timo

Good question. But not a valid one in GR. Time dilatation and length contraction can be considered as an effect of changing your base-vectors when switching from one coordinate system to another. But in GR vectors are only defined in a point of spacetime. So you -usually- can compare only vectors in the same point. And the effect of graviational time dilatation (whatever that´s supposed to be) is of course the same for all observers in one point. Also note that the statement "in a gravitational field, time slows down" is vague at best. It sais that Eigentime increases slower with the time-coordinate of your coordinate system. The time-coordinate of your coordinate system, however, has little to no physical meaning.

Good post Aardvark and there is not much in it I could refuse in general. Just a few remarks: - If I come to the doctor with bad general health I´ll certainly be given priority over an older person being in better shape unless the doc is a complete idiot. Not much point in debating over this point, however. I know what you mean and I think you also know what I mean. - I simply feel like the importance of "one should know his roots" is generally overrated. I certainly do not deny that some people have a strong interest in knowing. Maybe you don´t know your biological parents and know what you´re talking of. But the point that practically noone takes a gene-test on his parents if he´s told that they are his biological ones seems as a hint to me that it just isn´t of fundamental importance for one´s life (and honestly: Who would like to find out his parent´s actually aren´t the biological ones?). Perhaps I´m biased in this matter as I personally know someone who doesn´t know her biological parents and has not raised any interest in them so far although meeting them -at least the mother- would be possible. One last thing: I would expect that people who are considering using the mennotincluded-service think about the point that their children will never meet their father more than both of us do (more than me at least).

>> How the second car look the first car ? The driver of car 2 would look at the first car with condused disbelief because he never saw a car doing 600 km/h before, I think. But that´s only if he even recognizes what kind of object just speeded by. Now seriously: They both would see the other one coming towards them with a speed of 620 km/h.

Gravity is spacetime-curvature. Spacetime-cuvature is cause by mass (loosely speaking). Mass is described by the stress-energy-momentum tensor (the point that stress and momentum are also included is why I said "loosely speaking" in the last sentence). Curvature is a confusing term. It´s an attibute of space-time at every point. There are several equivalent definitions for it. However, it´s not nessecarily what you expect of an attribute called "curvature". For example, the surface of a cylinder is bent but not curved. Also, the definitions of curvature does not need the spacetime to be embedded in another space - it would be a bit contradictory if the universe (everything there is) was embedded in something greater. So as Aechylus said: Those pictures where a 2D surface is embedded in 3D are just visual aids. Example showing curvature: Take a unit sphere (an apple, for example) and a vector (pencil). Put your vector in the point (0,-1,0) (note that vectors in GR have a point they belong to). Now rotate your pencil along the equator (z=0) to the point (0, 1, 0). Rotate it along the circle (x=0) to your original point afterwards. Compare with the original pencil´s direction. Try the same on a cylinder. Funny, eh?

Yes, positron. Easily made typo. Also note that "mass" refers to the relativistic mass and not the more commonly used rest-mass (or to rest-mass only as an approximation).

"Diode" as I know it is an element of an electrical curcuit consisting of two attached differently dotated semi-conductors. Don´t worry if you didn´t get that sentence, it´s probably not important for you. But what might be important for you is the effects diodes have in electrical cicuits: - They (loosely speaking) only allow current to pass in one direction. - Some of them emmit light (of a color depending on the materials used) when a current flows through them. Those are also called Light Emmiting Diodes or in short: LEDs. There´s another thread here about LEDs, but I guess that´s also not what you are looking for. - Generally, the effect of light being emmited when a cuirrent flows can be reversed in a way that a current starts to flow when the diode is exposed to light. I don´t know if you can use all LEDs for this reverse process or if it has to be a special type of diode (I suspect the latter). I´m also not entirely sure if such elements would be called "diode" at all (but I think so - even think to remember the name "photo-diode" from my old electronic set). My bet would be that this are the diodes you are looking for as I can imagine them being used in digital cameras. To sum it up: In you case a diode is probably an electrical element that creates a current when light falls onto it.

Electron positron annihilation produces way too much enery. The resulting photon wouldn´t be visible to the human eye. The problem where to get the anti-matter from would be another reason why you wouldn´t use this effect. I´m too overcharged with tasks right now (2 people on ICQ) to give further explanations. This link might help: http://electronics.howstuffworks.com/led1.htm Found it on google, there are probably other and better pages to be found easily there. EDIT: I remember having answered questions about photons and their production (photons are the particles of light, in case you don´t know that) in this forum before. Maybe a look in this thread might also help you: http://www.scienceforums.net/forums/showthread.php?t=4990

I´d guess there are reasons for the numbers used in the Ringworld novels and I´d bet these numbers -obtainability of an artificial planet way bigger than earth left aside- work quite well. Can´t give them here as I don´t own the books and read them something like 10-15 years ago, though. It´s a bit dissapointing that Kingpin didn´t bother explaining his/her question so far. My bet was that the question was about the different metrics for describing a rotational black hole (Kerr-metric) and a non-rotational one (Schwarzschild-metric). But as this is the Classical Mechanics section and not Relativity (hint@yourdadonapogostick), Gilded´s 1st post sums it up quite well (except that the distribution of the mass in space also plays a role).

