# The show of the duel

#### Saw

Gold Member
This is a recapitulation of a subject that has incidentally appeared in another thread (Light shone in a train bouncing off mirrors…) that started over another issue (why does light take the direction of the source but not its speed?), although there may be a connection between the two things.

The subject is to judge if the duel described by Brian Greene in the Fabric of Cosmos (when explaining the relativity of simultaneity) is “fair”, in its original version and with a variation.
There is a duel on a train, between duellers that we will call Back and Front, situated at the tail and the tip of the car, respectively. For their duel, they employ laser guns, of identical construction.

There are two referees: Althonhare, on the train, and myself, Saw, on the ground. The signal for the commencement of the duel is given at the precise instant when Althon and myself are lined up. Of course, it is impossible that the two referees occupy the same position in space and so the perception of this alignment would require light travelling some distance from one to the other. But we stipulate that the distance is so small that its consequences are negligible. We all agree that the alignment (which is a single event) is simultaneous in both frames.

In order to give the duellers the signal for shooting, a pile of gunpowder, set midway between them (i.e., where Althon, on the car, and myself, on the ground, are standing at that instant), explodes and thus sends flashes of light in both directions.

Was the duel fair?

Each referee has a different opinion:

* Althon is certain that light from the flare reaches the duelers simultaneously, so he raises the green flag and declares it a fair draw.

[More technically: In the train frame, the light pulses travel equal paths in their respective directions and do it at the same speed. So the two distant events (= arrival of the signals to shoot) are SIMULTANEOUS.)

* According to the author, I wildly squeal foul play, claiming that Back got the light signal from the explosion before Front did. I explain that, because the train was moving forward, Back was heading toward the light while Front was moving away from it. This means that the light did not have to travel quite as far to reach Back, since he moved closer to it; moreover, the light had to travel farther to reach Front, since he moved away from it. Since the speed of light, moving left or right from anyone’s perspective, is constant, I am supposed to claim that it took the light longer to reach Front, since it had to travel farther, rendering the duel unfair.

[Idem: In the ground frame, light towards Back travels a shorter path, since its target is heading towards it, while light towards Front travels a longer path, since its target is escaping away. As both pulses travel at the same speed, the one that hits Back arrives earlier than the one that hits Front. The two distant events are NOT SIMULTANEOUS.]

Who is right? The matter is not trivial, because the judge who has made a mistake will be sanctioned by the competent supervision body and he may not be able to exercise his profession any more.

Fortunately (?), Einstein comes to the rescue of both with a salomonic opinion:

“Einstein’s unexpected answer is that they both are (…) they simply have different perspectives on the same sequence of events. The shocking thing that Einstein revealed is that their different perspectives yield different but equally valid claims of what events happen at the same time. Of course, at everyday speeds like that of the train, the disparity is small – Saw claims that Front got the light less than a trillionth of a second after Back- but were the train moving faster, near light speed, the time difference would be substantial” (literal quotation, I just changed the names).

But I am a prudent referee. I try to do my job properly. I know that the words of the law (you shall raise a green flag if the duel is “fair”) have a practical purpose and I must interpret them in the light of this practical purpose. The duel is “fair” if the two duellers are given equal opportunities of hitting each other, in practical terms. So my opinion that “Back got the signal earlier” is only relevant for the matter to be judged if it gives Back a real advantage (or disadvantage!).

To this effect, I consider the following scenarios:

(a) Back sees the signal earlier and fires earlier as well. Can his laser pulse hit Front before the latter receives the signal?

(b) If not, they have both received the signals and shot their laser pulses. The latter will also take some time to reach their targets. During this time interval, theoretically, any dueler could try the usual trick: fire and stand aside, before being wounded. There is very little time for this deed, especially if we talk about laser guns, but we can imagine the distance arbitrary long for this purpose. Is this time interval longer for Back than for Front or vice versa?

(c) If not, we imagine Back badly wounded, kneeling on the floor of the car, but brave enough to fire a second shot. Can it reach Front before the latter has the opportunity to fire his own second shot…?

You can think of other scenarios if you wish. The more, the better. That is the point of the exercise.

Then I call a group of experts on SR and ask their answers to the practical questions. If the answers, as I expect, are negative (there is no breach of the principle of equal opportunities), then I do not “wildly squeal foul play” as Brian Greene suggests. Instead, I gently and gallantly raise a green flag, in agreement with Althonhare’s opinion.
JesseM confirmed that the answers to questions (a) and (c) is:

No. If two events are simultaneous in any frame, that must mean there is a spacelike separation between the two events, meaning neither event lies in the other event's future light cone.
And to question (b):
Same time for both, since they're both at rest in the train frame, and they both fire at the same moment in this frame.
We still discussed for a while whether the question “is the duel fair?” is purely legal or also physical, but I think in the end we agreed that:

- The legal problem is the same as the physical problem. The role of the law is to solve problems, just as the role of phsyics is to solve problems.

