Schrödinger's hat

I haven't used prism, but this would be more the type of thing I'd use something command line based for. Such as the already suggested R (its plots look fairly basic at first glance, but they have many advanced features once you dig). Octave also has a number of plotting libraries available in which it wouldn't be too hard to code something like that (if it wasn't already built in -- I've certainly done graphs with differend line widths and graphs with color as a parameter before). Sage can probably do such things, too. Although both Sage and Octave are more aimed at being general purpose computation/maths environments than displaying data. Tools like these can be a bit scary if you haven't had any experience with command line or programming before, but they are extremely useful once you get into them.

I say space-like, as in like the thing we measure with metre sticks, because we have to treat them differently from the time-like dimension under transformations. This is true of both the Galilean transformation (which does not involve length contraction etc) and the Lorentz transformation. I assert only that (whatever they are ontologically) they behave in a way that matches the behavior of certain mathematical objects. The careful wording is because these mathematical objects behave slightly differently (although you can compare, and in some cases -- according to relativity --they will contribute to one another). One example is the interval we keep talking about [math] (\Delta s)^2 = (\Delta ct)^2 - (\Delta x)^2 - (\Delta y)^2 - (\Delta z)^2[/math]. You have to treat one dimension differently from the others. Within the strict bounds of GR (rather than some interpretation of the theory) this is because the contribution due to one part of the theory (the part that changes our coordinate system) is different to the other part (the part from their momentum-energy). This statement is like pointing to someone who does electrodynamics on a macro scale, and saying, "Ha! See! Scientists don't believe in atoms!" because the words they use in papers and textbooks talk about matter as if it were continuous. Some people elevate space-time to thingyness and say that it's an entity which is expanding. It's a useful conceptual tool and as long as your definition of what type of thing it is does not contradict the data (and the mathematical model you are using based on that data) you are free to interpret how you wish. I'm sorry, this isn't a discussion on ontology, that was the other thread. Trying to ask what something is before you've established how it works, or whether it is the same type of thing as another is premature. The topic here is whether or not SR implies reality is subjective. I don't see the what of time and space being relevant compared to the how. I gave these definitions to distinguish time from -- say -- a unicorn. These definitions are sufficient to talk about it, and to get some time from reality into our mental/mathematical model. Other than that I have no idea what they are, nor does your definiton of empty volume and duration between events seem in any way meaningful to me. This is probably because I have a different definition of the words than you. This is why I supplied a pragmatic definition in the first place. Perhaps you can agree with a slightly weaker definition? Time is at least the thing that is measured with clocks. Space is at least the thing that is measured with meter sticks. Whether or not they apply to the tie-dyed rabbit pelt or empty volume and duration. Upon reflection on my response to tar, I would also like to make these a bit more specific. Time is at least the thing that is measured with stationary clocks -- whether or not moving clocks measure the same thing. Space is at least the thing that is measured with stationary meter sticks -- whether or not moving meter sticks measure the same thing. With the same caveat that no Ontology is implied. Whether or not it is classifyable as science (I know of no testable predictions -- but that may have changed), M-theory is completely irrelevant to the discussion at this point. You're trying to skip a century of work of some of the world's brightest minds, when you refuse to even accept the predicates of SR for the sake of argument for long enough to comprehend the theory. Well this is actually useful for moving the discussion forward. How do you define the reference frame from which all measurements should be taken?

