Posted: Tue Aug 23, 2005 7:15 am Post subject: The Axiomatization of Physics – Step 1
The Need for Axiomatization: Einstein's Crazy Caricature of a Lopsided Space and Time
My first great leap in understanding time came as a great flash of insight into a whole new paradigm. In an instant I saw how to conceptualize special relativity with moving rulers. The idea is precise and clear if not amazingly obvious. It can be generalized to higher dimensions. It is a completely new representation of the "endlessly many spaces" of an n-dimensional spacetime. Take n to be 4 if you like.
There was always something in me that resented the traditional, lopsided explanations of special relativity. The popular explanations sounded downright kooky to me. If you knew better, you too would become greatly annoyed by the shallowness of sensationalized physics and the confusion it causes for many people.
When I use the word 'crazy' to describe conventional special relativity as taught by physicists, I don't mean that what they teach is scientifically wrong or inconsistent. I simply mean that the visual images they create and impose on the mind are unnatural, freakish, intellectually unsatisfying and very unsettling to a sensitive spirit. Must the physics of spacetime be taught, as it is now, to insult and disturb every mind that perceives and greatly loves the beauty of perfection, order and harmony?
I regret to inform you that this paper did not pass my tests. I am not saying that it is wrong, but it is posed in a language that is too technical and demanding, and I do not want to expose my students to that.
Cordially,
G. 't Hooft
To all appearances, if taken at face value, Professor ’t Hooft believes that students are harmed by challenging language. They shouldn't be exposed to what a thorough understanding of a subject really requires. Heaven forbid if any of them should ever stumble upon a link to a highly technical and demanding paper on special relativity. Perhaps my paper was too demanding for the Nobel Laureate? Perhaps physicists have idolized Einstein for so long and, in effect, have canonized the sacred oracles of the ancient opinions to so high and elevated a level that it is now impossible for them to grasp old concepts in new ways?
Physicists that I have interacted with are so perfectly content with their blinding traditions and convoluted ways of thinking that they seem repulsed and confused by a natural, mathematical approach to special relativity. Is their senseless opposition and bewilderment due to their practice and belief in dumbing down a subject to pictures and words instead of training young minds to think logically with math and equations?
A chimera is "an imaginary monster made up of grotesquely disparate parts." Einstein's contribution to special relativity was the simple yet grisly and tortured way he combined into one package the ideas of Larmor [1], Lorentz [2], and Poincaré [3]. Einstein successfully assembled and stitched together previously existing parts into an ugly composite theory. So it's ignorant to say that Einstein's formulation of special relativity is an original and strikingly brilliant idea requiring undying devotion. The original clashing pieces of the puzzle, certainly unresolved in Einstein's writings, are still all plainly visible.
Nevertheless, Einstein's explanation of the grotesquely disparate parts, in combination, reconciled several discordant facts of preexisting physics. It was the obvious next step to understanding relativity correctly and formulating it beautifully as a geometric group theory [4]. An introductory version of that perfect revision would highlight the essential invariants of special relativity as a physically intuitive model. Progress has been made. Spacetime diagrams are strained and delusive. The new Shubertian clock model of spacetime is very enlightening. [5].
It's proper to contrast the old, permanently lopsided view with the harmonious, always even, forward perspective. Einstein's contribution to special relativity is accurately construed as a forced contrivance. The obsession with riddles, diversions and paradoxes, considered important before, is no longer relevant. The enchanting chimera of religious relativistic physics is dead and should be buried.
Quote:
chimera: n.
1: (Greek mythology) fire-breathing she-monster with a lion's head and a goat's body and a serpent's tail; daughter of Typhon [syn: Chimera, Chimaera]
2. An imaginary monster made up of grotesquely disparate parts.
3. A grotesque product of the imagination.
4. A fanciful mental illusion or fabrication.
I have no preconceived notion of relativity. Whatever ideas I have absorbed from the copious literature have been just as you said, confused, apparently contradictory and inconsistent.
So it is no problem for me to want to see or entertain a different viewpoint. The reason I think your version of relativity is not catching on, is that you also seem to have a blind-spot: What is plain to you as a graduate of higher mathematics, with your elegant training in geometry, topology, and algebra is hardly plain to anyone else without this very specialized training.
There is no point in gloating over Hooft's failure to understand your version of things, or his failure to recognize it as a better way of learning relativity. The thing is, whether or not Hooft can 'rethink' or 'relearn' things all over again is a moot point, really. He doesn't need to 'protect' his students from your explanations and theories, because they are protected automatically by the complexity of it.
