theoretical physics Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain and predict natural phenomena. This is in contrast to experimental physics, which uses experi ...

, shape dynamics is a theory of

gravity In physics, gravity () is a fundamental interaction which causes mutual attraction between all things with mass or energy. Gravity is, by far, the weakest of the four fundamental interactions, approximately 1038 times weaker than the str ...

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 General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics. ...

and

quantum mechanics Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, q ...

. Shape dynamics is dynamically equivalent to the canonical formulation of general relativity, known as the

ADM formalism The ADM formalism (named for its authors Richard Arnowitt, Stanley Deser and Charles W. Misner) is a Hamiltonian formulation of general relativity that plays an important role in canonical quantum gravity and numerical relativity. It was fir ...

. Shape dynamics is not formulated as an implementation of

spacetime In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why diffe ...

diffeomorphism invariance In theoretical physics, general covariance, also known as diffeomorphism covariance or general invariance, consists of the invariance of the ''form'' of physical laws under arbitrary differentiable coordinate transformations. The essential idea is ...

, 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 In physics, canonical quantum gravity is an attempt to quantize the canonical formulation of general relativity (or canonical gravity). It is a Hamiltonian formulation of Einstein's general theory of relativity. The basic theory was outlined b ...

. 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 Quantum gravity (QG) is a field of theoretical physics that seeks to describe gravity according to the principles of quantum mechanics; it deals with environments in which neither gravitational nor quantum effects can be ignored, such as in the v ...

. 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 Julian Barbour (; born 1937) is a British physicist with research interests in quantum gravity and the history of science. Since receiving his PhD degree on the foundations of Albert Einstein's general theory of relativity at the University ...

Background

Mach's principle has been an important inspiration for the construction of

, 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

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.

Relation with general relativity

Shape dynamics possesses the same dynamics as general relativity, but has different gauge orbits. 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.

Problem of time in shape dynamics

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. 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.

Arrow of time in shape dynamics

Recent work by Julian Barbour, Tim Koslowski and Flavio Mercati 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.

References

Theoretical physics

Background

Relation with general relativity

Problem of time in shape dynamics

Arrow of time in shape dynamics

Further reading

References