space-like intervals represent real distances, remember it's [math](\Delta s)^2 = (\Delta (ct))^2 - (\Delta x)^2[/math] A negative interval squared (imaginary interval) merely represents more distance than time. So this would work if you were considering a world-line (ie. particle which is 1 dimensional at any given time) You wouldn't be able to use formulae for things like length contraction and time dilation, but these are simply derived from the Lorentz transform. I do not know how to think about non-pointlike objects in this context. Perhaps they would just fall apart because the interactions that bind them would not be able to travel between events in the object's 'worldline'? The distance from an object inside an event horizon to one outside is just undefined (rather than imaginary) as far as I know. You could probably consider a geodesic running the other way though. GR is a bit beyond me, but I believe the concept of distance gets a little shaky once you consider gravity, you have to talk about path length if you want any meaningful results. Well, I was restricting my thinking to SR in isolation (ie. events happen in Minkowski space) rather than special relativistic physics. I'm sure tachyons would cause paradoxes all over the place. Agreed None of our theories hint as to this being possible, and we've seen no evidence which hints that such a mechanism exists. It would also cause paradoxes and allow time travel in some sense or another. So we put it in the bin with all the other ideas that we have no reason to consider right now, until we find a good reason to do so. Actually, there's one exception ...kinda...in GR. It's not really moving faster than light, kind of side-stepping the limit by bending spacetime just so. It has its own problems (such as requiring infinite energy -- for one -- and possibly requiring that you can already move at FTL speeds ), and would probably have all the same paradox issues.

Photons don't change (they have different energies in different frames, but in any given frame they aren't changing) -- well, the wavefunction of the light changes over time, but let's not get into quantum for now -- any photon with a given energy-momentum and spin is completely indistinguishable from every other photon with those properties. They also have no size that I know of. At any rate, the concept of travelling at the speed of light is outside the realm of special relativity. It's a bit like asking, 'how green is a second?'. The answer is simply outside any model we have.

As far as I know there is nothing in special relativity which prevents objects that move faster than the speed of light, nor should a physicist balk at seeing imaginary numbers (for one, they come up when solving simple things like a mass-spring-damper system) as long as the observables you predict are real. We already have quantities that act like a speed faster than light and ways to deal with them, we call them space-like intervals (of which the concept of distance is a sub-set). Such objects are known as tachyons, and are generally considered not to exist due to there being zero evidence for them, or for theories that produce them. If they did exist they would be able to carry information back in time, this would break causality (and I think also the laws of thermodynamics). On top of this, you get other weird things, like the slower they move the more energy is required, and they cannot move slower than the speed of light. I also have no idea how an interaction between tachyons and normal matter would work. I also hear that they are incompatible with newer physics (somewhere between QED and string theory I hear talk of models producing tachyons and their rejection due to the fact this would cause runaway inflation) although I do not understand the rationale behind this. This would be fine, if you were somehow able to hop over the speed of light I'm pretty sure you'd just find that the light moved at the speed of light in the opposite direction.

So, applying the above argument to "forshortening" if there is no objective "reality" independent of various angless, then as discussed at length in the former thread, the distance between earth and sun varies with angles, shortening radically with high angles close to the orbital plane, for instance. Likewise, earth itself, measured as above, is flattened in the direction of view from which it is measured. My argument then as now is that one astronomical unit stays the same length and earth stays the same shape and size, in the objective world of what is, independent of extreme platforms (angles) from which they are measured. To claim otherwise, i.e., that "there is no preferred angle... those measurements are equally correct" means that one perspective is just as accurate as another, and therefore, there is no objective universe. It all depends on how we look at (and measure) it. So, accordingly the "at ninety-degrees" frame with what is being measured will be the accurate angle and yield the actual size and shape and distance between objects (measured from one of the end points.) I know this is perspective "heresy" but I believe it is sound philosophy of science. It is not an attempt to "debunk perspective" but to bring clarity to the confused subjective idealism of perspective regarding angle as "subject."

Nearly there. Remember that you can't just square everything and get the same expression, so your last step should be: ((x-1) *(x+1)^5)^(1/2) or √(x-1)*√((x+1)^5) Can you see how to get from there to the last step? (might be easier to go from the one above [math]\sqrt{(x-1)(x+1)^{5}}[/math] Can you somehow get a [math]x^2-1[/math] term in there? Also you can use latex on this forum by putting it in between [math] and [/math] tags.