- The legal problem is not to establish some arbitrary conventional rules for the game. The rules, on top of being agreed upon, must ensure that both duellers have “equal opportunities” to shoot and avoid being shot. For example, after receiving the signal for shooting and before being shot, the duellers may stand aside, duck down or try any other tricks. Do they dispose of the same time for this purpose?

- This legal need can be translated into a well-defined physical expression:
"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"
.

(I initially contested the definition, but I was wrong. In the end I agree that it was perfectly appropriate.)

In other words, the law requires that the duellers can do equal number of “tricks” to win. Physics answers that that means that their clocks register the same number of “ticks”, but in particular SR specifies that the relevant ticks are “proper” ticks, those of the duellers or of assistants situated where they receive the signals and the shots.

These are the calculations assuming that the relative velocity of the train wrt the ground is 0.5 c and that the train is 2 ls long in its rest frame (1.732 ls in the ground frame):

- In the train frame: Back and Front get the light signals simultaneously (after 1s) and their shots take 2 s to reach their opponents. So the proper time interval is 2s for both duellers. The duel is fair.

- In the ground frame, the duel is also fair because the proper time interval, although another one, is also the same for both duellers:

* Back got the light signal earlier (0.577 s) because he was heading towards the signal (thus making its path shorter), which looks like an advantage, but then it also happens that he is shot earlier (2.886 s), for the same reason, because he was heading towards the laser (thus making its path shorter), which is a disadvantage. The interval is 2.886 - 0.577 = 2.309 s.

* Front got the light signal later (1.732 s) because she was racing away from the light (thus making its path longer), which looks like a disadvantage, but then it also happens that he receives the shot earlier (3.732 s), for the same reason, because she was escaping from the bullet (thus making its path longer). The interval is 3.732 – 1.732 = 2.309 s.

Then we engaged in a challenging discussion about what it means that, as observed in the ground frame, the duellers get the light signals non-simultaneously.

My final understanding, as referee in the ground, is very prudent: the physical fact about it is that, if the clocks of my assistants (situated by Back and Front when they received the light signals) had been synchronized following the Einstein convention, they would show different readings; but this is just a piece of the puzzle; if I want to rule, I have to compare the clock readings at reception of the light signal with the clock readings of other assistants witnessing the reception of the shots; the latter are also non-simultaneous; the difference is the same for both duellers = they disposed of the same proper ticks to do their proper tricks.

Thus I rule, in accordance with the advice of SR experts, that the duel was fair.

Now we face a variation of the same duel: the signals for shooting are still light signals but the duellers shoot mechanical bullets.

What I have studied about SR tells me that the solution must be identical, since mechanical objects obey the same laws of physics:

- In the train frame: just like the light signals travel equal paths at the same speed (c) and arrive simultaneously at the duellers, their bullets also travel the same paths at the same speed and hit simultaneously the duellers.

- In the ground frame, you can reach the same conclusion through two routes:

* If I apply the relativistic formula for the addition of velocities to the light signals, I will discover that they still travel wrt me at c. And if I apply it to the bullets, I will find a differential element. Each bullet travels at a different speed wrt me, but this fact does not change the judgment: the bullet from Back travels wrt me faster than the bullet from Front, but less fast than it would if we had applied the Galilean addition formula (which compensates for the fact that it departs earlier) and the bullet from Front travels less slowly than it would with the Galilean formula (which compensates for the fact that it departs later).

* If, with the aid of my assistants, I measure directly the speed of the light signals and the bullets, I will find that they travel at exactly the speeds that stem from the above formulas.

However, I have doubts: due to some reasoning about the difference between what is “real” and “conceptual” (which I will not repeat here, it’s just too long), I fear that, in this new case, the proper time intervals (during which the duellers must do their tricks) should be different. I don’t know who would have the advantage and, in any case, the difference would be tiny. But this would make the duel, by definition, unfair, in the opinion of both referees.

Nevertheless, I conclude: if the experts tell me that the solution to the thought experiment I describe in the next post is “yes” (and maybe you give some sort of explanation for that, it doesn’t have to be very profound, I want to finish my job!), I will rule as proposed by SR.

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#### Saw

Gold Member
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 travelling 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?
Will, in these circumstances, light and matter travel in exact harmony in both directions?

If yes, is there any standard explanation of the cause for that, maybe related to the way that matter is accelerated?

#### ZikZak

It sounds like you're understanding the situation pretty well. It's not clear exactly what you're asking in the second duel, but:

When light travels through a medium, it travels at less than c. Now, in vacuum, it's perfectly legitimate to say "light travels at c" without referring to any reference frame because c is an invariant. But if the light ray is traveling in a medium at less than c, you now must specify the reference frame in which it travels at speed c'. In this case, when you use the material with refractive index n, the light ray travels at speed c'=c/n, in the rest frame of the material. In a different frame, the relativistic velocity addition formula must then be used. Indeed, this is exactly what the 1851 Fizeau Water Experiment demonstrates.

