md65536

1) It's a constant. It's not variable. It's not a matter of perception or a choice of curvature. See http://en.wikipedia.org/wiki/Pi#Infinite_series to see that it's an exact value and not dependent on experimental measurement. 2) On a plane, a circle is all the points equidistant to a center point. So you can reliably take a fixed length (measuring stick, string, etc) and rotate it around a fixed point to create a circle. Even on some curved surfaces, like the earth, you can do this because the line traced out will all be on a single plane. I'm not sure what surface properties are required to ensure this; it wouldn't generally work on a "wobbly" surface. On a curved surface, the circle's radius wouldn't be the same as the length of your stick/string. Not really though.

Okay that makes sense and complicates my oversimplistic view. Certainly there's more to this than I understand. There must be an effect from knocking molecules out of the way, but I don't think that that explanation can completely nullify the effect I propose. If it could, then a material that outgasses would also have no net force acting on it, because the force of the outgassing would be balanced by having air molecules knocked out of the way. Perhaps it's true with weak enough outgassing, and perhaps it's true in the radiometer.

The heated air molecules tend to knock slower colder molecules away, which is equalizing the pressure among any pockets of air. The warmer air on the black side is less dense, with higher energetic particles, and the colder air on the silver side is more dense but slower particles... the force on the surfaces of the vane from air pressure would also equalize, otherwise there is a pressure difference that would quickly equalize. Yes, it makes sense that the state of air pressure in the device would not have a net force on the wind vane, but that doesn't rule out my hypothesis. There are two things here: 1 is the heating of the air through conduction, and 2 is the forces imparted by the heated air. Even if (2) is shown to have no effect on the surfaces, it doesn't consider the effect of (1). If there is a significant force acting on the edge of the vanes due to molecules colliding with it, could this force be removed by making the vanes vanishingly thin? Is the thermal transpiration force proportional to the surface area of the edge, so the effect could be experimentally tested by measuring the effect with vanes of different thickness?

Keep at it!, and the skepticism should be replaced by understanding, at least gradually. Hopefully the science resonates with you too, not just the quotes.

I looked up an explanation of why a Crookes radiometer spins in light, and the accepted answer that I've seen---"thermal transpiration"---sounds like bull. Even if the effect is real, the force on the edges of the vanes would have to be small, and nowhere do I see even a suggestion that anyone has ever calculated the force. In fact it's claimed that "The correct solution to the problem was provided qualitatively": http://math.ucr.edu/home/baez/physics/General/LightMill/light-mill.html I came to a different conclusion (as have many; there are a lot of incorrect explanations). Assuming that the black surfaces are heated by the light, and are hotter than the ambient air temperature, then the black surfaces will heat the air via conduction. This involves air molecules arriving at the surface with relatively low energy, and leaving with higher energy ie. faster than they arrived. By conservation of momentum or Newton's third law, there must be a force pushing on the surface of the vane. This seems so simple, but is it wrong? More generally, it seems that for any conduction of heat, there must be a force acting on each of the hot and cold sides, pushing them away from each other. Is this true?

Two ways I can think of to take this one step at a time: Work forwards. Don't start with the unintuitive conclusions of SR, as with the animation examples. Instead start with the postulates of relativity. Take small steps figuring out what they mean, and accepting that they are consistent with reality. Then take steps toward understanding the logical consequences. Or work backwards. First accept that SR works in theory, even if you don't think it matches reality. Take an in-depth look at an example, and figure out how "what observers see" fits, according to SR. When you see that every example you try fits, it's easier to accept that SR is mathematically consistent, and then easier to accept that it is consistent with reality. On the other hand, taking small steps in trying to understand an example of SR while also denying time dilation etc, is like trying to take small steps through a brick wall.

