gib65

Here's something that confuses me about length contraction:

Suppose you had a toy train Tn1 on a straight track at position P1 (at Tn1's center). You turn on the voltage and watch Tn1 accelerate. It accelerates at rate A1 for an amount of time T1, covering a distance D1, and arriving at position P2 (again, at its center). (Note that Tn1 doesn't stop or slow down on or before P2, it just arrives there).

Now you remove Tn1 and place another train Tn2 on the track, only this time you position Tn2 at position P3 which is 20 meters ahead of P1 (where D1 > 20 meters). Again, you switch the voltage on. Tn2 accelerates at A2 for time T2, covering a distance D2, after which it arrives at position P4.

Now, given that the conditions of the track do not change throughout this experiment and that Tn1 is identical to Tn2, it would be fair to say that A1 = A2 and that if T1 = T2, then D1 = D2 and P2 - P1 = P4 - P3.

So far, none of this denies that Tn1 and Tn2 length contracted as they accelerated. We can say that Tn1 contracted by an amount L1 and that Tn2 contracted by an amount L2, and that L1 = L2.

But now, you conduct a third experiment. You repeat the previous two experiments in tandem: both Tn1 and Tn2 are placed on the same track at positions P1 and P3 (respectively). The track is turned on. One would expect both trains to accelerate at A1=A2 and if allowed to continue for time T1=T2, then they should each individually cover distance D1=D2 to arrive at position P2 and P4 (respectively). In other words, nothing from the previous two experiments should change just because we allow the two trains to accelerate at the same time.

I would expect both to still contract by L1=L2, but the part that has me confused (still) is that the space between them will also contract. What this seems to mean to me (from my most likely inadequate understanding) is that, from the point of view of a person standing still watching the trains go by, the Tn1 will look like it is "catching up" to Tn2--either that or Tn2 is "slowing down" (or at least not accelerating as fast) compared to Tn1--they are closing the gap between them, in other words.

This is what it would look like, at least, from the point of view of someone standing still watching the trains go by.

So, that person would have to measure that A1 > A2.

Maybe that's just the case. Maybe that's just what happens according to relativity. But it seems really odd that simply adding an extra train to the track would change the results of the experiment. Something's not right here.

At this point it would be nice to hear from gib65

It's not so much a paradox as something which seems (to me) to be missing something.

In a nutshell, I stated it like this in my OP:

If everyone has to observe the same events happening, then either a) everyone has to observe that the stop-clock indeed stopped, in which case your assistants would have to conclude that the flashes went off at different times (even though they observed the stop-clocks being synchronized), or b) everyone has to observe that the stop-clocks did not stop, in which case you would have to conclude that the flashes went off at different times (even though you observed the stop-clocks being synchronized).

So the clocks start out synchronized, and at the end of the experiment, someone has to conclude that they became unsynchronized. This is not a paradox, but it seems to suggest that somewhere certain laws of physics were violated (I don't think this is actually the case--the laws of physics being violated, I mean--which is why I say something seems to be missing in my understanding). Being initially synchronized and treated them same way throughout the whole thought experiment, you would think that exactly the same things would happen to the two clocks--they wouldn't enter into different states from each other--but they do enter into different states--one clock ends up behind the other--and so it seem (to a novice like me) like the laws of physics were different for one clock than for the other.

But I'm gathering that the consensus is that they two clocks are not treated the same--one is placed at the rear of the train and the other is placed at the front--and according to some on this board (and elsewhere--I have this question posted on another science forum), this can result in different effects happening to the two clocks during the acceleration phase of the train's journey (apparently, this is true even for someone on the train). The front clock will appear to run behind the rear clock relative to someone at rest.

But this is the part I'm trying to wrap my head around.

A few people have said it has to do with the simultaneity of when each part of the train starts accelerating. I don't think this has anything to do with physical lag, by which I mean the fact that an engine at the front of the train will start moving first and all other cars will start to move slightly after (because it takes time for the energy to travel down the length of the train and thereby pull each car)--kind of like how at a traffic light that turns green, not all cars start moving simultaneously but rather one after the other. I don't think this is it because you could just as well have the engine at the rear of the train pushing the other cars.

