Institute for Advanced Studies, Princeton New Jersey 08540
Reviews of Modern Physics, Vol. 51, No. 3, July 1979
(c) 1979 American Physical Society
Edited for Internet Science Education Project by Jack Sarfatti.
Quantitative estimates are derived for three classes of phenomena that may occur in an open cosmological model of Friedmann type.
(1) Normal physical processes taking place with very long time-scales.
(2) Biological processes that will result if life adapts itself to low ambient temperatures according to a postulated scaling law.
(3) Communication by radio between life forms existing in different parts of the universe.
The general conclusion of the analysis is that an open universe need not evolve into a state of permanent quiescence. Life and communication can continue for ever, utilizing a finite store of energy, if the assumed scaling laws are valid.
(*) This material was originally presented as four lectures, the “James Arthur Lectures on Time and its Mysteries” at New York University, Autumn 1978. The first lecture is addressed to a general audience, the other three to an audience of physicists and astronomers.
CONTENTS
Lecture I. Philosophy
Lecture II. Physics
A. Stellar evolution
B. Detachment of planets from stars
C. Detachment of stars from galaxies
D. Decay of orbits by gravitational radiation
E. Decay of black holes by the Hawking process
F. Matter is liquid at zero temperature
G. All matter decays to iron
H. Collapse of iron star to neutron star
I. Collapse of ordinary matter to black hole
Lecture III. Biology
Lecture IV. Communication
References
LECTURE I. PHILOSOPHY
A year ago Steven Weinberg published an excellent book, The First Three Minutes, (Weinberg, 1977), explaining to a lay audience the state of our knowledge about the beginning of the universe. In his sixth chapter he describes in detail how progress in understanding and observing the universe was delayed by the timidity of theorists.
This is often the way it is in physics – our mistake is not that we take our theories too seriously, but that we do not take them seriously enough. It is always hard to realize that these numbers and equations we play with at our desks have something to do with the real world. Even worse, there often seems to be a general agreement that certain phenomena are just not fit subjects for respectable theoretical and experimental effort. Alpher, Herman and Gamow (1948) deserve tremendous credit above all for being willing to take the early universe seriously, for working out what known physical laws have to say about the first three minutes. Yet even they did not take the final step, to convince the radio astronomers that they ought to look for a microwave radiation background. The most important thing accomplished by the ultimate discovery of the 3 K radiation background (Penzias and Wilson, 1965) was to force all of us to take seriously the idea that there _was_ an early universe.
Thanks to Penzias and Wilson, Weinberg and others, the study of the beginning of the universe is now respectable. Professional physicists who investigate the first three minutes or the first microsecond no longer need to feel shy when they talk about their work. But the end of the universe is another matter.
I have searched the literature for papers about the end of the universe (Rees, 1969; Davies, 1973; Islam, 1977 and 1979; Barrow and Tipler, 1978). This list is certainly not complete. But the striking thing about these papers is that they are written in an apologetic or jocular style, as if the authors were begging us not to take them seriously. The study of the remote future still seems to be as disreputable today as the study of the remote past was thirty years ago.
I am particularly indebted to Jamal Islam for an early draft of his 1977 paper which started me thinking seriously about the remote future. I hope with these lectures to hasten the arrival of the day when eschatology, the study of the end of the universe, will be a respectable scientific discipline and not merely a branch of theology.
Weinberg himself is not immune to the prejudices that I am trying to dispel. At the end of his book about the past history of the universe, he adds a short chapter about the future. He takes 150 pages to describe the first three minutes, and then dismisses the whole of the future in five pages. Without any discussion of technical details, he sums up his view of the future in twelve words:
The more the universe seems comprehensible, the more it also seems pointless.
Weinberg has here, perhaps unintentionally, identified a real problem. It is impossible to calculate in detail the long-range future of the universe without including the effects of life and intelligence. It is impossible to calculate the capabilities of life and intelligence without touching, at least peripherally, philosophical questions.
If we are to examine how intelligent life may be able to guide the physical development of the universe for its own purposes, we cannot altogether avoid considering what the values and purposes of intelligent life may be. But as soon as we mention the words value and purpose, we run into one of the most firmly entrenched taboos of twentieth-century science. Hear the voice of Jacques Monod (1970), high priest of scientific rationality, in his book Chance and Necessity:
Any mingling of knowledge with values is unlawful, forbidden.
