Fun stuff: Short and simple note on Black Holes.


In the series: Note 5: Short and simple note on Black Holes.

Version: 0.9
Date: June 9, 2020.
Status: Ready. But information will be added, to make it more informative.
By: Albert van der Sel.


Here, I like to discuss some nice theories on Black Holes.

Ofcourse, it won't be in minute detail. It's just a very short, and very simple overview
of some key elements. Although my text will be extremely simple, it sure is fantastic stuff to study.


It would help if you have a general idea about our own Galaxy (milky way), that is, that there
exists a "disk" with spiral arms, a nucleus, and a sort of spherical Halo around the disk. See note 1.
Also, knowledge of some key points on the theories of stars, would be helpful. See note 3.



Chapter 1. Introduction:

Chapter 1 is essentially the same stuff as was already shown in note 1.
But that was a nice intro, so I believe.
After that we will see some other theoretical features of Black Holes, neutron stars, white dwarfs.

The next phrase is not entirely correct, but if you would approach a black hole (ofcourse still being
at a large distance), it would resemble a pitch black "spherical" object.
It's indeed often said, that at the "event horizon" (or "Schwarzschild radius"), gravity is so high
that even light cannot escape anymore.

Some folks argue (or have good reasons) to rephrase that to: SpaceTime near the Horizon is so streched,
so that light is so extremely far redshifted, that it becomes "unobservable".
Then it only "looks like" as if Gravity is so strong that even light cannot escape anymore.

Indeed, talking about Black Holes can be confusing at certain moments. However, it remains true
that from the "outside", near the Horizon, gravitational effects, like strong Spacetime curvature, time dilation,
and other effects we expect to happen according General Relativity, seems all to be true.

It's certainly also true that GTR predicts time dilation when an object is getting nearer to a Mass-energy distribution.
For an outside observer, the clock inside the object is running slower and slower. For an observer riding along
with the object, nothing unusual happens, and time looks "proper".
The key point of "time dilation", is often attributed as the source for effects just outside of the "horizon".

Many different theoretical studies have been done, and still are intensively active, on the "nature"
of a Black hole. Classically, one important tool has always been Einstein's General Theory of Relativity (GTR).
But, many modern insights and theories have emerged the last decades.

1.1 An example of a modern approach:

One important other approach, using elements from Thermodynamics, surely produced
additional insights. For example, take a look at the Bekenstein-Hawking relation below:

S = kB * A
-------
4 * Lp
    (equation 1)

The most important elements here, are "S" (entropy) and "A" (surface area of the Black Hole).
We can ignore the other parameters (kB and Lp), for now.

The notion of "Entropy" (S) is often interpreted as a measure of the number of microstates that "sits"
behind a particular macrostate.

Many folks also interpret this as a measure of "information". The remarkable thing about the
upper formula is, that this Black Hole entropy seems to be proportional to the Surface Area "A"
of the Black Hole (at the horizon).

Note the remarkable fact, that this view is unrelated to the interior of the Black Hole.
It's just the area defined by the horizon, which plays a role.

It's indeed very remarkable ! Contrary: If a Black Hole would be a "singularity", then many folks often
argued that when matter is sucked in into the Black Hole, information would be lost forever.
Indeed, for many physicists a true infinitely small "singularity" was never much appealing.
However, the Bekenstein-Hawking relation might suggest that information (in whatever form)
might get "stored" (?) at (or near) the Horizon (the surface "A").
This model is actually rather close to the socalled "Firewall model" of Black Holes.

In a way, the more microstates a system has (larger Entropy), the more information it caries.
You might wonder why this is so. An analogy from ordinary datatransmission (networking) might help.
A monotone signal has no intrinsic information. But if you "modulate" it, by varying the frequency
or amplitude, then it can carry information. As it were, the number of "states" have increased.


You might thus go as far, as saying that when a Black Hole aggresively (*) sucks up matter,
then "A" would increase, and so does it's entropy, and so does the "information".

All in all, I have not mentioned a lot here, but I tell you that it is quite a different approach
compared to the idea of an infinite small singularity, within a "Schwarzschild radius" (or event horizon)
which "surrounds" it.

(*): Many folks say that a Black Hole does not sucks up matter. Matter may swirl towards a Black Hole,
as a cloud called an accretion disk of ionized plasma. It's heated up, and will emit light and other radiation
like X-rays. As it comes closer and closer to the Horizon, it may orbit it for many times, while still
heating up intensely. But then what happens next? Ultimately, the plasma may be jettisoned away as "jets",
perpendicular to the accretion disk. But parts may also reach the Horizon. See also a later section.


1.2 The semi-classical approach, using Relativity Theory:

It's also important to take a look at the (semi-) Classical approach using General Relativity.
It's still very important, since many physicist uses GTR, or derived theories (like EiBI gravity),
which still up to this day, produces stunning results.

In a nutshell, the semi-classical approach goes a bit like this:

SpaceTime is a 4D continuum, and Einstein uses a lot of differential Geometry (a sort of Math),
to search for his answers. One main result is, that SpaceTime "warps" or "is curved" when Mass-Energy
is near. Curved SpaceTime can be associated with Gravity.
When there is no Mass-Energy, or we are very far from it, SpaceTime is "flat" (and no Gravity).

This is all reflected in his General "Field Equations".

Einstein's field equations provide for a framework, but not a specific solution like a "metric",
for example, to calculate a distance in a certain SpaceTime.

Schwarzschild found a solution, that is a "way" to calculate distances, and a general description of SpaceTime,
which conforms to Einstein's equations.

