Relativistic Force Transformations

[jackqpublic]

I'm trying to visualize the relativity example given in the book. The one with 2 generals signing a treaty at ends of a train (the light in the middle goes on and they both sign when light reaches them). The observers claim that one of the generals signed first. However, the generals claim that they signed at the same time. Both are relatively correct.

What if instead of signing, the train was set up with a ball on a plank pefectly balanced in the middle:

0
------------------------------
|

And instead of signing, the generals would drop a weight on their side of the plank when the light reached them. According to the generals, they would drop the weight at the same time and the ball would remain balanced. What would the stationary observers see?

Thanks

Edit: My "diagram" is not formatting correctly the "0" and the "|" are supposed to be in the middle of the plank "---..."

 

[Ich]

What would the stationary observers see?

A balanced ball, and a bent plank. Bending first at one end later at the other, but the bends traveling at different velocities through the plank and arriving at the same time at the ball.

 

[Saw]

jackqpublic, you got the right answer to your question from Ich.

But after reading the story in The Elegant Universe, I would like to add a comment. Someone may think this is philosophy; it can be labeled that way, but anyhow it's relevant: it's about knowing what you're doing and what you are doing it for when making physics.

The example is similar to the duel show that Greene uses in The Fabric of Cosmos, for the same purpose, to "illustrate" the relativity of simultaneity. But here Greene is even clearer as to the message: relying on the opinion of ground observers, the countries of the two generals reinitiate the war because the nationals of the general who is deemed to have signed first feel duped!; so we see that an "important political problem" receives different solutions from different perspectives; this is "startling", says Greene, but it is "the essence of Einstein's discovery".

Well, to me this is thoroughly misleading. Not a good way to illustrate, i.e., to make apparent the gist of SR. I think the proper way to do it would have been precisely the opposite: show how, in spite of the discrepancy as to the simultaneity of the signatures between train and ground observers, it is obvious that the signing procedure was OK and those who claimed the contrary were utterly wrong. What do you think?

 

[Saw]

Googling a little, I've found similar examples with similar resolutions. For instance, in http://www.phys.vt.edu/~takeuchi/relativity/practice/problem03.html, the problem is a race between the hare and the tortoise:

The hare and the tortoise decide to have another race. But instead of running in the same direction toward the finish line, they decide to run in opposite directions toward finish lines located at equal distances from the starting line (…)

The race takes place and the referee sees both animals finish at the same time in the frame at rest with respect to the ground. Assuming that the two animals move at a constant velocity from start to finish, what is the result of the race as seen from the frames of the two moving animals?

The “solution” given by the author is:

Both the hare and the tortoise think that they won.

If you follow the line of simultaneity of the hare, the space-time point of the hare crossing the finish line is simultaneous with a space-time point on the tortoise's trajectory which is only about two thirds of the way to its finish line. So in the hare's frame of reference, it won the race.

The exact same argument applies to the tortoise's frame of reference so it too thinks it won the race.

Again, this looks like a bad illustration of the meaning of SR. If this were an exercise in an exam, the student who “naively” provided that answer should fail. If the hare and the tortoise think what is stated above, they are both wrong. I haven’t worked out the details of the reasons yet (that is what I would like to discuss if someone takes the challenge), but it seems intuitively obvious to me that the three involved frames should concur on the same solution: nobody “wins” the race, it’s a tie. Or not…?

 

[ZikZak]

Saw,

You are attempting to claim that there is something privileged about the train frame in the first example (post #3) and the referee's frame in the second example (post #4), such that what happens in those frames is "Really" what happened. You are attempting to claim that there are privileged, absolute frames: the exact opposite of what Relativity asserts. What is so special about the referee that makes him right and the hare and tortoise "utterly wrong"? Answer: nothing at all.

 

[atyy]

"At equal distances from the finish line"

Are the distances the same in all frames?

Do they start at the same time in all frames?

 

[Saw]

Saw,

You are attempting to claim that there is something privileged about the train frame in the first example (post #3) and the referee's frame in the second example (post #4), such that what happens in those frames is "Really" what happened. You are attempting to claim that there are privileged, absolute frames: the exact opposite of what Relativity asserts. What is so special about the referee that makes him right and the hare and tortoise "utterly wrong"? Answer: nothing at all.