So the light ray in medium and the bullet will always be aligned. And incidentally, since they are always aligned in the rest frame, they must always be aligned in every frame by invariance of coincidence.

#### Dale

Mentor
The rules, on top of being agreed upon, must ensure that both duellers have “equal opportunities” to shoot and avoid being shot.
I don't think so. I think if they agree on the rules then the rules are fair. I mean, maybe one dueller would prefer the flash to be a little further away in order to avoid the possibility that the flash affects his vision. Maybe one dueller is so much better than another that he is willing to accept some handicap. Maybe one dueller is so bent on exacting revenge that he considers accepting a handicap to be a reasonable price for the chance. I say if the duellers agree on the rules then they are fair, regardless of other concerns especially relativistic ones.

#### Saw

Gold Member
When light travels through a medium, it travels at less than c. Now, in vacuum, it's perfectly legitimate to say "light travels at c" without referring to any reference frame because c is an invariant. But if the light ray is traveling in a medium at less than c, you now must specify the reference frame in which it travels at speed c'. In this case, when you use the material with refractive index n, the light ray travels at speed c'=c/n, in the rest frame of the material. In a different frame, the relativistic velocity addition formula must then be used. Indeed, this is exactly what the 1851 Fizeau Water Experiment demonstrates.
I’ve revised what Einstein says in his 1916 book about the Fizeau’s experiment.

First, it sems that Einstein accepts that there is some quantitative difference between the outcome of the relativistic and Fizeau’s formula. He claims his relativistic formula is superior because it is based on a better conceptual model, so the error would be for Fizeau. Is this understanding right? If so, can you clarify to me the nature of the difference?

Fizeau’s experiment is, yes, the same as the one posed here:

- Fizeau’s tube is for Enstein the railway embankment, which we call in our duel the “ground”.
- The water flowing inside the tube in Fizeau’s experiment plays the part of the imaginary material that I’ve invented, which is at rest with “the train”.
- The light signal is travelling for Fizeau through the water and for us through the imaginary material and along the train.
- The bullet is travelling in full harmony with the slowed down light signal, regardless its direction.

Then you say:

- The train is travelling at v (0.5 c in our example) wrt the ground.
- The light is travelling:
* In the train frame = the frame of the imaginary material, “at speed c'=c/n”, n being the refractive index of that material. Let us say that c’ = 0.9 c.
* In the ground frame, the speed of light is determined by the relativistic addition formula. Let us focus on the light signal to Front. Under the Galilean addition formula, this light signal would travel wrt the ground at v + c’ = 0.5 + 0.9 = 1.4c. Under the relativistic formula, it would travel wrt the ground at (v + c’)/(1+vc’) = (0.9 + 0.5)/(1+0.5*0.9) = 0.966 c. Also the speed of the bullets (which travel in harmony with the light signal) would be 0.966 c wrt the ground.

Well, all this would mean that the speed of light is:

- In vacuum, invariant for all observers as if it didn’t acquire the motion of the source; on the contrray, it is as if it acquired the motion of each observer.
- In materials, more and more varying for different observers as we increase the density of medium, until ideally, when it reached an infinitely small speed in a highly dense medium, it would behave virtually as if it had “acquired the motion of the source”.

But please note that it also means that the speed of the bullet:

- As it becomes higher wrt to the train, it also becomes less and less variant for different observers, until ideally, very close to the speed of light, it would become virtually invariant for all observers.
- As it becomes lower wrt the train, it also becomes more and more variant for all observers, recovering its normal motion pattern of acquiring the speed of the source…

For the first fact, there is a connection with a physical fact: light travels through a less dense or denser medium, although it is not evident why travelling through a medium should make it less invariant: one can understand that it is slower but not that it is measured as more or less variant by different observers.

But for the second fact, there is no link at all with any physical explanation: the only thing we find in Einstein’s text is a very vague reference to the “electromagnetic nature of matter” and a referral to Lorentz’s investigations.

How do you feel about all this?

#### Mentz114

Gold Member
How do you feel about all this?
It's very boring.

Also distasteful. Duels are violent and in no way related to justice of except the mythical lawless wild west kind of 'justice'.

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#### Saw

Gold Member
It's very boring.

Also distasteful. Duels are violent and in no way related to justice of except the mythical lawless wild west kind of 'justice'.
Mentz114, I have explained (maybe not well enough) why there is here a physical problem:

"will the clock readings of the duellers show the same proper time intervals between firing and being shot?"

and why the explanation of SR (which may be right after all) appears to suffer from a logical gap, which has been also illustrated with a thought experiment, which in turn may have served many people to visualize how an often forgotten aspect of SR (the relativistic formula for addition of velocities) works.

