Schrödinger's hat

That was the other term that should show up when you use the quotient rule from your answer. Don't have time right this second to work it from the beginning, but it looks like the type of thing a trig substitution would work on. Another option would be to try a different variable. Rearrange for y, or find some other parameter that takes you along the curve.

So you're saying that no matter what is emitting the light, it will always arrive at its destination moving 186,000 according to the people at the destination?

You didn't really answer my question. What would someone moving towards the sun at 180,000 mps measure?

Good good. I was getting a little worried there. If I was that clueless on a separable ODE, something somewhere had gone terribly terribly wrong.

With respect to? A galaxy? A planet? Everyone?

The original integral was [math]\int \frac{x}{x^2 - 1} dx[/math] I added the 2 and the half by multiplying by 1=2/2, take the 1/2 out the front then you have 2x.

Pardon? I don't quite understand (was this even directed at me?).

Not quite following your reasoning. If you were to expand that w/ product rule the function in the integral would be one term, the other would be [math]\left(\frac{2}{3}\left(\frac{1}{2}+\frac{x^4}{64}+\frac{4}{x^4}\right)^\frac{3}{2}\right)\frac{d}{dx} \left(\frac{1}{\frac{x^3}{16} - \frac{16}{x^5}}\right)[/math]

Check your integral? Show us the rest of the working there or explain the process by which you got from the second to last to the last line. I think that may be at fault (I can't see anything else wrong).

For the right terms you'll have to go to your textbook or someone versed in biology, but I can explain the concepts. You have a gene, call it Color, it has the yellow (y) form and the green (G) When we breed a YY with a gg (plant that breeds true yellow w/ true green) we get the following possibilities | G G y yG yG y yG yG Because each parent had two of the same gene, there was only one possibility. We can also know that the green pods are dominant because the all the hybrids had green and not yellow pods. For the F1 cross with another F1, we have four possibilities, I'll fill in the first two for you: | y G y yy yG G Same principle for d Each row in the square represents a possibility from one parent each column represents a possibility from the second parent For the names you'll have to do a bit of research, pretty sure the pea colors thing should be in most textbooks (or try wiki).

So what you're saying, Iggy, is we'll never catch the dragon bus and no matter how hard we try we'll never get as high close as we once were?

Don't stress about it, we all make silly errors. It doesn't seem correlated with intelligence at all (you should see my mathematician friends trying to do arithmetic sometimes -- esp after an all nighter). If it's bad enough to be a problem then the only ways I know of to improve it are to do lots and lots of practise or marking assignments. Even then the effects are only temporary. If you want some practise exercises one place to get auto-generated (basic arithmetic and algebra) ones is the practise section of khan academy. Alternatively just go through your textbook and do all the exercises and get algebra practise that way (doesn't really matter if they're calculus exercises, there's still algebra practise). http://www.khanacademy.org Double check that (what happened to that two?). There should be a square or square root somewhere. Tip: Always put the function you get back through your ODE to see if it works. Also, if you want to know how to integrate something like that, use u substitution. The general theme is look for something that looks like: [math]\int\frac{du(x)}{dx}f(u(x)) dx[/math] Then we can use the chain rule to turn it into: [math]\int f(u)du[/math] So we see that the derivative of [math]x^2-1[/math] is [math]2x[/math] To get a 2 we can multiply by 2/2, then take the 1/2 out of the integral to get: [math]\frac{1}{2}\int\frac{2x}{x^2 - 1} dx[/math] Set [math]u=x^2-1[/math] which leaves us with: [math]\frac{1}{2}\int\frac{1}{u} du[/math] or [math]\frac{1}{2}\ln{u}+C=\frac{1}{2}\ln({x^2-1})+C[/math] You may want to use a few more steps (or if you don't follow I can explain further -- this is a bit terse).

