Concept of Curvature

Question

i. By means of appropriate examples in two, three and four dimensions, explain the concept of curvature. 

 

ii. What criteria could be used to determine of a space (or spacetime) is flat or curved?

Answer 1

i) The concept of curvature in two dimensions could be a circle. Although it only needs one parameter to define its curve, the distance between two points on the circle would use the two dimensional Euclidean distance. To go between two points on a circle the distance traveled differs from the Euclidean distance. For three dimensions you could use the surface of the Earth. Again to go between two points the shortest distance is more than the Euclidean distance. In four dimensions you could consider the Schwarzschild black hole. In this case going between two points would also include going forward in time. The distance between two points radially would differ from the Euclidean distance between them. The classical example of the infinite path length of a person falling into a black-hole also demonstrates the curvature of the black-hole.

ii) A good criteria for curvature would be if the Euclidean equation for distance doesn't hold locally the space is curved.