Show that the proper time, Delta au , for a radial infall from r=R to r=0 is finite for a timelike geodesic. Assume that the particle is "released" at r=R, thus its conserved energy can be written by using the relation u_(a)u^(a)=-1 for radial motion only. Use this result to then obtain d(r)/(d) au . Solve this for d au and finally integrate from r=R ro r=0 to obtain
Delta au =(pi R^((3)/(2)))/(2sqrt(2M))
Then compute the infall time in seconds for R=10M where M=M_(o.).
2. Show that the proper time, , for a radial infall from r = R to r = 0 is finite for a
timelike geodesic. Assume that the particle is "released" at r = R, thus its conserved
energy can be written by using the relation uu = -1 for radial motion only. Use this
result to then obtain dr/d. Solve this for d and finally integrate from r = R ro r = 0
to obtain
T =TTR3/2 2V2M
Then compute the infall time in seconds for R = 10M where M = Mo
