BLACK HOLES AS THE FINAL STATE IN THE EVOLUTION OF MASSIVE BODIES

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BLACK HOLES AS THE FINAL STATE IN THE EVOLUTION OF MASSIVE BODIES

Brandon Carter

To cite this version:

Brandon Carter. BLACK HOLES AS THE FINAL STATE IN THE EVOLUTION OF MASSIVE BODIES. Journal de Physique Colloques, 1973, 34 (C7), pp.C7-39-C7-47. �10.1051/jphyscol:1973707�.

�jpa-00215353�

JOURNAL DE PHYSIQUE Colloque C 7 , suppkment au no 11-12, Tome 34, Nouembre-Dkcembre 1973, page C7-39

BLACK HOLES AS THE FINAL STATE IN THE EVOLUTION OF MASSIVE BODIES

B R A N D O N CARTER,

D e p a r t m e n t o f A p p l i e d M a t h e m a t i c s a n d Theoretical Physics, Cambridge, England

R6sumC. - O n sait depuis 1931 qu'un corps dont la massed epasse une valeur critique (la limite d e Chandrasekhar), un peu plus grande que la masse du soleil, ne peut pas rester en equilibre gravitationnel a long terme, apres avoir epuise son energie thermique et son energie de turbulence et de rotation differentielle. A moins qu'elle ne perde assez de matiere pour amener sa masse en dessous d e la limite critique, il semble qu'une etoile massive doive finalement subir une contraction dramatique jusqu'a la formation d'un (( piege noir )) dans lequel le champ gravitationnel est si fort que mEme la lumiere ne peut pas echapper. I1 y a beaucoup de raisons de croire qu'une assez grande proportion d'etoiles massives perdent une partie de leur masse par des processus d'ejection, soit continus, soit explosifs. Cependant, il serait etonnant qu'il n'y ait pas une proportion iniportante d'ktoiles dans lesquelles subsisterait un noyau central de masse suffisante pour former un piege noir.

D'aprks notre comprehension actuelle de la theorie d'Einstein, le champ a l'exterieur du piege noir evoluerait vers un etat d'equilibre bien defini. L a forme de cet etat final est donne par la solution de Kerr et ne depend que de la masse et d u moment cinetique du piege noir. Nous venons d'acquerir une assez bonne idke theorique des proprietes mkcaniques d'un piege noir a u voisinage d'un Ctat d'equilibre. 11 y a une bonne analogie entre un pikge noir et un corps viscoelastique ordinaire.

I1 est evident par sa nature que la detection observationnelle d'un piege noir serait trks difficile.

Un piege noir peut se reveler indirectenlent par I'effet de son champ gravitationnel sur des corps voisins, comme par exemple dans un systeme binaire. Malheureusement cet effet ne suffit pas necessairement a distinguer un piege noir de plusieurs autres especes de corps de masse inferieure a la masse critique, mais dont la luminosite est relativement faible. Une autre maniere par laquelle un piege noir pourrait se reveler d'une f a ~ o n plus caracteristique est la radiation produite par la matiere environnante tombant sur lui. Ces deux critkres semblent &tre verifies par la source CygnusX-1. I1 est probable que cette source soit le premier cas convaincant de la decouverte obser- vationnelle d'un pikge noir.

Abstract. - It was first realised by Chandrasekhar that n o object above a critical mass not much larger than that o f the sun can remain in equilibrium under its own gravitational field in the long run after its thermal energy differential rotation ' I cetera have died away. It would appear, that unless it loses sufficient material to drop below this critical mass, any large star must ultimately collapse t o form a tr black hole )) in which the gravitational field is so strong that even light cannot escape. There is in fact much evidence that many large stars d o indeed lose much of their mass by both continuous and mass ejection processes. Nevertheless it would be very surprising if there were not a considerable proportion in which there remains a central core suffi- ciently large t o form a black hole.

According to our present understanding of Einstein's theory of gravity, the field outside the black hole should be expected to settle down towards a well defined equilibrium state. The form of this final state is described by the solution of Kerr ; and it depends only o n the mass and angular momentum of the hole. We are beginning to have a fairly good theoretical understanding of the properties of a black hole in o r near equilibrium. It turns out that its mechanical behaviour is closely analogous to that of a n ordinary visco-elastic body.

