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July 1963 Popular Science
 [Table of Contents]
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 When I read this account of relativistic
space travel by Dr. Wernher von Braun in a 1963 issue of Popular Science
magazine, I was a bit dubious about numbers he presented because they seemed to
me to assume a fixed value of relativistic speed during the entire round trip. It
is an example of the
Twin Paradox. Time dilation, as predicted by Einstein's theory
of Special Relativity, is barely noticeable at 10% the speed of light (0.1c), and
is not really significantly noticeable until around 30% the speed of light (0.3c),
so I suspected that maybe he had simplified the answer. His space ship example reached
that speed at 3½ months with a steady 1 G acceleration rate. Not wanting
to go to the effort of figuring it out for myself, I asked ChatGPT 4.0 to assess
the article and render an opinion. That result is presented on the right side of
this page. This was just one of several questions posed to him. This is the second
of two articles submitted by Dr. von Braun; the first being in the
January
issue.
Dr. Wernher von Braun Lays the Cards on the Table: Can We
Ever Go to the Stars?
ChatGPT 4.0 Assessment
Dr. Wernher von Braun's explanation of relativistic spaceflight provides a qualitative
description of how time dilation and relativistic effects influence long-distance
space travel. Below is a step-by-step verification of his calculations based on
the principles of special relativity.
Step 1: Relativistic Hyperbolic Motion Equations
For a spacecraft accelerating at a constant 1g (9.8 m/s2) in
its own frame, the relativistic equations of motion are:
- Distance traveled:
x= (c2/a0)
* (cosh(a0 * τ/c) - 1)
- Time elapsed in the stationary observer's frame:
t = (c/a0) * sinh(a0 * τ/c)
- Velocity:
v = c * tanh(a0 * τ/c)
Where, x is the distance traveled in the Earth frame, τ is the
proper time experienced by the astronaut, t is the time
elapsed on Earth, and v is the spacecraft's velocity.
Step 2: Journey to 1,000 Light-Years
We calculate the proper time τ experienced by the astronaut for a journey to
a star 1,000 light-years away, assuming constant acceleration for half the trip
and constant deceleration for the second half.
First, we convert units:
- Distance:
x= 1,000 light-years = 9.46*1018 meters
- Speed of light:
c = 3*108 m/s
- Acceleration:
a0 = 9.8 m/s2
To find the proper time τ for half the journey (x/2):
x/2
= (c2/a0) * (cosh(a0*τ/c) - 1)
Substituting:
4.73*1018 = (3*108)2/9.8 * (cosh(9.8τ/3*108)
- 1)
Simplifying:
4.73*1018 = 9.18*1015 * (cosh(3.27*10-8τ)
- 1)
Solving for cosh:
cosh(3.27*10-8τ) = 1 + (4.73*1018/9.18*1015)
cosh(3.27*10-8τ) = 515.6
Taking the inverse hyperbolic cosine:
3.27x10-8 * τ = cosh-1(515.6)
Using cosh-1(515.6) ≈ 7.644
τ = 7.644 / (3.27*10-8) ≈ 2.34*108 seconds
Converting to years:
τ ≈ 7.42 years
The full journey time (round trip) is double the proper time for one-way travel:
τround trip ≈ 2*7.42 ≈ 14.84 years
To find the Earth time t for the journey:
t = (c/a0) * sinh(a0 * τ/c)
Substituting τ = 7.42 years:
sinh(7.644) ≈ 1048.3
t = (3.06*107) * 1048.3 ≈ 3.21*1010 seconds
Converting to years:
t ≈ 1,017 years
Step 3: Corrections or Confirmations
Von Braun's calculation of 13.2 years for the astronaut's proper time aligns
closely with our calculated value (14.84 years round trip, or 7.42 years each way).
The difference is due to approximations or simplifications in his narrative. The
Earth time of 2,000 years is higher than ChatGPT's calculated value of 1,017 years.
Dr. von Braun might have assumed longer acceleration phases or other idealized
conditions.
Conclusions
Dr. von Braun's explanation is qualitatively correct, and the calculated proper
time for the astronaut aligns closely with his approximation of 13.2 years for one-way
travel. The Earth time (approximately 1,017 years) is slightly less than his stated
2,000 years, likely due to simplifications in his narrative. The relativistic time
dilation calculations fully account for the acceleration and deceleration periods,
consistent with Einstein's theory of special relativity.
