#57805 Twin Paradox

[vorpal blade]

I have to confess that the many (and I think contradictory) explanations of the twin paradox do not make any sense to me. I suppose I'll be hated by the physics profession for saying that. It's not like this is new to me. I've been studying it for about forty years now, and this is sort of in my educational field. I've carefully gone over the supposed experimental proofs and found them to be severely lacking in rigor. The theoretical explanations seem like verbal and mathematical sleight of hand.

I get the asymmetry of the problem. Truly one twin accelerates while the other does not, and acceleration is an absolute difference, not a relative one. The traveling twin has to come back, causing him to decelerate and accelerate again. So, sure, you can always claim that there is no paradox. One of the things that bothers me is that what happens after the period of acceleration? So for a couple of days the special theory does not apply. How does a couple of days turn in to a time difference of several years? It is more complicated than that, but no matter how they explain it I find it doesn't make sense.

No criticism for Laser Jock in how he answered this. Consider it a rant or pet peeve.

[C is for]

My little sister read this question yesterday and started going crazy. Trying to understand relativity ruined her day. So she's with you, vorpal. She just doesn't necessarily buy in.

[Tao]

While physics is first among my intellectual pursuits, I'd have to give it to you that it can fail spectacularly when it comes to the clear transferring of ideas from instructor to recipient. All too often this leads to the next generation of instructors teaching by their limited and/or biased grasp of the topic.

I am no different in this than the worst of them.

But if you'd like I can take a stab at explaining my views on the matter. If so, I'd ask: what do you think of your understanding the space-time continuum?

(and dang, I wish I still had my fist flash animation where I tried to put into images my thoughts on this matter.....)

[vorpal blade]

In my view space is fixed, absolute, unchangeable, boundless (has no end), does not change with our relative motion, has no shortcuts or tunnels, and eternal. Objects in that universe may change, move, or whatever, but occupy a fixed portion of that space at any given instant of time. That portion does not vary with how we look at it, or how fast we are moving relative to it. Time is an irreversible measure of change that cannot be altered by man.

That's how I look at it. I am aware that physicists believe that time and space are variable, flexible, and relative.

Allow me to illustrate some of what does not make sense to me in the explanation of the twin paradox. In the thought experiment a spaceship leaves earth and blasts off to a point in space that is, in the space-time framework of earth at a given distance, say 10 light years away. Right away we see that the spaceship is accelerating and the earth is not, so we can't say that the earth is moving away relative to the spaceship in the same way the spaceship is moving away from the earth. However, lets suppose that the spaceship reaches its near light speed velocity in a couple of days. Pretty fast acceleration. I haven't calculated just how fast, but I suppose we could make it tolerable and still achieve near light speed. Theoretically.

Now, after the spaceship achieves this high velocity it stops accelerating, starts to coast, and takes stock of the situation. If only a few days have passed on the earth, less than that has passed on the spaceship, I do believe, according to the Special Theory of Relativity. It is now coasting and is relatively still close to the earth. From this point on the earth can be said to be moving away from the spaceship in exactly the same way the spaceship is moving away from the earth. So if time has slowed for the people on the spaceship, as determined from the earth's space time frame, then it is also true that time has slowed for the people on earth, as determined by the people in the space ship. This supposedly a real difference, not just a perception of a difference. There is no way to tell, absolutely, which one is moving at this point. There is no absolute standard or framework, according to the Special Theory of Relativity. So the reality must be that in each case, and in its own space time framework the other is actually, physically and really aging more slowly.

I, of course find this more than strange and contradictory. How can each age more slowly than the other? The explanation is often given that until you bring them back together there is no conflict. Each measures the other as aging more slowly.

Now, remember that you cannot tell, after the acceleration has stopped, whether earth or the spaceship is moving. I can't see any explanation of how the universe is suppose to keep track of which one accelerated in the past. No record is kept. At this point in time (just a few days later, as measured either in the spaceship or on the earth) we can't tell which one accelerated. That information is lost.

According to the theory the guys on the spaceship now make a startling discovery. The point they were headed to is suddenly much, much closer than they measured it to be just a few days earlier. Of course they have the Special Theory of Relativity to explain it, but it seems more than strange to me that because they accelerated the distance between the earth and the point they were headed to actually, physically, and in reality changed. Somehow, without moving either the earth or the distant point, their actions altered the universe and the relationship between points. I find that ridiculous, but so far not necessarily in contradiction to the theory.

With the point in space suddenly much closer they take only a couple of years to get there, as determined by the people on the spaceship. On earth it appears that they take more than ten years to get there. On the spaceship they view the earth as moving away from them at almost the speed of light, so they calculate that the people on the earth are aging more slowly than those on the spaceship. Not only calculate, but according to all measurements they can make they really are aging more slowly. A couple of years later, when they get to that distant point, they figure that earth has aged much less than they have.

Now the spaceship slows down in a couple of days and comes to rest in the same space time framework as that of earth. Suddenly earth looks like it is ten light years away. Also, suddenly earth has aged not less than two years, which every measurement and calculation told them, but more than ten years since they left. How is it possible that in a couple of days the earth could age more than eight years under their very eyes? Remember, that when they were close to the earth and just coasting to the distant point it appeared that the earth was aging more slowly as it zoomed away from them. How did the act of slowing down in a couple of days make the earth age for them? And why is it that the amount the earth aged in a couple of days depended on an event that happened a couple of years ago, from their point of view?

If you imagine the scientists start their observations after the period of initial acceleration how are they supposed to know how much the earth will age when they start to accelerate in the direction of the earth? It doesn't depend on their rate of acceleration. It doesn't depend on the distance of earth and spaceship. It depends solely on the duration of time since the earth and the spaceship were not moving relative to each other, and on the fact that it was the spaceship that did the accelerating. So the degree of aging of the earth within a couple of days, as seen by the spaceship, depends on how long ago something happened in the past, which seems counter to other principles in physics, including the principle of relativity. We could be in for a rude shock if we blasted off to a star which ages ago happened to be at rest with the earth. We would suddenly find the star much older than it was before we left earth, and how much older would depend on how long ago the star and the earth were together.

Basically the calculations all talk about the relative motions but when it comes right down to it there is a preferred space time framework, and that is the framework of the earth. That is the absolute framework used, though the theorists deny it. They feel justified in using this framework because the ship accelerated, and not the earth, but they ignore all of the coasting time when the Special Theory should apply equally to both earth and spaceship.

I find it difficult to clearly explain what I object to in a widely accepted theory that makes no logical or rational sense, when I try to stay within the framework of the theory.

[Tao]

hmmm. I think I can see where some of the confusion may be coming from. There are multiple different effects that would come into play, each simple enough individually, but taken all at once are rather inundating. These being Lorentz transformations (The point they were headed to is suddenly much, much closer than they measured it to be just a few days earlier. Of course they have the Special Theory of Relativity to explain it, but it seems more than strange to me that because they accelerated the distance between the earth and the point they were headed to actually, physically, and in reality changed. Somehow, without moving either the earth or the distant point, their actions altered the universe and the relationship between points.), simultaneity (A couple of years later, when they get to that distant point, they figure that earth has aged much less than they have.), and Doppler shifting (How is it possible that in a couple of days the earth could age more than eight years under their very eyes?).

