joigus

I will assume you've not been swimming in Lake Chad. No expert here, but I agree with most popular advice: See your doctor.

OK. To the risk of sounding ridiculous, I will try to enunciate my own negative principle of physics. And let's all have a laugh. Very much in the spirit of other negative principles (which, remember, are the most robust principles of physics): 1) There are no perpetuum mobile machines (of the 1st and 2nd kind) --> 1st, 2nd pple. of thermodynamics 2) There is no way to distinguish (at one point) whether you're falling in a gravitational field or at rest, or uniformly moving (Pple. of Relativity + Equiv. Pple. integrated in one) 3) There are no quantum xerox machines (you can't clone quantum states) |a> --> |a>|a> as output ... (there are others) In the same spirit, what about this?: 4) There are no perfect vacuum cleaners (there is no way to conceive the vacuum as a featureless scenario by removing elements from the physics in the equations until you end up with nothing) You must remove the equations of physics themselves. Can you do that? Theoretical physicists are so used to racking their brains formulating a plausible vacuum, that they would find your arguments very out of touch with physics indeed, @michel123456.

How do you vary zero, one, or infinity? I do see something on the left.

I concur with Strange. HUP has to do with mean square deviations from average value, not measurement effects. You can apply as a bound to preparations (non-disruptive, or filtering measurements). This has been discussed elsewhere in the forums. There are implications of non-commutativity and 2nd-kind measurements (disruptive measurements), but it's more subtle, and I'd rather not go into that. Too many different cases involved. Even preparations are more subtle. Classical quantum mechanics books get it wrong, on account of being too simplistic. It's generally discussed that you can prepare an electron with an arbitrarily accurate state of px, py, pz (because in the framework of the theory they're commuting). That's a theoretical fiction. You can collimate electron beams with a selected value of, say pz, but you cannot guarantee that px, py are exactly zero, if nothing else, for the very simple experimental reason that you must make your beams go through diffraction windows in the perpendicular direction in order to filter them. I think that lies at the core of why people like M. Berry and others are finding a richer structure in electron beams than the naive academic picture that they're plane waves. Among other things, they can generally encapsulate orbital angular momentum. There are parallel developments in optics (Laguerre-Gaussian, Hermite-Gaussian beams...). I'm not 100 % sure that what I'm saying is totally watertight as to electrons. I'm particularly interested to read @Strange and @swansont's take on this. Sorry if these comments stray too off topic.

My opinion is yes, GR would have been found without Einstein. It would have taken a different order of consecutive realisations about different aspects of the theory. The field equations would have been named the Hilbert equations for gravity, probably. Formulated in a more mathematical language, and taken some decades to infer everything about photons red-shifting and bending, etc. And had Einstein been a woman, it would have taken a decade longer for everybody else to realise that she was right. Had it been a guy in Papua New Guinea, we still wouldn't know. And had it been a woman in Papua New Guinea, we would never know.

T'was a joke, Studiot.

There's also "local hero", "local customs", which involve time and memory.

There's no action at a distance. Wow! That is a NEW* idea. ------- *NEW=Not Even Wrong

That's a good question. Maybe the mental operation of removing everything (stars, galaxies, black holes, etc.) does not ultimately make any real sense. It wouldn't be that surprising to me. The thing is that when you get to a sufficiently sophisticated formulation of the physics, and you remove the things (the terms in the equations) that you can identify with the "contents" of space-time, it just doesn't give you a flat, featureless, inanimate thing, so to speak. I gives you something that's not what you would expect. Does that hold water? I don't know. But it holds whatever it is that it holds. Maybe it's the idea of removing everything that doesn't hold water, and we have to accept that we've been very naive all over again, in a very unexpected way.

And there you are! Another different meaning for the words "local" and "global". 😫

No. Not tensors. Don't worry. Just take the electrostatic potential and forget about the vector potential. That's what I meant. The mathematical problem is the same. For example, eq. (4) in: http://users.wfu.edu/natalie/s13phy712/lecturenote/lecture27/lecture27latexslides.pdf Like Duda said, electrostatic scalar potencial. That's what I meant by dropping indices. Take the scalar part and leave the rest alone. Where \delta means "evaluate at argument equals zero and cancel the integral sign". That's called Dirac's delta function.

Duda is right AFAIK. You cannot picture solutions to inhomogeneous eq. by propagating the profile of the static source, which seems what you're naively doing in your link. Inhomogeneous eq. behaves differently. You need the Green function. Then you have retarded solutions in terms of the density. Or you could just plot Lienard-Wiechert-like potentials by relating constants and eliminating indices.

One of the most important lessons of modern physics is that there is no simple way to define vacuum, or empty space-time. In GR "vacuum" is filled with structure, or has room for it. So is QFT's "vacuum". Whether that's pointing to an important philosophical statement or not, I do not know. Is our notion of an empty scenario inconsistent in itself, or does it resist a simple definition? May be. But the theories are correct in every other single instance. Maybe they're trying to tell us something.

