Thread: General Relativity vs. thermodynamics

Let us assume a spherical body of substantial gravitating mass. Let us assume also a spherical shell, thermally insulated on the outside and of much larger radius than the body, surrounding the body and concentric with it. Let us assume that the mass of the body is such that a clock at its surface runs at 90% of the rate of a clock located at the spherical shell. Let us further assume that niether of these objects is actively generating heat, but that the system has come to thermal equilbrium. If the temperature as recorded by a thermometer located at the shell is 2000K, what will a thermometer located at the surface of the inner body read?

Both thermometers will read the same. But I suspect the setup you wanted was if you had 2 thermal conductors of equal conductivity leading out of the system 1 from the core and 1 from the shell and those two conductors were the only way for any heat to escape from the system. Then the conductor attached to the shell would release more heat per unit time in the external time frame as the coefficient of conductivity contains a time dependant function so the core conductor's coefficient of conductivity is related to the slower time curve of the greater gravitational field near the core. Therefore less heat per unit time in the external time frame can be conducted out compared to the shell conductor.

Well, actually no, although the setup you have proposed is yet another interesting approach to the problem. I was rather thinking of the fact that the radiation from the core is red-shifted with respect to that of the shell, so that when it arrives at the shell, it should be at a lower apparant temperature than when it left the core. Likewise, the radiation from the shell would be blue-shifted as it approached the core, appearing to be hotter than when it left. The conclusion would seem, therefore, to be that when equibrium is achieved, the temperatures are different.

Originally Posted by Atomic-S

Well, actually no, although the setup you have proposed is yet another interesting approach to the problem. I was rather thinking of the fact that the radiation from the core is red-shifted with respect to that of the shell, so that when it arrives at the shell, it should be at a lower apparant temperature than when it left the core. Likewise, the radiation from the shell would be blue-shifted as it approached the core, appearing to be hotter than when it left. The conclusion would seem, therefore, to be that when equilibrium is achieved, the temperatures are different.

Red or Blue shifting only apply if the object is moving away(red) or moving towards (blue) an observer at speed. It has little or no apparent temperature difference between the two since you already mentioned that both the shell and core were equal even if giving off heat. What the shell gives off in heat to space so does the core give to the shell, they would act as one in order to keep equilibrium.
What is the clock for ?? To tell the temp ?? Or how long it takes to equalize the temp ???

Red or Blue shifting only apply if the object is moving away(red) or moving towards (blue) an observer at speed.

It also applies when there exists a major difference in gravitational altitude (potential) between the 2 points.

What is the clock for ?? To tell the temp ?? Or how long it takes to equalize the temp ???

The clock is to show the time dilation between the two points. The time dilation has the effect of altering the frequency of radiation as it passes from one to the other.