md65536

Post #19 does that. Another thing to consider: The total mass stays constant during the experiment, while r decreases. So the final acceleration relative to initial acceleration should be greater in the smaller mass with smaller r setup AB. If you begin with M_total/r^2 being just slightly greater in the CD setup, it should start with a higher acceleration and be overtaken by AB. My intuition says units don't matter??? G will be different with different units and so will the time the experiment takes, but not the outcome? So for total mass MAB, if MAB / rAB^2 is greater than or equal to MCD / rCD^2, the smaller mass will easily touch first. If you want to ensure that the larger masses touch first, an easy way (with overkill) is to make it so the end (maximum) acceleration is greater in that case than in the AB case. Ie. let r_final equal the distance between the centers of the two masses when touching, rather than the initial distance, and make the CD side have a greater M/r_final^2.

Yes, I've edited my wording to be closer to this. The mass M_2 of the object being acted on by a gravitational force F_12 cancels out in the acceleration equation above, since F12 = M2 * a12

Still, mathematically it is possible to deal with it---just make each particle half the size of the last and you can fit an infinite number of them in twice the space of the largest particle---it doesn't matter anyway because the masses were defined as point masses, and maths can handle much that is not physically possible. I'm stumped again here, but there's a quote from wikipedia that I think might be required to fully think through this: "As with most mathematical paradoxes, they generally reveal surprising and counter-intuitive mathematical results, rather than actual logical contradictions within modern axiomatic set theory." It seems a paradox is certain, but does that require a logical contradiction? If not, does the paradox imply a violation of physical laws in the "infinite collision" strawman? And even if so, is there even a paradox at all in Norton's Dome? And does that also imply a contradiction or violation of a law? Maybe, maybe the evidence was all presented in this thread and I don't understand it, but I think this is all way too much to just hand-wave through.

It should be only the total mass that matters. The force is proportional to the product of the two masses, but the acceleration of a mass acted on by a given force, is inversely proportional to its own mass. So only the mass of the gravitational object affects how fast another mass falls toward it. One mass falls toward the other, and the other falls toward the first. The closing acceleration will be the sum of the two's accelerations. It should be the same, with the same r (distance between the centers of the two masses) and same total mass, regardless of how the mass is distributed between the two. You should be able to manipulate the initial acceleration to favor either AB or CD, by balancing only two variables: r and total mass. I'm not sure if it's possible to have one begin accelerating faster, and have the other catch up and touch first, by also manipulating the gap.

I've been thinking about the "infinite series of perfectly elastic collisions" strawman that was introduced... According to the description of the system, at any time there is one and only one mass moving. It's either the first mass or the most recently hit mass. After a time of 2d/v, all of the masses have collided. At that time there is one mass moving, and nothing else to block its path, so that mass will exit the system. I think you would have a tough time proving that all of the masses have stopped moving. You would need to conclude that, before being able to justify any of the claims of physical laws being broken. At best you might prove that there is a contradiction. I don't think you could conclude without contradiction that all of the masses have stopped. Then technically, by the principle of explosion, you can derive that a physical law has been violated, but you could also derive that no physical law has been violated. Anyway, that aside, without finding a contradiction yet, I suspect that it might be possible to reasonably speak of the last member of an infinite set without any "the math can't handle it" nonsense. An infinite set is no problem, and ordering it is no problem (this set is well-ordered). There's a well-defined first member, so why not also a last? If we label all of the masses by the order in which they collided, it cannot be that any finitely numbered mass exits the system. But that's no problem because there's an infinite number of masses. It's given that the infinite number of collisions happen in a finite amount of time, the math handles that fine. All of any finite number of masses has a mass that comes after it. Not until counting an infinite number of masses, can you have a mass that has no mass after it. That would be the last mass. I can't derive a contradiction given the description of the system, can anyone else? The masses could also be labelled by their position on the x-axis. The last mass would have to have a position of 2d. There is no nearest number to that value, and any n'th mass, where n is a natural number, would have a definite location that is not equal to 2d. But that's no problem because there are an infinite number of masses. The location of the last mass would have to equal the sum of all of the infinite number of spaces between all preceding masses, and the sum n from 0 to infinity of d/(2^n) is 2d (the math can handle it). ---Edit: Then again, that makes no sense. If you count backward from a last mass, you'd need for there to be an infinite number of them before you reached a definite location < 2d, and none of the masses at 2d are in the defined set of masses. So a contradiction seems inevitable. I guess math can't handle physics after all.--- When someone says "the math can't handle it", without pointing out a flaw in the maths, other than that something doesn't make sense to them, it's usually the case that the flaw is in understanding.

