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?
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?
Let's answer the problem for a 1D universe. You have a spaceship that is effectively a coke-can shape, with a single flat face facing the positive x-direction and another identical face facing the negative x-direction. All photons are traveling either in the positive or negative direction, all going $c$, both according to the spaceship and according to an observer stationary relative to the CMB.
Take the spaceship speed to be $v$ relative to the CMB in the positive x-direction. The photons that hit the front face of the spaceship have a blue-shifted frequency $v'=v \sqrt{ (c+v)/(c-v)}$. Reverse sign for the red-shifted frequency of the photons hitting the back face.
I would argue that for this 1D universe, photons hit the front of the spaceship at the same rate as they hit the back, but I believe this would be incorrect for a 3D (or even 2D) world. This reasoning is difficult to work out. Consider the CMB photons to be distributed according to a regular lattice. The moving spaceship considers one axis of this lattice (the only one in 1D) to be length-contracted, but this doesn't change the fact that it is still a regular lattice. All photons travel the same speed according to the spaceship, so in 1D, the front and back are subjected to the same number per unit time. In 3D, however, there are photon directions that would hit the back of a stationary spaceship, but would hit the front of a moving spaceship. Therefore, it is incorrect to extrapolate this 1D assumption to 3D, so in that respect my answer will be incorrect for 3D.
The rate of energy deposited on the front versus the back follows easily if we assume all photons are absorbed. I'll use $n$ for the number of photons deposited per unit time. Again, in 1D $n$ is the same for both faces, but this is wrong in 3D.
$$\dot{Q}_f = n h v' = n h \sqrt{ \frac{c+v}{c-v} }$$ $$\dot{Q}_b = n h \sqrt{ \frac{c-v}{c+v} }$$
The force on both faces is the number absorbed per unit time times their momentum.
$$F_f = n \frac{h v'}{c} = \frac{n h}{c} \sqrt{ \frac{c+v}{c-v} }$$
Net force in 1D:
$$ F = \frac{n h}{c} \left( \sqrt{\frac{c+v}{c-v}} - \sqrt{\frac{c-v}{c+v}} \right)$$
This is in the reverse direction, which isn't a good thing. Ideally, we would like to engineer a situation that's more favorable to our spaceship. An obvious choice would be to make the back face reflect photons and the front face fully absorb them, you then get twice the forward thrust and the same reverse thrust. If your spaceship was at 0 degrees Kelvin, this would work... until you got up to 1/3 c, after which point the blue-shift overcomes the benefit and it still imparts a net force in the reverse direction.
I propose two example solutions of how the CMB could be used to a net advantage:
Proposal 1: Just make the entire front transparent. I have no idea how you could do this, but there is nothing physically preventing it. You spacecraft is completely passive and just a 2-way mirror. If someone knows of a principle preventing this let me know in the comments.
Proposal 2: For a more clunky solution, make the front face an absorptive face, the rear face a reflective face at the CMB temperature, an absorptive surface at the temperature its at, and use a heat pump to lower the temperature of the front face and raise the temperature of the rear face. If you consider 3D dynamics, you could potentially release the reverse thrust in a directional way to get better thrust than if you just emitted your on-board matter-energy in the form of photons from the rear face. This solution would be limited by the efficiency of your heat pump.
At hyper-relativistic speeds $v\approx c$, the energy and force from the back face is negligible compared to the front-face, so even theoretically there's no benefit you can get from the CMB whatsoever, so just give up.
