Light shone in a train bouncing off mirrors

[jefswat]

That is also conceptually wrong. Time is neither absolute nor relative, because it does not exist. Time, I agree, is what is measured as such. The rest of the concept that is usually intermixed with measured time in common language is the idea that things happen, “reality”. But I agree that in science reality must be kept as the north (if only we could plug it into the equations!) and then we must work and do mathematics with measured time, that is to say, the periodic motion of “objects” (whether mechanical or electromagnetic) within some portion of space that encapsulates it.

That may or may not be the case. But either way it has proven itself a very useful concept

 

[Saw]

That may or may not be the case. But either way it has proven itself a very useful concept

I agree. Really, this sort of statements (“time does not exist”) look too radical and may create controversy.

What I simply mean is:

If we take, as usual, the statement that two events are “simultaneous”:

- I look at one side and what do I find? The ORIGIN of the statement = The value measured by a clock. This is a reality, an event, a fact, a truth as big as a cathedral and of course it exists.
- I look at the other side and what do I find? This value, after combination with other values, leads to the PREDICTION of another event, which is also factual and true and of course it exists.
- And in the middle? Any other event or fact or truth? No, in the middle there is nothing.

So the statement “do you agree that this judgment of simultaneity is true and correct?” must be translated into “do you agree that this measurement value, after adequate combination with other values in accordance with certain formulas, must lead to the prediction of a certain fact?”

If the values that you are able to measure happened to be homogeneous for all observers, regardless their states of motion, the formula (the Galilean Transformation) would be easy to draft and of course it would predict facts.

If the value that, in practice, you obtain varies for different observers, the formula (the Lorentz Transformation) is more complicated to achieve, but if it is well drafted and you apply it judiciously, it will have equal predictive power.

So there is nothing “onthological” here. It is not a question of saying simultaneity is relative or absolute in radical terms. It is a practical issue. Hence of course time, in spite of being a mere concept, is practical, because if it is not practical, it is not time, it is bad measurement or bad mathematics.

 

[JesseM]

Ok, if you wish to continue the analysis (after the turmoil created by Althon), I'll leave aside subtleties on words and put things in a less baroque manner.

What was the goal? To discuss about what “the difference in judgments about simultaneity” means, just in case it is in the interest of physics.

1) First, “simultaneous” in a given frame means that:

if we have two clocks at either end of the train which have been synchronized in the train's frame using Einstein's convention, they will both read the same time when the lasers are fired next to them, but if we have two clocks on the ground which have been synchronized in the ground frame using Einstein's convention, and both clocks happen to be right next to the two duellers at the moment each fires his laser, then these two clocks will show different times when the lasers are fired next to them.

You specify the origin of the clock readings, how they have been obtained (the clocks have been synched through the Einstein convention and have ticked afterwards at the corresponding rate).

(A couple of footnotes, about obvious things, which I note just in case they are useful:

- That is important to remember, because it is part of the physical content of the definition. We do not know how variations of that convention might affect the outcome (it would depend on the nature of the variations), but let us just note it.

Well, the physically important thing about this convention is that if all inertial observers design their coordinate systems in this way and then figure out the correct equations for the laws of physics as expressed in their coordinate systems, they'll all get identical equations. This is a real physical symmetry in the laws of physics, "Lorentz symmetry" or "Lorentz invariance"; we can imagine alternate laws of physics where this wouldn't be true. Observers could choose a different simultaneity convention which would result in a different type of coordinate system, but the equations in this coordinate system would have to look different than they do when expressed in the standard SR inertial coordinate systems.

- The observers get different values, but it’s also true that the measurements are events and so they happen in all frames and all frames agree that they happen. Furthermore, one frame can predict the quantity of the other’s measurement.)

Yes, this is true.

2) Second, we must make use of the measurements of the observers for a purpose.

(Footnote: What purpose? For me, the aim of physics is “to solve problems”, like whether a duel is fair or not.

I agree the purpose is to solve problems, but I'd say it's to solve problems about well-defined physical questions like the time interval on a given clock, "fairness" is kind of a nebulous idea...if you have already defined your idea of "fairness" in purely physical terms, like "the duel is fair if each dueller experiences the same proper time between firing their own gun and the laser from the other guy's gun reaching their position", then you can use the laws of physics to judge if the duel is fair.

3) Third, your measurements serve their purpose by combination with other measurements.

The judgments about simultaneity alone do not serve any purpose. For example, in our case, we have to combine them with another measurement: we must determine if “both duellers have an equal amount of time on their own clocks between firing their own gun and being hit by (or dodging) the other guy's laser”. Once we do it, the trick is done. In my frame, the proper time of Back when shot – the proper time of Back when shooting = the proper time of Front when shot – the proper time of Front when shooting. So the duel is fair.

(Footnote: Both referees agree that the other has correctly applied the formula and obtained, ultimately, the right solution. Maybe you could comment on technicalities of this operation that I might have missed.)

Conclusion: both judgments of simultaneity are right, in the sense that, after due consideration of their origin and due combination with other measurements, they serve beautifully the common practical purpose.

Did I do my homework? Does this look more reasonable?

I think so...I might paraphrase this by saying that coordinate-dependent judgments like judgments about "simultaneity" in a given frame are only useful insofar as they are used in calculations about coordinate-invariant physical results like the proper time between two events on a given clock. Would you say this is a reasonable paraphrase?

Also, to respond to a bit of your most recent post:

So there is nothing “onthological” here. It is not a question of saying simultaneity is relative or absolute in radical terms. It is a practical issue. Hence of course time, in spite of being a mere concept, is practical, because if it is not practical, it is not time, it is bad measurement or bad mathematics.

I agree that physics cannot solve the ontological question of whether there is an absolute present. However, if all the laws of physics are Lorentz-symmetric, this does imply that there can never be any physical basis for saying one frame's judgments about simultaneity are more "correct" than any other's. So for a philosopher, this might at least be said to lend weight to the idea that there is no such thing as absolute simultaneity, just by the Occam's razor argument that we should try to avoid postulating extra metaphysical entities that have no relevance to any empirical observations. We can't prove that there's no physically undetectable "metaphysically preferred frame" whose judgments about simultaneity are "correct" in some absolute metaphysical sense, but we also can't prove that there aren't physically undetectable gremlins sitting on the shoulder of every human on the world; if there is no pressing philosophical argument for why we should believe in such entities, one can argue that it's simpler to assume they don't exist.

 

[altonhare]

Speed is not a vector though. You claim that if different frames disagree on which of two objects has a larger value of X, this would be a logical contradiction; does this not apply when X=speed even though you think it applies when X=velocity?

This is a similar objection to the one you raised earlier with length, width, and height. I resolved it by clarifying that X must, of course, be an explicit statement containing full information. In the LWH example I said the observers were "being sloppy" if they just said "X is 4 meters long, Y is 5 meters long, therefore Y is longer than X". In reality 4m "long" is not the only relevant measurement, the observer must specify all the conditions involved in the measurement. Therefore "X is 4 m long and Y is 5 m long when X is 8 m from Y" is a full statement if X and Y are the only entities involved. This observer concludes "Y is longer than X when they are 8 meters apart". Another observer might say "X is 3 m long and Y is 2 m long when X is 7 m from Y". This one concludes "X is longer than Y when they are 7 meters apart". These aren't contradictory, they're just different.

Similarly an observer is just being sloppy if s/he declares a binary, qualitative conclusion based on only the scalar speeds. It's not an explicit statement containing all the information. In both cases, the LWH one and the speed one, the observers that come to contradictory conclusions erroneously neglect to incorporate relevant information into their conclusion. Therefore their conclusion is unjustified.

Wait, when you say it's "no different" that means you believe there is an ontological truth about which of two event "really" has a greater x-coordinate, independent of our choice of coordinate system?

There are no ontological contradictions and this scenario is no different, as I pointed out.

I was saying the scenario you pointed out was no different than any other in the sense that it contained no ontological contradictions.

As I tried to illustrate in the example, observers do not come to any absolute conclusions about length, width or height. They come to conclusions about if X is longer than Y "when X is this distance from Y". Comparing the different conclusions is comparing apples to oranges. One conclusion says X is longer than Y when they are 5 m apart, another says Y is longer than X when they are 8 m apart. To keep numbers out of it they could simply say things like "X is longer than Y when X is further from Y than Z". Nobody will arrive at a contradiction if they are explicit and specific.

Yes, and my argument is that certain quantities are inherently frame-dependent, and thus there is no objective frame-independent reality about which of two objects has a greater velocity, the answer will depend on which of these artificial constructs we happen to use.

Quantities may be frame-dependent, but two observers will never disagree about which has a greater velocity, which has greater extent in a specific direction, etc. The quantities may vary up and down but never can they cross over such that qualitative conclusions contradict.

So: do you think there is an objective, coordinate-independent truth about whether point A and point B share the same x-coordinate?

No, I don't believe in "ghostly axes", as I stated before. I only believe that there is an objective reality (A is A) and, as such, there should be no true contradictions regardless of how you examine something. Observers only contradict because they have not been specific and explicit in their conclusions or because their premises/assumptions are wrong.

If each observer uses the procedure I discuss above, then there can in fact be situations where different frames disagree about which of two objects is longer, even if they agree on the orientations of their x-axis, y-axis, and z-axis.

Justify this.

And again, do you think your claim about observers never disagreeing about which of two objects has a greater length should also apply to questions of which of two events has a greater x-coordinate? In this case, as I said we don't even need to think about relativity to see that different coordinate systems can easily disagree on this.

They will not disagree or contradict if they include all the relevant info in their conclusion.

Any mathematical description of something is a way of imagining it, even if we can't form a visual picture of it.

Visualization is the only way to explain and understand a phenomenon. Mathematics is a way of describing some phenomenon. In particular mathematics can only describe dynamic concepts/processes.

I can't visualize colors of light outside the visible spectrum but I can form a mathematical model of such light in terms of its frequency,

You don't visualize colors at all because color is not a standalone object. Color is a concept you understand via comparison. An object is something you visualize by itself. If every entity were the same color do you think we'd still say something like "it's red"? No, we'd only have a conception of "color" by comparison.

It makes no sense to talk about visualizing colors in the first place, and even less sense to talk about visualizing colors that you can't see. Color is defined in terms of sight. It's like saying I can't visualize a table that isn't a table or I can't visualize a shape that isn't a shape.

There's no problem with not being able to see something, but it should be visualizable i.e. have shape. This is the only objective criterion for determining if something is "physical".

