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An "absolute rotation" "with respect to an inertial reference frame" is an oxymoron

I'll give you that, I meant to say "inertial rotation". But it's just word play. My point still stands.

The Sagnac Effect has NEVER been derived from the postulates of Special Relativity, which is exactly what is expected if one understands the experimental results summarized above.

You can easily derive it from the equations of SR, as summarized here. Also, you stated earlier that Lorenzian Relativity uses the exact same math as SR, so you're contradicting yourself here.

Wikipedia incorrectly states that Laue derived the Sagnac Effect from the postulates of Special Relativity.

Irrelevant. It can still be explained with SR.

Spacetime? Do you mean a "4D Pseudo Riemannian manifold" claimed to have something to do with physical reality? If so, such an abstraction is just a mathematical idea that exists only in the mathematician's mind. I am a realist, and thus interested in real matter in real motion.

Poor you. These manifolds are the basis for General Relativity. GR perfectly predicts things like the motion and orbital decay of two orbiting neutron stars, or lensing effects due to gravity. All experimental data agree with GR to a very high degree.

You probably also don't believe in quantum mechanics then, where all fermions are made up of spinor fields.

Perhaps you may be interested in what whistleblower and Nobel Laureate Robert Laughlin has admitted about this

Yay, another pointless quote/history lesson. Sounds like something Chris Reeve would say. Which one were you again?

All the experimental data that you are referring to is an extraordinary confirmation of the original version of Relativity, which is based on the physical existence of aether, which in turn provides an absolute frame of reference for all optical experiments.

No it isn't. The absolute framework of aether doesn't exist. Science doesn't agree with you.

The original interpretation of Relativity, i.e. what we may call "Lorentzian Relativity", is based on a qualitative interpretation where the speed of light is variable, time is absolute, and there is a preferred frame for light that is typically undetectable due to confounding properties of nature

So it's basically philosophy, since you just admit it's not measurable.
Meanwhile all our experiments confirm that the speed of light is constant, and has always been constant since the beginning of time (at least since the earliest light/time we can detect).

Which of the two conceptual interpretations is the wisest choice? I prefer the purely realistic, non-paradoxical, humanly intuitive one.

Obviously the correct interpretation is the one we can measure. That's by definition the "realistic" one. In all real frames of reference we can measure in (which excludes your imaginary ones), the speed of light is constant.

Just in case that you are interested in history, my choice is the same choice that most physicists accepted until 1919, when Eddington claimed on totally inadequate evidence that an eclipse verified General Relativity. It did not.

History doesn't change the actual laws of nature, so it's irrelevant.
You're also ignoring the many, much more accurate experiments that were done in the 100 years that followed and that verified GR with very high precision. Stop living in the past.

The limit of small m in that formula is still a non-zero number, because classical gravity acts entirely through mass. It's assumed that the relatively small mass, m, is still a positive number. F=0 when m=0. "Massless" is not the same as "relatively small mass".

This page describes early attempts to explain gravity bending light by classical methods, but your reasoning doesn't work.

"While a satellite may be a moon, the Moon is not a satellite ..." - Spocks world

see http://www.mathpages.com/home/...

I'm not exactly sure what you are asking.

We see a series of invariant quantities when we translate, rotate or boost from one flat geodesic to another. Those invariants may not be conserved where geodesics are curved. There are several ways to look at this. Firstly, *local* Lorentz invariance is the only promise of GR; there are no global promises (and in fact, the observable universe is enormously curved at cosmological scales ("lengths" along any of the four axes, where c=1). Secondly, GR has a series of more general symmetries than the Poincaré group provides SR. One of the important ones is, by Noether, the conservation of energy-momentum: the right hand side of the equation can give *all* its energy to the left, or vice-versa, in a generally covariant way.

Inertia was built into GR because it was a useful way to narrow the possible set of coordinates with which to study GR as it applies in this universe (or toy models of subsets thereof). The existence of inertia is a postulate of most models (note, not just metrics or the GR framework), rather than something derived from relativity. There's a turn-your-head-upside-down exploration of this at http://www.mathpages.com/home/...