You appear to be talking about a preferred reference frame -- a concept not present in special relativity. I need a bit more information before I can respond sensibly. So you are disputing the postulate of relativity? (Namely that the laws of physics are the same in all inertial reference frames -- inertial reference frames being those like the one where the flashlight thinks it's still.) Or that the speed of light is a law of physics? (ie. the speed of light will be measured as the same from an apparatus which is in a different inertial reference frame). If either of these are the case, how do you define the frame in which C=3*10^8m/s? The one where the CMBR is isotropic? Pick a random galaxy and take the frame where its redshift is zero? Non-moving compared to the centre of the milky way? The frame of Sol? These are all different (and vary by up to 500km/s or so iirc), and Earth's surface's velocity relative to these objects changes on a yearly and daily basis. If there is a preferred frame then doing experiments with moving instruments (like doing one in the morning and one in the afternoon) and assuming that they are stationary should yield different results. If you could try to predict what would happen according to your model what would happen according to instruments on the flashlight were one to assume it was stationary that would be useful. My response was based on the assumption that the flashlight can be considered stationary, and according to its instruments light moved at the same speed as it does according to instruments in other frames. This is one of the many counter-intuitive elements of SR. The galaxy clocks are slow in the flashlight frame, and the flashlight clock is slow in the galaxy frame. Neither time dilation nor length contraction make sense in isolation, it is only when you consider time dilation, length contraction and the simultenaety of relativity together that the paradoxes disappear -- as they are all consequences of the Lorentz transform.

Every time I see a physics analogy since seeing this, I think of it. Thought I'd share.

There's also a thing called an anion which is a solution to certain quantum physical equations which does not exist in three dimensions (or higher...I think?). They have certain properties which make people think they may be useful, so some physicists are trying to create really flat potential wells (sort of squeezing some stuff with lasers) to try and create them.

I need only assume that there are 3 space-like degrees of freedom, and one time-like degree of freedom. No ontology is implied, merely that it can be described as a vector space. I see no way in which 'empty volume' is any more of a descriptor of what space is than just calling it space. Same goes for time. What do you mean duration? Then what do you mean by ____ Eventually we just get back to 'some measure of change' or 'something measured with clocks. If you wouldn't mind, I'd prefer to take pragmatic definitions space: 'the thing we measure with meter sticks' and time: 'the thing we measure with clocks' Whatever they happen to be. Spacetime can be 'the thing, or things that we measure with clocks and meter sticks'. Whether space and time are somewhat interchangable (thus 'thing'), or completely separate entities ('things') If you read my response to tar, or any other derivation of the simultenaety of relativity you'll notice that once the assumption of constant speed of light is accepted, moving clocks no longer measure the same thing as stationary clocks. Will you at least concede that it's a dimension in the mathematical sense, in that it's a degree of freedom which is independant (again, mathematical definition -- as in orthogonal) from the three spatial degrees of freedom, and that it is required to describe events? This is what I (and most physicists -- when they are not discussing the ontology of space and time at least) mean when I say dimension. It relates in that any scenario we can set up in a laboritory -- or see in the world around us -- agrees with the results from this abstraction. Because the speeds and energies involved are so large, the real situations where the results of relativity differ significantly from classical physics all involve things that are very small, or very large and far away. We then induce that the same laws would apply to medium things, or close, large, things if they were moving fast. Just as one would induce that the sun still rose and set if one was underground for a number of days. Surely we have to establish what whether space and time are the same thing before we can even begin to come up with an ontology for them? No-one here that I've noticed has said anything about an ontology for spacetime. They merely use 'spacetime' as shorthand for 'a possibly unknown entity or non-entity wherein events occur such that they can be described by a four dimensional vector space' (it is also implied that one appends a 'that transforms under the lorentz group' to that if one is talking to a physicist, but we'll leave that bit off now for your sake). We get a bit tired of describing everything from scratch every time, which is why we come up with these 'sequences of vocalisations which may also be represented as a series of glyphs' or 'words'. I hope you'll let me use them without having to come up with an ontology for them first. No-one said anything at all about events being different in different reference frames. In fact the very point of the entire matter is that they aren't -- and this is the entire point (along with constancy of speed of light) of relativity and science in general. Let's say I have a vector pointing from the event in which bob trips over and skins his knee at the south pole, to alice sitting on santa's lap at the north. They happen at the same time. It might go[math] [x,y,z,t] = [0m,13000000m,0m,0s][/math] (note, all these numbers are rounded to 2 sig figs or so) No matter which coordinate system you use it is still the same vector. I could describe it as [math][0km,13000km,0km,0s][/math] if I wanted, or in miles or whatever. I could also describe it from a coordinate system which is rotated, it might go [math][13000000m,0m,0m,0s][/math] or [math][9000000m,0m,9000000m,0s][/math] depending on how I rotate my frame of reference. None of these is any more correct than the others. There is no absolute angle which all measurements are correct in. Now if I boost my frame of reference to 0.866 times the speed of light (a very similar operation), I might get (according to special relativity): [math][26000000m,0m,0m,-0.075s][/math] So the events happen further apart and at different times. If I take some different events, so they are at the same time I am no longer measuring the same thing. Say I take a slightly earlier event, where bob is in the process of tripping over, I get: [math][26000000m - 0.866c*0.075,0m,0m,0s] = [6500000m,0m,0m,0s][/math] This is the origin of length contraction. @capnrefsmmat Awww, he's (mis)quoting you to disprove me. Given our recent interactions I find that not the least bit hilarious. Without going back and re-reading his posts, I will posit that Capn was trying to get across the concept that there is no correct reference frame. They are all equal. We can define things like proper lengths of collections of events that we call a solid object, but if you had something like....three objects passing each other such that they were in a straight line at some point, and were to ask the question 'what is the length of the line' there is no single reference frame which you can single out as the one in which that is a proper length. You say you agree to constant c, but you steadfastly refuse any of the trains of logic which lead from there to the lorentz transform. Again, I'd like you to state the assumptions you're operating on, because as far as I can see some of them are contradictory. Perhaps I can help you along -- which of these are you assuming? The Galilean transform will give correct results when changing from one frame of reference to another, There is a well defined stopped reference frame from which all measurements can be taken Measurements taken from this stopped reference frame will differ somehow from those taken in another in a way that measurements from two different moving reference frames do not. The speed of light is a law of physics The laws of physics are the same in all non-accelerating reference frames

