joigus

I see. Of course it does. Terribly wrong. The point I was trying to make is that the darkest aspect of faith is possibly that so many people declare themselves believers just for fear of being rejected by their particular social group, their family, and closest friends. That's why I think it's so important that closet atheists come out. And that as much people as possible be helped to take this bold step. And that atheists raise awareness about this situation, that avenues for researching this phenomenon be promoted, etc. We simply cannot be certain of what it means when people say they are believers. That's what's most terrifying. We may think it's faith, a matter of convincing them with arguments. But maybe we're getting it all wrong. Maybe it's a much more worrying psychological phenomenon that we're up against. How many are there whose belief is declared only on the grounds of social fear? That's the big question. I think understanding that millions of people are hostages rather than acting out of convictions is very important. I also think a very serious question that atheists must ask themselves is "what can we do to help those people?" Trying to connect with what @CharonY said on Thursday: If we atheists want to have a good reason to form common-interest groups, and rather than wasting time in trying to convince hard-core believers, why not rise to a higher moral ground and address the ones who need us most? Those would be the ones who don't believe, but are afraid to say so.

Ditto.

Ditto.

Is that officially a joke?

Mmm. I don't think faith is a good benchmark of anything. It's too vague a concept. How many among the believers are really 100% certain about anything concerning their faith? I'm convinced that if an experiment were possible to set up measuring the degree of certainty that people personally obtain from their religion, it would show very poor levels in general. Most people "believe" only because it's a necessary step to be accepted in their community. In that sense, they believe in believing in god, as some illustrious atheists have said. A declaration of faith is just a declaration of faith. It's not faith. I agree. And I abide by it. That's why I'm so uncertain about so many things.

You also need general covariance, and the assumption that the field equations are a 2-rank tensorial, instead of higher rank, I think. They sound quite innocent assumptions, but they are really meaty. From there, the form of the Einstein tensor is almost forced (except for cosmological term) if you want the equations to be consistent with covariant conservation of the matter tensor. Personally, I find Einstein's reasoning quite transparent. Absolutely brilliant, but transparent. Feynman's, in general, I find more difficult to see the compelling character of some steps. I also need refreshing part of this material, I must confess. I wish @Markus Hanke or @Mordred expressed their opinion.

My answer --before I read your post-- was going to be "Yes, I believe religion exists."

Feynman arrived to that formulation of GR from a quantum elementary-particle batch of arguments. He also arrived at the homogeneous Maxwell equations from Heisenberg's uncertainty principle. He made astonishing connections. The problem with Feynman is there are big jumps in the logical reasoning supported by an incredibly far-sighted intuition and very deep knowledge of physics. So it's very difficult to know whether he's making a compelling logical argument or somehow incorporating a very well-digested idea from his toolkit, using analogies across the communicating vessels of mathematically-similar theories... Trying to answer your question, I'm not aware of any other way that's been practiced to arrive at GR.

It corrupts knowledge for obvious reasons. Taking a one-sided (non-objective) view of facts can only lead to misconceptions. As a consequence, it also corrupts the ethics of societies. Example: The banking-system's ethos: We want to take full advantage of predictable fluctuations in the market that are invisible to the general public, while taking full advantage of the general public when fluctuations become unpredictable --by taxing them to cover for our losses. We hide the flaws and shortcuts of such system behind a thick layer of red-tape, legalese, and marketing lingo. The more analytically intricate the system of concepts is, the higher the risk for pitfalls, both ethically and conceptually.

I totally concur with Markus that you've read too much into it. The relative error for macroscopic reading variables can be made as small as you want, making measurements of atomic parameters, extremely precise; meaning as close to the UP bound as one wants. Nothing more.