In the course of reviewing this, I feel as though my explanations are rather lacking, either due to my hiatus from the topic, or just general fuzzy-ness on my part regardless. Feel free to point out any major faux pas.

First off, spacetime as I see it is the literal description of time as a fourth dimension. If taken as axiomatic, I believe relativity can be logically deduced entirely thereby. (I realize it isn't tautilogical, therefor not an ideal axiom, but hey, this way, you've only got one proposition to consider, as the rest fall in line.)

Since our minds cannot effectively handle 4D topics, we can simply delete one of our more mundane dimensions for the purpose of thought experiments. (This holds in common experience as well, a paper or computer screen cannot effectively handle 3D, so we borrow gradients of the vertical dimension to represent depth.) For our purposes, I prefer to treat 3D objects as 2D and offer time as the 'new' depth. Thus, we can imagine one moment of our 3D world as a single 2D image, sandwiched between its immediate past and future. Passing through these moments reinforces our perception of time, just as flipping through a flipbook or watching a film grants the perception of motion (and time) to otherwise static pages or film cells.

While trying to describe my thoughts on this to a sibling, I fell upon an idea presented in Fahrenheit 451, where mention is made of bilbords being modified to be rendered effective with the vastly increased speed of traffic: if such billboards were placed in such a manner that passers-by would see 24 per second, they would be indistinguishable from a film. (24 fps being the standard for video-playback, I believe.) The problem is, we prefer not to think of ourselves passing through time, but rather like to consider ourselves stationary. Good enough, we can consider ourselves standing still and the billboards representing our perception of the passage of time are now blowing past us at 24 fps.

Whew, long winded ramble for mostly set-up.

Mmkay. On to Lorentz transformations. As long as you just putter around watching billboards pass you by, your perception wont ever be challenged. But what happens if you hop in your theoretical hot-rod (hey, as long as we're making stuff up, might as well dream big, no?) and take off down the road? At slower speeds, nothing would be noticeable, but as you approached the speed-limit, you'd start to catch up to individual billboards as they pass you by. get going fast enough and you'll get that framerate down to 12 fps, and time will have halved for you. (Also of interest: if you consider the 'billboards' as windows of information, then your passing them by is also the process by which others receive information about you. Thus, it isn't that hard to see that someone going at such as speed to be passed by 12 frames per second as opposed to the regular 24 would 'imprint' twice as much upon those frames, giving an explanation for mass accretion without matter creation. You can also deduce length contractions, but to be honest, it's not coming to me at the moment, I'd probably wrap up for the evening and segue into simultaneity at a later date.)

Summation of long rambling thought experiment: one facet of the 'twin paradox' can be explained by the time dilation and length contraction inherent in near lightspeed travel. Ignoring acceleration altogether and not trying to do any signals sent back would looks like this: twin on earth considers the trip of 10 ly distance at 87% of the speed of light = 11.49 years down and the same coming back, so he expects to see his sister in 23 years (why are they always assumed to be the same gender? shake it up, it makes keeping track easier, imo). Realizing that his sister would be moving at relativistic speeds, he realizes time dilation and length contraction of the moving frame are going to come into play. Thus he figures his sister will experience sqrt{1 - (.87^2)/(c^2)} = .5 time (that 87% wasn't random btw) and will have aged 5.75 years down and the same for the trip back, 11.49 total. He also notices that her ship will have contracted into a much shorter vessel, making its' Queen Mary II accommodations look like those of U-boat.

She, on the other hand, now considers her situation: her ship is stationary, while the rest of the universe travels around her at .87c. Thus the Earth darts away and the 'destination planet' charges toward her at .87c Due to length contraction, the entire universe gets stuffed on the proverbial bunk leaving the distance between the Earth and the approaching planet at 5 ly; half of what the textbook reads and the trip only takes 5.75 years, for a total of 11.49 years.

(If you want to throw a red flag and say that we need to treat the Earth the same as the ship in it's heading out and turning back from her perspective, you may do so, up until the turning around part. The outgoing twin and earth-bound twins are inertial references, thus interchangeable and indistingushable mathematically, but once you factor in a turn you are no longer in one inertial reference frame. This can be dodged by adding a third inertial party headed back and passing the destination point at the same point in spacetime as the female twin but realize that simultaneity comes into play, and each will have their own view of what time is 'proper'. I won't even try to touch that tonight, as I'm already well beyond normal sleep deficit. {Also note you cannot have any two relative speeds sum greater than c, thus comparing outgoing twin and returning twin's speeds cannot get you .87c +.87c =1.74c...})

[Damasta]

I haven't read everyone's posts, so I don't know how relevant this is. But I thought I'd throw in that some scientists (read here) reject the concept of a space-time continuum. They assert that the universe is made up of "space-time blocks".

[vorpal blade]

First off, spacetime as I see it is the literal description of time as a fourth dimension. If taken as axiomatic, I believe relativity can be logically deduced entirely thereby. (I realize it isn't tautilogical, therefor not an ideal axiom, but hey, this way, you've only got one proposition to consider, as the rest fall in line.)

I think classical physics is the result, not relativistic physics. The key assumption Einstein made was that the speed of light is a constant. Not just a constant in an inertial space-time frame, but across space-time frames. Einstein was a product of his time. In some ages light has been thought of as a particle, at other times as a wave. In the late nineteenth century the long standing theory of light as a wave was being challenged by evidence that the supposed medium in which the light-wave traveled, the ether, couldn't really exist in a conventional way. Consideration was given to going back to Newton's particle theory of light. Einstein hit upon the idea that using the Lorentz transformations, and interpreting the transformation equations as representing reality, the conflict between particle theory and wave theory could be resolved. The causality in this accommodation was that the mathematical theory, the model, came to be regarded as a higher truth than our actual physical experience with reality. Experience was abandoned in favor of "what falls out of the equations."

Mmkay. On to Lorentz transformations. As long as you just putter around watching billboards pass you by, your perception wont ever be challenged. But what happens if you hop in your theoretical hot-rod (hey, as long as we're making stuff up, might as well dream big, no?) and take off down the road? At slower speeds, nothing would be noticeable, but as you approached the speed-limit, you'd start to catch up to individual billboards as they pass you by. get going fast enough and you'll get that framerate down to 12 fps, and time will have halved for you.

I'm sorry, this analogy just isn't doing anything for me. If you go faster you will see more billboards per second, not less. Going twice as fast as the standard 24 fps results in 48 fps. But I really don't see the connection here. Time is not altered.

(If you want to throw a red flag and say that we need to treat the Earth the same as the ship in it's heading out and turning back from her perspective, you may do so, up until the turning around part. The outgoing twin and earth-bound twins are inertial references, thus interchangeable and indistingushable mathematically, but once you factor in a turn you are no longer in one inertial reference frame. This can be dodged by adding a third inertial party headed back and passing the destination point at the same point in spacetime as the female twin but realize that simultaneity comes into play, and each will have their own view of what time is 'proper'. I won't even try to touch that tonight, as I'm already well beyond normal sleep deficit. {Also note you cannot have any two relative speeds sum greater than c, thus comparing outgoing twin and returning twin's speeds cannot get you .87c +.87c =1.74c...})

This is really the crux of the usual methods of resolving the twin paradox, the handwaving that comes when you abandon the relativity in the situation and make a case for an absolute time frame because the twin in the spaceship does the deceleration and acceleration back toward earth. I cannot deny that this makes the case not symmetrical and thus not truly open to considering them exactly the same. My point is that the time of turn around and return acceleration can be made insignificant in both the space-time frame of the ship and the space-time inertial frame of the earth. If we ignore those brief interruptions then we should be able to use the Special Theory of Relatively (which has come to mean only in inertial time frames, though in Einstein's first paper he used an example of an accelerating space-time frame -- a spinning disk.) So for most of the trip the Special Theory should apply, and thus the girl twin is younger than her brother who stayed on earth, and also the brother is younger than the girl who traveled in the space shipped. So the theory is in contradiction with itself.

