An “analytical-metaphysical” take on Special Relativity!

[AnssiH]

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

 

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

[math]T_r = \frac{2L_0 cos(\theta)}{v_?(1-sin(\theta)^2)}=\frac{2L_0}{v_?}\frac{1}{\sqrt{1-sin(\theta)^2}}[/math]

 

I managed to understand it all, all the way to the expression [imath]T_r = \frac{2L_0 cos(\theta)}{v_?(1-sin(\theta)^2)}[/imath], but I don't know the algebraic steps to get to the final expression showing the symmetry to the earlier situation: [imath]\frac{2L_0}{v_?}\frac{1}{\sqrt{1-sin(\theta)^2}}[/imath]

 

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

 

-Anssi

[AnssiH]

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

 

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:

 

About the presentation form:

 

Note, that [imath]x,y,z,\tau[/imath] 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 [imath]x,y,z,\tau[/imath]) 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 [imath]x,y,z,\tau[/imath] form, are necessary for any self-coherent world model

 

Then pay attention to:

 

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 [imath]x,y,z,\tau[/imath] space).

 

The "relationships exposed before this thread" is referring to the end of the OP of "Derviation of Schrödinger Equation":

http://hypography.com/forums/philosophy-of-science/15451-deriving-schr-dingers-equation-my-fundamental.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 "[imath]x,y,z,\tau[/imath]" 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]

I managed to understand it all, all the way to the expression [imath]T_r = \frac{2L_0 cos(\theta)}{v_?(1-sin(\theta)^2)}[/imath], but I don't know the algebraic steps to get to the final expression showing the symmetry to the earlier situation: [imath]\frac{2L_0}{v_?}\frac{1}{\sqrt{1-sin(\theta)^2}}[/imath]

 

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:

[math]sin(\theta)=\frac{opposite}{hypotenuse},\;\;\;cos(\theta)= \frac{adjacent}{hypotenuse}\;\;and\;\;tan(\theta)=\frac{opposite}{adjacent}[/math]

 

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

[math]cos(\theta)=\sqrt{1-sin(\theta)^2}[/math]

 

which provides the conversion

[math]T_r = \frac{2L_0 cos(\theta)}{v_?(1-sin(\theta)^2)}=\frac{2L_0 \sqrt{1-sin(\theta)^2}}{v_?(1-sin(\theta)^2)}=\frac{2L_0 }{v_?\sqrt{1-sin(\theta)^2}}[/math]

 

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 [imath]\sqrt{2}[/imath] 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 [imath]cos(\theta)=\sqrt{1-sin(\theta)^2}[/imath] from looking at how you were using it, and so I had actually already written down:

 

[math]T_r = \frac{2L_0 \sqrt{1-sin(\theta)^2}}{v_?(1-sin(\theta)^2)}[/math]

 

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]

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!

[math]T_r = \frac{2L_0 \sqrt{1-sin(\theta)^2}}{v_?(1-sin(\theta)^2)}[/math]

 

Note that [imath]\sqrt{1-sin(\theta)^2}[/imath] is exactly the square root of [imath](1-sin(\theta)^2)[/imath]. Another way to say the same thing is to note that [imath]\sqrt{1-sin(\theta)^2}[/imath] times [imath]\sqrt{1-sin(\theta)^2}[/imath] is exactly the same as [imath](1-sin(\theta)^2)[/imath]. If you make that substitution into the above equation, you have

[math]T_r = \frac{2L_0}{v_?}\frac{\sqrt{1-sin(\theta)^2}}{\left(\sqrt{1-sin(\theta)^2}\right)\left(\sqrt{1-sin(\theta)^2}\right)} =\frac{2L_0}{v_?\sqrt{1-sin(\theta)^2}}[/math]

 

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

[math]T_r = \frac{2L_0 cos(\theta)}{v_?(1-sin(\theta)^2)}=\frac{2L_0 \sqrt{1-sin(\theta)^2}}{v_?(1-sin(\theta)^2)}=\frac{2L_0 }{\sqrt{v_?(1-sin(\theta)^2)}}[/math]

 

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]

Note that [imath]\sqrt{1-sin(\theta)^2}[/imath] is exactly the square root of [imath](1-sin(\theta)^2)[/imath]. Another way to say the same thing is to note that [imath]\sqrt{1-sin(\theta)^2}[/imath] times [imath]\sqrt{1-sin(\theta)^2}[/imath] is exactly the same as [imath](1-sin(\theta)^2)[/imath]. If you make that substitution into the above equation, you have

[math]T_r = \frac{2L_0}{v_?}\frac{\sqrt{1-sin(\theta)^2}}{\left(\sqrt{1-sin(\theta)^2}\right)\left(\sqrt{1-sin(\theta)^2}\right)} =\frac{2L_0}{v_?\sqrt{1-sin(\theta)^2}}[/math]

 

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

[math]T_r = \frac{2L_0 cos(\theta)}{v_?(1-sin(\theta)^2)}=\frac{2L_0 \sqrt{1-sin(\theta)^2}}{v_?(1-sin(\theta)^2)}=\frac{2L_0 }{\sqrt{v_?(1-sin(\theta)^2)}}[/math]

 

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.

 

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.

 

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.

 

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.

 

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 [imath]L'=L_0\sqrt{1-sin^2(\theta)}[/imath] (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 [imath]\alpha[/imath]). Note that [imath]sin(\theta)[/imath] 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 [imath] \tau [/imath] direction as [imath] 2L_0 [/imath] which is the same as the length of the rest clock and so it is how far the rest clock will move in the [imath] \tau [/imath] direction I don‘t understand this unless your trying to show the reflection taking place before the point at [imath] 2L_0 [/imath]. 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 [imath] \tau [/imath] direction we will conclude that the moving clock will move the same distance in the [imath] \tau [/imath] 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 [imath] \tau [/imath] axis. So if it does move [imath] 2L_0 [/imath] in the [imath] \tau [/imath] 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 [imath] \tau [/imath] 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 [imath] 2L_0 [/imath] in the [imath] \tau [/imath] 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 [imath] \tau [/imath] 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]

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.

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.

... 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!”

I don‘t understand this unless your trying to show the reflection taking place before the point at [imath] 2L_0 [/imath].

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).

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 [imath] \tau [/imath] direction.

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

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).

... 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]

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 [imath] \tau [/imath] direction as [imath] 2L_0 [/imath] which is the same as the length of the rest clock and so it is how far the rest clock will move in the [imath] \tau [/imath] direction I don‘t understand this unless your trying to show the reflection taking place before the point at [imath] 2L_0 [/imath]. 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 [imath] \tau [/imath] direction we will conclude that the moving clock will move the same distance in the [imath] \tau [/imath] 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 [imath] \tau [/imath] 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 [imath]v_?[/imath], 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 [imath]2L_0[/imath] in total distance.

