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Relativity and FTL Travel
by Jason W. Hinson ([email protected])
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PART I: Special Relativity
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This is PART I of the "Relativity and FTL Travel" FAQ. It
contains basic information about the theory of special relativity. In
the FTL discussion (PART IV of this FAQ), it is it is assumed that the
reader understands the concepts discussed below, while it is not
assumed that the reader has read parts II and III of this FAQ as they
are "optional reading". Therefore, if the reader is unfamiliar with
special relativity in general (and especially if the reader is
unfamiliar with space-time diagrams) then he or she should read this
part of the FAQ to understand the FTL discussion in PART IV.
For more information about this FAQ (including copyright
information and a table of contents for all parts of the FAQ), see the
"Relativity and FTL Travel--Introduction to the FAQ" portion which
should be distributed with this document.
Contents of PART I:
1. An Introduction to Special Relativity
1.1 Reasoning for its Existence
1.2 Time Dilation and Length Contraction Effects
1.3 Introducing Gamma
1.4 Energy and Momentum Considerations
1.5 Experimental Support for the Theory
2. Space-Time Diagrams
2.1 What are Space-Time Diagrams?
2.2 Basic Information About the Diagrams we will Construct
2.3 Constructing One for a "Stationary" Observer
2.4 Constructing One for a "Moving" Observer
2.5 Interchanging "Stationary" and "Moving"
2.6 "Future", "Past", and the Light Cone
1. An Introduction to Special Relativity
The main goal of this introduction is to make relativity and its
consequences feasible to those who have not seen them before. It
should also reinforce such ideas for those who are already somewhat
familiar with them. This introduction will not completely follow the
traditional way in which relativity came about. It will begin with a
pre-Einstein view of relativity. It will then give some reasoning for
why Einstein's view is plausible. This will lead to a discussion of
some of the consequences this theory has, odd as they may seem.
Finally, I want to mention some experimental evidence that supports
the theory.
1.1 Reasoning for its Existence
The idea of relativity was around in Newton's day, but it was
incomplete. It involved transforming from one frame of reference to
another frame which is moving with respect to the first. The
transformation was not completely correct, but it seemed so in the
realm of small speeds.
(NOTE: By "frame of reference" we sort of mean the "point of
view" of a particular observer. However, it doesn't necessarily mean
what the observer would actually see. Rather it involves what the
observer says is actually true from his point of view. For example,
today I might SEE a ray of light that came from an event which
occurred 1 light-year away from me. I would then say that in my frame
of reference, the event occurred one year ago--NOT today (when I see
the event)--because I DO take into account the time it took for the
signal to reach me. So, that is the way things really are in my frame
of reference, and that is the concept we will hold to when talking
about where and when something happens in some frame of reference. As
another example, when I say that event A is simultaneous with event B
in my frame of reference, I do not mean that I SEE them occur at the
same time, but rather that after time delays for the signals from the
two events are taken into account, I find that they REALLY were
simultaneous in my frame of reference. It will be important to
remember this later in our discussion.)
Now, here is an example of the Newtonian idea of transforming
from one frame of reference to another. Consider two observers, you
and me, for example. Let's say I am on a train that passes you at 30
miles per hour. I throw a ball in the direction the train is moving,
and the ball moves at 10 mph in MY point of view. Now consider a mark
on the train tracks. As I am holding the ball (before I throw it),
you see it initially moving along at the same speed I am moving (the
speed of the train). Then I throw the ball, and the ball is able to
reach the mark on the track before I do. So to you, the ball is
moving even faster than I (and the train). Obviously, it seems as if
the speed of the ball with respect to you is just the speed of the
ball with respect to me plus the speed of me with respect to you. So,
the speed of the ball with respect to you = 10 mph + 30 mph = 40 mph.
This was the first, simple idea for transforming velocities from one
frame of reference to another. It tries to explain a bit about
observations of one observer relative to another observer's
observations. In other words, this was part of the first concept of
relativity.
Now I introduce you to an important postulate that leads to the
concept of relativity that we have today. I believe it will seem
quite reasonable. I state it as it appears in a physics book by
Serway: "the laws of physics are the same in every inertial frame of
reference." What it means is that if you observe any physical laws
for a given situation in your frame of reference, then an observer in
a reference frame moving with a constant velocity with respect to you
should also agree that those physical laws apply to that situation.