The usual way out of this is to say that I cannot analyze it like that because of the asymmetry. This appears to me to be an ad hoc explanation that has no basis in the theory. One learns the "proper" way to do the calculations because one knows the result one wants to arrive at. Alternate calculations which would otherwise be allowed by the theory are dismissed because they give embarrassing and contradictory results. The theory is held as dogma, and every effort is made to bend the rules to support it.

Simultaneity and the Doppler shift are often thrown in, but they don't add anything to the discussion. They just obscure the facts. Reference is often made to the Minkowski diagrams, which are convenient for highlighting the mathematics while obscuring the underlying reality. The diagrams can show either that the traveling twin or the stay at home twin ages more slowly, but the diagrams showing the contradictory result are not allowed by those wishing to prove their point. But, if you think you can convince me that I don't really understand simultaneity, the Doppler shift, and Minkowski diagrams, and they really do explain things, feel free to give it a try.

Let's consider a simple starting claim by the Special Theory of Relativity. The twin accelerates in her spaceship and after a couple of days has reached a speed that approaches the speed of light. Let us suppose that that speed is such that the distance between the earth and star A, which was 10 light years in the space-time frame of the earth and star A, is now just 2 light years away in the space-time frame in which the spaceship now finds itself. The typical way to look at this from the point of view of Special Relativity is that the space ship changed space-time frames. That is okay with me up to a point. However, I would look at space-time frames as abstractions, not physical reality. They are mathematical ways to look at the situation, but not the underlying reality. So, my question is this, how is it that in a couple of days the spaceship went from being 10 light years away to being just 2 light years away. The physical reality, if the Special Theory is correct, is that the star and the spaceship moved closer to each other by 8 light years. I would like to point out that moving objects 8 light years in just a few days is much, much faster than the speed of light, which is a contradiction of the assumption of the Special Theory of Relativity.

I believe the answer usually given is that the star and the earth and the spaceship (to a large extent) do not actually move relative to their points in the space-time frame. The distance between points in that space-time frame just shrunk. I consider this sophistry, or to consider the mathematical model to be a higher truth that the physical reality. It is simple to me. One day the star and the earth (or the spaceship, which hasn't moved far in a couple of days) were ten light years apart. A couple of days later they are 2 light years apart. According to the long standing definition of velocity then we divide the distance moved by the time it took to move it, and we achieve the velocity. In this case hundreds of times the speed of light. The relativistic theorists claims this is simplistic and I can't do that. I'd like to ask exactly where in the theory does it specify that I can't do that? I'm only disallowed to do that because it gives an embarrassing result.

However, there are more serious inconsistencies to be consider, which can't as easily be dismissed as "in relativity we don't measure velocity in that manner."

[vorpal blade]

I read the article, Damasta, but I didn't see the relevance to the discussion. Interesting that they seem to have quantized the continuum. An interesting thought.

[Tao]

First off, spacetime as I see it is the literal description of time as a fourth dimension. If taken as axiomatic, I believe relativity can be logically deduced entirely thereby. (I realize it isn't tautilogical, therefor not an ideal axiom, but hey, this way, you've only got one proposition to consider, as the rest fall in line.)

I think classical physics is the result, not relativistic physics. The key assumption Einstein made was that the speed of light is a constant. Not just a constant in an inertial space-time frame, but across space-time frames.

Agreed. I start with the understanding of spacetime as 4D, as it is the way I personally had my eureka moment and was able to connect the dots from there.

Einstein hit upon the idea that using the Lorentz transformations, and interpreting the transformation equations as representing reality, the conflict between particle theory and wave theory could be resolved. The causality in this accommodation was that the mathematical theory, the model, came to be regarded as a higher truth than our actual physical experience with reality. Experience was abandoned in favor of "what falls out of the equations."

Ehh... I wouldn't say that was what Einstein was shooting for with special relativity. Then again, the whole wave/particle duality is still a little fuzzy in my mind, I've a bias against the Copenhagen interpretation of quantum physics that has been the norm for explaining it.

Mmkay. On to Lorentz transformations. As long as you just putter around watching billboards pass you by, your perception wont ever be challenged. But what happens if you hop in your theoretical hot-rod (hey, as long as we're making stuff up, might as well dream big, no?) and take off down the road? At slower speeds, nothing would be noticeable, but as you approached the speed-limit, you'd start to catch up to individual billboards as they pass you by. get going fast enough and you'll get that framerate down to 12 fps, and time will have halved for you.

I'm sorry, this analogy just isn't doing anything for me. If you go faster you will see more billboards per second, not less. Going twice as fast as the standard 24 fps results in 48 fps.

Ah, in order to make things simpler (perhaps a futile attempt), we start with you the viewer as stationary, with the images approaching from behind and proceeding ahead. A bit of a twist, but makes the later acceleration simpler to factor in.

But I really don't see the connection here. Time is not altered.

Yeah, I worried about that. I'm not that proficient at explaining the visualizations that I've developed, so I try to follow the path that led me to those visuals, which is (obviously) rather convoluted. Still, I find it useful, as it gives me a 3D example of relativistic effects in spacetime.

(If you want to throw a red flag and say that we need to treat the Earth the same as the ship in it's heading out and turning back from her perspective, you may do so, up until the turning around part. Blah blah blah...)

This is really the crux of the usual methods of resolving the twin paradox, the handwaving that comes when you abandon the relativity in the situation and make a case for an absolute time frame because the twin in the spaceship does the deceleration and acceleration back toward earth. I cannot deny that this makes the case not symmetrical and thus not truly open to considering them exactly the same.

Couldn't the same be done with Galileo's ship? (A juggler tossing a ball on a ship catches it just the same as one on land, even though a stationary viewer would have tracked a significant arc in the path.) If you have a sudden change of velocity of one party (the juggler was forced to walk the plank) Galileo's conclusions wouldn't apply. By changing the direction of the ship (regardless of how swiftly or slowly you do so), you've lost the platform from which your measurements were made.

My point is that the time of turn around and return acceleration can be made insignificant in both the space-time frame of the ship and the space-time inertial frame of the earth. If we ignore those brief interruptions then we should be able to use the Special Theory of Relatively (which has come to mean only in inertial time frames, though in Einstein's first paper he used an example of an accelerating space-time frame -- a spinning disk.) So for most of the trip the Special Theory should apply, and thus the girl twin is younger than her brother who stayed on earth, and also the brother is younger than the girl who traveled in the space shipped. So the theory is in contradiction with itself.

it's not the acceleration at all, you can set the example up with no acceleration whatsoever and still get the same results. Say instead of twins you're working with three separate timekeepers: One stationary relative to the standard flow of the universe, one at constant high velocity and one at the same velocity relative to the stationary in a direction opposing the second. Start the experiment when the second passes the first: effectively 'leaving Earth'. the destination/turnaround point being where the ships pass each other. Considering the three frames of reference gives comparable results.