But there are also very interesting collaries to his findings. It became clear (in this semi-classical theory),
that any mass has a "critical radius", meaning that if you would extremely compress that mass
under that treshold, it would fully collapse into a Black Hole. Intially, the interpretation
was such, that at that "critical radius", gravitation would be so strong that even light could
not escape anymore.

In somewhat "better" words, the "mass-density" would be so high, that the surrounding SpaceTime would be so
streched, that even light would so redshifted, that it does not show anymore.

For that critical radius, called Rs, Schwarzschild found that the relation
to any Mass "M" is:

Rs = 2 G M / c2     (equation 2)

Where "G" is the gravitational constant, "c" is the speed of light, and "M" is the mass inside
the critical radius "Rs".

For any mass "M", a corresponding critical Schwartzschild Radius rs (or "R") can be calculated,
which defines the Horizon, and effectively says when then mass becomes a "black hole".
For example, if the Sun's Total mass were to be compressed within (about) 2 miles (the Rs for that Mass)
then it would become a black hole. For Earth, we need to compress our planet within 1 cm.

All in all, also the semi-classical theories provided the framework to describe Black Holes.

I must be extremely carefull in not suggesting of the existence of any "structure" inside the Black Hole.
It's simply not known.


-The classical model seems to point at an infinitely small "singularity" within the critical radius (Horizon).
-The conjectures of Bekenstein/Hawking seem to suggest a sort of storage of information at or near the Horizon.

At least, a singularity seems to be in conflict with Quantum Theories.

There is a tiny problem perhaps in using phrases as "length stretches", or "SpaceTime stretches"
and that sort of statements.
We know from Special Relativity, that the SpaceTime distance between "events" must be constant.
That requirement has not dropped.
The only thing we can rightfully say is that SpaceTime gets very curved as you approach the critical radius,
And it will even be asymptotic at, or very near, the critical radius.
This means, according to most physicists, SpaceTime is so streched, that light will be infinitely redshifted.
Indeed, also according to GTR, the clock will go slower and slower, nearer and nearer to the Horizon.

In many discussions, it is often said that a remote observer may see a spacecraft to get "spagettified",
as it would get nearer and nearer to the critical radius. The observation however, is indeed correct.
It's correct for a remote observer, since light and time seems to "freeze" for that observer,
when the object get's very near the Horizon.
For that spacecraft itself, it's probably true that other events will take place.

Lastly, I must say that many treatments of Black Holes, consider Mass, Charge, and Angular momentum
at the initial state, and then looks (theoretically) what happens when a Black Hole forms,
when for example a massive star collapses at the end of her life.
Different models go around, like e.g. the Kerr Black Hole, and others.

Even rather exotic models were published. It never found much support in the community (as it seems
to me), but maybe you like to Google on the "Fuzzball" black hole theory.
It's a valid model, or maybe I should say, a valid attempt to explore Black Holes.

1.3 Now, what has been observed in reality, so far? Here are just a few examples.

-Cygnus X-1:

Also from my youth, I remember that it was strongly suspected that Cygnus X-1 was likely
to be a black hole. It's a black hole near a (normal) large blue star, and the black hole is
rather brutally sucking material away from it, leading to intense X-ray radiation.
The intense radiation, is produced due to the large acceleration of the material towards
the black hole, in a rotating disk, spiraling towards the black hole.
The radiation was detected during the '60's/'70's of the former century.
Today, almost nobody doubts that we actually have a black hole in this system.

It then would be the first one, that was ever discovered.

It's about 6000 ly away, and the black hole itself is likely to have a mass of (only) 0.8 Solar Mass.

-Massive Black Hole in the Center of our Milky way.

Gradually, it became clearer and clearer, that a very massive Black Hole, sits in the center
of our own spiral galaxy (the Milkyway). It's located at Sagittarius A, the center of the Milky way.

Even stronger: Today, It is generally assumed that having a massive Black Hole at it's center, is a
rather common feature of Spiral Galaxies in general !

For the Milky way: many clues were acumulated. For example, the socalled "S" stars have very close
orbits near that massive black hole, and have enormous speeds in higly rosettic orbits (elliptic with precession).
Using that data, it can be inferred that this Black Hole has a mass of around 4 million Solar Masses.
If you see the data on the orbit of star "S2" around the Black Hole, you can hardly believe it.

It is indeed fenomenal, that the orbit is exactly for what General Relativity predicts.

For more info: see one of the links below.

-Massive (and not so massive) Primordial Black Holes.

Interestingly, rather recently, more and more articles appear from astronomers who make a case for abundant
massive "primordial" black holes.
For example, in the "arxiv" library, many recent articles can be found.
However, the theory is not new, but it seems that it received a renewed interest from the community.

It's indeed highly remarkble. In essence, the abundant number of massive "primordial" black holes
came into existence, relatively short after the Big Bang. A number of astronomers believe
that they were actually a motor around Galaxy formation.

Note: For about Galaxy formation after the Big Bang: it was often assumed that small fluctuations during
the Inflationary period, were actually the source for later Galaxy formations.
But, it seems now, that an alternative is considered.


-Detected Gravitational waves due to Black Hole collisions.

Already predicted by Einstein's GTR, finaly, in september 2015, the first "direct" observation
of gravitational waves was performed by the the LIGO and Virgo Scientific Collaboration.
Only after extremely careful examinations, the result was announced to the public, in februari 2016.

Since Relativity Theory plays in a Continuum "background", all relative motions (especially accelerations)
of Mass-Energies must produce distortions in SpaceTime which propagate, not unlike
the usual ElectroMagnetic radiation (like radio waves). However, those are generally extremely weak
and connot currently be detected.

