md65536

I tried, thinking there must be a simpler solution than I gave...

I think most people who post here are amateurs, as I am. I don't think anyone is offended by questions about trying to understand relativity (as long as they don't involve purposefully ignoring answers people have already given and don't include "Relativity must be wrong, because..."). Also, no one understands it perfectly or has considered every aspect, so questions can be helpful to everyone.

Did my answer in steps to not spoil it all at once...

Edit: I agree with Janus' reply, some of my post is redundant. Figuring it all out first using only inertial frames, without specifying the timing of the train's acceleration, would probably be helpful. That answer applies to the case where the train is accelerated in such a way that the rest length of the train remains the same after acceleration. You have the trains remaining the same only in the track frame. Someone else had the backs of the trains synchronized. All of these situations are possible. People will fill in their own details and interpretations unless everything is unambiguously specified. If they're moving in unison in a particular frame, they will maintain their distance in that frame. Remember you have to specifically coordinate them to do this, and it is not always physically possible (Bell paradox; a rope or train that can't be infinitely stretched will break at a high enough gamma). Note: It is only the lengths between synchronized points that remain the same. If you synchronize all points (fronts and backs of both trains all synchronized) in the track frame then the trains and gap remain the same length (but must be physically stretched). If you synchronize only the middle points (as I think you might have specified) and let the trains maintain their rest length (this was not specified, so is left open to different configurations), then the trains will contract and the gap will indeed become greater in the track frame. Move inertially, I mean maintain constant velocity, or uses only one rest frame. Yes switching frames means changing relative velocity or accelerating. Err... I don't think this line of reasoning will help you until you've figured some other things out first. Frames don't have a single clock. Clocks measure proper time (the time at the clock, not the time at other locations). An event is a single point in space and time, and has a definite proper time according to a clock that passes through the event. A point of the train passing through a certain point on the track is an event, and all observers agree on the proper time that that happens. Now, you can have other clocks on the train and on the tracks, and different observers can disagree on which clocks are ahead or behind relative to the proper time of the given event. But again to say which is ahead or behind I think you'll have to specify the details of how you've coordinated the clocks... and where those clocks are located relative to the event. I'll work through an example to help myself figure out what I'm talking about... Suppose a train is 100m long in its rest frame and is heading East, and passes a station that's 100m long in the track's frame. Have 3 clocks on the train, front middle and back, synchronized in the train's frame. Have 3 clocks at the station, West end, middle, and East end, sync'd in the track's frame. Say the middle clocks pass each other at exactly 12 noon according to the station's middle clock, and 10:00 according to the train's middle clock (edited to avoid introducing misconceptions). According to the train, all 3 clocks on the train read "10:00". The station is contracted, so the front of the train has already passed the East end of the station, and the back of the train has not yet reached the West end of the station. According to the station, all 3 clocks in the station read "12:00" at the moment the middle clocks pass. The train is contracted, so the front of the train has not yet passed the East end of the station, and the back of the train has already passed the West end of the station. To sort out this situation, note that the front clock of the train reads 10:00 only after it has passed the East of the station. According to the station, this hasn't happened yet: the train's front clock does not yet read 10 and is running behind. Similarly, the train's rear clock had struck 10 earlier before reaching the station; it is ahead. Symmetrically, according to an observer on the train, the station's West clock is behind, and the East clock is ahead. Eg. the West clock reads 12 some time after the back of the train has passed, and that hasn't happened yet.

If you force various points to remain synchronized in the track frame, the distances between those points will remain the same in the track frame. Effectively you'll be stretching the distance between points (the gap will be increased in the moving train's rest frame) to exactly counteract length contraction. This is set up like Bell's paradox. The answers to your questions can probably be found in an explanation of the paradox. If you don't want to deal with the details of relativity of simultaneity, just give the train a fixed rest length, and let it remain moving inertially throughout the experiment. Don't worry about how it accelerated. If you want to have the train switch inertial frames, I think you are going to have to factor in the details of relativity of simultaneity, and you may need to decide on a few more details than you're giving. I think it's fairly common that people want to figure out one aspect of relativity that they don't get, and they completely avoid another aspect like it's too complicated to consider. It's like trying to figure out how 2+3=5 without considering the 3... "how does 2 add up to 5, relativity makes no sense!"

Which I think is equivalent to two clocks at different gravitational potentials in a uniform gravitational field, which also run at different rates relative to each other.