This is precisely the point I was trying to stress when I talked about introducing a very long object through a narrow opening. One foreshortens the object by rotating it with respect to the opening's diameter. If one doesn't, one can spoil both pieces. How real is that? Another example from phenomenology: When high-energy particles approach a nucleus, you can picture the nucleus as a cluster of more or less static objects from the POV of the approaching particle --time dilation in action. Also, particles with extremely high momentum appear to see the world to be two-dimensional (only 2 degrees of freedom), the so-called infinite-momentum approximation. Something that is perfectly explained by one of their dimensions to have become so hiper-dilated that efectively it has disappeared from the dynamics.

I'm there too. It's all a ying-yang thing.

I recognize no truth, but degrees of certainty.

What external forces do you wish to consider? So far it's the first time you mention any external forces. If you consider external forces, the problem changes completely and it's the first case I've discussed which applies. Make up your mind, please. The rest of your points only show you should study classical mechanics thoroughly before you make such assertions. I have a busy afternoon. Maybe later.

If you had an external field that acted on the COM coordinates, then you would have a potential energy depending on such coordinates. The Lagrangian is, \[L=T-V\] with T the kinetic energy and V the potential energy. The generalized momentum for the COM coordinates is, \[\boldsymbol{P}_{\textrm{cm}}=\frac{\partial L}{\partial\dot{X}_{\textrm{cm}}}\] and the evolution equation for X_cm is, \[\frac{d}{dt}\left(\frac{\partial L}{\partial\dot{X}_{\textrm{cm}}}\right)=\dot{P}_{\textrm{cm}}=\frac{\partial L}{\partial X_{\textrm{cm}}}=-\frac{\partial V}{\partial X_{\textrm{cm}}}\] Instead, you have, \[-\frac{\partial V}{\partial X_{\textrm{cm}}}=0\] So the equation of motion for the COM coordinates is, \[\dot{P}_{\textrm{cm}}=\frac{d}{dt}\left(\frac{\partial L}{\partial\dot{X}_{\textrm{cm}}}\right)=0\Rightarrow P_{\textrm{cm}}=\textrm{const.}\] Doesn't depend on internal details. You could have a whole civilization of tiny beings living inside. It's not gonna change anything. I'm sorry.

My opinion is that you cannot seriously believe in god if you've studied science in any length. Specially biology. But many scientists believe in believing in god. That is, they decide that it's a good social deal to keep saying they believe in god and, if pressed, talk about an abstract god, as in "god is the order in the cosmos" or something like that. Just to escape hostility from believers. Scientists discuss science even when the gathering has finished and the discussions keep going while they go back home, or to their respective hotel rooms. But I've never seen anybody discuss theology when they go back home from the church, the synagogue or the mosque. Religious people will leave you alone if you just say you're a believer. For all they care your "god" could be a telepathic giant cat living in another planet and handling the universe from there. As long as you say "I believe."

I have overlooked nothing that is relevant. Ficticious forces inside the ship can never result in acceleration of the COM. No one can write equations for variables that haven't been specified. What are the degrees of freedom of the system? Your new dynamical system looks very different now. It reminds me of a conveyor belt now.

I couldn't have put it more clearly. This big picture Ghideon is talking about saves you a lot of needless work. And believe you me, when you're working on difficult problems you need to have these tools handy, so you can save yourself a lot of workload. My background is that of a theorist. In theoretical physics you want to use powerful mathematical tools that allow you to treat problems that otherwise would be intractable. When you go back to simpler problems, like this, it feels as if you had X-rays to see the physics in them. Your system won't move.

Yes, your analysis is correct. The Hamiltonian formalism is not so efficient for this discussion, however. Noether's theorem is relevant in the Lagrangian formalism. The Lagrangian can be written in terms of the COM coordinates plus a couple of angle variables. There are obvious constraints between the axial rotation angle of what I've called the "nut" and the axial rotation angle of what I've called the "bolt" and the linear motion of both moving pieces, but there's no potential energy involving the COM coordinates (actually, no potential energy at all), so the problem is a one-liner in the Lagrangian formalism: \[\dot{P}_{\textrm{cm}}=\frac{d}{dt}\left(\frac{\partial L}{\partial\dot{X}_{\textrm{cm}}}\right)=0\Rightarrow P_{\textrm{cm}}=\textrm{const.}\] End of story. The beauty of the Lagrangian formalism is that you don't have to think about forces ever again, if your professional situation allows for that. Engineering works do require the Newtonian analysis very often, though. Whenever you have a constraint, you include it in a very straightforward way without thinking about forces. Internal forces "vanish" into constraints. Fictitious forces can also be dealt with very easily. They are relevant when there are external fields. They are all summarized in the constraints and they just don't appear in the formulation (they have to do with the use of curvilinear coordinates), if what you want to describe is the COM motion, which I think is the OP's primary interest.

