hang-over" meme "wants" us to view time as really being like space, only more so, such that experiential, empirical time, change, dynamics, etc., are all merely human, sensuous delusions, and such that true reality is "statical", "changeless" or "immutable", and "eternal".
Goedel's thought was deeply infected with this quasi-Parmenidean meme, leading him, for example, to interpret his "time-travel solution" to the Einstein Equations of General Relativity accordingly, viewing it as a kind of neo-Zenoan, neo-Parmenidean, <<reductio ad absurdum>> denial of the very existence of time, as well as to adopt a kind of "Mathematical Platonism" regarding mathematical [idea-]objects, regarding these objects, and the truths about them, as hailing from an "immutable"/"change-less", "transcendental", "immaterial", "eternal" and "eternally statical", domain, one akin to Plato's <<arithmoi eidetikoi>> "causal heaven".
The initial concept of "space-time", and the [x, y, z, ict] "four-vector" formalism for that '"time-space"', worked out by Hermann Minkowski, one of Einstein's teachers, for the Special Theory of Relativity, re-stimulated that meme in Einstein, and led Einstein to what Einstein himself designated as his greatest scientific blunder -- his rejection of the startling, but empirically correct, prediction, immanent in the tensor equations -- in the system of ten "simultaneous" nonlinear partial differential equations -- that express the system of Einstein's General Theory of Relativity, the prediction of a spatially self-expanding universe of time/energy-matter/space, which led Einstein, in a panic, to superimpose a static equilibrium upon those equations, by the value he chose for the "cosmological constant" that he then added to those equations.
He thus allowed "a mere observationalist", Hubble, to "score the coup" of announcing to the world the world-historic discovery of the self-expanding nature of universal space[-time], a "coup" which Einstein could have been the one to score, years earlier, simply by holding to the manifold integrity of his work, of his vast, General-Relativistic, discovery, by proudly and publicly announcing the prediction of the universal non-Parmenidean non-equilibrium that was immanent in his General-Relativistic equations.
It is, of course, a deep irony of human, scientific history, and, perhaps, a manifestation of the "cunning of human history", that Einstein's "cosmological constant" is today being used to provide an approximate, empirical model of the option, regarding the self-expansion of the universe, that -- given the "Parmenidean over-hang" still vitiating modern, "mathematico-science" -- no "serious" physicist even considered, but yet the option that has now been discovered, empirically, to be actual.
The "serious" physicists considered a -- quasi-Parmenidean -- "steady state" universe, one that filled-in with new matter as space expanded, so as to maintain a constant density of matter ["Steady State" hypothesis].
The "serious" physicists considered an -- again, quasi-Parmenidean -- "net positive-curvature" universe, one that expanded, but ever-more slowly, until the expansion stopped, and then even reversed, into a universal contraction, eventually reproducing the "infinitely" dense and contracted state from which the previous expansion had begun, thus positing an "eternally cyclical" universe of expansion followed by contraction, followed by expansion followed by contraction again. Perhaps not Parmenidean enough for Parmenides, who favored absolute universal stasis, but '[quasi-]Parmenidean enough for these "modern" physicists.
The "serious" physicists even considered a -- still quasi-Parmenidean -- "net negative curvature" universe, one that expanded "eternally", but ever-more slowly, never enough to stop expanding, let alone enough to reverse expansion, into contraction, in all finite time, but one which slowed/decelerated its expansion "asymptotically" -- such that it would "reach" stoppage only "at" t "=" "oo", a time which never arrives, and, in the "mean"-"time", grow only ever more rarefied in its mass-energy density.
Wrong, wrong, and wrong again!
Parmenides, or even "quasi-Parmenides", has a dismal track record as a guide to science!
What the "serious" physicists never considered -- but what expanded empirical observation has recently shown to be the actual -- is a -- contra-Parmenidean -- universal space that self-expands at a self-accelerating rate.