Observationally black holes are, by their nature, very difficult to detect directly. The presence of a black hole would be revealed by its gravitational e f e c t o n a nearby directly visible body, as in a binary system, but this effect would not necessarily distinguish a black hole from any other kind of object above this critical mass but with relatively low luminosity. However a black hole could also be revealed in a more characteristic manner by radiation produced by infalling matter from the surrounding medium. Both criteria are satisfied by the recently discovered X-ray source Cygnus X-1, which therefore appears to be the first convincing case of the observational detection of a black hole.

1. Introduction. - T h e t e r m black hole h a s been escape f r o m it. T h e flowering o f G e n e r a l Relativistic i n t r o d u c e d i n t o physics a n d a s t r o n o m y very recently black h o l e theory d u r i n g t h e last few years h a s been to describe a region o f spacetime where t h e gravi- inspired primarily by t h e ideas o f Wheeler in America t a t i o n a l field i s s o s t r o n g t h a t n o t even light c a n a n d Penrose in England. However o n this historically

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1973707

C7-40 BRANDON CARTER

motivated occasion it is appropriate to draw attention to the fact that the idea that matter could be hidden in what are now called black holes was originally put foreward in terms of the Newtonian theory of gravity and light by a Frenchman, namely Laplace.

In Newtonian theory, a particle projected with velocity u from the surface of a spherical body of mass M and radius R can escape to arbitrarily large distances only if

(in units such that Newton's constant G is unity).

Using the Newtonian theory in which light is consi- dered as consisting of particles ejected with a constant velocity c, Laplace thus deduced that a spherical body will be invisible from large distances whenever its radius is less than the critical value

any experimental evidence for believing in Einstein's theory rather than other rival relativistic gravitation theories which could lead to considerable differences in the strong fields near a black hole.) On the other hand there has been tremendous progress in our understanding of the microscopic structure and general properties of matter, about which almost nothing was known in Laplace's time. This has enabled us to develop a remarkably complete theo- retical picture of the evolution of astronomical bodies from diffuse gas clouds to the eventual formation of black holes of various masses. I shall devote the greater part of this talk to outlining this theoretical picture, which has been confirmed at many stages by a great wealth of accumulated astronomical obser- vation. It is to be emphasized that this picture is not dependent on any particular relativistic theory of gravity, and that as far as the bare essentials in the following description are concerned Newtonian gra- vitation theory (in conjunction with a relativistic description of the local properties of matter) will (in units such that c is unity), i. e. whenever it lies

be adequate.

within what is today known as the Schwarzschild

the following discussion I shall use (unrationa- radius.

lised) fundamental units in which not only c and G In his book, Expos S~stej?l (lu are set equal to one but where the Dirac form fi considered the conditions under which a real astro- of Planck7s constant is also set equal to one, in order physical body be hidden within its Schwarz- to concentrate attention on the theoretical essentials.

schild radius. The critical limit condition (2) can be The basic theory be formulated in terms of just expressed in terms a three fundamental parameters, 'namely the electroma- mean density of the object by gnetic (fine structure) coupling constant e2 -- 11137

(where we introduce the symbol

-

In the intervening period theoretical understanding had made great advances on two main fronts. On one hand the combined work of Maxwell and Einstein had produced a relativistic theory of light and gravi- tation, and hence a much deeper understanding of the detailed properties of black holes, an aspect which I shall discuss later on. (It should be noted that the more detailed predictions must in any case be regarded as very tentative. since there is not yet

the gravitational structure constant rn; -- and the mass ration tn,/tn, 111 800 where e is the charge of the electron and where m, and m, are the masses of the electron and the proton respectively.

2. Cold equilibrium states and the Chandrasekhar limit. - Provided that neither its temperature nor its pressure is too high, matter can be treated as being composed of a combination of light negatively charged electrons and heavy positively charged nuclei, with baryon number of order 102 a t most. At suffi- ciently low temperatures the matter will settle down to equilibrium as a crystalline or molecular solid, in which the positive and negative charges arrange themselves naturally in such a way that their mutual interaction is predominantly attractive, so that the mean potential energy per baryon will be given in rough order of magnitude by - e2 n'I3 where n is the baryon number density (which is equal, within a factor of 2 or so, to the electron number density).

