An “analytical-metaphysical” take on Special Relativity!

[Bombadil]

Originally Posted by Doctordick

Your use of the word “orientation” here bothers me! An object was defined to be a collection of elements which could be considered an entity unto itself for sufficient time to examine or use that fact. The only orientation of interest is its momentum direction in the four dimensional space being used here. No “orientation” is being maintained; the only thing being maintained is the fact that the elements making up that object have to remain in the vicinity of one another. Since they are all moving at the same speed (v_?) that fact has important logical consequences.

Perhaps my use of the word orientation was improperly chosen and I have been having a slight misunderstanding about just what an object in general is considered, and, when I was referring to a construct here:

Originally Posted by Bombadil

This can be accomplished by being able to consider the construct separately of the universe that is of any influences resulting from the Dirac delta function, effecting the construct outside of it can be ignored.

I should have used the word object instead. If so this is likely just a minor misunderstanding on my part of understanding just what an object is. I have been under the impression that an object is a collection of elements such that all of the elements that it is composed of will maintain the same order over changes in t so that it would be static. While this is certainly a subset of possible elements it may not be the entire set of objects.

Originally Posted by Doctordick

Well, of course, that is exactly the phenomena under examination.

Yes but it is part of assuming the existence of objects and need not be considered an assumption of its own which it could very easily be taken to be.

Originally Posted by Doctordick

Somehow I don't think you are following the deduction at all. If you followed what I was doing, you wouldn't ask such a question.

Maybe I’m just not fully understanding how the sphere was arrived at but it seems that it may be related to the fact that the solution to the wave equation in odd dimensions is determined entirely by the boundary values of the initial conditions while in even dimensions the entire sphere has an effect on the equation. The point being if we don’t assume the existence of objects might not an expanding solid ball be more of the proper description of the equation then the expanding sphere that you used.

Maybe this would be more closely related to why the universe appears three dimensional. Either way I agree that the topic should probably be dropped as it seems to be outside of the topic being discussed. I just wanted to give some idea of where my question was coming from.

From the remainder of your post it seems that either I’m not asking the right questions or you don’t have any problem with how I‘m understanding the topic at hand. With that in mind I’m going to return to the original post and see what there is that I have a problem following and when I come up with some new questions I will post them unless you bring my attention to something that you think I should pay particular attention to first.

[AnssiH]

I walked through the rest of the OP very carefully, and found no errors, except that I stumbled on the very last step:

Originally Posted by Doctordick

Thus it follows that the observed period of the rest clock (as seen by the moving observer) is,

I managed to understand it all, all the way to the expression , but I don't know the algebraic steps to get to the final expression showing the symmetry to the earlier situation:

Once again I expect it's something quite simple... :P

-Anssi

[AnssiH]

Hi Bombadil, I think I can provide some helpful comments as well;

Originally Posted by Bombadil

I have been under the impression that an object is a collection of elements such that all of the elements that it is composed of will maintain the same order over changes in t so that it would be static. While this is certainly a subset of possible elements it may not be the entire set of objects.

Yes, "object" here refers to some set of elemental entities that are moving in the same direction, i.e. they stay together for sufficient amount of time, for us to be able to point our finger at them and label it as an object. Perhaps it would be possible to randomly choose some set of muons, moving to different directions, and call that "an object", but you can see, it would be quite useless picture of reality if we chose to see it this way.

At any rate, your comment implies that you perhaps did not pick up the relevant point of such a definition of "object". It wasn't about whether we could define "a set of muons" as an object or not, but rather that what modern physics calls "a mirror", is under this presentation form understood as a set of elemental entities moving, for the most part, along the tau axis. That is NOT to imply, that reality is a space full of elemental entities flying around and then we'd try to pick up partially stable collections.

You should perhaps read the part titled "About the presentation form" very carefully from post #48:

Originally Posted by AnssiH

About the presentation form:

Note, that parameters, and all the other definitions related to the fundamental equation, are there for the sole purpose of being able to investigate relationships between things that we have defined in our head.

I.e. that the analysis is using a tau dimension to communicate this issue, is of little significance here. It does NOT mean that there exists an ontologically real tau dimension.

It does not even mean, that there literally exists a tau dimension as such in our minds. The presentation form (of ) is not important, what is important is the exposed relationships between defined things; While we do not consciously think about the world in terms of a tau dimension, the timewise relationships that are expressed in the form, are necessary for any self-coherent world model

Then pay attention to:

Originally Posted by AnssiH

Note that the relationships exposed before this thread, were shown to map perfectly onto the relationships given by modern physics, which yields us a way to see exactly how does a defined macroscopic object such as "a clock" map into this picture (of space).

