Time without end: Physics and biology in an open
universe*
Freeman J. l3yson
Institute for Advanced Study, Princeton, ¹iL)Jersey 08540
Quantitative estimates are derived for three classes of phenomena that may occur in an open cosmological
model of Friedmann type. (l) Normal physical processes taking place with very long time-scales. (2)
Biological processes that will result if life adapts itself to low ambient temperatures according to a
postulated scaling law. (3) Communication by radio between life forms existing in different parts of the
universe. The general conclusion of the analysis is that an open universe need not evolve into a state of
permanent quiescence. Life and communication can continue for ever, utilizing a finite store of energy, if
the assumed scaling laws are valid.
CONTENTS
I ecture I. Philosophy
Lecture II. Physics
A. Stell ar evolution
B. Detachment of planets from stars
C. Detachment of stars from galaxies
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
Lectur e III. Biology
Lecture IV. Communication
References
447
449
450
450
450
451
451
451
452
452
452
453
457
460
LECTURE I ~ PHILOSOPHY
*This material was originally presented as four lectures,
the "James Arthur Lectures on Time and its Mysteries" at
New York University, Autumn 1978. The first lecture is
addressed to a general audience, the other three to an audience
of physicists and astronomers.
A year ago Steven Weinberg published an excel.lent
book, The First Three Minutes, (Weinberg, 1977),
explaining to a lay audience the state of our knowl. edge
about the beginn. ing of the universe. In his sixth chapter
he describes in detail how progress in understanding
and observing the universe was delayed by the timidity
of theorists.
"This is often the way it is in physics —our mistake
is not that we take our theories too seriously, but that
we do not take them seriously enough. It is always hard
to realize that these numbers and equations we play
with at our desks have something to do with the real
world. Even worse, there often seems to be a general
agreement that certain phenomena are just not fit subjects
for respectable theoretical and experimental effort.
Alpher, Herman and Gamow (1948) deserve
tremendous credit above all for being willing to take the
early universe seriously, for working out what known
physical laws have to say about the first three minutes.
Yet even they did not take the final step, to convince
the radio astronomers that they ought to look for
a microwave radiation background. The most important
thing accomplished by the ultimate discovery of the
8 'K radiation background (Penzias and Wilson, 1965)
was to force all of us to take seriously the idea that
there seas an early universe. "
Thanks to Penzias and Wilson, Weinberg and others,
the study of the beginning of the universe is now respectable.
Professional. phys icists 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 and found very f'ew (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 eschatol. ogy, 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 impossibl. e 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 l.ife may be. But as soon as we
mention the words value and purpose, we run. into on. e
Reviews of Modein Physics, Vol. 51, No. 3, July 1979 Copyright 'l 979 American Physical Society
Freeman J. Dyson: Time without end: Physics and biology in an open universe
of the most firmly entrenched taboos of twentiethcentury
science. Hear the voice of Jacques Monod
(1970), high priest of scientific rationality, in his
book Chance and Necessity:
"Any mingling of knowledge with values is unlawful,
forbidden. "
t =To(g —sing),
and R is the radius of the universe given by
R =cT,(1 —cosP).
The whole universe is represented in terms of the
coordinates (g, }t)by a finite rectangular box
0& g& 2m, 0& g& m.
(2)
This universe is closed both in space and in time. Its
total duration is
where To is a quantity that is in principle measurable.
If our universe ig described by this model, then To
must be at least 10' years.
The simple model of a uniform zero-pressure open
universe has instead. of (1) the metric
ds' =R'[dP' —dX' —sinh'gdQ'], (6)
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 battl. e 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 steinberg were still.
fighting Bishop Wilberforee'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 questi. ons boldly a.nd
. 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
wil. l be rough and simple but always quantitative. The
aim is to establ. ish numerical bounds within which the
destiny of the universe must lie. I shall make no further
apology for mixing philosophical. speculations with
mathematical equations.
The two simplest cosmological models (Weinberg,
1972) describe a uniform zero-pressure universe
which may be either closed or open. The closed universe
has its geometry described by the metric
ds' =R'[dP' —dg' —sin'gdQ'],
where X is a space coordinate moving with the matter,
P is a time coordinate related to physical time t by
and the coordinates (g, y) extend over an infinite range
0&/&~, 0&it&~ (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 Bees' 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 ean
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 tha.t only a part of it would collapse and another
part of it would expand forever? I do not know the
answer to this question. If it turns out that the universe
is cl.osed, we shall still ha,ve 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 =Rg, (10)
where No is a number of the order of 10'0.
