consider the classical electromagnetic field strength tensor $F^{\mu\nu}$

I am wondering how to think about getting the right sign when defining the electric field components

is it like: $F^{0\nu} = F^{0i} = \partial^0 A^i - \partial^i A^0 = \partial_0 A^i + \partial_i A^0 = -E^i$? where I am using $(+, -, -, -)$ signature. where the final equality is our usual definition of the electric field. I think this is wrong because the indices get messed up in the penultimate equality...

if I am computing $\lvert E \lvert^2$ in tensor notation, is this $E^i E^i$? I think this should be the case. As opposed to $E_i E^i$.

@SillyGoose you usually do not compute $|E|^2$ in tensor notation

The relevant metric tensor is the spatial part of the metric tensor

So in flat space $\delta_{ij} E^i E^j$

okay i see

bleb I am getting that the electromagnetic energy density is negative...

With some appropriate tensor manipulation you can find out that $F^{\mu\nu} F_{\mu\nu}$ is the same as $E^2 - B^2$

By using identities related to the definitions of the Levi-Civita tensor and projections

right i get that it is equal to $2(E^2 - B^2)$

When trying to figure out the direction in which certain particles with certain momenta are moving, it is necessary to be using the contravariant components $p^\mu=(+|E|,p^i)$ instead of the covariant components. The Lorentz invariant phase is $k_\mu x^\mu=k_t x^t+k_i x^i=\eta^{tt}(k^t x^t-k^i x^i)$; the momentum is $\hat p_\mu=-i\nabla_\mu$. Between these definitions you can figure out the derivative part of the Faraday tensor. However, I do not know how you are defining the 4-potential $A$

man chasing down negative signs ;_;

Do you actually need to chase down the negative signs? Is it not acceptable to just put in the minus signs to make them correct as you need them?

@SillyGoose lmfao what is this

There are some bits that are internal constraints and so you have to get them correct. However, there are also quite a lot of bits that are conventions, and people can choose different conventions.

Hello Everyone...

Is there any physical meaning of magnitude and direction of angular momentum and torque?

@naturallyInconsistent NO

I guess I would be happier if I did but I can't

Pls see highlighted step, i don't understand how they put $\rho_{j} = R_{\perp}$

@Slereah I remember you posting ages ago a picture of your bookshelf