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Interstellar Travel: The Wait Calculation and the Incentive Trap of Progress
1. INTRODUCTION TO GROWTH
1.1 Keeping the Momentum Going?
Analysts of growth are divided on whether growth will save
human civilisation or destroy it. Fred Hoyle observed as long
ago as 1964 [1] that if technological growth reaches a certain
point and then fails recovery may be impossible.
It has often been said, if the human species fails to make
a go of it here on Earth, some other species will take
over the running. In a sense of developing high
intelligence, this is not correct. We have or will have,
exhausted the necessary physical prerequisites so far as
this planet is concerned. With coal gone, oil gone, high-
grade metallic ores gone, no species however competent
can make the long climb from primitive conditions to
high-level technology. This is a one-shot affair. If we
fail, this planetary system fails so far as intelligence is
concerned. The same is true of other planetary systems.
On each of them, there will be one chance, and one
chance only.
Many modern economists, fearful of Hoyle’s observation,
tend to side with E.C. Prescott, co-Nobel prize winner for
Economics in 2004 and a great proponent of growth who
remarked that the greatest challenge facing the world was,
‘…keeping the momentum going’ [2]. It is now fair to ask
whether keeping the momentum of growth going will take
humanity to the stars.
There are three major threats to the momentum of growth:
economic collapse, disease and natural disasters. From current
trends, all these fears have diminished, and it is quite probable
that civilisation on Earth has already passed the danger points
of collapse and is safe from permanently debilitating setbacks
to its expansion.
JBIS, Vol. 59, pp.xxx-xxx, 2006
INTERSTELLAR TRAVEL: THE WAIT CALCULATION
AND THE INCENTIVE TRAP OF PROGRESS
ANDREW KENNEDY
The Chronolith Project, calle Goles 48, bajo A, Seville 41002, Spain.
Email: drewk@eresmas.com
This paper describes an incentive trap of growth that shows that civilisations may delay interstellar exploration as long as
voyagers have the reasonable expectation that whenever they set out growth will continue to progress and find quicker means
of travel, overtaking them to reach and colonise the destination before they do. This paper analyses the voyagers’ wait
calculation, using the example of a trip to Barnard’s Star, and finds a surprising minimum to time to destination at a given rate
of growth that affects the expansion of all civilisations. Using simple equations of growth, it can be shown that there is a time
where the negative incentive to travel turns positive and where departures will beat departures made at all other times. Waiting
for fear future technology will make a journey redundant is irrational since it can be shown that if growth rates alter then
leaving earlier may be a better option. It considers that while growth is resilient and may follow surprising avenues, a future
discovery producing a quantum leap in travel technology that justifies waiting is unlikely.
Keywords: Interstellar travel, incentive trap, the wait calculation, expansion of civilisation
Past prognosticators on the perils of growth have all been
wrong. Paul Ehrlich, among many in the 1960s and 1970s,
predicted civilisation’s decline in the 21st century due to the
explosive growth in world population [3]. The Club of Rome in
1972 predicted widespread trouble for the world economy by
the year 2000 [4]. Many Commentators have written that re-
sources, especially oil, are already failing and that world growth
will inevitably decline from current levels. For example, Duncan
believes that world oil production has already peaked [5,6] and
Deffeyes believes it will peak between 2004–2008 [7].
If we expand our viewpoint a little from the purely local to
the global, we can observe that global long-term growth is far
more tenacious than is generally accepted. In spite of the dire
warnings, most commentators accept that, at this point in our
history, exponential growth continues to define the progress of
almost any particular sphere of activity or field of learning.
This is especially true of science itself where 80–90% of all
scientists who have ever lived are alive now [8].
Fears of a world economic collapse have all but vanished
[9]. Economic recessions last less and less time; they are meas-
ured in months and years rather than in decades [8]. The
Kondratev cycles (40-50 years) of decline of the past centuries
are no longer observed [8]. The average recession time (Juglar
Cycle) from the last century lasted between 8–10 years [8]. The
depression that followed the 20s boom was the last depression
to last this length of time. Further recessions occurred in the
70s, 80s, and 90s and most recently during 2001–2002, but
they are timed in months rather than years [8].