Zikzak, thanks for commenting. Before anything else, I apologize if my style is sometimes too categorical. That's professional deformation... I think I'm right, although I may be wrong, of course...

Said this, I do not claim that the problems posed (we can call them the Peace Treaty problem -the one narrated in The Elegant Universe- and the Race problem -the one about the hare and tortoise) can only be solved from a "privileged" frame. What I claim is that there is a single solution and that the same can be guessed by each frame on the basis of their respective measurements. It just happens that in a certain frame the solution is expressed more synthetically. So its only "privilege" is shortness of speech. It's a "light" privilege, a privilege in a weak sense, if you wish.

"At equal distances from the finish line"

Are the distances the same in all frames?

Atty, when the author says "At equal distances from the finish line", he means in the referee’s or ground frame. In other frames…? You can see a spacetime diagram of the Race in the link I posted above and for sure you can interpret it better than me. I suppose that both the hare and the tortoise measure that the ground patch from origin to finish line is shorter than as measured in the ground (coordinate length), but for example the hare reckons that the tortoise has to traverse a longer distance in hare’s frame before reaching the point of hare’s frame that will be aligned with the tortoise when the latter reaches the ground finish line. But it’s difficult to see… Did I get it right?

Do they start at the same time in all frames?

Yes, if you assume that referee, hare and tortoise synchronize their clocks at the start of the race.

 

[atyy]

Atty, when the author says "At equal distances from the finish line", he means in the referee’s or ground frame. In other frames…? You can see a spacetime diagram of the Race in the link I posted above and for sure you can interpret it better than me. I suppose that both the hare and the tortoise measure that the ground patch from origin to finish line is shorter than as measured in the ground (coordinate length), but for example the hare reckons that the tortoise has to traverse a longer distance in hare’s frame before reaching the point of hare’s frame that will be aligned with the tortoise when the latter reaches the ground finish line. But it’s difficult to see… Did I get it right?
Yes, if you assume that referee, hare and tortoise synchronize their clocks at the start of the race.

OK, I see, it's clearer what situation we're discussing when I look at the diagram. I don't know off the top of my head if you got it right - but my gut reaction is since we have constant velocity after the start, why not define the race as who has the greatest acceleration at the start of the race - that should be frame invariant?

BTW, didn't you and JesseM come up with some other frame invariant notion of fairness in your long thread about the duel? Would something similar be possible here?

 

[atyy]

How about defining it by whoever's proper time between start and intersection with the worldlines of the respective endpoints is greatest?

Edit: Maybe that should be "least"?

 

[Saw]

OK, I see, it's clearer what situation we're discussing when I look at the diagram. I don't know off the top of my head if you got it right - but my gut reaction is since we have constant velocity after the start, why not define the race as who has the greatest acceleration at the start of the race - that should be frame invariant?

BTW, didn't you and JesseM come up with some other frame invariant notion of fairness in your long thread about the duel? Would something similar be possible here?

Yes, it should be possible. My point is even that it is not even possible, but mandatory. I mean, either you find a frame-invariant concept that gives the solution (in which case only that solution is right) or the problem has no measurable solution (in which case discrepant opinions are not valid as solutions).

For this search, we have to look at the "spirit" of the problem. This sounds ghostly or nebulous, but in fact it is quite "material" or at least (if we want to embrace energy as well) "physical". In a race you want to reward the runner who has more... what? Chemical energy stored in its muscles? If for simplicity we imagine, as you do, that the runners only accelerate once and then travel inertially on wheels with negligible friction and so at constant speed, then the winner will be the one who, thanks to those powerful muscles, acquires, as you point out, a higher acceleration in that first effort. If we are told that hare and tortoise travel at the same velocity wrt the ground, it's because they managed to get identical accelerations. So the solution is a tie.

A different thing is how you reach that conclusion. In the ground frame, we look at the clocks of referee assistants located at the finish lines, see equal times (simultaneity) and infer: "accelerations were identical". But please note that "simultaneity in the ground frame" is just the clue; the reason would be the second, the identity of the initial jumps due to the identity of muscular strengths. If this solution is good, in the frames of hare and tortoise we should reach it as well, albeit on the basis of other clues...

