absolute space

[Pete]

XGen,
Your experiment is interesting, but it is useless to claim that it will produce a certain result if you haven't actually done it. We can sit here all day and say why we think it will have a certain result... but that doesn't demonstrate anything.

Accepted theory (backed up by other experiments) suggests that the result from your experiment will be different to what you claim it will be. If you do the experiment and can show that your claimed result is the actual result, great! Until then, it's just hot air.

 

[Xgen]

XGen,
Your experiment is interesting, but it is useless to claim that it will produce a certain result if you haven't actually done it. We can sit here all day and say why we think it will have a certain result... but that doesn't demonstrate anything.

Accepted theory (backed up by other experiments) suggests that the result from your experiment will be different to what you claim it will be. If you do the experiment and can show that your claimed result is the actual result, great! Until then, it's just hot air.

Before to make an experiment we should postulate limitations and goals of this experiment. Otherwise it make take us 1000 years to correct inprecisions and wrong interpretations of poorly maded experiments. Of course you are free to disagree to my opinion but you should provide logical and mathematicals proofs for it.

L is the length of the leg in question
c is some value (with dimensions of speed) that is assumed >> v
v is the speed of the apparatus WRT the ether
θ is the angle that the leg in question makes with the direction of motion through the ether
t is the round-trip time of flight of the light in the leg

t = (2L/c)[1-(v/c)2cos2θ]

So, for two legs perpendicular to each other, the time of flight difference Δt is:

Δt = t2 - t1 = (2L2/c)[1-(v/c)2cos2θ2] - (2L1/c)[1-(v/c)2cos2θ1]

Assuming for the time being that L is invariant and equal for both legs and that the legs are perpendicular gives:

Δt = (2L/c){[1-(v/c)2cos2θ] - [1-(v/c)2cos2(θ + π/2)]} = (2L/c){[1-(v/c)2cos2θ] - [1-(v/c)2sin2θ]}

Can you please put some diagrams for it. Why
t = (2L/c)[1-(v/c)2cos2θ] ?

There is something that I dont understand for MM and related experiments. The source of light S that enters the beamsplitter, is this continual source of light or it is impulse source (like laser)? Because if it is continual it is very bad. They had blurred the time-shifts. Also it is very bad if the source is isotropic (i.e irradiates light in all directions). Then different photons with have different time-shifts from both plates will reach interferometer. The source should irradiates only in one direction , to the beamsplitter and light should be collinear beam. I suupose that they had made this.

However even if this two flaws had been corrected the entire device is not good for positioning in absolute space because as I explained before light is moving in many directions and in this way averaging of the light paths and times is maded. Accualy I found an article about the Miller experiment that show that MM experiment have at least two serious flaws and results from Miller experiment are quite different, see it at

http://www.nov55.com/mrl.html

But I dont know for more recent experiment.

I think considering light in terms of photons makes this device impractical/senseless from the start. This device relies on the wave nature of light and disregards the particle nature. Is this a reason why you have a problem with the device?

This is one from the reasons. The other is that the light movement is bi-directional.

You are forgetting about the perpendicular leg of the apparatus...

OK, lets consider the diagram for the MM experiment. You see, I am not claming that there is no relativity (time-dilation and lengh-contraction). The idea for absolute space is not with contradiction to relativity. I will demonstrate you how the null-result of MM can be explained. Lets suppose that at some moment θ is 0 which means that Earth is moving exactly parallel to one of the arms of the apparatus. Lets it velocity be v. Lets consider what distances will pass light in the frame of the absolute space for both arms. For the arm uni-directional with v it is clear that:

d1 = [c/(c-v) + c/(c+v)]L = 2L/(1-v^2/c^2)

