Ask Ethan: What Happens When A Black Hole's Singularity Evaporates?
It's hard to imagine, given the full diversity of forms that matter takes in this Universe, that for millions of years, there were only neutral atoms of hydrogen and helium gas. It's perhaps equally hard to imagine that someday, quadrillions of years from now, all the stars will have gone dark. Only the remnants of our now-vibrant Universe will be left, including some of the most spectacular objects of all: black holes. But even they won't last forever. David Weber wants to know how that happens for this week's Ask Ethan, inquiring:
What happens when a black hole has lost enough energy due to hawking radiation that its energy density no longer supports a singularity with an event horizon? Put another way, what happens when a black hole ceases to be a black hole due to hawking radiation?
In order to answer this question, it's important to understand what a black hole actually is.
Nicole Rager Fuller/NSF
The anatomy of a very massive star throughout its life, culminating in a Type II Supernova when the core runs out of nuclear fuel.
Black holes generally form during the collapse of a massive star's core, where the spent nuclear fuel ceases to fuse into heavier elements. As fusion slows and ceases, the core experiences a severe drop in radiation pressure, which was the only thing holding the star up against gravitational collapse. While the outer layers often experience a runaway fusion reaction, blowing the progenitor star apart in a supernova, the core first collapses into a single atomic nucleus — a neutron star — but if the mass is too great, the neutrons themselves compress and collapse to such a dense state that a black hole forms. (A black hole can also form if a neutron star accretes enough mass from a companion star, crossing the threshold necessary to become a black hole.)
NASA/ESA Hubble Space Telescope collaboration
When a neutron star accretes enough matter, it can collapse to a black hole. When a black hole accretes matter, it grows an accretion disk and will increase its mass as matter gets funneled into the event horizon.
From a gravitational point of view, all it takes to become a black hole is to gather enough mass in a small enough volume of space that light cannot escape from within a certain region. Every mass, including planet Earth, has an escape velocity: the speed you'd need to achieve to completely escape from the gravitational pull at a given distance (e.g., the distance from Earth's center to its surface) from its center-of-mass. But if there's enough mass so that the speed you'd need to achieve at a certain distance from the center of mass is the speed of light or greater, then nothing can escape from it, since nothing can exceed the speed of light.
SXS team; Bohn et al 2015
The mass of a black hole is the sole determining factor of the radius of the event horizon, for a non-rotating, isolated black hole.
That distance from the center of mass where the escape velocity equals the speed of light — let's call it R — defines the size of the black hole's event horizon. But the fact that there's matter inside under these conditions has another consequence that's less-well appreciated: this matter must collapse down to a singularity. You might think there could be a state of matter that's stable and has a finite volume within the event horizon, but that's not physically possible.
In order to exert an outward force, an interior particle would have to send a force-carrying particle away from the center-of-mass and closer to the event horizon. But that force-carrying particle is also limited by the speed of light, and no matter where you are inside the event horizon, all light-like curves wind up at the center. The situation is even worse for slower, massive particles. Once you form a black hole with an event horizon, all the matter inside gets crunched into a singularity.
Wikimedia Commons user AllenMcC
The exterior spacetime to a Schwarzschild black hole, known as Flamm's Paraboloid, is easily calculable. But inside an event horizons, all geodesics lead to the central singularity.
And since nothing can escape, you might think a black hole would remain a black hole forever. If it weren't for quantum physics, this would be exactly what happens. But in quantum physics, there's a non-zero amount of energy inherent to space itself: the quantum vacuum. In curved space, the quantum vacuum takes on slightly different properties than in flat space, and there are no regions where the curvature is greater than near the singularity of a black hole. Combining these two laws of nature — quantum physics and the General Relativistic spacetime around a black hole — gives us the phenomenon of Hawking radiation.
Derek B. Leinweber
A visualization of QCD illustrates how particle/antiparticle pairs pop out of the quantum vacuum for very small amounts of time as a consequence of Heisenberg uncertainty.