The meat is in the second half of the second last paragraph. GR was explored starting with perturbations of the flat space metric, and it is normal still to think of free-falling objects in asymptotically flat space moving inertially (because, in part, "inertial frame of reference" is what more specialized theories of relativity -- SR and Galiliean, for instance -- call their special coordinate systems). However, it is reasonable to think of some aspects of geodesics in dynamical, curved spacetime as imparting an acceleration to test particles, even a "proper accelerometer" comoving with the test particle points nowhere in particular. One sees this used for real when introducing pseudogravitational fields to solve various GR problems, or even in real problems such as the apparent acceleration of the bulk of the Earth towards a test particle free falling on a geodesic its surface will intercept.

"Mass" can be extremely different conceptually in GR and in SR. There are various "mass in General Relativity" articles out there, few are especially easy to grok imho.

So one way to answer your first question "why don't we see discrepancies between gravitational and inertial mass?", the answer is variously [a] we can, it depends on how one states the equivalence principle (there are several EPs) [b] they're different in GR even if they are not always treated differently (especially in a region of asymptotically flat space), and mostly the equivalence is treated like a gauge symmetry or other symmetry, [c] the holographic principle can violate the EP dramatically in principle (this is relevant in the AMPS paradox, for example), as can various 3+1 gravity foliations, and probably other answers depending on which model, metric and coordinate systems one is using.

"you would have mass occasionally "escaping" from its "assigned" position and accumulating elsewhere"

Well, that's one problem with semiclassical gravity in the high energy limit -- an extremely compact massive excitation in a matter field may not localize well, so where does its gravity point to and how strong is it, or alternatively and equivalently, where does curvature tell it to "settle", and how does it "skip" from one such place to another (maybe it tunnels between "wells" ER style; or EPR style; maybe that's the same thing; the object doesn't know if most versions of the EP hold up )?

So, one way to look at Hawking radiation is precisely what you are asking: field content somehow escapes from its "assigned" position inside the black hole's horizon and lands on another geodesic which takes it elsewhere (perhaps to infinity). You can treat this as interactions among all the dynamical curved spacetime causing geodesics to materialize right at the in

No:

http://www.mathpages.com/home/...

Redundancy reduces MTBF, because the failure we're talking about is the failure of the system -- of the "assembly", not the failure of any given path.

There are ways to formulate a gravitational potential in 2 dimensions, but I think the idea is that if they existed gravitational waves (actually, all waves) would reverberate when propagating in 2 dimensions, and become distorted. Huygens' principle does not apply in 2 dimensions. See this link: http://www.mathpages.com/home/kmath242/kmath242.htm

I know what you mean.
My first thought reading through the post, was "Oh, maybe they put a sort of treadmill en-route to make the number of steps less than the required amount to reach home" and then I got to the "pull the legs off bit".

I guess I don't have the amorality in me, to make it as a real scientist.

What happened to giants of the community like Feynmann, and the way he treated his ants?

His comment was not nice, but you could have avoided it by not calling yourself a physicist without further explanation ("high-school physicist"?). But to answer your question, diagonalization was a mathematical technique that Georg Cantor used to show that the set of all real (rational + irrational) numbers is larger than the set of all counting numbers, even though they are both infinite. They are different "levels" of infinity. I think you can get it if you read this: http://www.mathpages.com/home/kmath371.htm

From what I read, while Newtonian mechanic can predict light bending for a small weighted photon, it don't predict light bending for zero mass photon.
QUOTE
However, there is a problematical aspect to this "Newtonian" prediction, because it's based on the assumption that particles of light can be accelerated and decelerated just like ordinary matter, and yet if this were the case, it would be difficult to explain why (in non-relativistic absolute space and time) all the light that we observe is traveling at a single characteristic speed. Admittedly if we posit that the rest mass of a particle of light is extremely small, it might be impossible to interact with such a particle without imparting to it a very high velocity, but this doesn't explain why all light seems to have precisely the same velocity, as if this particular speed is somehow a characteristic property of light. As a result of these concerns, especially as the wave conception of light began to supersede the corpuscular theory, the idea that gravity might bend light rays was largely discounted in Newtonian physics. (The same fate befell the idea of black holes, originally proposed by Mitchell based on the Newtonian escape velocity for light. Laplace also mentioned the idea in his Celestial Mechanics, but deleted it in the third edition, possibly because of the conceptual difficulties discussed here.)

light bending in newtonian physic

But then again I could misread the paragraph, the light bending prediction from newtonian physic was based on false premise.

