Positive gravitational subsystem energies from CFT cone relative entropies

Understand

The positivity of relative entropy for spatial subsystems in a holographic CFT implies the positivity of certain quantities in the dual gravitational theory.

• In this note, we consider CFT subsystems whose boundaries lie on the lightcone of a point $p$.
• We show that the positive gravitational quantity which corresponds to the relative entropy for such a subsystem $A$ is a novel notion of energy associated with a gravitational subsystem bounded by the minimal area extremal surface $\tilde{A}$ associated with $A$ and by the AdS boundary region $\hat{A}$ corresponding to the part of the lightcone from $p$ bounded by $\partial A$.
• This generalizes the results of arXiv:1605.01075 for ball-shaped regions by making use of the recent results in arXiv:1703.10656 for the vacuum modular Hamiltonian of regions bounded on lightcones.