E. Decay of black holes by the Hawking process

According to Hawking (1975), every black hole of mass M decays by emission of thermal radiation and finally disappears after a time

T = (G^2 M^3 / hbar c^4). (28)

For a black hole of one solar mass the lifetime is

T = 10^64 yr. (29)

Black holes of galactic mass will have lifetimes extending up to 10^100 yr.

At the end of its life, every black hole will emit about 10^31 erg of high-temperature radiation. The cold expanding universe will be illuminated by occasional fireworks for a very long time.

F. Matter is liquid at zero temperature

I next discuss a group of physical processes which occur in ordinary matter at zero temperature as a result of quantum-mechanical barrier penetration. The lifetimes for such processes are given by the Gamow formula

T = exp(S) T_0, (30)

where T_0 is a natural vibration period of the system, and S is the action integral

S = (2/hbar) INT (2MU(x))^(1/2) dx.

(31) Here x is a coordinate measuring the state of the system as it goes across the barrier, and U(x) is the height of the barrier as a function of x. To obtain a rough estimate of S, I replace (31) by

S = (8MUd^2/hbar^2)^(1/2), (32)

where d is the thickness, and U the average height of the barrier, and M is the mass of the object that is moving across it. I shall consider processes for which S is large, so that the lifetime (30) is extremely long.

As an example, consider the behavior of a lump of matter, a rock or a planet, after it has cooled to zero temperature. Its atoms are frozen into an apparently fixed arrangement by the forces of cohesion and chemical bonding. But from time to time the atoms will move and rearrange themselves, crossing energy barriers by quantum-mechanical tunneling. The height of the barrier will typically be of the order of a tenth of a Rydberg unit,

U = (1/20)(e^4 m/hbar^2), (33)

and the thickness will be of the order of a Bohr radius

d = (hbar^2/me^2), (34)

where m is the electron mass. The action integral (32) is then

S = (2Am_p/5m)^(1/2) = 27A^1/2, (35)

where m_P is the proton mass, and A is the atomic weight of the moving atom. For an iron atom with A = 56, S = 200, and (30) gives

T = 10^65 yr. (36)

Even the most rigid materials cannot preserve their shapes or their chemical structures for times long compared with (36). On a time scale of 10^65 yr, every piece of rock behaves like a liquid, flowing into a spherical shape under the influence of gravity. Its atoms and molecules will be ceaselessly diffusing around like the molecules in a drop of water.

G. All matter decays to iron

In matter at zero temperature, nuclear as well as chemical reactions will continue to occur. Elements heavier than iron will decay to iron by varoius processes such as fission and alpha emission. Elements lighter than iron will combine by nuclear fusion reactions, building gradually up to iron. Consider for example the fusion reaction in which two nuclei of atomic weight 1/2 A, charge 1/2 Z combine to form a nucleus (A,Z). The Coulomb repulsion of the two nuclei is effectively screened by electrons until they come within a distance

d = Z^(-1/2) (hbar^2/me^2) (37)

of each other. The Coulomb barrier has thickness d and height

U = (Z^2 e^2 / 4d) = 1/2 Z^(7/3) (e^4 m/hbar^2). (38)

The reduced mass for the relative motion of the two nuclei is

M = 1/4 AM_p. (39)

The action integral (32) then becomes

S = (1/2 A Z^(5/3)(m_p/m))^(1/2) = 30 A^(1/2) Z^(5/6). (40)

For two nuclei combining to form iron, Z = 26, A = 56, S = 3500, and

T = 10^1500 yr. (41)

On the time scale (41), ordinary matter is radioactive and is constantly generating nuclear energy.

H. Collapse of iron star to neutron star

After the time (41) has elapsed, most of the matter in the universe is in the form of ordinary low-mass stars that have settled down into white dwarf configurations and become cold spheres of pure iron.

But an iron star is still not in its state of lowest energy. It could release a huge amount of energy if it could collapse into a neutron star configuration. To collapse, it has only to penetrate a barrier of finite height and thickness.

It is an interesting question, whether there is an unsymmetrical mode of collapse passing over a lower saddle point than the symmetric mode.

I have not been able to find a plausible unsymmetric mode, and so I assume the collapse to be spherically symmetrical.

In the action integral (31), the coordinate x will be the radius of the star, and the integral will extend from r, the radius of a neutron star, to R, the radius of the iron star from which the collapse begins. The barrier height U(x) will depend on the equation of state of the matter, which is vey uncertain when x is close to r. Fortunately the equation of state is well known over the major part of the range of integration, when x is large compared to r and the main contribution to U(x) is the energy of nonrelativistic degenerate electrons

U(x) = (N^(5/3)hbar^2/2mx^2), (42)

where N is the number of electrons in the star.

The integration over x in (31) gives a logarithm

log(R/R_0), (43)

where R_0 is the radius at which the electrons become relativistic and the formula (42) fails. For low-mass stars the logarithm will be of the order of unity, and the part of the integral coming from the relativistic region x < R_0 will also be of the order of unity. The mass of the star is

M = 2Nm_p. (44)

I replace the logarithm (43) by unity and obtain for the action integral (31) the estimate

S = N^(4/3) (8m_p/m)^(1/2) = 120N^(4/3). (45)

The lifetime is then by (30)

T = exp(120N^(4/3))T_0. (46)

For a typical low-mass star we have

N = 10^56, S = 10^77, T = 10^(10^76) yr. (47)

In (46) it is completely immaterial whether T_0 is a small fraction of a second or a large number of years.

