Ask Ethan #15: The Universe’s Most Massive Black Holes Posted by Ethan on December 13, 2013 (143) More »

Ask Ethan #15: The Universe’s Most Massive Black Holes
Posted by Ethan on December 13, 2013
(143)
More »

Image credit: X-ray: NASA/CXC/SAO/A.Bogdan et al; Infrared: 2MASS/UMass/IPAC-Caltech/NASA/NSF.
“It is by going down into the abyss that we recover the treasures of life. Where you stumble, there lies your treasure.” –Joseph Campbell

Ever since we created a question/suggestion box here, we’ve been deluged by far more excellent questions than one person could possibly answer, but that doesn’t mean we aren’t trying! For this week’s Ask Ethan, our question comes from long-time fan and reader crd2, who asks:

As we look at the furthest quasars we see they have supermassive black holes, as large as 109 solar masses. By what mechanism are they able to reach such large sizes over so short a time scale?

Thanks for teaching me so much.

It turns out that the problem is even worse than you imagined, and it all goes back to the astrophysics of stars.


You might be used to the idea that stars come in a huge variety of sizes, colors, lifetimes and masses, and that these properties are all related to each other. The more massive a star is, the larger its fuel-burning core — operating under the principles of nuclear fusion — is, too. This means that more massive stars burn more luminously, have hotter temperatures, tend to be larger in radius, and also burn through their fuel more quickly.

Image credit: Morgan Keenan Spectral Classification by LucasVB, retrieved from Wikimedia Commons.

While a star like our Sun might take more than 10 billion years to burn through all the fuel in its core, stars can be tens or even hundreds of times more massive than our Sun is, and instead of billions of years, they can fuse all the hydrogen in their cores into helium in mere millions — or in extreme cases, possibly only hundreds of thousands — of years.

Image credit: Sakurambo at wikimedia commons.


What happens to those cores when they use up their fuel? You’ve got to realize that the energy released from those fusion reactions was the only thing holding the cores of these stars up against the tremendous force of gravity, which is consistently working to contract all the matter in this star down into as tiny a volume as possible. When those fusion reactions stop, the core contracts quickly. The speed is important, because if you compress something slowly, its temperature tends to stay constant but its entropy rises, while if you compress it quickly, its entropy stays constant but the temperature goes up!


In the case of a massive star’s core, that increased temperature means it can start fusing heavier and heavier elements, going from helium to carbon-nitrogen-and-oxygen to neon, magnesium, silicon, sulphur and eventually up to iron-nickel-and-cobalt in short order. (Note that these are mostly formed in increments of two, element-wise, due to helium nuclei fusing with the existing elements.) And when you reach iron-nickel-and-cobalt in the core — the most stable elements (on a per-nucleon basis) — there is no more fusion that occurs, and you get runaway core-collapse, resulting in a Type II supernova!


In a less-massive star that does this, you’ll get a neutron star at the core, while an even more massive star — with an even more massive core — won’t be able to stand up to gravity, creating a black hole at the core! A star about 15-20 times the mass of our Sun ought to produce a black hole at the center when it dies, and progressively more-and-more massive ones will produce even more massive black holes!
You could imagine huge numbers of massive enough stars producing black holes via this mechanism in a concentrated space, and then these black holes merging together over time. Or, a combination of mergers and feeding on stellar and interstellar matter, which we observe happening as well.


Image credit: Chandra X-ray Observatory (blue), Hubble Space Telescope (green), Spitzer Space Telescope (pink), & GALEX (purple).

Unfortunately, that wouldn’t get you there quickly enough to be consistent with our observations.

You see, if a star gets too massive, it won’t produce a black hole at its center! If you start looking at stars over about 130 solar masses, the interior of your star becomes so hot and energetic that the highest-energy radiation particles you create can form matter-antimatter pairs, in the form of positrons and electrons. This might not seem like a big deal, but remember what was happening inside the cores of these stars: the only thing holding them up against core collapse was the pressure created by the radiation resulting from nuclear fusion! When you start producing electron-positron pairs, you’re producing them out of the radiation present in the star’s core, which means you reduce the pressure in the core. This starts to happen in stars of about 100 solar masses, but once you get up to about 130 solar masses, this reduces the pressure enough that the core stars to collapse, and it does so quickly!

