Pathologies in Asymptotically Lifshitz Spacetimes
Keith Copsey, Robert Mann
TL;DR
This work reveals fundamental pathologies of asymptotically Lifshitz spacetimes: the standard flat Lifshitz ground state contains a naked null singularity at $r=0$ that cannot be cured by known string-theoretic corrections, and the initial-value problem is generically ill-posed as Lifshitz falloffs are not preserved under evolution. By formulating a Hamiltonian framework and analyzing both constraint equations and asymptotic expansions, the authors show that, under transverse symmetry, the energy is fixed by the asymptotic vector field and that maintaining Lifshitz behavior at all times severely restricts interior dynamics. Exact solutions illustrate interior singularities or horizons, while time-evolution analyses demonstrate that generic perturbations push the solution away from Lifshitz asymptotics unless highly fine-tuned. Collectively, these results challenge the viability of Lifshitz holography as a straightforward extension of AdS/CFT and motivate careful consideration of stability and interior regularity in string-theoretic realizations.
Abstract
There has been significant interest in the last several years in studying possible gravitational duals, known as Lifshitz spacetimes, to anisotropically scaling field theories by adding matter to distort the asymptotics of an AdS spacetime. We point out that putative ground state for the most heavily studied example of such a spacetime, that with a flat spatial section, suffers from a naked singularity and further point out this singularity is not resolvable by any known stringy effect. We review the reasons one might worry that asymptotically Lifshitz spacetimes are unstable and employ the initial data problem to study the stability of such systems. Rather surprisingly this question, and even the initial value problem itself, for these spacetimes turns out to generically not be well-posed. A generic normalizable state will evolve in such a way to violate Lifshitz asymptotics in finite time. Conversely, enforcing the desired asymptotics at all times puts strong restrictions not just on the metric and fields in the asymptotic region but in the deep interior as well. Generically, even perturbations of the matter field of compact support are not compatible with the desired asymptotics.