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
|
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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
| |
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Part C |
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Now find  , the t coordinate of the event in frame S'. Express your answer in seconds to three significant figures.
ANSWER: |
| = | 2.87
| |
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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: |
| = | 
|
|
|
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Part E |
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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: |
| = | 
|
|
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You should recognize this as the equation for length contraction. The time dilation equation can also be found from the Lorentz transformations. |
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