Wolfram Alpha:
Search by keyword:
Astronomy
Chemistry
Classical Mechanics
Classical Physics
Climate Change
Cosmology
Finance and Accounting
Game Theory
General Relativity
Group Theory
Lagrangian and Hamiltonian Mechanics
Macroeconomics
Mathematics
Microeconomics
Nuclear Physics
Particle Physics
Probability and Statistics
Programming and Computer Science
Quantum Computing
Quantum Field Theory
Quantum Mechanics
Semiconductor Reliability
Solid State Electronics
Special Relativity
Statistical Mechanics
String Theory
Superconductivity
Supersymmetry (SUSY) and Grand Unified Theory (GUT)
The Standard Model
Topology
Units, Constants and Useful Formulas
Introduction to Black Holes
---------------------------
The basic definition of a black hole is a mass where the
escape velocity is greater than the speed of light. From
classical physics we get the escape velocity for earth to
be:
v = √(2GM/R) where R is the radius of the earth.
Replacing v with c we get and rearranging we get:
R_{s} = 2GM/c^{2} ... A.
Very roughly, if the diameter of earth were .08 m it would
be a black hole. If we took a 1 kg mass, the force on the
surface of this compressed earth calculated using Newton's
law would be roughly 10^{19} N which is a force greater than
any other force known in physics.
The radius, R_{s}, at which light can begin to escape is
called the SCHWARZCHILD RADIUS and the surface of the
sphere formed with that radius is called the EVENT HORIZON.
Schwarzchild Metric
-------------------
In a similar manner to flat spacetime, we can construct an
invariant interval in curved spacetime called the SCHWARZCHILD
METRIC that describes the space time associated with a black
hole. The metric in spherical coordinates is derived from
Einstein's field equations (a complicated process not
attempted here!). It is (assuming c = 1):
dτ^{2} = (1 - R_{s}/R)dt^{2} - dR^{2}/(1 - R_{s}/R) - R^{2}dΩ^{2}
Or,
dτ^{2} = (1 - 2MG/R)dt^{2} - dR^{2}/(1 - 2MG/R) - R^{2}dΩ^{2}
Where R is the distance from center of the black hole to a
particular point in space. Note when R is very large this
equation reduces to the form of the flat space time metric.
dτ^{2} = dt^{2} - dR^{2} - R^{2}dΩ^{2}
The metric represents the relationship between the proper
time measured by the person, τ, and the time measured by
an observer. There are 2 interesting cases. As R approaches
R_{s}, the observed time, t, becomes longer and longer to
maintain the balance of the equation. At R_{s} = R the metric
blows up and to all intents and purposes, the observed time
becomes infinite. Therefore, a person accelerating towards
a black hole sees their velocity growing at an increasing
rate as they approach it. However, due to time dilation,
an observer would see the person actually travelling slower
and slower and never actually reaching the event horizon.
This can be seen in more detail in the following diagram.
Consider 2 people - Bob and Alice - travelling in a spaceship.
The path the rocket takes in spacetime is a hyperbola. At point
A, Bob 'ejects' Alice from the spaceship while it continues
on its hyperbolic path.
Between A and B, Alice and Bob can see each other with no issue.
Once Alice passes through B, however, things change. Alice will
continue to be able to see Bob, albeit at an ever increasing
distance, but Bob will no longer be able to see Alice. Alice
can see Bob because light from Bob is travelling at a 45 degree
angle indicated by the dotted lines. However, light leaving
Alice at C is not traveling at a 45 degree angle and would have
to exceed c to reach Bob at D.
What actually happens is that if Bob looks back at Alice as she
zooms away, and before she arrives at B, he will perceive her to
be moving at a slower and slower rate as she approaches point B.
In the limit it appears to take an infinite amount of time for
Alice to reach B.
The other interesting situation is when R = 0. This is more
complex to explain and will not be discussed here. It is
sufficient to say that the point R = 0 represents a SINGULARITY
where descriptions of space and time are physically not well
understood.
Entropy
-------
Assume we want to add 1 photon (equivalent to a 'bit' of
information) to a black hole. For such a photon to interact
with a black hole, its wavelength has to be roughly equal to
R_{s}. Thus, E = hf = hc/R_{s}. From Einstein, we can say that:
ΔE = Δmc^{2}
∴ Δm = ΔE/c^{2} = hc/R_{s}c^{2} = h/R_{s}c
Using equation A., the radius, R, changes with mass as follows:
ΔR = 2ΔmG/c^{2}
= 2Gh/c^{3}R_{s}
∴ ΔRR_{s} = 2Gh/c^{3} which has the dimensions of AREA!
The surface area of the Schwarzchild sphere is:
A = 4πR_{s}^{2}
∴ ΔA = 8πR_{s}ΔR
= 8π2Gh/c^{3} after substitution.