The rest of the text give you 100% reason, on the difficulty of measurement, up to the funny details that Einstein made an error initially and had a bending prediction identical as newtonian physic, and that really the measurement verification were more confirmed in 2004.

I've been looking for a calculation online which goes through in detail the calculation I sketched, so I don't have to rederive it myself. I found this page. The author derives it just with the line elements of the metric, without using the Christoffel symbols as I did.

The formula I gave appears below the text which reads, "Substituting this expression for dr into the above formula gives the proper local acceleration of a stationary observer", except he's using geometric units in which G=c=1. The preceding text is the derivation of that formula.

He also discusses at some length how the gravitational field can be finite at the horizon, yet the proper acceleration of a stationary observer can be infinite, as I stated. A freely falling observer will experience zero proper force at the horizon (or anywhere else where spacetime is not singular). But it requires an infinite boost to go from a freely falling frame to a stationary frame, when you're at the horizon, because the horizon is lightlike, not timelike: only light can be stationary there.

Well, the real advantage to this argument tack is that it is way more useful and productive than any conceivable (Yet Another...) discussion of Relativity could have been on Slashdot.

(If you want to understand Relativity and not just trade misconceptions, consider reading Reflections on Relativity, which can't be beaten for an online book. Bring your brain.)

Go learn about relativity.

I really support the parent on this one. The author shows a deep lack of respect for women, and for geeks. Here's my list, and I left a little room on it for your favorites.

Emmy Noether
Hedy Lamar
Marie Curie
Rosalind Franklin did all the x-ray diffraction heavy lifting for those punks watson and crick
Lise Meitner co-discovered the fission of uranium
Emilie du Chatelet http://www.mathpages.com/home/kmath595/kmath595.ht m
Mileva Maric einstien's ubergeek first wife, to whom some credit a lot of special relativity.
Hypatia mathematician, philosopher, martyr. http://www.agnesscott.edu/lriddle/women/hypatia.ht m

All should be on wikipedia.

You go girls!

On the other hand, query: Do high-energy (ie: high mass) photons have a gravitational effect? Or do the formulae only work given a rest mass?

This longhand method is what I had to learn in the '60s (pre-calculator dark age...) IIRC this was in the 5th or 6th grade, and forgotten before high school. As the linked article points out, the iterative method (Babylonion/Newton) is a much more efficient manual method. IMO this would have been better to teach kids even back then as it would provide a taste of numerical analysis and not some mindless rote mechanical method of arriving at an answer.

Where you go wrong in your post is you miss the key point of relativity. It is true that the Lorentz transform tells you how to go from one time dimension to another. What is not true is your assertion that the initial time dimension is privileged in the transform. The transform is fully symmetric, and in math, that's not just the observation that the two directions merely "look" the same, it is the observation that they are so thoroughly the same that there is no way to tell them apart. (Symmetry arguments are very a powerful tool in the mathematician's toolkit, one of the fundamental ones.)

What this professor is claiming is quite frankly relativity 101. For instance, it is directly addressed in Section 1-1 of Reflections on Relativity; right there in that last diagram is the idea of two distinct time axes with the only distinction between them being which one you happen to be the observer of. We're just barely out of the Preface, and in fact this book happens to develop the idea rather more slowly than some other references!

It takes a real genious to recognize that there is more than one time direction, and that it is "truly true" and not just mathematical sophistry or convenience. But the name of that genious is Albert Einstein, not Alex Mayer.

Is not one of the big problems with "gravitons" that gravity appears to act more or less instantaneously at great distances? And isn't that a little troubling from the "Action at a Distance is Big No-No" point of view?

It turns out that if you do a full analysis, when something like the Sun is moving at a constant rate, then the Earth will orbit in such a way that it almost looks like it is orbiting where the sun instantanously is, but the instantaneous-ness is an illusion. For a full and highly recommended mathematical treatment, see Reflections on Relativity.