But if we are to say that, relative to an observer at rest, the front of the train starts accelerating first (for whatever reason), then I'm wondering if I understand it correctly. Please tell me if I have this right: suppose that at the moment when the front train starts accelerating, both clocks read 12:00. Then the front train starts accelerating at a rate that pushes the front clock to tick away at twice the rate as the rear clock (relativistically speaking). It takes 5 seconds, according to the rear clock, for the front train to get up to speed and start coasting. At that point the rear clock will read 12:00:05 and the front clock will read 12:0010. (I'm avoiding making both trains accelerating at the same time in order to make things simpler--but I'm sure the same principles would apply even if the rear train started accelerating before the front trains stopped accelerating). Now it's the rear trains turn to start accelerating. It accelerates at the same rate for the same amount of time. This means its clock's rate will be doubled and will end up reading 12:00:15 by the time it starts coasting. However, the front train's clock will not read 12:00:15. Having achieved a high velocity, it's clock will tick away at a slower rate than that of the rear clock while it was at rest. In other words, it will be somewhere between 12:00:10 and 12:00:15. <-- They are out of sync. The rear clock now being ahead will result in its going off (emitting a flash of light) before the front clock.

Do I have this right?

If so, my only question is: is this a general rule of relativity? That in an accelerating reference frame, things closer to the rear of the reference frame (where "rear" is defined as positions further away from the direction of acceleration) will end up being "ahead" in time relative to things closer to the front, at least once the acceleration has stopped and the system begins coasting? And this, because there is no absolute simultaneity of when each part of the system starts to accelerating, things at the front being considered to start accelerating "first" relative to someone at rest?

This has nothing to do with acceleration or with the twin paradox. Please stop confusing people with your misinterpretations.

[math]t'=\gamma(t-vx/c^2)[/math] tells you that the coordinate time is a function of BOTH time AND location. This doesn't mean that "the clock in the front slows down more than the clock in the rear".

 

"One meter to the right of the origin", as an example, for x = +1m. Same as in physics 101.

So what does this mean? That time dilates more the further away something is from the direction of travel compared to something close to the front?

Yes, that's what I thought, but it still leave a question in the air:

What must the assistants conclude? They saw the stop-clocks being synchronized. Therefore, they must conclude that the one at the front slowed down relative to themselves more than the one at the rear. I know time dilation occurs as velocity increases, but is there an additional rule that says time dilates (towards the slower end) more for things at the front (or the end closer to the direction of travel) than for the rear?

A friend and I are in a debate about relativity. He thinks he's found a paradox that overturns SR. I'm a little more skeptical. Here's the thought experiment:

You have a train at rest. You set it up with a stop-clock at the exact center of the train. You are standing there at the exact center of the train with the stop-clock. In your hands you hold two more stop-clocks. You synchronize them. You hand them to two assistants. They casually walk, at the same time and at the same rate, to each end of the train and place each stop-clock there, then leave the train. The stop-clocks at the ends of the train are set to emit a flash of light at a pre-determined time. That pre-determined time is the same for both. The flashes of light are aimed at the stop-clock in the middle. When the stop-clock in the middle receives two flashes of light at the same time, and only at the same time, it will stop.

That's the setup.

Now the train gets moving. It reaches a velocity close to the speed of light. At some point in the journey, the stop-clocks at the ends of the train emit their light flashes.

The question my friend and I are debating is: does the stop-clock in the middle stop or not?

You would think that from the point of view of you travelling on the train with the middle stop-clock, you would see the stop-clock stop. The two stop-clocks which you observed being synchronized should remain synchronized since they are in the same reference frame as you. So you would think they would emit their light flashes at the same time relative to you, and you should therefore see the middle stop-clock stop.

But for anyone not on the train (like your two assistants who also observed the stop-clocks being synchronized), the middle stop-clock should not stop because even if the flashes of light are emitted at the same time, the front flash will reach the middle stop-clock before the rear flash. After getting off the train once it stops, bringing the middle stop-clock with you, and rendezvousing with your assistants, what will they see?

If everyone has to observe the same events happening, then either a) everyone has to observe that the stop-clock indeed stopped, in which case your assistants would have to conclude that the flashes went off at different times (even though they observed the stop-clocks being synchronized), or b) everyone has to observe that the stop-clocks did not stop, in which case you would have to conclude that the flashes went off at different times (even though you observed the stop-clocks being synchronized).

