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\^\ \ \ \ \ \ \ \ \ \ \ \ \ TIME WITHOUT END: PHYSICS AND BIOLOGY \^\,
\+
\^\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ IN AN OPEN UNIVERSE (*) \^\,
\+
\,
\+
\^\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ Freeman J. Dyson \^\,
\+
\^\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ Institute for Advanced Studies, \^\,
\+
\^\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ Princeton New Jersey 08540 \^\,
\+
\,
\+
\^\ \ \ \ \ Reviews of Modern Physics, Vol. 51, No. 3, July 1979 \^\,
\+
\6\^\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ C \11979 American Physical Society \^\,
\+
\,
\+
    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 
\+
conlusion \ 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. \,
\+
\,
\+
    \,
\+
\3    (*) This material was \ originally presented as four lectures, \/
\+\1
\3the "James Arthur Lectures on Time and its Mysteries" at New York 
\+\1
\3University, \ Autumn \ 1978. \ The \ first \ lecture \ is \ addressed to 
\+\1
\3a general audience, the other three \ to an audience of physicists \/
\+\1
\3and astronomers. \,
\+\1
\,
\+
\,
\+
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 \,
\+
    \,
\+\4
\1    A year ago \ Steven Weinberg published an \ excellent book, \4The \/
\+\1
\4First \ Three \ Minutes \ \1(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. \,
\+
    \,
\+
\3    "This is often the way it \ is in physics - our mistake is not \/
\+\1
\3that we take our theories too \ seriously, but that we do not take \/
\+\1
\3them seriously \ enough. It is \ always hard to \ realize that these 
\+\1
\3numbers and equations we play with at our desks have something to 
\+\1
\3do \ with the \ real world. \ Even worse, \ there often \ seems to \ be 
\+\1
\3a general \ agreement \ that \ certain \ phenomena \ are \ just not fit 
\+\1
\3subjects \ for \ respectable \ theoretical \ and experimental effort. 
\+\1
\3Alpher, Herman \ and Gamow (1948) deserve \ tremendous credit above 
\+\1
\3all for being \ willing to take the early \ universe seriously, for 
\+\1
\3working out what known physical laws \ have to say about the first 
\+\1
\3three \ minutes. Yet \ even they \ did not \ take the \ final step, to 
\+\1
\3convince \ the \ radio \ astronomers \ that \ they \ ought \ to look for 
\+\1
\3a microwave \ radiation \ background. \ \ The \ most \ important \ thing 
\+\1
\3accomplished \ by \ the \ ultimate \ discovery \ of \ the 3 K radiation 
\+\1
\3background (Penzias and \ Wilson, 1965) was to force \ all of us to \/
\+\1
\3take seriously the idea that there \4was \3an early universe." \,
\+\1
    \,
\+
    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 \/
\+
\4Chance and Necessity\1: \,
\+
\,
\+
    "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 \,
\+
\0\ \ \ \ \ \ \ \ \ \ \ \ (\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ )\,
\+\2\ \ \ \ \ \ 2\ \ \ \ 2\ \ \ 2\ \ \ \ \ 2\ \ \ \ \ \ 2\ \ \ \ \ 2
\1    ds\ \ = R\ \0|\1d\7j\ \ \1- d\7c\  \1- sin\  \7c \1d\7W\ \0| \1,                  (1)\,
\+
\0\ \ \ \ \ \ \ \ \ \ \ \ 9\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0\,
\+\1
\,
\+
where \7c \1is a space coordinate moving with the matter, \7j \1is a time \/
\+
coordinate related to physical time t by \,
\+
\,
\+
    t = T\ (\7j \1- sin \7j\1)                                   (2)\,
\+\2\ \ \ \ \ \ \ \ \ 0
\1\,
\+
and R is the radius of the universe given by \,
\+
\,
\+
    R = cT\ (1 - cos \7j\1).                                 (3)\,
\+\2\ \ \ \ \ \ \ \ \ \ 0
\1\,
\+
    The whole universe is represented in terms of the coordinates \/
\+
(\7j\1, \7c\1) by a finite rectangular box \,
\+
\,
\+
    0 < \7j \1< 2\7p \1,    0 < \7c \1< \7p                           \1(4)\,
\+
\,
\+
    This universe is closed both in \ space and in time. Its total \/
\+
duration is \,
\+
\,
\+
    2\7p\1T\  ,                                              (5)\,
\+\2\ \ \ \ \ \ \ 0
\1\,
\+
where T  \ is a quantity that \ is in principle \ measurable. If our 
\-\2\ \ \ \ \ \ \ 0\/
\+\1
universe is \ described by this \ model, then T  \ must be at \ least 
\-\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0\/
\+\ \ 10
\110   years. \,
\+
        \,
\+
    The simple model of a uniform zero-pressure open universe has \/
\+
instead of (1) the metric \,
\+
\0\ \ \ \ \ \ \ \ \ \ \ \ (\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \  \ \ \ )\,
\+\2\ \ \ \ \ \ 2\ \ \ \ 2\ \ \ 2\ \ \ \ \ 2\ \ \ \ \ \ \ 2\ \ \ \ \ 2
\1    ds\ \ = R\ \0|\1d\7j\ \ \1- d\7c\  \1- sinh\  \7c \1d\7W\ \0| \1,                 (6)\,
\+
\0\ \ \ \ \ \ \ \ \ \ \ \ 9\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \  \ \ 0\,
\+\1
\,
\+
where now \,
\+
\,
\+
    t = T\ (sinh \7j \1- \7j\1)                                  (7)\,
\+\2\ \ \ \ \ \ \ \ \ 0
\1\,
\+
    R = cT\ \)(cosh \7j \1- 1) ,                               (8)\,
\+\2\ \ \ \ \ \ \ \ \ \ 0
\1\,
\+
and the coordinates (\7j\1, \7c\1) extend over an infinite range \,
\+
\,
\+
    0 < \7j \1< +\98 \1,    0 < \7c \1< +\98                          \1(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). 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 
\+\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 10
\1out that the \ universe is closed, we shall \ still have about 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 metric (6) is \,
\+
\,
\+
    d = R\7j \1,                                            (10)\,
\+
\,
\+
with R given \ by (8), and \ the number of \ galaxies visible within \/
\+
the horizon is \,
\+
\,
\+
    N = N\ (sinh\ 2\7j \1- 2\7j\1)                                (11)\,
\+\2\ \ \ \ \ \ \ \ \ 0
\1\,
\+\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 10
\1where N\  is a number of the order of 10\ \ . \,
\-\2\ \ \ \ \ \ \ 0
\+\1
\,
\+
    Comparing (11) \ with (7), we \ see that the \ number of visible 
\-\/
\+\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 2
\1galaxies varies \ with t  at \ late times. It \ happens by a curious 
\+
numerical accident \ that the angular size \ of a typical galaxy at \/
\+
time t is \,
\+
\,
\+\2\ \ \ \ \ \ \ \ \ \ 5\ \ -1
\7    d \9= \110\ \ t\ \  rad,                                    (12)\,
\+
\,
\+
with t measured in years. Since (11) and (7) give \,
\+
\,
\+\2\ \ \ \ \ \ \ \ \ \ -10\ \ 2\ \ \ \ \ \ \ \ \ \ 2
\1    N \9= \110\ \ \ \ t\ ,       N\7d\ \ \9= \11 ,                       (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)] tht 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. \,
\+
\,
\+
    "All this the vast apparent \ provision in the starry mansions \/
\+
seem to \ promise: What ought we \ then not to do, \ to preserve our 
\+
natural \ birthright to \ it and \ to merit \ such inheritance, which 
\+
alas \ \ we \ think \ \ created \ all \ \ to \ gratify \ \ alone \ a race \ of 
\+
vain-glorious gigantic \ beings, while they \ are confined to \ this \/
\+
world, chained like so many atoms to a grain of sand." \,
\+
\,
\+
    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 (pr–©ez) 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 \9+ \18 kb 
\+\2
\1two \ billion \ years \ ago. \ This \ means \ that \ the position of the 
\+\2
\1capture resonance cannot have shifted by \ as much as 0.02 eV over 
\+\2\ \ \ \ 9
\12.10  \ yr. \ But \ the \ position \ of \ this \ resonance \ measures the 
\+\2
\1difference between the binding energies of the 149Sm ground state 
\+\2
\1and \ of \ the \ 150Sm \ compound \ state \ into \ which \ the neutron is 
\+\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 9
\1captured. These binding energies are each \ of the order of 10  eV 
\+\2
\1and depend in a complicated way upon the strengths of nuclear and 
\+\2
\1Coulomb \ interactions. The \ fact \ that \ the two \ binding energies 
\+\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 11
\1remained \ in balance \ to an \ accuracy of \ two parts \ in 10   over 
\+\2\ \ \ \ 9
\12.10  \ yr \ indicates \ that \ the \ strength \ of nuclear and Coulomb 
\+\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 18
\1forces cannot \ have varied by more \ than a few parts in \ 10   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 
\+\2
\1evidence of change was found does \ not, of course, prove that the 
\+\2
\1laws are \ strictly constant. In \ particular, it does \ not exclude 
\+\2
\1the \ possibility \ of \ a variation \ in \ strength \ of gravitational 
\+\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 18
\1forces with a time scale much shorter \ than 10   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 
\+\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 14
\1very low temperatures, \ within times of the order \ of 10   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 \,
\+
\,
\+\2\ \ \ \ \ \ \ \ \ \ \ \ \ -1
\1    T = (\7r\1V\7s\1)\ \                                          (14)\,
\+
\,
\+
where \ \7r \1is the \ density of \ stars in \ space, V the mean relative \/
\+
velocity of two \ stars, and \7s \1the 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 \,
\+
\,
\+\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ -41\ \ \ -3
\7    r   \1= 3.10\ \ \ \ km\ \                                   (15)\,
\+
\,
\+
    V   = 50 km/s                                       (16)\,
\+
\,
\+\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ 16\ \ \ 2
\7    s   \1= 2.10\ \ \ km\                                     (17)\,
\+
\,
\+\2\ \ \ \ \ \ \ \ \ \ \ \ 15
\1    T   = 10\ \ \ let.                                     (18)\,
\+
\,
\+
The \ time scale \ for an \ encounter causing \ serious disruption of 
\-\/
\+\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 15
\1planetary orbits will be considerably shorter than 10\ \  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 \,
\+
\,
\+
\0\ \ \ \ \ \ \ \ (\ \1GNM\ \0)\21/2\,
\+\1
    V = \0|\ \8---\ \0|\ \ \                                       \1(19)\,
\+
\0\ \ \ \ \ \ \ \ 9\ \ \1R\ \ \00\,
\+\1
\,
\+
    The cross \ section for a close \ encounter between two \ stars, \/
\+
changing their directions of motion b a large angle, is \,
\+
\,
\+
\0\ \ \ \ \ \ \ \ (\ \1GM\ \0)\22\ \ \ \0(\ \1R\ \0)\22\,
\+\1
\7    s \1= \0|\ \8--\ \0|\ \ \1= \0|\ \8-\ \0|\ \ \1.\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (20)\,
