Possible sign issue with amplitudes

[bleather]

We've found a possible issue with the amplitudes for spin weight $s = \pm 2$.

We found our amplitudes with an independent calculation disagreed by a sign, with the source term calculated independently of the toolkit. Tracing this error back, we found our source term calculated from the stress-energy tensor seemed to disagree with the toolkit by a minus sign.

To check this, I checked the toolkit amplitudes with archived amplitudes data computed by Scott Hughes (from Gremlin, I believe). I found a sign discrepancy of $-1$ between this data and the amplitudes from the toolkit. I've attached a zip file with Scott's data and notebook comparing the amplitudes with the master branch of the repository. We understand how this might not have been found because you wouldn't notice the sign difference when comparing fluxes.

Note the data in Scott's file rescales the fluxes by: ${\cal H} = \omega^{2}\dfrac{{\cal F}_{\ell m \omega}}{s}$

AmplitudeSignCheck.zip

[barrywardell]

I thought we checked this against data from @znasipak and/or @MvdMeent, although it's possible I'm mis-remembering. Can either of you confirm?

[barrywardell]

@Kevin-Cunningham may have also checked this at some point.

[znasipak]

It all comes down to a choice of convention in how you define $\psi_4$. Basically, does $\ddot{h}(r\rightarrow \infty) \simeq \psi_4$ or $\ddot{h}(r\rightarrow \infty) \simeq -\psi_4$? The Toolkit uses the latter, which I think is what Adam and Barry use in their recent review. This actually differs from my original code, which originally used the other convention.

[barrywardell]

I think we want to define $\psi_4$ consistent with the conventions in arXiv:2101.04592. Since we use a (-+++) convention the relationship between $\psi_4$ and $\ddot{h}$ includes a minus sign, but this does not affect the sign of $\psi_4$ itself, which should be the same for all standard conventions.

I have now carefully independently checked against a calculation in which I computed the s=-2 $\psi_{lm\omega}$ using Lorenz-gauge metric perturbation data and I agree that there is an overall minus sign difference compared to what there should be. This only affects the s=+/-2 cases. I suggest we change the sign of these for consistency with standard conventions.