This attraction is balanced by an effective repulsion between the electrons due to the exclusion principle which requires that they must have momenta whose average value is of the order of n1I3, which implies that there will be an average kinetic energy per baryon of order of ^tn;' 1 2 ~ ~ " Equilibrium will be achieved when the sum of the positive kinetic energy and the negative potential energy is minimised. It is clear that this occurs for a value of given by t ~ ' / ~

-

BLACK HOLES AS THE FINAL STATE 1N T H E EVOLUTION OF MASSIVE BODIES C7-41

(i. e. when the mass separation between electrons is of the order of the Bohr radius). Now (except in the most extreme conditions) the mass density p will be related to the baryon number density n by

We thus deduce that the equilibrium density of cold matter will be given in order of magnitude by

Of course the exact value will depend on the particular chemical constitution of the matter. Since there are geometric Iimitations on the efficiency with which the electrons are distributed over the available space (i. e. since there are in effect gaps between atoms) the above formula overestimates the density of materials such as solid hydrogen and liquid helium by a factor of order 10. F o r elements of higher atomic weight this correction factor is cancelled out by the results of concentration of charges in the nuclei, so that the formula (11) gives quite a good estimate of the mean density of a planet such as the earth (a few gm/cm3) which Laplace used as the basis of his estimate of the minimum size of a black hole.

Since he did not know of any states of matter with density higher than this it did not occur to Laplace that there could exist black holes with masses lower than the critical value

obtained by substituting (11) into (I). We now know that this estimate was unnecessarily conservative.

Laplace seems to have been encouraged to attach too much significance t o the characteristic planetary density (11) by the fact that the only large non pla- netary body about which he had information, namely the sun, has a mean density of the salue order of magnitude (close t o one gm/cm3). However in general, high temperatures or pressures can easily give rise t o equilibrium densities respectively very much smaller or very much larger than the value given by (11). It is essentially coincidental that the two effects cancel out in the case of the sun, although we shall see later that the mean density of an ordi- nary main sequence star can never differ from the value (11) by more than a factor of order lo2 either way.

Let us first consider the effects of high pressure, still restricting our attention to cold bodies for the time being. When pressure forces become large compared with the electrical attractions between particles, pressure ionisation will occur, so that the equilibrium density will be determined by balancing the pressure directly against the effective I-epulsive forces due t o the exclusion principle. Thus the pres- sure, P say, will be comparable to the kinetic energy

density, and will therefore be given by the degenerate electron gas formu!a

P

-

Now one can estimate the value P characteristic of the central regions of a self gravitating spherical body by balancing the corresponding characteristic pressure gradient -- PIR against a characteristic gravitational force density

-

p

-

^.

(Such an estimate would of course fail for a body with a central core which is so highly condensed that its equilibrium does not depend on the outer regions.

as for example in a highly evolved post-main sequence star. It will however give a reasonable order of magni- tude of a characteristic central temperature for simple objects such as main sequence stars, white dwarfs:

and neutron stars.) This leads to the estimate

for the characteristic density of a cold spherical body sufficiently massive for the long range gravitational attractions t o dominate the short range electrical attractions. Thus the formula (111) must be used in place of (11) whenever the latter gives a higher density, i. e. whenever the mass is greater than the critical value.

M

-

(whose significance was first pointed out by R. H.

Fowler). This is of the order of the niass of the planet Jupiter, which thus lies on the threshold separating the range of ordinary planetary bodies from the range of compressed degenerate electron gas stars.

The Inore massive end of this range includes the white dwarf stars in which the internal pressure has become so high that the electrons can remain dege- nerate even a t temperatures high enough for thermo- nuclear reactions t o take place. The range of stars governed by (111) terminates when the pressure becomes so high that the electrons beconie relativistic so that t o estimate the kinetic energy per particle the formula in,' ¹¹⁵¹³ must be replaced by n'I3, so that the pres- sure is given by the relativistic degenerate gas formula

in stead of ( 5 ) . Combining this with (6) we can deduce that the mass of a relativistic degenerate electron star is independent of the density being given by

It follows (:is was first discovered by Chandrasekh:~r) that this value, which is of the same order of magni-

C7-42 BRANDON CARTER

tude as the mass of the sun, represents an upper limit to the mass of a star which can be supported in a cold spherical equilibrium configuration by degene- rate electron pressure.

It has become clear from subsequent work (by Landau, Oppenheimer, and others) that Chandra- sekhar's limit gives the order of magnitude of the absolute maximum mass which can be supported against gravity in a cold spherical equilibrium state without forming a black hole by any possible kind of pressure. T o see why this is so let us consider how the pressure increases as the density of cold matter is increased above the critical value

at which the degenerate electrons become relativistic.