And Feynman improved upon it: "Science is imagination in a straight jacket."

It's not that we can't see what you see. It's rather that you're the only one here who can't see what everybody else can. Anybody who knows anything about physics has spent some time in their youth trying to make a toy model like the one you're proposing in their mind. It's a rite of passage. You must do all these checks in your mind if you want to understand better why you can't violate the momentum and angular momentum conservation laws. And then you must learn to unlearn them when you're studying general relativity. Because they don't always apply there. Do you know that there is a weak and a strong formulation of Newton's 3rd law concerning mutual directions? You seem not to be aware of it. It is the isotropy of space that forbids that action-reaction forces be in any other direction than the relative position vector for any two parts of the system. That's the basis for the strong statement of Newton's third law: Mutual forces are equal in magnitude and opposite in direction, and their direction coincides with the relative position vector. So that, \[\left(\boldsymbol{r}_{i}-\boldsymbol{r}_{j}\right)\wedge\boldsymbol{F}_{ij}=0\] for every pair, i, j. Otherwise, empty space would be anisotropic.

I know what you're trying to get at, and I'm trying to tell you it won't work and why. You can't transform internal angular momentum into COM momentum. You're trying to obtain translation from rotation. I know that from the very beginning. It won't work. I can invest a certain amount of time in telling you why in more detail, but I must assess carefully how much it's going to be worth it* in terms of a useful (hopefully for both of us and other interested users) dialogue in terms of elucidating meaningful physical concepts. The patronizing treatment ("well done!," "congratulations!," as if they were the advanced students that "understand better" what you're trying to do won't sell your idea any better. *Edit: Also, other users are handling this very well, so my help here is not much needed.

I think everybody has noticed that. You can argue that most everything has some kind of inner structure. They can't and they never do. They can decelerate, though, by means of radiation reaction. But it's the radiation field that pulls the electron almost to a standstill. Electrons, of course, are decelerated by their own radiation field, not by means of internal forces. You can argue that the electron "ejects" something (radiates photons). Nothing can either brake or accelerate itself without ejecting or absorbing something or having an external field producing these effects. It's a property of space-time (symmetries) not a property of any particular system you come up with. Your system looks very much to me like a frictionless nut turning around in a bolt. No new physics there that I can intuit. Transfer of mass doesn't apply to the system that you seem to be representing in your drawing --as noted by Swansont. Rockets obtain momentum by liberating exhaust mass to space. That's what makes the dm/dt term relevant. Your equations for force and torque do not apply to a system where there is mass transfer. Mass transfer does not apply to solid systems. The mass "stays there." Please, consider this possibility: What if you're about to embark on an ill-conceived project that will make you waste a number of years in something that can't work just because you won't listen to criticism? Your enthusiasm is praiseworthy, but you seem to be applying physical principles incorrectly.

Swansont's point is well taken. And it is, of course, pertinent. If you don't specify your variables on a diagram it's impossible to point out where the mistake is. What is possible to declare beyond any doubt is that there must be a mistake in your analysis, (Ghideon has pointed that out too) unless empty space be inhomogeneous or anysotropic at small scales. And the reason is the part of my analysis that you seem to have chosen not to address. Namely, Linear momentum appearing from nowhere requires space not to be symmetric. That's why I know it cannot be correct. If you were as kind as to provide more detail about the position of what exactly is it that r_A and r_R represent, maybe someone could, upon reflection, tell you were the mistake is.