In short, and ironically, Einstein's "cosmological constant" -- conceived by Einstein for a quite opposite -- for a quasi-Parmenidean -- purpose, once appropriately re-valued, has come to supply an early approximation to an empirically accurate model of the "dark" self-force of the universal -- and universally-self-accelerating -- self-expansion of our cosmos!
Empirical actuality seems to be "screaming to be heard", trying to "tell" "modern" scientists -- if only they were listening -- that that actuality is dynamical: far more dynamical -- and, indeed, far more "meta-dynamical"; even ever ontologically self-revolutionizing -- '''than was ever dreamt of in your philosophy, Horatio'''!
If "modern" scientists want a "philosophy", they had better start from Heraclitus!
All physical processes, regardless of their "meta-phase", occur in "forward time", per these F.E.D. hypotheses.
Now, at this point in their argument, F.E.D. recalls the so-called "phases" of ordinary matter, and their "phase transitions".
The phase of lowest "temperature" is the "solid phase".
Raising, sufficiently, the temperature of matter-energy that was in the "solid phase", to beyond its "next higher" temperature threshold, "transitions" that matter-energy, typically "raising" it from the "solid phase", into the "liquid phase".
Further raising the temperature, of matter-energy that was in the "liquid phase", to beyond its "next higher" temperature threshold, "transitions" that matter-energy, typically "raising" it further, from the "liquid phase", into the "gaseous phase".
Further raising the temperature, of matter-energy that was in the "gaseous phase", to beyond its "next higher" temperature threshold, "transitions" that matter-energy, typically "raising" it further, from the "gaseous phase", into the "plasma phase".
Next, they consider a possible [speculative] analogy, of the temperature "phases" of the ordinary kind of matter, to the velocity/momentum "states" of ordinary, and of extraordinary, kinds of matter-energy.
They re-describe the trichotomy described above as that of the "Tardyonic meta-phase", the "Luxonic meta-phase", and the "Tachyonic meta-phase" of matter-energy, and suggest that there may be exotic "field processes", analogous to "temperature rises", and "induce-able" by means of an appropriately-engineered "fields machine" [cf. Gary Hull], that would induce "meta-phase transitions" in bodies of matter-energy, moving them between any one of these "meta-phases" and its two "neighboring" -- one on either "side" -- "meta-phases", with the associated, meta-phase specific velocity characteristics ensuing.
Then they note, in the Einsteinian, special-relativistic momentum equation, that the V < c "Tardyonic meta-phase" is "qualified" by 1/E, i.e., by the Musean hypernumber E, that the V = c "Luxonic meta-phase" is "qualified" by the Musean hypernumber 1/(i+E), and that the V > c "Tachyonic meta-phase" is "qualified" by 1/i, i.e., by -i, leading them to re-characterize "Tardyonic meta-phase" matter-energy as that of "Epsilonic mass-energy", "Luxonic meta-phase" matter-energy as "Mixed meta-phase mass-energy", or "Hybrid meta-phase mass-energy", and "Tachyonic meta-phase" matter-energy as "Iotic mass-energy" [after "iota", the name for the Greek, ancestral version of the English letter "i"].
That is, "Tardyonic meta-phase", V < c matter-energy is "Epsilonic", "Tachyonic meta-phase", V > c matter-energy is "Iotic", and these two opposites "meet in the middle", or "mediate in the middle", and "mix", or "hybridize", so that "Luxonic meta-phase", V = c matter-energy is both "Epsilonic" and "Iotic" -- a "combination", or "hybrid", or "mixture", of the 1/E and of the 1/i "qualifiers", and of their associated physical qualities: 1/(i+E), or 1/(E+i) [the E and i hypernumber operators are additively commutative, even though they are multiplicatively anti-commutative].
¿But, where do they get off asserting that the V < c values of the Einsteinian, special-relativistic momentum equation, involve a Musean hypernumber 1/E "qualifier", and that the V > c values of that momentum equation involve a 1/i "qualifier"?
So far, we have only seen that the V = c values of that momentum equation involve a Musean hypernumber 1/(i+E) "qualifier".
