Scenario: a spaceship is travelling at a high fraction of $c$. The interstellar gas and CMB radiation has blueshifted significantly and we are facing a possible melting of the front radiation shield!
but the ship has good radiator area in the back so there might be a significant temperature difference between the back and the front shield.
Question: can we use this thermal differential to impulse the ship without expending more fuel? or all we can do is slow down the radiation drag?
Answer
Nice thought experiment!
The most optimistic scenario would be that all of the incident radiation on the front of the ship were light (that way there is no energy stuck in the form of rest-mass), and that it was all (somehow) captured (e.g. 100% efficient solar-panels).
It's easy (see: energy-momentum relation) to show that converting all of that energy to thrust would then exactly cancel-out the drag-force of radiation pressure. (You should try this as an exercise)
A heat engine, on the other-hand, is always going to be less efficient (see: heat-engine efficiency), and will thus be unable to recuperate the drag-losses.
But! What if your entire ship had (perfectly efficient) solar panels capturing CMB energy on all sides? Could you accelerate then?
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