Disease is a possible source of decline, although the reaction
of governments around the world to contain recent SARS and
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Andrew Kennedy
the theoretical threat from a mutated Avian flu outbreaks (in
2004–2005) shows that the impact of new diseases are most
likely to be well contained. The likelihood is slim, in any event,
that a disease pandemic would make much of a dent in the
annual population increase of 93 million a year. Even AIDS,
which attacks the sexually active portion of the population,
may kill 3+ million a year [8] for a while – representing perhaps
0.05% of the World’s population. Compared with 93 million
mouths added to the world’s population each year (at the turn of
the millennium) this is not very significant.
Even a pandemic on the scale of the first wave of the Black
Death in Europe braked growth very temporarily. The death
rate in Europe was probably about a third [8, 10], and it is
thought that its population did not rise to pre-1349 levels until
1600 [8]. Warfare came to a halt and trade was severely re-
duced. Yet a quick scan of history shows that a 150 years or so
afterwards, America had been discovered, improved ship-build-
ing and navigation (using the recent Western version of the
compass) made journeys to the Far East commonplace, ships
had sailed around the world (in AD1521), and Europe was
trading with China and India. The spread of credit-based bank-
ing and the invention of double entry bookkeeping had brought
investment funds into play, and warfare had picked up again
[11].
The years after the First World War are even more instruc-
tive. War action had killed off at least twenty million combat-
ants, mainly adults, and civilians world-wide [8] followed by
famine and deprivation in some areas, when an epidemic of flu
appeared - possibly the most virulent pandemic ever seen –
which killed off perhaps between 40-50 million (estimates
vary) world-wide between 1918–1919 [12]. A large portion of
these was among the 20-40 year age group, the largest group of
contributors to growth. Yet populations everywhere boomed in
the years afterwards, leading to the manic economic growth of
the 1920s. India, for example, lost at least 12–13 million to the
disease, but now is destined to become the most populous
country on Earth [13].
To take another example, the on-set of climactic conditions
that has been termed ‘The Little Ice Age’ began at the start of
the 14th century with widespread famines (1315-1317) and then
caused, a general ‘economic contraction’ [10]. In England, the
cycle of wet summers raised the wheat price to 6 times its
normal level and ‘men ate horses and dogs and even, it was
said, children’ [14]. However, by 1318, England was planning a
new invasion of Scotland. By 1330 with young Edward III on
the throne and in the midst of continuous warfare with the
nations of Britain and in France, nobles were accumulating
expensive armour, rich furred robes, jewelled belts, tapestries
and all the accoutrements of wealth. Castles were being turned
from military installations into palaces, books were being col-
lected and England’s imports of spices, wine, silks and furs,
rice and fruit stimulated the shipping industry [14]. The Bay of
Biscay was sometimes referred to as the ‘sea of the English’
[14] from the multitude of English ships sailing its waters.
Great churches were built or re-built in the new gothic style,
and the cathedral of St. Paul’s in London was finished. It was
the biggest church in Europe with a 500-foot tower and spire
[14].
Over a similar era in the New World, the Mayan and Toltec
empires collapsed completely at the end of the 12th century AD.
Yet the Aztec Empire rose in their place in the 15th century [10].
The Aztec capital, Tenochtitlán, at the time of the Spanish
conquest, was by then one of the great cities on Earth with a
population exceeding 140,000 people [8]. By contrast, in the
same era, Seville, the Spanish city that led the conquest of the
New World had at most 45,000 inhabitants [8].
Threats from worldwide natural catastrophes remain [10],
and Hempsell has analysed the historical record and predicts
the current chance of dying from one to be about 1 in 40 and
increasing into this century [15]. But even in early urban his-
tory where the frequency and impact of natural catastrophes on
a global scale may have been greater than previously thought
[10], society still recovers and overall growth is sustained in
the longer term.
In more recent times, events of the Second World War
illustrate the tenacity of growth. The intense bombardment of
Germany’s heartland could be considered similar to a large
natural disaster. Yet, Germany’s industrial output hardly slowed
until the last few weeks of the war. The tanks may have run out
of petrol in the Ardennes counter-offensive, but Germany still
managed to construct and put in the air the first jet aircraft, and
bombed London with the first intercontinental ballistic mis-
siles, having developed a plastics industry at the same time [8].
