  A general geometrical structure of the entanglement entropy for spatial
partition of a relativistic QFT system is established by using methods of the
effective gravity action and the spectral geometry. A special attention is
payed to the subleading terms in the entropy in different dimensions and to
behaviour in different states. It is conjectured, on the base of relation
between the entropy and the action, that in a fundamental theory the ground
state entanglement entropy per unit area equals $1/(4G_N)$, where $G_N$ is the
Newton constant in the low-energy gravity sector of the theory. The conjecture
opens a new avenue in analogue gravity models. For instance, in higher
dimensional condensed matter systems, which near a critical point are described
by relativistic QFT's, the entanglement entropy density defines an effective
gravitational coupling. By studying the properties of this constant one can get
new insights in quantum gravity phenomena, such as the universality of the
low-energy physics, the renormalization group behavior of $G_N$, the
statistical meaning of the Bekenstein-Hawking entropy.
