Question

Coordinates and metric 6.1 Hartle's Equation (7.7) shows an example where finite coordinate distance corresponds to infinite proper distance. In his Problem 7.1 you work out the proper distance from r' = 0 to, say, r1 at fixed d. It's a2 A.S=/ds= dr' r/2 s' lo which is infinitely greater than the coordinate distance of 1 Let's do the opposite, look at an infinite coordinate distance which is a finite proper distance. In Cartesian coordinates and flat space it takes you a fraction of a second to work out that the distance from (0,0) to (1,0) is 1. (Proper distance and coordinate distance are the same here.) But consider flat space in the coordinates u In x v = Y. (a) What is the coordinate distance between the two points mentioned above? (b) Write the metric in (u, v) coordinates.

Answer 1

u,u co-ordinates infinity a.) As we know eno is indefinite .so in - the coordinate distance is indifinite (14,,",) = (indefinite ,o) (94., 02) - (0,0) to coordinate distance indefinite an b) flat space time metue is ds²=-d+²+dx²+ dy²+ d2² but dx = e "duf dy=du. ds²= -4+²+ e?" du? + dw² + d2² x=e, y=r This is proper an example of distance proper infinite co-ordinate but finite distance in this case is 21 du du=of V20 laro =) cl) u=-0 2 - 0 ELT