This is not an introduction to relativity and I am not a scientist or a mathematician.
The discussion in this post is wild, wooly, connect-the-dots, and not well-organized. My recentest discussion involving the Four Causes, "Telos, entelechy, Aristotle's Four Causes, pleasure, & happiness" is better organized (but does not discuss special relativity or Thomistic beauty). Other discussions, ones that I ought for various reasons to revise, include:
"Compare to Aristotle, Aquinas, & Peirce", especially the passage beginning here;
"Mining Aristotle's Four Causes for order & balance"; and "Special relativity's light cone & the mind's temporal perspectives".
Now, I believe that social, political, or poetic forces are not major factors in determining theoretical physics. I hope readers will not take my mixing of subject matters in a Social Text way. It would be better to take it more in a Robin Williams way; imagine the comic using his capacity for swift switches of persona in order to point to similiarities between structured sets of ideas. Yet, when the similarities go rather farther than one might expect, they become something not merely playful. Why don’t I just read Joyce to get my fill of such serious play? Good point, except that my purpose is not primarily aesthetic.
Aristotle (Physics Bk. 2, Ch. 3 and Metaphysics Bk. 5, Part 2) said that there are four causes or reasons of change. He based the four on three principles.
Principle ~ ~ Cause
agent ~ ~ ~ ~ efficient cause, agent cause, an initiating cause
patient ~ ~ ~ matter, material, material cause, that out of which it is made and in which it stands
act ~ ~ ~ ~ ~ end, function, purpose, final cause, that for which it is made
act ~ ~ ~ ~ ~ form, formal cause, the shape or pattern in which it is made.
Aquinas distinguished act as action from act as form. Action isn’t really a different principle from act.
The agent drives or makes, makes something happen. The patient suffers, undergoes, supports, puts up with something’s happening. Generalizing:
Makes ice melt = Doesn’t let ice not melt.
Lets ice melt = Doesn’t make ice not melt.
(There’s a difference between positively helping and merely not obstructing, of course, but for simplicity’s sake we can ignore it here.) This is like necessity and possibility not only with regard to their intertransformability through judiciously inserted negators but also in a connection of meaning:
the ice is allowed to melt : the ice can melt.
Not that the agens-patiens formal transformation actually worked so very well. Thank goodness there’s the terra firma of 20th-Century mechanics for help at cross-idea puzzles. I arrived at the idea that the actum should be the action/energy and the passum the form/structure. But why shouldn’t the passum be the action/energy (like a “passion”), and the actum simply the form/structure? If we so ordained it, then senses of the words would tend to adapt themselves as long as the discussion remained in contact with real issues. There are four principles there, and they could be named in multiple ways, but for dearth of suitable words. Yet, if one has to pick one particular way, I prefer the agent-bearer-act-borne way shown below; it seems the way most accordant with long-time common notions with which one needs to work, and encounters with other ways would be occasions for making appropriate distinctions.
Substitute the Latin verb sustineo in appropriate forms. Sustinens, sustentum. Let sustinens equal patiens. I believe that a needed way to conceive the sustent (= sustentum) is as the borneness of forces or movements, agentia, their balance or stability, their structure, whether kinetic, static, or each in different regards (a structure, an anatomy, etc., also set of internal motions, a circulation, etc.) Such conception falls compellingly into systematic place with closely related conceptions correlated to the other three principles, as shown in the chart here. (If somebody thought to say that patiens needs no passive-voiced form because it already is the passive itself as such and is the passive’s core in concept and etymology alike, then the answer is that its passive-voiced form captures a kind of conception of agency — internally borne and balanced agency or the balance of internal agencies.)