Unless you are there to actually make it plain, no one will be able to embrace your better way of doing things.
Now for example, here am I, a ready and willing student who really wants to learn the subtleties and problems both with mastering relativity, and the mistaken ways it is explained or 'taught'. I am fully appreciative that it might take months or years of hard work to get anywhere in this field, *AND* I am willing to learn and invest the time and hard work to master it.
I may be deceiving myself, but I think I have the ability/intelligence to master almost anything given enough time, a good teacher, tools etc..
So I put this question to you: Can you teach me what you've got or not? I am not afraid of work, or embarrassed to admit my mathematical and other shortcomings. If you gave me assignments for evaluation or practise I would actually try to carry them out. Now I think I am quite clever, having designed and built the lowest distortion audio power amps ever made. I am not an ordinary Electrical Engineer. Nonetheless, I admit my education has many holes, amounting to a 'swiss cheese' effect when it comes to higher mathematics.
But what is essential to grasp your brilliant insight? What are the prerequisites? Why not experiment with me? I am willing.
Posted: Sat Aug 27, 2005 11:37 am Post subject: Special Relativity Tutorial
Rogue Physicist wrote:
There is no point in gloating over Hooft's failure to understand your version of things
It's not about a failure to understand. It's about the time required to go through a carefully reasoned paper verses the great ease to quickly recognize and approve what everyone has been parroting for the last 100 years.
Rogue Physicist wrote:
So I put this question to you: Can you teach me what you've got or not?
Sure. I believe that you are more than qualified. The answer is yes.
Rogue Physicist wrote:
But what is essential to grasp your brilliant insight? What are the prerequisites?
There are three essential prerequisites:
The first prerequisite is to believe that mathematics demands a struggle. Tenacity is required. Even the greatest mathematicians who know the specialized language fluently can't read through a carefully reasoned mathematical argument in a technical math paper as if it were a novel. They have to go through it carefully with a fine-toothed comb to make sure that every detail is correct. Please believe this; even brief mathematical arguments can have loopholes. Thinking is a time-consuming arduous process. There are no shortcuts.
Second: When reading mathematics, you have to be able to recognize what you don't understand and either have the wherewithal and courage to figure it out yourself or have the humility to ask questions.
The third prerequisite is a basic level of mathematical maturity. The paper does use fancy technical terminology, not to obfuscate but for precision. That's not mathematical maturity. There is no obstacle in learning technical lingo. The meaning of the math and jargon in special relativity is easy to define. With your help, future editions of my paper will have this thread as a glossary of technical words with detailed definitions. When we go through all that, I estimate that my paper is actually easier than all previously published books on special relativity. I believe that all you need is reasonable competence in high school algebra.
Rogue Physicist wrote:
I am not afraid of work, or embarrassed to admit my mathematical and other shortcomings. If you gave me assignments for evaluation or practise I would actually try to carry them out.
Why not experiment with me? I am willing.
It's a deal. I will elucidate and magnificently illuminate and clarify every detail. Henceforth, we are commandeering this thread for a complete tutorial. All that remains is for you to ask questions from what you encounter and I will answer, simplify and make each point clear as I walk you through the step-by-step logic of my paper. You will finish with a truly precise understanding of special relativity. Here is your first assignment: Start reading my approach to the Lorentz transformation and tell me what is the first sentence or equation of the paper that you doubt that you understand, or not understand at all.
If you need a technical term explained, I will explain it.
If it turns out that you could benefit from a review of high school algebra, I will be happy to search for some online tutorials for you.
In Einstein's model...moving clocks run slow and moving objects shrink in the direction of motion. ...
By explicit construction of moving clocks in our new universe, it will become obvious that moving clocks tick at the same rate as stationary clocks.
There is also a real desynchrony effect. (ibid, pg 2 para 2&3)
This is exciting: But I feel some vagueness in my own understanding of your statements here: point by point, what goes through my mind as I read these statements:
(1) In Einstein's model, I understood slowness of clocks as an effect of absolute acceleration, as opposed to say any uniform motion at almost any speed. Have I got Einstein, or the Lorentz xforms wrong? That is, (and perhaps it is a misinterpretation of either Special Relativity (SRT from now on) or just Einstein/his interpreters,) You could conceivably have something travelling at nearly the speed of light, and as long as it is not accelerating, its clock ticks normally(?). On the other hand, I wonder also about compression in the direction of motion: is that also only an 'acceleration' effect, or an effect claimed for all objects travelling at high uniform speeds regardless of the question of 'acceleration'? I ask clarification here because, if SRT is wrong on this, or if only someone's interpretation is wrong on this, it would sure help to know which, if only for contrast with what is to follow (the new universe picture).
