Hello, and welcome back to MPC! Last week, we spoke about a very important relationship: Einstein’s equivalence principle. Today, we are going to discuss a major implication of the equivalence principle.

Quick disclaimer! Recall that, a few weeks ago, I mentioned general relativity is significantly more complicated than special relativity and that we would, therefore, not be focusing on general relativity’s mathematics. In today’s post, we are going to be looking at a rather complicated idea in general relativity, but in a very simple way. This post will not be very rigorous. The goal of this post, and all posts on MPC, is to help the reader develop a conceptual understanding of modern physics. If you are looking for a source to help you write a report or study for an advanced-level test on general relativity, this post is not the place for you. However, if you are simply curious about the awesome phenomena that occur as a result of general relativity (or already know advanced general relativity and would like a new, light-hearted perspective on it), I hope that this post helps you!

Before we start to discuss anything, let’s review what we know about the equivalence principle. The main conclusion that we drew in last week’s post is that standing on the Earth’s surface is the same as standing in a rocket ship that is accelerating in the middle of outer space:

Figure 1: The equivalence principle

In other words, wherever you see an accelerating rocket ship, you can replace it with the Earth. Likewise, you can replace any pictures of the Earth with an accelerating rocket ship.

We can actually extend the equivalence principle — the relationship between Earth (or anything that “has” gravity) and a rocket ship is much deeper. Let’s “turn off” our rocket ship so it is just sitting in space:

Figure 2: A rocket ship sitting in space

If you were in this scenario, you would not feel anything — there is nothing bringing you to the floor of the rocket ship (because the rocket ship is not accelerating upwards). Can we equate this scenario to anything on Earth? We can! Imagine that our rocket ship is falling through the sky:

Figure 3: A rocket ship in free fall

**Note: Technically, you do not need to be in a rocket ship for what we are about to describe. However, it helps with the visualization.**

In Figure 3, you are accelerating downward, but so is the rocket ship. To put it another way, you are moving in the direction of the floor, but the floor is moving out from under you at the same rate. This creates a sensation that is equivalent to that in Figure 2 — a feeling of “nothing bringing you to the floor” (if you try to imagine this scenario, you will realize that the floor will always look as if it is the same distance away from you — you do not feel like you are being pulled to the ground). If Albert ever felt this sensation of “weightlessness” while inside of a rocket ship, he would not be able to tell if he were falling towards the Earth or in the middle of space:

Figure 4: An extension of the equivalence principle

Just like the relationship between the accelerating rocket ship and standing on Earth, anywhere that you see a rocket ship sitting in space, you can replace it with a rocket ship falling towards the Earth’s surface (and vice versa). As we will see, this extension of the equivalence principle is quite powerful.

Now, onto the fun stuff! Let’s imagine that a man, Albert, is standing in a “stationary” rocket ship in outer space:

Figure 5: Albert’s rocket ship sitting in space

**Note: The stick figure is no longer lopsided like it was in Figure 4. Do not get confused — Albert still feels “weightless” in Figure 5. He has been drawn in this position to make the concepts we are about to discuss easier to visualize.**

Let’s also imagine that Albert’s rocket ship has a small hole in it and light from a distant star is shining towards Albert’s rocket ship.

Figure 6: A particle of light entering Albert’s rocket ship

**Note: At the point where it is entering the rocket ship, the light has been drawn as a particle, a “photon.”**

Of course, as shown in Figure 6, the light from the star will be able to enter Albert’s rocket ship. Once it enters Albert’s rocket ship, though, how will it behave?

Some of you have probably heard that “light travels in straight lines.” In this case, when Albert’s rocket ship is just sitting in space, that is exactly what happens:

Figure 7: Light traveling horizontally through Albert’s rocket ship

Everything seems normal. Let’s now try to employ the equivalence principle. Albert’s rocket ship that is sitting in space is equivalent to Albert’s rocket ship falling towards Earth:

Figure 8: The extension of the equivalence principle

With this “new” scenario, the distant light will, once again, enter Albert’s rocket ship:

Figure 9: A particle of light entering Albert’s rocket ship

The light is going to take a certain amount of time (albeit a short amount of time) to reach the rightmost side of Albert’s rocket ship. In that time, how is the light going to move?:

Figure 10: There are many paths that the light could follow

**Note: “Starting point” refers to the moment when the light enters the rocket ship, and “ending point” refers to the moment when the light is aligned with the rightmost wall of the rocket ship.**

Light travels in straight lines, so it will just move horizontally, right?:

Figure 11: The light traveling “horizontally”

**Note: Notice that the light is still “inside” of the top rocket ship when it is aligned with the rightmost wall in Figure 11. However, do not forget that the rocket ship has actually moved downward to the “ending point” position by the time the light has reached this point, so the light is not actually inside of the rocket ship.**

This may seem like a logical path for the light to follow. However, this contradicts our equivalence principle. Remember that Albert cannot tell the difference between floating in a rocket ship in outer space and floating in a rocket ship that is falling towards the Earth. If the light did move as we drew it in Figure 11, it would look like it is traveling in a straight line from our perspective (outside of the rocket ship). To Albert, though, who is moving downward, its path would look something like this:

Figure 12: The path of the light according to Albert

So what? Well, if the light actually looked like it does in Figure 12 when Albert is falling towards the Earth and it looked like it does in Figure 7 when Albert is in outer space, Albert would be able to tell if he was falling towards the Earth or in outer space (just by looking at the path the light followed). According to the equivalence principle, this is not allowed.

How can we resolve this? We have to allow the light to travel in a straight line according to Albert. If we were to do this, we would get something like the following:

Figure 13: The correct path of the light

Notice that the path of the light seems to curve around the Earth (from our perspective outside of the rocket ship — to Albert, the path does not look curved). Albert’s falling rocket ship could be anywhere in the Earth’s sky and the light would have to be bent in order to satisfy the equivalence principle. This means that the Earth has to bend light everywhere within its gravitational field. In other words, the Earth itself “bends light”:

Figure 14: The Earth bending light

To put it more generally, anything with gravity (the Sun, the Moon, Mars, Jupiter, asteroids, etc.) bends light. Of course, the drawings in this post have been greatly exaggerated, but light really does bend as a result of gravity (just not as much as illustrated).

**Note: You may have realized that in Figure 13, the light approaching the rocket ship is also in a gravitational field, so it should be slightly bent as well. However, this example is just meant to provide a basic, conceptual understanding of the bending of light, so this detail has not been included in the drawing.**

Who cares? Gravity has the ability to make a lot of things “bend.” For instance, gravity can “bend” the path of an asteroid passing by:

Figure 15: The Earth bending the path of an asteroid

There is nothing strange about gravity bending the path of the asteroid in Figure 15. Who cares if gravity can bend light as well? Isn’t just the same idea?

That is what we are going to be talking about next week: why is the bending of light so important? Yes, right now it may not seem like anything special. But next week, we will discover that this simple fact completely reshapes how we view and analyze our universe. See you then!

For more information, be sure to check out this resource: http://theory.uchicago.edu/~marsano/ComptonLectures/Lecture6/Lecture6-handout.pdf

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