## Sunday, January 30, 2011

### Mastering Physics: ± Understanding Lorentz Transformations

Learning Goal: To be able to perform Lorentz transformations between inertial reference frames.
Suppose that an inertial reference frame S' moves in the positive x direction at speed with respect to another inertial reference frame S. In classical physics, the Galilean transformations relate the coordinates measured for an event in frame S to the coordinates measured for the same event in frame S'. Assuming that both frames have the same origin (i.e., at , ), the Galilean transformations take the following simple form:
,
.
The Galilean transformations are not valid at very large speeds. To transform between inertial frames when is close to the speed of light , we need to use the Lorentz transformations of special relativity. Again, assuming that both frames have the same origin, the Lorentz transformations take the following form:
.
These equations become more manageable with the introduction of the quantity
,
so that the Lorentz transformations become
,
.
Often, the space-time coordinates for an event will be given in the form , or just when the y and z coordinates are not important.

Part A
Consider an event with space-time coordinates in an inertial frame of reference S. Let S' be a second inertial frame of reference moving, in the positive x direction, with speed relative to frame S. Find the value of that will be needed to transform coordinates between frames S and S'. Use for the speed of light in vacuum.
Express your answer to three significant figures.
ANSWER:
 = 2.294
Part B
Suppose that S and S' share the same origin; that is, at , . Using the you calculated in Part A, find , the x coordinate of the event in frame S'.
Express your answer in meters to three significant figures.
ANSWER:
 = −6.650×108
Part C
Now find , the t coordinate of the event in frame S'.
Express your answer in seconds to three significant figures.
ANSWER:
 = 2.87
Suppose that you are stationary with respect to an inertial reference frame Z. A spaceship flies by you in the positive x direction with speed . Let Z' be the frame of reference associated with the spaceship; that is, the ship is stationary with respect to Z'. The frames Z and Z' have the same origin at . The proper length of the ship (the length of the ship as measured in the ship's frame, Z') is . In other words, a passenger on the ship measures the back of the ship to be at and the front to be at .

Part D
Find the factor that should be used to transform between frames Z and Z'.
Express your answer in terms of and .
ANSWER:
 =
Part E
At time , in your frame of reference Z, you measure the back of the spaceship to be at and the front of the ship to be at . Find an equation relating the length that you measure to the ship's proper length .
Express your answer in terms of and .
ANSWER:
 =
You should recognize this as the equation for length contraction. The time dilation equation can also be found from the Lorentz transformations.