## Spacetime

Phenomena such as time dilation and length contraction are not simply illusions. They are real effects. Our failure to appreciate this previously comes from a failure to appreciate the true nature of space and time. Classical physicists assumed that space and time were simply there, just like a blank piece of graph paper, and that the laws of physics were laid down on top of them. Einstein realized that the laws of physics were intimately entwined with space and time. We can understand the nature of this relationship by abandoning our normal three-dimensional world and replacing it with the four-dimensional world of space-time.

### 7.5.1 Four-vectors and Lorentz transformation

In space-time we simply treat time as another coordinate. To remind us that time is just another way of measuring distance, we sometimes write the time coordinate as ct, so that it has the same dimensions as the other coordinates. In this way, we could measure time in meters. What is a time of one meter? It is the time that light takes to travel one meter. (Note that we have previously used time as a measure of distance when we introduced the light-year.)

An interesting aside to this has come from the organizations that set international standards such as the meter and the second. It used to be that such units were defined independently, and c was just a measured quantity. The speed of light is now taken to have a defined value, where all decimal places beyond the most accurate measured value are taken to be zero. It now gives the conversion from meters to seconds. This means that we only need a standard for the second or the meter, but not both.

In space-time we speak of four-vectors to distinguish them from ordinary three-dimensional vectors. Any event is characterized by the four coordinates (ct, x, y, z). Observers in different iner-tial frames will note different coordinates for events, but the coordinates are related. If one inertial system is moving with respect to another at a speed v, in the x-direction, the coordinates in the transformations between the two coordinate systems are found by assuming they are linear in the coordinates, and must give the correct results for length contraction and time dilation. The result is (letting ( = v/c)

The reverse transformation is given by ct' = y (ct - (x) x' = y (x - (ct)

These relationships together are called the Lorentz transformation.

We interpret the Lorentz transformation as telling us that the rules of geometry are different in space-time than they are for ordinary space. To illustrate this point, we use a space-time diagram, like that shown in Fig. 7.8. For simplicity, we plot ct

0 0