This probably needs its own thread, but the conclusion is related, so... I disagree. If you have an infinite number of point masses arranged as such, then you have infinite mass with infinite density in the neighborhood of 2d. To be realistic, you could not assume Euclidean geometry or universal time. Realistically, you'd have a black hole which would absorb the momentum and move infinitesimally slowly toward 2d. From an outside perspective, the process would not have to complete in a local time of 2d/v, it could go on forever (local time). There's no violation of Newton's 1st law. But suppose we assume Newtonian physics still applies, admitting that the situation is not physically real. The neighborhood of infinite mass could still move infinitesimally slowly toward 2d, and conservation of momentum is not violated. Or a single mass could emerge from the system, it doesn't matter which one, and conservation of momentum is not violated. But suppose we assume that the impossible description of the system is still correct and neither of those cases happens. Then yes, you can have any law you want violated by specifying the system so that it is violated with some impossibility. If no ball exited, then the reverse direction needn't violate any laws either because the infinite mass can have finite momentum with infinitesimal velocity. The difference between that and Norton's Dome, is that the dome is a conceivably physically possible reality, and the dome isn't specified in a way that a law is required to be broken (though I guess that's not agreed upon).

Alright, let's drag this on. You disagreed with the statement "If only half of the law is violated, the law is not violated." You mean that if "A mass at rest must remain at rest" (only this half) is violated, then even if "the mass is acted on by an external force" (this half is not violated), then "A mass at rest must remain at rest unless the mass is acted on by an external force" is violated. If this is not what you mean, then what do you mean? The force that is present at r>0. But I would not use the word "causes". The cause is intentionally unspecified in the thought experiment. Yes. Your link in post #1 says so. http://www.pitt.edu/~jdnorton/Goodies/Dome/ Which of those is Newton's first law?

Okay excellent, I feel like we're making progress instead of just repeating mutually consistent statements. I feel like we're near a pinpoint of where we disagree. I disagree with you because "An object at rest must remain at rest" is not a law. Even if the situation requires that the statement must be true, there is no law that demands it. The applicable law only requires that "An object at rest must remain at rest unless no external force is acting on it." In the thought experiment the half-law remains true until some unexplained event (that is not a new force) makes it not true. In reality such an event may be possible (quantum randomness or whatever) or not (perhaps there is always a force that knocks things off balance). But it doesn't matter because even if the thought experiment is describing an impossible event, the event doesn't violate the full first law. There is a force everywhere on the dome that pushes the mass away from r=0, except at r=0. That is the force that satisfies the first law. Only while the mass is at r=0, is there no force, and Newton's law says it must remain at rest. And it does! For the entire time that the mass is balanced, Newton's law is followed. If the mass is not at r=0, there is a force, and Newton's law is satisfied (even if a common sense understanding of it is not). The law doesn't say anything about "initiating" motion. The law does not say that the external force that allows that the object does not remain at rest is one that does any initiating. The law allows the case where the "not being at rest" comes at the same time (or perhaps even before) the presence of the external force. A lack of initiation does not break the law. It violates common sense. It would be a mistake to have the law conform to common sense, and in this case I don't think your interpretation of the law is consistent with reality. I think that your interpretation of the law is too easily violated in nature at the quantum scale, and so it could not remain a law. Newton's law is not equivalent to "An external force causes an object to no longer remain at rest," any more than "An object not remaining at rest causes there to be an external force." Newton's law is time-reversible, and this is not. (probably wrong)

The closing distance is the same. Assuming they all start at rest, the rate of acceleration determines who touches first. The closing acceleration is proportional to total mass M1+M2 and inversely proportional to 1/r^2, where r is the distance between the centers of the masses. The question suggests that the mass is the more important factor, but there should be values where the r is more important.

Closer to what? If the A mass is negligible, A must traverse the entire distance to B. Yet, C and D will touch halfway between their original distance. Masses C and D are closer to their touch point. You could definitely rig the system with wording and unsaid assumptions. The masses might not have uniform density, mass A might be small only in size but just as massive, etc. But I'd assume the most straightforward case.

Because it fits the definition of unstable equilibrium. "If the system is displaced an arbitrarily small distance from the equilibrium state, the forces of the system cause it to move even farther away." http://en.wikipedia.org/wiki/Mechanical_equilibrium Remaining in equilibrium for an arbitrarily long time is consistent with the first law.

