Download III. Paradoxes in Special Relativity

Chapter 15
Spaceflight and Special Relativity
Brice N. Cassenti*
Rensselaer Polytechnic Institute, Hartford, CT 06120
I.
Introduction
T
HE Special Theory of Relativity is one of the foundations of modern physics. It provides a conceptual starting
point for the General Theory of Relativity and is a key ingredient of Quantum Field Theory. Yet many of its
predictions are counter-intuitive. These counter-intuitive predictions produce paradoxes that are not readily
explained; yet experimental results have verified the predictions of Special Relativity to an extraordinary degree.
The Special Theory of Relativity was proposed by Einstein in 1905 1 to reconcile experimental results that did
not agree with classical Newtonian mechanics. Special Relativity assumes that Maxwell’s equations for
electromagnetism are exactly correct by hypothesizing that the speed of light in a vacuum takes on the same value in
all inertial (constant velocity) reference frames. As a consequence of this assumption time cannot flow at the same
rate in all inertial reference systems, and this is at the heart of most of the paradoxes. An excellent introduction to
both Special and General Relativity can be found in Einstein’s own writings. Reference 2 is a detailed mathematical
treatment, while reference 3 is in a more popular presentation. Taylor and Wheeler have written an excellent
introduction to Special Relativity4. A detailed discussion of the paradoxes resulting from the special Theory of
Relativity can be found in reference 5.
The paradoxes are even more profound if speeds greater than the speed of light are considered. In cases where
objects travel faster than the speed of light there are reference frames where an effect can precede its cause. Thus
implying a form of time travel. Modern physics sometimes removes these effects by postulating that nothing can
move faster-than-light. But the General Theory of Relativity does not exclude the possibility, nor does the Special
Theory, and Quantum Mechanics may actually require it. Of course, it may be that it is just our poor understanding
of space and time, and that new concepts will clear up the confusion. The paradoxes that are presented are shown
to be primarily concerned with our concept of time. If our concept of time can be clearly defined then it may be
possible to resolve all of these paradoxes.
Paradoxes though do not mean that the theory is inaccurate or in error. Empirical evidence strongly favors the
Special Theory of Relativity, and has verified it to an extraordinary accuracy, and, hence Special Relativity can
provide a basis for sizing interstellar vehicles.
The sections to follow will introduce the Special Theory of Relativity, and then cover some of the well-known
paradoxes. It will close with some empirical results and a basis for the description of the motion of relativistic
rockets.
II.
Principle of Special Relativity
Special Relativity is based on the experimental fact that the speed of light in a vacuum remains the same in all
inertial frames regardless of the observer’s speed with respect to the light source. Many experiments were
performed in the late 1800s and early 1900s that verified that the speed of light did not vary with the relative speed
of the source, but a simple Galilean transformation of Maxwell’s equations indicated that the speed of light should
vary. Ad hoc theories were proposed to satisfy the observations. The most complete was that of Lorentz 1. Lorentz
proposed a transformation where both space and time were connected to the relative speed and, although Maxwell’s
equations were modified to account for motion with respect to the stationary ether, the effects were forced to cancel
*
Associate Professor, Department of Engineering and Science, 275 Windsor Street, Hartford, CT 06120
and Associate Fellow AIAA.
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exactly in order to agree with the experimental observations. Einstein, though, assumed that Maxwell’s equations
were exactly correct, and that the fault was in the Galilean transformation.
A. Postulates and Lorentz Transformation
The two postulates of Einstein’s Special Theory of Relativity are that: 1) physical laws hold in all inertial
reference systems, and 2) the speed of light (in a vacuum) is the same in all inertial reference frames. These two
postulates are all that is required to find the transformations that Lorentz found in his ad hoc manner.
Consider two inertial space-time reference systems
(primed and unprimed) as shown in Figure 1. The coordinates
( t , x ) are for the stationary system, and the coordinates
( t ' , x' ) are in the moving system. The origin of the primed
system is moving at the constant speed v along the x-axis.
Since Maxwell’s equations must remain linear in both systems
the coordinates ( t ' , x' ) should be a linear combination of the
coordinates ( t , x ) . Taking the origins of both systems to
coincide, the coordinates must be related by
ct '  Act  Bx
.