 

[Saw]

How about defining it by whoever's proper time between start and intersection with the worldlines of the respective endpoints is greatest?

Edit: Maybe that should be "least"?

Well, yes, that would be the same criterion as used in the duel example. Here the rule would be: whoever has the least proper time, wins; if both have the same proper times, it's a tie. Actually, it seems that links with the other criterion: the runner who acquired a higher acceleration in that first push would reach the finish line after accumulating less proper time...

 

[atyy]

Yes, it should be possible. My point is even that it is not even possible, but mandatory. I mean, either you find a frame-invariant concept that gives the solution (in which case only that solution is right) or the problem has no measurable solution (in which case discrepant opinions are not valid as solutions).

For this search, we have to look at the "spirit" of the problem. This sounds ghostly or nebulous, but in fact it is quite "material" or at least (if we want to embrace energy as well) "physical". In a race you want to reward the runner who has more... what? Chemical energy stored in its muscles? If for simplicity we imagine, as you do, that the runners only accelerate once and then travel inertially on wheels with negligible friction and so at constant speed, then the winner will be the one who, thanks to those powerful muscles, acquires, as you point out, a higher acceleration in that first effort. If we are told that hare and tortoise travel at the same velocity wrt the ground, it's because they managed to get identical accelerations. So the solution is a tie.

A different thing is how you reach that conclusion. In the ground frame, we look at the clocks of referee assistants located at the finish lines, see equal times (simultaneity) and infer: "accelerations were identical". But please note that "simultaneity in the ground frame" is just the clue; the reason would be the second, the identity of the initial jumps due to the identity of muscular strengths. If this solution is good, in the frames of hare and tortoise we should reach it as well, albeit on the basis of other clues...

I agree.

Well, yes, that would be the same criterion as used in the duel example. Here the rule would be: whoever has the least proper time, wins; if both have the same proper times, it's a tie. Actually, it seems that links with the other criterion: the runner who acquired a higher acceleration in that first push would reach the finish line after accumulating less proper time...

I think I like the proper time better, since it'll work for arbitrary accelerations, and then one only finds out the winner by looking at their intersections with the worldlines of the end points.

 

[Saw]

I think I like the proper time better, since it'll work for arbitrary accelerations, and then one only finds out the winner by looking at their intersections with the worldlines of the end points.

Well, yes, proper time is easy to spot, it's measured in both frames (hare's and tortoise's) and can be calculated, through the corresponding formula, on the basis of measurements obtained in any frame... Only I wished to highlight that if we choose it as reference, it's because there's a logical link between that measurement and the practical purpose of the problem. It's not a matter of arbitrary choice: "if you wish a unique solution, then SR offers you an invariant concept; but all concepts are equally valid; if you had whimsically chosen simultaneity, you'd have different solutions". No, the choice is narrowed down by that faculty that some books order you to dispense with, which is common sense. No need to accumulate exciting adjectives: "wondrous", "startling", "puzzling"... There's nothing in SR that your intuition cannot digest, just by changing the paradigm: in the past (classical relativity) "winning" always meant arriving "earlier" because all observers were supposed to measure simultaneity and time rates homogeneously; if experimentation proves that that is not the case, then "winning" starts meaning "earlier" in the ground frame or, what amounts to the same, "less proper time than your competitor", but in both cases those measurements or clues have problem-solving ability because they reflect what we really look for, i.e., who is a better runner.

But what about the Peace Treaty? I'll think of something and let you know.

 

[ZikZak]

I think you're just defining and redefining what it means to "win" the race now. But I do think discussing it both ways, in terms of both simultaneity and proper time would be useful in class.

 

[atyy]

I think you're just defining and redefining what it means to "win" the race now. But I do think discussing it both ways, in terms of both simultaneity and proper time would be useful in class.

Nah, it's standard in many races nowadays - each runner carries his own clock!

http://www.marathonguide.com/features/Articles/RaceTimingWithChip.cfm
http://running.about.com/od/racetraining/f/chiptime.htm

I have to admit that their concern was not special relativity though

 

[Saw]

I think you're just defining and redefining what it means to "win" the race now.

