Black Holes
Matthew Trimble
10/29/12
History
• Einstein Field Equations published in 1915.
• Karl Schwarzschild: physicist serving in
German army during WW1.
• Solved EFE for a non- rotating, spherical
source, and wrote a paper on quantum theory
while suffering from pemphigus on Russian
front.
• His solution is called the Schwarzschild Metric.
Schwarzschild Metric
Schwarzschild Metric
• Same metric can describe non- rotating black
holes.
• r*: Schwarzschild radius- the event horizon of
the black hole.
• r*= 2GM/c^2
• Observers outside cannot view events inside.
• 3km for the Sun, 9mm for the Earth.
Black Hole Formation
• Once matter is compressed smaller than r*,
collapse occurs, forming a black hole.
• The kind of pressures needed to do this are
typically found in Type II Supernova
explosions.
• r* is the point at which an object’s escape
velocity is c, meaning nothing can come back
out.
Vacuum Energy
• In empty space, pairs of particle-antiparticle
pairs appear and annihilate on the Planck
timescale: 5.39X10^-44 s
• Because this happens so quickly, this does not
violate the Uncertainty Principle.
• The short lifetime gives these particles the
name Virtual Particles.
Hawking Radiation
• Near a black hole, time is dilated enough that
these virtual particles last longer than the
Planck timescale.
• One particle can be released away from the
black hole, while the other falls in.
• By measuring positive energy particles, the
particle with negative energy had to fall into
the singularity, lowering the mass and energy
of the black hole.
Shrinking
• Because the blackbody temperature is
inversely proportional to the mass, the
Hawking Radiation causes the black hole to
shrink.
• This proportionality also means that a very
massive black hole radiates weakly, and can
easily overcome this loss through accretion.
Mini Black Holes
• With a very small M, these Hawking radiate
very quickly, meaning they will evaporate long
before they have a chance to accrete matter
and grow large.
Bekenstein-Hawking Entropy
• Derived using the blackbody temperature of
Hawking radiation.
• Entropy is also proportional to the number of
microstates.
• For a black hole, these microstates are the
number of ways a quantum black hole could be
formed.
• The B-H Entropy method agrees with M theory’s
prediction of the quantum states in a black hole.
Falling Inside a Black Hole
• Observer’s P.o.V: you freeze at the event
horizon, along with anything else the black
hole has every accreted.
• Your P.o.V: you’re time is the proper time (no
redshift), so you go right past r*.
• This is because the r*/r is a coordinate
singularity, not a physical singularity.
Eddington-Finkelstein Coordinates
• Singularity at r=r* vanishes.
• The ln|r-r*| term in the coordinates defines a
one way membrane.
• For advanced coordinates, particles can only
fall in.
• For retarded coordinates, particles can only
move out, theoretically defining a White Hole.
Conclusion
• Karl Schwarzschild was more dedicated to
physics than you ever will be.
• Black Holes are interesting objects that
require an abstract way of thinking in order to
explain them mathematically.
References
• http://en.wikipedia.org/wiki/Schwarzschild_metric
• http://en.wikipedia.org/wiki/Vacuum_energy
• http://en.wikipedia.org/wiki/Einstein_field_equation
s
• http://en.wikipedia.org/wiki/Karl_Schwarzschild
• “Relativity, Gravitation, and Cosmology”, Second
Edition, Ta-Pei Cheng
• PHZ4601 Lecture Notes, Fall 2012, Dr. Owens