D1 Coordinate Transformations before the STR
A physical event takes place at a certain time and at a certain place. Thus, in each inertial frame (coordinate system, reference frame) we can assign to an event a time coordinate as well as three local coordinates. In each coordinate system one point of 4D space-time belongs to an event.
We examine in this and the following two sections how the 4 coordinates assigned by an observer to an event in one inertial frame are correctly converted to the corresponding 4 coordinates assigned by a second observer of the same event in another inertial frame. The formulas describing this conversion are called coordinate transformations.
We assume (as previously) two coordinate systems - a black, non-prime system and a red, prime system, which are aligned to each other in a simple manner (same diagram as in B3):
The origin B of the red system moves with velocity v along the x-axis of black, and the x'-axis of red coincides with the x-axis of black. Thus A moves with velocity u = -v along the x'-axis of red. The two other spatial axes (y/y' and z/z') will always be parallel to each other. In addition, both red and black set their clocks to zero at the moment when they coincided. All other clocks that black possibly uses are synchronized within its frame with the master clock in A. And all 'red' clocks are synchronized with the red clock B within the red system.
We now consider coordinate conversions within the mechanics of Galileo and Newton: Since the clocks were set to zero by red and black at their meeting, both clocks (as well as all other clocks of red and black) will always agree. They agree to the degree of their accuracy, velocity of movevment has no influence on the synchronization or on the clock tick rate. Thus for each moment and in all places it applies
(1) t = t’
Also concerning the distances from the x-axis, which is identical to the x'-axis, one will find no differences. That is
(2) y = y’ and z = z’
There is something to convert only if one wants to convert an x'-coordinate of red to x or vice-versa. x' is the distance from the point of the event, projected on to the x/x' line, to the red origin B. Origin B has the distance v • t from the black origin A. Thus the x-coordinate of the event is calculated simply as
(3) x = x’ + v • t = x’ + v • t’ and correspondingly x’ = x - v • t
We have just derived the very simple Galileo transformations. We have crucially made use of Newton's concept of absolute time, which is the same for everyone, as well as his concept of absolute space, which permits the calculation of absolutely valid distances or lengths.
We will now demonstrate (as promised in A3) that the ‘classical’ addition of velocity results from this basis of Galileo and Newton (‘classic’ is always equivalent to ‘not relativistic’ in this context).
Assume C moves in the red system with the speed of w' in the x'-direction. The x'-coordinate of C is thus x' = a + w' • t', where a is any constant. This is the distance in the x'-direction from B to C. B is however at any given time t, as seen from black, at the point v • t. The point of C along the x-axis of black is thereby x = v • t + a + w' • t'. We thus have already used the absoluteness of space. Now we use Newton's absolute time and simply replace t' by t using (1). Thus we obtain x = v • t + a + w' • t = a + (v + w') • t. So the x-position of C increases with speed v + w' for black. Speeds simply add in classical mechanics.
In order to compare with the somewhat more complicated Lorentz transformations, which we will deduce in the next section, we present the Galileo transformations in the following table:
These coordinate transformations describe how in classical mechanics the coordinates (t, x, y, z), which black attributes to an event should be converted into the coordinates (t', x', y', z'), which red attributes to the same event - and vice-versa. If the two coordinate systems were less precisely aligned to each other, then naturally the lines 2, 3 and 4 in the small boxes would be somewhat more complicated. Nothing at all would change however in the first line, which reflects Newton's absolute time.
A somewhat childish suggestion: Go against the official direction of motion on an escalator (as you surely once did as a child) or on one of the long “moving sidewalks” one finds in airports. It is fun and also a direct way to experience the addition (or rather subtraction) of velocities.