If we consider that one photon constitutes one unit of entropy
then we can say that for ΔS photons:
ΔA = 8π2GhΔS/c^{3}
Or
ΔS = ΔAc^{3}/16πGh
Which can also be written in terms of the PLANCK LENGTH,
l_{P} = √(hG/2πc^{3}) = 1.616 x 10^{-35} m, as:
ΔS = ΔA/8l_{P}^{2}
This is the BEKENSTEIN FORMULA for a 'simple' black hole.
It basically says that the black hole entropy is proportional
to the AREA of its event horizon. Classically, entropy is
associated with volumes, so this is very different!
The Unruh Effect
----------------
The Unruh effect is a prediction of Quantum Field Theory.
It says that the vacuum when being accelerated is not the
empty state, but instead is filled with real particles. An
observer at rest sees the vacuum as a state with no real
particles present (on average there are fluctuations, but
they fluctuate around an average of zero particles present).
However, a uniformly accelerated observer has his own vacuum
which is different to the vacuum of the observer at rest.
From his point of view, the accelerating observer sees the
vacuum as containing a bath of real particles at a temperature
proportional to the acceleration. As a consequence, the
accelerating observer will observe blackbody radiation.
The Unruh effect also applies to detectors. For example,
all things being equal, a thermometer waved around in empty
space, will indicate a slightly higher temperature relative
to a its stationary value. The Unruh temperature is given
by:
T = ha/4π^{2}cK_{B}
Hawking Radiation
-----------------
Gravity and acceleration are related via the equivalence
principle. Close to the event horizon of a black hole,
virtual particle/antiparticle pairs resulting from vacuum
fluctuations are boosted by the intense curvature of
spacetime and become real particles. This process is
consistent with the Unruh effect. An observer near the
event horizon must accelerate to keep from falling in and
will see a bath of real particles that pop out of the horizon.
In contrast, an observer that free falls through the horizon
will see a quantum field state that looks like a vacuum state.
Thus, when quantum field theory is combined with curved
spacetime, the concept of a particle being present becomes
relative, not absolute. Now the bath of particles that the
accelerated observer sees contains energy and the only place
that this energy can come from is the black hole itself.
Normally, these particle/antiparticles annihilate each
other after a very short period of time and the total energy
is conserved. However, if some of these particles escape
before they can be annihilated, their antiparticles will
fall back through the event horizon. Since, the escaped
particles must have positive energy, the implication is
that the particles falling back in must have negative energy
if the total energy is to be conserved. This causes the
black hole to lose mass. The escaped particles constitute
HAWKING RADIATION.
Hawking radiation is black body radiation. Therefore, to
someone observing from a distance, it would appear that
the black hole is radiating, and therefore will have a
temperature. This temperature can be found from the laws
of thermodynamics. Temperature is defined in terms of
entropy and energy as:
k_{B}TΔS = ΔE
For a single photon ΔS = 1 and ΔE = hc/λ = hc/R_{s} = hc^{3}/2mG
Thus,
T = hc^{3}/2mGk_{B} = constant/m
A more rigorous proof leads to:
T = hg/4π^{2}cK_{B}
This, has the same form as the Unruh temperature.
It is worth noting that the formulas for entropy and
temperature both contain Planck's constant and therefore
demonstrate that both are quantum-gravitational effects.
Thus, the temperature of a black hole is finite and
inversely proportional to its mass. In other words, a
black hole has negative specific heat. A black hole of
about 5 solar masses would have a temperature of about
12×10^{-9} K. Since this is much less than the background
radiation temperature of about 2.7 K it would absorb
more radiation than it emits. The net influx of heat
energy adds to the mass of the black hole. As the mass
increases, the temperature falls further and the influx
of energy becomes greater and greater and so on.
For a 'small' black hole it is possible that the rate of
growth for the hole is less than the amount of emitted
Hawking radiation. This is saying that the temperature
of the emitted radiation is greater than the background
radiation temperature. Under these circumstances the
energy flow is in the opposite direction and the hole
'evaporates' away. This may explain why we never see
small black holes in reality. They form but evaporate
so quickly that we don't have time to observe them.
For example, a black hole with the mass of a car would
evaporate away in about 10^{-9} seconds.
Near the horizon there is a lot of 'quantum activity'.
There is chaos (entropy) and agitated motion. As a
result the temperature near (just outside) the event
horizon is extremely high. The black body temperature
discussed above is much lower than the horizon temperature
because the emitted radiation has been gravitationally
red shifted.
Bob and Alice Revisited
-----------------------
Previously we had said that Alice sailed through the
event horizon unscathed. Well, now we a have a conflict.
It would appear that Alice is not unscathed but gets
thermalized at the event horizon and is radiated out.
Which alternative is it? The answer is both. Bob sees
Alice thermalized near the horizon and and sees her
'bits' flying outward in the Hawking radiation. Alice
in her own frame of reference just falls through the
horizon unscathed. Since Alice and Bob can't communicate
there is no apparent contradiction. Bob thinks she's
dead while Alice can't communicate otherwise. Still,
something seems odd about this. How can we reconcile
these two scenarios. Let's consider the following.