That's the point, precisely. Solving a problem requires understanding the true meaning of the question: what does "win" mean, what is the reality that the word tries to catch? There's only one answer but that doesn't mean it's easy to find it. Anyhow, if the answer were (I'm not sure of that) "the winner is whoever arrives at the finish line with the least accumulated proper time" and that happened to be equivalent to "whoever arrives earlier in the referee's frame", that would not be dramatic.

We're used to accepting that when handling the old relative concepts, like velocity. If the police agent stops you for exceeding the speed limit, you will not dare to tell him: "but in my car's frame I was stationary!" You will understand that the "spirit" of the rule is not to travel too fast with regard to the Earth, because that is what creates danger for people on the Earth. If simultaneity becomes relative, it may happen the same: there's nothing wrong with the fact that the key to solving a certain problem is given by the simultaneity judgment of the ground (although it can also be reached through other routes, based upon other measurements that, in their own way, adequately interpreted, enlighten you analogously).

But I do think discussing it both ways, in terms of both simultaneity and proper time would be useful in class.

Let me know if you do!

 

[ZikZak]

Let me know if you do!

I almost certainly will when I teach it next semester. It's a good idea.

 

[atyy]

I almost certainly will when I teach it next semester. It's a good idea.

Oh, I now see why you don't like relativistic mass - since you actually have to not confuse some kids - unlike people like me who just hang out here for fun (I'm a biologist - relativity's not going to matter until the LHC creates enough black holes for radiotherapy!). Anyway, while we're being a bit serious, an important part of the set up in "proper time" conception is that the race must be "objectively fair". In this case, the endpoints must be defined as worldlines of objects moving inertially and stationary relative to and equidistant from the tortoise and hare "before" the start of the race - or something like that. The "before" is objective since we can define future and past on tortoise and hare worldlines coincident before the start event. It's this before that kind of picks out the referee's frame as special among the 3 - since actually considering before and after - it's ambiguous to assign an inertial frame to the hare and tortoise - their full worldlines including before and after define noninertial frames - only the referee's frame is inertial. Of course, the referee's frame is not truly special when the comparison is taken against all inertial frames. OK, that's imprecise, but I think the general idea should be ok.

 

[ZikZak]

Oh, I now see why you don't like relativistic mass - since you actually have to not confuse some kids

I more frequently have to UNconfuse the kids, who come pre-confused by high school teachers who have taught them to find the relativistic mass, and then use Newton's equations. *shudder*

unlike people like me who just hang out here for fun (I'm a biologist - relativity's not going to matter until the LHC creates enough black holes for radiotherapy!).

Ooh, cool! Can you eliminate tumors by sucking them into a black hole? ;)

 

[Saw]

I almost certainly will when I teach it next semester. It's a good idea.

Thanks, Zikzak. I wish I could be there. It'll be fun.

Oh, I now see why you don't like relativistic mass - since you actually have to not confuse some kids - unlike people like me who just hang out here for fun (I'm a biologist - relativity's not going to matter until the LHC creates enough black holes for radiotherapy!).

Well, I am a lawyer, but I am here for fun and also to do my job ;). You know, real problems are neither physical nor legal nor philosophical. They are just problems and reality knows nothing about administrative separations of knowledge. (I admit, and also regret!, my limitations in these discussions, however.)

Anyway, while we're being a bit serious, an important part of the set up in "proper time" conception is that the race must be "objectively fair". In this case, the endpoints must be defined as worldlines of objects moving inertially and stationary relative to and equidistant from the tortoise and hare "before" the start of the race - or something like that. The "before" is objective since we can define future and past on tortoise and hare worldlines coincident before the start event. It's this before that kind of picks out the referee's frame as special among the 3 - since actually considering before and after - it's ambiguous to assign an inertial frame to the hare and tortoise - their full worldlines including before and after define noninertial frames - only the referee's frame is inertial. Of course, the referee's frame is not truly special when the comparison is taken against all inertial frames. OK, that's imprecise, but I think the general idea should be ok.

