Length Contraction, Time Dilation and Lorentz Transformations

A very common question is "When should I use the simple length contraction or time formulas and when do I need to use the full Lorentz transformations?" Here are some tips:

Length contraction applies when you are talking about a distance that is independent of time (e.g. the distance between two objects that are fixed relative to one another or the distance between two ends of a single object). The "proper distance" in the formula is the distance in a frame where the two ends of the distance are not moving, and the formula says that the observed distance in another frame is smaller by a factor of gamma.

Note that length contraction does not apply if we are talking about the distance between two things that are moving relative to one another or to the distance between two events (rather than physical objects).

Time dilation applies when you are talking about two events that are at the same place in some frame (e.g. two ticks on the same clock). In this case,  the "proper time" in the formula is the time in this frame where the two events are at the same place, and the formula says that the observed time between these two events in some other frame is larger by a factor of gamma.

Note that time dilation does not apply if we are talking about the time between two events that are not in the same place in either frame.

When length contraction and time dilation do not apply (or even when they do), we can use Lorentz transformations.

1) Lorentz transformations relate the position and time of a SINGLE EVENT  in some frame S to the position and time in another frame S'. If you are applying the LT formula, ask yourself whether the x and t you are plugging in correspond to some specific event.

2) The main steps to using Lorentz transformations to solve a problem:

a) Identify one frame of reference and call it S. It is your choice, but usually it is simplest to call S the frame of reference in which you have the most information.

b) Identify another frame of reference and call it S'. Usually it is convenient to make this the reference frame corresponding to the information that you are trying to find in the problem.

c) Determine the velocity of frame S' relative to frame S.  The velocity v that appears in the Lorentz transformation formula or the velocity transformation formula
is always this velocity.
Make sure that you have defined the x direction so that this velocity is either in the positive or negative x direction.

d) Decide which events are relevant for the problem. Pick some event to be the origin of space and time coordinates (if the question doesn't do this for you). Determine the coordinates and times of the other events in one frame of reference. This might be information you are given, or information that you need to work out using ordinary kinematics formulae (e.g. distance = speed * time).  Finally, you can use the Lorentz transformation formulae to determine the positions and times of these events in the other frame of reference. This should be enough information to allow you to solve the problem.

3) The same steps a), b), and c) apply to using the velocity transformation formula or the formulae for transforming energy and momentum. Note that the velocity v in these formulae is still just the velocity of frame S' relative to frame S.