On-shell equation of the Lorentzian classicalized holographic tensor network

In the Lorentzian classicalized holographic tensor network (cHTN), we derive its relativistic on-shell equation from its Lorentzian action in the presence of a relativistic massive particle in the bulk spacetime: $-\sigma \mathrm{\hslash }\theta =M{c}^2$. Here, $\sigma$ is the von Neumann entropy of the cHTN per site in nats, $\theta$ is the real-proper-time expansion of the cHTN defined along the world line of the particle, and $M$ is the non-zero mass of the particle. We explain the physical properties, interpretation, and consequences of this equation. Specifically, from this equation we derive the properties of the on-shell proper acceleration of another massive particle in the bulk spacetime as those of the gravitational acceleration induced by the original massive particle.