Reference Frames.
Before we can make use of the transformation equations of relativity theory it is necessary to understand what is meant by a reference frame and how to construct one. As I write I am sitting in a rectangular room, and I have several clocks and a ruler. If I want to describe the location of an object I can conveniently define coordinate axes x, y and z to coincide with three edges of the walls, and define one corner of the room as position zero from which all other positions can be measured. There is nothing unique about the choice of axes. Any choice of positive x and y directions can be transformed into any other choice by rotation, but for any given x and y axes there are two possible positive z directions to choose from, and one can only be transformed into the other by a mirror reflection, not by a rotation. There is a convention as to which of the two possibilities is used, and this is a right-hand screw rule such that a rotation of a right-hand thread screw through 90 degrees from +x to +y will give motion of the screw in the +z direction. To put it another way, if I face a wall of my room, and the bottom left-hand corner is the point x = y = 0 and x is from left to right along the bottom edge of the wall, while y is upwards along the left hand side of the wall, then +z is the direction vertically out towards me from the bottom left corner.
A very important point must now be made. The transformation equations we are about to consider describe the relationship between two events, not between two objects. An object is something which exists more or less unchanged over a period of time. An event of the idealised sort being considered here happens at one point in space and at one point in time. An object may have a reference frame in which it is at rest. An event, however, can not be said to be at rest, or moving, or to be associated in any way with one reference frame more than any other. If I choose the origin of my reference frame to be the corner of my room, and place a clock there with a reading in seconds starting from zero at some instant, then I can define an event as being at the origin at the instant when the clock starts from zero, i.e. at x = y = z = t = 0.
We now consider another reference frame constructed by a different observer moving at velocity v relative to myself. For ease of analysis it is convenient to choose the same directions for x, y and z in the two frames, and to choose v to be in the +x direction, and also to use the same event as the origin of the second frame. This is illustrated at t = 0 in Fig.la and at a later time in Fig.1b. Only the x and y directions are shown. The event is not shown in Fig.1b, because of course it does not occur at the later time. The time also is not shown.
It is tempting to write t = t' = T where T is the reading on my clock at the later time. In Newtonian, or pre-relativistic physics this would be valid, but it is based on an assumption that there is a universal time and that observers moving with different velocities would agree on the time. Whether or not this is true is a matter for experiment to resolve.
