The Odd Case of Quantum Black
Holes
Black Holes can be considered as some of the most mysterious yet
fascinating objects in the universe. As they are usually portrayed
in science fiction these astronomical riddles are the omnivores of the
universe, consuming everything, even light, that dares to approach
their observable boundary, aka the ”Event Horizon”, never to be seen
again. In principle nothing can escape from them. Yet, as science has
often done in the past, this consensus has been partially disproven
by recent developments in quantum mechanics, as in its place arose
a new problem also known as the information paradox which still to
this day remains unresolved.
Introduction
1.1 The creation of a Black Hole
To understand what the problem is we must first take closer look to how black holes are formed. According to Einstein’s general theory of relativity if an object is compressed enough it can carve a region in space-time from which nothing can escape. But wait Jason! you might remark; Black holes are just stars that have died right? Well yes and no. Black Holes can indeed be massive stars that have ”died”, or to phrase it better stars that have went through a process called gravitational collapse in which the star literally contracts onto itself under the influence of its own gravity (why that happens now is an entirely different matter in and off itself and because I don’t want to blabber about something unrelated to the original subject of this article you can read more about it in this article by NASA:https://map.gsfc.nasa.gov/universe/rel stars.html),
but giant stars are not the only things that can turn into black holes.
Literally any object that is compressed down to what is called the Schwarzschild radius can turn into a black hole. The Schwarzschild radius is simply the radius of a sphere from which no information, no light, no particle can escape so that we can measure it (or so we thought, you’ll see what I mean later). The surface of that sphere is what we call the Event Horizon that I mentioned in the abstract. If the sun where to turn into a black hole it would have to be squeezed in a sphere with radius of about three kilometers! In the case that you would want to derive the Schwarzschild radius of anything you can simply use this equation:
The only problem with creating a black hole is the amount of energy required to contract an object within such a small space, since the force to counteract the repulsive quantum forces between its subatomic particles is too great. Such energy is only observed in the gravitational collapses of giant stars of about six or seven solar masses, making black hole candidates scarce in the present universe. But here’s the catch, it turns out stellar collapse is not the only way to create black holes.
1.2 Primordial Black Holes
We generally know that the universe is expanding, decreasing in density,
constantly, therefore it is safe to assume that in the past the
average density was way way higher than today and in fact so high
as to exceed the nuclear density of 2.3 × 1017 kg/m3
in the first microsecond
of the life of the universe. The highest value of density that
the universe could have started with is the so known Planck density,
about 1097 g/cm3
, a density so high that even the fabric of space-time
would break down. At these conditions black holes could have formed
as small as 10−35 m across, also known as the Planck length, and with
a mass of 10−8 kg, dimensions comparable to elementary particles.
For all this incredible information we have to thank Stephen Hawking
and Bernard J. Carr for their derivation of this mechanism for creating
black holes[2].
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The fact that black holes could be so small intrigued Hawking,
making him ponder about the possible quantum effects that could
come into play in such a small scale. Thus came his famous conclusion
of black holes emitting particles rather than just swallowing them[3].
2 Hawking Radiation
2.1 Emission and Evaporation
How can a black hole emit particles and why should it do that? A
strange question indeed and difficult to answer at that. To make
sense of this situation we must first take a look at some of quantum
mechanics most basic principles.
2.1.1 Vacuum Fluctuation
Heisenberg’s uncertainty principle states that one cannot know in
great detail and in the same time the exact position and speed of
a particle, or - from a more mathematical viewpoint - the product
of the uncertainty of both position and momentum should always be
greater than a certain value as seen in this inequality:
∆x∆p ≥
¯h /2
Where ¯h is the reduced Planck constant ( h /2π
or 1.054×10−34m2kg/s).
This relation has also been derived[4] in a energy/time form:
∆E∆t ≥
¯h /2
which means that in any point in space there must always be a minimal
change in energy no matter how small or ”empty” this space is. Such
a realization is very important because it necessitates that a vacuum
can never truly be a vacuum, as in containing nothing, meaning that
a small amount of energy should be created and later destroyed in
order to fill this empty space. Wow slow down there! What about
the conservation of energy? Energy cannot be created nor destroyed!
Worry not, everything is fine since Quantum Mechanics show us that
there is no violation of the law when we measure very short instances
of time, thus the uncertainty of energy is incredibly high.
2.1.2 Particle Pairs
Many of you might be baffled by the fact that particles can both
appear from and disappear into nothing but stay with me since this is
a very important property of particles and it directly leads us to why
black holes emit matter.
Einstein has generously provided us with a very important equation,
the famous
E = mc2
,
that shows us that energy and mass are
just two faces of the same coin. Therefore if energy can pop into existence
as the energy/time uncertainty principle dictates, then mass
can too. If in a small space where one would consider to be a true
vacuum suddenly appears a particle for a short moment and if that
particle happens to disappear after this time has passed then there is
a change in the energy in the space and so the uncertainty principle
is satisfied. This happens all the time in space, constantly particles
pop in existence only to disappear a moment later, filling space with
energy fluctuations. An important requirement for such an event to
unfold is that not only one particle appears but actually two, one made
of matter and one of antimatter. That way particles would annihilate
each other after a small amount of time. These particles are known
as virtual particles. Also we must clarify that one of two particles
has negative energy thus (stay with me) negative mass. This will be
important in the next part of the article.
