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Everything posted by Freeman

  1. OK, I have a stupid question, but if one had the equation: [math]\delta (g^{ij}(t+\delta t))/\delta (g^{ij}(t))[/math] wouldn't that be equivalent to [math]\partial_{t} g^{ij}(t)[/math] and thus definitionally [math] -R^{ij} [/math] via the definition of the Ricci flow?
  2. I told my professor "Uh...I think I should look this up for reference." He says "Well, there's only one text that comes to mind, but it's a rather hard introduction it's a book by Henneaux"! So I went to the library, and I checked it out, and it is well beyond something as simple as I'd like.
  3. Is there something a little more "user friendly" than Henneaux? It appears more heavy than I can handle at the moment.
  4. Hello, I didn't know where to put this (since it's not really fitting anywhere else), but I am kind of learning variational calculus and more specifically Classical Field Theory (Hamiltonian mechanics and the like) and I need a good book (or even a explanation) that explains second class constraints really well. I am familiar with Lagrangian and Hamiltonian mechanics, but I am a little rusty on the Poisson bracket (especially when using vectors with indices!). The reason I ask is because I was talking about a field theory (I can't remember it now, it was just a toy model) with my professor in his office hours, and he goes to the chalk board and says "Well, bing bing bing, you have this as a second-class constraint and zoooop you have this Poisson bracket and bing bing bing it doesn't vanish and looks really nasty. You'll be dealing with delta functions and more fun." And I sat there dazed as he did this; something told me I needed to read up a bit on it. Again, any help would be greatly appreciated!
  5. Cheers! Thanks for all the help everyone, it's really helped me better understand the use of variational techniques (and not just in general relativity )!
  6. I'm actually interested now, how would you go about building your own notebook? (With a quad core process that's liquid nitrogen cooled and nuclear powered and...)
  7. Cheers! One last question, I can't think straight at the moment but is the following acceptable: [math]\frac{\delta \partial_{t}X}{\delta X} = \partial_{t}\frac{\delta X}{\delta X}[/math] or is it more complicated than that?
  8. OK, so I had no clue where to put this, so here goes nothing. In general relativity, the variational methods used, I need to figure out the variation of the (three) christoffel symbol with respect to the (three) metric tensor: [math]\frac{\delta\Gamma^{i}_{bc}}{\delta g_{bc}} = ?[/math] the reason I ask is because I'm really looking for the variation of the Ricci tensor with respect to the metric (all of this is going on in three dimensions too, only the spatial ones) [math]\frac{\delta R_{ab}}{\delta g_{ab}} = \frac{\delta\Gamma^{c}_{ab;c}}{\delta g_{ab}} - \frac{\delta\Gamma^{c}_{ac;b}}{\delta g_{ab}}[/math] that's how I would figure it to be, so I'm wondering how would I go about this? Any help would be greatly appreciated!
  9. Pi is exactly three! :-D [math] \frac{\pi}{4} = \sum_{n=0}^{\infty} \frac{(-1)^{n}}{2n+1} [/math] or [math] \prod_{n=1}^{\infty} \frac{4n^2}{4n^2-1} = \frac{\pi}{2} [/math]
  10. I have been working with my friend in economics (always fun ) and we are trying to find polynomial roots using summations. The reason I ask is because we are modelling an economy as a system of equations, and if there is a surplus product then it becomes a polynomial. For example, we have an economy of two sectors (wheat and coal). We have the relationship 280 qr. Wheat + 12 t. coal --> 575 qr. Wheat 120 qr. Wheat + 8 t. coal --> 20 t. coal We then set up the value per unit Wheat as X and value per unit coal as Y, giving us: [Math]\frac{(1+r)(280X + 12Y)}{(1+r)(120X+8Y)} = \frac{575X}{20Y}[/Math] thus by reduction of the rate of profit ("1+r"), we multiply both sides out to receive [Math]20Y(280X + 12Y) = 575X(120X+8Y)[/Math] and by setting X=1 (because we assume f(Y)=X to get the value of coal in terms of wheat) we get a quadratic expression. In this case, we can just plug this into the quadratic equation and sha-zam, done. Yet what if we calculate out the value of the units wheat and coal in terms of the dated inputs? Intuitively, this would make no difference (from the economist's perspective). Yet mathematically, how would one portray this? Wouldn't it merely be an approximation of polynomial roots with a summation?