However, when Massive Black Holes gets nearer and nearer, they have an "interaction" (acceleration)
resulting in evenly proportional distortions (waves) in SpaceTime, which might be
detected, here on Earth.

And that's exactly what happened in 2016!

The label of the event: GW150914
The source: 2 black holes of approximately 30 and 35 Solar Masses, spiraled to each other
ever faster and faster, resulting in an incredible sort of "merger" of those two entities.
Distance: About 1.4 * 109 ly away from our Sun.
Peak signal: due to the merger, resulting in 3 Solar Masses energy conversion into Gravitational waves.

Note: ofcourse, since the event was calculated to be as far as 1.4 * 109 lightyears away from us,
we must understand that it actually happened about 1.4 * 109 years ago (indeed: that far in the past).

Tip: Why don't you Google on the LIGO detection equipment, and it's ability to measure extremely
small distortions in SpaceTime. I am quite sure you will knocked out of your socks!


-April 2019: The very first photo of the Massive black hole in M87.

In April 2019, the "Event Horizon Telescope" organization, published the first image
of a very massive black hole in the galaxy M87.

M87 is a galaxy, at a distance of about 55 * 106 ly.

Using a worldwide array of radiotelescopes, with telescopic devices distributed across the Globe,
it has proven to be possible to capture images of the Black Hole in the core of M87.

In that period, typically about 350 Terrabytes per telescope, per day, were processed by Supercomputers,
using smart algolrithms, ultimately resulting in very clear pictures.

The pictures essentially show extremely hot matter and gas, under the enormous gravitational pull
of that super massive black hole. On the pictures, the "shadow" of the Event Horizon is visible.
Also the twists of the matter and gas near the Horizon, due to extreme gravity, is shown.

The true Event Horizon is about 2.5 smaller than the Shadow it casts on the clouds.
It has been calculated to be around 7 * 109 km, which makes it quite comparable
to the size of our own Solar system. How about that !
The mass of this Black Hole is about 6.5 * 109 Solar masses.
Some links about the pictures of the Black Hole of M87:

Event Horizon Telescope
Picture and explanation M87 and it's supermassive Black Hole
Picture and explanation M87 and it's supermassive Black Hole
Nice article of National Geographic

Chapter 2. Standard approaches:

2.1 Semi-classical approaches:

There are quite a few of semi-classical approaches. I mean the term "semi-classical" in the following context:

-Ofcourse the theories of Maxwell, Newton and the like, are really classical theories.
They work brilliantly, but not (at all times) on the very small scale, or the very large scale.

-All sorts of conjectures, based on Einstein's theories, are not really classical, but they are
not entirely in the same realm as newer theories which aim to reconcile Quantum theories with Relativity.
Eisteins's GTR is absolutely brilliant ofcourse, but seems a bit limited (*) to describe SpaceTime and Gravity.

(*): Just to make sure: There is probably nobody around, who admires Einstein more than me.
Even if the Almighty would give me 500000000 years of time, then still I would not be able to produce GTR.
Yes, Einstein is "The Boss".

There are a few semi-classical approaches to find a solution to describe a Black Hole.

=> First, in principle, they are largely founded on Einstein's GTR.

=> Secondly, they are often called "No Hair" theories, since the main parameters of the object under study,
are Mass (M), Angular momentum or spin (J), and Charge (Q). The most determining one, is ofcourse the mass M.

If Black holes would only be characterized by M, J, and Q, then these are their hairs. But it seems to be
a very limited number of parameters (just three parameters), so it follows that, according to such theories,
black holes have "almost no hairs". They are fairly simple objects (according to such theories).

If one would take into account, a parameter like Temperature, the "number of hairs" increases,
and if added more parameters, we would arrive at "hairy" theories.

=> The semi-classical are based on a continuum SpaceTime, and no SpaceTime quantum structure
is taken into account.

But, maybe having "just" those three parameters, is not so bad at all. One way for a Black Hole to come into existence,
is when a super-massive star ends it's star-life in a Supernova explosion, which will let the core
collapse. Ofcourse the Mass of that core is obvious to take into account. But the star was spinning as well,
and that total Angular momentum should thus be a parameter too.
For electrical charge, the consensus among astronomers/physicists, is that it plays a minor role anyway.

Well, some of the really exiting stuff is probably not sufficiently (?) covered by the semi-classical approaches.
For example, what happens when something crosses the event-horizon? If that would be possible?
Is it a one-way only membrane, so that there is no communication whatsoever possible between the interior and exterior?
Or, like we have seen above, is only the Surface area important, like in a Firewall argument, as might be referred
from the Bekenstein-Hawking relation?


If you would say: "not really exiting stuff covered??? Nonsense!", then you are right. For example, one of the semi-classical
solutions is the "Kerr" black hole model, which also shows "SpaceTime dragging", and some quite curious
event Horizon Topologies. Also Penrose did considerable theoretical work, and amazing effects like "trapped" surfaces,
might be considered to be part of the Kerr theories (see below).

The following are well-known "semi-classical" solutions (also typed as "No Hair" solutions) to describe a black hole:
  • The "Schwarzschild" metric, which described the geometry of SpaceTime for an uncharged, spherically-symmetric,
    and non-rotating Black Hole, thus with J=0 and Q=0.

  • The "Reissner-Nordstrom" solution, for a non-rotating Black Hole, but which takes charge Q into account too.

  • The "Kerr" solution, describing the SpaceTime of a rotating but uncharged black hole.

  • The "Kerr-Newman" solution describing the SpaceTime of a rotating, and charged black hole.