(1)
x'  Cct  Dx
where c is the speed of light and A, B , C , D are constants
that depend only on the relative speed v. The quantities ct and
ct’ are used since they are distances and hence have the same
units as x and x’.
The speed of light, c, must be exactly the same in both coordinate systems. Hence, light moving through dx in
time dt in the unprimed coordinate system at speed c must also move through dx’ and dt’ in the primed system at
speed c. Then
c 2 dt 2  dx 2  c 2 dt ' 2 dx' 2  c 2 d 2 .
(2)
The quantity d is referred to as the proper time and represents time intervals of an object in its stationary frame.
Substituting equations (1) into equation (2) and equating the coefficients of dx2, dt2, and dxdt on both sides of
the resulting equation gives
A  D  cosh 
where
   v .
The function
 v 
B  C   sinh 
(3)
can be found by noting that a particle stationary in the primed system is
moving at v in the unprimed system. Using equations (1) and (3)
dx'
Cc  Dv
0
.
cdt '
Ac  Bv
(4)
Then
v
 tanh  .
c
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(5)
Equations (1) can now be written as
ct '   ct  x 
x'   x  ct 
where
  tanh 
and
  cosh   1 / 1   2
.
(6)
. Taylor and Wheeler4 refer to  as the velocity parameter.
The quantity  is also sometimes referred to as the rapidity. The transformation in equations (6) is commonly
known as the Lorentz transformation1.
We can now compare the coordinate systems by considering lines of constant ct’ and x’ (and constant ct and x)
as shown in Figure 2. Note that light emitted from the origin will travel along the diagonal in both coordinate
systems. Also note that the primed coordinate system is collapsing around the diagonal of the unprimed system and
at a relative speed equal to the speed of light, the primed coordinates will have completely collapsed. Equation (5)
indicates that v/c can only approach the speed of light if  is finite and real. For   1 ,  is imaginary, as well as
. The proper time also becomes imaginary as indicated by equation (2). Another way to look at faster-than-light
relative motion is that the spatial and temporal coordinates switch. This clearly means that objects can travel both
ways in time for an observation of faster-than-light particles. Another way to look at this is to consider what Figure
2 would look like for the primed system moving at more than the speed of light. The ct’ axis would be on the xcoordinate side, and the x’ axis would be on the ct coordinate side. Time in the primed system would be a spacelike coordinate to the observer in the primed system, and the space coordinate in the direction would be time-like in
the unprimed system. In Special Relativity physicists sometimes assume that nothing can go faster than light and
this completely removes those cases from consideration.
A. Lorentz Contraction and Time Dilation
Two significant consequences of Special Relativity are the apparent contraction of moving objects and the
apparent slowing of time in moving objects. The contraction of moving objects is easiest to predict once a method
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of measuring lengths is defined. We will define length as the Euclidean difference in coordinates at the same value
for the time coordinate. In our case this means the difference in the x-coordinate locations for a specific value of ct.
For a measurement of a rod stationary along the x-axis in the unprimed system we can set ct ' 0 in equations (6),
then in the primed system ct   x , and
x'  1   2 x
(7)
and stationary lengths in the unprimed system appear to be contracted in the primed system. Equation (7) is the
Lorentz contraction. Of course, a similar analysis will show that stationary lengths in the primed system appear
contracted in the unprimed system
Time dilation is the analogous situation for relative time measurements. Consider a stationary particle at the
origin of the unprimed system (i.e., x  0 ). Then equations (6) yield ct   x and
ct  1   2 ct '.
(8)
Hence time in the moving coordinate system appears to move slower. But for an observer in the primed system
watching a clock stationary at the origin of the unprimed system stationary (i.e., x' 0 ) we find from equations (6)
that
ct '  1   2 ct.
(9)
Hence observers in both frames will predict the other’s clocks are moving more slowly. If they continue to move at
a constant relative velocity (i.e., both systems are inertial) then they can only meet once, and there will be no
contradictions.
III.
Paradoxes in Special Relativity
A. Pole Vaulter Paradox
A popular paradox in Special Relativity is the “pole vaulter paradox”. In this scenario a pole vaulter carries his
pole horizontally at high speed. In our version of the paradox, the pole is 15 feet long according to the pole vaulter,
but he is moving so fast that an observer standing at a stationary barn that is 10 feet long sees the pole contracted to
half its length, which is now less than the length of the barn. The front door of the barn is open as the pole enters.
