I have to apologize in advance MacM, but these back and forth postings and responses are getting bigger and bigger each time, and there have been several other interesting threads opening up lately, so I'm only going to be able to respond to a selection of your questions and answers. If there's a particularly important point you feel I haven't adequately addressed, feel free to let me know.
You're saying that there's a reciprocity in the twins paradox that only GR is able to deal with unambiguously (without invoking your concept of "absolute motion"), and that's just not true in the slightest.
I'm sorry, but I can't make any sense of your equation. It looks like you're adding something finite to something infinitesimal. You should write it out in TEX format so you can display it as an actual equation, it's not that difficult. Here's a nice, concise cheat sheet that should have pretty much all the commands you need. Then when you want to write math statements, you just enclose it between tex statements in square brackets like this:
[this is where you'd write "tex"] (Math code as demonstrated on the cheat sheet) [this is where you'd write "/tex"]
I looked into it, here's a link discussing what Einstein did. I'll quote the relevant section.
Smart man, that one.
I use the same velocity "v" in both the trip to the navigational beacon and back for simplicity's sake. If I had chosen that the astronaut heads to the beacon with velocity "v", and returned with velocity "w", you would see "v" appear in the time dilation equation for the trip to the beacon, and "w" would appear in this equation for the return trip home. At no point in the dilation equations is there any reference whatsoever to what velocity the astronaut held in the past. The amount an object's time gets dilated at any instant, as measured by an observer, depends exclusively on their position and velocity with respect to this observer at that same instant.
Also I could go ahead and shift the problem so that the Earth and both twins start off moving at near light-speed as seen from another reference frame, frame "C". Then one of the twins takes off from Earth, navigates towards a beacon which has been placed at rest relative to Earth some distance away, and returns. Observer "C" will agree, and can prove through the monitoring of radio transmissions from known locations in space, that when the twins compare their clocks, they will have the exact same time discrepancy as they would have in the case where the Earth is considered to be at rest. There's no reference to who has to be considered moving at any given time. It could be that the Earth is flying backwards, and the astronaut decides to decelerate into a "stationary" frame for a bit, before accelerating to catch back up with their twin on Earth.
Suppose you let them drift apart for a bit. Furthermore, each astronaut carries a couple of powerful lasers that emit periodic pulses at a slight angle to each other, so that the other astronaut can triangulate on the distance from which each pulse was sent. That way, each astronaut can tell, from their own reference frame, how far away the other astronaut was when they sent their light pulse. Knowing how far away each pulse was, how many pulses there were and that they all travelled at the speed of light, each astronaut can measure how much time passed on the other astronaut's clock with every pulse, and compare it to what their own clocks read when each pulse was sent.
The answer to your question is neither "Yes" nor "No". The relativity of simultaneity comes into play here. Suppose "A" has set up a beacon at rest relative to them, and "B" is scheduled to pass this beacon at some point. Likewise, "B" has also set up a beacon in a symmetric situation. From A's perspective, B's clock will be ticking slow but the clock on B's beacon will be ticking fast, they won't be synchronized. Similarly, from B's perspective, A's clock will be ticking slow but A's beacon will be ticking fast. This seems to be your biggest problem with handling SR, you're only thinking about time dilation and not the fact that simultaneity is also a relative concept.
Like I say, the particle lifetimes measured in accelerators obey the laws of relativity, they do exactly what Einstein predicted they would do. Even before relativity was published there were strange effects being noticed, such as the nonlinearity of velocity addition. Here's a relativistic velocity addition effect Fizeau first noticed all the way back in 1851!
There's nothing wrong with the concept that the sun is flying towards me. If I'm on Earth one moment and then a few months later I see the sun rapidly coming up on me, I know it didn't just suddenly pick up a zillion terajoules of kinetic energy. In that case I know I must be in a rocket zooming towards the sun, at which point I look around and remember I'm an astronaut and recall that a few months ago it had felt like something was smashing me into my seat for a whole week.
