black holes large and small - astro.sunysb.edu · kip thorne, black holes and time warps:...

Black Holes

I What is a black hole?

I Who first thought of theconcept of a black hole?

I Black holes and generalrelativity

I The “no-hair” theorem

I The event horizon

I Is there a singularity?

I Rotating black holes andthe ergosphere

I Formation of black holes

I Hawking radiation, entropy,and black hole evaporation

I Orbits around black holes

I Observing black holes

Wikipedia

J.M. Lattimer Black Holes Large and Small

What is a Black Hole?

Standard definition: A region of space from which nothing, not evenlight, can escape.

I Where does the escape velocity equal the speed of light?

vesc =

√2GMBH

RSch= c

This defines the Schwarzschild radius RSch to be

RSch =2GMBH

c2' 3

M

M�km

I The event horizon marks the point of no return for any object.I A black hole is black because it absorbs everything incident on the

event horizon and reflects nothing.I Black holes are hypothesized to form in three ways:

I Gravitational collapse of a starI A high energy collisionI Density fluctuations in the early universe

I In general relativity, the black hole’s mass is concentrated at thecenter in a singularity of infinite density.


John Michell and Black Holes

The first reference is by a geologist and Anglican priest, John Michell, ina letter written to Henry Cavendish, of the Royal Society, in 1783.

He argued that a Sun with 500 times its radius and the same densitywould be so massive that it’s escape velocity would equal light speed.

He reasoned, from observations of radiation pressure, that light, like amass, has inertia. If gravity affects light like its mass equivalent, lightwould not be able to escape and would return to the surface.

He proposed using a prism to measure the gravitational weakening ofstarlight due to the surface gravity of the star.

Michell, around 1783, designed the experiment now attributed toCavendish which first accurately measured the force of gravity betweenmasses. This resulted in the first accurate values for G and M⊕.

He invented the torsion balance for the experiment but didn’t live to putit to use. His device passed to Henry Cavendish who performed theexperiment in 1797-8.


Michell’s Apparatus – The Torsion Balance

M⊕ = gR2⊕/G

Cavendish 1798, Phil. Trans. Roy. Soc. Lon., 469

κθ = LF = LGmM/r2

T = 2π

√I

κ= 2π

√mL2

2κ

G =2π2Lr2

MT 2θ

Harvard Lecture Demonstration


More About John Michell

Michell also tried to measure the radiation pressure of light, but when hefocused sunlight onto a compass needle, it melted.

Other astronomical contributions include a study of parallax, which heacknowledged was too small to be presently observable, but would be inthe future. He discussed how measurement of a star’s distance andapparent magnitude could be used to determine a star’s true luminosity.

He also proposed the explanation for twinkling of starlight.

In geology, he proposed that earthquakes were experienced as siesmicwaves of elastic compression travelling through the Earth, anddetermined the epicenter of the 1755 Lisbon earthquake.

He also first suggested that tsunamis were caused by earthquakes.

He defined the Mesozoic stratigraphic layer in the UK.


Other Early Work on Black Holes

I The mathematician Pierre-Simon Laplace made similar arguments in1796 in the first two editions of his book Exposition du systeme duMonde, although the discussion was removed from later editions.

I Karl Schwarzschild used Einstein’s newly developed theory of generalrelativity, in 1915, to find a solution applicable for a point mass oroutside a spherical mass. (A few months later, Johannes Droste,Lorentz’s student, independently derived it.)

I The solution had a bad behavior (a singularity) at the so-calledSchwarzschild radius, but it was only later (1939) that Oppenheimer,Tolman and Volkov interpreted this radius as the boundary of abubble where time stopped. This led to the idea of “frozen stars”.

I In 1958, David Finkelstein identified the Schwarzschild radius withthe event horizon, a perfect unidirectional membrane: causalinfluence can cross it in only one direction.

I He and Martin Kruskal extended the Schwarzshild solution into theinterior of the event horizon (so it could be applied to infallingobservers) by means of a coordinate transformation.