Yes there is. Remove relative motion from the picture, and have all observers share an inertial frame. Then you can work out the timing of light signals without worrying about relativistic effects. Different observers "see" different orders of events just due to delay of light. If you add in relative motion of observers, and keep invariance of speed of light, you'll have to deal with length contraction and time dilation and relativity of simultaneity. Countless (2+) threads seem to have the same pattern here. They start off with "Help me to understand relativity here" and after other people's explanations, turn to "Nevermind the explanation, I don't think relativity is right." I share the desire to figure this out on your own terms, using your own reasoning. However, I also know that it all becomes simpler after learning what other people have figured out about it, and the tools (proper time, definition of "event" etc) make it a lot easier to think about all this. Then, you don't have to fumble with concepts like becoming "unattached from the rest of the universe". I know because I've been there, trying to work with similar thinking. It is possible to figure this all out on your own terms, and retrace the discoveries of Einstein and Minkowski etc on your own, but it is easier to use their work and learn what SR says, before questioning what it means and whether it's right. By the way, I've looked at this from the perspective of "seeing the world exactly right" and it works, and I've come to the conclusion that special relativity is right, including time dilation and length contraction. Relativistic Doppler analysis might be what you want to look into, and should help with figuring out what the various observers see, consistent with relativistic effects.

Are you saying that the animations are wrong, because they incorrectly show what special relativity predicts? Or that the animations are wrong, because what SR predicts is wrong? Or that you don't understand the animations? Are you trying to understand how relativity works by studying these examples? Or are you trying to show that relativity is wrong, based on what you already know? Or... are you saying that the animations don't show the original example that was given? If the example is set up so that the observer on the train and the observer on the ground both see the lightning strikes simultaneously, then the observers must pass at that moment, for example if the observer on the train is moved back toward the rear of the train. The observer on the train still measures that the front lightning strike occurred earlier than the rear one (as the front one is farther away and light from it took longer to arrive), and the observer on the ground measures them as simultaneous (as they are equidistant). It's the same result regardless of observer position, even if "what is seen" is changed.

Perhaps separating "speculations" and "pseudoscience" might be beneficial. They are grouped together (both included in the Speculations forum description). Speculations is perched on the edge of Trash Can, with the threat of cutting off the conversation. If Speculations and Pseudoscientific Discussions were separated, with separate rules, then there would be not just the threat of killing bad discussions, but encouragment to improve them, as topics could be "promoted" to Speculations if they met the requirements of the stricter rules. However, I don't think it's worth it to try to make it great. This is an internet forum, and most of the topics in any section are from amateurs interested in science, or from people who are learning, and I don't see a lot of professionals actually doing original work here. It would be unrealistic to expect that of the Speculations forum. A "Good Speculations" section might go empty, perhaps only symbolic of the "eternal optimism" that they exist and are welcome. But if people are doing it right, they're probably doing it elsewhere than an open forum on the internet?

The first sentence is correct but not the rest. In the first sentence "in the same frame" would need to be understood as "*while* in the same frame", and "in retrospect" would not apply to what happened in other frames, before coming to relative rest. Coming to relative rest (sharing an inertial frame) does not change an observer's history, or sync it up with the other observer. Two reunited twins can still have different histories, having measured different orders of past events, etc. I would not recommend trying to figure out special relativity while adding additional complications of different observers and additional delays of light. It would be easier to learn what SR says of the train example, and when you got that, you can add other observers and see that everything remains consistent when additional delays of light are considered. Another thing to consider is that if all the observers you consider remain in a single frame, then a classical Galilean/Newtonian description of things works, in which everyone agrees on the timing and order of events. Yet, different observers can see events happen in different orders due to different delays of light. Differences in what is seen is considered separately from differences in what is measured. If you really must understand this all in terms of differences of observed delay of light, it might help to first understand all that can happen for different observers in a single inertial frame, before special relativity is even thrown in.

In my opinion, removing it would be like admitting that speculation no longer has a place in science, and that would be a huge loss. It's already at the bottom of the front page, as if it's of the lowest value. Just because there is rare (or no) valuable speculation, doesn't mean that the opportunity should be closed to all, discouraging any future conceivably valuable posts. Ideally all posters would be encouraged to "do science right", especially when it comes to speculation, but that's unrealistic because speculation attracts a lot of people who are neither interested in nor capable of learning the slow, difficult, right way. In my opinion, the best way to handle speculations is to encourage individual improvement, and not waste too much time forcing people to see what they don't want to see, and not treat speculations as a single coherent group of bad posts, so that a rare valuable post (intriguing or thought-provoking to others, or even just a path for the poster to learn) isn't assumed to be garbage and tossed aside.

