# Inertia

$$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$ $$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$ $$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$ $$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$$$\newcommand{\AA}{\unicode[.8,0]{x212B}}$$

## Symmetry and inertia

This story shows that translation symmetry is closely related to the relative nature of motion, as expressed by the principle of inertia. Riding in a train on a long, straight track at constant speed, how can you even tell you're in motion? You can look at the scenery outside, but that's irrelevant, because we could argue that the trees and cows are moving while you stand still. (The Martians say both train and scenery are moving.) The real point is whether you can detect your motion without reference to any external object. You can hear the repetitive thunk-thunk-thunk as the train passes from one piece of track to the next, but again this is just a reference to an external object --- all that proves is that you're moving relative to the tracks, but is there any way to tell that you're moving in some absolute sense? Assuming no interaction with the outside world, is there any experiment you can do that will give a different result when the train is in motion than when it's at rest? You could if translation symmetry was violated. If the laws of physics were different in different places, then as the train moved it would pass through them. “Riding over” these regions would be like riding over the pieces of track, but you would be able to detect the transition from one region to the next simply because experiments inside the train came out different, without referring to any external objects. Rather than the thunk-thunk-thunk of the rails, you would detect increases and decreases in some quantity such as the gravitational constant G, or the speed of light, or the mass of the electron.

We can therefore conclude that the following two hypotheses are closely related.

## The principle of inertia

The results of experiments don't depend on the straight-line, constant-speed motion of the apparatus.

## Translation symmetry

The laws of physics are the same at every point in space. Specifically, experiments don't give different results just because you set up your apparatus in a different place.

### Example 1: A state of absolute rest

Suppose that translation symmetry is violated. The laws of physics are different in one region of space than in another. Cruising in our spaceship, we monitor the fluctuations in the laws of physics by watching the needle on a meter that measures some fundamental quantity such as the gravitational constant. We make a short blast with the ship's engines and turn them off again. Now we see that the needle is wavering more slowly, so evidently it's taking us more time to move from one region to the next. We keep on blasting with the ship's engines until the fluctuations stop entirely. Now we know that we're in a state of absolute rest. The violation of translation symmetry logically resulted in a violation of the principle of inertia.

self-check:

Suppose you do an experiment to see how long it takes for a rock to drop one meter. This experiment comes out different if you do it on the moon. Does this violate translation symmetry?

Inertia is shared under a not declared license and was authored, remixed, and/or curated by LibreTexts.