Performing the quantum field theory calculation in curved space yields a surprising solution: that thermal, blackbody radiation is emitted in the space surrounding a black hole's event horizon. And the smaller the event horizon is, the greater the curvature of space near the event horizon is, and thus the greater the rate of Hawking radiation. If our Sun were a black hole, the temperature of the Hawking radiation would be about 62 nanokelvin; if you took the black hole at the center of our galaxy, 4,000,000 times as massive, the temperature would be about 15 femtokelvin, or just 0.000025% the temperature of the less massive one.
X-ray: NASA/UMass/D.Wang et al., IR: NASA/STScI
An X-ray / Infrared composite image of the black hole at the center of our galaxy: Sagittarius A*. It has a mass of about four million Suns, and is found surrounded by hot, X-ray emitting gas. However, it also emits (undetectable) Hawking radiation, at much, much lower temperatures.
This means the smallest black holes decay the fastest, and the largest ones live the longest. Doing the math, a solar mass black hole would live for about 10^67 years before evaporating, but the black hole at the center of our galaxy would live for 10^20 times as long before decaying. The crazy thing about it all is that right up until the final fraction-of-a-second, the black hole still has an event horizon. Once you form a singularity, you remain a singularity — and you retain an event horizon — right up until the moment your mass goes to zero.
E. Siegel
Hawking radiation is what inevitably results from the predictions of quantum physics in the curved spacetime surrounding a black hole's event horizon.
That final second of a black hole's life, however, will result in a very specific and very large release of energy. When the mass drops down to 228 metric tonnes, that's the signal that exactly one second remains. The event horizon size at the time will be 340 yoctometers, or 3.4 × 10^-22 meters: the size of one wavelength of a photon with an energy greater than any particle the LHC has ever produced. But in that final second, a total of 2.05 × 10^22 Joules of energy, the equivalent of five million megatons of TNT, will be released. It's as though a million nuclear fusion bombs went off all at once in a tiny region of space; that's the final stage of black hole evaporation.
NASA
As a black hole shrinks in mass and radius, the Hawking radiation emanating from it becomes greater and greater in temperature and power.
What's left? Just outgoing radiation. Whereas previously, there was a singularity in space where mass, and possibly charge and angular momentum existed in an infinitesimally small volume, now there is none. Space has been restored to its previously non-singular state, after what must have seemed like an eternity: enough time for the Universe to have done all it's done to date trillions upon trillions of times over. There will be no other stars or sources of light left when this occurs for the first time in our Universe; there will be no one to witness this spectacular explosion. But there's no "threshold" where this occurs. Rather, the black hole needs to evaporate completely. When it does, to the best of our knowledge, there will be nothing left behind at all but outgoing radiation.
ortega-pictures / pixabay
Against a seemingly eternal backdrop of everlasting darkness, a single flash of light will emerge: the evaporation of the final black hole in the Universe.
In other words, if you were to watch the last black hole in our Universe evaporate, you would see an empty void of space, that displayed no light or signs of activity for perhaps 10^100 years or more. All of a sudden, a tremendous outrush of radiation of a very particular spectrum and magnitude would appear, leaving a single point in space at 300,000 km/s. For the last time in our observable Universe, an event would have occurred to bathe the Universe in radiation. The last black hole evaporation of all would, in a poetic way, be the final time that the Universe would ever say, "Let there be light!"
Send in your Ask Ethan questions to startswithabang at gmail dot com!
Astrophysicist and author Ethan Siegel is the founder and primary writer of Starts With A Bang! Check out his first book, Beyond The Galaxy, and look for his second, Treknology, this October!
とても興味深く読みました：
\documentclass[12pt]{article}
\usepackage{latexsym,amsmath,amssymb,amsfonts,amstext,amsthm}
\numberwithin{equation}{section}
\begin{document}
\title{\bf Announcement 179: Division by zero is clear as z/0=0 and it is fundamental in mathematics\\
}
\author{{\it Institute of Reproducing Kernels}\\
Kawauchi-cho, 5-1648-16,\\
Kiryu 376-0041, Japan\\
\date{\today}
\maketitle
{\bf Abstract: } In this announcement, we shall introduce the zero division $z/0=0$. The result is a definite one and it is fundamental in mathematics.
