Lorentz Transformations
The relativity of simultaneity can be calculated using Lorentz transformations, which relate the coordinates used by one observer to coordinates used by another in uniform relative motion with respect to the first.
Assume that the first observer uses coordinates labeled t, x, y, and z, while the second observer uses coordinates labeled t', x', y', and z'. Now suppose that the first observer sees the second moving in the x-direction at a velocity v. And suppose that the observer's coordinate axes are parallel and that they have the same origin. Then, the Lorentz transformations show that the coordinates are related by the equations:
where c is the speed of light. If two events happen at the same time in the frame of the first observer, they will have identical values of the t-coordinate. However, if they have different values of the x-coordinate (different positions in the x-direction), we see that they will have different values of the t' coordinate; they will happen at different times in that frame. The term that accounts for the failure of absolute simultaneity is that v x/c2.
The equation t' = constant defines a "line of simultaneity" in the (x', t' ) coordinate system for the second (moving) observer, just as the equation t = constant defines the "line of simultaneity" for the first (stationary) observer in the (x, t) coordinate system. We can see from the above equations for the Lorentz transform that t' is constant if and only if t – v x/c2 = constant. Thus the set of points that make t constant are different from the set of points that makes t' constant. That is, the set of events which are regarded as simultaneous depends on the frame of reference used to make the comparison.
Graphically, this can be represented on a space-time diagram by the fact that a plot of the set of points regarded as simultaneous generates a line which depends on the observer. In the space-time diagram at the right, the dashed line represents a set of points considered to be simultaneous with the origin by an observer moving with a velocity v of one-quarter of the speed of light. The dotted horizontal line represents the set of points regarded as simultaneous with the origin by a stationary observer. This diagram is drawn using the (x, t) coordinates of the stationary observer, and is scaled so that the speed of light is one, i.e. so that a ray of light would be represented by a line with a 45° angle from the x axis. From our previous analysis, given that v = 0.25 and c = 1, the equation of the dashed line of simultaneity is t – 0.25x = 0 and with v = 0, the equation of the dotted line of simultaneity is t = 0.
It should also be mentioned that Lorentz came up with his ideas based on the assumption that the Aether existed.
Read more about this topic: Relativity Of Simultaneity