As agens and sustentum are specially akin, so are sustinens and actum as mass and energy. In special relativity a system’s heat, its internal energy, is quantitatively part of the rest mass of the system; and the entire rest mass is rest energy, is, in a sense, the real internal energy from the relativistic viewpoint. Every philosopher interested in the conception of matter should keep those things in mind. (With special relativity, energy becomes no more an arbitrary idioscopic quantity than is momentum or mass.) Matter involves internal energy, some of it getting used in internal useful work. Matter is process, much of it congealed, but not all. Matter is resource, right down through its microstructure. Matter, the earth, as a richness is an ancient common notion pre-echoing modern physics. This may or may not seem interesting, but it’s a little less uninteresting when you draw it out. The steady rhythms of normal work. “Housekeeping,”. Internal work, the molecules in their collective stochastic process, the matter as incubative and generative rather than spent — it is fully spent when it is fully converted into kinetic energy, not when it drifts around, cold and barren of biological activity. The fundamental mechanics of matter is molecular dynamics, a statistical dynamics of stochastic processes — steady, with closeness-proportionate dependence on intermediate-stage conditions. As sustinens represents such conceptions as those of process, steadiness, and homeostasis, so its passive-voice counterpart sustentum represents conceptions of structure (kinetic and/or static) and structural integrity. Thomists will (I hope!) quickly note that the four terms align with Aquinas’s three requirements for beauty plus an enriched version of a requirement dating back to Aristotle.
|distance & displacement, travel, speed & velocity, momentum, impulse, force||agens||due directive magnitude (magnitude distinguishes things beautiful (pulchra) from pretty (formosa)|
|proper time, proper “aging,” proper “aging” rate, rest mass, internal work & power (“proper” = in object’s own reference frame)||sustinens||harmony, due proportion, due rhythm|
|time dilation (amount & rate), energy, work, power||actum||radiance, claritas, vibrance|
|distance-with-direction untraveled (due to sublightspeed), slowness, “tied-up” momentum, internal impulses & forces||sustentum||wholeness, integritas|
Though all four requisites are here correlated to conceptions in mechanics, the jewel of it is that one finds ALL FOUR consistently correlated to conceptions in mechanics. The underlying unity of conception is the jewel; one does not need to hold beauty to be a creature of mechanical conceptions in order to use them for their clarity and coherence. And, of course, one does not need to hold that we need to agree about which things are beautiful; one needs to hold only that, if a thing be beautiful in appearance, it can be discussed in esthetic terms of its force or size, its processes, its culminations, and its structures, discussed in regard to the ways in which it rewards one’s impressionability by overall movement and variation, one’s capacity to estimate process, development, and apportionment, one’s capacity to discern and savor blooms and culminations, and one’s capacity to recognize form and structure. I will go so far as to offer a variation of James Joyce’s authorial stand-in Stephen Dedalus’s characterizations (in A Portrait of the Artist as a Young Man) of the stages of the apprehension of the beautiful, which were arrest, fascination, enchantment. In particular, the above-shown conception of integritas sive perfectio as structural integrity is truer to the Thomistic conception than is Joyce’ conception of it as being simple unity as bounded against the outside, and the conception of debita proportio sive consonantia as a keeping, a kind of processual homeostasis or constancy, not only suits the Thomistic and classical emphasis on rhythm in regard to proportion, but also helps distinguish integritas sive perfectio and debita proportio sive consonantia with welcome clarity.
Nam ad pulchritudinem tria requiruntur. Primo quidem, integritas sive perfectio, quae enim diminuta sunt, hoc ipso turpia sunt. Et debita proportio sive consonantia. Et iterum claritas, unde quae habent colorem nitidum, pulchra esse dicuntur. — Summae Theologiae, Prima Pars, Quaestio 39, Articulus 8
enim means "indeed" or "that is to say".
diminuta means "dashed to pieces" or "destructively violated" and should not be confused with the milder "deminuta."
turpia means "ugly," "base," "revolting," etc.
claritas means "clarity" but mainly in the sense of vividness, brightness, distinctness, and even loudness and fame.
nitidum means "bright," "sleek," "shiny," "glistening," "blooming."