\+\2\ \ \ \ \ \ \ \ \ \ \ 2
\0\ \ \ \ \ \ \ \ 9\ \1V\ \ \00\ \ \ \ 9\ \1N\ \00\,
\+\1
\,
\+
    The \ average \ time \ that \ a star \ spends \ between \ two \ close \/
\+
encounters is \,
\+
\,
\+\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 3
\1\ \ \ \ \ \ \ \ \ 1\ \ \ \ \0(\ \1NR\ \ \0)\21/2\,
\+\1
\ \ \ \ T = \8---\ \1= \0|\ \8---\ \0|\ \ \                                 \1(21)\,
\+
\7    \ \ \ \ r\1V\7s\ \ \ \09\ \1GM\ \ \00\,
\+\1
\,
\+\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 11
\1    If we are \ considering a typical large galaxy \ with N = 10öö, 
\-\/
\+\2\ \ \ \ \ \ \ \ 17
\1R\6\6t\6\1= 3.10   km, then \,
\+
\,
\+\2\ \ \ \ \ \ \ \ \ \ 19
\1    T = 10\ \ \ let.                                       (22)\,
\+
\,
\+
    Dynamical \ relaxation of \ the galaxy \ proceeds mainly through \/
\+
distant stellar encounters with a time scale \,
\+
\,
\+
\ \ \ \ \ \ \ \ \ \ \ T\,
\+\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 18
\1\ \ \ \ T\ \ = \8-----\ \ \1= 10\ \  let.                             (23)\,
\+\2\ \ \ \ \ N
\1    \ \ \ \ \ log N\,
\+
    \,
\+
    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 
\+\2\ \ \ \ \ \ \ \ \ \ \ 19
\1order of 10   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 \,
\+
\,
\+
\0\ \ \ \ \ \ \ \ \ (\ \1V\ \0)\25\ \1E\,
\+
    E\ \ = \0|\ \8-\ \0|\ \ \8-\  \1.                                    (24)\,
\+\2\ \ \ \ \ g
\0\ \ \ \ \ \ \ \ \ 9\ \1c\ \00\ \ \1P\,
\+
\,
\+
    Any gravitationally \ bound system of \ objects orbiting around \/
\+
each other \ will decay by \ this mechanism of \ radiation drag with \/
\+
a time scale \,
\+
\,
\+
\0    \ \ \ \ \ (\ \1c\ \0)\25\,
\+\1
\ \ \ \ t\ \ = \0|\ \8-\ \0|\ \1.P   .                                   (25)                   \,
\+\2\ \ \ \ \ g
\0\ \ \ \ \ \ \ \ \ 9\ \1V\ \00\,
\+\1
\,
\+
    For \ the earth \ orbiting \ around \ the sun, \ the gravitational \/
\+
radiation time scale is \,
\+
\,
\+\2\ \ \ \ \ \ \ \ \ \ \ 20
\1    T\ \ = 10\ \ \ let.                                      (26)\,
\+\2\ \ \ \ \ g
\1\,
\+
    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 \/
\+\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 8
\1V = 200 km/sec and P = 2.10\  yr, the time scale is \,
\+
\,
\+\2\ \ \ \ \ \ \ \ \ \ \ 24
\1    T\ \ = 10\ \ \ \ let.                                     (27)\,
\+\2\ \ \ \ \ g
\1\,
\+
    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 \,
\+
\,
\+\2\ \ \ \ \ \ \ \ \ 2\ 3
\1\ \ \ \ \ \ \ \ G\ M\,
\+
    T = \8----                                            \1(28)\,
\+\2\ \ \ \ \ \ \ \ \ \ 4
\9\ \ \ \ \ \ \ \ h\1c\,
\+
\,
\+
For a black hole of one solar mass the lifetime is \,
\+
\,
\+\2\ \ \ \ \ \ \ \ \ \ 64
\1    T = 10\ \ \ \ let.\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (29)\,
\+
\,
\+
    Black holes of galactic mass will have lifetimes extending up \/
\+\2\ \ \ \ \ 100
\1to 10    yr. At the end of \ its life, every black hole will emit \/
\+\2\ \ \ \ \ \ \ \ 31
\1about 10   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 \,
\+
\ \ \ \ \ \ \ \ \ \ \ \  \,
\+\ \ \ \ \ \ \ \ \ S
    T = e .T\  ,                                         (30)\,
\+\2\ \ \ \ \ \ \ \ \ \ \ \ 0
\1\,
\+
\,
\+
where T  \ is a natural vibration \ period of the \ system, and S is 
\-\2\ \ \ \ \ \ \ 0\/
\+\1
the action integral \,
\+
\,
\+
\ \ \ \ \ \ \ \ 2\ \0i\ (\ \ \ \ \ \ )\21/2\,
\+\1
    S = \8-\ \0|\ |\12MU(x)\0|\ \ \ \ \1dx .                            (31)\,
\+
\9\ \ \ \ \ \ \ \ h\ \0j\ 9\ \ \ \ \ \ 0\,
\+\1
\,
\+
    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 \,
\+
\,
\+\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ 2
\0\ \ \ \ \ \ \ \ (\ \18MUd\ \ \0)\21/2\,
\+\1
    S = \0|\ \8-----\ \0|\ \ \                                     \1(32)\,
\+\2\ \ \ \ \ \ \ \ \ \ \ \ 2
\0\ \ \ \ \ \ \ \ 9\ \ \9h\ \ \ \ \00\,
\+\1
\,
\+
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. \,
\+
\,
\+
    Asan \ 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, \,
\+
\,
\+\2\ \ \ \ \ \ \ \ \ \ \ \ 4
\1\ \ \ \ \ \ \ \ 1\ \ e\,
\+
    U = \8--\ --                                           \1(33)\,
\+\2\ \ \ \ \ \ \ \ \ \ \ \ 2
\1\ \ \ \ \ \ \ \ 20\ \9h\,
\+\1
\,
\+
and the thickness will be of the order of a Bohr radius \,
\+
\,
\+\2\ \ \ \ \ \ \ \ \ 2
\9\ \ \ \ \ \ \ \ h\,
\+\1
    d = \8--- \1,                                           (34)\,
\+\2\ \ \ \ \ \ \ \ \ \ 2
\1\ \ \ \ \ \ \ \ me\,
\+
\,
\+
where m is the electron mass. The action integral (32) is then \,
\+
\,
\+
\0\ \ \ \ \ \ \ \ (\ \12Am\ \ \0)\21/2\,
\+\ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ 1/2
\1    S = \0|\ \8----\ \0|\ \ \ \ \1= 27.A\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (35)\,
\+
\0\ \ \ \ \ \ \ \ 9\ \ \15m\ \ \00\,
\+\1
\,
\+
where m  \ is the proton mass, \ and A is the atomic \ weight of the 
\-\2\ \ \ \ \ \ \ p\/
\+\1
moving \ atom. For \ an iron \ atom with \ A = 56, S = 200, \ and (30) \/
\+
gives \,
\+
\,
\+\2\ \ \ \ \ \ \ \ \ \ 65
\1    T = 10\ \ \ let.                                       (36)\,
\+
    \,
\+
    Even the most rigid materials cannot preserve their shapes or \/
\+
their chemical \ structures for times long \ compared with (36). On 
\+\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 65
\1a time \ scale \ of \ 10   \ 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 \ b varius \ 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 \,
\+
\,
\+\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ 2
\9\ \ \ \ \ \ \ \ \ \ \ \ \ h\,
\+\2\ \ \ \ \ \ \ \ \ 1/2
\1    d = Z\ \ \ \ \8---                                        \1(37)\,
\+\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 2
\1\ \ \ \ \ \ \ \ \ \ \ \ \ me\,
\+
\,
\+
of each other. The Coulomb barrier has thickness d and height \,
\+
\,
\+\2\ \ \ \ \ \ \ \ \ 2\ 2\ \ \ \ \ \ \ \ \ \ \ 4
\1\ \ \ \ \ \ \ \ Z\ e\ \ \ \ 1\ \ \ \ \ \ e\ m\,
\+\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 7/3
\1    U = \8----\ \1= \8-\ \1Z\ \ \ \ \8---                               \1(38)\,
\+\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \  \ 2
\1\ \ \ \ \ \ \ \ \ 4d\ \ \ \ 2\ \ \ \ \  \9h\,
\+\1
\,
\+
The reduced mass for the relative motion of the two nuclei is \,
\+
\,
\+
    M = AM\ /4                                           (39)\,
\+\2\ \ \ \ \ \ \ \ \ \ p
\1\,
\+
The action integral (32) then becomes \,
\+
\,
\+
\0\ \ \ \ \ \ \ \ (\ \1A\ \ \ \ \ \ m\ \ \0)\21/2\,
\+\ \ \ \ \ \ \ \ \ \ \ \ \ 5/3\ \ p\ \ \ \ \ \ \ \ \ \ \ \ 1/2\ \ 5/6
\1    S = \0|\ \8-\ \1Z\ \ \ \ \8--\ \0|\ \ \ \ \1= 30.A\ \ \ .Z\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (40)\,
\+
\0\ \ \ \ \ \ \ \ 9\ \12\ \ \ \ \ \ m\ \ \00\,
\+\1
\,
\+
For two nuclei combining to form \ iron, Z = 26, A = 56, S = 3500, \/
\+
and \,
\+
\,
\+\2\ \ \ \ \ \ \ \ \ \ 1500
\1    T = 10\ \ \ \  let.\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \    (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 \,
\+
\,
\+\2\ \ \ \ \ \ \ \ \ \ \ \ 5/3\ 2
\1\ \ \ \ \ \ \ \ \ \ \ N\ \ \ \9h\,
\+\1
    U(x) = \8------\ \                                      \1(42)\,
\+\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 2
\1\ \ \ \ \ \ \ \ \ \ \ \ 2mx\,
\+
\,
\+
where N is the \ number of electrons in the \ star. The integration \/
\+
over x in (31) gives a logarithm \,
\+
\,
\+
\0\ \ \ \ \ \ \ \ (\ \1R\ \ \0)\,
\+\1
    log \0| \8--\ \0| \1,                                        (43)\,
\+
\0\ \ \ \ \ \ \ \ 9\ \1R\ \ \00\,
\+\2\ \ \ \ \ \ \ \ \ \ \ 0
\1\,
\+
where R  is the radius at which the electrons become relativistic 
\-\2\ \ \ \ \ \ \ 0\/
\+\1
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  will \ also be of the order of 
\-\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0\/
\+\1
unity. The mass of the star is \,
\+
\,
\+
    M = 2Nm\                                             (44)\,
\+\2\ \ \ \ \ \ \ \ \ \ \ p
\1\,
\+
    I replace \ the logarithm \ (43) by \ unity and \ obtain for \ the \/
\+
action integral (31) the estimate \,
\+
\,
\+
\0\ \ \ \ \ \ \ \ \ \ \ \ \ (\ \18m\ \ \0)\,
\+\2\ \ \ \ \ \ \ \ \ 4/3\ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ 4/3
\1    S = N\ \ \ \ \0|\ \8---\ \0|\ \1= 120.N\ \ \ \ .                       (45)\,
\+
\0\ \ \ \ \ \ \ \ \ \ \ \ \ 9\ \ \1m\ \ \00\,
\+\1
\,
\+
The lifetime is then by (30) \,
\+
\0\ \ \ \ \ \ \ \ \ \ \  (\ \ \ \ \ \ \ \ )\,
\+\2\ \ \ \ \ \ \ \ \ \ \  \ \ \ \ \ \ 4/3
\1    T = exp\ \0|\1120.N   \0|\1.T  .                             (46)\,
\+\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0
\0\ \ \ \ \ \ \ \ \ \ \  9\ \ \ \ \ \ \ \ 0\,
\+\1
\,
\+
For a typical low-mass star we have \,
\+
\,
\+\2\ \ \ \ \ \ \ \ \ \ 56
\1    N = 10\,
\+
\,
\+\2\ \ \ \ \ \ \ \ \ \ 77
\1    S = 10\,
\+
\2\ \ \ \ \ \ \ \ \ \ \ \ 76\,
\+\ \ \ \ \ \ \ \ \ \ 10
\1    T = 10\ \ \ \ \ \ let .                                   (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 correet 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 \,
\+
\,
\+
\0\ \ \ \ \ \ \ \ \ \ \ \ (\ \ \ \ \ \ \ \ )\,
\+\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 4/3
\1    T = exp\ \0|\1120.N\ \ \ \0|\1.T\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \    (48)\,
\+\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ B\ \ \ \ \ 0
\0\ \ \ \ \ \ \ \ \ \ \ \ 9\ \ \ \ \ \ \ \ 0 \,
\+\1
\,
\+
where N  \ is the number of \ electrons in a piece of \ iron of mass 
\-\2\ \ \ \ \ \ \ B\/
\+\1
equal to the \ minimum mass M  of a black \ hole. The lifetime (48) 
\-\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ B\/
\+\1
is \ the \ same \ for \ any \ piece \ of \ matter \ of \ mass greater than 
\+
M . Matter in \ pieces \ with \ mass \ smaller \ than M  is absolutely 
\-\2\ B\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ B
\+\1
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 . All that we know for sure is (V¨echno, co v¡me jistˆ, je)\,
\-\2\ \ \ \ B
\+\1
\,
\+
    0 \9< \1M\ \ \9< \1M\                                          (49)\,
\+\2\ \ \ \ \ \ \ \ \ B\ \ \ \ c
\1\,
\+
where \,
\+
\,
\+
\0\ \ \ \ \ \ \ \ (\ \9h\1c\ \0)\23/2\ \11\,
\+\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 33
\1    M\ \ =\0| \8--\ \0|\ \ \ \ \8--\ \1= 4.10\ \ \ g\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (50)\,
\+\2\ \ \ \ \ c\ \ \ \ \ \ \ \ \ \ \ \ \ 2
\0\ \ \ \ \ \ \ \ 9 \1G\ \ \00\ \ \ \ \1m\,