It seems to me that Charge is not really regarded as a very important parameter, So the first fundamental solution
by Schwarzschild (M, no J, no Q), and the later solution by Kerr (M, J, no Q) are very worthwile to discuss further.

2.1.1 The general expression for a "metric":

Now, the following equation might look rather unfamiliar to you, but it is a general expression for the metric
in any sort of symmetric, continuous sort of SpaceTime (flat, curved, whatever...):

ds2 = gμν dxμdxν     (equation 3)

The gμν is a Matrix, and the other terms are vectors.
Do not worry about the format of that equation. It will become clear later on.

A "metric" is simply a rather expensive, and luxury word for "distance" in Space (or distance in SpaceTime).
It describes a linesegment, or distance, and thus describes important features of the SpaceTime under consideration.

Let's take a look at such "metric", in a simple flat 3D space.

2.1.2 Example metric in flat Euclidean 3D Space:

In a flat Eucledian Space, like an "ordinary" 3D Space (R3), you may draw a Cartesian coordinate system.
Basically, such Cartesian coordinate system uses three perpendicular axes, the x-, y- and z-axis.
The whole purpose of such coordinate system, is to describe or pinpoint "points" in Space.
Once you have established that, you can say almost everything about that SpaceTime, like the distance between points,
or lines, or planes in that Space etc.. etc..

In a flat Space Eucedian Space, like an "ordinary" 3D Space (R3), you may draw a Cartesian coordinate system.
Basically, such Cartesian coordinate system uses three perpendicular axes, the x-, y- and z-axis.
The whole purpose of such coordinate system, is to describe or pinpoint "points" in Space.

A point in such Space, might be denoted by (x, y, z). It's also posssible to draw something that's called
a vector, from the Origin (center) of the coordinate system, to this random point (x, y, z).

The fact that such Eucedian Space is "flat", means this. Suppose you are on the x-axis. Suppose you walk in
the +x direction. So, you position might then be, as time passes, something like (1,0,0), then (2,0,0), etc..
Your position does not depend in any way, on "y" or "z". That is Δ x, as you move, has no relation with changes
on "y" or "z": those does not happen at all. Space is not curved, and a movement along one axis, does not
influence other coordinates. You will see this clearly in matrix form, in just a moment.

Distance in R3:

This is basically no more than applying the "Pythagorean theorem".

For example, in R3 we have the square of the distance between two points P=(x1, y1, z1), and Q=(x2, y2, z2):

|PQ|2 = (x2 - x1)2 + (y2 - y1)2 + (z2 - z1)2

Here, implicitly, it is assumed in this discussion, that the Coordinate system, or any point in it, is fully described by a linear
combination of the basis (or unit-) vectors (1,0,0), (0,1,0), and (0,0,1).

When we would consider the distance from some point (x,y,z) relative to the Origin of our coordinate system, we may simply write:

ds2=dx2 + dy2 + dz2   (equation 4)

The "ds" most often represents "very small distance", as if we would only consider very small variations.
Therefore, in true flat Space, you may view equation 2 to be equivalent to:

s2=x2 + y2 + z2   (equation 5)

However, I will stick most often to the "ds" (and "dx" etc..) notation.
In general, in Rn, of dimension "n" (with "n" axis):

ds2=dx12 + . . . + dxn2

Now, equation 3 might be very intruiging. But here is a very simple equivalent equation for Eucledian flat Space R3:

Suppose we review equation 3 again, however this time from a flat Space, and using plain matrices/vectors. Then:

ds2 = ┌ 1 0 0 ┐
│ 0 1 0 │
└ 0 0 1 ┘
┌ x ┐
│ y │
└ z ┘
┌ x ┐
│ y │
└ z ┘
= ┌ 1x+0y+0z ┐
│ 0x+1y+0z │
└ 0x+0y+1z ┘
┌ x ┐
│ y │
└ z ┘
= ┌ x ┐
│ y │
└ z ┘
┌ x ┐
│ y │
└ z ┘
= x2 + y2 + z2   (equation 6)

So, here we have an example of equation 3, but this time in simple flat 3D space.
The lenght of ds is simply an expression a la the Pythagorean theorem.
Please note that I used flat 3D space, to illustrate equation 3, which clearly holds in this case!
So, I hope that it demystifies equation 3 a little bit.

2.1.3 The Schwarzschild metric in 4D SpaceTime:

Schwarzschild did a thorough review of Einsteins equations. He also showed that a "critical radius" can be associated
with any amount of Mass (we will see that in a minute).
This "critical radius" is also called "event horizon", or simply "horizon", and often also "Schwarzschild radius".

For example, I you would compress all of the mass of the Earth in into the size of a small marble (9 mm), then the escape velocity
from "Earth" would be the speed of light (c). In that case, the "horizon" would be 9 mm.

If you would do the same for the Sun, you need to compress all mass into a sphere of less than 3 km.

Karl Schwartzschild calculated a "horizon" for any mass ("M") compressed inside a critical radius R.
So, the terms "horizon" and "critical radius" are the same, actually.

When you, from the "outside", would move towards that radius, the curvature of spacetime increases, and gravity strongly increases too.
Some folks find it better not to say that gravity increases, but that the curvature of spacetime increases.
Yes, but actually both should mean the same thing.

At the Schwarzschild radius, the gravity is so strong that even light cannot escape anymore.
And SpaceTime is curved asymptotical once you get very near the critical radius.
Again, many folks like to rephrase it to: curvature is so strong that light is so redshifted, we do not see it anymore.
Whether that last statement is really correct... No, probably not. We will see some considerations from Thermodynamics
as well, which will show that a Black Hole still radiates. And then ofcourse we have the Hawking radiation as well,
which we will see in a later part of this note.