Well, "looky here" --
1. For bodies of mass-energy residing/moving "in" the "Tardyonic meta-phase", V < c, the argument -- the difference -- inside the denominator-resident square-root function is always a positive number, "greater than" 0, or "to the right of" the origin-"point", and labeled by "0", on the "Real" number-line, because 1 [or, e.g., (c^2)/(c^2)], the "minuend" therein, is always greater than its "subtrahend" --
What the "serious" physicists never considered -- but what expanded empirical observation has recently shown to be the actual -- is a -- contra-Parmenidean -- universal space that self-expands at a self-accelerating rate.
In short, and ironically, Einstein's "cosmological constant" -- conceived by Einstein for a quite opposite -- for a quasi-Parmenidean -- purpose, once appropriately re-valued, has come to supply an early approximation to an empirically accurate model of the "dark" self-force of the universal -- and universally-self-accelerating -- self-expansion of our cosmos!
Empirical actuality seems to be "screaming to be heard", trying to "tell" "modern" scientists -- if only they were listening -- that that actuality is dynamical: far more dynamical -- and, indeed, far more "meta-dynamical"; even ever ontologically self-revolutionizing -- '''than was ever dreamt of in your philosophy, Horatio'''!
If "modern" scientists want a "philosophy", they had better start from Heraclitus!
All physical processes, regardless of their "meta-phase", occur in "forward time", per these F.E.D. hypotheses.
Now, at this point in their argument, F.E.D. recalls the so-called "phases" of ordinary matter, and their "phase transitions".
The phase of lowest "temperature" is the "solid phase".
Raising, sufficiently, the temperature of matter-energy that was in the "solid phase", to beyond its "next higher" temperature threshold, "transitions" that matter-energy, typically "raising" it from the "solid phase", into the "liquid phase".
Further raising the temperature, of matter-energy that was in the "liquid phase", to beyond its "next higher" temperature threshold, "transitions" that matter-energy, typically "raising" it further, from the "liquid phase", into the "gaseous phase".
Further raising the temperature, of matter-energy that was in the "gaseous phase", to beyond its "next higher" temperature threshold, "transitions" that matter-energy, typically "raising" it further, from the "gaseous phase", into the "plasma phase".
Next, they consider a possible [speculative] analogy, of the temperature "phases" of the ordinary kind of matter, to the velocity/momentum "states" of ordinary, and of extraordinary, kinds of matter-energy.
They re-describe the trichotomy described above as that of the "Tardyonic meta-phase", the "Luxonic meta-phase", and the "Tachyonic meta-phase" of matter-energy, and suggest that there may be exotic "field processes", analogous to "temperature rises", and "induce-able" by means of an appropriately-engineered "fields machine" [cf. Gary Hull], that would induce "meta-phase transitions" in bodies of matter-energy, moving them between any one of these "meta-phases" and its two "neighboring" -- one on either "side" -- "meta-phases", with the associated, meta-phase specific velocity characteristics ensuing.
Then they note, in the Einsteinian, special-relativistic momentum equation, that the V < c "Tardyonic meta-phase" is "qualified" by 1/E, i.e., by the Musean hypernumber E, that the V = c "Luxonic meta-phase" is "qualified" by the Musean hypernumber 1/(i+E), and that the V > c "Tachyonic meta-phase" is "qualified" by 1/i, i.e., by -i, leading them to re-characterize "Tardyonic meta-phase" matter-energy as that of "Epsilonic mass-energy", "Luxonic meta-phase" matter-energy as "Mixed meta-phase mass-energy", or "Hybrid meta-phase mass-energy", and "Tachyonic meta-phase" matter-energy as "Iotic mass-energy" [after "iota", the name for the Greek, ancestral version of the English letter "i"].
