One of the most common headlines I see in popular science articles is “Was Einstein Wrong?”. While it is certainly true that relativity is going to be replaced by some more advanced theory of physics at some stage, the basis of Einstein’s theory is unlikely to change.

Relativity breaks down at the centres of black holes and at the extreme conditions at the start of the Big Bang, but that doesn’t mean the foundations of the theory are unsound. The foundations of relativity theory are used to build all the new hypothetical theories that we want to supersede relativity one day. So the basics of relativity theory are not really in any doubt.

The reason for this is that when it’s boiled down, relativity theory can actually be derived from some very simple assumptions which virtually nobody denies. Here we will take a look at what these assumptions are.

## Coordinate Transforms

The fundamental claim of relativity is that the coordinate transforms that leave the laws of physics unchanged are the “Lorentz transforms”. Let me put this into English a bit more. What are coordinate transforms?

Suppose you’re standing still on a train station platform, and a train with your mate on goes whizzing past you. There are two coordinate frames in this situation. There’s the your “rest frame” which is the frame of you and the station, and there is the rest frame of your mate in the train, where he takes himself to be at rest and you to be moving.

What we want in physics is a way to translate between these two frames. The dictionary we use for the translation is called the coordinate transformations. In the days of Newton we used to think that the correct coordinate transformations were the Galilean transformations. These preserved Newton’s laws and are the most intuitive transformations for everyday life. They say that the time in each frame is equal, and that if the train is moving along the x axis then if you are at position x on the platform, your mate in the train will see you obeying the laws of physics, but as if you were moving at the speed of the train away from them.

These transforms seem like the obvious ones, it’s hard to see how it could be any different, but it turns out they are incorrect. These transformations preserve Newton’s Laws, but Newton’s Laws are not the fundamental laws of physics. This is the key statement of relativity, that the actual laws of physics remain the same under Lorentz coordinate transforms, not Galilean ones.

## Derivation of the Lorentz Transforms

Now let’s pretend that we don’t know what the Lorentz transforms actually are. It turns out that we can derive these transformation laws from some very simple facts that we assume about space and how coordinate transforms should work.

### Translational Invariance

What this means is that it shouldn’t affect the physics if we shift one of coordinate systems a bit in one direction. Take our previous example of you on the platform. It shouldn’t matter if we decide to put you at the origin of your own coordinate frame, or one meter from the origin.

In other words, if we perform an experiment in one place, then perform the experiment an inch to the left, the results of the experiments should be the same.

### Simple Velocities

Let’s use the train example. From the point of view of someone on the platform, if the train is moving at speed v, then we should observe the distance it travels in a time t to be x=vt.

### Multiple Transformations

Suppose that there is a third coordinate frame in our train example – somebody desperately running to try and catch the train. We assume that if we coordinate transform from your stationary frame, the the frame of your mate on the train, and then from that frame to the frame of the person running, the result should be the same as if we transformed straight from your stationary frame to the frame of the person running.

In other words if you transform from A to B to C, the result should be the same as transforming straight from A to C.

### Transforming Back

If you transform to the frame of the train moving at speed v, and then apply a transform from that frame to a different frame moving at -v, then you should get back to where you started.

### Isotropy

There are no privileged directions in space. It doesn’t matter in which direction you move.

It turns out that from these assumptions alone, one can derive the Lorentz transforms and hence the foundations of relativity.

(as a side note, from these assumptions you get the correct Lorentz transforms, and you arrive at the conclusion that there should be a constant speed in the universe, but this derivation doesn’t tell you what that speed should be. Experiment tells us that the constant speed is the speed of light, so we put this speed in and we get the Lorentz transforms. But it’s interesting to note that if you choose to set the constant speed to infinity, you get the Galilean transformations!)

## Why Einstein will Never be Wrong

So there you have it, the assumptions made in the derivation of the Lorentz transforms are really very simple and seem to be irrefutable. Since these assumptions seem correct, one arrives at the conclusion that the Lorentz transforms are the proper coordinate transforms that leave the laws of physics unchanged.

This is the central postulate of relativity, and all hypothetical theories in theoretical physics are built so that they are “Lorentz covariant” – meaning that the laws in these theories must be unchanged under the Lorentz coordinate transformations.

So while relativity will inevitably be superseded by some more accurate, advanced theory. It’s central claim about the form of the correct coordinate transforms is unlikely to change much or at all in future physics theories. Einstein’s central claim of relativity is unlikely to be wrong.

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