Here I show the Schwarzschild metric, in spherical coordinates (ct, r, θ, φ).
Why spherical coordinates are used, is that Schwarzschild considered a spherical (symmetrical) Mass distribution.
In such a situation, spherical coordinates are the preferred choice (easier than using (ct, x, y, z)).
Do not forget that Einstein's SpaceTime is 4 dimensional, that is: (x,y,z,ct), or expressed in angular coordinates: (ct, r, θ, φ).

As applied to a spherical mass distribution, the Schwarzschild matrix gμν of the metric will be shown below.
This then, is the gμν as we have seen in equation 3.

In the equation for d2 below, it must be understood that the whole Mass now is compressed within the Horizon,
or "critical radius", with "radius" R, with respect to the centre of the distribution.
The Schwarzschild metric, or Schwarzschild solution, is valid for all SpaceTime "outside" the critical radius.
In the metric below, "r" is the radius as counted from the center of the Mass distribution, and "R" is the event horizon.

gμν = ┌ -(1-R/r).0.........0.............0 ┐
│ 0........1/(1-R/r).0.............0 │
│ 0........0.........r2............0 │
└ 0........0.........0.....r2sin2(θ) ┘

Now, using the equation for the metric "ds2 = gμν dxμdxν" and performing
quite some calculus, Schwarzschild found that ds2 can be expressed as:

ds2 = -(1 - R/r) c2 dt2 + dr2 /(1 - R/r) + r2(dθ2 + sin2(dθ)dφ2)   (equation 7)

The equation also expresses the rate of curvature, as the rate of "ds" to "dr", where "dr" would be a measure
of distance in flat SpaceTime.

Equation 7, leads us to two singularities. If r → R, nearing from the outside, we can see asymptotic behaviour,
because of the term dr2 /(1 - R/r). If you look closely, then (1 - R/r) will go to "0", and
thus the whole term will go to infinity. Also, when r → 0, we have a singularity.

It might be confusing, but the Schwarzschild solution, might be interpreted as a solution for SpaceTime,
outside of "R". So, in such an interpretation, it's not valid as a description for the interior.
But..., you may encounter other interpretations as well !

As we have already seen before, from the metric it can be found that:

Rs = 2 G M / c2    

which means that any mass has a "critical radius", meaning that if you would extremely compress that mass
under that treshold (Rs), it would fully collapse into a Black Hole. Intially, the interpretation
was such, that at that "critical radius", gravitation would be so strong that even light could
not escape anymore.
That's the same equation as equation 2 above. If you would read some articles on this matter, many folks
are not happy with a phrase like:

"(1.) Gravitation would be so strong that even light could not escape anymore".

It should be quite the same as the phrase:

"(2.) The surrounding SpaceTime would be so streched, that even light would be unobservable".

But, the most common notion (or explanation) found, is:

"(3.) The gravitational time dilation is the cause that an object falling into a black hole appears to slow down
more and more, as it approaches the event horizon, eventually taking an infinite amount of time."


It's certainly so that GTR predicts time dilation when an object is getting nearer to a Mass-energy distribution.
For an outside observer, the clock inside the object is running slower and slower. For an observer riding along
with the object, nothing unusual happens, and time looks "proper".

2.1.4 The "Kerr" metric in 4D SpaceTime:

The impact of the discovery of the Kerr metric, was rather huge.
It's really true, that this time, a better realistic model of Black Holes was discovered.

While GTR was published in 1915, just only a few months later, Schwarzschild came with his solution.
However, it was felt all along, that it was a great achievement, but maybe, the model is a bit too lean (J=0).

Almost everything in the Universe is spinning, and so do stars.

Since a Black Hole may be the result of a supernova explosion of a very massive star, having Angular momentum
in the cakculation of the SpaceTime description, seems only natural.

Since the Schwarzschild metric, it took almost 50 years before Kerr arrived at a description of a Black Hole,
having Mass and Angular momentum.

One reason for such a long time, is the complexity of the derivation. It is for example certainly true,
that one can use an "optimized" coordinate system, that reduces the complexity in a relevant way.
If you look at the matrix in the Schwarzschild metric (see above), there are actually only diagonal elements.
But a similar matrix in the Kerr model, has off-diagonal elements too.

It's in hindsight, rather obvious that the system now is less symmetric. Using the Schwarzschild model,
we have a perfectly symmetric model, viewed from any direction.
However, when rotation is involved, we obviously have a special direction in space, namely the axis
of rotation.
What's even more, a collapsing star, will have an Angular momentum, but just as you can see with an
ice-skater, when the diameter gets smaller and smaller, the rotational speed is likely to increase.
Would it even be possible that it arrests the collapse? Anyway, we now see how powerfull an Angular
momentum can be. It definitely must be an item for the calculation.

Back to the metric, and it's matrix. Using a clever coordinate system, one can reduce those off-diagonal elements.
For example, Eddington-Finkelstein-, or Boyer-Lindquist, or Kerr-Schild coordinates, can make a difference.

It must be noted, that the choice for a coordinate system, does not (or should not) fundamentally change anything.
However, it can make a lot of difference for the complexity of the derivation.

Ofcourse, there is no use to mention the derivation in this simple note.

