kevin86 said:
How do I derive the equation for Simultaneityfrom one of the lorentz transformation.
If you can help me with that, please help with time dilation as well.
The textbook focused mainly on the mathematically derivation without using the lorentz transformations, and I cannot find any answers online.
Try the wikipedia article on the relativity of simultaneity
http://en.wikipedia.org/wiki/Relativity_of_simultaneity
Basically, events are simultaneous if they have the same t coordinate.
Suppose we have two frames: S1, with coordinates x and t, and S2, with coordinates x' and t'.
Then two events are simultaneous in frame S1 if they have the same t coordinate, i.e. t(event1) = t(event2).
Two events are simultaneous in frame S2 if they have the same t' coordinate, i.e. t'(event1) = t'(event2).
The two sets above are not the same.
Because the Lorentz transform gives t' and x' in terms of t and x, one can determine the equation of a 'line of simultaneity' in S2 in terms of x and t.
Let us find the set of events simultaneous with the origin. Then we have
t' = gamma * (t - v*x/c^2) = 0
This means that the equation of t'=0 is t = vx/c^2 , which defines the equation of the "line of simultaneity" of events in S2 in terms of the coordinates (t,x) of S1.
If you work out a more general example, you'll find that all lines of simultaneity have the same slope on the space-time diagram, which by the example above is slope = dx/dt = c^2 / v.