Chapter 6 derives the gravitational redshift by comparing light signals sent between…
PHYSICS · Multi-part · Fermi University Physics · Relativity & Gravitation · The equivalence principle and gravity as geometry · The equivalence principle and gravity as geometry
The Geometry of Gravity, Ch The equivalence principle under test
Problem
Chapter 6 derives the gravitational redshift by comparing light signals sent between two fixed heights in a static Newtonian potential Φ, reading off Δν/ν=ΔΦ/c2; this is the special case αLPI=0 of the general local-position-invariance relation Δν/ν=(1+αLPI)ΔΦ/c2. Chapter 5's Rindler treatment of an accelerating rocket derives, from special relativity alone, that clocks fixed at the nose and tail run at different rates. Chapter 6's Schild argument uses a nonzero photon redshift, together with energy conservation in flat spacetime, to conclude that spacetime is curved. Suppose a satellite clock comparison detects a genuine, statistically decisive nonzero value αLPI=1.1×10−6: a real effect, not a bound consistent with zero.
((a)) Does the Rindler-coordinate line element and the differential ageing it predicts between clocks fixed at the nose and tail of the accelerating rocket remain valid without amendment?
((b)) Does the formula Δν/ν=ΔΦ/c2, used to read the redshift straight off the Newtonian potential, remain valid without amendment?
((c)) Does Schild's conclusion, that a nonzero redshift forces spacetime to be curved, survive? State for each whether it survives unchanged, needs amendment, or is unaffected in its logical force, with a one-sentence reason, and say in one sentence what a nonzero αLPI does and does not tell us about how the Earth's mass produces Φ.