md65536

Yes (depending on the distance).

If a particle is oscillating side-to-side it wouldn't be moving up and down the gravity well. Yet, particles seem to accelerate at the same rate regardless of how they're oriented. How would you account for that?

TAR, I had figured that you probably never would have bother along the lines of reasoning you were on, if you already understood existing ways of resolving the paradox. But you said you were reading up on the paradox, so there's always the opportunity to incorporate existing solutions. Anyway I think that it's more useful to research what's already been done (whether with the paradox or linguistics) than to start from scratch, and better to be meticulous and systematic than being vague and ambiguous... I mean like focussing on the rules of linguistics rather than divining meaning from heaps of words. For example, is it possible to implement a Turing machine using the rules of linguistics? (I don't know anything about linguistics but I assume it's not.) If you could, then you could use linguistics to solve a HUGE set of classes of problems that can be solved using Turing machines. Or similarly, if you define the words "not" and "and", and have enough rules to organize arbitrarily long combinations of the words, you might be able to describe the output of any binary logic circuit using only linguistic rules. Then anything computable by a computer can be described in words using only those rules and the few defined words. This is the opposite of what you're aiming at I think. This might prove that a linguistic resolution is possible, but it might be super long and convoluted and not easily understood, only "technically" meaningful. But, if you build meticulously on what is known to be correct, you can end up with something correct. Like I said before I think, if you keep looking broader and adding more vaguely defined ideas, it just gets harder to find errors in reasoning, now not only in the "paradox" but also in everything you've described about it. Anyway, if you could "compute" a linguistic solution, that might not be enough anyway. If the design of the sentences comes from the math etc, I think that's still math, not just linguistics. (But then again!, could you build a linguistic machine to generate and evaluate all possible linguistic machines, or something like that? If so then you might produce computable results of math without ever needing the math.) Now I've going off on a ramble... TL;DR I think you'll find the answers in the details and in what's already been figured out by others, not in the unlimitedly broad meaning of it all.

Note that you gave an incorrect definition of infinity, and then showed that it ("your first claim" which you made up) was incorrect. Your description of infinity followed the rules of linguistics. Mathematically it was wrong. If you use only linguistics, and are free to choose whatever meanings of words you want, you can write out Zeno's paradox or others in ways that can be solved depending on the meanings you choose, or that can't be solved by choosing other meanings that don't follow the math/physics/etc.

I think this answers the question as I interpret it. If you double the total mass, the relative acceleration will double, regardless of how the additional mass is distributed between the two bodies. Basically, a 2nd Earth mass would fall toward a "fixed location Earth" at the same rate as any other object would, but the Earth isn't fixed and would also fall toward the other equal mass, doubling the total acceleration. A lot of the answers in this thread are assuming an Earth-sized object, even though only mass is mentioned, and I don't think the question intended to allow for the location of the masses to be moved to account for different sizes.

I guess that's one of the benefits/drawbacks of linguistics. There's so much room for expanding meaning and interpretation. Four people have given answers and no one agrees with anyone else. The meaning of the problem isn't even settled on. Physics gives a definite prediction and the corresponding math is explanatory and unambiguous (not that Zeno's model perfectly represents reality, or that the problem couldn't be modified to be modelled differently, eg with quantum time or something, but that's beside the point). I disagree that the gear analogy is related to the paradox. I don't see it. However since there's no physical paradox, what you said---(paraphrased) that the paradox is only in the understanding---must be true, so I guess if you resolve the paradox in a different way with a different meaning, even if it makes no sense to me, even if I think it's simply brushing aside the paradox, I suppose that still resolves it for you. If the presence of the paradox is subjective, depending on the understanding of it, then the presence of the "lingual solution" is too. Still I don't see it that way. The mathematical description of the problem, whether expressed in numbers or symbols or words, is unambiguous and so is the resolution.