Monod was one of the seminal minds in the flowering of molecular biology in this century. It takes some courage to defy his anathema. But I will defy him, and encourage others to do so. The taboo against mixing knowledge with values arose during the nineteenth century out of the great battle between the evolutionary biologists led by Thomas Huxley and the churchmen led by Bishop Wilberforce. Huxley won the battle, but a hundred years later Monod and Weinberg were still fighting Bishop Wilberforce’s ghost.
Physicists today have no reason to be afraid of Wilberforce’s ghost. If our analysis of the long-range future leads us to raise questions related to the ultimate meaning and purpose of life, then let us examine these questions boldly and without embarrassment. If our answers to these questions are naive and preliminary, so much the better for the continued vitality of our science.
I propose in these lectures to explore the future as Weinberg in his book explored the past. My arguments will be rough and simple but always quantitative. The aim is to establish numerical bounds within which the destiny of the universe must lie. I shall make no further apology for mixing philosophical speculations with mathematical equations.
The two simplest cosmological models (Weinberg, 1972) describe a uniform zero-pressure universe which may be either closed or open.
The closed universe has its geometry described by the metric
ds^2 = R^2 [dpsi^2 – dchi^2 – sin^2 chi dOmega^2], (1)
where chi is a space coordinate moving with the matter, psi is a time coordinate related to physical time t by
t = T_0 (psi – sin psi), (2)
and R is the radius of the universe given by
R = c T_0 (1 – cos psi). (3)
The whole universe is represented in terms of the coordinates (psi, chi) by a finite rectangular box
0 < psi < 2 pi, 0 < chi < pi. (4)
This universe is closed both in space and in time. Its total duration is
2 pi T_0, (5)
where T_0 is a quantity that is in principle measurable. If our universe is described by this model, then T_0 must be at least 10^10 years.
The simple model of a uniform zero-pressure open universe has instead of (1) the metric
ds^2 = R^2 [dpsi^2 – dchi^2 – sinh^2 chi dOmega^2], (6)
where now
t = T_0 (sinh psi – psi), (7)
R = c T_0 (cosh psi – 1), (8)
and the coordinates (psi, chi) extend over an infinite range
0 < psi < infinity, 0 < chi < infinity. (9)
The open universe is infinite both in space and in time.
The models (1) and (6) are only the simplest possibilities. Many more complicated models can be found in the literature. For my purpose it is sufficient to discuss (1) and (6) as representative of closed and open universes.
The great question, whether our universe is in fact closed or open, will before long be settled by observation. I do not say more about this question, except to remark that my philosophical bias strongly favors an open universe and that the observational evidence does not exclude it (Gott, Gunn, Schramm, and Tinsley, 1974 and 1976).
The prevailing view (Weinberg, 1977) holds the future of open and closed universes to be equally dismal. According to this view, we have only the choice of being fried in a closed universe or frozen in an open one.
The end of the closed universe has been studied in detail by Rees (1969). Regrettably I have to concur with Rees’ verdict that in this case we have no escape from frying. No matter how deep we burrow into the earth to shield ourselves from the ever-increasing fury of the blue-shifted background radiation, we can only postpone by a few million years our miserable end. I shall not discuss the closed universe in detail, since it gives me a feeling of claustrophobia to imagine our whole existence confined within the box (4).
Sarfatti note: Frank Tipler gives a new view of immortal life in a closed universe in his new book, The Physics of Immortality. The main idea is that conscious time can be infinite even if physical time is finite. And that super-advanced intelligence can use gravitational shear to generate infinite conscious time of virtual beings in a cosmic computer simulation. I only raise one question which may offer us a thin chance of survival. Supposing that we discover the universe to be naturally closed and doomed to collapse, is it conceivable that by intelligent intervention, converting matter into radiation and causing energy to flow purposefully on a cosmic scale, we could break open a closed universe and change the topology of space-time so that only a part of it would collapse and another part of it would expand forever? I do not know the answer to this question. If it turns out that the universe is closed, we shall still have about 10^10 years to explore the possibility of a technological fix that would burst it open.