Example pls, Kingpin. I don´t understand what you mean.

yes' date=' those: Do you know for sure who your biological father is? You cannot. As for "everyone should have a right to know their biological parents": Sounds nice, but seems inobtainable to me regardless of that service. In the end I don´t think that knowledge is too important (I haven´t done a gene-test to find out if my father is really my biological father, for example - it just isn´t that important to me). Being a natural scientist I have problems with the term "can be responsible". Problem one is that I wouldn´t know how to define responsibility in a system as complex as a human life. Second problem is that the word "can" to me reads as "i want to make that claim and relativize it so that noone can refuse it". Having no father figure can as well be responsible for being awarded the nobel prize in literature. The statement probably is a loose interpretation of a statistical correlation of "antisocial behavior" (however that´s measured) and "having no father figure" (easier to measure but also not trivial). While this correlation might exist, applying statistics to individuals is discrimination. - There are much more basic needs than having father and mother. - Who tells you those children won´t have father and mother (didn´t read the link so hit me if "this service is only for single women having no boyfriend or husband" is written there - I doubt it)? What about couples who can´t have children otherwise? Is a father less of a father if he isn´t the biological one? - What about couples who have children breaking up? Would you prefer a child having arguing parents who don´t like each other over the child living with only one of it´s parents?

Just out of interest and in case anyone knows this: Is the image above colored by means of image-processing? Or is it possible to get (usefull) colored pictures in an electron microscope by taking the direct information like assigning a color to certain intensities of the tuple (Secondary Electrons, Back Scattered Electrons) ? On topic: I doubt that universities offer "rent an electron microscope"-services. But if you manage to convince the rigt persons that your research is interesting they might let you use it for free under supervision (I´ve got no idea what you´re doing so I cannot judge if someone might consider it worth doing). Alternatively I´d bet that there are smaller companies specializing in doing tasks like quality control that use electron microscopes for their tasks. If you´re able to tell what you want to have measured in advance they might do this for you for a fee (but you´ll most certainly miss the fun of playing around with an electron microscope). In both cases you should really know in advance what you are going to do and how it works. Everyone will be very displeased if your sample doesn´t stand vacuum and ends up smearing on the walls of the mircroscope ...

Not really stupid but hard to tackle. For your question about gravitation at T=0 I can´t really give an answer because it´s hard for me to imagine such a system. But generally, the laws of physical interactions are unchanged by temperature - just the set of possible effects and their probabilities are (that´s very losely speaking, of course). To refer to your question about molecular motion stops/doesn´t stop: - Strictly speaking saying that temparature is a measure for motion is not even correct. You can consider spin-systems and threat them with thermodynamical methods (assign a temperature to them). But there is no motion involved at all. - In Classical Mechanics, which is a bad approximation for very low temperatures, T=0 would mean "stop of all motion" if a system where this term makes sense is considered. - In Quantum Mechanics I would have problems defining what I mean by motion of the particles (actually, I could think of a few ways but that discussion would lead way too far). However, for common systems (probably all) the kinetic energy will be >0 even at T=0. That´s probably what the guy from wikipedia meant by saying that molecular motion does not stop completely - which is, imho, a false statement.

Maybe it would help if you explained the experiment a bit better as I wouldn´t be so sure that everyone knows it. For example: Where is time measured at all? Or even more general: What and how do you measure? Another thing I´d like to mention: I don´t know how the others feel about your note you put on the end of your posts but to me it sounds pretty arrogant (are you an ongoing physicist, perhaps?). Maybe you should skip that and learn to tell usefull statements from crap. An ability that´s usefull quite often, anyways.

There´s another very simple explanation: Eigentime -the ammount of time that the respective twin ages- is the length of his trajectory through spacetime. Since they don´t take the same path through spacetime the length of their paths differ (in this case) and hence they have aged differently when they meet again. btw.: Didn´t bother reading the link provided above, so maybe it´s also said there - but I doubt it. EDIT: Read it. Quite an ok explanation although they used the "twin two is no inertial frame all the time" argumentation which I don´t like because the concept of inertial frames is confusing at least, as Companiero allready showed.

As allready mentioned the binary system is just a way to label the (natural) numbers. Now give each letter a number and you can store texts by storing the according numbers. Same for pictures: Give each color a number and store the numbers corresponding to the colors.

a) no. b) Also not the same as the E = mc² + (pc)² that you were complaining about. c) Now I remember why I usually ignore certain threads so don´t expect me to participate here further.

