Einstein gravity as a 3D conformally invariant theory (1010.2481v2)

Abstract: We give an alternative description of the physical content of general relativity that does not require a Lorentz invariant spacetime. Instead, we find that gravity admits a dual description in terms of a theory where local size is irrelevant. The dual theory is invariant under foliation preserving 3-diffeomorphisms and 3D conformal transformations that preserve the 3-volume (for the spatially compact case). Locally, this symmetry is identical to that of Horava-Lifshitz gravity in the high energy limit but our theory is equivalent to Einstein gravity. Specifically, we find that the solutions of general relativity, in a gauge where the spatial hypersurfaces have constant mean extrinsic curvature, can be mapped to solutions of a particular gauge fixing of the dual theory. Moreover, this duality is not accidental. We provide a general geometric picture for our procedure that allows us to trade foliation invariance for conformal invariance. The dual theory provides a new proposal for the theory space of quantum gravity.

Overview of "Einstein Gravity as a 3D Conformally Invariant Theory"

This paper introduces a novel interpretation of Einstein's general relativity that does not assume Lorentz invariance. Instead, it presents a dual description where the local scale of spatial metrics is irrelevant, resonating with conformal invariance principles. The duality draws symmetry parallels with Hořava–Lifshitz gravity, particularly in their high-energy limits, yet the paper proves the dual theory to be equivalent to Einstein gravity.

Key Contributions

1. Dual Description of Gravity: The authors provide a formal mechanism to trade foliation invariance inherent to general relativity for invariance under conformal transformations, specifically in 3D. This dual construct challenges traditional notions by suggesting that physical dynamics can be equivalently expressed through 3D conformal invariance instead of 4D Lorentz invariance.
2. Volume Preservation: In spatially compact cases, the dual description requires the preservation of spatial volume under conformal transformations, aligning with the principles underlying York's method for the initial value problem in general relativity. The dual theory dismisses local scale, focusing instead on the dynamical evolution of shapes.
3. Geometric Understanding and Gauge Fixing: A geometric framework is developed, underpinning the symmetry exchange process. The authors leverage Stueckelberg fields and gauge fixing algorithms to formalize how general relativity’s foliation invariance corresponds to an alternative symmetry structure based on conformal transformations.
4. Implications for Quantum Gravity: The authors propose that shape dynamics, the resultant theory from the duality, can redefine the theory space of quantum gravity. This impacts how we approach quantum gravity’s Hilbert space, suggesting potential directions for Loop Quantum Gravity and influencing theories reliant on 4D diffeomorphism invariant metrics.

Implications and Speculations

• Quantum Gravity Theory Space: This dual description offers a new avenue for exploring quantum gravity, potentially addressing some of the ghost problems in conventional general relativity quantizations by altering the fixed point structure seen in asymptotic safety approaches.
• Local Invariance and Shape Dynamics: The duality strengthens the conjecture that intrinsic scale-independent geometries might encapsulate the true dynamical degrees of freedom in gravity. This insight could inform approaches that utilize conformal invariance within quantum gravity paradigms, such as Loop Quantum Shape Dynamics.
• Connection to Hořava–Lifshitz Gravity: The dual theory’s symmetry group bears resemblance to Hořava–Lifshitz gravity’s theoretical constructs, which might bridge insights between non-relativistic quantum gravity frameworks and general relativity, especially concerning high-energy phenomena.

Future Directions

The paper sets a foundation for future explorations into alternative quantum gravity formulations and suggests further investigation into how shape dynamics might be quantized effectively. With the groundwork laid for appreciating the deep-seated connection between classical and conformal gravity frameworks, subsequent research could explore computational simulations and experimental probes of conformal invariance implications within astrophysical and cosmological contexts. Additionally, the exploration of Hilbert space constructions in shape dynamics could redefine conventional approaches in loop quantum gravity, potentially leading to novel interpretations of quantum spacetime.

In conclusion, this paper dissects and reconfigures the theoretical landscape of gravity, arguing that shape dynamics offers both a valid alternative and a complementary perspective to Einstein gravity, fundamentally altering how spacetime and geometry interplay in classical and potentially quantum realms.