The main thing I see comp-sci students struggle with is the unexpected maths when they start doing algorithms. Pay attention in math class, especially when you do limits/calculus and discrete (logic and set theory and all that). Then again this could be because I tutor maths and not programming -- they probably go to someone else when they are struggling with that. Something like this can be good for practising some of the skills you'll need (and it's pretty fun if you're a nerd like me), especially if you do it in a lower level language (C, assembly) or even some strange esoteric one (but maybe not Befunge). The more you think in a logically structured way (whether it be maths, programming some games or solving puzzles), the easier it will be. If you crack open a programming book (or the internet) and start programming something -- or program mario in your graphics calculator when you're bored in class -- you'll learn a lot, especially if you do a bit of wheel re-inventing along the way (such as choosing and implementing your own sorting algorithms rather than using some language's built in sort function).

And all your four-vectors (and tensors, I suppose, but let's not go there -- also their components may change, but they are still the same vector), four-acceleration, energy-momentum...and so on. Events. If I explode in one frame, I will explode in every frame, it's just a matter of when and where. Order of events along a time-like path, (or a space-like path not sure if it's mixed....probably still ordered?) We can keep going if you like?

For one, spacetime intervals. If I stick a clock on each pole of the earth, and synchronise them (to the same time-zone), then record the time and place they are in when they each show 3:47:00am December 12th 2011, then the quantity c^2t^2-x^2-y^2-z^2 will never change no matter which FOR I am in. Things that do depend on frames of reference: Time and space intervals. The two events (different clocks reading the same time) will happen at different times in different frames. They will happen at different places in different frames. Only the combination (time and space) is invariant. Think back to that example I gave you of two of a circle. If you do not consider the z axis when considering it from different angles (frames of reference) then it will get shorter. Your refusal to consider the combination of spacial and temporal interval as being the invariant is exactly absurd to anyone who understands relativity as refusing to acknowledge the z axis for measuring the shape of a circle of paper. But over and over again your present the two options: 1) Spatial intervals do not change between reference frames. 2) Reality is subjective. Everyone here has responded at some point or another with: 3) Reality is four dimensional. To which your response is: No it isn't, pick 1 or 2 Owl, you cannot attack length contraction in isolation, it is a direct logical consequence of the lorentz transform. No scientist got up one day and said 'oh, I know. I'll just say that lengths of objects get smaller if they're moving.' They reluctantly accepted it as a logical consequence of adding two assumptions to existing theory, (or arguably only the second): 1) The laws of physics are the same in all inertial reference frames, and 2) The speed of light is a law of physics. Whenever someone tries to walk you through the chain of logic that leads from these assumptions to length contraction you tell them they're performing mind games or just refuse to respond.