Before Alice crosses the horizon, Bob decides to take
a look at her to see if she is being thermalized. In
order to do this, however, he has to hit her with short
enough wavelength radiation to resolve her position
near the horizon. Unfortunately, the energy associated
with these waves is enough on its own to thermalize her.
In essence it is Bob looking at Alice that thermalizes
her!. Therefore, we can now reconcile Bob and Alice's
situation by treating as a quantum effect, namely, the
act of observing a system disturbs the it!
Therefore, it seems there are 2 distinct versions of
the same reality. The 3 dimensions that Alice sees inside
the black hole and the 2 dimensions associated with the
thin surface of hot material which thermalizes her and
radiates her back out. So the question becomes, "is
there a way of representing the 3 dimensions inside the
black hole with the 2 dimensions associated with surface
of the event horizon. This is exactly what the HOLOGRAPHIC
PRINCIPLE is. Information stored on the surface of the
event horizon can be visualized in terms of a volume in
the same way that an optical holograms can represent 2D
images in 3D.
Black Hole Information Paradox
------------------------------
Information comes in bits. Bits are indestructible. They
can be ejected but this results in the addition of heat to
the environment that causes a corresponding increase in
entropy. In fact, for a computer LANDAUER's PRINCIPLE
states that:
E = kTln(2)
Where E is the energy to erase 1 bit and T is the temperature
of the circuitry. This is one reason why computers need to
be cooled.
Information consumed by a black hole would seem to violate
the idea of information conservation because once the
information is inside the black hole it can never escape.
The only way it could escape is via Hawking radiation but
this would require that a duplicate copy of the information
inside the event horizon exists outside of it. However,
having two copies of the information would violate quantum
theory.
Another issue concerns the 'monogamy of entanglement'. In
the Hawking process the ingoing information is necessarily
entangled with the outgoing information. However, the
outgoing information also needs to be entangled independently
with all Hawking radiation emitted in the past. However,
entanglement involving both is not allowed.
The Holographic principle is an attractive solution to both
problems since information just outside the 2D event horizon
can be regarded as a representation of infomation in the 3D
interior.
AdS/CFT Correspondence
----------------------
The Holographic principle got a large boost in 1997 with
the formulation of anti-de Sitter/Conformal Field Theory
correspondence (AdS/CFT) by Maldacena. Maldacena's original
example of AdS/CFT showed the correspondence between a
particular type of gravity in 5 dimensionsal anti-de Sitter
space and a 4 dimensional quantum field theory. de Sitter
space is the vacuum solution of Einstein's field equations
with a positive cosmological constant (expanding universe).
It represents a universe with zero to slightly positive
scalar curvature (curved spherically). anti-de Sitter
space is the vacuum solution of Einstein's field equations
with a negative cosmological constant (contracting universe).
It represents a universe with negative scalar curvature
(hyperbolic space). Let's look at this qualitatively in
terms of the following model.
Hyperbolic space can be viewed as a disk as illustrated
below.
One can define the distance between points of this disk
in such a way that all the triangles and squares are the
same size and the circular outer boundary are infinitely
far from any point in the interior.
Now imagine a stack of hyperbolic disks where each disk
represents the state of the universe at a given time. The
result is a solid cylinder in which any cross section is
a copy of the hyperbolic disk.
This construction describes a hypothetical universe with
only 2 space and 1 time dimension. However, hyperbolic
space can have more than 2 dimensions and one can "stack up"
copies of hyperbolic space to get higher-dimensional models
of anti-de Sitter space.
The key feature of the anti-de Sitter space is the boundary
of the cylinder. One property of this boundary is that,
locally around any point, it looks just like flat spacetime.
AdS/CFT conjectures that this boundary can be regarded as
the "spacetime" for a conformal field theory. A conformal
field theory (CFT) is a quantum field theory which is invariant
under conformal transformations. This means that the physics
of the theory looks the same at all length scales. Conformal
field theories care about angles, but not about distances.
Notice that the boundary of the cylinder has fewer dimensions
than anti-de Sitter space itself. For example, in the 3
dimensional example illustrated above, the boundary is a 2
dimensional surface.
The upshot of all this is that inside the cylinder we have
a gravitational theory in D dimensions and on the boundary
we have a QFT in D - 1 dimensions. These theories are
conjectured to be exactly equivalent, despite living in
different numbers of dimensions. Therefore, one can perform
calculations in one theory and translate those calculations
into calculations in the other theory.
Following Maldacena's discovery, physicists have discovered
many different realizations of the AdS/CFT correspondence.
The theories involved are generally not viable models of the
real world, but they have certain features which make them
useful for solving problems in quantum field theory and
quantum gravity. Regardless, AdS/CFT correspondence surely
has a bearing on the black hole information paradox.
Caveats
-------
Despite these advances in understanding there are still
problems. Crudely speaking, when a black hole evaporates
it will eventually reach a point where so information has
departed the black hole that not enough remains at the event
horizon for holography to represent the interior. In this
scenario, an in falling observer would crash into the
'firewall' just aboves the horizon and would be fried to
a crisp! This would indicate that the picture of black
holes painted by General Relativity is dramatically wrong.