I am mainly interested in the open cosmology, since it seems to give enormously greater scope for the activities of life and intelligence.
Horizons in the open cosmology expand indefinitely. To be precise, the distance to the horizon in the open metric (6) is
d = R psi, (10)
with R given by (8), and the number of galaxies visible within the horizon is
N = N_0 (sinh 2 psi – 2 psi), (11)
where N_0 is a number of the order of 10^10.
Comparing (11) with (7), we see that the number of visible galaxies varies with t^2 at late times. It happens by a curious numerical accident that the angular size of a typical galaxy at time t is
delta ~ 10^5 t^(-1) rad, (12)
with t measured in years. Since (11) and (7) give
N ~ 10^(-10) t^2, N delta^2 ~ 1, (13)
it turns out that _the sky is always just filled with galaxies_, no matter how far into the future we go. As the apparent size of each galaxy dwindles, new galaxies constantly appear at the horizon to fill in the gaps. The light from the distant galaxies will be strongly red-shifted. But the sky will never become empty and dark, if we can tune our eyes to longer and longer wavelengths as time goes on.
I shall discuss three principal questions within the framework of the open universe with the metric (6).
(1) Does the universe freeze into a state of permanent physical quiescence as it expands and cools?
(2) Is it possible for life and intelligence to survive indefinitely?
(3) Is it possible to maintain communication and transmit information across the constantly expanding distances between galaxies?
These three questions will be discussed in detail in Lectures 2, 3 and 4.
Tentatively, I shall answer them with a no, a yes, and a maybe.
My answers are perhaps only a reflection of my optimistic philosophical bias. I do not expect everybody to agree with the answers. My purpose is to start people thinking seriously about the questions.
If, as I hope, my answers turn out to be right, what does it mean? It means that we have discovered in physics and astronomy an analog to the theorem of Goedel (1931) in pure mathematics.
Goedel proved [see Nagel and Newman (1956)] that the world of pure mathematics is inexhaustible; no finite set of axioms and rules of inference can ever encompass the whole of mathematics; given any finite set of axioms, we can find meaningful mathematical questions which the axioms leave unanswered.
I hope that an analogous situation exists in the physical world.
If my view of the future is correct, it means that the world of physics and astronomy is also inexhaustible; no matter how far we go into the future, there will always be new things happening, new information coming in, new worlds to explore, a constantly expanding domain of life, consciousness, and memory.
When I talk in this style, I am mixing knowledge with values, disobeying Monod’s prohibition. But I am in good company. Before the days of Darwin and Huxley and Bishop Wilberforce, in the eighteenth century, scientists were not subject to any taboo against mixing science and values. When Thomas Wright (1750), the discoverer of galaxies, announced his discovery, he was not afraid to use a theological argument to support an astronomical theory.
Since as the Creation is, so is the Creator also magnified, we may conclude in consequence of an infinity, and an infinite all-active power, that as the visible creation is supposed to be full of siderial systems and planetary worlds, so on, in like similar manner, the endless immensity is an unlimited plenum of creations not unlike the known…. That this in all probability may be the real case, is in some degree made evident by the many cloudy spots, just perceivable by us, as far without our starry Regions, in which tho’ visibly luminous spaces, no one star or particular constituent body can possibly be distinguished; those in all likelyhood may be external creation, bordering upon the known one, too remote for even our telescopes to reach.
Thirty-five years later, Wright’s speculations were confirmed by William Herschel’s precise observations. Wright also computed the number of habitable worlds in our galaxy:
In all together then we may safely reckon 170,000,000, and yet be much within compass, exclusive of the comets which I judge to be by far the most numerous part of creation.
His statement about the comets may also be correct, although he does not tell us how he estimated their number. For him the existence of so many habitable worlds was not just a scientific hypothesis but a cause for moral reflection:
In this great celestial creation, the catastrophy of a world, such as ours, or even the total dissolution of a system of worlds, may be possibly be no more to the great Author of Nature, than the most common accident in life with us, and in all probability such final and general Doomsdays may be as frequent there, as even Birthdays or mortality with us upon the earth. This idea has something so cheerful in it, that I know I can never look upon the stars without wondering why the whole world does not become astronomers; and that endowed with sense and reason should neglect a science they are naturally so much interested in, and so capable of enlarging their understanding, as next to a demonstration must convince them of their immortality, and reconcile them to all those little difficulties incident to human nature, without the least anxiety.
