I studied theoretical physics and read Tolman’s book almost 30 years ago before switching to programming. After that I have been following the topic as a hobby.
As I understand, the math behind a particular generalization of thermodynamics that Tolman had used was sound. But there are other ways to try to merge the notion of temperature with GR that are considered more compatible with current attempts to merge gravity and Quantum Mechanics. We do not know which one is right.
In Tolman’s approach one can not have thermal equilibrium in a stationary thermally isolated gravitation body if the temperature is constant. Isolated here is important. Think as if a star is surrounded by a membrane that no heat can penetrate and consider the resulting temperature profile.
Extremely simplifying to the point of risking to give wrong impression the gravity has inherent energy gradient itself and to get isolated stationary system without energy flow the temperature must show a gradient as well.
Tolman himself did not try to apply that to black holes, but it is straightforward to apply his equations in that case and that gives the absolute zero temperature at the event horizon. Again extremely simplifying in Tolman’s approach a black hole is a gigantic freezer from a point of view of external observer. That neatly solves the apparent disappearance of entropy. There is none as approaching the event horizon from a point of the external observer means loosing temperature. So entropy is lost before the horizon and there is no information paradox.
Edit: for a star in a thermally isolated membrane Tolman’s approach gives that with a thermal equilibrium the temperature drops towards the center as the gravity is stronger there. Although that may sound counterintuitive, the idea is that stronger gravity slows down the time and from a point of an external observer that means colder temperature. And at black hole horizon time stops completely as seen by the external observer so the temperature is zero.
As I understand, the math behind a particular generalization of thermodynamics that Tolman had used was sound. But there are other ways to try to merge the notion of temperature with GR that are considered more compatible with current attempts to merge gravity and Quantum Mechanics. We do not know which one is right.
In Tolman’s approach one can not have thermal equilibrium in a stationary thermally isolated gravitation body if the temperature is constant. Isolated here is important. Think as if a star is surrounded by a membrane that no heat can penetrate and consider the resulting temperature profile.
Extremely simplifying to the point of risking to give wrong impression the gravity has inherent energy gradient itself and to get isolated stationary system without energy flow the temperature must show a gradient as well.
Tolman himself did not try to apply that to black holes, but it is straightforward to apply his equations in that case and that gives the absolute zero temperature at the event horizon. Again extremely simplifying in Tolman’s approach a black hole is a gigantic freezer from a point of view of external observer. That neatly solves the apparent disappearance of entropy. There is none as approaching the event horizon from a point of the external observer means loosing temperature. So entropy is lost before the horizon and there is no information paradox.
Edit: for a star in a thermally isolated membrane Tolman’s approach gives that with a thermal equilibrium the temperature drops towards the center as the gravity is stronger there. Although that may sound counterintuitive, the idea is that stronger gravity slows down the time and from a point of an external observer that means colder temperature. And at black hole horizon time stops completely as seen by the external observer so the temperature is zero.