That is, "Tardyonic meta-phase", V < c matter-energy is "Epsilonic", "Tachyonic meta-phase", V > c matter-energy is "Iotic", and these two opposites "meet in the middle", or "mediate in the middle", and "mix", or "hybridize", so that "Luxonic meta-phase", V = c matter-energy is both "Epsilonic" and "Iotic" -- a "combination", or "hybrid", or "mixture", of the 1/E and of the 1/i "qualifiers", and of their associated physical qualities: 1/(i+E), or 1/(E+i) [the E and i hypernumber operators are additively commutative, even though they are multiplicatively anti-commutative].
¿But, where do they get off asserting that the V < c values of the Einsteinian, special-relativistic momentum equation, involve a Musean hypernumber 1/E "qualifier", and that the V > c values of that momentum equation involve a 1/i "qualifier"?
So far, we have only seen that the V = c values of that momentum equation involve a Musean hypernumber 1/(i+E) "qualifier".
Well, "looky here" --
1. For bodies of mass-energy residing/moving "in" the "Tardyonic meta-phase", V < c, the argument -- the difference -- inside the denominator-resident square-root function is always a positive number, "greater than" 0, or "to the right of" the origin-"point", and labeled by "0", on the "Real" number-line, because 1 [or, e.g., (c^2)/(c^2)], the "minuend" therein, is always greater than its "subtrahend" --
(V^2)/(c^2)
-- precisely because V < c.
Now, given that a "factor" of 1 can always be "factored out" of a positive "Real" number difference, such as --
Now, given that a "factor" of 1 can always be "factored out" of a positive "Real" number difference, such as --
( 1 - (V^2)/(c^2) )
-- or equivalently, such as --
((c^2)/(c^2) - (V^2)/(c^2) )
-- we can equivalently rewrite this Momentum equation, for V < c, as --
p(V) = MV/( ( (1)( 1 - (V^2)/(c^2) ) )^(1/2) )
-- and, taking the square-root of that (1) factor, ( (1)^(1/2) ), to be equal to E, we then obtain --
p(V) = MV/( E x ( ( 1 - (V^2)/(c^2) )^(1/2) ) )
-- and since, as we have noted above, the Mudea hypernumber value +E = E/1 = E, the E in the denominator can "rise up" to become, equivalently, E in the numerator --
p(V) = EMV/( 1 - (V^2)/(c^2) )^(1/2) ).
F.E.D. conclusion: "Tardyonic" matter-energy is "qualified" by the Musean hypernumber E in the numerator, or, equivalently, by that hypernumber E in the denominator, or by 1/E = E/1 overall, and the "Tardyonic meta-phase" of matter-energy is therefore also the "Epsilonic meta-phase" of matter energy, iff these quantifiable, "convolute", arithmetical "qualifiers" do turn out to have qualitative physical significance.
Now notice that we can go on in this way, repeatedly factoring 1, out of the "positive" --
p(V) = MV/( ( (1)( 1 - (V^2)/(c^2) ) )^(1/2) )
-- and, taking the square-root of that (1) factor, ( (1)^(1/2) ), to be equal to E, we then obtain --
p(V) = MV/( E x ( ( 1 - (V^2)/(c^2) )^(1/2) ) )
-- and since, as we have noted above, the Mudea hypernumber value +E = E/1 = E, the E in the denominator can "rise up" to become, equivalently, E in the numerator --
p(V) = EMV/( 1 - (V^2)/(c^2) )^(1/2) ).
F.E.D. conclusion: "Tardyonic" matter-energy is "qualified" by the Musean hypernumber E in the numerator, or, equivalently, by that hypernumber E in the denominator, or by 1/E = E/1 overall, and the "Tardyonic meta-phase" of matter-energy is therefore also the "Epsilonic meta-phase" of matter energy, iff these quantifiable, "convolute", arithmetical "qualifiers" do turn out to have qualitative physical significance.
Now notice that we can go on in this way, repeatedly factoring 1, out of the "positive" --
( 1 - (V^2)/(c^2) )
-- argument of the square-root function --
( 1 - (V^2)/(c^2) )^(1/2) ).