We do not know whether every collapse of an iron star into a neutron star will produce a supernova explosion. At the very least, it will produce a huge outburst of energy in the form of neutrinos and a modest burst of energy in the form of x rays and visible light.

The universe will still be producing occasional fireworks after times as long as (47).

I. Collapse of ordinary matter to black holes

The long lifetime (47) of iron stars is only correct if they do not collapse with a shorter lifetime into black holes. For collapse of any piece of bulk matter into a black hole, the same formulae apply as for collapse into a neutron star. The only difference is that the integration in the action integral (31) now extends down to the black hole radius instead of to the neutron star radius. The main part of the integral comes from larger values of x and is the same in both cases. The lifetime for collapse into a black hole is therefore still given by (46). But there is an important change in the meaning of N. If small black holes are possible, a small part of a star can collapse by itself into a black hole.

Once a small black hole has been formed, it will in a short time swallow the rest of the star. The lifetime for collapse of any star is then given by

T = exp(120N_B^(4/3)) T_0, (48)

where N_B is the number of electrons in a piece of iron of mass equal to the minimum mass M_B of a black hole. The lifetime (48) is the same for any piece of matter of mass greater than M_B. Matter in pieces with mass smaller than M_B is absolutely stable. For a more complete discussion of the problem of collapse into black holes, see Harrison, Thorne, Wakano, and Wheeler (1965).

The numerical value of the lifetime (48) depends on the value of M_B. All that we know for sure is

0 <= M_B <= M_c, (49)

where

M_c = (hbar c/G)^(3/2) m_p^(-2) = 4.10^33 g (50)

is the Chandrasekhar mass. Black holes must exist for every mass larger than M_c, because stars with mass larger than M_c have no stable final state and must inevitably collapse.

Four hypotheses concerning M_B have been put forward:

(i) M_B = 0. Then black holes of arbitrarily small mass exist and the formula (48) is meaningless. In this case all matter is unstable with a comparatively short lifetime, as suggested by Zeldovich (1977).

(ii) M_B is equal to the Planck mass

M_B = M_PL = (hbar c/G)^(1/2) = 2.10^(-5) g. (51)

This value of M_B is suggested by Hawking's theory of radiation from black holes (Hawking, 1975), according to which every black hole loses mass until it reaches a mass of order M_PL, at which point it disappears in a burst of radiation. In this case (48) gives

N_B = 10^19, T = 10^(10^26) yr. (52)

(iii) M_B is equal to the quantum mass

M_B = M_Q = (hbar c/Gm_P) = 3.10^14 g, (53)

as suggested by Harrison, Thorne, Wakano, and Wheeler (1965). Here M_Q is the mass of the smallest black hole for which a classical theory is meaningful. Only for masses larger than M_Q can we consider the barrier penetration formula (31) to be physically justified. If (53) holds, then

N_B = 10^38, T = 10^(10^32) yr. (54)

(iv) M_B is equal to the Chandrasekhar mass (50). In this case the lifetime for collapse into a black hole is of the same order as the lifetime (47) for collapse into a neutron star.

The long-range future of the universe depends crucially on which of these four alternatives is correct.

If (iv) is correct, stars may collapse into black holes and dissolve into pure radiation, but masses of planetary size exist forever.

If (iii) is correct, planets will disappear with the lifetime (54), but material objects with masses up to a few million tons are stable.

If (ii) is correct, human-sized objects will disappear with the lifetime (52), but dust grains with diameter less than about 100 mu will last for ever.

If (i) is correct, all material objects disappear and only radiation is left.

If I were compelled to choose one of the four alternatives as more likely than the others, I would choose (ii).

I consider (iii) and (iv) unlikely because they are inconsistent with Hawking's theory of black-hole radiation.

I find (i) implausible because it is difficult to see why a proton should not decay rapidly if it can decay at all. But in our present state of ignorance, none of the four possibilities can be excluded.

The results of this lecture are summarized in Table I. This list of time scales of physical processes makes no claim to be complete. Undoubtedly many other physical processes will be occurring with time scales as long as, or longer than, those I have listed.

The main conclusion I wish to draw from my analysis is the following:

So far as we can imagine into the future, things continue to happen. In the open cosmology, history has no end.

TABLE I. Summary of time scales.

Closed Universe

Total duration 10^11 yr

Open Universe

Low-mass stars cool off 10^14 yr

Planets detached from stars 10^15 yr

Stars detached from galaxies 10^19 yr

Decay of orbits by gravitational radiation 10^20 yr

Decay of black holes by Hawking process 10^64 yr

Matter liquid at zero temperature 10^65 yr

All matter decays to iron 10^1500 yr

Collapse of ordinary matter to black hole

[alternative (ii)] 10^(10^26) yr

Collapse of stars to neutron stars or black holes [alternative (iv)] 10^(10^76) yr