Validity of Schwarzschild Metric in Real BHs

[MattRob]

So, I've been reading through "Exploring Black Holes: Introduction to General Relativity" by Wheeler and Taylor, and I've had some ideas I wanted to pursue and do some research in regarding trajectories within the event horizon.

In this, I'd like to have the mathematical tools to investigate the validity of various claims. I know PF isn't a place to do research - these aren't those questions I aim to address with my work, these are different questions which I know others already know the answer to better than I do:

The book brings up the Kerr Metric but doesn't go too deeply into it, and also raises the point of the Cauchy Horizon, which, due to the fact that real black holes are probably best described by the extreme case of a maximally spinning black hole, and that the Kerr Metric is no longer valid at (or within, I assume?) the Cauchy Horizon, and that in such a maximally spinning black hole, the Cauchy horizon is at the same radius as the usual Event Horizon, the Kerr Metric would then seem invalid for describing anywhere within the event horizon of a real black hole (as approximated by the maximally spinning case).

The problem is, the questions I'd like to answer require mathematical analysis that will reveal some properties of a trajectory inside a real black hole.

So, if the Kerr is explicitly stated to be invalid in this region,

1) am I better off asking these questions and analyzing them with the Schwarzschild solution? Even though the Schwarzschild describes a non-spinning black hole, perhaps at least some of the qualitative features I'm after will carry over, and remain true in the case of a spinning black hole? Or would I be better off using the Kerr solution, as it retains more validity, even if not an accurate description of the Cauchy Horizon?

I'm limited in my mathematical knowledge at the moment to single variable calculus and linear algebra (matrices and the sort), otherwise I could, of course, just use more general equations instead of relying on specific solutions/metrics, such as the Kerr and Schwarzschild.

And two other big questions;
2) Is the Cauchy Horizon a region near which the Kerr solution is invalid, or is it a region within which the Kerr solution is invalid? In other words, could I still use the Kerr solution to answer my questions as long as I'm not near the Horizon or the Singularity (or is the singularity the Cauchy Horizon)?
3) Is there a metric I could use, knowing only single variable calculus, that is valid everywhere inside a spinning black hole?

Thanks very much!

 

[DEvens]

I have not read the book your are referring to. And I have never studied the Kerr metric in any detail.

But the way to get a feel for black holes (in my humble opinion) is to study Oppenheimer-Snyder dust clouds. You start with a spherically symmetric, uniform density, cloud of dust, at rest. Dust is a "technical term" meaning matter with zero pressure. This is not as unreasonable as it seems at first. A galaxy sized mass will disappear behind its own horizon with a density similar to that of air. So the approximation that dust will have zero pressure is not massively unphysical.

So you start with this cloud. And you let it evolve. The keen thing is, it collapses uniformly. The density remains uniform, though increasing with time. It increases uniformly all through the cloud.

So that turns the system into a single dimensional differential equation. And it happens to be one that has an exact solution. Outside the mass there is a Schwarzschild metric. Inside there is a somewhat different metric. So you can evolve the cloud from its original condition of at-rest with uniform finite density, to behind its own horizon, to completely collapsed into a singularity. And you can see how things evolve forward, with this exact solution, at each point.

I am trying to recall the text I used to study this stuff. I think it was Weinberg's book on gravity, but not sure. This was a while ago, being my report for 4th year undergrad, which I handed in in 1982.

 

[PeterDonis]

real black holes are probably best described by the extreme case of a maximally spinning black hole

Why do you think this? Many real black holes are not spinning much at all, as far as we can tell. Even the ones that are spinning rapidly (for example, the ones that are theorized to power quasars) will not be maximally spinning; that's an idealized case that is not expected to actually be realized (because there is no way to "spin up" a black hole to that point; it's an unattainable limit).

 

[MattRob]

Why do you think this? Many real black holes are not spinning much at all, as far as we can tell. Even the ones that are spinning rapidly (for example, the ones that are theorized to power quasars) will not be maximally spinning; that's an idealized case that is not expected to actually be realized (because there is no way to "spin up" a black hole to that point; it's an unattainable limit).

Gravitation, by Misner, Thorne, and Wheeler is quoted in the book;

(Page 885): "Most objects (massive stars; galactic nuclei; ...) that can collapse to form black holes have so much angular momentum that the holes they produce should be 'very live' (the angular momentum parameter a = J / M nearly equal to M; J nearly equal to M²).

