Scalar wave separation in Kerr spacetime ($\Psi =e^{-i\omega t+im\varphi }R(r)\Theta (\theta )$)

For the massless Klein-Gordon equation on a Kerr black hole, the covariant wave operator separates into radial and angular ordinary differential equations. With $K=(r^2+a^2)\omega -am$ and angular separation constant $\Lambda$, the radial equation is $(\Delta R^{\prime }{)}^{\prime }+[K^2/\Delta -a^2{\omega }^2+2am\omega -\Lambda ]R=0$. The angular equation is $(\sin \theta {\Theta }^{\prime }{)}^{\prime }/\sin \theta +[a^2{\omega }^2{\cos }^2\theta -m^2/{\sin }^2\theta +\Lambda ]\Theta =0$. No extra radial factor is included in the displayed ansatz.

A scalar spheroidal harmonic is a regular angular solution of $(\sin \theta S^{\prime }{)}^{\prime }/\sin \theta +[c^2{\cos }^2\theta -m^2/{\sin }^2\theta +\Lambda ]S=0$; with $e^{im\varphi }$ it is smooth at both axes. For real $c$, regularity gives discrete eigenvalues ${\Lambda }_{\ell m}(c)$. At $c=0$ it reduces to the angular part of a spherical harmonic, with $\Lambda =\ell (\ell +1)$. The sign of the $c^2$ term distinguishes the convention used here.