Assignment 1

Full Marks: 25

Due date: 06/02/18

1. Consider a spacetime metric, ds2 = gµνdxµdxν = (1−rg sin θ/r)c2dt2−

(1 + rg sin θ/r)dr2 − r2dθ2 − r2 sin2 θdφ2, where rg is a constant and c isspeed of light. Find out the equations of motion of a point particle inthis spacetime and then proceed for obtaining the temporal and spatialcomponents of its velocity at an arbitrary spacetime point. 6+6

2. Consider a lightning occurring in the equatorial plane at a point ‘A’ ofradius 1000 km from the center of gravity in the above spacetime and anobserver at ‘B’ of radius 900 km sees it. What would be the frequency shiftof light off lightning when the points ‘A’ and ‘B’ are at rest and rg = 100km? Interpret the result. 4

3. For the above metric, find out the components of tetrad/vierbein.Establish whether the corresponding spacetime is flat or curved due togravity. 4+5