Does a gravitating body have to undergo gravitational collapse?

[Agerhell]

But can it really escape? Photons going in outward direction are redshifted but photons going inward are blueshifted. So it seems that inside of the body can not really cool down.
Of course photons will escape from surface of the body but then there are photons coming from the rest of the universe that would be highly blushifted.

I find your temperature discussion interesting. If we assume that we have an extremely dense planet such as that light originating from the surface of that body is redshifted by a factor of two (twice as long wavelenghts) as compared to light originating from a place were the gravitational effects are miniscule. What actually would happen is that people living on the surface of that planet will think that the temperature of the universe is not 2.7 Kelvin but 5.4 Kelvin. This as all electromagnetic waves from space will appear blueshifted by a factor of two which as Wien's displacement law shows will make the people on the planet think the temperature is twice as high. Do note that even if the light appears blueshifted on the surface of the planet it will not magically gain any energy from somewhere traveling down to the planet.

So the people on the planet will measure their temperature as 5.4 Kelvin, in thermal balance with the cosmic background radiation, but as the thermal radiation from the planet reaches a distant observer the observer will think that the planets surface temperature is only 2.7 Kelvin, in balance with the cosmic background radiation, because of the redshift. So both the distant observer and the planets inhibitants will think that the planet has the same temperature as the cosmic backgrund radiation, however they will disagree on the temperature of the cosmic background. Was this what you were wondering?

There are some problems with this explanation. For instance Stefan-Boltsmanns law states that the radiated power should go as the temperature to the power of four... That means that the people at the surface of the planet, that measures their temperature as 5.4 Kelvin would expect them to radiate 16 times as much energy then if the temperature had been 2.7 Kelvin. If the Stefan-Boltzmanns law holds for the people on the planet then the distant observer should be surprised that the planet radiates 8 times as much energy as a black-bodyradiator at 2.7 Kelvin should... Hmmm now I am somewhat confused...

 

[PeterDonis]

This simply does not make sense. You can't have zero pressure if you have non-zero kinetic energy (I mean "internal" KE, not KE of bulk motion).

Maybe you mean pressure from repulsion like degeneracy pressure?

No, I was talking about ordinary "kinetic" pressure. The key thing is that, in the simple form of the virial theorem that you stated, all the particles have to move on geodesics; that is, they must all be in free fall. That's what I meant by "moving solely under the influence of gravity". But if there is pressure present, then at least some of the particles can't be moving on geodesics; they can't be in free fall. The particles that make up the Earth, for example, are not in free fall (except, possibly, at the very center).

Conversely, if we consider, say, the Solar System, with the Sun and each of the planets, moons, asteroids, comets, etc. as a "particle", then to the extent we can view the system as a whole as a "fluid" at all, it is a fluid with zero pressure, because all of the "particles" are in free fall. None of them are "pushing on" any others the way individual pieces of the Earth push on each other. Perhaps it is better to say, in cases like the Solar System, that the model of the system as a "fluid" breaks down, rather than that it is a "fluid" with zero pressure. The reason the latter term is used is that the stress-energy tensor of the system as a whole, as used for example in matter-dominated FRW models in cosmology, is the same as the stress-energy tensor of a fluid with zero pressure.

 

[PeterDonis]

Stefan-Boltsmanns law states that the radiated power should go as the temperature to the power of four... That means that the people at the surface of the planet, that measures their temperature as 5.4 Kelvin would expect them to radiate 16 times as much energy then if the temperature had been 2.7 Kelvin. If the Stefan-Boltzmanns law holds for the people on the planet then the distant observer should be surprised that the planet radiates 8 times as much energy as a black-bodyradiator at 2.7 Kelvin should... Hmmm now I am somewhat confused...

The gravitational redshift/blueshift factor goes as the inverse square root of the radius. The radiated power goes as the inverse area of the sphere as a function of radius, i.e., as the inverse square of the radius. So the radiated power, measured at any given radius, goes as the fourth power of the temperature at that radius, adjusted for redshift/blueshift. Everything fits together fine.

 

[Agerhell]

The gravitational redshift/blueshift factor goes as the inverse square root of the radius. The radiated power goes as the inverse area of the sphere as a function of radius, i.e., as the inverse square of the radius. So the radiated power, measured at any given radius, goes as the fourth power of the temperature at that radius, adjusted for redshift/blueshift. Everything fits together fine.

I missed the time dilation. As the distant observer and the planets inhabitants have different measures of time as well as of frequency, a certain amount of energy radiated out from the planet per time unit in this particular example will be perceived as only a fourth of that amount of energy per time unit for the distant observer.

Does the planets inhabitants and the distant oberserver somehow disagree on the planets surface area, is that what you are saying? If you have some "gravitational length contraction" which is the same factor as the redshift and the time-dilation then the puzzle would be solved...

 

[PeterDonis]

I missed the time dilation. As the distant observer and the planets inhabitants have different measures of time as well as of frequency, a certain amount of energy radiated out from the planet per time unit in this particular example will be perceived as only a fourth of that amount of energy per time unit for the distant observer.

Does the planets inhabitants and the distant oberserver somehow disagree on the planets surface area, is that what you are saying? If you have some "gravitational length contraction" which is the same factor as the redshift and the time-dilation then the puzzle would be solved...

No, the surface area of the planet looks the same to the inhabitants and the distant observers. There is no puzzle; you agreed to the resolution yourself in the first paragraph of the above quote: "a certain amount of energy radiated out from the planet per time unit in this particular example will be perceived as only a fourth of that amount of energy per time unit for the distant observer." I.e., the observed energy radiated matches the fourth power of the observed temperature: one-fourth the energy for twice the distance for radiated power; 1/sqrt(2) the temperature for twice the distance due to gravitational redshift. One-fourth is the fourth power of 1/sqrt(2).

 

[zonde]

There are some problems with this explanation. For instance Stefan-Boltsmanns law states that the radiated power should go as the temperature to the power of four... That means that the people at the surface of the planet, that measures their temperature as 5.4 Kelvin would expect them to radiate 16 times as much energy then if the temperature had been 2.7 Kelvin. If the Stefan-Boltzmanns law holds for the people on the planet then the distant observer should be surprised that the planet radiates 8 times as much energy as a black-bodyradiator at 2.7 Kelvin should... Hmmm now I am somewhat confused...

Hmm, I think that most straight forward explanation is hidden in units of http://en.wikipedia.org/wiki/Stefan%E2%80%93Boltzmann_constant" . In base units it has seconds cubed in denominator. So it should be reduced (divided) by time dilation factor cubed. That can account for apparent reduction of intensity by factor of 2^3.