The usual way out of this is to say that I cannot analyze it like that because of the asymmetry. This appears to me to be an ad hoc explanation that has no basis in the theory. One learns the "proper" way to do the calculations because one knows the result one wants to arrive at. Alternate calculations which would otherwise be allowed by the theory are dismissed because they give embarrassing and contradictory results. The theory is held as dogma, and every effort is made to bend the rules to support it.

Ehh, I'd be likely to agree with you about this in just about any other arena. Considering the million dollar prize awaiting the person who proves Einstein wrong, I'm hesitant to agree to a conspiracy of closed minds. Also taking a look at similar paradoxes that are still under debate, like the relativistic buoyancy paradox makes me question it further. I don't know. I've just learned about some of these recently.

[quote="vorpal blade"Simultaneity and the Doppler shift are often thrown in, but they don't add anything to the discussion. They just obscure the facts.[/quote] I'd agree with you for doppler shift and acceleration, but as I understand it, simultaneity is pretty vital to understand. Remember that the thought experiment is based in a 4D setting, where time is mutable. Knowing that my 'now' and your 'now' are skewed (albeit predictably) is rather significant. The ladder paradox shows that fairly well.

[quote="vorpal blade"Let's consider a simple starting claim by the Special Theory of Relativity. The twin accelerates in her spaceship and after a couple of days has reached a speed that approaches the speed of light. Let us suppose that that speed is such that the distance between the earth and star A, which was 10 light years in the space-time frame of the earth and star A, is now just 2 light years away in the space-time frame in which the spaceship now finds itself. The typical way to look at this from the point of view of Special Relativity is that the space ship changed space-time frames. That is okay with me up to a point. However, I would look at space-time frames as abstractions, not physical reality. They are mathematical ways to look at the situation, but not the underlying reality. So, my question is this, how is it that in a couple of days the spaceship went from being 10 light years away to being just 2 light years away. The physical reality, if the Special Theory is correct, is that the star and the spaceship moved closer to each other by 8 light years. I would like to point out that moving objects 8 light years in just a few days is much, much faster than the speed of light, which is a contradiction of the assumption of the Special Theory of Relativity.

I believe the answer usually given is that the star and the earth and the spaceship (to a large extent) do not actually move relative to their points in the space-time frame. The distance between points in that space-time frame just shrunk. I consider this sophistry, or to consider the mathematical model to be a higher truth that the physical reality. It is simple to me. One day the star and the earth (or the spaceship, which hasn't moved far in a couple of days) were ten light years apart. A couple of days later they are 2 light years apart. According to the long standing definition of velocity then we divide the distance moved by the time it took to move it, and we achieve the velocity. In this case hundreds of times the speed of light. The relativistic theorists claims this is simplistic and I can't do that. I'd like to ask exactly where in the theory does it specify that I can't do that? I'm only disallowed to do that because it gives an embarrassing result.

However, there are more serious inconsistencies to be consider, which can't as easily be dismissed as "in relativity we don't measure velocity in that manner."[/quote]

Whoa. You don't hold any of the Lorentz transformations to be valid? Hmmm. That's interesting.... I'll see what I can draw up....

[vorpal blade]

Einstein was raised with the wave theory of light. The problem scientists were having at the time Einstein arrived on the scene was with the speed of light. According to the wave theory the speed of light is a constant in the medium in which it propagates. The speed of sound is a constant in air. It doesn't matter how fast the train is approaching, the speed is constant in air, so the sound kind of stacks up in front of the train as it approaches you. When you hear the train you hear it as a higher pitch because of this stack up of waves--a shortening of the sound wavelength--in front of the train. As the train passes you the pitch is lower because the sound waves, still propagating with a constant speed in air--which is stationary with you on the ground near the train track--are made longer as the train which generates the sound waves is moving away at the same time it is generating the sound waves. You could call this a Doppler shift. This has been explained better than I have done it in the textbooks.

So, the scientists were looking for the medium in which light waves were propagated. In classical physics you can't have waves propagating without a medium. It is precisely the disturbance of the medium which gives meaning to the propagation. The scientists found that no matter how they measured the speed of light it was always a constant--just as though the ether (the name for the medium in which light was assumed to travel) was at rest with the earth. They tried to look at distance stars, which they determined were moving with respect to earth by reason of a red shift in the spectrum, and the earth was at rest with the ether. It didn't matter whether it was summer or winter (when the earth was not traveling in a different direction with respect to the distant star, the ether was always at rest with the earth. It was conjectured that the earth somehow dragged the ether around with it, but other experiments seemed to refute that. Michelson and Morely shortly before Einstein created his theory, performed a famous experiment which was interpreted to mean that the ether, if there was an ether, was at rest simultaneously with the moving object and with the stationary object. This seemed contradictory to the wave theory of light. Einstein "saved" the wave theory by postulating that the speed of light was a constant, as measured in the space-time frame of the source of the light, and it was exactly the same speed when measured by observers moving relative to the source, and in their own space-time frame.

In ordinary physics this is absurd. If you throw a ball on a train the person sees the ball travel away from him at the same rate as he would see the ball travel away from him if he threw the ball with the same force on the ground. On the ground, however, an observer sees the person traveling at the speed of the train, and he sees the ball travel at the speed of the train plus the speed of the thrown ball relative to the ball thrower. According to Einstein's theory, if the ball is a burst of light, which travels away from the man at the speed of light, then the ball or burst of light is seen by the observer to travel at a slower speed (relative to the man on the train) than the person on the train sees it, in order that the light has not changed speeds simply because the source is on the train. The man on the train sees the speed of light as moving away from him at 186,000 miles a second, and the man on the ground sees the light travel away from the man at 186,000 miles per second minus the speed of the train. So Einstein saved the wave theory by sacrificing some pretty important concepts of physics.

The main problem I have with your moving billboards is, while convenient to visualize time being sped up or slowed down, we know that it is just an illusion. You can speed up or slow down the rate of projection of a film, and you get the illusion of fast motion or slow motion, but you know it is just an illusion. Many scientists for some time interpreted Einstein in just that way. There was the illusion of time slowing down, but it wasn't real. I wouldn't care much about the twin paradox if it were just an illusion. The twin on the rocket ship sees the twin on earth aging more slowly, and the twin on the earth sees the twin on the rocket ship aging more slowly. However, it is just an illusion. When they get back together they find they are exactly the same age as one another and they have a good laugh over the illusion. However, that is not the way it is generally taught today. We are told it is not an illusion, but one twin is actually older than the other when they are brought together again. So the analogy of the billboards is a false one if you are trying to show real time dilation, rather than just the illusion of time dilation.

So far I don't think we are getting any closer to understanding each other. I have more to say.

[vorpal blade]

The usual way out of this is to say that I cannot analyze it like that because of the asymmetry. This appears to me to be an ad hoc explanation that has no basis in the theory. One learns the "proper" way to do the calculations because one knows the result one wants to arrive at. Alternate calculations which would otherwise be allowed by the theory are dismissed because they give embarrassing and contradictory results. The theory is held as dogma, and every effort is made to bend the rules to support it.