Proper Time in Relativity
In the context of relativity, proper time is the time measured
by a clock that is at rest relative to the observer and travels along the same path
as the observer. It is the time experienced directly by an observer or a particle
moving along a given worldline. Proper time is a key concept
in both special and general relativity.
Definition
Proper time (τ) is defined mathematically as the integral of the spacetime
interval along the worldline of an object:
τ = ∫ √(1 - v2/c2) dt
Where:
- v is the instantaneous velocity of the object relative to a chosen
reference frame.
- c is the speed of light.
- t is the time measured in the reference frame of an external observer.
The proper time represents the true elapsed time as experienced by the moving
observer and is invariant, meaning all observers in relativity agree on its value,
even if they measure time differently in their own frames.
Key Characteristics
1. Frame Dependent:
Proper time depends on the object’s path through spacetime, not on the coordinates
of an external observer. An observer in their own rest frame always measures proper
time with their own clock.
2. Time Dilation:
In special relativity, a moving clock experiences time dilation relative to a
stationary observer. The proper time elapsed on the moving clock is shorter than
the coordinate time elapsed in the stationary observer's frame:
Δτ = Δt √(1 - v2/c2)
This means that for high velocities, the proper time experienced by the moving
observer is significantly less than the time experienced by a stationary observer.
3. General Relativity:
In general relativity, proper time also accounts for gravitational time dilation.
The formula becomes more complex, incorporating the effects of spacetime curvature.
Examples
Astronauts in Relativistic Space Travel:
If an astronaut travels at a significant fraction of the speed of light, the
time they experience on their spacecraft (proper time) will be less than the time
measured on Earth (coordinate time).
Twin Paradox:
In this thought experiment, the twin traveling at relativistic speeds experiences
less proper time than the twin who remains stationary, illustrating the difference
between proper time and coordinate time.
Conclusion
Proper time is the most direct and "personal" measure of time for an observer
traveling along a particular path in spacetime. It is a fundamental quantity in
relativity that highlights how time depends on motion and gravitational influences,
differing from the absolute, universal time of Newtonian mechanics.
Worldline in Relativity
In the context of relativity, a worldline is the path that an
object traces through spacetime as it moves. It represents the history of the object's
position in both space and time, providing a complete description of its motion.
The concept of a worldline is central to both special and general relativity, where
time is treated as a fourth dimension, alongside the three spatial dimensions.
Definition
A worldline is a curve in four-dimensional spacetime, where the coordinates of
the object are given as a function of its proper time (τ):
(t(τ), x(τ), y(τ), z(τ))
Here:
- t(τ) is the time coordinate as a function of proper time.
- x(τ), y(τ), z(τ) are the spatial coordinates as functions
of proper time.
The worldline describes the continuous trajectory of the object in spacetime
from its past to its future.
Key Characteristics
1. Timelike Worldlines:
These represent the paths of objects with mass that move slower than the speed
of light. The spacetime interval along a timelike worldline is positive, and such
paths are associated with proper time experienced by the object.
2. Lightlike (or Null) Worldlines:
These represent the paths of massless particles, such as photons, which travel
at the speed of light. The spacetime interval along a lightlike worldline is zero.
3. Spacelike Worldlines:
These describe hypothetical paths that exceed the speed of light. Such worldlines
are not physically realizable in relativity and are mostly used in theoretical contexts.
Worldlines in Special Relativity
In special relativity, worldlines are straight lines in a flat spacetime diagram
for objects moving with constant velocity. For objects undergoing acceleration,
the worldlines become curved, reflecting their changing velocities.
Worldlines in General Relativity
In general relativity, spacetime can be curved by the presence of mass and energy.
The worldlines of objects are influenced by this curvature, causing them to follow
geodesics, which are the "straightest possible" paths in curved spacetime.
Significance
Worldlines provide a graphical and mathematical way to visualize and analyze
motion in spacetime. They are used to study causality, interactions, and the relationship
between events. For instance:
- The intersection of two worldlines represents an interaction or event involving
two objects.
