Einstein-Cartan geometry is a type of geometric framework that extends Einstein's theory of general relativity to include the intrinsic spin of particles, as described by Cartan's theory of spinors. It allows for a more comprehensive understanding of the curvature of spacetime and the behavior of matter and energy within it.
CDT, or Causal Dynamical Triangulations, is a quantum gravity theory that combines the principles of general relativity with those of quantum mechanics. Einstein-Cartan geometry is used in CDT to describe the discrete spacetime structure, or "building blocks," on which the theory is based. It allows for the incorporation of both curvature and spin into the model.
The main difference is the incorporation of intrinsic spin into the theory. In general relativity, gravity is described solely by the curvature of spacetime, while in Einstein-Cartan geometry, the spin of particles also plays a role in the behavior of gravity. Additionally, Einstein-Cartan geometry allows for torsion, or the twisting of spacetime, which is not included in general relativity.
One potential implication is the ability to better understand the behavior of matter and energy at a quantum level in the presence of gravity. Einstein-Cartan geometry also allows for the possibility of describing spacetime as discrete building blocks, rather than a continuous fabric, which could have implications for our understanding of the fundamental nature of the universe.
Einstein-Cartan geometry in CDT is one of many approaches being explored in the search for a theory of quantum gravity. It offers a unique perspective by incorporating both curvature and spin into the model, which could potentially lead to a more comprehensive understanding of the behavior of matter and energy at a quantum level in the presence of gravity. Its use in CDT also allows for the exploration of discrete spacetime structures, which could have implications for the unification of quantum mechanics and general relativity.