Also, your use of the term "non-contradictory" is non-standard. You have yet to show how SR leads to any actual contradictions. All you've shown is that it contradicts your arbitrary "rule". So what?

I have shown that in no instance do observers disagree on qualitative issues if they are explicit and specific. Except in the case of "relativity of simultaneity". In this case two observers disagree that an event was simultaneous. In all other instances observers either agree or are comparing apples with oranges.

What this tells me is that spatial locality of two entities is what's important, not temporal. If temporal locality is unimportant then we can do away with the t coordinate altogether and just talk about relative motion. Now instead of talking about AB being "simultaneous" we just say that A and B came in contact.

What's difficult about this proposal is that we have to include the relative motion of the internal machinery of the clock and the photons and this process is not well understood.

That might well be true. So?

So as scientists peculiarities intrigue us and beg us to look deeper! Maybe it's just a coincidence, maybe there's no significance at all to the observation that the "relativity of simultaneity" is the only qualitative contradiction. But maybe so.

Fundamental disagreement is far more severe. All parties involved agree on everything except the final outcome. I define red to be 600nm wavelength(I made that up). We measure light of 600 nm. I think its red, you think its blue but we both agree that 600nm wavelength is red light. obviously the person who thinks its blue is just not thinking clearly. This is the kind of disagreement that leads to fatal flaws in theories and contradictions that can't be solved. It is also the type of disagreement that after 100 years, SR still have not been proven to have.

Does anyone else have a problem understanding this? Its basic philosophy.(Thats actually where I stole it from)

I see what you're talking about. In a "fundamental disagreement" one person is contradicting themselves.

Fred: I define red as 600nm on this device
Billy: I agree.
Fred: The device says 600 nm, therefore the light is red.
Billy: I disagree.

In this case, there's no disagreement because Billy is simply wrong. When a person invokes a self-contradiction the other person is justified in declaring them "wrong" and sending them back to the drawing board.

If nobody contradicts themselves, there are no "fundamental disagreements" as you called it.

Really? A is going faster than B. A is going MUCH faster than B. A is barely going faster than B. I make no mention of numbers.

You may not have in your sentence, but "much" is a subjective term. To grant it any rigorous meaning you have to define it. If you don't define it then the second statement carries the same meaning as the first. How will you define it? If A is going 10 m/s faster than B does that qualify as "much"? In order to define it unambiguously you will have to mention numbers.

In my ball example, I measure the ball to be going north at 10 m/s. You measure the ball to be going south at 15 m/s. If you still don't feel that north and south are qualitative, replace them with up and down. One observer says up at 10 m/s, the other says down at 15 m/s.

There will be no contradiction, each one just has to actually define clearly what in the world they mean by "north, south, up, and down". Once they do they will find no qualitative disagreement.

I think you need to start new and give us your theory as it stands now. You have changed your mind so many times that I don't know what you believe anymore and consequently, my arguments may be outdated and therefore irrelivent.

When observers come to qualitative, binary conclusions they will not disagree unless A) They made a mistake, such as not integrating all the relevant information (such as directionality) or B) They are talking about non colocal simultaneity

A is just human error. B intrigues me.

What else can be the final outcome, other than what you seek from the beginning = solve practical problems where real things that “exist” are involved and interact with others, producing events that “happen” and causing us practical concerns

I'm not only concerned with calculating the final answer, I want to dig out whatever significance I can find. The fact that there is only one instance where I can find a qualitative contradiction (not due to human error of course) in different frames makes me think.

That is also conceptually wrong. Time is neither absolute nor relative, because it does not exist.

In one case we have "simultaneous" events AB that everyone agrees on in every frame. In another case we have spatially separate events AC and BD that some disagree on the simultaneity of. This leads me to believe that it is spatial separation that is important, i.e. just relative motion/location. It is when entity's are colocal that "something" happens, not necessarily when they are cotemporal. Since AC and BD can be "cotemporal" in some frame but not in another, and this makes no difference in what actually happens, the "time" parameter actually seems superfluous. Colocal is always significant, cotemporal sometimes is, so why do we need the latter? What matters is, are X and Y colocal or not?

In other words, relativity of simultaneity is necessary for predicting what may happen, but has no impact itself on what may happen.

I disagree. The declaration of whether spatially separate collisions AC and BD were simultaneous or not is completely extraneous to calculating the "final result". All the observers need to calculate the final result are the pertinent numbers. In fact, one of the ways to interpret my arguments so far is that "non co-local simultaneity" is an invalid concept and spatially separate events should not be declared either way, because it has no meaning. Extending this further to colocal entities AB, we no longer need to specify a particular time (and by extension that they were "simultaneous"). The fact that these two entities are colocal depends only on their spatial arrangement and defines this "event". I conclude that the time parameter, while it may be useful for us right now because we don't understand the relative motions involved at the most fundamental level, is superfluous.

So there is nothing “onthological” here. It is not a question of saying simultaneity is relative or absolute in radical terms. It is a practical issue. Hence of course time, in spite of being a mere concept, is practical, because if it is not practical, it is not time, it is bad measurement or bad mathematics.

The ontological implication is that time is unlike length, width, and height. That it is not a central component to reality but rather a useful parameter.

 

[JesseM]

Speed is not a vector though. You claim that if different frames disagree on which of two objects has a larger value of X, this would be a logical contradiction; does this not apply when X=speed even though you think it applies when X=velocity?

This is a similar objection to the one you raised earlier with length, width, and height. I resolved it by clarifying that X must, of course, be an explicit statement containing full information.

Full information about what? About every physical aspect of the problem? Certainly specifying the velocities of two objects doesn't specify everything physical about them (it doesn't specify their positions for example). And what's more, when you say both must agree about which velocity is "greater", you only seem to be talking about one aspect of the velocity and ignoring other aspects like the angles of the two velocity vectors.

It's not even clear what you mean by "greater" when talking about two velocity vectors--you obviously aren't just talking about the norm of each vector since that would just be the speed which is always positive, but before you argued that an object with negative velocity on the x-axis had a "smaller velocity" than one at rest. If we have only a single spatial dimension, then all velocity vectors are parallel to each other so we can just talk about positive or negative velocities on this axis, but what if we have 2 or three spatial dimensions and non-parallel velocities? You can take the component of each velocity vector which lies parallel to a particular axis and then the components will each be either positive or negative, but in this case, which has the greater velocity depends on what axis you want to use.

For example, suppose we have an x-y-z coordinate grid, and we break down two velocity vectors into their x and y and z components. For example, object A has Vx = 5 meters/second and Vy = -3 meters/second and Vz = 0 meters/second, while object B has Vx = 4 meters/second and Vy = 12 meters/second and Vz = 0 meters/second. Obviously object B has a greater speed in this frame, but which has a "greater velocity"? The x-component of A's velocity is greater than B's, but the y-component of B's velocity is greater than A's. So do you claim there is some absolute truth about whether A or B has a "greater velocity" here, where "greater velocity" does not just mean "greater speed" (i.e. greater norm of the velocity vector)? If so, what is it? Please give me a specific answer to this question about whether A or B has a greater velocity here.

Also, even if we choose to focus on a particular axis like the x-axis, you didn't address my point that simply by rotating the x-axis by 180 degrees, a velocity vector which was previously in the -x direction of the old coordinate system can now be in the +x direction of the new one, whereas simply changing the orientation of a frame's x-axis without changing the frame's velocity won't cause you to change your judgments about which objects are at rest. So if in the first coordinate system object A has a negative velocity on the x-axis and object B is at rest, then by rotating the x-axis 180 degrees, now object A will have a positive velocity on the x-axis while B will still be at rest. If you claim there is some objective truth about which object has the greater velocity along the x-axis, then once again it seems you must believe in some sort of ghostly "true" x-axis.

Quantities may be frame-dependent, but two observers will never disagree about which has a greater velocity, which has greater extent in a specific direction, etc.

Your comment about "greater velocity" seems unclear as I discussed above. And in SR different coordinate systems do disagree about which of two objects has a greater extent in a specific direction, because of length contraction. Even if you think there is an absolute truth about which frame's judgment is "really" correct, do you disagree that according to the standard way of defining SR coordinate systems, disagreements between coordinate systems about which of two objects has a greater "length" are quite possible?

No, I don't believe in "ghostly axes", as I stated before. I only believe that there is an objective reality (A is A) and, as such, there should be no true contradictions regardless of how you examine something.

But you don't believe that certain quantities, such as speed, represent "objective realities", and thus you don't believe there need to be objective truths about which object has the greater speed--is that right? If so, why can't you accept the possibility that quantities like velocity or length may also fail to represent "objective realities"?

If each observer uses the procedure I discuss above, then there can in fact be situations where different frames disagree about which of two objects is longer, even if they agree on the orientations of their x-axis, y-axis, and z-axis.

Justify this.

The "procedure I discussed above" was just the standard one for constructing inertial coordinate systems in SR, and the result is that the coordinates assigned to the same event by different observers are related by the Lorentz transformation. Do you not understand that under the Lorentz transformation, even if two coordinate systems have their spatial axes oriented the same way, if the two coordinate systems are in motion relative to one another they can disagree about which of two objects has a greater length? If so I can give you a numerical example, if that's what you're asking for. But if you're asking me to justify something else, please be specific about what it is.

You don't visualize colors at all because color is not a standalone object. Color is a concept you understand via comparison. An object is something you visualize by itself. If every entity were the same color do you think we'd still say something like "it's red"? No, we'd only have a conception of "color" by comparison.

It makes no sense to talk about visualizing colors in the first place, and even less sense to talk about visualizing colors that you can't see. Color is defined in terms of sight. It's like saying I can't visualize a table that isn't a table or I can't visualize a shape that isn't a shape.

There's no problem with not being able to see something, but it should be visualizable i.e. have shape. This is the only objective criterion for determining if something is "physical".

But by "visualizable" do you mean actually being able to form a visual image of the shape in your mind's eye, or do you just mean that the shape can be defined using the language of mathematics? If we have a cartesian coordinate system with 3 spatial axes, the shape of any object can be described mathematically by giving some equations which tell you which points lie on its surface; for example, the equation x^2 + y^2 + z^2 = 1 describes the surface of a sphere with radius 1 centered on the origin, all points which lie on its surface (and only those points) will have coordinates that satisfy that equation. Similarly we can describe the shape of objects in a hypothetical 4D space with equations of 4 variables, like the surface of a "hypersphere" which has equation x^2 + y^2 + z^2 + w^2 = 1. We can't picture such 4D objects visually because our brains have become adapted to find 3D space intuitive, but I imagine if you could somehow take the brain of a newborn and feed it sensory inputs from a simulated 4-dimensional body in virtual reality, as it grew up it would be able to visualize 4-dimensional shapes. Visualization isn't a very rigorous criterion in any case because it depends on the contingent details of our biology and history, whereas mathematics allows us to define the notion of "shape" in a completely rigorous way that doesn't depend on what we can visualize (and arguably a blind person can't 'visualize' any shapes at all, although I suppose they could imagine what it would feel like to run their hands over it).

