"The Schwarzschild metric (spacetime geometry) is not the same thing as Schwarzschild coordinates."
I know..
"Your description of these other coordinates as "more exotic" is not a good one. However, your assumptions about the book's expectations are of course up to you and I'm not going to...
"If this argument is valid (and I'm not saying it isn't), it would apply equally well to Schwarzschild coordinates, since those have not yet been covered in the book either."
The book assumes some knowledge of the Schwarzschild metric. I contend it is therefore totally reasonable to assume...
Thanks. But I was not aware of either of those coordinate systems. And I doubt they were they intended ones to start from in this case, because I think it would be absurd for this book to have assumed we would know this.
Also, whether or not the geodesic is falling from rest at infinity was not...
I have been learning a bit about Fermi normal coordinates in Eric Poisson's "A Relativist's Toolkit". Problem 1.10 in this book is to express the Schwarzschild metric in Fermi normal coordinates about a radially infalling, timelike geodesic.
I know that in the Fermi normal coordinates (denoted...
For (a) and (b), since the geodesic is not affinely parametrised, we have that ##t^a\nabla_a t^b = f(\lambda) t^b##, for some function f.
As a results, for (a) I get that ##t^a \nabla_a \epsilon = 2 f(\lambda) \epsilon##. And for (b) I get that ##t^a \nabla_a p = f(\lambda) p##. (I can write...
This should be
$$
\sigma^{k \ \gamma}_{\theta}\delta^{\beta}_{\alpha} - \sigma^{k \ \beta}_{\alpha} \delta_{\theta}^{\gamma} = i \epsilon_{ijk}\sigma^{i \ \gamma}_{\alpha} \sigma^{j \ \beta}_{\theta}.
$$.
Thanks for the suggestion, but I still couldn't show the two sides to be equal. I have...
So I think I have reduced the above to trying to conclude that:
$$
\sigma^{k \ \beta}_{\alpha} \delta_{\theta}^{\gamma} - \sigma^{k \ \gamma}_{\theta}\delta^{\beta}_{\alpha} = i \epsilon_{ijk}\sigma^{i \ \gamma}_{\alpha} \sigma^{j \ \beta}_{\theta}.
$$
If anyone has any suggestions, it would...
I am trying to reproduce the results of a certain paper here. In particular, I'm trying to verify their eqn 5.31.
The setup is N = 4 gauge quantum mechanics, obtained by the dimensional reduction of N = 1 gauge theory in 4 dimensions. ##\sigma^i## denotes the ith pauli matrix. ##\lambda_{A...
Suppose ##\lambda_A## and ##\bar{\lambda}_A## are fermions (A goes from 1 to N) and ##\{ \lambda_{A \alpha}, \bar{\lambda}_B^{\beta}\} = \delta_{AB}\delta_{\alpha}^{\beta}##.
Let ##\sigma^i## denote the Pauli matrices.
Does it follow that ##[\bar{\lambda}_A \sigma^i \lambda_A, \bar{\lambda_B}...
Thank you.
I mistakenly used that ##E_p = p^2 + m^2##, instead of ##E_p = \sqrt{p^2 + m^2}##, so both your power on ##E_p## and your factor of ##2## are correct. I should have noticed dimensional mismatch. I think I've been less diligent since things were set to ##1##. But this is not a good...
Thank you. That is the kind of thing I was hoping for. I think that justifies to me the final integration by parts with the ##a_p^{\dagger}## and ##a_p##.
How about this step:
?
Assuming I have the correct steps.
If anyone has a more efficient method, please let me know.Also, I would appreciate more detail on why the boundary terms are zero. Both for the initial place where I was stuck above when we are taking the derivative off the delta functions, and then also for...