Picturing SpaceTime
Take a piece of paper and draw two intersecting straight lines on it.
Add an arrowhead on one end of each and label one with a t (and henceforth we’ll call that line the t-axis) and the other with an x (to be called the x-axis).
We are going to consider the t-axis as a representation of all the events in the life of an observer O and lines parallel to the x-axis as representing the set of all events that are considered by O to be simultaneous with the point where that line meets the t-axis.
In order to make the correspondence explicit we need to identify points on the axes with particular events in space-time by choosing arbitrary units of time and distance and making these correspond to some chosen length unit on our piece of paper. So, considering the time and displacement both to be zero at the point (called the Origin) where the axes cross, we’ll mark a point between there and the t arrowhead as corresponding to one time unit and a point between the origin and the x arrowhead as corresponding to one displacement unit. We could use any units for this mapping, but in order to have something concrete in mind, and for a bit of simplification later on, let’s say that one cm on the t-axis corresponds to one second of time and one cm on the x-axis to one light-second of distance (ie about 3/4 of the distance from the Earth to the Moon). With these choices, a line parallel to the t-axis through the point one cm from the t-axis corresponds to the events that happen one light second away from the observer in one particular direction, and a line parallel to the x-axis at a distance one cm in the direction of the t-arrow corresponds to all events which the Observer considers to have happened exactly one second after the Origin event.
The Light Cone
Now consider the set of events on the world line of a light ray transmitted from the origin in the positive x direction. Since light travels at the rate of one light-second per second, these points are all at equal distances from the x and t axes and so are on the bisector of the angle between them. [If the units had been different, then it wouldn’t be the bisector but rather the line on which the ratio of distances from t and x axes was equal to the speed of light in terms of the corresponding units – eg in m and seconds it would be the line on which x=ct=299 792 458t, so in this case the angles would not be equal but there would still be a definite and fixed relationship between them. (In particular the ratio of their sines would have to equal c.)]
Similarly a light ray travelling in the opposite direction has a world line that (in a picture with equally plotted units corresponding to c=1) bisects the angle between the positive t and negative x axes and so is perpendicular to that of the forward ray. [In the case of more general units the split would again be such that the ratio of sines is c and for c>1 the opening angle of the forward cone would be greater than 90 degrees.]
Relativity of Simultaneity
Now consider a second observer [math]O_2[/math] who meets the first (who from now on we’ll call [math]O_1[/math]) at time zero moving by with a constant relative velocity. Then [math]O_2[/math]’s world line is just another straight line pointing into the forward light cone. But IF [math]O_2[/math] also sees the speed of light as the same as [math]O_1[/math] then the light lines must divide the angle between [math]O_2[/math]’s t and x axes in the same ratio as for [math]O_1[/math]. So [math]O_2[/math]’s x-axis is different from [math]O_1[/math]’s. Or in other words, the set of events that [math]O_2[/math] sees as simultaneous with their meeting is different from the ones [math]O_1[/math] sees as simultaneous – which is what is meant by the claim that the idea of simultaneity depends on, or is relative to, the choice of observer.
The Lorentz Transform
In order to discuss exactly how the different versions of simultaneity are related we need to work more explicitly in terms of the coordinates.
If we have already identified a scale on either the [math]t_1[/math] axis or the [math]t_2[/math] axis then we can determine the other as follows:
If we let [math]\alpha[/math] be the value on the [math]t_2[/math] axis where it is crossed by the line [math]t_1 = 1[/math], then by symmetry and similar triangles the line [math]t_2 = \alpha[/math] will meet the [math]t_1[/math] axis at the point where [math]t_1 = \alpha^{2}[/math]. Taking the geometric mean of these two points on the [math]t_1[/math] axis gives the point on the [math]t_1[/math] axis at which [math]t_2=1[/math], and so from which the line of constant [math]t_2[/math] meets the [math]t_2[/math] axis at [math]t_2=1[/math].
If we let [math]\gamma[/math] be the value on the [math]t_2[/math] axis from which the line of constant [math]t_2[/math] meets the [math]t_1[/math] axis at [math]t_1=1[/math], then by symmetry and similar triangles the line of constant [math]t_1[/math] from that point will meet the [math]t_1[/math] axis at the point where [math]t_1 = \gamma^{2}[/math]. Taking the geometric mean of these two points on the [math]t_1[/math] axis gives the point from which the line of constant [math]t_1[/math] meets the [math]t_2[/math] axis at [math]t_2=1[/math].
If the point on the [math]t_1[/math] axis with [math]t_1 = 1[/math] is on the line with [math]t_2=\gamma[/math], then by symmetry, the point on the [math]t_2[/math] axis with [math]t_2 = 1[/math] is on the line with [math]t_1=\gamma[/math]
If [math]O_2[/math] is seen by [math]O_1[/math] to be moving with a constant velocity v, then in terms of [math]O_1[/math]’s coordinates [math]O_2[/math]’s world line has the equation [math]x_1=v t_1[/math]. Assuming that they have both chosen the same particular far off object to define their positive x directions (as opposed to each defining the positive direction as pointing towards the other), and that they agree on their relative speed, the world line of [math]O_1[/math] will be given in terms of [math]O_2[/math]’s coordinates by [math]x_2=-v t_2[/math].