It is an illusion because if the Sun were to somehow, say, instantly and magically disappear, the Earth would continue to orbit the Sun until the news got to the Earth at light speed.

For more information, consult the rest of that very good book.

Why is this new? Richard Feynman talked about ants long time ago. Even as far back as when he was a kid, as he discusses in his book Surely You're Joking Mr. Feynman (which has the text of the book, and this section, about 1/3 of the way down). First non-lamer post.

Unfortunately we frequently read in the newspapers about how someone has succeeded in transmitting a wave with a group velocity exceeding c, and we are asked to regard this as an astounding discovery, overturning the principles of relativity, etc. The problem with these stories is that the group velocity corresponds to the actual signal velocity only under conditions of normal dispersion, or, more generally, under conditions when the group velocity is less than the phase velocity. In other circumstances, the group velocity does not necessarily represent the actual propagation speed of any information or energy. For example, in a regime of anomalous dispersion, which means the refractive index decreases with increasing wave number, the preceding formula shows that what we called the group velocity exceeds what we called the phase velocity. In such circumstances the group velocity no longer represents the speed at which information or energy propagates.

btw the site is missing the "slingshot" using other planets. this uses the fact that a hyperbolic orbit leaves the planet at the same speed as it entered, *in the planets frame*. so in the spaceships initial frame the speed when leaving can be larger or smaller. (i restrained from explaining further, isnt complicated but find it difficult to write clearly about in this form)
(http://www.mathpages.com/home/kmath114.htm seems good, but its missing how to aproach the planet to head out a certain direction (if you know a better sites, post!))

I've always thought they use the methods like in the given website for a first estimation, and then use simulation to determine more exactly the orbit. Is this true? they may use Monte Carlo simulation (simulate slightly differently many times) to account for errors that get into thrust direction and strenght, but these are probably small.

i know that explicit analytical formula's for trayectories for more then three planets are not known (or dont exits in form of normal arithmetic). (all of which have non neglible mass)
so everything yielding an analytical result uses aproximations. (are these really good enough?)

also the website doesnt say anything about how the position of the planets and spacecraft are measured, by earth or spacecraft. and how all this data is processed. (this isnt the fault of the website, its just about the mathematics of it)

Well, at the speed of light... yes, things going at the speed of light experience nothing that can be called the progression of time.

But matter can't travel that fast, only things without mass. So, there is the interesting question of what you have that you would call a "bike" or "you".

Physics does not break at the speed of light, but intuitive physics is dead. Relativity is a strain on it at any high speed but just forget lightspeed.

(As I always do when this topic comes up, if you want a crack at understanding this stuff for real, try Reflections on Relativity, free online.)

Not the theories of Relativity, the Principle. There is a difference. Einstein's theories of Relativity solved an increasingly important conflict between physicists beliefs that the Principle of Relativity was true (an intuitive belief) and their inability to put solid math around the way the Universe works.

The first chapter of this work should help. Basically, the principle of relativity is that physics is the same for all inertial reference frames; Einstien put that together with the fact that light appears to travel the same speed for all observers. Galilean relativity doesn't work with that; it has other contradictions inherent in it (it can't answer the Zeno paradox, again, see the linked work), but it takes longer to notice. There are other relativity theories that haven't panned out, either.

Pardon the pedantry, it's intended to be educational.

The experiments you mention do not violate special relativity, nor do they violate Maxwell's laws of electromagnetism (which rely on relativity). The signal velocity does not exceed the speed of light.

An argument against the above is here. It's far better than my argument, but it just seems too crazy for an object to traverse the infinite future and come back to the present by falling into a black hole. There'd be two copies of the object in existance.

Think of it as the space itself around your spaceship being pulled into the Black Hole faster than the speed of light. In order to escape, you have to be able to exceed the local speed of light just to stay still on the surface that is being pulled out from under you, which you're not allowed to do, no matter how you accelerate.

That's a simplified, strictly local picture of what's going on, and it'd be marked as "wrong" on a physics test. But without going into the details, it's probably the best you can do.