There speaks the eighteenth century. But Steven Weinberg says,
The more the universe seems comprehensible, the more it also seems pointless.
If Weinberg is speaking for the twentieth century, I prefer the eighteenth. LECTURE II. PHYSICS
In this lecture, following Islam (1977), I investigate the physical processes that will occur in an open universe over very long periods of time. I consider the natural universe undisturbed by effects of life and intelligence. Life and intelligence will be discussed in lectures 3 and 4.
Two assumptions underlie the discussion.
(1) The laws of physics do not change with time.
(2) The relevant laws of physics are already known to us.
These two assumptions were also made by Weinberg (1977) in his description of the past. My justification for making them is the same as his. Whether or not we believe that the presently known laws of physics are the final and unchanging truth, it is illuminating to explore the consequences of these laws as far as we can reach into the past or the future. It is better to be too bold than too timid in extrapolating our knowledge from the known into the unknown. It may happen again, as it happened with the cosmological speculations of Alpher, Herman, and Gamow (1948), that a naive extrapolation of known laws into new territory will lead us to ask important new questions.
I have summarized elsewhere (Dyson, 1972, 1978) the evidence supporting the hypothesis that the laws of physics do not change. The most striking piece of evidence was discovered recently by Shlyakhter (1976) in the measurements of isotope ratios in ore samples taken from the natural fission reactor that operated about 2 billion years ago in the Oklo uranium mine in Gabon (Maurette, 1976). The crucial quantity is the ratio (149Sm/147Sm) between the abundances of two light isotopes of samarium which are not fission products. In normal samarium this ratio is about 0.9; in the Oklo reactor it is about 0.02. Evidently the 149Sm has been heavily depleted by the dose of thermal neutrons to which it was exposed during the operation of the reactor. If we measure in a modern reactor the thermal neutron capture cross section of 149Sm, we find the value 55kb, dominated by a strong capture resonance at a neutron energy of 0.1 eV. A detailed analysis of the Oklo isotope ratio leads to the conclusion that the 149Sm cross setion was in the range 55+-8 kb two billion years ago. This means that the position of the capture resonance cannot have shifted by as much as 0.02 eV over 2.10^9 yr. But the position of this resonance measures the difference between the binding energies of the 149Sm ground state and of the 150Sm compound state into which the neutron is captured. These binding energies are each of the order of 10^9 eV and depend in a complicated way upon the strengths of nuclear and Coulomb interactions. The fact that the two binding energies remained in balance to an accuracy of two parts in 10^11 over 2.10^9 yr indicates that the strength of nuclear and Coulomb forces cannot have varied by more than a few parts in 10^18 per year.
This is by far the most sensitive test that we have yet found of the constancy of the laws of physics.
The fact that no evidence of change was found does not, of course, prove that the laws are strictly constant. In particular, it does not exclude the possibility of a variation in strength of gravitational forces with a time scale much shorter than 10^18 yr. For the sake of simplicity, I assume that the laws are strictly constant. Any other assumption would be more complicated and would introduce additional arbitrary hypotheses.
It is in principle impossible for me to bring experimental evidence to support the hypothesis that the laws of physics relevant to the remote future are already known to us. The most serious uncertainty affecting the ultimate fate of the universe is the question whether the proton is absolutely stable against decay into lighter particles. If the proton is unstable, all matter is transitory and must dissolve into radiation.
Some serious theoretical arguments have been put forward (Zeldovich, 1977; Barrow and Tipler, 1978; Feinberg, Goldhaber, and Steigman, 1978) supporting the view that the proton should decay with a long half-life, perhaps through virtual processes involving black holes. The experimental limits on the rate of proton decay (Kropp and Reines, 1965) do not exclude the existence of such processes. Again on grounds of simplicity, I disregard these possibilities and suppose the proton to be absolutely stable. I will discuss in detail later the effect of real processes involving black holes on the stability of matter in bulk.