  11. From my understanding, grav probe b was trying to prove the equivalence principle by use of gyros. But doesn't the Ricci tensor "forbid" the torsion of the manifold (spacetime)? That was my impression.
  12. Freeman

    What is Universe ?

    I hate to disagree with everyone's definition here, but I have to disagree with everyone's definition The universe is a collection of processes, not things! To say that it is merely an assortment of objects and things is confessing "Yeah, I believe in a Newtonian universe." General Relativity, on the other hand, argues that the universe is defined by what goes on within it. It is more like a tennis game rather than the tennis court.
  13. I've been studying Misner, Thorne, and Wheeler's Gravitation and a thought had occurred to me: if black holes had angular momentum, why wouldn't this cause torsion in spacetime? If this is true, wouldn't we need to reject Einstein's field equation and "get" a "new one"? Maybe I jumped to the conclusion too soon thinking "Aha, angular momentum of a black hole affects spacetime, thus spacetime would be 'twisted' or at least victim of some sort of torsion." I was also considering this in terms of quantized spacetime, which may have caused the problem(!).
  14. My friend asked me to help him on his math for his economics, but I got rather lost since I do not know too much about Linear Algebra. His problem was that he had an economy (this is, for those who are curious, from Piero Sraffa) such that the inputs equal the outputs. The economy was: 280 qr. Corn + 12.t. iron --> 400 qr. corn 120 qr Corn + 8 t. iron --> 20 t. iron The solution to me was simple, just subtract from both sides the corresponding units to derive the exchange values of 10 qr. Corn = 1 t. iron. That much was correct, but I think by coincidence. Where my friend and I got lost was with a surplus. Suppose the economy became 280 qr. Corn + 12.t. iron --> 575 qr. corn 120 qr Corn + 8 t. iron --> 20 t. iron I thought "Aha! Use an Eigenvalue...or eignevector...or some eigenthing!" However, that's the only thing I've heard of in Linear Algebra. I do know that Piero Sraffa (that madman who came up with this Linear nightmare) said to use a rate of profit scalar, r, and multiply both sides by "1+r". The problem alas was that he never stated how to find the thing! I tried mathworld, but all I got was that I think I need some sort of right eigenvector. I'm more lost than found If some bright young feller would help, that would be fantastic.
  15. I first came upon them in Three Roads to Quantum Gravity by Lee Smolin, then I dug them up in the internet. I have read his book Road to Reality and from his description, it's just a more complex spinor. Rather than working with two axes, one works with four? Is that it? Or am I way off?
  16. OK, I am having a problem with this economic theory by Piero Sraffa. He says that production affects value, that's not the hard part. Here is his first example of a hypothetical economy: 280 qr. wheat + 12 t. iron --> 400 qr. Wheat 120 qr. wheat + 8 t. iron --> 20 t. iron I figured, hey what the heck, just set the value of a ton of iron to x and a qr. of wheat to y. 280y + 12x = 400y 120y + 8x = 20x Then subtract and we get: 12x = 120y 120y = 12x Thus 10 qr. wheat is exchangeable for 1 t. iron. But now he introduces the production with a surplus of 175 qr. wheat. There is a "rate of profit" scalar "r" that is [math]0\geq r\geq1[/math]and then we multiply both (1+r)(inputs). He then says OK given (1+r)(280y + 12x) = 575y (1+r)(120y + 8x) = 20x then magically comes to the conclusion r=0.25 and that 15y=x. I don't understand that little minor part.