\+\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p
\1\,
\+
\,
\+
is the Chandrasekhar mass. Black \ holes must exist for every mass \/
\+
larger than M , \ because stars with mass larger \ than M_c have no 
\-\2\ \ \ \ \ \ \ \ \ \ \ \ \ c\/
\+\1
stable final state and must inevitably collapse. \,
\+
\,
\+
Four hypotheses concerning M\  have been put forward: \,
\-\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ B
\+\1
\,
\+
(i) M  = 0. Then black holes of \ arbitrarily small mass exist and 
\-\2\ \ \ \ \ B\/
\+\1
the \ formula (48) \ is meaningless. \ \ In this \ case all \ matter is 
\+
unstable \ with a comparatively \ short lifetime, \ as suggested \ by \/
\+
Zeldovich (1977). \,
\+
\,
\+
(ii) M\  is equal to the Planck mass \,
\-\2\ \ \ \ \ \ B
\+\1
\0\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (\ \9h\1c\ \0)\21/2\,
\+\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ -5
\1    M\ \ = M\ \ \ = \0|\ \8--\ \0|\ \ \ \ \1= 2.10\ \ \ g                     (51)\,
\+\2\ \ \ \ \ B\ \ \ \ PL
\0\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 9\ \1G\ \ \00\,
\+\1
\,
\+
This value of \ M\  is suggested by Hawking's \ theory of radiation 
\-\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ B\/
\+\1
from black holes (Hawking, 1975), \ according to which every black 
\+
hole loses mass \ until it reaches a mass of \ order M\ \ , at which 
\-\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ PL
\+\1
point it \ disappears in a burst \ of radiation. In \ this case (48) \/
\+
gives \,
\+
\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 26\,
\+\ \ \ \ \ \ \ \ \ \ \ 19\ \ \ \ \ \ \ \ \ 10
\1    N\  = 10\ \ \ , T = 10\ \ \ \ \ \ let.                        (52)\,
\+\2\ \ \ \ \ B
\1\,
\+
\,
\+
(iii) M\  is equal to the quantum mass \,
\-\2\ \ \ \ \ \ \ B
\+\1
\9\ \ \ \ \ \ \ \ \ \ \ \ \ \ h\1c\,
\+\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 14
\1    M\ \ = M\ \ = \8---\ \1= 3.10\ \ \ g                            (53)\,
\+\2\ \ \ \ \ B\ \ \ \ Q
\1\ \ \ \ \ \ \ \ \ \ \ \ \ \ Gm\,
\+\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p
\1\,
\+
as \ suggested by \ Harrison, Thorne, \ Wakano, and \ Wheeler (1965). \/
\+
Here \ M  \ is \ the \ mass \ of \ the \ smallest \ black \ hole for which 
\-\2\ \ \ \ \ \ \ Q
\+\1
a classical theory is meaningful. Only \ for masses larger than M  
\-\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ Q
\+\1
can \ we \ consider \ the \ barrier \ penetration \ formula \ (31) to be \/
\+
physically justified. If (53) holds, then \,
\+
\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 32\,
\+\ \ \ \ \ \ \ \ \ \ \ 38\ \ \ \ \ \ \ \ \ 10
\1    N\  = 10\ \ \ , T = 10\ \ \ \ \ let.                         (54)\,
\+\2\ \ \ \ \ B
\1\,
\+
\,
\+
(iv) M  is equal to the Chandrasekhar mass (50). In this case the 
\-\2\ \ \ \ \ \ B\/
\+\1
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 \,
\+\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 11
\1\ \ Total duration \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \   10\ \  yr \,
\+
\,
\+
Open Universe \,
\+\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 14
\1\ \ Low-mass stars cool off \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \  10\ \  yr \,
\+
\,
\+\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 15
\1\ \ Planets detached from stars \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 10   yr \,
\+
\,
\+\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 19
\1\ \ Stars detached from galaxies \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 10   yr \,
\+
\,
\+\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 20
\1\ \ Decay of orbits by gravitational radiation \ \ \ 10   yr \,
\+
\,
\+\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 64
\1\ \ Decay of black holes by Hawking process \ \ \ \ \ \ 10   yr \,
\+
\,
\+\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 65
\1\ \ Matter liquid at zero temperature \ \ \ \ \ \ \ \ \ \ \ \ 10\ \  yr \,
\+
\,
\+\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1500
\1\ \ All matter decays to iron \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 10\ \ \ \  yr \,
\+
\,
\+
\ \ Collapse of ordinary matter to black hole \,
\+\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 26
\+\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 10
\1\ \ \ \ [alternative (ii)] \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 10\ \ \ \  yr \,
\+
\,
\+
\ \ Collapse of stars to neutron stars \,
\+\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 76
\+\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 10
\1\ \ \ \ or black holes [alternative (iv)] \ \ \ \ \ \ \ \ \ \ 10\ \ \ \  yr \,
\+
\,
\+
\,
\+
LECTURE III. BIOLOGY \,
\+
\,
\+
    Looking at \ the past history \ of life, we \ see that it \ takes \/
\+\2\ \ \ \ \ \ \ \ \ 6\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 7
\1about \ 10  years \ to evolve \ a new species, \ 10  years \ to evolve 
\+\2\ \ \ \ \ \ \ \ \ \ \ \ 8\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 9
\1a genus, \ 10  \ years \ to \ evolve \ a class, \ 10  \ years \ to evolve 
\+\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 10
\1a phylum, and less than 10   years to evolve all the way from the 
\+
primeval slime to Homo Sapiens. If life continues in this fashion 
\+
in the future, \ it is impossible to set any \ limit to the variety 
\+
of physical forms that life \ may assume. What changes could occur 
\+\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 10
\1in the \ next 10   years to \ rival the changes of \ the past? It is 
\+\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 10
\1conceivable \ that in \ another 10   \ years life \ could evolve away 
\+
from flesh and blood and become embodied in an interstellar black \/
\+
cloud (Hoyle, 1957) or in a sentient computer (Capek, 1923). \,
\+
    \,
\+
    Here \ is a list \ of deep \ questions concerning \ the nature of \/
\+
life and consciousness. \,
\+
\,
\+
(i) Is the basis of consciousness matter or structure? \,
\+
(ii) Are sentient black clouds, or sentient computers, possible? \,
\+
(iii) Can we apply scaling laws in biology? \,
\+
    \,
\+
    These are \ questions that we do \ not know how to \ answer. But \/
\+
they are not in principle \ unanswerable. It is possible that they 
\+
will \ be \ answered \ fairly \ soon \ \ as \ a result \ of \ progress \ in \/
\+
experimental biology. \,
\+
    \,
\+
    Let me spell out more explicitly the meaning of question (i). \/
\+
My \ consciousness \ is \ somehow \ associated \ with \ a collection of 
\+
organic molecules \ inside my head. \ The question is, \ whether the 
\+
existence of my consciousness depends \ on the actual substance of 
\+
a particular set of \ molecules or whether it only \ depends on the 
\+
structure \ of \ the \ molecules. \ In \ other \ words, if I could make 
\+
a copy of \ my brain with \ the same structure \ but using different \/
\+
materials, would the copy think it was me? \,
\+
    \,
\+
    If \ the answer \ to question \ (i) is \ "matter", then \ life and \/
\+
consciousness can never evolve away from flesh and blood. In this 
\+
case the answers \ to questions (ii) and (iii) \ are negative. Life 
\+
can \ then continue \ to exist \ only so \ long as \ warm environments 
\+
exist, with \ liquid water and a continuing \ supply of free energy 
\+
to \ support a constant \ rate of \ metabolism. In \ this case, since 
\+
a galaxy has only a finite supply of free energy, the duration of 
\+
life is finite. As the universe expands and cools, the sources of 
\+
free energy that life requires for its metabolism will ultimately \/
\+
be exhausted. \,
\+
    \,
\+
    Since \ I am a philosophical \ optimist, I assume \ as a working \/
\+
hypothesis that \ the answer to question \ (i) is "structure". Then \/
\+
life \ is free \ to evolve \ into whatever \ material embodiment best 
\+
suits its purposes. \ The answers to questions (ii) \ and (iii) are 
\+
affirmative, and a quantitative discussion \ of the future of life 
\+
in \ the \ universe \ becomes \ possible. \ If \ it \ should happen, for 
\+
example, that \ matter is ultimately stable \ against collapse into 
\+
black \ holes only \ when it \ is subdivided \ into dust grains a few 
\+
microns in \ diameter, then the \ preferred embodiment for \ life in 
\+
the \ remote future \ must be \ something like \ Hoyle's black cloud, 
\+
a large assemblage of dust \ grains carrying positive and negative 
\+
charges, organizing itself and communicating with itself by means 
\+
of electromagnetic \ forces. We cannot imagine \ in detail how such 
\+
a cloud could \ maintain the state of \ dynamic equilibrium that we 
\+
call life. But \ we also could not have \ imagined the architecture \/
\+
of a living cell of protoplasm if we had never seen one. \,
\+
    \,
\+
    For \ a quantitative description \ of \ the \ way life \ may adapt \/
\+
itself to a cold environment, I need to assume a scaling law that 
\+
is independent \ of any particular \ material embodiment that \ life 
\+
may find \ for itself. The \ following is a formal \ statement of my \/
\+
scaling law: \,
\+
    \,
\+
\3    Biological Scaling Hypothesis. If \ we copy a living creature, \/
\+\1
\3quantum state \ by quantum state, \ so that the \ Hamiltonian of the \/
\+\1
\3copy is \,
\+\1
\,
\+\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ -1
\1    H\ \ = \7l\1.U.H.U\ \                                       (55)\,
\+\2\ \ \ \ \ c
\1\,
\+
\,
\+
\3where \ \1H \3is \ the \ Hamiltonian \ of \ the \ creature, \ \1U \3is a unitary \/
\+\1
\3operator, \ and \ \7l \3is \ a positive \ \ scaling \ factor, \ and \ if \ the \/
\+\1
\3environment is \ similarly copied so that \ the temperatures of the 
\+\1
\3environments of the creature and \ the copy are respectively \1T \3and 
\+\1
\7l\1T\3, \ then \ the \ copy \ is \ alive, \ subjectively \ identical \ to the 
\+\1
\3original creature, with all its \ vital functions reduced in speed \/
\+\1
\3by the same factor \7l\3.\,
\+\1
    \,
\+
    The \ structure of \ the Schroedinger \ equation, with \ time and \/
\+
energy appearing \ as conjugate variables, makes \ the form of this 
\+
scaling \ \ hypothesis \ plausible. \ \ It \ is \ \ at \ present \ a purely 
\+
theoretical hypothesis, not susceptible to any experimental test. 
