# What is Inertia?

### Introduction

TEST In the collection of his adventures, “Surely You’re Joking Mr Feynman”, Nobel laureate Richard Feynman recounts an anecdote regarding his father illustrating the difference between knowing the name of something and actually understanding something. After an example on knowing the different name of birds in different languages, he says,

One day, I was playing with an “express wagon,” a little wagon with a railing around it. It had a ball in it, and when I pulled the wagon, I noticed something about the way the ball moved. I went to my father and said, “Say, Pop, I noticed something. When I pull the wagon, the ball rolls to the back of the wagon. And when I’m pulling it along and I suddenly stop, the ball rolls to the front of the wagon. Why is that?” “That, nobody knows,” he said. “The general principle is that things which are moving tend to keep on moving, and things which are standing still tend to stand still, unless you push them hard. This tendency is called ‘inertia,’ but nobody knows why it’s true.”

Perhaps he was being a touch facetious in order to illustrate his point, but in fact we do understand inertia. Surprisingly, for such an apparently simple phenomenon, the explanation is a deep one and takes us to a fundamental understanding of space and time.

### Inertia and Newton’s Laws

Inertia is defined as the resistance of a physical object to any change in the state of its motion i.e. objects tend to keep moving in a straight line at a constant speed. This behaviour is described by Newton’s First Law:

1. An object will remain at rest or continue with uniform motion unless acted on by an external force

A key insight here is that a particle moving by inertia offers no resistance to its motion. In contrast, if a particle is forced to deviate from inertial motion - i.e. it is accelerated - then the particle offers a resistance to this acceleration. So if a particle travelling inertially is subject to some force, it offers a reaction against that force. This gives us Newton’s 2nd and 3rd Laws:

1. $F = m a$. The force is proportional to the acceleration, through the inertial mass.

2. For every force exerted on an object, the object exerts back an equal and opposite force.

### The Einsteinian View

In GR, the motion of an object subject to no external forces (i.e. resistance, friction, etc.) is described by the geodesic equation.

We can read this equation simply as acceleration (LHS) = curvature (RHS), where the degree of curvature is quantified by the Christoffel Symbol $\Gamma$. A particle following a geodesic is in free-fall. It moves only by inertia, that is once it is moving, it keeps moving, on a path determined by this equation.

In flat Minkowski spacetime inertial motion is a straight timelike wordline. In a curved spacetime, a free particle follows a curved timelike worldline.

### But what about acceleration?

Someone might now furrow their brow and say “But wait - isn’t inertial motion associated with zero acceleration? A particle in free fall is obviously accelerating, so how can you describe its motion as inertial?”

This is a seismic shift from Newtonian to Einsteinian thinking. In Newtonian physics, a ball sitting on a table surface is not accelerating. If this ball then falls off the edge of the table, it accelerates.

Conversely, the Einsteinian view is that a ball sitting on the table surface is being accelerated, since its motion is being prevented from following a free-fall geodesic due to the external force exerted on it by the table. When dropped the ball is in free-fall and the acceleration is zero. To see that the acceleration is zero is illuminated by the fact that a falling observer does not feel his own weight - Einstein’s supposed ‘happiest thought’.

“Okay…but clearly the ball when dropped IS accelerating! It’s speed is increasing! What gives?”

The acceleration of a dropped ball is only an apparent acceleration as distinct from an absolute acceleration.

The absolute acceleration is the acceleration experienced when a particle is prevented from travelling on its geodesic due to deformation of its worldline. A body undergoing an absolute acceleration will ‘feel’ the force acting on it. Conversely, the apparent acceleration results just from the curvature of spacetime. The classical example of this is two people on the surface of the Earth, standing at the equator. To them, the surface of the Earth appears flat and they don’t know that they are standing on a curved surface. If they both then walk due north they will observe that they come closer together, in contrast to what would be expected where they on a flat surface where parallel lines never meet. IF they don’t know about the curvature of the surface they’re on, they might attribute their convergent motion to some invisible force acting on them. The same is identically true of our dropped ball, except that the acceleration is due to the curvature of a 4D spacetime. Such a phenomenon is known as geodesic deviation. We can further illustrate this through the geodesic equation. Were the ball on a flat spacetime then the curvature is zero, i.e. $\Gamma = 0$, and so,

i.e. zero acceleration.

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