Which will happen? And whatever the answer to this is, what does this imply about the timing of events that are initially synchronized?

Thanks everyone for your answers.

I guess I'm still confused. I'm just wonder why "the middle" should be considered to be exactly mid-way between the two trains. I mean, why not somewhere around Jupiter? (assuming Jupiter is in the direction of their travel). Why doesn't all length contract around a "center" which is placed light-years away from Earth, in which case the trains, once they get up to speed, will be positioned way out in space?

I think I *might* know the answer to this, but you tell me: all events that happen for one observer must happen for all observers--so if a passenger on one of the trains experiences trains begin at some point on the track, pass by a station, and stop at another station, those events must happen for everyone (including observers at the stations). So the question isn't where length contraction ends up positioning the trains, but when each event happens. This includes when each point on each train passes by each point on the track (say, for example, somewhere between the first station and the last station). That the trains seem to be contracted from the point of view of someone at the first station just means that the time at which the front of the train passes by a point on the track close to the station will be judged to be closer in time to that when the rear of the train passes by that same point (closer in time, that is, compared to someone on the train). If these two events--the front of the train and the rear passing by a certain point--occur closer in time than they would according to a Newtonian perspective, then one can only conclude that the length of the train must also be less than it would according to a Newtonian perspective.

In other words, it's a matter of there being no pre-determined time at which certain points on the trains must pass by certain points on the track (no pre-determined time, that is, on which all observers will agree), and length contraction is simply a consequence of this. Am I in the ball park?

Hello,

I've got questions about caffeine tolerance. Is there a minimum amount that one can take per day without tolerance building up? Say, 50mg per day? Or does tolerance start to build up no matter how little one takes?

Hello,

I've been trying to understand the quantum fluctuation theory of the Big Bang lately. That's probably not the technical name of the theory, but it's the one that says the Big Bang is the result of a quantum fluctuation in the void--the spontaneous appearance of some particle or force or something that very quickly lead to a run-away inflation of spacetime and matter and energy.

First of all, is there any credence to this theory among scientists?

Second, what exactly "fluctuated"?

Third, if from the Big Bang was create space a time, what time was there for any "fluctuation" to occur in such that it could give rise to time and everything else in it?

Please keep in mind that I'm a novice when it comes to these matters and I don't understand the technical jargon that goes along with it (which includes the math). Thanks.

Interesting answers,

To be honest, I was expecting an answer involving filters--as in, when you want to study the sun, and really look at it, we have to use filters that tend to block out light at the low and high ends of the spectrum, leaving only the middle which happens to be yellow. Could this also be true?

I was having a conversation with a girl from Japan today, and it lead to a very interesting question: why do we think the sun is yellow?

The conversation started when someone mentioned the color of the sun being yellow, and she (the girl from Japan) interjected and said "No, it's orange". Eventually, she explained to us that in Japan, they actually think of the sun as orange, and children there, when they color drawings of the sun, will use orange.

It lead to speculations that during evening, when you look at the sun, it does look orange. So why do we in the west have a noon-time bias when it comes to the color of the sun? Furthermore, it stands to question how we can even know what the color of the sun is at high noon. I mean, one should never look directly at the sun, but in the evening it's not so bad, and when you do, you see orange. But who has ever looked directly into the sun at high noon without getting temporarily blinded? The blinding happens almost instantly, too quick to take any notice of what color the sun is.

So if we can't actually inspect the sun with our own eyes at high noon, how do we know it's yellow?

Preserve some kind of representation of how it was affected by its neighboring neurons in the past (insofar as that representation is useful to its proper functioning).

Can a neuron "remember" being stimulated? Can it remember how long ago, or what pattern of stimulation it received (for example, three times in one millisecond and then two times in the following millisecond)? If so, can it use this information to figure out whether or not to fire?

I am also not sure about your use of # operator - if x # y means x raised to itself y times then surely it means x^(x^y) 2^2^2 = 16

Oh, yeah, well I kind of meant it in the same sense as when we say that a x b means "a added to itself b times" when we really mean "a added to itself b-1 times". Is a + a the same as a being added to itself twice or is it being added just once?