Comparing (ll) with (7), we see that the number of
visible galaxies varies with t at late times. It happens
by a,curious numerica, l accident that the angular
size of a typical gal.axy at time t is
with R given by (8), and the number of galaxies visible
within the horizon. is
N =N, (sinh2$ —2g),
where now
t = T( i sgnh—g),
R = cTO(cosh' —1),
(7)
(8)
10't 'rad,
with t measured in years. Since (11)and ('7) give
(12)
Rev. Mod. Phys. , Vol. 51, No. 3, July 1979
Freeman J. Dyson: Time without end: Physics and biology in an open universe
it turns out that the sky is alggays just filled Ilia
galaxies, no matter how far into the future we go. As
the apparent size of each galaxy dwindles, new galaxies
constantly appear Bt the horizon to fill in the gaps.
The light from the distant galaxies will be strongly
red-shifted. But the sky wil. l 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
indef initely?
(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.
U, as I hope, my answers turn out to be right, what
does it mean'P It means that we have discovered in
physics and astronomy an analog to the theorem of
Godel (1931)in pure mathematics. Godel proved [see
Nagel and Newman (1956)]that the world of pure
mathematics is inexhaustible; no finite set of axioms
and rules of inference can ever encompass the whole
of mathematics; given any finite set of axioms, we
can find meaningful mathematical questions which the
axioms leave unanswered. I hope that an analogous
situation exists in the physical world. If my view of
the future is eorreet, 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 cl.oudy 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 the 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
worMs 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 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, thatI know I ean never look
upon the stars without wondering why the whole world
does not become astronomers; and that men endowed
with sense and reason should neglect a science they are
naturally so much interested in, and so capable of
enlarging their understandding, 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, chain. ed 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, then 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 wil. l 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 Bs we can
reach into the past or the future. It is better to be too
Rev. Mod. Phys. , Vol. 51, No. 3, July 1979
450 Freeman J. Dyson: Time without end: Physics and biology in an open universe
bold than too timid in extrapol. ating 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 (~49Sm/'4'Sm) between the abundance s 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 ' 'Sm has
been heavily depleted by the dose of thermal neutrons
to which it was exposed during the operation of the reactor.
If we measure in a modern reactor the thermal
neutron capture cross section of '~ Sm, we find the
value 55 kb, dominated by a strong capture resonance
at a neutron energy of 0.1 eV. A detailed analysis of
the Oklo isotope ratios leads to the conclusion that the
' 'Sm cross section was in the range 55+ 8 kb two billion
years ago. This means that the position of the
capture resonance cannot have shifted by as much as
0.02 eV over 2.10' yr. But the position of this resonance
measures the difference between the binding
energies of the '4'Sm ground state and of the "Sm
compound state into which the neutron is captured.
These binding energies are each of the order of 10'
eV and depend in a complicated way upon the strengths
of nuclear and Coulomb interactions'. The fact that the
two binding energies remained in balance to an accuracy
of 2 parts in 10"over 2.10' yr indicates that
the strengths of nuclear and Coulomb forces cannot
have varied by more than a few parts in 10'8 per year.
This is by far the most sensitive test that we have yet
found of the constancy of the laws of physics. The fact
that no evidence of change was found does not, of
course, prove that the laws are strictly constant. In
particular, it does not exclude- the possibility of a
variation in strength of gravitational forces with a
time scale much shorter than 10' yr. For the sake
of simplicity, I assume that the laws are strictly
constant. Any other assumption would be more com--
plicated 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 halflife,
perhaps through virtual processes involving black
The longest-lived low-mass stars will exhaust their
hydrogen fuel, contract into white dwarf configurations,
and cool down to very low temperatures, within times
of the order of 10'~ 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 evolu-
tion.
B. Detachment of planets from stars
The average time required to detach a planet from a
star by a close encounter with a second star is
T = (pV&) ', (14)
where p is the density of stars in space, V the mean
relative velocity of two stars, and & the cross section
for an encounter resulting in detachment. For the
earth-sun system, moving in the outer regions of the
disk of a spiral galaxy, approximate numerical values
al e
p = 3.10 "km '
V = 50km/sec,
0 = 2.10"km,
T =10"yr.
(15)
(16)
(17)
(18)
The time scale for an encounter causing serious disruption
of planetary orbits will be considerably shorter
han 10~5 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 consists of N stars of mass M in a volume of
radius R, their root-mean-square velocity will be of
order
V = [t."Nile/R]'" . (19)
The cross section for a close encounter between two
stars, changing their directions of motion by a large
angle, is
o' = (GM/V')' = (R/N)' (20)
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 time scales.
Classical astronomical processes come first, quantummechanical
proces ses later.
Note added in Proof Sin. ce 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
Schr am m, 1979).
