Schrödinger's hat

I'll see if I can find another analogy and stretch it as far as it will go: A warm-blooded mammal. If you made it not warm-blooded it wouldn't be a mammal anymore, as that is one of the properties of a mammal It is possible that we could get a mouse, and modify it so it was cold-blooded, but it wouldn't be a mammal (or a mouse) anymore. It'd be a different thing. We'd also have to change other things to make it so our modified creature was still alive. We could also find an existing cold-blooded animal a bit like a mouse, maybe a lizard of some kind. It could share other properties (propensity to forage, size....the analogy is starting to wear thin about here). It would also have some other properties that differed (egg-laying, scales etc) The same way an electron without charge isn't an electron anymore. There's something vaguely similar (it has the same spin, and the same flavour) an electron-neutrino, but it has other properties that must be different (mass, isospin etc). Otherwise it wouldn't be a valid particle ("stay alive" so to speak).

Well technically ^0.5 is many valued and as such makes the resulting expression not a function unless we take 'the positive solution of ^0.5' Okay, now I got it. It was just a matter of figuring out exactly how I was allowed to cheat . Although my answer is possibly a bit more convoluted than it should be.

Of course. One more question: Are we allowed to assume a^0.5 will be the positive square root, as per convention?

Ah, now it's more interesting, and a bit of a head scratcher. The only way I can think of still involves something along the lines of [math] \lim_{a\rightarrow\infty} (\pi-3)(g(x))^a[/math] Alternatively, I could cheat and pretend that by * you meant complex conjugate...

I'm assuming you're including things like the step function in piecewise functions. Still seems either too obvious or too obtuse. What about floor/ceiling/round? Perhaps there is some form of succinct way of stating what types of things we are allowed to add?

This seems a little bit ill-defined. What are the restrictions and rules? I mean, it'd be a bit trivial with the heaviside step function. Or just: [math] f(x) = x^2 | x \neq 3[/math] [math] f(x) = \pi \, | x = 3 [/math] (note: I don't pretend that these are the intended answer, they just don't appear to be excluded by what you said)

Good to see that repeating ourselves over and over again isn't completely in vain. Next time you'll probably make much faster progress if you take the attitude 'these scientists are crazy, but let's take their assumptions as true for now, and see where it leads'.

Change in internal energy

Well for starters, LISP comes with operators out of the box, they just work a little differently -- (+ 3 4 5) will evaluate as 12. This seems to me to be a bit of an extreme view, akin to saying imperative languages don't have functions (ie. the only imperative language by that definition would be assembly without macros). I was under the impression that functional to imperative was more of a spectrum (in the style of the use of the language as well as the language itself) than a hard line: At one end we have lisp, prolog, or pure lambda calculus. Then maybe Haskell and a few others. I don't really have any experience with these so I don't know quite where to put them There's a large middle ground, I'd put pyhton, javascript and so on here, as they are amenable to both functional and imperative styles. Then we'd have C, fortran, Java etc. And at the extreme imperative end, assembly. This is probably a bit simplistic as stack/array based languages don't really fit on this scale anywhere.

In functional languages, everything is defined as a function ie. this is that. They tend to be well suited to recursion (defining something in terms of itself). You might have something like: factorial 0 = 1 factorial n = n * factorial (n - 1) Then call it with something like: ans = factorial 5 Functions are first class objects, they can be treated just like variables. Whereas in an imperative language you have to describe the process to get the result as a series of instructions, or operations ie. to get this, do that. You might have something like: void factorial(n, dest) { dest = 1; for (int i=1; i < n; i++) { dest *= i; } return; } Then call it with something like: factorial(5, ans); This is even more imperative in style than you would typically write it in a C-like language, but illustrates the distinction.

Congrats. Also your post explaining it was very well put. @Owl this fullgod frame tar speaks of is what we mean when we speak of a preferred reference frame.

W is work, Q is heat. Usually you'll use: [math] \Delta U = W + Q [/math] Where W is the work done and Q is the heat flow. Which says the change in the internal energy is the work done on the system plus the heat put into it.

The field only updates at the speed of light.

I'm not sure, but I'd think that GPS (with a good beacon) over that sort of distance would be more accurate than a laser. Changes in refractive index over that distance would need to be considered for that level of precision

Yes, and I'd like to give you this, but I need some cooperation. The train of logic is starts at some form of thought experiment involving light beams. Again, this requires patient application of logic. I will happily go through any combination of the assumptions (constancy of speed of light as measured by different observers, no preferred frame, universal and well defined now etc) and describe the logical consequences of taking each as true or false, but I need some indication you understood what I was saying. Let's drop this for now and focus on why relativity predicts what it does and examining the premises of our arguments, no progress has been made any other time you have claimed this, I doubt responding to it again would have any effect. Length contraction of objects is a direct logical consequence of (or logically equivalent to, at least) the relativity of simultaneity. It is much easier to explain relativity of simultaneity, so let's discuss that. Yes, we have a name for the things that are part of objective, non-frame-dependant reality. We call them invariants. Not all measurements are invariants. This picture you paint is completely consistent with a relativistic world view. Again you are presenting two options: Distances are fundamentally representitive of reality and invariant under change of frame. Distances are fundamentally representitive of reality and vary under change of frame Ignoring the possibility of a third: Distances are not fundamentally representitive of reality and vary under change of frame. There is another quantity (interval) which is almost indistinguishable from distance when used to describe objects moving at low speed. Describing reality in terms of intervals will give you a result that does not depend on frame of reference. Again, I qualified that block with 'according to relativity'. Must you insist that I qualify every statement? I have said that I will show you why relativity posits this, along with the lines of reasoning based on different assumptions that are consistent with experiment. It will require sevaral posts and some level of cooperation. It is not about light delay. Two objects moving relative to one another have a different now (over most/all of space). Two objects in the same place and time have the same here and now. Just repeating this over and over again is useless. Let's go through the logic. I do not wish to assume that. I was going to examine every combination of assumptions I could think of.

