In Star Trek: The Next Generation, Romulan warbirds use artificial controlled singularities as their primary power source.
Is there an in-universe explanation how this actually works and/or how the Romulans generate energy from a singularity?
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The TNG Writer's Manual (4th Season) makes it clear that Romulan spacecraft generate energy by means of harnessing the x-ray emissions from a captive quantum singularity.
In the TNG : "Timescape" we see both the exterior and a partial interior of a Romulan engine core. The singularity itself is missing but this still allows us to make some reasonable assumptions based on what remains:
The first answer is the formation of virtual particles in the space surrounding the singularity (as proposed in the real world by Dr. Stephen Hawking). These virtual particles can be interacted with special fluids or targets to generate ions for thrust or power for Romulan Warbirds.
The second answer is the 'Seeping effect' of energy described in many science fiction universes. To create a singularity, you need certain amount of energy (insane amount of energy if you consider our Universe, but achievable in Star Trek universe), this makes the singularity artificially made one. The formation of a singularity actually opens up a portal through space and time, (as we have seen in Star Trek reboot). The energy (in the form of gravitons seeping in from the other passage must be great enough to be successfully converted to thrust and power for Romulan Warbirds. Like the immense flow of water through a hole in the sink, the energy seeps in and can be harnessed.