The "relationships exposed before this thread" is referring to the end of the OP of "Derviation of Schrödinger Equation":
http://hypography.com/forums/philoso...ndamental.html

Note that the fundamental equation by itself is quite useless, as all we know is that any valid worldview should satisfy its constraints. The algebraic mapping from the fundamental equation to Schrödinger's Equation is what allows us to say something useful ABOUT MODERN PHYSICS.

After having shown how Schrödinger's Equation maps to the fundamental equation, a comparison with Schrödinger Equation yields a way to express other definitions from modern physics; "Energy", "Momentum" and "Mass" (look at Energy, Momentum and Mass operators at the end of the OP)

That is what allows us to say how something like "a mirror" and "a massless oscillator" (photon) from "modern physics" maps to "" presentation.

Whether we could define different sorts of objects than what modern physics does (such that dissolve all around the universe almost immediately), is somewhat a side issue.

And btw I would like to comment that, just the fact that the fundamental equation yields Schrödinger's Equation, and consequently the definitions of mass etc, and a way to define something that behaves like a "photon" puts quite a considerable weight on this analysis. (Photons are extremely elusive entities from the perspective of modern physics, both when expressed in terms of relativity and in terms of QM, and I would hope everyone would understand how so all by themselves rather than just read it from a book)

Anyway, how do you find the math itself? Spotted any errors?

-Anssi

[Doctordick]

Originally Posted by AnssiH

I managed to understand it all, all the way to the expression , but I don't know the algebraic steps to get to the final expression showing the symmetry to the earlier situation:

Once again I expect it's something quite simple... :P

From one perspective it is simple; from another it is complicated. The problem in a nutshell is your lack of familiarity with mathematics. Take a look at Trigonometry - Wikipedia, the free encyclopedia. You will find the definitions of the trigonometric functions. They are defined as functions of an angle in a right triangle (a right triangle has three angles, one of which is ninety degrees). Pick one of the remaining angles (one of the two which are not ninety degrees). Then the various trigonometric functions are defined in terms of the ratios of various sides of the triangle (with respect to a specific angle, the three sides are called the hypotenuse, the adjacent and the opposite. From those definitions, one can define the various trigonometric function (below are the defintions of "sine", "cosine" and "tangent") usually written as follows:

we won't worry about the rest (trigonometry has several more defined functions but they can all be defined in terms of those three (in fact, note that tan=sin/cos).

At any rate, put those definitions together with the Pythagorean theorem (an interesting proof) and one has

which provides the conversion

which, as you say, is pretty simple.

I am still trying to make sure my Dirac deduction is error free. It is a bit longer than I had originally intended but I think it will be interesting reading. To anyone who likes to think about things anyway. Regarding Turtles comment “where's the beef”, I found out that the the ancient Pythagorean school knew that the square root of two was irrational but hid it for many years. Perhaps the fact that modern physics is a tautology is just another one of those cases of hidden knowledge. Apparently the ancients drowned the guy who discovered could not be represented by a ratio of integers. By the way, I have looked at Fuller's Synergetics and, as far as I can see, it would make an excellent door stop. If you find anything in there worth discussing let me know; I trust your judgment.

Have fun -- Dick

[AnssiH]

Hmm, yeah, I had actually figured out that from looking at how you were using it, and so I had actually already written down:

But now I have no idea how to do that final step... (When I first tried this, I actually decided this can't be a good route and didn't even mention I'd tried it )

So, I still don't know why that very final step is valid... I guess it must be something incredibly simple since you just stated it :P

-Anssi

[Doctordick]

Originally Posted by AnssiH

So, I still don't know why that very final step is valid... I guess it must be something incredibly simple since you just stated it :P

Yeah, “incredibly simple” is a pretty good description. This time I know you will kick yourself!

Note that is exactly the square root of . Another way to say the same thing is to note that times is exactly the same as . If you make that substitution into the above equation, you have

On second thought, maybe I should be the one kicking myself. When I looked back at your post, you quoted me as saying

which is “WRONG”! I checked my post and found that your quote is correct; that is indeed what I said. So I went back and looked at the original post. There it is correct! It is also correct in your original post which I was answering (if you look, you will see that you had the v_? outside the square root sign).

I have just about finished my Dirac's equation deduction. I am just adding a few comments at the end. Trying to make sure I quote the same equations is getting me mentally muddled so I just quit. I'll get back on it tomorrow.

Have fun -- Dick

[AnssiH]

Originally Posted by Doctordick

Note that is exactly the square root of . Another way to say the same thing is to note that times is exactly the same as . If you make that substitution into the above equation, you have

Well so it is... Didn't really spot that route at all... :I But that concludes my walk through of this thread!