Also: http://www.khanacademy.org/ http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html

Well, when the derivative is zero you have an extreme of your function. Can you solve your derivative for zero? Hmm, pi is a constant, it doesn't change. I'll give you another nudge along the way. We want: [math]\frac{d}{dx}-3\cos{(4x-\pi)}=0[/math] First, carry the constant out the front, it doesn't matter. We recognise we want to use the chain rule, so we can set [math]u=4x-\pi[/math] and identify the rest as it as: [math]-3 \frac{d}{du}\cos{u}\frac{du}{dx}=0[/math] (If you need more explanation of how/why this works, do ask) Splitting our problem into two problems we hopefully already know how to solve: First, the derivative of [math]\cos{u}[/math] with respect to u (you can safely ignore what u might depend on, and all the rest of the problem for this step) Second, the derivative of [math]4x-\pi[/math] with respect to x. One thing that might help is to remember that derivatives are linear, the derivative of the sum is the sum of the derivatives. Hopefully you can do: [math]\frac{d(4x)}{dx}[/math] easily enough. And you asked what the derivative of [math]\pi[/math] (with respect to x) is. Think about what 'derivative' means. How fast does the value of [math]\pi[/math] change as we change x?

I think you meant off where I highlighted the on. Moving ahead with that assumption: How far and how long it takes depends on your reference frame. According to our poor traveller, he hasn't moved at all, so no distance. As for the time according to the traveller, or proper time, it wouldn't be very long. I don't understand GR well enough to tell you exactly how long, but it'd be roughly the same as falling into a star of similar mass.

This seems a lot like a homework question. We generally don't provide complete answers to such things. Perhaps you could make a start and we'll guide you through the rest. Some questions to which the answers should get you started: Do you know what the derivative of the cosine function is? Do you know what the chain rule is? Do you know what it means when the derivative of a function is zero If you have more general questions (ie. you need an explanation of the chain rule) feel free to ask.

Newton.

I know I've solved this in the past, but there's one case the following approach does not account for. Ah, worked it out again rather than trying to remember:

I just thought I might mention again: Don't electrocute yourself. Perhaps the zombieSquirrel label is quite appropriate then.

No this is a simple scientific statment. The definition of now I am using is simply a class of coordinates, it has the same ontological status as a line of longitude, or an altitude. Sadly I have no word for this concept which is distinct from the philosophical entity 'now' (whether we be talking about presentism or some extension of relativity to include the concept). We can discuss philosophy of science only once we agree on the science. I did not state 'according to the model of relativity' in that sentence because I thought it was clear from the context. At any rate, you're still conflating light delay and the simultenaety of relativity. These are distinct concepts. Let's say I'm sitting here, typing on my computer, and there is a solar flare happening now, so I'll see it in eight minutes. There's also an alien surfing the solar flare in its version of extreme sports, moving at a high fraction of c. For the sake of this argument ignore the motion of earth relative to the sun, any contribution from this is insignificant. According to relativity: To me, the solar flare, the surfing, and the typing are happening now. To some laboritory on the sun which is not moving relative to the sun, the solar flare, the surfing and the typing are happening now. To the surfer, the solar flare is happening now, but the typing is not. Note that the sun-lab and I measure the same now (everywhere in space) because we are not moving relative to one another. The surfing and the solar flare always happen at the same time because they are in the same place. But the surfer measures a different now to me. This is a direct logical consequence of all observers measuring a constant speed of light. It is quite possible to argue that all but some special set of observers are mistaken in what they measure as 'now', but: If all observers measure all light to be moving at 3*10^8m/s relative to them, then they MUST measure different sets of events to be happening at the same time if they move at different velocities. Exactly how this measurement is reflective of reality is open to philosophical discussion. Whether or not this measurement will be made is not, and would not be even if there were a preferred frame of reference or special relativity were wrong in a number of other ways. If this is not obvious to you (and it is not obvious to most, I would possibly even say all, people), then respond to my post about the jousters and I will endeavour to show you why.

The difference is quite subtle. I could have two rods which are 1m long in a frame, one of them is moving at 0.866c in that frame, the other is still. But if I switch to a frame where the first rod is still (according to SR) the formerly moving (now still) rod will now be 2m and the formerly still rod will be 0.5m. This is because the measure 'length' was not representitive of the true shape of the rod. The rest length is a different concept, the rest length of one rod is and always will be 1m, and 2m for the other -- no matter the frame. Rest length is a similar concept to length, but it specifies a frame, ie. "The length in the frame where this object is still." For distances that aren't related to a single object, rest length is not defined. You are welcome to criticize science, but you have to do it in a scientific manner. If you are talking about philosophy of science, then your philosophy must be consistent with science. We have been trying to show you that (our reading of) your stated assumptions lead(s) to a contradiction. Things that have the same velocity as me, and are in my now share the same now as me no matter where they are. So do things that are at my here and share my now (regardless of their velocity). Things that are spatially distant and moving at a different velocity do not share my now. If you could please respond to my diagram and outline of the jousters situation, I will show you the logic that leads to this conclusion from stated assumptions (which you will be welcome to challenge as they are stated).

You are forgiven....this time. But be warned that if you do it again there will be dire consequences.