# The Triplet Paradox and the Alternative Lorentz Transformation

An experiment with cesium clocks carried onboard two aircraft as they circumnavigated the globe in opposite directions provided an important confirmation of the time-dilation effect of the special theory of relativity (STR). A key element in the discussion of this event was the fact that since a clock located on the Equator is not at rest in an inertial system because of the Earth’s rotation, it could not be used directly as a reference in applying Einstein’s formula. It is pointed out that this line of argumentation implies that if the Earth were not rotating, the onboard clocks

would both be at rest in inertial systems after the airplanes had reached their cruising altitude and had stopped accelerating. On this basis it could be concluded that a person traveling on one of these airplanes would be aging more rapidly than his twin on the other because the latter is in relative motion to him. One can make exactly the same argument for the other twin, however, and this leads to an obvious contradiction since they cannot both be aging faster than the other.

To resolve this issue (Triplet Paradox), it is necessary to distinguish between different inertial systems based on their state of motion. In particular, it is argued that there is a uniquely defined objective rest system (ORS) in both cases (i.e., with and without the Earth rotating) from which the prescriptions of the theory must be applied. This interpretation is put on a sound basis by introducing an alternative Lorentz transformation (ALT) that not only satisfies both of Einstein’s original postulates of STR but also assumes that the rates of clocks are always strictly

proportional to one another regardless of their position in space or state of relative motion. The success of the Global Positioning System (GPS) technology provides detailed experimental verification of the latter assumption and therefore rules out the original Lorentz transformation (LT) as a physically valid set of equations relating the space-time measurements of observers in different rest frames.