Thermal radiation during black hole collapse

[nickyrtr]

My question is, when a compact star collapses to form a black hole, how does it manage to dispose of the heat necessary to make this transition?

Naively, it would seem that the object can never cool itself fast enough due to increasing gravitational redshift of its thermal radiation. Each radiated photon's energy is reduced by a factor of 1/(1+z) and the apparent rate of photon emission is reduced by 1/(1+z), so the total energy flux out of the object is reduced by a factor of 1/(1+z)^2.

Furthermore, the solid angle through which each surface element's radiation can escape to space decreases with redshift, as more and more emitted light is bent back inward toward the object. If I'm figuring right, that solid angle is reduced by a factor of 1/(1+z)^2 in the limit of high z. That makes a total factor of 1/(1+z)^4 reducing the radiative energy flux out of the object.

Once z is high enough that energy flux out of the object is equal to incident energy flux from distant stars and the cosmic background, it would seem that the object cannot cool itself any more. Unless there is an endothermic reaction of some sort taking place, I don't see how the star could contract further without expelling heat into space, so it would seem that the collapse cannot proceed beyond this point. Thus z remains finite and an event horizon never forms.

Having read some of Abhas Mitra's papers and critiques of them, the consensus appears to be that black hole formation is NOT forbidden by thermodynamics, so what is wrong with the above argument?

 

[Naty1]

You may have a fine point I'm not interpreting, but let's start this way:

I don't see how the star could contract further without expelling heat into space, so it would seem that the collapse cannot proceed beyond this point.

The point, I believe, is the opposite: after sufficient energy and heat is expelled from a stellar object, after billions upon billions of years when it's nuclear fuel is largely used, then gravity has sufficient strength to cause collapse of the remaining stellar constituents.

On the other hand, degenerate matter might prevent the formation of a black hole with insufficient mass. The Chandrasekhar limit limits the mass of bodies made from electron-degenerate matter, about 1.4 solar masses. As white dwarfs are composed of electron-degenerate matter, no nonrotating white dwarf can be heavier than the Chandrasekhar limit.

Once an event horizon forms, heat is expelled slowly via Hawking (thermal) radiation...a quantum mechanical effect...with a black body like radiation temperature inversely proportional to the black hole mass. Oddly, perhaps, smaller black holes are hotter.

Wikipedia explains Hawking radiation:
"Physical insight on the process may be gained by imagining that particle-antiparticle radiation is emitted from just beyond the event horizon. This radiation does not come directly from the black hole itself, but rather is a result of virtual particles being "boosted" by the black hole's gravitation into becoming real particles."

http://en.wikipedia.org/wiki/Hawking_radiation#Overview

 

[nickyrtr]

Naty1, thanks for replying, but I think I didn't explain my question very well. I'm talking about a compact star massive enough to overcome neutron degeneracy pressure. The textbook explanation of this object's fate is that it collapses within its own Schwarzschild radius (r_s) and becomes a black hole, with an event horizon and singularity.

If the object collapses gradually, it remains in hydrostatic equilibrium, so the object's pressure, temperature, density and radius are all uniquely determined by its internal energy. Therefore, in order for its radius to change, its internal energy must change. Assuming also there are no available nuclear/particle reactions, the only way the object's internal energy can change is by exchanging radiation with space. Therefore, in order for the object to contract (or expand) it must emit (or absorb) radiation.

Now as the object's radius contracts closer to r_s, the radiation it emits is increasingly redshifted, as seen by a distant observer, and a greater fraction of it is bent backward and reabsorbed by the object, never escaping to space. Thus the rate at which the object can shed internal energy approaches zero as its radius approaches r_s. At the same time, radiation is falling into the object, from starlight and the cosmic microwave background. The energy flux on the object from these faint sources is small, but it is still finite, therefore the object's net rate of energy loss becomes zero at some radius r > r_s.

All this would say that an event horizon never forms, not even after an infinite time, but the consensus in the literature is that a black hole CAN form in finite time. Unfortunately, I found I could not understand the general relativistic derivations for this result, which would probably reveal the mistake in the above argument. I am hoping that by posting it here, someone better versed in GR can identify the error.