As soon as the electrons have energies large compared with their rest-mass, it becomes energetically favou- rable for them t o combine with the protons in the nuclei to form additional neutrons (since the diffe- rence between the two masses of a neutron and a proton is comparable with the mass of an electron).

As a result the electron number density becomes small compared with the baryon number density 11, so that the pressure drops significantly below the value given by the formula (8). It follows that the masses of the cold spherical equilibrium states with mean densities greater than the critical value (10) will actually decrease as the density is increased (which implies that these states cannot be stable). The pres- sure will not go u p sufficiently rapidly for the mass to start to increase again (with a consequent restoration of stability) until the mean density is comparable with the density in an atomic nucleus. At this stage the individual nuclei merge to form a continuous neutron fluid, so that there will be a neutron dege- neracy pressure which will be given by the formula

(obtained analogously t o the electron degeneracy formula (9, using the approximate equality of the neutron and proton masses). It follows from (6) that the corresponding neutron star densities are given by

p -- ^,?I; M ~ . (12)

Just as the white dwarf star range terminates when the degenerate electrons become relativistic, so does the neutron star range when, in an analogous manner, the degenerate neutrons become relativistic. Since the pressure of a relativistic degenerate gas does not depend on the mass of the particles, the same formula (9) applies to neutrons as t o electrons. It follows that the degenerate neutron star range terminates a t an upper mass of the same order of magnitude as the Chandrasekhar limiting value, (9). The corres- ponding limiting density will be given by the value

which thus satisfies the black hole limit condition (I).

This means that the radius R of a neutron star a t the upper end of the allowed mass range will be compa- rable with its Schwarzschild radius, i. e. of the order of a few km.

It should be pointed out that a t densities of the order of the limiting value (13) (which is greater by a factor of order 10' than the density of ordinary nuclear matter) there will be strong repulsive forces between nuclei which will tend t o increase the pressure significantly above the value given by (ll), while on the other hand hyperons will be created, which will tend t o reduce the pressure. Our physical knowledge is not yet sufficient to calculate these effects accurately, so that the exact neutron star mass limit remains uncertain by a factor of two o r more, and it is not yet known whether it is greater o r less than the white dwarf mass limit. Nevertheless the uncertainties are not important enough to effect the validity of the formulas (9) and (13) as order of magnitude estimates.

At densities above the critical value (13) our under- standing of the properties of matter breaks down completely. Fortunately however we d o not need to consider such densities since, by (I), any spherical body whose mass is greater in magnitude that the Chandrasekhar limiting value (9) will clearly be already within its Schwarzschild radius if its density is compa- rable or greater than the critical value (13).

This completes our demonstration of the theorem that onlj, cold spl~erical equilibrium states with mass greater it1 order of rnagnitwde than the C/~andrasekhar limit nzust be a black hole. The above reasonning has of course been heuristic rather than rigorous.

A rigorous proof could only be given subject t o par- ticular assumptions about the theory of gravity and for Einstein's General Relativity such a proof has in fact been given by Harrison, Thorne, Wakano and Wheeler. However the cruder heuristic argument given here has the advantage that it clearly applies to an)) plausible theory of gravity (as well as to the Newtonian theory used by Laplace, even though this can no longer be regarded as plausible in the present context).

None of the order of magnitude estimates in the present section would be significantly affected by such small departures from spherical symmetry as could be produced by rigid rotation. Nevertheless a body of arbitrarily large mass could reach a temporary equilibrium without forming a black hole if diffe- rential rotation or high internal temperature were allowed. However clearly neither of these effects can persist in the long run. Thus there is nothing to prevent the ultimate formation of black holes with masses as small as the Chandrasekhar limiting value, given in order of magnitude by (9) (i. e. by the inverse of the gravitational structure constant) which is smaller than the Laplace limit (4) by a factor (e2 m,/m,)3.

It was pointed out by Wheeler that it would in principle be possible to create a black hole with mass

BLACK HOLES AS THE FlNAL STATE IN THE EVOLUTION O F MASSlVE BODIES C7-43

even smaller than the Chandrasekhar limit by applying an external pressure. In a relativistic theory the speed of light hypersurfaces are characteristic surfaces limiting the rate at which causal influences and phy- sical particles can be propagated. Therefore in any relativistic theory (although not in the Newtonian theory used by Laplace) an object which finds itself within a black hole is literally trapped - it can never get out. Thus the creation of a black hole is an irre- versible process. It follows that even though external pressure would be required to force a body below the Chandrasekhar mass limit within its Schwarzschild radius, the body would necessarily remain within the black hole so formed even after the source of external pressure was removed.