The results however, are just fantastic. Let's review a few of those:

  1. The solution shows two Horizons. An inner and and an outer Horizon. However, consensus exists
    that only one is physically significant (the inner one).
  2. The Kerr black hole, defines an "ergo surface", or "ergo sphere" outside the Horizon, which is completely
    dominated by gravitational "frame dragging". It means that anything, like a "test particle",
    follows the rotation of the Black Hole.
  3. It's theoretically possible, purely based on the theory, that an object enters the ergo sphere,
    and steal energy, or mass-energy, from the Angular energy of Black Hole. The mathemetical properties
    of the ergo sphere show that this would be indeed a possibility. However, the physical significance is not at all
    sure at this moment. This phenomenom is called "The Penrose Process".
    However, if such process is repeated very often, the Angular energy of the Black Hole goes to zero, and the
    framedragging and the ergo sphere cease to exist.
  4. It's quite spectacular, but the Kerr model seems to point that the internal Singularity, in fact,
    is a ring, thus the singularity is shaped as a Ring.

Now you may ask: what is the physical relevance of the Schwarzschild model and the Kerr model?
That is not easy to answer. First, some considerations are still left out, like Thermodynamics.
Also, we do not see sufficient integration with Quantum Physics.

However, the models are relevant! Ask any astronomer, and you will see that even observations
are held against (most notably) the Kerr model.

2.2 Thermodynamics, and ideas of Bekenstein and Hawking:

The list of scientists who have contributed to Black Hole theories, is huge. Really large.
Studies in the past were nummerous, and modern Studies are ongoing, and the amount of monthly articles
in the Arxiv libary, as always, looks explosive.

So, there seems to be no special reason to single out Bekenstein and Hawking.
Although both contributed enormously to Black Hole theories.

The thing is too..., I like to say something in this section about Thermodynamics, and some
of the ideas of Bekenstein, and Hawking, for example, about "Hawking radiation".
Remember, section 2.1 was largely based on Einstein's General Theory of Relativity.
And now, we go back using rather classical Thermodynamics (with some Quantum corrections).
But I must admit, that what I say below, is not "entirely" classical.

Personally, I like equation (1), and the reasonably plausible idea of "information" stored
at the Horizon, and that entropy/information is linear with the Surface area "A".

Questions remain ofcourse.

-There is important Physics at work in white dwarfs, and neutron stars, like the "Pauli exclusion principle",
or the "Heisenberg relations". What happened to those, when the Black Hole formed?

-I have always (literally always) wondered what happened to the elementary particles.
For the "lighter sister", the Neutron star, there are good explanations (later more on this).
For Black Holes too, there are ideas, but that is for a later section.

-And SpaceTime itself, near or at the Horizon, or even "inside", does Superstring or Loop Quantum
tell us something of importance?

Indeed insanely cool stuff. However, I like "to go back" in this section, to some points of
a rather classical Theory, namely Thermodynamics.
There now exists enormous amounts of arguments and proposals with respect to "the state of information"
near or at Black Holes. But first we need to see some classical arguments at work.

The strange thing is (or not so strange?), Black Holes, as observed from the outside (ofcourse),
behave pretty much as expected from the Physics of Thermodynamics.
Since this is a simple note (because I am simple), I will only list some keypoints of the findings.
But the whole thing is rather curious.

Again, please note that the stuff below is rather a sort array of keypoints, but does not
coherently describes a "theory".

2.2.1 Some arguments of Bekenstein:

From various articles of Bekenstein himself, or comments by others thereof, it rather easy to find the following
keypoint which is advocated by Bekenstein and others.

A Black Hole's interior and "contents" (?) are shut off to an outside observer.
One can relate the associated Entropy of a Black Hole, to the radius of the (observable) Horizon, and thus
to the related Surface area of the Black Hole. The relation is:

S = kB * A
-------
4 * Lp

However, Planck's length Lp is the following:

Lp = √ (ħ G / c3) = (about) 1.6 x 10-35 m.

where c is the speed of light, ħ is the socalled reduced Plancks constant, and G is the universal gravitational constant.
So, the lenght of Planck is "build" from very fundamental constants from physics. It's the smallest (theoretical) distance
known in Physics.

We thus can rewrite equation above as:
S = c3 * A
---------
4 * ħ * G

You may actually (ignoring all constants) simply say that S:

S ≡ ¼ A

(By the way, Bekenstein conjectured that the constant factor must be close somewhere to ¼, while
Hawking later found the precise value.)

Ok, but now what? Well, Bekenstein and others, this way advocate that the Horizon might be seen as
a collection of extremely small area's (with the size of the Planck scale), which all add up to the entropy "S".
This view might be supported by the existence of Lp in the formula listed above.
For a picture describing this view, you might want to take a look at the following figure:

Scholarpedia: Event Horizon linked to a Planck Area

This is by no means a rigorously proven fact. In better words: "It seems like a good pointer to investigate further".

What is the physical significance? The above sounds very cool, but the physical significance is not established.
However, it is tempting to assume that the "information" at the Horizon, is stored as the most fundamental,
and smallest possible surface area's. Speculative: one Planck area = one bit.

2.2.2 Hawking radiation:

=>Can Black Holes only absorb, and never emit anything?

That's a very fundamental question, and actually also determines if a Thermodynamic approach would be possible.
If Black Holes only absorb, and never emit, then Thermodynamics do not seem to have much of a chance.
In that case, it might also follow that the "temperature" is "0" (?).

=>What If a Black Hole indeed is thermal, with radiation?

In such case, many speak of an "information paradox" or an apparant "loss of information".
In case of Quantum Mechanical arguments, it might be seen that there exists a problem:

-The wavefunction encodes the information of a Quantum System. The evolution is determined by "Operators".
The "sudden" death of such information is viewed as a possible problem. Others however, simply view
it as the equivalent of a "pertubative" measurement, which (in the Copenhagen view), leads to
a collapse of the statevector anyway.

