Exploring the Applicability of Birkhoff’s Theorem in Jackiw-Teitelboim Gravity

We present a comprehensive and technically rigorous analysis of the status of Birkhoff’s theorem in Jackiw–Teitelboim (JT) gravity, a paradigmatic two-dimensional model for studying semiclassical gravitational dynamics. While Birkhoff’s theorem is well established in four-dimensional general relativity–asserting the uniqueness and staticity of vacuum solutions under reflection symmetry remains subtle due to the absence of propagating gravitational degrees of freedom. In this work, we systematically investigate the space of symmetry under radially symmetric configurations in JT gravity using both conformal and Schwarzschild-like gauges. Through analytical techniques and integral transformations, we explore the conditions under which vacuum solutions remain time-independent, identifying classes of metric-dilaton configurations that either uphold or violate Birkhoff-type behavior. Our findings reveal that the theorem holds only in restricted cases, depending critically on the separability of the conformal factor and the structure of the dilaton potential. These results clarify longstanding ambiguities surrounding symmetry and dynamics in two-dimensional gravity and establish JT gravity as a controlled setting for probing the breakdown of classical gravitational theorems in lower-dimensional and holographic contexts. This analysis contributes to a deeper understanding of the interplay between symmetry, integrability, and geometry in quantum gravity and strongly coupled systems.