(2) What disturbs me about both Einstein (SRT) and Newton Mechanics (NMT from now on), is that both talk about a priviledged set of 'inertial frames'. In both these theories, there is a priviledged subset (although infinite in number) of frames for which the laws of physics 'simplify', namely 'inertial' frames. Both theories to me hoist themselves by their own petard, since neither offer a rationale for this, or any objective way of determining the 'true' set of such frames. (Although I think Newton is more honest, in just declaring 'Absolute Space' by fiat, and locking a set of frames to it (by experiment?) empirically.)
(3) In my mind, logically, (like Mach I suppose), there should not be such a set of frames. If I arbitrarily chose an 'accelerating' frame, I should be able to define another infinite subset of frames moving uniformly to it, in which all my laws of physics should hold just as simply. It should only be a point of view that some objects will turn out to be 'accelerating' in one frame, and moving uniformly in another. To me this should apply, whether or not I choose 'Galilean' or 'Lorentzian' transforms in the first place, for which I presume or assert that a simplified set of physical laws hold.
(4) In both universes you speak of 'moving clocks', and again I (like Mach) squirm since I cannot conceive of any motion detectable or definable other than relatively. Whether particles, or macro-objects, all detectable motion seems relative. It is only when 'force'or some form of conservation of something is added that any uniqueness of frames seems possible, and I am utterly suspicious that all such 'sets' defined create their own illusionary 'forces', which should vanish in another frame.
(5) 'desynchronity' I think I understand: the clocks actually get 'out of sync' either via practical procedures for comparing them, or physically due to motion (not acceleration?)...
How badly have I misunderstood page 2? Please understand I am not in any way being facetious or flippant here. To me at least, the questions I have listed above seem reasonable. Either they have answers, or I as a student am missing some vital information or perceptual organization. I have not dared to read Section 2 yet, with the rulers, because I feel a bit paralyzed already here.
The Lorentz Xform has always been perceived as a rule for translating the perspective of one observer's sense of space and time to the perspective of the other observers in relative motion. Forget about your faith in Lorentz & Einstein!
(1) Okay, I have always been disturbed by Einstein's introduction of 'observers' into physical laws, and the seemingly poor explanations of both 'clocks' and 'measuring sticks', and supposed processes of measurement.
(2) I have heard various 'explanations' of the Lorentz xforms, including "rotations of 4-vectors" in space. I am unsatisfied with any explanations so far.
(3) I have no faith in Einstein, not because I don't like him, but because I suspect he isn't the best interpreter of science. As for Lorentz, what little I know of him tells me he was clever, but wrong. Poincaire seems more trustworthy and more like the real originator of 'Lorentz xforms'.
So far, so good, I think. Now, I have peeked ahead and read over the rest of the paper, skipping the algebra, and searching for a logical structure for the argument. It all looks elegant, but there are key points I need clarification on.
Lets start with section 2: 'The Shubertian Clock'.
I love the two sliding rulers. I take it that these rulers are non-deformable in length, and the speed of their motion is irrelevant to the discussion. In this way, I can see that a 'clock' can be made by attaching a 'reading window' to a ruler anywhere, and looking through it to the other ruler.
And I can see that in 'Absolute Time', in my interpretation, all clocks tick at the same time in perfect synch, regardless of the speed or acceleration of the rulers. These clocks are nice, because they automatically and mechanically retain synch. In this scheme, there seems no need to abandon notions of absolute planes of simultaneity or 'Absolute Time'. It can remain as Newton defined it, impervious to influence.
Now that I have had a bit of time again to think about the moving rulers, I have a few questions:
Quote:
"Imagine...only...two possible states of motion..."
Imagine one of (the rulers) sliding on the other at a constant velocity.
...just rely on our own best guess (for the meaning of constant velocity)..." (pg 2 para 5,6)
(1) The notion of a 'state' of motion is new to me, but suggests almost by implication 'constant velocity', according to common sense. You designate such by a simple Greek letter, and I suppose the single 'state' of motion could be rather abstract or as complex as one might desire, provided some unambiguous description is available. But I get the feeling that is too general for this discussion. What I think you mean is that by calling it a (single) 'state', you are suggesting almost automatically a constant velocity and a simple description. Hoping you will confirm something here.