At the present epoch it is difficult to envisage forces sufficiently strong to create black holes significan- tly below the Chandrasekhar mass limit. However Hawking has pointed out that sufficient pressures could have existed in the early stages of the big bang at the time of creation of the universe. Therefore turbulence or other forms of inhomogeneity at that time could conceivably have led to the formation of small black holes. Since the formation of a black hole is irrever- sible, such black holes would continue to exist today, although they would not necessarily have remained small. (They could have formed larger black holes by coalescence or accretion.)

Although there are no strong reasons for believing in the existence of sn~all black holes of the Wheeler- Hawking type, the theorem demonstrated in this section provides overwhelming reasons for taking seriously the idea that black holes above the Chan- drasekhar mass limit must be formed frequently as the final states in the evolution of large stars, when their differential rotation and thermal energy has been dissipated. I shall give a brief outline of the essentials of the theory of stellar evolution towards a final black hole state in the following section. Since dif- ferential rotation does not in fact seem to play an important role (due not so much to friction as to the ubiquitous effects of magnetic and convection) the discussion will be restricted to the thermal evolution of spherical stars.

3. Thermal evolution towards a final black hole state. - The basic process of stellar evolution can be thought of as consisting of the gravitational contrac- tion of an approximately spherical body at a rate which is controlled by the luminosity, i. e. by the rate of radiation of the thermal energy produced by the release of gravitational and nuclear energy.

The basic trend of this process can be represented as a steady movement from left to right in the log Mllog p diagram, which terminates when the body reaches a final state which will either be an ordinary cold equilibrium configuration or a black hole depending roughly on whether the mass is greater or less than the Chandrasekhar limit. The detailed evolutionary track

can be more complicated than this however, due to the possibilities of steady accretion or mass loss, not to mention the possibilities of fragmentation during the initial stages and of explosion during the final stages.

Let us start by considering the position in the diagram of a star whose central regions have tempe- ratures given in order of magnitude by some parti- cular characteristic value T say, which we shall take to be measured directly in energy units (i. e. in units such that Boltzman's constant k is set equal to one).

Let us suppose that T is sufficiently large for the ther- mal contributions to the pressure to be large compared with the exclusion principle contributions as given by (5) or (S), i. e. let us suppose the star is sufficiently hot to be non-degenerate. Then the characteristic pressure of the central regions will be given by

The first possibility occurs when the contribution of relativistic particles (which will mainly be photons under most conditions) is dominant. The second possibility occurs when the contribution of non- relativistic particles (whose number density will be of the same order of magnitude as the baryon number density n ) is dominant. It follows from the gravi- tational equilibrium condition (6) that the charac- teristic density p of a hot (i. e. non-degenerate) star with characteristic central temperature T will be given by

Thus we see that it is true generally for hot (i. e.

non-degenerate) stars as well as for cold ones, that

- 2

the inverse gravitational coupling constant, tn, , determines the order of magnitude of the upper limit to the range of masses within which the hot non- relativistic gas pressure law applies. It follows that there is a general tendency to instability in hot stars whose masses are of larger order of magnitude than this Chandrasekhar limit. This results from the fact that in an adiabatic expansion or compression of a non-degenerate relativistic (i. e. radiation pressure dominated) gas the ratio n / T 3 (which represents the baryon/photon number density ratio) is preserved, so that the pressure P will vary in proportion to p,,,.

Since this is the same power of p that appears in the pressure equilibrium eq. (6), it is clear that such a star can be only marginally stable, due to being able to vary (adiabatically) through a range of equilibrium states of differing densities while keeping its mass fixed. For states whose masses are only one or two powers of ten greater than the mass of the sun (which of the same order as the Chandrasekhar limit) the residual contribution of non-relativistic gas pressure is sufficient to tip the balance decisively in favour of

BRANDON CARTER

FIG. I. - Log MILog p diagram for the equilibrium states of an isolated non-rotating body of mass M and mean density p.

