- The paper demonstrates that conformal invariance of the Weyl curvature tensor arises from decomposing the Riemann tensor into trace and traceless parts.
- It formulates explicit conditions for conformal transformations leading to Ricci-flat and Einstein-matter spacetimes using modern gauge theory techniques.
- The study constructs a Weyl-covariant gravity model, showcasing local scale invariance and potential pathways for unifying gravity with other fundamental interactions.
An Overview of Weyl Geometry
This paper presents a comprehensive paper on the properties of Weyl geometry, building on the foundational work established in Riemannian spacetimes. It begins by exploring the conformal invariance of the Weyl curvature tensor, establishing that it is the trace-free part of the Riemann curvature tensor. The decomposition of the Riemann curvature into trace and traceless components facilitates this understanding and underscores its significance in conformal transformations.
One of the key contributions of this work is the formulation of conditions for the existence of conformal transformations to a Ricci-flat spacetime and extension to spacetimes satisfying the Einstein equation with matter. Additionally, the paper introduces a theory of Weyl-covariant gravity constructed around a curvature-linear action. In this framework, vacuum solutions are presented as conformal equivalence classes of Ricci-flat metrics within an integrable Weyl geometry. Notably, this theory remains invariant under local dilatations but not under the complete conformal group, distinguishing it from other potential formulations of conformal gravity.
The paper further revisits the pioneering work of H. Weyl who proposed incorporating additional symmetries in Riemannian geometry to unify electromagnetism with gravity. This theoretical pursuit, though ultimately unsuccessful in its original form, gave rise to U(1) gauge theory and deepened the understanding of Weyl geometry itself. Weyl geometries permit variations in both orientation and length of vectors under parallel transport, thus providing a broader generalization of Riemannian geometries. The subsequent development of integrable Weyl geometries, where the Weyl vector is derived from the gradient of a function, reveals connections to scale invariant general relativity.
Three rationales are presented for studying general relativity through the lens of integrable Weyl geometry:
- Local Scale Invariance: General relativity is inherently invariant under global changes of units. Formulating it within an integrable Weyl geometry extends this invariance to a local scale.
- Conformal and Projective Geometries: By leveraging the paths of both light pulses and massive test particles, a conformal connection on spacetime can be aligned with a Weyl geometry in the limit of near-light velocities, as demonstrated in the work by Ehlers, Pirani, and Schild.
- Higher Symmetry Approaches: Gravitational theories that embrace the full conformal group naturally align with formulations based on integrable Weyl geometry rather than exclusively Riemannian structures, potentially offering additional insights or realizations of general relativity.
The paper employs modern techniques from gauge theory to elucidate the structure of Weyl geometry, systematically walking through derivations and their implications for curvature, metric, and geometric connections. The decomposition of the Riemann curvature is shown to reveal the Weyl curvature tensor's conformal invariance, and explicit transformations of the Ricci and Schouten tensors ensure this invariance is maintained.
Furthermore, the discussion extends to the theoretical formulation of Weyl-covariant gravity grounded in alternative symmetries like the biconformal group. Actions nonlinear and linear in curvature are examined, including those that render the underlying theory potentially renormalizable due to scale invariance properties.
Implications and Future Directions
This paper advances the theoretical framework of Weyl geometry by demonstrating its mathematical elegance and potential applications to gravitational models. The nuanced understanding of conformal invariance and scale covariant gravity paves the way for deeper exploration in both classical and quantum realms of gravitational theory. In particular, the notions explored here might inform future attempts to coherently integrate gravity with other fundamental interactions, especially in contexts where symmetry and invariance play crucial roles.
The simplifications introduced for exploring Weyl geometries in any spacetime dimension might catalyze additional research into higher-dimensional theories of gravity or even string theory, where such geometric concepts frequently find applicability. Furthermore, the nuanced modifications to Dirac's approach to scalar-tensor theories open paths for reconsideration and potential reconciliation of long-standing problems within those frameworks.
Overall, this investigation offers significant insights into the subtleties and potential of Weyl geometry as a rich field of paper, continuing to inspire theoretical exploration and possibly informing our understanding of the universe at its most fundamental level.