Black HolesASTR 2110

Sarazin

Calculation of Curved Spacetime near Merging Black Holes

Test #2 Monday, November 13, 11 - 11:50 am Ruffner G006 (classroom) Bring pencils, paper, calculator You may not consult the text, your notes,

or any other materials or any person You 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 questions No essay questions

Black HolesASTR 2110

Sarazin

Calculation of Curved Spacetime near Merging Black Holes

Black Holes Gravity crushes star to a point = singularity Surrounded by ``surface’’ from which

nothing can escape = event horizon

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

− r dθ( )2− r sinθ dφ( )2

RS ≡2GMc2 = 3 km M

M"

#$

%

&'

¤

Black Hole Physics Nothing Escapes Event Horizon

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

backwards

ds( )2= c 1− RS

rdt

"

#$

%

&'

2

−dr

1− RSr

"

#

$$$$

%

&

''''

2

− r dθ( )2− r sinθ dφ( )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!!

Infall into Black Hole

infalling person

outside observer

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

Black Hole Physics Orbits near Black Hole

In general, complex. Consider circular orbits Newton: 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: 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 Holes Kerr 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 )

Spinning Black Holes Kerr Metric

Exterior Solution Event Horizon Oblate spheroid “ergosphere”

Can escape, but must rotate with black hole

top view

side view

Spinning Black Holes Kerr Metric

Interior Solution Ring singularity Inner Horizon!? But interior solution is

unstable

Spinning Black Holes Kerr Metric

Worm holes?? Unstable è closes

as soon as anything enters it!

white hole

black hole

Closed wormhole

Energy from Black Holes Penrose Mechanism

Rotating BH Throw something out

backwards Eout ≈ Ein + Δmc2 !! May not be important in

nature (too complicated), but …

Solution to Mankind’s Problems Biggest problems:

Get rid of garbage, pollution

Make energy Build city around

rotating BH

Energy from Black Holes Natural Method

Drop matter into BH carelessly Bump into, rub against other

matter Friction è KE è heat è light Virial 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 m mass m

Black Hole Physics Black Hole Surface Areas

When BHs merge, mass can increase or decrease Total surface area of BH horizons always

increases in GR 2nd Law of BH Dynamics Entropy always increases

S = c3k4G

A Entropy

T = c3

8πkGM Temperature

The Death of StarsASTR2110

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 supergiant Inert carbon/oxygen core

Low Mass Star - Death

AC

D

E

F

G

H

MS B

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

Low Mass Star - Death

AC

D

E

F

G

H

MS B

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

Low Mass Star - Death

AC

D

E

F

G

H

MS B

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

Planetary Nebula

Planetary Nebula

Planetary Nebula

Planetary Nebula

Planetary Nebula

Planetary Nebula

Planetary Nebula

Planetary Nebula

Planetary Nebula

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

WD cools, fades away

Low Mass Star - Death

AC

D

E

F

G

H

MS B

SupernovaeASTR2110

Sarazin

SN1987A - X-ray and Optical

High Mass Star (>8 M�)

Left as red supergiant with Fe core and many burning shells

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

High Mass Star - Death

Fe Core Fe 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 + νe , lots of neutrinos

Neutron Core Core 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 Supernova

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

Very, very dense

Energetics Virial Theorem: ENS = (1/2) PE

During infall, ΔE = ΔPE ≈ 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

Heavy Elements If explosion is efficient, most of materials outside of iron core are ejected.

Also, fusion during explosion makes more heavy elements

Supernova make and/or disperse most of the heavy elements in the Universe!!

But, iron core -> neutron star, core-collapse SN don’t disperse much iron

Final Outcome? Explosion very complex, but . . .

1.  If nearly all outer layers are ejected: neutron star, M ≈ 1.4 M�

2.  If enough of outer layers remain or fall back so that M ≳ 3 M� è Black Hole

Uncertain theoretical problem, but if initial mass of star is

1.  8 M� ≲ Mstar ≲ 20 M� -> neutron star

2.  20 M� ≲ Mstar -> black hole

Core-Collapse Supernova

•  Core of massive star (50 M8) collapses, forms black hole

•  Rotating material from star falls onto BH

•  Black holes shoots out jets of gas at 0.99995 c

•  Jets blast through star, blowing it up

•  If jets point in our direction, we see a gamma-ray burst

Alternative:Collapsar Model = Supernova Due to Jets

from a Black Hole

Collapsar Model = Supernova Due to Jets from a Black Hole