When the back end of the pole enters, the front end has not reached the back door, and the front door closes. Hence
to the observer the pole is entirely enclosed by the barn. When the front of the pole meets the back door, the back
door is opened and the pole leaves the barn. Of course to the pole vaulter the barn appears half as long (i.e., 5 feet)
and the pole vaulter will note that the pole was never completely in the barn. The pole vaulter will also note that the
back door opened before the front door closed. Obviously, the pole cannot both be longer and shorter than the barn.
The paradox is solved by tracking the front and back end of the pole and barn in both the pole vaulter’s and the
observer’s system. The transformations will show that the order the doors open and close is reversed in each system
agreeing with the observations. The results will also show that the time interval when the doors open and close is
shorter the time it would take light to traverse the distance between the doors of the barn in the observer’s coordinate
system, and is also shorter than the time it would take light to traverse the length of the pole in the pole vaulter’s
coordinate system. Hence, closing the front door cannot influence opening the back door if the speed of light is the
maximum speed in the universe.
This leads to the primary contradiction of Special Relativity with our common low speed experience. Space
and especially time are not what we picture. Newtonian physics exists in a universe where time is the same for all
observers, and Special Relativity says it cannot be if the speed of light is the same in all inertial (non-accelerating)
reference systems.
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B.
Twin Paradox
The best-known paradox in Special Relativity is the twin paradox4,6. In this paradox the problem involves time
directly. Consider two fraternal twins, Alice and Bob preparing for a long space voyage. Bob is in mission control
while Alice will do the traveling. Alice departs at a very high speed so that Bob sees Alice’s clocks running at half
speed, but, of course, Alice sees Bob’s clock running at half speed also. At the half way point Alice reverses speed,
so that on the return each sees the other’s clock as running at half speed. When Alice stops to meet Bob, they both
cannot think that each has aged half as much, thus he paradox. Certainly the Lorentz transformations apply to Bob,
since at no time did he experience any acceleration, and hence, Alice will have aged half as much as Bob.
Obviously the problem is with the acceleration Alice experiences. It can be shown that while Alice changes her
speed she will see Bob’s clock run faster. All three of her accelerations will exactly compensate for the slowing
during the constant velocity portions of the trip, and both Bob and Alice will agree that each of their clocks are
correct when they meet. Actually, there are many ways to resolve the paradox. Weiss 6 has an excellent discussion.
The section to follow on constant acceleration rockets can be used to show that each will agree with the clock
readings on Alice’s arrival.
C.
Faster-than-Light Travel
Objects that move faster than the speed of light can reverse the order of events in some inertial systems. Figure
3 illustrates the appearance of two objects in two inertial systems. The object moving from A to B is moving less
than the speed of light in the unprimed system. The projection of the two points A and B onto the ct-axis clearly
shows that A precedes B. We can also project the points A and B onto the ct’-axis, which again clearly shows that
A precedes B.
If we now consider the path from C to D, the slope of the line shows that in the unprimed system the speed of
an object going from C to D is greater than the speed of light. Again projecting onto the ct-axis shows that C
precedes D in the unprimed system, but in the primed system a projection onto the ct’-axis shows that D now
precedes C. Hence events that are observed for an object moving faster than the speed of light can have effects that
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precede the cause in some inertial systems. For example a dish that breaks going from C to D in the unprimed
system will magically re-assemble itself when observed in the primed system. The simplest resolution of the
paradox is to again assume that physical objects cannot move faster-than-light, but, of course, this may not be the
actual explanation.
D.
Instant Messaging
Consider the case illustrated in Figure 4. Two observers Art (A) and Don (D) are stationary in the unprimed
coordinate system. There are two other observers, Brenda (B) and Cathy (C), in the primed system, moving at a

constant velocity v with respect to the unprimed system. Art hands off a message to Brenda as she goes by.
Brenda then instantaneously transmits the message to Cathy. Cathy hands the message to Don, who instantaneously
sends the message to Art. The loop in Figure 4 shows that Art receives the message before he hands it off to
Brenda. Art “now” has two messages. “After” completing the loop again Art has four, and the process continues to
double infinitely many times, creating quite a puzzling paradox. The problem again lies with speeds greater than the
speed of light. The objects that are moving faster-than-light are messages, which need not consist of a physical
mass. Nevertheless the information in the messages exceeds the speed of light.