Lines labelled by Roman numerals are desined by the equations labelled by the same Roman numerals in the text. There can be no equilibrium states in the vertically shaded part of the diagram which is bounded by the black hole locus, I (since this part of the diagram would represent bodies lying within their Schwarzschild radii). Also there can be no equilibrium states in the diagonally shaded part at the diagram (which represents states which would have to be held together by external pressure).

stability. However there are many effects which can easily tip the balance the other way at various stages in the formation and evolution of stars which are significantly larger than this. T o start with radiation pressure may blow off the outer layers of a large mass condensing out of a diffuse cloud, thus making it difficult for large stars ever to form. Even if they do come into existence such stars have a strong ten- dency both to continuous mass loss and to fragmen- tation due to pulsational instabilities. Although these processes are very complicated and by no means well understood in detail, it is fairly clear that the /let result o f grailitatio~~al conclensatio~i from a large rliJirse cloud iil~tst be the ,for-~ization oj'relatiuely ~ t n a l l borli~s

W ~ O S L ' /i7asses ~tlill tlot differ 0~1 1,1at7y orrlers of' 171ag1ii- t~rcle fio~il the 11o11 relntioistic pr-c.ss~rrc liniit 111,~. When a gravitationally bound body has formed with mass safely below this limit it will be stable against any further fragmentation.

As a star contracts it will pass through three distinct temperature regimes which are particularly significant.

The first occurs when the temperature of the condens- ing gas cloud becomes comparable with the Rydberg ionisation energy, i. e. when T

-

contraction will take place at a more controlled rate.

The corresponding ionization isotherm in the mass density diagram is given by

this isotherm branches off from the cold body locus at the Fowler pressure ionisation limit.

The second temperature regime is that at which thermonuclear energy release takes place. Since the reaction rates are highly temperature sensitive they can autonlatically adjust so that the energy release exactly replaces the thermal energy lost by radiation, thus halting the contraction. The greater part of the energy release occurs in the hydrogen burning (main sequence) stage in which a star normally spends the greater part of its evolutionary lifetime (except possibly for a final stage of cooling when it is already close to a degenerate final state). The temperature at which thermonuclear reactions take place is deter- mined essentially by the energy required for tunnelling through the Coulomb repulsion barrier between the nuclei. This is given by the analogue of the Rydberg energy with the nuclear mass in place of the electron mass, from which it follows that thermonuclear reaction rates will become appreciable at tempe- ratures given in order of magnitude by T

-

Since the reactions are produced mainly by the nuclei in the high energy tail of the Maxwell energy

BLACK HOLES AS THE FINAL STATE IN T H E EVOLUTION O F MASSIVE BODIES C7-45

distribution, this formula rather overestimates the temperature for hydrogen burning ; it gives a better estimate for the burning of heavier nuclei, for which this tail effect is cancelled by correction f. ~ ~ c t o r s allowing for the larger mass and charge numbers.

The corresponding isotherm in the mass density diagram is given by

The third temperature regime of special importance is the temperature a t which electron positron pair creation, i. e. T

-

This isotherm is entirely confined to the relativistic pressure range M 2 m i 2 , and it intersects the cold body equilibrium locus a t the Chandrasekhar limit point. Any star which reaches this isotherm must either explode o r else collapse to form a black hole or a neutron star. Even if the outer layers explode, the central core which remains may still be larger than the white dwarf limit in which case it must still collapse to form a black hole o r a neutron star. On the other hand a star which is safely below the Chandrasekhar limiting mass can evolve relatively peacefully to a final white dwarf state without ever reaching the pair creation isotherm.

This completes our outline of stellar evolution theory. T o summarise what is already a summary, the basic picture is that stars form by condensing out of diffuse clouds, undergoing various fragmentation a n d mass loss processes until they become main sequence stars with masses in the range within a few order of magnitude of the limiting value (9). Subse- quently (and after further continuous or expIosive mass loss in the case of the more massive stars) they will end u p as white dwarfs, o r neutron stars with

- 2

masses comparable with o r below the limit, ^{H Z ,} , o r else they will end up as black holes with masses above, but within a few orders of magnitude of this limit.

It would be desirable to be able to make detailed statistical predictions of the relative proportions of white dwarfs, neutron stars and black holes which should be expected to exist in the universe at the present time. Unfortunately however this is quite beyond the scope of present-day astrophysics. T o start with there is n o reliable theory of stellar birth statistics, i. e. of the proportions of stars formed in different mass ranges. Moreover although some par- ticular niodels have been worked out in considerable numerical detail, there is not yet any sound general understanding of the final explosive stages of stellar evolution even for an isolated star. For close binary systems (which are the ones with offer the best hope of observational detection) the situation is even more complicated due to the possibility of mass transfer.