If we do so, we obtain the following "cycle" of alternating, "oscillating", "qualification" of p(V), ... by
( 1 - (V^2)/(c^2) )^(1/2) ).
If we do so, we obtain the following "cycle" of alternating, "oscillating", "qualification" of p(V), ... by
r = 1, then by E = ((1)^(1/2)), then by r again, then by E again, then by r again, . . ., a "cycle", or "oscillation", which, if these arithmetical "qualifiers" do turn out to have physical significance, oscillates with a "frequency" which is completely indeterminate from the rest of the equation, and which is completely independent of the t argument in P = p( V ) = p( v(t) ), thus suggesting an "oscillation of qualities" -- an oscillation back-and-forth, between two related qualities, that of the E "arithmetical qualifier", and that of the r "arithmetical qualifier", and one reminiscent of the de Broglie "matter-waves" -- one that is intrinsic to matter-energy of, or "in", the "Tardyonic meta-phase" --
p(V) =
p(V) =
EMV/( 1 - (V^2)/(c^2) )^(1/2) ) =
EMV/( ( (1)( 1 - (V^2)/(c^2) ) )^(1/2) ) =
EMV/( E x ( ( 1 - (V^2)/(c^2) )^(1/2) ) ) =
EMV/( E x ( ( 1 - (V^2)/(c^2) )^(1/2) ) ) =
EEMV/( 1 - (V^2)/(c^2) )^(1/2) ) =
1MV/( 1 - (V^2)/(c^2) )^(1/2) ) =
1MV/( 1 - (V^2)/(c^2) )^(1/2) ) =
MV/( ( (1)( 1 - (V^2)/(c^2) ) )^(1/2) ) =
MV/( E x ( ( 1 - (V^2)/(c^2) )^(1/2) ) ) =
EMV/( 1 - (V^2)/(c^2) )^(1/2) ) = . . ..
3. For bodies of mass-energy residing/moving "in" the "Tachyonic meta-phase", V > c, so that the argument -- the difference -- inside the denominator-resident square-root function is always a negative number, "less than" 0, or "to the left of" the origin-"point", labeled "0", on the "Real" number-line, because 1 [or, e.g., (c^2)/(c^2) ], the "minuend" therein, is always less than its "subtrahend" --
MV/( E x ( ( 1 - (V^2)/(c^2) )^(1/2) ) ) =
EMV/( 1 - (V^2)/(c^2) )^(1/2) ) = . . ..
3. For bodies of mass-energy residing/moving "in" the "Tachyonic meta-phase", V > c, so that the argument -- the difference -- inside the denominator-resident square-root function is always a negative number, "less than" 0, or "to the left of" the origin-"point", labeled "0", on the "Real" number-line, because 1 [or, e.g., (c^2)/(c^2) ], the "minuend" therein, is always less than its "subtrahend" --
( (c + delta-c)^2 )/(c^2)
-- a fraction which is always greater than 1, because (c + delta-c) is > c, i.e., precisely because --
(c + delta-c) = V > c.
Now, given that a "factor" of -1 can be "factored out" of a "negative" "Real" number difference, such as --
( 1 - ((c + delta-c)^2)/(c^2) )
-- or equivalently, such as --
( (c^2)/(c^2) - ((c + delta-c)^2)/(c^2) )
-- leaving a "positive" version of the formerly "negative" value of that difference, we can equivalently rewrite this Momentum equation, for V > c, as --
p((c + delta-c)) =
( (c^2)/(c^2) - ((c + delta-c)^2)/(c^2) )
-- leaving a "positive" version of the formerly "negative" value of that difference, we can equivalently rewrite this Momentum equation, for V > c, as --
p((c + delta-c)) =
(M x (c + delta-c))/( (-1)( -1 + ((c + delta-c)^2)/(c^2) )^(1/2) )
-- and, taking the square-root of that (-1) factor, ( (-1)^(1/2) ), to be equal to i, we then obtain --
p((c + delta-c)) =
-- and, taking the square-root of that (-1) factor, ( (-1)^(1/2) ), to be equal to i, we then obtain --
p((c + delta-c)) =
(M x (c + delta-c))/( i x ( ( -1 + ( (c + delta-c)^2 )/(c^2) )^(1/2) )
-- and since, as we have noted above, 1/i = -i, the +i in the denominator can "rise up" to become, equivalently, -i in the numerator --
p((c + delta-c)) =
-- and since, as we have noted above, 1/i = -i, the +i in the denominator can "rise up" to become, equivalently, -i in the numerator --
p((c + delta-c)) =
-iMV/( ( -1 + (V^2)/(c^2) )^(1/2) ).