Do you agree that this is a fair restatement of yours, that says the same thing but is true in every interpretation?: "The mass is at r=0 and there is no force on it. Newton's law says it must therefore remain at rest (unless acted on by a force)." 1. This is true. 2. The "(unless acted on by a force)" can be ignored because it is already given. 3. The ignored part can only be ignored as long as there is no force. 4. Half of the law doesn't remain in effect longer than the full law. It isn't true that "the mass must remain at rest" if there is a force acting on the mass. 5. If the mass is not at r=0, there is a net force on the mass. 6. Since the mass is initially at rest and must remain at rest, if the mass is not at r=0 then there is a violation of the first half of the law. 7. The first half of the law (the mass must remain at rest) is only violated when the mass is not at r=0, in which case there is a net force and the full statement of law is technically satisfied. 8. If only half of the law is violated, the law is not violated. Do you disagree with any of these statements? I believe there is a trick in your statement regarding point 4. You're ignoring half of Newton's first law because it's redundant, and expecting that the first half must "remain" true even when the ignored second half no longer requires it.

Trying again... Yes, this is false; Newton's first law does not say all of that. Yes, if the first part ("The mass is at r=0 and there is no force on it") is true, Newton's law says that the second part ("so it must remain at rest") is true. If the second part is true, then the first part remains true. The first part is initially true. Yes, there is a contradiction if only one part is true and the other is false. What Newton's law does NOT say is that both parts cannot be false together. And certainly Newton's law does not say that it must remain at rest indefinitely or for some amount of time EVEN in the case that a force is introduced (as is the case if the mass is not at r=0). If the mass is not at r=0, both parts are false. I don't accept that previous applications of the law still apply even if the conditions which would violate the law no longer hold. ---- Edit: Here's yet another way to explain how I'm looking at this. Assume the first part is true: "The mass is at r=0 and there is no force on it" What Newton's law says is "So it must remain at rest unless there is a force acting on it." Since we've already established that there is no force, the "unless" part can be removed, and your statement is true. However, if you change the situation so that the mass is no longer at r=0, the condition that no force acts on it is no longer true. You can not still say that Newton's law still says that "it must remain at rest", because what Newton's law actually says is "it must remain at rest unless a force is acting upon it". The issue is... for how long, or until when, does "remains" apply? By a strict reading of the law, it applies "unless" a force acts upon it. The "remains" requirement disappears if the "unless" is satisfied. It does not have to last.

[accidentally erased this comment with an edit when I meant to add a new reply]

If you don't require the force before the mass is away from r=0 then where is the problem? While the mass is at r=0, there is no net force and the mass is at rest. While the mass is not at r=0, there is a net force. There is no net force at r=0. The net force is everywhere else. If a force causes the mass to move off of r=0, and no other forces are present in the system, then the mass must not be at r=0 when it has moved. Yes, you're right that the mass remains at rest while no net force is applied! But there is a net force that is applied in any other case. Are any of these statements false?

Yes, it is the lack of a classical cause that's the (only) problem. The force that the mass experiences at any other point is enough to move it off its original point. It does not have to feel the force before it can begin to move, at least not to comply with the first law. The first law only says that the net force and not remaining at rest go together, not that one causes the other and not that one must come before the other. Maybe I'm missing where it says that, maybe you're mixing up what the law actually says with a common sense understanding of what the law means. Here's another way to explain my point of view: In one state, the mass is at rest on the balance point and no net force is applied. The first law is not violated. In another state, the mass is not at rest and not at the balance point and a net force is applied. The first law is not violated. Say the mass goes from the first state to the second... what causes that? Frankly it doesn't matter, it is not included in the thought experiment. What if some quantum mechanical effect, like uncertainty of location and momentum, causes the change in state? Is uncertainty of position a force? No. Does it violate the first law??? No. Does it cause the mass to no longer be at rest? No, the forces everywhere else on the dome does. The cause doesn't matter, it's not a part of the first law. If it is, I don't see it. The word "remains" isn't qualified with "until some causal force", it's only "unless" the presence of a force. It makes sense to me because nothing material to the thought experiment is omitted. We're not discussing a prediction of whether or not the mass actually does move off the point (the original link mentions 2 solutions, one for each case), and we don't need to discuss "why" it might move.