Finally, in addition to these essentially mathematical problems, there is the difficulty that we have insufi- cient theoretical physical understanding of the pro- perties of high density neutron-hyperon matter to calculate the exact upper limit for neutron star mass.

Despite there uncertainties there are strong reasons for believing in the widespread existence of black holes. Stars whose masses are comparable with or smaller than the Chandrasekhar limit, have evolu- tionary lifetimes comparable with or longer than the present age of tlie universe. However the more massive stars, i. e. those which are potentially capable of forming black holes, have very much shorter evolutionary lifetimes, which implies that the number which have existed in the past and already burnt out must be large compared with the number which exist at present. Since it is hard to believe that mass loss in the final stages is sufficiently efficient to leave cores which are always below the Chandrasekhar limit, one would therefore expect that black holes of the order of a few tens of times this liniit should occur commonly within the galaxy, possihly consti- tuting a significant fraction of the invisible ((( missing ))) mass of the galaxy.

The reasons for believing in existence of the much more massive black holes of the kind proposed by Lynden-Bell are much less compelling, since we have no generally acceptable theory either of galaxy formation and evolution, in general, or of quasars in particular. Attempts such as those of Fowler and others to explain quasars as superniassive stars, (occurring where the thermonuclear reaction isotherm approaches the black hole limit) suffer from dificulties due to the tendency to instability referred t o pre- viously. However if such objects d o exist one would certainly expect that they would collapse t o form massive black holes of the size originally envisaged by Laplace.

4. General relativistic theory of black holes. -

During the last decade there has been a large amount of theoretical work on black holes within the frame-

C7-46 BRANDON CARTER

work of Einstein's General Theory of Relativity.

A theorem of Penrose (1965) and subsequent work by Hawking makes it clear that gravitational collapse will inevitably give rise to some form of spacetime singularity where the predictive power of the Einstein theory breaks down. However their work also encou- rages belief in what Penrose has called the ((cosmic censorship hypothesis ⁾⁾ which is a conjecture that the singularities which arise are always hidden within the black holes, so that Einstein's theory is fully capable of dealing adequately with all the externally observable part of spacetime, i. e. the part outside the black holes.

Subject to this hypothesis it has been possible t o give rigourous proofs of a number of important properties of black holes. In particular it has been shown that the surface area of the horizon (i. e. the null hypersurface bounding the black hole) can never decrease, and that when two black holes coalesce the area of the combined black hole is greater than the sum of the areas of the two separate black holes before they come together. The area of a black hole plays a role rather analogous to that of the entropy in an ordinary thermally conducting viscous self- gravitating fluid body which is thermally insulated from its surroundings. The rigidly rotating (uncharg- ed) equilibrium states of such a body will depend on only two parameters, namely the total mass energy M and the total angular momentum J, and in a reversible (entropy conserving) transformation their variations will be related by d M ⁼ R d J where Q is the angular velocity. It turns out that the same is true of an (uncharged) black hole, except that the condition for reversibility is conservation of surface area instead of entropy. Moreover in a transformation of an viscous fluid body the local rate of entropy generation is proportional to the square of the rate of shear in the material ; analogously in a non- reversible transformation of a black hole caused purely by infalling gravitational radiation (with no addition of matter) the rate of increase in area is proportional to the square of the rate of shear of the null generators of the horizon forming the boun- dary of the black hole. These results (which are mainly due to Bardeen, Christodoulou, Hartle, Hawking, and myself) are described in more detail in reference [7].

Although it has not been rigourously proved, it seems fairly clear that even though it may have been formed by a fairly violent collapse, the resulting oscillations will be damped out rapidly by gravitational radiation so that the black hole will settle down to a final sta- tionary equilibrium within a timescale of the order of the time required for light to travel a distance equal to the relevant Schwarzschild radius. It has been proved that the final vacuum state must be axisym- metric, and there are strong reasons for believing that this state must be given by the two parameter solution of Einstein's equations found by Kerr in

1963, for which the area A angular velocity Q are given in terms of the basic parameters M and J by

where J is restricted by the condition

When the possibility of an external electromagnetic field is taken into account there are more conjectures and fewer proofs. However there are strong reasons for believing that if the black hole has n o total charge then it can have no magnetic field. If it does have a total charge, one expects that it should have a magne- tic dipole moment proportional to its angular mornen- tum, with the same gyromagnetic ratio as occurs in the simple Dirac theory of the electron. Moreover the condition that the final state be axisymmetric still applies in the presence of an external field.