No, he KNOWS that the detectors are moving, so are all the stars. He's staying still. Good thought experiment, but you can't rely on a moving clock to measure time if you assume the speed of light is constant. Now consider what happens from the light's perspective. According to the light it is still, and the buoys are moving. All of our mathematics and physics must work in this case just as well as the other case or they do not obey the postulate of relativity (laws of physics are the same in all inertial reference frames). At some point bouy 1 passes it and displays 12:00:00 (let's say they pass really close so your reading is accurate, or you have some other means of synchronisation). Some time later buoy 94 passes, reading 12:00:01 and the flash goes off. Now: You have a beam of light going at c in either direction, one of the buoys is moving towards you at 0.5c, the other away. Let's start with the approaching one: Let's say the distance to the buoy is 1 unit when the light flashes. We don't know how big a unit is yet (unless we make some assumption about length contraction, but we can use it nonetheless. The light hits it when [math]ct=1u+0.5ct[/math] or [math]t=\frac{1}{2}[/math]u/c time units in the future. We know that the clock reads 12:00:01.5 The light hits the other buoy some time later, when [math]ct=1u-0.5ct[/math] or [math]t=\frac{2}{3}[/math] u/c time units in the future. We now know that the two clocks displayed the same timestamp at different times, so the thing they measure can't be the time of the headlight. We can do further thought experiments (or if we're clever use the data from this one) to figure out what exactly they are measuring (a combination of space a and time) in this frame.

I whipped up a quick demo for this here -- bear in mind it's still rather alpha so the interface is rather clunky, if it doesn't work on your browser or acts funky I'd like to know so I can fix it. Click on twins' paradox, un-pause it and watch the demos, should explain better than words, especially if you look at the space-time diagram below. They will both agree that the travelling twin is the younger one when he gets back. The key is that the stationary twin did not accelerate. When the travelling twin went from moving at 0.866c (the speed to age half as much) to -0.866 c, the event he considered 'now' changed dramatically. When he was travelling away, twin a was aging younger than him, and had only aged 12.5 years at the point twin b decided to turn around. Then when twin b turned around, the twin a that he considered to be in his present was suddenly 12.5 years older than him. The twin a that is 12.5 years younger than him is suddenly 25 years in his past. Velocity doesn't add linearly in a relativistic world. If twin b and c both leave a at 0.866c, then from twin b's point of view a will be moving at 0.866c and aging at half the rate. twin c will be moving closer to 0.99c and aging at 1/7th the rate.

Yup! That is if you meant [math]-(i^2)[/math] (which is what your notation would imply [math](-i)^2[/math] is also -1

Expanding it in series form doesn't seem to help -- at least with anything I know, other than a numerical approach I don't have a clue. Wolfram alpha confirms that it's rather ugly. http://www.wolframalpha.com/input/?i=solve%28x*%28csc%28x%29+%2B+1%29+%3D+pi%29+for+x

Not quite sure exactly where you went wrong, but there's a much shorter way, at least to get as far as power per area (watts per square metre rather than candela). If you want candela you have to take into account the sensitivity of the human eye. The logic for getting flux over one square metre from the luminosity of the sun is [math] \frac{L_{sun}}{A_{earth\, orbit}} \times 1 m^2[/math] Or for power per square metre [math]\frac{L}{A}[/math] Also the figure found for luminosity disagrees with yours. [math]\frac{3.85\times10^{26}}{4\pi \times1.496^2\times10^22}[/math] This comes to about 1300Watts To get luminous intensity from this you'll have to do an integral, or look up the ratio between power per square metre and luminous intensity for a black body at around the sun's temperature. I can explain this step in more detail if you need it.