\+
To \ avoid misunderstanding, \ I should emphasize \ that the scaling 
\+
law does not apply to the change of the metabolic rate of a given 
\+
organism as a function of \ temperature. For example, when a snake 
\+
or a lizard \ changes its temperature, \ its metabolic rate \ varies 
\+
exponentially rather than linearly with T. The linear scaling law 
\+
applies to an ensemble of copies of a snake, each copy adapted to 
\+
a different temperature. It does \ not apply to a particular snake \/
\+
with varying T. \,
\+
\,
\+
    From \ this point \ on, I assume \ the scaling \ hypothesis to be \/
\+
valid \ and examine \ its \ consequences \ for the \ potentialities of 
\+
life. The \ first consequence is \ that the appropriate \ measure of 
\+
time \ as experienced \ subjectively \ by \ a living creature \ is not \/
\+
physical time t but the quantity \,
\+
\,
\+\ \ \ \ \ \ \ \ \ \ \ \ \ \ t
\0\ \ \ \ \ \ \ \ \ \ \ \ \ i\,
\+\1
    u(t) = f.\0|\ \ \7y\1(t') dt' ,                             (56)    \,
\+
\0\ \ \ \ \ \ \ \ \ \ \ \ \ j\,
\+\1\ \ \ \ \ \ \ \ \ \ \ \ \ \ 0
\,
\+
where \7y\1(t) is \ the temperature of the creature \ and f = 1/300" is \/
\+
a scale factor which it is convenient \ to introduce so as to make 
\+
u(t) \ dimensionless. I call \ u(t) "subjective \ time". The \ second \/
\+
consequence \ of \ \ the \ scaling \ law \ is \ \ that \ any \ creature \ is 
\+
characterized by a quantity Q which \ measures its rate of entropy 
\+
production per unit of subjective time. If entropy is measured in 
\+
information units \ or bits, and \ if u is measured \ in "moments of 
\+
consciousness", then Q is a pure \ number expressing the amount of 
\+
information that must be processed \ in order to keep the creature 
\+
alive \ long \ enough \ to \ say \ "\3Cogito, \ ergo \ sum\1". \ I call Q the 
\+
"complexity" \ of \ \ the \ creature. \ For \ \ example, \ a human \ being 
\+
dissipates about 200 \ W of power at a temperature of \ 300 K, with 
\+
each \ moment \ of \ consciousness \ lasting \ about a second. A human \/
\+
being therefore has \,
\+
\,
\+\2\ \ \ \ \ \ \ \ \ \ 23
\"    Q = 10\ \ \ \1bits.                                      (57)\,
\+
    \,
\+
    This \ Q is \ a measure \ of \ the \ complexity \ of \ the molecular \/
\+
structures involved \ in a single act of \ human awareness. For the \/
\+
human species as a whole, \,
\+
\,
\+\2\ \ \ \ \ \ \ \ \ \ 33
\1    Q = 10\ \ \ bits.                                      (58)\,
\+
\,
\+
a number which \ tells us the \ order of magnitude \ of the material \/
\+
resources required for maintenance of an intelligent society. \,
\+
\,
\+
A creature or a society with \ given Q and given temperature theta \/
\+
will dissipate energy at a rate \,
\+
\,
\+\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 2
\1    m = k.f.Q.\7y\   \1.                                     (59)\,
\+
\,
\+
    Here \ m is the \ metabolic rate \ measured in \ ergs per second, \/
\+
k is Boltzmann's constant, and \ f is the coefficient appearing in \/
\+
(56). It \ is important that \ m varies with the \ square of \7y\1, \ one 
\+
factor \7y \1coming from the relationship between energy and entropy, 
\+
the \ other \ factor \ theta \ coming \ from \ the \ assumed temperature \/
\+
dependence of the rate of vital processes. \,
\+
    \,
\+
    I am \ assuming that \ life is \ free to \ choose its temperature \/
\+
\7y\1(t) so \ as to maximize \ its chances of \ survival. There are \ two 
\+
physical constraints \ on \7y\1(t). The first \ constraint is that \7y\1(t) \/
\+
must \ always be \ greater than \ the temperature \ of the \ universal 
\+
background radiation, \ which is the \ lowest temperature available \/
\+
for a heat sink. That is to say \,
\+
\,
\+\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 28
\7    y\1(t) > a/R   ,  a = 3.10\ \ \ deg cm\ ,                 (60)\,
\+
\,
\+
where R is \ the radius of the \ universe, varying with t according \/
\+
to \ (7) \ and \ (8). \ At \ the \ present \ time \ the condition (60) is 
\+
satisfied with a factor of 100 to spare. The second constraint on 
\+
\7y\1(t) is \ that a physical mechanism must \ exist for radiating away \/
\+
into space \ the waste heat generated \ by metabolism. To formulate 
\+
the second constraint quantitatively, \ I assume that the ultimate 
\+
disposal of waste heat is by radiation and that the only relevant \/
\+
form of radiation is electromagnetic. \,
\+
\,
\+
There is an absolute upper limit \,
\+
\,
\+\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 2
\1\ \ \ \ \ \ \ \ \ \ \ \ \ \ Ne\,
\+\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 2\ 3\ \ \ \ \ 3
\1    I(\7y\1) < 2\7g\ \8---\ \9h\ \1c\ .(k\7y\1)\                             (61)\,
\+
\ \ \ \ \ \ \ \ \ \ \ \ \ \ m\,
\+
\,
\+
on \ the \ power \ that \ can \ be \ radiated \ by \ a material \ radiator \/
\+
containing N electrons at temperature \7y\1. Here \,
\+
\,
\+\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 3
\0\ \ \ \ \ \ \ \ \ \ \ \ (\ \ \ \1x\ \ \ \ \0)\,
\+\1
\7    g \1= max \0|\ \8------\ \0|\ \9= \11,42                           (62)\,
\+\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ x
\0\ \ \ \ \ \ \ \ \ \ \ \ 9\ \1e\ \ - 1\ \00\,
\+\1
\,
\+
is the \ height of the \ maximum of the \ Planck radiation spectrum. \/
\+
Since \ I could not \ find (61) \ in the \ textbooks, I give \ a quick 
\+
proof, \ following the \ Handbuch \ article \ of Bethe \ and Saltpeter 
\+
(1957). \ The formula \ for the \ power emitted \ by electric \ dipole \/
\+
radiation is \,
\+
\,
\+
\0\ \ \ \ \ \ \ \ \ \ \ \ \ \ i\,
\+\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 4/2\ \ \ 3\ \ \ \ \ \ 2
\1    I(\7y\1) = \9S\ \ \0|\ \1d\7W \9S S\ \7r\ \ w\ \ \ \ p\1c\ \ \0|\1D\ \ \0|\                \1(63)\,
\+\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ i\ \ ij\ \ \ \ \ \ \ \ ij
\ \ \ \ \ \ \ \ \ \ \ p\ \ \0j\ \ \ \ \2i\ j\,
\+\1
\,
\+
Here p is \ the polarization vector \ of a photon emitted \ into the \/
\+
solid angle \ d\7W\1, \3i \1is the \ initial and \3j \1the \ final state of \ the \/
\+
radiator, \,
\+
\,
\+
\ \ \ \ \ \ \ \ \ 1\ \ \ \ \ \0(\ \ \1E\ \ \ \0)\,
\+\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ i
\7    r\ \ \1= \8-\ \1exp\ \0|\1- \8--\ \7y\0|                                 \1(64)\,
\+\2\ \ \ \ \ i
\1\ \ \ \ \ \ \ \ \ Z\ \ \ \ \ \09\ \ \1k\ \ \ \00\,
\+\1
\,
\+
is the probability that the radiator is initially in state \3i\1, \,
\+
\,
\+
\ \ \ \ \ \ \ \ \ \ E\ \ - E\,
\+\2\ \ \ \ \ \ \ \ \ \ \ i\ \ \ \ j
\7    w\ \ \ \1= \8-------                                       \1(65)\,
\+\2\ \ \ \ \ ij
\9\ \ \ \ \ \ \ \ \ \ \ \ \ h\,
\+\1
\,
\+
is the frequency of the photon, \ and D   is the matrix element of 
\-\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ ij\/
\+\1
the radiator dipole \ moment between states i and j. \ The sum (63) \/
\+
is taken only over pairs of states (i,j) with \,
\+
\,
\+
    E\  > E\ \ .                                           (66)\,
\+\2\ \ \ \ \ i\ \ \ \ j
\1\,
\+
Now there is an exact sum rule for dipole moments, \,
\+
\,
\+\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 2
\1\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1\ \ \ \ \ \9d\1D\ \ \ \9d\1D\ \ \ \ \ \ \ \ Ne\ \9h\,
\+\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 2
\9    S\ \7w\ \ \ \0|\1D  \0|\ \ \1= \8--\ \1<D \8--\ \1- \8--\ \1D>\ \ \ = \8----            \1(67)\,
\+\2\ \ \ \ \ \ \ ij\ \ \ ij\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ jj
\ \ \ \ i\ \ \ \ \ \ \ \ \ \ \ \ \ \ \12i\ \ \ \ \9d\1t\ \ \ \9d\1t\ \ \ \ \ \ \ \ \ 2m\,
\+
\,
\+
But we have to be careful in using (67) to find a bound for (63), \/
\+
since some of the terms in (67) are negative. The following trick 
\+
works. In \ every term of \ (63), \7w   \1is \ positive by (66), \ and so 
\-\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ ij\/
\+\1
(62) gives \,
\+
\,
\+
\0\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (\ \1k\7y\ \0)\23\ \0!\ \ \ (\ \9h\7w\ \ \ \ \0)\ \ \ \ @\ \ \ \ (\ \ \ \ \ \ \ )(\ \1k\7y\ \0)\23\,
\+\ \ \ \ \ \ \ \ 3\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ ij
\7    r\ w\ \ \ \1< \7gr\ \0|\ \8--\ \0|\ \1.\0|\1exp\0|\ \8---- \7y\0| \1- 1\0| \1= \7g\0|\7r\ \ \1- \7r\ \0||\ \8--\ \0|\,
\+\2\ \ \ \ \ i\ ij\ \ \ \ \ i\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ j\ \ \ \ i
\0\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 9\ \9h\ \ \00\ \ 1\ \ \ 9\ \ \1k\ \ \ \ \00\ \ \ \ 2\ \ \ \ 9\ \ \ \ \ \ \ 09\ \9h\ \ \00\,
\+\1
\,
\+
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (68)\,
\+
\,
\+
Therefore (63) implies \,
\+
\,
\+
\0\ \ \ \ \ \ \ \ \ \ \ \ (\ \1k\7y \0)\23\ \ \ \0i\ \ \ \ !\ \ \ \ \ (\ \ \ \ \ \ \ )\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ @\,
\+\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 3\ \ \ \ \ \ 2
\1    I(\7y\1) < \7g\0|\ \8--\ \0|\ \1.\9S\ \0|\ \1d\7W \0|\ \9S\ S\ \0|\7r\ \ \1- \7r\ \0|\1.\7w\ \ \1.2\7p\1c\ .\0|\1D\ \ \0|\ |\1.\,
\+\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ j\ \ \ \ i\ \ \ ij\ \ \ \ \ \ \ \ ij
\0\ \ \ \ \ \ \ \ \ \ \ \ 9\ \9h\ \ \00\ \ \2p\ \0j\ \ \ \ 1\ \2i\ j\ \09\ \ \ \ \ \ \ 0\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 2\,
\+\1
\,
\+
                                                        (69)\,
\+
\,
\+
Now the summation \ indices (i,j) can be exchanged \ in the part of \/
\+
(69) involving \7r\ \1. The result is \,
\-\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ i
\+\1
\,
\+
\0\ \ \ \ \ \ \ \ \ \ \ \ (\ \1k\7y\ \0)\23\ \ \ \0i\ \ \ \ !\ \ \ \ \ \ \ \ \     \ \ \ \ \ \ \ \ \ \ @\,
\+\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 3\ \ \ \ \ \ 2
\1    I(\7y\1) <\ \7g\0|\ \8--\ \0|\ \1.\9S\ \0| \1d\7W\ \0| \9S S\ \7r\ \1.\7w  \1.2\7p\1c\ .\0|\1D\ \ \0|\ | \1,  (70)\,
\+\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ j\ \ ij\ \ \ \ \ \ \ \ ij
\0\ \ \ \ \ \ \ \ \ \ \ \ 9\ \9h\ \ \00\ \ \2p\ \0j\ \ \ \ 1\ \2i\ j\ \ \ \ \ \     \ \ \ \ \ \ \ \ \ \02\,
\+\1
\,
\+
with \ the summation \ now extending \ over all \ (i,j) whether \ (66) \/
\+
holds or \ not. The sum \ rule (67) \ can \ then be used \ in (70) and \/
\+
gives the result (61). \,
\+
    \,
\+
    This \ proof of \ (61) assumes \ that all \ particles other \ than \/
\+
electrons \ have \ so \ large \ a mass \ that \ they \ are negligible in 
\+
generating radiation. \ It also assumes \ that magnetic dipole \ and 
\+
higher multipole \ radiation is negligible. \ It is an \ interesting 
\+
question whether \ (61) could be \ proved without using \ the dipole \/
\+
approximation (63). \,
\+
\,
\+
    It may at \ first sight appear strange that \ the right side of \/
\+\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 3\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 4
\1(61) \ is proportional \ to \7y  \ \1rather than \ \7y \1, since the standard 
\+
Stefan-Boltzmann formula \ for the power radiated \ by a black body 
\+\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 4
\1is \ proportional to \ \7y \1. \ The \ Stefan-Boltzmann formula \ does not 
\+
apply \ in \ this \ case \ because \ it \ requires \ the \ radiator to be 
\+
optically thick. The maximum radiated \ power given by (61) can be \/
\+
attained only when the radiator is optically thin. \,
\+
    \,
\+
    Afer \ this \ little \ digression \ \ into \ physics, \ I return \ to \/
\+
biology. \ The second \ constraint on \ the temperature \ theta of an 
\+
enduring form of life is that the rate of energy dissipation (59) 
\+
must not \ exceed the power \ (61) that can \ be radiated away \ into 
\+
space. \ This \ constraint \ implies \ a fixed \ lower \ bound \ for the \/
\+
temperature, \,
\+
\,
\+
\ \ \ \ \ \ \ \ \ Q\ \ \ \ \ Q\,
\+\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ -28
\1    k\7y \1> \8- \7e \1= \8-\ \110\ \ \ \ \ erg ,                           (71)\,
\+
\ \ \ \ \ \ \ \ \ N\ \ \ \ \ N\,