(2) Only one state of motion is actually referred to from this point on, or described, as far as I can tell. Perhaps the second state is simply "no relative motion", between the rulers at any rate. In that case all readings would be constant and independant of time. (confirm, or what have I missed?) I don't want to confuse actual 'state(s)' of motion with simple viewpoints of observers.
(3) Motion can only be described or conceived as occuring "between" two objects (even if one is just an observer). Next comes the problem of measuring the motion. Here, we have created an automated 'clock'. But to my mind this places some constraints upon the markings of the rulers as well, these being the only way we have of detecting motion in this new one-dimensional 'universe'. In order to relate actual speed (and constancy of velocity) to the true motion, the marks must be equally spaced, and the scales must be identical in size/scale. At least this seems the simplest scheme for accurately determining the true motion and synchronizing the readings between clocks/observers/frames.
So I come up with three(?) separate constraints:
(a) The rulers are rigid, and the space they travel in is Euclidean. They undergo no detectable deformation in length either at once or in partial areas of their length, unless they both experience the same deformity at identical spacial positions. (Imagine the rulers moving through a distorted lens, but maintaining sync point by point.)
(b) The rulers can be marked in some equally spaced fashion, and the scales of each ruler kept identical and stay fixed that way. You could imagine numbered billiard-balls or repelling electrical charges moving in a chain.
(c) There would be some way of communicating the readings between observers on different rulers, that can be correlated to some absolute Time.
Again, have I missed anything?
Quote:
I know what you're thinking. These clocks aren't properly synchronized. ... In our universe, a continuous stream of numbers are steadily flowing past...you can set your watch by them.
You see, actually, I was thinking the opposite: The clocks, far from being improperly synchronized are impossible to unsynchronize! What have I got wrong here? I don't see the clocks ever having any mechanism to get out of sync, or deviate from a simple addition or subtraction of a constant, or the reversal of a -+ sign.
Now I come to the first bit of slippery math: On page 3, para 4 to 7, I am unfamiliar with the symbols and the functional forms etc. It looks good, but it isn't self-evident by inspection for me at any rate. The text is clear however:
Quote:
The main property of constant velocity is that equal distances are covered in equal times.
This makes perfect sense to me, but again I find a subtle edge to it:
An observer reading a Shubertian Clock has no way of knowing or verifying any distortion of the ruler (or frame) he is sitting on. His perception of constant velocity is purely based upon equal distances going by on the other ruler/frame. That is, he has no way of knowing if the Shubertian clocks for the observers on the ruler moving by are behaving or have any relation at all to his clocks or time perception.
Quote:
The theory of relativity is all about defining time consistently for all observers in all possible states of motion.
Here I find a great difficulty, but I will try to sort it out:
(a) We are talking about Special Relativity (SRT), not General Relativity (GRT). I must assume this because the entire discussion is about the Lorentz transform.
(b) 'all possible states of motion' in our specific New Universe is only one (or two) possible states previously defined, in other words by fiat we have limited discussion to constant velocity (in fact one velocity, presumably described by a constant).
From independant reading, I have found opinions about the scope of SRT such as the following:
Quote:
...during these periods B is in a non-inertial reference frame and the laws of SRT do not apply.(!) ....Special Relativity does not tell us how to deal with the transition from one inertial frame to another (!)...
Relativity An Introduction to Spacetime Physics by Steve Adams 1997
Quote:
In SRT, as in Newtonian Mechanics, inertial frames enjoy a priviledged status. The principle of relativity applies only to them. An inertial frame is one where Newton's law of inertia holds: a body subject to no net external force remains at rest or moves in a straight line with uniform velocity. All inertial frames move uniformly relative to one another. Understanding Relativity by Leo Sartori 1996
It seems in your paper you are not challenging this basic viewpoint.
But rather just questioning some paradoxical ideas posed by interpreters of SRT. Is this correct?
Posted: Fri Sep 02, 2005 10:24 am Post subject: Special Relativity Tutorial
Abstract
I am only capable of clarifying one issue at a time. It would be a great help if you would only ask me one question at a time, and that we stick with it until it's completely resolved. I get overwhelmed easily when given too much to do. You now have a large number of directions to select from, out of my condensed response.
Section 1
Rogue Physicist wrote:
In Einstein's model, I understood slowness of clocks as an effect of absolute acceleration, as opposed to say any uniform motion at almost any speed.
Lorentz contraction and time dilation are explained online by Einstein himself. You wanted Einstein's point of view. Acceleration is not involved. These effects are due to relative motion. See chapter 12 of the 1920's book, Relativity: The Special and General Theory: The Behaviour of Measuring-Rods and Clocks in Motion.