F.E.D. conclusion: "Tachyonic" matter-energy is "qualified" by -i in the numerator, or by i in the denominator, or by +1/i overall, and the "Tachyonic meta-phase" of matter-energy is therefore also the "Iotic meta-phase" of matter energy, if these quantifiable, "convolute", arithmetical "qualifiers" have qualitative physical significance.
Note that there is, for this "meta-phase", no equivalent "oscillation" between the +1/i and the r qualifiers, because the "factoring out" of the -1 "factor" from the difference argument of the "denominated" square-root function leaves the remaining, other "factor" of that argument "negated" by the factor of -1, and thus turned into a "positive" value.
One could thus "factor out" a "factor" of ((+1)^(1/2)) = E from the remaining, "positive", argument-factor, thereby obtaining an indeterminate oscillation, back-and-forth between a "qualifier" of -i in the numerator, and a "qualifier" of Musean hypernumber -Ei = +iE in the numerator, for the "Tachyonic", or "Iotic", "meta-phase" of matter-energy, again, iff these quantifiable, "convolute", arithmetical "qualifiers" have qualitative physical significance.
Now, the core of F.E.D. 's hypothesis regarding this possibility for an "inter-stellar drive" hereby is their speculation that there may exist a "non-celeritarian", or "non-velocitarian", "field-state-based" pathway -- a pathway that is not by way of "celerity", by way of "acceleration", or by way of increasing velocity directly, "propulsively", for an initially "Tardyonic" body of matter-energy, to pass from --
F.E.D. conclusion: "Tachyonic" matter-energy is "qualified" by -i in the numerator, or by i in the denominator, or by +1/i overall, and the "Tachyonic meta-phase" of matter-energy is therefore also the "Iotic meta-phase" of matter energy, if these quantifiable, "convolute", arithmetical "qualifiers" have qualitative physical significance.
Note that there is, for this "meta-phase", no equivalent "oscillation" between the +1/i and the r qualifiers, because the "factoring out" of the -1 "factor" from the difference argument of the "denominated" square-root function leaves the remaining, other "factor" of that argument "negated" by the factor of -1, and thus turned into a "positive" value.
One could thus "factor out" a "factor" of ((+1)^(1/2)) = E from the remaining, "positive", argument-factor, thereby obtaining an indeterminate oscillation, back-and-forth between a "qualifier" of -i in the numerator, and a "qualifier" of Musean hypernumber -Ei = +iE in the numerator, for the "Tachyonic", or "Iotic", "meta-phase" of matter-energy, again, iff these quantifiable, "convolute", arithmetical "qualifiers" have qualitative physical significance.
Now, the core of F.E.D. 's hypothesis regarding this possibility for an "inter-stellar drive" hereby is their speculation that there may exist a "non-celeritarian", or "non-velocitarian", "field-state-based" pathway -- a pathway that is not by way of "celerity", by way of "acceleration", or by way of increasing velocity directly, "propulsively", for an initially "Tardyonic" body of matter-energy, to pass from --
[ V < c ]
-- through --
[ V = c ]
-- to --
[ V > c ].
That is, increasing velocity would not be the "cause" of a starship's transition from the "Tardyonic
meta-phase", through the "Luxonic meta-phase", [in]to the "Tachyonic metaphase".
