Entropic force versus temperature force

We introduce the cavity enclosing a source mass $M$ to define the temperature force. Starting with the Tolman temperature in the stationary spacetime, we find a non-relativistic temperature $T_{non}= T_\infty(1-Φ/c^2)$ with the Newtonian potential $Φ$. This temperature could be also derived from the Tolman-Ehrenfest effect, satisfying a relation of $T=T_{\infty}e^{-Φ/c^2}$ with the local temperature $T$. Finally, we derive the temperature force $\vec{F}_{tem}=mc^2(\vec{\nabla} \ln T )$ which leads to the Newtonian force law without introducing the holographic screen defined by holographic principle and equipartition law for entropic force.

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