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The latter could imply some sort of internal property being the causal agent, such as a mental state. And, before you get to it, there actually are people who think that electrons have mental states. It's called panpsychism.

Clocks don't measure time. Clocks measure states relative to other states. They measure other clocks.

Are you talking forward or backward? Forward, none. Backward, the incident ball in the forward version. In the setup, there is a defined first ball in the forward version. So, in the backward version, there is no defined last ball, but there is a defined first ball. Actually, we can add 1 more ball to your infinite collection and have a defined last ball at both ends. If we label the location of the first ball at x=0, then the last ball is at exactly x=2.

AFAIK, in the entire literature, no one has shown any laws to be broken. It violates Lipschitz continuity, but that isn't a problem. It will be guaranteed that you have a unique solution if your equation is Lipschitz continuous, but failing to be Lipschitz continuous does not entail a failure to have a unique solution. Lipschitz continuity isn't a requirement for Newtonian Mechanics anyway. In fact, this paper argues that there will always be problem cases if you require Lipschitz continuity in Newtonian Mechanics and it is not the case that making such a requirement "can accommodate forcing the particle at rest at the apex to remain there at rest". Now, the question we need to ask is whether the Lipschitz discontinuity is due to the co-ordinate singularity or due to the situation itself. That, I don't know. One could reformulate it in a different co-ordinate system (perhaps with cylindrical co-ordinates and having the force in terms of r), though I'm not sure this has been done to date.

emphasis mine I'm not sure that's true. There's the same amount of balls the entire time. There's the same number of balls moving at any one time. Since there's the same amount of balls and the masses of the balls are constant, the mass is the same throughout the interval. Where is the violation here? It seems to me that it is just a violation of common sense relying on common sense's known trouble handling infinity. Yeah, a ball emerges from the end, but there was always a ball moving. There's no point in the interval at which there is no motion which you can use as a starting point in the time reversal to show momentum showing up from nowhere.

Yeah, that's my quandry about violation of the first law. If it satisfies the second, I'm not sure how it could fail to satisfy the first. If at every instance in an interval of time F=ma holds, there is no subinterval such that there is acceleration without a force. If that's what he was implying, then he was implying that physics can't handle motion at all.

There's NEVER an adjacent point. Between any two points lies another.

As overtone suggested, your nitpick isn't really relevant. Slide it. Whatever. As it slides, it loses KE to PE. If the KE to start is exactly equal to mballghpeak, then the sliding ball will remain at rest on top of the dome. Ran backward, this is still the ball sitting on top of the dome for an unspecified amount of time and spontaneously sliding down. Furthermore, since the dome is radially symmetric, this can happen at any angle (once you define a zero angle). There are infinitely many solutions such that the ball ends up on top, so there are infinitely many solutions, so the argument goes, that have the ball spontaneously sliding down the dome. I'll go ahead and read that paper of Earman's tonight, but I probably can't tell you anything about it until monday since I'll be away at a wedding all week. If you're interested in what he said, the paper is linked in my initial post in this thread.

The idea behind the dome is to try to show that there are cases of indeterminism in Newtonian physics. There are previous known examples (such as space invaders), but they break energy conservation. The claim here is that Norton's dome shows indeterminism without breaking any rules. While I tend to agree with you on this, Friedman and Earman have been said to have shown that the first law is a special case of the second when formulated in a four-dimensional mechanics, so it's redundant. From everything else I've read of Earman (and it's a lot; he's one of my intellectual mancrushes), I'm willing to give him the benefit of the doubt on this until I have time to read the paper to which I just linked. For those who are unpersuaded, Norton provides an appeal to time reversability in pages 11 and 12 of the original paper. Basically, since Newtonian Mechanics is time-reversable, any event happening in one direction through time is also has a possible reversal. Eggs can, though I'm not aware of any reports of this happening, unsplatter and jump back on the counter according to Newtonian metaphysics. Norton here doesn't give a rigorous mathematical treatment of time reversal, but rather a qualitative one simply to pump intuition to his side. Consider the same dome and the same ball, but this time with the ball at the bottom. Now, roll the ball up the dome. If you roll too hard, then the ball reaches the peak then rolls down the other side. If you don't roll the ball hard enough, then it fails to reach the peak and rolls back down. If, however, you roll it in the goldilocks zone, the ball stops exactly on the peak and stays there forever. Running that in reverse, according to Norton, gives the solution that the ball can sit atop the dome for an unspecified amount of time and spontaneously roll down the dome. For those who want to read more about it, Norton has a longer more recent paper here, though it doesn't really address Swansont's issue.

Norton gives a simple rundown of his dome here.