Ehh, I'd be likely to agree with you about this in just about any other arena. Considering the million dollar prize awaiting the person who proves Einstein wrong, I'm hesitant to agree to a conspiracy of closed minds. Also taking a look at similar paradoxes that are still under debate, like the relativistic buoyancy paradox makes me question it further. I don't know. I've just learned about some of these recently.

I have a couple of books on my book shelf at work that claim to have disproved Einstein. You've probably never heard of them. A number of important scientists have spent years arguing that Einstein was wrong. The usual response is for a number of other scientists with all the weight of authority they can muster attack the dissenting scientists. You are made to feel like you just don't understand Einstein. That you are some kind of a kook or odd ball. Your intelligence and academic credentials are called into question. You are treated in a condescending way. For most of us going against the stream is just not worth it.

When you first learn about what Einstein's theories really mean you are in no position to argue. The priests of the profession tell you to go home and study it. When you tell them it doesn't make sense, like the preacher who preaches that God is a loving person without body, parts and passions, who sits on the top of a topless throne, whose circumference is nowhere, the preacher tells you, "That is the beauty of it!" You are not expected to understand it because such things are out of our normal range of experience. Evolution hasn't prepared us to cope with relativistic speeds.

You are further told that hundreds of experiments have proved the theory without a doubt. It is extremely daunting to challenge the weight of a century of experimental proof. Most of us are intimidated by that. Almost none of us actually go and read the original experimental results, as I did several years out of school. I've studied dozens of these experiments.

You would have to examine each experiment one by one to see what was done, but in general let me point out that every proof is merely held to the standard of whether or not it is consistent with the theory. This is typical of any theory. The problem is that at first the spectacular successes of the theory merely showed that light had a particle nature, when everyone thought it was a wave. Or that light had a wave nature. Einstein's theory neatly takes into consideration that light can be considered either a wave or a particle. So almost any experiment will be consistent with the theory. What was needed, and is still needed, is an alternate theory to challenge Einstein. Then, perhaps, an experiment could be devised to show that Einstein's theory was not in agreement with the alternate theory. But as long as Einstein's theory is only compared with incorrect theories his theory looks good.

I have to say that there is really no "million dollar prize" for proving Einstein wrong. Only the end of your professional career if you try. It isn't easy to prove the theory wrong. You would have to do something like the twin paradox to prove it, and we can't travel that fast. Experiments have been done with fast moving particles, but the only foundation of particle physics, the understanding of what is happening, the interpretations of the equipment, the reactions, the movement, and so forth is built on the assumption that relativity is correct. Fudge factors are built in to bias the interpretation of the results in favor of the theory. And this is all at a level that the scientist is unaware of. Everything is indirectly observed by interpreting things which were put together interpreting things in a certain way. It would be an impossible task to single-handedly rewrite particle physics without relativity, just to see what biases have been built into it.

Some of the experimental proofs were doctored to give the results predicted by the theory. In some cases it has taken court orders using the freedom of information act to expose this fact. In this world the honors, the "million dollar prize" goes to the one who can first come up with a really convincing proof that Einstein was correct. Many scientist really know that the experiments we have seen fall short of convincing. But you would never know that by reading the books on relativity. Most scientist probably look at a challenge to Einstein as a kind of belief that the world is flat. They pity you for your belief, and are anything but eager to bestow a million dollar prize for showing them that the world is flat.

It isn't a conspiracy in the traditional sense of the world. Just the elite priests of the profession anxious to keep their standing, enamored by a theory that transcends common understanding, who won't give a dissenting voice the time of day, unless it comes from a well respected scientist, and then they do attack in a closed-minded way. I don't know if my explanation is convincing, but it has been my personal observation for the past forty years.

"Simultaneity and the Doppler shift are often thrown in, but they don't add anything to the discussion. They just obscure the facts.

I'd agree with you for doppler shift and acceleration, but as I understand it, simultaneity is pretty vital to understand. Remember that the thought experiment is based in a 4D setting, where time is mutable. Knowing that my 'now' and your 'now' are skewed (albeit predictably) is rather significant. The ladder paradox shows that fairly well.

I forget what the ladder paradox is all about. I am aware that simultaneity is another causality to Einstein's theories, but simultaneity is only an issue when we are trying to determine whether two events, separated in space, occurred at the same time. This is extraneous to the discussion if we always concern ourselves with what is happening at a given point in space. When I say that one twin move from one space-time frame to another and "suddenly" finds the situation radically different from what was previously observed, I realize that it takes time to make the measurements, learn the facts, and receive the information. But that is not important. You enter the new space-time frame and look at the clock staring you in the face, which was previously set up in a non-relativistic way, and you know for a fact what the time is and the change of time should be, or you thought it would be, before you changed to that space-time frame. I'll try to make that more clear in a later posting by using an example. What people often do is throw in the simultaneous concept when I don't really care what is happening back on earth, but what is happening on the distance star which is in an inertial frame with the earth.

Whoa. You don't hold any of the Lorentz transformations to be valid? Hmmm. That's interesting.... I'll see what I can draw up....

True, I think all of relativity is nonsense. What i would like to do is to show that it is not only nonsense, but inconsistent nonsense.

[Tao]

Ah. Forgive me. I had presumed by your introduction that you had understood Lorentz transformations and had struggled with Einstein's use of them as the basis for special relativity. I admit I'm a little baffled as to what kind of worldview would develop that could describe the experimental results any better. Granted, you can hand-wave many tests as fudged or biased, but what about things that originally had nothing to do with SR that still needed to take it into account, like GPS? Is the government in on this conspiracy as well, fudging the seconds around on GPS satellites in order to correct them with Lorentz transformations? While definitely a possibility, I'd think I'd sit with Occam on this one.

I also don't know where you're getting the idea that Einstein "saved the wave theory by sacrificing some pretty important concepts of physics." The theory of special relativity was pretty much the death knell to the many convoluted ideas of the aether-based light waves. Remember, the Michaelson-Morely experiments were prior to Einstein's time, his was the clearest explanation of the evidence that was piling up against wave-theorists. The Young double slit experiment had already seemed to kill Newton's light particle theory, and it was largely Einstein and his work that combined the two into what we currently argue over in quantum physics classes today.

While his theory did call into question many Galilean assumptions already held dear, so did Galileo's theory's wreak havoc on Aristotelian physics. Remember the whole leaning tower of Pisa event was a thought experiment, had Galileo actually done it, the precolumbian paparazzi would have noticed that heavier object hit the ground first, just as Aristotle explained, leaving Galileo in an Italian pickle. Galilean physics is counter intuitive until extenuating factors are taken into consideration, in this case, air resistance. So too, is relativity often considered counter intuitive until a basic understanding of all affected circumstances is achieved. A major reason I am a physics major is due to the ease at which this understanding came to me, it has been many years since relativity was anything other than intuitive to me. Quantum physics still holds many oddities that I've not been able to build up from an axiomatic base, therefor much of it is still confusing as all get out for me.

I'd say one major reason SR is the 800 pound gorilla in physics today is largely due to the failure of any other theory panning out. I agree with you that attempting to reason with physicists and proving Einstein wrong (or even proving him right by non-standard methods) is all too often an exercise in futility, until you can establish enough of a mathematical foundation for your argument that arguments can be proved or disproved logically.