On second thought, maybe I should be the one kicking myself. When I looked back at your post, you quoted me as saying

which is “WRONG”! I checked my post and found that your quote is correct; that is indeed what I said. So I went back and looked at the original post. There it is correct! It is also correct in your original post which I was answering (if you look, you will see that you had the v_? outside the square root sign).

That may have been throwing me off partially too, but still, when I'm looking at the papers where I tried to do that algebra today, I wasn't really on the right track at all :P Probably would have noticed the error if I had first at least gotten the idea of that substitution... Oh well...

-Anssi

[Bombadil]

Hi AnssiH.

Originally Posted by AnssiH

Yes, "object" here refers to some set of elemental entities that are moving in the same direction, i.e. they stay together for sufficient amount of time, for us to be able to point our finger at them and label it as an object. Perhaps it would be possible to randomly choose some set of muons, moving to different directions, and call that "an object", but you can see, it would be quite useless picture of reality if we chose to see it this way.

Actually I think much of my trouble is in understanding just what is necessary to call something an object. For instance saying that the elements that an object is composed of stay together long enough to point our finger at somehow doesn’t seem to be adequate as it says nothing about how it will behave. Suppose for instance that it expanded whenever it was moving. Clearly such a thing could not be called an object or be used as a clock. What is sufficient is saying that the collection of point can be considered separately of the rest of the universe but this really says nothing about where the points are, and to be of much use we will have to solve the equation for our object and it will need some sort of property so that we can use it as a clock.

Originally Posted by AnssiH

Note that the fundamental equation by itself is quite useless, as all we know is that any valid worldview should satisfy its constraints. The algebraic mapping from the fundamental equation to Schrödinger's Equation is what allows us to say something useful ABOUT MODERN PHYSICS.

It also tells us something about modeling any other system. As it says that any system could be approximated by quantum mechanics or at least the Schrödinger equation and who knows where the fundamental equation might lead with more general models, especially considering that DoctorDick is the only one that understands it well enough to work with it (as far as I know). But I have to wonder why no one has shown any interest in this aspect of the presentation maybe they just think that it is too general a model to be of any use or maybe they just don‘t understand or believe this is possible.

Originally Posted by AnssiH

Anyway, how do you find the math itself? Spotted any errors?

I haven’t been spotting the errors, or if I have been I have been giving them so little thought as not to point them out something that I really need to be careful not to do if it is the case.

Now back to the first post
Firstly the goal of your diagrams and geometric description of the Lorenz transformation is to gain some geometric insight into the consequences of the Lorenz transformation being part of our explanation.

Originally Posted by Doctordick

Note that the length of the moving clock is shown to be L'. This has been done because we know that the symmetry discussed in the previous section must require the Lorentz contraction to be a valid on any macroscopic solution if interactions with the rest of the universe may be neglected (up to this point the model was scale invariant): i.e., when we solve the problem in the moving clocks system we want the length of the clock as seen by the observer in that moving frame to be L₀. We use the scale freedom in our model to set that length (as seen from the rest system) to be L'; then and only then can we seriously call the clocks identical. This will require (the inverse of the relativistic transformation deduced earlier: i.e., in order to get the length of the moving clock in the primed coordinate system we have to multibly by ). Note that is exactly the apparent velocity of the moving clock divided by the velocity of the elemental entities, v_?, which actually has nothing to do with time. Since all velocities are v_?, it follows directly that d₁ + d₂ = S. Please note that everything so far is being graphed as seen in the frame of the rest clock: i.e., S=v_?T_m, where T_m is the period of the moving clock as seen from the rest frame.

Notice that the following geometric figure is embedded in the previous diagram.

The first thing that you do is set the length of the clock in its rest frame equal to the length of your standard clock this way we can call them identical and they differ by only a factor of scale due to one of them moving.

Now you have the distance that the rest observer says the moving clock moves in the direction as which is the same as the length of the rest clock and so it is how far the rest clock will move in the direction I don‘t understand this unless your trying to show the reflection taking place before the point at . The only way that I can see that we might come to this conclusion is since both observers will observe the other clock to move the same distance in the direction we will conclude that the moving clock will move the same distance in the direction as the rest clock. But I can’t see how we get around the issue of total distance traveled as the moving clock moves along both the x and axis. So if it does move in the direction it seems that it should move a larger total distance in the rest frame which we already decided can’t happen, although this seems to be the case for both observers. Or maybe I’m just misreading your diagram as it seems that from any frame the other clock should appear to be running slower, that is moving less distance in the direction.