So it heats up, and it also contains a huge number of positrons, which annihilate with normal matter, producing gamma-rays which also heat up the core even further! Eventually, you create something so energetic in the core that the entire star is blown apart in the most spectacular type of supernova we’ve ever observed: a pair-instability supernova! This not only destroys the outer layers of the star, but the core as well, leaving absolutely nothing behind!

Without sufficiently large black holes formed in very short order in the Universe, we still might get supermassive black holes like the ones we find at the center of our own galaxy, which — from the gravitational orbits of stars around it — weighs in at a few million solar masses.

But that wouldn’t get you up to the billions of solar masses found in, for example, this relatively nearby galaxy (as you can see from its ultrarelativistic jet, below): Messier 87.


The thing that crd2 alludes to in his question is that supermassive black holes on this order — with many billions of solar masses — are found at very high redshifts, which means they’ve been around, and very big, in the Universe for a very long time!

You might think that we could have just started off the Universe with black holes of this magnitude, but that’s simply inconsistent with our picture of the young Universe from the matter power spectrum and from the fluctuations in the cosmic microwave background. Wherever these supermassive black holes came from, it’s unlikely that they were primordial in nature, but they’re certainly present in even very young galaxies!

So if normal stars can’t do it, and the Universe wasn’t born with them, where do they come from?

It turns out that stars can get even more massive than the ones we’ve talked about, and when they do, there’s a new hope. Let’s go back to the first stars that formed in the Universe — out of the primordial hydrogen and helium gas that existed back then — just a few million years after the Big Bang.

There’s plenty of suggestive evidence that the stars that formed back then were from huge regions, not like the star clusters containing a few hundred-or-thousand stars in our galaxy, but containing millions (or even more) of stars when they’re born. And if we look to the largest star forming region we have locally — the Tarantula Nebula located in the Large Magellanic Cloud — we can get a clue as to what we think is going on.


This region of space is nearly 1000 light years across, with the massive star-forming region in the center — R136 — containing about 450,000 solar masses worth of new stars. This entire complex is active, forming new, massive stars. But at the center of this central region, you can find something truly remarkable: the most massive star known (so far) in the entire Universe!

Image credit: NASA, ESA, & F. Paresce (INAF-IASF), R. O'Connell (U. Virginia), & the HST WFC3 Science Oversight Committee.
Image credit: NASA, ESA, & F. Paresce (INAF-IASF), R. O’Connell (U. Virginia), & the HST WFC3 Science Oversight Committee.

The largest star in here is 265 times the mass of our Sun, and that’s a very remarkable place to be. You see, remember what I told you about pair-instability supernovae, and how they destroy stars over 130 solar masses, leaving no black hole behind? That’s true, but it’s only true up to a point; that story is only true for stars with masses above 130 solar masses and below 250 solar masses. If we get even more massive than that, we begin to create gamma rays that are so energetic that they cause photodisintegration, where these gamma rays cool down the interior of the star by blowing the heavy nuclei back apart into light (helium and hydrogen) elements.

Image credit: © Swinburne University of Technology, edits by me.
Image credit: © Swinburne University of Technology, edits by me.

In a star with more than 250 Solar Masses, it simply collapses entirely into a black hole. A 260 solar mass star would create a 260 solar mass black hole, a 1000 solar mass star would make a 1000 solar mass black hole, etc. And so if we can make a star that exceeds that limit here, in our own isolated little corner of space, then we certainly made these objects when the Universe was very young, and we probably made a good-sized number of them. And over time, they’ll merge!


And if you can get an initial region kicked off with a massive black hole of a few thousand solar masses after just a few million (or few tens of millions of) years, the rapid merger and accretion of these collapsed, star-forming regions make it unthinkable that these early, large black holes wouldn’t merge with one another and grow. In short order, they’d be forming increasingly larger and larger black holes at the centers of these objects: the Universe’s first large galaxies!
Image credit: The National Astronomical Observatory of Japan.
Image credit: The National Astronomical Observatory of Japan.

And that continued growth-over-time could easily result from some naïve estimates into a black hole of many hundreds of millions of solar masses for a Milky Way-sized galaxy. It’s not hard to imagine that more massive galaxies — or nonlinear effects — could ramp that up into the billions of solar masses without a problem. And although we don’t know for sure, that’s where we think, to the best of our knowledge, that supermassive black holes come from!