Hmm… The problem of the duel (the story proposed by Brian Greene in The Fabric of Cosmos), which we analyzed in the thread The show of the duel, only involved inertial frames and in spite of that it had a clear solution. I summarize the display and the reasoning.

The duellers are Back and Front, located respectively at the back and front of an inertial train. The referee on the train, when meeting another referee on the platform, sends light pulses to the duellers, who shoot laser rays on reception of the flashes.

The spirit of the problem (the practical result that we seek when we stipulate “the duel must be fair”) consists of, basically, two rules:

1. There is a forbidden trick: no dueller must be able to hit his adversary while the latter has not yet seen her own flash.
2. The duellers may try different tricks to win the duel: dodging, bending down, bringing up a shield…; both duellers must be able to carry out the same number of tricks between the two relevant moments (receiving the flash and receiving the shot from the opponent).

Solution to 1:

In train frame: that cannot happen because the flashes arrive simultaneously at the duellers.
In ground frame: that cannot happen because the distance between the two events is space-like and so the events are not causally connected. In less technical terms, that cannot happen because when Back, for instance, receives her flash (earlier in the ground frame), the flash to Front is already on its way, by definition, and SR postulates that Back is unable to send any projectile that travels faster than light.

Solution to 2:

In train frame: if the flashes reach the duellers at the same time, so do the shots (assuming, for convenience, that the duellers have equal reaction times); so Back and Front can do the same number of tricks.
In ground frame: if Back receives her flash earlier, she also receives the shot aimed at her earlier; vice versa for Front; one thing compensates the other and both duellers dispose of equal coordinate time intervals to do their tricks.

If you now refine the question and ask specifically, how many tricks can Back and Front carry out, then the answer is “proper time” between the two relevant moments, which is obtained:

In train frame: by Back and Front, by reading their respective clocks or, if you wish applying the formula dt^2 – dx^2, where dx^2 is = 0.
In ground frame: by combining dt’ (as the difference between the reading of the clock located where the dueller sees the flash and where he or she is shot) and dx’ (distance between those two points) in the same formula, which gives identical result.

Thus, for example, if bringing up a shield takes Back 2.1 s (when she's in any frame) and the available proper time interval on the train during the duel is 2 s, all frames infer that she will not manage to do it.

So the duel is fair for both frames and the number of actions that the duellers can (hypothetically) carry out during the duel is guessed by both frames, although the train frame (which is of course inertial) does have, as commented, a certain "linguistic" advantage: for it, the path to knowledge is shorter; it just has to check that the flashes warning the duellers that they can shoot are simultaneous to infer that the duel is fair and it just has to read the proper time of its clocks to guess the number of tricks or opportunities that each dueller disposes of.

 

[atyy]

As long as everything is specified using geometric invariants - which is how relativity defines "reality" - it makes no difference whether one uses inertial or non-inertial frames. The only difference is that in transforming between inertial frames, it doesn't hurt if you forget to transform the metric, but if you transform between non-inertial frames, it hurts if you forget to do so.

 

[Saw]

As long as everything is specified using geometric invariants - which is how relativity defines "reality" - it makes no difference whether one uses inertial or non-inertial frames. The only difference is that in transforming between inertial frames, it doesn't hurt if you forget to transform the metric, but if you transform between non-inertial frames, it hurts if you forget to do so.

That's interesting. It'd be helpful if you could elaborate on that. In what sense would we be here transforming between non-inertial frames?

 

[Saw]

ZikZak, one question for you:

Suppose we want to retain the criterion that the winner is whoever shows more muscular strength. In physics that would be "force", to be measured as F = ma, m being the mass of the runner.

Suppose that this F is measured in the ref's frame (the ground) and also in some inertial frames passing by and which have, wrt the ground, the same velocity as will be acquired by the runners as result of their efforts. Let's call those frames hare and tortoise.

In relativistic terms, what would this F be as measured by each of those frames (ref, hare and tortoise), for each runner? Thanks in advance, if you can take that trouble.

 

[atyy]

That's interesting. It'd be helpful if you could elaborate on that. In what sense would we be here transforming between non-inertial frames?

Well, my point is not that we have to use non-inertial frames, rather that it makes no difference if we choose to.