On the other hand, if I'm a meteor floating through space for a couple of billion years, and one day I notice an entire solar system rushing up on me at hundreds of thousands of kilometres an hour, there's nothing wrong with that scenario and there's no reason the meteor should think it somehow got propelled up to massive speeds.
If I understand what you're saying correctly, they used muon anisotropy to measure the speed of the Earth relative to the CMB. In order to do this, I believe they would have had to know the average lifetime of a muon from watching their decays here on Earth. So your logic seems circular because you're then claiming the experiment was then able to measure muon lifetimes, whereas this seems as if it would have been a necessary input at the very start.
I don't plan to get too heavily involved in this debate, and it's starting to get pretty cluttered so I don't know how much longer I'll stick around here. But I think I've narrowed down MacM's problem to misconceptions about the relative simultaneity of time and how systems of clocks are set up in different reference frames. I see that you mentioned this issue as one of your grievances, and it's something I'll probably try to focus on rather than getting sidetracked by all the choo choo trains, airplanes and rockets.
Please clarify.
You're saying that there's a reciprocity in the twins paradox that only GR is able to deal with unambiguously (without invoking your concept of "absolute motion"), and that's just not true in the slightest.
It does simplify calculation but the final time dilation is inertial velocity differential * duration + integration of velocity differentail * duration of the acceleration period.
I'm sorry, but I can't make any sense of your equation. It looks like you're adding something finite to something infinitesimal. You should write it out in TEX format so you can display it as an actual equation, it's not that difficult. Here's a nice, concise cheat sheet that should have pretty much all the commands you need. Then when you want to write math statements, you just enclose it between tex statements in square brackets like this:
[this is where you'd write "tex"] (Math code as demonstrated on the cheat sheet) [this is where you'd write "/tex"]
I would need to see a bonafide example of his having done that. To my knowledge he was stimmied by the paradox until he developed GR. Perhaps he may have made some comment just before publishing GR but as my link states he used GR to resolve the issue.
I looked into it, here's a link discussing what Einstein did. I'll quote the relevant section.
In his famous work on special relativity in 1905, Albert Einstein predicted that when two clocks were brought together and synchronized, and then one was moved away and brought back, the clock which had undergone the traveling would be found to be lagging behind the clock which had stayed put. Einstein considered this to be a natural consequence of special relativity, not a paradox as some suggested, and in 1911, he restated and elaborated on this result in the following form:
If we placed a living organism in a box ... one could arrange that the organism, after any arbitrary lengthy flight, could be returned to its original spot in a scarcely altered condition, while corresponding organisms which had remained in their original positions had already long since given way to new generations. For the moving organism the lengthy time of the journey was a mere instant, provided the motion took place with approximately the speed of light. (in Resnick and Halliday, 1992)
Smart man, that one.
Sorry but you missed the boat on this one. t' = t(1-v^2/c^2)^0.5
The shift in velocity is "v" the velocity you use to compute the affect. It is a shift because it is different than the inertial rest velocity which becomes "0".
I use the same velocity "v" in both the trip to the navigational beacon and back for simplicity's sake. If I had chosen that the astronaut heads to the beacon with velocity "v", and returned with velocity "w", you would see "v" appear in the time dilation equation for the trip to the beacon, and "w" would appear in this equation for the return trip home. At no point in the dilation equations is there any reference whatsoever to what velocity the astronaut held in the past. The amount an object's time gets dilated at any instant, as measured by an observer, depends exclusively on their position and velocity with respect to this observer at that same instant.
Also I could go ahead and shift the problem so that the Earth and both twins start off moving at near light-speed as seen from another reference frame, frame "C". Then one of the twins takes off from Earth, navigates towards a beacon which has been placed at rest relative to Earth some distance away, and returns. Observer "C" will agree, and can prove through the monitoring of radio transmissions from known locations in space, that when the twins compare their clocks, they will have the exact same time discrepancy as they would have in the case where the Earth is considered to be at rest. There's no reference to who has to be considered moving at any given time. It could be that the Earth is flying backwards, and the astronaut decides to decelerate into a "stationary" frame for a bit, before accelerating to catch back up with their twin on Earth.
