Freeman

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?

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.

Is there something a little more "user friendly" than Henneaux? It appears more heavy than I can handle at the moment.

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!

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 )!

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...)

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?

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!

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]

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?

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.

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.

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(!).

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.

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?