The speed of light is the ratio of these two things in your own frame. If you want to make your measurements agree with someone in another frame you have to include your (and their) measurements of time and space. Much as you have to include measurements of how far in the x direction, the y direction and the z direction to get a true measure of the distance from one side of the paper circle to the other. Then we agree, but you are presenting a false dichotomy. 1) The world is three dimensional and relativity is correct -- thus the world is subjective. 2) The world is three dimensional (and objective) and therefore relativity is incorrect Whenever anyone presents option 3) The world is objective and four dimensional you just respond with 'I think it isn't'. It is obvious that assuming the model on which relativity is based is incorrect will lead you to finding paradoxes if you assume the results of that model are true. Now we're getting somewhere (hur hur, see what I did there). The notion of now has no physical meaning in a relativistic world. You can define a set of events and call it delta t=0, but that's doesn't make them more or less important than any other random set of events. All relativity leaves us with is: The past (things that can effect us). This is defined by one's light cone (the set of events that are closer than ct away from you, where t is the amount of time since the event happened). Here and now (this event right here, right now, that is being effected by the past and effects the future). This is what people talk about when they say the present is only defined locally. The future (things we can effect). If you assume 'now is now everywhere' you are going to reach many paradoxes. It's not denying the well established description, it's extending it. Earth science doesn't deal with high velocities so they do not have to consider the four dimensional nature of the universe. Exactly. It is objectively a four-dimensional shape. Not any type of sphere or sphereoid. Only when you restrict yourself to looking at a thin slice does it look like a sphere or oblate sphereoid. Objective reality is four dimensional. You are right to be bewildered. Most modern physics is highly counter-intuitive, and this discussion got right into the thick of things rather than starting at the beginning. You can extend your understanding and imagination in a number of ways. The most common seems to be 'just shut up and calculate', whereby the student of physics is made to calculate results, and do experiments which depend on these results over and over again until it starts to become intuitive to them. This method is rather inefficient. Another is to make analogies to the Euclidean and three dimensional equivalents of certain concepts. I'll eventually get around to writing something on this. Unfortunately everything I know of written on the subject requires at least some facility with mathematics. Perhaps someone else here can suggest a good book. Setting C to 1 and using that system of units (which is not SI) is very common among mathematicians and computational/theoretical physicists -- much to the chagrin of anyone who has to do experiments with their results. Ignore the fact that distance is defined by time for now, this is a result of the constancy of the speed of light. The constant thing that measures space can be a ruler, or a tape etc. The constant thing that measures time can be a clock (or oscillation of some radiation) etc. If you keep your clock still relative to your ruler, and use them to measure the speed of a beam of light you will always get the same result no matter where you are or how fast you're moving (if your clock and ruler accelerate or are in a strong gravitational field you won't always get the same result -- it'll depend on the acceleration). If you are in a different frame to the ruler and the clock you'll get the same result for the speed of the beam of light, but you'll disagree on the position parts of the apparatus were in, and the times at which the other frame made their measurements. If you want a full explanation from the beginning I'm happy to provide it, but I do not think it is in the scope of this thread. Also there are plenty of explanations around which people smarter than me have put more effort into.

I'm not entirely sure this is the right time of night for me to be dividing things, but isn't that: [math] \int\frac{1}{y}dy = \int \frac{x}{x^2 - 1} dx [/math] Which will have a somewhat different solution (if you're stuck, I'd probably try u substitution first). As to simplifying the exponential, I'd write that something along the lines of [math]Ae^{\frac{x^2}{4}(x^2-2)}[/math] Which is still a bit more messy than is usual for contrived exercises. Looking at the integral formed above the logs will probably cancel and you'll get a nice polynomial.

Saying 'saying so doesn't make it so' doesn't make it not so either. Your point is? Mine is that the 4D spacetime picture of things is entirely consistent with the maths and with the experiments we've performed. If you want an example of an experiment you can try at home which does not work if we replace the relativistic frame change equations with ones which keep earth as a sphere, try this: Get a battery, a length of thin wire, and a nail. Wind the wire into a coil. Connect the battery Lift the nail with your electromagnet. We know of no reason that the moving current should effect the nail without taking relativity into account. It was this very effect that led people to coming up with the Lorentz Transforms some time before Einstein was around, they just didn't know why they disagreed with the Galilean transforms. No it's not. A slice of Earth at any given time is a near-sphere. Earth at all times is more like a helix (if you're looking at it from, say, the sun, or Pluto). If you slice it right you can get an oblate sphereoid. We just call it a 3D sphere because that's good enough. You're almost there! Now if earth (locally) were a 4D hypercylinder and you rotated it then took a slice, it'd look like an oblate sphereoid. Length contraction is a logical consequence of the Lorentz transform, so is time dilation. If Maxwell's equations are correct then the Lorentz transforms are, too. The Lorentz transforms are also the only linear transformation of coordinates that maintain a constant speed of light(and one or two other conditions, like Newtonian physics being a rough approximation). There are probably other transformations, but Occam's razor told us to use this set. You say this is obviously absurd. The entire universe -- and by extension all of physics -- is absurd. Get used to it. The scientific community (and all science students since) begrudgingly accepted such things because it is the simplest logical explanation for what we observed. I am quite familiar with ontology. I was explaining why you got a tautology when you asked a scientist to define it. No-one I know of has a better one. Your talk of durations between events is just as tautological. When a scientist talks of time (unless he's a crazy theoretical physicist like abj -- they may have an actual definition), it's just filler for 'the amount of __unknown thing/process/material or medium__ that causes the radiation from a Caesium atom to oscillate 9,192,631,770 times'. 'time dilates' is just shorthand for clocks (and all observable processes) moving slower. This is exactly the same as pointing to a 1m ruler when asked what a metre is, or holding up your hands when asked what space is. One thing we do know, however, is that the mathematics (which accurately predicts many things) indicates that metre sticks for one person will sometimes measure time for another, and clocks in one frame measure distance in another. Oh, and I re-iterate: Please state your assumptions, or at the very least answer this question: Is there a global and well defined present?

