Is It Possible To Pull Something Out Of A Black Hole?
The Universe is out there, waiting for you to discover it Opinions expressed by Forbes Contributors are their own.
Ethan Siegel , Contributor
Ethan Siegel , Contributor
Andrew Hamilton / JILA / University of Colorado
In a Schwarzschild black hole, falling in leads you to the singularity, and darkness. But in a charged, Reissner-Nordstrom black hole, the light can eventually catch up to you as you fall in, tunneling you to another location in the Universe. Unfortunately, Reissner-Nordstrom black holes probably don't physically exist.
Once something falls into a black hole, it can never get out. No matter how much energy you have, you can never move faster than the speed of light, and yet you'd need to in order to exit of the event horizon once you've crossed inside. But what if you tried to cheat that little rule by tethering a tiny object that just dipped inside the event horizon to a much larger, more massive one that was destined to escape? Could you pull something out of a black hole that way, or any other way? The laws of physics are restrictive, but they should tell us whether it's possible or not. Let's find out!
AllenMcC. of Wikimedia Commons
Flamm's paraboloid, shown here, represents the spacetime curvature outside the event horizon of a Schwarzschild black hole.
A black hole isn’t merely an ultra-dense, ultra-massive singularity, where space is curved so tremendously that anything that falls in can’t escape. Although that’s what we conventionally think of, a black hole is more accurately the region of space around this objects from which no form of matter or energy — not even light itself — can escape. This isn’t as foreign or exotic as you might think: if you took the Sun, exactly as-is, and compressed it down to a region of space just a few kilometers in radius, a black hole is exactly what you’d wind up with. Although our Sun is in no danger of undergoing such a transition, there are stars in the Universe that will wind up producing a black hole in this very fashion.
NASA, ESA, and E. Sabbi (ESA/STScI); Acknowledgment: R. O’Connell (University of Virginia) and the Wide Field Camera 3 Science Oversight Committee
The star forming region 30 Doradus, in the Tarantula Nebula in one of the Milky Way's satellite galaxies, contains the largest, highest-mass stars known to humanity. The largest, R136a1, is approximately 260 times the Sun's mass.
The most massive stars in the Universe — stars with twenty, forty, a hundred, or even, at the core of the super star cluster shown above, up to 260 times the mass of our Sun — are the bluest, hottest, and most luminous objects out there. They also burn through the nuclear fuel in their cores the most quickly of all stars: just one or two million years instead of many billions like the Sun. When these inner cores run out of nuclear fuel, the nuclei at the core are subject to tremendous gravitational forces: forces so strong that, without the incredible pressure from the radiation of nuclear fusion to hold them up, they implode. In less extreme cases, the nuclei and electrons have so much energy that they fuse into a mass of neutrons, all bound together. If the core is more massive than a few times the mass of the Sun, those neutrons will be so dense and so massive that they themselves will collapse, leading to a black hole.
Mark A. Garlick
An illustration of an active black hole, one that accretes matter and accelerates a portion of it outwards in two perpendicular jets, may describe the black hole at the center of our galaxy in many regards. But nothing from within the event horizon could ever get out.
That’s the minimum mass of a black hole, mind you: a few times the mass of the Sun. Black holes can grow much larger than that, though, by merging together, by devouring matter-and-energy, and by sinking to the centers of galaxies. At the center of the Milky Way, we’ve identified an object that’s some four million times the mass of the Sun, where individual stars are seen orbiting it, but where no light of any wavelength is emitted.
Other galaxies can have even more massive black holes that are thousands of times the mass of our own, with no theoretical upper limit to how large they can grow. But there are two interesting properties about black holes that are going to lead us to the answer of whether anything tethered can escape. The first is what happens to space the more massive a black hole gets. The definition of a black hole is that no object can escape from its gravitational pull in a region of space, no matter how quickly that object accelerates, no matter even if it moves at the speed of light. That border between where an object could and an object couldn’t escape is what’s known as an event horizon, and every black hole has one.
Ute Kraus, Physics education group Kraus, Universität Hildesheim; background: Axel Mellinger
The black hole at the center of the Milky Way, along with the actual, physical size of the Event Horizon pictured in white. The visual extent of darkness will appear to be 5/2 as large as the event horizon itself.
