In the framework of quantum gravity, the idea of superposing different shapes of spacetime, or superposing different quantum states of spacetime, is indeed a challenging concept. The concept you're referring to, where spacetime can be in a superposition of different shapes or geometries, is often discussed in the context of the superposition principle in quantum mechanics.
In quantum mechanics, the superposition principle states that a physical system can exist in a combination, or superposition, of multiple states simultaneously. However, applying this principle to spacetime and gravity introduces several difficulties. One of the main challenges is that general relativity, which describes gravity as the curvature of spacetime, does not naturally incorporate superposition.
In general relativity, the shape or geometry of spacetime is determined by the distribution of matter and energy within it. The curvature of spacetime is related to the stress-energy tensor, which is a source term in Einstein's field equations. This implies that the geometry of spacetime is determined by the matter and energy content at a particular point, rather than being a superposition of different geometries.
The problem arises when we try to reconcile quantum mechanics with general relativity, leading to the quest for a theory of quantum gravity. In such a theory, the nature of spacetime itself would need to be described quantum mechanically. However, there is currently no widely accepted theory of quantum gravity that has been conclusively established.
In some approaches to quantum gravity, such as loop quantum gravity or certain string theory formulations, discrete or quantized structures of spacetime are considered. These theories propose that spacetime has a fundamental granularity at very small scales, where the notion of smooth and continuous geometry breaks down. In these frameworks, the superposition of different discrete configurations of spacetime can be mathematically defined.
However, whether curved spacetimes form a vector space in the traditional sense is not straightforward. Vector spaces have well-defined properties, such as closure under addition and scalar multiplication, and these properties may not hold for superpositions of different curved spacetimes. The mathematical framework to describe superpositions of curved spacetimes in quantum gravity is still an active area of research and subject to ongoing theoretical investigations.
In summary, while the idea of superposing different shapes of spacetime is intriguing and has been explored in various approaches to quantum gravity, the precise mathematical and conceptual framework for describing these superpositions is still an open question. The field of quantum gravity aims to provide a consistent framework that unifies quantum mechanics and general relativity, but a complete understanding of how to treat superpositions of curved spacetimes is still an area of active research.