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Again the simplest way to remove the paradox is to assume that nature does not allow any cause to propagate
faster then the speed of light. But there is another explanation. The infinite number of messages arriving at A is
clearly a non-linear process and will not satisfy the linear relationships assumed in Special Relativity. This nonlinear process may be related to the collapse of the wave function in Quantum Mechanics, but a nonlinear process
would violate the postulates of Quantum Mechanics. The non-linear theory could result in one of the infinite
number of messages being chosen, and resolve the paradox. That is, the collapse of the wave function in Quantum
Mechanics could be the mechanism that resolves the paradox and may clear up the conceptual problems with
entanglement7.
A better illustration of the collapse of the wave function is the case where Art can randomly choose between
two messages one labeled Up and the other Down. Art randomly chooses one of the two. For example, Art chooses
Up and hands it off to B. Instantaneously Art will have three messages. Two of these are Up and the third is Down.
A random choice now is more likely to be Up. Of course instantly there will be an infinite number of messages
consisting of the actual choice for the collapse of the wave function, and the other choice will effectively not appear
at all. Hence, the collapse of the wave function could be explained as a self-referential closed space-time loop6.
IV.
Empirical Foundations
Paradoxes in the Special Theory of Relativity seem to imply that the theory may not be physically consistent, but
the Special Theory of Relativity has been demonstrated to be accurate to an extraordinary precision. The text by
Bergmann8 contains an entire chapter on the experimental verification of the Special Theory of Relativity. Born 9
has a more popular description of the Special Theory of Relativity but it does contain references to experimental
verification scattered throughout his chapter on Special Relativity. A readily accessible summary10 contains
numerous references to experiments and their results.
All of the experiments so far performed indicate that the Special Theory of Relativity is an extremely accurate
representation of the nature of the universe. Of course, with over a century of experimental verification it will be
difficult to publish any experimental violations, nevertheless searches for violation do continue (e.g., see the article
by Kostelecky11 which contains a popular description for the continuing search for experimental violations).
Several of specific topics are summarized below.
A. Michelson-Morley Experiment
The Michelson-Morley experiment is the classic experiment referenced for the verification of the fundamental
tenets of the Special Theory. It consists of a Michelson interferometer where a single light beam is split into two
beams moving at right angles. When the beams are brought back together interference fringes appear according to
the path lengths of the split beams and the speed of light in each direction. Variations in the travel times can then be
measured along different directions. Since the Earth has a preferred direction for its orbital velocity about the sun,
any change in the speed of light due to the motion of the Earth will be recorded. The most recent experiment shows
that there is no change with direction for the velocity of light 10 to within 2 parts in 10,000.
B. Half-lives of Relativistic Particles
One of the predictions of the Special Theory is that changes in the time observed by individuals moving at
relativistic velocities. Cosmic rays are a natural source of high energy (relativistic) particles and in fact the muon,
which is produced when high energy cosmic rays collide with atoms in the upper atmosphere, has been observed at
ground level indicating that its life has been significantly extended beyond its nominal 2 sec lifetime. Experiments
done in particle accelerators are much more accurate. Experiments have shown accuracies in time dilation
predictions to better than 0.1 percent12.
C. Doppler Shifts
The Doppler shift on the spectra of quasars, which are moving away at relativistic velocities, is a very accurate
test of the theory of Special Relativity for radial motion but the more important tests are those that involve
transverse motion. Here laboratory tests have verified the theory with accuracies of better than 3x10 -6 relative to the
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predicted value9. These tests are particularly important since the predictions of Special Relativity are at odds with
all non-relativistic (Galilean) theories.
D. Cerenkov Radiation
Cerenkov radiation is produced when charged particles move through a material at speeds faster than the speed
of light in the material. This is analogous to the motion of an aircraft through the air at faster than the local speed of
sound in the air. When these particles exceed the local speed of light in a material a “light boom,” instead of a sonic
boom, is created from radiation emitted by the particle as it slows down. The emitted light propagates perpendicular
to the wave front of the “light boom.” Also the intensity of the light increases with increasing frequency and so is
preferentially blue. In a material the local speed of light is reduced by a factor equal to the index of refraction. For
example, if particles move through water, which has an index of refraction of 4/3, with more than 3/4 of the speed of
light then Cerenkov radiation will be observed. Such speeds are easily exceeded in cosmic ray and in elementary
particle physics experiments. In fact, Cerenkov radiation is widely used in elementary particle physics experiments
to differentiate relativistic particles of the same momentum but different masses 11and the measurements are
completely consistent with the Special Theory of Relativity.