What's odd, though, is immediately afterwards it gives a sample problem, giving an estimation of the angular momentum of the sun in conventional units at $J_{conv} = 1.91*10^{41}$ kg * m²/s.

Our equations are set up so that $a$ is expressed in meters. We need to divide by the sun's mass, then, in kg multiplied by the speed of light to account for the fact that in our system of units c = 1.

$a = \frac{ J_{conv} }{M_{kg} c } = 320.5$ meters
And using the sun's mass in geometric units as $M = \frac{GM_{kg}}{c^{2}} = ~1475$ meters
So that the dimensionless ratio $a / M = \frac{J}{M^{2}} = ~0.22$

0.22 doesn't seem like something well-approximated by the maximally spinning case, which makes that quote somewhat odd.

I think I'll do some preliminary analysis in my work using the Schwarzschild metric, since that will probably be easier, and once I understand exactly what I'm doing and the approaches to take, then I'll come back with the Kerr solution's added complexity to account for an angular momentum component, $a$, and focus my work on the region in-between the event horizon and the Cauchy Horizon.

I'm still curious about whether the Cauchy Horizon is a region within which the Kerr solution is no longer valid, or if it's a region simply near which the Kerr solution doesn't accurately describe spacetime. What little I could find on it describes it as a boundary in-between space and closed time-like geodesics, so it seems as though the latter would be the case.

 

[PeterDonis]

Gravitation, by Misner, Thorne, and Wheeler is quoted in the book;

Bear in mind that MTW was published in 1973. A lot of observational data has been collected since then. As I understand it, currently we would expect some holes (like the ones in active galactic nuclei--but it's not clear, for example, that the hole at the center of our galaxy is in this category) to have large angular momentum, and others (like the ones formed by gravitational collapse of stellar cores that are over the maximum mass limit for neutron stars) to not have large angular momentum. (Although there is a wrinkle in the latter case: if both stars of a binary become black holes, even if the holes individually don't have much angular momentum, when they spiral into each other and merge they are likely to form a hole with a lot of angular momentum, since the angular momentum contained in the stars' orbital motion is now contained in the hole.)

I'm still curious about whether the Cauchy Horizon is a region within which the Kerr solution is no longer valid, or if it's a region simply near which the Kerr solution doesn't accurately describe spacetime.

It's the latter; in fact, IIRC, some physicists think that the Kerr solution isn't a good description pretty much everywhere inside the outer horizon (i.e., the event horizon, not the Cauchy horizon).

The problem is stability against small perturbations. This is actually an issue even for non-rotating holes, i.e., for holes whose exterior is described by the Schwarzschild geometry, and it's simpler to explain for that case. Suppose we have a hole that, as far as we can tell from the outside, has exactly zero angular momentum and is exactly spherical. If we were to fall inside the hole, we would find the spacetime curvature getting stronger and stronger as we got further and further inside. As the curvature gets stronger, we would find that it is not exactly spherical; there are unavoidable fluctuations because it is extremely unlikely that all the matter that ever fell into the hole was distributed with exact spherical symmetry, and these fluctuations get amplified as you get further inside the hole. So once we are far enough inside, the perfectly spherical Schwarzschild geometry will not be a good description; the actual geometry will be a chaotic mess.

Similar remarks apply to a Kerr hole, but the rotation in the Kerr case makes it even more likely that fluctuations will become significant, which means that the region where the Kerr geometry is no longer a good description is likely to start sooner when you fall into such a hole. So well before you would even reach the Cauchy horizon in the idealized case, the interior geometry would no longer be Kerr, even approximately.

Another factor is that, if we take the exact Kerr geometry and ask what happens to a small bit of radiation that falls into the hole, we find that, as it approaches the Cauchy horizon, gravitational blueshift increases its frequency without bound. So even in an "ideal" Kerr hole, where there were (improbably) no fluctuations at all in the matter that fell in over time, the exact Kerr geometry would still be destroyed before you got to the Cauchy horizon, by the blueshifted radiation.

 

[stevebd1]

While it focuses mainly on the Kerr-Newman metric (a BH with spin and charge) you might find the latter part of the following paper (from page 33 onwards) useful-

http://casa.colorado.edu/~ajsh/phys5770_08/bh.pdf

It also talks about mass inflation (or infinite blueshift) at the Cauchy horizon from about page 25 (though this is for a charged BH).

You might also find the following link interesting also-

http://jila.colorado.edu/~ajsh/insidebh/waterfall.html