### Properties of nature

We say that a potential law of physics is **homogeneous** if it doesn't change under a translation of coordinates, and **isotropic** if it doesn't change under a rotation of coordinates.

We expect all valid laws of physics to have these two properties. From Newton's perspective, this would have been all that was required. It would not have been necessary to consider translations in time, since time was an invariant quantity by definition.

In special relativity however, we must insist on the stronger condition that a law of nature has to **Lorentz invariant** - the law cannot change under Lorentz transformations.

### Candidate laws

We have to be careful to express laws of nature in equations involving quantities and sets of quantities that satisfy Lorentz invariance. We can consider some invalid laws of nature and see where they break down.

In the following discussion we say a *collision* between two particles is simply an unspecified change in the velocity of one and/or other of the particles.

Suppose that a law of nature stated that collisions between particles occur whenever their $x$-coordinates coincided.

For example, in figure 6.1, a collision is shown to occur since $$x_A - x_B = 0$$

Is this a valid law? No, since it is not *isotropic*. We could rotate the axes by $45^{\circ}$ say to new coordinate positions, $\vec{x}'_A, \vec{x}'_B$ where the $x$-components don't now coincide, $$x'_A - x'_B \neq 0$$

It is, however, *homogeneous*, since the difference, $x_A - x_B$ won't change under a translation.

Nb. We could even think up a law that isn't homogeneous. For example, a collision occurs only when two particles have a particular value for their $x$-component, $x_A = x_B = x_0$, say. Then, a collision won't occur if the origin is translated.

In order to satisfy **isotropy**, we need to describe laws in terms of (three-dimensional) vectors. If the law is that a collision will occur only if all three components of the position coincide, then this will still be true under both translations and rotations. That is, when $$\vec{x}_A - \vec{x}_B = 0$$

We can state this more generally. Suppose we have some expression that results in a vector, $\vec{w}$ say, and that in some coordinate frame, $$\vec{w} \rightarrow (w_x, w_y, w_z) = 0$$ then this will be true in all frames, $$\vec{w} \rightarrow (w'_x, w'_y, w'_z) = 0$$ thus we can write the expression in a frame-invariant way, $$\vec{w} \equiv 0$$

Can we think of a law that's homogeneous and isotropic, but not Lorentz invariant? Yes, suppose that a collision between two particles can occur only if their time components coincide (they exist at the same time).

Straightaway we can see the problem - simultaneity. A collision in one frame will not be a collision in a moving frame.

To satisfy Lorentz invariance, we need to say that all four components of the positions of the particles in space-time must coincide before a collision can occur. That is, when $$X_A^\mu - X_B^\mu = 0$$

This will remain true under translations, rotations and boosts (Lorentz transformations).

### Four-vector equations

We have seen that if we can express laws of nature in terms of four-vector equations, then these equations are the same in all reference frames.

We can say this in two ways:

- If an expression, resulting in a four-vector, is
*zero*in one frame, then it will be*zero*in all frames, $$A^\mu = 0 \Leftrightarrow A^{\mu'} = 0$$ - If two vector expressions are
*equal*in one frame, then they will be*equal*in all frames, $$A^\mu = B^\mu \Leftrightarrow A^{\mu'} = B^{\mu'}$$

Nb. We can and will generalise this notion to *tensor* equations, where tensors are generalised four-vectors with arbitrary numbers of indices.