Stellar displacement energy identity (${\omega }^2\int \rho |\mathbf{\xi }{|}^2=-\frac{1}{4\pi G}\int |\nabla {\varphi }_1{|}^2+\int \left(\frac{|p_1{|}^2}{\gamma p}+\rho N^2|{\xi }_r{|}^2\right)$)

For a nonrotating self-gravitating star, let $p_1,{\rho }_1,{\varphi }_1$ be Eulerian perturbations and $\Delta p=-\gamma p\nabla \cdot \mathbf{\xi }$ the adiabatic Lagrangian perturbation. Integration of the linear momentum equation against ${\mathbf{\xi }}^{\ast }$, using hydrostatic balance and the perturbed Poisson equation, yields the displayed Rayleigh quotient. The gravitational integral is over all space and the other integrals over the star. The compression denominator is $\gamma p$, not $\gamma \rho$. Surface terms vanish for compactly supported trials or the physical free-surface and exterior potential conditions.

Choose a smooth radial bump $f$ supported inside a shell where the stellar buoyancy frequency has $N^2<0$. The displayed fluid displacement has $\nabla \cdot \mathbf{\xi }=(\rho g/\gamma p){\xi }_r$, hence its Eulerian pressure perturbation vanishes identically. Its buoyancy contribution to the stellar displacement energy identity is strictly negative and self-gravity is nonpositive. The variational principle therefore proves convective instability with no unsupported assumption that incompressibility alone removes pressure work. Large spherical-harmonic degree also makes perturbed self-gravity small.