Black Holes: An IntroductionHistorical Highlights 1783: Rev. John Mitchell imagines the existence of...
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Black Holes: An Introduction Larry Price Astronomy Club 11/3/08
Transcript of Black Holes: An IntroductionHistorical Highlights 1783: Rev. John Mitchell imagines the existence of...
Black Holes: An IntroductionLarry Price
Astronomy Club
11/3/08
Historical Highlights
1783: Rev. John Mitchell imagines the existence of black holes.
~1907-1916: Einstein develops general theory of relativity.
1916: Karl Schwarzschild discovers an exact solution to Einstein’s equations.
1930: Subrahmanyan Chandrasekhar considers the end state of the collapse of stars.
1967: John Wheeler coins the term “black hole”.
The Reverend’s Idea
Objects can be so massive (have a strong enough gravitational field) that not even light can move fast enough to escape.
Aside I: Energy
Kinetic energy: Energy associated with the motion of an object.
K =1
2mv
2
Potential Energy: Energy stored in a system.
Kinetic energy:
Ug = !GmM
r
Aside II: Conservation of Energy
Energy cannot be created or destroyed!
Kinetic energy:
(K + U)initial = (K + U)final
Aside III: Escape Velocity
What speed is required to escape the gravitational pull of an object?
Kinetic energy:
Aside III: Escape Velocity
What speed is required to escape the gravitational pull of an object?
Kinetic energy:
Conservation of energy says:
(K + Ug)initial = 0
or
vescape =
!
2GM
r
The Reverend’s IdeaRevisited
What if the escape velocity is the speed of light?
Then
M =rc2
2Gr =
2GM
c2or
A Subtlety in the Argument
Why would light respond to gravity the same waynormal matter does?
Newton: Light is made of tiny particles (“corpuscles”).
Einstein: Gravity is really curved space and time.
General Relativity in a Nutshell
Gab = 8!Tab
Einstein’s equations:
General Relativity in a Nutshell
Gab = 8!Tab
Geometry Matter=
Einstein’s equations:
General Relativity in a NutshellSolutions to Einstein’s equations tell us how to measure distances invariantly.
Flatspace:
(!s)2 = !(c!t)2 + (!x)2 + (!y)2 + (!z)2
(think Pythagorean theorem)
General Relativity in a Nutshell
(!s)2 = !(c!t)2 + (!r)2 + r2(!!)2 + r2 sin2 !(!")2
Another way of writing flatspace:
A Black Hole Solution
(!s)2 = !
!
1 !
2GM
c2r
"
(c!t)2 +(!r)2
1 !
2GM
c2r
+r2(!!)2 + r2 sin2 !(!")2
(the Schwarzschild solution)
A Tour of Schwarzschild
(!s)2 = !
!
1 !
2GM
c2r
"
(c!t)2 +(!r)2
1 !
2GM
c2r
+r2(!!)2 + r2 sin2 !(!")2
r > 2GM
The “exterior” solution.
nothing funny here
A Tour of Schwarzschild
(!s)2 = !
!
1 !
2GM
c2r
"
(c!t)2 +(!r)2
1 !
2GM
c2r
+r2(!!)2 + r2 sin2 !(!")2
r > 2GM
The “exterior” solution.
A Tour of Schwarzschild
(!s)2 = !
!
1 !
2GM
c2r
"
(c!t)2 +(!r)2
1 !
2GM
c2r
+r2(!!)2 + r2 sin2 !(!")2
The “interior” solution.
r < 2GM
becomes positive
becomes negative
A Tour of Schwarzschild
(!s)2 = !
!
1 !
2GM
c2r
"
(c!t)2 +(!r)2
1 !
2GM
c2r
+r2(!!)2 + r2 sin2 !(!")2
The “interior” solution.
r < 2GM
A Tour of Schwarzschild
(!s)2 = !
!
1 !
2GM
c2r
"
(c!t)2 +(!r)2
1 !
2GM
c2r
+r2(!!)2 + r2 sin2 !(!")2
The event horizon.
r = 2GM
vanishes
blows up
A Tour of Schwarzschild
(!s)2 = !
!
1 !
2GM
c2r
"
(c!t)2 +(!r)2
1 !
2GM
c2r
+r2(!!)2 + r2 sin2 !(!")2
The event horizon.
r = 2GM
A Tour of Schwarzschild
(!s)2 = !
!
1 !
2GM
c2r
"
(c!t)2 +(!r)2
1 !
2GM
c2r
+r2(!!)2 + r2 sin2 !(!")2
The singularity.
r = 0
blows up
blows up
A Tour of Schwarzschild
(!s)2 = !
!
1 !
2GM
c2r
"
(c!t)2 +(!r)2
1 !
2GM
c2r
+r2(!!)2 + r2 sin2 !(!")2
The singularity.
r = 0
More General Black Holes
Black holes can only have three types of “hair”.
Mass
Charge
Angular momentum
Mandatory Movie
