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For the scalar wave separation in Kerr spacetime, continue to use $A=r^2+a^2$ and $s=\sin \theta$. First verify the determinant in the hint. Direct multiplication of the covariant $t,\varphi$ components givesHence the block determinant is $-\Delta {\sin }^2\theta$. Inverting this block givesThe other inverse components are $g^{rr}=\Delta /\Sigma$, $g^{\theta \theta }=1/\Sigma$, and $\sqrt{-g}=\Sigma s$. The covariant wave operator on a scalar consequently has the divergence formLet $m_{\varphi }$ denote the azimuthal mode number, to distinguish it from the axial vector $m$. Insert the mode $\Psi =e^{-i\omega t+i{m}_{\varphi }\varphi }R(r)\Theta (\theta )$ in the massless Klein-Gordon equation. Single-valuedness makes $m_{\varphi }$ an integer. The $t,\varphi$ derivatives giveWith $K(r)=A\omega -a{m}_{\varphi }$, division by the mode factor gives, on patches where $R\Theta \ne 0$,The radial and angular expressions must be opposite constants. Defining the separation constant as $\Lambda$, we obtain the two ordinary differential equationsThese equations also hold at zeros of a mode by continuity, without dividing there. Regular angular solutions are scalar spheroidal harmonics, with discrete $\Lambda ={\Lambda }_{\ell m_{\varphi }}(a\omega )$. For $a\omega =0$ the angular equation becomes the associated Legendre function equation, with $\Lambda =\ell (\ell +1)$ and $\ell \ge |m_{\varphi }|$, providing a useful check of the signs and normalization. The radial function here is exactly $R$ in the chosen ansatz, without an additional factor of $1/r$.

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.

A separation constant arises when an equation is a sum of functions of independent variables. If $F(r)+G(\theta )=0$ on a product region, each function is constant, with opposite signs. Boundary or regularity conditions can then select discrete values, as for scalar spheroidal harmonics.