 

But their displacement along [imath]\tau[/imath] ends up being exactly [imath]2L_0[/imath]

 

So if it does move [imath] 2L_0 [/imath] in the [imath] \tau [/imath] 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 [imath] \tau [/imath] 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

[Bombadil]

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.

 

It seems to me that it really depends on ones reason for using the word object. If it is because it makes it easy to refer to something or using it in day to day use then I would agree in using it as referring to any particular thing, on the other hand if the purpose is to construct an exact clock for the purpose of studying the effect of simultaneity as you have done, then it seems to me that a more conservative stance needs to be taken. For instance, I don’t think that in day to day life or for that matter any experiment that has been performed or will be, that a perfect clock which is what we need to study the effects of simultaneity will ever be constructed as there will always be outside variables that’s influence is considered so small as to be ignored. On the other hand isn’t the whole point of your definition of a object to define something that we can study without any outside interference no matter how small, or else don’t we have to go back and ask, what happens if the universe has a preferred frame of reference as we have no defense for ignoring those influences if it does (which is something that we are not ready to do yet). So on day to day experience I think it is safe to loosen up what we consider an object considerably as otherwise there is no example of what we might consider an object to be. But don’t we have to use an object to construct a clock?

 

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

 

Maybe I should have said that the distance that the object traveled must remain the same as the distance that the oscillator travels. Actually this brings up at least one question that seems that it may be of interest, which is if the oscillator has a different speed will we notice it. From how I am understanding things right now it will always be measured to travel the same speed. Will this mean that if the oscillator moves at a different speed, that the objects we are considering still travel the same distance, I don’t know maybe DoctorDick can shed some light on this but I suspect that we just aren’t far enough along to meaningfully answer this question yet.

 

Now back to the diagram,

 

 

I think there is actually a slightly deceptive thing that is going on here. That is, for a rest clock a rest observer will use it to measure the distance moved in the [imath] \tau [/imath] direction and is what will be considered time to the rest observer (if we define time to be when objects interact) that is he can use it as t but only by defining it to be t. While in the diagram of the moving clock what the rest clock will actually measure is the length of the diagonal line the first mirror moves along. This is not due to it measuring the t axis (which can only be done by defining a frame of reference in which the t axis can be measured) but rather due to our defining it as a rest frame and so using it to define the measurements used in the diagram. This seems to me to be a slightly different issue and it seems important to notice the difference.

 

The point being, in order for things to interact they must exist at the same t and x location, even though t is not technically a dimension of you coordinate system. The location on the [imath] \tau [/imath] axis has no influence on interactions due to the infinite uncertainty on the [imath] \tau [/imath] axis, as you say the elementals are smeared out in the [imath] \tau [/imath] direction.

 

So the task is to find when the total distance the oscillator has moved (which is actually in your diagram the length of the line you have marking the x axis starting at the first mirror) has the same length as the line marking the distance the second mirror has moved and when at this point they both have the same x position.

 

At this point the way that I come up with to do this is to first note that the triangles marked A and B are at least similar to each other. That is, one of them can be scaled so that both of the triangles are identical. I won’t go through the details of how I would do this, but this can be shown by comparing the angles between the triangles.

 

After this is done it is simply the question of how far along the x axis the oscillator will travel before it has traveled the same distance as the clock and has the same x location. By comparing the corresponding sides of the triangles we see that the side of triangle A that the oscillator travels along corresponds to the side of triangle B that the mirror travels along likewise the side of triangle A that the mirror travels along corresponds to the side of triangle B that the oscillator travels along, considering that they are right triangles we can conclude that triangle A and B will be identical when the mirror and oscillator can interact.

 

A similar argument will show that triangles a and b are identical. Combining these two proofs will show that the moving clock will still measure the distance the clock moves along the [imath] \tau [/imath] axis even when viewed from the rest frame.

 

There is a point being made here that should not be over looked in your diagram, that is at point 2 when the oscillator hit’s the first mirror, the moving observer will say that the oscillator has taken the same amount of time as it will take to return to the first mirror. While the rest observer will measure that time as measured from his clock to be [imath] d_1 [/imath] he will then say that [imath] d_2 [/imath] is the time that the oscillator will take to return to the first mirror. The point being that events that the rest observer will considers to be happening simultaneously on the x axis will not be considered simultaneous for the moving observer.

 

It seems that this point could also be made by taking the midpoint of the two mirrors and considering when two oscillators released from this point one heading right the other left will hit the mirrors. The moving observer will consider the oscillators to hit the mirrors simultaneously but the rest observer will say that the one traveling in the same direction as the clock will take considerably longer to get to the mirror then the one traveling the opposite direction as the clock. In fact if I’m understanding your diagram correctly any two events that take place on a line that is at a right angel to the path of the first mirror will be considered to be simultaneous events for the rest observer but the moving observer will consider the events to be simultaneous only if they take place on a line parallel to the x axis.

[AnssiH]

It seems to me that it really depends on ones reason for using the word object. If it is because it makes it easy to refer to something or using it in day to day use then I would agree in using it as referring to any particular thing, on the other hand if the purpose is to construct an exact clock for the purpose of studying the effect of simultaneity as you have done, then it seems to me that a more conservative stance needs to be taken. For instance, I don’t think that in day to day life or for that matter any experiment that has been performed or will be, that a perfect clock which is what we need to study the effects of simultaneity will ever be constructed as there will always be outside variables that’s influence is considered so small as to be ignored. On the other hand isn’t the whole point of your definition of a object to define something that we can study without any outside interference no matter how small, or else don’t we have to go back and ask, what happens if the universe has a preferred frame of reference as we have no defense for ignoring those influences if it does (which is something that we are not ready to do yet). So on day to day experience I think it is safe to loosen up what we consider an object considerably as otherwise there is no example of what we might consider an object to be. But don’t we have to use an object to construct a clock?

 

The important point is that "a photon and two mirrors" is about the simplest construction that modern physics takes as "a clock". What is being analyzed is our expectations for the behaviour of such a construction, given that its parts behave according to the definitions given by modern physics/definitions necessary for self-coherence.

 

What is shown in the final analysis is that we can plot any situation, in terms of those defined entities, according to any (moving) coordinate system, and still set the speed [imath]v_?[/imath] as isotropic across all coordinate systems in all directions, as making that move only transforms our definition of simultaneity and geometry. (It makes some aspects of our representation of reality simpler, but it should be understood how we make that move only in our head). That is exactly what yields the idea of isotropic C in modern physics (something that at least the mainstream publications state as a fact of nature, and that is exactly what raises all those interesting ontological questions about relativity).

 

Note that it is not an issue of constructing a "perfectly accurate clock" when it comes to analyzing relativistic time relationships. It is a matter of looking very carefully at the definitions behind massless oscillators (photons), definitions behind massive things (mirrors), the observable behaviour between them (we don't see how the photon moves between the mirrors, we see the final cycle count), and the impact that a self-coherent transformation from one coordinate system to another has got onto those definitions (onto our expectations for them to remain valid after the transformation).