A linguistic description of a mathematical result is still the result of math. The results of calculation can't always be obtained through linguistics alone. But this isn't about math vs linguistics. I think everyone involved has admitted that the paradox resolution can be described in words. The problem I think is more about ignoring analysis of the details of the paradox (avoiding math is just one way to do that). Instead of pinpointing the problem (in words or math), TAR is doing the opposite, considering Kant and the nature of thought and looking at the big picture, settling on a "solid basis, upon which to build a lingual understanding of a situation"... and it doesn't work... so maybe he can go broader... smash the impossible concept into atoms and spread them around and look for the problem there. I think that's what TAR's lingual solution is: to restate a paradox with less meaning and more convolution (it all depends on how the brain works, etc), until it is vague enough that there is no longer anything left that is both meaningful and puzzling.

Where is the math inappropriately applied? What situation is locked in? Are we talking about an accurate mathematical representation of the paradox, or something else? Math describes the situation and resolves the paradox, I can't see anything inappropriate about that. The math certainly doesn't "lock in" something incorrect.

Yes, the horizon is moving at the speed of light. Locally this is c. From far away, it has a coordinate speed of 0 (it is "frozen"). I think that it means that whatever speed you're going when you fall in (eg. if you start from infinity and are close to c relative to the singularity, or if you're trying to escape and fall in slower), the horizon will pass by you at c. Yes I think you're right that passing the horizon would be nothing fancy. It is only the non-inertial observer, who is trying to hover near the horizon, that sees things as weird. I think it measures extreme length contraction. Then "a lot of light" is contracted near the horizon (whereas the in-falling inertial observer doesn't experience this, and sees no special burst of light as the horizon passes it). Also the rest of the universe should appear faster, brighter, and bluer, but I think only to the hovering observer? It sounds right that the horizon must be a definite place, but it *is* a light-like surface. I think to resolve this it must be that the horizon is stationary at a definite location because it is "frozen" due to infinite time dilation, and this applies to any stationary outside observer. But locally (ie. when falling in) it is not frozen, and light-like things move at a coordinate speed of c. ("Frozen" here means only in the sense of it being a "frozen star", with a coordinate speed of 0 due to infinite time dilation.)

What would happen if you had say a very dense planet, with a very dense thin ring around it (like neutron star density or something). Then add an accretion disk of "normal" matter. I imagine the matter would form a thick even shell around the dense thin ring, if gravitational pull was effectively always toward the nearest point on the ring. If you knocked out the center planet with a collision of another massive object, while the ring-world is solidifying but before it hits the planet, couldn't you end up with such a toroid? Edit: Or would the formation of the core ring be impossible if it was that significantly massive?

But in this case it literally does add up. It is reasonable, and resolved, mathematically. You have not even demonstrated that your "lingual solution" can convey the meaning (or an understanding) of the problem, let alone resolve it. Those aren't the words that are the problem. You probably can describe the paradox without math, and resolve it in words. Every time the warrior catches up to the tortoise, the latter has moved on. Or to make it more purely linguistic: Every time the warrior's position is made to mean the same as the tortoise's position, the tortoise's position has been made to mean something different. Or something like that. The paradox is that the warrior's position will never mean the same thing as the tortoise's, even though we know they must mean the same thing in the end. The solution is that "never" in this context means not after any (finite) number of iterations. It does not mean "not after any (finite) amount of time". A linguistic confusion of the implied meaning of the word "never" causes the paradox, and sorting out the meaning can resolve it. But is that just a description of the math, in words? Is there a linguistic understanding of numbers of iterations and the meaning of "finite" etc without the math? Does the solution make sense without considering the math? The meaning of "An infinite number of iterations as described can be completed in a finite time" is proven and understood in the math; are the words alone satisfactory? I feel that the more words you've used, the more you've wandered away from the meaning and understanding of the problem. That's not a math vs words thing... I think it's analogous to writing pages of numbers and connecting them with relations, saying "The mathematical answer is in here somewhere, I've just got to write the numbers until the solution presents itself." You can question the concept of meaning, etc, but I don't see you getting any closer to describing anything related to Zeno's paradox. So I think that your attempt at reasoning through the paradox, with a goal of shunning the math, has not been successful.

