What is the equivalence principle?

Folks, I recently read the website of the Flat Earth Society. I’m serious! It’s a most remarkable collection of… nonsense. Maybe most remarkable is how it throws together physical facts that are correct – but then gets their consequences completely wrong! This is most evident when it comes to flat earthers’ elaborations on Einstein’s equivalence principle.


The equivalence principle is experimentally extremely well-confirmed, yes. But flat earthers misconstrue evidence for the equivalence principle as “evidence for universal acceleration” or what they call the “universal accelerator”. By this they mean that the gravitational acceleration is the same everywhere on earth. It is not. But, you see, they believe that on their flat earth, there is no gravity. Instead, the flat earth is accelerating upwards. So, if you drop an apple, it’s not that gravity is pulling it down, it’s that the earth comes up and hits the apple.

The interesting thing is now that flat earthers’ claim Einstein said you cannot distinguish upward acceleration from downward gravity. That’s the equivalence principle, supposedly. So, you see, Einstein said it and therefore the earth is flat.

You can read on their website:

“Why does the physics of gravity behave exactly as if the earth were accelerating upwards? The Universal Accelerator answers this long-standing mystery, which has baffled generations of scientists, by positing that the earth is accelerating upwards.”

Ingenious! Why didn’t Einstein think of this? Well, because it’s wrong. And in this video, I will explain why it’s wrong. So, what is the equivalence principle? The equivalence principle says that:

“Acceleration in a flat space-time is locally indistinguishable from gravity.”

Okay, that sounds somewhat technical, so let us go through this step by step. I assume you know what acceleration is because otherwise you would not be watching a physics channel. Flat space-time means you are dealing with special relativity. So, you have combined space and time, as Einstein told us to do, but they are not curved; they’re flat, like a sheet of paper. “Locally” means in a small region. So, the equivalence principle says: If you can only make measurements in a small region around you, then you cannot tell acceleration apart from gravity. You can only tell them apart if you can make measurements over a large enough distances.

This is what Einstein’s thought experiment with the elevator was all about. I talked about this in an earlier video. If you’re in the elevator, you don’t know whether the elevator is sitting on the surface of a planet and gravity is pulling down, or if the elevator is accelerating upward.

The historical relevance of the equivalence principle is that it allowed Einstein to make the step from special relativity to general relativity. This worked because he already knew how to describe acceleration in flat space – you can do that with special relativity. In general relativity then, space-time is curved, but locally it is flat. So you can use special relativity locally and get general relativity. The equivalence principle connects both – that was Einstein’s great insight.

So, the equivalence principle says that you cannot tell gravity from acceleration in a small region. That sounds indeed very much like what flat earthers say. But here’s the important point: How large the region needs to be to tell apart gravity from acceleration depends on how precisely you can measure and how far you are willing to walk. If you cannot measure very precisely, you may have to climb on a mountain top. You then find that the acceleration up there is smaller than at sea level. Why? Because the gravitational force decreases with the distance to the center of the earth. That’s Newton’s 1/R² force. Indeed, since the earth is not exactly a sphere, the acceleration also differs somewhat between the equator and the poles. This can and has been measured to great precision.

Yeah, we’ve know all this for some while. If the acceleration we normally assign to gravity was the same everywhere on earth, that would contradict a huge number of measurements. Evidence strongly speaks against it. If you measure very precisely, you can even find evidence for the non-universality of the gravitational pull in the laboratory. Mountains themselves, for example, have a non-negligible gravitational pull. This can, and has been measured, already in the 18th century. The gravitational acceleration caused by the ground underneath your feet has also local variations at constant altitude just because in some places the density of the ground is higher than in others.

So, explaining gravity as a universal acceleration is in conflict with a lot of evidence. But can you instead just give the flat earth a gravitational pull? No, that does not fit with evidence either. Because for a disk the gravitational acceleration does not drop with 1/R². It falls more slowly with the distance from the disk. Exactly how depends on how far you are from the edge of the disk. In any case, it’s clearly wrong.

The equivalence principle is sometimes stated differently than I put it, namely as the equality of inertial and gravitational mass. Physicists don’t particularly like this way of formulating the equivalence principle because it’s not only mass that gravitates. All kinds of energy densities and momentum flow and pressure and so on also gravitate. So, strictly speaking it’s not correct to merely say inertial mass equals gravitational mass.

But in the special case when you are looking at a slowly moving point particle with a mass that is very small compared to earth, then the equality of inertial and gravitational mass is a good way to think of the equivalence principle. If you use the approximation of Newtonian gravity, then you would describe this by saying that F equals m_i times a, with m_i the inertial mass and a the acceleration, and that must be balanced with the gravitational force that is m_g, the gravitational mass of the particle, times the mass of earth divided by R^2, where R is the distance from the center of earth which is, excuse me, a sphere. So, if the inertial mass is equal to the gravitational mass of the particle, then these masses cancel out. If you calculate the path on which the particle moves, it will therefore not depend on the mass.

In general relativity, the equivalence of inertial and gravitational mass for a point particle has a very simple interpretation. Remember that, in general relativity, gravity is not a force. Gravity is really caused by the curvature of space-time. In this curved space-time a point particle just takes the path of the longest possible proper time between two places. This is an entirely geometrical requirement and does not depend on the mass of the particle.

Let me add that physicists use a few subtle distinctions of equivalence principles, in particular for quantum objects. If you want to know the technical details, please check the information below the video for a reference.

In summary, if you encounter a flat earther who wants to impress you with going on about the equivalence principle, all you need to know is that the equivalence principle is not evidence for universal acceleration. This is most definitely not what Einstein said.

If this video left you wishing you understood Einstein’s work better, I suggest you have a look at Brilliant dot org, who have been sponsoring this video. Brilliant offers online courses on a large variety of topics in mathematics and science, including physics. They have interactive courses on special relativity, general relativity, and even gravitational physics, where they explore the equivalence principle specifically. Brilliant is a great starting point to really understand how Einstein’s theories work and also test your understanding along the way.

To support this channel and learn more about Brilliant, go to brilliant.org/Sabine, and sign up for free. The first two-hundred people who go to that link will get twenty percent off the annual Premium subscription.

Thanks for watching, see you next week.