 

 

The commentary about clocks not measuring time is, in this context, just a stab at trying to promote some mental hygiene. I'm sure all physicists understand that there exists outside variables that affect the measurement of any clock, and those influences just cannot be taken into account in any practical manner. The problem is, that you tend to still stick with the terminology where you keep implying that "clocks measure time", while implying at the very next sentence that "things can interact only when they exist in the same place at the same time".

 

To be exactly correct, clocks show a reading that is governed by the electromagnetic phenomena (also defined by modern physics). And the meaning of that reading is of course very different from that "other time", whose purpose is to tell us whether two airplanes will collide with each others (whether they will meet in the same place at the same "time").

 

I would very much agree with DD, that it allows you to shed some naive realistic ideas regarding "what time is" when you explicitly understand the distinction between an evolution parameter associated with our expectations, and the measurement which electromagnetic constructions called "clocks" give us.

 

Note that the former definition of "time" does not only tell us whether the airplanes would collide, but it is in also an integral part of our understanding of how clocks work in microscopic sense. Understanding time as an evolution parameter to our expectations, is necessary part of understanding where that observable measurement of that clock is coming from (incl. time dilation).

 

Aaaand still about your comment regarding the "preferred frame". Before the exists any notion of "space", there must exist definitions for "persistent objects" (it is their existence that gives any meaning to space). Without defining any persistent entities, there is no meaning to any inertial frames, preferred or otherwise. I.e. we are talking about features that exist in our world model, in our head. Also put this together with the fact that it was proven that isotropic speed of light is a consequence of self-coherent object definitions (one can always assume their frame is the one where light moves at speed C), you see that the question of preferred frame is moot at this junction. In terms of our world model, we have all the options still available.

 

For example, there is that new thread about nonlocality, and if one understands this thread, they also understand that is is entirely possible to suppose non-local (super-luminal) speed of information, so to explain Bell experiments that way, without any violation to relativity. In fact, you could use Bell experiments to establish a convention about preferred frame. That doesn't mean it would be the only way to explain Bell experiments, nor that it would prove something about ontological reality (but I'm sure many people would cling onto it like a new religion).

 

And btw, Erasmus' response in that nonlocality thread could be taken to imply that the explanations that preserve special relativistic locality are the only valid ones, which I'm afraid is not true :I They are certainly valid, but not the only possibilities.

 

=Bombadil]

So if it does move [imath]2L_0[/imath] in the [imath]\tau[/imath] 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.

 

Maybe I should have said that the distance that the object traveled must remain the same as the distance that the oscillator travels.

 

Yes, and the mirrors will travel the same total distance as the oscillator, in the [imath]x,y,z,\tau[/imath] space. The mirrors move [imath]2L_0[/imath] along the [imath]\tau[/imath] direction, so in total distance in [imath]x,y,z,\tau[/imath] space they move more than [imath]2L_0[/imath], and so does the oscillator. It is taken to move at exactly the speed [imath]v_?[/imath] against the chosen coordinate system, but not against the mirrors. (I.e. if the mirrors were moving almost at the speed of light in terms of a coordinate system, we would take it the photon in between is barely catching up).

 

Actually this brings up at least one question that seems that it may be of interest, which is if the oscillator has a different speed will we notice it. From how I am understanding things right now it will always be measured to travel the same speed. Will this mean that if the oscillator moves at a different speed, that the objects we are considering still travel the same distance,

 

It is important here to exactly pay attention to how things have been defined in terms of how one things is related to another, and consequently what sorts of aspects are measurable.

 

The oscillator was defined as a massless entity, so we know it is not moving along the [imath]\tau[/imath] direction at all, by definition. The confidence of being able to define an entity like that at all, comes from the earlier step where the definition of mass was already found. I.e. if it was moving slower than [imath]v_?[/imath] along [imath]y[/imath], it wouldn't be seen as a photon, and the construction wouldn't be taken as a "clock" at all.

 

If you asked that because you are thinking about whether an observer could find out if the photon is moving at speeds other than C against their own frame (i.e. "which observer is correct?"), the answer is no... and to explain how, is to repeat almost the whole OP :)

 

I don’t know maybe DoctorDick can shed some light on this but I suspect that we just aren’t far enough along to meaningfully answer this question yet.

 

I think we are well far enough :)

 

Now back to the diagram,

 

 

I think there is actually a slightly deceptive thing that is going on here. That is, for a rest clock a rest observer will use it to measure the distance moved in the [imath] \tau [/imath] direction and is what will be considered time to the rest observer (if we define time to be when objects interact) that is he can use it as t but only by defining it to be t.

 

It is NOT said there that he should use the measurement of his clock as "t" (the evolution parameter of the explanation, telling whether things can interact), even though they perfectly co-incide in this analytical example of perfect conditions. On the contrary, it turns out that EVEN THOUGH the evolution parameter and what the rest clock displays will co-incide exactly (ignoring outside influences), it would be an error to take them as meaning the same.

 

Like I commented above, normally physicists use the kind terminology where they mix up "t" and "clock display" just the way that you refer to "slightly deceptive"... I think you are starting to see the problem :)

 

So, what is under investigation is the measurement of 2 clocks; whatever difference they display, is commonly taken as "time dilation".

 

So yes, [imath]2L_0[/imath] represents one clock cycle of the rest clock.

 

Obviously, you can associate [imath]2L_0[/imath] with [imath]t[/imath] via [imath]v_?[/imath]

 

[math]\Delta t_{RestClockCycle}=\frac{2L_0}{v_?}[/math]

 

I.e. that's how much [imath]t[/imath] parameter advances during 1 clock cycle of the rest clock.

 

[imath]S[/imath] represents the distance that the moving clock moves in the [imath]x,y,z,\tau[/imath] space of the rest observer, while its oscillator makes a cycle. GIVEN that the rest observer supposes the oscillator of the moving clock also moves at speed [imath]v_?[/imath] against his coordinate system.

 

And again you can associate [imath]S[/imath] with [imath]t[/imath] via [imath]v_?[/imath]

 

[math]\Delta t_{MovingClockCycle}=\frac{S}{v_?}[/math]

 

So, if you know the difference between [imath]S[/imath] and [imath]2L_0[/imath], you know the difference between 1 clock cycle of the rest clock and 1 clock cycle of the moving clock (in terms of how the situation has been plotted by the rest observer; in terms of his [imath]t[/imath]).

 

If you remember the first parts of the OP (or if you are otherwise fluent with trigonometry), you already know that the difference between [imath]S[/imath] and [imath]2L_0[/imath] can be easily figured out from the angle [imath]\theta[/imath] as:

 

[math]S = 2L_0 \frac{1}{\sqrt{1-\sin (\theta )^2}} = \frac{2L_0}{\cos (\theta )}[/math]

 

(Just put that middle expression there so that it would be more obvious that this is actually quite exactly the standard relativistic transformation).

