Tim Koslowski, Perimeter Institute
Title: Shape dynamics
PDF of the talk (500k)
Audio [.wav 33MB], Audio [.aif 3MB].
Newton’s Method of Defining Motion
Newton’s method, still present in many theoreticians’ intuition, takes space to be real like a perfectly smooth table top (suppressing one space dimension) that extends to infinity in all directions. Imagine three particles that in two instants form slightly different triangles (1 and 2). The three sides of each triangle define the relative configuration. Consider triangle 1. In Newtonian dynamics, you can locate and orient 1 however you like. Space being homogeneous and isotropic, all choices are on an equal footing. But 2 is a different relative configuration. Can one say how much each particle has moved? According to Newton, many different motions of the particles correspond to the same change of the relative configuration. Keeping the position of 1 fixed, one can place the centre of mass of 2, C2, anywhere; the orientation of 2 is also free. In three-dimensional space, three degrees of freedom correspond to the possible changes of the sides of the
triangle (relative data), three to the position of C2, and three to the orientation. The three relative data cannot be changed, but the choices made for the remainder are disturbingly arbitrary. In fact, Galilean relativity means
that the position of C2 is not critical. But the orientational data are crucial. Different choices for them put different angular momenta L into the system, and the resulting motions are very different. Two snapshots of relative configurations contain no information about L; you need three to get a handle on L. Now we consider the alternative.
Dynamics Based on Best Matching
The definition of motion by best matching is illustrated in the figure. Dynamics based on it is more restrictive than Newtonian dynamics. The reason can be ‘read off’ from the figure. Best matching, as shown in b, does two things. It brings the centers of mass of the two triangles to a common point and sets their net relative rotation about it to zero. This last means that a dynamical system governed by best matching is always constrained, in Newtonian terms, to have vanishing total angular momentum L. In fact, the dynamical equations are Newtonian; the constraint L = 0 is maintained by them if it holds at any one instant.
Figure 1. The Definition of Motion by Best Matching. Three particles, at the vertices of the grey and dashed triangles at two instants, move relative to each other. The difference between the triangles is fact, but can one determine unique displacements of the particles? It seems not. Even if we hold the grey triangle fixed in space, we can place the dashed triangle relative to it in any arbitrary position, as in a. There seems to be no way to define unique displacements. However, we can bring the dashed triangle into the position b, in which it most nearly ‘covers’ the grey triangle. A natural minimizing procedure determines when ‘best matching’ is achieved. The displacements that take one from the grey to the dashed triangle are not defined relative to space but relative to the grey triangle. The procedure is reciprocal and must be applied to the complete dynamical system under consideration.
So far, we have not considered size. This is where Shape Dynamics proper begins. Size implies the existence of a scale to measure it by. But, if our three particles are the universe, where is a scale to measure its size? Size is another Newtonian absolute. Best matching can be extended to include adjustment of the relative sizes. This is done for particle dynamics here. It leads to a further constraint. Not only the angular momentum but also something called the dilatational momentum must vanish. The dynamics of any universe governed by best matching becomes even more restrictive than Newtonian dynamics.
Best Matching in the Theory of Gravity
Best matching can be applied to the dynamics of geometry and compared with Einstein's general relativity (GR), which was created as a description of the four-dimensional geometry of spacetime. However, it can be reformulated as a dynamical theory in which three-dimensional geometry (3-geometry) evolves. This was done in the late 1950s by Dirac and Arnowitt, Deser, and Misner (ADM), who found a particularly elegant way to do it that is now called the ADM formalism and is based on the Hamiltonian form of dynamics. In the ADM formalism, the diffeomorphism constraint, mentioned a few times by Tim Koslowski, plays a prominent role. Its presence can be explained by a sophisticated generalization of the particle best matching shown in the figure. This shows that the notion of change was radically modified when Einstein created GR (though this fact is rather well hidden in the spacetime formulation). The notion of change employed in GR means that it is background independent . In the ADM formalism as it stands, there is no constraint that corresponds to best matching with respect to size. However, in addition to the diffeomorphism constraint, or rather constraints as there are infinitely many of them, there are also infinitely many Hamiltonian constraints. They reflect the absence of an external time in Einstein's theory and the almost complete freedom to define simultaneity at spatially separated points in the universe. It has proved very difficult to take them into account in a quantum theory of gravity. Building on previous work, Tim and his collaborators Henrique Gomes and Sean Gryb have found an alternative Hamiltonian representation of dynamical geometry in which all but one of the Hamiltonian constraints can be swapped for conformal constraints. These conformal constraints arise from a best matching in which the volume of space can be adjusted with infinite flexibility. Imagine a balloon with curves drawn on it that form certain angles wherever they meet. One can imagine blowing up the balloon or letting it contract by different amounts everywhere on its surface. In this process, the angles at which the curves meet cannot change, but the distances between points can. This is called a conformal transformation and is clearly analogous to changing the overall size of figures in Euclidean space. The conformal transformations that Tim discusses in his talk are applied to curved 3-geometries that close up on themselves like the surface of the earth does in two dimensions. The alternative, or dual, representation of gravity through the introduction of conformal best matching seems to open up new routes to quantum gravity. At the moment, the most promising looks to be the symmetry doubling idea discussed by Tim. However, it is early days. There are plenty of possible obstacles to progress in this direction, as Tim is careful to emphasize. One of the things that intrigues me most about Shape Dynamics is that, if we are to explain the key facts of cosmology by a spatially closed expanding universe, we cannot allow completely unrestricted conformal transformations in the best matching but only the volume-preserving ones (VPCTs) that Tim discusses. This is a tiny restriction but strikes me as the very last vestige of Newton's absolute space. I think this might be telling us something fundamental about the quantum mechanics of the universe. Meanwhile it is very encouraging to see technical possibilities emerging in the new conceptual framework.
