Then the time invariance of the metric implies conservation of the quantity g ( v , d t ) = v 0 / T 2 {\displaystyle g(v,dt)=v^{0}/T^{2}} , where v 0 {\displaystyle v^{0}} is the time component of the 4-velocity v {\displaystyle v} of the body. At the infinity g ( v , d t ) = 1 {\displaystyle g(v,dt)=1} , so v 0 = T 2 {\displaystyle v^{0}=T^{2}} , or, in coordinates adjusted to the local time dilation, v l o c 0 = T {\displaystyle v_{loc}^{0}=T} ; that is, time dilation due to acquired velocity (as measured at the falling body's position) equals to the gravitational time dilation in the well the body fell into. Applying this argument more generally one gets that (under the same assumptions on the metric) the relative gravitational time dilation between two points equals to the time dilation due to velocity needed to climb from the lower point to the higher.