That is all quite related to physics. Your comment is not. If I and many people appreciate this forum, it is because it is serious and posters are not teenagers who clatter it with nonsensical out-of-the point remarks. I would appreciate if you could delete your post, so that I can delete this and thus not waste people's time forcing them to read such "improper" things.

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#### Dale

Mentor
- In the ground frame, the duel is also fair because the proper time interval, although another one, is also the same for both duellers:

* Back got the light signal earlier (0.577 s) because he was heading towards the signal (thus making its path shorter), which looks like an advantage, but then it also happens that he is shot earlier (2.886 s), for the same reason, because he was heading towards the laser (thus making its path shorter), which is a disadvantage. The interval is 2.886 - 0.577 = 2.309 s.

* Front got the light signal later (1.732 s) because she was racing away from the light (thus making its path longer), which looks like a disadvantage, but then it also happens that he receives the shot earlier (3.732 s), for the same reason, because she was escaping from the bullet (thus making its path longer). The interval is 3.732 – 1.732 = 2.309 s.
This calculation is incorrect. The proper time interval is invariant, so it is also 2 s in the ground frame (that is what invariant means). You are calculating the coordinate time in the ground frame, not the proper time.

You don't need to do any calculations. Once you have specified the rules in terms of frame-invariant quantities then you know that, by definition, all frames will agree.

#### JesseM

"will the clock readings of the duellers show the same proper time intervals between firing and being shot?"
Isn't it pretty obvious that they will, just based on considering things in the rest frame of both duellers, and assuming their guns are physically identical and they both fire simultaneously in this frame? Once you've figured out the proper times in this frame, you know they must be the same in every other frame, since different frames never disagree about frame-invariant quantities like the proper time on a worldline between two events on that worldline.

#### Saw

Gold Member
This calculation is incorrect. The proper time interval is invariant, so it is also 2 s in the ground frame (that is what invariant means). You are calculating the coordinate time in the ground frame, not the proper time. You don't need to do any calculations. Once you have specified the rules in terms of frame-invariant quantities then you know that, by definition, all frames will agree
Thanks a lot. I had made a mental note about a problem here, but forgot to pursue it. Yes, in the ground frame, those values are of the clocks of different assistants of mine: one witnesses the arrival of the signal to Back, but only another one, situated more to the right (the train moves rightwards wrt he ground), witnesses the event when Back is wounded. The same happens with Front, logically. That means different clocks, with different worldlines, so we are talking about coordinate time.

But then I try to find a “proper time interval” in the ground frame and don’t find it. How do you get the proper 2s in the ground frame? Or do you mean that the ground agrees that the proper time interval measured in the train is what it is measured in the train?

On the other hand, the truth is that the coordinate time intervals measured by my assistants on the ground are also equal for both duellers. However, JessesM’s definition of fairness had focused on proper times. Why? In the end, the duel is fair if both duellers have equal chances to do their “tricks”. So if I, from the ground, could by chance only measure coordinate time intervals (see the question above), for me, it would also be a valid duel, because what matters for my purpose is that both duellers have equal chances to kill each other, not that there are more or fewer chances that they kill each other… But what if that was the question? What if a judge asked: is this situation likely to end up with a death? Would proper or coordinate time prevail? What is more “real”, the “invariant” quantity (the proper time interval), which is the only one that is identical cross-frame? But I don’t want to ask too many questions at the same time…

#### Saw

Gold Member
Isn't it pretty obvious that they will, just based on considering things in the rest frame of both duellers, and assuming their guns are physically identical and they both fire simultaneously in this frame?
Ok. I will explain why I have my doubts. Unfortunately, I can't summarize it. This is the reason:

Before the SR experts, I had interviewed a panel of classical scientists, who still adhere to the Galilean/Newtonian ideas.

They had argued that a homogeneous duel (either mechanical signals and mechanical shots or light signals plus laser shots) is fair, but an inhomogeneous one (light signals and mechanical shots is not). They justified it as follows:

a) Homogeneous duel:

a.1) Mechanical signals (e.g., balls) and shots (e.g.: bullets):

Before being projected, all elements share the same “state of motion” of the train. Any motion that is imparted to them is added or “superimposed” to that original state of motion. Therefore, relative to the train, everything happens as if the duellers were on the ground. Nothing changes and the duel is as fair as it would be on the ground. They call this the principle of relativity and it looks quite logical to me.

a.2) Light signals and laser shots:

Light has its own state of motion, which is not affected by the motion of the train. They justify it because light, just like sound travels through the air, would travel through a medium called the aether. Only wrt to that medium would light’s speed be c. Because of this, they explain that, if, for example, the aether is at rest with the ground and the train has been accelerated towards Front, the light signal will reach Back earlier (as it travels at c+v with regard to him) and only later Front (wrt to whom it travels at c-v).