On the contrary, the action of the starship's "engine", by manipulating "exotic force-fields", in changing the starship's own "field-state", would be the "cause" of these successive "meta-phase transitions".
The "quantum jump-like" increases in the velocity of the starship, from "sub-luminal velocities", through "luminal velocities", to "super-luminal velocities", would be the "effect" of that "cause".
That is, this "non-velocitarian pathway" would "translate" a body of "Tardyonic meta-phase" matter-energy -- say a starship, and its contents -- by way of a "field machine" [cf. Gary Hull] that would "heat it up", so to speak, passing beyond the critical point where the "meta-phase transition", from the "Tardyonic meta-phase", to the "Luxonic meta-phase", is breached, so that the spatial translation velocity of the body/starship would "automatically" transit, or "jump", to Vb > c, in the "vector direction" in which that body/starship was already traveling, as that "heat-up" threshold was crossed.
Next, that body's/starship's in-board "field machine" would further "heat up" that body/starship, passing beyond the critical point where the "meta-phase transition", from the "Luxonic meta-phase", to the "Tachyonic meta-phase", is breached, so that the spatial translation velocity of the body/starship would "automatically" transit, or "jump", to Vb > c, or Vs > c, in the "vector direction" in which that body/starship was already traveling, as that "heat-up" threshold was crossed.
The starship would "rest" in the "Tachyonic meta-phase", in, e.g., transiting between two mutually "distant" stellar/planetary systems, for the main duration of its voyage.
When the starship achieved sufficient proximity to its destination stellar/planetary system, it would begin to "down-shift", reversing its previous "meta-phases progression", as it was "cooled down", so to speak, by its in-board "field machine", first transitioning from the "Tachyonic meta-phase" back to the "Luxonic meta-phase", and then from the "Luxonic meta-phase" back to the "Tardyonic meta-phase", which is the "meta-phase" that it would "occupy" upon landing, e.g., on the surface of one of the planets in its destination stellar/planetary system.
The "fields engine" of such an "inter-stellar drive", based upon the foregoing hypothetical principles -- iff those principles hypotheses turn out to be physically significant, and empirically validated -- would be a "exotic fields generation machine", capable, ultimately, of generating a "tunable celerity", or a "tunable velocitation", in the "Tachyonic meta-phase", or "Iotic meta-phase", i.e., one "tunable" though a whole "continuum" of "superluminal" velocity "factors", s in R+, such that s > 1, and --
Vs = sc > c
On the contrary, the action of the starship's "engine", by manipulating "exotic force-fields", in changing the starship's own "field-state", would be the "cause" of these successive "meta-phase transitions".
The "quantum jump-like" increases in the velocity of the starship, from "sub-luminal velocities", through "luminal velocities", to "super-luminal velocities", would be the "effect" of that "cause".
That is, this "non-velocitarian pathway" would "translate" a body of "Tardyonic meta-phase" matter-energy -- say a starship, and its contents -- by way of a "field machine" [cf. Gary Hull] that would "heat it up", so to speak, passing beyond the critical point where the "meta-phase transition", from the "Tardyonic meta-phase", to the "Luxonic meta-phase", is breached, so that the spatial translation velocity of the body/starship would "automatically" transit, or "jump", to Vb > c, in the "vector direction" in which that body/starship was already traveling, as that "heat-up" threshold was crossed.
Next, that body's/starship's in-board "field machine" would further "heat up" that body/starship, passing beyond the critical point where the "meta-phase transition", from the "Luxonic meta-phase", to the "Tachyonic meta-phase", is breached, so that the spatial translation velocity of the body/starship would "automatically" transit, or "jump", to Vb > c, or Vs > c, in the "vector direction" in which that body/starship was already traveling, as that "heat-up" threshold was crossed.
The starship would "rest" in the "Tachyonic meta-phase", in, e.g., transiting between two mutually "distant" stellar/planetary systems, for the main duration of its voyage.