What might surprise you is that the curvature of space is much smaller at the event horizon around the most massive black holes, and is most severe (and largest) around the least massive ones! Think about it this way: if you “stood” on the event horizon of a black hole, with your feet right at the edge and your head some 1.6 meters farther away from the singularity, there would be a force stretching — spaghettifying — your body. If that black hole were the one at the center of our galaxy, the force that stretches you would be only 0.1% the force of gravity here on Earth, while if Earth itself were turned into a black hole and you stood on that, that stretching force would be some 1020 times as strong as Earth’s gravity!
ESO, ESA/Hubble, M. Kornmesser
Even something as massive as a star, if brought too close to a black hole, will find itself stretched-and-compressed into a long, thin filament: spaghettified. The effects on a human being are equally severe if the black hole is low enough in mass.
If these stretching forces are small at the edge of the event horizon, they’re not going to be much larger inside the event horizon, and so — given the strength of the electromagnetic forces that hold solid objects together — perhaps we’ll be able to do exactly what was suggested: dangle an object outside the event horizon, cross it momentarily, and then pull it safely back. But would that be possible? To understand this, let’s go back to what happens at the very border between a neutron star and a black hole: just at that mass threshold.
ESO/Luís Calçada
A neutron star is one of the densest collections of matter in the Universe, but there is an upper limit to their mass. Exceed it, and the neutron star will further collapse to form a black hole.
Imagine you’ve got a ball of neutrons that’s spectacularly dense, but where a photon on the surface can still escape off into space and not necessarily spiral in to the neutron star itself. Now, let’s place one more neutron on that surface, and suddenly the core itself can’t hold up against gravitational collapse. But rather than thinking about what’s happening at the surface, let’s think about what’s happening inside the region where the black hole is forming. Imagine an individual neutron, made up of quarks and gluons, and imagine how the gluons need to travel from one quark to another within a neutron in order to exchange forces.
Wikimedia Commons user Qashqaiilove
The force exchanges inside a proton, mediated by colored quarks, can only move at the speed of light; no faster. Inside a black hole's event horizon, these light-like geodesics are inevitably drawn to the central singularity.
Now, one of these quarks is going to be closer to the singularity at the center of the black hole than another, and another will be farther away. For an exchange of forces to happen — and for a neutron to be stable — a gluon will have to travel, at some point, from the closer quark to the farther quark. But even at the speed of light (and gluons are massless), that’s not possible! All null geodesics, or the path an object moving at the speed of light will travel along, will lead to the singularity at the center of the black hole. Moreover, they will never get farther away from the black hole’s singularity than they are at the moment of emission. That is why a neutron inside of a black hole’s event horizon must collapse to become part of the singularity at the center.
Ask The Van / UIUC Physics Department
Once you cross the threshold to form a black hole, everything inside the event horizon crunches down to a singularity that is, at most, one-dimensional. No 3D structures can survive intact.
So now, let’s come back to the tether example: you've got a small mass tethered to a large ship; the ship is outside the event horizon but the mass dips inside. Whenever any particle crosses the event horizon, it’s impossible for any particle — even light — to escape from it again. But photons and gluons are the very particles we need to exchange forces with the particles that are still outside the event horizon, and they can’t go there!
This doesn’t necessarily mean that your tether will snap; it more likely means that the rushing ride towards the singularity will pull your entire ship in. Sure, the tidal forces, under the right conditions, won’t tear you apart, but that’s not what makes reaching the singularity inevitable. Rather, it’s the incredible attractive force of gravitation and the fact that all particles of all masses, energies and velocities have no choice but to head towards the singularity once they cross the event horizon.
Bob Gardner / ETSU
Anything that find itself inside the event horizon that surrounds a black hole, no matter what else is going on in the Universe, will find itself sucked into the central singularity.
And for that reason, I’m sorry to say, there is still no way out of a black hole once you cross the event horizon. You can cut your losses and cut off what's already inside, or you can stay connected and let everything get sucked inside. The choice is up to you, but let this be a lesson to everyone who has dreams of someday flying by a black hole: keep your hands and feet inside!
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!
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\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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