Carter Plunging Fix

[fjebaker]

Along with

• Sign Flip Times fjebaker/CarterBoyerLindquist.jl#5
  this now fixes calculation errors in the plunging region redshift.

Plots

For M=1.0, a=-1.0 at tolerance 1e-9:

[fjebaker]

Fixes #18

[phajy]

Can you add a contour that shows the ISCO (r =~ 9 r_g in this case)? I'd expect g to drop inside the ISCO on the near side as the radial velocity increases. (Apologies for jumping in mid-conversation; might be misinterpreting the figures!)

[fjebaker]

@phajy I wasn't convinced by these plots either, which is why I've left the PR unmerged...

With the ISCO contour:

There's a slight convention error that I noticed last week, which may be causing this:

• Sign Convention fjebaker/GeodesicBase.jl#4

Switching the direction of the r component of the velocity (and in Carter's method, the sign of the radial potential) when calculating the redshift yields:

[fjebaker]

The only problem is that doesn't feel right:

The current photon velocity vector for the redshift is the reverse of the calculated 4-vector, since we trace forwards through time, but calculate backwards.

AccretionFormulae.jl/src/redshift.jl

Lines 176 to 180 in d784a62

	# reverse signs of the velocity vector
	# since we're integrating backwards
	p = @inbounds @SVector [
	-v[1], -v[2], 0, -v[4]
	]

But to get the 2nd plot in my last comment requires

p = @inbounds @SVector [
    -v[1], v[2], 0, -v[4]
]

(and similar for the Carter method) which feels like it's going the wrong way?

[phajy]

Thanks for adding the contour. I hadn't realised where the outer radius of the disk was which caused me a bit of confusion but now it is obvious! Yes, we're integrating backwards to need to reverse the photon direction but the disk gas velocity should be OK without needing to be reversed.

Ah, one other oddity I've thought of that arises from integrating backwards from the observer. A ray propagating close enough to the (spinning) black hole should co-rotate with the black hole (whichever direction it is spinning in if a = ±1). Does the symmetry of the equations change with t -> -t (e.g., do we need a -> -a or something like that)?

[fjebaker]

@phajy I tried using the a -> -a transform within the plunging, and the resulting plot is pretty unphysical.

Comparing the redshift plot in Cunningham (1975) fig 1, the images obtained with the latter

p = @inbounds @SVector [
    -v[1], v[2], 0, -v[4]
]

seem to match the profiles for a=0 and a=0.988, which implies that under time-reversal, the radial velocity component does not change. However, I can't seem to provide some intuitive argument for why this would be the case.

[fjebaker]

Updated the signs to reflect the second case -- we'll come up with a justification later, but this will at least let Dan and Kyle explore physical quantities within the ISCO for arbitrary spin.