Holographic heat engines for Schwarzschild black holes

We construct reversible black hole heat engines in asymptotically flat spacetime using quasi-local gravitational thermodynamics. We enclose a Schwarzschild black hole in a finite spherical cavity and identify the working substance with the dual thermal system on the cavity boundary. The surface pressure and boundary area define a thermodynamic pressure-volume pair in the dual system, while all coupling constants of the gravitational theory remain fixed. We derive exact efficiencies for the Carnot, Otto, Diesel, Brayton, and Stirling engines and compare them numerically. These efficiencies probe the quasi-local equations of state of the black hole. The regenerated Stirling efficiency remains remarkably close to the Carnot bound and approaches it in the high-temperature limit.