2.1.3 Radiation and Black Hole Temperature
If virtual particles come into existence extremely close to the surface
of a black hole, even closer than the photon sphere (a spherical cortex
on which light literally can orbit the black hole), then there is a chance
that the particle pair will not annihilate itself but rather the particles
will part ways, one escaping the black hole and the other falling into
it. If the second one happens to be the one with negative mass then
the net mass of the black hole decreases, and since we cannot see
the negative mass particle as it has fallen into the hole we would
only see the bizarre effect of the black hole shrinking accompanied
by an emission of a particle than in actuality was just the part of
the pair that survived. Some of you may point out the fact that
in the grand scheme of things, there should be an equal chance of
both the negative mass particle and the positive mass particle to fall
into the black hole. That is not the case and the reasoning for the
phenomenon takes us back to the world of Thermodynamics. The
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second law of Thermodynamics states that the entropy of an isolated
system (Entropy being the measure of disorder in the universe [5])
must always increase or stay at an equilibrium. Thus in a statistical
view of the system there is a higher chance that the negative mass
particle will plunge into the black hole, as this action will increase the
entropy of both the system and consequently the universe.
Nevertheless, what we must take from all this is that we see that the black hole has the ability to radiate particles in a way very reminiscent of to what we know as temperature. Hawking in his work to study the effects of quantum mechanics on the surface of a black hole he derived a formula[3] also that shows that the temperature of a black hole is inversely proportional to the mass of the black hole, meaning a small black hole emits more energy than a big one. The formula he derived is this:
which shows us that its proportional to the hole’s mass. As the hole
decreases in size it evaporates energy faster and in greater amounts until
it gets so small that the amount of energy that needs to be released
is so great and it has to be released in such a small amount of time that
the hole literally explodes with more power than a million-megaton
nuclear bomb! Of course these formulas are only approximations since
a lot more things come into play when dealing with real life black holes
like the radiation from the Cosmic Microwave Background or random
matter falling into the black hole.
2.2 Leading to the information paradox
Hawking’s work is incredibly important and praise worthy as he connected
three seemingly unrelated areas of physics, these being Relativity,
Quantum Mechanics and Thermodynamics. Unfortunately this is
the part that I introduce the problem that I mentioned in the abstract,
the infamous information paradox, that took the physics society by
storm.
3 The Information Paradox
3.1 Relativity versus Quantum Mechanics
3.1.1 The essence of information
In physics the term information is not really something tangible, but
it must not be mistaken for not being well defined. Physical information
can be defined as the complete wavefunction of a particle or just
all of the properties of the particle (these being charge, spin, mass
etc.). This definition can be expanded to an arrangement of particles,
denoting to the way that the individual particles are connected and
interact with each other.
A very good example of this way of thinking
has been recently introduced by the YouTube channel Kurzgesagt in
their recent video ”Why Black Holes Could Delete The Universe The
Information Paradox”[6]. In their video they say that if you arrange a
bunch of carbon atoms in a certain way you will get coal but if you arrange
them in a different way you get diamond, therefore information
is just a property of the arrangement of these atoms.
A very important law that also has to be stated is the conservation
of information, that is that information cannot be destroyed (some-
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thing that was derived by the quantum field theory and Liouville’s
theorem). It can be separated to small pieces that are very hard to
measure accurately (like burning a piece of paper and then trying to
reconstruct the original by measuring every single change that underwent
with every single molecule), or it can be stored somewhere that
is not accessible by the laws of physics i.e. the surface of a black hole.
3.1.2 The contradiction
The paradox that arose by the conjecture of hawking radiation stems
from the fact that information is ”lost” when falling into a black hole
according to relativity. But if the black hole evaporates its mass away
completely what happens to the mass inside? Information cannot be
destroyed according to quantum mechanics but that is what would
happen if a black hole where to evaporate completely. What is going
on here? Hawking strongly believed in his theory and supported it
firmly, suggesting that information is indeed lost. A conviction of this
scale is frightening since it can mean that our current understanding
of physics is so deeply mistaken that we would need to scrap all of the
efforts of thousands of physicists across history out of the window and
force us to start over.
3.2 Other Solutions
Only recently physicists have come up with possible ways to cut this
gordian knot, with propositions like the black hole leaving a remnant
of information after its death or creating an entirely new universe
briefly before its death to store the information or it just leaks out
the information over time. Still all of those hypotheses are yet to be
proved since we have not yet seen a black hole evaporate, as we do not
have the proper equipment for such a feat in our current technological
state.
References
[1] M.L. Kutner. Astronomy: A Physical Perspective. Cambridge
University Press, 2003.
[2] B. J. Carr and S. W. Hawking. Black holes in the early Universe.
Monthly Notices of the Royal Astronomical Society, 168:399–416,
aug 1974.
[3] Stephen W Hawking. Particle creation by black holes. Communications
in mathematical physics, 43(3):199–220, 1975.
[4] J S Briggs. A derivation of the time-energy uncertainty relation.
Journal of Physics: Conference Series, 99(1):012002, 2008.
[5] J. Gribbin, M. Gribbin, and J. Gribbin. Q is for Quantum: An
Encyclopedia of Particle Physics. Touchstone, 2000.
[6] Kurzgesagt In a Nutshell. Why Black Holes Could Delete The
Universe The Information Paradox. youtu.be/yWO-cvGETRQ,
August 2017.
