Abstract
We investigate the conditions imposable on a scalar field at the boundary of the so-called Lifshitz spacetime which has been proposed as the dual to Lifshitz field theories. For effective mass squared between −(d + z − 1)2 /4 and z 2 − (d + z − 1)2 /4, we find a one-parameter choice of boundary condition type. The bottom end of this range corresponds to a Breitenlohner-Freedman bound; below it, the Klein-Gordon operator need not be positive, so we cannot make sense of the dynamics. Above the top end of the range, only one boundary condition type is available; here we expect compact initial data will remain compact in the future.
Highlights
Peeling off a different factor for each piece of the metric, chosen to ensure that the pieces remain finite as the radius approaches infinity
We investigate the conditions imposable on a scalar field at the boundary of the so-called Lifshitz spacetime which has been proposed as the dual to Lifshitz field theories
In order to preserve the asymptotic conditions, it becomes necessary to turn off some possible modes when doing holography. [8] sets boundary conditions in a different manner, by first requiring that all divergences be cancellable by local counterterms. [9] builds a stressenergy tensor at the boundary for spacetimes with z = 2, again imposing a set of boundary conditions a priori. [10] studies perturbations in a particular Hamiltonian formulation, again imposing boundary conditions and limiting the available solutions. [11] considers the effect of the null energy condition on causality at Lifshitz boundaries
Summary
We will review the procedure described in [12]. We are interested in studying the behavior of fields φ solving particular wave equations on a spacetime. The procedure described in [12] finds the consistent boundary data imposable for a field φ solving a wave equation on a static stably causal spacetime, such that these boundary data allow “good dynamics”, as defined in the introduction, for the field φ in the full spacetime. Analyze the boundary conditions obeyed by φt for a given extension AU These steps provide a precise means for extending the operator A to an action AU on the spatial boundary of Σ, which is well-defined since it is a single time-slice. We proceed to explore each of these steps in more detail
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