The million dollar prize I referred to for proving Einstein wrong is the Nobel prize awaiting the individual or team that can develop and prove a unification theory between relativity and quantum physics. They currently don't mesh, (at least, I personally am not convinced at some of the more far fetched attempts to make them fit. Perhaps this is more due to my lack of understanding, perhaps not.) and something has got to give if a unifying theory is to pan out. I personally feel the breakthrough will come in a fundamental shift in our perception of a basic concept like time or gravity. (Honestly, I've always imagined time as quantized, and wondered what effects that would have at a quantum level...)

{edited at the realization that I come across extremely cocky.. whoa there Tao, more sleep, less talk...}

tl,dr: Our current understanding of relativity and/or quantum mechanics is going to have to change, and the person(s) doing so will have one heck of a time proving their point. That being said doesn't make the individual theories any less internally consistent (nor even counter intuitive) than Galileo's discoveries.

[Tao]

If you'd like, we can see where we get working towards understanding, but I'd think it'd be much easier for all involved if we first figured out what we agree on, then work from there. You seem established on Galilean physics, and while my flip comment about Newton was exactly that, I'd be lying to say that it needs to be assumed we both agree that the fundamental theory of calculus holds true, much less Newton's theories on gravitation.

[vorpal blade]

It's a little disturbing when I keep telling you I do not think it is a conspiracy, and you keep telling me that I am saying it is a conspiracy. How can we communicate at this rate?

The experiments were showing that the ether-based theories of wave propagation were not going to work, as you point out. Einstein was able to save the wave theory by dispensing with a fundamental aspect of classical wave theory. He proposed that there is no need to identify the traditional ether as some sort of medium at rest in a particular reference frame. Light propagated away from any source as though it propagated in a medium traveling with the source. However, seen by an observer in a different inertial frame, one traveling at a given velocity with respect to the light source, light seemed to be traveling as though in a medium at rest with the observers inertial frame. Without this kind of "explanation" the wave theory was going to be in real trouble. However now it was possible to keep the wave theory. The medium in which it traveled was always at rest with respect to every observer, regardless of their relative motion. Since this is consistent with a ballistic theory of light, and also consistent with a wave theory of light scientists didn't need to give up previous experimental evidence, just an interpretation of it. Eventually the need to make reference to a medium in which light propagated became less important, but was still the way it was taught in optical sciences in the 1980s. They still talked about dielectric constants of empty space. There are still a number of scientists who believe everything can be explained in terms of the wave theory of light.

I consider the nature of the universe as being of fixed space and time a pretty fundamental principle of physics.

Well, I'll give it another go tomorrow.

You consider the Special Theory of Relativity intuitive? You are a hard case.

[Tao]

It's a little disturbing when I keep telling you I do not think it is a conspiracy, and you keep telling me that I am saying it is a conspiracy. How can we communicate at this rate?

While you stated that it "isn't a conspiracy in the traditional sense of the world. Just the elite priests of the profession anxious to keep their standing, enamored by a theory that transcends common understanding, who won't give a dissenting voice the time of day, unless it comes from a well respected scientist, and then they do attack in a closed-minded way." That's pretty much how I view a conspiracy, a collective closed-minded bias against open-minded investigation. And what's more, I agree with you that this is all too often an accurate portrayal of the sciences. At least in Copernicus' day we could blame the Church, nowadays, closed-minded scientists don't have that luxury, they just teach their students and ignore anything they disagree with.

The experiments were showing that the ether-based theories of wave propagation were not going to work, as you point out. ... There are still a number of scientists who believe everything can be explained in terms of the wave theory of light.

While I usually would file wave/particle duality under quantum physics, I agree with what I understand your thrust to be in this paragraph. Would you say you hold primarily with the wave theory? I've no particular feelings one way or the other, (again, I've my own beef with Copenhagen as it has been explained to me) but how would a pure wave theory account for the quantization of light as seen in photon reception? (perhaps stick with Planck's quantization of emission/absorption? But then, even he abandoned it to wave/particle duality, if I understand correctly.) Perhaps off-topic, perhaps not.

I consider the nature of the universe as being of fixed space and time a pretty fundamental principle of physics.

I can understand the appeal there, no doubt about it. Focusing on space alone, what are your thoughts on the big bang, or at least, on Hubble's realization that all observable galaxies seem to be receding from us? Also, shifting into general relativity, what are your thoughts on gravitational lensing?

You consider the Special Theory of Relativity intuitive? You are a hard case.

Hard case or head case, I tend to evoke one reaction or the other... Nonetheless it is true, SR came quite naturally to me and GR followed suit after some roundabout work. (heh, some rather interesting notes doodled on the back of pass-along cards...) But I do take some pride in the fact that I've been able to explain my views to numerous non-physicists (among them my sister and some of my Philosophy classmates, no intellectual slouches, just not necessarily mathematically minded) and end with their grasp of SR sitting comparable with those in my physics classes. (Side note- never ever correlate Physics with Metaphysics in ear-shot of a Physics professor. Philosophy professors may not mind physics chat in their classes, but the reverse isn't always the case. Whoo did I catch a lecture for that...)

[vorpal blade]

It's a little disturbing when I keep telling you I do not think it is a conspiracy, and you keep telling me that I am saying it is a conspiracy. How can we communicate at this rate?

While you stated that it "isn't a conspiracy in the traditional sense of the world. Just the elite priests of the profession anxious to keep their standing, enamored by a theory that transcends common understanding, who won't give a dissenting voice the time of day, unless it comes from a well respected scientist, and then they do attack in a closed-minded way." That's pretty much how I view a conspiracy, a collective closed-minded bias against open-minded investigation. And what's more, I agree with you that this is all too often an accurate portrayal of the sciences. At least in Copernicus' day we could blame the Church, nowadays, closed-minded scientists don't have that luxury, they just teach their students and ignore anything they disagree with.

When I think of a conspiracy I think of people who plot to achieve a certain objective. I think of a secret agreement to perform an unlawful act. What I am talking about is a group of people who are mostly mislead by false and misleading "proofs" or "arguments." These "proofs" and "arguments" are not generally intentionally given with the intent to deceive, but by people who have also been mislead or duped, probably accidentally. Scientists have a tendency to find what they are looking for, and stop looking when they have found it. This is not a conspiracy to conceal the truth, just a common limitation of human nature. I believe there are a few bad characters who know they are doctrining their data to fit the prevailing theory, but most of them just sit around puzzling how to interpret their results, and when they find a plausible explanation that fits the theory they say, "A ha! That's it!" and stopped looking for alternate explanations. I'm reminded of a scientist who analyzed some photographs of some spectrum and came to the conclusion that there was a blue shift. Then Einstein came along and his theory became more popular. The scientist went back to his photographs and suddenly saw that he could interpret the data to fit Einstein's theory predicting a red shift. I don't think he meant to deceive anyone, there was no conspiracy, just the way humans work. These theories become religious tenets of scientists. You wouldn't say the Baptists conspire to withhold the truth, though a few of them might if it made their church look bad. More likely you'd just saying they were biased. No secret agreement to work with others to conceal the truth.

