Causal structures and holography

Abstract

We explore the description of bulk causal structure in a dual field theory. We observe that in the spacetime dual to a spacelike non-commutative field theory, the causal structure in the boundary directions is modified asymptotically. We propose that this modification is described in the dual theory by a modification of the micro-causal light cone. Previous studies of this micro-causal light cone for spacelike non-commutativite field theories agree with the expectations from the bulk spacetime. We describe the spacetime dual to field theories with lightlike non-commutativity, and show that they generically have a drastic modification of the light cone in the bulk: the spacetime is non-distinguishing. This means that the spacetime while being devoid of closed timelike or null curves, has causal curves that are ``almost closed''. We go on to show that the micro-causal light cone in the field theory agrees with this prediction from the bulk.

Figures (2)

• Figure 1: Light cone structure in the non-distinguishing spacetime (\ref{['ndgeo']}). From point ${\bf P}$ we can reach point ${\bf R}$ by a usual causal curve. To reach point ${\bf Q}$ which is spacelike separated in the Minkowski light cone, we need to use the asymptotic structure of the spacetime. By using curves that venture out further in $r$ we can arrange for the causal curve to return arbitrarily close to the starting point. As a result the future of the point ${\bf P}$ is the part of the spacetime above the constant $u$ plane shown above.
• Figure 2: Feynman diagrams contributing to ${\cal M}^{(2)}$. Due to the non-commutative interaction we have both planar and non-planar graphs.