What about, say, Norton's Dome? (try this http://en.wikipedia.org/wiki/Norton's_dome )

....in practice. In theory, however, it's perfectly deterministic. We'd just have to make our description a LOT more complicated. You have to factor in the initial shape and orientation of the paper. You have to factor in the pressure change caused by the air displacement (roughly--low pressure on top and high pressure below, but is it even? ). You need to factor in all of these quantities that are normally ignored because this problem is generally irrelevant. Indeed. Don't think I'm correcting you, just clarifying.

And, you still fail to grasp the point. If you don't like people being an ass to you, then people probably don't like you being an ass to them. This was an attempt to get you to see that. But perhaps you're beyond saving. No, it's not. You're still wrong. Even in the context of the twin paradox you're still wrong. We only know there is no gravitational field in the twin paradox because we stipulated that there wasn't one. Having an accelerometer doesn't make us know that one is accelerating. The readings on the acceleratometer would be perfectly consistent with an observer in a gravitational field. Your claim wasn't indexed to the twin paradox, but rather was an absolute universally generalized claim. Even in the context of the twin paradox, you're still wrong. "No, acceleration (proper acceleration , more correctly said) is absolute. You can measure it with an accelerometer, so you know exactly who's accelerating and who's not.". Accelerometer readings, despite the name, do not generally imply acceleration over gravity. The twin paradox is entirely consistent with one of the inertial twins passing through a gravitational field. Their trajectory curves such that they meet their twin and they 'feel the acceleration'. You're still wrong. So, what we have here is: 1) You still fail to comprehend the point of the exchange. 2) You still fail to comprehend that you are in fact incorrect.

Err , I am under the impression that you do not even understand my initial post. It has absolutely nothing to do with being able to tell the difference between the effects of a gravitational field and of acceleration. Really? Let's pull the quote again: "You can measure it with an accelerometer, so you know exactly who's accelerating and who's not" has "absolutely nothing to do with being able to tell the difference between the effects of a gravitational field of acceleration."? How, exactly is that? Maybe you don't even understand your initial post. Either that or you are so terrible at communicating that it's a suprise that you can even use your keyboard (Gee, isn't being a dick to people fun? It's no wonder you like to do it so much). You said "you can measure it with an accelerometer SO you know exactly who is accelerating and who's not". That word "so" there is used in such a way to imply that the second clause is somehow causally dependent upon the first clause. It is precisely because we can measure acceleration with an accelerometer that we know who is and isn't moving. Yet, that's not actually true, since the reading could quite possibly be from a stationary observer in a gravitational field. That's what your initial post said. Do you understand it yet? Not very becoming for a moderator. Ah, so you understand that berating people for inaccuracy is inappropriate. Now, given that you've been told so many times before to stop and that you finally now appear to understand that it's inappropriate, I'm going to be heavy-handed with the warning points in the future. If I see you berating people, you'll get a warning point. If I see you berating people to the point of people wanting to leave the site, it will not be hard at all for me to get enough moderator support to send you on an involuntary vacation.

So what I quoted from you wasn't quite correct. Perhaps I should berate you now. That's how we play here? We berate people trying to learn over inaccuracies until they want to leave?

How can you tell the reading on your accelerometer from proper acceleration from a reading on your accelerometer from being in a gravitational field?

Gravity, on a flat Earth, would be almost parallel to the surface instead of normal to it. Objects would be dragged along the surface towards the center until the reach the center (or stop due to friction) where the net gravitational force is zero. Assuming friction isn't enough to stop the object before it reaches the center, there will be damped harmonic motion back and forth across the center until the object comes to a rest.

Didn't the fashion police ban hair gel like 15 years ago?

The universe as we know it is the result of a symmetry-breaking phase transition.

Well, let's see. We can't count the data in his Principia, since that's what was used to derive the formula. The first actual test of Newton's gravity was the Cavendish experiment which took place 111 years after Newton's Principia was published and only 40 years before stellar parallax was first observed. So, pushing the first evidence back only 40 years doesn't do much to defeat the point that this is an area of science where the actual acceptance of the theory is long before any evidence came around. That's especially true since Newton's gravity isn't inconsistent with geocentrism. Remember that Ptolemaic and Tychic geocentrism both describe the same data. What Newtonian gravity predicts isn't strictly speaking an elliptical orbit. In a reference frame at rest with respect to the sun, the trajectory is an ellipse. If you chance the reference frame and coordinate system such that the frame is at rest with respect to the Earth, you get something different altogether. Remember that: The three systems are observationally interchangeable with respect to the planets. Where they differ is the parallax. What Newtonian gravity did was finally give the heliocentric proponents the beginnings of the metaphysics they've been missing for 134 years. Even then, it doesn't exactly come with a mechanism for the action at a distance. So, Newtonian gravity removed one of the disadvantages it had, but it was still disconfirmed repeatedly until 1838. It was then that the heliocentrists could point to the inconsitency between geocentrism and observation. Stellar parallax and geocentrism just don't go together.