Rogue Physicist wrote:
'desynchronity' I think I understand: the clocks actually get 'out of sync'
That's exactly correct. My point in beginning with rational sounding terminology is that Einstein's popular phrase "moving clocks run slow" is confusing. It suggests an instantaneous comparison between moving clocks, whereas, the truth is that instantaneousness doesn't exist.
As I've stated in the abstract, relativity can't be understood with words. You need to master the concepts behind the words mathematically.
Section 2
Rogue Physicist wrote:
In order to relate actual speed (and constancy of velocity) to the true motion, the marks must be equally spaced, and the scales must be identical in size/scale. At least this seems the simplest scheme for accurately determining the true motion and synchronizing the readings between clocks/observers/frames.
But of course: All marks are equally spaced and properly numbered. All my rulers are standard, identical copies and indistinguishable in every respect, except for the fact that they all move at different velocities.
Rogue Physicist wrote:
The rulers are rigid, and the space they travel in is Euclidean.
It's better to say that the lines (rulers) are Euclidean spaces. As a mathematician, I define 4-dimensional spacetime as an infinite number of moving 3-dimensional Euclidean spaces that are continually passing through each other.
Rogue Physicist wrote:
I love the two sliding rulers. … I can see that a 'clock' can be made by attaching a 'reading window' to a ruler anywhere, and looking through it to the other ruler.
That's a beautiful picture. Yes, it's extremely inviting to visualize clock time with a 'reading window' at each point on both frames of reference.
Rogue Physicist wrote:
The notion of a 'state' of motion is new to me, but suggests almost by implication 'constant velocity', according to common sense. … What I think you mean is that by calling it a (single) 'state', you are suggesting almost automatically a constant velocity and a simple description.
Correct.
'Constant velocity' is routinely conceptualized with the misleading notion of simultaneity and instantaneousness. I wanted to sidestep the fallacy. It was necessary therefore to define 'constant velocity' from the primitive, uncontested and irreducible concept of there being different kinds of motion. Please notice that I achieved my goal without invoking the existence or fabrication of a cosmic everywhere present "now."
Rogue Physicist wrote:
Only one state of motion is actually referred to from this point on, or described, as far as I can tell.
Each ruler represents a state of motion, even though one frame of reference may be regarded as not moving. My use of 'state' is specialized physics lingo and in this context simply means 'a distinct motion.' I would like the reader to intuit the general concept of 'state' because I've worked on many principles that need it and I see that it may be useful in later work.
'Observers' play no critical role in SRT. They only exist to entertain the reader and to travel from here to there and do experiments. Placing oneself completely into the problem and figuring out how you would measure things conveys an understanding of the principles of physics in mathematical universes.
Rogue Physicist wrote:
In that case all readings would be constant and independent of time
We only concern ourselves with distinct states of motion. That means that there's always a nonzero u_ij that measures the proper velocity of frame j from the perspective of frame i.
Rogue Physicist wrote:
"I know what you're thinking. These clocks aren't properly synchronized. ... In our universe, a continuous stream of numbers are steadily flowing past...you can set your watch by them." - Shubert.
You see, actually, I was thinking the opposite: The clocks, far from being improperly synchronized are impossible to unsynchronize! What have I got wrong here?
The clocks are calibrated to tick at identical rates. Synchronization means that spatially separated clocks read the same time. Those are two different concepts. It's important to understand that there are an infinite number of clocks. There's a clock at each point. The clock time T at x is a function T(x,x').
There are two essential points that failed, I believe, to grab your attention. I'm not assuming the existence of a cosmic everywhere present "now." I never reach the conclusion that instantaneousness exists.
Just because we begin with the postulate that all clocks are calibrated to tick at identical rates doesn't mean that we are guaranteed to not arrive at a weird model of spacetime.
Rogue Physicist wrote:
he has no way of knowing if the Shubertian clocks for the observers on the ruler moving by are behaving or have any relation at all to his clocks or time perception.
It's an axiom that (between any two sliding rulers L1 and L2) the number of notches per second, measured from one ruler, equals the same number of notches per second on the other ruler.
It's essential to know how to measure u_ij.
Rogue Physicist wrote:
"Special Relativity does not tell us how to deal with the transition from one inertial frame to another (!)..." - Steve Adams.
I gave specific rules about jumping from one frame to another. Forget about the misdirection of other writers. My rules are clear and consistent.
