The equivalence principle and local frames

An ideal accelerometer measures the magnitude of four-acceleration $a^\mu=u^\nu\nabla_\nu u^\mu$. With our $(+,-,-,-)$ signature its nonzero acceleration vector is spacelike, so its reading is $\sqrt{-a^\mu a_\mu}$. An observer held above Earth reads approximately g. A small freely falling instrument reads zero, ignoring air resistance, spin coupling, and finite-size effects. Its coordinate altitude can nevertheless have a nonzero second derivative.

Coordinate acceleration depends on the chart. One may choose coordinates in which an orbiting object changes spatial position rapidly, or coordinates moving with that object. Proper acceleration is instead determined along the worldline and survives a change of chart. This makes it the right diagnostic for asking whether a rocket engine, a floor, or another nongravitational mechanism is forcing an observer away from free fall.

At any event in a smooth spacetime, choose an orthonormal tetrad: four basis vectors whose inner products reproduce the Minkowski metric. In suitable local coordinates, the metric equals the flat metric at that event and its first derivatives vanish. The connection can therefore vanish there. Second derivatives generally remain, encoding curvature. An extended neighborhood cannot in general be made exactly Minkowskian merely by renaming its coordinates.

A freely falling laboratory approximates special relativity when its size and observation time are small compared with curvature scales. Along a chosen geodesic, a suitable transported frame makes this approximation systematic. Two separated falling particles can still accelerate relative to each other. Their relative motion is a tidal measurement, so it distinguishes genuinely curved geometry from a uniformly accelerated laboratory in flat spacetime. Local equivalence is powerful precisely because its scope is stated.

In a weak, static gravitational field, take the Newtonian potential to satisfy $|\Phi|/c^2\ll1$. A clock held at fixed spatial coordinates has $d\tau\simeq(1+\Phi/c^2)dt$. A higher clock near Earth, where the potential is less negative, accumulates slightly more proper time per unit of this shared stationary coordinate time. Comparing clocks requires a specified signal protocol or a reunion; there is no direct universal clock at every location.

Near Earth over a height small compared with its radius, the potential difference is approximately $g\Delta h$. The resulting fractional clock-rate difference is approximately $g\Delta h/c^2$. This is not caused by a clock mechanism becoming defective: ideal atomic clocks measure proper time along different worldlines. If the clocks move relative to the stationary coordinates, kinematic time dilation contributes too, and must be included before interpreting the measured difference as gravitational.