I say the clocks don't stop. The clocks were synchronized in the track frame before the train started moving. They won't be sync'd in the moving train frame unless the parts of the train are started in a peculiar way to achieve this result, with either parts of the train having different velocity profiles or the train being deformed in its own frame when moving. The clocks were synchronized in the track frame. They'll stay synchronized only if they accelerate at the same time in that frame. However if that is the case, the moving train will not be length-contracted shortened in the track frame (it will be stretched in its own frame as the front started moving too long before the back did, but stretched plus length-contracted to maintain its rest length in the track frame, as in Bell paradox). If this is the case, the problem can be solved without even considering relativity, just delay of light will tell you that in the track frame the simultaneously emitted signals won't reach the moving middle of the train simultaneously. In the track frame, the middle of the frame closes the distance to where the front signal was emitted, to intercept it before the rear's signal. In the train's frame, the front clock is ahead and the signal is emitted before the rear signal.

Strange is right. Remember the twin paradox. When the accelerating twin changes inertial frame, the coordinate time of the distant twin that it accelerates toward jumps forward (only relative to a local clock). The coordinate time of a twin that it accelerates away from jumps backward. The "jump" is merely a change in relative simultaneity. From the other thread... if a train is a certain length at rest on the tracks and all parts of it accelerate similarly and it is again the same length in its new rest frame, it must be that from the track's perspective, the back of the train accelerates first and the front last. But from the moving train perspective, the front accelerated first and the back last. The observer in the middle was in both frames... so how can that be? Just before accelerating, the middle observer understands that she will accelerate first before the front of the train. Suppose with sync'd clocks the middle observer will accelerate at time t=1, and the front at time t=2. The middle will accelerate toward the stopped front of the train, so the coordinate time of the front of the train will "jump" ahead relative to the accelerating middle, let's just suppose it jumps ahead 2 units of time to t'=3. Now an instant after time t=1, the middle observer is now in the moving frame, and the front of the train has already started moving at its time t'=2, and continued to move up until its current time t'=3. The clocks are no longer in sync. The distance to the location on the tracks where the front of the train started from is now length contracted and is no longer half the length of the train... how can the train still be the original length? Well, the front of the train has been moving between its time t'=2 and t'=3, which occurred in an instant for the instantly accelerated middle, and the front is now ahead of that point, at the right distance. To get the exact details you'd have to do the math. As always all the details work out consistently. Meanwhile the observer at the front of the train observers local time passing normally, while other coordinate times jump (backward). Since the clocks are no longer in sync after the train accelerated, they won't emit the flashes at the same time.

It sounds like you understand it. There's no "center of contraction" similar to how there's no center of the universe yet it is expanding in all directions. The contraction (expansion) is uniform... the same at any point. Yes, I would say that picking a center is similar to picking the time that an arbitrary point on an infinitely long train passes an arbitrary point on infinitely long tracks. You could choose different points and it happens at a different time. You could as easily say that relative simultaneity is a consequence of length contraction and time dilation, or that all three are a consequence of the constant speed of light (or vice versa). They all fit together, no matter which one you start with.

Careful because there is no universal meaning of "at the same time". It will be different for someone beside the tracks compared to someone on the moving trains. For example... This is true for an observer at rest relative to the tracks, if the trains are 100 m long with 100 m between in the moving trains' frame. If all parts of the trains start at the same time according to an observer on the tracks, they'll remain 100 m long with 100 m between them in the track's frame. Is this helpful or confusing? Unfortunately you can't investigate all the details of a thought experiment like this one without being precise about whose frame measurements are specified in. The question can be simplified by considering only inertial frames and ignoring when the trains start moving. If the trains are moving with gamma=2 and are 100 m long with 100 m in between them in their own frame, then they'll be 50 m long and 50 m between them in the track's frame. The answer is that all lengths in the direction of motion are contracted. (Just for complication, how would a train have to begin moving in order to get this situation? You can imagine the trains starting at rest in the track frame, and an inertial "ghost train" that eventually lines up with the moving trains. From the track the ghost trains are 50 m long and separated by 50 m, so the end of the last train will have to start moving first to line up with the ghost train, then the start of the second train, etc until finally the start of the forward train begins moving last. Meanwhile from the perspective of the moving trains's frame, the ghost trains are 100 m with 100 m between, while the trains at rest relative to the track are 50 m long with 50 m between. From this frame, the front of the forward train must begin moving first, and the end of the trailing train must begin moving last. Fun! Also: It would be a frame in which the tracks and the moving train have the same speed but in opposite directions, in which the trains started moving at the same time. They're length contracted the same whether at rest relative to the tracks, or traveling in the opposite direction as the tracks.)