For in fact for beauty three things are required. First certainly, integrity or perfection, indeed things which have been dashed to pieces [or destructively violated], by this very fact are ugly [base, disgusting, “gross”]. And due proportion or consonance. And again clarity [or brightness], whence things which have bright [or glistening or blooming] color are said to be beautiful.— Summa Theologica, First Part, Question 39, Article 8. (tr. mine).
|Joyce’s Dedalus’s account.|
1. Arrest by a form in its wholeness and unity as apprehended by demarcation by a bounding line drawn separating the image from world. You see a thing.
2. Fascination in relations, part balanced against part, rhythms, in their harmony, of the image as apprehended by analysis. You see a thing.
3. Enchantment (“...of the heart”) with the whatness (“quiddity”), the essence or essential quality, in its radiance, of/in the image, as apprehended by a necessary synthesis. You see what thing.
|Current tetrastic account.|
1. Arrest in a presentative onset (of thing or apprehension of it), in its due directive magnitude, standing out located, oriented, and orientative. As if to say, “So be it?”
Echoes: extremization, graph theory, optimization, inverse optimizational problems & sensitivity, forces.
2. Fascination in an elaborative process and proportionings, (of thing or apprehension of it) in its harmony, due proportion, due rhythm. As if to say, “So be it.”
Echoes: measure, enumeration, probability, statistics, matter.
3. Takenness or enrapturement in a revelative culmination (of thing or apprehension) in its radiance and vibrancy. As if to say, “Is it?”
Echoes: deriving a curve, calculation, information, cybernetics & decoding, life.
4. Attachment, devotion, to/in a structuring, an establishmental solidification (of thing or apprehension of it) in its wholeness and integrity. As if to say, “It is.”
Echoes: limits, ordering, logic, philosophy, intelligent life.
The agent initiates decisively, the matter is a kind of means, the act is the end, and the structure is the evidence, the embodiment of a recognition, and perhaps the fulfillment of associated expectations as to signs that would confirm the result. This semiotic and evidential borneness, balance or stability of internal forces, this supportedness, or check, goes beyond the end, though not in the sense of leaving the end behind somehow. Íchnê or entelecheíai (pace Aristotle) go beyond the teleological just as people go beyond vegetables, philosophy goes beyond cybernetics, logic goes beyond information theory, and structures of order and of applicability of mathematical induction go beyond algebra (= calculation theory) Forces, matter, life, intelligent life. Beginnings, means or middles, ends, checks. Archaí, mésa, télê, and íchnê or élenchoi in a certain sense. Four whole modes of causation or conditions-dependence — (1) sensitive and chaotic; (2) statistical and stochastic; (3) cybernetic and corrective toward ends; and (4) intelligent, logical, evidential, semiotic and observational, evolving and constructive on the basis of checks, checkings. (1) Inverse optimization, (2) statistical theory, (3) ampliatively inductive information theory, and (4) philosophy. My placement of inverse optimization (as a field of theoretically oriented research) as appropriate to the study of sensitive and chaotic processes is something of a research-classificatory prediction. And lately I have heard that a complex systems researcher Crutchfield wrote (2002) “every state change in a complex system can have a different rule attached”; if it turns out that those rules can be optimizational, then my prediction inches closer to fulfillment. This ties in with my post “Logical quantities, categories of research, and categories.”
* * *
Structure. Supports, checks, and balances. Can the movements and the impulses which would be imparted in the removal of restraints, and which are balanced and vector-summed to zero within the system, be expressed also as a positive quantity? If the system’s linear momentum and the linear energy associated with it can be expressed, both of them, as positive quantities, then why not internal movement (restrained or unrestrained) just like the rest mass (rest energy) associated with it? And, after all, though they contribute zero to the system’s net momentum, still the internal movements and restraints, right down through the most elementary structures, are still there and are structural elements of the system. I believe that the answer is yes. It is a magnitude with direction but it is not Lorenz-covariant (i.e., not part of the relevant 4-vector, the 4-momentum) and is not vector-additive, which shouldn’t be surprises, for, otherwise, physicists would be using it. Kinetic energy is a directionless quantity but isn’t invariant, scalar-additive, or part of the 4-momentum -- but, on the other hand, kinetic energy doesn't bear the burden of being a directional quantity lacking vector-additivity. Still it’s good to have the hook of a positive quantity on which to place the conception of momentum “tied up” in a system; the relevant formulas relating the quantity to others bear out mathematically the regularity of their conceptual relations, determine which kinematic and other related quantities are akin to it, and also point out that this “internal” quantity, unlike rest mass, varies dependently on the observer’s reference frame. Also, the “progression” of kinetic quantities in their mathematical properties from theory-friendly to theory-unfriendly may, for whatever it’s worth philosophically at least, have analogies elsewhere. One can find an echo or two of such a thing elsewhere, but it would be useful to have a logical understanding of such things.
