Local Accelerations

Einstein formulated the general theory of relativity (GTR) in 1915 based on his Equivalence Principle. In prerelativistic terms, a uniform gravitational field of strength g may be exactly simulated inside a rigid laboratory in a completely gravity-free region of space by accelerating this laboratory with a constant acceleration g m/s² relative to an inertial frame. By releasing two small test masses, their behavior reveals the physical environment.

Suppose an unseen massive body is near the rigid laboratory. What behaviors of the two small test masses will reveal its presence?

Answer

The presence of the massive body can be determined by the trajectories of the two test masses upon their release. In the simple case in which the laboratory is not moving with respect to the massive body, when released equidistant from the object but separated from each other, the two test masses will move toward each other faster than their mutual gravitational acceleration as they fall toward the body. In addition, if they are separated vertically so that one test mass begins closer to the massive body than the other, their vertical separation distance will change as they fall. In a uniform gravitational field their separation distance would remain fixed in value in each test.

One can extend this problem to consider a rotating massive body. Can observers inside a spaceship determine by “local” measurements only that is, without looking outside if they are in the field of a rotating central mass, or if they are just moving with velocity V on a Schwarzschild background metric? Yes they can; by using at least four test particles inside their spaceship and having the capability to measure their relative accelerations, they can succeed in determining all the components of the Riemann tensor and decide whether they are in the spacetime curvature of a rotating central mass. Note that gyroscopes do not help here because one would need to check their alignment with stars outside, which is forbidden. The challenge here is to measure a new effect, called intrinsic gravitomagnetism, introduced by the GTR, that the space-time geometry and the corresponding curvature invariants are affected and determined by both mass-energy and mass-energy currents relative to other mass that is, by mass-energy currents that cannot be eliminated by a Lorentz transformation. See the Ciufolini and Wheeler reference below for the details.

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