Theory of director fluctuations about a hedgehog defect in a nematic drop

We present calculations of eigenmode energies and wave functions of both azimuthal and polar distortions of the nematic director relative to a radial hedgehog trapped in a spherical drop with a smaller concentric spherical droplet at its core. All surfaces interior to the drop have perpendicular (homeotropic) boundary conditions. We also calculate director correlation functions and their relaxation times. Of particular interest is a critical mode whose energy, with fixed Frank constants, vanishes as the ratio $\mu =R_2/R_1$ increases toward a critical value ${\mu }_c$, where $R_2$ is the radius of the drop and $R_1$ that of the inner droplet, and then becomes negative for $\mu >{\mu }_c$. Our calculations form a basis for interpreting experimental measurements of director fluctuations relative to a radial hedgehog state in a spherical drop. We compare results with those obtained by previous investigations, which use a calculational approach different from ours, and with our experimental observations.