However because lenght is contracted in the direction of movement d1 becomes:

d1 = 2Lsqrt((1-v^2/c^2)/(1-v^2/c^2) = 2L/sqrt((1-v^2/c^2)/

Lets now see what happens with the perpendicular arm. It can be shown that the shortest path between the starting point of the beam (source S at the moment it is emitted from the beamsplitter) and the end point (source S at the moment when the beam enters interferometer) is :

d2(min) = 2sqrt[L^2 + (d/2)^2] , where d is the distance that S had traveled, d = vdt = d2(min)v/c, from here:

d2(min) = 2[sqrt(L^2 + d2(min)^2 (v/c)^2/4)] ,

d2(min)^2 = 4L^2 + d2(min)^2*(v/c)^2

d2(min) = 2L/sqrt(1-v^2/c^2)

So it turns out that d1 and d2 are equal. I think that it can be proved for arbitrary angle θ.

However at the case with PASD relativity is not an obstacle because both photons are with opposite directions and relativity is same for them. I think that relativity effects are frequently perceived incorectly, and this principle for simultaneity (or whatever it is called) is not interpreted correctly. I dont know how it had been postulated at the beggining from Einstein but if it means that everything is happening simultaneously in all directions independently from the velocity v in absolute space , it is wrong. In this way the frames of reference are something like chunks of absolute spaces which are not related in any way. May be I am wrong about that, but I just can not believe that such absurd thing had been widely accepted and that it is based on so poorly realized ezperiment as MM. I think that the general conclusions in SR can be received without the principle for simultaneity.

I don't see how an interpretation of the HUP proves anything, much less that everything moves at the speed of light or any other speed for that matter.

If space and time are discretized the speed becomes quantum quantity and everything can move for one quantum of time only with velocity c or velocity 0. Axxeleration also becomes quantum quantity and for one quantum of time it can be c or 0.

So do you say that photons do not go with velocity c in all reference frames?

I say that the absorption events are not happening at the same time. Observations for them - may be.

When photons move with c in all reference frames, they moves with c and in what I had defined as absolute space. Just because they move with c events can not hapen at the same time. If for example photons moves with c-v against v and with c+v in the direction of v, it would be fullfiled what you claim - that both events are simultaneous in the moving frame of reference. How you will explain that photons are absorbed at the same times but still are moving with c.

If I had make some mistake up to now I will apologize to you all, but first you must show me where I am wrong.

 

[errandir]

Before to make an experiment we should postulate limitations and goals of this experiment.

I totally agree. This was in fact done in the case of the MM experiment, among countless others.

Can you please put some diagrams for it. Why
t = (2L/c)[1-(v/c)2cos2θ] ?

I would show diagrams, except that, from some of the quotes that follow, I think you have a good idea of of basic diagram for the apparatus. This is basically the generalization to an arbitrary angle from the plane of motion, as you have requested (in your previous post). There is actually a slight difference, though. In the most popular version of these formulae, the perpendicular beam is actually <i>not perpendicular</i>. My formulae assume that the beams <i>are perpendicular</i>. This makes the equations slightly different, but, I am not going to waist any more time discussing the MM experiment until we get something else straight (see some of the following responses).

There is something that I dont understand for MM and related experiments. The source of light S that enters the beamsplitter, is this continual source of light or it is impulse source (like laser)?

I don't know all of the hairy detials about the MM exp. (and frankly, I don't really care all that much), but I'm almost positive that the source was continuous <i>and</i> like a laser. I don't understand why you think a laser is not continuous. What is an "impulse source?"

Because if it is continual it is very bad. They had blurred the time-shifts.

Can you explain this further? If you mean what I think you mean, then you are correct to say that the time shifts were blurred in the sense that they could only resolve their measurements to some fraction of a light fringe. However, this "blur" was far less than the amount of discrepancy that they found with the prediction of the ether theory. The first time Michelson ran the experiment on his own, he concluded that his resolution was in fact "too blurry" to make any conclusion. But, when he reran the experiment with Morley, they made drastic improvements to the apparatus and to their technique, which eliminated this ambiguity. Later, up until the 1930's this was somewhat of a popular experiment to improve upon and then run, with a trend of increasing confidence in the results.

Also it is very bad if the source is isotropic (i.e irradiates light in all directions).

