Transcript Document

GENERAL RELATIVITY &
BLACK HOLES
by
Robert Nemiroff
Michigan Tech
Physics X: About This Course
• Officially "Extraordinary Concepts in Physics"
• Being taught for credit at Michigan Tech
o Light on math, heavy on concepts
o Anyone anywhere is welcome
• No textbook required
o Wikipedia, web links, and lectures only
o Find all the lectures with Google at:
 "Starship Asterisk" then "Physics X"
o
http://bb.nightskylive.net/asterisk/viewforum.php?f=39
GENERAL RELATIVITY: INTERVALS
Static, symmetric spacetime:
ds2 = g11 dr2 + g22 dθ2 + g33 dΦ2 +g44 dt2
ds2 = 0 is a "null path" where photons fly
General Relativity: Gravity near a mass
Principle Equation: The Schwarzschild metric
• Describes gravitation near
o point mass
o non-spinning
o not charged
o not gaining or losing mass
General Relativity: Curved Space
• Flam's Parabaloid
o visualizing space near point mass
o fake vertical dimension
o distances same as near black hole
General Relativity:
Gravity near a mass
The Schwarzschild metric
• r = radial distance from point mass (actually C/ 2 π)
• C = (Capital C) = circumference of orbit measured at r
• τ = time measured at r (called the proper time)
• t = time measured at infinity
• d = differential = "small change in"
• c = speed of light
• rs = Schwarzschild radius
• θ = angle from north pole
• φ = "spin angle" from prime meridian
General Relativity: Gravity near a mass
Schwarzschild radius:
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This is the radius of the event horizon
G = Newtonian gravitational constant
M = mass (could be measured in kilograms)
c = speed of light (or the maximum speed in vacuum)
rs ~ 3 km (M / Msun)
For the Earth, rs ~ 1 centimeter
Every object could be condensed inside its Schwarzschild radius
if we could push hard enough.
Black Holes: No Hair Theorem
A classical (GR) black hole is completely defined by its:
• Mass
• Spin
• Charge
As such, black holes are said to have "no hair."
Special cases:
• Spin=0 and Charge=0: Schwarzschild Black Hole
• Charge=0: Kerr Black hole
• Spin=0: Reissner–Nordström black hole
• Nothing=0: Kerr-Newman black hole
Black Holes: Event Horizon
The sphere of no return (no spin - Schwarzschild solution). The radius is the
Schwarzschild radius: rs.
Only a coordinate singularity: can go past there (but not back). In classical GR,
no knowledge comes to us from inside the event horizon. (Part of the No Hair
Theorem.)
Inside rs,
• there may or may not be a singularity (depends on QM).
• time and space reverse.
o it is only possible to move toward the center
Black Holes: Ergosphere
Rotating black holes have an spheroid
outside of their event horizon:
the ergosphere.
Properties:
• Inside - must rotate with the black hole faster than c (as tracked by a distant
observer)
• Outside - still must rotate with black hole
o called "frame dragging"
o person frame dragged feels less rotation
• It is possible to escape from ergosphere
• Penrose process; Blandford-Znajek process
o steals energy from rotating black holes
o could power extraterrestrial civilizations
o might power some quasars
Black Holes: Thermodynamics
Past GR, black holes are now thought to emit particles, and hence
have an equivalent temperature.
The surface area of a black hole event horizon is proportional to its
temperature.
High mass black holes are very cold -- do not lose much power to
evaporation.
Low mass black holes are very hot -- they can evaporate completely.
Black Holes: Information paradox
Can you store information in a black hole?
Let's say you borrow your king's favorite flash drive only to
mistakenly drop it into a black hole. You see the drive hovering just
outside of the event horizon. Can you go get it back?
1. Yes, don't be a wimp, just go get it.
2. No, it will fall into the black hole regardless.
3. Might the king be appeased by a nice fruit basket?
Black Holes: Information paradox
2. No, it will fall into the black hole regardless.
Actually, you will have a brief time when you can grab it back. After
that, though, it's theoretical curtains for the flash memory drive, even
if you can still see its image. Chasing after the image may well bring
you too close to black hole yourself to have a reasonable chance of
escaping. Lowering yourself into the black hole on a thick rope will
only lead to the rope breaking.
Black Holes: Information paradox
OK, the jump drive is gone. Can you get the information back from
the jump drive? If so, maybe you could copy that onto another jump
drive and avoid having to buy the king a nice fruit basket.
1. Yes, if you can see the drive, you can see the information on it,
and you can get this information back.
2. No, once the drive is irretrievably lost, so is the information.
3. I don't know and I've learned to live with uncertainty.
Black Holes: Information paradox
3. I don't know and I've learned to live with uncertainty.
No one is sure, although the present consensus appears to be that
the information might be somehow retrievable. For more
information, see the lecture on "Black Hole Evaporation".
Black Holes: Information paradox
Can you get information back from black hole?
• No:
o Good: Preserves no hair theorem
o Bad: Violates quantum unitarity?
• Yes:
o Information leaks out during evaporation
o Information explodes out at final evaporation explosion
o Information stored in Planck-sized remnant
o Information stored in break-away baby universe
o Information stored in past-future correlations
Black Holes: White Holes
GR equations of BH are time symmetric.
So matter can also physically come out of a black hole.
White holes have gravity just like black holes.
Some think white holes ARE just black holes.
White holes are candidates for the endpoints to worm holes.
Black Holes: Worm Holes
Also known as Einstein-Rosen Bridges
Practical existence is controversial:
Black Holes: Worm Holes
Could connect a black hole to a white hole and allow objects entering
the black hole to exit the white hole.
Need negative mass-energy to keep throat open.
Negative mass-energy common in cosmic strings, dark energy?
Might allow
• Faster than light communication
• Time travel
General Relativity: Black Holes
Black hole twin paradox
• very similar to special relativistic twin paradox
• one twin leaves and hangs out near a black hole
• other twin stays home in normal space
• BH twin returns -- finds home twin old and gray