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In theoretical physics, **shape dynamics** is a theory of gravity that implements Mach's principle, developed with the specific goal to obviate the problem of time and thereby open a new path toward the resolution of incompatibilities between general relativity and quantum mechanics.

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.^{[1]} An important consequence of shape dynamics is the absence of a problem of time in canonical quantum gravity.^{[2]} 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.^{[3]}

An important development in this theory was contributed in 2010 by Henrique Gomes, Sean Gryb and Tim Koslowski, building on an approach initiated by Julian Barbour.

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.^{[4]} 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.^{[5]} 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.^{[6]} 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^{[7]} 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.^{[8]}

Shape dynamics possesses the same dynamics as general relativity, but has different gauge orbits.^{[9]} The link between general relativity and shape dynamics can be established using the ADM formalism in the following way: Shape dynamics can be gauge fixed in such a way that its initial value problem and its equations of motion coincide with the initial value problem and equations of motion of the ADM formalism in constant mean extrinsic curvature gauge. This equivalence ensures that classical shape dynamics and classical general relativity are locally indistinguishable. However, there is the possibility for global differences.^{[10]}^{[11]}^{[12]}^{[13]}

The shape dynamics formulation of gravity possesses a physical Hamiltonian that generates evolution of spatial conformal geometry. This disentangles the problem of time in quantum gravity: The gauge problem (the choice of foliation in the spacetime description) is replaced by the problem of finding spatial conformal geometries, leaving an evolution that is comparable to a system with time dependent Hamiltonian.^{[14]} The problem of time is suggested to be completely solved by restricting oneself to "objective observables," which are those observables that do not depend on any external clock or rod.^{[15]}

Recent work by Julian Barbour, Tim Koslowski and Flavio Mercati^{[16]} demonstrates that Shape Dynamics possesses a physical arrow of time given by the growth of complexity and the dynamical storage of locally accessible records of the past. This is a property of the dynamical law and does not require any special initial condition.

- Mercati, Flavio (2014). "A Shape Dynamics Tutorial". arXiv:1409.0105 [gr-qc].
- Mach's principle

**^**Barbour, Julian (2012). "Gravity as Machian Shape Dynamics" (PDF).*fqxi talk*.**^**Koslowski, Tim. "Tim Koslowski's homepage". Retrieved 2012-11-18.**^**Koslowski, Tim (2013). "Shape Dynamics and Effective Field Theory".*International Journal of Modern Physics A*.**28**(13): 1330017. arXiv:1305.1487. Bibcode:2013IJMPA..2830017K. doi:10.1142/S0217751X13300172. S2CID 118614894.**^**Merali, Zeeya (2012). "Is Einstein's Greatest Work All Wrong—Because He Didn't Go Far Enough?".*Discover magazine*. Retrieved 2012-04-10.**^**Barbour, Julian; Bertotti, Bruno (1982). "Mach's principle and the structure of dynamical theories" (PDF).*Proceedings of the Royal Society A*.**382**(1783): 295–306. Bibcode:1982RSPSA.382..295B. doi:10.1098/rspa.1982.0102. S2CID 123089455.**^**Anderson, Edward; Barbour, Julian; Foster, Brendan; Kelleher, Bryan; Ó Murchadha, Niall (2005). "The physical gravitational degrees of freedom".*Classical and Quantum Gravity*.**22**(9): 1795–1802. arXiv:gr-qc/0407104. Bibcode:2005CQGra..22.1795A. doi:10.1088/0264-9381/22/9/020. S2CID 119476891.**^**Gomes, Henrique; Gryb, Sean; Koslowski, Tim (2011). "Einstein Gravity as a 3D Conformally Invariant Theory".*Classical and Quantum Gravity*.**28**(4): 045005. arXiv:1010.2481. Bibcode:2011CQGra..28d5005G. doi:10.1088/0264-9381/28/4/045005. S2CID 119215598.**^**Perimeter Institute (2011). "What if size really doesn't matter?" (PDF).*annual report 2011*.**^**Gomes, Henrique; Koslowski, Tim (2012). "The Link between General Relativity and Shape Dynamics".*Classical and Quantum Gravity*.**29**(7): 075009. arXiv:1101.5974. Bibcode:2012CQGra..29g5009G. doi:10.1088/0264-9381/29/7/075009. S2CID 119208720.**^**Gomes, Henrique; Koslowski, Tim (2012). "Frequently asked questions about Shape Dynamics".*Foundations of Physics*.**43**(12): 1428–1458. arXiv:1211.5878. Bibcode:2013FoPh...43.1428G. doi:10.1007/s10701-013-9754-0. S2CID 118434969.**^**Gomes, Henrique (2014). "A Birkhoff Theorem for Shape Dynamics".*Classical and Quantum Gravity*.**31**(8): 085008. arXiv:1305.0310. Bibcode:2014CQGra..31h5008G. doi:10.1088/0264-9381/31/8/085008. S2CID 119261085.**^**Gomes, Henrique; Herczeg, Gabriel (2014). "A Rotating Black Hole Solution for Shape Dynamics".*Classical and Quantum Gravity*.**31**(17): 175014. arXiv:1310.6095. Bibcode:2014CQGra..31q5014G. doi:10.1088/0264-9381/31/17/175014. S2CID 119208372.**^**Herczeg, Gabriel (2015). "Parity Horizons, Black Holes and Chronology Protection in Shape Dynamics".*Classical and Quantum Gravity*.**33**(22): 225002. arXiv:1508.06704. Bibcode:2016CQGra..33v5002H. doi:10.1088/0264-9381/33/22/225002.**^**Koslowski, Tim (2012). "Observable Equivalence between General Relativity and Shape Dynamics". arXiv:1203.6688 [gr-qc].**^**Barbour, Julian; Koslowski, Tim; Mercati, Flavio (2013). "The solution to the problem of time in Shape Dynamics".*Classical and Quantum Gravity*.**31**(15): 155001. arXiv:1302.6264. Bibcode:2014CQGra..31o5001B. doi:10.1088/0264-9381/31/15/155001. S2CID 119251890.**^**Barbour, Julian; Koslowski, Tim; Mercati, Flavio (2014). "Identification of a gravitational arrow of time".*Phys. Rev. Lett*.**113**(18): 181101. arXiv:1409.0917. Bibcode:2014PhRvL.113r1101B. doi:10.1103/PhysRevLett.113.181101. PMID 25396357. S2CID 25038135.