 

[jefswat]

There will be no contradiction, each one just has to actually define clearly what in the world they mean by "north, south, up, and down". Once they do they will find no qualitative disagreement.

If you are the observer that sees the ball going 15 m/s down and see your counter part going 25 m/s down, wouldn't you agree that your counterpart would measure the ball going at 10 m/s up relative to him? Don't you agree that in this case there is no quantitative disagreement once the observers clearly define the context of their measurements to one another.

Another example: Imagine looking at a 2 dimensional square in three space. There are two observers, one(1) looks on perpendicular to the plane the square is in. The other observer(2) is looking at an angle so he sees an odd looking rectangle. (2) would necessarily compute a smaller area than (1) since his view is screwed up. Say observer (2) gets .5 m² and (1) gets 1 m². You may think that (1) has the correct area, but notice. If (1) did some math to determine how his answer would change if he were looking from (2)'s position, don't you agree that he would get .5 m² there by agreeing that (2) measured the correct value based on his view? So the clear contridiction in their quantitative measurements is reconciled once you transform properly to between the two frames in this case. Agree so far?

 

[atyy]

If we take, as usual, the statement that two events are “simultaneous”:

- I look at one side and what do I find? The ORIGIN of the statement = The value measured by a clock. This is a reality, an event, a fact, a truth as big as a cathedral and of course it exists.
- I look at the other side and what do I find? This value, after combination with other values, leads to the PREDICTION of another event, which is also factual and true and of course it exists.
- And in the middle? Any other event or fact or truth? No, in the middle there is nothing.

Maybe you will like "many-fingered time"? Like in Zeh, p187, http://books.google.com/books?id=4yUu-simAhMC&printsec=frontcover#PPA187,M1, or Nikolic, http://arxiv.org/abs/hep-th/0501046.

 

[Saw]

Maybe you will like "many-fingered time"? Like in Zeh, p187, http://books.google.com/books?id=4yUu-simAhMC&printsec=frontcover#PPA187,M1, or Nikolic, http://arxiv.org/abs/hep-th/0501046.

Uhf... Too advanced for me...

 

[Saw]

I agree the purpose is to solve problems, but I'd say it's to solve problems about well-defined physical questions like the time interval on a given clock, "fairness" is kind of a nebulous idea...if you have already defined your idea of "fairness" in purely physical terms, like "the duel is fair if each dueller experiences the same proper time between firing their own gun and the laser from the other guy's gun reaching their position", then you can use the laws of physics to judge if the duel is fair.

Miscommunication between people who have different educational backgrounds and are used to different specialised languages is unavoidable, but both specialists should be talking about the same thing. If a good lawyer gave you an account of what “fairness” means for legal purposes, you would immediately recognize in that picture your own physical definition. Likewise, if you develop in full that definition before the court, the judge should understand you as if she were reading an Act. There is no possible option, because ultimately both understandings, if well constructed, aim at the same target.

But there is still something I would like to clarify. I am the layman who has the practical problem and you are the specialist who must sell a convincing solution to me. You cannot demand that I accept as solution that the game is fair only because it conforms to a certain technical rule internal to a theory, the theory must also appear to me consistent and reasonable. Of course, if I ever dared to raise objections, I should not be imprecise, ignorant or obtuse. First, I must precisely define my needs, which practical problem I present to you. Second, I must make an effort to understand the basics of your theory, otherwise you would feel like talking to a wall. Third, even if there are still some obscure points in the back of my mind, as long as the overall picture looks reasonable, I will rule on the basis of what you propose.

This is what has happened so far. I mean by fairness that the two duellers have equal opportunities. The outcome of the duel must depend exclusively on their skills and how they use them. We discard that there are external distortions, but at least the rules of the game must be fair: the fact that the train moves wrt me and that the signals for shooting are light signals does not entail different opportunities for the duellers. Furthermore, based on my experience, I have identified a number of situations where, for me, the game would not be fair and asked specific questions about them. You have given your opinion and I have understood it. So I rule without any doubt that the game was fair.

But imagine that in another case, where a new configuration is introduced, I had concerns, because it seems to me that the rule "the duel is fair if each dueller experiences the same proper time between firing their own gun and the laser from the other guy's gun reaching their position" might not meet my needs.

Here the simile of the referee is not useful any more. Think of me as the legislator. I must establish some handy rule for referees to follow automatically. Would you agree that, in that case, the role of the scientist is to overcome my doubts and convince me that his rule must become the law? Would you agree that my needs are your needs and my purpose is your own purpose?

I agree that 99% of the subjects of the human law cannot be solved this way, because actually the needs are too nebulous and the word “fair” is a good example of that. But here we have a simpler case where “fairness” has been precisely defined and so it seems that the human law must be equal to the physical law. Eventually, it might happen that the physical law does not solve the problem to the maximum degree of certainty, but in that case I would also adopt your rule as the law, because that is the best we can do, and we would still have tried to achieve our purpose as best as we could. Ok?

I'll comment on the other points later...

 

[Saw]

Well, the physically important thing about this convention is that if all inertial observers design their coordinate systems in this way and then figure out the correct equations for the laws of physics as expressed in their coordinate systems, they'll all get identical equations. This is a real physical symmetry in the laws of physics, "Lorentz symmetry" or "Lorentz invariance"; we can imagine alternate laws of physics where this wouldn't be true. Observers could choose a different simultaneity convention which would result in a different type of coordinate system, but the equations in this coordinate system would have to look different than they do when expressed in the standard SR inertial coordinate systems.

Please tell me if this is a reasonably approximate paraphrase in layman terms:

On the basis of the measurements obtained this way, observers draw their coordinate systems, that is to say, they make a pictorial representation of what happens, a geometry. The geometry itself would suffice: you can visualize whether events will happen or not. Another way to express it is through mathematical equations. All these languages do the trick: (i) you can predict on the basis of the values in your coordinate system which events will happen and (ii) you can transform from the “dialect” of one coordinate system into another.

The only thing I miss here is some emphasis on the physical characteristics of the instrument. Nowadays:

- Time rate is best measured with atomic clocks, whose “ticker” is electromagnetic radiation oscillating in round-trips (a more sophisticated version of the “light clock”?). Thus the second is defined as “the time needed for a cesium-133 atom to perform 9,192,631,770 complete oscillations”.

- Distant clocks are synchronized with the Einstein convention: the trip time attributed to the distant clock is the time (ideally, measured by an atomic clock) that light takes to reach the distant clock and return, divided by 2.

- The metre is defined as “the distance traveled by light in free space in 1⁄299,792,458 of a second”, as measured preferably by an atomic clock. Is this measurement also made after a round-trip? I presume so.

I find little literature over the subject. But it seems both space and time are measured in a manner that has three features:

(i) the protagonist is light (or any electromagnetic wave);
(ii) the quantitative assessment is done after a round trip; and
(iii) the measurements are intertwined.

Could we thus say that we talk about “Lorentz symmetry” because we use a “Lorentz instrument”?

 

[Saw]

Also, to respond to a bit of your most recent post: I agree that physics cannot solve the ontological question of whether there is an absolute present.

Well, I do not really claim that there “is” an “absolute present” as a real entity, although it is true that I may have led you to think so due to imprecise language. I will explain below.

However, if all the laws of physics are Lorentz-symmetric, this does imply that there can never be any physical basis for saying one frame's judgments about simultaneity are more "correct" than any other's.

I agree. I would even eliminate the “if”. Based on the evidence that you, experts, have provided to me, I long ago convinced myself that the laws of physics are Lorentz-symmetric.

So for a philosopher, this might at least be said to lend weight to the idea that there is no such thing as absolute simultaneity, just by the Occam's razor argument that we should try to avoid postulating extra metaphysical entities that have no relevance to any empirical observations. We can't prove that there's no physically undetectable "metaphysically preferred frame" whose judgments about simultaneity are "correct" in some absolute metaphysical sense, but we also can't prove that there aren't physically undetectable gremlins sitting on the shoulder of every human on the world; if there is no pressing philosophical argument for why we should believe in such entities, one can argue that it's simpler to assume they don't exist.

I thank you for the idea to put at the beginning of the post my personal word definitions. This sounds like declaring the “variables” beforehand in programming languages. Their name is just a helpful reminder, what is important is the role or function that they play in the logic of the code. I will explain what I understand by “real” and “conceptual” and afterwards you can name these “variables” as you wish.

“Real” = matter particles and photons (or wave-particles or whatever they are), that is to say, the actors of the play, which “exist”, and the interactions between them or events, which “happen”. (The distinction, I admit, is somehow artificial, since everything is moving inside matter, causing continuous events, “things are events”, “nouns are verbs”, but it can serve as a start). This is the fundamental thing. All observers should agree on this, no matter their different perspectives (e.g.: position or state of motion).

“Concepts” = mere logical tricks that we invent to solve practical problems. We have some facts (eg: the measurements) and we want to predict other facts that are interesting for some practical purposes. How do we establish the link between the former and the latter? We apply logic and conclude that the measurements (of facts) necessarily lead to the predictions (of facts).

The confusion arises only if we get trapped in our own idea and start thinking of the “concept” as if it were a photon or an oxygen atom or a pretty lady, and then we ask each other: “do you agree that absolute or relative simultaneity exists?” I am sorry. I will not answer that question, unless you be more precise. There may be other ways to put it, but I have decided to reserve the word “exists” or “is” for the “real” things, like a photon or an event. I do not apply it to a “concept”. Yes, somehow, concepts exist, in my mind, and they are very close to reality, since they play with real facts as input and produce predictions of facts as output. But when referring to a concept, one must not forget what it serves for: a concept is valid if it relies on real facts and logically predicts real facts. Thus I would request that the question is reformulated as follows. For example: “do you agree that this concept is useless because its input is not found in reality?” or “do you agree that this concept is treacherous, because it derives bad outputs on the basis of a faulty logic?”

I think you will fully agree with this “code” even if we may have different preferences for the names of the “variables”, wouldn’t you?

Now let us be more practical. I would paraphrase your comments in my own words, as follows:

1) The concept of absolute simultaneity is useless, because all measurements of simultaneity are always frame-dependent.