I am now ready to begin the discussion of physical processes that will occur in the open cosmology (6), going successively to longer and longer timescales. Classical astronomical processes come first, quantum-mechanical processes later.
_Note added in proof._ Since these lectures were given, a spate of papers has appeared discussing grand unification models of particle physics in which the proton is unstable (Nanopoulos, 1978; Pati, 1979; Turner and Schramm, 1979).
A. Stellar evolution
The longest-lived low-mass stars will exhaust their hydrogen fuel, contract into white dwarf configurations, and cool down to very low temperatures, within times of the order of 10^14 years.
Stars of larger mass will take a shorter time to reach a cold final state, which may be a white dwarf, a neutron star, or a black hole configuration, depending on the details of their evolution.
B. Detachment of planets from stars
The average time required to detach a planet from a star by a close encounter with a second star is
T = (rho V sigma)^(-1), (14)
where rho is the density of stars in space, V the mean relative velocity of two stars, and sigma the cross section for an encounter resulting in detachment. For the earth-sun system, moving in the outer regions of the disk of a spiral galaxy, approximate numerical values are
rho = 3.10^(-41) km^(-3), (15)
V = 50 k/sec, (16)
sigma = 2.10^16 km^2, (17)
T = 10^15 yr. (18)
The time scale for an encounter causing serious disruption of planetary orbits will be considerably shorter than 10^15 yr.
C. Detachment of stars from galaxies
The dynamical evolution of galaxies is a complicated process, not yet completely understood. I give here only a very rough estimate of the time scale. If a galaxy of N stars of mass M in a volume of radius R, their root-mean-square velocity will be of order
V = [GNM/R]^(1/2). (19)
The cross section for a close encounter between two stars, changing their directions of motion b a large angle, is
sigma = (GM/V^2)^2 = (R/N)^2. (20)
The average time that a star spends between two close encounters is
T = (rho V sigma)^(-1) = (NR^3/GM)^(1/2). (21)
If we are considering a typical large galaxy with N = 10^11, R = 3.10^17 km, then
T = 10^19 yr. (22)
Dynamical relaxation of the galaxy proceeds mainly through distant stellar encounters with a time scale
T_N = T (log N)^(-1) = 10^18 yr. (23)
The combined effect of dynamical relaxation and close encounters is to produce a collapse of the central regions of the galaxy into a black hole, together with an evaporation of stars from the outer regions. The evaporated stars achieve escape velocity and become detached from the galaxy after a time of the order of 10^19 yr. We do not know what fraction of the mass of the galaxy ultimately collapses and what fraction escapes. The fraction escaping probably lies between 90% and 99%.
The violent events which we now observe occurring in the central regions of many galaxies are probably caused by a similar process of dynamical evolution oprating on a much shorter time scale. According to (21), the time scale for evolution and collapse will be short if the dynamical units are few and massive, for example compact star clusters and gas clouds rather than individual stars. The long time scale (22) applies to a galaxy containing no dynamical units larger than individual stars.
D. Decay of orbits by gravitational radiation
If a mass is orbiting around a fixed center with velocity V, period P, and kinetic energy E, it will have energy by gravitational radiation at a rate of order
E_g = (V/c)^5 (E/P). (24)
Any gravitationally bound system of objects orbiting around each other will decay by this mechanism of radiation drag with a time scale
t_g = (c/V)^5 P. (25)
For the earth orbiting around the sun, the gravitational radiation time scale is
T_g = 10^20 yr. (26)
Since this is much longer than (18), the earth will almost certainly escape fom the sun before gravitational radiation can pull it inward. But if it should happen that the sun should escape from the galaxy with the earth still attached to it, then the earth will ultimately coalesce with the sun after a time of order (26).
The orbits of the stars in a galaxy will also be decaying by gravitational radiation with time scale (25), where P is now the period of their galactic orbits. For a galaxy like our own, with V = 200 km/sec and P = 2.10^8 yr, the time scale is —–>
T_g = 10^24 yr. (27) This is again much longer than (22), showing that dynamical relaxation dominates gravitational radiation in the evolution of galaxies.
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
