The idea that the number of spacetime dimensions is not dependent on the choice of coordinates in string/M-theory can be understood through the concept of background independence. Background independence is a fundamental principle in modern theoretical physics that suggests the laws of physics should be independent of the specific background structure or coordinates used to describe spacetime.
In string/M-theory, the theory is formulated in a way that is background independent. This means that the theory is constructed in a manner that does not assume a fixed background geometry or a predetermined number of dimensions for spacetime. Instead, the theory allows for the possibility of different solutions with varying numbers of spacetime dimensions.
To see why the number of dimensions is not dependent on the choice of coordinates, we can consider the concept of compactification. In string/M-theory, additional dimensions beyond the usual three spatial dimensions (x, y, z) and one time dimension (t) can be "curled up" or compactified to form tiny, curled dimensions that are not directly observable at our energy scales.
The specific shape and size of these compactified dimensions can vary in different solutions. The choice of coordinates used to describe these dimensions is arbitrary and does not affect the underlying physics. The theory remains consistent and background independent regardless of the choice of coordinates.
In this framework, the number of dimensions is not fixed but can be determined dynamically through the geometry and topology of the compactified dimensions. The theory allows for different solutions with different numbers of spacetime dimensions, and the choice of coordinates does not impact this fundamental aspect of the theory.
It's important to note that while this explanation provides an understanding of the concept within the framework of string/M-theory, the topic is highly complex and involves advanced mathematical concepts. String/M-theory is an active area of research, and many aspects are still being explored and refined by physicists.