  17. You mean so it will ask the question over and over again, right? import objectdraw.*; import java.awt.*; public class New_bot { public Bot(//String[] args maybe?) { EasyReader input=new EasyReader(); String question; int joey=1; System.out.println("Hi, I am the P.A.I.B.(Plaine Artificial Intellegence Bot!)"); while(joey>0) { } public void Entered_Text(String j) { j=j.toLowerCase(); //all lower case String reply; if (j.indexOf("hi")!=-1) reply="Hello!"; else if (j.indexOf("hello")!=-1) reply="Hi!"; else reply="I don't know. I am still learning. Try some other question."; return reply; joey++; } } } That's the lazy way of solving the problem I don't know about programming this online, but if you use a boolean for checking if the page has loaded for the first time for the user or not and have it say "Welcome back." or something of the sort, but this is overkill more likely and done with more ease with PHP. As far as making an applet (oh God!) I know how to do it, but it's been a while. If you use that class and then you need to put it in the folder you are go to be using. Then (uh...) you need to make a code somewhat like the following: import objectdraw.*; import java.awt.*; public class New_Bot extends WindowController { public void begin() { Bot joe; joe = new Bot(); } } Then you would probably want to include methods of some sort for some reason. You would probably want to double check all of this, since I haven't dealt with this in a great deal of time. I hate applets
  18. I've been discussing with an undergrad in math (he's a physics major) and he mentioned casually that he'd like to study Loop Quantum Gravity at grad school. Since I have only read a little on LQG, I really didn't know what to tell in terms of where to apply. So, I googled Ashtekar and found out he teaches at Penn State University. That is the only place I could think of for studying LQG but I don't want to tell my friend "Hey, you're doomed to go only there!" Where else is LQG dominant? Caltech or MIT maybe?
  19. I am programming in java, and I am curious whether I could create a public static method to make public static methods. Is this feasible or impossible?
  20. My head is spinning from Penrose's treatment of twistors. Isn't it essentially a special sort of spinor? [math]Z^{/alpha} = (\omega^{A}, \pi_{A'})[/math] where Z is a twistor. Also, why is the linear and angular momentum used? Can't I use, say, something else that satisfies: [math] \omega^{A} = i r^{AA'}\pi_{A'}[/math]; [math]\frac{\omega^{A}}{\pi_{A'}} = ir^{AA'}[/math] or am I on the wrong track totally?
  21. I understand the Ricci tensor is the trace of the Riemann tensor, but I don't fully understand the Riemann tensor; it's the tangent tensor of the curved area? What is the ricci tensor, then, exactly in layman's terms? The reason I ask is because I am trying to understand the lefthand side of Einstein's field equation; I understand the stress energy tensor (as explained in the Feynman Lectures in Physics). What exactly does the Einstein tensor ([math]G_{ab} = R_{ab} - \frac{1}{2}g_{ab}R[/math] where [math]R = g^{ab}R_{ab}[/math]) signify in layman's terms?
  22. Ha! Forgot about that superposition problem...so there really is no set volume of a single photon beyond the cube of wavelength? Hmm...
  23. I was discussing with a few friends the idea of photons in general relativity...essentially that light is a geodesic. However, it got me thinking to loop quantum gravity that would deal with a photon rather than light itself...wouldn't the photon definitionally take up space? Obviously it does, but how much space? Further, isn't light energy? And as Einstein pointed out, energy has mass. So wouldn't it distort spacetime only slightly that has no real significance in everyday life? But none of my texts answer these questions...and google isn't answering either. So my question is twofold: 1. What is the volume of a photon? 2. What is the density of energy for photons (or light, whichever)?
  24. The money is in the fields of biochemistry...so go with the origin of life.
  25. I am sorry to say that I do not understand the metric tensor one bit. Isn't it the Kronecker Delta in Euclidean space? And wouldn't that be a collection of row vectors?
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