joigus

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

Nor to me.

Very droll. Is it something I said? Don't go there. Now seriously. You need something to move through space-time to even start talking about gravity. Even if you just have imaginary points trying to follow the closest-to-a-straight-line trajectories you can find, you do have gravity. But Gravity can be defined even if there are no sources (energy-momentum tensor). Those are empty space-times. They can be defined theoretically. Out of this context, I don't know what you're talking about.

Maths is the language of physics. Common language and diagrams are useful complementary tools. That's just how it is. Physics cannot be formulated in just words (however philosophically sophisticated) or pictures. It just can't. I'm afraid I must insist on @Strange's question: What else is required?

Of course it's more than that. I don't see the "only" bit in my statement about how observers cannot get rid of time in their view of the universe. Volume is also. All observers have volume. That doesn't imply that "volume" is an only-observer kind of concept. You can formulate gravity in different dimensions. Not only 4. 4 is special as concerns topology, not Einstein's equations. Although going down to fewer dimensions than 4 makes gravity more tractable, or trivial, if you will. In 3 dimensions the Ricci tensor codifies all possible degrees of freedom to deal with gravity. In 2 dimensions the Ricci scalar is enough. And in 1 dim there is no gravity because there can be no intrinsic curvature. But I'm not quite sure what you mean by, Can you elaborate? Just like observers. I would distinguish pure speculations as rigid frames of thinking that ignore the basis of generally agreed-upon physics from open-ended lines of reasoning that strive to incorporate what's known and try to grope a bit further. That's what I was trying to do here. I wouldn't be surprised that some of the more adroit users here find some kind of overlapping with what I've said. Although I think disagreement among equally adroit users would be just as likely.

Was thinking about this some 24 hours ago and wondering what had become of you and your problem. I'll keep an eye on it. It's interesting.

I agree. I wasn't trying to put the blame on anybody. I was just interested in the definitions. Different countries seem to have different thresholds for what is justifiable. But you can never gladly ignore the social cauldron in which those concepts are formed.

Interesting. I thought the criterion for "justifiable" would be self-defence. But no: That's a bit loose. If you point your gun at a guy who's holding up a shop, tell him to raise his hands, and he does, and you shoot him in cold blood. Would that be considered as the killing of a felon during the commission of a felony, thus justifiable? Another example: http://www.bjreview.com.cn/forum/txt/2009-04/28/content_193066.htm I tend to agree with this kind of thinking, for this and other similar problems. +1