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: $-sigma hbar theta=mc^2$.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.