A. Stellar evolution
Rev. Mod. Phys. , Vol. 51, No. 3, July 1979
Freeman J. Dyson: Time without end: Physics and biology in an open universe
The average time that a star spends between two close
encounters is
namical relaxation dominates gravitational radiation
j.n the evolution of galaxies.
T =(pV&) '=(NB'/GM)''.
If we are considering a typical large galaxy with
1V=10", Q =3.10' km, then
T = 10~9 yr.
(21)
(22)
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 d-isappears after a time
Dynamical relaxation of the galaxy proceeds mainly
through distant stellar encounters with a time scale
Ts =T(log%) '=10"yr. (23)
T = (G'M/@c') .
For a, black hole of one solar mass the lifetime is
(28)
The combined effect of dynamical relaxation and close
encounters is to produce a collap'se of the central
regions of the galaxy into a black hole, together with
an evaporation of stars from the outer regions. The
evaporated stars achieve escape velocity and become .
detached from the galaxy after a time of the order
10"yr. We do not know what fraction of the mass of
the galaxy ultimately coll.apses and what fraction
escapes. The fraction escaping probably lies between
90%%uo and 99%%uo.
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
operating 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 J.ong 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 I', and kinetic energy E, it will
lose energy by gravitational radiation at a rate of order
E~ = (V/c)'(E/P) . (24)
Any gravitationally bound system of objects orbiting
around each other will decay by this mechanism of
radiation drag with a time scale
T, = (c/V) V . (25)
For the earth orbiting around the sun, the gravitational
radiation. time scale is
Tg =10 yr ~ (26)
T~ =10 yr. (27)
Since this is much longer than (18), the earth will
almost certainly escape from 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
(28).
The orbits of the stars in. a galaxy will also be decaying
by gravitational radiation with time scale (25),
where 2 is now the period of their galactic orbits. For
a galaxy like our own, with V = 200 km/sec and
I' =2.10' yr, the time scale is
F. Matter is liquid at zero temperature
I next discuss a group of physical processes which
occur in ordinary matter at zero temperature as a
result of quantum-mechanical barrier penetration. The
lifetimes for such processes are given by the Gamow
formula
T = exp(S}T,, (30)
where To is a n.atural vibration period of the system,
and S is the action integral
s = yys) fgsss(. ))'s. (31)
Here x is a coordinate measuring the state of the system
as it goes across the barrier, and U(x) is the
height of the barrier as a function of x. To obtain a
rough estimate of S, I replace (31)by
S = (8MUd'/h')' ', (32)
where d is the thickness, and U the average height
of the barrier, and M is the mass of the object that is
moving across it. I shall consider processes for which
S is large, so that the lifetime (30) is extremely long.
As an example, consider the behavior of a lump of
matter, a rock or a planet, after it has cool.ed to zero
temperature. Its atoms are frozen. into an apparently
fixed arrangement by the forces of cohesion and chemical
bonding. But from time to time the atoms will
move and rearrange themselves, crossing energy
barriers by quantum-mechanical tunneling. The height
of the barrier will typically be of the order of a tenth
of a. Rydberg unit,
U = (1/20)(e'm/I ), (33)
and the thickness will be of the order of a, Bohr radius
d = (5'/me'), (34)
where m is the electron mass. The action integral (32)
is then
T =1064 yr
Black holes of galactic mass will have lifetimes extending
up to 10' yr. At the end of its life, every black
hole will become for a short time very bright. In the
last second of its existence it will emit about 10"erg
of high-temperature radiation. The cold expanding
universe will be illuminated by occasional fireworks
for a very long time.
This is again much longer than (22}, showing that dy- S = (2A.m /5m)' ' = 27@ ' (35)
Rev. Mod. Phys. , Vol. 51, No. 3, July 1979
Freeman J. Dyson: Time without end: Physics and biology in an open universe
where m~ is the proton mass, and A is the atomic
weight of the moving atom. For an iron atom with
A =56, 8 =200, and (30) gives
T =10"yr. (36}
Even the most rigid materials cannot preserve their
shapes or their chemical structures for times long
compared with (36). On a time scale of 10"yr, every
piece of rock behaves like a l.iquid, flowing into a
spherical shape under the influence of gravity. Its
atoms and molecules wil. l be ceaselessly diffusing
around like the molecules in a drop of wa.ter.
G. All matter decays to iron
In matter at zero temperature, nuclear as well. as
chemical rea.ctions will continue to occur. Elements
heavier than iron will decay to iron by various
processes such as fission and alpha emission. Elements
lighter than iron will combine by nuclear fusion
reactions, building gradua. lly up to iron. Consider for
example the fusion reaction in which two nuclei of
atomic weight &&, charge ~Z combine to form a nucleus
(/I, Z). The Coulomb repulsion of the two nuclei
is effectively screened by electrons until they come
within a distance
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 very uncertain when x
is close to x. Fortunately the equation of state is
well known over the major part of the range of integration,
when x is large compared to y and the main
contribution to U(x) is the energy of nonrelativistic
degenerate electrons
U(x) = (N 'II'/2mx2),
where N is the number of electrons in the star.
