Casual Dynamical Triangulation (CDT) is a framework used in theoretical physics, specifically in the field of quantum gravity, to describe the microscopic structure of spacetime. It is one of the approaches to formulating a theory of quantum gravity, which aims to reconcile quantum mechanics with Einstein's theory of general relativity.
In general relativity, spacetime is a smooth and continuous manifold that is subject to the curvature caused by the presence of matter and energy. However, at very small distances or high energies, such as those encountered near the Planck scale, it is expected that the structure of spacetime becomes fundamentally discrete or "quantized." This discreteness is necessary to reconcile the principles of quantum mechanics with gravity.
In CDT, the approach is to discretize spacetime by representing it as a collection of simple building blocks called simplices. These simplices are geometric objects with well-defined shapes and sizes, which can be thought of as higher-dimensional analogues of triangles. By assembling these simplices in a particular way, CDT provides a discrete approximation of the underlying spacetime geometry.
The key distinction between CDT and Einstein's theory of general relativity lies in the treatment of time. In CDT, spacetime is treated in a way that gives a preferred direction to time, unlike general relativity, where time is treated on an equal footing with the three spatial dimensions. This preferred direction arises from the specific construction of the discrete spacetime lattice in CDT, where the notion of causality is preserved.
This preferred direction of time in CDT is often referred to as "causal," as it implies that the ordering of events in time is well-defined and influences how the geometry of spacetime is constructed. By imposing causality on the discrete spacetime lattice, CDT allows for the study of the dynamics of spacetime and the emergence of a continuum description at larger scales.
It's important to note that CDT is still a theoretical framework under development, and there is ongoing research and exploration into its properties and implications. While it offers a different perspective on the structure of spacetime compared to general relativity, its full compatibility and connection to Einstein's theory of relativity, particularly in the context of quantum gravity, are still subjects of active investigation.