  We analytically study the lightcone limit of the conformal bootstrap
equations for 4-point functions containing global symmetry currents and the
stress tensor in 3d CFTs. We show that the contribution of the stress tensor to
the anomalous dimensions of large spin double-twist states is negative if and
only if the conformal collider physics bounds are satisfied. In the context of
AdS/CFT these results indicate a relation between the attractiveness of AdS
gravity and positivity of the CFT energy flux. We also study the contribution
of non-Abelian conserved currents to the anomalous dimensions of double-twist
operators, corresponding to the gauge binding energy of 2-particle states in
AdS. We show that the representation of the double-twist state determines the
sign of the gauge binding energy if and only if the coefficient appearing in
the current 3-point function satisfies a similar bound, which is equivalent to
an upper bound on the charge flux asymmetry of the CFT.
