Black holes are fascinating entities that typically arise from gravitationally collapsing bodies, such as stars at the end of their life cycle or star collisions. Despite their violent origin, black holes relax to a stationary state, which can then be fully described by a mere handful of variables such as their mass, charge and angular momentum. Interestingly, like ordinary matter systems such as ideal gases, black holes also obey an equation of state. The physical parameters describing stationary black-holes satisfy what is known as the Komar relation E=2TS where E is related to the Hamiltonian of the Einstein-Hilbert action, T is the Hawking temperature and S is the Bekenstein-Hawking entropy. 

Most of the effort in literature has been to understand the microscopic statistical origin of black-hole entropy. While this may be explained by a host of other theories (referred to as the "universality problem" of black hole entropy),  the entanglement approach is further capable of reproducing the results of black hole thermodynamics. The entanglement approach requires us to look at a quantum scalar field residing in a background space-time with a horizon, such as that of a black hole. Upon lattice-regularization, the field degrees of freedom are replaced by a network of coupled harmonic oscillators.

One can then bipartition the system close to the space-time horizon and describe what is known as entanglement mechanics, by defining the following measures :

On numerically simulating the entanglement mechanics of the horizon, we obtain a one-to-one correspondence between i) entanglement entropy and Bekenstein-Hawking entropy (well-known analogy in literature) and ii) entanglement energy and Komar energy (novel result). The correspondence is such that the Komar relation and the generalized Smarr formula of black hole thermodynamics are recovered exactly.

The Smarr formula for asymptotically flat and non-flat space-times is generally derived using Komar integral relations or via the first law of thermodynamics with the help of scaling relations, both of which make use of Killing potential. Entanglement mechanics is an independent approach towards the derivation of the generalized Smarr formula, which essentially describes the equation of state for black holes. The intrinsically quantum phenomenon of entanglement pertaining to the field near a horizon, gives rise to not just the thermodynamic quantities associated with the space-time, but also the exact relation connecting the same.