  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. As part of our
analysis, we give an analytic expression for the extremal surface in pure AdS
associated with any such region $A$. We note that its form immediately implies
the Markov property of the CFT vacuum (saturation of strong subadditivity) for
regions bounded on the same lightcone. This gives a holographic proof of the
result proven for general CFTs in arXiv:1703.10656. A similar holographic proof
shows the Markov property for regions bounded on a lightsheet for non-conformal
holographic theories defined by relevant perturbations of a CFT.
