conventions.md

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Naming Conventions

We aim to have consistent naming conventions across the Toolkit. Where possible the following conventions should be used. In all cases the symbols used below should be understood to be adimensionalized, e.g., for the orbital frequencies $\Omega \equiv M\Omega$.

General

We use geometrized units such that $G=c=1$ and metric signature $(-+++)$.

Binary components

• $M$ - the mass of the primary

• $a$ - the magnitude of the spin on the primary

• $\mu$ - the mass of the secondary

• $\sigma$ - the magnitude of the spin on the secondary

• $\eta = \mu/M$ - the mass ratio

Spin-weighted spheroidal harmonics

• $\lambda$ - the spin-weighted spheroidal-harmonic eigenvalue. Our definition for $\lambda$ is consistent with that of Teukolsky, S. A., Astrophys. J. 185, 635–647 (1973) and that of Sasaki, M. & Tagoshi, H. Living Rev. Relativ. (2003) 6: 6.
• $\gamma$ - the spheroidicity ($=a\omega$ for perturbations of Kerr spacetime)

Geodesic motion

• $\mathcal{E}$ - the orbital energy

• $\mathcal{L}$ - the orbital angular momentum

• $\mathcal{Q}$ - the orbital Carter constant

• $p$ - the semi-latus rectum

• $e$ - the orbital eccentricity

• $x_\text{inc}$ - the orbital inclination

• $\Omega_\alpha$ - the orbital frequencies w.r.t. Boyer-Lindquist time

• $\Upsilon_\alpha$ - the orbital frequencies w.r.t. Mino time

• $\omega_\alpha$ - the orbital frequencies w.r.t. propertime

• $(t,r,\theta, \varphi)$ - the usual Schwarzschild or Boyer-Lindquist coordinates

• $\tau$ - propertime

• $\lambda$ - Mino time

Mode calculations

• $l$ - the harmonic index
• $m$ - the azimuthal index
• $n$ - the radial overtone index OR for quasi-normal modes the overtone index
• $k$ - the polar overtone index
• $\omega$ - the mode frequency