 

[stevebd1]

Hello nickyrtr. You seem to be implying that a neutron star might voluntarily collapse into a black hole with the aid of cosmic radiation. A rapidly rotating neutron star might voluntarily collapse into a BH as the spin reduces (over millions of years) as the oblate spheriod shape reduces back to a sphere, reducing the internal volume and increasing the density but in most cases, neutron stars collapse into BH's because they accrete matter from an adjacent star which pushes them over the 2.5-3 sol mass limit. Most stellar black holes observed are part of a binary system-

http://en.wikipedia.org/wiki/Stellar_black_hole

 

[Dmitry67]

If the object collapses gradually, it remains in hydrostatic equilibrium, so the object's pressure, temperature, density and radius are all uniquely determined by its internal energy.

Collapse is a violent process. Collapsing object is not in an equilibrium - even more, you can not (even theoretically) define static observers inside the black hole. It is slow only for the observes outside - but not because of any equilibrium but because of the time dilation. For an observer, standing on a surface of a neutron star the process would be sudden and it takes just a little bit longer then it takes for light to cross the diameter of the star. For the neutron star (10km) it is abour 30microseconds...

Regarding the duration of BH collapse (how BH can ever form as everything is frozen in time at the horizon?) it is a different (and long) story...

 

[nickyrtr]

Collapse is a violent process. Collapsing object is not in an equilibrium - even more, you can not (even theoretically) define static observers inside the black hole. It is slow only for the observes outside - but not because of any equilibrium but because of the time dilation. For an observer, standing on a surface of a neutron star the process would be sudden and it takes just a little bit longer then it takes for light to cross the diameter of the star. For the neutron star (10km) it is abour 30microseconds..

If I understand correctly, you mean that kinetic energy is not small compared to thermal and gravitational energy, so the object surface is carried past r_s by its own inward momentum. Does this mean that the object cannot collapse into a BH if the star matter has too much internal friction? (more friction meaning the object state is closer to equilibrium)

 

[skeptic2]

Regarding the duration of BH collapse (how BH can ever form as everything is frozen in time at the horizon?) it is a different (and long) story...

I would very much like to hear the long story. Perhaps you would consider continuing the discussion at https://www.physicsforums.com/showthread.php?t=274512&page=6

 

[Naty1]

I have two books for reading this summer: Black Holes and Time Warps by Stephen HAwking and Kip Thorn, and The Black Hole War by Leonard Susskind.

A quick read in a short section of chapter 3 of the former matches my posted description above...

I would think either of those two books will answer you question...

If the object collapses gradually, it remains in hydrostatic equilibrium,

That sounds rather self contradictory..in any case I do not know what you mean.

 

[Dmitry67]

If I understand correctly, you mean that kinetic energy is not small compared to thermal and gravitational energy, so the object surface is carried past r_s by its own inward momentum. Does this mean that the object cannot collapse into a BH if the star matter has too much internal friction? (more friction meaning the object state is closer to equilibrium)

No friction can "save" the collapsing object.

The object surface is not 'carried'... Well, there are some nice diagrams which show the structure of the spacetime inside the black hole.

Remember that in GR gravitational force is an 'imaginary' force, a results of a curvature of spacetime. The spacetime inside BH is so curved towards the center, so the famous Minkovsky 'hourglass' is turned ore then 45 degrees. For a freely falling observer inside the black hole, the center of the black hole is not 'somewhere', but in the FUTURE.

So, as no friction, pressure etc can prevent matter from reaching 4pm today, nothing can prevent it from falling into singularity.

here are some nice pictures with these lightcones: http://www.phy.syr.edu/courses/modules/LIGHTCONE/schwarzschild.html

 

[nickyrtr]

Naty1, let me be more precise. By "collapsing gradually" I mean

M(dR/dt)² << NkT

Where M is the compact star's mass, N is its number of particles, R is its radius, t is time as measured by an observer on the star's surface, and T is temperature as measured by the same observer.

