Calculating the binding energy of a black hole

[avrndef]

Hi.

I tried to calculate the gravitational binding energy of a black hole, but I suspect that I did not do it correctly.

I used the following formulae:

black hole radius (Swarzschild radius):

$$R(m)=\frac{2Gm}{c^{2}}$$

gravitational potential energy for separating two masses from distance r to infinity:

$$dU=G\frac{m_{1}m_{2}}{r}$$

and then integrated them:

$$U=G\int^{M}_{0}\frac{mdm}{R(m)}=G\int^{M}_{0}\frac{c^2}{2G}dm=\frac{Mc^2}{2}$$

Mass-energy equivalence:

$$M_U=\frac{U}{c^2}=\frac{M}{2}$$

The result is that the binding energy is equivalent to half the mass of the black hole.

I think I have erred somewhere for two reasons: (1) the potential energy formula I have might apply only in a Newtonian universe and (2) Wikipedia states the exact same result for neutron stars - "The gravitational binding energy of a neutron star with two solar masses is equivalent to the total conversion of one solar mass to energy." As the radius of a black hole is less than that of a neutron star (of the same mass), then the binding energy should be greater.

Have I erred, and where? What's the right way of doing this?

 

[starthaus]

Hi.

I tried to calculate the gravitational binding energy of a black hole, but I suspect that I did not do it correctly.

I used the following formulae:

black hole radius (Swarzschild radius):

$$R(m)=\frac{2Gm}{c^{2}}$$

gravitational potential energy for separating two masses from distance r to infinity:

$$dU=G\frac{m_{1}m_{2}}{r}$$

and then integrated them:

$$U=G\int^{M}_{0}\frac{mdm}{R(m)}=G\int^{M}_{0}\frac{c^2}{2G}dm=\frac{Mc^2}{2}$$

Mass-energy equivalence:

$$M_U=\frac{U}{c^2}=\frac{M}{2}$$

The result is that the binding energy is equivalent to half the mass of the black hole.

I think I have erred somewhere for two reasons: (1) the potential energy formula I have might apply only in a Newtonian universe and (2) Wikipedia states the exact same result for neutron stars - "The gravitational binding energy of a neutron star with two solar masses is equivalent to the total conversion of one solar mass to energy." As the radius of a black hole is less than that of a neutron star (of the same mass), then the binding energy should be greater.

Have I erred, and where? What's the right way of doing this?

$$r$$ and $$R(m)$$ are two very different entities. You have inadvertently substituted one for the other.

 

[bcrowell]

If this question were to have a well-defined answer, then it would have to be of the form kM, where M is the Schwarzschild parameter that corresponds to the Newtonian field in the distant-field limit, and by an appropriate no-hair theorem (I guess Birkhoff's theorem in this case) k would have to be a universal constant.

Whether k=1/2 or some other value ... I don't think that's a well-defined question. Newtonian mechanics has a clear distinction between kinetic energy and potential energy, but GR doesn't have any such clear categories.

I don't think the WP statement is relevant here. They're talking about the binding energy relative to separating the two neutron stars -- without changing the structure of the neutron stars.

 

[avrndef]

Thanks for the answers.

For the main point:

As far as I know, in a Newtonian bound system the binding energy is radiated away during the formation of the system. The mass of the resulting bound system is the mass of its original components minus the mass-equivalent of the binding energy (which is called the mass deficit or mass defect (?)).

If my original question was ill-defined in general relativity then I will ask a slightly different question: if we let some matter (e.g. a very large, hollow shell) gravitationally collapse and form a black hole, then some of the mass of the original matter should end up in the black hole and some mass should be radiated away. Is this correct? Is it possible to calculate how much of the original mass is radiated away?

Now for some nitpicking:

I don't think the WP statement is relevant here. They're talking about the binding energy relative to separating the two neutron stars -- without changing the structure of the neutron stars.

Although this may indeed be irrelevant, I think they are talking about the energy required to take apart a single neutron star of two solar masses, bit by tiny bit, and then to move all the bits to infinity. The statement can be found at http://en.wikipedia.org/wiki/Neutron_star#Properties - second paragraph from the top. (I hope that posting this link doesn't violate any policy...)

$$r$$ and $$R(m)$$ are two very different entities. You have inadvertently substituted one for the other.

In relativity these things may indeed be different, I don't know enough about it to tell. In a Newtonian universe they should be substitutable. Let's say a dust mote of mass $$dm$$ lies on a sphere of mass $$m$$ and radius $$R(m)$$. It should be possible to treat the dust mote and the sphere as two point masses separated by the distance of $$R(m)$$. As such we should be able to use the potential energy formula

$$dU=G\frac{m_1m_2}{r}$$

with the substitutions

$$m_1=m; m_2=dm; r=R(m)$$

to get

$$dU=G\frac{mdm}{R(m)}$$

Thank you for your time.