Rotating and Charged Black Holes

I In 1963, Roy Kerr found an analytic solution for the spacetime forrotating black holes.

I In 1965, Ezra Newman found an analytic solution for charged and/orrotating black holes. However, it is not possible for black holes toattain a significant charge except in a complete vacuum.

I In 1967, John Wheeler coined the name black hole.

I Israel, Carter and Robinson evolved the no-hair theorem, whichstates that a black hole is completely described by just the mass,spin rate and electric charge. These are the only properties of ablack hole visible from the outside.

I The charge and spin of a black hole are limited:

Q2 +

(J

M

)2

≤ M2

I Violations of this limit lead to naked singularities which are notcloaked by event horizons. The cosmic censorship hypothesis forbidsthis from happening; it is supported by numerical simulations.


The Event HorizonThe boundary in spacetimethrough which matter andlight can only pass in onedirection. Thus,information cannot beextracted from inside it.

General relativity predictsthat spacetime is deformedsuch that particle paths arebent towards the mass. Atthe horizon, no paths leadaway from the black hole.

Time slows down near ablack hole, compared to adistant observer. This leadsalso to gravitationalredshift, as anticipated byMichell.

Wikipedia


The Singularity

The event horizon is not a true singularity, but the center is.

A non-rotating hole has a point singularity; a rotating one has a torussingularity. All the mass is contained within the singularity, which hasinfinite density.

Within a non-rotating hole, observers are inexorably carried into thesingularity. In the case of a charged or rotating hole, it is possible for anobserver to avoid the singularity and to re-emerge through the horizoninto another spacetime, or even into one’s own past. These possibilitiescan occur only if the black hole has perfect symmetry, which won’t occurwhen the observer falls in.

There is also a boundary, known as the photon sphere, where photonsmoving tangentially to it are trapped in a circular orbit. This lies outsidethe event horizon. For a non-rotating black hole, Rph = 3GMBH/c2. Theorbit is unstable; any perturbation will cause ejection or injection.

The appearance of singularities is usually perceived as as a breakdown ofgeneral relativity. However, it occurs in a domain in which quantummechanics ought to be important. A unified theory of gravity andquantum mechanics is not yet possible.


The ErgosphereA rotating black hole has a region,outside the event horizon, where itis impossible to “sit still”. Theergrosphere is an oblate spheroid.

Any rotating object in generalrelativity “drags” an observer(frame dragging), but inside theergosphere you’d have to travelfaster than c to avoid it.

If radiation or matter collides withinthe ergosphere, a projectile couldexit the ergosphere with moreenergy than it entered with. Theother projectile loses energy andfalls in the event horizon. ThisPenrose process permits energyextraction from a black hole, whichslows it down and lowers its mass.

A related process, theBlandford-Znajek mechanism,operates in the presence of alarge magnetic field. Thisprocess could be important inpowering active galactic nucleiand some gamma ray bursts.

Wikipedia


The Blandford-Znajek Mechanism

Kip Thorne, Black holes and time warps: Einstein’s outrageous legacy, Norton, 1994


Black Hole Thermodynamics and Evaporation

Stephen Hawking showed in 1974 that black holes can emit thermalradiation, which is caused by quantum mechanical tunneling through theevent horizon.

The radiation rate and spectrum is that of a perfect blackbody

L = 4πR2SchσT 4

BH

where the black hole temperature is

TBH =~c3

4GkB MBH.

Therefore, over time, black holes evaporate. However, black holes abovea certain mass (3× 10−8 M�, i.e., the Moon’s mass) have TBH < 2.7 Kand gain more mass by swallowing the cosmic background radiation thanthey lose by evaporation; they will grow with time.

Since L ∝ M−2BH the time-to-evaporation is τBH ∝ M3

BH . For a 1M� hole,TBH ' 10−7 K and τBH ' 1074 s.