I think I need to exit this thread, because I'm arguing obsessively and hypocritically. My main point is just a personal pet peeve with the forums, that many people who are justifiably harshly critical of arguments that challenge accepted or assumed answers, often tend to abandon that harshly critical thinking when arguing for such accepted or assumed answers. I think I've been a pain in the butt, sorry studiot. I still think that the value of tau has a more natural relation to circles, and the understanding of the meaning of tau or pi becomes clearer when that relationship is examined, but despite acknowledging merit in a choice of the value of tau over pi, I don't think it's worth either changing everything nor having 2 "standard" ratios. I think the best option for people who think the value of tau is clearer or more illustrative than pi, is to define it in terms of pi (instead of treating it as a standard), before using it in their work.

That's fine. Plus I agree with you on issue 3). However none of your 3 issues addresses the real issue brought up in OP's link. You have missed the main issue and ignored it as silly or irrelevant. Also, it is fine to ask for a balanced view, but nobody is bringing forth the other side of the view. It is unfair to call for a balanced view, implying that there is a good argument supporting your side, and then expect someone else to present it. That "good argument in support of pi over tau" may not exist. But okay, some points in support of pi have been mentioned, including that it is the established constant and it has a more natural relationship to diameter, which was probably how circles were more commonly defined when the ratio was first investigated. However, my personal main issue with your arguments is that you're supporting the easy side of an argument with bad reasoning. That we should keep pi as it is, is the default side of the argument, easy to justify. You're trying to shut down the harder side of the argument---the side which requires going out on a limb a bit and trying to convince people of a major change of mind---using bad reasoning. "The world works on diameters" is bad reasoning. "This is a silly waste of time" is bad reasoning. The arguments seem designed to end critical thought about an issue that you don't want to bother thinking about, and perhaps assume that anyone who does is a detrimental crackpot???

There are already ways to tell when it's overhead now: http://n2yo.com/?s=25544 http://www.n2yo.com/passes/?s=25544 also tells you when it will be visible. Applicable to this thread, http://www.n2yo.com/whats-up/whats-up-now.php might be useful for identifying a satellite when you see it. As for a light on a satellite, I vaguely remember a story from a year or two ago about a satellite getting a 60-watt bulb, to be used for calibrating light measurements through the atmosphere or something. However, if you consider that a visible satellite is reflecting sunlight, possibly over a considerable surface area, I imagine you'd need an impractically powerful light to easily identify a satellite from Earth with the naked eye?

Satellites at low orbit speeds will move across the sky at four degrees per minute only if observed from the center of their orbit, which is not what you're seeing unless you're at the center of the Earth. Since your whole sky makes up a tiny portion of their orbit, you see them move at a much higher angular velocity relative to you. To account for the parallax... when you say "little different" how different are you talking about? What height do you calculate, and what margin of error? One way I imagine a "nearby" (Earth orbit) moving object being the same distance at zenith and on the horizon is if it has a very eccentric orbit, and is farther from the Earth at zenith. But I'd guess that measurement error is more plausible.

I don't get whether you're arguing that pi is the better constant to use, or just that it should be kept because so many people use it. (I guess there are many people measuring pipe diameters and multiplying by pi to get circumference, and they'd be confused if they had to divide by 2, or offended if the constant they were using wasn't the "official circle constant" used by mathematicians or something like that.) I have no doubt that the practical importance of diameters led to the historical choice of pi instead of tau. Just for the sake of argument, imagine that tau had been chosen instead, and all along we'd been using 6.28 as the constant that everybody knows. Then imagine that someone came on these boards and said "We should be using half of that value instead, because more people deal with diameters than radius. We should change the value used in mathematics, because it is more practical and convenient for the greater number of people measuring pipe." Would you support this argument and say that it has merit? If yes, then I accept your argument but disagree. If no, then I think your argument that pi is better in mathematics is false, and that catering to the people who measure diameter is not a valid justification, and that the real justification is that pi should be kept because it's already ubiquitous.