Japan, still a partly feudal society when it waged war in 1940,
had its industrial base destroyed by 1945. 50 years on, before
the stock market falls of the 1990s, it owned the most expensive
real estate on the planet, and the Tokyo Stock Exchange had
overtaken that of New York in the trading of securities [8].
It would seem that technological growth is getting harder
and harder to slow down. Such growth will not only consume
the Earth rapidly but also require the re-building of the Solar
System into Dyson spheres, or something like them. Sagan and
Shklovskii calculated in 1966 that, with an average annual
growth rate of 1/3 of a percent, in 2,500 years energy demand
will outstrip the total solar radiation falling on the Earth by a
factor of 100,000 [16].
Though, as our argument suggests, and as many economists
believe, as long as growth continues unabated, Human civilisa-
tion will find ways to circumvent this problem. This is a safe
prediction, even if the precise sequence of steps cannot be
predicted.
As far as space travel is concerned, however, growth presents
a new problem.
2. THE INCENTIVE TRAP
Let us imagine that technological growth has provided an inter-
stellar space ship to travel to Barnard’s Star (6 light years
distant). The voyagers are prepared to leave behind their birth-
place forever to voyage far and long into space to visit, even
colonise, a distant planet. This is the moment to pause and
consider the following:
Civilisation is growing faster each day. Scientific progress
will produce ever-faster means of travel. It may even solve the
riddle of travel at light speeds or beyond. If that happens while
the ship is still travelling, people setting out later will get to the
destination ahead of it, making the first trip a wasted sacrifice.
It is clear that if the time to wait for the development of
faster means of travel is far greater than the length of the
journey, then the voyagers should go ahead and make the
journey. But if the likely future travel time plus the length of
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Interstellar Travel: The Wait Calculation and the Incentive Trap of Progress
wait is equal or less than the current journey time then they
should definitely wait.
Is it possible to calculate precisely how long the voyagers
should wait for any desired journey time?
2.1 A Growth Equation
Using a classic doubling equation to describe the effects growth
would have on achievable velocity of travel:
speed of travel, v = v0 2t/h (1)
Where t is the waiting time interval, h is the time between
doublings of speed.
Then, the waiting time can be found for making the journey
in a time comfortable for the voyagers.
Re-arranging equation 1 gives:
t = h ((log v – log v0)/log 2)
and here is an example:
Current technology say, enables the travellers to set off now
to Barnard’s Star, 6 light years distant, at a given velocity and
reach it in 12,000 years (v0 = c/2000).
Conceivable future technologies could make the journey
quicker [17, 18, 19, 20, 21, 22]. As a bench mark, Project
Daedalus, conceived by the BIS in the 1970s, was designed to
achieve a speed of about 1/10 the speed of light, making the trip
to Barnard’s Star in about 50 years [23]. However, this journey
was planned as a fly-by not a landing since carrying the extra
propellant required for deceleration was unfeasible. We can
consider that were decelerations to orbital velocity possible the
journey time would take twice as long.
If technology growth is likely to double every 100 years the
speed at which this journey could be made, then, using equation
–1, it would seem that a voyager need only wait 690 years or so
to make the journey in 100 years or less (i.e. at a speed of 6/100
speed of light). In other words, the star could be reached in well
under a thousand years from now simply by waiting. Total time
to destination is 690 years of wait + 100 years of travel = 790
years.
Aware of what growth could do to the available velocity of
travel, could others wait longer and travel even more quickly?
Generalising equation —1 thus,
Let t0 be the travel time to destination now.
And T be the time it takes at some later time after a time t of
waiting.
1/T = (1 / t0 ) 2t / h
(2)
t0 / T = 2 t / h
The full time to destination t + T must be less than or equal
to t0 otherwise there’s no point waiting. By plotting total time to
destination from now against the wait time (see fig.1), an
important result is revealed.
For a doubling time of 100 years, the minimum time to
destination is ~782 years achieved after ~637 years of waiting.
