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