..So, as no friction, pressure etc can prevent matter from reaching 4pm today, nothing can prevent it from falling into singularity.

here are some nice pictures with these lightcones: http://www.phy.syr.edu/courses/modules/LIGHTCONE/schwarzschild.html

I thought this was only true after the event horizon has formed. If an event horizon does not yet exist, it is still possible that some force could stop the collapse, correct?

 

[Dmitry67]

If an event horizon does not yet exist, it is still possible that some force could stop the collapse, correct?

Yes, correct

 

[skeptic2]

If the neutron degeneracy pressure could no longer support a neutron star and it began to collapse, even if the radius were greater than Rs, what force could halt the collapse?

 

[nickyrtr]

If the neutron degeneracy pressure could no longer support a neutron star and it began to collapse, even if the radius were greater than Rs, what force could halt the collapse?

The object's own thermal pressure could resist collapse, or so I was wondering.

That's how main sequence stars support themselves against collapse; heat lost by emitting radiation is balanced by heat generated by nuclear reactions, so the star's temperature (and hence its radius) remains nearly constant.

In the case of an overmassive neutron star, hardly any heat is generated by internal reactions, but as R approaches Rs, it can hardly emit any radiation either. Also, when the emitted radiation flux becomes very weak, it is balanced by incoming radiation from elsewhere in the universe, so the object is emitting no net radiation at all. By analogy to the main sequence star, this would seem to suggest that the object no longer contracts.

However, Dmitry's point (if I understand it) is that a collapsing, overmassive neutron star is not in dynamical equilibrium, unlike a main sequence star. That would mean that a constituent neutron's motion is more like free fall than thermal, brownian motion.

My followup question, then, is this: does black hole formation require a constraint on neutron-neutron interactions? e.g., that the net interaction force on an infalling neutron is much less than the gravitational force on the same particle. This is what I meant by asking about "internal friction".

 

[stevebd1]

If I understand correctly, you mean that kinetic energy is not small compared to thermal and gravitational energy, so the object surface is carried past r_s by its own inward momentum. Does this mean that the object cannot collapse into a BH if the star matter has too much internal friction? (more friction meaning the object state is closer to equilibrium)

This has probably already been implied but it's worth noting that a black hole can initially form within the core of the neutron star first where the matter is more compressed (i.e. the entire mass as a whole does not need to fall within the specific Schwarzschild radius for the neutron star). The core of the neutron star might be GQP which can be anything upwards of 2e+18 kg/m^3. If an average density of ~2e+26 kg/m^3 is attained within a 2 metre diameter sphere within the core of the neutron star then the core will collapse into a black hole, the event horizon will then work its way through the layers of the NS to the surface.

 

[nickyrtr]

This has probably already been implied but it's worth noting that a black hole can initially form within the core of the neutron star first where the matter is more compressed (i.e. the entire mass as a whole does not need to fall within the specific Schwarzschild radius for the neutron star). The core of the neutron star might be GQP which can be anything upwards of 2e+18 kg/m^3. If an average density of ~2e+26 kg/m^3 is attained within a 2 metre diameter sphere within the core of the neutron star then the core will collapse into a black hole, the event horizon will then work its way through the layers of the NS to the surface.

Ah, so then radiation is not the only relevant form of heat transfer. The contracting core could also cool by conduction or convection, exchanging heat with the star's upper layers. Perhaps these forms of heat exchange are not as attenuated by extreme gravity as radiation?

 

[Dmitry67]

In the case of an overmassive neutron star, hardly any heat is generated by internal reactions, but as R approaches Rs, it can hardly emit any radiation either. Also, when the emitted radiation flux becomes very weak, it is balanced by incoming radiation from elsewhere in the universe, so the object is emitting no net radiation at all. By analogy to the main sequence star, this would seem to suggest that the object no longer contracts.

However, Dmitry's point (if I understand it) is that a collapsing, overmassive neutron star is not in dynamical equilibrium, unlike a main sequence star. That would mean that a constituent neutron's motion is more like free fall than thermal, brownian motion.