5. Observations and their significance. - Since it is (by definition) invisible a black hole cannot be detected directly, but only through its external gra- vitational field (or conceivably by its external elec- tromagnetic field in the unlikely event of a collapsing star carrying a significant charge with it).

It has been suggested that the presence of a large number of black holes in a globular cluster could be detected through the effect that they would have in increasing the density and hence the brightness of the central regions, due to the fact that being more massive than most of the stars of the cluster (which are below the Chandrasekhar limit) they would tend t o concentrate in the central regions. So far attempts to detect this effect have given negative results. An observation of this kind in any case gives information rather about the birth and death statistics of massive stars than about black holes as such.

Since the processes leading t o the formation of black holes are not particularly sensitive t o the theory of gravity, one would need to be able to make detailed observations of individual black holes t o obtain a worthwhile check on Einstein's theory.

The situation in this respect is rather like the situation which arises in relation to gravitational waves.

The mere detection of such waves does not give a check on the theory. Moreover their non-existence at the strengths predicted by Einstein's theory (or alternatively their existence a t the very much greater strengths which Weber has claimed t o have detected) would pose major problems of a general astrophysical rather than merely a gravitational theoretical nature, of the same kind as would be posed by the absence (or the presence in excess numbers and sizes) of black holes.

The only chance of detecting an individual black

BLACK HOLES AS T H E FINAL STATE IN T H E EVOLUTION O F MASSIVE BODIES C7-47

hole with a few times the Chandrasekhar mass would be in a binary system, or if it were in a situation where it could accrete matter rapidly enough for the gra- vitational energy released to be detectable in the form of X-rays. Until recently searches of binary systems proved inconclusive since it is dificult to be sure that a n invisible partner is a black hole rather than some other kind of massive but not very luminous body.

(Since there are good reasons for believing that black holes are common it would however be reaso- nable t o suppose that a considerable proportion of such doubtful cases are in fact black holes.)

However the recently discovered source Cygnus X-1 appears to be a fairly clear cut case, consisting of a close binary system which is also a source of X-rays of the kind one would expect to be emitted by matter falling onto a compact body, and in which both partners have masses of the order of ten times the Chandrasekhar limit. Recent theoretical investi- gations of mass flow from a large star onto a compact (black hole o r neutron star) partner in a close binary configuration have been carried out recently by Shakura and Sungaev, Rees and Pringle, Davidson and Ostriker and others. Systems of this kind in which the compact partner of the X-ray binary is below

the critical mass limit, and hence presumably a neu- tron star, have alsc been discovered, namely Centaurus X-3 and Hercules X-1. These two sources pulsate with frequencies of the order of seconds, due (it is generally believed) to the effects of a non-axisymmetric magnetic field rotating with the neutron star. Since an uncharged black hole cannot have a magnetic field (and since even in a charged black hole the field is axisymmetric) one is led to the prediction that the source Cygnus X-l should not pulsate in this way, a prediction which is in fact confirmed by obser- vation.

Unfortunately the confirmation of this prediction cannot be regarded as a very discriminating test of General Relativity, since this is the kind of result one would expect t o obtain in many related gravi- tation theories, such as that of Brans and Dicke. In order to test Einstein's theory specifically it would be necessary to check much more detailed properties of the Kerr solutions. Very much detailed observations will be needed for this than those possible a t present.

Even though the relativistic effects in neutron stars are less extreme than in black holes, it is not unlikely that, being more tangible, the former may offer better opportunities for precise tests.

References

[I] LAPLACE, P. S., EXPOS du SystZme du Monde, (1798). [5] ZELODOVICH, Y. B. and NOVIKOV, I. D., Relativistic Astro [2] CHANDRASEKHAR, S., Introduction to the Study of Stellar physics, (Chicago, 1971).

Structure, (Chicago, 1939).

[3] SCHWARZSCHILD, M., Structure and Evolution of the Stars, [6] HAWKING, S. W. and ELLIS, G. F. R., The Large Scale

(Princeton, 1957). Structure of Spacetime, (Cambridge, 1973).

[4] HARRISON, B. K., THORNE, K. S., WAKANO, M. and WHEELER,

J. A., Gravitation Theory and Gravitational Collapse. [71 DEWIIT. B. and C., BLACK HOLES, Ed. (Gordon and Breach,

(Chicago, 1965). 1973).