\bigskip
\section{Introduction}
%\label{sect1}
By a natural extension of the fractions
\begin{equation}
\frac{b}{a}
\end{equation}
for any complex numbers $a$ and $b$, we, recently, found the surprising 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 result is a very special case for general fractional functions in \cite{cs}.
The division by zero has a long and mysterious story over the world (see, for example, google site with division by zero) with its physical viewpoints since the document of zero in India on AD 628, however,
Sin-Ei, Takahasi (\cite{taka}) (see also \cite{kmsy}) established a simple and decisive interpretation (1.2) by analyzing some full extensions of fractions and by showing the complete characterization for the property (1.2). His result will show that our mathematics says that the result (1.2) should be accepted as a natural one:
\bigskip
{\bf Proposition. }{\it Let F be a function from ${\bf C }\times {\bf C }$ to ${\bf C }$ such that
$$
F (b, a)F (c, d)= F (bc, ad)
$$
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 } $
$$
F (b, 0) = 0.
$$
}
\medskip
\section{What are the fractions $ b/a$?}
For many mathematicians, the division $b/a$ will be considered as the inverse of product;
that is, the fraction
\begin{equation}
\frac{b}{a}
\end{equation}
is defined as the solution of the equation
\begin{equation}
a\cdot x= b.
\end{equation}
The idea and the equation (2.2) show that the division by zero is impossible, with a strong conclusion. Meanwhile, the problem has been a long and old question:
As a typical example of the division by zero, we shall recall the fundamental law by Newton:
\begin{equation}
F = G \frac{m_1 m_2}{r^2}
\end{equation}
for two masses $m_1, m_2$ with a distance $r$ and for a constant $G$. Of course,
\begin{equation}
\lim_{r \to +0} F =\infty,
\end{equation}
however, in our fraction
\begin{equation}
F = G \frac{m_1 m_2}{0} = 0.
\end{equation}
\medskip
Now, we shall introduce an another approach. The division $b/a$ may be defined {\bf independently of the product}. Indeed, in Japan, the division $b/a$ ; $b$ {\bf raru} $a$ ({\bf jozan}) is defined as how many $a$ exists in $b$, this idea comes from subtraction $a$ repeatedly. (Meanwhile, product comes from addition).
In Japanese language for "division", there exists such a concept independently of product.
H. Michiwaki and his 6 years old girl said for the result $ 100/0=0$ that the result is clear, from the meaning of the fractions independently the concept of product and they said:
$100/0=0$ does not mean that $100= 0 \times 0$. Meanwhile, many mathematicians had a confusion for the result.
Her understanding is reasonable and may be acceptable:
$100/2=50 \quad$ will mean that we divide 100 by 2, then each will have 50.
$100/10=10 \quad$ will mean that we divide 100 by10, then each will have 10.
$100/0=0 \quad$ will mean that we do not divide 100, and then nobody will have at all and so 0.
Furthermore, she said then the rest is 100; that is, mathematically;
$$
100 = 0\cdot 0 + 100.
$$
Now, all the mathematicians may accept the division by zero $100/0=0$ with natural feelings as a trivial one?
\medskip
For simplicity, we shall consider the numbers on non-negative real numbers. We wish to define the division (or fraction) $b/a$ following the usual procedure for its calculation, however, we have to take care for the division by zero:
The first principle, for example, for $100/2 $ we shall consider it as follows:
$$
100-2-2-2-,...,-2.
$$
How may times can we subtract $2$? At this case, it is 50 times and so, the fraction is $50$.
The second case, for example, for $3/2$ we shall consider it as follows:
$$
3 - 2 = 1
$$
and the rest (remainder) is $1$, and for the rest $1$, we multiple $10$,
then we consider similarly as follows:
$$
10-2-2-2-2-2=0.
$$
Therefore $10/2=5$ and so we define as follows:
$$
\frac{3}{2} =1 + 0.5 = 1.5.
$$
By these procedures, for $a \ne 0$ we can define the fraction $b/a$, usually. Here we do not need the concept of product. Except the zero division, all the results for fractions are valid and accepted.