Thanks for asking such a great question, and congratulations to crd2 on winning the fourth (of five) Year In Space 2014 Wall Calendar as part of our giveaway here on Starts With A Bang! You can always get your own here (and at a discount if you say you were sent there by us), but you have one more chance if you’d like to win one on this very blog: submit your question here, and if we choose you for our next Ask Ethan column, you win!http://scienceblogs.com/startswithabang/2013/12/13/ask-ethan-15-the-universes-most-massive-black-holes/

Ask Ethan #15: The Universe's Most Massive Black Holes – Starts With ...
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2013/12/13 - reminds me of an Aesop Fable ... I'm not even getting into the parts of the math that require division by zero (a singularity)causing the metric to “blow up” (not a very scientific concept, mathematicians may explode, but ...... Photograph of participants of the first Solvay Conference, in 1911, Brussels, Belgium.http://scienceblogs.com/startswithabang/2013/12/13/ask-ethan-15-the-universes-most-massive-black-holes/

\documentclass[12pt]{article}
\usepackage{latexsym,amsmath,amssymb,amsfonts,amstext,amsthm}
\numberwithin{equation}{section}
\begin{document}
\title{\bf Announcement 300: New challenges on the division by zero z/0=0\\
(2016.05.22)}
\author{{\it Institute of Reproducing Kernels}\\
Kawauchi-cho, 5-1648-16,\\
Kiryu 376-0041, Japan\\

%\date{\today}
\maketitle
{\bf Abstract: } In this announcement, for its importance we would like to state the
situation on the division by zero and propose basic new challenges.

\bigskip
\section{Introduction}
%\label{sect1}
By a {\bf natural extension} of the fractions
\begin{equation}
\frac{b}{a}
\end{equation}
for any complex numbers $a$ and $b$, we found the simple and beautiful result, for any complex number $b$
\begin{equation}
\frac{b}{0}=0, 
\end{equation}
incidentally in \cite{s} by the Tikhonov regularization for the Hadamard product inversions for matrices and we discussed their properties and gave several physical interpretations on the general fractions in \cite{kmsy} for the case of real numbers.

The division by zero has a long and mysterious story over the world (see, for example, Google site with the division by zero) with its physical viewpoints since the document of zero in India on AD 628, however,
Sin-Ei Takahasi (\cite{kmsy}) established a simple and decisive interpretation (1.2) by analyzing the extensions of fractions and by showing the complete characterization for the property (1.2):

\bigskip

{\bf Proposition 1. }{\it Let F be a function from ${\bf C }\times {\bf C }$ to ${\bf C }$ satisfying
$$
F (b, a)F (c, d)= F (bc, ad) 
$$ 
for all
$$
a, b, c, d \in {\bf C }
$$
and 
$$
F (b, a) = \frac {b}{a }, \quad a, b \in {\bf C }, a \ne 0.
$$
Then, we obtain, for any $b \in {\bf C } $ 
$$
F (b, 0) = 0.
$$
}

Note that the complete proof of this proposition is simply given by 2 or 3 lines.

\medskip
We thus should consider, for any complex number $b$, as (1.2); 
that is, for the mapping
\begin{equation}
w = \frac{1}{z},
\end{equation}
the image of $z=0$ is $w=0$ ({\bf should be defined}). This fact seems to be a curious one in connection with our well-established popular image for the point at infinity on the Riemann sphere. Therefore, the division by zero will give great impacts to complex analysis and to our ideas for the space and universe.

However, the division by zero (1.2) is now clear, indeed, for the introduction of (1.2), we have several independent approaches as in:

\medskip
1) by the generalization of the fractions by the Tikhonov regularization or by the Moore-Penrose generalized inverse, 

\medskip
2) by the intuitive meaning of the fractions (division) by H. Michiwaki,

\medskip
3) by the unique extension of the fractions by S. Takahasi, as in the above,

\medskip
4) by the extension of the fundamental function $W = 1/z$ from ${\bf C} \setminus \{0\}$ into ${\bf C}$ such that $W =1/z$ is a one to one and onto mapping from $ {\bf C} \setminus \{0\} $ onto ${\bf C} \setminus \{0\}$ and the division by zero $1/0=0$ is a one to one and onto mapping extension of the function $W =1/z $ from ${\bf C}$ onto ${\bf C}$, 