 

The OP is a more comprehensive explanation of exactly that bit, and the video clip that I made should be fairly simple representation of the same thing.

 

The point being, in order for things to interact they must exist at the same t and x location, even though t is not technically a dimension of you coordinate system. The location on the [imath] \tau [/imath] axis has no influence on interactions due to the infinite uncertainty on the [imath] \tau [/imath] axis, as you say the elementals are smeared out in the [imath] \tau [/imath] direction.

 

Yes and "t" as an evolution parameter of the explanation is such that it tells you when things collide; in those animations what you are seeing is just one "t" per video frame; they are not spacetime diagrams.

 

So the task is to find when the total distance the oscillator has moved (which is actually in your diagram the length of the line you have marking the x axis starting at the first mirror) has the same length as the line marking the distance the second mirror has moved and when at this point they both have the same x position.

 

At this point the way that I come up with to do this is to first note that the triangles marked A and B are at least similar to each other.

 

You might want to take a look at my post #37 where I asked about the same thing. http://hypography.com/forums/philosophy-of-science/18861-an-analytical-metaphysical-take-special-relativity-4.html#post269406

 

(And sounds like your analysis was correct, at least after superficial skimming)

 

There is a point being made here that should not be over looked in your diagram, that is at point 2 when the oscillator hit’s the first mirror, the moving observer will say that the oscillator has taken the same amount of time as it will take to return to the first mirror. While the rest observer will measure that time as measured from his clock to be [imath] d_1 [/imath] he will then say that [imath] d_2 [/imath] is the time that the oscillator will take to return to the first mirror. The point being that events that the rest observer will considers to be happening simultaneously on the x axis will not be considered simultaneous for the moving observer.

 

Absolutely correct. It is a consequence of both observers taking the speed [imath]v_?[/imath] as the same to all directions in their own coordinate system, and completely analogous to the postulate about isotropic speed in relativity (with exactly the same consequences, regarding their idea of simultaneity).

 

The point being, "they can do it, but that ability tells us nothing about reality".

 

There is commentary about that issue towards the end of the OP, and perhaps it is easier in aligning this with the view of modern physics if you take a look at my post #48

 

http://hypography.com/forums/philosophy-of-science/18861-an-analytical-metaphysical-take-special-relativity-5.html#post271521

The part titled "About simultaneity and standard perspective of relativity"

 

-Anssi

[Bombadil]

To be exactly correct, clocks show a reading that is governed by the electromagnetic phenomena (also defined by modern physics). And the meaning of that reading is of course very different from that "other time", whose purpose is to tell us whether two airplanes will collide with each others (whether they will meet in the same place at the same "time").

 

I have to wonder though, if it is governed by the electromagnetic phenomena or if it is not more accurate to say the electromagnetic phenomena must obey the Lorenz transformation to remain consistent. As we have not shown that electromagnetic phenomena are needed to make a clock (measure displacement along the [imath] \tau [/imath] axis) or that there aren’t other phenomena that are governed by the Lorenz contraction. In short it seems that what has been shown is that any massive element is governed by the Lorenz transformation.

 

Aaaand still about your comment regarding the "preferred frame". Before the exists any notion of "space", there must exist definitions for "persistent objects" (it is their existence that gives any meaning to space). Without defining any persistent entities, there is no meaning to any inertial frames, preferred or otherwise. I.e. we are talking about features that exist in our world model, in our head. Also put this together with the fact that it was proven that isotropic speed of light is a consequence of self-coherent object definitions (one can always assume their frame is the one where light moves at speed C), you see that the question of preferred frame is moot at this junction. In terms of our world model, we have all the options still available.

 

Yes all options are open to us including the option to make a universe in which the Lorenz transformation is as useless as we want as DoctorDick points out here

 

There is a big difference between “having to ignore” and “ignoring”. If a decent approximation can be achieved by “ignoring” then, to the same extent that the approximation is decent, the Lorenz transformations must be decent. And an object (which by definition presumes there is no impact upon its fundamental structure from the rest of the universe) requires the Lorenz transformation to be a central characteristic of any flaw free explanation. And, in order for that explanation to be flaw free, it cannot be changed by moving into one of those regions where objects can not exist. The explanation is a flaw-free explanation of “the universe” not just part of the universe.

 

Right now I’m only concerned with this as an after thought as I think that we will all agree that we are not interested in such possibilities right now. I just think that we need to remember that this is one of those options that are open to use if we chose to use it.

 

What I’m wanting to know is what is the minimum that we need to do so that we know that the Lorenz transformation is useful to us. Saying that collections of elements exist that can be considered separately of the rest of the universe will definitely be sufficient for us to say the Lorenz transformation will be useful in describing the behavior of such collections of elements in that if we are in the frame of one of these object we will need the Lorenz transformation to move to a new reference frame.

 

The only complaint that I have with saying that an object is any thing that we can point our finger at is that originally the definition of an object was “(an object being defined to be a coherent collection of elemental entities which can be regarded as an entity unto itself)” quoted from the opening post there is of course some room for interpretation here as a entity was not defined after words but I see little reason to try and do so.

 

In truth I really don’t have a problem with considering an object to be any thing that stays together long enough to point my finger at, perhaps saying any collection of elements that we can give meaning to would work also, I’m not particular about what we use as long as we agree that this definition is not the same as the one originally given unless someone can show that it is the case.

 

I really don’t care how we define any word within reason as long as we don’t use definitions that are not compatible with each other or where it is clear that every one will agree that they aren’t compatible and what is meant. I just want to stay consistent and suspect that there are a few spots in this thread that I have not been with the definitions that were originally given.

 

I think that we can all agree that for any clock that is of interest right now we can and are ignoring anything that will have any influence on it, we simply don’t consider anything else to exist.

 

For example, there is that new thread about nonlocality, and if one understands this thread, they also understand that is is entirely possible to suppose non-local (super-luminal) speed of information, so to explain Bell experiments that way, without any violation to relativity. In fact, you could use Bell experiments to establish a convention about preferred frame. That doesn't mean it would be the only way to explain Bell experiments, nor that it would prove something about ontological reality (but I'm sure many people would cling onto it like a new religion).

 

Yes it seems that the problem really would be a strait forward task to know when the information would arrive one would simply choose a frame to use to know when the signal would arrive and then ignore the [imath] \tau [/imath] reading on any other clock until we where interested in changing to that coordinate system. And be careful to be consistent with ones definitions as one is not truly using the value of t which would make things simpler. The point being is that the reading on a clock says nothing about when a signal arrives at a location only what the reading on a clock will be when it arrives.

