High School The Singularity and Hawking radiation

[DreamLord101]

Hello everyone!
Im a newcomer, a teenager who has countless doubts with respect to relativity, quantum theory etc. But these two questions bother me the most:

1) Hawking radiation states that when the separation of a particle (eg. a photon) into charged particles happens in the event horizon, and one charged particle gets sucked in, to conserve energy, the black hole reduces some of its mass, and eventually, it evaporates and nothing remains in place.
But from what I have heard, a black hole has the singularity which has infinite mass. So when the charged particle is sucked in, shouldn't it have no effect as the singularity has infinite mass!I mean, infinity minus something is still infinity! A rather stupid question, but please clear my doubt!

2) Secondly, is the singularity in the centre of a proton sized black hole exactly similar to the singularity in the centre of a super massive black hole?
Answers would be greatly appreciated, thanks.

 

[Nugatory]

There are several things going on here.
- Hawking radiation happens at the event horizon of the black hole, which is nowhere "near" (we need the scare-quotes, but right now we don't need to go into why we need them) the singularity at the "center".
- Hawking radiation does not involve the separation of particles into pairs, one member of which gets sucked in and the other which escapes; that's a sort of "lies to children" oversimplification for people who haven't been through some serious college-level math and physics. Unfortunately, the explanation of what really does happen is fairly involved (you'll find links to Hawking's paper in some other threads here) which is why that oversimplification is so popular.
- The singularity does not have infinite mass. You may be misunderstanding popular descriptions that say that because that finite mass is packed into a single point with zero volume the singularity has infinite density. However, even that is somewhat misleading. If you take the equations of general relativity at face value and assume that they apply even under the most extreme conditions, then you do conclude that there is nothing to prevent all the mass of the black hole from collapsing down to a single "point" (sorry, that's another scare-quote) with infinite density. However, it is very likely that some other physical effects that we haven't been able to study yet comes into play under those extreme conditions and something else happens. What that might be is a subject of ongoing research (also many more threads here - search for 'quantum gravity"). This leads into the answer to your second question:

2) Secondly, is the singularity in the centre of a proton sized black hole exactly similar to the singularity in the centre of a super massive black hole?

According to the equations of general relativity, yes. A point is a point, whether the mass of a proton or the mass of a galaxy is concentrated there. However, we don't know and we won'tknow until we have a satisfactory theory of what does happen in the extreme conditions at the "center" of a black hole.

 

[PeterDonis]

A point is a point

The singularity inside a Schwarzschild black hole is not a point; it's a spacelike line. (More precisely, it's the limit of a spacelike surface composed of an infinite sequence of 2-spheres, as the radius of the 2-spheres goes to zero.)

 

[pervect]

Hello everyone!
Im a newcomer, a teenager who has countless doubts with respect to relativity, quantum theory etc. But these two questions bother me the most:
1) Hawking radiation states that when the separation of a particle (eg. a photon) into charged particles happens in the event horizon, and one charged particle gets sucked in, to conserve energy, the black hole reduces some of its mass, and eventually, it evaporates and nothing remains in place.
But from what I have heard, a black hole has the singularity which has infinite mass.

The singularity may have an infinite _density_. But in non-exacting physicists terminology (as opposed to a more conservative fully mathematical view with all the I's dotted and T's crossed), we can say that the singularity has a finite mass, which is equal to the mass of the black hole.

Energy conservation in GR turns out to be trickier than one would expect but in this particular example (a single black hole, presumed to be well way from any other massive objects), there isn't any issue with energy conservation. The mass of the black hole is converted into energy, which radiates away.

I won't go into the technical details, but I will mention there are at least four different concepts of what i blithelyly caled "the" mass of the black hole, the ADM mass, the Bondi mass, the Komar mass, and the Schwarzschild mass parameter. If you get more interested in what we mean by "the mass" of a black hole, it would be worth doing some google & research about these four terms, if you have the background.

For an insight into the mathematical side of the issue, consider the analogous case of a point charge, where we have an "infinite charge density", but a finite total charge. If you want to see some of the deeper issues in this seemingly simple view, I'd recommend Baez's insight article here on PF, "Struggles with the Continuum" <<link>> , for a hint of what I've glossed over on the mathematical side.

 

[Nugatory]

The singularity inside a Schwarzschild black hole is not a point; it's a spacelike line. (More precisely, it's the limit of a spacelike surface composed of an infinite sequence of 2-spheres, as the radius of the 2-spheres goes to zero.)

Yes... perhaps I should have repeated the scare-quotes, but I had already wrapped them around the word once.

 

[PAllen]

I won't go into the technical details, but I will mention there are at least four different concepts of what i blithelyly caled "the" mass of the black hole, the ADM mass, the Bondi mass, the Komar mass, and the Schwarzschild mass parameter. If you get more interested in what we mean by "the mass" of a black hole, it would be worth doing some google & research about these four terms, if you have the background.

An observation is that all of these are the same for the Schwarzschild metric. Note also that for this case you must use the surface integral form of the Komar mass, as the volume integral is undefined. Obviously, they are not defined in general, and in many cases none are defined (e.g for cosmological solutions, the metric is neither stationary nor asymptotically flat, so none of these are defined).

 

[DreamLord101]

The singularity may have an infinite _density_. But in non-exacting physicists terminology (as opposed to a more conservative fully mathematical view with all the I's dotted and T's crossed), we can say that the singularity has a finite mass, which is equal to the mass of the black hole.

Energy conservation in GR turns out to be trickier than one would expect but in this particular example (a single black hole, presumed to be well way from any other massive objects), there isn't any issue with energy conservation. The mass of the black hole is converted into energy, which radiates away.

I won't go into the technical details, but I will mention there are at least four different concepts of what i blithelyly caled "the" mass of the black hole, the ADM mass, the Bondi mass, the Komar mass, and the Schwarzschild mass parameter. If you get more interested in what we mean by "the mass" of a black hole, it would be worth doing some google & research about these four terms, if you have the background.

For an insight into the mathematical side of the issue, consider the analogous case of a point charge, where we have an "infinite charge density", but a finite total charge. If you want to see some of the deeper issues in this seemingly simple view, I'd recommend Baez's insight article here on PF, "Struggles with the Continuum" <<link>> , for a hint of what I've glossed over on the mathematical side.

Thanks you very much, I'm 14 (!) so I'm assuming the math is extremely tough!

 

[Nugatory]

Thanks you very much, I'm 14 (!) so I'm assuming the math is extremely tough!

It's not inherently hard, but it is based on math that is based on math that is based on math that you haven't gotten to yet.