Under $E=mc^2$, 1kg of matter has $9\times 10^{10}$ joules of energy. So, if I had just the light shining from $9\times 10^8 $ 100 Watt light bulbs inside a perfectly reflective box, would that light have the same amount of inertia as the 1kg of matter? This is a follow up to my similar question earlier this week about the gravity of light.
Is the gravity of light equal to the gravity of mass under $E=mc^2$?
Answer
First we might want to think how the energy of "free photon gas" changes according to an observer that changes his frame. One laser beam is traveling to the west and other similar one is traveling to the east. That is our "free photon gas". The observer is first at rest relative to the center of mass of the gas, then the observer accelerates to the east.
Well, the energy changes the same way as the energy of material objects. The details of the change are that one beam loses energy and the other beam gains more energy than the other one loses. Relativistic Doppler shift formula can be used to calculate the change.
Now let's consider light in a box. The box is a long one inside of which one laser beam is traveling to the west and another similar one is traveling to the east. The box and the beams have the same length.
In this case the photons that keep moving to the west gain energy and photons that keep moving to the east lose energy, the amount of change is the same as in the "free light" case for those particular photons. BUT according to the accelerating observer the west beam gains photons from the east beam, and the east beam loses photons to the west beam. So we can see that this trapped light gains more energy than the free light does. So trapped light gains more momentum too. So it has more inertia. In boosts things with large inertia change their momentum a lot.
So from all of that we conclude: One joule of trapped light has more inertia than one joule of free light - which has the same amount of inertia as one joule of matter.
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