The Twin Paradox

The twin paradox (or clock paradox), proposed by Albert Einstein, is a thought experiment that examines the different perception of time between two observers in different states of motion. The protagonists are two twins, one of whom takes a trip in a spaceship at speeds close to the speed of light. The other stays on Earth. Upon return, the traveler is younger than the Earthbound twin due to the effects of the special theory of relativity. But from the traveler's point of view, the one moving away is the one left on Earth, and the twin on the ship would be the one aging faster. How is the paradox resolved?

The Special Theory of Relativity

In 1905, an unknown German physicist named Albert Einstein published an article that would radically change the meaning of concepts like space and time. In his "Zur Elektrodynamik bewegter Körper" - the name of the original article - Einstein revolutionized the world by postulating what we now know as special theory of relativity. This theory rests on the principle of relativity and the constancy of the speed of light in any inertial reference frame. It established an equivalence between "mass" and "energy" and redefined the concept of "spacetime".

From it came predictions and, of course, some curiosities. One of the most puzzling is that an observer sees a moving body as having a shorter length than when at rest. Another is that the duration of events affecting a moving body is longer relative to the same event measured by an observer in the body's rest frame. Leaving the mathematics aside, the special theory of relativity tells us that time slows down with velocity.

The Twin Paradox

This gives rise to the famous 'twin paradox'. Imagine two twin brothers, one of whom boards a spaceship and takes a trip to a nearby star, while his brother stays on Earth. The spaceship, as we know, cannot travel faster than light, but it has a propulsion system that allows it to move at a considerable fraction of that speed. The trip lasts a few years, and when the traveler reaches his destination, he begins the return.

The special theory of relativity states that time passes more slowly aboard the ship, since it slows down with velocity. This means that, for the traveler, time passes more slowly than for his brother on Earth. In other words, the astronaut ages more slowly.

When the trip ends, when the twins meet again on Earth, both have aged. However, due to the effects of the special theory of relativity, the traveler is younger than his brother. So far there is no paradox or contradiction. The problem appears when we take into account that velocity has no absolute meaning, but is relative.

Indeed, if we are on a moving train and walk toward one of its ends, what is our velocity? As posed, the question makes no sense. Before we can answer, we need to know relative to what we measure our velocity. Our velocity relative to the train is very different from the velocity relative to the tracks, and also different from the velocity relative to another train. In other words, we need to define a reference frame relative to which we measure our velocity.

Returning to our twins, the above explanation is formulated from the point of view of the twin who stays on Earth. He sees his brother moving at a significant speed relative to his reference frame, and time passes more slowly for his traveling brother. But if we analyze the problem from the point of view of the twin aboard the spaceship, taking the ship itself as the reference, it is the Earth that moves at great speed relative to him. This means that it would be his brother, the one on Earth, who experiences time dilation.

On return, the traveler should find that his brother is younger than him. And here the paradox appears: both expect to see the other brother younger than themselves. Obviously, either they are the same age, or one is younger than the other, but they cannot both be younger than the other simultaneously.

Not a Symmetric Problem

This contradiction caused physicists quite a headache, and Einstein himself dealt with it. Currently there are several ways to explain this paradox, none of which is free of a good dose of mathematics and physical formulations. First, we must consider that we are dealing with a problem that, although it may seem, is not symmetric. The spaceship undergoes accelerations when starting the trip, when braking at its destination, etc. Therefore, when we adopt the ship as the reference frame, we are not using an inertial reference frame and the special theory of relativity does not apply in it. In fact, due to limitations like this, Einstein developed the general theory of relativity, which does apply to non-inertial systems.

One consequence of this theory is that the effects of a gravitational field and an acceleration are indistinguishable. If you are locked in a spaceship without windows accelerating at 1 g, you could not tell whether you are on the surface of the Earth or in space. The other effect of the general theory of relativity is time dilation that occurs in the presence of a gravitational field, which becomes more important the greater its intensity.

All this means that when experiencing accelerations and decelerations, time dilations also occur. In the case of the twins, although each sees the other's time pass more slowly during the moments when the ship travels at constant velocity, during the accelerations and decelerations of the spaceship the time of the traveling twin passes more slowly than that of his brother.

This reasoning does not suffice to resolve the paradox, since if the acceleration and deceleration of the spaceship does not exceed 1 g (9.8 m/s2), the time dilation due to the general theory of relativity will be greater in the case of the brother who stays on Earth. As if this were not already confusing enough, posing the problem this way the final difference between the ages of the twins would depend only on the acceleration of the ship, regardless of its final velocity or the duration of the journey. Fortunately, the special theory of relativity has an answer for this.

The Resolution

Let's imagine a variation of the experiment. Now, a space traveler moving at constant velocity passes very close to the Earth, at which moment his clock and a clock on our planet are synchronized. The ship continues its journey (without changing direction or velocity) until it reaches another planet, and continues onward.

Since both reference frames (the Earth's and the passing ship's) are inertial, we can apply the special theory of relativity without problems. In doing so, we will discover that the ship's clock will be behind Earth's clock, regardless of which reference frame we use. The reason is that, when posing the paradox, we have only taken into account dilation, but we have left out another closely related effect: the spatial contraction mentioned at the beginning.

When moving at high velocity, not only does time slow down on board, but space contracts in the direction of motion. In the previous example, an observer on Earth would see that the spaceship is much shorter than it measures at rest. The traveler on the ship, from his point of view, observes how the rest of the universe moves and contracts in the direction of his motion. For him, the Earth and the other planet are much closer together, and they are not exactly spherical.

As the distance between the Earth and the planet contracts, the traveler takes less time to cover that space and ages less in that time. The paradox, in reality, is only possible due to an error in its formulation. All this may seem counterintuitive. If you still do not understand it, this master class will make it more than clear.

As you can see, not only did the ancient Greeks pose paradoxes. Modern physicists have also sweated quite a while trying to solve problems that, although they have a much more complex formulation, equally challenge common sense.