My followup question, then, is this: does black hole formation require a constraint on neutron-neutron interactions? e.g., that the net interaction force on an infalling neutron is much less than the gravitational force on the same particle. This is what I meant by asking about "internal friction".

No heat is needed for the White Dwarfs, neutron stars (and quark- and strangelets- stars) to be stable. The gravitational force is balanced by the pressure of the degenerate matter. The pressure of the degenerate matter does not depend on the temperature at all (until that temparature is not high enough to remove the degeneration)

I still don't understand your question. Before the formation of the horizon, d9ifferent forces can slow down or stop the collapse. After it is inevitable.

Also, note that the S.radius is proportional to mass M, while the volume is poportional to M**3. Hence, you can create a super-massive black hole you don't need to have any super-duper condensed matter at all. Theoretically, you can make a BH from water, air, etc...

 

[skeptic2]

If matter, from the inertial frame of a distant observer, cannot cross the event horizon due to both time dilation and space contraction, doesn't the event horizon itself halt the collapse into a black hole?

 

[nickyrtr]

No heat is needed for the White Dwarfs, neutron stars (and quark- and strangelets- stars) to be stable. The gravitational force is balanced by the pressure of the degenerate matter. The pressure of the degenerate matter does not depend on the temperature at all (until that temparature is not high enough to remove the degeneration)...

Either degeneracy or heat can support a star against collapse. A main sequence star is supported mainly by heat, whereas a white dwarf or neutron star (or other proposed quark stars, etc.) is supported mainly by degeneracy. However, in all cases, the total pressure is a sum, something like this:

P_total = P_radiation + P_thermal + P_degeneracy - P_gravity

If this total pressure is positive, the star surface is accelerated outward. If negative, the star surface is accelerated inward. The Chandrasekhar limit, then, could be written as:

P_gravity > P_degeneracy

As you say, degeneracy pressure does not depend on temperature, and since gravity also does not depend on temperature, the above inequality is not affected by temperature change.

But now consider the contribution of thermal and radiation pressure. Gravity must also overcome these if P_total < 0, and these terms vary with temperature. Has it been shown that thermal and radiation pressure cannot overcome gravity as an object's radius approaches R_s? If so, can the argument be made within the Schwarzschild framework, or is a more complete GR treatment required?

 

[Dmitry67]

Again, it is a long story. In our world we say 'it exists', 'it existed', 'it will exist' based on our intuitive (classical) vision of the flat spacetime. But that fails if we talk about the BH.

So does BH exist NOW? The problem is that the answer depends on how NOW is defined.

There is some logic in a claim 'for me, a distant observer, BH never forms'.
But if you claim that 'hence, there are no BH' it is easy to prove that this is wrong. Take a spaceship and fly into the black hole. You will hit a singualrity in a finite time

Regarding the "halt of the collapse because of the time dilation" I don't think it is a good description of what happens. Time dilation is just an observational effect. Falling observers do not feel it. Imagine that all events are recorded on a tape (like it was in VHS era) and you slow down the tape, watching a movie of, say, car crash. Does slowing down the tape and stopping it prevent a crash, recorded on it?

 

[Dmitry67]

But now consider the contribution of thermal and radiation pressure. Gravity must also overcome these if P_total < 0, and these terms vary with temperature. Has it been shown that thermal and radiation pressure cannot overcome gravity as an object's radius approaches R_s? If so, can the argument be made within the Schwarzschild framework, or is a more complete GR treatement required?

Well, if very hot White Dwarf is so close to its limit (or neutron star is close to the critical mass) then when these objects cool down the pressure slightlydecrease, and object (suddenly) collapses

The same might happen if you add some mass from the outside...

 

[stevebd1]

In GR, pressure contributes to gravity-

$$g=\rho+\frac{1}{c^2}(P_x+P_y+P_z)$$

Another remarkable feature of Einstein's equation is the pressure term: it says that not only energy density but also pressure causes gravitational attraction. This may seem to violate our intuition that pressure makes matter want to expand! Here, however, we are talking about gravitational effects of pressure, which are undetectably small in everyday circumstances...

