PHYSICS · Subjective · Fermi University Physics · Relativity & Gravitation · The equivalence principle and gravity as geometry · The equivalence principle and gravity as geometry
Relativity and Gravitation, Ch Gravity as curved spacetime: redshift and local inertial frames
Problem
A transmitter stands at the foot of a rigid, unchanging tower on the Earth and a receiver at the top, a height h=250 m above it. Both are permanently at rest in the tower frame, and the field is static. The transmitter sends short pulses exactly 1.000 s apart as read by its own clock. Take g=9.81ms−2, constant over the tower, and work to first order in gh/c2. Exactly one of the following statements is correct. Which? (a) Because neither transmitter nor receiver moves in the tower frame there can be no first-order shift at all; any shift seen in such an experiment is the ordinary Doppler shift caused by the Earth's orbital motion. (b) Drawn in Minkowski geometry, the two pulse worldlines are carried into one another by a translation in t, which leaves the geometry unchanged, so the arrival gap at the top would be exactly 1.000 s; but the redshift derived twice over requires the gap to be longer by (gh/c2)×1.000s=2.7×10−14s, so a flat spacetime cannot describe a static tower in a gravitational field. (c) The clash is removed by keeping spacetime flat and adopting a force law in which light loses energy as it climbs; only the law of force, not the geometry, has to change. (d) The clash does show that spacetime is curved, but curvature is meaningful only if spacetime is embedded in a further spatial dimension for it to bend into.