All the rest of your objections that you might have at this point, like "If space is being pulled into the hole, where is all that 'space' coming from?" stem from the fact that to really understand black holes, you have to understand relativity. A good online resource that covers what you need to know is here. Here's the first black hole section (although they come up again in later chapters), although you'll find it is radically out of context if read on its own I point to that for completeness, you won't be able to understand it properly without reading huge chunks of the book before it (and that's labelled in the TOC as page 413!). You will also find that it is really freaking hard to understand, though, if you really try, probably not impossible. But I should point out I've already dedicated over three weeks of evenings to that book and only really gotten to chapter three, and I probably need to go back and start from the beginning again to really nail things down in my head before trying to move on. If it's easy to understand, it's wrong. (Unless you're an absolute mathematical genious.)

Think of it as the space itself around your spaceship being pulled into the Black Hole faster than the speed of light. In order to escape, you have to be able to exceed the local speed of light just to stay still on the surface that is being pulled out from under you, which you're not allowed to do, no matter how you accelerate.

That's a simplified, strictly local picture of what's going on, and it'd be marked as "wrong" on a physics test. But without going into the details, it's probably the best you can do.

All the rest of your objections that you might have at this point, like "If space is being pulled into the hole, where is all that 'space' coming from?" stem from the fact that to really understand black holes, you have to understand relativity. A good online resource that covers what you need to know is here. Here's the first black hole section (although they come up again in later chapters), although you'll find it is radically out of context if read on its own I point to that for completeness, you won't be able to understand it properly without reading huge chunks of the book before it (and that's labelled in the TOC as page 413!). You will also find that it is really freaking hard to understand, though, if you really try, probably not impossible. But I should point out I've already dedicated over three weeks of evenings to that book and only really gotten to chapter three, and I probably need to go back and start from the beginning again to really nail things down in my head before trying to move on. If it's easy to understand, it's wrong. (Unless you're an absolute mathematical genious.)

The idea of "escape velocity" only applies to ballistic objects. It an object has more initial kinetic energy than the potential energy needed to escape the gravitational field of the object they are moving away from, it'll eventually be any arbitray distance from the massive object, *without any additional forces being applied*. If you have a source of thrust greater than the gravitational attraction, you can move away from a massive object at any arbitrarily slow speed you like. The reason rockets don't do that is because it's more efficient to burn their fuel quickly. So they convert their potential energy (rocket fuel) to kinetic energy (speed) fairly quickly, and no longer have a source of thrust. Then the kinetic energy is converted back into gravitational potential energy, according to the acceleration due to the gravitational force. The point at which all kinetic energy is exhausted (speed is zero), they start falling back towards the mass.

The above is all basic newtonian mechanics that it's very common for people to get wrong.

Here's a link the explains why the amount of thrust you'd need to hover at the event horizon is infinite:
http://www.mathpages.com/rr/s7-03/7-03.htm

Obviously to accelerate away from the black hole when inside the event horizon would require more thrust than to hover at the event horizon, so it'd also be infinite.

If 10 billion light years worth of protons travelling from a galaxy had a mass, does it's own emitted energy pull it away from the bang? and do other stars' emitted energy push away at the accellerating galaxies?

You're suffering from what I call the Big Number Fallacy; while the number of photons may be large, the amount of matter is so much larger it completely swamps it.

More to the point, conservation of energy and mass<->energy equivalency says that when a star emits a photon, it loses that energy in mass. Obviously, stars are not routinely boiling away due to photon energy losses, or indeed, energy losses at all. Not enough mass-energy is floating around as photons to affect anything.

What was that experiment confirming Earth's 'tearing' effect of gravity the sattelite?

I think you're referring to the "frame dragging" experiment, which is almost completely unrelated, except inasmuch as they are both related to relativity.

I know it's fun to play word games with the shiny Physics and Cosmology terms, but if you really care, you need to learn the real stuff, not merely keywords. I rather liked this; the fact that it's seriously tough shit is a good sign, if you get my drift. If it's easy, you're just playing word games.

At the moment, this is all I can find.

Frame-dragging isn't the same thing as aether drag. Aether drag can account for Michelson-Morley, but fails other observational tests, such as stellar aberration. Frame-dragging doesn't affect M-M very much.

The speed of light is constant in any inertial frame. (It doesn't have to be constant in non-inertial frames; see, for instance the Sagnac effect.)

