Delta1212

Atoms are made up of particles. Fundamental particles are not made of atoms. Photons are not made up of atoms and do not contain quarks.

Distance is time multiplied by average speed. Average speed for a uniformly accelerating object starting from rest is half its final velocity. Final velocity would be acceleration times the time.

The point was that if he creates something that he can't lift, then he can't lift it and is therefore not omnipotent. If he can't create something he can't lift, then there is something that he can't make and is therefore not omnipotent.

If one observer was closer to Andromeda, the two observers are no longer co-located which means you can't determine which moment for each observer is simultaneous with which moment for the other any longer, and you have no basis left for comparing differences in simultaneity with Andromeda.

With the caveat that, obviously, I don't know exactly why this is, I would hazard a guess that the fact that none of the tested birds died might indicate some part of why. A major reason that diseases tend to evolve to be less severe over time is that the more rapidly and consistently the disease kills its host, the less opportunity it has to spread and replicate. If the disease is exceedingly unlikely to kill the host, even in more virulent forms, this restriction is, if not eliminated (because this is obviously laboratory conditions), then at least greatly lessened. If the more severe symptoms help the bacteria spread, and don't greatly increase the likelihood of the host dying, then I don't see why it wouldn't evolve to be more severe. Again, that's pure speculation on my part.

If I understand you correctly, I believe you've got it, yes. The two co-located observers will agree that "now" on Earth is the same, and they will see the same light from Andromeda, but they will disagree on what is "currently" happening in Andromeda, but which they won't be able to see for about 2.5 million years.

That was the best way I can explain it. If you still aren't able to grasp the mistake you're making then I'm going to have to let someone else make the attempt because apparently what I'm saying isn't getting through.

The formula is being used to calculate the difference in time on Andromeda that two observers calculate as being simultaneous with "now" on Earth. The time only needs to be calculated for one of the observers because Wolfram has set the time for one observer to 0, rendering the result for the other equal to the difference between the two. To break it down: [math]t_e[/math] is the time on Earth considered "now" conveniently, as they say in the link, set to 0. [math]d[/math] is the distance from Andromeda to the observer. [math]v[/math] is the relative velocity of the observer with Andromeda which, for simplicity's sake and so that they only have to calculate the simultaneous Andromeda time for one observer, they have set to 0 for an observer at rest with Earth. This obviously isn't true to reality, and doesn't have to be done this way, but it doesn't impact the basic paradox and makes the math simpler. [math]t_a[/math] is how much further in the future on Andromeda an observer with velocity [math]v[/math] at distance from Andromeda [math]d[/math] will calculate to be simultaneous to [math]t_e[/math] than an observer also at distance [math]d[/math] but at rest with respect to Andromeda. If you leave Earth and Andromeda traveling at their respective velocities (which is fine) and plug Andromeda's velocity into [math]v[/math], then you have found the difference in the time that an observer on Earth at rest with respect to Earth calculates to be simultaneous on Andromeda with "now" with the time that an observer also on Earth but at rest with respect to Andromeda calculates to be simultaneous with "now." This is, of course, an example of the Andromeda paradox, but the paradox doesn't require such extreme speeds to work and certainly doesn't require the exact relative velocity of the Andromeda galaxy. If you want to illustrate the paradox using the more typically used walking speeds without assuming Earth and Andromeda are at rest, you would need to calculate [math]t_a[/math] once for [math]v[/math] equals Andromeda's velocity for the observer standing still on Earth and once for [math]v[/math] equals Andromeda's velocity plus the walking speed of the observer, and then subtract the first from the second. By setting Andromeda's speed relative to Earth to 0, they set the value of [math]t_a[/math] for the first case to 0, and since subtracting 0 from the [math]t_a[/math] calculated by the walking observer would have no effect, they can strike out all of the math from the stationary observer and find the difference between the two observers' calculated simultaneous times on Andromeda solely by calculating the time for the walking observer. So yes, the Wolfram mathematical formalism does include two observers, but they've zeroed one out to simplify the math.

I figured as much but thought I'd ask to be sure.

You're correct, that was an imprecisely chosen word on my part, though I did note in the second paragraph that the observation would have to be made retrospectively since it would need to wait for the light to arrive.

Perhaps it would help if you could link to an example that doesn't use Wolfram's unorthodox approach?

We don't travel at a significant fraction of c in order to get to Mars. Mars is orders of magnitude closer than the nearest star and doesn't require nearly the same speeds to get there in a reasonable time.

No. What you're describing is true, and is down to the same effect as the Andromeda paradox, but the paradox is a very specific case of this effect. I do think I may have found the bit on the Wolfram site that is responsible for the confusion, however. "[math]t_a[/math] is the time advance on Andromeda, which can be considered simultaneous with an event on Earth occurring at , most conveniently set equal to 0." This looks like it might be saying that [math]t_a[/math] is the difference between the time of the as measured by the Earth and that observed on Andromeda. It's not, however, the advanced time on Andromeda as observed by Andromeda but rather the advance of time on Andromeda as observed by the other, walking, observer on Earth. That is, if [math]t_e[/math] is set to 0, then a velocity of 0 (the observer on Earth at rest) will give you a [math]t_a[/math] of 0. That is Andromeda time "0" representing the moment that the stationary Earth observer beloved is simultaneous to Earth's "now." If you plug in the velocity of a person walking right past this stationary Earth observer, the velocity will be that of the person working, and [math]t_a[/math] will give you the time on Andromeda that this walking observer measures to be simultaneous to Earth's "now," a time that will be in advance of the time that the stationary observer measures. The Andromeda paradox is a demonstration of the fact that observers that are very close together moving at only slightly different speeds can have a significant difference in the time they measure to be simultaneous with "now" at very large distances. So that someone walking past you in the street may believe that the events going on "now" in Andromeda are days off from the events that you believe are going on in Andromeda "now" (retrospectively since we have to wait for the light to reach us, obviously) solely because of the slight difference in your relative velocities and the very great distance from here to Andromeda.