E. Tachyon Searches
Tachyons are hypothetical particles that move faster than the speed of light and, if they existed, could produce
causality paradoxes. A brief popular description on tachyons can be found in reference 15. For a particle with a
velocity exceeding the speed of light, equations (11) imply that both the energy and the momentum become
imaginary (i.e., a multiple of  1 ). This is consistent with a rest mass that has an imaginary value. There is
nothing in the Special Theory of Relativity that precludes such particles from existing, but reported searches have
been consistent with the observation that tachyons do not exist. The experimental limits on the imaginary massenergy of the tachyon particles13 have been reported as less than 104 eV.
Even if tachyons do exist, the Feinberg reinterpretation principle 14 could be used to show that tachyons will not
violate causality. Recall that the order for cause and effect is dependent on the relative velocity of the observers.
Causality would be violated if a tachyon could send information into its own past. The Feinberg reinterpretation
principle states that sending a negative energy tachyon backward in time (which could violate causality) would
appear as positive energy tachyons moving forward in time. According to the Feinberg reinterpretation principle,
the creation and absorption of tachyons cannot be differentiated by observers undergoing relative motion. Hence, if
the Fienberg reinterpretation principle holds, it would be possible to use tachyons to send energy forward in time,
but it would not be possible to communicate backward in time 15.
V.
Relativistic Rockets
Although Special Relativity applies to constant velocity processes, it can be extended to accelerating objects for
applications to interstellar spaceflight. The extension only requires that we constantly update the reference velocity
to an accelerating system. Before proceeding we must find the relativistic form of the momentum and energy so that
2
2
we may apply the conservation of momentum to relativistic rockets. If we multiply equation (2) by m0 c , where
m0 is the mass in the frame at rest. Then
2
2
 dt 
 dx 
m02 c 4  m02 c 4    m02 c 2   .
 d 
 d 
(10)
The first quantity on the right hand side is the relativistic energy, E, and the second is the relativistic momentum,
p1,4. Using the coefficients in equation (6), the relativistic energy and the three dimensional momentum form a four
dimensional vector with components
E  m0 c 2 ,
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(11a)
and a total momentum
pc  m0 c 2 .
(11b)
For small speeds (   1 and cosh   1   / 2 ) equation (5) yields
2
1
 1 
E  m 0 c 2 1   2   m 0 c 2  m 0 v 2 .
2
 2 
(12)
The second term is the kinetic energy and the first is the rest mass energy.
A.
Constant Acceleration Rocket
Many authors have considered relativistic rockets, (for example, in references 16 and 17), and all are based on
the conservation of momentum and energy. Using the velocity parameter the momentum is given by
p  m0 c sinh  .
(13)
Then
dp
d
 m0 c cosh 
.
d
d
In the accelerating frame
  0 and cosh   1 .
(14)
Using Newton’s second law
dp
d
 m0 c
 m0 a ,
d
d
(15)
where a is the acceleration felt in the accelerating (rocket) frame of reference. Then the acceleration is simply
given by
ac
d
.
d
(16)
Using equation (2) it follows that for all inertial reference frames
dt
 cosh 
d
.
dx
 c sinh 
d
(17)
For constant acceleration a, equation (16) can be integrated to yield

where
  0 at   0 .
a
,
c
Substituting equation (18) into equations (17) and integrating,
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(18)
c
 a 
sinh  
a
 c 
,
2
c 
 a  
x
cosh    1
a 
 c  
t
where
(19)
t  0 , and x  0 at   0 . Dividing the last of equations (17) by the first yields
dx
 c tanh  ,
dt
which reproduces equation (5). For small proper times,
(20)
 , equations (19) and (20) become
t 
a 2
.
x
2
dx
 a
d
(21)
Equations (21) are the non-relativistic expressions for constant acceleration.
Equations (18) through (20) can now be used to obtain results for a spacecraft accelerating to relativistic speeds.