Does anybody else find this fascinating--that if both operands are 2 for any "growth" operator, then the result will always be 4. By a "growth" operator, I mean +, x, power, etc.--and operation whose result is greater than its operands (as opposed to "shrinking" operators like -, /, square root, etc.).

2 + 2 = 4

2 x 2 = 4

2 ^ 2 = 4

2 # 2 = 4 (where we could suppose x # y means x raised to itself y times).

Wouldn't this trend go on indefinitely? Couldn't we say that 2 @ 2 = 4 where @ is any "growth" operator whatever?

And wouldn't a similar rule apply where 4 % 2 = 2 where % is any "shrinking" operator whatever?

NOTE: I'm aware that I've defined "growth" operations poorly--for example, 10 x 0.5 = 5 (5 is hardly greater than 10)--but I think it's good enough to get the idea across that +, x, ^ seem to be one group of operators that have a common feature in their results, and similarly for -, /, square root.

Thanks, everyone, for your replies.

There's a tendency for the evolution of mathematics to branch. If we assume the basic line of mathematics to be basic arithmetic, then one of the earliest branchings would have to be algebra. More recent developments might include calculus, combinations and permutations, trigonometry, set theory, logic and computer science, and so on. One thing I'm curious about is whether there is a mathematics of patterns. You know, repetitions and regularities. Is there such a thing?

Thanks both.

Does anyone know the name of these formulas?

What are the formula for applying Einstein's relativity in mathematics? Suppose, for example, that I wanted to resolve the twin paradox. Couldn't I do it by applying one of the formula of relativity theory and calculate how much time goes by for each twin, and therefore which one ages more? What would these formula be?

I still don't see why an eather is needed at all. We would only need an eather to expain EM waves if EM waves were just like regular energy waves (the kind you find in water waves). What wave/particle duality tells us is that although the wave model of light and other particles works for some applications, it doesn't work for everything all the time. What this tells us is that it is a mistake to think of light waves as exactly like ordinary energy waves. They're similar, but not quite the same thing. Therefore, we have no right to draw the conclusion that an eather is necessary (as we might think it is for normal energy waves). Maybe we'll find out that it is necessary after we deepen our understand of EM waves, but for now, there's no reason to think an eather is needed.

Isn't this explained by wave/particle duality of matter/energy? If you can't understand how energy travels through a void, think of it as a particle. If you can't understand how a particle can travel as a wave, think of it as energy.

Just locally you can set up a foliation as [math]M = \Sigma\times R[/math], where [math]\Sigma[/math] is a 3 dimensional manifold. You generally won't have any canonical way to this, but you should be aware of the notion of globally hyperbolic space-times. Anyway, a point on [math]\Sigma[/math] for a specified "time" is just that, a point. From a four dimensional point of view you have a line that is parametrised by "time". (Locally anyway)

I'm not sure what that means. A point can't grow as a function of time?

There is something smaller than a point. We got Singularity.

A singularity is smaller than a point?

From what I understand, space and time can curve, warp, stretch, and maybe even fuse and fission. Can space and time also scale? Can a volume of space grow or shrink? If so, then can a geometric point in space grow to the size of a super-massive black hole?

This is what I think *might* be happening with black holes. When a star becomes massive enough to form a black hole, its center of mass not only becomes a new event horizon but the space within it inflates. In inflates in such a way that there is literally nothing inside it.

This might explain a few of the paradoxes surrounding black holes. From the point of view of someone far away from the black hole, it takes an eternity for anything to reach the event horizon. It is often wondered what happens to someone once they fall through the event horizon, what things look like from their perspective. But if the black hole is really just a geometric point blown up to the size of a massive star, then there is nothing inside the black hole. There is no space smaller than a geometric point. The event horizon, therefore, represents the end of space. It takes an eternity for someone falling towards the black hole to reach the event horizon because it takes an eternity to reach the end of space.

Not sure what this implies about the point of view of the one falling towards the black hole. Maybe space deflates the closer you get (or the faster you approach?) the event horizon such that when you actually reach it, it returns to being the size of a standard geometric point--or maybe the black hole flattens along the axis on which the person is falling. In any case, what happens to someone once they reach the event horizon is that they "land"--that is, they reach the centers of the black hole which, once again, coincides with the event horizon--and there they stay frozen in time (presumably dead).

Does anyone here think there are any merits to this theory?