Consider all of a system’s momenta that are associated with kinetic energies and potential energies in the system, all the momenta that would be there in kinetic form if all restraints were dissolved leaving a cloud of massless particles. In summing them, much of the quantity will be cancelled out by directional oppositions which balance momenta, and (assuming that the system is not under external restraint or in an accelerational field) the remainder will be net momentum attributed to the system as a whole due to its motion in the frame of an observer at rest — “external” momentum. Those “canceled-out” momenta, kinetic and potential, are still there in a sense; the movements potential and actual which they quantify are still there. What if one wishes to sum all those momenta and the impulses which would be imparted in the complete removal of restraints, right down to the structure of the fundamental particles? Sounds very hard to do. Now, by collisions with suitable antiparticles, all of the particles and their bonds can be converted to kinetic energy such that, as expressed in relativistic light-units (where lightspeed = 1 so that we can streamline and drop the c ’ s as is sometimes done), for each particle (now at lightspeed) the kinetic energy e = the total energy E = the momentum p. (Putting the c's back in, it's
Note: The signal speed limit is not arbitrary in conception and leads to the conceptual unification of space and time. With it, energy and mass are time quantities just as momentum is a distance quantity. The signal-speed limit’s presence is not arbitrary and its quantitative value is arbitrary only in terms of conventional idioscopic measurement units and in terms of its particular (i.e., perhaps changeable) idioscopic proportion to other physically fundamental quantities.
p, momentum = E−(E−p) ~ ~ ~ e, kinetic energy = E−m
~ ~ ~ ~ total energy, E = m+e = p+(E−p) =
m, rest mass = E−e ~ ~ ~ ~ ~ ~ “tied-up” momentum = E−p
So, the total energy can be seen as comprised of two modes of momentum just as it is comprised of mass and energy. It is a further reflection of the unity of space and time and helps let one see that all these quantities, momentum, energy, mass, are really the same thing variously structured in terms of reference frames and questions of direction of motion. The bad news is, again, that, though E−p has both magnitude and direction, it is not Lorentz-covariant (it’s not part of the relevant 4-vector, the 4-momentum) and it is not vector-additive. E−p has not been the toast of special relativity, much less general relativity. It doesn’t even have a letter of its own, much less a name. I’m not sure what kind of physicist would find the quantity mention-worthy, let alone useful.
Nevertheless, there it is, hidden away.
1. Momentum is a kinetics version of the quantity force which was conceived in the effort to quantify the amount of efficient causativeness, the agens cause.
2. Mass, and its current consensus heir rest mass (increasingly called simply “mass”), was conceived in order to quantify the amount of matter, the patiens cause.
3. Energy was conceived in order to quantify the amount of work done or doable, the actum.
4. E−p really is as if it had been conceived in order to quantify the amount of formal cause — if the formal cause had been decidedly conceived of as structure. I arrived at
In order to think of form other than as lengths in arrangement or non-metric generalizations thereof, instead think of it as movements, forces, in balance. To think of it as an alternative to unbalanced movements and forces, think of it in terms of the displacement (distance with direction) that is untraveled because the potential travel is tied up in the structure. This conception of form as “static” structure relates the form in terms of potential kinetics and potential dynamics with that which is in, around, or even distant from the form. The conception itself need not be static, since structures can be quite complex, and can change and evolve, yet still have staticity in significant respects within significant time frames. The conception has its esthetic side, as a structure can consist in apparent tension and apparent potential spring, and apparent weight and apparent potential fall, thus conjuring up an imaginative play of movement and action in a meaningful dream-halo or dream-corona around the capped tensions and pressures.