Again, I'm not sure about this detail, but I'm pretty sure the light source was collimated and highly directive. Otherwise, it would not make any since at all. Regardless of how they have done it, when I ran the experiment, I used a HeNe source (which emits a coherent laser). In principle, I'm almost positive that the source should be this type.

... I found an article about the Miller experiment that show that MM experiment have at least two serious flaws ...

I'll have to read this after I'm done posting.

<i>EDIT:
Right off the bat, I did not recognize the journal: APEIRON. Also, most of the citations were from journals that I did not recognize. Furthermore, call me a bigot if you like, but APEIRON seems to have a somewhat questionable review board, with most members being from the Balkans or Mexico. I didn't see one member from the US, Germany, England, or Japan, for instance, though I did notice one from Russia.

In light of these things, I did not read the article in detail. But it did bring to light a certain issue that I should investigate further, I suppose: I did not see reference to the small-time-scale rotation of the apparatus in the MM exp. The article seems to focus on systematic errors induced by large-time-scale averaging, which I don't think should even be an issue in the MM exp. Like I said, I'll have to look into this, perhaps, when I have spare time.</i>

The idea for absolute space is not with contradiction to relativity.

Say WA--! Are you talking about Galilean relativity then? When I say relativity, I mean "Einsteinian" or "modern" relativity.

I will demonstrate you how the null-result of MM can be explained.
...
... because lenght is contracted in the direction of movement d1 becomes:
...
... it turns out that d1 and d2 are equal.

This is exactly the Lorentz contraction to which I alluded. I absolutely do not wish to discuss this. It demonstrates that you should probably re-study you late 19th centuray science history and also (modern) relativity from the basics (i.e. the fundamental postulates and the motivation from the inconsistency between Maxwell's equations and Newton's laws).

... both photons are with opposite directions and relativity is same for them.

What do you mean? Can you please try to rephrase this?

I think that relativity effects are frequently perceived incorectly, and this principle for simultaneity (or whatever it is called) is not interpreted correctly.

Maybe you should consider this complaint reflexively.

... I just can not believe that such absurd thing had been widely accepted and that it is based on so poorly realized ezperiment as MM.

Of what absurd thing do you speak. Keep in mind that scientists did not simply stop performing experiments over 100 years ago.

I think that the general conclusions in SR can be received without the principle for simultaneity.

I think that we probably have different ideas about what is the "principle of simultaneity." Would you care to elaborate?

If space and time are discretized the speed becomes quantum quantity ...

Well, OK, I can't argue with this. I don't know anything about quantizing space and time, though I am suspicious that this is not legitimate. At any rate, there seem to be other more pressing issues than this. But, if you would like to explain the mechanism for this discretization, I'm all ears.

When photons move with c in all reference frames, they moves with c and in what I had defined as absolute space. Just because they move with c events can not hapen at the same time. If for example photons moves with c-v against v and with c+v in the direction of v, it would be fullfiled what you claim - that both events are simultaneous in the moving frame of reference. How you will explain that photons are absorbed at the same times but still are moving with c.

Wow, I didn't understand any of that. Please rephrase.

... you must show me where I am wrong.

All right, now I'm going to have to assume that you do not mean exactly what you say with this. But, just out of curiousity, how much do you think that you are paying us? If it's more than nothing, I would sure like to know when I'm going to get my cut.

 

[Crisp]

If I had make some mistake up to now I will apologize to you all, but first you must show me where I am wrong.

After the proof by induction, the proof by first principles, now: the proof by not being proven wrong. Reminds me of "the proof by general consensus" in other threads.

 

[errandir]

... now: the proof by not being proven wrong.

Oh, come on. Don't you know this is the way science is done these days by the young, green wannabes, and therefore we should all learn to adapt, lest we become old, not-green, and obsolete. It's not like this is the first thread on a physics forum that has engaged this procedure.

 

[Xgen]

Well, first I would like to make some remarks. It appears that modern theory for relativity deals with not what really is happening but with what is observed. And that is two quite different things.