Rogue Physicist wrote:
"In SRT, as in Newtonian Mechanics, inertial frames enjoy a privileged status. The principle of relativity applies only to them. An inertial frame is one where Newton's law of inertia holds: a body subject to no net external force remains at rest or moves in a straight line with uniform velocity. All inertial frames move uniformly relative to one another."
It seems, in your paper, you are not challenging this basic viewpoint.
You are correct. I am assuming the simplest theory of physics imaginable:
1. Standing on a rotating platform is different than not twirling around. Children know this from their experience on the playground.
2. If our universe were empty, except for an incredibly long non-rotating ruler and a camera, and if the camera was programmed to take a picture of the ruler's etched markings at equal time intervals (assume that the camera is always at a fixed distance from the ruler), then the center of every picture would be of the same point, or each picture would record different points, all equally spaced from each other.
Rogue Physicist wrote:
I cannot conceive of any motion detectable or definable other than relatively.
We could define motion on a merry-go-round in terms of the acceleration felt by an observer but physics on a spinning platform is far more difficult than special relativity in one spatial dimension. Let's stick to the subject.
How do I define a privileged set of 'inertial frames' in SRT?
By subscribing to the axiomatic method. I believe in the mathematization of physics. I begin by building model universes composed of two and then three moving rulers for the purpose of gradually progressing to the achievable level of an infinite number of rulers. There isn't much difference conceptually between a three-ruler universe and the universe we actually inhabit.
My strategy is ancient. I am able to cite all the history of mathematics as precedent. Euclidean geometry is based on a few undefined terms. The meaning of point and line isn't defined in geometry, for instance. Just enough is stated to give a precise outline of an idea that can be explored by reasoning mathematically. Likewise, my approach to special relativity begins with undefined terms and essential elements. The few rulers I start with are privileged. By default, they are my inertial frames of reference. The basic facts I set forth about my rulers and how rulers in motion correlate with experiential time determine everything in my spacetime models. Another undefined term I use in my formulation of physics is wristwatch. I can't think of a more worthy goal in science than to axiomatize all of physics (Hilbert's sixth problem).
Rogue Physicist wrote:
If I arbitrarily chose an 'accelerating' frame, I should be able to define another infinite subset of frames moving uniformly to it
In the axiomatic view of my approach (although I didn't write it up formally as an axiomatic system), you're very much entitled to conceive of all my inertial reference frames as accelerating uncontrollably and collectively in a specific direction. It doesn't change anything. A universe in SRT is defined as a fixed collection of conceivable inertial frames with consistent relationships between these frames. All theorems can be built on allowable thought experiments with wristwatches in motion, particles in collision, etc. If you want to entertain motion that differs from inertial motion, then, there is a consistent principle to discover and difficult mathematics to solve. Congratulations, you now grasp a more sophisticated universe.
First let me thank you for your quick and detailed reply, which obviously you put alot of effort into. I will read it over a few times before seeing what questions are left over, or what direction to continue the inquiry from here.
Posted: Wed Sep 07, 2005 1:52 am Post subject: Second Pass:
It still seems reasonable to me that no matter how well you satisfy other readers, if I myself don't understand your paper, there has still been a small communication failure of some kind. I frankly admit I don't yet understand it. I frankly admit I have great holes in my mathematical education. I am an experimentalist rather than a theorist. Nonetheless I really want to understand your paper, and SRT in general. And I can't accept the idea that other people can understand something so easily, while I personally fail so badly at it. So I don't want to give up, just because we are coming from different backgrounds, or I have inadvertently gotten off on the wrong foot.
Even if you have rejected my last post, I still think there was something in it worth saving, namely a couple of questions about your approach. But lets leave it for now, if you feel that directly digging into your paper is a better approach here.
Step by step then, line by line if need be, here is my next difficulty in your paper:
Quote:
The time T at position x ( when x' flies by and is directly overhead) is a function of x and x'. This is essential. T is the time at x when x' touches x. Let's write this as T = T(x,x').
I confess I have no idea what this means. Essential it may be. But it is impenetrable to someone like me with much more than a high-school education, but not a formal math diploma.
Please explain what this means, if possible. The terms, and the unknown function are truly unknown to me.
Posted: Sun Sep 11, 2005 1:56 pm Post subject: The Axiomatization of Physics – Step 1
The Axiomatization of Physics – Step 1
RP, I appreciate your honesty. I'm very confident that I can clarify my presentation and answer all your questions.
I'm aware that my approach is difficult and that it can be greatly improved. I realize that the math and arguments in my paper are tightly compressed. Don't let that get you down. I'm certain that SRT will become clear to you when I expand the presentation with sharply focused details. I only need to highlight all my axioms and explain the mathematics.
