The Einstein equations with position-dependent Planck length

This paper derives modified Einstein field equations G_ab + Λg_ab = (8πG₀/μ)T_ab from Jacobson's (1995) thermodynamic derivation of general relativity, with one modification: the Planck length is allowed to depend on position through the dynamics of compact extra dimensions in a Kähler compactification. Two scaling laws from the compactification geometry — the local gravitational constant G_local = G₀/μ and the local gravitational Planck constant ℏ_grav = ℏ₀μ1/3, where μ is the compact volume ratio — combine in the Clausius relation δQ = TdS to yield the modified field equations. The modification appears only on the right-hand side of the equations, ensuring gravitational waves propagate at speed c in vacuum and that the standard 2:1 ratio between null and timelike geodesic deflections is preserved across all acceleration regimes. The Newtonian limit recovers the MOND interpolating function with no free parameters beyond a₀ = cH/6. The Bianchi identity yields a modified conservation law ∇_a(Tᵃᵇ/μ) = 0, identifying compact dimension compression energy as a gravitating component. Originally submitted to Classical and Quantum Gravity on March 16, 2026.