Black Holes

If the mass of a star exceeds MOV, and if it does not lose mass during its evolution it can no longer reach any stable final state. The force of gravity will dominate over all other forces, and the star will collapse to a black hole. A black hole is black because not even light can escape from it. Already at the end of the 18th century Laplace showed that a sufficiently massive body would prevent the escape of light from its surface. According to classical mechanics, the escape velocity from a body of radius R and mass M is

This is greater than the speed of light, if the radius is smaller than the critical radius

The same value for the critical radius, the Schwarzschild radius, is obtained from the general theory of relativity. For example, for the Sun, RS is about 3 km; however, the Sun's mass is so small that it cannot become a black hole by normal stellar evolution. Because the mass of a black hole formed by stellar collapse has to be larger than MOV the radius of the smallest black holes formed in this way is about 5-10 km.

The properties of black holes have to be studied on the basis of the general theory of relativity, which is beyond the scope of this book. Thus only some basic properties are discussed qualitatively.

An event horizon is a surface through which no information can be sent out, even in principle. A black hole is surrounded by an event horizon at the Schwarzschild radius (Fig. 14.10). In the theory of relativity each observer carries with him his own local measure of time. If two observers are at rest with respect to each other at the same point their clocks go at the same rate. Otherwise their clock rates are different, and the apparent course of events differs, too.

Near the event horizon the different time definitions become significant. An observer falling into a black

Axis of rotation

Axis of rotation

Fig. 14.10. A black hole is surrounded by a spherical event horizon. In addition to this a rotating black hole is surrounded by a flattened surface inside which no matter can remain stationary. This region is called the ergosphere

hole reaches the centre in a finite time, according to his own clock, and does not notice anything special as he passes through the event horizon. However, to a distant observer he never seems to reach the event horizon; his velocity of fall seems to decrease towards zero as he approaches the horizon.

The slowing down of time also appears as a decrease in the frequency of light signals. The formula for the gravitational redshift can be written in terms of the Schwarzschild radius as (Appendix B)

2GM Rs

Here, v is the frequency of radiation emitted at a distance r from the black hole and vTO the frequency observed by an infinitely distant observer. It can be seen that the frequency at infinity approaches zero for radiation emitted near the event horizon.

Since the gravitational force is directed towards the centre of the hole and depends on the distance, different parts of a falling body feel a gravitational pull that is different in magnitude and direction. The tidal forces become extremely large near a black hole so that any material falling into the hole will be torn apart. All atoms and elementary particles are destroyed near the central point, and the final state of matter is unknown to present-day physics. The observable properties of a black hole do not depend on how it was made.

Not only all information on the material composition disappears as a star collapses into a black hole; any magnetic field, for example, also disappears behind the event horizon. A black hole can only have three observable properties: mass, angular momentum and electric charge.

It is improbable that a black hole could have a significant net charge. Rotation, on the other hand, is typical to stars, and thus black holes, too, must rotate. Since the angular momentum is conserved, stars collapsed to black holes must rotate very fast.

In 1963 Roy Kerr managed to find a solution of the field equations for a rotating black hole. In addition to the event horizon a rotating hole has another limiting surface, an ellipsoidal static limit (Fig. 14.10). Objects inside the static limit cannot be kept stationary by any force, but they must orbit the hole. However, it is possible to escape from the region between the static limit and the event horizon, called the ergosphere. In fact it is

Fig. 14.11. The arrow shows the variable star V1357Cyg. Its companion is the suspected black hole Cygnus X-1. The bright star to the lower right of V1357 is n Cygni, one of the brightest stars in the constellation Cygnus

Fig. 14.11. The arrow shows the variable star V1357Cyg. Its companion is the suspected black hole Cygnus X-1. The bright star to the lower right of V1357 is n Cygni, one of the brightest stars in the constellation Cygnus

possible to utilize the rotational energy of a black hole by dropping an object to the ergosphere in such a way that part of the object falls into the hole and another part is slung out. The outcoming part may then have considerably more kinetic energy than the original object.

At present the only known way in which a black hole could be directly observed is by means of the radiation from gas falling into it. For example, if a black hole is part of a binary system, gas streaming from the companion will settle into a disc around the hole. Matter at

Ergosphere Radius

Fig. 14.12. Scale drawings of 16 black-hole binaries in the Milky Way (courtesy of J.Orosz). The Sun-Mercury distance (0.4AU) is shown at the top. The estimated binary inclination is indicated by the tilt of the accretion disk. The colour of the companion star roughly indicates its surface temperature. (R.A.Remillard, J.E.McClintock 2006, ARAA 44, 54)

Fig. 14.12. Scale drawings of 16 black-hole binaries in the Milky Way (courtesy of J.Orosz). The Sun-Mercury distance (0.4AU) is shown at the top. The estimated binary inclination is indicated by the tilt of the accretion disk. The colour of the companion star roughly indicates its surface temperature. (R.A.Remillard, J.E.McClintock 2006, ARAA 44, 54)

the inner edge of the disc will fall into the hole. The accreting gas will lose a considerable part of its energy (up to 40% of the rest mass) as radiation, which should be observable in the X-ray region.

Some rapidly and irregularly varying X-ray sources of the right kind have been discovered. The first strong evidence for black hole in an X-ray binary was for CygnusX-1 (Fig. 14.10). Its luminosity varies on the

time scale of 0.001 s, which means that the emitting region must be only 0.001 light-seconds or a few hundred kilometres in size. Only neutron stars and black holes are small and dense enough to give rise to such high-energy processes. CygnusX-1 is the smaller component of the double system HDE 226868. The larger component is an optically visible supergiant with amass 20-25 M0. The mass of the unseen component has been calculated to be 10-15 M0. If this is correct, the mass of the secondary component is much larger than the upper limit for a neutron star, and thus it has to be a black hole.

Today 20 such systems are known, where the compact component has a mass larger than 3 M0, and therefore is probably a black hole. As shown in Fig. 14.12 these can be of very different sizes. Nearly all of them have been discovered as X-ray novae.

Many frightening stories about black holes have been invented. It should therefore be stressed that they obey the same dynamical laws as other stars - they are not lurking in the darkness of space to attack innocent passers-by. If the Sun became a black hole, the planets would continue in their orbits as if nothing had happened.

So far we have discussed only black holes with masses in the range of stellar masses. There is however no upper limit to the mass of a black hole. Many active phenomena in the nuclei of galaxies can be explained with supermassive black holes with masses of millions or thousands of millions solar masses (see Sect. 18.4 and 19.9).

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Responses

  • LALLI
    Which stars on the main sequence will become black holes?
    2 months ago

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