 

[starthaus]

In relativity these things may indeed be different, I don't know enough about it to tell. In a Newtonian universe they should be substitutable. Let's say a dust mote of mass $$dm$$ lies on a sphere of mass $$m$$ and radius $$R(m)$$. It should be possible to treat the dust mote and the sphere as two point masses separated by the distance of $$R(m)$$. As such we should be able to use the potential energy formula

$$dU=G\frac{m_1m_2}{r}$$

with the substitutions

$$m_1=m; m_2=dm; r=R(m)$$

to get

$$dU=G\frac{mdm}{R(m)}$$

Thank you for your time.

Yes, it is as wrong as before.

 

[avrndef]

Yes, it is as wrong as before.

As I have apparently horribly misunderstood something basic, could you (or someone) please help me figure it out? What is $$r$$, what is $$R(m)$$ and why can't one be substituted for the other?

 

[starthaus]

As I have apparently horribly misunderstood something basic, could you (or someone) please help me figure it out? What is $$r$$, what is $$R(m)$$ and why can't one be substituted for the other?

$$r$$ is the radial coordinate, as such, it is variable (and does not depend on mass)
$$R(m)$$ is a constant characteristic to each gravitating body

 

[avrndef]

$$r$$ is the radial coordinate, as such, it is variable (and does not depend on mass)
$$R(m)$$ is a constant characteristic to each gravitating body

Thanks, this cleared up things nicely.

I got the gravitational potential energy formula from Wikipedia and it turns out I copied it down carelessly. Wikipedia uses a capital R, not a lowercase r:

$$U=-G\frac{m_1m_2}{R}$$

(found in http://en.wikipedia.org/wiki/Gravitational_potential_energy#Gravitational_potential_energy)

I (mis)used the lowercase r notation not to mean the radial coordinate, but the final (or initial) distance between $$m_1$$ and $$m_2$$.


The notation R(m) I used for the Schwarzschild radius, which should be correct. I integrated over the accretion process of the black hole, and as its mass increases from m=0 to m=M then R(m) would increase as well.


All this isn't actually very relevant to my question, as bcrowell said that GR doesn't clearly distinguish between potential and kinetic energies. So:

If you let a hollow shell collapse and form a black hole, what fraction of its mass is radiated away and what fraction ends up as the black hole?

How does one calculate this?

 

[George Jones]

If you let a hollow shell collapse and form a black hole, what fraction of its mass is radiated away and what fraction ends up as the black hole?

How does one calculate this?

Do you have experience with calculations in general relativity? If you do, then, for a model of radiating, collapsing star, you should look at Vaidya spacetime.

 

[avrndef]

Do you have experience with calculations in general relativity? If you do, then, for a model of radiating, collapsing star, you should look at Vaidya spacetime.

I am a layman. I liked - and paid attention in - physics and math classes (high school physics + 2 physics classes in the university). I know little about GR and less about the math behind it; I wouldn't recognize a tensor if it hit me in the face. I'm willing to learn, however.

A radiating, collapsing star is very interesting, although I doubt I can make much sense of the math (for now). As far as I know there were some relativity textbooks posted somewhere in the forums, and can you give me a link to a book/paper explaining Vaidya spacetime?

Perhaps modeling a collapsing shell in the Schwarzschild solution would be simpler?

Thanks.

 

[George Jones]

For thin shells and Vaidya spacetimes, I recommend A Relativists Toolkit: The Mathematics of Black-Hole Mechanics by Eric Poisson, and Exact Space-Times in Einstein's General Relativity by Jerry B. Griffiths and Jiri Podolsky. Prerequisite for both of these books is knowledge equivalent to that obtained in a standard introductory general relativity course.

 

[bcrowell]

Avrndef, it sounds like you're trying to calculate things that you don't know enough GR to be able to calculate. Some books you could try:

-Gardner, Relativity Simply Explained (cheap paperback, nonmathematical; gives an excellent conceptual foundation)
-Taylor and Wheeler, Spacetime Physics (SR, not cheap)
-Taylor and Wheeler, Exploring Black Holes (GR, not cheap; uses about the level of math that you seem to be comfortable with)
-Crowell (me), General Relativity (free online at http://www.lightandmatter.com/genrel ; starts with your math level and develops a few more mathematical tools)

 

[avrndef]

Avrndef, it sounds like you're trying to calculate things that you don't know enough GR to be able to calculate.

That's indeed the truth.

Thanks for the books, I'll take a look at them.