2) Your definition of reality is in itself a concept. You are introducing in it the idea of simultaneity. If someone says that certain things exist right “now” and those things are some distance away, he has to measure it through physical means. If two observers measure the time coordinate of an event, they will get relative values. So that is not part of what is valid for all observers.

I basically agree with those comments (if they reflect at all what you mean) but have only some partial objections, which may be relevant for our practical problem (was the duel fair?) in the variation that I have commented before: the duellers receive light signals and shoot normal bullets.

 

[Saw]

2) Your definition of reality is in itself a concept. You are introducing in it the idea of simultaneity. If someone says that certain things exist right “now” and those things are some distance away, he has to measure it through physical means. If two observers measure the time coordinate of an event, they will get relative values. So that is not part of what is valid for all observers.

I once read this opinion: “Absolute simultaneity exists. Just clap your hands and… is the universe there? It is, of course”.

I think the author had a point, but he created much confusion by introducing a spurious visitor, a concept (simultaneity), in the definition of reality. This mistake may be unavoidable to a good extent, because we think with concepts but, if possible, it should be avoided. At least, we shouldn’t need simultaneity to understand what reality is.

Let us forget about time “for a while” and make a mental effort to describe a physical situation without its support.

You are blind. Someone throws to you a ball. Unfortunately, the knowledge that the ball is approaching will not magically appear in your mind. We are also physical objects ourselves and we can only learn anything about our peers by physically interacting with them. So you must wait until the ball touches your chest and this information is transmitted to your brain. Now you conclude: “the ball has interacted with me, ergo it exists, it is real”. It is regrettable that you had to wait for the interaction to reach this conclusion, because the ball existed anyhow, even if it had not yet interacted with you, but we are physical and that is the way it is. At least, the important thing is that you have realized that the ball exists, reality exists!

Now the ball bounces off your chest. Distance is created between you and the ball. Does it exist? It might have been blown off, but then it would be still somewhere in another form (mass-energy is conserved). Anyhow, like in our example of the duel, we can imagine that all external influences (including gravity) have been removed. So the ball will keep moving forever in a straight line as a good inertial frame. Thus the fact that there is distance between you and the ball does not change anything. It should not make you change your mind. If you had concluded that the ball existed when it interacted with you, there is no valid reason why you should start thinking that it does not exist after the interaction.

The “impossible negative proof argument” (I cannot prove there aren't physically undetectable gremlins sitting on the shoulder of every human on the world) is not applicable here: we haven’t ever, ever physically detected any such gremlins, but we have physically detected the ball. So the assumption that it exists is a logical inference of a physically detected fact.

What if you want to physically check that the ball exists? Strictly speaking, you would have to interact with the ball, in one of these two ways: an assistant of yours runs into the ball, interacts with it, and he travels to your place to deliver his message (yes or no, it does or it doesn’t) or you send a messenger (e.g.: a light beam) to reflect against the ball (if it comes back, it is because the ball was there; if not, it has disappeared or changed its trajectory). In the ideal display that we have chosen, the answer should be always positive, by sheer logic, because we have eliminated from the stage any external distortion factor.

Certainly, there may be “internal” distortion factors: if the ball is a muon, has it disintegrated or not? Well, in that case, the answer would even be beyond the reach of logic. I would still claim that the ball exists in some other form (mass-energy conservation principle), but I could not argue on the basis of logic that it exists in its initial form (a muon). However, the stage of the duel has been arranged so that such kind of concern will not arise, since the actors are more enduring and the distance is small: a photon or a bullet will not disintegrate in the short trip required to traverse the length of the car or the distance between the mid-point of the car and one edge. Thus logic is still helpful in this particular context… for what purpose?

Yes, all this looks very interesting from a spiritual point of view: you may feel sure that reality (a Being) is there, in some form or another, and that a ball is there, keeping its form, in a certain pre-arranged physical display, but can you use that certainty for the purpose of solving problems? Is that useful for the physical problem that is waving its hand before you and me?

In our current investigation (there are light signals and the shooters shoot bullets; I am the referee on the ground and I have to rule if the duel is fair, guided by physicists), is it relevant at all?

It is my impression that the answer is definitely YES. It is PHYSICALLY relevant, since it helps to define the problem in purely PHYSICAL terms. The problem was “equal opportunities”, in the legal jargon. What is its translation into the language of physics? Have I said anything at all that that makes you doubt about your initial definition?

 

[Saw]

Your definition was:

"the duel is fair if each dueller experiences the same proper time between firing their own gun and the laser from the other guy's gun reaching their position"

Since my doubts arise with bullets, we would have to adapt it like this:

"the duel is fair if each dueller experiences the same proper time between firing their own gun (=receiving their own light signal) and the bullet from the other guy's gun reaching their position"

For example, we first calculate the “proper time” of Back (= difference between his own clock reading at reception by his retina and brain of the signal and his own clock reading when he is wounded). We do the same for Front. Right? If those two intervals are identical, we conclude the duel is fair…

But is it so automatic? Most probably the “proper time intervals” of both duellers will be identical. But I am not sure whether that will prove that the duel is fair.

I had said: I, the judge, want to create a temporary sanctuary for discussion, where we only talk about “reality”, so as to define the problem itself in terms of pure “reality”, without being contaminated by any “conceptual” prejudice.

If you now enter the room with a clock in your hands, a mild judge will remind you of the rules of the discussion, but a tough one might accuse you of contempt to the court.

It is a question of methodological approach. The reading of the clocks is a measurement. In the case of clocks, a measurement of change: a ticker moves within a box or counter and every time the ticker touches a wall of the box or, if you wish, completes a full oscillation by hitting back the place of departure, some mechanism in the box perceives it and counts one unit of change. That is one thing. Another thing is whether this fact is logically linked to the resolution of the problem. That is a judgment. If the judgment is logically sound, we can say that the events of the instrument (or set of instruments whose readings are combined in a complex reading through equations) have mirrored the events of the reality we are interested in and thus we can construct the following idea: for the purpose at hand, it is “as if” my clock were a perfect mirror of what happens in the car, it is “as if” my clock contained the film of the story. Please look at it and you will see a video of the duel. Yes, a well constructed “concept” is a faithful picture of reality. Consequently, for shortness, in common language we talk as if it were the reality it is aiming at reproducing. But a bad concept is mocking at reality, it is caricature of reality. And that is precisely the question we are trying to ascertain. Therefore, you cannot introduce the instrument and its concept in the definition of the problem, since then you are begging the question.

Think of this. You are virtually telling me: “I know that a certain fact will happen for sure, so please phrase your question so as to ask whether that particular fact will happen, since that way I will ensure that the answer will be positive and I will confirm my original prediction that the game is fair…” That is not fair!

Therefore, I would prefer to keep pushing to define the problem in terms of the “real” events or physical interactions that happen inside the car. (It is not that the clock readings are not real; they are, but they are not the "real-life" events whose judgment has been brought before the court.)

“Opportunities” is the occasion or possibility to do things to harm and avoid being harmed.

These things that the duellers can do (e.g.: send other projectiles to intercept the bullet, produce a shield to stop it, step aside or duck down…) are instances of interactions with the environment. We can call them the tricks. The more you have, the better.

Second, the number of tricks that you can carry out is determined by two other interactions:

- You can start doing your tricks when you interact with the signal, but the signal is produced by another interaction, its creation by the referee through some physical operation.
- You cannot do any more tricks when you receive the shot, which is also generated by another physical interaction.

In conclusion, I would propose the following definition of the problem in physical terms:

"this particular duel (light signals, shooting bullets) is fair if each dueller can do the same number of physical interactions to avoid being shot, no matter if there is a difference between the physical nature of the signals for shooting (light) and the signals for ceasing to shoot and die… (bullets)"

Or something like that, I don’t know. I would need some feedback… Does it make any sense for you? For me, it is roughly a good legal definition, although I doubt about the details of the wording. Do you find that it is a reasonable physical definition, as well?

 

[atyy]

If you now enter the room with a clock in your hands, a mild judge will remind you of the rules of the discussion, but a tough one might accuse you of contempt to the court.

Let both duellers have identical pacemakers.

 

[Saw]

Let both duellers have identical pacemakers.

Yes, that is insightful. Let us situate the discussion on the physical characteristics of the "real-life" events, on the one side, and on the physical characteristics of the "instrumental" events, on the other side. To the extent that the latter match with and mirror the former, we'll have a good theory. I know that the answer is affirmative, since SR is at the root of moderns science and modern technology. Otherwise I would not be writing in my computer! But it seems that approach, on top of being physical, is pedagogical. It permits better communication between scientists and laymen. It helps a referee, like myself, render a fair judgment, with a reasonable motivation. Otherwise my resolution would be annulled due to lack of motivation!

 

[Saw]

Before I forget this idea, just a hint (I’m thinking aloud): maybe, even if there IS a difference between the behaviour of the measured objects in real life, you can measure and plot that behaviour with a light instrument, because that is fully IRRELEVANT for the measurement purposes; but maybe, anyhow, we have to accept that the best instrument is only almost perfect, it leaves a tiny amount of uncertainty… We will see. What we should do now is allow into the court room the technicians with their different instruments and analyze how they work, starting with the classical team and their mechanical clocks, in order to ascertain their pros and cons.

 

[Phrak]

Back in the wordy days of physics, I don't think Einstein took one tenth as many words solidify what seems to be argued and unargued here. Did I get lost on the philosophy page?

 

[matheinste]

Hello all.

With regards to simultaneity. While browsing past threads I came accros Simultaneity which has the secondmost number of replies (280) and the thirdmosty munber of views (16,900). In #18 Dalespam says:-

--- The point is that simultaneity is an artificial construct arising from the definition of a coordinate system, not something objectively real in its own right. Fundamentally it appears that the universe doesn't care about simultaneity, only about causality. Two simultaneous events cannot be causally connected, so what does it matter if one happened before the other? On the other hand, a cause should always come before an effect, and this is exactly what we see in relativity. A cause will preceed the effect in all reference frames, and for the rest it doesn't really matter. -----

I that is a very good way of putting it.

In such scenarios as those under discussion, simultaneity seems to matter in the sense that we have introduced a human element of fairness/right and wrong.

