lecture16 black holes - university of virginia · black hole physics “black holes have no hair”...

Black HolesASTR 2120

Sarazin

Calculation of Curved Spacetimenear Merging Black Holes

Test #1Monday, February 24, 11 - 11:50 amASTR 265 (classroom)Bring pencils, paper, calculatorYou may not consult the text, your notes,

or any other materials or any personYou may bring a 3x5 card with equations~2/3 Quantitative Problems (like

homework problems)~1/3 Qualitative Questions

Multiple Choice, Short Answer, Fill In the Blank questionsNo essay questions

Test #1 (Cont.)Equation/Formula Card:You may bring one 3x5 inch index card with

equations and formulae written on both sides.DO NOT LIST pc, AU, M¤, L¤, R¤

DO NOT LIST the stellar classifications OBAFGKMDO NOT INCLUDE ANY QUALITATIVE MATERIAL

(text, etc.)Give name on card

Test #1 (Cont.)Material:Chapters 7 (review), 13, 14, 15, 17, 18, 19.3, 23.3Basic Stellar Properties, Binary Stars, the Sun,

Stellar Spectra and Atmospheres, Stellar Interiors, Nuclear Energy, Stellar Evolution, Stellar Remnants, General Relativity(qualitative only)

Homeworks 1-4Know pc, AU, M¤, L¤, R¤ , OBAFGKMNo problem set week of Feb. 18-24 to allow study

for test

Test #1 (Cont.)

Review Session:Discussion sessionFriday, February 21, 2-3 pm

Black HolesASTR 2120

Sarazin

Calculation of Curved Spacetimenear Merging Black Holes

Schwarzschild Metric forBlack Hole

Karl Schwarzschild (1916):Spherical coordinates. Non-rotating black hole

ds( )2= c 1− 2GM

c2rdt

"

#$

%

&'

2

−dr

1− 2GMc2r

"

#

$$$$

%

&

''''

2

− r dθ( )2− r sinθ dφ( )2

ds( )2= c 1− RS

rdt

"

#$

%

&'

2

−dr

1− RSr

"

#

$$$$

%

&

''''

2


RS ≡2GMc2 = 3 km M

M"

#$

%

&'

¤

Black Hole PhysicsNothing Escapes Event Horizon

Radius and Time Reverse Roles!!Radius always goes down, time can go

backwardsds( )2= c 1− RS

rdt

"

#$

%

&'

2

−dr

1− RSr

"

#

$$$$

%

&

''''

2


r < RS ⇒RSr>1⇒ 1− RS

r imaginary, 1− RS

r

"

#$

%

&'

2

< 0

(dt)2 term negative (normally positive)(dr)2 term positive (normally negative)r↔ t radius and time reverse roles!!

Black Hole PhysicsNothing Escapes Event Horizon

"Now, here, you see, it takes all the running you can do, to keep in the same place. If you want to get somewhere else, you must run at least twice as fast as that!”

-- The Red Queen

Black Hole PhysicsCosmic Censorship?

There are no “naked singularities”,uncovered by a horizon XXX Rated

“The Naked Singularity”

See unnatural acts performed

“Really Hot” -Penthouse

Black Hole PhysicsNothing Really Happens @ Event Horizon

Infalling person:falls through horizon, nothing happens

Outside observer:first speed up, then slow down, redshift, fade, hang

forever just above



Outside observer:first speed up, then slow down, redshift, fade, hang forever

just above

Infall into Black Hole

infalling person

outside observer

infall slows, v è 0, ∞ redshift, fades away, hovers forever just above horizon

Black Hole PhysicsOrbits near Black Hole

In general, complex. Consider circular orbitsNewton: any radius possible

GR: No stable circular orbits r ≤ 3 RS

Black Hole Physics“Black Holes Have No Hair”

No matter what they swallow, only retain:

Black Hole Physics“Black Holes Have No Hair”

No matter what they swallow, only retain:Mass M, Charge Q, Spin Angular Momentum J

Interstellar matter, very high electrical conductivity è “shorts out” charge

Astrophysical Black Holes:Mass M, Spin Angular Momentum J

Spinning Black HolesKerr Metric (Boyer-Linquist Coordinates)

ds( )2= 1− RSr

ρ2

"

#$

%

&' c dt( )2

−ρ2

Δdr2 − ρ dθ( )2

− r2 +α 2 +RSrα

2

ρ2 sin2θ"

#$

%

&' sinθ dφ( )2

+2RSrα sin2θ

ρ2 cdt dφ

RS =2GMc2 Schwarzschild radius

α ≡JMc

(units of length)

ρ2 ≡ r2 +α 2 cos2θ (units of length)Δ ≡ r2 − RSr +α

2 (units of length2 )

J < G M2 / c

Spinning Black HolesKerr Metric

Exterior SolutionEvent HorizonOblate spheroid “ergosphere”

Can escape, but must rotatewith black hole

top view

side view


Interior SolutionRing singularityInner Horizon!?But interior solution is

unstable


Worm holes??

white hole

black hole


Worm holes??


