Physics in the vicinity of black holes

Andrej Čadež, Massimo Calvani, Andreja Gomboc, Claudio Fanton,Uroš Kostić

Plan of the talk

• Black hole solutions

• Light-like and time-like orbits

• Astrophysical evidence

• Acreetion disks and line profiles

• Effects due to curved space-time and tidal interaction

Schwarzschild solution

Transformation to Novikov coordinates; Introduce:

ds2 12Mrdt2 11 2Mr dr2 r2d2 Sin2d2

T 2MR 122M 2M2

R2 1

12R2 132ArcCos2MR2 1

Rr2M

1 and find that the metric

ds2 dT2R2 1R2R2dR2 2d2 Sin2d2

Can be transformed into the above, where the coordinates are related as shown in next picture

Constant Schwarzschild coordinates are shown as colored curves and constant Schw. Time as black lines

6M 10M 14M 18M 22M26MR

10

20

30

40

T

Orbits in exterior Schwarzschild space-time

2 3 5 7 10 20 40 60rc2

GM

-0.4

-0.2

0.2

0.4

0.6

Ueffc2

Effective potential

Ueff 2

2r21 2G M

rc2 G M

r

space likeorbits:

2drd2 l2

2 r21 2G M

rc2 G M

r

1 c2

2 122c4

light likeorbits 0, 0, Limitl:12drd2 2

2r21 2G M

rc2 G M

r12c2

Angular momentum

l r2dd

Types of lightlike geodesics in the outer region of the Schwarzschild space-time

constants of motion: angular momentum, two components of orbit normal

Timelike geodesics are similar, except that there are also precessing Kepler type orbits

constants of motion: energy, angular momentum, two components of orbit normal

All equations of orbits are analyticaly solvable in terms of elliptic functions (Čadež&Kostič,Phys.Rev.D72,104024(2005))

L/=10• Left:fixed small angular momentum, increasing energy>c2

• Right:fixed >c2 energy, decreasing angular momentum

• Down bound <c2 energy

Kerr metric

Geodesic in the outer region of Kerr space-time are similar to those in the Schwarzschild space-time, except that orbital angular momentum is

coupled to the angular momentum of the black hole, which induces a precession of the orbit about the black hole spin axis with the angular

velocity proportional to 1/r3. As a result orbits can no longer be considered planar. For light-like orbits only two constants are known:

angular momentum and Carter’s constant, which suffices to analytically express only the projection of the orbit equation on the r- “plane” (Fanton,Calvani,deFelice,Čadež:PASJ49,159-169(1997)), the

coordinate and the time are not known to be expressible analytically.

Astrophysical evidence

• Stellar mass black holes are very small

• In order to find them, people were looking for binaries with one dark and very massive (more than 3 solar masses) component. A few were found, but it was very difficult to confirm the mass of the dark component, and, in particular, to exclude the possibility that the dark component is a neutron star.

• Quasars, the superluminous galaxies, that were known to have very small (at most a few light years or light months) central engines were theorized to be powered by massive (up to 109 Solar Masses) black holes. The first strong hint of the existence of galactic black holes came from Hubble space telescope:

RS 2G Mc2

3kmMSonceM

Akrecijski disk okrog središča galaksije NGC6251

(posneto 1999)

Središče galaksije NGC9822a

Aktivne galaksije so podobne kvazarjem in iz njih brizgajo podobni ultrarelativistični

curki, vidni v radijski svetlobiSpodaj radijska slika, levo optična

Aktivna galaksija M87 radijsko in optično; doplerskahitrost kroženja v disku

Akrecijski diski okoli kompaktnih zvezd – tako si predstavljamo

Radijska slika SS433

Disks and jets are ubiquitous, but to prove that they are formed around a black hole, one must show that disk material is orbiting at a velocity close to speed of light and we do not have the resolution to distinguish

between the approaching and receding part of the disk. When observing the disk at coarser resolution, the spectral contibutions from

approaching and receding parts blend into a Doppler broadened spectral line. Doppler broadenings of a few 1000km/sec were observed in some active galaxies, but still far from the near speed of light velocity expected in a relativistic dics around the black hole. The reason is that

the temperature in the disk increases toward the center and optical lines can no longer be produced in such hot regions. So must look at

X-ray spectra. First observations by ASCA (launched 1993)

Assume thin, optically thick disk and deduce radial emisivity law;line shapes are consistent with the emitting region very close to the

black hole

Čadež, Calvani,New Astronomy 5,69,2000

x

The X-ray line shape does not strongly select between different

viable models

Y

Zaresna črna luknja

Radijska slika galakticnega centra (NRAO, Jusef Zadek) levo in infrardeči

blišč v SgrA* (Genzel et. All, Nature 425,934,2003 )

b

Galactic center stars

Many flares in infrared and in X-rays have been observed in the Galactic center since, since the rate is close to 1 per day. A typical energy release per flare is of the order 1039.5erg = 3.5 1018gc2. Flares often exhibit periodic modultions on a time scale of 15 to 20 minutes and last a few thousand seconds. Can this modulation have something to do with the motion of a small source in the vicinity of the black hole?

How would one see a point source of light falling down the black hole? (Sorry, movies must be played outside Powerpoint)

Tidal capture of a solar mass star by a galactic black hole (rg/c=20s)

A simple fit to the infrared flare is possible if one assumes that an object of the size of an asteroid is critically captured by the black hole. This requires the object to continuously loose angular momentum and

energy until it reaches the critical angular momentum for capture

The effective potential as a function of time, felt by a gravitating body that is tidaly interacting with another body (the black hole); during the process energy is dissipated by tides and angular momentum is transfered between spin and orbit (Hut’s theory is well understood for classical stars, but still needs closer

scrutiny in connection with black holes)