The best relativity resource I've found is an in-depth online book called "Reflections on Relativity". Be warned, it's the real deal; read it slowly and carefully. Intro calculus should suffice to get you through chapter one (which took me about two weeks spare time in the evening to read and digest), after that it gets tougher, although I'm finding the subjects don't build on each other so much after that so you can skip something you can't follow and keep going. (On the other hand, I only just finished Chapter 2, of 9.)

About the only thing I can tell you, short of linking the book as I did or quoting it more extensively than Slashdot will allow, is that nearly everything physics fanboys think they know is wrong. Don't rely on Star Trek for your physics, get the real deal; it'll only take as much time as a few episodes of Star Trek and you'll feel much better about your expanding horizons :-)

Fahrenheit is measured at standard pressure. It is not subject to the fluctuations you cite.

Now while celsius seems to make more sense based around the freezing and boiling points of water, notice that fahrenheit is as well:

0 degrees is the freezing point of an equal ice/salt mixture
32 degrees is the freezing point of water
212 is the boiling point of water

Now, 96 degrees (for the mean human body temperature) was originally the top bound (base-12 system, making eight segments). The boiling point of 212 was introduced later bumping 96 up to 98.6. However, since we're talking about base-12 here you can see that with a zero of 32 degrees and a high of 212 there are 180 degrees (360/2) of fidelity which are in 15 equal segments of 12 (12*15+32 = 212). So it really has the same characteristics of celsius, but it is using a different base.

Now one thing that is interesting is that 0 - 100 degrees fahrenheit is roughly the extents of normal temperatures in which we live and are able to function normally within. These map approximately to -18 to 38 degrees celsius. Now which system is more logical? The boiling point of water is not as important to me as my body temperature is, since our bodies cannot tolerate temperatures much beyond our internal temperature.

Also the celsius system has roughly half the fidelity of the fahrenheit system (56%). That means that when reporting temperatures I have a better idea what temp it is since there is less room for error. This could be avoided by providing temperatures with decimal points, but this isn't neccessary with the Fahrenheit system.

I don't really have much of a beef with celsius other than the lower useful bandwidth of whole numbers in the (meaningful) range of human habitability. That and base-12 is very handy. Speaking of twelve, check out this interesting link that shows the sequence of 12 repeating numbers for the iteration of -1/(sin(x)cos(x).

>>> I don't see how this is 100% simultaneous. Let's say one pair breaks down.

It is simultaneous simply because as long as you don't perturb it, a pair of entangled particles constitute a single quantum system, a single "particle" if you want. As soon as you touch one of them, the pair is broken and now suddenly you can imagine having two separate particles popping up in space at certain distance between them.

>>> We call that event A in the observer X's IRF. Then another pair breaks down - event B. Let's use A as our reference point. An observer Y in a different IRF will say that event B did not happen at the same time oberver X said it did. There will be a difference in the time elapsed.

No. The state vector collapse is not an observable event in itself. Only the measurement is an observable event. All you can do (as observer X) is to measure both particles and hope/presume that the collapse already hapened - but you have no 100% guarantee that of course, unless you setup the experiment in certain ways.

>>> Particle A and B are entangled. Let them be 1 lightyear apart. When the state changes, observers of each particle send a signal to each other. The observer at A gets the signal about B one year later. The observer at B gets the signal about A one year later. I don't see how anything really changes by adding in a bunch of IRFs.

Of course, the outcome of the measurement depends on the state of the entangled pair and the type of the measurement. But there will be always a certain non-linear correlation between the two measurements done by observers at A and B. This means that the measurements were already correlated before the observers could possibly communicate. How would you explain this correlation? The actual measurement happened one year in each IRF before getting the data from the other observer.

Here is a more detalied explanation.

Actually the moon is slowly expanding its orbit. It is moving farther and farther from the earth and one day the earth will no longer have a moon. Check it out here. A brief explanation on our falling moon!

But by the time we don't have a moon, we'll have a giant space station up there that will take its place. And then everyone will be quoting "That's no moon, that's a space station."

Quoting mathpages: The fifth element, i.e., the quintessence, according to Plato was identified with the dodecahedron. He says simply "God used this solid for the whole universe, embriodering figures on it".

