madness said:...What do you mean by coordinates here? Presumably transforming to a frame which is accelerating with respect to the flat frame would result in curvature...
Curved space refers to the concept of space being curved or distorted, as described by Einstein's theory of general relativity. Coordinate choices, on the other hand, refer to the different ways of representing this curved space mathematically, such as using different coordinate systems or reference frames.
Distinguishing between curved space and coordinate choices is important because it helps us understand the underlying geometry of space and how it can be described and measured mathematically. It also allows us to make accurate predictions and calculations in areas such as physics and astronomy.
One way to differentiate between curved space and coordinate choices is by looking at the mathematical equations themselves. Curved space will be represented by equations that involve curvature terms, such as the Ricci tensor or Ricci scalar, while coordinate choices will be reflected in the choice of coordinate system, such as Cartesian, polar, or spherical coordinates.
Yes, curved space and coordinate choices can coexist. In fact, coordinate choices are necessary for describing curved space mathematically. Different coordinate systems can be used to represent the same curved space, with each coordinate system providing a different perspective or way of measuring the curvature.
Yes, there are many practical applications for understanding the distinction between curved space and coordinate choices. This knowledge is essential for fields such as physics, astronomy, and engineering, where accurate calculations and predictions in curved spaces are crucial. It also has implications for understanding the structure of the universe and the behavior of objects in space.