Please continue reporting what you don't understand. I want this thread to be a realistic tutorial. What follows is a better, clearer, reorganized approach. If I'm still not being detailed enough on any point, please say so.
Abstract: The aim of The Axiomatization of Physics – Step 1, is to define physics and then axiomatize an increasingly complex hierarchy of mathematical models of spacetime. Spacetime in high dimensions begins by defining time in one spatial dimension.
There is an easy to understand foundation to build upon in one dimension. It generates the simplest models of spacetime conceivable.
Preliminaries
You know what the real numbers are and what a line is. It's useful to merge these two ideas. Henceforth, whenever we talk about a line, think of this very simple picture: every real number corresponds to a unique point on the line.
Real numbers have an order to them; one is less (<) than another, and so forth. Think of them as arraigned on the line in their natural order. What we now have, at this stage of construction, is called the real number line.
There's one especially obvious structure that we may attach to a number line if we want to model the real world. That indispensable structure is the notion of distance between points. If x and y are points on the line gamma, then the distance between x and y is denoted by d(x,y). The distance is defined by the absolute value: d(x,y) = |y-x|. We have now arrived at a one-dimensional Euclidean space.
Physics should evolve from a small number of axioms
"If geometry is to serve as a model for the treatment of physical axioms, we shall try first by a small number of axioms to include as large a class as possible of physical phenomena, and then by adjoining new axioms to arrive gradually at the more special theories. … The mathematician will have also to take account not only of those theories coming near to reality, but also, as in geometry, of all logically possible theories. He must be always alert to obtain a complete survey of all conclusions derivable from the system of axioms assumed." David Hilbert, International Congress of Mathematicians at Paris in 1900.
Note the harmony between Hilbert's view of physics and my definition: Physics is the mathematical study of all conceivable universes. A universe is a mathematical model that describes spacetime, matter, energy and their interactions. Think of each model universe as filling one page in the atlas of all possible universes.
Why axiomatize physics?
Quote:
In the history of physics, ideas that were once seen to be fundamental, general, and inescapable parts of the theoretical framework are sometimes later seen to be consequent, special, and but one possibility among many in a yet more general theoretical framework. … Examples are the earth-centered picture of the solar system, the Newtonian notion of time, the exact status of the laws of thermodynamics, the Euclidean laws of spatial geometry, and classical determinism. In view of this history, it is appropriate to ask of any current theory "which ideas are truly fundamental and which are 'excess baggage'." J.B. Hartle, Classical physics and Hamiltonian quantum mechanics as relics of the Big Bang, Physica Scripta T 36 (1991), 228-236.
The reductionist approach — explaining physical phenomena in terms of simple, mathematically precise, quantities — has been extraordinarily successful in almost all areas of physics. It goes against everything we have learned about nature to propose a theory in which complicated macroscopic objects, whose precise definition must ultimately be arbitrary, are fundamental quantities. A. Kent, Against many-worlds interpretations, Int. J. Mod. Phys. 5 (1990), 1745-1762.
A great physical theory is not mature until it has been put in a precise mathematical form, and it is often only in such a mature form that it admits clear answers to conceptual problems. A. S. Wightman, Hilbert's sixth problem: mathematical treatment of the axioms of physics, in: Proc. Sympos. Pure Math., Vol. 28, AMS, 1976, pp. 147-220.
It's now time to construct the simplest universe imaginable. The symbol for it is Xi_2. This universe consists of two one-dimensional Euclidean spaces. For ease of understanding, you may picture these two abstract lines as pristine, frictionless rulers gamma and gamma'. Can you imagine gamma' sliding on gamma at a constant velocity?
This universe has infinitely many infinitesimal observers. They are assigned to specific points on gamma and gamma'. Observers aren't relevant to spacetime but they are useful in explaining physics.
Every observer has a name. For example, observer O(2.1) is assigned to point x=2.1 on gamma and observer O'(-3.0) is assigned to the point x'= -3.0 on gamma'.
Remember the convenience of algebra. The symbols x and x' are variables. x represents a general point on gamma. x' represents a general point on gamma'. Notice, then, the algebraic way of making an infinite number of assignments: A mathematician would put it like this: Observer O(x) is assigned to point x on gamma. Observer O'(x) is assigned to point x' on gamma'.
There are only four things that an observer can do (until he or she is granted greater privileges).
An observer knows the numerical value of the point he or she is assigned to.
An observer can read the time on his or her own wristwatch.