Deciding if two spatially separated events are simultaneous is merely a case of applying the agreed definition. A problem which arises in some of these proposed puzzles is that "making two events happen simultaneously" can only be done by making them both causally connected to a single event, such as the throwing of a switch, to set the chain of events in motion. The decision on the simultaneity of two spatially saparated events is an artificial construct and "constucting" the simultaneity of events is engineered. You cannot engineer a situation where things happen without having control of them, and this implies a causal connection. So I think it may be fair to say that unless the events happen to be simultaneous by chance, you cannot "construct" the simultaneity of two events unless the simultaneity is engineered from a single event/cause, or by some other mechanism constucted by a conscious entity. The last proviso is added as a get out clause should my belief that simultaneity cannot be manufactured in other ways than from a single event causally connected to both the events that are required to be simultaneous is wrong.

Note that in Eistein's train and embankment thought experiment the lightning strikes just happen to be simultaneous, they are not "consructed" to be so.

Matheinste.

 

[Saw]

With regards to simultaneity. While browsing past threads I came accros Simultaneity which has the secondmost number of replies (280) and the thirdmosty munber of views (16,900).

In #18 Dalespam says:

- --- The point is that simultaneity is an artificial construct arising from the definition of a coordinate system, not something objectively real in its own right. Fundamentally it appears that the universe doesn't care about simultaneity, only about causality. Two simultaneous events cannot be causally connected, so what does it matter if one happened before the other? On the other hand, a cause should always come before an effect, and this is exactly what we see in relativity. A cause will preceed the effect in all reference frames, and for the rest it doesn't really matter. -----

I that is a very good way of putting it.

Welcome back, Matheinste, and thanks for the quotation. I am just a little wordier, maybe due to my idiosyncrasy, but I fully agree with this sentence:

“simultaneity is an artificial construct arising from the definition of a coordinate system, not something objectively real in its own right. Fundamentally it appears that the universe doesn't care about simultaneity, only about causality.”

But my doubt is still the practical problem I have posed.

In such scenarios as those under discussion, simultaneity seems to matter in the sense that we have introduced a human element of fairness/right and wrong.

Well, I think I have phrased the problem in a manner that is purely practical and physical. See my post #163.

Did I get lost on the philosophy page?

Rather, I believe that refusing to answer this problem would amount to closing the doors of physics to a physical issue, for a sort of metaphysical reason = thinking that a concept hits at reality, due to some mysterious axiom, and refusing to discuss the logic of that axiom. Because, if you have been patient enough to read the whole lot of words, are you sure that the answer is clear? If so, why don’t you share it with us? If you were in my court room, I wouldn’t throw you out of the court, because I think that a physical problem is a legal problem. Now I am in your forum. Would you throw me out of it because you think that a practical, every-day life problem is not a physical problem?

 

[atyy]

It is my impression that the answer is definitely YES. It is PHYSICALLY relevant, since it helps to define the problem in purely PHYSICAL terms. The problem was “equal opportunities”, in the legal jargon. What is its translation into the language of physics? Have I said anything at all that that makes you doubt about your initial definition?

Something that is "physical" is something that is agreed on by all reference frames, ie. frame invariant quantities. The "proper time" is one such invariant. In SR, the lengths of 4-vectors are invariant.

However, what is invariant differs from theory to theory. In Newtonian physics, it is the lengths of 3-vectors that are invariant.

Which invariants are more real? That is a matter for experiment, and so far it seems that SR's reality is more real that Newtonian reality.

Within SR, there is the concept of an ideal clock, and experiments indicate that atomic clocks are physical instantiations of the theoretical ideal clock. Are biological processes ideal clocks? Strictly speaking, I don't think we know. However, it seems a reasonable assumption to make, and it is a standard assumption that is made in the SR solution of the twin paradox.

 

[Saw]

Let both duellers have identical pacemakers.

Something that is "physical" is something that is agreed on by all reference frames, ie. frame invariant quantities.

The "proper time" is one such invariant. In SR, the lengths of 4-vectors are invariant. However, what is invariant differs from theory to theory. In Newtonian physics, it is the lengths of 3-vectors that are invariant.

Which invariants are more real? That is a matter for experiment, and so far it seems that SR's reality is more real that Newtonian reality.

Ok, you are telling me that “equal number of ticks” (the proper time interval invariant) = “equal number of tricks” = “equal number of actions that each dueller can do to avoid being shot”. Right?

I just ask: for which physical reason will that happen? You say: this is a true prediction because it is confirmed by experiments. I agree that this is the way for physics to operate, since Galileo released it from pure speculation…

But then the distinction between the behaviour of the light signals and the behaviour of the bullets is totally blurred out. What do we do then with all the introduction to SR: the difference between mechanical objects (which take the motion of the source) and light (which doesn’t)…? Has SR showed that, in the end, the motion patterns of matter and light are fully equivalent, the only difference being that light is much faster?

Imagine that the referee sends, at the same time, light and mechanical signals to the duellers and that the latter shoot with both bullets and laser beams. It just happens that the light travels both ways through a tube of a certain “imaginary” material. Light slows down when crossing certain media, as a function of their index of refraction. The bullets are as fast as can be achievable and our imaginary material has an imaginary refraction index such that, when we observe the light traveling to Back, such light is slowed down to the point of being always aligned with the corresponding bullet. If we observe the light in the other direction or in its way back, will it also be aligned with the bullet?

What is SR’s answer to this thought-experiment?

If the answer is yes, is there any hint at the reason? For example, I’ve sometimes read that SR proved that, in the end, "matter moves like light", since all internal forces holding our atoms and molecules together are electromagnetic. I’ve also read that the same applies to other interactions, like weak interactions (eg: decay of muons), thus showing that all fundamental forces of nature are substantially alike and may one day be unified, since they follow the same rules that have been obtained analyzing the properties of light. Is it so?

 

[atyy]

Ok, you are telling me that “equal number of ticks” (the proper time interval invariant) = “equal number of tricks” = “equal number of actions that each dueller can do to avoid being shot”. Right?

Yes, that's right, but that's not what I was telling you. If you work in an inertial frame, say the frame on the train, you can define "fair" in whatever way you want. For example, you could define it so that it is fair even though Front always dies. Now if you are worried that your definition of fairness is frame dependent, then what you have to do is translate your fairness criteria in terms of relativistic invariants. If you specify your criteria in terms of relativistic invariants, referees in all frames will always agree on whether some set of events in spacetime match that criteria. I believe this is what JesseM was doing when he proposed a set of criteria in terms of proper time.

But then the distinction between the behaviour of the light signals and the behaviour of the bullets is totally blurred out. What do we do then with all the introduction to SR: the difference between mechanical objects (which take the motion of the source) and light (which doesn’t)…?

Hmmm, that's not the way I learned SR, but it seems right to me.

Imagine that the referee sends, at the same time, light and mechanical signals to the duellers and that the latter shoot with both bullets and laser beams. It just happens that the light travels both ways through a tube of a certain “imaginary” material. Light slows down when crossing certain media, as a function of their index of refraction. The bullets are as fast as can be achievable and our imaginary material has an imaginary refraction index such that, when we observe the light traveling to Back, such light is slowed down to the point of being always aligned with the corresponding bullet. If we observe the light in the other direction or in its way back, will it also be aligned with the bullet?

My understanding is that light in a medium is just like a bullet. SR makes a distinction between light in a vacuum and everything else. For everything else, like a bullet or light in a medium, if you run fast enough you will catch up with it. For light in a vacuum, you can run as fast as you want, and it will always go away from you at "the speed of light".

If the answer is yes, is there any hint at the reason? For example, I’ve sometimes read that SR proved that, in the end, "matter moves like light", since all internal forces holding our atoms and molecules together are electromagnetic. I’ve also read that the same applies to other interactions, like weak interactions (eg: decay of muons), thus showing that all fundamental forces of nature are substantially alike and may one day be unified, since they follow the same rules that have been obtained analyzing the properties of light. Is it so?

Matter is massive so it moves slower than light. When light interacts with matter, things are a complicated jumble such that light seems to slow down. Light in vacuum is special. However, it is experimentally true that all the laws of physics governing light and matter share the same symmetry of Lorentz invariance.

 

[Saw]

Yes, that's right, but that's not what I was telling you. If you work in an inertial frame, say the frame on the train, you can define "fair" in whatever way you want. For example, you could define it so that it is fair even though Front always dies. Now if you are worried that your definition of fairness is frame dependent, then what you have to do is translate your fairness criteria in terms of relativistic invariants. If you specify your criteria in terms of relativistic invariants, referees in all frames will always agree on whether some set of events in spacetime match that criteria. I believe this is what JesseM was doing when he proposed a set of criteria in terms of proper time.

Yes, JesseM’s definition was a substantial improvement vis-à-vis Greene’s. The physical solution to a physical problem cannot be frame-dependent. But I wanted to push a step further. It might theoretically happen that this result (the number of proper ticks for each dueller is identical) is always measured because the instrument is not precise enough and does not catch deviations from that rule. It might happen that, under normal, every-day circumstances, the proper times intervals are always identical, but, under extraordinary circumstances we are not accustomed to, we discover that the intervals are not identical. Hence if we do not want to be blinded by prejudices, we must look at the physical causes. I can list one thousand examples of scientists who asked themselves these “why questions” and thus logically predicted that the experimental results would be different with better instruments or under different circumstances and such statements were later confirmed by better observations.

The classical example of Galileo’s discovery of the principle of equivalence is illustrative. Should we throw Galileo out of a physics forum? By the way, Galileo reasoned with logic and, yes, he was quite wordy.

Therefore, I say: let us look at the causes. We have, on the one hand, the “warnings” (light signals) and, on the other hand, the “shots” (bullets). Tell me “something”, something at least about the physical nature of the interactions that cause the motion of the light signals and of the bullets. I do not need that we find here the ultimate explanation of those interactions. I just want to be told if, for you, the nature of the interaction is in both cases such that, for motion purposes, light signals and bullets behave identically. If the answer is yes, we can still discuss a little more why. If the answer is no, there may still be a (smalller) room for discussion.

Please look at the problem as follows: SR is telling me that, even if different observers observed the “warnings” to be behaving unequally, they also observed the “shots” to behave unequally, so that one thing compensates the other and in the end they all agree that the duellers disposed of the same number of “ticks” to do their “tricks”. I have no problem with that if the “warnings” and the “shots” are both made of the same thing (light). But when the “warnings” are light and the “shots” are bullets, if you wish to maintain the same solution, you must then hold that light and matter, at least in so far as motion pattern is concerned, are analogous or analogous under certain circumstances. It may be so. But please say it expressly.

Hmmm, that's not the way I learned SR, but it seems right to me.

Maybe you were taught, as an introduction to SR, Galileo’s description of the ship, where “nothing changes” = the laws of physics are the same =

You will discover not the least change in all the effects named, nor could you tell from any of them whether the ship was moving or standing still.

