Einstein Gravity from Conformal Gravity (1105.5632v2)

Abstract: We show that that four dimensional conformal gravity plus a simple Neumann boundary condition can be used to get the semiclassical (or tree level) wavefunction of the universe of four dimensional asymptotically de-Sitter or Euclidean anti-de Sitter spacetimes. This simple Neumann boundary condition selects the Einstein solution out of the more numerous solutions of conformal gravity. It thus removes the ghosts of conformal gravity from this computation. In the case of a five dimensional pure gravity theory with a positive cosmological constant we show that the late time superhorizon tree level probability measure, $|\Psi [ g ]|^2$, for its four dimensional spatial slices is given by the action of Euclidean four dimensional conformal gravity.

• The paper demonstrates that imposing Neumann boundary conditions in four-dimensional conformal gravity recovers Einstein gravity solutions without ghost states.
• It employs analytic continuation in asymptotically de-Sitter and Euclidean anti-de Sitter spacetimes to establish a direct link between the two gravitational theories.
• The study highlights computational advantages at tree-level and explores implications for black hole thermodynamics and partition functions.

Overview of "Einstein Gravity from Conformal Gravity"

This paper explores a novel approach to bridging the gap between conformal gravity and Einstein gravity, specifically within the context of asymptotically de-Sitter and Euclidean anti-de Sitter (EAdS) spacetimes. The primary assertion is that by implementing a Neumann boundary condition within four-dimensional conformal gravity, one can derive the classical wavefunction of the universe that aligns with the solutions predicted by Einstein gravity, thereby eliminating the problematic ghosts typically associated with conformal gravity.

Key Concepts

Conformal gravity is a gravitational theory that exhibits invariance under Weyl transformations, characterized by an action constructed from the square of the Weyl tensor. Despite its appealing symmetry properties, conformal gravity has historically been undermined by the presence of ghost states due to its higher-order equations of motion. The paper suggests that adding simple boundary conditions can effectively reduce the viable solutions to only those corresponding to Einstein's solutions, thereby avoiding the ghost problem.

Main Results and Claims

1. Boundary Conditions and Einstein Gravity Solutions: The authors demonstrate that applying Neumann boundary conditions selects solutions matching those from Einstein gravity. This boundary condition suppresses additional conformal gravity solutions, ensuring that ghosts do not contribute to the semiclassical wavefunction.
2. Conformal Gravity and Superhorizon Measures: For five-dimensional de Sitter (dS) or AdS settings, the paper argues that the wavefunction of the four-dimensional slices is determined by the Euclidean conformal gravity action. This finding derives from an analytic continuation approach and establishes a profound link between four-dimensional conformal gravity and the semiclassical limits of higher-dimensional gravitational theories.
3. Computational Advantages: It is posited that conformal gravity might offer computational advantages, particularly at the level of tree diagrams. The possibility of leveraging conformal gravity's symmetries for efficient calculations in AdS or dS settings merits further exploration.
4. Black Holes and Partition Functions: The paper extends these ideas to explore black hole solutions within the framework of conformal gravity, noting potential implications for partition functions and thermodynamic properties.

Implications and Future Directions

The implications of this work are significant for both theoretical and computational aspects of gravitational theory. If conformal gravity can effectively reproduce the predictions of Einstein gravity at tree-level, it holds potential for simplifying calculations involving gravitational interactions without sacrificing physical accuracy.

Moreover, the proposed equivalence offers a fresh perspective on the interplay between different formulations of gravity, hinting at possible reconciliations between classical and quantum descriptions of spacetime. Theoretically, the paper raises intriguing questions about the nature of gravity in various dimensions and suggests new avenues for exploring quantum gravity theories using conformal symmetry.

Future developments may investigate the full quantum implications of this approach, particularly with respect to N = 4 supersymmetric conformal gravity, which is conjectured to be finite. Moreover, a rigorous elucidation of the quantum corrected wavefunction might clarify whether conformal gravity can provide a viable framework for a quantum theory of gravity.

This paper lays foundational work for re-examining gravitational phenomena through the lens of conformal gravity, particularly in non-trivial spacetime geometries, offering a potentially wide array of applications in theoretical physics.