Pseudo Force

In reading Feynman’s Lectures, I encountered a peculiar notion called pseudo force. It really is a simple concept, but it made me view the world in a tiny bit different way.

So what is a pseudo force?

Feynman gives an clear explanation of it that cannot be mistaken. Directly quoting his explanation from his lecture, let us say there are Joe and Moe. Joe is at position x, y anx z. Moe is at:

with s being the displacement from Moe’s system relative to Joe’s. Would the laws of motion look the same for both systems?

If s is a constant,

and thus the laws of motion would surely look the same. Even if

acceleration being the second derivative of s with respect to t would be 0 and the laws of physics would look the same in both systems. However, when s is more complicated, like

then

would not be zero but instead would be a. This gives an awkward situation where the laws of motion in Moe’s system is different from the ones in Joe’s:

This means that Moe’s system is accelerating with respect to Joe’s.

An example of such pseudo force is “centrifugal force”. When spinning an object with a rope tied to it, the object feels the force for being thrown outward. Simple enough. In order for the object to spin in a circle, an acceleration is required because the direction of the motion is changing although the speed is constant. The acceleration is a centripetal acceleration and the force is called centripetal force. This pseudo force thing now might seem like a just different way of viewing this force from an different angle. Acceleration of non-inertial frame of reference with respect to inertial frame of reference gives rise to the pseudo force.

What was really interesting for me was thinking about gravity as a pseudo force. It is noted in the lecture that one very important feature of a pseudo force is that it is always proportional to the masses. Well, the same is true for gravity. So can we think of gravity as a kind of a pseudo force? Yes, according to Feynman. And according to Einstein as well.

Einstein said that gravitation and acceleration are indistinguishable. In describing this in a crude way, he gives an example of the world where it is a two-dimensional world. Suppose we know nothing but two-dimensional world but the actual world is three-dimensional. Let us say we are living on a sphere, and perceiving that we live in two-dimensional world, we shoot a ball from point A to B. This would look like it is going in a linear direction to us, but with another ball shot the other direction, the two balls would seem like they are attracting each other. This is due to the fact that the world is actually three-dimensional and the world is a sphere where the shortest distance from A to B is around the surface of the sphere. In reality, the balls do not attract each other at all but due to the geometry of the world we live in, they seem to us that they do. Gravity can be viewed to work the same way.

Now, it is impossible for me to understand how gravity really works from this simple example, but the concept of pseudo force definitely got me started thinking about some of the different ways we could think about gravitational force. When I find out more about how gravity really works and what it really is, I will write about it.