Growing Black Hole Metric Approximation: 2MG/c^2

[stevendaryl]

This is a question inspired by the "Golf Ball" thread, which is no longer open for comments, I guess.

For a black hole of constant mass, the metric external to the black hole can be written in Schwarzschild metric, which is characterized by the constant $M$, and the corresponding radius $2 M G/c^2$. What I'm wondering is whether in a situation where there is a tiny (compared to the mass of the black hole) stream of mass falling into the black hole, is it a good approximation to the time-dependent external metric to simply replace $M$ by $M(t)$? Or would the effect of infalling matter make a more complicated change to the metric?

 

[Dale]

I don't know the answer to that. I was hoping that one of the participants of that thread would know and we could continue the discussion.

I suspect that there is no analytical solution for that scenario as proposed, but if you replaced the golf balls with collapsing spherically symmetric shells then there should be an analytical solution.

 

[aleazk]

Perhaps the Vaidya metric gives you something similar to what you are looking for.

 

[Dale]

Perhaps the Vaidya metric gives you something similar to what you are looking for.

That does look good, maybe not for the golf-ball metric, but certainly this seems like the right metric to use for Hawking radiation discussions.

I wonder what it means by "The emitted particles or energy-matter flows have zero rest mass and thus are generally called "null dusts", typically such as photons and neutrinos, but cannot be electromagnetic waves". How can something possibly be a photon but yet cannot be electromagnetic waves? And how can something have zero rest mass and you use a neutrino as an example?

 

[Matterwave]

I would suspect that the non-spherical nature of the problem would make things much more difficult. It seems to me that the M(t) approach would not work since the solution is still spherically symmetric; but the stream of golf balls come from one direction, the black hole would necessarily gain momentum from these golf balls in a particular direction...and that seems very hairy to try to solve.

 

[PeterDonis]

I wonder what it means by "The emitted particles or energy-matter flows have zero rest mass and thus are generally called "null dusts", typically such as photons and neutrinos, but cannot be electromagnetic waves".

It is an approximation, as I understand it, that basically amounts to geometric optics (i.e., very short wavelength compared to the characteristic length scale of spacetime curvature) plus spherical symmetry. A true EM wave can't be spherically symmetric, but in the short wave approximation it can be modeled as a spherically symmetric flux of massless particles.

 

[PeterDonis]

how can something have zero rest mass and you use a neutrino as an example?

I suspect nobody has updated the page to reflect that we now know neutrinos to have nonzero (but very small) masses.

 

[PAllen]

The problem with the golf ball problem was that, as initially posed you had a possibly infinite chain of golf balls, and no other matter. I think it was Pervect who first observed that believing this could be modeled, after a while, as golf balls falling into a near SC metric was not plausible - there is no spherical symmetry. I can post here what I PMed to Dalespam:

http://books.google.com/books?id=zp...te matter cylinder general relativity&f=false

which suggests that such initial conditions would produce a naked singularity of a particular type. Reading further, even a sufficiently (but inconceivably) large finite string of golf balls might produce a naked singularity. These are the authors to whom Hawking conceded the initial version of his cosmic censorship hypothesis. He then revised it to note that these solutions rely on axial distributions of matter with such perfect symmetry that there are no ripples or bulges that break the symmetry. In the real world, they would evolve quickly away perfect symmetry. So Hawking added to the Cosmic Censorship a mathematical formalization of 'stable against small perturbations':

A naked singularity forming for an 'open set' of initial conditions. That rules out these solutions, and the question remains unresolved, with no recent progress I'm aware of.

 

[rjbeery]

The problem with the golf ball problem was that, as initially posed you had a possibly infinite chain of golf balls, and no other matter. I think it was Pervect who first observed that believing this could be modeled, after a while, as golf balls falling into a near SC metric was not plausible - there is no spherical symmetry. I can post here what I PMed to Dalespam:

http://books.google.com/books?id=zp...te matter cylinder general relativity&f=false

which suggests that such initial conditions would produce a naked singularity of a particular type. Reading further, even a sufficiently (but inconceivably) large finite string of golf balls might produce a naked singularity. These are the authors to whom Hawking conceded the initial version of his cosmic censorship hypothesis. He then revised it to note that these solutions rely on axial distributions of matter with such perfect symmetry that there are no ripples or bulges that break the symmetry. In the real world, they would evolve quickly away perfect symmetry. So Hawking added to the Cosmic Censorship a mathematical formalization of 'stable against small perturbations':

A naked singularity forming for an 'open set' of initial conditions. That rules out these solutions, and the question remains unresolved, with no recent progress I'm aware of.

This was admittedly a shortcoming in my setup. We could grind each golf ball to a fine dust which fell into the black hole in a spherically symmetric manner, I don't care. I was less interested in the hard mathematical analysis and more interested in the physical observations which GR theory says could be made.

 

[cr7einstein]

I think that even though you could change M with M(t), there would be a more complicated variation to the metric. For instance, you may have to consider MOMENTARY entropy changes as well. Then there's Hawking radiation and the aggravation of quantum fluctuations with introduction of new particles. The horizon's expansion has to bee considered too. There should be complex metric perturbations, on the whole. I've only just glanced at it, but maybe Chandrashekhar's book The Mathematical Theory of Black Holes will help...