## 3.0 - Fun with the Special Theory of Relativity

### The Infamous Twin Paradox

Since SR dictates that two different observers each have equal right to view an event with respect to their frames of reference, we come to many not-so-apparent paradoxes. With a little patience, most of the paradoxes can be shown to have logical answers that agree with both the predicted SR outcome and the observed outcome. Let's look the most famous of these paradoxes - The Twin Paradox.

Suppose two twins, John and Hunter, share the same reference frame with each other on the earth. John is sitting in a spaceship and Hunter is standing on the ground. The twins each have identical watches that they now synchronize. After synchronizing, John blasts off and speeds away at 60% the speed of light. As John travels away, both twins have the right to view the other as experiencing the relativistic effects (length contraction and time dilation). For the sake of simplicity, we will assume that they have an accurate method with which to measure these effects. If John never returns, there will never be an answer to the question of who actually experienced the effects. But what happens if John does turn around and return to the earth? Both would agree that John aged more slowly than Hunter did, thus time for John was slower than it was for Hunter. To prove this, all they have to do is look at their watches. John's watch will show that it took less time for him to go and return than Hunter's watch shows. As Hunter stood there waiting, time passed faster for him than it did for John. Why is this the case if both were traveling at 60% the speed of light with respect to one another?

The first point to understand is that acceleration in SR is a little tricky (it's actually handled better in Einstein's Theory of General Relativity - GR). I don't mean to say that SR can't handle acceleration, because it can. In SR, you can describe the acceleration in terms of locally "co-moving" inertial frames. This allows SR to view all motion to be uniform, meaning constant velocity (non-accelerating). The second point is that SR is a "special" theory. By this, I mean that it is applicable in situations where there is no gravity, hence where space-time is flat. In GR, Einstein unifies acceleration and gravity so actually my previous statement is redundant. Anyway, the lack of gravity in SR is why it is called "Special Relativity". Now, back to the paradox… While both did view the other as shrinking and slowing down, the person that actually underwent the acceleration to reach the high speed is the one that aged less. If you dig deeper into the world of SR, you will realize that it's not really the acceleration that is important; it's the change of frame. Until John and Hunter returned to a frame of reference where their relative motion was zero (where they are standing beside each other) they would always disagree with what the other said he saw. As strange as this seems, there really isn't a conflict - both did observe that the other was experiencing the relativistic effects. One technique that is used to show the dynamics of the Twin Paradox is a concept is called the Relativistic Doppler Effect.

We'll look at the Doppler Effect in the next section.