Thermodynamics and Energy

F = ma as a Rindler Modular-Work Identity in Quantum Field Theory

Authors: Russell S Clark, Russell S Clark II
Comments: 16 Pages.

For the Minkowski vacuum and the right Rindler wedge, the Bisognano—Wichmann theorem fixes the vacuum modular flow to be Lorentz-boost flow. For a translated finite-energy excitation, this implies an exact, acceleration-independent identity for the vacuum-relativeone-sided modular energy,d/dp∆⟨KW⟩ = 2πℏcEW(p),where p is displacement normal to the horizon and EW(p) is the vacuum-subtracted energycontained in the wedge. If the excitation is concentrated well inside the wedge, Ew(p) ≃ E.Selecting a Rindler observer of proper acceleration a converts the dimensionless modulargenerator into the proper-time Hamiltonian Ha = kBTUKw, with TU = ℏa/(2πckB). Hence d/dp∆⟨Ha⟩ = a/c2Ew(p).Under a specified quasistatic control protocol, this is the external holding-force identity. Atthe reference orbit, a wedge-contained narrow state of invariant mass M obeys |Fhold| = Ma.The entanglement interpretation is more limited. For a finite state, the exact regulatedrelation isd/dp∆⟨Ha⟩ = Tu x d∆Sw/dp +kBTu x d/dp x Srel(ρW,p∥Ωw),so Tu x dSw alone is not generally equal to work. The relative-entropy term is second orderonly for infinitesimal state variations. In that linear-response sector, the entanglement firstlaw yields d(δSw)/dp = 2πkBδE/(ℏc), and for δE = Mc2 this reproduces the coefficient in Verlinde’s local entropy—displacement ansatz. The result is therefore an exact modular-work identity and a linearized entropic representation of Rindler holding force, not a derivation of acceleration, general equations of motion, or gravity.
Category: Thermodynamics and Energy