An observer can perceive the numbers that fly by on the other ruler when those numbers touch (collide, meet, pass through) his or her assigned number.
An observer has a memory and can do mathematics.
It follows that observer O(x) can note the time T1 on his or her own wristwatch when the point x'_1 on the other ruler passes by and also the time T2 when the point x'_2 passes by. Proper velocity u_xx' is defined by the equation u_xx' = (x'_1 – x'_2) / (T2 – T1). This number is assumed to be constant. No matter which observer on gamma performs this experiment or when, everyone will agree on the value of this constant. The notation u_xx' represents the proper velocity of gamma' as measured by any observer on gamma.
When we progress to the universe Xi_3, which has three states of motion, the notation gets prettier. There, each state is indexed by a number i = 1, 2, or 3.
With this superior notation, u_ij is the proper velocity of frame j as measured from frame i.
The Indistinguishability Postulate: u_ij = -u_ji
Are you with me so far? Now is the time to ask questions before I move on to answering your question.
Some of my questions will be dumb, so bear with me:
Note 1: On the Set of Real numbers and the numberline
Okay, I admit talking about the numberline makes me nervous. Keep in mind, that I am coming from a varied educational background, which can't help but colour my understanding re: mathematics. I don't want to complicate things, because I also like to equate simplicity with elegance, when it works. Your initial discussion of the numberline leaves some gaps, but is quite lucid, and holds together well.
Here are the things that cross my mind as I read this bit:
(1) To me the numberline itself is a continuum. But the set of Reals isn't. It is just a set of numbers. I believe I read somewhere that the old definitions of a line as a collection of points is actually absurd, as are concepts of area and volume (extension in space). That is, there is no way to get to a line by gathering an infinite number of points together.
But this may not matter, since you have lept ahead with a handy definition of distance that has the appearance of solving all the dilemmas:
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The distance is defined by the absolute value: d(x,y) = |y-x|. We have now arrived at a one-dimensional Euclidean space.
Wow! I feel this actually works, except, I have one remaining problem now:
It also seems obvious to me that |y-x| is something *NEW*. It is not a point, and it is not a Real Number, and it is not a location. It is a scalar quantity, but is an entirely new entity to our previously defined universe. This 'thing' called distance is an abstract thing, which has no direct relation to the numberline or the Real numbers marked on it.
I just don't feel right glossing over the fact that we have created a new abstract entity that has no physical direct connection to our Universe. I believe it is just as real as any 'Real' on the numberline, and just as real as any point in space, but clearly it is something new in nature, and 'invisible' as it were. It only exists in the minds of our observers somehow, and yet it is a part of their reality.
But I can move on, since I haven't the mathematical training to do much more with it except note that it is remarkable.
The discussion on the Axiomatization of physics is very interesting, but I am not sure exactly what to do with that either. It seems like a powerful approach, but since I am not a real mathematician, I can hardly understand the issues discussed.
The definition of an observer, and what he can and can't do is by far the most practical bit so far. This really gave me something to hold onto, and I felt like I understood this part. And I can suspend my disbelief enough for the purpose of supposing little one-dimensional observers with these powers. I don't need an explanation of how or why they exist. And since I am comfortable with the hypothetical beings, I think I can handle giving them new powers as needed.
Now once again, the math makes me nervous:
Quote:
It follows that observer O(x) can note the time T1 on his or her own wristwatch when the point x'_1 on the other ruler passes by and also the time T2 when the point x'_2 passes by. Proper velocity u_xx' is defined by the equation u_xx' = (x'_1 – x'_2) / (T2 – T1). This number is assumed to be constant. No matter which observer on gamma performs this experiment or when, everyone will agree on the value of this constant. The notation u_xx' represents the proper velocity of gamma' as measured by any observer on gamma.
This looks meaty, but needs to be broken down for me to really get it:
As I struggle with it, it occurs to me that the 'Proper Velocity' = delta x/delta t is a ratio. In the past, I always thought of such quantities, be they vectors or scalars, as absolute amounts, (although admittedly set by choice of scales or units). Now here for the first time I am confronted with a 'velocity' defined as a ratio. I don't know, was I asleep in high school or away that day? Possibly. I just don't feel like I really understand what we have done here properly.
Finally, your new notation for extra rulers is a nice tantalizing bit, but doesn't help me now. And your Indistinguishability Postulate: u_ij = -u_ji seems to make sense, but I am not sure what I am agreeing to by going with this:
I hope this gives you an idea of where my mind wanders during your lectures.
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