Not speaking from experience, I can imagine some of the major draw is fantasy and a sense of belonging. With fantasy, you can escape reality and make your own. In your mind you can make your alternate reality as important and fascinating and magical as you want. So even if the cartoon is not a richly developed world, or aimed at your demographic, you can build it out in your mind and get attached to what you've imagined. Then once you've escaped into this alternate reality---which you might have done partly out of not feeling a sense of importance or belonging in the real world---and you find like-minded people who accept you and your interests, the sense of importance and community in the fantasy world becomes a real feeling. Maybe for some it starts as a slight interest, and then they get pushed away from people who think they're weird toward people who embrace and accept them. Then, the more you get in to a fantasy world, the bigger it gets for you, and the more you feel disinterested or out of place in the "real" world. This applies to any fantasy world, I think. What bronies see from inside that world is a lot bigger than what we see from the outside. It's probably very different for different people, but I see the two aspects of a fantasy world and of seeking acceptance of quirky interests as appealing.

You can experiment by holding weights. Swing them and your body will sway in the opposite direction. This already happens even without any additional weight. It's not that one part of you moves while the other is fixed; both move, but the massive parts not as much. You can detect slight backward motion of your body when punching forward. Here's an experiment: Stand steady with your back to a wall, as close to touching it as possible. Punch both hands quickly straight forward, and feel the force with which your back pushes against the wall.

But all evidence is consistent with "universal gravitation", that all masses attract each other. If you introduce another massive body, a test mass will gravitate toward it the same as it would to any other similar mass. Does your speculation predict that a mass will gravitate differently to its "source", contrary to universal gravitation? If not, how do you differentiate a "source" and everything else that effects gravitation?

I disagree. The proper conclusion is that there was thrust measured with both setups. You can conclude that there is no net thrust due to the configuration of one setup vs the other, but you can't dismiss the measurement entirely until you identify what's causing it, even if you assume it must be due to some bias or error.

I'm trying to follow this conversation but I don't get it. Is there an example of what you'd want it to do? I can't see why anyone would expect a GPS by default to give altitude relative to the tide level. If it did, would that mean that the readings on land would also fluctuate with the tides? Or that your altitude coordinate would be different depending on if you were on sea or land? I can see nautical GPS applications for that. A lot of GPS devices run custom software, and I wouldn't doubt that someone has incorporated tide data into some software (even something general, might have some kind of "sea mode" setting). If not, it should be possible to automate that. But that should be handled after getting the position from the satellites, it shouldn't be built in to the satellite system.

It's not filled with long waves that occupy space. Light is quantized into packets that have a point location. Wherever you measure "where" light energy is, it will be measured as individual photons each with a single point location. A sea of photons in space would be entirely empty space with photons in it, taking up no volume. Another way it's not like a sea or a volume filled with matter is that the photons don't interact. They won't spread out to fill a space, they'll just go where they're going. I would not agree with this (as an amateur), though I think many people would. The wave-particle duality doesn't mean that light "is" a wave and it "is" a particle, only that it consistently has properties of either, and it only certainly has those properties when measured. So, when you're not measuring the particle properties of a photon, there's really no point in saying that it "is" a particle even where it's not being measured. You could easily just say that yes, if they were measured there would be photons all over between Sun and Earth at all times. But a sure statement about photons in between measurements can cause problems. For example, the sun is about 8 minutes away, so you could say that there are a bunch of photons traveling at different points along an 8-minute journey between the Sun and you. But if you accelerate to near c you can shorten that journey to say 1 minute. Then there is only 1 minute worth of photons, but they're still traveling at the same speed. What happened to the other 7 minutes worth? You might imagine relative time being "adjusted", so those en-route photons are brought back into the Sun and now haven't left yet. BUT I think this is pointless. This is an example of a confusing interpretation based on the assertion that the photons must exist as particles along their entire (8-minute) journey. On the other hand, if you don't measure any photons in between Sun and Earth, you only have to say that they "are" where they're measured, and the existence of photons beyond how they are measured becomes irrelevant philosophy. So I would not agree or say anything certain about the existence of photons anywhere between where they're emitted and detected (absorbed or measured). Or consider a different example. Suppose two people agree that there are photons constantly traveling along straight lines from Sun to Earth. Now put a double-slit between Sun and Earth. Suppose one person says "Each photon must go through either one slit or the other" while the other disagrees and says "Each photon must go through both slits simultaneously." It depends on interpretation, and neither interpretation is meaningfully measured. Similarly, agreeing on the behavior of photons without the double-slit depends on interpretation, even if it's easier to agree due to no obvious problems with intuition.