When the starship achieved sufficient proximity to its destination stellar/planetary system, it would begin to "down-shift", reversing its previous "meta-phases progression", as it was "cooled down", so to speak, by its in-board "field machine", first transitioning from the "Tachyonic meta-phase" back to the "Luxonic meta-phase", and then from the "Luxonic meta-phase" back to the "Tardyonic meta-phase", which is the "meta-phase" that it would "occupy" upon landing, e.g., on the surface of one of the planets in its destination stellar/planetary system.
The "fields engine" of such an "inter-stellar drive", based upon the foregoing hypothetical principles -- iff those principles hypotheses turn out to be physically significant, and empirically validated -- would be a "exotic fields generation machine", capable, ultimately, of generating a "tunable celerity", or a "tunable velocitation", in the "Tachyonic meta-phase", or "Iotic meta-phase", i.e., one "tunable" though a whole "continuum" of "superluminal" velocity "factors", s in R+, such that s > 1, and --
Vs = sc > c
-- where --
c + delta-c = c + (s - 1)c = sc
-- so that --
delta-c = (s - 1)c.
One must quickly inquire, at this point, as to whether or not this "fields engine" -- the "field-generation machine" itself, that would supposedly induce these "meta-phase transitions" -- could survive its own "transitions" -- let alone the starship that houses it as a whole; let alone the human passengers, and the cargo, of that starship.
There would be many questions to ask -- and many physical experiments to be performed -- in the pursuit of the question as to whether or not such an inter-stellar drive could become "real" -- and practical.
We will leave off of those considerations for now, for the purposes of this blog entry, and, in the next blog entry of this sub-series, turn to validating the "hypernumber computations" upon which the F.E.D. inter-stellar drive hypotheses are based.
We will then, finally, consider whether or not the results of this "hypernumber [re-]qualification" of the Einsteinian special-relativistic momentum equation, are in agreement with the results of F.E.D.'s canonical "dialectical, metrical meta-number re-qualification" of singularity-prone, "purely-quantitative" [and usually NONlinear] differential equations, using the "metrical qualifiers", or "dimensional qualifiers" -- i.e., the "metrical units/<<monads>>", or the "dimensional units/<<monads>> -- of F.E.D.'s "Mu" --
One must quickly inquire, at this point, as to whether or not this "fields engine" -- the "field-generation machine" itself, that would supposedly induce these "meta-phase transitions" -- could survive its own "transitions" -- let alone the starship that houses it as a whole; let alone the human passengers, and the cargo, of that starship.
There would be many questions to ask -- and many physical experiments to be performed -- in the pursuit of the question as to whether or not such an inter-stellar drive could become "real" -- and practical.
We will leave off of those considerations for now, for the purposes of this blog entry, and, in the next blog entry of this sub-series, turn to validating the "hypernumber computations" upon which the F.E.D. inter-stellar drive hypotheses are based.
We will then, finally, consider whether or not the results of this "hypernumber [re-]qualification" of the Einsteinian special-relativistic momentum equation, are in agreement with the results of F.E.D.'s canonical "dialectical, metrical meta-number re-qualification" of singularity-prone, "purely-quantitative" [and usually NONlinear] differential equations, using the "metrical qualifiers", or "dimensional qualifiers" -- i.e., the "metrical units/<<monads>>", or the "dimensional units/<<monads>> -- of F.E.D.'s "Mu" --
Rq/MQN = Rq/MU
-- systems of dialectical arithmetics.
For more
information regarding these
Seldonian insights, please see --
For partially pictographical, ‘poster-ized’ visualizations of many of these Seldonian insights -- specimens of ‘dialectical art’ – as well as illustrated
books by the F.E.D. Press, see --
https://www.etsy.com/shop/DialecticsMATH
¡ENJOY!
Regards,
Miguel Detonacciones,
Voting Member, Foundation Encyclopedia Dialectica [F.E.D.];
Elected Member, F.E.D. General Council;
Participant, F.E.D. Special Council for Public Liaison;
Officer, F.E.D. Office of Public Liaison.
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