But, for illustration, take the entire worldline of the hare - including before and after the hare's acceleration evernt. We define the worldline to be the x=0 line. We define t=0 on the worldline to be at the acceleration event, and t otherwise to be the proper time elapsed. To define x, t coordinates of an arbitrary event P away from the worldline, we define TPH as the earliest hare proper time at which a light ray from P can reach the hare wordline, and THP as the latest proper time at which a light ray from the hare worldline can reach P. Then we define P's x coordinate is as (TPH-THP)/2 and P's t coordinate as (TPH+THP)/2. In this way, we have set up coordinates for events anywhere in spacetime - which is noninertial, since it is based on an accelerated worldline.

I've taken the above method from Dolby and Gull's http://arxiv.org/abs/gr-qc/0104077 . One thing to be checked that I haven't is how big a patch of spacetime the coordinates are well-defined.

 

[Saw]

Atyy, I re-read the article (I think I had downloaded it another time you mentioned it) and I'm afraid it'd take me some light-years to understand it. I'm not proud of that, I wish I could grasp it some day... Anyhow, in the meantime, I've thought of another test of the idea that the race is a tie if both runners arrive at the same time to the finish lines as measured in the ref's frame:

- let the runners, at arrival at their destinations, stop;

- it doesn´t matter if they rest for some "time", as long as it's the same time for both (different frames will measure that this rest time is of different size, but they'll all agree that it's equal for both competitors, which is what really matters);

- after rest time, they jump back with equal strength as applied at the start of the race and acquire the same speeds (but opposite directions) wrt the referee and

- according to SR, they'll arrive simultaneously at the origin, as judged by all frames.

This bears some resemblance to the solution in the duel example:

- If any frame measures that one runner arrived earlier at the finish line in the go-trip (i.e., its jump yielded a better result), such frame must also admit that such advantage benefits the other competitor in the return-trip.

- Vice versa: if for any frame a runner arrived later in the go-trip (i.e., its effort was handicapped by circumstances), such frame must also accept that the handicap is for the other competitor in the return trip.

- If the runners happen to return at the origin at the same time, as the theory predicts, it's because the advantage sets off the handicap.

Hence the two competitors had equal abilities (whatever the source: genetics, hard work in the gym...) and must share the price.

Makes sense?

But please keep on elaborating on your own route, if you wish. That's SR, in my opinion: ideally, if well followed, all paths lead to Rome; in practice, due to conceptual difficulties, I usually end up in Móstoles (a city of the industrial belt of Madrid...).

 

[Saw]

Saw:
I looked and it looks like the thread has died. But I don't disagree with the position you were taking there: you were basically saying that (1) the "solution" (i.e., the "right" specification of who wins the race, for example) has to involve quantities that everyone can agree on, so there's no ambiguity, and (2) which quantities you choose has to be guided by the "spirit" of the problem. I agree with both of those statements. I was focusing on (1) in my comments, but I didn't mean to imply that (2) isn't important too.

Thanks for taking the trouble. On the other hand, the thread is open. For example, I am still curious about the question I asked in post #23. If we tried to fix as "specification" the forces applied by the runners in their initial jumps (ma), what would be the measurement of that force by different inertial frames, let's say the ones on which the runners "jump"?

 

[PeterDonis]

If we tried to fix as "specification" the forces applied by the runners in their initial jumps (ma), what would be the measurement of that force by different inertial frames, let's say the ones on which the runners "jump"?

The invariant way to specify force in relativity is

$$F = \frac{d P}{d \tau}$$

where P is the object's 4-momentum and $\tau$ is the object's proper time. In the object's rest frame (before the force is applied), for sufficiently small forces (such that the object's velocity after the force is applied is small enough compared to the speed of light), we can approximate the above by using the Newtonian formula $F = ma$. However, making the velocities that small would make the problem trivial, since the whole point is to consider relativistic effects.

The http://en.wikipedia.org/wiki/Four-force" goes into how to translate the above equation into a given frame. Basically, you just need to apply $\gamma$ factors in the appropriate places; I don't have time to post the details, but they're not hard to work out. Unfortunately, the result isn't very interesting. The numerical value of the two forces varies from frame to frame, but all three frames agree on their relative magnitudes.