Also I’m not understanding how you have found the reflection points of the oscillator. Maybe this has something to do with not understanding why the moving clock moves in the direction. Although clearly the moving observer will conclude that the reflection of the oscillator will reflect off of the first mirror after the clock has moved half of its total distance in the direction as he will consider it to be at rest, while the rest observer will conclude that the same reflection will have to take place at some latter time as a result of the oscillator having moved further in the x direction while the oscillator was moving toward its first reflection point. Likewise it will take less time to return as the oscillator will have less distance to travel. This has substantial consequences when simultaneity is considered as it means the two observers won’t agree on what events are simultaneous. This seems to be a considerable understatement to me which I will try to elaborate on latter when I better understand your diagrams.

[Doctordick]

Originally Posted by Bombadil

Suppose for instance that it expanded whenever it was moving. Clearly such a thing could not be called an object or be used as a clock.

I think you are being overly conservative in your concept of “an object”. In my head, an object is anything I can talk about as an entity unto itself. I may not ordinarily think of an explosion as an object but I have certainly heard it discussed as an object, particularly if it happened to be an issue in a photograph.

Originally Posted by Bombadil

What is sufficient is saying that the collection of point can be considered separately of the rest of the universe but this really says nothing about where the points are, and to be of much use we will have to solve the equation for our object and it will need some sort of property so that we can use it as a clock.

If you look back at my post, you will discover that I defined a clock to be an object with some very specific properties. Or are you saying you can define a clock without defining something which can be conceived as an object? If that is the case, I again think you are being overly conservative in your concept of “an object”. I go along with Anssi, that an object is anything which maintains an independent existence long enough to point my finger at it.

Originally Posted by Bombadil

... maybe they just don‘t understand or believe this is possible.

You have hit the nail right on the head; I have been told many times, by many great authorities, that I am a “crack pot”: “what I have claimed to have done, can not be done!”

Originally Posted by Bombadil

I don‘t understand this unless your trying to show the reflection taking place before the point at .

You are confusing the geometric construction with the significant events. Remember, in this mental model, everything is spread out in the tau direction from minus infinity to plus infinity and different positions in tau are unobservable; however, if everything is moving at a constant velocity we can “presume” it moves a distance v_?t in time “t” so, even if we can't observe a motion in the tau direction, we can conclude it is taking place. Furthermore, notice the vertical line between the point of the left hand arrow and the point near the small letter b. That is the line from minus infinity to plus infinity which denotes the moment the photon reaches the left hand mirror (the moving observer will perceive that interaction as taking place exactly 2L₀ after it started on that same mirror as measured with his clock).

Originally Posted by Bombadil

Or maybe I’m just misreading your diagram as it seems that from any frame the other clock should appear to be running slower, that is moving less distance in the direction.

Yes, you are misreading everything as you are totally omitting the fact that everything is totally smeared out in the tau direction.

Originally Posted by Bombadil

Also I’m not understanding how you have found the reflection points of the oscillator.

It is that point where the x position of the photon (smeared out in the tau direction) is exactly the same as the x position of the right hand mirror (smeared out in the tau direction). Once again, you are totally omitting the fact that everything is totally smeared out from minus to plus infinity in the tau direction! Take a look at the video's Anssi made (they are now in the original post of this thread).

Originally Posted by Bombadil

... while the rest observer will conclude that the same reflection will have to take place at some latter time as a result of the oscillator having moved further in the x direction while the oscillator was moving toward its first reflection point.

Time is defined only by “interactions” along the path of the entity interacting. Things can interact only if they are in the same place at the same time! Since everything is “totally smeared out in the tau direction” it doesn't make any difference what the value of tau is. If you want to know “when” the interaction takes place, you must examine their paths from the last time they interacted. If they interact again, the actual length of those paths must be the same (including the tau component). Since they are both “totally smeared out in the tau direction”, they will interact anytime their x positions are the same (or their x,y,z positions were the same if we were dealing in all three dimensions).

You will never pick up on my diagrams until you can comprehend the consequences of being “smeared out in the tau direction”. We are dealing with entities which are momentum quantized in the tau direction. The Heisenberg uncertainty principle guarantees that their position in the tau direction is unknowable!

Have fun -- Dick

[AnssiH]

Originally Posted by Bombadil

Actually I think much of my trouble is in understanding just what is necessary to call something an object. For instance saying that the elements that an object is composed of stay together long enough to point our finger at somehow doesn’t seem to be adequate as it says nothing about how it will behave. Suppose for instance that it expanded whenever it was moving. Clearly such a thing could not be called an object or be used as a clock. What is sufficient is saying that the collection of point can be considered separately of the rest of the universe but this really says nothing about where the points are, and to be of much use we will have to solve the equation for our object and it will need some sort of property so that we can use it as a clock.