Whatever. Can someone tells me what in modern theory for relativity rejects the existing of absolute space? Why absolute space is imposible to exist according to you.

I will try to explain why there should exist absolute space.

Lets for example have three frames - K1,K2 and K3, moving with velocities v12,v23 and v13 WRT each other. Since velocities are different (because that is the only thing that makes the frames distinguishable) it can be found such a frame in which the total velocity to all other frames is smallest , if for example v12 >v13 , and v23 >v13, then (v12+v13) < (v12+v23) and (v12+v13) < (v23 + v13) , so K1 is with the smallest total velocity. You can easy see that for all other combinations there is frame with minimal total velocity.

Lets have four frames - K1,K2,K3,K4 . We apply the following procedure. We first take 3 from them, for example K1,K2,K3 and remove the one with the biggest total velocity, for example K3. Now we remain again with 3 frames - K1,K2 and K4, and again a frame with the smallest total velocity can be found. Then finally a frame that moves with a smallest total velocity to ALL frames will be picked and this is an imediate prove that absolute space should exist.

errandir,

thank for your profound reply but I am not going to discuss trivial things with you about what is Gallilean and Einstein relativity and so on, this can be read in books. I think however that this Einstein relativity is quite old-fashioned and I dont understand why you call it "modern". I can not pretend that I had studied very well GR and that I know all the terminology in it (I found it for rather boring) but my own conception for relativity tells me that both photons can not be absorbed at the same time. I didn't found similar experiments in the net so I think that the topic is interesting. If you like to consider this device seriously then lets continue. If you think that I am, not educated enough (this is probably true) then we can finish this discussion. You had shown that you are quite reasonable and well informed person so your respectfull presence will be evaluated in the future. My discussion with you was very enlighting. Thank you. (also if you dont have spare time my least intension is to "get you cutted', so better keep working and dont lose time and money with uneducated persons like me).

 

[Crisp]

Well, first I would like to make some remarks. It appears that modern theory for relativity deals with not what really is happening but with what is observed. And that is two quite different things.

Incorrect. In science, the empirical fact i.e. what you observe in experiments, is considered to be "reality". Special relativity as a theory predicts what someone would observe, i.e. how he would experience reality. It requires some bending of the mind to get this idea.

Lets have four frames - K1,K2,K3,K4 . We apply the following procedure. We first take 3 from them, for example K1,K2,K3 and remove the one with the biggest total velocity, for example K3. Now we remain again with 3 frames - K1,K2 and K4, and again a frame with the smallest total velocity can be found. Then finally a frame that moves with a smallest total velocity to ALL frames will be picked and this is an imediate prove that absolute space should exist.

Of all the "absolute space" and "aether must exist" threads on this forum -- and believe me, as the oldest posting member I have seen many -- I must admit that this is the best argument I have seen so far, great work!

However, aren't you using the classical addition of velocities here, assuming that what is the slowest frame for one is also the slowest for another ? Two observers would agree on their mutually respective velocities (if I move with velocity +v with respect to you, you move with velocity -v with respect to me), but is this true when you include a third observer ? I am pretty sure it is not, because the relativistic velocity addition formula also includes the velocity of this third observer.

I have to think on it if this argument is applicable, but I think the problem lies there.

Bye!

Crisp

 

[errandir]

Lets for example have three frames - K1,K2 and K3, moving with velocities v12,v23 and v13 WRT each other.

So far so good.

... it can be found such a frame in which the total velocity to all other frames is smallest ...

Can you define "total velocity," please?

... if for example v12 >v13 , and v23 >v13, then (v12+v13) < (v12+v23) and (v12+v13) < (v23 + v13) ...

I don't see how you got the last inequality. The third inequality follows trivially from the second, I agree. But the last inequality has no logical connection to any of the previous three that I can see.

We first take 3 from them, for example K1,K2,K3 and remove the one with the biggest total velocity, ...