While an inventive thought experiment, you neglected to mention that the reflecting beam of light possesses a wavelength. When the separation between the two mirrors drops below this wavelength, your assumption about beam reflection from a surface breaks down.

Why not have the distance between the mirrors as (1/(n^2) + 2w) where w is the wavelength of the beam of light?

Right on the first point, but the last point is not at all clear to me. Special relativity shows that simple-minded Galilean transforms cause electrodynamics to break, but that a Lorentz transform preserves these relationships. Could you elaborate on why you think special relativity is involved here?

See here. Zeno's achilles argument isn't the only part of Zeno's paradox.

How did this get a 5?

That's what I want to know ;)

Calculus does in fact completely solve the problem of summing the infinite number of ever-smaller timeslices that Achilles takes to pass the tortoise.

No it doesn't. The mirror thought experiment is an extension of Zeno's achilles argument, and calculas obviously doesn't explain that! How can the direction of the light beam be determined by the last mirror in the sequence if there is no last mirror?

And you have no understanding of relativity if you think that Zeno's Paradox implies relativity. Special relativity is nothing more and nothing less than exploring the complete implications (ignoring acceleration and gravity) of the fact that the speed of light is a constant regardless of your velocity. Zeno didn't know the speed of light was a constant, and without that piece of information there are no grounds to postulate special relativity.

I think we're talking about two different things there. Zeno had four arguments, of which the Achilles argument is only one of. Essencially, Zeno was arguing that reality must be discrete, and yet it must be continuous (Achilles argues for discreteness). The Stadium arguement involves too bodies passing each other at speed (already this starts to sound familiar, no?). See here for more information on that (it does a better, and more accurate job at explaning that then I ever could!)

An "infinite number" of mirrors can't exist, so not being able to determine what they would do is not a failing of physics; it's a failing of your thought experiment.

Um... Well, firstly, you don't really understand the concept of a thought experiment. For instance, it's impossible to ride a light beam and for us to travel at the speed of light (well, so far as we know), but by reasoning out what would happen under such circumstances, Einstein formulated the Special Theory of Relativity. The 'infinite mirrors' extention of Zeno's Achilles argument is proof by contradiction.

There is a tradition of using calculas to solve this problem, and my explanation of the method behind the Achilles example was, well, iffy :)

That said, take the mirrors example (or a similar set up like Zeno's maze). That's essencially the same problem, but set up so that the direction of the moving object (ie. the light beam) depends on the last mirror. Calculas doesn't explain this.

Having shown that reality is discrete (though I find it dubious that this proves all reality is discrete), he then goes on with his 'Arrow' thought experiment to say that if there's no difference between a moving object and still object at a discrete interval in time, what makes the moving one move and the still one stay still? The paper seems to argue that this is because the discrete intervals are between certain times- that the invervals aren't infinitely small.

Again though, that's fairly obvious, and it all seems a bit iffy to me. I'm not even sure I've argued it all correctly without mistakes, though at least no-one has yet called me a moron for disparging the great name of Zeno. Check out here for a more in depth explanation.

You are regurgitating dogma:

"The first two arguments are usually interpreted as critiques of the idea of continuous motion in infinitely divisible space and time. They differ only in that the first is expressed in terms of absolute motion, whereas the second shows that the same argument applies to relative motion. Regarding these first two arguments, there's a tradition among some high school calculus teachers to present them as "Zeno's Paradox", and then "resolve the paradox" by pointing out that an infinite series can have a finite sum. This may be a useful pedagogical device for beginning calculus students, but it misses an interesting and important philosophical point implied by Zeno's arguments."

Read the rest at:
Zeno and the Paradox of Motion

At this point, the distance between the mirrors is zero, which means the light must be standing still. If light cannot stand still, this must mean the distance between the mirrors is not zero, and hence does the example not apply (because it makes the assumption that mirrors can have zero distance).

Infinity isn't a number, and something infintesimately small is not zero. However far you go with the mirrors, the light is still bouncing. Assuming a light beam has no width to speak of, there is no point at which the light beam is not bouncing. And yet the distance the light beam travels is finite, even if the mirrors are infinite! Therefore the light will emerge in a finite space of time, and at every point will be bouncing and zigzagging along. You're right that it's impossible, of course, but that's why it's a paradox!