As an example, consider the conservation of momentum. Say that
there are two balls coming straight at one another. They collide and
go off in opposite directions. Conservation of momentum says that if
you add up the total momentum (which for small velocities is given by
mass times velocity) before the collision and after the collision,
then the two should be identical. Now, let this experiment be
performed on a train where the balls are moving along the line of the
train's motion. An outside observer would say that the initial and
final velocities of the balls are one thing, while an observer on the
train would say they were something different. However, BOTH
observers must agree that the total momentum is conserved. One will
say that the momentum before and after were both some number, while
the other will say that the momentum before and after were both some
other number. They will disagree on what the actual numbers are, but
they will agree that the law holds. We should be able to apply this
to any physical law. If not, (i.e., if physical laws were different
for different frames of reference) then we could change the laws of
physics just by traveling in a particular reference frame.
A very interesting result occurs when you apply this postulate to
the laws of electrodynamics. What one finds is that in order for the
laws of electrodynamics to be the same in all inertial reference
frames, it must be true that the speed of electromagnetic waves (such
as light) is the same for all inertial observers. Simply stating that
may not make you think that there is anything that interesting about
it, but it has amazing consequences. Consider letting a beam of light
take the place of the ball in the first example given in this
introduction. If the train is moving at half the velocity of light,
wouldn't you expect the light beam (which is traveling at the speed of
light with respect to the train) to look as if it is traveling one and
a half that speed with respect to an outside observer? Well, this is
not the case. The old ideas of relativity in Newton's day do not
apply here. What accounts for this peculiarity is time dilation and
length contraction.
1.2 Time Dilation and Length Contraction Effects
Now, I give an example of how time dilation can help explain a
peculiarity that arises from the above concept. Again we consider a
train, but let's give it a speed of 0.6 c (where c = the speed of
light which is 3E8 m/s--3E8 means 3 times 10 to the eighth). An
occupant of this train shines a beam of light so that (to him) the
beam goes straight up, hits a mirror at the top of the train, and
bounces back to the floor of the train where some instrument detects
it. Now, in my point of view (outside the train), that beam of light
does not travel straight up and straight down, but makes an up-side-
down "V" shape because of the motion of the train. Here is a diagram
of what I see:
Diagram 1-1
/|\
/ | \
/ | \
light beam going up->/ | \<-light beam on return trip
/ | \
/ | \
/ | \
/ | \
---------|---------->train's motion (v = 0.6 c)
Let's say that the trip up takes 10 seconds in my point of view. The
distance the train travels during that time is:
(0.6 * 3E8 m/s) * 10 s = 18E8 m.
The distance that the beam travels on the way up (the slanted line to
the left) must be
3E8 m/s * 10s = 30E8 m.
Since the left side of the above figure is a right triangle, and we
know the length of two of the sides, we can now solve for the height
of the train using the Pythagorean theorem (note "^2" means "raised to
the power of two"):
Height^2 * (18E8 m)^2 = (30E8 m)^2
so
Height = [(30E8 m)^2 - (18E8 m)^2]^0.5 = 24E8 m.
(It is a tall train because we said that it took the light 10 seconds
to reach the top, but this IS just a thought experiment.) Now we
consider the frame of reference of the traveler. The light MUST
travel at 3E8 m/s for him also, and the height of the train doesn't
change because relativity contracts lengths only in the direction of
motion. Therefore, in his frame the light will reach the top of the
train in
24E8 m / 3E8 (m/s) = 8 seconds,
and there you have it. To me the event takes 10 seconds, while
according to him it must take only 8 seconds. We measure time in
different ways.
You see, to me the distance the light travels is longer than the
height of the train (see the diagram). So, the only way the observer
on the train could say that the light traveled the height of the train
while I say that the light travels a longer distance is if we either
(1) have different ideas for the speed of the light because we are in
different frames of reference, or (2) we have different ideas for the
time it takes the light to travel because we are in different frames
of reference. Now, in Newton's days, they would believe that the
former were true. The light would be no different from, say, a ball,
and observers in different frames of reference can observer different
speeds for a ball (remember our first example in this introduction).
However, with the principles of Einstein's relativity, we find that
the speed of light is unlike other speeds in that it must always be
the same regardless of your frame of reference. Thus, the second
explanation must be the case, and in my frame of reference, the clock
of the observer on the train is going slower than mine.
As I mentioned above, length contraction is another consequence
of relativity. Consider the same two travelers in our previous
example, and let each of them hold a meter stick horizontally (so that
the length of the stick is oriented in the direction of motion of the
train). To the outside observer, the meter stick of the traveler on
the train will look as if it is shorter than a meter.
