###### By Don Lincoln, Ph.D., Fermi National Accelerator Laboratory (Fermilab)

## Let’s think about the implications of the idea that a person seeing a clock move experiences more time than a person seeing the clock as stationary. But doesn’t that run afoul of Einstein’s first premise, which is that the laws of physics are the same for all observers, and, further, any observer can claim to be stationary?

### The Moving External Observer

The train experiment in relativity assumes that the outside observer can see the train moving. But what if it is the other way round? The scenario used is that a person on the tracks sees another person moving on a train car, and the train car guy’s clock ran slower than the clock of the guy on the track. But we could change the perspective.

We could say that the person in the train car is stationary and that the person on the track is moving off to the left. In that case, it would be correct to say that the guy on the track experiences less time, and the person in the train experiences more time. So that’s a problem.

From one perspective, the person in the train is experiencing more time than the person on the tracks, while from the other perspective, the person on the tracks is experiencing more time than the person on the train. However, both of these can’t be true.

This is a transcript from the video seriesUnderstanding the Misconceptions of Science. Watch it now, on The Great Courses Plus.

### The Lorentz Transformation Special Case

The time dilation equation is actually a special case of the more general Lorentz transforms. If you compare the Lorentz transform for time to the time dilation equation, they look pretty similar, but the Lorentz transform has an extra term: (v^{2}/c^{2})*x.

This means that the time dilation equation is a special case of the Lorenz time transform. But, it only works if *v* equals 0 or *x* equals 0. If v=0, that just means the two observers aren’t moving with respect to one another, so you don’t need relativity at all. If the position is *x*=0, it means that this is the location of the unprimed observer.

And this is a key point. It implies that the time dilation equation only applies at the location of the unprimed observer!

Learn more about Einstein’s rejection of black holes.

### Transforming Perspectives

So, let’s start with the two observers. Unprimed is moving off to the right at velocity *v*, and primed is watching him go. We can convert the time experienced by each observer using the Lorentz transform. And there are really only two noteworthy locations: the location of the unprimed observer, which we’ll call Location 1, and the location of the primed observer, which we’ll call Location 2.

To use the Lorentz transforms, we need to know the *x* and *t* for an observer. For the unprimed observer, location 1—his location—isn’t moving and time is being experienced. So we can say that according to the unprimed observer, (x, t) = (0, t).

And, if we use the Lorentz transforms, we see that the *x* and *t* seen by the primed observer is just (x_{prime}, t_{prime}) = (γvt, γt).

Looking at the time side, that just means that t_{prime} = *γ*t.

Location 2 is a little trickier. According to the unprimed observer, the primed observer is moving off to the left. The unprimed observer sees the location of the primed observer as ‘*-vt*’. Accordingly, we can write the position and time of location 2 as seen by the unprimed observer, and it is (x, t) = (–vt, t).

We can again use the Lorentz transforms to determine how the primed observer sees location 2. The result is (x_{prime}, t_{prime}) = (0, t/γ).

The position makes sense, since the primed observer thinks they are unmoving and is always at location x_{prime} = 0.

But the time conversion is a little weirder. It says that at location 2, t_{prime} = t/γ.

That’s divided, not multiplied, and exactly the opposite of what we saw with the time dilation equation.

### Does the Primed Observer Always Experience More Time?

Since *γ* is greater than 1 at relativistic speeds, this means that here the primed observer experiences more time than the unprimed observer at Location 1. And, at Location 2, the primed observer experiences less time than the unprimed observer. The unprimed observer experiences time identically at location 1 and 2, but the primed observer experiences time differently at both locations.

Now, that sounds like another paradox, but there is a subtle point that must be taken into account. The unprimed observer isn’t seeing those locations moving, while the primed observer is. So, this isn’t a paradox. There’s a difference between the two individuals, and that is why one sees the time at both locations as the same and the other sees the time at both locations as different.

Remember that the seeming paradox was that both individuals could claim to be unmoving, and that their time was faster than a moving person. But that’s because we didn’t take the location into account. If you take the location into account, we find that both observers say that the time experienced where they are is shorter than the time experienced by the other person and longer where the other person is. Since both observers make the same claim, there is no paradox.

### The Twin Paradox

There is another seeming paradox that is often mentioned in relativity, called the twin paradox. In the twin paradox, a set of twins does an experiment. One stays on Earth while the other flies off to a distant star at very high speeds and returns. According to relativity, the twin who went to the star and back will return much younger than his stay-at-home sibling.

This also seems paradoxical because the person in the spaceship could say that they’re stationary and the Earth zoomed away and returned. And of course, since both twins could say that they’re stationary, it seems that both could claim to be the one who aged more.

The answer to the twin paradox is somewhat more complex than what we’ve done here, but it’s very similar. You simply properly apply the Lorentz transforms, and you find that the paradox isn’t paradoxical.

### Common Questions about Time Dilation

**Q. What implication does the fixed speed of light have for a primed observer in relativity?**

If the speed of light is the same for all observers, whether primed or unprimed, it implies a dilation of time for an unmoving primed observer.

**Q. According to relativity, what kind of observer will experience time dilation?**

According to theory of relativity, an unmoving primed observer will observe time dilation only at the location he presently occupies.

**Q. What paradox presents itself with regard to time dilation?**

Since any observer can consider himself or herself to be the sole unmoving point, all can claim to experience time dilation.

**Q. What is the popular idea of the ‘twin paradox’ in relativity?**

There is a seeming paradox that is often mentioned in relativity, called the twin paradox. In the twin paradox, a set of twins does an experiment. One stays on Earth while the other flies off to a distant star at very high speeds and returns. According to relativity, the twin who went to the star and back will return much younger than his stay-at-home sibling. Paradoxically, the twin on the spaceship could also claim to have aged more because he can consider himself to be stationary.