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/c2
- 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 objects 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 theorys

prediction of the quantum states in a black hole.

Falling Inside a Black Hole

- Observers P.o.V you freeze at the event

horizon, along with anything else the black hole

has every accreted. - Your P.o.V youre 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 rr vanishes.
- The lnr-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_equati

ons - http//en.wikipedia.org/wiki/Karl_Schwarzschild
- Relativity, Gravitation, and Cosmology, Second

Edition, Ta-Pei Cheng - PHZ4601 Lecture Notes, Fall 2012, Dr. Owens