Jacobson (1995) demonstrated that the Einstein field equations emerge as an equation of statefrom the first law of thermodynamics applied to local Rindler horizons, assuming the Bekenstein-Hawking entropy-area relation S = A/4 and the Unruh temperature T = ℏa/2πckB . His derivationassumes local thermodynamic equilibrium: all vacuum modes contribute fully to the horizon entropy.We show that this assumption fails at very low accelerations, where the Rindler horizon distancec2 /a approaches the Hubble radius RH = c/H0 . The finite size of the observable universe imposesan infrared cutoff on the vacuum mode spectrum, reducing the effective entropy below S = A/4.Propagating this modified entropy through Jacobson’s thermodynamic machinery yields, in the non-relativistic weak-field limit, a modified inertial mass mi = mg 1 − (2c2 /aΘ)2 , where Θ = 2c/H0 isthe Hubble diameter. This is McCulloch’s Quantized Inertia. The critical acceleration at which thecorrection becomes order unity is a0 = cH0 ≈ 6.9 × 10−10 m/s2 —the same order as the empiricalMOND scale (aMOND≈ 1.2 × 10−10 m/s2 ), with the O(1) numerical prefactor depending on the0detailed mode-counting between horizons. This scale is derived here with no free parameters fromthe geometry of nested horizons. Standard inertia is recovered at high accelerations. We thusdemonstrate that Quantized Inertia is not an independent hypothesis but a boundary correction toJacobson’s established framework—the natural consequence of applying horizon thermodynamics ina universe of finite extent.
Keith Brodie (Mon,) studied this question.