Although the journey takes longer (145 years), those who left at
this minimum would beat anyone who wanted to wait to make
the journey in 100 years. A doubling time of 50 years brings the
minimum time to destination to ~441 years after ~371 years of
waiting and a shorter travel time of 70 years.
It should be noted that v0 = c/2000, (150 km sec-1) is an
optimistic figure and assumes a current technological level that
we may not quite yet possess. As an illustration, NASA’s mis-
sion to Pluto, ‘New Horizons’, leaving in January 2006 is being
launched by an Atlas V-551 at an injection speed of ~ 12 km
sec–1 (48,000 km hr-1), although the ‘slingshot’ maneuver around
Jupiter could raise cruising speed by as much as 4 km sec-1 [24].
It is expected to pass Pluto at a speed of ~14 km sec–1. Taking
approximately this mission velocity as v0 = c/20,000 as a lower
base velocity, we find the waiting time to minimum has risen to
~969 years.
2.2 The Minimum Wait Time
Figure 1 shows that at a given growth rate there is a waiting
time longer than which will not get the voyagers to the destina-
tion any quicker. A departure at this point beats any later
departure since, even though growth will continue to produce
faster velocities, the time of waiting is too long to make up with
any faster velocity. After the minimum, the incentive trap ceases:
those who leave later arrive later.
The Daedalus project requires, among other things, the min-
ing of thousands of tons of 3He from Jupiter’s atmosphere [23].
We have no idea how this might be done right now, but if the
Daedalus project turns out to be the plan in use, the travel to
and going into orbit around Barnard’s star in a voyage of a 100
years, and taking a velocity of travel doubling time, h = 100
years, the above calculation shows that we have to wait ~ 969
years before it will done, making the minimum time from now
(where v0 = c/20,000 ) to the Barnard Star destination ~ 1069
years.
Faster growth (for example using a doubling time of h = 50
years), while shortening the waiting time, does not avoid the
minimum in the curve. It only makes it sharper, exacerbating
how the small differences of waiting convert into longer jour-
neys.
This equation does not guarantee that Project Daedalus or
any scheme conceived now will come to fruition. Another
scheme entirely novel may take its place. The equation does not
favour any imaginative exercise in prediction. It merely de-
scribes an achievable velocity of travel produced by continuous
growth. The implications of this will be discussed below.
2.3 Compound growth
Equation —1 has a fixed time interval and does not incorporate
the acceleration inherent in compounded technological growth.
Let us use a simple compound equation
v = v0 (1 + r)t(3)
Where r = is the % yearly growth increment,
Using this equation, we can easily model the rate of increase
of the average speed of travel available to the ordinary citizen
over the last century. At the end of the 19th century, car enthusi-
asts had taken internal combustion engine cars to 80 km/hr [8].
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Andrew Kennedy
Taking r =0.01 we find speed has grown to ~135 km/hr in 1950.
At the turn of this century, where t = 100 years, this equation
produces an available speed of 225 km/hr, the cruising speed of
an elite sports car.
We can see how long this same rate of growth would take to
make the interstellar journey to Barnard’s Star (6 light years) in
100 years or less using our current realistic base interstellar
velocity as above of c/20000.
v = 6c / 100; v0 = c/20000; where c = speed of light
Eliminating c and re-arranging we have
t = (log 6/100 – log 1/20000)/log (1.01)
which gives waiting time to minimum, t = 712 years (see
Fig. 2).
For v = c and v0 = c/20000, this modest compounded growth
if it continues in this simple fashion might produce travel at the
speed of light in t = 950 years.
As above, we can generalise this equation thus:
t0 / T = (1 + r)t(4)
where T = travel time after a time t of wait; r is the average %
yearly rate of change. r may increase or decrease slightly from
year to year, but in the long term, these differences will be
absorbed into an average increment.
2.4 The Waiting Calculation
From this simple calculation, we can see that any civilisation
may prefer to wait until growth produces a travel time that
approaches the minimum since this will also be the minimum
expended energy. Successive generations will have less and
less time to wait for the minimum, but, given that the average
long-term rate of growth does not change appreciably, any
attainable velocity will lie on the same curve and the point in
time where the minimum occurs does not change.