-But even worse: suppose a "pure state" (a Ket) enters a Black Hole. Then something happens which is not
fully understood yet (meaning: what happens exactly with matter, in or at, a Black Hole?)
But, if a Black Hole indeed radiates, which is thermal, then this thermal information represents
a Mixed state. In QM terms: a pure state is a Ket, with in principle is known information, and a Mixed state
can only be represented by a density matrix. You thus might say that "something" threw away information.
The problem sits in the fact that the Black Hole performed (?) a "non-unitary transformation".

Is it really a paradox? The point is, that most say "yes", but still a rather relevant number of physicists say "no".
Folks who say no, have arguments that any thermal system can be associated with entropy, and that's valid
for a Black Hole as well. This opens up a wide array of possibilities and viewpoints.

Possibly this is too simple: if you would assume, that as time goes on, the Entropy of the Black Hole
increases, then the paradox is solved.
Not entirely. The point is, that if radiation takes place, there will still be missing information.
What get's out, is different from information content, than what went in before.
The whole point ofcourse, is that the Black Hole resembles a sinkhole with respect to information.
It's a tough problem, if only to get it right, in what exactly is meant by it.

-Some say that recently, the dispute was indeed settled. However, is that really true, that is, exists
consensus among the involved scientists, active in that field? I must say that it all seems a bit quirky
at this moment. Personally, I have not seen a convincing argument.

But let's go to Hawking radiation...

According to Quantum Field theories, the Vacuum (or empty space) "does" fluctuations that permeate all of spacetime.
So, thus also near, or very near, the Event Horizon.
It allows the creation of particle-antiparticle pairs of virtual particles, or virtual photons.

Closer to Home, there are good pointers for the existence of Vacuum fluctuations, like the Lamb shift,
or the Casimir effect.

The Energy-Time Heisenberg uncertainty relation, might give a good description of the effect, but is not really
explaining it (that is: giving the cause, if any cause would exist, of the effect).

It might look a bit at what was going on with a Kerr Black Hole, at the ergosphere.
It has been shown that a testparticle, going through the ergosphere, might steal away energy
from the Angular energy of the Black Hole (the Penrose process).

When a virtual pair, due to vacuum fluctuations, comes into existence near the Horizon,
one member might go "down", while the other escapes to free space. Since momentum must be conserved,
the free particle's momentum gain, and Black Hole loss, must be equal.
In effect: The Black Hole looses energy, while radiation escapes.

Hawking further found that any black hole would emit particles, and it would do so with a corresponding
thermal spectrum TH:

TH = ħ κ/2π

When scouting the many articles, there are several arguments about negative energies, in the ergosphere,
which complicate matters quite a bit. I will ignore that in this simple text

Hawking further makes a case, that very small black holes, if they would exist, would evaporate in short times.
We probably can easily agree to this theorem, since a very small black hole, would be relatively quick
be overwhelmed by the vacuum fluctuations, and quickly radiates away.

Chapter 3. A few words on White dwarfs and Neutron Stars:

It's recommended that you would take a look at note 3 first: "Note 3: short and simple intro on Stars".

3.1 Before we consider White Dwarfs and Neutron Stars, we need to know:

In the classical model, atoms are "pretty much" empty:

Already in 1908, Rutherford found that atoms are quite empty. That might sound strange,
but in an experiment, he bombarded a thin gold foil with socalled α particles (Helium nuclei).
Only a very small fraction diffracted or bounced back, while the large majority just flew right through it,
without any obstruction.
Not much later, the Rutherford/Bohr atom model was deviced, where atoms have a very small nucleus (with protons
and neutrons), and a wider electroncloud surrounding it.

Ofcourse, we know that later, Quantum Mechanical improvements were found, enhancing the theory.
For example, discrete energy levels, and several other quantum numbers were added.

Take a look at the following. It's just for illustrational purposes:

-The classical radius of the Hydrogen atom is about: 0.5310-12 m (not exact).
-The classical radius of a Carbon atom is about 70 x 10-12 m (not exact).
-The classical radius of the Nucleus a Carbon atom is about 2.7 x 10-15 m (not exact).
-The classical radius of a proton is about: 0.87 10-15 m.

For us, it's important to know the following: suppose that atoms (like Carbon or Oxigen) are fully ionized,
and all of the electrons are (so to speak) "detached", we can incredably increase the density of matter,
by "having" the nuclei closer together.

Stars, and the balance between outward- and inward pressure:

In every star, there exists a balance between the inward pressure due to Gravitation, and outward pressure
due to nuclear fusion processes in the star's core.

It is so, that younger stars, generally, and initially (and then for a long time afterwards) have nuclear fusion
going on, with Hydrogen (H) nuclei, into Helium (He) nuclei.

But when the Hydrogen starts to run out, the original outward pressure, caused by fusion, diminishes,
and the gravitational pressure will win over that outward pressure. While the temperature and pressure increases,
conditions are met to other to have fusion with He, an later on, with, selective, heavier elements.

In how far such process repeats, depends on the Mass of the star. Generally, for medium mass stars (< 1.4 Solar mass)
it probably ends when Carbon and Oxygen is created.
The star might end with a Nova (explosion), at which the core collapses and may result in a White Dwarf.

For heavy stars (> 1.4 Solar Mass), it might go in various ways.
In that last phases, ever more heavier elements are formed due to fusion of lighter elements.
When iron (Fe) is formed, a special situation exists with respect to nuclear fusion.
It could well be, that a considerable amount of the core, consists of Iron nuclei.
The end of life, of very heavy stars, is a Supernova. Again, various types are possible.
In any case, the core collapses, and may result in a Neutron star, or even a Black Hole.

