The Schrödinger equation, which is a fundamental equation in quantum mechanics, describes the behavior of quantum systems in terms of wave functions. It is a non-relativistic equation and does not explicitly incorporate the effects of gravity or the nature of spacetime.
However, when we consider the fundamental principles of quantum mechanics and general relativity, which is the theory of gravity in the framework of Einstein's theory, there arises a need for a theory of quantum gravity. The main motivation behind quantum gravity is to reconcile the principles of quantum mechanics and general relativity in a consistent framework.
Quantum mechanics successfully describes the behavior of particles at microscopic scales, while general relativity describes the behavior of gravity and the structure of spacetime on cosmological and macroscopic scales. Yet, when we attempt to apply both theories simultaneously, significant conceptual and mathematical difficulties arise. In particular, the mathematics of quantum mechanics and general relativity do not easily mesh together, and the two theories appear incompatible in certain situations, such as the extreme conditions found in the early universe or at the center of black holes.
A theory of quantum gravity seeks to unify these two fundamental theories by providing a framework that encompasses both quantum mechanics and general relativity. It aims to describe the behavior of gravity and the structure of spacetime at the quantum level. Such a theory would be necessary to understand phenomena that involve both quantum mechanical and gravitational effects, such as the behavior of particles in the vicinity of black holes or the very early universe.
While several approaches to quantum gravity have been proposed, such as string theory, loop quantum gravity, and others, a complete and experimentally validated theory of quantum gravity is still an ongoing area of research and remains a significant challenge in theoretical physics.