Now, DON'T BE FOOLED! One of the first concepts which can get
into the mind of a newcomer to relativity involves a statement like,
"if you are moving, your clock slows down." However, the question of
which clock is REALLY running slowly (mine or the one on the train)
has NO absolute answer! It is important to remember that all motion
is relative. That is, there is no such thing as absolute motion. You
cannot say that it is the train that is absolutely moving and that I
am the one who is actually sitting still.
Have you ever had the experience of sitting in a car, noticing
that the car beside you seems to be moving backward, and then
realizing that you are "actually" moving forward. Well, the only
reason you say that "actually" you were moving forward is because you
are considering the ground to be stationary, and it was you who was
moving with respect to the ground rather than the other car. Before
you looked at the ground (or surrounding scenery) you had no way of
knowing which of you was "really" moving. Now, if you did this in
space (with space ships instead of cars), and there were no other
objects around to reference to, then what would be the difference in
saying that your space ship was the one that was moving or saying that
it was the other space ship that was moving? As long as neither of
you are undergoing an acceleration (which would mean you were not in
an inertial frame of reference) there is no absolute answer to the
question of which one of you is moving and which of you is sitting
still. You are moving with respect to him, but then again, he is
moving with respect to you. All motion is relative, and all inertial
frames are equivalent.
So what does that mean for us in this "train" example. Well,
from the point of view of the traveler on the train, he is the one who
is sitting still, while I zip past him at 0.6 c. Since he can apply
the concepts of relativity just as I can (that's the postulate of
relativity--all physical laws are the same for all inertial
observers), and in his frame of reference I am the one who is in
motion, that means that he will think that it is MY clock that is
running slowly and that MY meter sticks are length contracted.
So, there is NO absolute answer to the question of which of our
clocks is REALLY running slower than the other and which of our meter
sticks is REALLY length contracted smaller than the other. The only
way to answer this question is relative to whose frame of reference
you are considering. In my frame of reference his clock is running
slower than mine, but in his frame of reference my clock is running
slower than his. This lends itself over to what seem to be paradoxes
such as "the twin paradox". Understanding these paradoxes can be a
key to really grasping some major concepts of special relativity. The
explanation of these paradoxes will be given for the interested reader
in PART II of this FAQ.
1.3 Introducing Gamma
Now, the closer one gets to the speed of light with respect to an
observer, slower ones clock ticks and the shorter ones meter stick
will be in the frame of reference of that observer. The factor which
determines the amount of length contraction and time dilation is
called gamma.
Gamma is defined as (1 - v^2/c^2)^(-1/2). For our train (for
which v = 0.6 c in my frame of reference), gamma is 1.25 in my frame
of reference. Lengths will be contracted and time dilated (as seen by
me--the outside observer) by a factor of 1/gamma = 0.8, which is what
we demonstrated with the difference in measured times (8 seconds
compared to 10 seconds). Gamma is obviously an important number in
relativity, and it will appear as we discuss other consequences of the
theory (including the effects of special relativity on energy and
momentum considerations).
1.4 Energy and Momentum Considerations
Another consequence of relativity is a relationship between mass,
energy, and momentum. Note that velocity involves the question of how
far you go and how long it takes. Obviously, if relativity affects
the way observers view lengths and times relative to one another, one
could expect that any Newtonian concepts involving velocity might need
to be re-thought.
Consider momentum (which in Newtonian mechanics is defined as
mass times velocity). With relativity, this value is no longer
conserved in different reference frames when an interaction takes
place. The quantity that is conserved is relativistic momentum which
is defined as
p = gamma * m * v
where gamma is defined in the previous section.
By further considering conservation of momentum and energy as
viewed from two frames of reference, one can find that the following
equation must be true for the total energy of an unbound particle:
E^2 = p^2 * c^2 + m^2 * c^4
where E is energy, m is mass, and p is the relativistic momentum as
defined above.
Now, by manipulating the above equations, one can find another
way to express the total energy as
E = gamma * m * c^2
Notice that even when an object is at rest (gamma = 1) it still has an
energy of
E = m * c^2
Many of you have seen something like this stated in context with the
theory of relativity. It says that mass itself contains energy.
It is important to note that the mass in the above equations has
a special definition which we will now discuss. As a traveler
approaches the speed of light with respect to an observer, the
observer sees the "mass" of the traveler increase. (By "mass", we
mean the property that indicates (1) how much force is needed to
create a certain acceleration and (2) how much gravitational pull you
will feel from that object in Newtonian gravitation). However, the
mass in the above equations is defined as the mass measured in the
rest frame of the object. That mass is always the same. We sometimes
define the properties of mass as they are seen by the observer (I will
call these properties "observed mass") as being gamma * m. Thus, we
could also write the total energy as
E = (observed mass) * c^2
That observed mass approaches infinity as the object approaches the
speed of light with respect to the observer.