Future generations may approach this minimum point with
heightened anticipation if they only have the capability to make
a single launch, since leaving at any other time than the mini-
mum is risky. Many of the trips to the more remote stars
planned for the future, using the technological techniques that
are expected to be available, will take place at velocities of
around 0.02c [25] and may thus take thousands of years rather
than hundreds. Consequently, there is the fear that leaving
earlier than the minimum will mean that future generations will
not only arrive at these destinations earlier, having had an
easier trip, but may squander all the fruits of the landfall before
the original voyagers arrive.
If the civilisation has the capability to make several launches,
then they could make use of the spread of arrival times to
encourage individuals to leave on the basis that others would
either be there first to welcome them or be following close
behind bringing with them the future technologies.
Civilisations are not obliged to launch at the minimum, but
waiting until the minimum is the most efficient way to explore.
Other than this, a departure time to choose may be simply a
matter of psychology. Few will want to travel willingly into
space unless they can participate in the landfall. One can read-
ily imagine travellers giving up, say, 30 years of their life on a
journey – even in hibernation - as long as they can be rewarded
with the arrival. Longer travel times, although they imply a
happier landfall (if later travellers have arrived first), may not
be so attractive to the voyager.
In this respect, the wait calculation is crucial. Either side
of the minimum, voyagers will arrive later than those who
set off at the minimum. At the minimum wait time, growth
will not catch the voyagers up during their journey. They
will arrive to an unsettled destination, expecting others to
follow, but not knowing if the vanguard of civilisation will
appear on their horizon before much time has passed. If they
leave before or after the minimum and find their destination
still unsettled when they arrive, they will know that growth
has slowed or stopped and that they will be alone for some
time.
But there are other anxieties. Historically, humans tend to
eradicate earlier or more primitive cultures. If the first voyag-
ers, in order to make sure that there is a welcome committee,
leave too early, they could arrive too late. Rather than end up as
brave colleagues to the pioneer party, they would be an awk-
ward presence in a world that has advanced far beyond their
experience. They will be historical curiosities, hardly at the
forefront of social change. They would have little or no training
in the advanced culture, little to contribute, and little scope for
being independent. They would probably be a burden, and may
even be, because of a world-view developed in an earlier
epoch, a thorn in the side of the authorities.
Fig. 1 A plot of total time to destination from now against travel
time in years using v0 = c/2000.
Fig. 2 Using the compounded growth equation and plotting total
time to destination from now against the waiting time in years
where v0 = c/20000
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Interstellar Travel: The Wait Calculation and the Incentive Trap of Progress
If we trust in the continued expansion of civilisation, the
risks of being overtaken are real. All civilisations, including
Earth, are caught in this incentive trap of growth. And it is an
important consideration since it is perfectly possible that the
colonisation of space will be a competitive venture between
disparate cultural groups. In which case, getting the waiting
calculation wrong may be fatal since, after the minimum has
passed, those who leave later arrive later.
2.5 What Rate of Growth is Likely?
An increase in the velocity of interstellar travel of 1% per
annum may be unrealistic since it leads so quickly to light
speeds. The chances are very high that the Galaxy would be
teeming with travellers if any civilisation could sustain this rate
of growth. In fact, the appearance of the first interstellar travel-
ler arriving on Earth would be a very bad sign, because we
would know that the vast rates of growth that got the traveller
here are poised to swamp humanity with further travellers
within a very short period of time. It seems unlikely, therefore,
that there have been isolated cases of contact with ETIs in the
historical or even recent past since there has been no follow-up
given this expansion rate of a civilisation. Interstellar travel
does not, therefore, appear to be commonplace. There must be
delaying factors to growth.
So what rate of velocity growth can be used? Let us consider
that the rate of Human expansion is intimately tied to the use of
resources. In this respect then, it should be more closely related
to the integrated long-term return on investment rather than to
any short-term usage of energy. However, more work will need
to be done on establishing the precise long-term relationship
between the energy resources used and power output available
for interstellar travel, but to illustrate the argument we will take
the recent simple annual percentage growth in energy con-
sumption to represent the same rate of growth of power avail-
able to interstellar flight.