What do you mean "there cannot"? If time is discrete, Zenon's paradox does not apply, because it talks about timeslices smaller than what the actual ones would be.

Sorry; I meant you cannot have a discrete slice of time if time is continuous. However, I've said elsewhere that the paper in the article seemed, well, dubious. I'm not saying I agree with the paper, or that the paper is of any import as the article seems to suggest. Just that the original poster misunderstood what the paper was proposing.

Zeno's paradox does not claim Achilles can never catch up with the tortoise; making such a claim would require talking about infinite time -- Zeno's paradox does only talk about the time before Achilles catches up with the tortoise, hence the correct conclusion is "Achilles cannot possibly catch up with the turtle in the timeframe before he catches up with the turtle".

Well, the quote for Zeno's Achilles paradox I have is: "The slower will never be overtaken by the quicker, for that which is pursuing must first reach the point from which that which is fleeing started, so that the slower must always be some distance ahead."

So, the total correction is 44 us per day--the clocks must be slower by about half a part per billion. That's one second difference every sixty years.

He has no taken into account Benford's Law in determining transaction amounts, and is falsely assuming a flat distribution.

All that needs to happen now is to fit just one more VW Bug into the Library of Congress...and the level of concurent coincidences will reach Kevin Bacon-esque levels.

The Dullness of 1729

http://www.mathpages.com/home/kmath312.htm

...purports to prove, by logical means, that change (motion) and plurality are impossible. Contrary to common belief, the paradox is not resolved by the concept of limit.

Read More...

• Gain insight into the minds of the ancients (Plato would not let anyone into his school who hadn't mastered the geometry of the Elements),
• Improve your reasoning skills (Abraham Lincoln read Euclid when he decided to supplement his education later in life), and
• Be exposed to some of the most beautiful things that mathematics - or any academic pursuit - has to offer ("Euclid alone has looked on beauty bare." --Edna St. Vincent Millay)

The electrons move (as a result of an applied voltage) at what is known as the drift velocity. A example in copper is also available.

Current doesn't stop (your "current move, current not move" parenthetical). Current is not a thing, but is a description of a situation: moving charge is a current. An Ampere is defined as one Coulomb of charge passing a reference plane in one second.

How fast a signal propagates down a wire is its group velocity.

The "friction" mentioned by the original poster I interpret to be a flawed understanding of how resisivity works. Electrical signals travelling through resistive materials are attenuated, not slowed down, due to the resistance. Changes in velocity are due to changes in the dielectic constant.

"It is impossible to divide a cube into two cubes, a fourth power into two fourth powers, and in general any power except the square into two powers with the same exponents,...I have discovered a truly wonderful proof of this, but the margin is too narrow to hold it."

He was a really clever guy, but that was really far out... =)
The difference of Fermat and this "inventor"-guy of course beeing that Fermat is/was a very merited scientist, and his credibility made it possible for him to sneak this one past.
Follow this link to check it out in more depth.
I found the Fermat reference really fun, but perhaps it's just us (ex) math types...

From the ranging experiments, scientists know that the average distance between the centers of the Earth and the Moon is 385,000 kilometers with an accuracy of better than one part in 10 billion. Laser ranging has also made possible a wealth of new information about the dynamics and structure of the Moon. Among many new observations, scientists now believe that the Moon may harbor a liquid core. The theory has been proposed from data on the Moon's rate of rotation and very slight bobbing motions caused by gravitational forces from the Sun and Earth.

Ranging has also determined that the length of an Earth day has distinct small-scale variations of about one thousandth of a second over the course of a year, caused by the atmosphere, tides, and Earth's core. In addition, precise positions of the laser ranging observatories on Earth are slowly drifting as the crustal plates on Earth drift. The observatory on Maui is seen to be drifting away from the observatory in Texas.

A lifespan specified only by accidents should have a Poisson probability distribution. The probability of your demise as a function of time is a decreasing exponential with appropriate scaling. While there will be an "expected" lifespan, this will be very different from today's more bell-curve-like distribution, with most people dying much younger and a lucky few living a very long time, and very few people dying exactly at 600 years old.