The integration over x in (31) gives a logarithm
log(Z/f~, ),
(42)
where Bo is the radius at which the electrons become
relativistic and the formula (42) fails. For low-mass
stars the logarithm will be of the order of unity, and
the part of the integral coming from the relativistic
region x&B,will. also be of the order of unity. The
mass of the star is
M = 2Nmp. (44)
I replace the logarithm (43) by unity and obtain for the
action integral (31) the estimate
S =N 5(8m/, /m)'' =120%+5. (45
d=Z ' (I1/me) (37)
of each other. The Coulomb barrier has thickness d
and height
The lifetime is then by (30)
T = exp(120%~5)T, .
For a typical low-mass star we have
(46)
U = (Z'e'/4d) = ,'Z'/'(e'm/h ) . — (38) N =1056 S =1077, T =10'"' yr (4'I)
The reduced mass for the relative motion of the two
nuclei ls
M= 4/Imp.
The action integral (32) then becomes
(~1Z5/3(m /m))1/2 3a41/2Z5/5
For two nuclei combining to form iron, Z = 26,
A. =56, S =3500, and
In (46) it is completely immaterial whether T5 is a
small fraction of a second or a large number of years.
We do not know whether every col.lapse of an iron
star into a neutron star mill produce a supernova explosion.
At the very l.east, it wil. l produce a, huge outburst
of energy in the form of neutrinos and a modest
outburst of energy in the form of x rays and visible
light. The universe will still be producing occasional
fireworks after times as long is (47).
T =10 yr. (41)
On the time scale (41)„ordinary matter is radioactive
and is constantly generating nuclear energy.
H. Collapse of iron star to neutron star
After the time (41) has elapsed, most of the matter in
the universe is in the form of ordinary low-mass stars
that have settled down into white dwarf configurations
and become cold spheres of pure iron. But an iron
star is still not in its state of lowest energy. It could
release a huge amount of energy if it could collapse
into a neutron star configuration. To collapse, it
has only to penetrate a barrier of finite height and
thickness. It is an interesting question, whether there
is an unsymmetrical mode of collapse passing over a
lower sa,ddle 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 y, the radius of a neutron star, to &, the
l. Collapse of ordinary matter to black holes
(48)
where N~ is the number of electrons in a piece of iron
The long lifetime (47) of iron stars is only correct
if they do not collapse with a shorter lifetime into
black holes. For collapse of any piece of bulk matter
into a black hole, the same formulae apply as for collapse
into a neutron star. The only difference is that
the integration in the action integral (31) now extends
down to the bla, ck 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 eases. 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
wil. l. in a short time swallow the rest of the star. The
lifetime for collapse of any star is then given by
T =exp(120&e 5)T5,
Rev. Mod. Phys. , Vol. 51, No. 3, July 1979
Freeman J. Dyson: Time without end: .Physics and biology in an open universe . 453
of mass equal to the minimum mass M~ of a black hole.
The lifetime (48) is the same for any piece of matter
of mass greater than M~. Matter in pieces with mass
smaller than M~ is absolutely stable. For a more
complete discussion of the problem of collapse into
black holes, see Harrison, Thorne, Wakano, and
Wheeler (1965).
The numerical value of the lifetime (48) depends on
the value of M~. All that we know for sure is
0~M~ ~M~,
where
M, = (hc/G)'"m~' = 4.10"g
(49)
(50)
M~ =Mp„——(Sc/G) =2.10 5
g. (51)
This value of M~ is suggested by Hawking's theory of
radiation from black holes (Hawking, 1975), according
to which every black hole loses mass until it reaches
a mass of order Mp&, at which point it disappears in
a burst of radiation. In this case (48) gives
Nz ——10~9 T = 10~+ yr .
(iii) Ms is equal to the quantum mass
M, =Mo=(@c/Gm, ) =3.10"g,
as suggested by Harrison, Thorne, Wakano, and
Wheeler (1965). Here Mo is the mass of the smallest
black hole for which a classical description is meaningful.
Only for masses larger than M can we consider
the barrier penetration formula (31) to be physically
justified. If (53) holds, then
N~ =10, T =10'0 yr. (54)
(iv) Ms is equal to the Chandrasekhar mass (50). In
this case the lifetime for collapse into a black hole is
of the same order as the lifetime (47) for collapse
into a neutron star.
The long-range future of the universe depends
crucially on which of these four alternatives is correct.