\+
\,
\+
\ \ \ \ \ \ \ \ 137\ \9h\1f\,
\+\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 2
\7    e \1= \8---\ --\ \1mc\                                       (72)\,
\+
\ \ \ \ \ \ \ \ 2\7g\ \ \1k\,
\+
\,
\+
\ \ \ \ \ \ \ \ Q\ \7e\ \ \ \1Q\,
\+\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ -12
\7    y \1> \8-\ -\ \1= \8-\ \110\ \ \ \ deg.                              (73)\,
\+
\ \ \ \ \ \ \ \ N\ k\ \ \ N\,
\+
\,
\+
    The ratio \ (Q/N) between the complexity \ of a society and the \/
\+
number of \ electrons at its \ disposal cannot be \ made arbitrarily \/
\+
small. For the present human species, with Q given by (58) and \,
\+
\,
\+\2\ \ \ \ \ \ \ \ \ \ 42
\1    N = 10\ \                                             (74)\,
\+
\,
\+
being the number of electrons in the earth's biosphere, the ratio \/
\+\2\ \ \ \ \ \ \ -9
\1is \ \ 10  . \ As \ \ a society \ improves \ \ in \ mental \ \ capacity \ and 
\+
sophistication, \ the \ ratio \ is \ likely \ to \ increase rather than 
\+
decrease. Therefore (73) and (59) imply a lower bound to the rate 
\+
of energy \ dissipation of a society of \ a given complexity. Since 
\+
the total store \ of energy available to a society \ is finite, its 
\+
lifetime is also finite. We \ have reached the sad conclusion that 
\+
the slowing down of metabolism described by my biological scaling 
\+
hypothesis \ is \ \ insufficient \ to \ allow \ \ a society \ to \ survive \/
\+
indefinitely. \,
\+
    \,
\+
    Fortunately, life \ has another strategy with \ which to escape \/
\+
from \ this \ impasse, \ namely \ hibernation. \ Life \ may \ metabolize 
\+
intermittently, but may continue to radiate waste heat into space 
\+
during \ its periods \ of hibernation. \ When life \ is in its active 
\+
phase, \ it \ will \ be \ in \ thermal \ contact \ with \ its radiator at 
\+
temperature theta. \ When life is \ hibernating, the radiator \ will 
\+
still be \ at temperature \7y \1but the \ life will be at \ a much lower \/
\+
temperature so that metabolism is effectively stopped. \,
\+
    \,
\+
    Suppose \ then that \ a society spends \ a fraction g(t) \ of its \/
\+
time in the active phase and a fraction [1-g(t)] hibernating. The 
\+
cycles of activity and hibernation should be short enough so that 
\+
g(t) and \7y\1(t) do not vary \ appreciably during any one cycle. Then 
\+
(56) and (59) \ no longer hold. Instead, subjective \ time is given \/
\+
by \,
\+
\,
\+\ \ \ \ \ \ \ \ \ \ \ \ \ \ t
\0\ \ \ \ \ \ \ \ \ \ \ \ \ i\,
\+\1
    u(t) = f \0|\ \ \1g(t').\7y\1(t') dt'                         (74)\,
\+
\0\ \ \ \ \ \ \ \ \ \ \ \ \ j\,
\+\1\ \ \ \ \ \ \ \ \ \ \ \ \ \ 0
\,
\+
and the average rate of dissipation of energy is \,
\+
\,
\+\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 2
\1    m = k.f.Q.g.\7y\                                       \1(75)\,
\+
\,
\+
The constraint (71) is replaced by \,
\+
\,
\+
\ \ \ \ \ \ \ \ \ \ \ Q\ \7e\,
\+\1
\7    y\1(t) > \8-\ -\ \1g(t) .                                   (76)\,
\+
\ \ \ \ \ \ \ \ \ \ \ N\ k\,
\+
\,
\+
    Life keeps in \ step with the limit (61) \ on radiated power by \/
\+
lowering its duty cycle in proportion to its temperature. \,
\+
\,
\+
    As \ an \ example \ of \ \ a possible \ strategy \ for \ a long-lived \/
\+
society, we can \ satisfy the constraints (60) and \ (76) by a wide \/
\+
margin if we take \,
\+
\,
\+
\7\ \ \ \ \ \ \ \ \ \ \ y\1(t)\ \ \ \0(\ \1t\ \ \0)\1-\7a\,
\+\1
    g(t) = \8----\ \1= \0|\ \8--\ \0|\ \  \1,                            (77)\,
\+
\7\ \ \ \ \ \ \ \ \ \  \ y\ \ \ \ \ \09\ \1t\ \ \00\,
\+\2\ \ \ \ \ \ \ \ \ \ \ \ \ 0\ \ \ \ \ \ \ 0
\1\ \ \ \ \ \ \ \ \ \ \  \,
\+
where \7y  \1and \ t  are the present temperature \ of life the present 
\-\2\ \ \ \ \ \ \ 0\ \ \ \ \ \ \ 0\/
\+\1
age of the universe. The exponent alpha has to lie in the range \,
\+
\,
\+
    1/3 < \7a \1< 1/2 ,                                     (78)\,
\+
\,
\+
and for definiteness we take \,
\+
\,
\+
\7    a \1= 3/8.                                            (79)\,
\+
    \,
\+
Subjective time then becomes by (74) \,
\+
\,
\+
\0\ \ \ \ \ \ \ \ \ \ \ \ \ (\ \1t\ \ \0)\21/4\,
\+\1
    u(t) = A \0| \8--\ \0|\ \ \  \1,                                (80)\,
\+
\0\ \ \ \ \ \ \ \ \ \ \ \ \ 9\ \1t\ \ \00\,
\+\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0
\1\,
\+
where \,
\+
\,
\+\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 18
\1    A = 4f.\7y\ \1.t\  = 10\ \                                  (81)\,
\+\2\ \ \ \ \ \ \ \ \ \ \ \ 0\ \ 0
\1\,
\+
is \ the \ present \ age \ of \ the \ universe \ measured \ in moments of \/
\+
consciousness. The average rate of energy dissipation is by (75) \,
\+
\,
\+
\0\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (\ \1t\ \ \0)\2-9/8\,
\+\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 2
\1    m(t) = kfQ.\7y\ \ \0|\ \8--\ \0|\ \ \ \                             \1(82)\,
\+\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0
\0\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 9\ \1t\ \ \00\,
\+\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0
\1\,
\+
    The \ total \ energy \ metabolized \ over \ all \ time \ from t\  to 
\-\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0\/
\+\1
infinity is \,
\+
\,
\+\ \ \ \ \ +\98
\0    i\,
\+\1
\0\ \ \ \ |\ \ \ \1m(t) dt = BQ                                    (83)\,
\+
\0\ \ \ \ j\,
\+\1\ \ \ \ \ t
\2\ \ \ \ \ \ 0\,
\+\1
\,
\+
\,
\+\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 4
\1    B = 2AK.\7y\  \1= 6.10\ \ \ erg.                            (84)\,
\+\2\ \ \ \ \ \ \ \ \ \ \ \ \ 0
\1\,
\+
    \,
\+
    This \ example shows \ that it \ is possible \ for life \ with the \/
\+
strategy of \ hibernation to achieve \ simultaneously its two \ main \/
\+\3
\1objectives. \ First, \ \ according \ to \ (80), \ \ \3subjective \ time \ is 
\+\1
\3infinite\1; \ although the \ biological clocks \ are slowing \ down and 
\+
running intermittently \ as the universe \ expands, subjective time 
\+\3
\1goes \ on forever. \ Second, according \ to (83), \ \3the total \ energy 
\+\1
\3required for \ indefinite survival is finite\1. \ The conditions (78) 
\+
are \ sufficient \ to \ make \ the \ integral \ (83) convergent and the \/
\+
integral (74) divergent as t \9L \1+\98\1. \,
\+
    \,
\+
    According \ to \ (83) \ and \ (84), \ the \ supply \ of \ free energy \/
\+
required \ for \ the \ indefinite \ survival \ of \ a society \ with the 
\+
complexity (58) \ of the present human \ species, starting from the \/
\+
present time and continuing forever, is of the order \,
\+
\,
\+\2\ \ \ \ \ \ \ \ \ \ \ \ \ 37
\1    BQ =\ 6.10\ \ \ \ erg,                                   (85)\,
\+
\,
\+
about \ as much \ energy as \ the sun \ radiates in \ eight hours. The \/
\+
energy \ resources \ of \ a galaxy \ would \ be \ sufficient to support 
\+\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 24
\1indefinitely a society with a complexity about 10   times greater \/
\+
than our own. \,
\+
    \,
\+
    These \ conclusions \ are \ valid \ in \ an \ open cosmology. It is \/
\+
interesting to \ examine the very different \ situation that exists 
\+
in a closed cosmology. \ If life tries to survive \ for an infinite 
\+
subjective time in a closed cosmology, speeding up its metabolism 
\+
as \ \ the \ universe \ \ contracts \ and \ \ the \ background \ \ radiation 
\+
temperature rises, \ the relations (56) \ and (59) still \ hold, but \/
\+
physical time t has only a finite duration (5). If \,
\+
\,
\+
\7\ \ \ \ t \1= 2\7p\1T\  - t  ,                                     (86)\,
\+\2\ \ \ \ \ \ \ \ \ \ \ 0
\1    \,
\+
the background radiation temperature \,
\+
\,
\+
\ \ \ \ \ \ \ \ \ \ \ \ \ \  1\,
\+
\7    y\ \ \ \ \ \1= a\ \8----                                      \1(87)\,
\+\2\ \ \ \ \ R(t)
\1\ \ \ \ \ \ \ \ \ \ \ \ \ \ R(t)\,
\+
\,
\+\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ -2/3
\1is proportional to \ \7t     \1as \7t \9L \10, by virtue \ of (2) and (3). If \/
\+
the temperature \ \7y\1(t) of life \ remains close to \ \7y  \1as as \ \7t \9L \10, 
\-\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ R\/
\+\1
then the \ integral (56) is finite \ while the integral of \ (59) is 
\+
infinite. \ We \ have \ an \ infinite \ energy \ requirement to achieve 
\+
a finite \ subjective lifetime. \ If \ \7y\1(t) \ tends to \ infinity more 
\+
slowly \ than \7y \1, \ the total \ duration of \ subjective time remains 
\-\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ R
\+\1
finite. \ If \7y\1(t) \ tends to \ \ infinity more \ rapidly than \ \7y \1, the 
\-\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ R
\+\1
energy \ \ requirement \ \ for \ \ metabolism \ \ remains \ infinite. \ The 
\+
biological clocks \ can never speed \ up fast enough \ to squeeze an \/
\+
infinite subjective time into a finite universe. \,
\+
    \,
\+
    I return with a feeling of relief \ to the wide open spaces of \/
\+
the \ open universe. \ I do not \ need to \ emphasize the partial and 
\+
preliminary character of the conclusions that I have presented in 
\+
this lecture. I have only delineated in the crudest fashion a few 
\+
of the physical \ problems that life must encounter \ in its effort 
\+
to survive \ in a cold universe. \ I have not addressed \ at all the 
\+
multitude of questions that arise as soon as one tries to imagine 
\+
in detail the architecture of a form of life adapted to extremely 
\+
low temperatures. \ Do there exist functional \ equivalents in low-
\+
temperature \ systems for \ muscle, nerve, \ hand, voice, \ eye, ear, \/
\+
brain, and memory? I have no answers to these questions. \,
\+
\,
\+
    It is \ possible to say a little \ about memory without getting \/
\+
into detailed architectural problems, since memory is an abstract \/
\+
concept. The capacity of a memory can be described quantitatively \/
\+
as \ a certain number \ of bits \ of information. \ I would like \ our 
\+
descendants \ to \ be \ endowed \ not \ only \ with \ an infinitely long 
\+
subjective lifetime \ but also with a memory \ of endlessly growing 
\+
capacity. \ To \ \ be \ immortal \ with \ a finite \ \ memory \ is \ highly 
\+
unsatisfactory; it seems hardly worthwhile \ to be immortal if one 
\+
must ultimately erase all trace of one's origins in order to make 
\+
room for new \ experience. There are two forms \ of memory known to 
\+
physicists, \ analog \ and \ digital. \ All \ our \ computer technology 
\+
nowadays \ is based \ on digital \ memory. But \ digital memory is in 
\+
principle limited \ in capacity by \ the number of \ atoms available 
\+
for \ its construction. \ A society with \ finite material resources 
\+
can \ never \ \ build \ a digital \ memory \ \ beyond \ a certain \ finite 
\+
capacity. \ Therefore digital \ memory \ cannot \ be adequate \ to the \/
\+
needs of a life form planning to survive indefinitely. \,
\+
        \,
\+
    Fortunately, there \ is no limit in \ principle to the capacity \/
\+
of an analog memory built out \ of a fixed number of components in 
\+
an expanding \ universe. For example, a physical \ quantity such as 
\+
the angle between \ two stars in the sky can \ be used as an analog 
\+
memory unit. \ The capacity of \ this memory unit \ is equal to \ the 
\+
number \ of significant \ binary digits \ to which \ the angle can be 
\+
measured. \ As \ the \ universe \ expands \ and \ the stars recede, the 
\+
number \ of \ \ significant \ digits \ in \ the \ \ angle \ will \ increase 
\+
logarithmically with time. Measurements of atomic frequencies and 
\+
energy levels can also in \ principle be measured with a number of 
\+
significant \ figures \ proportional \ to \ \ (log \ t). \ Therefore \ an 
\+
immortal \ civilization should \ ultimately find \ ways to \ code its 
\+
archives in an analog memory \ with capacity growing like (log t). 