The experiments were showing that the ether-based theories of wave propagation were not going to work, as you point out. ... There are still a number of scientists who believe everything can be explained in terms of the wave theory of light.

While I usually would file wave/particle duality under quantum physics, I agree with what I understand your thrust to be in this paragraph. Would you say you hold primarily with the wave theory? I've no particular feelings one way or the other, (again, I've my own beef with Copenhagen as it has been explained to me) but how would a pure wave theory account for the quantization of light as seen in photon reception? (perhaps stick with Planck's quantization of emission/absorption? But then, even he abandoned it to wave/particle duality, if I understand correctly.) Perhaps off-topic, perhaps not.

I'm not sure what the truth is here. If you make an experiment that tests the particle nature of light you find it has a particle nature. If you perform an experiment to test the wave nature of light you find it has a wave nature. My preference is for a particle theory. It goes into what I believe to be the nature of light as a spirit particle. We have no idea what the nature is of spirit particles. Somehow the particle of light has the ability to act like a wave. I have no idea how that works, except it seems to involve an element of intelligence.

Physics was my minor at BYU. I then went on to U. of U. and studied advanced graduate level physics. In the 1980s I spent a couple of years studying the nature of light from the professors of optical sciences at the University of Arizona. A few of my professors expressed the belief in class that the experimental evidence for the quantum nature of light could be explained or derived by wave theory. They were disappointed that so many others had gone to quantum optics, and expressed their contempt for the quantization of light. However, they did not take the time in class to elaborate on that view, and I never went to ask them about it.

I consider the nature of the universe as being of fixed space and time a pretty fundamental principle of physics.

I can understand the appeal there, no doubt about it. Focusing on space alone, what are your thoughts on the big bang, or at least, on Hubble's realization that all observable galaxies seem to be receding from us? Also, shifting into general relativity, what are your thoughts on gravitational lensing?

I don't have a strong opinion on the big bang theory or on gravitational lensing. I think these things could be discussed in terms of a particle theory, and possibly related to the means of creation.

[FauxRaiden]

I was actually the one that posted the relativity question.

I was bored in between classes here at the BYU and was reading up on the theories relating to light travel (does that make me a geek?). Anyway it was messing with my head big time trying to understand the relation between time and decay which is why I asked the question.

Frankly, I still don't understand it but I don't think any amount of explanation will make me understand it. However, I can see some sort of relation between gravity and decay but...I don't know. It was just kind of a for fun reading deal.

[vorpal blade]

FauxRaiden, this is something I sort of obsess over, from time to time, and have since high school (which was a long time ago.) Nice to know where the question came from.

[vorpal blade]

So, I have a simple example of what seems like inconsistent nonsense to me. It is so simple, and so many smart people have studied the Special Theory of Relativity, that there must be a simple solution that everyone has already seen through it and gone on to other things. If someone would be so kind to point out the flaw in my reasoning you could put me out of my misery, so to speak. At least temporarily. In this problem I will try to use correctly the calculations of relativity.

Suppose that earth has established a series of space stations which are exactly one light year apart. They are positioned in a straight line. These space stations are a rest in the same inertial space-time frame as the earth. The last of the ten space stations is located near star X, located of course 10 light years away from earth. Star X is not moving with respect to earth. There is no question of the distance between the space stations. Light signals are sent from one station to the next, and it is established that it takes exactly two years round trip for a signal to go to the nearest space station and return. Clocks are set up and synchronized so that when it is December 31 in the year 3000 on earth the clocks read the same date and time on each space station. All of this may not be necessary, but hopefully sets the stage for my little problem.

Let us suppose that on January 1, 3001 a space ship leaves earth carrying a baby girl, the twin of a baby boy left behind on earth. The space ship also carries a clock and it is set initially to be the same as the clock on earth and all the space stations. The space ship rapidly accelerates and in a couple of days, as measured on earth, the space ship reaches the velocity of 4/5 the speed of light (chosen for calculation simplicity). When it reaches that speed the engines are shut off, there is no more acceleration, and the space ship coasts until just before the end of the journey.

According to the usual interpretation of General Relativity the clock on the space ship is no longer synchronized with the clock on the earth. But when the space ship has been underway for just a couple of days, as measured on earth, the clock on the space ship won’t disagree with the clock on the earth by more than a few days. Near the start of the journey, when the space ship just begins it’s coasting, the clock on earth and the clock in the space ship read approximately the same value. To the people on the space ship the earth is moving away at a constant velocity, and star X is approaching with the same velocity.

Let me consider what is happening from the point of view of the people on the earth and in the space stations. Because the space ship is traveling at 0.8c there is a time dilation on the ship as seen from the inertial reference frame of earth and the space stations. The space ship travels the distance between earth and the first space station, one light year, in time t1 = 1ly/0.8c = 1.25 years. So the space ship passes the first space station approximately a quarter of a year into the year 3002. According to the Special Theory of Relativity time is slowed down on the space ship, as seen from a space station, so as the space ship zooms past space station 1 (without changing velocity) the people in the space station read the clock on the space ship as t1 multiplied by the square root of 1 minus the velocity of the space ship squared divided by the speed of light squared. If I’m doing my math correctly they read that the clock in the space ship says that 1.25*0.6 = 0.75 years has passed since they left earth. It appears that on the space ship they are only three quarters into the year 3001.

And so it goes as the space ship passes each space station. The clock at each space station reads 1.25 years later than the clock read at the previous space station, while the clock on the space ship is marking time at 0.6 the rate of the space station clocks. Just before the space ship starts to decelerate the clock at the last space station is almost at 12.5 years after the space ship left earth, or half way through the year 3013. Meanwhile the clock on the space ship is observed from the last space station to have aged only 0.6x12.5 = 7.5 years and reads that it is mid-year 3008. The space ship decelerates in a couple of days and lands on the 10th space station, near the star X. Those on the space station see that their clock reads mid 3013 and read that the clock on the space ship is mid 3008. The baby twin girl is now 7.5 years old, while it can be verified that the baby twin boy left back on earth is now 12.5 years old. So says the clock on space station 10 which is synched to earth.

Okay, now let’s consider the events as seen on the space ship. The space ship blasts off from earth and in approximately two days reaches the cruise velocity of 0.8c. When they shut off the engines they feel no acceleration, and it appears that earth is moving away from them at 0.8c while star X approaches at speed 0.8c. Now, however, it appears that star X is not 10 light years away, but the distance has been constricted according to the Special Theory of Relativity by the factor square root of 1 minus the velocity squared divided by the speed of light squared. In other words the distance to star X is 6 light years. Similarly the distance to the first space station is 0.6 light years and the distance to the second space station from earth is 1.2 light years, as seen from the space ship.

With a distance of just 0.6 light years to the first space station, and traveling at the speed of 0.8c, the time it takes to reach the first space station, as measured on the space ship’s clock, is just 0.6lr/0.8c = 0.75 years. When it passes the first space station the clock on the space ship reads that it is three quarters of the year into the year 3001. This is exactly what the people on space station 1 saw when they looked at the space ship’s clock as it zoomed past. When the space ship has traveled the distance of 6 light years to the final space station the clock on the space ship reads that 6ly/0.8c = 7.5 years have passed. This is also what the people on the last space station see on the clock of the space ship when it lands.