Now, we shall consider the zero division, for example, $100/0$. Since
$$
100 - 0 = 100,
$$
that is, by the subtraction $100 - 0$, 100 does not decrease, so we can not say we subtract any from $100$. Therefore, the subtract number should be understood as zero; that is,
$$
\frac{100}{0} = 0.
$$
We can understand this: the division by $0$ means that it does not divide $100$ and so, the result is $0$.
Similarly, we can see that
$$
\frac{0}{0} =0.
$$
As a conclusion, we should define the zero divison as, for any $b$
$$
\frac{b}{0} =0.
$$
See \cite{kmsy} for the details.
\medskip
\section{In complex analysis}
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 $w=0$. 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.
However, we shall recall the elementary function
\begin{equation}
W(z) = \exp \frac{1}{z}
\end{equation}
$$
= 1 + \frac{1}{1! z} + \frac{1}{2! z^2} + \frac{1}{3! z^3} + \cdot \cdot \cdot .
$$
The function has an essential singularity around the origin. When we consider (1.2), meanwhile, surprisingly enough, we have:
\begin{equation}
W(0) = 1.
\end{equation}
{\bf The point at infinity is not a number} and so we will not be able to consider the function (3.2) at the zero point $z = 0$, meanwhile, we can consider the value $1$ as in (3.3) at the zero point $z = 0$. How do we consider these situations?
In the famous standard textbook on Complex Analysis, L. V. Ahlfors (\cite{ahlfors}) introduced the point at infinity as a number and the Riemann sphere model as well known, however, our interpretation will be suitable as a number. We will not be able to accept the point at infinity as a number.
As a typical result, we can derive the surprising result: {\it At an isolated singular point of an analytic function, it takes a definite value }{\bf with a natural meaning.} As the important applications for this result, the extension formula of functions with analytic parameters may be obtained and singular integrals may be interpretated with the division by zero, naturally (\cite{msty}).
\bigskip
\section{Conclusion}
The division by zero $b/0=0$ is possible and the result is naturally determined, uniquely.
The result does not contradict with the present mathematics - however, in complex analysis, we need only to change a little presentation for the pole; not essentially, because we did not consider the division by zero, essentially.
The common understanding that the division by zero is impossible should be changed with many text books and mathematical science books. The definition of the fractions may be introduced by {\it the method of Michiwaki} in the elementary school, even.
Should we teach the beautiful fact, widely?:
For the elementary graph of the fundamental function
$$
y = f(x) = \frac{1}{x},
$$
$$
f(0) = 0.
$$
The result is applicable widely and will give a new understanding for the universe ({\bf Announcement 166}).
\medskip
If the division by zero $b/0=0$ is not introduced, then it seems that mathematics is incomplete in a sense, and by the intoduction of the division by zero, mathematics will become complete in a sense and perfectly beautiful.
\bigskip
section{Remarks}
For the procedure of the developing of the division by zero and for some general ideas on the division by zero, we presented the following announcements in Japanese:
\medskip
{\bf Announcement 148} (2014.2.12): $100/0=0, 0/0=0$ -- by a natural extension of fractions -- A wish of the God
\medskip
{\bf Announcement 154} (2014.4.22): A new world: division by zero, a curious world, a new idea
\medskip
{\bf Announcement 157} (2014.5.8): We wish to know the idea of the God for the division by zero; why the infinity and zero point are coincident?
\medskip
{\bf Announcement 161} (2014.5.30): Learning from the division by zero, sprits of mathematics and of looking for the truth
\medskip
{\bf Announcement 163} (2014.6.17): The division by zero, an extremely pleasant mathematics - shall we look for the pleasant division by zero: a proposal for a fun club looking for the division by zero.
\medskip
{\bf Announcement 166} (2014.6.29): New general ideas for the universe from the viewpoint of the division by zero
\medskip
{\bf Announcement 171} (2014.7.30): The meanings of product and division -- The division by zero is trivial from the own sense of the division independently of the concept of product
\medskip
{\bf Announcement 176} (2014.8.9): Should be changed the education of the division by zero
\bigskip
\bibliographystyle{plain}
\begin{thebibliography}{10}
\bibitem{ahlfors}
L. V. Ahlfors, Complex Analysis, McGraw-Hill Book Company, 1966.
\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}
S. Koshiba, H. Michiwaki, S. Saitoh and M. Yamane,
An interpretation of the division by zero z/0=0 without the concept of product
(note).