 

However there might be some mathematical difficulty in describing this that I am overlooking caused form the fundamental equation being the equation of an expanding sphere that is expanding slower then such a signal would be traveling. But I think that we might be going off topic again.

 

If you asked that because you are thinking about whether an observer could find out if the photon is moving at speeds other than C against their own frame (i.e. "which observer is correct?"), the answer is no... and to explain how, is to repeat almost the whole OP :)

 

No, I asked it because Doctordick has made sure that it is understood that the speed of the oscillator (speed of light) is not assumed constant and he has said that he will use this fact latter but he has also shown that the speed of the oscillator will be measured to be the same in every frame so I’m wondering what possible advantage assuming a non constant speed of light might have let alone just what would be meant by a non constant speed as it will always be measured to be the same when measured form any reference frame.

 

It is NOT said there that he should use the measurement of his clock as "t" (the evolution parameter of the explanation, telling whether things can interact), even though they perfectly co-incide in this analytical example of perfect conditions. On the contrary, it turns out that EVEN THOUGH the evolution parameter and what the rest clock displays will co-incide exactly (ignoring outside influences), it would be an error to take them as meaning the same.

 

 

Yes the point that I’m trying to make is that no matter what frame we choose to use to make our diagram, the length of S in the diagram will be exactly what the reading on the clock that we are using for the rest frame in our diagram is multiplied by a scaling factor only needed to use the right units. My understanding is that we have been using the same units for every measurement to keep things simple.

 

So, if you know the difference between [imath]S[/imath] and [imath]2L_0[/imath], you know the difference between 1 clock cycle of the rest clock and 1 clock cycle of the moving clock (in terms of how the situation has been plotted by the rest observer; in terms of his [imath]t[/imath]).

 

Actually considering that the only triangles of interest are right triangles with any two more pieces of information we can solve for any information that we want uniquely with the exception of both being angles (scale becomes an issue). All that we would need is the use of the Pythagorean theorem and the use of the trigonometric functions that have been used so far and their inverse functions. But there’s really no need to derive the necessary transformations we would probably just be wasting time at this point. Just thought I’d point that out though.

[Doctordick]

Bombadil, I have utterly no idea as to how to reach you. Your interpretation of what I am doing, what I am proving, is so far left of the ball that I simply can not comprehend your intellectual distortions. After this post, I will leave arguments with you to Anssi (if he wants to take on the job). You need to start from scratch an think things out carefully. With regard to your earlier post

... on the other hand if the purpose is to construct an exact clock for the purpose of studying the effect of simultaneity as you have done, then it seems to me that a more conservative stance needs to be taken.

What, you don't think my clock fulfills my definition of object? Or perhaps you figure to reference time by some other means. Think about it.

For instance, I don’t think that in day to day life or for that matter any experiment that has been performed or will be, that a perfect clock which is what we need to study the effects of simultaneity will ever be constructed as there will always be outside variables that’s influence is considered so small as to be ignored.

To the extent that the relativistic transformations may be ignored? I think not!

... or else don’t we have to go back and ask, what happens if the universe has a preferred frame of reference as we have no defense for ignoring those influences...

You miss the whole point. The issue is to produce our expectations; that is a very definite consequence of what we include and what we ignore. Relativity certainly ignores many influences (for the most part, any influence we don't understand). Again, you seem to be walking off in that same direction I have complained about over and over and over again. What I initially proved was that every conceivable explanation of anything can be interpreted in a manner such that it obeys my fundamental equation. No where do I ever even suggest that it is a mechanism for generating explanations! It does however provide a constraint on self consistent explanations.

... otherwise there is no example of what we might consider an object to be.

What is going on here? You can not comprehend a collection of events which satisfy my definition of an object? Go back and read the post; I defined an object for the sole purpose (at this time) that I could create one: i.e., my clock!

...which is if the oscillator has a different speed will we notice it.

And how the devil do you intend to notice such a thing? From the fundamental equation, every element moves with exactly the same speed, v_?. The “apparent” velocity obviously consists of the x,y and z components and, as tau is a totally fictitious coordinate, it follows that the fourth component of the velocity is fixed by the requirement that [imath]|v_?|=\sqrt{v_x^2+v_y^2+v_z^2+v_\tau^2}[/imath]. If you solve that for [imath]v_\tau[/imath] you have [imath]|v_\tau|=\sqrt{v_?^2-v_x^2-v_y^2-v_z^2}[/imath]. Any massless object has no momentum (and thus no velocity) in the tau direction. Its apparent velocity will thus be v_?; now all you have to do is have some way of defining time. It all comes right back to defining a clock!

From how I am understanding things right now it will always be measured to travel the same speed.

You can not define speed without defining time first; and that is why I defined a clock. If you can't follow that you are missing the entire proposition.

This is not due to it measuring the t axis...

There is no “t axis”; t is defined by the period of the clock: i.e., the time necessary for the oscillator to complete a round trip is defined to be a fixed time.

So the task is to find when the total distance the oscillator has moved...

No, the total distance any element has moved between interactions (which occur at the same time) is always d=v_?t by definition of the fundamental equation. If you can't understand that, you have no hope of understanding my presentation.

The point being that events that the rest observer will considers to be happening simultaneously on the x axis will not be considered simultaneous for the moving observer.

No, the point is that “simultaneity” is a fictional concept anytime the two “simultaneous” events are not in exactly the same position. “Simultaneity” is only defined by interaction and interactions only occur when [imath]\delta(\vec{x}_i-\vec{x}_j)\neq 0[/imath].

In fact if I’m understanding your diagram correctly...

Understanding that diagram correctly seems to be beyond your ken. You keep omitting the fact that everything is smeared out in the tau direction! We can't draw in that fact as doing so removes our ability to talk about positions in the tau direction. (Anssi's video does a better job because it shows “time” correctly.) Anytime one makes a specific drawing (i.e., displaying an actual interaction where time has the same value) that tau position (in the drawing) is only valid for the interacting elements under examination. Notice that tau is set the same as per the initial start point (the left hand mirror and the photon). Simultaneity is defined by the fact that an interaction has taken place. Notice that at the second interaction the tau values of the two elements are nothing like one another but, since all elements are completely smeared out in the tau direction, we can spot interactions by combining the fact that they occur at the same y point after the same length of time. Both the rest observer and the moving observer see specific interactions occurring at the same time but not at the same tau. You don't appear to be capable of handling the fact that Everything is smeared out in the tau direction. Anssi's video handles the smearing directly, an effect which simply can not be handled in a static drawing. And yet you even seem to be able to ignore it there.

 

Until you can get your head around the meaning of the drawings, you will never comprehend what is and is not a valid argument concerning these interactions. Unless I get at least an impression that you are beginning to understand, I am simply going to ignore your posts. And your complaints in your last post are totally without merit.