Oops, thanks for the correction! The lesson might be that proper use of math trumps intuition every time. Another way to see that I was wrong, is if we placed a flag in space at that time at Earth's location but at rest in Mouse's frame, the distance between Mouse and flag would be the rest distance I was speaking of. However, though Earth passes the flag at that moment in Mouse's frame, it wouldn't pass at the same moment in Cat's frame. Is the 0.5 years because Cat spends half its year traveling and half at rest relative to Earth?

Which comment did I misrepresent?

This was already mentioned. You can derive the area of a circle, for example, by splitting it into infinitesimal triangles of height r and base dt, all around the circumference. The triangles have an area of 1/2 r dt, and getting rid of the factor of 1/2 makes it look simpler but it doesn't make it more understandable. http://en.wikipedia.org/wiki/Area_of_a_disk#Triangle_proof Or, integrating the area of rings of size 2pi t dt, and the factor of 1/2 come from integrating t dt. http://en.wikipedia.org/wiki/Area_of_a_disk#Onion_proof There's a natural factor of 1/2 in the area of a circle. The internal angles of a triangle add up to pi... or 180 degrees, half a turn. Getting rid of the 1/2 doesn't help here either. Are there any examples where the occurrence of pi is more easily understood, or more "natural" than if it were expressed using tau? Where the use of tau would obfuscate compared to the use of pi? And is there any such contradicting evidence? Pi has been established as a convention and is firmly entrenched and would be a major pain to switch away from, well beyond the caring of most people. That's a good argument and is probably the reason pi will win for the foreseeable future. But the argument that pi is inherently or naturally better hasn't been supported here, while examples have been given supporting tau.

Excellent answers, better than what I had in mind. As usual the maths can verify any intuitive understanding and drive further understanding. For example, while "nearby acceleration" doesn't make a big difference, it doesn't mean that nearby motion can be ignored. For example if Cat waited only 1 nanosecond before each chase, it would stay within a meter of Mouse, but Mouse would still age at twice the rate. Each chase would contribute only a tiny amount to the difference in age, but there would be quadrillions of them over a year. Or in other words, with high relative speed and a long duration there will be time dilation. Each time that Cat and Mouse separate (inertially), and then Cat accelerates toward Mouse, is essentially an iteration of the Twin Paradox, with Mouse being the inertial twin. What happens in between happens when they're together, for a negligible time, and contributes nothing to the time dilation. I can't explain it better than that but Toffo has and Iggy shows why it is. For additional question (1): If Cat spends one day of proper time receding from Mouse at v, then approaches at -v, it will take one day to catch up. As Janus pointed out, in this variation Mouse remains inertial when Cat catches up, and simply lets Cat pass by it. Where are they? 1 year proper time has passed for Cat, and 2 years for Mouse. According to Mouse, Earth has receded for 2 years at 0.866c and they are 1.732 LY away from Earth. According to Mouse, Earth has also aged at half the rate (1 year). For an Earthbound observer, it is Mouse that has aged at half the rate. After 1 year of proper time for Cat, which is 2 for Mouse, Earth ages 4 years. Mouse has spent 4 years at 0.866c and is 3.464 LY away. From Cat's perspective, it is at the same location as Mouse but currently traveling at 0.990c relative to Earth (composition of velocities). If we take the distance according to Mouse of 1.732 LY, consider that a rest distance in Mouse's frame (okay to do I think?), then for Cat this is length contracted by a factor of gamma (=2, since Cat is traveling at v=0.866c relative to Mouse), so it is 0.866 LY from Earth in its frame. For question (2): In (1), Mouse has only experienced the initial acceleration away from Earth, while Cat keeps accelerating on each chase. In (2), Mouse and Cat each accelerate to a velocity of 0.990c (composition of velocities) relative to their previous rest frame, on each chase. The proper acceleration is comparable, though technically Mouse accelerated less on its first leg; whoever accelerated last experienced the most overall proper acceleration. Technically when Cat catches up to Mouse (at the end of 1 year), that's still Cat. The choice of direction that Mouse takes each time doesn't make a difference in the relative timing and distance between Cat and Mouse, but makes a huge difference relative to Earth. In (2), they've essentially been accelerating away from Earth to near-c, 183 times I think, over the Cat's year. Before a Cat-month is up, the Lorentz factor relative to Earth is in the millions. According to Earth, Cat and Mouse's clocks are so dilated and they're traveling at very close to c, the universe will probably end before Cat ages a full year. Meanwhile according to Cat, Earth time has ground to a halt, but after 1 year Cat is still traveling at its fastest speed and the distance to Earth is extremely length contracted... Earth has been receding at very near c for a year but it remains (slightly) less than one light year away. For Mouse: each time it accelerates it is traveling faster away from Earth than Cat is, yet it still ages 2 years while Cat ages 1, and it is slightly less than 2 LY away from Earth. I may have made some mistakes. Theory and its maths comes from observations, and intuitive understanding comes from all 3, but trying to answer using minimal math can be unreliable.