I think virtual particles are not constrained by the speed of light (or they can be interpreted as being able to exceed c) but they don't define the speed of light. No a photon doesn't experience time. It doesn't have a valid frame of reference and you can't ride on a photon. You can't describe sensible observations at that speed, it is an analogy at best. By analogy, distance in the direction of travel is contracted to 0 and can be traversed in an instant. A neutrino experiences time at the usual rate (1 second per second). Ie. it ages normally, or "its clock" ticks normally. Generally other clocks would be slowed to a near stop.

Neutrinos are somewhere around there (very wide ballpark figure), and they're objects. So I think this situation happens all the time.

The horizon is a light-like surface, and it doesn't have a rest frame, so it doesn't make sense to compare an object's speed relative to it. Similarly, it doesn't make sense to say that we're all traveling at c relative to a beam of light. It makes no sense conceptually or mathematically, and trying to imagine it will confuse. Just as a pulse of light passes an object at c, the horizon passes an in-falling object at c, but in neither case is the object traveling at c relative to anything. If it doesn't make sense, consider that how things look near the horizon is very different depending on if you're inertial or trying to escape. At the horizon, escaping is equivalent to trying to outrun light (conceptually and mathematically impossible), and approaching that approaches infinite length contraction, severely distorting space relative to the inertial in-falling observer.

There's no transfer of energy. Acceleration toward a mass would be converting gravitational potential energy into kinetic energy, and is conserved. The mass of a BH is "mass-energy", and it is conserved. Gravitational energy can be transferred in the form of gravitational waves but that's not applicable to acceleration of masses directly toward each other. If I understand it correctly, you can think of the two masses in terms of two independent static gravitational fields, and (I think) they don't change if the masses are accelerating at a constant rate. However if you change their direction (like if the BH is orbiting the other mass or some other mass) then you have changes to the field that need to be propagated in the form of gravitational waves (emitted energy in all directions which is mostly lost, rather than transferred to the other mass). BUT I think I'm missing something because the acceleration toward a BH depends on the distance to it, so how is that change in acceleration handled?

I just meant it as advice. Ideas generally aren't right or wrong. Even impossible ideas can lead to useful creative solutions. The advice is to estimate the magnitude of the problem before figuring out the details of a possible solution. For example, using values from google searches I estimate the cost of putting 1 mm depth of ocean surface into low Earth orbit with current methods (rockets) would cost about 300 times the value of all resources on Earth. So, organizing who would do it is not the problem that needs to be solved. Another example: You mention Earth's fleet increasing in time. Using the 6 micron current estimate, how many microns of depth would you estimate would be significant relative to other contributions to sea level rise, and how many times or orders of magnitude bigger would the fleet have to be to contribute that depth?

Yes, I meant how would you propose that it might be done. A useful skill, probably too rare at any age let alone 15, is the ability to estimate things. Figure out how much sea level rise you're talking about here, and google "ocean area", and multiply to figure out the volume of water. How many kilograms is that? Google/research how much it costs to to lift a kg to orbit (let alone lunar orbit). Then you'll see how much this idea costs with current technology. Or computing it in terms of energy needed is also a good measure. Thinking about ideas like this and calculating them is a good thing, because if current technology is not up to it, it can inspire new ideas for doing it better. But in this case you'd have to find a solution that is many many many magnitudes better than what exists for getting stuff out of Earth's gravitational well. It costs a LOT to send a person into space, never mind the entire fleet of boats on Earth (or 6 microns worth of sea level), never mind glaciers and icecaps. But other problems, like "What part of Earth would I flood?" might lead to possible solutions, and it might be a good idea too. There might be somewhere where a dam and diverted river could flood a desert enough to save some cities??? Or at least, if we can't think of any way to stop icecap melt and global flooding, we might as well at least think of ways to mitigate the effects. Better to figure out the magnitude of the problem before forming a committee to regulate a solution.