If (iv) is correct, stars may collapse into black holes
and dissolve into pure radiation, but masses of
planetary size exist for ever. 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
wil. l disappear with the lifetime (52), but dust grains
with diameter less than about 100 p, 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 alteris
the Chandrasekhar mass. Black holes must exist
for every mass larger than M„because stars with
mass larger than M, have no stable final state and must
inevitably collaps e.
Four hypotheses concerning M~ have been put for-
ward.
(i) Ms =0. Then black holes of arbitrarily small
mass exist and the formula (48) is meaningless. In
this case all matter is unstable with a comparatively
short lifetime, as suggested by Zeldovich (1977).
(ii) Ms is equal to the Planck mass
TABLE l. Summary of time scales.
Closed Universe
Total duration j p11 yr
Open Univers, e
Low-mass stars cool off
Planets detached from stars
Stars detached from galaxies
Decay of orbits by gravitational radiation
Decay of black holes by Hawking process
Matter liquid at zero temperature
All matter decays to iron
Collapse of ordinary matter to
black hole [alternative {ii)]
Collapse of stars to neutron stars
or black holes [alternative (iv)]
1p'4 yr
1p15 yr
1p" yr
1p20 yr
lp64 yr
1p" yr
1p'"' yr
1yP26
j pro~6
yr
natives 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 cppclusion I wish to draw from my analysis
is the following: So far as we ean imagine into the
future, things continue to happen. In the open cosmology,
history has no end.
LECTURE II I. SIOLOGY
I ooking at the past history of life, we see that it
takes about 10' years to evolve a new species, 10
years to evolve a genus, 108 years to evolve a class,
10' years to evolve a phylum, and less than 10' years
to evolve all the way from the primaeval slime to
Homo Sapiens. If l.ife 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 in the next 10' years to rival
the changes of the past'P It is conceivable that in
another 10' years life couM 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'P
(ii) Are sentient black clouds, or sentient computers,
possible'P
(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.
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Freeman J. Dyson: Time without end: Physics and biology in an openuniverse
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 me7
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
exampl. e, that matter is ultimately stable against collapse
into bl.ack 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 cl.oud 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:
Binlogicat Scaling Hypothesis. Ifue copy a lining
creature, quantum state by quantum state, so that the
Hamittonian of the copy is
where His the Hamiltonian of the creature, Uis a
unitary operator, and &is a positive scaling factor,
and if the environment is similarly copied so that the
temperatures of the environments of the creature and
the copy are respectively 'T and ~T, then. the copy is
alive, subjectively identical to the original creature,
with a/I its vital functions reduced in speed by the same
factor A. .
The structure of the Schrodinger 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 l.inear scaling law appl. ies 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 potential.
ities of life. The first consequence is that the
appropr iate measure of time as experienced sub jectively
by a living creature is not physical time t but
the quantity
where B(t) is the temperature of the creature and
f= (300 deg s-ec) ' is a scale factor which it is convenient
to introduce so as to make u dimensionless. I
call u "subjective time. " The second consequence of
the sealing 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 "Cogito, ergo sum. " 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
Q = 10 bits . (5'7)
This Q is a measure of the complexity of the mol. ecular
structures involved in. a single act of human awareness.
For the human species as a whole,
Q =10'3 bits, (58)
a number which tells us the order of magnitude of the
material resources required for the maintenance of an
inte ll.igent society.
A creature or a society with given Q and given temperature
6 will dissipate energy at a rate
(59)
Here m is the metabolic rate measured in ergs per
second, h is Boltzmann's constant, and fis the coefficient
appearing in (56). It is important that m
varies with the square of 0, one factor 6 coining from
the relationship behveen energy and entropy, the other
factor 6 coming from the assumed temperature dependence
of the rate of vital proces ses.
I am assuming that life is free to choose its temperature
B(t) so as to maximize its chances of survival
There are two physical constraints on B(t). The first
constraint is that B(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
Rev. Mod. Phys. , Vol. 51, No. 3, July 1979
Freeman J. Dyson: Time without end: Physics and biology in an open universe
8(t)&aR ', a =3.10',deg cm, (60)
where A 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 8(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
I(8)& 2y(Ne'/'mk'c')(k 8)' (61)
on the power that can be radiated by a material. radiator
containing N electrons at temperature 0. Here
y =max[x'(e" —1) 'j =1.42
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 Salpeter (1957). The formula for the power
emitted by el.ectric dipol. e radiation is
(62)
&&t) =
p fdo p p»&~;, /2«')ID (63)
Here p is the polarization vector of a photon emitted
into the solid angle dQ, z is the initial and j the final
state of the radiator,
p; =Z 'exp(-E, /k8) (64)
is the probabil. ity that the radiator is initially in state i,
with the summation now extending over all (t, 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 (61) is proportional to 8' rather than 8, since
the standard Stefan-Boltzmann formula for the pomer
radiated by a black body is proportional to 0 . 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.