The correct equation is E² = (mc²)² + (pc)²

We had a similar experience with our photos from last year´s sylvester. On both group photos we shot one of the girls had something that looked like a lightning bolt impacting in her head. We hab absolutely no idea (but a lot of abstruse theories ) what it was until we came by the place we made the photo the next day. Actually, it was a tiny branch that was directly in front of the camera which then appeared lightning-like due to being directly in front of the flashlight. Too bad I don´t have the photos anymore so I can´t show you what I mean. But I´m not a photographer so I can´t really tell you what the orbs on your pictures are.

Countable and uncountable are mathematical terms describing a 1st approx of the cardinality of a set. A set is countable if there´s on a bijection on a subset of the natural numbers. It´s uncountable if there´s not. Might sound stupid to you but it´s elementary (1st semester) math and also quite important for integration in the sense of Lebesque-integration (sets with a measurement of zero can be left out of the integration - countable sets usually have a measurement of zero) and other scientific areas (Poincare´s theorem about the return of orbits would come to my mind, spontaneously). Still drunk

The amount of integers between 0 and 1 is zero. The amount of rational numbers between 0 and 1 is countable (infinite). The amount of real numbers between 0 and 1 is uncountable (one could say: Even more infinite). There is no "between 0 and 1" for complex numbers unless further specification. The question is not valid for imaginary numbers because 1 isn´t an imaginary number. What numbers do you talk about? I´m drunk.

Yes. The dirac-delta is also not a function in a mathematical sense. Think of it as a ... aehrm ... method (wanted to write "function" but that would have sounded curious in this context) that gives you the value of a function at a certain variable in the sense of [math] \int_{-\infty}^{+\infty} dx f(x) \delta(x-x') = f(x') [/math] In your example above that would be f(x)=1 and x'=0. From a technical standpoint it´s more or less the same as the normal delta which is, for example, used to extract certain terms from sums. Just that the sums are integrals here (which are also quite similar to sums). The missing complex conjugation was the minor problem as it´s obvious. I don´t know if it´s really worth remembering these conditions as for most problems you usually simply assume you have a complete base. At least I didn´t know the exact conditions (and the way they´re usually written down) out of my head - would have had less trouble with your question if I had, though

The 1st equation simply sais that the states [math] \{ \psi_n \} [/math] form an orthonormal base (ignoring that you forgot the complex cunjugation on the 1st factor). This, however, does not say it´s a base for the whole vector space so it does not show completeness. The 2nd equation is flawed, at least because for x' = x: [math] \sum_n \psi_n(x) - \psi_n(x) = \sum_n 0 = 0 \neq \delta(0) [/math]. I´ll try to figure out the correct expression because I´m also interested in that but it´ll take a minute. ---------------------------------------------- Hopefully final EDIT: The 2nd equation is (staying within your notation) supposed to be: [math] \sum_n \psi_n(x) \psi^*_n(x') = \delta(x-x') [/math] (*) The base [math] \{ \psi_n(x) \} [/math] is complete if any state [math] \phi(x) [/math] can be written as: [math] \phi = \sum_n c_n \psi_n(x) [/math]. Or in words: You must be able to write any vector as a linear combination of base-vectors. Proof that a set of states satisfying (*) is a complete base: [math] \phi(x) = \int dx' \delta(x-x') \phi(x') [/math] [math] = \int dx' \left(\sum_n \psi_n(x) \psi^*_n(x') \right) \phi(x') [/math] [math] = \sum_n \psi_n(x) \int dx' \psi_n^*(x') \phi(x') [/math] [math] = \sum_n <\psi_n | \phi> \psi_n(x) [/math] [math] = \sum_n c_n \psi_n(x) [/math] Remarks: a) To sum it up: The first equation sais that the base is orthonormal. The second one sais the the base is complete. The first equation is certainly a hidden condition in above proof but I´m too lazy to double-check it to find out where. b) Please take more time to check your equations next time. Both had more or less serious flaws in it and it can take others (me in this case) quite some time to find them. Just because you saved two minutes on double-checking your post. c) One of the more interesting questions, I´ve read here.

If at all it´s [math] |v|^2 = c^2 = c^2v_t^2 - v_x^2 - v_y^2 - v_z^2 [/math] (assuming time is measured in seconds and space in meters). It actually IS common to work without units or lenght-dimensions for time' date=' especially in theoretical physics and mathematics. But you should know what you are doing, then . [math] v_t [/math] in above would be "coordinate time" per "eigentime". The units depend on what units you chose for them.

Call the "letters" "digits" if that´s more familiar for you. I should indeed have taken the time to look that word up before posting to avoid unnecessary confusion. But in the end the point is just that you have a finite set of symbols (ten for decimal, two for binary) that you name the infinite number of natural numbers with. What does the thing in parentheses mean?

@Mad Mardigan: Your table is flawed from line "11=..." on.