\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. Vol. 27, No 2 (2014), pp. 191-198, DOI: 10.12732/ijam.v27i2.9.
\bibitem{msty}
H. Michiwaki, S. Saitoh, M. Takagi and M. Yamada,
A new concept for the point at infinity and the division by zero z/0=0
(note).
\bibitem{s}
S. Saitoh, Generalized inversions of Hadamard and tensor products for matrices, Advances in Linear Algebra \& Matrix Theory. Vol.4 No.2 (2014), 87-95. http://www.scirp.org/journal/ALAMT/
\bibitem{taka}
S.-E. Takahasi,
{On the identities $100/0=0$ and $ 0/0=0$}
(note).
\bibitem{ttk}
S.-E. Takahasi, M. Tsukada and Y. Kobayashi, Classification of continuous fractional binary operators on the real and complex fields. (submitted)
\end{thebibliography}
\end{document}
For the procedure of the developing of the division by zero and for some general ideas on the division by zero, we presented the following announcements in Japanese:
\medskip
{\bf Announcement 148} (2014.2.12): $100/0=0, 0/0=0$ -- by a natural extension of fractions -- A wish of the God
\medskip
{\bf Announcement 154} (2014.4.22): A new world: division by zero, a curious world, a new idea
\medskip
{\bf Announcement 157} (2014.5.8): We wish to know the idea of the God for the division by zero; why the infinity and zero point are coincident?
\medskip
{\bf Announcement 161} (2014.5.30): Learning from the division by zero, sprits of mathematics and of looking for the truth
\medskip
{\bf Announcement 163} (2014.6.17): The division by zero, an extremely pleasant mathematics - shall we look for the pleasant division by zero: a proposal for a fun club looking for the division by zero.
\medskip
{\bf Announcement 166} (2014.6.29): New general ideas for the universe from the viewpoint of the division by zero
\medskip
{\bf Announcement 171} (2014.7.30): The meanings of product and division -- The division by zero is trivial from the own sense of the division independently of the concept of product
\medskip
{\bf Announcement 176} (2014.8.9): Should be changed the education of the division by zero
\bigskip
\bibliographystyle{plain}
\begin{thebibliography}{10}
\bibitem{ahlfors}
L. V. Ahlfors, Complex Analysis, McGraw-Hill Book Company, 1966.
\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}
S. Koshiba, H. Michiwaki, S. Saitoh and M. Yamane,
An interpretation of the division by zero z/0=0 without the concept of product
(note).
\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. Vol. 27, No 2 (2014), pp. 191-198, DOI: 10.12732/ijam.v27i2.9.
\bibitem{msty}
H. Michiwaki, S. Saitoh, M. Takagi and M. Yamada,
A new concept for the point at infinity and the division by zero z/0=0
(note).
\bibitem{s}
S. Saitoh, Generalized inversions of Hadamard and tensor products for matrices, Advances in Linear Algebra \& Matrix Theory. Vol.4 No.2 (2014), 87-95. http://www.scirp.org/journal/ALAMT/
\bibitem{taka}
S.-E. Takahasi,
{On the identities $100/0=0$ and $ 0/0=0$}
(note).
\bibitem{ttk}
S.-E. Takahasi, M. Tsukada and Y. Kobayashi, Classification of continuous fractional binary operators on the real and complex fields. (submitted)
\end{thebibliography}
\end{document}
Title page of Leonhard Euler, Vollständige Anleitung zur Algebra, Vol. 1 (edition of 1771, first published in 1770), and p. 34 from Article 83, where Euler explains why a number divided by zero gives infinity.
私は数学を信じない。 アルバート・アインシュタイン / I don't believe in mathematics. Albert Einstein→ゼロ除算ができなかったからではないでしょうか。
1423793753.460.341866474681。
Einstein's Only Mistake: Division by Zero
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