- The slope of a worldline in a spacetime diagram corresponds to the object's
velocity.
- Parallel worldlines represent objects at rest relative to each other in special
relativity.
Conclusion
A worldline encapsulates the full history of an object's motion through spacetime,
making it an essential concept for understanding relativistic physics. Whether in
flat or curved spacetime, worldlines allow physicists to analyze how objects move
and interact under the influence of relativity.
Relativistic effects become noticeable when an object approaches speeds where
its velocity ν is a significant fraction of the speed of light c. A common rule
of thumb is that these effects are appreciable when ν ≥ 0.1c, or 10%
of the speed of light. This threshold is based on the Lorentz factor, γ, defined
as: γ = 1/√(1 - c2/ν2).
When ν = 0.1c, the Lorentz factor is:
γ= 1/√t(1 - 0.12) ≈ 1.005
This small increase of about 0.5% in γ compared to its non-relativistic value
(γ=1) means relativistic effects are starting to emerge but are not yet dramatic.
For practical purposes, relativistic effects become increasingly important as v
approaches 0.3c or higher, where γ≈1.048, and deviations from classical physics
become more apparent.
In applications like particle physics or astrophysics, relativistic corrections
are crucial for ν > 0.1c. In contrast, for everyday scenarios where ν << c,
these effects are negligible.
Dr. von Braun (right) with J. R. Dempsey, president of General
Dynamics/Astronautics.
How far to a star? A scale drawing, with sun and earth 1/8" in.
apart, would take more than a half-mile of paper to show the distance to the nearest
star.
By speeding nearly as fast as light, you could go to a star and return in your
lifetime - to find centuries had gone by on earth meanwhile
Q. Will we ever be able to travel to solar systems beyond our own?
A. The past 20 years should have taught us to use the word "impossible"
with utmost caution. Nevertheless, human travel beyond our own solar system is a
staggering concept. Even the most reckless optimists do not expect it to come about
in our generation - or the next.
Light, traveling at 186,000 miles per second, needs 8.3 minutes to span the 93
million miles between the sun and earth. Light takes 5 1/2 hours to travel from
the sun to Pluto, outermost planet of our solar system. But it takes 4.3 years to
reach Alpha Centauri, the nearest fixed star (4.3 light-years away); 470 years to
Polaris; and 27,000 years to get to the center of our galaxy - a lens-shaped island
in space, a little less than 100,000 light-years in diameter, made up of an estimated
total of some 200 billion suns.
Q. Can we build a rocket powerful enough to travel so far?
A. We have to impart to an object a veracity of slightly more
than 25,000 feet per second to place it in a low orbit around the earth, About 36,000
feet per second is needed to hurl it to the moon - which is still within range of
the earth's pull - and just a trifle more to kick it completely out of the earth's
gravitational field. If we accelerate it up to a terminal speed of 56,000 feet per
second (in such a fashion that it leaves the earth in the same direction in which
the earth is orbiting at 107,000 feet per second around the sun), it will enter
a parabolic flight path and escape from our solar system.
***
Phew ... and now, back to something easy like designing rockets!
It's no small feat to make understandable the eerie problems
of "relativistic" space flight to stars thousands of light-years away. So PS echoes
this impromptu postscript that Dr. von Braun jotted on his letter forwarding this
month's manuscript to Editor Bob Crossley.
From the point of view of power requirements, a needed velocity of 56,000 feet
per second (38,000 m.p.h.) may not sound too bad. Just one extra stage on top of
the Saturn V, our Apollo moon rocket, could impart that speed to an object of about
8,000 pounds. But as the object coasted, its power spent, on its "uphill" path out
of the pull of the sun's gravity, its speed would gradually diminish almost to zero.
Millions of years would elapse before it reached one of the nearest fixed stars.
To reduce travel time to figures compatible with the life span of man, travel
speeds must approach the speed of light.
Not even nuclear-fission or nuclear-fusion processes are adequate to produce
such speeds. For all their dramatic display of power, they convert only a tiny fraction
of the mass involved into energy. It would be necessary to devise a rocket mechanism
wherein the entire mass, M, of the injected "propellant" is converted into radiation
energy, E, according to Einstein's famous equation:
E = MxC2. The exhaust of such a "photon rocket" would be a beam of
radiation, and the exhaust velocity would of course be equal to the velocity of
light, C.