\+
Such \ a memory \ will \ put \ severe \ constraints \ on \ the \ rate \ of 
\+
acquisition of permanent new knowledge, \ but at least it does not \/
\+
forbid it altogether. \,
\+
\,
\+
\,
\+
LECTURE IV. COMMUNICATION \,
\+
\,
\+
    In this \ last lecture I examine the \ problem of communication \/
\+
between two \ societies separated by a large \ distance in the open 
\+
universe with metric (6). I assume that they communicate by means 
\+
of electromagnetic signals. Without \ loss of generality I suppose 
\+
that \ society A, \ moving along \ the world-line \ \7x \1= 0, transmits, 
\+
while \ society B, \ moving along \ a world-line with \ the co-moving 
\+
coordinate \ \7x \1= \7h\1, receives. \ A signal transmitted \ by A when the 
\+
time coordinate \ \7j \1= \7x \1will be received \ by B when \7j \1= \7x \1+ \7h\1. \ If 
\+
the transmitted \ frequency is omega, the \ received frequency will \/
\+
be red-shifted to \,
\+
\,
\+
\7\ \ \ \ \ \ \ \ \ \ \ w\ \ \ \ \ \ \ \1R\,
\+\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ A
\7    w\1' = \8-----\ \1= \7w \8--                                   \1(88)\,
\+
\ \ \ \ \ \ \ \ \ 1 + z\ \ \ \ \ R\,
\+\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ B
\1\,
\+
    R\ \ = cT\ (cosh \7x \1- 1)                                (89)\,
\+\2\ \ \ \ \ A\ \ \ \ \ 0
\1\,
\+
    R\ \ = cT\ (cosh (\7x \1+ \7h\1) - 1)                          (90)\,
\+\2\ \ \ \ \ B\ \ \ \ \ 0
\1    \,
\+
    \,
\+
    The bandwidths \ B and B' will \ be related by \ the same factor \/
\+
(1\6\6t\6\1+\6\6t\6\1z). \ The proper \ distance between \ A and B at \ the time \ the 
\+
signal is received \ is d  = R \7h\1. However, the \ area of the sphere 
\-\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ L\ \ \ \ B\/
\+\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 2
\7x \1= \7h \1at the same instant is 4\7p\1d , with \,
\-\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ T
\+\1
\,
\+
    d\  = R\  sinh \7h \1.                                    (91)\,
\+\2\ \ \ \ \ T\ \ \ \ B
\1\,
\+
    If A transmits F photons per steradian in the direction of B, \/
\+
the number of photons received by B will be \,
\+
\,
\+
\7\ \ \ \ \ \ \ \ \ \ \ s\1'\,
\+
    F' = F \8--                                           \1(92)\,
\+\2\ \ \ \ \ \ \ \ \ \ \ \ 2
\1\ \ \ \ \ \ \ \ \ \ \ d\,
\+\2\ \ \ \ \ \ \ \ \ \ \ \ T
\1\,
\+
where \7s\1' \ is the effective \ cross section (£‡inn˜ \ pr–©ez) of the \/
\+
receiver. \,
\+
    \,
\+
    Now the cross section of a receiver for absorbing a photon of \/
\+
frequency \ \7w\1' \ is \ given \ by \ a formula \ similar \ to \ (63) in the \/
\+
previous lecture \,
\+
\,
\+\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 2
\1\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 4\7p\ w\,
\+\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ ji\ \ \ \ \ \ 2
\7    S\1' = \9S\ S\ \7r\ \ \8------\ \0|\1D  \0|\ \1.\7d\1(\7w\ \  \1- \7w\1')               (93)\,
\+\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ i\ \ \ \ \ \ \ \ \ \ ij\ \ \ \ \ \ ji
\ \ \ \ \ \ \ \ \ i j \ \ \ \ \ \9h\1c\,
\+
\,
\+
with D   \ again a dipole matrix \ element between states \ i and j. 
\-\2\ \ \ \ \ \ ij\/
\+\1
When this is integrated over all \7w\1', we obtain precisely the left 
\+
side \ of the \ sum rule \ (67). The \ contribution from \ negative \7w\1' \/
\+
represents induced emission of a photon by the receiver. I assume 
\+
that the receiver is incoherent with the incident photon, so that \/
\+
induced emission is negligible. Then the sum rule gives \,
\+
\,
\+\ \ \ \ \ +\98\ \ \ \ \ \ \ \ \ \ \ \ \ \ \22\ 2
\0\ \ \ \ i\ \ \ \ \ \ \ \ \ \ \ \ \ \ \12\7p\ \1e\,
\+
\0    |\ \ \7S\1' d\7w\1' = N' \8-----  \1,                             (94)\,
\+
\0\ \ \ \ j\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \1mc\,
\+\ \ \ \ \ 0
\,
\+
where \ N' is \ the number \ of electrons \ in the \ receiver. If \ the \/
\+
receiver is tuned to the frequency omega' with bandwidth B', (94) \/
\+
gives \,
\+
\,
\+
\7    S\1'.B' \9< \1N'.S\                                        (95)\,
\+\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0
\1\,
\+\2\ \ \ \ \ \ \ \ \ \ \ 2\ 2
\1\ \ \ \ \ \ \ \ \ 2\7p\ \1e\,
\+\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 2
\1    S\  = \8-----\ \1= 0,167 cm\ /s  .                         (96)\,
\+\2\ \ \ \ \ 0
\1\ \ \ \ \ \ \ \ \ \ mc\,
\+
    \,
\+
    To avoid confusion of units, \ I measure both omega' and B' in \/
\+
radians \ per \ second \ rather \ than \ in \ Hertz. \ I assume \ that an 
\+
advanced \ civilization will \ be able \ to design \ a receiver which \/
\+
makes (95) hold with equality. Then (92) becomes \,
\+
\,
\+
\ \ \ \ \ \ \ \ \ FN'S\,
\+\2\ \ \ \ \ \ \ \ \ \ \ \ \ 0
\1    F' = \8-----   \1.                                      (97)\,
\+\2\ \ \ \ \ \ \ \ \ \ 2
\1\ \ \ \ \ \ \ \ \ d\ B'\,
\+\2\ \ \ \ \ \ \ \ \ \ T
\1    \,
\+
    I assume that the transmitter \ contains N electrons which can \/
\+
be \ driven in \ phase so \ as to \ produce a beam \ of radiation with 
\+\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ -1/2
\1angular \ spread \ of \ the \ order \ N    . \ If \ the \ transmitter \ is \/
\+
considered to be an array \ of N dipoles with optimum phasing, the \/
\+
number of photons per steradian in the beam is \,
\+
\,
\+
\ \ \ \ \ \ \ \ 3N\ E\,
\+
    F = \8--\ --  \1,                                        (98)\,
\+
\ \ \ \ \ \ \ \ 8\7p\ \9h\7w\,
\+\1
\,
\+
where E is \ the total energy transmitted. \ The number of received \/
\+
photons is then \,
\+
\,
\+
\ \ \ \ \ \ \ \ \ 3NN'ES\,
\+\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0
\1    F' = \8--------                                       \1(99)\,
\+\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ 2
\1\ \ \ \ \ \ \ \ \ 8\7p\9h\7w\1d\ B'\,
\+\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ T
\1\,
\+
    We \ see at \ once from \ \ (99) that \ low frequencies \ and small \/
\+
bandwidths \ are desirable \ for increasing \ the number \ of photons 
\+
received. \ But \ we \ are \ interested \ in \ transmitting information 
\+
rather \ than \ photons. \ To \ extract \ information efficiently from 
\+
a given number of photons we \ should use a bandwidth equal to the \/
\+
detection rate, \,
\+
\,
\+
\ \ \ \ \ \ \ \ \ F'\ \ \ \ \ \ \ F'\,
\+
    B' = \8--\ \1, B = \8--  \1,                                 (100)\,
\+
\7\ \ \ \ \ \ \ \ \ t\ \ \ \ \ \ \ \ t\,
\+\2\ \ \ \ \ \ \ \ \ \ B\ \ \ \ \ \ \ \ A
\1\,
\+
where \7t \1is the duration of the \ reception, and \7t  \1is the duration 
\-\2\ \ \ \ \ \ \ B\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ A\/
\+\1
of the transmission. With this \ bandwidth, F' represents both the 
\+
number \ of photons \ and also \ the number \ of bits \ of information 
\+
received. It is convenient to express \ \7t  \1and \7t  \1as a fraction of 
\-\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ A\ \ \ \ \ \ B
\+\1
the \ radius of \ the universe \ \ at the \ times of \ transmission and \/
\+
reception \,
\+
\,
\+
\ \ \ \ \ \ \ \ \ \ \ R\ \ \ \ \ \ \ \ \ \ \ R\,
\+\2\ \ \ \ \ \ \ \ \ \ \ \ A\ \ \ \ \ \ \ \ \ \ \ B
\7    t\ \ \1= \7d \8--\ \1, \7t\ \ \1= \7d\ \8--\ \1.                             (101)\,
\+\2\ \ \ \ \ A\ \ \ \ \ \ \ \ \ \ \ B
\1\ \ \ \ \ \ \ \ \ \ \ c\ \ \ \ \ \ \ \ \ \ \ c\,
\+
\,
\+
The condition \,
\+
\,
\+
\7    d \9< \11                                               (102)\,
\+
\,
\+
then puts a lower \ bound on the bandwidth B. \ I shall also assume \/
\+
for simplicity that the frequency omega is chosen to be as low as \/
\+
possible consistent with the bandwidth B, namely \,
\+
\,
\+
\7    w \1= B, \7w\1' = B'\ .                                    (103)\,
\+
\,
\+
Then (99), (100), (101) give \,
\+
\,
\+\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 2
\0    \ \ \ \ \ ( \1NN'\7d\ \1E\ \ \ \ \ 1\ \ \ \ \ \ \0)\21/3\,
\+\1
\ \ \ \ F' = \0| \8------\ ----------\ \0|\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \1(104)\,
\+\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 2
\0\ \ \ \ \ \ \ \ \ 9 \11 + z\ \ sinh\ \ \7h \1E\ \ \00\,
\+\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ c
\1\,
\+
where by (96) \,
\+
\,
\+\2\ \ \ \ \ \ \ \ \ \ \ \ \ 2
\1\ \ \ \ \ \ \ \ \ 8\7p\9h\1c\ \ \ \ 4\,
\+\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 2\ \ \ \ \ \ \ -5
\1    E\ \ = \8-----\ \1= \8--\ \1137.mc\ \ = 3.10\ \  erg.               (105)\,
\+\2\ \ \ \ \ c
\1\ \ \ \ \ \ \ \ \ \ 3S\ \ \ \ \ 3\7p\,
\+\2\ \ \ \ \ \ \ \ \ \ \ \ 0
\1\,
\+
    We see from \ (104) that the quantity of \ information that can \/
\+
be \ transmitted from \ A to B with \ a given expenditure \ of energy 
\+
does \ not decrease \ with time \ as the \ universe expands and A and 
\+
B move \ apart. The \ increase in \ distance is \ compensated by \ the 
\+
decrease in the energy cost of each photon and by the increase of \/
\+
receiver cross seetion with decreasing bandwidth. \,
\+
    \,
\+
    The received signal is (104). We \ now have to compare it with \/
\+
the \ received \ noise. \ The \ background \ noise \ in the universe at 
\+
frequency \ omega \ \ can \ be \ described \ by \ \ an \ equivalent \ noise 
\+
temperature T , so that the \ number of photons per unit bandwidth 
\-\2\ \ \ \ \ \ \ \ \ \ \ \ \ N
\+\1
per steradian \ per square centimeter \ per second is \ given by the \/
\+
Rayleigh-Jeans formula \,
\+
\,
\+
\ \ \ \ \ \ \ \ \ \ \ kT\ \7w\,
\+\2\ \ \ \ \ \ \ \ \ \ \ \ \ N
\1    I(\7w\1) = \8------  \1.                                    (106)\,
\+\2\ \ \ \ \ \ \ \ \ \ \ \ \ 3\ \ 2
\1\ \ \ \ \ \ \ \ \ \ \ 4\7p\ \9h\1c\,
\+
\,
\+
    This \ formula \ is \ merely \ a definition \ of \ T  , which is in 
\-\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ N\/
\+\1
general a function of omega and t. I do not assume that the noise 
\+
has a Planck \ spectrum over the whole \ range of frequencies. Only 
\+
a part \ of \ the \ noise \ is \ \ due \ to \ the \ primordial \ background 
\+
radiation, which \ has a Planck spectrum with \ temperature \7y \1. The 
\-\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ R
\+\1
primordial noise temperature \7y  \ \1varies inversely with the radius 
\-\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ R\/
\+\1
of the universe, \,
\+
\,
\+
    k\7y\ \1R\,
\+\2\ \ \ \ \ \ R\ \ \ \ \ \ \ \ \ \ 29
\8\ \ \ \ ----\ \1= \7L \1= 10\ \ \ ,                                   (107)\,
\+
\9\ \ \ \ \ h\1c\,
\+
\,
\+