Very good point. +1 (Sorry, can't give more rep-points today, I owe you one). "Local" and "non-local" are used with at least somewhat different meanings in different contexts. One of them, as you point out, is "local" as opposed to "global". This "local" as opposed to "global" has to do with properties at a point or at the vicinity, as opposed to properties of the whole tapestry, so to speak. In field theories the latter always (AFAIK) are integrals of the field variables. For example, in GR a very famous one is the genus of the manifold (the number of holes). It's to do with the integral of the Ricci scalar to the whole manifold. The value of R itself at a point would be a local property. But they're related. A local PDE would be one in which all the variables involved are expressed in terms of their values at one point. It's a point by point statement. If you force to be involved arbitrarily high order of the spacial derivatives, that's another way of invoking very far away phenomena at point x. A useful way of understanding it, I think, is this "grading of the concept of locality". Imagine a world so local that's even more local than ours: nothing can propagate: \[\frac{\partial}{\partial t}\varphi\left(x,t\right)=f\left(x,t\right)\] Your evolution eq. does not involve any spacial derivatives at all. In that case, the configuration at point x and at point x' are not connected. Physical quantities evolve at every point independently. Next step is propagating: the time derivative is involved with the spacial derivatives. You can assume first order, second, etc. in spacial derivatives. Everybody calls this local, but it's "less local" only in the sense that field variables get affected in far-away points if you wait long enough. You could always call a theory in finite order of spacial derivatives "local". You would only have to extend the set of initial data to higher and higher order spacial derivatives. Your field variables would be now phi, phi', phi'', etc. The problem is when the order of spacial derivatives is unbounded. Then there is no way that you can re-define your state as local in any reasonable sense. Your field variables are sensitive to arbitrarily-high-order inhomogeneities in the spacial variables. You would have to provide all the derivatives, which amounts to providing the function in all space. This graded explanation of locality is not standard, but I think it clarifies (or could clarify) how the relation between spacial inhomogeneity and time evolution is related to the intuitive concept of what local evolution must be. I made a mistake here. There are no dtn terms in the expansion. The arbitrarily high speed is implied somewhere else. But I'm sure you're right. I'll think about it later. Maybe somebody comes up with the right idea.

Well, with the characterization of non-locality that I was talking about, you could still have non-local influences. Maybe your function (I'm assuming it's a source, or perhaps a coupling) is zero at x (the field-point you're reading your fields in terms of). But f', f'', f''' don't have to be zero. Take for example the inversion of the Earth's magnetic field. Some crazy theory could come up tomorrow, saying that the Earth's magnetic field is extremely sensitive to the seventh-order derivative of the local density of matter at some point in the Andromeda galaxy. Almost anything that you can implement by an explicit dependence on time, you could equally well do with some crazy assumption of the form f(x+a,t) with a translating you field point x to the Andromeda. It would be very difficult to disprove quickly. That's what I think anyway. But I remember you once pointed out how local conservation laws (the continuity equation) are ubiquitous in physics (something like that, I don't remember the precise point now). The reasons for rejecting the idea would rather be Ockam-based, I think. Also, anything that happens here, if it involves energy, must have come from the surroundings... The possibility of non-locality opens a really frightful can of worms, IMO... Edit: Hopefully interesting related note... Some decades ago people became heavily involved in models that explained the wave packet reduction in terms of a non-local modification of Schrödinger equation. Something along the lines of, \[-\frac{\hbar^{2}}{2m}\nabla^{2}\varphi\left(x,t\right)+f\left(x+a,t\right)\varphi\left(x,t\right)=i\hbar\frac{\partial}{\partial t}\varphi\left(x,t\right)\] with a special non-local potential that acts at some point (where the measurement is performed) and kills the wave function at distant points. The Coleman-Hepp model I think is the most famous one. Bell proved* that this is not possible unless you're willing to sacrifice unitarity (infinite evolution times, singularities in the Hamiltonian, horrible things like those). Something that, IMHO, should have been obvious from the start, as the Copenhaguen rule for normalising the state violates linearity, and a simple projection without normalisation (giving up the normalisation factor), violates the isometric character (probability conservation). But, as nobody presses this point anymore, I never use this argument anymore. Sorry for the off-topic excursion. Edit2: People said very crazy things about non-locality several decades ago, and they kind of got away with it. *Edit 3: Bell proved that for the Coleman-Hepp model.

@cladking Common categories are not Aristotelian (classical) categories. The concept of cat comes from the clustering together by family resemblance of particular instances of what we call cats, not by the definition of closed (mathematical) equivalence classes. There's even a mathematical theory for the concept you're groping towards: fuzzy sets. Overall, your discourse sounds cathartic, more than based on thought out concepts. You sound dissatisfied and you seem to want to voice your dissatisfaction. You should try some common-interest group based on emotions, rather than a scientific / philosophical battleground for your complaints. That's my advice, anyway. Edit: Here's an example of your "cats"

Read some Wittgenstein. And then some modern cognitive scientists. They've already developed the point you're trying to make.