Again, we're going to need a definition for "total velocity" and a qualification for it to be the biggest.

... I am not going to discuss trivial things with you about what is Gallilean and Einstein relativity and so on, ...

Well, they are trivial to some people. However, I sincerely (and without derision) believe that you may be confused about that distinction. Einstein's relativity (1905) strictly and explicitly forbids the existence of absolute space. If you think that these theories are trivial, then how can you have overlooked this basic premise.

I think however that this Einstein relativity is quite old-fashioned and I dont understand why you call it "modern".

I call it modern because I have never heard of an experimental disproof performed between 1905 and 2004 (that I have found to be valid).

I can not pretend that I had studied very well GR ...

I don't think we need to make this an issue. No problem.

I didn't found similar experiments in the net so I think that the topic is interesting. If you like to consider this device seriously then lets continue.

Ultimately, this is the main point. I would sincerely like to hear about the experiment when you have run it. But, until you do, I would also like to know that you realize the firm ground on which SR stands within its admitted limitations.

 

[Quantum Quack]

I am sorry guys and maybe my pc is playing up but I have not posted to this thread prior to this, AN EARLIER post quotes me as the author and I can not make claim to that.

"Originally Posted by Quantum Quack

Michelson and Morley discovered that velocity of light is equal in all directions. But I dont think that it means that absolute space do not exist, just the opposite. The ether they were ....." etc etc

 

[Crisp]

Then finally a frame that moves with a smallest total velocity to ALL frames will be picked and this is an imediate prove that absolute space should exist.

Darn, I missed an obvious one here.

The "total" velocity of yourself with respect to yourself is always zero, hence the smallest it can get. Conclusion: the absolute frame of reference you talk about is always your own. And this is true for every frame of reference.

Contradiction.

Bye!

Crisp

 

[Xgen]

Can you define "total velocity," please?

total velocity of a frame = Sum(|vi|)/N , for i = 0 to N, where N is the number of the frames, |vi| - the absolute value of the magnitude of the velocity between both frames. The direction of the velocity do not matter.

The total velocity remains finite even if N is infinity because |vi| can vary from 0 to c.

I don't see how you got the last inequality. The third inequality follows trivially from the second, I agree. But the last inequality has no logical connection to any of the previous three that I can see.

Take a sheet of paper and a pencil. Draw three points - A,B and C , such that they do not lie on a line, and distances AB,BC and AC are different. Now connect them. The triangle ABC express what I explained before. AB correspond to v12, AC to v13 and BC to v23. The A, B and C corresponds to the 3 frames. The total velocity of A is (|v12| + |v13|) and corresponds to the AB+AC, of B - AB + BC, and of C - AC + BC. Now it is clear that if AB diff BC diff AC then the 3 frames can be "sorted", the absoluteness of a frame is measured with the value of the total velocity, the frame with minimal total velocity is most absolute.

If we have more then 3 frames? Now we have polyhedral and the total velocity for every point is the sum of all distances in which this point participate. Accualy not always only one point will have minimal total velocity, so if two points have equal total velocity they are equal absolute.

Accualy if new frames are added we will get closer to the absolute space. The reverse of the absolute space is the most relative space - the frame with the biggest total velocity. Photons accualy are examples for this space. They are moving WRT everything with velocity of c, so if we add one photon to our group of frames it will be the winner for the most relative frame. A black hole accualy is the body which is closest to the absolute space.

As you can see the idea is very simple.

Darn, I missed an obvious one here.

The "total" velocity of yourself with respect to yourself is always zero, hence the smallest it can get. Conclusion: the absolute frame of reference you talk about is always your own. And this is true for every frame of reference.

If you have only one frame it is the most absolute, yes.

If you have two frames they are equally absolute. You need at least three frames to determine different levels of absoluteness and relativity.

Accually if the concept of absolute space and measures of absoluteness for the frames are used, many problems in SR and GR will be solved very easy (I mean twin-paradox and etc.).

