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From Wikipedia:

In theoretical physics, shape dynamics is a theory of gravity that implements Mach's principle.

In General relativity curvature of Spacetime causes gravity, in String theory it is Graviton, what causes gravity in Shape Dynamics?

Also, this article claims Gravitational waves disprove Mach's principle, is it true?

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Shape dynamics is a classical theory of gravity where solutions are described by time evolution of spatial (three dimensional Riemannian) conformal geometries.

In shape dynamics time is a distinct parameter on which the spatial geometry depends, while in GR there is a unified concept of spacetime. The basic degrees of freedom of Shape dynamics are Riemannian metric and its canonically conjugate momentum. In this regard shape dynamics resembles ADM formalism of general relativity, but the structure of constraints and gauge symmetries are different for the two approaches. Gauge symmetries of gravitational field in shape dynamics consist of spatial diffeomorphisms and local Weyl transformations.

Despite having different structures and different degrees of freedom shape dynamics largely provides the same predictions as GR. Specifically, in the vicinity of a generic solution point shape dynamics and general relativity are locally equivalent, by a suitable gauge-fixing it is possible to reconstruct spacetime metric in the vicinity of that point from a shape dynamics solution and this metric would satisfy the usual Einstein field equations. However, such gauge-fixing may not be possible to achieve globally, so global properties of solutions for the two theories may be different. For example, black hole solutions in shape dynamics differ from GR counterparts at and inside the horizon (see e.g. here).

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  • $\begingroup$ So in General Relativity there is 4-dimensional spacetime but Shape dynamics uses 3 spatial dimensions + 1 time dimension? $\endgroup$
    – user311753
    Sep 1 at 12:16
  • $\begingroup$ @ArAj: Yes, in SD the role of time being conceptually different and without diffeomorphisms mixing spatial coordinates and time (like we have in GR). $\endgroup$
    – A.V.S.
    Sep 1 at 14:22
  • $\begingroup$ In Shape dynamics Gravity is caused by curvature of 3 spatial dimensions? $\endgroup$
    – user311753
    Sep 2 at 8:42
  • $\begingroup$ Not just the 3D curvature! In addition to curved 3D metric ($g_{ij}(x)$) we also need conjugate momenta ($p^{ij}(x)$). Just like one cannot specify the state of a particle with just its position, you need also velocity (or momentum). $\endgroup$
    – A.V.S.
    Sep 2 at 12:26

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