After this little digression into physics, I return to
biology. The second constraint on the temperature 6)
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,
k8& (Q/N)e =(Q/N)10 '8 erg,
e = (137/2y)(8f/k)mc',
8& (Q/N)( /~) = (Q/N)» "d.g.
(71)
(72)
(73)
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
&dq& =tt '(E& —E&) (65) N =10" (74)
is the frequency of the photon, and D;,. is the matrix
element of the radiator dipole moment between states
i and j. The sum (63) is taken only over pairs of states
(t,j) with
E; &E'~.
Nom there is an exact sum rule for dipole moments,
(66)
g &d;;~D;z[' = (I/2i)(DD —DD), ,
= (Ne'5/2m) . (67)
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),
is positive by (66), and so (62) gives
p; (u~)~ & yp; (k 8/@)'(exp(h (d;,/k 8) —1)
=y(p~ —W)(I 8/@)'. (68)
Therefore (63) implies
(e) y1(ke/&h)'P fdQ P P (»-»)(~;,/2«*)(D;,
(69)
Now the summation indices (i,j) can be exchanged in
the part of (69) involving p&. The result is
l(e)& y( !8/ )p)f))d~ ~ p»&&t.,/2«)(D;;I*,
P j
(70)
being the number of electrons in the earth's biosphere,
the ratio is 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 l.ower bound to the rate of energy dissipation
of a society of given complexity. Since the total store
of energy available to a society is finite, its lifetime
is also finite. %e have reached the sad conclusion
that the slowing down of metabol. ism described by my
biological scaling hypothesis is insufficient to allom
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
When life is hibernating, the radiator will still be at
temperature 0 but the life mill 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 8(t) do not vary
appreciably during any one cycle. Then (56) and (59)
no l.onger hold. Instead, subjective time is given by
Rev. Mod. Phys. , Vol. 51, No. 3, July 1979
456 Freeman J. Dyson: Time without end: Physics and biology in an open universe
and the average rate of dissipation of energy is
m =kfQglP .
The constraint (71) is replaced by
&(t) & (Q/~)(e/k)g (t) .
(75)
1 1
3 2' (78)
and for definiteness we take
o. =3/8.
Subjective time then becomes by (74)
u(t) =A. (t/t, )'",
where
(79)
A =4f&ot0=10~8 (81)
is the present age of the universe measured in moments
of consciousness. The average rate of energy
dissipation is by (75)
m(t) =kfQ & (t2O/t, ) '~'. (82)
The total. energy metabolized over all time from tp to
infinity is
mt t=B
tp
B=2Akop =6.104erg.
(83)
(84)
This example shows that it is possible for life with
the strategy of hibernation to achieve simultaneously
its two main objectives. First, according to (80),
subjective time is infinite; although the biological
clocks are slowing down and running intermittently as
the universe expands, subjective time goes on forever.
Second, according to (83), the total energy required
for indefinite survival is finite. The conditions (78) are
sufficient to make the integral (83) convergent and the
integral (74) divergent as t- ~.
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
BQ =6.103~ erg, (85)
about as much energy as the sun radiates in eight
hours. The energy resources of a galaxy would be
sufficient to support indefinitely a society with a complexity
about 10' times greater than our own.
These conclus'ions are valid in an open cosmology.
Life keeps in step with the limit (61) on radiated power
by lowering its duty cycle in proportion to its temperatur
e.
As an example of a possible strategy for a long-lived
society, we ean satisfy the constraints (60) and (76) by
a wide margin if we take
g(t) =(~(t)/ll, ) =(t/t. }
"
where 6p and tp are the present temperature of life and
the present age of the universe. The exponent a has to
l.ie in the range
7 2 HATT
the background radiation temperature
~s(t) =a(&(t))"'
(86)
is proportional to v
'' as T- 0, by virtue of (2) and
(3). If the temperature 6(t) of life remains close to
8„as v- 0, 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
6(t) tends to infinity more slowly than Os, the total
duration of subjective time remains finite. If 0(t} tends
to infinity more rapidly than 0~, the 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 del.ineated 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'P 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 l.ike 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
worthwhil. e to be immortal if one must ultimately
erase al.l 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.
Fortunatel. y, 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
It is interesting to examine the very different s ituation
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
contrac ts and the backgr ound radiation temperature
rises, the relations (56) and (59}still hold, but physical
time t has only a finite duration (5). If
Rev. Mod. Phys. , Vol. 51, No. 3, July 1979
Freeman J. Dyson: Time without end: Physics and biology in an open universe 457
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 (logt). Therefore
an immortal civilization should ultimately find ways
to code its archives in an analog memory with capacity
growing like (logt). 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.