Black Hole Entropy

Hawking showed in 1971 that the total area of the event horizons ofmerging black holes cannot decrease. This seems analagous to thesecond law of thermodynamics which states that the total entropy of asystem cannot decrease.

Bekenstein postulated that the area is equivalent to the entropy of ablack hole:

SBH =kB c3

4~GABH =

4πGkB

~cM2

BH

Hawking’s discovery is known as the second law of black holethermodynamics.

There are analgous zeroth and first laws of black hole thermodynamics ifmass acts as energy, area as entropy, and surface gravity as temperature:

gBH = constant, dMBH =gBH

8πdABH ; T = constant, dE = TdS

Gerard ’t Hooft and Leonard Susskind proposed the holographic principle,anything happening within a volume can be described by data on itsboundary, based on the connection between entropy and black hole area.


How Are Black Holes Created?

I Black holes are the end productof gravitational collapse.

I At the end of it’s life, a star hasa degenerate core in which mostof the pressure is not thermalbut due to the Fermi ExclusionPrinciple: no two fermions canoccupy the same momentumand spin state simultaneously.

I Electrons (fermions) pile on topof each other with greater andgreater energies, producingpressure that resists gravity.

I Chandrasekhar showed that theupper mass limit for adegenerate electron core isabout 1.4 M�, the so-calledChandrasekhar Mass MCh.

I A star with less than 5 or so M�sheds its outer envelope and is leftwith a degenerate core of less thanMCh, which cools into a stable whitedwarf.

I Larger stars have more massive cores.Once nuclear burning ceases, thesecores are unstable and collapse.

I Collapse halts due to the repulsivenuclear force: a proto-neutron star isborn and a shock is produced thatmay ultimately eject the star’senvelope.

I But a neutron star also has amaximum stable mass, estimated tobe between 2 and 3.2 M�. Remnantswith more mass will undergo asecond collapse, forming a black hole.


Detection of Stellar Black Holes

Most stellar black holes are members of X-ray binaries in which matter isaccreted from a donor star. Infalling matter onto the black hole releasesgravitational potential energy (up to about 30% of its rest mass energy).

The maximum rate of spherical accretion is given by the Eddington limit.This is where radiation pressure balances gravity. The radiative flux fromaccretion falls as 1/r2 as does gravity, so the Eddington limit dependsonly on the source’s mass M and the opacity κ of the accreting matter:

LEdd =4πcGM

κ= 1.3× 1038

(M

M�

)erg s−1.

With the blackbody formula L = 4πR2σT 4,the effective temperature of emission is

Teff =

(cGM

σκR2

)1/4

' 2× 107 K.

This corresponds hν = kB Teff ' 1.6 keV X-rays. Wikipedia


Determination of Black Hole Masses in Binaries

Kepler’s 1–2–3 Law

G (MX + MC )

a3=

(2π

Porb

)2

Companion velocity

vC =2π

PorbaC sin i

Center of mass

aMX = (MX + MC )aC

Combine these:

v3C Porb

2πG=

(MX sin i)3

(MX + MC )2= f

Note that MX > fC .

For Cyg X-1, i < 60◦,Porb = 5.6 days,f = 0.244± 0.005 M�, andthe mass of the B0 supergiantcompanion HDE 226868 isMC > 20 M�, leading toMX > 7 M�.

http://www.spacetelescope.org/extras/posters/cygnus−x1/


Confirmed Stellar Black Hole Masses

Casares (2006)

neu

tron

star

s

b

b

bM33 X7

b J1650-500

b


Intermediate Mass Black HolesIntermediate mass black holes havemasses 30 < M/M� < 30, 000. Todate, among the suspects:

I GCIRS 13E, which is orbiting0.4 light-years from thesupermassive black hole (SagA∗) at the Galaxy’s center. Ithas a mass of 1300 M� andlies within a cluster of 7 otherstars, a possible remnant of aglobular cluster stripped byinteractions with Sag A.Runaway stellar collisions mighthave produced this black hole.

I M82 X-1, about 500 M�,which is orbited by anevaporating red giant.