\medskip
and

\medskip

5) by considering the values of functions with the mean values of functions.
\medskip

Furthermore, in (\cite{msy}) we gave the results in order to show the reality of the division by zero in our world:

\medskip

\medskip
A) a field structure containing the division by zero --- the Yamada field ${\bf Y}$,

\medskip
B) by the gradient of the $y$ axis on the $(x,y)$ plane --- $\tan \frac{\pi}{2} =0$,
\medskip

C) by the reflection $W =1/\overline{z}$ of $W= z$ with respect to the unit circle with center at the origin on the complex $z$ plane --- the reflection point of zero is zero,
\medskip

and
\medskip

D) by considering rotation of a right circular cone having some very interesting
phenomenon from some practical and physical problem.

\medskip

In (\cite{mos}), many division by zero results in Euclidean spaces are given and the basic idea at the point at infinity should be changed. In (\cite{ms}), we gave beautiful geometrical interpretations of determinants from the viewpoint of the division by zero. The results show that the division by zero is our basic and elementary mathematics in our world.

\medskip

See J. A. Bergstra, Y. Hirshfeld and J. V. Tucker \cite{bht} for the relationship between fields and the division by zero, and the importance of the division by zero for computer science. It seems that the relationship of the division by zero and field structures are abstract in their paper.

Meanwhile, J. P. Barukcic and I. Barukcic (\cite{bb}) discussed recently the relation between the divisions $0/0$, $1/0$ and special relative theory of Einstein. However, their logic seems to be curious and their results contradict with ours.

Furthermore, T. S. Reis and J.A.D.W. Anderson (\cite{ra,ra2}) extend the system of the real numbers by introducing an ideal number for the division by zero $0/0$. 

Meanwhile, we should refer to up-to-date information:

{\it Riemann Hypothesis Addendum - Breakthrough

Kurt Arbenz
https://www.researchgate.net/publication/272022137 Riemann Hypothesis Addendum - Breakthrough.}

\medskip

Here, we recall Albert Einstein's words on mathematics:
Blackholes are where God divided by zero.
I don't believe in mathematics.
George Gamow (1904-1968) Russian-born American nuclear physicist and cosmologist remarked that "it is well known to students of high school algebra" that division by zero is not valid; and Einstein admitted it as {\bf the biggest blunder of his life} [1]：
1. Gamow, G., My World Line (Viking, New York). p 44, 1970.

For our ideas on the division by zero, see the survey style announcements 179,185,237,246,247,250 and 252 of the Institute of Reproducing Kernels (\cite{ann179,ann185,ann237,ann246,ann247,ann250,ann252,ann293}).

\section{On mathematics}
Apparently, the division by zero is a great missing in our mathematics and the result (1.2) is definitely determined as our basic mathematics, as we see from Proposition 1. Note its very general assumptions and many fundamental evidences in our world in (\cite{kmsy,msy,mos}). The results will give great impacts on Euclidean spaces, analytic geometry, calculus, differential equations, complex analysis and physical problems. See our announcements for the details.

The mysterious history of the division by zero over one thousand years is a great shame of mathematicians and human race on the world history, like the Ptolemaic system (geocentric theory). The division by zero will become a typical symbol of foolish human race with long and unceasing struggles. Future people will realize this fact as a definite common sense.

We should check and fill our mathematics, globally and beautifully, from the viewpoint of the division by zero. Our mathematics will be more perfect and beautiful, and will give great impacts to our basic ideas on the universe.

\section{Albert Einstein's biggest blunder}
The division by zero is directly related to the Einstein's theory and various 
physical problems
containing the division by zero. Now we should check the theory and the problems by the concept of the RIGHT and DEFINITE division by zero. Now is the best time since 100 years from Albert Einstein. It seems that the background knowledge is timely fruitful.

\section{Computer systems}
The above Professors listed are wishing the contributions in order to avoid the zero division trouble in computers. Now, we should arrange new computer systems in order not to meet the division by zero trouble in computer systems.

\section{General ideas on the universe}
The division by zero may be related to religion, philosophy and the ideas on the universe, and it will creat a new world. Look the new world.

