Exploration Part 3:
The Mass-Energy Equivalence
A Gedanken Experiment
Imagine a rocket equipped with a photon cannon fires two identical photons–one in the
direction of motion of the rocket with respect to an external observer and one in the
opposite direction. A detailed analysis of what happens, as seen by observers in both
the rocket frame and external frame (lab frame) leads to an interesting result!
The Rocket Frame View
Photons carry both momentum and energy according to the expressions: p 
E  hf .
h

and
Your turn
1. If photons carry momentum, what is the total change in momentum as measured
in the rocket frame? Explain your answer.
The Lab Frame View
In the rocket frame, momentum is conserved as the two photons carry equal amounts of
momentum in opposite directions. In the lab frame, you would observe something quite
different! Consider the photons that are fired by the rocket. If detectors are set up in the
forward and reverse direction then they will "see" very different photons.
To assist you, open the toolbox on the right and run the applet "massEnergy". You can
control the velocity of the rocket relative to the lab frame and observe the behaviour of
photons fired from the photon cannon as seen in each frame.
Your turn
2. What will happen to the wavelengths of the photons as recorded by the two
detectors in the lab frame? Use the relativistic Doppler shift formula to calculate
the wavelength of the photons as received by the forward and reverse detectors.
3. Do the photons carry equal and opposite momentum as seen from the lab
frame? Explain your answer.
Reconciling the Two Views
If momentum is conserved in the rocket frame it must also be conserved in the lab
frame! In other words, p  0 for both frames. A lab frame observer, however, sees
both a red-shifted (backward directed) and a blue-shifted (forward directed) photon.
So, if the photons, as seen from the lab frame, have different wavelengths, they cannot
have equal and opposite momenta! How is this puzzle reconciled? With the idea that
from the lab frame there is a small amount of missing momentum.
Your turn
4. By applying the Relativistic Doppler formula, determine the difference in
momentum between the blue and red-shifted photons.
The resolution of this puzzle is to conclude that as small amount of mass must have also
been carried away by the photons! To see this, return to p  0 and write this as
  mv    m v  m  v  . The term Δv must equal zero because the behaviour of the
rocket must be something that observers in both frames agree upon. The rocket's speed
did not change otherwise the rocket frame observer would experience a "lurch" or force
when the photon cannon was fired.
The missing mass term Δmv can be equated to the difference in momentum between
the blue and red photons according to the expression:
Start with the blue and red-shifted wavelengths:
blue
v2
v2
1

c 2  and  
c2  .

0
red
v
v 0
1
1
c
c
1
The difference in wavelength is:
red  blue
v


2
2
v2  1
1 
c  and, if v << c, 1  v  1 .
 1 2 



 0
0
c2
c  1 v 1 v 
v2
1 2
c
 c
c
From the rocket frame the difference in momentum carried by the two photons is:
p photons 
h
blue

h
red
   blue 
 h  red
.
 blue red 
Your turn
5. Show by algebraic manipulation of the above formulas that for low velocity (v <
0.1c) the missing momentum is
p photons  h
2v
0 .
c
If the missing momentum is carried off as mass then it follows that which becomes
mv 
E
2hf 0 v
and simplifying you get m  2 .
2
c
c
Your turn
6. Justify the last steps in previous two equations that were used to conclude that
m 
E
c2
How did Einstein resolve the puzzle? He put forward a breathtaking idea—the missing
momentum was carried by the mass of the photons and the photons have mass only
because they have energy. In other words, mass and energy are no longer independent
ideas but, in a very deep sense, equivalent!