md65536

Because pi (or tau) is a ratio, whose value has meaning. You lose the meaning by choosing a value of 1 for the constant.

+/- v here is always the velocity of Cat and Mouse relative to each other. When I say Mouse reverses direction I mean that it travels at v relative to Cat, in the direction that Cat has just come from. I didn't realize it at the time, but this is not worded intuitively, once you calculate its change in velocity consistent with everything else that's written. "Mouse reverses" might better be replaced by "Mouse moves in the opposite direction". > Also when you say that the mouse reverses direction, it is traveling at 0.866c towards the Earth relative to the Earth. No, I didn't make it clear enough but none of the velocities are relative to Earth. I mean that if Mouse was approaching at speed |v| along Cat's positive x-axis, then after they meet it recedes at speed |v| along Cat's negative x-axis. All times are according to local clocks... Cat waits 24 of its own hours. The "additional questions" don't change any of the details of the main question, except for specifying particular directions instead of "random". Does this clarify enough? Sorry, as usual I was sloppy with the details. With this would you like to change your answer? The first part of your answer I think was interpreted correctly but I think the answer is wrong (8 days to catch up, I don't think is right according to any of the 3 observers). By the way: The puzzle should be solvable using relativistic reasoning alone, essentially without math. I know there are some people on these forums who want to understand SR and/or the twin paradox intuitively without the math... well if such an understanding is possible then this puzzle should showcase it. That is, if the solution I have in mind is correct.

The Lorentz factor when v = 0.866c.

I don't really know enough about pi to care as much as others (and I think it would be a waste to use two Greek letters for essentially the same constant), but... Imagine if you bought eggs in packages of 24 half-eggs and you'll see that counting eggs in units of "1 egg" makes the most sense. If a unit circle had a diameter of 1 then I'd probably think that pi makes more sense than tau.

And pi = tua/2 still has a factor of 1/2 around. Try reading this before assuming the idea is worthless: http://www.math.utah.edu/~palais/pi.pdf There are several examples of how tau is a more natural fit in various equations. Do you have a counter example where using 2pi would obscure how and why the constant fits into the equation? Is there an example where the occurrence of pi makes intuitive sense but a version using 2pi would not? Sometimes the factor of 1/2 makes the occurrence of pi clearer. For example, A = pi r^2, or 1/2 tau r^2. The one with pi seems simpler, however it hides the factor of 1/2 that naturally comes up when deriving the area with integration, similar to how a factor of 1/2 makes intuitive sense in a formula for triangle area. The only convincing argument against tau that I've seen here is that the use of pi is an established convention and it would be too much trouble to change that all (and to make an electron's charge called "positive", and any other such misleading conventions). And I agree with that. I think that most people who don't really deal with pi much just think of it as a weird constant that just happens to have a strange value. Perhaps it would make more intuitive sense to people if everything was based on tau instead (meaning "one turn" as used in the pdf above). OR perhaps that intuitive sense only comes by really considering the meaning of tau vs. pi, and would be missed if one or the other is simply used without consideration of why.

It's irrational not an integer, the question doesn't make sense. I use it for the usual stuff... Riemann zeta functions and Gaussian integrals etc. Ok truthfully: the last time I remember using pi was dealing with angles between vectors, in radians. Of which there are 2pi = tau per revolution. (Or, the maximum normalized angle is pi, so it's not like using one of pi or tau is always better than the other.)

It's not about diameter vs radius. Pi occurs in plenty of formulas that don't use diameter. Many of them use radius, and nobody advocates using diameter instead of radius in all formulas simply "because the world works in diameters", or because diameter is easier for the plumbers.

http://www.scienceforums.net/topic/74683-acceleration-is-not-important-in-the-twin-paradox/ Note that it is 20 pages long and I don't think full agreement was ever reached, and the title of the thread was admitted to be misleading and incorrectly worded, so hopefully it doesn't cause more confusion than good!

I ask you to give/make me a pile of gold/money by tomorrow. Will I convince you of proof of no you? What god is it that grants wishes? Even if you demand proof of what a supposed god is actually claimed to do, if that god "works in mysterious ways" there is always that loophole to excuse it from providing proof.