Let's just assume what owl said is right for a moment and see where it gets us (I know, I know, most of you have done this thought experiment long ago, but there are some here reading that haven't), either we come to a logical contradiction, something that disagrees with experiments (such as Michelson Morley) a new prediction, or relativity is illogical. . Let's start with this one: The speed of light is constant -- Owl, could you elaborate on this? Maybe I can clarify my thought experiment: I'm going to use earth time/distance, and ship time/distance just to be clear. Even if they turn out to be the same. We're on earth sitting still. There's an object moving in this world, say a space-ship, at 0.9 times the speed of light. Let's say it's one light second long. A light in the back of the ship turns on. The light will move 1 light second in one second, but the ship moves 0.9 light seconds in that time. So one second after the earth-time the light turns on, the light beam is 0.9 light seconds away from the front of the ship. We've put some equipment or a scientist on the ship, it makes the same measurement. Turns the light on, sees which detectors have lit up 1 ship-second later Owl, according to your model. Will the front of the light beam be: a) 0.1 light seconds along the ship when the moving scientist takes his measurement, or b)1 light second along -- at the front of the ship? Also Owl: If you come on the chat we can discuss this much faster, and I can get an idea of what your model means.

Uhm, I assume you meant C++ rather than C+ (or perhaps C#?). C++ is an extension of C to include more modern features (such as objects). C# is more recent still. I don't know much about it, but I gather it is somewhat like java. It's less that java and C++ are new/old and more that they have different goals. There is still active development of C/C++ compilers (don't know about the language standard itself, but this is less important. I think the languages may have everything they should need already). Java is based on a virtual machine, and has a lot of high level features like garbage collection (it cleans up memory for you when you're done with it). A full explanation quite involved, but if you don't want to go into detail: Basically it does a lot of stuff for the programmer, often making his life easier. The down-side of this is there is more overhead (the minimum amount of stuff you need running to make your program work), and for some tasks it is slower than C or C++ (probably not in any way you or I would notice, your program also has to be written to take advantage of the extra speed of a compiled language). C was invented first, because when computers had very limited resources and the overhead of a virtual machine wasn't often worth the advantages.

There may be one other way out, some kind of non linear re-definition of either 'now', distance, or velocity (or how they relate) with respect to position. I struggle to think of how this would work conceptually, and cannot really be bothered to do the maths for the vague possibility that there is some hugely convoluted theory that is consistent with experiment for one or two additional logical steps before it breaks down. Also, you can go the lorentzian relativity route and say that moving objects actually do get smaller because they are a doppler shifted wave in a (completely undetectable and otherwise irrelevant) aether. This means that when you are stopped relative to this undetectable reference frame what you observe is the true now and true size of everything, and all other observers see things incorrectly because they --and all their instruments -- are doppler shifted. As far as I know, this will work until the point where you try and think about gravity -- or look at Occam's Razor too closely.

I weep for your students and teaching assistants.

Okay, I'm assuming there can still be things, devices and such in your world. So let's set up a thought experiment. There's an object moving in this world, say a space-ship, at 0.9 times the speed of light. Let's say it's one light second long. A light in the back of the ship turns on. According to what you said, it will take 1 second for the light to reach the front of the ship. According to some stationary (moving at 0.9 times the speed of light with respect to the ship) instruments (in your model) the ship is still one light second long. The light at the back turns on. One second later, the front of the ship has moved 0.9 light seconds. Now either: The light moves at 1.9 times the speed you stated according to the stationary equipment, or the light does not reach the front of the ship in 1 second.

I don't get 11/6 unless I've dropped a minus sign (which happens frequently enough). But the method is easy enough. It's a bit easier to see if you take 1/x^2 out of the brackets. [math] \frac{1}{x^2}\sqrt{ \frac{x^4}{2}+\frac{x^8}{64} + 4}[/math] [math] \frac{1}{x^2}\sqrt{ (\frac{1}{8}x^4 + 2)^2}[/math] Remove the square root and you're away