Oh yeah! So the correct principal curvatures are ##k_1 = -1/R_0, k_2 = -1/R_0, k_3 = -1/R_0##?
If this is the case then the Gaussian curvature will be ##k=k_1 k_2 k_3 = -1/R_0^3## which has the correct sign. If ##R_0 = 1## then ##k=-1## which is the assumption from the start.
Actually, I made a mistake interpreting the vectors ##\partial_b \mathbf{e}_a## as lower index, i.e., ##(\partial_b \mathbf{e}_a)_\mu##. Actually, they are upper index vectors ##(\partial_b \mathbf{e}_a)^\mu##. So when computing ##L_{\psi \psi}##, it is as you did, i.e., ##L_{\psi \psi} = n_\mu...
Your last term should be ##-\frac{w}{R_0} R_0 \cosh \psi## due to the metric ##g=diag(-1,1,1,1)## right? The minus sign in ##g## is what makes it minus. That's the reason the issue comes out.
Indeed, if I remove ##\cosh 2\psi## then I get ##k_3 = -\frac{1}{R_0}##. However, I'm not yet sure where I went wrong. The necessary quantities to calculate ##L## are listed above, although I did not explicitly type out the vectors ##\partial_b \mathbf{e}_a##, but that is just the derivative of...
I would like to calculate the principal and Gaussian curvature of the spatial part of the Friedmann-Robertson-Walker (FRW) metric; specifically, the negative Gaussian curvature ##k=-1##. The FRW metric is,
\begin{equation*}
ds^2 = -dt^2 + R(t)^2 \left( \frac{dr^2}{1-k r^2} + r^2 d\Omega^2...
I agree it is a valid concern, I think it is convenient to write it that way since we can think of a vector as the inner product of its components and basis vectors. Although, as to the precise mathematical justification, I'm not sure. One thing is that it is quite a common notation being used...
Oh! I see what's wrong. I should have written,
\begin{equation*}
\mathbf{r} = r'^i e'_i = e'_i r'^i = e_j (G^T)^j_{\; i} G^i_{\; k} r^k
\end{equation*}
Now, the matrix notation and the index notation match, is this correct?
Also, with regards to the idea that you have not seen, please take a...
Given a vector ##\mathbf{r} = r^i e_i## where ##r^i## are the components, ##e_i## are the basis vectors, and ##i = 1, \ldots, n##. In matrix notation,
\begin{equation*}
\mathbf{r} = \begin{bmatrix} e_1 & e_2 & \ldots e_n \end{bmatrix} \begin{bmatrix} r^1 \\ r^2 \\ \vdots\\ r^n \end{bmatrix}...
I'm studying CFT, and I find the lecture notes and books really confusing and devoid of explanations (more details).
In a scale transformation ##x' = \lambda x##, the field ##\phi(x)## should also be affected by the scale transformation, i.e., ##\phi'(x') = \phi'(\lambda x) = \lambda^{-\Delta}...
I'm reading Group Theory by A. Zee , specifically, chapter I.3 on rotations. He used the passive transformation in analyzing a point ##P## in space. There are two observers, one labeled with unprimed coordinates and the other with primed coordinates. From the figure below, he deduced the...
It is currently the material that I'm studying, and I got curious whether it can act as a hydrogen sensor. As for the diffusion rate, I'm not yet sure, that is why I'm gathering resources in order to investigate further.
I was reading a paper [Appl. Phys. Lett. 91, 181910 (2007)] that has fabricated a Palladium (Pd) based sensor for hydrogen. One of the key points is that it is known that Pd is quite sensitive to hydrogen which is why the paper utilized a Pd array that may act as a sensor for detecting hydrogen...
Nice question, this issue also baffles me. I've taken QFT before, it seems to me that there is no particular book/resource that deals specifically with the "math" of QFT. It's apparent to me that the math used in QFT is quite diverse and there is no obvious structure that tells you, e.g. it's...
I see, your response clarified even further some concepts. I'll summarize again what I understood up until now. Please correct me if I'm wrong.
Elements of the Lie group are not the same as elements of the Lie algebra. So, the mechanism of representations for Lie groups and Lie algebras is not...