So far this is strange, but not necessarily nonsense or self-contradictory. Here comes that part of the story. The question is, what do the people on the space ship see when they look at the clocks at the space stations they pass? Well, from the point of view of the space ship time is passing more slowly on earth and on the space stations that it is on the space ship, according to the Special Theory of Relativity. That is because to them it is the earth which is moving, not the space ship. How much is the speed of the clocks reduced on earth and on the space stations as seen by the space ship? The earth and space stations are traveling at speed 0.8c past them, so the time multiplication factor is 0.6. Just as the clock on the space ship was seen from the space stations as going the factor 0.6 slower, by the principle of relativity the clocks on the space station are moving slower as seen by the space ship by the factor 0.6 relative to the ship’s clock. Hence when the space ship passes space station 1 the ship’s clock reads that 0.75 years have passed on the ship’s clock, but only 0.75x0.6 = 0.45 years have passed on the clock of space station 1, as seen from the space ship.

This seems like nonsense to me. How can it be that those on space station 1 read their clock to show it is a quarter of the way into the year 3002, while at the same time and in the same place those on the space ship read the same clock to say that it is less than half way through the year 3001? I don’t see how there can be an issue of simultaneity in this, because we are talking about an event that is clearly happening simultaneously. Had the space ship collided with the space station you could not have said the space station was destroyed months after the space ship was destroyed, could you? The space ship rockets into (or past) space station 1 without stopping and clocks are read at one point in space.

To continue, by the time the space ship approaches the last space station the clock on the space ship is nearing mid-year 3008, while the space ship sees the clock on the space station as nearly mid-year 3005 (0.6*7.5 years after the start in January 3001). After a couple of days of deceleration the people on the space ship get out and compare clocks. They have been observing the passage of time on the space stations, and just a couple of days before landing they could see and read the clock on space station 10 to be nearly mid-year 3005. If Einstein’s theory is correct they should be seeing the clock on space station 10 to say mid-year 3005. However, if Einstein’s theory is correct the people on space station 10 look at their own clock and read that it is mid-year 3013. What are the people of the space ship going to see when they land? A clock that reads mid-year 3005 which is what their observations have been and what Einstein’s theory predicts? Or a clock that reads mid-year 3013, which is what the people on the space station have been seeing, and which is what Einstein’s theory predicts? How can Einstein’s theory predict two conflicting clock readings? Did the clock on the space station age 8 years while the people in the space ship took two days to decelerate and land on the space station? This too seems a contradiction of the theory.

So there you have it. Can you resolve the paradox for me?

[Tao]

Can you resolve the paradox for me?

Heh, I can give it a go, but I must admit I'm not sure how best to go about it. First off, you do know that the acceleration that keeps creeping into your examples throws this into General Relativity and even deeper math, when it can easily be done away with? Just assume each vehicle (Earth included) is already at velocity at the beginning of the thought experiment.

-.8v | | ←|
.8v |→ | |
0v |O | |
└--------10 ly-------┘└--------10 ly-------┘

Ouch. That really doesn't work once posted... In fact, since I'm not up to getting this post in Courier (or even anything that will accept multiple blank spots without autodeleting them..., you can see a fairly good visual representation of this idea in this video.

Anyway, I'll point out where I think the hangup is, and see where everything pans out tomorrow, if you'd like, I'll crunch the GR numbers on your example step by step, explaining where we might be getting our wires crossed. But, unless I'm mistaken (a very real possibility at this hour) I think I can see one tripwire already.

So, I have a simple example of what seems like inconsistent nonsense to me. It is so simple, and so many smart people have studied the Special Theory of Relativity, that there must be a simple solution that everyone has already seen through it and gone on to other things. If someone would be so kind to point out the flaw in my reasoning you could put me out of my misery, so to speak. At least temporarily. In this problem I will try to use correctly the calculations of relativity.
...
...
...
So far this is strange, but not necessarily nonsense or self-contradictory.

Yup. With you so far, most everything seems in line.

How can it be that those on space station 1 read their clock to show it is a quarter of the way into the year 3002, while at the same time and in the same place those on the space ship read the same clock to say that it is less than half way through the year 3001? I don’t see how there can be an issue of simultaneity in this, because we are talking about an event that is clearly happening simultaneously. Had the space ship collided with the space station you could not have said the space station was destroyed months after the space ship was destroyed, could you? The space ship rockets into (or past) space station 1 without stopping and clocks are read at one point in space.

Mmkay. This is where the SR example can be more difficult to explain, but the GR one requires more of a leap of faith in my opinion. Simultaneity really shows up in the non-accelerating example. In the accelerating example (one obviously more realistic, the same effects crop up in the act of shifting out of inertial reference frames. I get the feeling that while strange, the length contraction seems to be an acceptable oddity to you, am I correct? Moving at this speed warps the very woof of the universe. (sorry bad puns...) It is when you get down to time that things start to boggle the mind. The key factor to consider is that spatial dimentions and the temporal one are for most all purposes identical. Just as the length contraction isn't restricted to the space from Earth to Star X but the entire universe is effectively reduced by 3/5ths along that axis. So too must we treat time. the problem is, if you want to compare two timelines that are progressing at different rates, they are only going to match up at one point. (The image of gears set at a certain ratio comes to mind. Not sure if the image is complete enough in my head to get across properly; how they can always mesh at the point of contact, but their arcs are such that x steps back for one will be significantly different for the other. I'll think on it and see if I can understand it enough myself to explain to anyone, or scrap it later) If you've chosen to anchor your perception at the passing of the first satellite, looking back in time would show that you no longer agree that the satellite clock's Dec 31 3000 and Earth's Dec 31 3000 mesh. In most examples given, that point is held as the common throughout, effectively leaving all the dilation to occur further on down the line.

Thus, if you want to leave acceleration out of it, and demand that Dec 31 3000 is the point at which you are setting all clocks to match, yes, an observer in a different inertial reference frame at relativistic speeds will continue to see his clock and the clocks around him grow more disparate, as the time dilation from your anchor point continues to compound. While I've treated each example as if the twin was moving, as long as everything remains inertial, the mismatch is going to continue apace regardless of point of view, (just as a 3/5ths gear ratio will give the same measurable mis-match as a 5/3rds one...)

In your more believable GR setting, with a negative acceleration dropping our travelers back to the reference frame of the Earth and star X as I understand it you are essentially correct, one of the effects of switching from a frame moving at 3/5ths time to full time would involve the catching up of a fair amount of slack. The speed at which you accelerate at either end makes no difference to the end result (other than OSHA standards for your passengers, of course.) You could make it a 1g acceleration with a flip point in the middle, allowing bone density to be preserved and bad sci-fi novels to be set, but in the end, accelerating against the flow of your reduced time equates to a temporal pile-up as you realign yourselves with another reference frame.

The goofy triangular graphs are an attempt to show this in one coherent image, notice that as the world line shifts in respect to the dilation of time, a 'pile-up' naturally occurs. (On some this is more readily seen, on others, it looks horribly forced, completely missing the point of the image, in my eyes. Then again, so do maps when we try to fit a global image onto a rectangular piece of paper...)

Anyway, if I've not horribly botched this up, I'll take a better crack at it later. I'd better catch some shuteye.

***Ooooh! maybe the image of the two different time lines as rubber bands.... with markings.... go to bed! grrr...