The Physics of De Sitter Black Hole Heat Capacity

[PeteSampras]

I am reading the paper https://arxiv.org/pdf/0801.4591.pdf ,

says "Fortunately, the fact that the heat capacity of the cosmological horizon be positive permits to foresee the evolution of the space in absence of any external source. Taking the correct signs for the temperatures one can notice that, for a given value of M, T+ > T++. Therefore, during their interaction due to its positive heat capacity the cosmological horizon would increase its temperature, and so its radius. Conversely, the black hole horizon would become even hotter because of its negative heat capacity and shrink. In this way, there should be a net flux of energy from the black hole horizon into cosmological horizon. "

where T+, and T++ are the temperature of the black hole and cosmological horizon, respectively.

Why the fact that T+>T++ and C++>0 implies that the cosmological horizon radii increase?, and
Why the fact that T+>T++ and C+<0 implies that the black hole horizon radii shrink?,

I think in a scheme as the figure.

 

[PeterDonis]

Why the fact that T+>T++ and C++>0 implies that the cosmological horizon radii increase?

The cosmological horizon radius increases if its temperature increases because C++> 0. That has nothing to do with the relative temperatures of the two horizons.

Why the fact that T+>T++ and C+<0 implies that the black hole horizon radii shrink?

The black hole's horizon radius shrinks if its temperature increases because C+<0. That has nothing to do with the relative temperature of the two horizons.

The only role that T+>T++ plays in the logic is to make heat flow from the black hole to the cosmological horizon. It plays no role in determining how the radius of each horizon changes.

 

[PeterDonis]

The cosmological horizon radius increases if its temperature increases because C++> 0.

The black hole's horizon radius shrinks if its temperature increases because C+<0.

To expand on this a bit: we have a situation in which T+ > T++, which means heat flows from the black hole to the cosmological horizon. That means the heat flow into the cosmological horizon, which I'll call Q++, is positive. Then the fact that C++ is also positive means that T++ increases.

Also, the heat flow into the black hole, which I'll call Q+, is negative, because heat is flowing out of the black hole. Then the fact that C+ is also negative means that T+ increases.