In a recent thread, the question came up of whether the presence of gravitational time dilation implies spacetime curvature. My answer in that thread was no: This was based on the obvious counterexample of observers at rest in Rindler coordinates in flat Minkowski spacetime; two observers at different Rindler ##x## coordinates will be gravitationally time dilated relative to each other, yet the spacetime they are in is flat. However, it occurred to me that there is an argument in the literature, originally due to Schild and described in MTW (which is where I encountered it), which purports to show that gravitational time dilation does imply spacetime curvature. I will give the argument here as it is given in section 7.3 of MTW: Consider two observers at rest in the gravitational field of the Earth, one at height ##z_1## and the other at height ##z_2 > z_1##. The lower observer sends two successive light pulses to the upper observer. This defines four events in spacetime as follows: E1 and E2 are the emissions of the two light pulses by the lower observer, and R1 and R2 are the receptions of the two light pulses by the upper observer. These four events form a parallelogram in spacetime--it must be a parallelogram because opposite sides are parallel. The lower and upper sides, E1-E2 and R1-R2, are parallel because the two observers are at constant heights; and the light pulse sides, E1-R1 and E2-R2, are parallel because the spacetime is static, so both light pulses follow exactly identical paths--the second is just the first translated in time, and time translation leaves the geometry of the path invariant. However, the lower and upper sides of this parallelogram have unequal lengths! This is because of gravitational time dilation: the upper side, R1-R2, is longer than the lower side, E1-E2. This is impossible in a flat spacetime; therefore any spacetime in which gravitational time dilation is present in this way must be curved. The problem is that the above argument would seem to apply equally well to a pair of Rindler observers in Minkowski spacetime! The worldlines of observers at rest in Rindler coordinates are orbits of a timelike Killing vector field, so two successive light pulses from a Rindler observer at ##z_1## in Rindler coordinates to a second observer at ##z_2 > z_1## should be parallel, and so should the worldlines of the observers themselves. So we should have a parallelogram in the same sense, but with two opposite sides unequal--which should imply that Minkowski spacetime must be curved! So the question is: how do we reconcile these apparently contradictory statements?