Boundary stress tensor and asymptoticallyAdS3non-Einstein spaces at the chiral point

Abstract

Chiral gravity admits asymptotically ${\mathrm{AdS}}_{3}$ solutions that are not locally equivalent to ${\mathrm{AdS}}_{3}$; meaning that solutions do exist which, while obeying the strong boundary conditions usually imposed in general relativity, happen not to be Einstein spaces. In topologically massive gravity (TMG), the existence of non-Einstein solutions is particularly connected to the question about the role played by complex saddle points in the Euclidean path integral. Consequently, studying (the existence of) nonlocally ${\mathrm{AdS}}_{3}$ solutions to chiral gravity is relevant to understanding the quantum theory. Here, we discuss a special family of nonlocally ${\mathrm{AdS}}_{3}$ solutions to chiral gravity. In particular, we show that such solutions persist when one deforms the theory by adding the higher-curvature terms of the so-called new massive gravity. Moreover, the addition of higher-curvature terms to the gravity action introduces new nonlocally ${\mathrm{AdS}}_{3}$ solutions that have no analogues in TMG. Both stationary and time-dependent, axially symmetric solutions that asymptote ${\mathrm{AdS}}_{3}$ space without being locally equivalent to it appear. Defining the boundary stress tensor for the full theory, we show that these non-Einstein geometries have associated vanishing conserved charges.