Shape dynamics, invented by Julian Barbour, is a theory of spacetime that applies an action principle, called Jacobi’s principle, and a procedure called best matching, to a configuration space of conformal 3-manifolds, called shape space, to recover the Hamiltonian formulation of general relativity, called the ADM formalism. Shape dynamics is a “space-first” theory, with time arising in a secondary fashion via Jacobi’s principle. The ADM formalism produces a spacelike foliation of spacetime, whose leaves are parameterized by the “time coordinate.”
Causal theory, invented by Rafael Sorkin, is a “time-first” theory. Time is viewed as a proxy for causality, which is taken to be fundamental. Spatial separation is viewed, as in relativity, as indicative of causal disjunction, and is taken to be secondary. Several different versions of causal theory exist, of which the best-known “pure causal” version is Rafael Sorkin’s causal set theory. Dynamical triangulation is another well-known “causal” theory, but is really a hybrid theory with both spatial and temporal structure built in at the fundamental level. My own version of causal theory is summarized here. It is based on the causal metric hypothesis, which states that the fundamental structure of spacetime (and optimistically, all of physics!) is determined, up to scale, by causal relations. Sorkin’s motto “order plus number equals geometry” may be regarded as a special case of the causal metric hypothesis.
Shape dynamics and causal theory both rely heavily on binary relations. In shape dynamics, the relations are symmetric, and are viewed as encoding spatial structure. In causal theory, the relations are antisymmetric, and are viewed as encoding causal (i.e., temporal) structure. The following question occurred to me while discussing these two theories with Daniel Alves, Lawrence Crowell, Sean Gryb, and Flavio Mercati during the 2012 FQXi essay contest: is there a shape/causal duality principle relating appropriate versions of shape dynamics and causal theory? Of course, a continuum theory like standard shape dynamics is not directly comparable to a locally finite theory like causal set theory, but these theories can be reformulated in comparable contexts.
Shape dynamics and causal theory have seemingly complementary features; each takes to be fundamental what the other takes to be secondary. If some version of shape/causal duality exists, neither idea need be at fault. Between the two theories, I tend to be biased in favor of causal theory, largely on philosophical grounds. While I see no good reason to regard spatial relations as fundamental, I view causality as a very compelling foundational principle. However, shape dynamics does have certain advantages. Julian Barbour tells me that just as “nothing comes from nothing,” so “not much comes from not much,” and causal theory is “not much” in his view. It is true that the structure of causal theory is very parsimonious, and to propose recovering known physics thereby requires considerable optimism. Of course, these are only two of many theories, and both may be incorrect or inadequate. However, a shape/causal duality would lend greater credence to both theories, while opening up interesting lines of research.