Worm holes??Unstable è closes

as soon as anything enters it!

Closed wormhole

Energy from Black HolesPenrose Mechanism

Rotating BHThrow something out

backwardsEout ≈ Ein + Δmc2 !!

May not be important in nature (too complicated), but …

Solution to Mankind’s ProblemsBiggest problems:

Get rid of garbage, pollution

Make energyBuild city around

rotating BH

Energy from Black HolesNatural Method

Drop matter into BH carelesslyBump into, rub against other

matterFriction è KE è heat è lightVirial Thm.Heat ≈ (1/2) |PE| ≈ (1/2) GMm/RS

≈ (1/2) GMm / (2GM/c2)Eout = Heat ≈ mc2/4 !!!Accretion by BHs is biggest

important energy source in Universe

mass mmass m

Black Hole PhysicsBlack Hole Surface Areas

When BHs merger, mass can increase or decreaseTotal surface area of BH horizons always

increases in GR2nd Law of BH DynamicsEntropy always increases

S = c3k4G

A Entropy

T = c3

8πkGM Temperature

The Death of StarsASTR2120

Sarazin

Cat’s Eye Planetary Nebula

Low Mass Star (1 M�)Left as AGB (red supergiant) with C/O core, giant H envelope

H envelope, ~ few AU

C/O core, ~ 109 cm, degenerate & inert

Asymptotic Giant Branch = Supergiant (I)Red supergiantInert carbon/oxygen core

Low Mass Star - Death



AC

D

E

FG

H

MSB

Low Mass Star (1 M�) - Cont.Inert core, must die

Envelope very extended, cool.

Hydrogen recombines: H+ + e- ® H + 13.6 eV

Gravitational binding energy per atom = G M mH / R » 10 ev (M / M�)(R/AU)-1

H shell ejected, (1/2) mHv2 ~ few eV, v ~ 20 km/s

Planetary Nebula


AC

D

E

FG

H

MSB

Planetary Nebula & White Dwarf

Degenerate core revealed, becomes a white dwarf

C/O for Sun

other composition for other masses

Core initially very hot, Teff ~ 100,000 K, lots of UV, ionizes planetary nebula

Fast wind from WD sweeps envelope into shell


AC

D

E

FG

H

MSB



C/O for Sun




Planetary Nebula



C/O for Sun




WD cools, fades away


AC

D

E

FG

H

MSB

SupernovaeASTR2120

Sarazin

SN1987A - X-ray and Optical

High Mass Star (>8 M�)Left as red supergiant or WR starwith Fe core and many burning shells

A) Inert Fe core, many burning shellsB) Either Red Supergiant or blue WR star

High Mass Star - Death

Fe CoreFe core, no fuel left, inert core, supported by electron degeneracy pressure

Fe core ~ Fe white dwarf (outer envelope extended, weight not so large)

Mcore ≳ 1.4 M� Chandrasekhar mass ®implodes!

Heat, Fe nuclei ® p, n

p+ + e- ® n + ne , lots of neutrinos

Neutron CoreCore is now mainly neutrons

Core collapses until R ~ 10 km, neutrons become degenerate, degeneracy pressure

Neutron core bounces, hits infalling gas at v ~ c/3

Shock from bounce plus neutrinos blow out envelope of star

Core-Collapse Supernova!!

Core-Collapse Supernova

Core-Collapse SupernovaCore becomes a neutron star (at least initially)

Very, very dense

EnergeticsVirial Theorem: ENS = (1/2) PE

During infall, DE = DPE » PE

So, Ebinding = (1/2) |PE| » (1/2) G M2/R » G(1.4 M� )2 / (10 km) is released

Etot » 3 x 1053 ergs released in supernova!!

Released from dense, NS region, only neutrinos can escape!

Etot » 3 x 1053 ergs in neutrinos, main energy of core-collapse supernova!!

Energetics (Cont.)Shock and neutrinos blow off envelope of star at ~ 10,000 km/s

Eexplsion » 1051 ergs in kinetic energy

Explosion heats material to 109 K initially. Cools rapidly, but very bright initially

L (optical, max) » 1010 L� !!

Would fade rapidly, but radioactivity keeps bright for months

Elight » 1049 ergs in light!!

Energetics (Cont.)Amazing result: most of energy comes out in neutrinos

lecture16 black holes - university of virginia · black hole physics “black holes have no hair”...

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