I was thinking about misleading intuition of the role of acceleration in the twin paradox, so I came up with this puzzle to test intuition about it: There are 2 observers, Cat and Mouse, starting at relative rest. Mouse instantly accelerates and takes off in a random direction, traveling with a speed of v. After some time, Cat instantly accelerates and chases, approaching at a speed of -v relative to Mouse. When Cat reaches Mouse, Cat remains inertial, while Mouse again takes off in some direction at speed v relative to Cat. Again after some time, Cat chases at relative speed -v. This repeats. If [math]v=\frac{\sqrt{3}}{2}c[/math], so that [math]\gamma=2[/math], and this goes on for 1 year according to Cat's clock, how much time has passed on Mouse's clock (assume they're reunited at that point)? Is additional information needed to answer the question? Additional questions: 1) If they started on Earth, and Mouse reverses direction each time Cat catches up, and if Cat waits one day each iteration before chasing, roughly where will they end up at the end of the year? 2) In the case of question (1), who experiences more proper acceleration? If instead of heading backwards, Mouse heads off always in the same direction, at v, then who experiences more proper acceleration? How does this modification affect the relative clock timing?

That's what I was trying to speak about. We say photons exist as particles, but the only time we can say anything concrete about their existence is when we measure them. I was thinking (in the case of gravitons at least)... couldn't it be possible that they exist not in "lumps" but only interact or are measured in lumps? That is, whenever you look at them, the observation is lumpy, where the lumpyness (particleness) is a property of the observation, not the thing observed? But then I second guess myself because what is the point of speaking of something's existence outside of its measurable effects? In the case of quantum gravity could that make a difference? Could it be a continuous thing that affects mass continuously, but when you measure it you would theoretically get quantized measurements? Or is it nonsense to speak of effects of gravity and measurements of gravity as if they're different things? Edit: I guess you already answered the question using the photon analogy. But in that analogy, why do we say photons exist as particles instead of only interact as such? Does it even matter if we said one or the other (since there's no measurable difference)? And would that be the same in the case of gravitons?

Yes, it's "relativity of simultaneity" that you're missing. You've set it up so that they're symmetrical (what A sees of B is exactly what B sees of A). Yes, each measures the other's clock running slowly, and when they meet the clocks are the same. The solution to that paradox is that they don't measure their "starting time" as simultaneous. Observer A measures that B started earlier, and B measures that A started earlier. They don't agree on the simultaneity of their starting. As others have pointed out, for an observer C who observes the symmetry, A and B start simultaneously. No... acceleration doesn't affect a local ideal clock. If they meet at a point and then enter a common inertial frame, it doesn't matter whose frame that is, the result will be the same. That is, if they come together and A instantly accelerates to match B, or if B instantly accelerates to match A, the result will be the same. We kind of got into a big argument about all this in another thread, so I'll just point out: "Which observer accelerates" makes a big difference while A and B are apart.

True, but with a "coffee break" they're both together in a common frame, and they'll be able to say for certain that one clock is ahead of the other, or that the two clocks are in sync. (Actually "together" is the important part for agreement, they will agree on whose clock is faster even if they just pass through the same point. They'll agree at that moment. Actually... "together" OR "relatively at rest in a common inertial frame" is good enough for agreement.) It depends on previous conditions which you didn't specify (when and how were the clocks reset or last synchronized, and/or what were the spacetime paths that each took before getting onto their inertial collision-course trajectories). You've set it up so that A and B are symmetrical, so assuming that everything else is set up symmetrically (paths, synchronization method, whatever), then their clocks will be the same when they meet. Or you could set it up so that one of their clocks is faster (a necessarily asymmetrical setup). Note that if they meet/collide at a given point, the result will be the same whether they come to rest in a frame that averages their velocities, or if they come to rest in A's frame, or in B's frame. All 3 cases will give the same answer for "whose clock is faster?" given some particular starting conditions.

Is that reasonably expected? Or is it just a wild possibility? This is above my head, forgive me if I'm rambling incoherently and off topic, but... In quantum theory is it only observable aspects that must be quantized? And the effects of gravity do not need to be observed locally (eg. an observer in freefall doesn't need to know if it's in empty space or approaching a mass)? Could a quantized measurement of gravity (ie. a graviton) be an aspect of the observation or measurement of spacetime curvature, and not an aspect of spacetime itself? (ie. having no existence outside of observation.) Edit: Oh wait... the only time gravitons would come into play anyway is when the gravitational field changes, which the observer would (or could) detect or "feel", which would constitute an observation... so there would be no point in separating observable effects from anything else... ?