Both of these. Space and time are the same quantity. As far as spacetime being malleable, I do not understand General Relativity well enough to discuss it with you. As far as I can gather the discussion has only been about flat spacetime. If you wish for the reasoning behind this, either open up any relativity textbook, or come to Brisbane, Australia and we can discuss it over coffee. It's not the kind of thing that can be explained easily in a forum post. None of us are saying that reality depends on frames of reference. We are saying that measurements of spaces, distances, and times depend on frames of reference. We're saying that distances change in different reference frames because distances are not a true measure of reality. Maybe another Euclidean example will help. Get (or imagine) a piece of paper in front of you on the desk/table/floor. Cut it into a circle this is your two dimensional Earth. The surface of your table/desk is space, which has the x and y dimensions. The vertical dimension is time. Now, tilt your circle on an axis parallel to x so it's almost 90 degrees. Measure its length along the x axis You'll find it's almost 0. If you restrict yourself to only making x and y measurements of where the edges of the circle are you find that the circle is now an ellipse. For the people on the circle, the table contracted the same way. Reality hasn't changed to make the circle an ellipse (or to make the desk far shorter), it's just that you are no longer measuring something that reflects reality (you need to measure [math]\sqrt{x^2 + y^2 + z^2}[/math] not just [math]\sqrt{x^2 + y^2}[/math]. Down to the presence of a minus sign Boost (change in velocity) is exactly like rotation. This process is almost exactly mirrors (to be a precise Euclidean version of length contraction you'd need to use a cylinder, and you'd find that length would dilate) what you describe by taking measurements of earth from different reference frames without taking into account the times at which different points (note that 4d points are events) appear. You need to measure [math]\sqrt{c^2t^2 - x^2 - y^2 - z^2}[/math] (interval -- sometimes the thing inside the square root is negative, so we usually use interval squared, not interval) not [math]\sqrt{x^2+y^2+z^2}[/math] @owl, this description and model is -- to the best of my knowledge -- the mainstream one and is based on certain assumptions (namely the constancy of the speed of light, and liberal use of occam's razor to exclude things like lorentzian relativity). As you have not stated a set of assumptions for this discussion, people have been using what is largely regarded as scientific fact in their arguments. If you disagree with this (and are not starting a discussion on the premise that they are/might be false) then the onus is on you to disprove it. If you hold some assumptions (ie. there is a universal, well defined Now) to be more or less important than other assumptions (ie. the speed of light is constant), you are going to come to completely different conclusions. Of the two assumptions, one is known not to come into contradiction with any known experimental result. The other leads to something like Lorentzian relativity. I do not know how well this holds up in discussions of gravity, but it does give some valid results in special relativistic scenarios (note that in this theory things actually do change size if they are moving, but they do it in such a way that you can't tell which one is moving). It is also rejected by mainstream science because it is more convoluted (involves an un-measurable stationary aether). This is the same answer you will get from a scientist if you ask any other question. We can break things down to their parts, but when you ask 'what is time' or 'what is an electron' eventually we just have to point at one. There often comes a time when we can break a concept down further, but then you will just ask 'what is _smaller part_' and we're back to square one. I think you have made it abundantly clear that you don't. I also think that you should make an effort to understand the concept of minkowski spacetime (even if you consider it wrong) before discussing alternative interpretations of the mathematics. This is not the only assumption you're operating on. You appear to be operating on a number of assumptions, some of which are contradictory, or at least contradictory with assumptions that everyone else on the thread is operating on. @everyone else: This is a philosphy thread, not a science thread we should stop bashing our heads together and figure out what exactly it is that owl is trying to discuss, rather than trying uselessly to teach him concepts from relativity over and over. @everyone Let's lay down some base assumptions. Here are some possibilities. I am not stating that these are true, just that some of the participants may have been using them with or without everyone (including the poster) realising. 1) The speed of light is constant in all inertial reference frames 2) There is a global and well defined now 3) Three coordinates (x,y,z) are sufficient to describe shapes 4) Time dilation is a proven phenomenon 5) Length contraction is a proven phenomenon 6) Aether theories have been disproven for long enough that nobody cares about them 7) Space is the same type of thing as time and we live in a 4 dimensional (3,1) hyperbolic universe 8) Spacetime is flat 9) Gravity doesn't exist 10) There is a global and well defined coordinate system 11) Reality is objective My posts as far as I recall have assumed 1, 4, 5, 6, 7, 8, 9, 10, and 11. 8, 9 and 10 are not actually true, but they greatly simplify discussion.