with \ R given \ by \ (8). \ I assume \ that \ the total noise spectrum \/
\+
scales in the same way with radius as the universe expands, thus \,
\+
\,
\+
    T\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \9h\7w\,
\+\2\ \ \ \ \ N
\8\ \ \ \ --\ \1= f(x)  , x = \8--\ \ \1,                              (108)\,
\+
\7\ \ \ \ y\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \1k\7y\,
\+\2\ \ \ \ \ R\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ R
\1\,
\+
with \ f a universal \ function of \ x. When \ x is of \ the order \ of \/
\+
unity, the \ noise is dominated \ by the primordial \ radiation, and \/
\+
f(x) takes the Planck form \,
\+
\,
\+
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ x\,
\+
    f(x) = f\ (x) = \8------\ \ \1,  x \9= \11 .                   (109)\,
\+\2\ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ x
\1\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ e\ \ - 1\,
\+
\,
\+
    But \ there \ will \ be \ strong \ deviations \ from (109) at large \/
\+
x (due \ to \ \ red-shifted \ starlight) \ and \ at \ \ small \ x (due \ to 
\+
nonthermal radio sources). Without going into details, we can say 
\+
that f(x) \ is a generally decreasing \ function of x and \ tends to \/
\+
zero rapidly as x \9L \1+\98\1. \,
\+
\,
\+
The total energy density of radiation in the universe is \,
\+
\,
\+\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 4\ 4
\1\ \ \ \ 4\7p\ \0i\ \ \ \ \ \ \ \ \ \ \ \ \ \  \1k \7y\,
\+\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \  \  R
\8    --\ \0|\ \1I(\7w\1).\9h\7w \1d\7w \1= \8------\ \1I\ ,                        (110)\,
\+\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 2\ 3\ 3
\1\ \ \ \ c\ \ \0j\ \ \ \ \ \ \ \ \ \ \ \ \ \ \7p\ \9h\ \1c\,
\+
\,
\+
with \,
\+
\,
\+\ \ \ \ \ \ \ \ \ +\98
\0\ \ \ \ \ \ \ \ i\,
\+\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 2
\1    I = \0|\ \ \ \1f(x).x\ \ dx .                                (111)\,
\+
\0\ \ \ \ \ \ \ \ j\,
\+\1\ \ \ \ \ \ \ \ \ 0
\,
\+
    The \ integral \ I must \ be \ convergent \ at \ both \ high and low \/
\+
frequencies. Therefore we can find a numerical bound b such that \,
\+
\,
\+\2\ \ \ \ \ 3
\1    x\ .f(x) < b                                         (112)\,
\+
\,
\+
for all x. \ In fact (112) probably holds with \ b = 10 if we avoid \/
\+
certain discrete frequencies such as the 1420 MHz hydrogen line. \,
\+
    \,
\+
    The number \ of noise photons \ received during the \ time \7t  \1by 
\-\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ B\/
\+\1
the receiver with bandwidth B' and cross section \7S\1' is \,
\+
\,
\+
    F\  = 4\7pS\1'B'\7t\ \1I(\7w\1') .                                (113)\,
\+\2\ \ \ \ \ N\ \ \ \ \ \ \ \ \ \ B
\1\,
\+
\,
\+
    We substitute from (95), (96), (100), (103), (106), and (108) \/
\+
into (113) and obtain \,
\+
\,
\+
\ \ \ \ \ \ \ \ \ 2r\,
\+\2\ \ \ \ \ \ \ \ \ \ \ 0
\1    F\ \ = \8---\ \1fN'F' ,                                    (114)\,
\+\2\ \ \ \ \ N
\7\ \ \ \ \ \ \ \ \ g\,
\+\2\ \ \ \ \ \ \ \ \ \ B
\1\,
\+
where \,
\+
\,
\+\2\ \ \ \ \ \ \ \ \ \ 2
\1\ \ \ \ \ \ \ \ \ e\,
\+\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ -13
\1\ \ \ \ r\ \ = \8---\ \1= 3.10\ \ \ \ cm ,                             (115)\,
\+\2\ \ \ \ \ 0\ \ \ \ \ 2
\1    \ \ \ \ \ mc\,
\+
\,
\+
and \,
\+
\ \ \ \ \ \ \ \ \ \ hc\ \ \ \ R\,
\+\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ B
\7    g\ \ \1= \8----\ \1= \8--\                                      \1(116)\,
\+\2\ \ \ \ \ B
\1\ \ \ \ \ \ \ \ \ k\7y\ \ \ \ \ L\,
\+\2\ \ \ \ \ \ \ \ \ \ \ R'
\1\,
\+
is the \ wavelength of the primordial \ background radiation at the \/
\+
time of \ reception. Since F' \ is the signal, \ the signal-to-noise \/
\+
ratio is \,
\+
\,
\+
\7\ \ \ \ \ \ \ \ \ \ \ \ g\,
\+\2\ \ \ \ \ \ \ \ \ \ \ \ \ B
\1    R\ \ \ = \8------  \1.                                     (117)\,
\+\2\ \ \ \ \ SN
\1\ \ \ \ \ \ \ \ \ \ 2fN'r\,
\+\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0
\1\,
\+
    In this \ formula, f is the \ noise-temperature ratio given \ by \/
\+
(108), N' is the number of \ electrons in the receiver, and r , \7g  
\-\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0\ \ \ B
\+\1
are given by (115), (116). Note that in calculating (117) we have 
\+
not \ given the \ receiver any \ credit for \ angular discrimination, 
\+
since \ the \ cross \ section \ \7S\1' \ given \ by \ (95) is independent of \/
\+
direction. \,
\+
    \,
\+
    I now summarize \ the conclusions of \ the analysis so \ far. We \/
\+
have \ a transmitter and \ a receiver on \ the world-lines \ A and B, \/
\+
transmitting and receiving at times \,
\+
\,
\+
    t\ \ = T\ .(sinh \7x \1- \7x\1) ,  t  = T .[sinh(\7x \1+ \7h\1) - (\7x \1+ \7h\1)]\ ,\,
\+\2\ \ \ \ \ A\ \ \ \ 0\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ B\ \ \ \ 0
\1\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \    (118)\,
\+
\,
\+
According to (89) and (101), \,
\+
\,
\+
\ \ \ \ \ \ \ \ \ \ \ dt\ \ \ \ \ \ \ \ \ \ \ \ \ \ dt\,
\+\2\ \ \ \ \ \ \ \ \ \ \ \ \ A\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ B
\7    t\ \ \1= \7d\ \8---\ \1,    \7t\ \ \1= \7d \8---\ \1.                        (119)\,
\+\2\ \ \ \ \ A\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ B
\1\ \ \ \ \ \ \ \ \ \ \ d\7x\ \ \ \ \ \ \ \ \ \ \ \ \ \ \1d\7x\,
\+\1
\,
\+
    It is \ convenient to think \ of the transmitter as permanently \/
\+
aimed \ at \ the \ receiver, \ and \ transmitting \ intermittently with 
\+
a certain \ duty cycle \ \7d \1which may \ vary with \ \7x\1. When \ \7d \1= 1 the \/
\+
transmitter is on all the time. The number F' of photons received \/
\+
in the time \ \7t  \1can then be considered as \ a bit rate in terms of 
\-\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ B
\+\1
the variable \7x\1. In fact, F' dxi is the number of bits received in 
\+
the interval d\7x\1. \ It is useful to work \ with the variable \7x \1since \/
\+
it maintains a constant difference eta between A and B. \,
\+
    \,
\+
    From (100), (101), (103), (107), and (108) we derive a simple \/
\+
formula for the bit rate, \,
\+
\,
\+
    F' = \7L\1x.\7d \1.                                         (120)\,
\+
\,
\+
    The \ energy \ E transmitted \ in \ the \ \ time \ \7t  \ \1can \ also \ be 
\-\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ A\/
\+\1
considered as \ the rate of energy \ transmission per unit interval \/
\+
d\7x\1. From (104) and (120) we find \,
\+
\,
\+\2\ \ \ \ \ \ \ \ \ 3
\7\ \ \ \ \ \ \ \ L\,
\+\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 2\ \ \ \ \ 3
\1    E = \8---\ \1(1 + z).sinh\ (\7h\1).x\ .\7d\1.E\ \ .\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (121)\,
\+\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ c
\1\ \ \ \ \ \ \ \ NN'\,
\+
\,
\+
    We are still free to choose the parameters x [determining the \/
\+
frequency omega by \ (108)] and \7d\1, both of which \ may vary with \7x\1. \/
\+
The only constraints are (102) and the signal-to-noise condition \,
\+
\,
\+
    R\ \ \ \9> \110 ,                                          (122)\,
\+\2\ \ \ \ \ SN
\1\,
\+
\,
\+
the \ signal-to-noise ratio \ being defined \ by (117). \ If I assume \/
\+
that (112) holds with b=10, then (122) will be satisfied provided \/
\+
that \,
\+
\,
\+
\0\ \ \ \ \ \ \ \ (\ \1G\ \0)\21/3\,
\+\1
    x > \0|\ \8-\ \0|\ \ \  \1,                                      (123)               \,
\+
\0\ \ \ \ \ \ \ \ 9\ \1r\ \00\,
\+\1
\,
\+
with \,
\+
\,
\+
\ \ \ \ \ \ \ \ 200.r\ \ \ \ N'\ \ \ \ \ \ \ \ \ \ \ N'\,
\+\2\ \ \ \ \ \ \ \ \ \ \ \ \ 0\ \ \ \ \ \ \ \ \ \ \ -9
\1    G = \8------\ -----\ \1= 10\ \ \ \8-----\ \ \1,                    (124)\,
\+
\7\ \ \ \ \ \ \ \ \ \ g\ \ \ \ \11 + z\ \ \ \ \ \ \ \ 1 + z\,
\+\2\ \ \ \ \ \ \ \ \ \ \ p
\1\,
\+
\ \ \ \ \ \ \ \ R\ \ \ \ cosh \7x \1- 1\,
\+\7\ \ \ \ \ \ \ \ \ L
\1    r = \8--\ \1= \8----------\ \1.                               (125)\,
\+
\ \ \ \ \ \ \ \ R\ \ \ \ cosh \7x \1- 1\,
\+\2\ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ p
\1\,
\+
\,
\+
    Here \7g \1, R  \ and \7x  \1are the present \ values of the background 
\-\2\ \ \ \ \ \ \ \ \ \ p\ \ \ p\ \ \ \ \ \ \ p\/
\+\1
radiation wavelength, \ the radius of \ the universe, and \ the time 
\+
coordinate \7j\1. It is noteworthy that the signal-to-noise condition 
\+
(123) may be difficult to satisfy at early times when r is small, 
\+
but gets \ progressively easier as \ time goes on \ and the universe 
\+
becomes quieter. To avoid an \ extavagant expenditure of energy at 
\+
earlier times, I choose \ the duty cycle delta to \ be small at the \/
\+
beginning, increasing gradually until it reaches unity. \,
\+
\,
\+
    All the requirements are satisfied if we choose \,
\+
\,
\+
\0\ \ \ \ \ \ \ \ \ \ \ \ ((\ \1G\ \0)\21/3\ \  \11\ \ \ \0)\,
\+\1
    x = max \0{|\ \8-\ \0|\ \ \ \1, \8----\ \0} \1,                         (126)\,
\+\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1/2
\0\ \ \ \ \ \ \ \ \ \ \ \ 99\ \1R\ \00\ \ \ \ \ \7x\ \ \ \ \00\,
\+\1
\,
\+
\0\ \ \ \ \ \ \ \ \ \ \ \ (\ \1r\ \ \ \ \ \ \ \ \ \ \ \0)\,
\+\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ -3/2
\7    d \1= min \0{\ \8-\ \7x\ \ \ \ \ \1, 1 \0} \1,                           (127)\,
\+
\0\ \ \ \ \ \ \ \ \ \ \ \ 9\ \1G\ \ \ \ \ \ \ \ \ \ \ \00\,
\+\1
\,
\+
so that \,
\+
\,
\+\2\ \ \ \ \ 3\ \ \ \ \ -3/2
\1    x\ \7d \1= \7x\ \ \ \                                          \1(128)\,
\+
\,
\+
\,
\+
for all \ \7x\1. The transition \ between the two \ ranges in (126) \ and \/
\+
(127) occurs at \,
\+
\,
\+
\7    x \1= \7x\ \ \9= \1log G ,                                    (129)\,
\+\2\ \ \ \ \ \ \ \ \ T
\1\,
\+
since \ \7x \1increases logarithmically \ with r by \ (125). With \ these \/
\+
choices of x and \7d\1, (120) and (121) become \,
\+
\,
\+
\0\ \ \ \ \ \ \ \ \ \ \ \ \ \  ((\ \1r\ \0)\22/3\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \0)\,
\+\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ -3/2\ \ \ \ -1/2
\1    F' = \7L\1.min\ \0{|\ \8-\ \0|\ \ \ \1.\7x\ \ \ \ \ \1, \7x\ \ \ \ \ \0} \1,              (130)\,
\+
\0\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 99\ \1G\ \00\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0\,
\+\1
\,
\+\2\ \ \ \ \ \ \ \ \ 3
\7\ \ \ \ \ \ \ \ L\,
\+\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 2\ \ \ \ \ \ \ \ -3/2
\1    E = \8---\ \1(1 + z).sinh (\7h\1).E\ .\7x\ \ \ \ \ \1.                 (131)\,
\+\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ c