Yes, but it's a theoretical maximum. And even then, it's only a theoretical maximum given certain assumptions (eg. transistor logic and one transistor per atom). Those are probably reasonable assumptions, but theoretical maximums can be overcome if the assumptions change, and that sometimes happens. Certainly that's a high number for today. That could potentially be a benefit in more authentically simulating a human brain. And there may be massive shortcuts that allow a huge amount of simulated detail to be processed in parallel, or shared among objects. Some people imagine simulating all the atoms in a system individually, but perhaps it's possible to do a few calculations that can be used by all of them. Perhaps there is a huge amount of brain work that can be done once and used by all the simulated brains. I can't imagine what any such short-cut might be, but by the time we're simulating brains (if ever), I'm sure people (or sims!) will discover tons of news ways to do it better and cheaper than we can imagine possible today.

I specifically avoided taking sides on whether a god exists or not, because neither side is proven. I should have more clearly separated "delusion" and "belief in something unknown", but I think what I wrote is true whether a god exists or not. Some degree of delusion is healthy (I argue it's "necessary"), other degrees are unhealthy, whether it's about religion or anything else. I'm thinking along the lines of "obsessive mental preoccupation" or something, and I'm talking about people who can't stop facing reality. This is really only a guess, and I'm no expert of course, but I think that the normal, "non-broken" ability to quiet your thoughts (even just to allow for sleep) is a type of avoidance of all the pressing issues that could occupy your mind, and is SIMILAR to suppressing thought with religion. Moderate consumption of alcohol can be healthy, same with religion! I specifically think there is no great difference between atheists and theists in this. Relaxing a racing mind is healthy, moderate use of external reality-suppressants can be healthy, and moderate use of religion can be healthy. Here's an example which will hopefully make my point clear: Suppose someone likes to relax by reading fiction. Are they necessarily broken? They are avoiding reality! Suppose their behavior is partly inspired by a fictitious character. Are they broken then? I don't see the difference between that and letting religion shape one's reality, whether the religion is "true" or not, with the caveat that extreme escape of reality with any of these things can be unhealthy to the point of breaking someone. Conversely, refusing to "waste" time or potential, and only constantly and fully facing measurable reality, is not healthy. A possible difference that comes from whether or not a person realizes they're avoiding reality, I think doesn't matter in small degree. Addition: Looking back over the thread, if by "believing that the sky fairy will make it better isn't going to improve things nearly as well as actually doing something about it," you meant only in the case where belief prevents necessary action in life, then I agree (falls under "extremism" I think), but that's not what was being argued.

That doesn't make it a false generalization. But Leibniz's argument doesn't prove that god exists. Given the assumption that god exists and is the source of everything and doesn't contend with itself and whatever else, the argument concludes that the truths of religion and philosophy are consistent. With the proper formal details, that can be a valid logical statement. Once again you're taking what you see is the right side of an argument, and justifying it with bad reasoning.

Sure, but nobody needs to spend every moment of their lives doing the one thing that will most improve things at the moment. Nobody does. That's the type of delusion I'm talking about... our necessary ability to selectively forget about things, and to be satisfied with mediocrity. Eg. the belief that what I do on a day-to-day basis is important relative to all the better things I could be doing instead. Nobody will know everything, probably not ever. It is necessary for one's mental health to be satisfied with not knowing everything, and with achieving less than the ultimate. Yes it's a shame that a lot of people aim too low and there is wasted potential, but a lot of those people live an adequately healthy life. Dr. Jack Shephard was an over-achieving surgeon but he was broken because he had a torturous obsession with saving people --- being broken and doing a lot to improve things aren't mutually exclusive. Someone who spends Sundays in church and looks to religion to provide ethics, is not certainly broken. Parents who lose a child because they decided to pray away the diabetes are surely broken. I don't think belief in a god is the deciding factor, but extremeness of that belief might be.

Well, I'm assuming that they don't! I'm talking about ideal meter sticks here, and assuming that they are completely contained in a local frame of reference. If a physical ruler is used, I'm assuming it is small enough (perhaps less than 1m in extreme gravity) that it is completely in local flat spacetime (or at least within any small margin of error you need). So I think that tidal forces can be ignored. And apart from that, 1m is locally 1m for any observer, so a meter stick should not change in a gravitational field. However, distant meter sticks are not necessarily 1m. The problem I think I'm trying to work through is that there's not really a single, obvious way to compare distant meter sticks. You can't bring them side-by-side while still maintaining the difference that occurs at a distance. You can decide to compare how they "look", or measure using timing of light signals, but then you get different answers depending on how you measure, corresponding to the variety of definitions of distance that there are... Anyway when I say "meter stick" I'm referring to "one unit of length as measured by a local observer" (at the stick's location).