“So you claim that Back has an advantage, don’t you?”, I ask.

“Only momentarily, since you have to take into account that Back receives a fast signal but also a fast shot, just like Front receives a slow signal but also a slow shot. One thing compensates the other and in the end both duellers dispose of the same time to do their tricks.”

I check with some real numbers and, actually, with a train being 2 ls long, both duellers would dispose of a time interval of 2.66 s to do their tricks. That is fair.

b) Inhomogeneous duel (light signals and mechanical shots)

With respect to this configuration, the classical scientists argue that the duel is not fair:

- The light signal would arrive at Back earlier and later at Front.
- And this advantage for Back (firing earlier) would not be cancelled out by the correlative disadvantage of receiving Front’s shot earlier, since the bullets, unlike the light signals, do travel wrt the train at the same speed in both directions.

“Hmm”, I object, “are you sure of that? I think the next team, the SR team, will claim that the referee of the train should measure that the two signals are simultaneous for the duellers, because of the relativity of simultaneity or something like that…”

“Time and simultaneity are absolute”, they object.

“Please, don’t tell me fairy tales. Time is just a concept. In the end, if you really want to be sure about how the light signals and the mechanical shots behave, you have to compare their qualities with those of other real objects, with which you make them participate in a sort of competition. How do you measure the motion of objects?”

They explain the following:

Speed is the rate of change of distance over time. We divide this concept into two components: “distance” and “rate of change”. Different frames measure different distances, though with homogeneous sticks. But different frames directly measure the same rate of change. How? Imagine a ball bouncing between the top and the bottom of a spaceship, far from any gravity source (thus we can imagine that it moves inertially from top to bottom). It can act as a clock, in order to measure that “rate of change”: different observers disagree on the distance that the ball traverses on their respective X axes, though they share the same Y axis; however, the difference as to the X axis would only be relevant if the ball went out of the clock and hit the observers on their eyes; but, for our current purpose, which is to measure “change”, such difference is irrelevant, since what we are interested in now is the “number of ticks” and the “ticks counter” (the floor of the spacechip) has, by definition, for all observers, the same state of motion as the ticker, which forcefully means that all observers will measure the same number of ticks…

That looks ingenious, but I still make some objections:

- What happens when the ticker oscillates? The answer is that, with every collision, it loses kinetic energy and slows down, maybe erratically and in any case it is doomed to stop, more sooner than later…
- Is the ball fast? No, it is very slow, which doesn’t enable us to obtain small units for the sake of precision in measurements…
- How do you measure long distances when you cannot lay one stick after the other…?

Since all the answers to these practical questions are unsatisfactory, I dismiss the classical lot and receive the SR team. Although they start by telling me that “time is relative”, I directly ask them the practical question: how do you measure time and distances/length?

The answer is they use light for all purposes: they measure time rate with light clocks (atomic clocks) and synchronize clocks and measure length and distances with light signals, following the Einstein convention.

The choice of light seems definitely preferable to me:

For measuring the rate of change, it is a perfect ticker (fast and massless, so immune to collisions).
For the measurement of length, it may reach anywhere in the cosmos.
For synch purposes, it might even serve to synchronize clocks between different frames, since its motion is not affected by the motion of the source or the reflecting surface…

Yes, the problem is that it seems to move “on its own”, it doesn’t share the state of motion of the source and hence it will render different measurements for each observer, but there will be no problem with these intermediary disagreements if we find a conversion rule that leads to final agreement on events, on what happens.

I am shown the equations for transforming intervals and coordinates. The mathematical derivation looks logical. You take as start-point the Galilean Transformations, you keep c and v as invariants, but you allow x and t to change… The resulting formulas seem to work: each observer attributes to the other TD, LC and lack of synchronization, but the disagreement in each piece of the puzzle does not impede final agreement on certain events.

But what events? All the thought experiments take as ticker or protagonist a light signal. In fact, if the protagonist had been a ball, the thought experiments would have ended soon with a classical solution: someone would have said that the matter ticker takes the motion of the source and the fun would be over… Therefore, it seems to me that the mathematical reasoning “only” allows for this prediction: it guesses the x and t values of light signals in your coordinate system on the basis of the x’ and t’ values of any other frame’s light signals in his coordinate system… = it precisely guesses the events of other measurement instruments, but does it exactly predict the events of a bullet?

I had dismissed the classical advisors because their measurement instruments are inaccurate, but their prediction about what “happens” with the bullets, at least on a one-way trip, based on the idea that matter takes the motion of the source, seemed spotless: if we hire bouncing bullets of an elastic material as a ticker between the walls of the train car, we’ll make a mistake; but if we see them as “real life” objects, it still seems that they suffer the same push in both directions, which means there is no advantage in this respect for any dueller; instead, in their one-way to the duellers, the light signals “behave” differently for each observer, so one of them has an objective advantage. This might lead to the conclusion that the clock of one of the duellers (maybe Back’s, maybe Front’s) registers a higher difference for the “Interval” that we have taken as reference (the interval between firing and being shot).