Both have been questioned.

0.1 lt-yr

New Scientist: Hazel Muir,Brad Hansen, Milos Milosavljevic

Gemini Observatory/AURA


Intermediate Mass Black Holes

In addition, it is suspected that many globular clusters might harborintermediate mass black holes. They are detectable fromlarger-than-normal velocities of stars near the cluster centers.


Supermassive Mass Black Holes

Supermassive black holes havemasses from 105 − 1010 M�.

Possibly all galaxies, including theMilky Way, have supermassiveblack holes at their centers.

Compared to stellar holes, tidalforces in their vicinity areconsiderably weaker.

They are detectable due to thefast Keplerian motion of nearbygas, stars and water masers.

There is a strong correlationbetween the velocity dispersion ofgalaxies and their central holemasses. The tightness indicates astrong feedback between holegrowth and bulge mass.

They are believed to be the”central engine” of active galacticnuclei, like Seyfert galaxies andquasars.

Wikipedia


The Center of the Milky Way

Wikipedia


Spitzer Infrared View of Galactic CenterNASA/JPL-Caltech/S. Stolovy (SSC/Caltech)


Supernova Remnants Near the Galactic Center

http://cassfos02.ucsd.edu/public/tutorial/MW.html


300 Light-Years Around the Galactic Center

Hubble/Spitzer


Stars Near the Galactic Center

ESO/S. Gillessen et al.

Stellar orbits

-�0.025 lt-yr


Measuring the Mass of Sgr A∗

Fitting positions and radialvelocities of star S2 yields a mass(4.1± 0.6)× 106 M�. The error isprimarily due to uncertainties in thedistance to the Galactic center.

S2

S2

6

?0.01lt-yr

630AU


Black Hole Cannibalism in NGC 3393

Two supermassive black holes(3× 107 M� and 106 M�)discovered orbiting at a separationof 490 light-years in the galaxyNGC 3393, about 1.6× 108

light-years distant. They willcollide in about 10,000 years.

X-ray: NASA/CXC/SAO/G. Fabbiano et alOptical: NASA/STScI

NASA/Hubble


Star Swallowed by Black Hole?

Swift

NASA/CXC/M.Weiss

GRB110328A wasn’t a normal GRB.It has persisted for months and originatesfrom a galactic core 3.8 billion light-yearsdistant. No previous evidence of X-ray orUV emissions. The burst is a jet from thetidal disruption of a star and we arelooking down the barrel.


Formation of Supermassive Black HolesA supermassive black hole ofabout 106 M� has been inferredto exist in a blue compact dwarfgalaxy Henize 2-10 which has aconcentrated region of extremestar formation.

It is actively accreting mass andemitting X-rays.

Although this dwarf galaxy isnearby, it resembles galaxiesthought to exist in the infantUniverse which don’t havesubstantial spheroidalcomponents.

Therefore, it appearssupermassive black hole birthprecedes the buildup of galaxyspheroids.

Reines et al., Nature, 2011


Can We Directly Detect an Event Horizon?

I Event horizon of Sgr A∗ has largest angular sizeof any black hole in the Universe, 10µ arc-sec.

I VLBI observations of Sgr A∗ show disc emissionon a scale of 37µ arc sec, already evidence forthe existence of an event horizon.

I Submillimeter VLBI observations might haveenough time resolution to detect periodicity inemissions from hot spots at the innermoststable circular orbit, which will allow thedetermination of the black hole spin.

I A proposedsubmillimeter VLBI“Event HorizonTelescope” will produceimages of the Galacticcenter showingsilhouettes predicted bygeneral relativisticlensing.

I The same techniquesare applicable to M87.

Simulated image7-telescope array

Fish & Doeleman (2010)

Model of 345 GHzemission

A. Broderick

Simulated image13-telescope array

Fish & Doeleman (2010)


NGC 3393


black holes large and small - astro.sunysb.edu · kip thorne, black holes and time warps:...

Documents