If the receiver is tuned to the frequency u' with
bandwidth B', (94) gives
P 'B' ~N'S„
S,= (2&'e'/m c)= 0.167 cm' s ec (96)
To avoid confusion of units, I measure both (d' 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
LECTUR E IV. COIVIMUNICATION E' = (FN'So/d B') (97)
(d (dQ~
1+x (88)
ft„=cT,(cosh( —1),
fly = cT,(cosh()+q) —1) .
(89)
(90)
The bandwidths B and B' will be related by the same
factor'(I+a). The proper distance between A and 8 at
the time the signal is received is d~ =&~g. However,
the area of the sphere g =q at the same instant is
43'd gq with
dg =Qg sinh'g (91)
If A transmits I"photons per steradian in the direction
of B, the number of photons received by B will be
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 al.ong the world-line g =0,I
transmits, while society B, moving along a world-line
with the co-moving coordinate g =q, receives. A
signal transmitted by A when the time coordinate
f = g will be received by 8 when Q = )+q. If the transmitted
frequency is (d, the received frequency will be
red-shifted to
I assuage that the transmitter contains N electrons
which can be driven in phase so as to produce a beam
of radiation with angular 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
E' = (3N/8m)(R/h (u), (98)
where E is the total energy transmitted. The number
of received photons is then
E' = (3NN'ES o/8vkud' B') (99)
We see at once from (99) that low frequencies and
small bandwidths are desirabl. e 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,
B' = (E'/r~), B = (E'/r~), (100)
r„=(m„/c), r, = (m, /c) . (101)
where v~ is the duration of the reception, and v ~ the
duration of the transmission. Vfith this bandwidth, X
represents both the number of photons and also the
number of bits of information received. It is convenient
to express 7.„and 7~ as a fraction of the
radius of the universe at the times of transmission and
reception
(E+1/d2 ) (92)
The condition
Q' =
Q Q pg(4m2(uj;/AC)(D)~)25((u; —(u') (93)
where Z' is the effective cross section of the receiver.
Now the cross section of a receiver for absorbing a
photon of frequency co' is given by a formula similar
to (63) in the previous lecture
(102)
then puts a lower bound on the bandwidth B. I shall also
assume for simplicity that the frequency (d is chosen
to be as low as possible consistent with the bandwidth
B, namely
m =B, cu' =B'. (103)
Q 'd(u' =¹(2w'e'/mc), (94)
where N' is the number of electrons in the receiver.
with D;z again a dipole matrix element between states
i and j. When this is integrated over all. (d, we obtain
precisely the left side of the sum rule (67). The contribution
from negative (d' 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
Then (99), (100), (101)give
NÃr 52
(1+ a)(sinh2q)E, (104)
where by (96)
E,= (8mhc2/38o) = (4/3w)137mc2 =3.10 ' erg . (105)
We see from (104) that the quantity of information that
can be transmitted from A to 8 with a given expenditure
of energy does not decrease with time as the universe
Rev. Mod. Phys. , Vol. 51, No. 3, July 1979
458 Freeman J. Dyson: Time without end: Physicsand biology in an openuniverse
expands and A and 8 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 section with dec reas ing bandwidth.
The received signal is (104). We now have to compare
it with the received noise. The background noise
in the universe at frequency co can be described by an
equivalent noise temperature T~, so that the number
of photons per unit bandwidth per steradian per square
centimeter per second is given by the Rayleigh-Jeans
formula
S =(2~,/~, )fN'S',
where
r, = (e'/mc') = 3.10 "cm,
(114)
(115)
A.~ = (kc/k B~l ) = A (116)
is the wavelength of the primordial background radiation
at the time of reception. Since I" is the signal,
the signal-to-noise ratio is
f(&u) = (kT„~/4n'hc') . (106)
R 3~ ——(A.a /2fN'xo) .
This formula is merely a definition of T~, which is in
general a function of u 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 0~. The primordial noise
temperature 6~ varies inversely with the radius of the
universe,
(kBsR/kc) =A =10",
In this formula, f is the noise-temperature ratio given
by (108), N' is the number of electrons in the receiver,
and 3 o, Aa 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 Z',
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
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„=TO(sinh( —g), ta = T(sin h(&+q) —(g+q)).
(118)
(T~/B~) =f(x), x = (k~/k B~), (108)
According to (89) and (101),
with fa 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
f(x) =f~(x) = x(e"—1) ', x - 1 .
But there will be strong deviations from (109) at large
x (due to red-shifted stariight) 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- ~.
