We derive the exact Klein-Gordon equation for a massive scalar field in the Clausius-Mossotti (CM) metric ds² = W^ (1/3) c²dt² − W^ (−1/3) (dr² + r²dΩ²), where W = (1−β²) / (1+2β²) and β² = 2GM/ (rc²). Three structural properties distinguish this equation from the Schwarzschild case: (i) the spatial Laplacian is exactly flat — no metric function appears inside the radial derivative, a direct consequence of the CM identity gₜt × |gᵣr| = 1; (ii) the effective particle mass μₑff = μ W^ (−1/6) increases near the gravitating source, opposite to GR where the effective mass decreases; (iii) the frequency term carries W^ (−2/3), encoding gravitational time dilation into the wave's temporal oscillation alone. Numerical bound-state eigenvalues at gravitational coupling strengths αg = 0. 1, 0. 3, and 0. 5 show that CM bound states are systematically less tightly bound than GR, with the ground state shift reaching +18. 5% at αg = 0. 5 and CM supporting three bound states compared to GR's four. The weaker binding is traced to the 4π+2π=6π geometric decomposition: geodesics use the ratio gₜt/|gᵣr| (exponents add, giving full GR), while the d'Alembertian uses √ (−g) ×gʳr (exponents cancel, losing spatial confinement). Zero free parameters are introduced.
Mandeep Singh (Tue,) studied this question.