Twin Paradox

The So-Called "Twin Paradox"

Imagine you have a twin, and if you already have a twin, this portion of the vizualization will prove quite easy for you.

Now let's assume, for the sake of simplicity, that you an your twin live on a space station far away from any massive objects. Your twin has been picked for a mission that will take them at half the speed of light. There is a clock aboard the ship, measuring the length of the trip in years. You will have an exact copy of this clock and the two of you will set them to precisely the same times before she leaves.

When your twin comes back 30 years later, you notice her clock has recorded only 26 years. But how can this be??

Considering Time Dilation

Let's consider the start of this trip. When your twin leaves, she is moving at a constant speed relative to you, and to her, you are moving at a constant speed as well. Therefore, if you could observer each other's clocks, you would notice that her clock is moving slow, and she would notice that your clock is slow as well and that it is moving at the exact same rate that you observed her's moving at. This is due to Time Dilation, a well proven consequence of special relativity.

So, during this first leg of the trip, your clocks should be logging the same times. The issue arises when your twin turns around. To understand why, we need to familiarize ourselves with Einstein's two postulates of special relativity.

The Two Postulates of Special Relativity

The two postulates are

1. First postulate (principle of relativity)

The laws by which the states of physical systems undergo change are not affected, whether these changes of state be referred to the one or the other of two systems of coordinates in uniform translatory motion.

This means that the laws of physics are the same in all inertial frames. Essentially, assuming an observer is moving at a constant speed, or is at rest, they should note the same difference in time or length relative to one another.

1. Second postulate (invariance of c)

As measured in any inertial frame of reference, light is always propagated in empty space with a definite velocity c that is independent of the state of motion of the emitting body.

This means that no matter what an observer is doing, no matter how fast they -or- the source of the light are going, everyone will always measure the same rate for light. This is why it is denoted $c$, for constant.

Applying These Postulates to The Twin Paradox

Now that we understand the postulates, let's see how this affects the you and your twin's situations.

You are stationary on the space station, relative to your twin, and to your twin is stationary relative to the space station. However, when your twin reaches the first half of her journey, she necessarily must turn around. This is where the discrepancy between your two frame of references occurs, because when she turns, she accelerates.

By accelerating her ship, your twin ceases to be an inertial observer. This acceleration won't simultaneously occur on the space station either. Therefore, your twin has left the frame of reference that allowed the two of you to clock identical times (remember postulate one). Your clock will continue to move slow, relative to her, but her clock will have sped up, relative to you.

Now we understand why your twin has aged four years less than you have, additionally, we might understand why this is not really a paradox. Rather, it is a natural consequence of physics.