The Twin Paradox

Twin Paradox as seen by the Twin left on the Earth:

The Earth Twin can measure the distance from the Earth to the Start with an instrument equivalent to a meter stick, implying that the distance measured by the Earth Twin is a Proper Length, . Thus, the time that the Twin on the Earth will wait to see back the brother traveling in the spaceship is . Therefore, if the twins were  old before starting the trip, the age of the Twin left on the Earth as measured by the Twin on Earth is .

The Earth Twin measures the evolution of the trip with several clocks, implying that the time measured by the Earth Twin is an Improper Time. In addition, the Twin in the Spaceship needs only one clock to study the evolution of the entire trip, implying that the time measured by the Spaceship Twin is a Proper Time. Therefore, the Twin on the Earth, that knows special relativity, assigns duration of the trip for the Twin on the spaceship as .

The age of the Spaceship Twin as measured by the Earth is .

Twin Paradox as seen by the Twin traveling in the Spaceship:

The distance to be traveled by the Spaceship Twin as measured by him cannot be made with a meter stick or equivalent. Therefore, the distance measured by the Spaceship Twin is an Improper Length, . Thus, the Spaceship Twin argue that the time that he will need to wait to see his twin brother is the same as the time that will take him to travel the previous distance both ways, . Therefore, the age of the Twin on the spaceship as measured by the Spaceship Twin after the trip is .

From the point of view of the Twin in the Spaceship, he needs only one clock to study the evolution of the entire trip, implying that the time measured by the Spaceship Twin is a Proper Time. Equivalently, the Earth Twin measures the evolution of the trip with several clocks, implying that the time measured by the Earth Twin is an Improper Time, . Thus, accordingly with the Twin in the Spaceship, the age of the Twin on the Earth is

Notice that both agree in the mutual ages after the trip, solving the Paradox.

 by Luis F. Sáez, Ph. D. Comments and Suggestions: LSaez@dallaswinwin.com