Associated sets of quantities, all units expressed using light-units (miles and light-miles — or — light-years and years — or — etc.) such that lightspeed c = 1, the c's are dropped from the formulas (as is sometimes done), and velocity v ends up expressed as a fraction of lightspeed, e.g, if v = half a light-year per year, then v = 1/2 without need for mentioning the units of measurement.
|Time minus proper time,|
|Distance-with-direction traveled, ||Δn|
CHANGES OF SEPARA- TION
|Proper time unpassed due to time dilation that comes with speed,|
v = Δd/Δt
RATES OF SEPARA- TION CHANGE
|“Unaging” rate, |
When d>t, then τ is an imaginary-number quantity and its square a negative, so the equation still works. But more usual is to replace “τ”with the expression “σ” for the positive amount of the spacelike spacetime interval as for instance in:
|Lasting, duration, time passed,|
Δt/Δt = 1 =
the timelike spacetime interval
Time minus displacement (time minus distance-with-direction)
proper time passed
Displacement (distance-with-direction) untraveled due to slowness
“Aging” rate, the Fitzgerald Ratio, reciprocal of the Lorentz Factor γ
Slowness in a direction
|Time needed to pass length through a given point at rest,|
length over velocity,
p = Ev
e = E−m =
Length, L1 =
E = m+e
by “aging” rate (length multiplied by the Lorentz Factor γ)
Note: Sometimes “m0” is used to represent rest mass, while “m” is used for total mass (total mass is proportionate to total energy E).
(not part of the
Forces ~ ~ ~ ~ momentum
Matter ~ ~ ~ ~ mass
Life ~ ~ ~ ~ ~ ~ energy
Intelligence ~ ~ tied-up momentum
The implicit prediction is that “tied-up” momentum, E−p, is actually important to the understanding of intelligent life and is so in some way that should seem fairly obvious once it is pointed out. Maybe we don’t count as intelligent life? It’s one thing to talk about the importance of structure in a qualitative way. People keep building infrastructure. But E−p is a quantity. Moreover, at everyday speeds on Earth, E−p hardly differs from rest mass quantitatively. Some sort of relativistic phenomena in or among neurons doesn’t “seem fairly obvious,” though it would be, as they say, kinda cool. Well, the correlation of energy with life is not in terms of energy only as the kinetic energy of the system’s travel for an observer at rest. Internal energy, external energy, it’s all important for life. So I suppose I should think of the tied-up momentum mostly in terms of more or less static structure (not just a big commotion), without minding too much whether it is internal or external, a structure static but continually growing and evolving, a web of webs, network of networks, extending through all artificial physical structures, vibrating with waves of fine information and logic traveling all around, along with decision-making, skilled work, feeling and sentiment, and imagination and knowledge, in an increasingly small world, till anyone is brought virtually face to face with anyone. And at the same time it is the ultimate shell or husk, record of everything, the evolution that doesn’t forget. I don’t know.
It occurs to me that somebody at peirce-l the other day said something to the effect that the nervous system appears to consist largely in restraints of actions, reflexes, impulses. I’ll have to dig that up, and check that which I’ve just said. Checks and balances.
Momentum, Lorentz-covariant (as part of the energy-momentum 4-vector) and vector-additive.
Rest mass, Lorentz-invariant.
~ ~ ~ Total energy, not Lorenz-invariant, but Lorentz-covariant as part of the energy-momentum 4-vector).
Kinetic energy, not Lorentz-covariant.
Tied-up momentum, not Lorentz-covariant and not vector-additive.
"As far as the laws of mathematics [language] refer to reality, they are not certain; and as far as they are certain, they do not refer to reality."
- Einsteing [added]
In reality, there are no fourfolds, three-folds, or two-folds.
In using words to describe reality, any division, naming, or categorizing, are purely whim. You _make_ this model work, in part due to your view of "time." In reality, there is only now.