\1\ \ \ \ \ \ \ \ NN'\,
\+
    \,
\+
    Now \ consider the \ total number \ of bits \ received at B up to \/
\+
some \ epoch \ \7x \1in \ the \ remote \ future. \ According to (130), this \/
\+
number is approximately \,
\+
\,
\+\7\ \ \ \ \ \ \ \ \ \ x
\0\ \ \ \ \ \ \ \ \ i\,
\+\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1/2
\1    F\ \ = \0|\ \ \1F' d\7x \1= 2\7L\1.\7x\ \ \ \ \1,                           (132)\,
\+\2\ \ \ \ \ T
\0\ \ \ \ \ \ \ \ \ j\,
\+\1\ \ \ \ \ \ \ \ \ \ 0
\,
\+
and increases \ without limit as \ \7x \1increases. On the \ other hand, \/
\+
the \ total energy \ expended by \ the transmitter \ over the \ entire \/
\+
future is finite, \,
\+
\,
\+\ \ \ \ \ \ \ \ \ \ +\98\ \ \ \ \ \ \ \ \ \ \ \23
\0\ \ \ \ \ \ \ \ \ i\ \ \ \ \ \ \ \ \ \ \ \ \7L\,
\+\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ h\ \ \ \ \ \22\ \ \ \ \ -1/2
\1    E\ \ = \0|\ \ \ \1E d\7x \1= 2 \8---\ \1e\ .sinh\ (\7h\1).\7x\ \ \ \ \1.E\ \ .        (133)\,
\+\2\ \ \ \ \ T\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ c
\0\ \ \ \ \ \ \ \ \ j\ \ \ \ \ \ \ \ \ \ \ \ \1NN'\,
\+\ \ \ \ \ \ \ \ \ \ 0
\,
\+
        \,
\+
    In \ (133) \ I have \ replaced \ the \ red \ shift \ (1\6\6t\6\1+\6\6t\6\1z) \ by its \/
\+\7\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ h
\1asymptotic \ value \ e  \ as \ \7x\6\6t\6\9L\6\6t\6\98\1. \ I have \ thus \ reached the same 
\+
optimistic \ conclusion concerning \ communication as \ I reached in 
\+
the previous lecture about bilogical survival. It is in principle 
\+
possible \ to \ communicate \ forever \ with \ a remote \ society in an \/
\+
expanding universe, using a finite expenditure of energy. \,
\+
    \,
\+
    It is \ interesting to make some \ crude numerical estimates of \/
\+
the magnitudes of \ F  and E . By (107), \ the cumulative bit count 
\-\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ T\ \ \ \ \ \ T\/
\+\1
in every communication channel is the same, of the order \,
\+
\,
\+\2\ \ \ \ \ \ \ \ \ \ \ 29\ \ -1/2
\1    F\ \ = 10\ \ .\7x\ \ \ \ \ \1,                                   (134)\,
\+\2\ \ \ \ \ T
\1\,
\+
a quantity \ of \ information \ amply \ sufficient \ to \ encompass the \/
\+
history of a complex civilization. To estimate E , I suppose that 
\-\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ T
\+\1
the transmitter and the receiver \ each contain 1 kg of electrons, \/
\+
so that \,
\+
\,
\+\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 30
\1    N = N' = 10\ \ \ .                                     (135)\,
\+
\,
\+
Then (133) with (105) gives \,
\+
\,
\+\2\ \ \ \ \ \ \ \ \ \ \ 23\ \ \7h\ \ \ \ \ \22
\1    E\ \ = 10\ \ .e\ .sinh\ (\7h\1)\   erg .                       (136)\,
\+\2\ \ \ \ \ T
\1\,
\+
\,
\+\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 9
\1    This is of the order of 10  W yr, an extremely small quantity \/
\+
of \ energy \ \ by \ astronomical \ standards. \ \ A society \ which \ has 
\+
available to it the energy \ resources of a solar-type star (about 
\+\2\ \ 36
\110   \ W yr) could \ easily provide \ the energy \ to power permanent 
\+\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 22
\1communication channels \ with all the \ 10   stars that \ lie within 
\+
the \ sphere \7h \1< 1. \ That is \ to say, \ all societies \ within a red \/
\+
shift \,
\+
\,
\+
    z = e - 1 = 1.718 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \      (137) \,
\+
\,
\+
of one \ another could remain \ in permanent communication. \ On the \/
\+
other hand, direct communication between two societies with large 
\+
separation would be prohibitively expensive. Because of the rapid 
\+
exponential growrth of E  with \7h\1, the upper limit to the range of 
\-\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ T\/
\+\1
possible direct communication lies at about \7h \1= 10. \,
\+
\,
\+
    It is easier to transmit information to larger distances than \/
\+
\7h \1= 10 without great expenditure \ of energy, if several societies \/
\+
en route \ serve as relay \ stations, receiving and \ amplifying and 
\+
retransmitting the signal in turn. \ In this way messages could be 
\+
delivered over \ arbitrarily great distances \ across the universe. 
\+
Every society \ in the universe \ could ultimately be \ brought into \/
\+
contact with every other society. \,
\+
    \,
\+
    As I remarked in the first lecture [see Eq. (11)], the number \/
\+\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 2
\1of galaxies \ that lie within \ a sphere \7h \1< \7j \1grows \ like e \7j \1when 
\+
\7j \1is large. So, when we try to establish linkages between distant 
\+
societies, there will be a severe problem of selection. There are 
\+
too many galaxies at large distances. \ To which of them should we 
\+
listen? \ To \ \ which \ of \ them \ should \ \ we \ relay \ messages? \ The 
\+
moreperfect our technical means of communication become, the more 
\+
difficulty \ we shall \ have \ in \ deciding which \ communications to \/
\+
ignore. \,
\+
\,
\+
    In conlusion, I would like to emphasize that I have not given \/
\+
any \ definitive proof \ of my \ statement that \ communication of an 
\+
infinite quantity \ of information at \ a finite cost in \ energy is 
\+
possible. To \ give a definitive proof, I would \ have to design in 
\+
detail a transmitter and a receiver and demonstrate that they can 
\+
do what I claim. I have not even tried to design the hardware for 
\+
my \ communications \ system. \ All \ I have \ done \ is \ to \ show that 
\+
a system \ performing according \ to \ my \ specifications is \ not in 
\+
obvious \ contradiction \ with \ the \ \ known \ laws \ of \ physics \ and \/
\+
information theory. \,
\+
\,
\+
    The \ universe that \ I have explored \ in a preliminary \ way in \/
\+
these lectures \ is very different from \ the universe which Steven 
\+
Weinberg had in \ mind when he said, "The \ more the universe seems 
\+
comprehensible, the \ more it also seems \ pointless." I have found 
\+
a universe \ growing \ without \ limit \ in \ richness and complexity, 
\+
a universe of \ life surviving forever and \ making itself known to 
\+
its \ neighbors across \ unimaginable gulfs \ of space \ and time. Is 
\+
Weinberg's universe or mine closer \ to the truth? One day, before \/
\+
long, we should know. \,
\+
        \,
\+
    Whether the details of my calculations turn out to be correct \/
\+
or \ not, \ I think \ I have \ shown \ that \ there are good scientific 
\+
reasons \ for \ taking \ seriously \ the \ possibility \ that \ life and 
\+
intelligence \ can succeed \ in molding \ this universe \ of ours \ to 
\+
their own \ purposes. As Haldane (1924) \ the biologist wrote fifty 
\+
years ago, \ "The human intellect \ is feeble, and \ there are times 
\+
when \ it does \ not assert \ the infinity \ of its \ claims. But even \/
\+
then: \,
\+
\,
\+
\ \ \ \ \ \ \ \ Though in black jest it bows and nods, \,
\+
\ \ \ \ \ \ \ \ I know it is roaring at the gods, \,
\+
\ \ \ \ \ \ \ \ Waiting the last eclipse." \,
\+
\,
\+
\,
\+
REFERENCES \,
\+
\,
\+
* Alpher, R.A., \ R.C. Herman, and \ G. Gamow, 1948, \ Phys.Rev. 74, \/
\+
  1198. \,
\+
* Barrow, J.D., \ and F.J. Tipler, \ 1978, "Eternity is \ Unstable", \/
\+
  Nature (Lond.) 276, 453. \,
\+
* Bethe, \ H.A., and \ E.E. Salpeter, \ 1957, "Quantum \ Mechanics of \/
\+
  One- and Two-Electron Systems", in Handbuch Phys. 35, 334-348. \,
\+
* Capek, \ K., \ _R.U.R._, \ translated \ by \ Paul Selver (Doubleday, \/
\+
  Garden City, N.Y.) \,
\+
* Davies, P.C.W., 1973, Mon.Not.Roy.Astron.Soc. 161, 1. \,
\+
* Dyson, F.J., \ 1972, _Aspects of \ Quantum Theory_, edited \ by A. \/
\+
  Salam \ and \ \ E.P. \ Wigner \ (Cambridge \ \ University, \ Cambridge, \/
\+
  England), Chap.13. \,
\+
* Dyson, F.J., 1978, "Variation of Constants", in _Current Trends \/
\+
  in \ the Theory \ of Fields_, \ edited by \ J.E. Lannutti \ and P.K. \/
\+
  Williams (American Institute of Physics, New York), pp163-167. \,
\+
* Feinberg, \ \ G., \ \ M. \ \ Goldhaber, \ \ and \ \ G. \ \ Steigman, \ 1978, \/
\+
  "Multiplicative Baryon Number \ Conservation and the Oscillation \/
\+
  of \ Hydrogen into \ Antihydrogen", Columbia \ University preprint \/
\+
  CU-TP-117. \,
\+
* Goedel, K., Monatsh.Math.Phys. 38, 173. \,
\+
* Gott, \ J.R., III, \ J.E. Gunn, \ D.N. Schramm, \ and B.M. Tinsley, \/
\+
  1974, Astrophys.J. 194, 543. \,
\+
* Gott, \ J.R., III, \ J.E. Gunn, \ D.N. Schramm, \ and B.M. Tinsley, \/
\+
  1976, Sci.Am. 234, 62 (March, 1976). \,
\+
* Haldane, J.B.S., _Daedalus, or, \ Science and the Future_ (Kegan \/
\+
  Paul, London). \,
\+
* Harrison, B.K., K.S. Thorne, M. Wakano, and J.A. Wheeler, 1965, \/
\+
  _Gravitation Theory \ and Gravitational Collapse_ \ (Univesity of \/
\+
  Chicago, Chicago), Chap.11. \,
\+
* Hawking, S.W., 1975, Commun.Math.Phys. 43, 199. \,
\+
* Hoyle, F., 1957, _The Black Cloud_ (Harper, New York). \,
\+
* Islam, J.N., 1977, Q.J.R.Astron.Soc. 18, 3. \,
\+
* Islam, J.N., 1979, Sky Telesc. 57, 13. \,
\+
* Kropp, W.P., and F. Reines, 1965, Phys.Rev. 137, 740. \,
\+
* Maurette, M., 1976, Annu.Rev.Nucl.Sci. 26, 319. \,
\+
* Monod, J., \ _Chance and Necessity_, translated \ by A. Wainhouse \/
\+
  (Knopf, New York) [_Le Hasard \ et la Necessite_, 1970 (Editions \/
\+
  du Seuil, Paris)]. \,
\+
* Nagel, E., and J.R. Newman, 1956, Sci.Am. 194, 71 (June, 1956). \,
\+
* Nanopoulos, \ D.V., \ 1978, \ _Protons \ are \ not Forever_, Harvard \/
\+
  Preprint\ HUTP-78/A062. \,
\+
* Pati, \ J.C., 1979, \ _Grand Unification \ and Proton \ Stability_, \/
\+
  Univesity of Maryland Preprint No. 79-171. \,
\+
* Penzias, A.A., and R.W. Wilson, 1965, Astrophys.J. 142, 419. \,
\+
* Rees, M.J., 1969, Observatory 89, 193. \,
\+
* Shlyakhter, A.I., 1976, Nature (Lond.,) 264, 340. \,
\+
* Turner, M.S., and D.N. Schramm, 1979, _The Origin of Baryons in \/
\+
  the Universe and the \ Astrophysical Implications_, Enrico Fermi \/
\+
  Institute Preprint No. 79-10. \,
\+
* Weinberg, S., \ 1972, _Gravitation and \ Cosmology_ (Wiley, New \   \/
\+
  York), Chap.15. \,
\+
* Weinberg, S., 1977, The First Three Minutes (Basic, New York). \,
\+
* Wright, T., 1750, _An Original \ Theory or New Hypothesis of the \/
\+
  Universe_, facsimile reprint with \ introduction by M.A. Hoskin, \/
\+
  1971 (MacDonald, London, and American Elsevier, New York). \,
\+
* Zeldovich, Y.B., 1977, Sov.Phys.-JETP 45, 9. \,
\+
\,
\+
Literatura: \,
\+
\,
\+
[X1]  http://www.aleph.se/Trans/Global/Omega/dyson.txt \,
\+

\=