There's no instant where the mass is not at rest and not subject to a net force. Every instant that the mass is at rest, it is subject to no net force. Every instant that the mass is not at rest, it is subject to a net force. Unless you're saying there's an instant where the mass is in motion but hasn't yet moved. Or rather I think you mean that there's an instant where the mass has been moved off its initial point but is not yet subject to the net force that applies at its new position.* You're saying there's no force to move the mass off its balance point. But the force is everywhere around that point. If the mass has moved, it is subject to a net force that is already part of the system. I don't accept your objection that an additional force is needed to knock the mass off its balance, because that is an answer to the question of "why" does the mass not remain at rest, and the first law only concerns "what" is observed. I don't know how you can say that the law is instantaneously violated, yet the law can't be applied instantaneously. That seems like a way of saying "it is so but it can't be proven". Can your objection be expressed mathematically? When I consider the math, I see no violation. * tl;dr... I think this is the heart of the matter. You're saying that the force that knocks the mass off its balance must come before the force that is applied at the mass's new location. I'm saying that's merely classical causality, and is not encoded in the first law. The causal condition can be violated without the first law being violated. I've looked at the first law to see where I might be wrong and I don't see the causal condition that you're using.

The forces cancel only at the one point. You're looking at "causes", I'm looking at "durations" in which the first law may be violated, and there is none. The "some other force" is present everywhere on the dome except the one point. I repeat: In any possible duration in which the mass is not at rest, it is acted upon by an external force. The law is not violated. (Going back to my original argument, which was maybe not clearly thought out... this is why I say they analyze the thought experiment at an instant, because there is no violation in any duration.) I think what is happening here is that the thought experiment is being analyzed in a purely mathematical way, and because of that we purposefully ignore the physics that are easy to ignore, the stuff that's not intuitive to assume, like uncertainty or randomness etc. YET we mistakenly keep the physics that we intuitively assume is true even in the purely mathematical view, such as causality. But I don't think you need to assume causality; Newton's first law doesn't state that the force must come before the change in momentum, in a strictly causal manner. For example, a bizarre interpretation in which only a force can move the mass off the point, and in which the first law is not violated (to my understanding)... if you look at the moment that the mass has moved off the point, it becomes subject to a net force because there is a net force everywhere off that point. In quantum mechanics, I don't think you are able to say for certain that one event preceded the other on the smallest time and distance scales, so you can have a situation where the force that is there when the mass has moved off the point is the force that moves the mass off the point in the first place! This is certainly bizarre because it violates our common sense understanding of causality, but such is the nature of quantum mechanics... yet it doesn't violate the first law. The absurdities that can happen in a moment with quantum mechanics do not apply in a classical duration. --- I'm not saying that this is what is really happening when/if the mass moves off the dome, but I'm saying this is the type of assumption that is making you see a violation of the first law where there isn't one.

But no one is suggesting an instantaneous violation of the law, you're simply assuming there must be one if the mass moves while no additional external force is introduced (besides gravity, since it has no effect at the mass's initial rest location). Intuitively, the law says what you're saying, that the force has to come first before the object is no longer at rest. However, if it only applies to non-zero durations, then that is not necessarily so. The law does not explicitly make a statement about causality. Consider any duration. If the mass remains at rest over the duration, then it remains in a spot where there is no net force, and there is no violation of the first law. If the mass does not remain at rest, it is at some other location at some point in the duration, at where there is a net force due to the dome shape and gravity, so the first law is not violated in that duration either. Intuitively we want to think of a duration in which the mass is no longer at rest, but before any force allows it to slide off the dome, but mathematically there is none. If anything maybe there's some violation of causality, but regardless, it seems to me that if you simply take out a cause (ie force) to get the mass off balance, you don't end up with a violation of the first law. The law is on the condition, "unless acted upon by an external force," which seems to suggest causality... but in this case, any duration in which the mass is not at rest includes an external force acting on it, so the condition is satisfied.

I disagree. If you're right that you can't apply the first law instantaneously, then there should be no possible case where an instant on its own violates the law (since it can't be applied), so any instant would be vacuously consistent with the law. Perhaps they falsely suggest the analysis supports the first law using their new version (though I don't see it that way), but a law that doesn't apply isn't inconsistent. If you insist on non-zero durations, there is none in which the mass doesn't remain at rest but is in the absence of a net external force. So it is consistent with the first law still. The only way it could be violated is in an instant. In this case they're using an unrealistic simplified model (whether they're omitting the cause of the movement off balance, or indeterministic effects). The explanation of any movement lies outside of the analysis. But I think that's fine since they're not trying to predict/explain what happens and why, they're only showing that their simplified model on its own doesn't violate Newton's laws.

