Modern Physics: 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 $t=t'=0$ , $x=x'=0$ ), the Galilean transformations take the following simple form:

$t'=t, \quad x'=x-ut$ ,

$y'=y, \quad z'=z$ .

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:

$t'=\frac{t-\frac{u}{c^2}x}{\sqrt{1-\frac{u^2}{c^2}}}, \quad x'=\frac{x-ut}{\sqrt{1-\frac{u^2}{c^2}}}$

$y'=y, \quad z'=z$ .

These equations become more manageable with the introduction of the quantity

$\gamma = \frac{1}{\sqrt{1-\frac{u^2}{c^2}}}=(1-u^2/c^2)^{-1/2}$ ,

so that the Lorentz transformations become

$t'=\gamma\left(t-\frac{u}{c^2}x\right), \quad x'=\gamma(x-ut)$ ,

$y'=y, \quad z'=z$ .

Often, the space-time coordinates for an event will be given in the form $(t,x,y,z)$ , or just $(t,x)$ when the y and z coordinates are not important.

Part A

Consider an event with space-time coordinates $(t=2.00\;{\rm s},\; x=2.50\times 10^{8}\; {\rm m})$ in an inertial frame of reference S. Let S' be a second inertial frame of reference moving, in the positive x direction, with speed $2.70\times10^{8}\;{\rm m/s}$ relative to frame S. Find the value of gamma

that will be needed to transform coordinates between frames S and S'. Use $c = 3 \times 10^8\; {\rm m/s}$ 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 $t=t'=0$ , $x=x'=0$ . Using the gamma

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

$\rm m$

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

$\rm s$

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 $t=t'=0$ . The proper length of the ship (the length of the ship as measured in the ship's frame, Z') is l_0

. In other words, a passenger on the ship measures the back of the ship to be at $x'=0$ and the front to be at $x'=l_0$ .

Part D

Find the

factor that should be used to transform between frames Z and Z'.

Express your answer in terms of

and

ANSWER:

$\frac{1}{\sqrt{1-\frac{v^{2}}{c^{2}}}}$

Part E

At time $t=0$ , in your frame of reference Z, you measure the back of the spaceship to be at $x=0$ and the front of the ship to be at $x=l$ . Find an equation relating the length that you measure