I'd suggest adding The Copernican Revolution by Kuhn to your reading list. It's about this very topic. That's because it's been disproven. Actually, when it comes to actual evidence, this happened relatively recently. But we've abandoned geocentrism for other epistemic reasons long before that. Geocentrism isn't nearly as dumb as people might tell you, if you don't have a telescope. Go outside. You can watch the sun, day by day, rise in the east and set in the west. It makes an arc. Go out at night, and you see the same thing with the moon. You see what appears to be all of the 'fixed stars' moving together tracing arcs in the sky. That's legitimately how the universe looks. This is the observational data. It's no wonder most ancient people with cosmologies had the stars embedded in a sphere that rotates about an axis running through the Earth. It's no wonder that these people described the sun as going around the Earth. The problem is the planets. They don't behave at all. They plane (Greek for 'wander'). But, it turns out, you can in fact chart their motion by having circles within circles. With enough circles, you can describe the motion of any planet as seen from Earth. Aristotle's physics (all they had at the time) said that that's how stuff past the moon worked. It goes in circles. The planets just happen to go in circles that themselves go in circles. What we have is the observational data meeting the physics, though in a more complicated manner than was desired. This model works just as well as the heliocentric model for the planets. There's no planetary motion that heliocentrism can accurately describe that geocentrism can't. Heliocentrism, however, scores epistemology points because of its relative simplicity. If you're to describe the motion of the planets, the geocentric way uses far more terms than the heliocentric way. There are also observations that 'fall out' of heliocentrism that are more or less fine tuned on geocentrism. That's why the heliocentric model eventually won out, but it's a bit more complicated than that. The planets aren't the only thing out there. In the geocentric model, the Earth is at the center of the stellar sphere, whereas in the heliocentric model, the Earth moves around in it. This means that our distance to the stars is the same on the geocentric model, but it varies on the heliocentric model. This introduces an observational difference between the two. The heliocentric model has to predict stellar parallax whereas the geocentric model predicts no stellar parallax. Parallax is the same phenomena that is apparent when looking at something like an analog clock from an angle. Looking at the hands from different angles has the hand align with different things. The same thing happens with stars in the heliocentric model. So, when Galileo advances the telescope technology, we should see it, right? Well, we looked, and it wasn't there. That's a great observational support for geocentrism. Heliocentrism may have simplicity, but geocentrism not only has physics on its side, but also the evidence. The heliocentrics, however, simply say that they weren't wrong about heliocentrism, but rather they were merely wrong about the distance to the fixed stars. They keep pushing it back and back. On the one hand, heliocentrism has the simplicity, but geocentrism has the evidence and the physics. There's a different kind of geocentrism that Tycho Brahe proposed which keeps the evidence of geocentrism but rejects the physics. This allows him to have a geocentric system with the simplicity of heliocentrism. In Brahe's system, the Earth is still the center, but all of the other planets go around the sun and the sun orbits the Earth. It's not until 1838 that stellar parallax is observed. That's right, Wisconsin is already officially a state before we have any observational evidence favoring heliocentric models over geocentric ones. So, geocentrism is finally falsified through evidence. So, the reason you don't see those people talk about it is that it has been long rejected and is inconsistent with observation. Geocentrism simply can't be true with stellar parallax. Stellar parallax is pretty bulletproof. Ptolemy wasn't the only geocentric model in town. Brahe's system is geocentric, but doesn't use epicycles.

There's also the option of doing an HPS degree. It's the area of philosophy specifically dealing with science.

Let's just use a nonrotating massive body since it's simpler than an Alcubierre metric. Nonrotating spherical bodies have a Schwartzchild metric: [math]c^2\tau^2=(1-\frac{r_s}{r})c^2dt^2-(1-\frac{r_s}{r})^{-1}dr^2-r^2(d\theta^2+\sin^{2}\theta{d}\phi^2)[/math] Let's see how far you have to get to get a flat space. [math]r_2=2G\frac{M}{c^2}[/math], so: [math]c^2\tau^2=(1-2G\frac{M}{{c^2}r})c^2dt^2-(1-2G\frac{M}{{c^2}r})^{-1}dr^2-r^2(d\theta^2+\sin^{2}\theta{d}\phi^2)[/math] So, we want to get [math]{1-2G\frac{M}{{c^2}r}}=1[/math]. That's obviously not going to work since it needs an infinite radius. So, there's no distance away that will not be affected. The question is how little do you want it affected?

Thanks to everyone who helped out with the GoFundMe. Yesterday, we were finally able to take him home. He came home with a little implant to feed him directly into his stomach as well as with an oxygen tank. Unfortunately, this morning, he broke his implant. We rushed him to the hospital and got a new one installed. Then, after I went to work, he broke that one. Right now, he's back in the hospital awaiting surgery to get a different type of implant.

It depends on whether or not you consider the spacetime itself to have energy. [math]\bigtriangledown{T^{\mu\nu}}=0[/math], so there is conservation of 'stuff'-energy.