Sisyphus

Both animations are totally correct. In both frames, light moves at C, as it must, always. Since it is moving at C, it has a velocity of C relative to anything that is at rest: Earth in the first animation, the ship in the second. And clearly, it won't have a velocity of C relative to anything that is not at rest. No, that's for translating between frames. In one frame, velocities add normally. It's just that no velocity can be greater than C. Not so. The speed limit is C. That makes the maximum closing or separation speed 2C, for objects moving in opposite directions.

Don't forget that the "edge" pieces would also be curved, having a quarter circle cross section. Since there are 4 of each kind, you can just do it as four cylinders. The answer as I see it would be: 2wh + 2wd + 2hd + wπ + hπ + dπ + 4π/3

In a computer? Seems like binary would just make it harder. However, in base googolplex, the long form is simply "10."

What's the definition of magic? As Arthur C. Clarke famously said, any sufficiently advanced technology is indistinguishable from magic. And it seems like it would also be true that any sufficiently analyzed "magic" would just be a natural phenomenon.

In every frame of reference, light is moving at C. Therefore, in a given frame of reference, it is moving at C relative to something which is at rest in that frame. In that same frame, it is not moving at C relative to something which is not at rest.

Of course it does. Where do you think it's implied that it doesn't?

Because most of the traditions of Christmas don't originate with Christianity. It's a mixture of all sorts of different religions and traditions, ancient and modern. But no, of course that doesn't compromise Christian beliefs, which I'm assuming is the point ydoaps was making. The atheist who celebrates Christmas is celebrating traditions that originally come from religions he doesn't believe in, and so is the Christian. And it doesn't matter. Why would it?

I don't know anyone who's offended by "merry Christmas." I know lots of people who get in a giant tizzy over "happy holidays," though, since apparently acknowledging the existence of non-Christian holidays (or the fact that there are several important holidays in a short period of time) is anti-Christian somehow. And it's that attitude, if anything, that can be offensive to non-Christians. To answer the OP, atheists should celebrate whatever they feel like. They don't have rules like that. I would also advise them to be as tolerant as possible of others forcing religious stuff onto them, since it's better for their blood pressure that way.

I'm pretty sure space expands everywhere, actually. It's just that gravity holds structures together anyway. But in any case, if it were true that there was expansion only between structures, the photon still wouldn't contract unless the space it was traveling through was contracting. If it was just not expanding, then nothing would happen.

Which reference frame are the yellow lines the speed of light for? It looks like a third frame.

The speed of sound isn't dependent on the speed of the source, either. Only the medium it's traveling through. If you're only looking at one frame of reference (and you are), then the situation is the same. If you listen to a plane traveling at mach 0.5 away from you, then returning at the same speed, you'll hear it receding for 3 times as long as you hear it approaching.

But that hasn't made any difference. There are no Lorentz transformations in the OP, and a diagram of sending letters would look the same. "On day 75, you receive your friend's letter that he sent on day 50, telling you he was returning. All the letters he sends on the return journey you receive in the next 25 days."

Clearly it's politically incorrect in small town Texas to have a gay-straight alliance club. And as a result, the school administrators took away everyone's rights to have extracurricular clubs. I don't think that's what Jackson meant, though.

It's not even necessary to invoke relativity here. Let the diagrams in the OP represent not spaceships sending light signals, but airplanes sending sound signals through the air. It looks the same, because it's not a relativity problem. It's a completely mundane observation about signals from a moving source, namely that by the time they arrive, the source is no longer in the same location as when they were sent. (Duh.)

When I say I'm not comfortable, I don't mean that it's wrong. I mean that I'm worried it will lead to misunderstandings. For example, the title of this thread references the twin paradox and Minkowski diagrams, neither of which have much bearing on the original post, which could just as easily be about listening to an airplane fly overhead. What do you hope I'm thinking?

Sure, I suppose. When Earth receives the signal from when the spaceship was farthest away at day 50, it will be day 75. I'm not really comfortable with focusing on the "observed position," though. The observer doesn't really think the spaceship is there - he knows his information is 25 days out of date. I'm also not really comfortable with using spaceships and light signals, etc. Not because it's inaccurate, but because it might give the impression that that is what the twin paradox is all about. So far in this topic, relativity has been irrelevant. You could do the same thing with the sound of an airplane, or postcards from a world traveling friend. You receive the postcard days after it was sent. There is no confusion that your friend is still in the post office mailing it as you are reading it. You know the signal has taken time, and your friend has had more adventures in the meantime, that you will have to wait for the next postcard to hear about.

No, the Earth observer will calculate the same velocity for both legs of the journey. That he receives the signals from the return journey in a shorter amount of time is just a consequence of the doppler effect. It's not really moving faster. Again, this is true of any situation where a moving body is sending signals faster than they are moving. You can do the same thing with the sound of a speeding car driving away and back again - you'll be hearing the driving away part of the journey for longer than the returning part, even though it is moving the same speed in both directions. (And in fact, if the car was moving faster than sound, you wouldn't hear the return journey until after it had already arrived!)

If you don't know the speed of the spaceship, then all you can know is that it will be some amount of time less than 50 days. None of this so far has anything to do with time/space dilation or the twin paradox. You could set up the same thing with a traveler sending letters through the mail.

Yes, it does. To echo Janus, it does not matter when the event is observed - it matters when the observed event occurs. At day 75, the Earth observer observes that the spaceship has changed direction at day 50. He knows that it takes 25 days for the signal to reach him, from that distance of 25 light-days. (Obviously, all times and distances are in the Earth's rest frame. They will be different in the traveling twin's outgoing and incoming rest frames.)

These statements are contradictory. Events which are simultaneous in one reference frame are not simultaneous in any other reference frame. You are not just disputing an interpretation. Whether you realize it or not, you are disputing the demonstrable fact that relativity makes correct predictions. Nobody in this thread is making any such claim.

Did you read what you linked to, michel? Don't just pick out key words. It's a description of an analogy. In this analogy, L is the spacetime interval. Length and duration are the x and y coordinates. Read the rest of that section, please. 0.

Zero spacetime interval, of which seconds and meters are both measures. For simplicity, let's just say zero light minutes. Yes. In our rest frame, it is an event 10 minutes in the future in time and 8.3 light minutes distant in space. In other frames of reference, those values will be different. The spacetime interval between them is about 5.5 light minutes - this value is independent of reference frame.

Since it seems that Iggy is not here at the moment: The spacetime interval between an event now on Earth and an event on the Sun 8.3 minutes in the past or future as measured in our rest frame is zero. The spacetime interval between an event now on Earth and an event on the Sun 10 minutes in the past or future as measured in our rest frame is 5.5 light-minutes, or 5.5 minutes, depending on how you want to look at it, those being equivalent measures of spacetime. An observer traveling at a constant velocity such that they are present for both events will experience 5.5 minutes of time in between them.

Yes, between events. The Earth and the Sun are not events. They are objects, moving relative to one another, with durations of billions of years.

"Units of space expanding" is another way of saying the same thing. Either way, the distance will be greater at the later time. Light will take more than 1G years to travel the distance.