Consider that, in energy terms, the power required to pro-
duce a given rate of travel is proportional to the square of the
velocity. Hence, a doubling of the velocity of travel requires a
quadrupling of power. Thus the velocity growth rate will be
proportional to the root of power production, v
p1/2.
The world energy consumption rose at an annual rate of
1.4% between 1990–2202, and it is expected to rise to about
2% between 2002 and 2025 [26].
So, taking r = 0.014 as the actual annual increment in the
velocity of travel we have:
v = v0 (1.014) t/2 (5)
In which case, where v0 = 1 / 20000, minimum time to
Bernard’s Star under these requirements for growth requires a
wait from now of ~966 years, a figure very little different to that
produced by equation –1.
2.6 Relativity Effects
However, as human growth and expansion produces ever-higher
travel speeds, eventually the constraints of Relativity enter the
picture. But is this significant?
At relativistic speeds, the mass rises and hence the power
required to accelerate a spaceship rises by a factor of
(1 – v2 / c2)1/2 (6)
However, these relativistic effects are modest at speeds quite
close to the speed of light. At 99% the speed of light, the mass
has risen only by a factor of 7 or so.
In general, if the time to a particular power level and hence to a
velocity level at a given velocity of travel is retarded by relativ-
istic considerations, then we can summarise the situation for
any particular trip using equations —5 and —6 like this:
where v = n.c / T and v0 = n.c / t0
we have
t0 / T. (1 – n2 / T2)1/2 = (1 + r)t/2
t = (2.log (t0 ) – (2.log T + log (n2 / T2)) / log (1 + r)
By plotting time to destination against this new waiting time
in fig.3 we see that the relativity component does not change
the basic curve derived from equation —5 Long before relativ-
ity effects are in evidence – say when v > 0.8c, giving a journey
time of (6/8) = 7.77 years - the minimum time to destination (at
r = 0.014) has already been reached, which, for the Barnard
Star journey requires a wait of ~ 966 years (plus a journey time
of 145 years) with a minimum time to destination of ~1111
years.
For near destinations and at these speeds (~4% the speed of
light), there is no significant time-dilation to improve the expe-
rience, this journey can be made only with hibernation or with
so-called ‘generation’ ships which contain a small group of
people whose children or grandchildren will see the destina-
tion. With these figures, the effort put into reaching the veloci-
ties required by Project Daedalus (~c/10), or even faster con-
cepts, will be not be worth it.
If the power growth rate ever slows then the minimum time
to destination increases rapidly (see Fig. 4). A long term aver-
age power growth of 5% pa produces a minimum time to
Barnard’s Star from now of ~327 years from now. But if the
annualised rate slows as civilisation comes up against resource
limits for example, then making any kind of useful interstellar
journey becomes increasingly remote. A growth rate of ½ % pa.
would produce a minimum time to destination of ~ 2287 years
with a journey time of 400 years.
An important result of equation –7 is that, since achieved
velocity grows with time, the minimum time to destination occurs
at an increasing velocity for further destinations (Fig. 5). Thus, the
incentive trap will always be present for far destinations unless
these velocities can be achieved. At a given rate of growth, a
journey planned with any velocity either side of the plotted line
(Fig.5) will be subject to the incentive trap. This does not mean the
journey cannot be made, rather it shows that the incentive trap will
be an inevitable component of the decision whether or not to go.
3. CAN THE INCENTIVE TRAP BE AVOIDED?
3.1 Is the Disincentive to Travel Irrational?
The incentive trap occurs when interstellar voyagers have rea-
son to believe that being overtaken poses a risk or implies a
waste of effort. During the initial stages of growth, there is no
incentive for voyagers to leave at any particular time since
6
Andrew Kennedy
technological development will overtake them if they do leave.
This dis-incentive to travel continues until a minimum, when
for a given rate and destination, waiting for growth to produce
higher travel velocities will not allow voyagers to overtake an
earlier launch.
The wait equation describes this minimum. This is the
point at which the negative incentive to leave changes to a
positive one; where the incentive to set out on the interstellar
journey is the strongest.
