Coordinate checks

[damonge]

@xzackli after a quick look at your notebooks I'm wondering if we've corrected for the following:

• The theta vs. dec sign. This should (I think) come down to changing t21 -> -t21.
• The fact that t11 corresponds to t_theta_theta and not t_phi_phi (i.e. the 1st coordinate is what one would naively consider to be y)

[xzackli]

re:flipping the sign of t21, here are the principal eigenvectors for a small cutout of the sky. I don't have intuition for the tidal field, so I'm not totally sure which one is correct. Here, black is matrix with (t11, t12, t12, t22) and red is the matrix with (t11, -t12, -t12, t22).

[damonge]

OK, awesome plot!

So my main point was that the tidal tensor is roughly defined as t11 = dT/dtheta/dtheta, t22 = dT/dtheta/dphi and t12 =dT/dtheta/dphi. Depending on how the inertia tensor of the galaxy shapes was defined (i.e. the definition of the position angle), this might imply a shift in the sign of t12 (if dec instead of theta was used, since ddec = -dtheta).

Another related issue is whether the 1st coordinate of the inertia tensor is theta/dec or phi/ra (i.e. which component should be match with t11, which one with t22 etc.).

So maybe, to make sure we understand everything, it'd be good to write down exactly what you figured out the position angle to be, since that essentially defines the inertia tensor. Maybe we can start some quick and dirty latex notes to make sure we're on the same page. I'll try to start them tomorrow if I have time.

[xzackli]

Independent of the galaxy shape, I think there is a right and wrong way to choose the sign of t12 if you choose coordinates for the vector field's vectors. For example, not changing the sign on t12 and then plotting the vector field yields a field which is not rotationally symmetric around the brightest sources.

Contrast with the correction,

It's probably because plotting forces you to choose coordinates for the space that the vector field lives in (i.e. the tangent space). I have chosen when plotting these vector fields to represent x increasing in the -phi direction, and y increasing in the -theta direction, as you would expect when lying down on the ground with your head pointed north. I take the first and second components of the principal eigenvector, and interpret them as x and y.

Finally, the effect of transposing x and y vectors is actually to rotate the field by 90 degrees from if you were to make t12 negative.