 

[Crisp]

If you have two frames they are equally absolute. You need at least three frames to determine different levels of absoluteness and relativity.

Equally absolute ? I thought the whole point of absolute frames was to have one basic frame of reference, what's the point of having equally absolute frames. If you have two absolute frames, then they are relatively equal aswell, no ? :bugeye:

 

[errandir]

|vi| - the absolute value of the magnitude of the velocity between both frames.

What is the "velocity between both frames?" Do you mean the velocity of one frame WRT another/the other frame? That's what it looks like you mean, so I'll continue responding under this assumption.

Take a sheet of paper and a pencil. Draw three points - A,B and C , such that they do not lie on a line, and distances AB,BC and AC are different. Now connect them. The triangle ABC express what I explained before.

Done. However, one example of a drawn triangle does not prove an algebraic relationship. You need to draw all generally representative examples, of which I can think of two.

AB correspond to v12, AC to v13 and BC to v23.

If this is so, then you have allowed me to draw a triangle that does not agree with your inequalities. You need to specify the triangle more strictly. Just to let you know, my triangle has AB = 5.0", AC = 3.5", and BC = 4.4". And another thing: the specification that you have given for the triangle seems to allow for infinite velocity, unless you are somehow identifying c with infinity.

total velocity of a frame = Sum(|vi|)/N , for i = 0 to N, where N is the number of the frames, ...

The total velocity of A is (|v12| + |v13|) ...

These seem to be contradictory. Surely you meant to divide by 2?

Accualy if new frames are added we will get closer to the absolute space.

"Closer?" How far away can we be?

The reverse of the absolute space is the most relative space - the frame with the biggest total velocity. Photons accualy are examples for this space. They are moving WRT everything with velocity of c, ...

Do you mean this to be the complement? I don't follow what you mean by reverse.

A black hole accualy is the body which is closest to the absolute space.

How close is it?

As you can see the idea is very simple.

And I can't make sense of it! I must be an idiot.

If you have only one frame it is the most absolute, yes.

And it exists in a trivial space that only has one point. There are approximately an infinite number of frames in any real space that comes to mind.

... many problems in SR and GR will be solved very easy (I mean twin-paradox and etc.).

The twin paradox is already solved "very easy." Let's hear about the "etc.," shall we. Oh, and let's not get into GR; I don't see the need in this discussion.

 

[Xgen]

What is the "velocity between both frames?" Do you mean the velocity of one frame WRT another/the other frame? That's what it looks like you mean, so I'll continue responding under this assumption.

Yes, what else it can be.

Done. However, one example of a drawn triangle does not prove an algebraic relationship. You need to draw all generally representative examples, of which I can think of two.

If this is so, then you have allowed me to draw a triangle that does not agree with your inequalities. You need to specify the triangle more strictly. Just to let you know, my triangle has AB = 5.0", AC = 3.5", and BC = 4.4". And another thing: the specification that you have given for the triangle seems to allow for infinite velocity, unless you are somehow identifying c with infinity.

Every frame correspond to a point and every velocity to a line with finite lenght (because velocity is always smaller then c) . I can not think about more simplest illustration. To every 3 frames with velocities v1,v2 and v3 I can always map a triangle. I am not interested from velocities directions only from theirs magnitude so the orientation of the triangle sides have nothing to do with velocities directions. Is there some problem with it? Why you think that velocities can be infinite? The triangle can not have infinite side.

These seem to be contradictory. Surely you meant to divide by 2?.

Yes Sorry.

"Closer?" How far away can we be?

By closer I mean "moving with smaller velocity WRT ..."

Do you mean this to be the complement? I don't follow what you mean by reverse.

I thought that it is clear. The winner for the frame with smaller total velocity is the most absolute frame. The winner for the frame with biggest total velocity is the most relative frame.

The twin paradox is already solved "very easy." Let's hear about the "etc.," shall we. Oh, and let's not get into GR; I don't see the need in this discussion.

