The origin of entropy production in spacetime thermodynamics

We find that the ambiguity term of approximate Killing vector field is responsible for the entropy production term. Without the ambiguity term, pure Einstein theory and $f(R)$ satisfy the relation of thermodynamic equilibrium. Considering such an ambiguity term of approximate Killing vector field, we can get the entropy production term and the entropy in $f(R)$ with a form defined by Jacobson. In pure Einstein theory, the shear term is the only geometric contribution of entropy production term, while in $f(R)$ it can also contribute. We believe our approach and conclusion can be generalized to other gravity theory.