Recent content by Lyalpha

And by proportinial, I mean proportional.

My professor asks, "Double check a formula that specifies how Riemann tensor is proportional to a curvature scalar." in our homework. The closet thing I can find is the relation between the ricci tensor and the curvature scalar in einstein's field equation for empty space.

I'm in an intro course and my shaky ability to solve differential equations is apparent. How would you go about solving \ddot{r}-r\ddot{\theta}=0 \ddot{\theta}+\frac{1}{r}\dot{r}\dot{\theta}=0 It might be obvious. They're the geodesic equations for a 2d polar coordinate system (if...

Yes but why do we use the square root of the determinate of -g to transform between volume elements in different frames? I'd like to see the math of it.

I see that it is \sqrt{|-g|}d^{4}x but I'm not sure why it needs to be multiplied by the square root of the determinant of -g. It must be the Jacobian of g_{μ\nu} or something right? So I guess I'm asking how do you calculate the Jacobian of the metric.

But isn't \eta^{n}_{n} equal to \eta^{0}_{0} + \eta^{1}_{1} + \eta^{2}_{2} + ... + \eta^{D}_{D} where \eta^{0}_{0} = -1 and \eta^{1}_{1}, \eta^{2}_{2}, \eta^{D}_{D} are 1?

I'm starting with a conformally flat space with the interval ds^{2} = e^{2φ} η_{ab} dx^{a} dx^{b} , where \eta_{ab} is a fundamental metric tensor of D-dimensional flat space. I'm trying to find the usual stuff (Christoffel symbols, curvature tensor, ricci tensor, curvature scalar). When it...

But what if it's the fundamental metric tensor? Where g_{00} is -1 and the rest are 1? Would I get D-2 ?

I asked the question because I'm getting g^{λ}_{λ} after some contractions of a certain curvature tensor. For Example I have a term g^{μ}_{σ} and I'm contracting the tensor by setting μ and σ to λ. The term will now equal the dimension D?

ok thanks

A^{\nu} = g^{μ \nu}A_{μ} A_{μ} = g_{μ\nu}A^{\nu} A_{μ} = g_{μ\nu}g^{\nu\rho}A_{\rho} g_{μ\nu}g^{\nu\rho} = g^{\rho}_{\mu} g^{\rho}_{\mu} = 1 for μ = ρ g^{\rho}_{\mu} = 0 for μ \neq ρ taken from Dirac's book

My understanding of the Einstein Summation convention is that you sum over the repeated indices. But when I look at the metric tensor for a flat space I know that g^{λ}_{λ} = 1 But the summation convention makes me think that it should equal the trace of the matrix g_{μσ}. So it should...