The only complaint that I have with saying that an object is any thing that we can point our finger at is that originally the definition of an object was “(an object being defined to be a coherent collection of elemental entities which can be regarded as an entity unto itself)” quoted from the opening post there is of course some room for interpretation here as a entity was not defined after words but I see little reason to try and do so.

Can you understand the meaning of the word “coherent”? Can you point your finger at something which is not coherent? I would like to see that. And, you see little reason to try and define “entity”? How about “anything you can refer to” (or for that matter, point your finger at). Although if I can't think of it as an entity unto itself, I don't know how I could point my finger at it. By the way, if you want to insist that “you can point to any object” I would like to see you point at a lithium nucleus (I am fairly sure there is one near you). That is certainly an object nuclear physicists are seriously concerned with.

In truth I really don’t have a problem with considering an object to be any thing that stays together long enough to point my finger at, perhaps saying any collection of elements that we can give meaning to would work also, I’m not particular about what we use as long as we agree that this definition is not the same as the one originally given unless someone can show that it is the case.

If it's incoherent collection, just how can it “stay together”? “Any collection of elements that we can give meaning would work”? In other words, you would call any incoherent collection of fundamental elements “an object”? I am afraid I would want at least a little persistence, at least coherence over a period long enough to be worth thinking about.

 

An object is any thing (a thing? and yet it is not a thing you can not refer to it as an entity?) that stays together (it stays together with out being coherent in any way?) long enough (persistent without being coherent?) to point your finger at? I don't think you have any comprehension of what I am saying. Perhaps this (something I posted a number of years ago) is a better expression of my definition of an object

For example "an object" is best defined as a collection of elements whose behavior with one another on an internal elemental level is of no real consequence to it's interaction with the rest of the significant elements in the explanation at the time scale of interest: i.e., the idea of an object is a very valuable concept.

The point being is that the reading on a clock says nothing about when a signal arrives at a location only what the reading on a clock will be when it arrives.

That is right! The reading on a clock has nothing to do with “time” (the fact that two objects can interact). It is only an approximation usable together with a definition of simultaneity! The two go hand in hand.

I am overlooking caused form the fundamental equation being the equation of an expanding sphere that is expanding slower then such a signal would be traveling.

What you are overlooking is the fact that you can not even talk about fast or slow without defining a way of measuring time: i.e., you need to define a clock before you can even hope to measure time. It is fundamentally an unmeasurable variable and saying that "clocks measure time" points you towards a fundamentally incoherent concept of time.

...I’m wondering what possible advantage assuming a non constant speed of light might have...

You are wondering about things which do not come up until one gets into general relativity and if you can't understand my presentation of special relativity, you will never follow general relativity.

 

So I leave your problems to you and Anssi. If he wants to think about them. The exercise won't hurt him except time wise.

 

Have fun -- Dick

[AnssiH]

To be exactly correct, clocks show a reading that is governed by the electromagnetic phenomena (also defined by modern physics). And the meaning of that reading is of course very different from that "other time", whose purpose is to tell us whether two airplanes will collide with each others (whether they will meet in the same place at the same "time").

 

I have to wonder though, if it is governed by the electromagnetic phenomena or if it is not more accurate to say the electromagnetic phenomena must obey the Lorenz transformation to remain consistent. As we have not shown that electromagnetic phenomena are needed to make a clock (measure displacement along the [imath] \tau [/imath] axis) or that there aren’t other phenomena that are governed by the Lorenz contraction. In short it seems that what has been shown is that any massive element is governed by the Lorenz transformation.

 

I made that comment from the direction of modern physics, which takes the stance that everything (even a lump of clay) is governed by electromagnetic phenomena. And that being the case, the definitions behind electromagnetism are to be taken into account in a careful analysis of how a clock works, even if we weren't talking about DD's epistemological analysis. And that already means that the reading on the clock is a different thing than what could be taken as an evolution parameter to our worldview (i.e. a parameter having to do with how we think the electromagnetic phenomena itself evolves over... well, "time")

 

Uh, if that last paragraph just seems very confusing, it's because it is... It is really very difficult to dodge all the ambiguity here... As an attempt to be brief, I added the comment "(also defined by modern physics)", so the reader wouldn't forget that the definitions of "electromagnetism" by themselves are also something that need to fall in between the symmetry arguments, i.e. they shouldn't be just taken as something that ontologically exist.

 

And yes, electromagnetic phenomena is certainly following Lorentz transformation just like anything else...

 

You should view Lorentz transformation simply as a procedure that preserves some desired velocity, over a transformation from one inertial frame to another (from one moving coordinate system to another). That transformation became necessary in DD's analysis, when the form of the fundamental equation required that the probabilities propagate at fixed speeds; and if one wishes to plot some situation in terms of different coordinate systems in self-coherent manner and still satisfy the fundamental symmetry requirements, they must use Lorentz transformation... (And that has got nothing to do with what sorts of collections of elemental entities one might see as "objects")

 

One way to put it is, if you have 2 observers moving to different directions, and both making predictions over the same system, they can both choose to understand that system in terms of probabilities propagating at the same fixed speed against their own coordinate system, as long as they understand what sort of transformation is involved there (Lorentz transformation).

 

That transformation does preserve the given fixed speed, but it also affects their notion of simultaneity and their notion of geometry.

 

Standard presentation about relativity would say "thus speed affects time and length", but certainly you can understand, that when some arbitrarily chosen observer changes its perspective on you ("you" being that system under inspection), that hardly changes your ontological construction (length or timewise speed) one way or another. I.e. we are talking about an epistemological effect rather than an ontological one.

 

At any rate, Lorentz transformation is discussed in the beginning of the OP NOT because it would directly yield an epistemological explanation for the validity of relativistic time dilation. It is discussed there, because it is one necessary step in the analysis of how we form valid/coherent expectations over the behaviour of the kind of construction, that modern physics calls "a clock". That is, when we plot our expectations for that construction across different coordinate systems.

 

If we did not use Lorentz transformation properly in there, we would find that two physicists would make different (observable) predictions about the same system, depending on the coordinate system they chose to plot the situation in.

 

I think this whole thing becomes far easier when you drop the idea that the clock is being moved around in different cases, and just think about the situation where it is only the observer changing the frame in which he plots the situation. (These situations need to be symmetrical anyway, in special relativity where we don't consider acceleration yet)

 

Yes all options are open to us including the option to make a universe in which the Lorenz transformation is as useless as we want as DoctorDick points out here

 

Right now I’m only concerned with this as an after thought as I think that we will all agree that we are not interested in such possibilities right now. I just think that we need to remember that this is one of those options that are open to use if we chose to use it.

 

What I’m wanting to know is what is the minimum that we need to do so that we know that the Lorenz transformation is useful to us.

 

The requirement was brought up via the fundamental equation being "a linear wave equation with wave solutions of fixed velocity", so I'd say the minimum is then in understanding the algebraic steps from the symmetry arguments to that characteristic of the fundamental equation.

