It’s All About Relativity

“It is impossible to travel faster than light, and certainly not desirable, as one’s hat keeps blowing off.” – Woody Allen

If you’ve ever studied a bit of Special Relativity, you will have encountered its two key postulates:

(Wikipedia)

1. First postulate (principle of relativity)

The laws of physics are the same in all inertial frames of reference.

2. Second postulate (invariance of c)

As measured in any inertial frame of reference, light is always propagated in empty space with a definite velocity c that is independent of the state of motion of the emitting body. Or: the speed of light in free space has the same value c in all inertial frames of reference.

I once asked a class I gave a lecture to which of those two postulates they thought was most important. Apart from those that wisely abstained, everyone agreed that they thought the light postulate was most important.

This however is a common misconception. It arises because of how mind-bending it is to discover one day that there is a maximum speed in the universe. That however fast you’re travelling, a light pulse will always travel at a speed c away from you.

On the flip-side, it seems much more mundane to notice that things behave the same way whether you’re sat in a laboratory, or on a smooth train moving at a constant speed. But really, this is the key postulate of relativity theory. It may not be too extreme to claim that the observation that the laws of physics are the same no matter with what speed you are moving is the most profound observation in the history of Physics.

In fact, we can get most of the way to deriving the constancy of the speed of light just from the Principle of Relativity alone. In other words, the speed of light being constant is just a consequence of the laws of physics being the same in all inertial frames.

Coordinate Transforms

To understand why the Relativity Principle is much more important than the light postulate, we need to introduce some terminology.

When we use the word “inertial frame” in Physics we’re referring to something like a room moving at a constant speed with a set of Cartesian coordinates attached to it – almost like a map of a room.

At a train station for example, there are two relevant “inertial frames”. The train station itself is an inertial frame, moving at speed 0, if you’re standing on it. A train moving past the station at a constant speed is then another inertial frame moving relative to yours at some speed v.

Now a coordinate transform is something we use when we want to see what something looks like in another frame. Say you’re on the station holding a clock, a coordinate transform would tell you how a person on the moving train would see your clock behave.

We used to think that the correct coordinate transform to describe this scenario looked like this:

If the train is moving at speed v, then the observer on the train would see the clock you’re holding move backwards at a speed v relative to them.

I hope nothing I’ve said there seems controversial, this was what we used to think the coordinate transforms were based on common sense.

It turns out however that Relativity is all about why these transformations are actually incorrect!

Can’t we Transform However we Like?

An astute reader may be thinking – “Can’t we transform to whatever coordinate system we like?”, “Why is there such a thing as an incorrect transformation?”. While it’s true that however we transform, the behaviour of the system should not be affected (think about it, the way the Earth looks should not be affected by the way we decide to draw our maps!), there are certain transformations that preserve the form of the laws of physics.

It is these transformations that are interested in. The Galilean transforms above preserve the form of Newton’s Laws of motion. Both you on the platform, and the person watching you on the train would be able to use Newton’s Laws to describe the motion of the clock if the correct transformations were the Galilean ones.

But it just so turns out that although Newton’s Laws are an excellent approximation, the actual laws of physics are preserved not by Galilean Transforms, but by a different set of transforms known as the Lorentz Transforms.

We can derive the form of these transformations using only intuitive assumptions about space and the Relativity Principle.

The Derivation of the Lorentz Transforms

If we make certain assumptions about the nature of space and motion, how does this constrain the transformations that may be the correct ones?

Here are some assumptions that we make:

• If you perform an experiment at one point in space (call it A) and then perform the same experiment next to point A there should be no difference in the outcome of the experiment
• If one frame moves with velocity v, it’s origin appears to be at x=vt
• If you transform from a frame A to a frame B and then to a frame C, there must be a transform that can take you straight from A to C (If you have two trains moving past a station at different speeds, you should be able to transform between any of them and the station)
• If you transform to a frame moving at v, and then transform from that frame to a frame moving at -v, you must return to the same frame
• Spatial Isotropy – there are no special directions in space
• Synchrony Convention – choose a method of synchronising clocks
• Relativity Principle

If you make these assumptions you end up with what are called the Ignatowski Transformations.

If you examine the form of these transformations, you find that they imply the existence of an invariant speed!

Note that we’ve discovered there must be an invariant speed in the universe, without any considerations about light!

The Lorentz Transforms are obtained by applying the Relativity Principle to Electromagnetism. We know from Maxwell’s equations that you can calculate that the speed of an electromagnetic wave (light) is c. The Relativity Principle tells us that Maxwell’s equations hold in any frame, and so we come to the conclusion that the speed of light must be c in any frame.

We have thus found our invariant speed. We plug this invariant speed into the Ignatowski transforms to obtain the Lorentz transforms, which will be familiar to any student of physics:

The Lorentz transforms form almost the entire basis for Special Relativity, all strange effects you may have heard about like time dilation, length contraction and the twin paradox arise solely because the laws of physics are invariant under the Lorentz transformations, not the Galilean ones.

To conclude then, the constancy of the speed of light is just a consequence of applying the relativity principle to electromagnetism. It’s hugely profound and interesting that light behaves like this, but the true content of Relativity is that the laws of physics are the same in all inertial frames, not that the speed of light is constant.