Conclusion: I would rule that the duel is by definition unfair, unless I am told that SR says something more, namely it has discovered that, with regard to the issue of taking (A) or not taking (B) the motion of the source, both matter and light behave “alike”… but following which model, A or B?

If it were A (also light takes the motion of the source), we wouldn’t need SR and this has been ruled out, at least for light, by experiments.
If it were B, it could not be simply B, because the relativistic formula for the addition of velocities [(v - w)/(1-vw), assuming c=1 and velocities are measured as a fraction of c] only departs from the classical formula (v + w) and assimilates matter to light to the extent that the product “vw” is a substantial fraction of 1. So (although this is still confusing to me) I draw this funny conclusion: the bullet takes the motion of the source when it is at rest in the train, but when it is shot it starts not taking the motion of the source and moving “on its own”, this effect being smaller the lower that v (speed of the train) or w (speed of the bullet wrt the train) are, and the bigger the higher that v or w are…

That is why I posted the question that you can find above about the thought experiment of light travelling through an imaginary very dense material, in harmony with the bullets.

#### JesseM

For the first fact, there is a connection with a physical fact: light travels through a less dense or denser medium, although it is not evident why travelling through a medium should make it less invariant: one can understand that it is slower but not that it is measured as more or less variant by different observers.

But for the second fact, there is no link at all with any physical explanation: the only thing we find in Einstein’s text is a very vague reference to the “electromagnetic nature of matter” and a referral to Lorentz’s investigations.
Maybe the problem here is that you're imagining the velocity addition formula somehow depends on physical properties of the bullet? It doesn't, it can be derived directly from the Lorentz transform based only on knowing the coordinates of a moving object in one frame and then figuring out what the equivalent coordinates would be in another frame; so, if two observers in a classical Newtonian universe chose to use coordinate systems that were related to one another by the Lorentz transformation, they would find coordinate velocities in one system related to coordinate velocities in another exactly the same way, as long as you're using this coordinate transformation you don't have to worry about any physical properties of the moving object whatsoever. Of course, in a classical Newtonian universe the distances and times in these two coordinate systems would not match up with distances and times as measured by normal rulers and clocks at rest with respect to each observer, but you're always free to use a "weird" coordinate system if you choose. And in relativity of course coordinate systems based on normal rulers and clocks (and the Einstein clock synchronization convention) will naturally be related to one another by the Lorentz transform, and the laws of physics will take the same form in different inertial coordinate systems constructed this way, so in this context it is the most "natural" coordinate transformation.

#### atyy

a.2) Light signals and laser shots:

Light has its own state of motion, which is not affected by the motion of the train. They justify it because light, just like sound travels through the air, would travel through a medium called the aether. Only wrt to that medium would light’s speed be c. Because of this, they explain that, if, for example, the aether is at rest with the ground and the train has been accelerated towards Front, the light signal will reach Back earlier (as it travels at c+v with regard to him) and only later Front (wrt to whom it travels at c-v).
For a train at constant velocity, the Newtonian panel will find that the train is an aether frame.

#### Dale

Mentor
But then I try to find a “proper time interval” in the ground frame and don’t find it. How do you get the proper 2s in the ground frame?
http://en.wikipedia.org/wiki/Proper_time" [Broken] is defined as dτ² = dt² - dx²/c² which is proportional to the spacetime interval ds² = -c²dt² + dx² and is real for timelike intervals. All frames agree on these quantities.

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#### Saw

Gold Member
http://en.wikipedia.org/wiki/Proper_time" [Broken] is defined as dτ² = dt² - dx²/c² .
Ok, and if I measure distances in light seconds, I understand that c becomes = 1 and thus the formula for proper time simplifies to dτ² = dt² - dx².

So in the duel the “proper time interval” between the two relevant events (when a dueller fires and when he is shot) is:

* In the train frame: since here the two events occur at the same place (dx = 0), the expression reduces to dτ² = dt² and this in turn to dt = the difference between the readings of the clock of the dueller at the time of each event (when he fires and when he is hit by the other’s shot).

In particular, both duellers shoot at t1 = 1 and are shot at t2 = 3, so the interval is for both of them 2s.

* In the ground frame:

- As to Back:
. An assistant of mine observes that Back fires at t1 = 0.557 s, Back moves on and another assistant of mine observes that Back is shot at t2 = 2.886 s. The difference = coordinate time = dt = 2.309 s.
. The distance between the two events = dx = 1.1545 s.

