Box 9.2 falling from rest in schwarzschild spacetime the geodesic equation for any spacetime is (see equation 8.14); (9.19) in order to evaluate this, we need to know the four-velocity nt" of an object at rest. the spatial components of u" are zero, but the time component ur e dt/dt 1 here because the coordinate time t in this situation is not the same as the time τ mea- sured by the object's clock, even if the object is at rest. we can most easily evaluate what ur is in these coordinates using the tensor equation u-u-u"guvu" =-1, which applies in all coordinate systems. in this case, since the spatial components of 11μ are all zero, you can show that this implies that u, = (1-20m)-1/2 (this is a valuable technique that we will use often to evaluate the t component of (9.20) the four-velocity of an object at rest in a given spacetime.) exercise 9.2.1. verify equation 9.20. now return to equation 9.19. first, note that when the metric is diagonal (as it is in this case), the definition of the metric inverse garg 8/ simply implies that the metric inverse is also diagonal and each diagonal element of goß is simply equal to vbap. note als our object is nonzero, the only nonzero term in the implicit sums over μ and v will be the one where μ = v = . using this information, you can easily show that the geodesic equation for γ = r in this case implies that o in this case that since only the t component of the four-velocity of dr gm (9.21) exercise 9.2.2. verify equation 9.21.