\bigskip

We are standing on a new generation and in front of the new world, as in the discovery of the Americas.

\bigskip

\bibliographystyle{plain}
\begin{thebibliography}{10}

\bibitem{bb}
J. P. Barukcic and I. Barukcic, Anti Aristotle—The Division of Zero by Zero. Journal of Applied Mathematics and Physics, {\bf 4}(2016), 749-761.
doi: 10.4236/jamp.2016.44085.

\bibitem{bht}
J. A. Bergstra, Y. Hirshfeld and J. V. Tucker,
Meadows and the equational specification of division (arXiv:0901.0823v1[math.RA] 7 Jan 2009).

\bibitem{cs}
L. P. Castro and S. Saitoh, Fractional functions and their representations, Complex Anal. Oper. Theory {\bf7} (2013), no. 4, 1049-1063. 

\bibitem{kmsy}
M. Kuroda, H. Michiwaki, S. Saitoh, and M. Yamane,
New meanings of the division by zero and interpretations on $100/0=0$ and on $0/0=0$,
Int. J. Appl. Math. {\bf 27} (2014), no 2, pp. 191-198, DOI: 10.12732/ijam.v27i2.9.

\bibitem{ms}
T. Matsuura and S. Saitoh,
Matrices and division by zero $z/0=0$,
Linear Algebra \& Matrix Theory (ALAMT)(to appear).

\bibitem{msy}
H. Michiwaki, S. Saitoh, and M.Yamada, 
Reality of the division by zero $z/0=0$. IJAPM International J. of Applied Physics and Math. {\bf 6}(2015), 1--8. http://www.ijapm.org/show-63-504-1.html

\bibitem{mos}
H. Michiwaki, H. Okumura, and S. Saitoh,
Division by Zero $z/0 = 0$ in Euclidean Spaces.
International Journal of Mathematics and Computation 
(in press).

\bibitem{ra}
T. S. Reis and J.A.D.W. Anderson,
Transdifferential and Transintegral Calculus,
Proceedings of the World Congress on Engineering and Computer Science 2014 Vol I
WCECS 2014, 22-24 October, 2014, San Francisco, USA

\bibitem{ra2}
T. S. Reis and J.A.D.W. Anderson,
Transreal Calculus, 
IAENG International J. of Applied Math., {\bf 45}(2015): IJAM 45 1 06.

\bibitem{s}
S. Saitoh, Generalized inversions of Hadamard and tensor products for matrices, Advances in Linear Algebra \& Matrix Theory. {\bf 4} (2014), no. 2, 87--95. http://www.scirp.org/journal/ALAMT/ 

\bibitem{ttk}
S.-E. Takahasi, M. Tsukada and Y. Kobayashi, Classification of continuous fractional binary operations on the real and complex fields, Tokyo Journal of Mathematics, {\bf 38}(2015), no. 2, 369-380.

\bibitem{ann179}
Announcement 179 (2014.8.30): Division by zero is clear as z/0=0 and it is fundamental in mathematics.

\bibitem{ann185}
Announcement 185 (2014.10.22): The importance of the division by zero $z/0=0$.

\bibitem{ann237}
Announcement 237 (2015.6.18): A reality of the division by zero $z/0=0$ by geometrical optics.

\bibitem{ann246}
Announcement 246 (2015.9.17): An interpretation of the division by zero $1/0=0$ by the gradients of lines.

\bibitem{ann247}
Announcement 247 (2015.9.22): The gradient of y-axis is zero and $\tan (\pi/2) =0$ by the division by zero $1/0=0$.

\bibitem{ann250}
Announcement 250 (2015.10.20): What are numbers? - the Yamada field containing the division by zero $z/0=0$.

\bibitem{ann252}
Announcement 252 (2015.11.1): Circles and
curvature - an interpretation by Mr.
Hiroshi Michiwaki of the division by
zero $r/0 = 0$.

\bibitem{ann281}
Announcement 281(2016.2.1): The importance of the division by zero $z/0=0$.

\bibitem{ann282}
Announcement 282(2016.2.2): The Division by Zero $z/0=0$ on the Second Birthday.

\bibitem{ann293}
Announcement 293(2016.3.27): Parallel lines on the Euclidean plane from the viewpoint of division by zero 1/0=0.

\end{thebibliography}

\end{document}