What Is Shape Dynamics?
Shape Dynamics (SD) is a reformulation of Einsteins theory of gravity that emphasizes the role of spatial conformal geometry over the traditional spacetime picture. Instead of describing gravity as curvature of a fourdimensional manifold, SD treats it as the evolution of the shape of space itself, with a timeless underlying structure.
The program was initiated in the early 2000s by researchers seeking a theory that is manifestly invariant under local scale (conformal) transformations of spatial metrics. The resulting framework retains the same dynamical content as General Relativity (GR) for a wide class of solutions while offering a different, sometimes more intuitive, perspective on the nature of time and the degrees of freedom of the gravitational field.
Core Principles
Shape Dynamics rests on three tightly coupled ideas:
- Conformal Invariance: The fundamental objects are spatial metrics defined up to a local rescaling (a shape) rather than a specified size. This means that two metrics that differ only by a smooth, pointwise factor describe the same physical configuration.
- Relational Time: Time is not an external parameter but emerges from change. The theory adopts a Machian stance, where the ordering of configurations provides a notion of temporal succession.
- Equivalence to GR onShell: When the constraints of SD are satisfied, the resulting dynamics reproduce the same solutions as Einsteins equations for globally hyperbolic spacetimes.
These principles are encoded in a Hamiltonian formulation that replaces the Hamiltonian constraint of GR with a conformal constraint. The resulting constraint algebra is simpler, which often makes calculations more tractable.
Relation to General Relativity
While SD and GR are empirically equivalent in many regimes, they differ in conceptual emphasis and technical structure. Below is a concise comparison:
| Aspect | General Relativity | Shape Dynamics |
|---|---|---|
| Fundamental Symmetry | Diffeomorphism invariance of spacetime (4d) | 3d spatial diffeomorphisms + local conformal invariance |
| Constraints | Hamiltonian + momentum constraints | Momentum constraint + conformal constraint |
| Role of Time | Coordinate time is a gauge choice; proper time is physical. | Time emerges from change; no absolute time coordinate. |
| Canonical Variables | Spatial metric \(g_{ij}\) and its conjugate momentum \(\pi^{ij}\). | Conformal class of \(g_{ij}\) and a scalefree momentum. |
One practical advantage of SD is that its constraint algebra is free of the infamous structure functions that complicate the canonical quantization of GR. This simplicity has motivated several attempts to formulate a quantum version of gravity within the shapedynamical framework.
Physical Implications
Adopting a shapecentric viewpoint leads to several intriguing consequences:
- ScaleFree Cosmology: In a universe described purely by shape, the cosmological expansion can be interpreted as a change of spatial conformal geometry rather than a stretching of distances. This opens the door to novel explanations for the observed acceleration without invoking dark energy.
- Black Hole Interiors: The conformal symmetry of SD may regularize singularities by allowing the metric to crush to zero size without diverging curvature invariants. Some models suggest that black hole interiors could be smoothly continued through a shapedynamical bounce.
- Quantum Gravity Pathways: Because the Hamiltonian constraint is replaced by a linear conformal constraint, the canonical quantization process avoids the problem of defining a selfadjoint Hamiltonian operator. Early proposals based on loop variables and groupfield techniques have shown promise.
Shape Dynamics does not deny the success of General Relativity; it reexpresses it in a language where scale is a gauge redundancy. Julian Barbour (paraphrased)
Current Research and Outlook
Over the past decade, several research groups worldwide have been expanding the foundations and applications of Shape Dynamics. Some active areas include:
- Cosmological Perturbations: Understanding how primordial fluctuations evolve in a conformally invariant setting, with implications for the Cosmic Microwave Background.
- Numerical Simulations: Developing algorithms that exploit the simpler constraint algebra to simulate gravitational collapse and binary mergers.
- Quantum Implementations: Constructing Hilbert spaces that respect the conformal constraint, and exploring connections to the AdS/CFT correspondence.
- Philosophical Foundations: Analyzing the relational notion of time and its compatibility with various interpretations of quantum mechanics.
While many questions remainparticularly regarding the treatment of matter fields and the handling of global topologyShape Dynamics continues to attract interest as a fresh lens on the gravitational interaction.
