# Who explained the twin paradox?

## The problematic twins

#### An article by Markus Pössel

In Einstein's special theory of relativity, it does not make sense to speak of “time” in the singular. How time passes depends on movement, and the prime example of this is the oft-cited hypothetical twins.

### Gemini on the move

One of them stays on earth while the other goes on a round trip with a rocket that is almost as fast as light:

At the end of the trip, when the two twins face each other again, the twin who stayed at home is significantly older than the rocket traveler. For example, it may be that only two years have passed on the rocket's on-board clocks, the twin that has traveled subjectively only experienced two years of travel time and aged around this time, while at home on Earth a full 30 years passed between takeoff and return are. When the twins meet again at the end, you can clearly see this age difference.

So far, so unusual, but undeniable reality: You can actually carry out such a round trip experiment not with rockets, but with elementary particles in particle accelerators and determine: the "internal clock" of an elementary particle in the circular accelerator is significantly slower than that of a particle of the same type next to it the accelerator is at rest (compare the page The Relativity of Space and Time in the chapter Special Theory of Relativity by Einstein for Beginners).

### A question of equality

If one tries to explain the phenomenon with the so-called time dilation, a fundamental and also very unusual effect of the special theory of relativity, the scenario becomes a twin problem and even an apparent twin paradox. This effect affects an observer (more precisely: inertial observer), for example on a space station floating freely in space. For such an observer it follows from the special theory of relativity in general: For clocks moving relative to him, be it the clocks of a second space station flying past, be it the clock of a rocket circling wildly, this observer finds in comparison that they go slower than his own. When he compares the time displays, he comes to the conclusion: During every second that passes on his own watch, less than one second passes on the moving clocks. The slowdown not only applies to clocks, but all processes on the moving space station or in the rocket are slowed down to exactly the same extent from our observer's point of view. A five-minute egg cooked in the space station flying past still only takes five minutes compared to the egg timer next to the saucepan. But measured on the clock of our observer, who sees the egg and egg timer fly by, the cooking process and the advancement of the egg timer pointer by five minutes both take significantly longer than five minutes.

The decisive property of time dilation is that it is reciprocal in certain situations. With two observers, each in his free-floating space station, who fly past each other at constant speed, goes for everyone of the two slows down the time in the other space station. (For those who already consider this to be a contradiction, we recommend the advanced topic The Dialectic of Relativity.)

With the help of time dilation, which is often abbreviated to "Moving clocks go slower", one can try to explain the twin effect. No wonder that the traveling twin ages more slowly - those who stayed at home can finally argue with the time dilation: Moving clocks run more slowly, and so less time has passed on the clock of the moving twin than on that of the one who stayed at home when the two meet on return meet again.

All well and good - but why can't you reverse the argument? Movement is relative, one could argue. What's stopping the twin in the rocket from considering themselves dormant? From this point of view, it would be the twin on earth that would move, first move away and then approach again, and according to "moving clocks go slower" the rocket twin could deduce that on the clock of the earth twin to the end Meeting would have to pass less time than on his own. Each of the twins, one might think, has the same right to argue that the other must age more slowly than himself. And yet after the rocket returns, if the twins can compare their watches directly, only one can be right. A contradiction - a twin paradox?

### The special role of the inertial systems

To resolve the apparent contradiction, it is necessary to take a closer look at which observers the time dilation applies to and which not. In the above description it was only mentioned in brackets: The fact that clocks move more slowly applies precisely from the point of view of inertial observers, of observers whose space stations are inertial systems. What this means is indicated by the examples chosen above of the space stations floating freely in space: In an inertial system there is perfect weightlessness. An object that is not acted on by external forces remains immobile in the interior of such a space station or moves at a constant speed on a straight path.

So it is necessary to check for each of the twins: Is he an inertial observer - and therefore observes the time dilation of the special theory of relativity on all objects moved relative to him?

An unfortunate complication: The twin on earth is not an inertial observer, because he is in a gravitational field: objects that he lets go fall to the ground at an accelerated rate. This can be taken into account in two ways: On the one hand, within the framework of Einstein's theory of gravity, the general theory of relativity, one can work out how the gravitational field influences the clocks of the twin who stayed at home. The result: Compared to the age differences to the almost light-fast twin, the influence is negligibly small, and the twin who stayed at home can at least approximately invoke the time dilation to explain the other twin's remaining younger. If that is not enough for you, you can easily change the situation and leave the twin who stayed at home far away in space in a free-floating space station waiting for the other twin to return, so that he is definitely in an inertial system. This twin is therefore just as certain to use the time dilation formula: Judging from its reference system, moving clocks run more slowly.

What about the rocket-traveling twin? He is also not in an inertial system. If it were to simply fly free, in a straight line, and away from Earth (or the space station) at a constant speed, it would eventually fly on and on and never return. In order to come back, an acceleration phase is inevitably necessary, be it braking and then accelerating towards the earth, be it flying a U-turn. In both cases, the rocket occupant feels the acceleration, just as most readers should know it from everyday driving: when braking, a force pulls him out of the seat, when accelerating again, it pushes him into his seat, and when turning he feels pulled sideways. The acceleration is also felt by every object that the rocket occupant in his cabin lets go of, and this shows that due to the inevitable acceleration phase, the twin in the rocket is not continuously in an inertial system. He must therefore not apply the formula for simple time dilation at all.

On closer inspection, the two twins are not equal - the one in the rocket inevitably has to expose himself to accelerations if he wants to return to the other twin. This also resolves the apparent paradox: only the conclusion of the twin who stayed at home, that the rocker twin's moving clock goes slower and that this twin is younger when they meet, is justified. (The in-depth topic Twins and Wanderers is devoted to the question of how one should imagine the effect of acceleration - does it directly affect the rate of the clocks in the rocket? -