There are a number of important situations in which ρ does not dominate over P. In a neutron star, for example, which is held up by the degeneracy pressure of the neutronium it consists of, pressure and energy density contribute comparably to the right-hand side of Einstein's equation. Moreover, above a mass of about 2 solar masses a nonrotating neutron star will inevitably collapse to form a black hole, thanks in part to the gravitational attraction caused by pressure.

http://math.ucr.edu/home/baez/einstein/einstein.pdf page 7

which means as pressure increases, the gravity increases and collapse becomes inevitable.

 

[nickyrtr]

Well, if very hot White Dwarf is so close to its limit (or neutron star is close to the critical mass) then when these objects cool down the pressure slightlydecrease, and object (suddenly) collapses...

But then the collapsing object will get hot again, will it not? When we rapidly compress any matter here on Earth, it gets hotter. Intuitively, it seems the collapsing object would behave the same; as it collapses, its temperature increases. Therefore, thermal/radiation pressure will increase, slowing the collapse. Is the thermal/radiation pressure enough to stop the collapse? I don't know, but I hope that some of you readers can answer this question.

 

[nickyrtr]

In GR, pressure contributes to gravity-

$$g=\rho+\frac{1}{c^2}(P_x+P_y+P_z)$$

Thank you, I think this is the answer I was looking for.

 

[skeptic2]

Regarding the "halt of the collapse because of the time dilation" I don't think it is a good description of what happens. Time dilation is just an observational effect. Falling observers do not feel it. Imagine that all events are recorded on a tape (like it was in VHS era) and you slow down the tape, watching a movie of, say, car crash. Does slowing down the tape and stopping it prevent a crash, recorded on it?

I think there are two different effects that are closely related and easily confused. These effects can be shown in the reference to light cones that you posted. http://www.phy.syr.edu/courses/modul...arzschild.html In the bottom diagram that shows the foolish observer falling into a black hole, if he, while still outside the horizon, were to transmit his signals at regular intervals (from the perspective of a distant observer), the distant observer would see the signals arriving with increasing intervals. At the same time (so to speak), the infalling observer would be experiencing increasing time dilation. In order for him to send the signals at regular intervals from the time frame of a distant observer, he would have to send them with decreasing intervals in his own time frame. It is easy to confuse the increasing intervals of the signals received by the distant observer with the time dilation of the infalling observer.

In addition, as the infalling observer approaches the event horizon, space is contracted and a distant observer would see him cover less and less distance in more and more time, if of course he could still see him. The effect of seeing an infalling object slow down and freeze at the horizon is due not only to the effect shown in the diagram but also to time dilation and space contraction near the horizon.

Infalling observers, as you pointed out, do not perceive either the time dilation or space contraction except as they look outward. As they approach the horizon ever so closely, time would be dilated so extremely that the black hole would literally evaporate beneath them as they continue to fall in. How then can anything cross the horizon?

 

[Dmitry67]

Infalling observers, as you pointed out, do not perceive either the time dilation or space contraction except as they look outward. As they approach the horizon ever so closely, time would be dilated so extremely that the black hole would literally evaporate beneath them as they continue to fall in. How then can anything cross the horizon?

So we are talking about BH + Hawking radiation?

Then no, falling observers hit the singularity, because in their frame BH evaporates very slowly.

For an outside observer BH evaporates (in a finite time) right at the moment when a foolish observer finally falls in (in a finite time). But I can not provide you additional details. For such advances things we need a real guru :)

 

[Naty1]

Ah, so then radiation is not the only relevant form of heat transfer. The contracting core could also cool by conduction or convection, exchanging heat with the star's upper layers. Perhaps these forms of heat exchange are not as attenuated by extreme gravity as radiation?

Sounds like a question: How does heat transfer in a plasma? A few quick searches in Wikipedia did NOT turn up anything...but
http://en.wikipedia.org/wiki/Plasma_(physics)#Temperatures

has some interesting information. If you are really interested I'd search for things like "plasma heat transfer".