This question is rather non-sensicle I gather you are asking which element, after hydrogen fuels our sun? Do you mean chronologically or percentage? As I understand it the amount of helium fusion going on on Sol at the moment is negligible. Question 2 is also a bit ambiguous One could say that a beetle is a bit larger than an ant and for (some kinds of ant and beetle) the ratio of their lengths would match the ratio of Jovian radius to Solar. Alternatively you could talk about mass, the sun is 1000 times as big (massive) as Jupiter which would definitely be none of the above. This is why we invented units and numbers Other questions/topics you might want to explore: How much matter does the sun turn into energy every second? How far away is it? How old is it/how long before it runs low on hydrogen and expands?

Perhaps a chrome drawbar or doornob or similar?

Just look at your eigenvalues and think about the behavior if you perturbed your system slightly from your fixed point. If one or more of your eigenvalues is has a positive real part, there exists the possibility that you'd perturb your system in such a way that the distance from your fixed point will increase. (i suppose this is roughly equivalent to a saddle) If they are all negative then no (small) perturbation will cause the system to leave that point If they are all positive then the point is unstable If some of your eigenvalues are the same you will get degeneracies and so on

In the past I've found the wiki maths/physics articles to be obtuse to the point of being completely useless for learning any new concepts. They made great reference once the concept was understood. I suppose there are two explanations: 1) I am reaching the level with maths where I understand the new concepts with less dumbing down; 2) Someone who is better at explaining has been doing a lot more writing.

I suppose you could call any degree of freedom a dimension, but this generally one is talking about something measured in the same units as space (time and space are often both measured in metres or seconds -- or natural units -- by relativists). If you're talking about the fifth (and higher) dimension as it is used by string theorists, then I only have the vaguest idea of what it means for one dimension to be smaller than the others. As I understand it, only one of the dimensions they refer to is time-like, the rest are all space-like. Going from three spatial dimensions to four is just like going to three from two. Take the example of making a cube. You get one square, lay it out in your two dimensions (x and y) Place another square on each edge fold the net you have made at 90 degrees in a direction that is not x or y -- call it z -- at each join. then place a square on top. If you had access to four dimensions you could make a hyper-cube the same way. Get a cube put another cube on each face (6 more cubes) fold the net you have made 90 degrees in a direction that is not x, y, or z at each join place another cube in the gap One other way to build intuition is to look at projections of 4d http://www.urticator.net/maze/ http://sourceforge.net/projects/dtris/ If you have four spacial dimensions and time -- that makes five. Apply the same process again to get six etc etc

Wiki articles on such things are improving. That diagram for parallel transport is the best representation of the concept I've seen

If it were some force of gravity between objects, then different clocks would be effected different ways. On top of that, when the clock is in free fall there won't be a force between different components (other than tidal forces). I don't quite follow. If you're saying that a moving object would produce a gravitational force in its own frame, then this would imply a preferred reference frame and provide an easy way to disprove all of relativity. As to the gravitational field around an object that is moving in a frame, you would probably have to ask someone more knowledgable to me. It will change the field, but I'm not exactly sure how. Photons do carry energy-momentum. If you were to somehow keep 10^17 joules of photons which were all travelling in different directions together in a tight ball, it would have a similar gravitational effect to around about a kilogram of matter (at least until they spread out a few nanoseconds later).

Yes, engineering would probably be the best place for this. (or physics if you want to discuss theory). Not quite, it's sort of a combo of both. Could be interesting. @zombieSquirrel: 1) Don't electrocute yourself. 2) Don't electrocute yourself. If you're using high voltage, find some way of isolating the high voltage parts of the circuit so that you only turn them on when you're a safe distance away. If you want to use the projectile to bridge the contacts, make sure the current runs the right way as you will also be making a rail gun. Also, there's often a reason why people haven't done these things before. Perhaps draw up some diagrams and we can think about how the magnetic fields will interact (probably orthogonal?).

One confounding factor is angular momentum. Neutron stars can spin so fast that the angular momentum can provide some (small) measure of support. Naively I would think this would alter the mass required for black hole formation.