The problem is that nobody knows how to build a photon rocket. Certain subatomic
processes are known, such as the joining of an electron (a small negatively charged
particle) with a positron (an equally small positively charged particle), that directly
transform matter into energy according to Einstein's equation. But so far, physicists
have been unable to devise any large-scale processes for this transformation.
There are also tremendous engineering obstacles. By definition, a photon rocket
converts its propellant stream into an extremely powerful light beam. To bundle
this beam, some sort of mirror is needed. Even if it had a reflectivity of 99 percent,
better than our best existing ones, that one percent of absorbed radiation energy
would instantly melt the mirror - considering the billions of kilowatts converted
into the power carried away by the light beam.
Q. Is it true that it is impossible to exceed the speed of light?
A. Yes. But as we shall see, this is partly a matter of definition.
Suppose we had overcome the "minor" problems just described, and we did have a rocket
capable of "beaming away" 100 percent of the mass of its propellant with an exhaust
velocity equal to the speed of light. What could we do with it?
If the rocket had a mass ratio (the ratio between its fully fueled and empty
weight) of 3, it could reach 80 percent of the speed of light. With a mass ratio
of 10, its terminal velocity would be about 98 percent; and with a mass ratio of
1,000 (about what we have today in some of our chemical multistage planetary rockets),
we would hit 99.9998 percent of the speed of light.
Again we refer to Dr. Einstein. His Theory of Relativity (which has stood the
test of many critical experiments, and has been universally accepted by the scientific
community) shows that the inertia of an object's mass approaches infinity as the
object approaches the speed of light. Hence it would take infinite power to accelerate
an object beyond the "light barrier."
But, amazingly enough, the same theory states that a stellar astronaut could
still travel to a star 1,000 light-years away and return within his adult life.
Q. How could an astronaut travel 2,000 light-years in a lifetime?
A. "Time dilation" would help him to stay young. For many people,
the strange phenomenon called time dilation is a hard pill to swallow. The flow
of time appears to us completely unaffected by physical conditions. Whether we sleep
or work, sit at a desk or in a speeding jetliner, our wrist watch seems to tick
away at the same pace. So does our heart.
But the fact is that this cherished piece of "everyday experience" is valid only
in the realm of relatively low velocities in which we slowpokes live.
A meson (an unstable subatomic particle ), when traveling at a velocity close
to the speed of light, has a clearly longer decay time than its 2.1-microsecond
"half-life" at lower speeds - when an earth-fixed observer does the timing. But
if the observer were flying along with the meson, the half-life of 2.1 microseconds
would not seem to be affected by the particle's speed, since the observer's watch
would be subjected to the same time dilation as the meson itself.
The Theory of Relativity tells us that the pace of time becomes slower and slower
for an object approaching the speed of light, compared with time's rate of passage
for a stationary observer. At the speed of light itself - an upper limit that no
object can ever reach - time would come to a complete standstill. If an object could
go so fast, it could cover vast distances while, for a man flying along with it,
no time would elapse - neither for his watch nor for his heartbeat, which controls
his life span.
This strange effect makes it possible for a stellar astronaut to travel from
the earth to a fixed star 1,000 light-years away, in what he would think was 13.2
years. For the trip back he would need another 13.2 years. If he didn't spend any
additional time at his destination, he would thus have been away from the earth
for 26.4 years. The trouble is that, during his absence, more than 2,000 years would
have elapsed on earth. Thus, upon return, he might wind up in a zoo.
Q. What would a trip to a star be like?
A. Let us assume we have a photon rocket capable of a continuous
acceleration of 1 G. Suppose, too, that our mass ratio is large enough to get us
very close to the speed of light; carry us to a star 1,000 light-years away; and
slow us down again to normal speeds, so we can visit one of the star's planets.
The rocket is also to be capable of flying us back to earth - possibly by "refueling"
during the stay at that distant solar system.