- As to Front:
. An assistant of mine observes that Front fires at t1 = 1.732 s, Front moves on and another assistant of mine observes that Front is shot at t2 = 4.041 s (not 3.732 as I had said before). The difference = coordinate time = dt = 2.309 s.
. The distance between the two events = dx = 1.1545 s.

Both for Back and Front, dτ² = dt² - dx² = (2.309)2 – (1.1545)2 = 5.3 – 1.3 = 4, so τ = 2 s.

Thanks for the correction again.

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#### Dale

Mentor
Both for Back and Front, dτ² = dt² - dx² = (2.309)2 – (1.1545)2 = 5.3 – 1.3 = 4, so τ = 2 s.
You got it exactly.

#### Saw

Gold Member
you're always free to use a "weird" coordinate system if you choose. And in relativity of course coordinate systems based on normal rulers and clocks (and the Einstein clock synchronization convention) will naturally be related to one another by the Lorentz transform, and the laws of physics will take the same form in different inertial coordinate systems constructed this way, so in this context it is the most "natural" coordinate transformation.
Well, the “ball clock” I talked about in the introduction is “weird”, because it has major physical shortcomings: the ticker is slow and its collisions with the walls of the instrument make it lose kinetic energy. Therefore, if we used this instrument and drew our coordinate systems with the values measured with it and related the x readings of different frames with the Galilean Transformations, we would make imperfect predictions.

But would you agree that if those physical shortcomings did not exist, the “ball clock” would render homogeneous time measurements for all observers and hence the classical coordinate system and the GTs would be preferable, because they are simpler? If those physical cons did not exist, wouldn’t the pros of the instrument make it preferable, as well as the geometry and mathematics it is associated with?

I say all this after assuming that the light instrument is preferable because it does its job more efficiently. In this sense, yes, it is more “natural”. I just want to point out that the choice of the instrument (which no doubt has certain physical characteristics) conditions the choice of the diagram and of the corresponding mathematics... But all the rest is unclear. I'll think of an example.

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#### JesseM

Well, the “ball clock” I talked about in the introduction is “weird”, because it has major physical shortcomings: the ticker is slow and its collisions with the walls of the instrument make it lose kinetic energy. Therefore, if we used this instrument and drew our coordinate systems with the values measured with it and related the x readings of different frames with the Galilean Transformations, we would make imperfect predictions.

But would you agree that if those physical shortcomings did not exist, the “ball clock” would render homogeneous time measurements for all observers and hence the classical coordinate system and the GTs would be preferable, because they are simpler? If those physical cons did not exist, wouldn’t the pros of the instrument make it preferable, as well as the geometry and mathematics it is associated with?
You didn't really specify the details of how the ball clock would work in different frames. Does each observer have his own distinct ball bouncing up and down in his own rest frame? Is each observer's ball bouncing the same distance from floor to ceiling, and at the same speed, relative to the coordinates of SR inertial frames? If all this is true, then ball clocks will show time dilation just like light clocks. Since all the laws of physics are Lorentz-symmetric, all physical clocks will exhibit time dilation, the only way this wouldn't be the case would be if some of the laws of physics were not Lorentz-symmetric (like if we lived in a Newtonian universe where the fundamental laws were Galilei-symmetric).
Saw said:
I just want to point out that the choice of the instrument (which no doubt has certain physical characteristics) conditions the choice of the diagram and of the corresponding mathematics...
No, that is definitely a major misunderstanding, if there were any physical instrument that didn't function identically in the different inertial frames related by the Lorentz transformation, this would be a violation of the first postulate and would therefore prove relativity wrong. There is no reason to believe that such a thing is possible in reality, and if anyone could show it was they would make headlines and probably win the Nobel prize.

#### Saw

Gold Member
You didn't really specify the details of how the ball clock would work in different frames. Does each observer have his own distinct ball bouncing up and down in his own rest frame?
Yes. There are two observers armed with ball clocks, moving relative to each other as inertial frames.

Is each observer's ball bouncing the same distance from floor to ceiling,
Yes, and since this distance is in the axis perpendicular to the direction of relative motion between the two observers, there is no issue of length contraction here. Even if the observers are SR observers, they’ll agree that this y distance is the same for both of them.

and at the same speed, relative to the coordinates of SR inertial frames?
Well, if we introduce speed, we complicate things, because speed is distance divided by time and then we must wonder… “time”, but measured how? It will have to be measured by another observer, armed with another instrument. But which instrument? One like this very instrument? Then the question is what kind of time this other instrument measures, homogeneous or inhomogeneous for all observers...? We go into a vicious circle.

I prefer to talk, instead of time, about “change”. Motion can be regarded as the fact that distance is changing (growing or shrinking) between two objects. So the mind decomposes this phenomenon into its two elements and measures them independently. After a while, you look at the object in question and observe how much distance it has traversed and at the same time you observe how much “change” has happened. You measure the change with some other object that is ticking or oscillating inside a box (every tick or toc is a unit of “change”). Then you put the two things together and say that, for example, two kilometres of distance fit within one minute of change. That is why speed is a division. You distribute the space units between the units of change.

