Theory of relativity

The equivalence principle

Within a sufficiently small region, a uniform gravitational field is locally indistinguishable from acceleration of the reference frame; this principle motivates general relativity.

Imagine being inside a sealed elevator. At rest on Earth, the floor supports you; accelerating through empty space, the floor pushes you in much the same way. Over a small enough region, the two situations give the same mechanical results.

a_{\text{grav}} \leftrightarrow a_{\text{frame}}

Definition: Local statement

Einstein's equivalence principle says that in a sufficiently small freely falling laboratory, nongravitational physics takes the form of special relativity. “Local” matters: a nonuniform gravitational field produces tidal effects that no change of frame can remove over a finite region.

Compare trajectories in an accelerating frame with motion near a mass. Enlarge the observed region to see tidal effects emerge.

From force to geometry

An observer held at rest in a gravitational field has proper acceleration, whereas an ideal freely falling observer feels no weight. In general relativity, free bodies follow spacetime geodesics; tidal effects reveal curvature rather than a uniform force that can be transformed away.

Example: Accelerating elevator

An elevator in empty space accelerates upward at $a=9.8\,\mathrm{m/s^2}$ . An object is released relative to the cabin. What is its acceleration relative to the cabin?

Solution

In the inertial frame of empty space, the object has no force and no acceleration while the cabin accelerates upward. Relative to the cabin it accelerates downward at $9.8\,\mathrm{m/s^2}$ , as in the equivalent gravitational field.

Example: Estimating a tidal acceleration

In a Newtonian field, $g(r)=GM/r^2$ . Two test masses are separated radially by $L\ll r$ at distance $r$ from the source. Estimate their acceleration difference to see why a finite freely falling laboratory cannot remove every gravitational effect.

Solution

To first order, $\Delta g\simeq (dg/dr)L=-2GM L/r^3$ . Its magnitude is the radial tidal acceleration: free fall removes the common acceleration at the laboratory’s center, not the field gradient.

Quick check

In a sufficiently small experiment, can mechanical measurements distinguish a uniform gravitational field from cabin acceleration?

Why does equivalence hold only locally?

References

Albert Einstein (1953). The Meaning of Relativity
Sean Carroll (2019). Spacetime and Geometry