 

But you can understand this thread if you allow yourself to take that characteristic on faith for the time being, and then just think about how that necessity - of preserving a given fixed velocity - brings us straight into Lorentz transformation.

 

Saying that collections of elements exist that can be considered separately of the rest of the universe will definitely be sufficient for us to say the Lorenz transformation will be useful in describing the behavior of such collections of elements in that if we are in the frame of one of these object we will need the Lorenz transformation to move to a new reference frame.

 

The only complaint that I have with saying that an object is any thing that we can point our finger at is that originally the definition of an object was “(an object being defined to be a coherent collection of elemental entities which can be regarded as an entity unto itself)” quoted from the opening post there is of course some room for interpretation here as a entity was not defined after words but I see little reason to try and do so.

 

In truth I really don’t have a problem with considering an object to be any thing that stays together long enough to point my finger at, perhaps saying any collection of elements that we can give meaning to would work also, I’m not particular about what we use as long as we agree that this definition is not the same as the one originally given unless someone can show that it is the case.

 

I really don’t care how we define any word within reason as long as we don’t use definitions that are not compatible with each other or where it is clear that every one will agree that they aren’t compatible and what is meant. I just want to stay consistent and suspect that there are a few spots in this thread that I have not been with the definitions that were originally given.

 

Yes I'm sure this thread has got its fair share of ambiguity, and I'm guessing I am probably unable to spot some of them myself simply because of already having had some idea about the issues behind this thread... And in this context, I could well take the assertions...

 

“an object being defined to be a coherent collection of elemental entities which can be regarded as an entity unto itself”

and

"an object to be any thing that stays together long enough to point my finger at"

and

"an object to be any collection of elements that we can give meaning to"

 

...to all refer to the exact same idea. :shrug:

 

That being said, yes now I think you are mis-interpreting something that was probably said in an ambiguous manner... There was a comment in the beginning of the OP:

 

"...if the data belonging to a given observation could be divided into two (or more) sets having negligible influence on one another, those sets could be examined independently of one another: i.e., these collections would end up being constrained by exactly the same relationship which constrained the original universe. This is to say that these subsets (or “objects”) could be analyzed as a universes unto themselves..."

 

That is the same as saying that we don't need to take into account the entire known universe all the time in our thinking, but instead it is possible to understand a small portion of the data at a time, if you break it into specific collections of elemental entities. Such collections that pretty much stick together without being poked at by the rest of the universe all the time, and that are not scattered all around the universe.

 

I think you are getting bogged down too much into this if you try to think of what sorts of collections really could be seen as an object, in order to be completely liberal. Like I said couple of times before, the important point of this turns out to be, what sorts of collections modern physics takes as "a clock".

 

I think that we can all agree that for any clock that is of interest right now we can and are ignoring anything that will have any influence on it, we simply don’t consider anything else to exist.

 

You could say that yes.

 

Yes the point that I’m trying to make is that no matter what frame we choose to use to make our diagram, the length of S in the diagram will be exactly what the reading on the clock that we are using for the rest frame in our diagram is multiplied by a scaling factor only needed to use the right units. My understanding is that we have been using the same units for every measurement to keep things simple.

 

Hmmm.... I have to say I do not understand what you are saying there... :I

 

Actually considering that the only triangles of interest are right triangles with any two more pieces of information we can solve for any information that we want uniquely with the exception of both being angles (scale becomes an issue). All that we would need is the use of the Pythagorean theorem and the use of the trigonometric functions that have been used so far and their inverse functions. But there’s really no need to derive the necessary transformations we would probably just be wasting time at this point. Just thought I’d point that out though.

 

...or there. Sorry :P

 

-Anssi

[Doctordick]

Hmmm.... I have to say I do not understand what you are saying there... :I

 

[math]\cdots[/math]

 

...or there. Sorry :P

I think that what he is doing is presuming that the very act of thinking in terms of plotting points in a Euclidean geometry requires positions to be defined: i.e., in his mind, units of length have already been defined and he totally overlooks the fact that one cannot define lengths without defining a ruler. The fundamental equation presumes a measure has been defined but does not actually establish the mechanism of the measure; something I think he has trouble conceiving of. The equation specifies plots of fixed scale presuming a mechanism can be established: i.e., any solution can be universally scaled without effecting the solution at all.

 

Have fun -- Dick

[Bombadil]

If no one sees any improvement in this post over my previous posts let me know or if no one replies I’ll take that the same way and plan on rereading most all of the thread if not all of it, instead of just what seems to be of most importance, before posting again and see if I can pick up anything from doing that.

 

What, you don't think my clock fulfills my definition of object? Or perhaps you figure to reference time by some other means. Think about it.

 

No I cant say that I have ever doubted that your clock was an object. Actually I think I have been misdirecting my attention away from why you defined an object. I have been under the impression that the reason for defining an object was because you knew that you could define an object so that it would obey the relation [imath] r=v_?t=\sqrt{x'^2+y'^2+z'^2+\tau'^2} [/imath] and I have been overlooking the fact that you defined an object so that you could simply construct a vary useful object called a clock.

 

Taking a closer look at the relation [imath] r=v_?t=\sqrt{x'^2+y'^2+z'^2+\tau'^2} [/imath] it is actually satisfied by any element or collection of elements as long as the Dirac delta function [imath] \delta(\vec{x}_i -\vec{x}_j) [/imath] can be ignored and so has no effect on the elements outside of the object. I point this out because I have been considering anything that satisfies this relation to be an object. The problem is that any collection of elements that might have been of interest up to this point and many that aren’t could be considered an object actually now that I think about it, I can’t think of anything that would not obey this relation. Because if we just look at the elements over a small enough change in t (the evolution parameter) we can consider any collection of elements to obey this relation.

 

After some thought on the matter I have to wonder if considering an object to be any collection of elements that obey the relation [imath] r=v_?t=\sqrt{x'^2+y'^2+z'^2+\tau'^2} [/imath] is an entirely rational stance as there are defiantly collections of elements that could be considered an object under this definition that would not be considered an object for any other reason then that they satisfy this relation. You could for instance consider completely unrelated things to be part of the same object.

 

Yes the point that I’m trying to make is that no matter what frame we choose to use to make our diagram, the length of S in the diagram will be exactly what the reading on the clock that we are using for the rest frame in our diagram is multiplied by a scaling factor only needed to use the right units. My understanding is that we have been using the same units for every measurement to keep things simple.

 

Hmmm.... I have to say I do not understand what you are saying there... :I

 

Oh yah, sorry about that. I think that I have been overlooking a few things there. Firstly that units are an issue only of an explanation. We will never have a need to define a unit of measure only how they relate to each other and how they are defined as we will never actually make a measurement.