You are more familiar with GR then me. Find out by yourself. My point here is that there are instances when more then just one or two frame of reference are needed. If we have only 2 we can not specify which is more absolute. And we can not specify which one is the rest frame. We need a third frame to do this. It is like the trains-paradox. Two trains are standing at a station. At some moment one from the trains starts to move. But an observer in the other train will not be able to say (if he do not have some additional information) if his train if moving or not ? He need to have a view to the station - the absolute frame in this case (or he needs to call with his phone to a person that is standing at the station).

 

[errandir]

To every 3 frames with velocities v1,v2 and v3 I can always map a triangle. I am not interested from velocities directions only from theirs magnitude so the orientation of the triangle sides have nothing to do with velocities directions. Is there some problem with it?

I agree that they map into a triangle. But there is still a problem with the last inequality that you presented in one of your earlier posts. I would just like to hear one of two things: 1) "whoops, that last inequality was a typo/mistake/etc." or 2) "here are the other assumptions that I am making that lead to this inequality..."

Why you think that velocities can be infinite? The triangle can not have infinite side.

I don't think that they can be. I was just taking your triangle specifications as stated and identifying that as an unresolved specification. So, do you simply limit the length of the sides to some value, c?

You are more familiar with GR then me.

How many ways do I need to indicate that I don't care about GR in this discussion? I don't understand why you feel the need to keep mentioning it.

My point here is that there are instances when more then just one or two frame of reference are needed. If we have only 2 we can not specify which is more absolute.

But this isn't an issue. There are an infinite number of distinct frames of reference. And there is always only exactly one of these that is directly accessible to a given observer. SR doesn't deny this.

... an observer in the other train will not be able to say (if he do not have some additional information) if his train if moving or not ? He need to have a view to the station - the absolute frame in this case ... .

It suspiciously appears as though you have absolutely no appreciation for SR (1905). The station is NOT an absolute frame in SR. Now, what it looks like you're trying to do is to artificially complicate SR to create a need to simplify it.

 

[Xgen]

I agree that they map into a triangle. But there is still a problem with the last inequality that you presented in one of your earlier posts. I would just like to hear one of two things: 1) "whoops, that last inequality was a typo/mistake/etc." or 2) "here are the other assumptions that I am making that lead to this inequality..."

If you mean that earlier I writed (v12+v13) < (v12+v23) and (v12+v13) < (v23 + v13), I did mean accualy modules of the velocities. Sorry if that is so big problem for you.

I don't think that they can be. I was just taking your triangle specifications as stated and identifying that as an unresolved specification. So, do you simply limit the length of the sides to some value, c?

Absolutely. My triangle can not have infinity side because it will no more be triangle (maybe two parallel lines). But it can accually have a side 0. It is when two frames are in rest WRT each-other.

I will demonstrate you how in the case with both trains Earth happens to be the most absolute frame. Lets have 2 trains moving with velocity 40 km/h one agains other, and since they are moving on the Earth of cource, we have three frames of reference - two for the both trains and one for the Earth. Lets find out the total velocities. The total velocity of the first train is (|40| + |80|)/2 = 60. For the second train is the same. For the Earth is (|40|+|40|)/2 = 40. So in this case the Earth is the frame with smallest velocities and should be refered as the most absolute frame. The two trains are equally relative.

Same is and with twin paradox. Lets suppose for simplicity that the second twin was not sended into space but stays on Earth but is moving around it with velocity closer to c. How to determine which twin is resting and which is moving? Since the Earth is in rest WRT first twin, we take two points from it and assign them as different frames with zero velocity. Then they two will be winners for frames with minimum total velocity and should be refered as the most absolute frames then the frame of the second twin. You will probably ask why we dont take two points from the second twin and why not assign them frames? Then just the reversed result will be obtained. However the second twin is with neglectible size compare to Earth. Also it is stated that he moves between two points on the Earth. If we continue with this logic we may come to something similar to Ahiles and turtle paradox, and to the final conclusion that space and time should be discrete. From a pure continual point of view this paradox can not be solved because we have only two frames.