Gross underestimation.

Exactly. In the Taylor expansion that I wrote before that infinite speed would be implicit in a2 /dt2, a3/dt3 etc. (powers of velocity) as compared to the values of the d(n)f's. You can make this propagation as fast as you want in principle. If there were such a thing, it would reflect instantly in the values of the fields everywhere. To the extent that I'm aware, nobody has taken this idea seriously, but every now and then there are claims that some quantity or other could have a non-local definition lying somewhere. I think the word non-local is one of the most abused terms during the last decades. Some rigour in the definitions is necessary so that everybody understands what they're talking about. But that's really my two cents about the matter...

I understand your objection. The clarification is in one subtle self-correction I made about one of my statements that deserves to have been overlooked on account of my sloppiness. Here it is: where the f's are the dynamical variables, not involving t explicitly. It's classical mechanics I'm talking about, with trajectories qi(t), then it'd be ds2=gijdqidqj When you have a system of fields, the trajectories would be configurations of your fields. So you would have a trajectory in field space. With propagating fields it probably wouldn't work, but with topological fields I'm reasonably sure it would (topological fields are highly constrained). In a way, they would work much as coordinates do. I can develop the point in case you're interested, but won't press it otherwise. Yes. But circular reasoning is not a death sentence for a fledgling idea for a theory. IMHO, tautologies are necessary to start formulating a theory. Let me explain. Galilean relativity principle: What is an inertial system? One in which Newton's laws are satisfied. Where are Newton's laws satisfied? Only in inertial systems. Get out of the tautology: Identify as legitimate forces only those that can be attached to physical sources (densities). Those cannot be globally removed by re-framings. You're left with forces that can't be globally removed by re-framing and leave out those that can: ficticious forces. Mass and force What is mass?: ratio between force and acceleration What is force?: mass times acceleration Get out of the tautology: force (in most interesting cases) is a universal function of position (and perhaps velocity in a very special way as a pseudo vector) I have this (maybe not general enough, etc.) intuition that all good theories start with a tautology and then make auxiliary assumptions to get out of it and make them productive. I don't know what you think about that. But your points are certainly well taken. And very sharp, as usual. +1

Very good answers you're getting here. Let's hope crackpottery doesn't make an appearance.

Very interesting question. +1. The concept of locality in field theory that I'm familiar with is better characterised with a precise mathematical definition. A typical non-local evolution equation would be, \[f\left(x+a,t\right)=L\varphi\left(x,t\right)\] where L is some differential operator, \varphi is your field and f is a source. The values of the field depend on distant values of the source. Any change in f would affect your field instantaneously. This can also be characterised by the dependence of the evolution on arbitrarily high orders in the spacial derivatives of the source (it could be the field itself). If you want to express the evolution in a local reading (values at x, and not at x+a), you would have, \[f\left(x+a,t\right)=f\left(x,t\right)+af'\left(x,t\right)+\frac{1}{2!}a^{2}f''\left(x,t\right)+\cdots\] You could have more complicated patterns of non-locality. For example, if your source term were of the form, \[\int_{-a}^{a}dx'f\left(x'-x,t\right)=L\varphi\left(x,t\right)\] There is usually a parameter like a here, which tells you how far away this range of non-local influence is. People talk about non-locality in relation with Bell's theorem, but they are confusing this concept with that of non-separability, which is very different. Edit: Another possibility for your source term: \[f\left(x,t\right)=\int_{-\infty}^{\infty}daf\left(a\right)F\left(x-a,t\right)\] Range a: the influence is exponentially suppressed by an a-dependent factor. f(a) falls off to a certain range. Edit 2: I'm not so sure about what you mean here. I would have to think about it.

Sorry. You're right. You do make a point. I was under the influence of the last couple of comments I've had to answer to, which were quite pointless. Thank you. +1 You do make a good point here.

Told you. Thinking is hard, and you have opted for a simplified version of it. You've thrown away tens of thousands of years of human knowledge right there. Hardly the point.

Empty means vacuum Einstein equations (T=0). That's what it means. There are solutions to the Einstein field equations that are non-trivial. The matter term is zero. Gravity gravitates, did you know? So gravity itself (curvature) can perturb space time and deviate from the globally constant metric (flat spacetime). You must know these things if you don't want to be involved in a non-ending conversation about words that have a very precise technical meaning. The vacuum Einstein field equations \[R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu}=0\] Do not imply, repeat, not imply that: \[g_{\mu\nu}=\left(\begin{array}{cccc} -1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0\\ 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 1 \end{array}\right)\] globally. That's where you're wrong. Edit: As MigL has said too, sorry I didn't notice. +1 Edit 2: This is because the theory is non-linear. This is getting ridiculous, really. Words have a special meaning. I suppose you're trying to tackle it from philosophy or common language. That's not how it works. Then there's another non-sensical conversation going on about quantization of time I won't address for obvious reasons. There's a reason why Pauli introduced the words "not even wrong". Exactly. +1