No Arabic abstract
We describe a new form of retrocausality, which is found in the behaviour of a class of causal set theories, called energetic causal sets (ECS). These are discrete sets of events, connected by causal relations. They have three orders: (1) a birth order, which is the order in which events are generated; this is a total order which is the true causal order, (2) a dynamical partial order, which prescribes the flows of energy and momentum amongst events, (3) an emergent causal order, which is defined by the geometry of an emergent Minkowski spacetime, in which the events of the causal sets are embedded. However, the embedding of the events in the emergent Minkowski spacetime may preserve neither the true causal order in (1), nor correspond completely with the microscopic partial order in (2). We call this disordered causality, and we here demonstrate its occurrence in specific ECS models. This is the second in a series of papers centered around the question: Should we accept violations of causality as a lesser price to pay in order to keep realist formulations of quantum theory? We begin to address this in the first paper [1] and continue here by giving an explicit example of an ECS model in the classical regime, in which causality is disordered.
Causal set quantum gravity is a Lorentzian approach to quantum gravity, based on the causal structure of spacetime. It models each spacetime configuration as a discrete causal network of spacetime points. As such, key questions of the approach include whether and how a reconstruction of a sufficiently coarse-grained spacetime geometry is possible from a causal set. As an example for the recovery of spatial geometry from discrete causal structure, the construction of a spatial distance function for causal sets is reviewed. Secondly, it is an open question whether the path sum over all causal sets gives rise to an expectation value for the causal set that corresponds to a cosmologically viable spacetime. To provide a tool to tackle the path sum over causal sets, the derivation of a flow equation for the effective action for causal sets in matrix-model language is reviewed. This could provide a way to coarse-grain discrete networks in a background-independent way. Finally, a short roadmap to test the asymptotic-safety conjecture in Lorentzian quantum gravity using causal sets is sketched.
We consider pseudoconvexity properties in Lorentzian and Riemannian manifolds and their relationship in static spacetimes. We provide an example of a causally continuous and maximal null pseudoconvex spacetime that fails to be causally simple. Its Riemannian factor provides an analogous example of a manifold that is minimally pseudoconvex, but fails to be convex.
A usual causal requirement on a viable theory of matter is that the speed of sound be at most the speed of light. In view of various recent papers querying this limit, the question is revisited here. We point to various issues confronting theories that violate the usual constraint.
We show that the causal properties of asymptotically flat spacetimes depend on their dimensionality: while the time-like future of any point in the past conformal infinity $mathcal{I}^-$ contains the whole of the future conformal infinity $mathcal{I}^+$ in $(2+1)$ and $(3+1)$ dimensional Schwarzschild spacetimes, this property (which we call the Penrose property) does not hold for $(d+1)$ dimensional Schwarzschild if $d>3$. We also show that the Penrose property holds for the Kerr solution in $(3+1)$ dimensions, and discuss the connection with scattering theory in the presence of positive mass.
We develop causality theory for upper semi-continuous distributions of cones over manifolds generalizing results from mathematical relativity in two directions: non-round cones and non-regular differentiability assumptions. We prove the validity of most results of the regular Lorentzian causality theory including causal ladder, Fermats principle, notable singularity theorems in their causal formulation, Avez-Seifert theorem, characterizations of stable causality and global hyperbolicity by means of (smooth) time functions. For instance, we give the first proof for these structures of the equivalence between stable causality, $K$-causality and existence of a time function. The result implies that closed cone structures that admit continuous increasing functions also admit smooth ones. We also study proper cone structures, the fiber bundle analog of proper cones. For them we obtain most results on domains of dependence. Moreover, we prove that horismos and Cauchy horizons are generated by lightlike geodesics, the latter being defined through the achronality property. Causal geodesics and steep temporal functions are obtained with a powerful product trick. The paper also contains a study of Lorentz-Minkowski spaces under very weak regularity conditions. Finally, we introduce the concepts of stable distance and stable spacetime solving two well known problems (a) the characterization of Lorentzian manifolds embeddable in Minkowski spacetime, they turn out to be the stable spacetimes, (b) the proof that topology, order and distance (with a formula a la Connes) can be represented by the smooth steep temporal functions. The paper is self-contained, in fact we do not use any advanced result from mathematical relativity.