But in the next paragraph he also specified a physical cause for that description:

The cause of all these correspondences of effects is the fact that the ship's motion is common to all the things contained in it, and to the air also. That is why I said you should be below decks; for if this took place above in the open air, which would not follow the course of the ship, more or less noticeable differences would be seen in some of the effects noted.

I am far from postulating that light moves through a medium (a hypothetical aether). Maybe it does not take the motion of the source for another reason, namely because it “self-accelerates” itself (the electric field creates a magnetic field, which in turn creates an electric field) and this self-acceleration follows a pattern that is independent of the motion of the emitting or reflecting matter. Both models may be analogous for practical purposes. But if you believe that any of them is applicable to the light “warnings”, then you have to explain if it is also applicable to the mechanical “shots”.

Does SR postulate that the mechanical shots follow a Galilean pattern of motion when they are at rest in a frame and, as their speed increases, their motion pattern progressively conforms to light’s motion pattern?

My understanding is that light in a medium is just like a bullet. SR makes a distinction between light in a vacuum and everything else. For everything else, like a bullet or light in a medium, if you run fast enough you will catch up with it. For light in a vacuum, you can run as fast as you want, and it will always go away from you at "the speed of light".

Matter is massive so it moves slower than light. When light interacts with matter, things are a complicated jumble such that light seems to slow down. Light in vacuum is special. However, it is experimentally true that all the laws of physics governing light and matter share the same symmetry of Lorentz invariance.

It is clear that in vacuum light is faster than anything else, while in media it may not be. But the question is: As the light warning is slowed down by the medium, does it mean that it progressively adapts its motion pattern to conform to the mechanical bullet’s motion pattern?

All these questions look quite physical to me. Or shall we come back to the old days of physics where they were left in the hand of philosophers? Shall I, the referee, decide on my own, without the advice of the experts?

 

[Saw]

Atyy, needless to say that some ironical comments in my latest post were not addressed to you, but to the opinion that this just philosophy...

 

[Dale]

In #18 Dalespam says
...
that is a very good way of putting it.

Thank you!

While browsing past threads I came accros Simultaneity which has the secondmost number of replies (280) and the thirdmosty munber of views (16,900).

That was indeed a very long thread! I congratulate you for your browsing stamina.

 

[Dale]

The physical solution to a physical problem cannot be frame-dependent.

Are you still talking about the fairness of the duel? How is "fairness" physical? That seems like a big stretch.

"Fairness" is even less physical than simultaneity, so what would be wrong with it being frame-variant as long as the duellers agree on the rules. In any case, if someone is so stupid as to duel are they going to be smart enough to understand the relativity of simultaneity anyway?

It might happen that, under normal, every-day circumstances, the proper times intervals are always identical, but, under extraordinary circumstances we are not accustomed to, we discover that the intervals are not identical.

In what specific extraordinary circumstances are you suggesting that the proper time could ever be frame varying?

 

[atyy]

It might happen that, under normal, every-day circumstances, the proper times intervals are always identical, but, under extraordinary circumstances we are not accustomed to, we discover that the intervals are not identical. Hence if we do not want to be blinded by prejudices, we must look at the physical causes. I can list one thousand examples of scientists who asked themselves these “why questions” and thus logically predicted that the experimental results would be different with better instruments or under different circumstances and such statements were later confirmed by better observations.

Yes, SR is an experimentally verified description of nature, and may be wrong at some level. However, we haven't reached that point yet, eg. Mattingly, http://relativity.livingreviews.org/Articles/lrr-2005-5/ .

There is also the caveat that maybe complex biological clcoks do not behave like ideal clocks. However, it is a reasonable assumption, since biological and atomic clocks are made of the same stuff, and it is the standard assumption in the twin paradox. Nonetheless, it is an assumption.

Therefore, I say: let us look at the causes. We have, on the one hand, the “warnings” (light signals) and, on the other hand, the “shots” (bullets). Tell me “something”, something at least about the physical nature of the interactions that cause the motion of the light signals and of the bullets. I do not need that we find here the ultimate explanation of those interactions. I just want to be told if, for you, the nature of the interaction is in both cases such that, for motion purposes, light signals and bullets behave identically. If the answer is yes, we can still discuss a little more why. If the answer is no, there may still be a (smalller) room for discussion.

Please look at the problem as follows: SR is telling me that, even if different observers observed the “warnings” to be behaving unequally, they also observed the “shots” to behave unequally, so that one thing compensates the other and in the end they all agree that the duellers disposed of the same number of “ticks” to do their “tricks”. I have no problem with that if the “warnings” and the “shots” are both made of the same thing (light). But when the “warnings” are light and the “shots” are bullets, if you wish to maintain the same solution, you must then hold that light and matter, at least in so far as motion pattern is concerned, are analogous or analogous under certain circumstances. It may be so. But please say it expressly.

But I don't believe that different observers observe relativistic invariants unequally. The proper time is a relativistic invariant. Yes, with respect relativistic invariants, light and matter behave analogously.

I know there is a different way of looking at it involving a frame-dependent concept like simultaneity, and having all sorts of frame-dependent quantities cancel out in different frames, but that is very difficult and always makes my head spin.

I am far from postulating that light moves through a medium (a hypothetical aether). Maybe it does not take the motion of the source for another reason, namely because it “self-accelerates” itself (the electric field creates a magnetic field, which in turn creates an electric field) and this self-acceleration follows a pattern that is independent of the motion of the emitting or reflecting matter. Both models may be analogous for practical purposes. But if you believe that any of them is applicable to the light “warnings”, then you have to explain if it is also applicable to the mechanical “shots”.

The reason is that all the known laws of physics have Lorentz invariance.

http://pdg.lbl.gov/2008/reviews/rpp2008-rev-qcd.pdf
http://pdg.lbl.gov/2008/reviews/rpp2008-rev-standard-model.pdf
http://arxiv.org/abs/gr-qc/9512024

It is true that we don't know how to calculate from the standard model many everyday phenomena. However, there are links. For example, from QCD we can get that quarks make up protons. From QED we know that protons and electrons bind into hydrogen atoms. From QED we can transition into relativistic QM from which we can transition to non-relativistic QM from which we can transition into condensed matter physics.

However, it is the Lorentz invariance, rather than any detailed reasoning that is important. For example, suppose I have spherically symmetric ball. If I turn it upside down, it will look the same. Do I need an explanation from the standard model about how the atoms in the ball interact with each other and with my non-spherically symmetric hands which inverted the ball, and with my non-spherically symmetric eyes and brain? Or can I just argue that the ball is spherically symmetric?

Does SR postulate that the mechanical shots follow a Galilean pattern of motion when they are at rest in a frame and, as their speed increases, their motion pattern progressively conforms to light’s motion pattern?

It is clear that in vacuum light is faster than anything else, while in media it may not be. But the question is: As the light warning is slowed down by the medium, does it mean that it progressively adapts its motion pattern to conform to the mechanical bullet’s motion pattern?

I don't understand what "their motion pattern progressively conforms to light’s motion pattern" means. The shots always obey Lorentzian relativity even at low speeds. Light in vacuum, and light in media and bullets always obey Lorentzian relativity.

 

[Saw]

Are you still talking about the fairness of the duel? How is "fairness" physical? That seems like a big stretch.

"Fairness" is even less physical than simultaneity, so what would be wrong with it being frame-variant as long as the duellers agree on the rules. In any case, if someone is so stupid as to duel are they going to be smart enough to understand the relativity of simultaneity anyway?
In what specific extraordinary circumstances are you suggesting that the proper time could ever be frame varying?

Hello DaleSpam. I had missed your posts. We were not used to so authoritative visits here, in this chaotic thread, since long ago. Maybe I should write a recapitulation. I'll try in the next post.

 

[altonhare]

Full information about what? About every physical aspect of the problem? Certainly specifying the velocities of two objects doesn't specify everything physical about them (it doesn't specify their positions for example). And what's more, when you say both must agree about which velocity is "greater", you only seem to be talking about one aspect of the velocity and ignoring other aspects like the angles of the two velocity vectors.

It's not even clear what you mean by "greater" when talking about two velocity vectors--you obviously aren't just talking about the norm of each vector since that would just be the speed which is always positive, but before you argued that an object with negative velocity on the x-axis had a "smaller velocity" than one at rest. If we have only a single spatial dimension, then all velocity vectors are parallel to each other so we can just talk about positive or negative velocities on this axis, but what if we have 2 or three spatial dimensions and non-parallel velocities? You can take the component of each velocity vector which lies parallel to a particular axis and then the components will each be either positive or negative, but in this case, which has the greater velocity depends on what axis you want to use.


For example, suppose we have an x-y-z coordinate grid, and we break down two velocity vectors into their x and y and z components. For example, object A has Vx = 5 meters/second and Vy = -3 meters/second and Vz = 0 meters/second, while object B has Vx = 4 meters/second and Vy = 12 meters/second and Vz = 0 meters/second. Obviously object B has a greater speed in this frame, but which has a "greater velocity"? The x-component of A's velocity is greater than B's, but the y-component of B's velocity is greater than A's. So do you claim there is some absolute truth about whether A or B has a "greater velocity" here, where "greater velocity" does not just mean "greater speed" (i.e. greater norm of the velocity vector)? If so, what is it? Please give me a specific answer to this question about whether A or B has a greater velocity here.

By "full information" I mean the observers incorporate all their data into their qualitative conclusions. If their qualitative conclusions conflict the only reason is that they neglected to measure a critical attribute.

About the velocity question. We must be careful about what we mean when we invoke "direction". Observers in rotated coordinate frames concluded that the object had this X extent in a direction parallel to a line through another object while Dx away from that object and Y extent in a direction parallel to a line through another object while Dy away from that object. The two other objects were necessary to define the observer's coordinate system and the distances were critical components. Different rotated observers had data containing apples and oranges because their extents were all reported along with Dx's and Dy's, which were not the same.

The same occurs when an observer uses a rotated coordinate system to measure velocity. The velocity is measured in a direction defined by some reference object. An observer that "turns around" will be talking about velocity in the direction of x2 while the original observer will be talking about velocity in the direction of x1. They are, again, comparing apples and oranges. This is not to say, of course, that they cannot combine their quantitative data and resolve the issue, it just means they cannot draw logically comparable qualitative conclusions without quantitatively accounting for "reference frame". In this case they are not really qualitatively comparing so much as normalizing their data to have the same units.

