Einstein and Gravitation — Question 1

Question 1

A freely falling accelerometer does not measure the Newtonian gravity magnitude

In a small laboratory near a spherical planet, an ideal spring accelerometer measures non-gravitational force per unit mass. Compare a supported laboratory with one in free fall. Work in the weak-field limit and neglect tidal differences across the apparatus.

1. Find accelerometer readings in the two laboratories when local Newtonian gravity is $g$.

2. Explain which laboratory approximates a local inertial frame and why a zero reading does not establish that spacetime is flat.

Question 1 - Solution

A freely falling accelerometer does not measure the Newtonian gravity magnitude

(a) Derivation and calculation

Take upward as positive. A supported laboratory satisfies $$N-mg=0,\qquad a_{\mathrm{meter}}=N/m,$$ so $$\boxed{a_{\mathrm{meter,supported}}=g}.$$ In free fall, $$N=0,\quad\ddot z=-g,\qquad\boxed{a_{\mathrm{meter,free}}=0}.$$ The accelerometer measures non-gravitational force per unit mass, not the downward coordinate acceleration.

(b) Result and interpretation

For an approximately uniform field, use a falling origin $z_f$ with $\ddot z_f=-g$. A free particle has $$\ddot z'=\ddot z-\ddot z_f=(-g)-(-g)=0.$$ This gives a local inertial description. For separated free particles, however, $$\delta\ddot{\vec r}\simeq(\delta\vec r\cdot\boldsymbol\nabla)\vec g$$ can remain nonzero: one zero accelerometer reading does not establish flat spacetime.

Physical check

The local approximation removes a common acceleration, while finite tidal differences can remain measurable.