Oh, the rock will be obliterated and the material from the ship will keep right on sailing away at high speed, but the ship won't be intact. The rock will "burn up" the ship just as much as the ship "burns up" the rock.

Ok, so what you seem to have demonstrated is that a moment in Andromeda that is observed to be simultaneous with "now" on Earth will not be observed to be simultaneous with the same time on Earth when observed from Andromeda's frame?

xyzt, what is it, exactly, that you think the Andromeda paradox states?

The Andromeda paradox is about the difference in what is considered to be "now" at a distant location between two co-located observers with small relative motion. There is no example of the Andromeda paradox that only has one observer.

But both observers are on Earth. Or are you just using 'earthling observer' to mean the one that is standing still (with respect to Earth) rather than the one walking past him?

You can keep repeating that, but it doesn't make you less mistaken in your assumption of what the paradox is about. Yes, all motion is relative, but the relative motion between Earth and Andromeda is irrelevant to the different between the "nows" of two observers moving with respect to one another on Earth. Neither is at rest with respect to Andromeda (unless Andromeda is treated as being at rest with respect to Earth, as was the case in the example that you linked to). If you want to go the route of "motion is relative" then yes, we all understand that you can say that Andromeda is moving toward the Earth, or Earth is moving toward Andromeda, but if two observers are just sitting on Earth, the Andromeda galaxy is not moving faster towards one than the other. If one of those two starts moving toward Andromeda, then, yes, you could say that Andromeda is moving slightly faster toward one than the other, just as easily as saying that one is moving slightly faster toward Andromeda than the other. But it is the relative difference between these velocities that causes the Andromeda paradox, not Andromeda's net velocity. The relative velocities being compared are between two observers that are very close and have a very small difference in speed, and the difference in what they consider to be simultaneous with "now" over very large distances, such as that of the distance between Earth and Andromeda. Andromeda's relative velocity is irrelevant. Only the very small difference between the velocities of the observers is what is being considered for this particular effect. It even says that in the explanation that you linked to, a fact which you have not addressed.

Except, again, taken straight from the page you cited, it's being assumed that Earth and Andromeda are fixed with regard to each other. Nobody is fixing an observer with regard to Andromeda and having them move with Earth. The relative motion between Earth and Andromeda is just being ignored because it isn't the cause of the Andromeda paradox. It's the, comparatively minor, relative motion between the two observers, who are both on Earth, not moving with respect to Earth, and who will see a major difference between what events on Earth are simultaneous to what events on Andromeda. It says this in the Wolfram description that you linked to. The v in the equation is the relative velocity between the two observers, not the relative velocity between Earth and Andromeda, as you stated in your initial explanation.

Yes, but in that explanation, it says to assume that the Earth and Andromeda remain at a fixed distance with no relative motion. You said that the v in the equation is the relative velocity between Earth and Andromeda. It is actually the relative velocity between observers co-located on Earth. Per the explanation on the page you were citing.

Unless it's an ark ship, it needs to be going at an appreciable fraction of the speed of light in order to reach another star before the crew is too old to do anything much. At that speed, I'm not sure how much good kevlar is going to do when a space rock hits you with the equivalent force of a small nuclear bomb.

The point that is being made is that the Andromeda paradox is about the comparatively small difference in velocity of co-located observers creating a great difference in what time can be considered simultaneous when comparing events at a very large distance. You could replace Andromeda with a galaxy that is at rest with regard to Earth at the same distance and you would still observe the Andromeda paradox in full effect. It's not about the relative motion of Andromeda with the observers; it's about the relative motion between the observers themselves.

I had the same thoughts about both the similarity to Dutch and the Scandinavian languages. I speak neither, but I can make out some Dutch and a lot of the Frisian version of the prayer just from the similarity to German. Probably. I rarely think in German anymore, but back when I was using it more frequently I'd sometimes slip into it and it'd be a few minutes before I realized I wasn't thinking in English. Oddly, I've dreamed in Spanish but never German, despite my German being generally better.

It's been a few years since I did Calc, but this doesn't look particularly complicated. 1 and 2 look good, although you're missing some parentheses in 2b that are important but the math is done as if they're there so it comes out right, and 2c you wrote the limit as x approaches 5 when it should have been 8, but again, you used 8 in the actual math so it came out fine. 3a looks good. 3b, you should redo. You forgot to bring down the 3(x+h) after multiplying out the rest of that line, so your f(x+h) is missing a big chunk for most of the problem. You also made a couple of small mistakes with a sign and factoring an h out of a 3x later in the problem, so it's probably best just to start it fresh.