Potential energy and the gravitational field of a collapsing cloud of gas

[Herbascious J]

Imagine an empty void of intergalactic space. In this space there is a cloud of diffuse gas, mostly hydrogen and helium. The gas is non-rotating and very cold just above absolute zero. There is nothing else around this cloud, and so it has a clear center of gravity, and no other objects influencing it gravitationally. The cloud has a precise mass that is well determined and does not change. The cloud has a gravitational field that can be detected by an observer far away from the cloud in the surrounding empty space. I am going to assume the gravitational field of this cloud is described by general relativity and so its gravitational field arises from its energy content.

Imagine the cloud begins to collapse under its own gravity. As it does so, the gas begins to move inward very quickly and heat up aggressively. Eventually, all this gas settles down into a compact object that might be spinning, but more importantly the object is considerably hotter. After this takes place, the object emits infra-red radiation in all directions and loses heat in the form of photons. Over a long period of time the body radiates away much of its heat and cools back down to a temperature similar to how it began when it was a diffuse cloud. The energy radiated away is lost to deep space and is no longer considered part of the local system being described and beyond the detection of any observers.

It would appear that the radiation is a loss of energy of the compact object, and because of this, the object will be slightly less massive after cooling down. My question is; does this mean that the object has a slightly weaker gravitational field after it cools down and looses energy? If so, how does this relate to the object’s gravitational field before it collapses while it is still a cool, diffuse, cloud of gas? I imagine this in two ways; one, as the cloud of gas collapses, the molecules speed up and begin to heat up and this makes the cloud more energetic, so the gravitational field of the cloud begins to increase due to all of this new energy in the form of momentum and temperature. This increase in gravitational field would be slight, but detectable by a far away observer in theory. As the object cools back down, after collapse, this increase in energy would lost through radiation and so its gravitational field would return to what it was before.

The second way I imagine this, is that as the cloud collapses, the gas heats up, but this does not change the over-all energy of the cloud because it is something like a closed system, and therefore the potential energy of the cloud in its diffuse, cool state contributes to the over all energy in the beginning, and therefore the gravitational field does not change for outside observers as the cloud collapses and heats up.

However, as the compact object cools as described above, we do have a loss of heat and energy from the system, so the gravitational field should then go down slightly. This is strange to me because the exact same amount of gas is constant through out this thought experiment, and in the end, the temperature returns to its initial level. The only difference is that it starts as a cloud and ends as a compact object. But somehow the gravitational field is slightly weaker. What is the correct interpretation of this thought experiment according to general relativity? Does potential energy have a gravitational field associated with it that contributes to the system over all? I hope that makes sense.

 

[PeterDonis]

Eventually, all this gas settles down into a compact object that might be spinning

No, it can't be, since by your specification the original cloud had zero angular momentum and the cloud is not exchanging angular momentum with anything else.

It would appear that the radiation is a loss of energy of the compact object, and because of this, the object will be slightly less massive after cooling down.

Yes.

does this mean that the object has a slightly weaker gravitational field after it cools down and looses energy?

This question is not well posed, because there is no single number that describes the "gravitational field" as "stronger" or "weaker". See further comments below.

A well posed question would be: does this mean the object has a smaller mass after it cools down and loses energy? The answer to that is yes. Once you allow radiation to escape to infinity, the overall spacetime is no longer that of an "isolated object", so your intuition that the overall mass of an "isolated object" cannot change no longer applies. The object has indeed lost mass because of the energy carried away by radiation.

The more technical way of stating the above is that, once you have radiation escaping to infinity, the Bondi mass of the spacetime is no longer equal to the ADM mass; it is less, because it does not include the energy contained in the radiation that escapes to infinity. The Bondi mass is what generates what you are intuitively thinking of as the "gravitational field" of the object itself (not including the radiation). The ADM mass, which does remain constant for this scenario, includes the energy contained in the radiation, and is what you would use to ground an intuition that "the overall mass does not change" for this scenario--the "overall mass" has to include the energy in the radiation, but once that radiation has escaped, someone measuring the mass of the object alone won't see that energy.

I imagine this in two ways; one, as the cloud of gas collapses, the molecules speed up and begin to heat up and this makes the cloud more energetic, so the gravitational field of the cloud begins to increase due to all of this new energy in the form of momentum and temperature.

No, this is not correct. During the collapse process itself, if we idealize it so that no radiation at all escapes to infinity until it is all done, does not change the externally measured mass of the system at all. All of the stuff you are thinking of, which increases the kinetic energy of the object, is canceled out by an equal decrease in the gravitational potential energy as the object becomes more compact. In the technical language I gave above, during the collapse, the ADM mass and the Bondi mass remain the same.

as the cloud collapses, the gas heats up, but this does not change the over-all energy of the cloud because it is something like a closed system, and therefore the potential energy of the cloud in its diffuse, cool state contributes to the over all energy in the beginning, and therefore the gravitational field does not change for outside observers as the cloud collapses and heats up.

This is correct. The externally measured mass of the cloud will not change during the collapse (if, as I said above, we idealize the collapse so no radiation is emitted while it is in progress).

However, as the cloud collapses and becomes more compact, it becomes possible for an external observer to reach regions where the "gravitational field" is stronger than they could reach before, because the object is more compact and so it is possible to get closer to its center while still remaining outside it, and "closer to its center" means "stronger gravitational field". And even after the final object has radiated away its excess energy, it will still be much more compact, so the "gravitational field" at its surface will still be stronger than the "gravitational field" at the surface of the original cloud.

So, as noted above, it doesn't make sense to ask whether the gravitational field after the collapse is done and all of the excess energy is radiated away is "stronger" or "weaker" than at the start. In one sense it's "weaker" because the mass of the object itself has decreased. But in another sense it's "stronger" because an external observer can now reach regions much closer to the center than before.

as the compact object cools as described above, we do have a loss of heat and energy from the system, so the gravitational field should then go down slightly.

Yes, this is correct.

This is strange to me because the exact same amount of gas is constant through out this thought experiment, and in the end, the temperature returns to its initial level.

Yes, but the final object is much more compact than the original cloud. That means it has significant (negative) gravitational potential energy (or "gravitational binding energy" might be a better term), whereas the original cloud had zero (or at least negligible) gravitational potential energy. That makes a negative contribution to the mass of the object (which, as noted above, is balanced by the positive energy carried away by radiation, so the overall energy in the spacetime, the ADM mass, is constant).

The only difference is that it starts as a cloud and ends as a compact object.

Yes, but that difference is significant. See above.

But somehow the gravitational field is slightly weaker.

Depends on how you look at it. See above.