General Relativity II: The Equivalence Principle

Hello, and welcome back to MPC! Last week, we discovered the mistake we had made when we originally analyzed the twin paradox. Specifically, we realized that Albert (the twin in the rocket ship) was accelerating during his journey and, therefore, we could not just use special relativity to analyze the situation. At the end of the post, I mentioned that, after the journey, Albert would be younger than Richard (and that they would both agree on this fact). Today, we will start to discuss why this is the case.

One of Einstein’s greatest impacts on physics (and the world) came from a simple realization he had: gravity and acceleration are indistinguishable. Now, for some of you, this may make perfect sense. Imagine that you are skydiving. As you are falling through the sky, you will notice that you are moving faster and faster towards the Earth’s surface:

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Figure 1: Acceleration while skydiving

What is pulling you towards the Earth? Gravity! That is one way of thinking about this idea, but the relationship between gravity and acceleration is much deeper. Imagine that you are sitting at a red light:

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Figure 2: Stopped at a red light

The light suddenly turns green. What are you going to do? You are going to step on the gas pedal to accelerate to the speed limit!:

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Figure 3: Acceleration while driving

In Figure 3 you are accelerating just like you were in Figure 1, but would you call your acceleration in Figure 3 gravity? Probably not — typically, we think of gravity as a force of attraction between two objects. You know, from your experience driving a car, that your acceleration is just the result of some chemical reaction (think about the fuel that you put in your car). However, let’s imagine a situation where you are in a new, autonomous vehicle that has all of its windows blacked out:

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Figure 4: Stopped at a red light (in a car with blackout windows)

Suddenly, you feel the same acceleration that you felt in Figure 3. You may logically conclude that your car is accelerating in the same way it accelerated in Figure 3 (through a chemical reaction):

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Figure 5: Acceleration while driving (in a car with blackout windows)

However, could you know for sure that this is the cause of your acceleration? Is it not also possible that some (very strong) person is pushing your car, causing your car to accelerate at just the “right” acceleration?:

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Figure 6: A second possible source of acceleration

Moreover, is it also possible that you are not on the Earth, but you are actually in outer space (after you entered the car, the car was somehow transported there)? This may sound strange, but bear with me! If you were in outer space, it is possible that the Earth’s gravity is causing you to accelerate at just the “right” acceleration:

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Figure 7: A third possible source of acceleration

Crazy? Yes. Possible? Yes!

**Note: You may be thinking: “If I am in a car with blackout windows, how can I even tell that I am accelerating?” Although the human body cannot sense constant velocity, it can sense acceleration. This is why you can, for example, “feel” a plane taking off (but not “feel” anything once you are in the sky). Another important thing to note (for those of you with more physics experience) is that, in Figure 7, we are “pretending” that gravity is not acting on the human. The proceeding example will not require this assumption, the purpose of the example in Figure 7 is to simply build an intuition.**

Using this line of reasoning, Einstein realized that there is no way to distinguish gravity and acceleration. Yes, from our own past experiences we can “distinguish” the two to an extent (we have the ability to say that Figure 1 shows gravity at work but Figure 3 just shows acceleration), but Einstein claimed that, at the end of the day, gravity and acceleration are equivalent. This idea is typically called the equivalence principle.

I hope that the previous car example has helped build your intuition for the equivalence principle. I would like to give one more example of this idea using a rocket ship.

Let’s go back to the skydiving example:

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Figure 8: Acceleration while skydiving (again)

As we discussed, you will see yourself approaching the surface of the Earth more and more quickly as time progresses. We tend to attribute this to gravity — the force pulling us towards the center of the Earth. However, from your perspective, would it not look as if the Earth (and everything on it) were being pulled towards you at a faster and faster rate?:

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Figure 9: Acceleration while skydiving (from a new perspective)

**Note: For simplification purposes, the Earth has been drawn as a flat surface moving towards you. In reality, you would see the entire Earth moving towards you.**

Let’s now imagine that, instead of the Earth, you were “skydiving” inside of a very large rocket ship. As was the case for the skydiver, we can look at this in two ways. In the first case, the rocket ship is on Earth and there is a gravitational force pulling you down (causing you to accelerate):

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Figure 10: Acceleration while skydiving (in a rocket ship)

In the second (and more interesting) case, the rocket ship is actually in the middle of space (there is no gravity acting on you) and is accelerating up towards you:

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Figure 11: A second possible source of acceleration while skydiving

**Note: The movement of the rocket ship towards you at specific intervals of time is not illustrated. Refer to Figure 9 to get an idea of what this would look like.**

The two perspectives (Figure 10 and Figure 11) are both completely valid! This rocket ship example is closer to the model that is typically used to describe the equivalence principle. Using this idea, we can actually show that simply standing on the Earth is the same as standing in an accelerating rocket ship:

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Figure 12: The equivalence principle

**Note: You may think that, standing on Earth, gravity is not acting on you. You do not feel anything when you just stand still, right? The truth is, gravity is acting on you, but a force from the ground is counteracting gravity so you do not feel it. If you are confused by the fact that these two scenarios are equivalent, try thinking about what would happen if you jumped in each scenario. If you think hard enough, you will realize that jumping “feels” and “looks” the same in both situations.**

That is it for this week. The ideas from this post may seem a little weird and abstract, but we will be putting them to good use in the next few blog posts. Next week, we will start discussing the implications of the equivalence principle. See you then!

For more information, be sure to check out these resources: http://www.einstein-online.info/spotlights/equivalence_principle, https://www.youtube.com/watch?v=NblR01hHK6U

(featured image: https://ak9.picdn.net/shutterstock/videos/2610602/thumb/1.jpg?i10c=img.resize(height:160))

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