In the train/embankment scenario we are not talking about rotated coordinate systems. The two observer's qualitative conclusions are directly comparable, they should both be apples. Both observers should conclude that the first apple is redder than the second apple. If they were using "rotated" coordinate systems then they would be asking each other if O1's apple is redder than O2's orange (not quite a perfect analogy here but I use it nonethless to avoid being overly pedantic).

In cases of length, colocal "simultaneity", and velocity observers comparing apples reach the same qualitative conclusions. In the case of noncolocal "simultaneity" they do not. It is my argument that apples are always comparable and, when we find that observers are coming to different qualitative conclusions when comparing apples, we need to reassess. It appears that noncolocal simultaneity is simply meaningless because it leads to logical contradiction. Local simultaneity does not need the "simultaneity" at all, we can simply say that A and B are local i.e. occupy the same location (or came in contact).

If you claim there is some objective truth about which object has the greater velocity along the x-axis, then once again it seems you must believe in some sort of ghostly "true" x-axis.

Indeed I do not believe this. In the most essential fundamental sense observers talk about motion of an object in a direction toward another object. They may pick 2 or 3 objects which they determine to be along perpendicular lines of site for convenience. The objective truth is that object X is moving faster than Y toward one (or more) reference object(s). If two observers are using different reference object(s) then they are not comparing apples.

Your comment about "greater velocity" seems unclear as I discussed above. And in SR different coordinate systems do disagree about which of two objects has a greater extent in a specific direction, because of length contraction. Even if you think there is an absolute truth about which frame's judgment is "really" correct, do you disagree that according to the standard way of defining SR coordinate systems, disagreements between coordinate systems about which of two objects has a greater "length" are quite possible?

Only if the observers are superficial and care not to actually think about what they mean by "length". They are referring to extent in a specific direction. I went over this a few times. In rotated coordinate systems observers are not comparing apples. Additionally they will know they are not comparing apples and will not be led astray to coming to contradictory conclusions.

But you don't believe that certain quantities, such as speed, represent "objective realities", and thus you don't believe there need to be objective truths about which object has the greater speed--is that right? If so, why can't you accept the possibility that quantities like velocity or length may also fail to represent "objective realities"?

It's not that speed isn't an "objective reality", it's simply not the whole story. Two observers would be foolish to come to qualitative conclusions based on excising certain data. Or you could argue that they are trying their best to incorporate everything relevant. If they are but they still arrive at a qualitative contradiction, the only answer is that they did not include everything relevant. Just as with the metal block and the brick. Two observers thought that, as long as their rulers were identical, they would come to the same conclusions about whether the brick or the metal was bigger. When they came to qualitatively contradictory conclusions did they throw up their hands and say,"That's just how it is sometimes!" or did they look for a reason?

In the brick/metal example the two observers left out temperature. In the "speed" example they leave out direction.

The "procedure I discussed above" was just the standard one for constructing inertial coordinate systems in SR, and the result is that the coordinates assigned to the same event by different observers are related by the Lorentz transformation. Do you not understand that under the Lorentz transformation, even if two coordinate systems have their spatial axes oriented the same way, if the two coordinate systems are in motion relative to one another they can disagree about which of two objects has a greater length? If so I can give you a numerical example, if that's what you're asking for. But if you're asking me to justify something else, please be specific about what it is.

So there are no rotations of coordinate axes. O1 and O2 are just watching A and B fly away (B moving faster relative to O1 and O2 at the outset and they are identical in size). Or O2 is standing on B. There is no way that, without rotating their axes, they will come to contradictory conclusions about which one is bigger. O2 can stand on B and turn around but must now use negative coordinates. i.e. O1 sees A and B flying away from him/her while O2 sees B approaching. O2 will not conclude on a length contraction of A relative to B but rather a length expansion:

Va1 = xa1*i + ya1*j
Vb1 = xb1*i + yb1*j

with (xa1²+ya1²)^1/2 < (xb1²+yb1²)^1/2

Va2 = (xb1-xa1)*i+(yb1-ya1)*j
Vb2 =0

In the length contraction formula the normed velocity of a will be less than the normed velocity of b for both observers, leading to the conclusion that B is shorter than A for both observers. When O2 turns around on B s/he is staring at "coordinate system" full of negative numbers. A will appear to be moving away from B, Va<0, the conclusion is that B is moving faster than A and that A is longer than B.

But by "visualizable" do you mean actually being able to form a visual image of the shape in your mind's eye, or do you just mean that the shape can be defined using the language of mathematics?

This could start venturing far off topic, but I mean the former. In this case it's an object, but it doesn't exist because it lacks location.

We can't picture such 4D objects visually because our brains have become adapted to find 3D space intuitive, but I imagine if you could somehow take the brain of a newborn and feed it sensory inputs from a simulated 4-dimensional body in virtual reality, as it grew up it would be able to visualize 4-dimensional shapes.

I don't buy any of this. If you're going to believe such things you may as well also believe in Santa Claus, the Flying Spaghetti Monster, and the Tooth Fairy.

Visualization isn't a very rigorous criterion in any case because it depends on the contingent details of our biology and history, whereas mathematics allows us to define the notion of "shape" in a completely rigorous way that doesn't depend on what we can visualize (and arguably a blind person can't 'visualize' any shapes at all, although I suppose they could imagine what it would feel like to run their hands over it).

Mathematics was created by humans and definitely depends on our biology/history directing our cognition.

Shape is a static concept. It is the primary, most essential quality of an object. I am not talking about the quantitative description of it. I'm talking about the primary quality a thing has independent of any other things. Other attributes we assign like color, age, smooth, rough, etc. are relational, they depend upon a comparison to another object. Shape is primitive. An object doesn't have shape by comparison with objects that lack shape because there are no objects that lack shape. An object has shape because if it didn't it would be nothing, i.e. it would not be an object. An object has shape even if it is the only object in the universe.

Before one can use mathematics to quantitatively describe/characterize an object, one must point to it, or at least a model of it. This is the only test. Without this crucial component the equations may refer to an object, but maybe not. I don't have a problem with equations by themselves, I have a problem with the proposing of physical interpretations/explanations which are literally unimaginable. These are the same kinds of interpretations and explanations provided by traditional religion. In fact, traditional religion (older times) at least proposed the anthropomorphic God, and other entities with shape, although the acts these hypothetical entities performed were no less than supernatural. Today religion has forsaken these "God objects" for "God the concept" i.e. "he" is everywhere and nowhere, etc. The point is, we are asked to accept it on faith rather than on the ability to visualize the real thing ourselves, which we are disbarred from. I don't care if you predict the weather perfectly for the next week or year, and have equations that show it quantitatively, if you tell me it was an unimaginable mechanism behind it you don't understand any better than I do. You've just done the requisite research to produce a good correlative model. I have no reason to believe humans are somehow precluded from understanding something about the universe, we have the capacity to understand anything.

If you are the observer that sees the ball going 15 m/s down and see your counter part going 25 m/s down, wouldn't you agree that your counterpart would measure the ball going at 10 m/s up relative to him? Don't you agree that in this case there is no quantitative disagreement once the observers clearly define the context of their measurements to one another.

There's never quantitative disagreement once observers apply the correct equations to the situation so that they are using common units (identical ticks of a clock and identical meter-sticks). This has nothing to do with what I'm talking about, which are qualitative conclusions. In fact, I took the effort before to distinguish between quantitative and qualitative.

Another example: Imagine looking at a 2 dimensional square in three space. There are two observers, one(1) looks on perpendicular to the plane the square is in. The other observer(2) is looking at an angle so he sees an odd looking rectangle. (2) would necessarily compute a smaller area than (1) since his view is screwed up. Say observer (2) gets .5 m² and (1) gets 1 m². You may think that (1) has the correct area, but notice. If (1) did some math to determine how his answer would change if he were looking from (2)'s position, don't you agree that he would get .5 m² there by agreeing that (2) measured the correct value based on his view? So the clear contridiction in their quantitative measurements is reconciled once you transform properly to between the two frames in this case. Agree so far?

Again, of course observers will quantitatively agree once they apply the correct equations. This has never been in doubt.

Hello all.

With regards to simultaneity. While browsing past threads I came accros Simultaneity which has the secondmost number of replies (280) and the thirdmosty munber of views (16,900). In #18 Dalespam says:-

--- The point is that simultaneity is an artificial construct arising from the definition of a coordinate system, not something objectively real in its own right. Fundamentally it appears that the universe doesn't care about simultaneity, only about causality. Two simultaneous events cannot be causally connected, so what does it matter if one happened before the other? On the other hand, a cause should always come before an effect, and this is exactly what we see in relativity. A cause will preceed the effect in all reference frames, and for the rest it doesn't really matter. -----

This is quite good, and is in the spirit of what I've been saying in the sense that the universe only cares about causality, i.e. did A hit B or not, not about simultaneity.
This (Dale's) is a very good way of putting it.

In such scenarios as those under discussion, simultaneity seems to matter in the sense that we have introduced a human element of fairness/right and wrong.

Deciding if two spatially separated events are simultaneous is merely a case of applying the agreed definition. A problem which arises in some of these proposed puzzles is that "making two events happen simultaneously" can only be done by making them both causally connected to a single event, such as the throwing of a switch, to set the chain of events in motion. The decision on the simultaneity of two spatially saparated events is an artificial construct and "constucting" the simultaneity of events is engineered. You cannot engineer a situation where things happen without having control of them, and this implies a causal connection. So I think it may be fair to say that unless the events happen to be simultaneous by chance, you cannot "construct" the simultaneity of two events unless the simultaneity is engineered from a single event/cause, or by some other mechanism constucted by a conscious entity. The last proviso is added as a get out clause should my belief that simultaneity cannot be manufactured in other ways than from a single event causally connected to both the events that are required to be simultaneous is wrong.

Note that in Eistein's train and embankment thought experiment the lightning strikes just happen to be simultaneous, they are not "consructed" to be so.

Matheinste.[/QUOTE]

I don't think you need the last proviso. Simultaneity is about the spatial locality of objects and nothing else. An event is simultaneous by definition and, indeed, the term "simultaneous" is generally superfluous except in human endeavors involving duels and trials and such. Nature doesn't seem to care about what we think of as "temporal simultaneity" but rather only about spatial locality and causality.

 

[JesseM]

By "full information" I mean the observers incorporate all their data into their qualitative conclusions. If their qualitative conclusions conflict the only reason is that they neglected to measure a critical attribute.

About the velocity question. We must be careful about what we mean when we invoke "direction". Observers in rotated coordinate frames concluded that the object had this X extent in a direction parallel to a line through another object while Dx away from that object and Y extent in a direction parallel to a line through another object while Dy away from that object. The two other objects were necessary to define the observer's coordinate system and the distances were critical components. Different rotated observers had data containing apples and oranges because their extents were all reported along with Dx's and Dy's, which were not the same.

