Shape dynamics

In theoretical physics Shape dynamics is a theory of gravity that implements Mach's principle. Shape dynamics is dynamically equivalent to the canonical formulation of General Relativity, known as the ADM formalism. Shape dynamics is not formulated as an implementation of spacetime diffeomorphism invariance, but as an implementation of spatial relationalism based on spatial diffeomorphisms and spatial Weyl symmetry. An important consequence of shape dynamics is the absence of a problem of time in canonical quantum gravity. The replacement of the spacetime picture with a picture of evolving spatial conformal geometry opens the door for a number of new approaches to quantum gravity.

Mach's principle has been an important inspiration for the construction of general relativity, but the physical interpretation of Einstein's formulation of general relativity still requires external clocks and rods and thus fails to be manifestly relational. Mach's principle would be fully implemented if the predictions of general relativity were independent of the choice of clocks and rods. Barbour and Bertotti conjectured that Jacobi's principle and a mechanism they called "best matching" were construction principles for a fully Machian theory. Barbour implemented these principles in collaboration with Niall Ó Murchadha, Edward Anderson, Brendan Foster and Bryan Kelleher to derive the ADM formalism in constant mean curvature gauge. This did not implement Mach's principle, because the predictions of general relativity in constant mean curvature gauge depend on the choice of clocks and rods. Mach's principle was successfully implemented in 2010 by Henrique Gomes, Sean Gryb and Tim Koslowski who drew on the work of Barbour and his collaborators to describe gravity in a fully relational manner as the evolution of the conformal geometry of space.

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