Not only that, but in some equations where a single pi shows up, there is a natural relationship with a half circle (see animation at "http://en.wikipedia.org/wiki/Euler's_identity" for example). And as studiot pointed out, "A non integer ratio or any other number makes less sense than unity as the basis for counting turns." -- we don't count turns by half-circles. Do you have any examples? Of interest would be examples where converting from pi to tau would not only result in an extra step (as with dividing a diameter by two to obtain a radius), but where the occurrence of pi actually makes more intuitive sense than an occurrence of tau, and the use of tau would make a formula harder to understand?

The answer's right and the reasoning is interesting and I hadn't thought of it that way. You say that when Mouse accelerates the speed remains constant... can the same thing be said for Cat when it accelerates? Yes, the second scenario is what I was describing for additional question (1). When I first wrote it out I hadn't realized that Mouse remains inertial after its first acceleration! With this version, it is easy to make an analogy to the twin paradox, with Mouse being the inertial twin. Then the puzzle becomes figuring out whether it matters if Mouse keeps accelerating between each iteration of the twin paradox procedure. One way to squeeze some sense out of "Mouse reverses direction" is to consider it like this: When Cat joins Mouse, Mouse could instantly accelerate to be at rest with Cat, and then instantly accelerate in the opposite direction. (And then seeing that the result of Mouse instantly leaving and returning to its old inertial frame is the same as if Mouse simply remained in that inertial frame.)

A warning though, I think that the "intuitive" solutions might do nothing to help make the paradox more understandable. Also, to verify or prove the intuitive solutions should take some maths. Personally, I think that an intuitive understanding comes from figuring out the details (using math) for all the different observers and seeing how they're all mutually consistent, but then hopefully that understanding can be applied in cases like this. I believe this is the correct answer. Is it possible to show that the other details don't matter (Mouse's own accelerations for example), or that "what Cat measures" is consistent?

You must understand though that these constants represent the relationship between "turns" or angles or circumference etc, and radius or diameter. As you argue for turns in units of 1, with radius in units of 1 the relationship is tau. But you also argue that the world works in diameters, and with diameter of 1 the relationship is pi. However, a unit circle remains defined with a radius of 1 and a diameter of 2, and as you say "any other number makes less sense than unity." So there is a discord between the choice of radius as the unit measurement, and pi as the constant. It seems to me that all along you've been trying to show that the entire discussion is pointless by bringing up examples of pointless arguments (like whether pi is even or odd and whether tau is more aesthetically pleasing in that regard), while ignoring the actual arguments supporting tau.

Calculate it from the perspective of an additional observer that remains inertial. In your example, you could still use "Earth" as the start/end point and an inertial observer. Calculate the aging of twin A relative to Earth. Do the same for B. Since A and B reunite at the start point (this is how you set it up, correct?), they do so "simultaneously" (according to anyone), A's and B's clocks can be meaningfully compared at that point.