The total energy density of radiation in the universe
1s
=5(dt„/dg), ~ =5(dt /d(). (119)
It is convenient to think of the transmitter as permanently
aimed at the receiver, and transmitting intermittently
with a certain duty cycle 5 which may vary
with (. When 5 =1 the transmitter is on all the time.
The number P' of photons received in the time T~ can
then be considered as a bit rate in terms of the variable
g. In fact, F'dg is the number of bits received in
the interval d(. It is useful to work with the variable
g since it maintains a constant difference q between
A and B.
From (100), (101), (103), (107), and (108) we derive
a simple formula for the bit rate,
4m (kB,) fI((d)@cod(d(gk33)
with
(110} (120)
The energy E transmitted in the time T~ can also be
considered as the rate of energy transmission per unit
interval dg. From (104) and (120) we find
I= x x2dx.
E= (A'/NN')(I +z)(sinh'q)x35E, . (121)
The integral I must be convergent at both high and low
frequencies. Therefore we can find a numerical. bound
b such that
x'f(x}& k
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 7~ by the receiver with bandwidth E' and cross
section Z' is
+SN (122)
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
We are still free to choose the parameters x [determining
the frequency m by (108)] and 5, both of which
may vary with g. The only constraints are (102) and
the signal-to-noise condition
9'~ = 4vZ'B'Ta I(u)') .
We substitute from (95), (96), (100), (103), (106), and
(108) into (113)and obtain
G =(20(b',/X~)N'(I+a) ' =10 '¹(I+a)',
3'=(R„/R~) =(cosh ( —1)/(cosh g~ —1).
(124)
(125)
Rev. Mod. Phys. , Vol. 51, No. 3, July 1979
Freeman J. Dyson: Time without end: Physics and biology in an open universe
x= max[(G/r)»', & ''],
5 =min[(r/G)& 3 ', 1],
(125)
(127)
so that
(128)
for all (. The transition between the two ranges in
(126) and (127) occurs at
g = (r- logG, (129)
since ( increases logarithmically with x by (125).
these choices of x and 5, (120) and (121)become
E'=Amin[(x/G)' $
' g '']
E= (A /NN')(1+z)(sinh2q)E, )
(130)
(131)
Now consider the total number of bits received at B
up to some epoch g in the remote future. According to
(130), this number is approximately
F~ = I'd =2A (132)
and increases without limit as g increases. On the
other hand, the total energy expended by the transmitter
over the entire future is finite,
E'& —— M =2 A N¹ e"sinh'q ~ 'E, . (133)
In (133) I have replaced the red shift (1+z) by its
asymptotic value e" as (-~. I have thus reached the
same optimistic conclusion concerning communication
as I reached in the previous lecture about biological
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 Er and Er. By (107),
the cumulative bit count in every communication channel
is the same, of the order
1029(»2 (134)
a quantity of information amply sufficient to encompass
the history of a compl, ex civilization. To
estimate E~, I suppose that the transmitter and the
receiver each contain 1 kg of electrons, so that
N =N' =10 (135)
Here A&, R& and (& are the present vq, ines of the background
radiation wavelength, the radius of the universe,
and the time coordinate Q. It is noteworthy that the
signal-to-noise condition (123) may be difficult to
satisfy at early times when z is small, but gets progressively
easier as time goes on and the universe
becomes quieter. To avoid an extravagant expenditure
of energy at early times, I choose the duty cycle 5 to
be small at'the beginning, increasing gradually until it
reaches unity.
All the requirements are satisfied if we choose
Then (133) with (105) gives
Er = 10' (8"sLnh g) el g .
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 10"W yr) could easily
provide the energy to power permanent communication
channels with all the 10"stars that lie within the
sphere @&1. That is to say, all societies within a red
shift
g =g —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
growth of E~ with q, the upper limit to the range of
possible direct communication lies at about g =10.
It is easy to transmit information to larger distances
than g =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 of galaxies that lie within a sphere q &g grows
like e' when Q is large. So, when we try to establish
l.inkages between distant societies, there will be a
severe problem of selection. There are too many
galaxies at l.arge distances. To which of them should
we listen? To which of them Should we relay messages?
The more perfect our technical means of
communication become, the more difficulty we shall
have in deciding which communications to ignore.
In conclusion, 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 trie6 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 al.so 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 shall
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
Rev. Mod. Phys. , Vol. 51, No. 3, July 1979
Freeman J. Dyson: Time without end: Physics and biology in an open universe
this univ ers e of ours to their own purpos es. As
Haidane (1924) the biologist wrote fifty years ago,
"The human intellect is feeble, and there are times
when it does Ilot assert the lnfinlbj of its clalIHs.
But even theri:
Though in black jest it bows and nods,
I know it is roaring at the gods,
%aiting the last eclipse. "
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