garys_2k said:
That's possible, but what would he see, please? I'm trying to settle this in my mind.
Then, what would he measure, and how? Would he "measure" by taking what he sees and applying some correction, or would measurement tools react differently than what he sees?
I'm not trying to pick nits, I really don't understand. Baby steps, please. Someone posted earlier that as soon as he stopped that he'd see a whole bunch of time fly by on earth, all in a huge rush. I doubt that's right. Is it?. I really think Able would see time proceeding normally on earth, at the same pace as time is proceeding for him, but with a 26 day offset.
OK. Some of the concepts fit together, like pieces of a puzzle, so it's hard to know what to start with first. Once you have the whole puzzle, it's easier to see how the pieces fit together.
Some people like to say that, when we are looking at Barnard's Star, which is about 6 LY away, we are seeing it as it was 6 years
ago.
That word, "ago," needs some investigation. All we
really know is that we're seeing it here, now. On observer near Barnard's Star would also see us 6 years
ago. So this leads to a seeming paradox, and we have to understand "ago" with respect to something that we might call "now."
In Galilean/Newtonian mechanics, this is easy. We just synchronize our clocks with an instantaneous signal, and that defines "now" for the both of us.
But we can't do that with Lorentzian/Einsteinian mechanics. So instead, we imagine a line of clocks reaching from here to Barnard's Star. Let's say that they're all synchronized by a distant supernova, at right angles to the line, far enough away that the difference in the distance along the line is insignificant.
This line of clocks is approximately what Dr Fendetestas called a "simultaneity line" for what we call "now." We extend this idea to fill all of space, or at least as much as we need, at known distances and known times, and we call this our reference frame. Fortunately, under SR, and with the examples in this thread, we only have to concern ourselves with one spacial dimension in the direction of travel, so we can just think of the line.
Already, there's a problem. If our local clock reads 12:00, we will see a clock one light-hour away as reading 11:00. But we make our measurements with respect to our entire frame, so we think of it as being 12:00 everywhere on our frame. So what we see isn't what we measure.
(The concept of a "simultaneity line" is a bit more complex; there will be a 12:00 simultaneity line and a 12:37 simultaneity line, and so on. So you can think of a bunch of simultaneity lines. But there will always be a "now" simultaneity line, and you can extrapolate the other clocks from your clock where you are. I'll omit the "now" in subsequent discussion and just assume that Abel always makes measurements for his now.)
We imagine our frame as moving with us, even accelerating. Maybe each clock has a rocket on it, and the rockets go on and off as a result of other distant supernova explosions.
When the rockets are off, with two reference frames, we have SR. We make our time dilation measurements or observations with respect to the
closest clocks in the other reference frame. Under SR, we will see all of our clocks moving at the same rate. Under acceleration, as explained earlier, we will see clocks "above" us going faster. (Define "above" as in the direction of acceleration.)
Someone posted earlier that as soon as he stopped that he'd see a whole bunch of time fly by on earth, all in a huge rush. I doubt that's right. Is it?
Now it should be understandable. To keep it simple, let's just consider the moment after Abel has mostly decelerated, just before he turns off the rockets. So his frame and the Earth frame are close enough to stationary relative to each other that we don't have to worry about SR time dilation. Abel does the measurements with respect to the clock in his reference frame closest to Earth. He will measure this clock as going very, very fast due to the acceleration. But since it's close to the Earth clock and not moving much relative to it, he will also have to measure the Earth clock as going very, very fast. So, according to his measurements, the Earth clock must be going very, very fast. So he will measure a bunch of time going by quickly. Of course, what he sees involves light getting to him, so the change will be much more gradual than what he measures, and it will take the whole trip back to finish.
If you recall, I said that you could work out the twin paradox either with SR or GR. If you read this post carefully, you will have noticed that there's a seeming "cheat," which is actually a little bit of SR. While Abel was just moving, his simultaneity line always seemed normal. He could look at the serial numbers of all his clocks, look up the distance in a book, compare the distance from the reading on the display and the speed of light, and everything would be OK. While accelerating, the rates of the clocks would go all wahoonie-shaped. So, after the acceleration, you might think that his book wouldn't work any more. This is quite correct.
From SR, you already know that while your simultaneity line looks horizontal (constant
t), the simultaneity lines of other moving frames would appear to be at an angle to yours, so you would disagree on what events are simultaneous, except locally.
If Abel got a new simultaneity line by resynchronizing all of his clocks to another supernova explosion, it would appear to be at an angle to his old simultaneity line (the new one would be horizontal). It would be at just the same angle as another inertial frame already going in the other direction.
This is why you can explain the Twin Paradox under SR. If you just switch reference frames, you throw your old book away and use the new captain's book.
Or you can do it in GR. I won't try to give the math, just the zen. When you're not accelerating, you can imagine all your simultaneity lines stacked up, forming a grid. This is your spacetime reference frame. It looks like a plain Cartesian grid, and it's called "flat spacetime" for this reason. Abel is looking at his clock, which he's holding right up to his eye, so he always sees himself as going straight up through time.
As Abel accelerates, his simultaneity lines according to his book seem to curve all around him. But he still thinks he's going straight up in time, at the same rate. When he turns the rockets off, spacetime will flatten out again all around him. But he will, in the meantime, have gone through time to another part of the curved grid where the simultaneity lines are at a different angle to the time line, so he will see his old simultaneity lines at a different angle now.
Einstein had a cute name for a curved frame. He called it a "mollusc," evoking the shape of a section of a snail's shell.