The same occurs when an observer uses a rotated coordinate system to measure velocity. The velocity is measured in a direction defined by some reference object. An observer that "turns around" will be talking about velocity in the direction of x2 while the original observer will be talking about velocity in the direction of x1. They are, again, comparing apples and oranges. This is not to say, of course, that they cannot combine their quantitative data and resolve the issue, it just means they cannot draw logically comparable qualitative conclusions without quantitatively accounting for "reference frame". In this case they are not really qualitatively comparing so much as normalizing their data to have the same units.

In the train/embankment scenario we are not talking about rotated coordinate systems. The two observer's qualitative conclusions are directly comparable, they should both be apples. Both observers should conclude that the first apple is redder than the second apple. If they were using "rotated" coordinate systems then they would be asking each other if O1's apple is redder than O2's orange (not quite a perfect analogy here but I use it nonethless to avoid being overly pedantic).

In cases of length, colocal "simultaneity", and velocity observers comparing apples reach the same qualitative conclusions. In the case of noncolocal "simultaneity" they do not. It is my argument that apples are always comparable and, when we find that observers are coming to different qualitative conclusions when comparing apples, we need to reassess. It appears that noncolocal simultaneity is simply meaningless because it leads to logical contradiction. Local simultaneity does not need the "simultaneity" at all, we can simply say that A and B are local i.e. occupy the same location (or came in contact).
Indeed I do not believe this. In the most essential fundamental sense observers talk about motion of an object in a direction toward another object. They may pick 2 or 3 objects which they determine to be along perpendicular lines of site for convenience. The objective truth is that object X is moving faster than Y toward one (or more) reference object(s). If two observers are using different reference object(s) then they are not comparing apples.

It seems you are changing your claim somewhat then. You are no longer claiming there is any objective truth about which of two objects has a "greater velocity", but just that there is an objective truth about which of two objects has a greater velocity towards some third "reference object". But if so, how are you defining "velocity towards the reference object"? Is it just the rate at which the distance between one object and the reference object is growing smaller, so it would only be negative if the object was moving away from the reference object? If so, consider the following example. Suppose in frame #1, object C is the "reference object" which is at rest in this frame, object A is approaching it at 0.6c in the +x direction, and object B is approaching at 0.8c in the -x direction. In this frame B would have a larger velocity towards the reference object according to the definition above. But now transform into frame #2 which has its x-axis oriented the same way but is moving at 0.6c relative to frame #1, in the same direction as object A. In frame #2 A is at rest while C is moving at 0.6c in the -x direction, and using the relativistic velocity addition formula, we find that in frame #2 object B has velocity (0.8c + 0.6c)/(1 + 0.8*0.6) = 0.946c in the -x direction. So in this frame, the distance between A and C is shrinking at a rate of 0.6c, while the distance between B and C is shrinking at a rate of 0.946c - 0.6c = 0.346c, meaning in this frame it is A that has a larger velocity towards the reference object using the definition above.

It's not that speed isn't an "objective reality", it's simply not the whole story. Two observers would be foolish to come to qualitative conclusions based on excising certain data. Or you could argue that they are trying their best to incorporate everything relevant. If they are but they still arrive at a qualitative contradiction, the only answer is that they did not include everything relevant.

And why can't I say that simultaneity is not the whole story either, and therefore it would be foolish to say that logic forces us to conclude that there must be a single truth about whether two events at different locations are simultaneous or not without more specification of the context (like what inertial frame we are using)?

The "procedure I discussed above" was just the standard one for constructing inertial coordinate systems in SR, and the result is that the coordinates assigned to the same event by different observers are related by the Lorentz transformation. Do you not understand that under the Lorentz transformation, even if two coordinate systems have their spatial axes oriented the same way, if the two coordinate systems are in motion relative to one another they can disagree about which of two objects has a greater length? If so I can give you a numerical example, if that's what you're asking for. But if you're asking me to justify something else, please be specific about what it is.

So there are no rotations of coordinate axes. O1 and O2 are just watching A and B fly away (B moving faster relative to O1 and O2 at the outset and they are identical in size). Or O2 is standing on B. There is no way that, without rotating their axes, they will come to contradictory conclusions about which one is bigger.

There is if O1 and O2 use coordinate systems that are related by the Lorentz transformation. If O1 assigns an event some coordinates x,y,z,t, then the Lorentz transformation tells us that O2 should assign it these x',y',z',t' coordinates:

x' = gamma*(x - vt)
y' = y
z' = z
t' = gamma*(t - vx/c^2)
where gamma = 1/\sqrt{1 - v^2/c^2}

Here we are assuming that the x',y',z' axes of O2's coordianate system are oriented parallel to the x,y,z axes of O1's coordinate system (no spatial rotation), that the origin of O2's system is moving at velocity v in the +x direction of O1 (which means the origin of O1's system is moving at velocity v in the -x' direction of O2), and that the origins of the two coordinate systems coincide at t=0 in O1's system and t'=0 in O2's system. Let's also assume for the sake of argument that v=0.6c in this example, which means gamma = 1/0.8 = 1.25.

In this case, consider two rods which are oriented along the x and x' axes of the two coordinate systems, with rod A being at rest in O1's frame and rod B being at rest in O2's frame. Suppose that in O1's frame, rod A is 10 light-seconds long while rod B is only 8 light-seconds long, and that the left end of both rods lies at the origin at t=0 in this frame. If we label rod A's left end as "AL" and the right end as "AR", then AL's x-coordinate as a function of time in this frame is x(t) = 0, and AR's x-coordinate as a function of time is x(t) = 10 (both are constants since A is at rest in this frame). Meanwhile, if we use "BL" and "BR" to label the left and right ends of rod B, then in O1's frame BL's x-coordinate as a function of time is x(t) = t*0.6 and BR's x-coordinate as a function of time is x(t) = t*0.6 + 8. Agreed so far?

But now suppose we want to know x'(t') for each of these four rod endpoints in the O2 frame. In this case I'd say that AL's function is x'(t') = t'*(-0.6) and AR's function is x'(t') = t'*(-0.6) + 8, meaning that at any given t' coordinate the distance between the two ends of A is 8 light-seconds. I'd also say that BL's function is x'(t') = 0 and BR's function is x'(t') = 10, meaning that the distance between the two ends of B is 10 light-seconds. You can check yourself that these functions are correct according to the Lorentz transform. For example, pick any event on the worldline of AR whose coordinates satisfy x(t) = 10 in the O1 frame, and then find the corresponding coordinates in the O2 frame, you'll find that the coordinates in the O2 frame do always satisfy x'(t') = t'*(-0.6) + 8. For example, try x=10 and t=20; in this case, the Lorentz transform gives:

x' = 1.25 * (10 - 0.6*20) = 1.25*(-2) = -2.5
t' = 1.25 * (20 - 0.6*10) = 17.5

And t'*(-0.6) + 8 = 17.5*(-0.6) + 8 = -10.5 + 8 = -2.5, so it does work. More generally, if you pick x=10 and t=T, where T can be absolutely any number, you get:

x' = 1.25 * (10 - 0.6*T) = 12.5 - 0.75T, which gives T = 16.666... - 1.333...*x'
t' = 1.25 * (T - 0.6*10) = 1.25*T - 7.5, which gives T = 6 + 0.8t'

Combining the two gives -1.333...*x' = 0.8t' - 10.666..., dividing both sides by -1.333... gives x' = -0.6*t' + 8.

You can check that the other functions for position as a function of time are equivalent too. So, this shows that the two frames disagree on which has a greater length even though their spatial axes are all parallel and pointing in the same directions.

Of course, context is important here too--the coordinates each observer assigns to things are based on their own system of rulers and clocks, and each observer says the other observer's rulers are shrunk relative to their own. So in a way you could say that comparing the claims of the two observers about which rod is longer is another "apples and oranges" comparison. But this is exactly the point, there is no objective truth about which rod is longer in any absolute sense, only different contextual truths that are defined relative to a particular coordinate system.

We can't picture such 4D objects visually because our brains have become adapted to find 3D space intuitive, but I imagine if you could somehow take the brain of a newborn and feed it sensory inputs from a simulated 4-dimensional body in virtual reality, as it grew up it would be able to visualize 4-dimensional shapes.

I don't buy any of this. If you're going to believe such things you may as well also believe in Santa Claus, the Flying Spaghetti Monster, and the Tooth Fairy.

You don't believe that a newborn brain hooked up to different types of sensory input might adapt to be able to make sense of it? For example, if you hooked it up to artificial eyes which could detect a broader spectrum of electromagnetic frequencies, you don't think it would experience more colors than we see? If you hooked it up to something like the echolocation system of a bat or dolphin, you don't think it would come to experience this just as intuitively as we experience vision? The brain is quite adaptable, after all. And if you accept any of these possibilities, I don't see what's so hard to accept about the idea of adapting to navigate 4-dimensional geometry, which is just as consistent mathematically as 3-dimensional geometry. Indeed, we can create A-life creatures in virtual worlds with neural networks that adapt to the task of controlling evolved virtual bodies within simulated 2D or 3D worlds (see this video for example), hopefully you'd agree that we could do something similar with evolved virtual creatures in a 4D simulation; of course such simple simulated neural networks probably have little in the way of inner experience, but unless you are some kind of dualist or vitalist who thinks that no possible AI could have true consciousness, there shouldn't be any fundamental reason why it would be harder to create an intelligent AI "native" to a simulated 4D world as opposed to a simulated 3D world.

Mathematics was created by humans and definitely depends on our biology/history directing our cognition.

I disagree that mathematics depends on our biology (although what areas of mathematics we find interesting may depend on biology)--do you imagine that intelligent creatures with a different biology might disagree that if you take one discrete object and add it to a collection of two other discrete objects, the result will be 3 discrete objects? ('discrete' here assumes no splitting or merging)

Shape is a static concept. It is the primary, most essential quality of an object.

Why do you believe that? We can have objects with simulated shapes in virtual environments, but clearly in this case the shape is just a secondary result of the web of causes and effects going on inside the computer as it performs its calculations (since after all we can run the same simulation on two different computers with different spatial relationships between the computing elements, but the pattern of cause and effect as both run the same program will be the same); why couldn't it be true in the real world too that what's fundamental is something more like a collection of "events" linked by a particular pattern of cause and effect? This would be closer to the relational view of space and time postulated by the philosopher (and co-inventor of calculus) Leibniz.