Note that equation (17) says that the rocket will cover distance dx in on-board time d. Hence, when the hyperbolic
sine of the velocity parameter is greater than one the crew will experience travel at speeds greater than the speed of
light. Although, the stationary observer will note that the rocket is always moving less than the speed of light
according to equation (20), there is no contradiction. Time intervals in the stationary observer’s frame, from
equation (17), are always larger than the crew’s changes in time. The speeds can become so large in the crew’s
frame of reference that they could circumnavigate our expanding universe in less than their working lifetime if they
accelerate continuously at one earth gravity. The conclusion is that relativity does not limit the speeds that can be
achieved. Of course, the observer on the ground will be long gone, along with his stellar system, long before the
circumnavigating crew returns.
We now have enough information to resolve the twin paradox. Time in the accelerating twin’s reference frame
can now be taken as the proper time, , and the readings of the clocks on return can be predicted for each twin. The
analysis shows no paradox.
There is more in the consideration of accelerations than just a resolution of the twin paradox. The case of
constant acceleration also points to an approach that will allow the inclusion of gravitational accelerations, since in
both cases the accelerations experienced do not depend on the mass, this will lead directly to the fundamental
principle of the General Theory of Relativity.
B. Photon Rocket
We can estimate the amount of energy required by considering a photon rocket. Since a photon is a particle of
light it has no rest mass, and, hence, m0  0 in equations (11). Then equation (10) in terms of energy and
momentum becomes
m02 c 4  E 2  p 2 c 2  0 ,
(22)
and, taking the positive root of equation (22),
E  pc .
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(23)
Consider the conservation of momentum in the rocket’s frame of reference. If dE is the energy of the photons
released in proper time d, then
dp 1 dE
d
,

 mc
d c d
d
(24)
where m is the current mass of the rocket in the reference frame moving with the rocket, but the conservation of
energy yields, in the accelerating rocket frame

d E  mc 2
d
  dE

d
dm
 0.
d
 c2
(25)
Substituting for the energy in equation (25) using equation (24), and integrating
m  mi e 
,
(26)
where mi, is the mass of the rocket at =0. If the final mass occurs at
MR 

mi
e f
mf
 f
, then the mass ratio is
.
(27)
Although, for reasonable final masses, the propellant mass required to circumnavigate the universe would be less
than the mass of the universe it would nevertheless be enormous.
It should be noted that the above could be readily extended to mass annihilation rockets (i.e., antimatter
rockets)17,18, and gives reasonable estimates for the performance requirements of interstellar antimatter rockets.
Antimatter rockets require not only the application of energy and momentum conservation, but the conservation of
baryon (nucleon) number must be also applied. A baryon number of plus one applies to protons and neutrons, while
antiprotons and antineutrons have a baryon number of minus one. The conservation laws can also be extended to
the Bussard interstellar ramjet19, where interstellar matter is collected by the moving spacecraft and used as
propellant. The conservation laws are applied to the interstellar ramjet in references 20 and 21.
VI.
Conclusion
Special relativity is based on sound empirical evidence and demonstrates to an extraordinary degree of accuracy
that the speed of light is constant in inertial reference frames. It is our concept of time that has had the most
dramatic change as a result of the Special Theory of Relativity, and, again, it is with our concept of time that
produces paradoxical results. This is especially true when objects are postulated to travel faster that the speed of
light. But these faster-than-light paradoxes in the Special Theory of Relativity are more than curiosities based on
our concepts of time and space. They can point the way to a more complete understanding of space and time. They
can even add insight into other paradoxes such as the collapse of the wave function in quantum mechanics, and may
even point the way to new physical theories that may allow unlimited access to the universe.
Although the Special Theory of Relativity is applicable only in inertial reference frames, it can be extended to
accelerating spacecraft for sizing interstellar missions. Only the conservation of momentum, energy and baryon
(nucleon) number are required. Sizing, using a photon rocket as an example, will show that although practical
interstellar flight will be difficult and expensive, it is not impossible.
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Within the constraints of the Special Relativity it is clearly impossible to exceed the speed of light, and
interstellar flight will always be handicapped by long travel times to outside observers. Our only hope of reducing
the long travel times may be through the General Theory of Relativity (e.g., warp drives) and/or Quantum
Mechanics.
References
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1Einstein,
2Einstein,
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American Institute of Aeronautics and Astronautics