Hello, and welcome back to MPC! Last week, we discussed how massive objects can completely warp space itself. As promised at the end of last week’s post, today we will focus on how massive objects affect time.

Before we get started, I highly recommend that you check out the two posts on time dilation in special relativity (found here and here) if you haven’t done so already. The ideas and arguments we will be discussing today are essentially analogous to those made in the linked posts. Additionally, as I have stated for the other general relativity posts, the goal of this post is to provide a conceptual understanding of time dilation in general relativity, not a rigorous one. It is my hope that this post provides you with a more intuitive way of thinking about this rather complicated phenomenon.

Recall our original discussion on the bending of light. Specifically, recall that we imagined someone, who we will call Albert, in a rocket ship falling towards the Earth:

Figure 1: Albert in his rocket ship

We can say that Albert is in Earth’s gravitational field. Essentially, all this sentence means is that, at the position where Albert is located, he is being influenced by Earth’s gravity (i.e. he is falling towards the Earth). In our discussion on the bending of light, we imagined that a photon was passing through Albert’s rocket ship:

Figure 2: A photon passing through Albert’s rocket ship (as seen by Albert)

**Note: To Albert, the photon appears to travel in a perfectly straight line. If this is confusing, I recommend checking out this post.**

How long will the photon spend in the rocket ship according to Albert (a value we will call t0)? Well, that depends on the width of the rocket ship, which we will call d0:

Figure 3: The distance the photon travels in Albert’s rocket ship (as seen by Albert)

**Note: For illustration purposes, the length d0 is shown as the distance between the center of the photon at the first and last instance it is fully inside of the rocket ship.**

Using Figure 3, the fact that the speed of light is constant (c; see this post), and our velocity equation, we can develop an equation:

That was not too bad! Do not forget, however, that we are interested in how gravity affects time. We have calculated t0, which is a time measurement taken within a gravitational field (it was taken by Albert who is in a gravitational field). If we want to see the effect that gravity has on this time measurement, we have to compare t0 to a time measurement taken “outside” of a gravitational field.

That’s where Richard comes in. We’ll say that Richard is standing very far away from the Earth, but can still see the light traveling through Albert’s rocket ship:

Figure 4: Richard in his rocket ship

Recall that gravity bends light, so Richard would see something along the lines of:

Figure 5: A photon passing through Albert’s rocket ship (as seen by Richard)

**Note: The bending of light in the figure is greatly exaggerated. Additionally, in Figure 2 we only have one rocket ship illustrated, but here we have two. This is because Richard can see the rocket ship moving through space (he is outside of it), while Albert cannot (he is inside of it, so he moves with it). The two rocket ships in Figure 5 represent what Richard sees when the photon first enters the rocket ship and when it is about to leave. If Figure 5 is confusing, I recommend checking out this post.**

As is illustrated in Figure 5, the light bends inside of Albert’s rocket ship (according to Richard; recall that gravity bends light). We will call the length of this curve d1:

Figure 6: The distance the photon travels in Albert’s rocket ship (as seen by Richard)

It should not be hard to believe that d1 is larger than d0 (some of you may have heard the saying “the shortest path is the straightest one”). For simplicity, we will say that d1 is simply 4 times as long as d0. In other words:

Despite the difference in distances traveled, the photon drawn in Figure 2 and the photon drawn in Figure 5 must travel at the same speed. Why? Because the speed of light is constant (c, see this post)! Using this fact, we can calculate the amount of time the photon spends in the rocket ship according to Richard (a value we will call t1). First, let’s use our definition of velocity:

We can plug in Equation 2:

Next, we can plug in Equation 1 and solve for t1:

We have found that t1 is 4 times as much as t0? What does this mean? Let’s imagine that Albert thinks the photon spends 1 second inside of his rocket ship (t0 = 1 second). How much time does Richard think the photon spends in Albert’s rocket ship? Using Equation 5:

4 seconds! In other words, Richard thinks that the photon spends more time in the rocket ship than Albert does. We can flip this sentence around and say that Albert thinks the photon spends less time in the rocket ship than Richard does.

What does this mean? Time runs slower for Albert than it does for Richard! How so? Imagine that Albert and Richard are both 20 years old and Albert thinks that the photon spends 1 year traveling through his rocket ship (it s a very, very, very large rocket ship). Furthermore, imagine that Albert raises his hand when the light enters the rocket ship and lowers his hand when the photon exits. How old are Albert and Richard when Albert’s hand finally lowers? Well, for Albert, only 1 year will have passed (t0 = 1 year), so Albert will be 21 years old. How about Richard? Once again, we can use Equation 5:

4 years have passed, so Richard is 24 years old. Another way to look at this is that, for every 1 year Albert ages, Richard ages 4 years. This is why we can say that time is running slower for Albert.

It is important to note that we made up that d1 is 4 times as much as d0. Nonetheless, d1 is certainly larger than d0. As a result, we can draw the general conclusion that time runs slower in gravitational fields (you can try replacing 4 with any number greater than 1 and you will see that Albert’s clock always runs slower than Richard’s). To phrase it differently, the bending of light (which causes d1 to be longer than d0) causes clocks in gravitational fields to run slower than those that are not in gravitational fields.

That’s all for this week! As a fun challenge, I recommend trying to solve the twin paradox (discussed in this post and this post) using what we learned today. Next week, we will be diving into something different: mass-energy equivalence. With mass-energy equivalence, we will be discussing one of the most famous equations of all time! I’m excited, and I hope you are too! See you then!

For more information, be sure to check out this resource: http://thescienceexplorer.com/universe/how-gravity-changes-time-effect-known-gravitational-time-dilation

(featured image: http://salvadordaliprints.org/Salvador%20Dali%20Clocks.jpg)