A sealed laboratory is dropped from a tall gantry and falls freely near the Earth's…
PHYSICS · Multi-part · Fermi University Physics · Relativity & Gravitation · The equivalence principle and gravity as geometry · Differential geometry of curved spacetime
Relativity and Gravitation, Ch Gravity as curved spacetime: redshift and local inertial frames
Problem
A sealed laboratory is dropped from a tall gantry and falls freely near the Earth's surface. Inside it, two small test masses are released at rest relative to the laboratory, L=2.0 m apart along a horizontal line, that is, at right angles to the line towards the Earth's centre. Air drag inside and outside the laboratory is ignored, the laboratory does not rotate, and the tidal field is treated as uniform over the laboratory (first order in the separation). Take the Earth's mass M⊕=5.972×1024 kg, its radius R⊕=6.371×106 m, G=6.674×10−11Nm2kg−2, and treat the laboratory as staying at r≈R⊕ throughout.
((a)) Estimate the magnitude of the tidal field component along the line joining the pair, ∣E⊥∣=GM⊕/r3, and say whether the pair drifts apart or together.
((b)) The experimenter can resolve a change of separation of δ=0.10 mm. For roughly how long does the laboratory count as an inertial frame at that resolution, and why is this longer than for a vertical pair of the same length?