That could work. Where would you put the water? We could build a few large dams, maybe flood a continent or two. Wait a minute, what is the problem you would have solved? The problem of sea level rise? Sea level rise is only a problem because if you raise the sea, then the sea goes onto what is currently land, and then it's under water. So you want to solve that problem by manually putting the water onto land? One way to store a significant amount of sea on land is in a frozen state, in very thick glaciers and ice sheets. Do you have a viable replacement? How much energy and cost would you estimate it would take to take some number of centimeters of global ocean surface out of the sea? How would you do it?

Perhaps a different perspective might help. All of the properties of a black hole (which by the no-hair theorem are only mass, charge, angular momentum, linear momentum, and location) are properties of (or available at?) the surface of the black hole. So while information about anything inside the surface is unavailable, these properties do not disappear. I'm not sure how this relates to the other answers (are the properties available only because they're "frozen" at the horizon?). But if you think of a black hole in terms of its surface, it is a thing with measurable mass etc... only the inside is unknowable to us.

Why do you say "instead"? Do you not accept that melting icecaps contribute to sea level rise? There are other factors too. Increase in water temperature expands it and I think is a major factor in sea level rise. Land masses shifting contributes, either up or down. Precipitation too... a couple years ago there was a measurable decrease in sea level while Australia was being flooded, and a lot of the water was temporarily on land [http://www.theguardian.com/environment/2013/aug/23/australian-floods-global-sea-level]. All of these different factors can be measured or estimated, and they all contribute a certain amount. It's not that there is just one reason for sea level rise. As far as I know, melting of ice on land is the biggest contributor and its effect on sea level is well understood.

Yes. This estimate http://what-if.xkcd.com/33/ figures that all the ships in the sea contribute about 0.006 mm of sea level.

Light follows a null geodesic (but not all geodesics are null, eg. a planet's orbit). Is what you wrote equivalent to saying that if you transported a short enough ("local") straight ruler along a null geodesic, the ruler never bends? The ends are always tangential to the null geodesic. Is that a reasonably good way to say it? That would mean there is no local acceleration of light anywhere along the geodesic. Light of course remains at a fixed local speed, but also does not change direction locally. However, a null geodesic can be curved when measured remotely (eg. gravitational lensing). Is it then correct to say that the only acceleration of light is coordinate acceleration (which is just called "acceleration" anyway?), but not proper acceleration? Edit: After thinking about it, I don't think the idea of a locally contained ruler makes sense. There must be a different way to say it... Light always travels in a straight line in flat spacetime, and spacetime is locally flat, and light follows that local flatness. A non-null geodesic is generally not locally flat?

Wrong how? The symbol => could mean "divide the result by 7 and express in base-7".

Probably, but I don't know or plan to figure out the math to show that. Still, it *is* a real effect even if relatively insignificant. What is a realistic twin paradox setup anyway? Everything depends on the details. Since I have no idea how big the effect is I wouldn't say one way or the other without working through an example. I'd rather avoid the issue by letting the ship length be negligible. Contrary to post 144, an observer and its clock are usually considered to have the same position, so that none of this matters. Not because the effect is small, but because the set up is described without the effect mattering. If someone else is interested in the effect, they could set it up differently, and calculate it.

This is generally true I think but it is avoidable. See: http://www.mathpages.com/home/kmath422/kmath422.htm This is mostly about Born rigid motion, where distance distortions are avoided, which partly supports your argument: "In other words, if we contrive to hold the spatial relations fixed during an acceleration, a phase shift is introduced between different parts of the object, just as, if the phase is held constant, there is spatial stretching." The second part implies there are cases where it it possible to have no phase shift. This all is only a problem if the accelerating twin has a non-negligible length, and in this case I think you need to specify the details of how it accelerates, or just how the different parts of the ship are coordinated. You can specify it some way so that there is a phase shift, or another way so that there is not. (Unless you specifically set it up to avoid either spatial or temporal distortions, you'll get both, so I think you're generally right.)