 

Also originally the fundamental equation was derived in a Euclidean coordinate system but I never considered that the only thing of impotence was continuity and that it was Euclidean. Even the fact that all axis’s have the same scale was set up for self consistency reasons although we never defined that scale. Meaning that when we define a clock we have defined a measure of the x,y,z coordinate system at the same time. We have not defined what we call these measurements but we have given how we can define these measurements. That is, our clock can just as easily be used as a measuring rod as it can be used to measure [imath] \tau [/imath]. Which is something that I have not been paying much attention to.

 

 

Actually considering that the only triangles of interest are right triangles with any two more pieces of information we can solve for any information that we want uniquely with the exception of both being angles (scale becomes an issue). All that we would need is the use of the Pythagorean theorem and the use of the trigonometric functions that have been used so far and their inverse functions. But there’s really no need to derive the necessary transformations we would probably just be wasting time at this point. Just thought I’d point that out though.

 

...or there. Sorry :P

 

Again I’v been overlooking the fact that you can’t define a measurement of one axis without defining a measurement of all of the axis’s. What I have really been wanting to do is find the value of [imath]\sin (\theta ) [/imath] with defining only a clock or measure of length. Something that I must say after considering what I am in fact suggesting, must conclude that it is quite impossible as, as soon as you define a measure of one axis you define the others. The point being that the speed of light is not a fundamental dimension but a consequence of defining a measure distance and defining a clock. As both are needed to measure speed or compare speeds.

 

No, the point is that “simultaneity” is a fictional concept anytime the two “simultaneous” events are not in exactly the same position. “Simultaneity” is only defined by interaction and interactions only occur when [imath]\delta(\vec{x}_i-\vec{x}_j)\neq 0[/imath].

 

So any one will consider events to be simultaneous if either their clocks say the same thing or if they believe that a signal from both events has traveled the same distance to reach them and they arrive at the same time. The problem is that both definitions require that either distance or what we mean by time have been defined.

 

That is in order to conclude that two signals have traveled the same distance we must define distance which implies a measure of [imath] \tau [/imath]. And in order to conclude that two clocks are set to the same reading we must define just what we mean by their reading and define how we can set them to the same reading. Which will require that we know how long a signal has been traveling to reach us or how far we have moved in the [imath] \tau [/imath] direction which again means we set a measure of distance and [imath] \tau [/imath].

 

Either method will also require that we know what the speed of the oscillator is measured to be. In the rest frame of the fundamental equation this is a straight forward task after a clock is defined as all elements will move according to the equation [imath] r=v_?t=\sqrt{x'^2+y'^2+z'^2+\tau'^2} [/imath] but in order to understand why a moving observer will measure the speed to be the same we must understand that in order for him to measure the speed of the oscillator he must measure the movement in two directions and assume that it travels the same speed in both directions. He will then use his measure of [imath] \tau [/imath] to finds its speed. We can also take the stance that since there is no way for the moving observer to know if he is in the rest frame or not, so he can assume that he is and the fundamental equation must still be valid.

 

Also the relation [imath] r=v_?t=\sqrt{x'^2+y'^2+z'^2+\tau'^2} [/imath] requires all elements to move the same distance over a change in the evolution parameter t so that the only reason that any objects can interact is because that they are totally smeared out in the [imath] \tau [/imath] direction otherwise the elements would be at totally different locations on the [imath] \tau [/imath] axis which would mean that they would never interact. This also has the effect of making it impossible to measure length along the [imath] \tau [/imath] axis so that all that we can measure is the distance moved in the [imath] \tau [/imath] direction (actual location has no meaning anyhow).

 

This leads us to conclude that the speed of the oscillator is fixed as once we have defined an object to be a unit rod we will use the displacement in the [imath] \tau [/imath] direction to define speed and since over any period in which the Dirac delta function can be ignored we can derive the Lorenz transformation. We conclude that no matter how we define distance the Lorenz transformations is a requirement of self consistency.

 

How our measurements change over time, and proving that two distances are the same, is an issue of our explanation not an issue of if Lorenz contractions will take place. Hence as long as we understand how our clock will change over time (a issue of no impotence to us as it is an issue of our explanation) in relation to previous measurements we can use it to measure distance and time.

 

In short what this is about is that originally when you derived the fundamental equation you assumed that all points could be mapped into a Euclidean space. You didn’t however define a coordinate system to use to define locations in the space. The fact is we are free to choose any thing we want to define our coordinate system but it must be part of the explanation . While we are free to choose anything we want we should choose something that will stay together and is for practical proposes stable. Otherwise our explanation will have to include this and could easily become incredibly complicated. This is I think your reason for defining an object.

 

Now we must define a clock and distance to give speed any meaning, actually we define length at the same time as we define a measure of [imath] \tau [/imath](a clock). Now in order to preserve the constant speed of all elements in the fundamental equation the Lorenz transformation becomes a consequence needed to remain consistent.

 

We can now either observe that a moving observer will have no way to conclude that he is not at rest and so will conclud that he is at rest, or that he can only measure a two way traveling of a signal and so will measure the same speed for an oscillator from his frame (I suspect that there are other ways to arrive at this conclusion). Either way he will have no way to tell his frame from any other frame so will arrive at the same conclusions that we do when observing other fames. The idea of simultaneity is then an idea made up to explain what the reading of a clock was when events happened. The idea of what events happen at the same time is entirely an issue of how you define length and time.

[Turtle]

...

Aaaand still about your comment regarding the "preferred frame". Before the[re] exists any notion of "space", there must exist definitions for "persistent objects" (it is their existence that gives any meaning to space). Without defining any persistent entities, there is no meaning to any inertial frames, preferred or otherwise. I.e. we are talking about features that exist in our world model, in our head. ...

-Anssi

 

i think hofstadter disagrees and has space and object as a simultaneity. neither figure or ground is "first". this following source is just a review of GEB mind you, but it goes to the facts of the matter. whether this nuance is of any consequence to your larger discussion is for you to decide of course, but your comment struck a chord and i have a weakness for turtles. ;) :shrug:

 

Gödel, Escher, Bach: an Eternal Golden Braid / Douglas R. Hofstadter

...A simple example is the dialogue preceding Chapter III. Called “Sonata for Unaccompanied Achilles”, this dialogue directly quotes only one character, Achilles, while he converses over the phone with his friend the Tortoise. They talk about the mathematical notion of figure and ground: how, by defining one subset of a given set, you implicitly define another subset of that same set -- the part that is not included in the first subset. In the visual arts, this is best exemplified by Escher's Mosaic lithographs, where the shapes that form the background for a group of black “phantasmagorical beasts” define another set of figures, in white. The musical example that Hofstadter uses is Bach's Sonatas for Unaccompanied Violin, where the listeners' imagination fill in “between the notes” as the violin plays, and one often imagines hearing the accompanying piano. But the form of the dialogue pulls the very same trick, as the reader can easily imagine the Tortoise answering Achilles at the other end of the line!