The clock in the gravity well would tick slower than clocks farther from the mass, so that wouldn't allow an organism to age faster. As for the second question, I'd say definitely yes, both SR and GR allow that. An object orbiting a black hole might find that the rest of the universe has aged 13.8 billion years while it has only aged 1 B. As I understand it, the fastest a clock can possibly tick is when it is inertial and in the absence of a gravitational field. This is pretty close to describing a clock used to measure cosmic age. (Perhaps a clock in a void ticks slightly faster???) It seems it would be a problem if anything was older than the universe: http://en.wikipedia.org/wiki/Cosmic_age_problem By the way, your questions probably only make sense in context of some choices of frame of reference, hence all the talk of cosmic age and universal 'now'. You'd complained that this isn't the best place to ask questions... perhaps so, but this isn't Yahoo Answers or a class with teachers who have answer keys, it's a forum where everyone can discuss.

It does, thanks, but I don't think it's right. I'm getting hung up on your explanation because you used the word "present" to describe the surface of the balloon, which is everywhere. How is that different from a universal 'now'? I think it's incorrect that the same 13.8 billion year age that is measured everywhere uses different distance scales in different places. I think the assumption of homogeneity on the largest scales is what allows us to speak of a universal age in the first place, and it implies that the scale everywhere (ignoring smaller scale "lumps") is the same. In the balloon analogy I think it would be spherical (representing large scale homogeneity), with lumps. Everywhere on the ideal sphere, it would be measured as 13.8 billion, on the same scale. The lumps would have different scales, and could have wildly different measurements, but "no universal now" means that you couldn't have a single balloon surface where every point including the lumps are all reading 13.8B (or any consistent surface at all that includes all the lumps, I think). I think the way out of this confusion is that the "present" balloon-surface/age-of-universe represents the large scale universe with a chosen reference frame, and the "no universal now" is talking about the lumps and different reference frames. They're talking about two different things. I tried googling this and found explanations of what I'm trying to describe, but not what you're describing. Do you have a reference? https://www.google.ca/#q=is+the+universe+the+same+age+everywhere%3F From http://physics.stackexchange.com/questions/11390/is-the-entire-universe-the-same-age :

Because "agreement" implies some form of common ground in the measurement. It doesn't mean much to say that everyone agrees on the age of the universe, just at different times. Everyone on Earth agrees that the sun is overhead, right??? (According to their own clocks, at noon.) No, agreement doesn't mean "measured at different times"... I don't think this is applicable because the age of the universe isn't measured using just any local clock. Any local clock can measure 10 billion years passing, but that doesn't mean every observer can say "The universe is 10 billion years old" when their local clock marks that time. This is because the age of the universe is measured according to "privileged" clocks. A clock orbiting a mass in a deep gravity well at relativistic speeds wouldn't be used to measure the age of the universe. By the way I think we're mixing up some concepts from some different threads, and I'm trying to see that the different concepts aren't actually contradictory. Not sure if I understand it yet. "No there is no universal now" means (as best as I can summarize) that there's no general concept of simultaneity in GR, and in SR there is simultaneity but it is generally not agreed upon in different frames. "Everyone agrees on the age of the universe" means that the universe is homogeneous, and everywhere you can find similar observers (including only inertial observers in low-strength gravitational field) whose clocks can all be said to tick at the same rate, in all of their frames. The "age of the universe" refers to the general homogeneous areas, not the "lumps" or the individual different possible inertial frames. This does mean choosing a "privileged" frame, but it's a reasonable choice because---assuming the universe is homogeneous---it is applicable almost everywhere. Eg. Earth's own clock has been moving in various orbits and in a gravitational field, but with low speed and low gravity, and a local clock agrees pretty closely with a "universal time", at least to the precision we can estimate the age of the universe. When speaking of the age of the universe, if you get mixed up between Earth time and universal time, it's not going to make a difference. However, deep in a gravity well, or orbiting at high speed, the difference could be significant. http://www.jb.man.ac.uk/~jpl/cosmo/RW.html "All observers agree on the age of the universe" means "All observers can agree on a set of clocks that can meaningfully be called synchronized (even in GR) and which can be set so that time t=0 represents the beginning of the universe." ... But definitely not "all observers agree on the locally measured time since the beginning of time."

Even cheaper solutions are probably out there. I doubt that temperature readings from the average person (and location) would be useful. It would be a nightmare to deal with and would concentrate in densely populated areas, etc. However I think it's likely that there are still places on Earth where additional measuring would be useful to scientists, but maybe not necessary enough that anyone has a budget for it. I'm sure that if some amateurs gathered data in the right places, and did it properly, it could be a useful addition to existing data sources.

I don't think that's true. Most historical temperature data does not have good thermometers everywhere they need to be. They don't throw out poor data, they use the best data they have, adjusted for uncertainty (ie. bigger error bars) and biases, etc. Often multiple data sources are combined, with accurate and inaccurate data combined in some intelligent manner.