And how do you count units of change? For a change to exist, you need something moving (the ticker) and a counter, which is the walls of the clock.

The ticker is a moving thing. It is true that the two observers moving wrt each other have different perspectives and use different coordinate systems. How can it be that they agree on what happens to the tickers of both of them?

Well, to start with, they do not disagree on everything. The motion of the ticker can be decomposed into two vectors: its motion through the Y axis (perpendicular to the relative direction of motion of the two observers) and through the X axis. For the observer at rest with the clock, there is only motion through the Y axis, for an external observer there is also motion through the X axis. But at least we have thus reduced the scope of the disagreement: for example, for me your ticker has lateral motion while mine hasn’t any; for you my ticker has lateral motion while yours hasn’t any; but we both agree that the two tickers have the same Y motion.

Is this discrepancy about the X motion relevant? It depends on what purpose we consider.

If we regard the ticker as a “real life” object, then the discrepancy is important. If the ticker accidentally escaped out of the clock through a hole (breaking into “real life”), it would hurt the local observer more or less hard, but it would hurt an external observer, approaching in another frame, much harder, because its speed and hence its energy content is different for him.

However, here we do not have that purpose. We are not analyzing the damaging capacity of the ticker for our eyes. We are considering only the “instrumental” role of the ticker, how many “changes” it measures, as counted by the owner of the clock, the observer at rest with it.

What is important for this purpose? Not the motion of the ticker wrt a foreign observer but wrt to its own clock:

- What was the initial state of motion of the ticker wrt its clock? In both cases, in the case of both tickers, nil. Before being set in motion, the ticker was fully at rest with the frame where it ticks.

- What is the state of motion it acquires? Both tickers are pushed by an agent located at their respective frames, so they receive the same push relative to their own frames.

- Who counts the ticks? Their own clock, which is supposed to share their own state of motion, except for the transversal motion, which is the same for both observers in spite of their different perspectives.

Therefore, both observers should agree that their tickers tick homogeneously. Their reasoning will be: “Ok, your ticker is pushed wrt to me harder or more weakly than wrt you, but I admit that here the relevant thing is how fast it will travel wrt you, since you are the counter and not me. Ok, your ticker has to traverse in my CS an X path that it doesn’t traverse in yours, but it is true that the floor and ceiling of your clock also traverse the same path, in harmony with the ticker, and after all you are the counter. So I agree that your ticker ticks wrt your counter the same number of ticks that mine ticks wrt to my own counter”.

Nevertheless, here we have eliminated the distorting effect of collisions when the ticker bounces off and this may disqualify the instrument for practical purposes.

And, yes, there may be another reason why these ideal instruments would not render homogeneus results for all observers: it may be that two apparently equal pushes (for example, the balls are shot from identical little cannons) do not generate the same result, the same motion wrt to the local frame, this being due to the electromagnetic nature of the interaction... But again that would be a physical reason at the heart of the option for Lorentzian diagrams and equations...

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#### JesseM

Well, if we introduce speed, we complicate things, because speed is distance divided by time and then we must wonder… “time”, but measured how? It will have to be measured by another observer, armed with another instrument. But which instrument? One like this very instrument? Then the question is what kind of time this other instrument measures, homogeneous or inhomogeneous for all observers...? We go into a vicious circle.
All that matters is that the devices you want to serve as clocks are constructed identically in each frame. For example, instead of worrying about speed, each observer could use physically identical springs which have been compressed an equal amount (as measured by their own rulers) to set the balls in motion from an initial state of rest, and in this case the Lorentz-invariance of the laws of physics ensures that these ball clocks will all have the same ratio of "ticks" to the ticks of a light clock at rest in the same frame. Likewise, there are various ways to construct physical clocks that don't involve looking at the time for some object (or light particle) to get from location to another, like clocks based on the decay rate of radioactive isotopes. If you aren't concerned too much about accuracy, you could even use biological rhythms like heart rate; these would experience time dilation too. There is no way that different observers could use identical physical procedures to construct clocks in their own frame and yet not have clocks in different frames show time dilation by the amount predicted by the Lorentz transformation.
Saw said:
And, yes, there may be another reason why these ideal instruments would not render homogeneus results for all observers: it may be that two apparently equal pushes (for example, the balls are shot from identical little cannons) do not generate the same result, the same motion wrt to the local frame, this being due to the electromagnetic nature of the interaction
As long as the frames' coordinates are the standard SR inertial frames related by the Lorentz transform, then identical cannons would have to fire the balls at the same coordinate speed according to relativity, anything else would violate the first postulate.

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