As we depart from the earth, the stars of the firmament will first appear in
their familiar yellowish hue. As our vehicle builds up speed toward our target star,
the Doppler effect will cause a striking change in this star's color. From its original
yellow, the light received from it will shift through green, blue, and violet, and
toward ultraviolet-in other words, to higher frequencies. Simultaneously the color
of the receding sun will slowly change from yellow to orange, red, and toward infrared-that
is, to lower frequencies.
This is easy to understand: A boat running against the waves is hit by them at
a higher frequency than a stationary pier is; a boat running with the waves, at
a reduced frequency.
After about three and a half months our stellar photon rocket has reached about
30 percent of the speed of light. The frequency of the sun's peak radiation output
now passes the border of the visible spectrum and moves into the infrared. As a
result, the sun dims rapidly, and soon becomes invisible. One month later, the destination
star likewise becomes invisible - the peak of its radiation intensity has shifted
into the ultraviolet.
As our velocity keeps increasing, two circular dark spots are formed around the
destination star and the sun, and keep growing in diameter. Between these blind
"bow and stern spots," the stars of the firmament appear as a multicolored array
of concentric circles, like a huge rainbow:
Near the black bow spot, the stars look violet. Further aft, they are blue and
green. Abeam, they shine in their original yellowish hue. Still farther aft, they
look orange, and the dark stern spot is surrounded by a ring of red stars.
Due to "relativistic" effects, the dark bow spot grows only to an opening angle
of 43 degrees. After we exceed 74 percent of the speed of light (11 months after
departure), it begins to contract again. But the stern spot around the sun continues
to grow steadily. Hence, as our traveling speed approaches the speed of light, the
visible portion of the firmament will become compressed into an ever-narrowing rainbow
around the invisible target star.
The opening angle of the yellow ring, in this rainbow, is a perfect yardstick
for the ratio between our traveling speed and the speed of light. In analogy to
the well-known Mach Number (ratio of the speed to speed of sound), this ratio is
sometimes called the Einstein Number.
In 6.6 years from the time of departure, our speeding photon rocket hits Einstein
Number 0.999998, and we are at the half-way point of our journey. However, on trying
to measure the remaining distance to our destination star (now emitting predominantly
x-rays), we find it only about a light-year away! In fact, without further power
application, we would pass it a year later - 7.6 years' "dilated ship's time" after
departure - if we were to refrain from slowing down for our forthcoming visit.
But in order to visit one of the star's planets, we have to turn our ship around
and use our photonic rocket thrust for braking. Of course our slowing down means
that we'll reach our target, not in another year, but much later. Only after another
6.6 years - 13.2 years after departure - will we near our target, at a relative
approach speed close to zero. During the second 6.6 years - that is, during the
retardation maneuver - all those celestial "rainbow" phenomena of the acceleration
period will take place in reverse. Upon arrival, the firmament will look like its
old self again.
If we had a telescope powerful enough to observe events on earth from our new
vantage point, we would find our home planet very much as it was when we left it.
But, being 1,000 light-years away, we are actually watching events that happened
on earth 1,000 years ago. (This is the non-dilated time that has elapsed on earth
since we left.) The amazing thing is that, due to the time dilation aboard our speeding
rocket, we have aged only 13.2 years during our outbound voyage.
Eerie as this may sound, it is all in perfect harmony with modern ideas of the
laws of space and time. (Men today have the same difficulty in accepting the concept
of relativistic time that our ancestors had in seeing how people "down under" in
Australia could walk head down without dropping off the globe. But that is because
our experience does not include very great distances and extremely high speeds.)
While the insights of modern physics permit us to dissect the anatomy of interstellar
flight, we must forego rash conclusions that any such flights are imminent, or feasible.
We cannot yet even define an adequate power source. If we had it, many problems
of using it would be beyond us. Other obstacles may be even more formidable. For
instance, what would happen to an interstellar rocket that hit even a small meteoroid,
if the collision were at nearly the speed of light?
In summary, with our present knowledge, we can respond to the challenge of stellar
space flight solely with intellectual concepts and purely hypothetical analysis.
Hardware solutions are still entirely beyond our reach and far, far away.
Dr. von Braun will consider answering questions from readers of Popular Science
in the magazine, but he cannot undertake to answer each one by mail. Letters to
him should be addressed in care of Popular Science, 355 Lexington Ave., New York
17, N. Y.
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