In short, when the nuclear fuel has depleted, the star will collapse further due to Gravity.
What happens next, is mainly determined by the mass of it's core.


So, a star might continue it's further existence either as a White Dwarf, Neutron Star, or even a Black Hole.
A determining parameter, is the Mass of the collapsing core. Here, the "Chandrasekhar limit"
plays an important role.

Heisenberg uncertainty relation, and the Pauli exclusion principle:

Heisenberg uncertainty relation:

One of the most used examples of the Heisenberg uncertainty principle, is when you consider the pair
of "non commuting" observables like the "position (x)" and "momentum (p)" of a quantum system, like particle.

In fact, as from the moment that you proposed a "wave packet" to express the "position" of a particle,
the "Heisenberg uncertainty principle" will be build in automatically. It arises from the wave properties
which are inherent in the QM description of nature.

The Heisenberg inequality relation for position (x) and momentum (p) is:

Δx . Δp > ℏ / 2

It means that the more precisely the position of some particle is determined, the less precisely its
momentum (or velocity) can be known, and the other way around.

It has been proven, theoretically and experimentally, for many quantum systems.

Please note that the derivation has nothing (!) to do with "precision" of measurements. It's just build
deep in the description of QM.

So, if you would be able to very precisely determine the position, the uncertainty in momentum (or velocity)
is large. Reversely, if you would be able to very precisely determine the momentum (or velocity), there exists
a large uncertainty in the position.

Pauli exclusion principle:

For the discussion of White Dwarfs and Neutron stars, this principle is of decisive importance.

The "Pauli exclusion principle" applies for "fermions", that is, the collection of the "usual"
elementary particles like electrons. It states that no two near particles may be in the same state
or configuration.
For example, look at the electrons in an atom. They are distributed in such a way, so that a given state
is occupied by only one at a time. Suppose two are in the same Energy level, then at least some other
quantum number must be different, like their spin.
This principle also holds for "position" (location) in space.

So, for example, in a neutron star, the enormous gravitational pressure tries to squeeze the neutrons together.
You might then reason that the neutrons "must" move speedily, to avoid the breach of the exclusion principle.
This indeed provides for an outward pressure, counteracting gravity.

3.2 The Chandrasekhar limit:

In 1931, Chandrasekhar proposed a model (later called the "Chandrasekhar limit"), which describes
the criterium on how to determine whether a star becomes a White Dwarf or a Neutron Star (or even a Black Hole).

It's partly based on GTR, and Quantum physics. The theorem is in accordence with observations, since
all White Dwarfs observed sofar, have a mass under 1.4 Solar Mass.
It also means that if the collapsing core is over 1.4 Solar Mass, it's likely that a Neutron star will
be the result.

The theorem is:

-If the core of a collapsing star is < 1.4 Solar Mass, it's likely it will become a White Dwarf.
-If the core of a collapsing star is > 1.4 Solar Mass, it's likely it will become a Neutron Star.

It can be proven mathematically. Or in better words, it can be derived from established physics.
It's not very usefull, I think, to repeat the derivation in this simple note.

So, at which "barrier" is the "solution" of a Neutron star not possible anymore?
This is likely be defined by the "Oppenheimer-Volkoff limit".

If a collapsing core > (about) 3 Solar Mass, the gravitational pressure at a certain point, will
theoretically become so high, that even neutrons cannot exist anymore. Or, in other words,
the star will collapse further and passes even beyond the Schwartzschild Radius, which is the boundary of a Black Hole.
Don't take the number of "(about) 3 Solar Mass" too literally. There is an intrinsic uncertainty in the value.

Remember that Schwartzschild found (see also section 1.2), that any Mass has a Radius defined, where,
if the mass is so much compressed that it passes that Radius, the collapse is total, and a Black Hole
will be the result, based on GTR.

For example, for our Sun, you need to compress the whole mass within a radius of about 3km.

Now, our Sun is likely to become a White Dwarf, since it's mass < 1.4 Solar Mass.
For heavier stars, that is, well over 3 x Solar Mass, the gravitational collapse will be so strong
that even neutrons will give up, and the core goes beyond it's Schwartzschild Radius.

Note: when scouting older articles, the figure of "3 x Solar Mass" is often mentioned. However, the more recent
an article, the higher the barrier seems to get.


3.3 White Dwarfs and Neutron stars:

Since Black Holes is the main topic of this note, the discussion below is indeed rather "skinny".

These objects have lead, and still does ofcourse, to facinating studies and results.
The studies are "large", meaning lots of material to discuss about.

A White Dwarf may have a size which is comparable to our Planet, but it's mass may vary ofcourse.
One thing is for sure: it's not above 1.4 Solar Mass. So, quite typical it maybe somewhere in the order
of the mass of our Sun. That fact alone is nothing else than amazing: Something as "heavy" as our Sun,
having the dimension of a small Planet.

As was argued above, when matter is fully ionized (electrons removed), we end up with nuclei, and a sort
of "gas" of electrons.
This gas of electrons is called "degenerate", and is not bound to nuclei (atomic nucleus) anymore.

When ligher- or medium mass stars collapse, at a certain point, the fusion chain has reached Carbon and Oxygen.
However, fusion processes stops, since the gravitational pressure will not result in the temperature needed
for fusion to even more heavier elements.

The gravitational collapse, stops, because of a (at first sight) rather peculiar counter force.
Again, the Pauli exclusion principle is the cause that the electrons move extremely quickly around,
in order "to obey" the Pauli and Heisenberg Quantum rules.
This results in an outward pressure, counteracting the Gravitational pressure.



That's all. Hope you liked it.