In addition to the equations not making sense, they are consistent with everything else that's known about photons, and there is no way in which such a reference frame makes physical sense. Think of what it means to have an observational frame of reference, and I suspect that anything you think of will not apply to a photon. All lengths contract to zero in the direction of travel, so they can't measure length. All trips would take zero time, so there is no way to measure travel time. A photon's state doesn't change from emission to absorption, ie. it doesn't age, ie. it measures no passage of time. A photon doesn't absorb other photons, so it can't observe anything while traveling. To accelerate to c requires infinite energy, unless you have no mass, but if you have no mass then you have no rest energy, so you can't be at rest and you can't have a rest frame. All these things fit together, and there's nothing that describes a photon being able to observe. The math models physical reality, and the math doesn't work for a photon's frame of reference, while there is also no such thing known or theoretical that makes sense to model.

I think I'm starting to get it. And you say this works if the separation is in any direction, it doesn't matter if it's in the direction of V? Say, as per OP, a separation only in the y-axis, that's the same as a separation only in the x-axis? Or in other words, your math with one spatial dimension works the same as with 2 or 3, and it doesn't matter how the separation vector and the velocity vector are oriented relative to the other?

Is the x-axis separation of the photons equal to 0 for all observers, including those whose velocity relative to the S frame is not in the same direction as V? Yes, if your equations are complete then I'm wrong. But don't your equations also hold that [math]\gamma(V)t_0=0[/math] in the case that x_0 = 0 and t_0 = 0? Which implies that if the photon emissions are simultaneous in frame S they're simultaneous in all frames?

The latter, because I'm having trouble following your math. I suspect that you're answering a different question but I might be wrong. If there is a y-axis separation, and x_0 = 0, and letting t_0 = 0 (the photons are emitted simultaneously in S frame), then the x separation is always 0 in your formula. Is that true for all observers, even ones that aren't limited to motion in the x direction? I think this is relevant, because it is fairly common for people to claim that there is an absolute frame of reference by ignoring any frames where their premises don't hold. I believe that the premise implied by OP that the x-separation is 0 in all frames is wrong, but your math seems to support it.

But what if the separation in space is not in the direction of V, such as in OP's setup?

In post #1 the photons are described as moving parallel to each other. The only separation is along the y-axis, not in the direction of travel. Good for you for using math, I admit once again that I'm deficient. However the math is useless if it describes something entirely different from the prose. If you wish to continue discussing your variation instead of OP's, why not start a new thread?

Do the photons have a comoving frame? Delta1212 understands what I was getting at. I was speaking only of OP's premise that the photons are "always" side-by-side. I agree it's not an essential part of the answer. If OP's thought experiment requires the photons to be separated and side-by-side in all frames, then it's a problem, but the experiment can be restated without that requirement.

Are the photons emitted simultaneously in all frames? If no, are they side-by-side in all frames? If no, is OP's premise in the thought experiment flawed? Isn't that relevant? Do photons have a comoving frame? Edit: I guess you could put the photons together and thus have them emitted simultaneously in every frame, and still come to the same conclusions as OP did if you try to consider an invalid frame in which the photons' relative speed is 0. So RoS isn't a necessary part of the answer and it's possible to have the photons stay together in every frame. I guess this would only come into play if you used the distance between the photons in a calculation that derives a contradiction if absolute simultaneity is assumed, which I don't think has been done here, so you're right that RoS need not be considered. That's a good point. No matter how distorted the bar is by relativistic effects, it remains "together" in any frame. OP could use that if "the photons remain side-by-side" was replaced with "the photons remain close to each other." However the bar is only an analogy, since the bar has a rest frame but photons don't. (The bar ends don't have an invariant "single speed" but the photons do.)

No, I'm not. You're right that they have the same speed, but if they're not emitted simultaneously then they won't be "several photons moving parallel to each other, side-by-side, always staying abreast to each other," as per OP. Unless their separation is negligible, they can't be emitted simultaneously in all frames. This is a correct application of RoS. For example, if there is separation and you allow observers arbitrarily close to the speed of light, you can shorten the travel time of light arbitrarily (due to length contraction), and contrive an observer for whom one photon reaches its destination before the other photon is emitted. In this case the photons don't travel together in any way.