In spite of this, there is a widespread belief that a depar-
ture at any time runs the risk of being undermined by sudden
leaps in technology that could make the early voyage redun-
dant regardless of the time it set out. In other words, there is
never a strong incentive to leave. Vulpetti voices this typical
expectation, ‘…While …a vehicle proceeds to nearby star(s),
probably humankind’s physical knowledge could exhibit
some leap(s) inducing breakthrough(s) in space propulsion
technology.’ [25]. He recognises that this expectation could
undermine the willingness to leave on an interstellar flight
and even the willingness to invest in technologies designed
for interstellar flight.
This fear, however, is irrational. It implies that the applica-
tion of scientific breakthroughs is independent of interconnect-
ing patterns of growth. This fear has two manifestations. Firstly,
that a single remarkable technological breakthrough may hap-
pen at any time independent of any growth factor, and secondly,
that overall growth itself may take sudden leaps upwards and
alter the future beyond recognition.
Both these fears are based on misconceptions of the nature of
growth. In order to appreciate these misconceptions, it is worth
dwelling on some random cultural and scientific illustrations.
3.2 Brief Reflections on the Nature of Growth
We have seen how overall growth is resilient, but its expression
is unpredictable. At the mid-way point of the 16th century,
England was in decline, riven by civil unrest, corruption and
lack of finance [27]. Spain had destroyed the vast American
nations of the Aztecs and Incas and had a monopoly on all trade
with the Americas and its endless stream of gold and silver.
Through marriage and the use of its crack mercenary troops, it
controlled half of all Europe. Although the Cabots, sailing out
of Bristol, England, had been the first Europeans to set foot on
the North American mainland (1497), no permanent English
colony would be set up in the Americas until the 17th century.
In 1556, no one would have predicted that two hundred and
twenty years later, the new American Nation would be speaking
English and breaking away from its mother country of Britain.
Science continually produces rogue results and highly specula-
tive theorisation, but these have to await the time in which sense
can be made of them. Max Planck introduced the idea of energy
quantum in 1900. Einstein independently solved the problem of
the photoelectric effect with quanta in 1905 [9], and by 1926 the
Quantum theory was in existence[9]. The best scientific minds of
the century had applied themselves to developing the theory and
yet it was not until almost 50 years later that the tricks of normali-
sation, discovered by Feynmann, Bethe and others, allowed actual
physical values to be extracted from the theory. Entanglement in
quantum physics has been around since at least 1935 [29] and is
only now (70 years later) finding applications in quantum comput-
ing and cipher creation in which it can theoretically be employed,
though it still cannot permit faster than light communication as we
understand it now. More than 100 years after the conception of the
quantum, a unified field theory still remains elusive.
There are leaps in knowledge certainly, but exploiting the
leaps is never as obvious as it might first appear. The principle
of nuclear fusion was understood in the 1930s when Hans
Bethe saw that the fusion of hydrogen nuclei produced energy
and was the source of the energy in stars. A fusion bomb was
exploded in 1952 [9]. But although massively expensive work
on projects designed to make fusion an energy source for this
century have been proceeding for over 50 years, no one can
even predict when fusion reactors will come on stream, such is
the amount of work still to do.
Fig. 3 Total time to destination from now against the waiting time
in years where v0 = c/20000 and r = 0.014.
Fig. 4 Plotting average rates of growth in % annual increments
against the minimum time to a destination of 6 light years where
v0 = c/20000.
Fig. 5 Taking the velocity at the minimum time to destination for
destinations between 5 and 500 light years for r = 0.014;
0.02;0.03;0.05, and where v0 = c/20000
7
Interstellar Travel: The Wait Calculation and the Incentive Trap of Progress
Some great technological discoveries take years to fulfil
their promise, if at all. In 1899, Belgian Camille Jenatzy report-
edly drove his electric car, “La Jamais Contente” (‘Never Satis-
fied’) to a world land speed record of more than 100 km per
hour. By 1900, more electric vehicles were registered in the
U.S. than steam or gasoline vehicles [28]. Yet today the per-
formance of electric cars is still a long way behind the internal
combustion engine vehicles. The gradual improvements and
slow accumulations of efficiencies and new technologies have
benefited the internal combustion engine and not the battery.