1 - Given "A" and "B" are at common inertial rest and "A" accelerates ways I think you would agree that "A" is accumulating less time than "B".
Yes/No?
Suppose you let them drift apart for a bit. Furthermore, each astronaut carries a couple of powerful lasers that emit periodic pulses at a slight angle to each other, so that the other astronaut can triangulate on the distance from which each pulse was sent. That way, each astronaut can tell, from their own reference frame, how far away the other astronaut was when they sent their light pulse. Knowing how far away each pulse was, how many pulses there were and that they all travelled at the speed of light, each astronaut can measure how much time passed on the other astronaut's clock with every pulse, and compare it to what their own clocks read when each pulse was sent.
The answer to your question is neither "Yes" nor "No". The relativity of simultaneity comes into play here. Suppose "A" has set up a beacon at rest relative to them, and "B" is scheduled to pass this beacon at some point. Likewise, "B" has also set up a beacon in a symmetric situation. From A's perspective, B's clock will be ticking slow but the clock on B's beacon will be ticking fast, they won't be synchronized. Similarly, from B's perspective, A's clock will be ticking slow but A's beacon will be ticking fast. This seems to be your biggest problem with handling SR, you're only thinking about time dilation and not the fact that simultaneity is also a relative concept.
The same is true of the affects of relative velocity. It may cause you to see, measure or bvelieve the remote resting clock is dilated but once you stop the motion the emperical data doesn't support that conclusion.
Like I say, the particle lifetimes measured in accelerators obey the laws of relativity, they do exactly what Einstein predicted they would do. Even before relativity was published there were strange effects being noticed, such as the nonlinearity of velocity addition. Here's a relativistic velocity addition effect Fizeau first noticed all the way back in 1851!
My point exactly. If you do not define the probe as having absolute motion you are in fact creating the sitution where the sun is moving to you.
There's nothing wrong with the concept that the sun is flying towards me. If I'm on Earth one moment and then a few months later I see the sun rapidly coming up on me, I know it didn't just suddenly pick up a zillion terajoules of kinetic energy. In that case I know I must be in a rocket zooming towards the sun, at which point I look around and remember I'm an astronaut and recall that a few months ago it had felt like something was smashing me into my seat for a whole week.
On the other hand, if I'm a meteor floating through space for a couple of billion years, and one day I notice an entire solar system rushing up on me at hundreds of thousands of kilometres an hour, there's nothing wrong with that scenario and there's no reason the meteor should think it somehow got propelled up to massive speeds.
I must add again the fact that a recent study found that there is a muon ansitrophy to earth and it was used to compute the solar system motion in the universe replicating the 350km/s motion found by CMB and other measurements.
That is the conclusion of the study was that it was more accurate to compute muon life time via it's absolute motion than relative motion just to earth!!!!!.
If I understand what you're saying correctly, they used muon anisotropy to measure the speed of the Earth relative to the CMB. In order to do this, I believe they would have had to know the average lifetime of a muon from watching their decays here on Earth. So your logic seems circular because you're then claiming the experiment was then able to measure muon lifetimes, whereas this seems as if it would have been a necessary input at the very start.
CptBork:
I'm not sure whether you have read the whole thread. I have covered all the issues you're attempting to discuss with MacM already. He is repetitive and never learns anything, so you're wasting your time. To waste a little less of it, you might like to review some of my interactions with MacM in this thread.
...
...
And that's just for starters.
Anyway, if you want to engage with him, good luck.
I don't plan to get too heavily involved in this debate, and it's starting to get pretty cluttered so I don't know how much longer I'll stick around here. But I think I've narrowed down MacM's problem to misconceptions about the relative simultaneity of time and how systems of clocks are set up in different reference frames. I see that you mentioned this issue as one of your grievances, and it's something I'll probably try to focus on rather than getting sidetracked by all the choo choo trains, airplanes and rockets.