Higher Derivative Gravity, Causality and Positivity of Energy in a UV complete QFT

Abstract

In this note we discuss the relation between the constraints imposed by causality in the bulk of $AdS$ and the condition of positivity of the energy measured in ideal calorimeters in a collider experiment in the dual CFT. We first extend the analysis in the literature and recover all bounds imposed by causality of the boundary theory in the bulk dynamics for all polarizations of the graviton and the gauge boson field. These results translate to specific bounds for the ratio of central charges $\frac{a}{c}$ in the dual CFT, already found by analyzing the energy one point function. Then, we generalize this discussion and we study shock wave backgrounds in which we make manifest the relation between causality in the bulk and the three point function in the dual field theory. We remark that particular care has to be given to the exponentiation procedure of the three point function when solving the classical equations of motion in the higher gravity theory, as it is not clear that every theory will present causality problems. Finally, we present a field theoretic argument explaining the positivity of energy condition in any UV complete QFT.