New technologies follow known laws of power use and
information spread and are obliged to connect with what al-
ready exists. Remarkable theoretical discoveries, if they end up
being used at all, play their part in maintaining the growth rate;
they do not make its plotted curve and the positive incentive to
leave redundant.
3.3 Can the Positive Incentive to Leave be Undermined?
In spite of the low probability of remarkable discoveries lead-
ing rapidly to advances in practical knowledge, the disincentive
to leave may still be held if there is the belief that the overall
long-term growth rates are prone to change greatly with time.
The disincentive to travel may continue on past the minimum
time to destination if there is strong evidence that future growth
is likely to rise significantly and produce even faster velocities
than earlier growth rates.
If long-term growth does exhibit a level of uncertainty,
prospective interstellar voyagers will be able to assign a variety
of statistical techniques to the growth curve, and will be able to
give a probability value to each point on their minimum time to
destination graph.
One of the simplest techniques is the use of variance and
standard deviations from a mean.
If we made an observation each year of the value of the rate
of growth, r, then the mean growth rate is simply,
Mr = (r1 + r2….+ rt) / t
And the standard deviation will be
“((r1 – M1)2
+( r2 – M2)2….+( rt –Mt)2) / t )
By using the statistical inference that the variability of the
average of a set of observations is inversely proportional to the
square root of the number of observations, we can see that the
probability of a growth rate departing from the mean should fall
as time goes on. The likely future growth rate becomes more
predictable rather than less.
For the two cases where long term growth falls or rises,
waiting too long is still irrational. Where growth falls, the
available velocity of future travel fails to grow, in which case
waiting puts off the arrival time at the destination considerably.
In the second case where growth rises, the minimum time to
destination will be less and occur earlier, giving a strong reason
to leave earlier than the time predicted by constant growth.
Given this simple illustration and the historical picture of
growth, we can be reasonably sure that neither great changes in
growth nor great leaps in technology will alter the existence of
the transition from the negative to the positive incentive to
travel. The perceived ever-present disincentive to travel men-
tioned by Vulpetti can be overcome even allowing for uncer-
tainties in growth rates.
4. DISCUSSION
The treatment of growth in this paper describes important
factors and constraints to consider when deciding to journey to
the stars.
It has been shown that by plotting minimum time to destina-
tion from now against waiting time for a growth rate in velocity
of travel, there is a point where the negative incentive to wait
turns into a positive incentive to leave. Leaving before the
minimum time allows future growth to overtake the voyagers,
leaving after the minimum time will mean the voyagers cannot
catch up those who left at the minimum.
It has been shown that taking reasonable estimates for growth,
an interstellar journey of 6 light years can best be made in about
635 years from now if growth continues at about 1.4% per
annum. At this point, the journey could be made in 145 years. It
is likely that genetic engineering and improved techniques of
hibernation may make this journey feasible in an individual’s
lifetime. Humans may also undertake this voyage if they can be
sure that their children will see the destination. But the voyage
times will rise if growth ever slows. At a growth rate of ½ %
pa., the minimum time to destination for the same journey will
require a journey of ~400 years, at a velocity of 1.5 % the speed
of light. This will require the construction of ‘world ships’ or
something close to them [25]. The incentive trap of growth
will certainly come into play and few voyagers would be will-
ing to risk such a long trip until they are sure they will not be
overtaken. Even so, the time to destination minimum can be
used to entice people to make long journeys by using a spread
of launches around the minimum so that welcome and back-up
can be engineered for any particular group of voyagers.
It is considered that the overall growth curve represents a
summation of many inter-related sectors of growth, and that the
probability of a discovery in any one sector contributing, on its
own, to a sudden radical departure from the overall rate is not
likely. While unpredictable changes can contribute to the over-
all growth rate, it would be wrong for interstellar voyagers to
delay a departure based on the hope that a technological break-
through will improve the journey times calculated on the basis
of the known growth rate, and that to ignore the positive
incentive described by the wait equation would be irrational.
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(Received 7 November 2005; 26 January 2006; 28 February 2006)
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