To chime in, and generalize from belief in a supernatural humanoid to belief in maybe *anything* that requires faith... Delusion can be beneficial and healthy, depending. In that sense, any belief that helps someone might be considered to make them "less broken". We don't need to constantly face every aspect of reality, and if we did we'd probably go mad ("broken"). Things like impending death, questions of the point of existence, etc, can healthily be avoided rather than obsessed over, and in that sense faith can be helpful to some. For others, a harsh examination of reality works better. But delusion can be bad as well. Many people start off on "spiritual journeys" and are so consumed by the initial beneficial feelings they get, that they become obsessed and devoted to achieving that drug-like high. Pushed too far, faith can become a net detriment, leading to neurotic behavior and feelings. For many, belief in a god is a benefit that is preferable to some other search for answers, and a source of hope, guidance, etc. For many, it is source of misery, guilt, etc etc. Some are less broken by it, some are more broken. The thread title is a false generalization.

I don't see how the last statement follows, unless you assume that relative velocities add like they do in Galilean relativity. See http://en.wikipedia.org/wiki/Velocity-addition_formula#Special_theory_of_relativity and note that if you use a value of c for either u or v, the end result is c. In other words, whatever the speed of the source of light, the speed of photons projected from it will still be c. The reason why this works is that the composition formula is derived from consistent equations that begin with the assumption that the speed of light is c regardless of the source. Better understanding might come from looking at how it is derived? It might help to know that the invariance of c isn't a prediction of SR, but an assumption of it. Based on experimental evidence, it was observed that the speed of light is c regardless of the motion of its source, in all measurements ever made and in all experiments. The equations of SR are a description of how that can be possible and be consistent, and of how other things like time and distance must then relate.

deltaTB/deltaTA would be the inverse of the Lorentz factor, which isn't equal to velocity. All speed measurements are similarly affected in that if a moving object B fires another object towards the observer, then the velocity of the projectile relative to A can be calculated using the "composition of velocities" formula. The formula works for all values up to and including c. If you consider a photon as a projectile and use c in that formula, the result is that both A and B calculate the relative speed of the photon is c.

Definitely looks like bulldozer marks. I did a bit of googling and found: http://en.wikipedia.org/wiki/New_Valley_Project http://wikimapia.org/27465/Sheik-Zayed-Canal It looks like they were building irrigation reservoirs like at coordinates 22.80089,31.478651. Without doing proper research, I suspect that the canal might have been planned on a straight path (leading past near your find), but had to be diverted for whatever reasons. It could be an abandoned site. That's some good work, Lou.

Note: One thing I've realized I got wrong in earlier posts is assuming that light cones are conical through all of spacetime, but I think that's only true locally (or in flat spacetime)??? Schwarzschild coordinates use the local coordinates of an observer at an infinite distance, right? The proper distance from any stationary external observer, to the EH... is it infinite? I've read that the proper time of a free falling observer, to reach the singularity, is finite; is that only because proper lengths "near" the EH become length-contracted to nothing as the EH passes the observer? If so then if you remained stationary outside "near" the EH, any stationary ruler no matter how long would not reach the EH. You could not "dip" a hand into the EH at low speed (ie. unless it's effectively falling in). Edit: Nope, I have no idea what I'm talking about here... https://en.wikipedia.org/wiki/Event_horizon says the proper distance from a stationary observer is finite.

I think that length contraction comes into play here. If you imagine 2 observers falling into a BH together, and just as approaching the EH one accelerates away (assuming it has unlimited energy resources) to remain stationary while the other continues free-falling, then the difference in speeds of the two observers must be near c? So the distances they measure will be severely length contracted. I might be wrong but I think the "stationary" observer will now see the EH as far away, as the length stops being contracted. The EH would again have to appear "frozen" to an observer stationary relative to it? An observer can't hover at the EH, but can it get arbitrarily close? Anyway, no matter how close or far, if you dipped your hand (or a long rope) in, you couldn't possibly get it back! In the link I posted, someone speculated that perhaps after crossing the EH, it would seem to recede from you at greater than c. So if you pointed a beam of light "behind" you, that light would never reach the EH. It is an event horizon even from within. If there is a direction toward the EH, the singularity is still also in that direction because that's what the light's going to hit!