Thermodynamics of Black Holes: Analyzing Carnot Cycles

[PeteSampras]

Hello,

I did read one paper about the Carnot cycle in a black hole. https://arxiv.org/pdf/1404.5982.pdf

After formula (15) this paper says:

1) "The vanishing of CV is the “isochore equals adiabat” result, specific to static black holes, making our Carnot cycles particularly simple to make explicit. We can put a Carnot cycle on the diagram by picking two isotherms for TH and TC, and then dropping two vertical lines between them to close the loop as we did in figure 2."

The vanishing of CV implies that there are isochores curves?

2) "Actually, an explicit expression for Cp would suggest that we ought to have a new engine that we can analyze simply, involving two isobars and two isochores/adiabats"

Cp non zero implies that there are isobars curves.

¿there is someone that understand these two arguments?

 

[PeterDonis]

The vanishing of CV implies that there are isochores curves?

That's not what the paper says. The paper says that, for a static black hole, isochoric (constant volume) curves are the same as adiabatic (constant entropy) curves. As the paper says, this is because, for a static black hole, entropy and volume (more precisely the "volume" that appears in the extended black hole thermodynamic equations given in the paper) both depend on the horizon radius, so either one being constant requires the other to be constant as well (since it means the horizon radius is constant).

Cp non zero implies that there are isobars curves.

That's not what the paper says. It says that if you have an explicit expression for $C_p$, it's easy to analyze an engine where two sides of the loop are isobars (constant pressure curves).