Maybe, maybe not, but you shouldn't stop too much at thinking about what sorts of elemental entity collections one might be able to see as an object. Like I said, the relevant bit here is, what sorts of collections would be considered "a mirror" and "an oscillator", i.e. the components defined by modern physics as the buildings blocks of "a clock".

We got the necessary information for answering that question at the end of the Schrödinger's thread, where it was shown how "mass" maps to this view.

Following those steps; an object that would behave like a mirror, would have to be a collection of entities staying together, and if we were to look at a mirror that is at rest in our coordinate system, it means in this representation that it is moving entirely in tau direction.

It also tells us something about modeling any other system. As it says that any system could be approximated by quantum mechanics or at least the Schrödinger equation and who knows where the fundamental equation might lead with more general models, especially considering that DoctorDick is the only one that understands it well enough to work with it (as far as I know). But I have to wonder why no one has shown any interest in this aspect of the presentation maybe they just think that it is too general a model to be of any use or maybe they just don‘t understand or believe this is possible.

Seems to be mostly the two latter ones... (If someone thinks "this is too general to be of any use", they certainly have missed the point of the whole presentation). I think Kuhn's "The structure of scientific revolutions" explains quite a bit about this.

I haven’t been spotting the errors, or if I have been I have been giving them so little thought as not to point them out something that I really need to be careful not to do if it is the case.

Yeah, I have to say I would be quite surprised to find a fatal mistake from the analysis at this point, as it does actually explain quite rationally very many thus far unexplainable properties of modern physics.

DD already replied to your questions about the analysis, but I thought maybe I will give an answer with my own words as well; perhaps that's helpful in resolving some inconvenient ambiguities

Now you have the distance that the rest observer says the moving clock moves in the direction as which is the same as the length of the rest clock and so it is how far the rest clock will move in the direction I don‘t understand this unless your trying to show the reflection taking place before the point at . The only way that I can see that we might come to this conclusion is since both observers will observe the other clock to move the same distance in the direction we will conclude that the moving clock will move the same distance in the direction as the rest clock. But I can’t see how we get around the issue of total distance traveled as the moving clock moves along both the x and axis.

Take a look at the video of the moving clock (click your way to youtube to see the clear HD version). It's a moving version of exactly the diagram you put to your quote.

It represents how the observer conceives the behaviour of the clock that is moving in his coordinate system, when he is supposing the speed of the elements of the clock to be exactly , against his coordinate system.

Look at how the oscillator is gaining on the mirrors in the y-direction. Notice how one total cycle takes longer this way than it would if the clock was "at rest".

I.e. by the time the oscillator has completed one full cycle, the mirrors have managed to move more than in total distance.

But their displacement along ends up being exactly

So if it does move in the direction it seems that it should move a larger total distance in the rest frame which we already decided can’t happen,

We have not decided that can't happen, you must have misinterpreted something at the OP.

although this seems to be the case for both observers. Or maybe I’m just misreading your diagram as it seems that from any frame the other clock should appear to be running slower, that is moving less distance in the direction.

Yes, the picture is symmetrical between observers, and I think people who understand relativity (I mean actually understand instead of just "know") should see that symmetry as blatantly obvious from the presentation, as some (seemingly problematic) aspects of that issue are somewhat analogous to how the symmetry works in standard relativity.

Look at it this way; The clock in the "moving clock" animation is the same clock as what was first shown at rest; all we did was we changed the coordinate system from which to plot the exact same situation. We didn't do anything to the clock really. Let's call that first clock "A".

If we place a new identical clock (B) that is at rest in that second coordinate system, we have a situation where that clock B is doing more clock cycles than clock A.

If we now move our perspective back to the original coordinate system where clock A is at rest, we do the same transformation to clock B, and end up to a situation where clock A is doing more clock cycles than clock B.

So if you understand the (absolutely self-coherent) transformation from one coordinate system to another, you already understand that particular symmetry between coordinate systems as well. To understand it in more detail, you just have to take into account how simultaneity is defined uniquely in each coordinate system, and how notion of simultaneity affects length measurements (to measure the length of a box, you must know where its front and back ends are at a given instant; where they are "simultaneously").

Pay particular attention to how everything here is simply a function of chosen coordinate system. I.e. something that happens entirely in our minds. I.e. why in relativity "each observer disagrees with each other's time measurements, length measurements and notion of simultaneity". What it is, it's a juggling with definitions, that are all related to each others, leading to particular symmetries between coordinate systems.

-Anssi