Further, note the case where the rest mass of an object is zero
(such is the case for a photon). Given the equation for the energy in
the form E = gamma*m*c^2, one might at first glance think that the
energy was zero when m = 0. However, note that massless particles
like the photon travel at the speed of light. Since gamma goes to
infinity as the velocity of an object goes to c, the equation E =
gamma*m*c^2 involves one part which goes to zero and one part which
goes to infinity, and it is not obvious what the energy would be.
However, if we use the energy equation in the form E^2 = p^2*c^2 +
m^2*c^4, then we can see that when m = 0 then the energy is given by E
= p*c. Now, a photon has a momentum (it can "slam" into particles and
change their motion, for example) which is determined by its
wavelength. It thus has an energy given by p*c, even though it has no
rest mass.
1.5 Experimental Support for the Theory
These amazing consequences of relativity do have experimental
foundations. One of these involves the creation of particles called
muons by cosmic rays (from the sun) in the upper atmosphere. In the
rest frame of a muon, its life time is only about 2.2E-6 seconds. Even
if the muon could travel at the speed of light, it could still go only
about 660 meters during its life time. Because of that, they should
not be able to reach the surface of the Earth. However, it has been
observed that large numbers of them do reach the Earth. From our
point of view, time in the muon's frame of reference is running
slowly, since the muons are traveling very fast with respect to us.
So the 2.2E-6 seconds are slowed down, and the muon has enough time to
reach the earth.
We must also be able to explain the result from the muon's frame
of reference. In its point of view, it does have only 2.2E-6 seconds
to live. However, the muon would say that it is the Earth which is
speeding toward the muon. Therefore, the distance from the top of the
atmosphere to the Earth's surface is length contracted. Thus, from
the muon's point of view, it lives a very small amount of time, but it
doesn't have that far to go. This is an interesting point of
Relativity--the physical results (e.g. the muon reaches the Earth's
surface) must be true for all observers; however, the explanation as
to how it came about can be different from different frames of
reference.
Another verification of special relativity is found all the time
in particle physics. The results of having a particle strike a target
can be understood only if one takes the total energy of the particle
to be E = Gamma * m * c^2, which was predicted by relativity.
These are only a few examples that give credibility to the theory
of relativity. Its predictions have turned out to be true in many
cases, and to date, no evidence exists that would tend to undermine
the theory in the areas where it applies.
In the above discussion of relativity's effects on space and time
we have specifically mentioned length contraction and time dilation.
However, there is a little more to it than that, and the next section
attempts to explain this to some extent.
2. Space-Time Diagrams
In this section we examine certain constructions known as space-
time diagrams. After a short look at why we need to discuss these
diagrams, I will explain what they are and what purpose they serve.
Next we will construct a space-time diagram for a particular observer.
Then, using the same techniques, we will construct a second diagram to
represent the coordinate system for a second observer who is moving
with respect to the first observer. This second diagram will show the
second observer's frame of reference with respect to the first
observer; however, we will also switch around the diagram to show what
the first observer's frame of reference looks like with respect to the
second observer. Finally, we will compare the way these two observers
view space and time, which will make it necessary to first discuss a
diagram known as a light cone.
2.1 What are Space-Time Diagrams?
In the previous section we talked about the major consequences of
special relativity, but now I want to concentrate more specifically on
how relativity causes a transformation of space and time. Relativity
causes a little more than can be understood by simple notions of
length contraction and time dilation. It actually results in two
different observers having two different space-time coordinate
systems. The coordinates transform from one frame to the other
through what is known as a Lorentz Transformation. Without getting
deep into the math, much can be understood about such transforms by
considering space-time diagrams.
A space-time diagram gives us a means of representing events
which occur at different locations and at different times. Every
event is portrayed as a point somewhere on the space-time diagram.
Because of relativity, different observers which are moving relative
to one another will have different coordinates for any given event.
However, with space-time diagrams, we can picture these different
coordinate systems on the same diagram, and this allows us to
understand how they are related to one another.
2.2 Basic Information About the Diagrams we will Construct
In the diagrams we will be using, only one direction in space
will be considered--the x direction. So, the space-time diagram
consists of a coordinate system with one axis to represent space (the
x direction) and another to represent time. Where these two principle
axes meet is the origin. This is simply a point in space that we have
defined as x = 0 and a moment in time that we have defined as t = 0.
In Diagram 2-1 (below) I have drawn these two axes and marked the
origin with an o.
For certain reasons we want to define the units that we will use
for distances and times in a very specific way. Let's define the unit
for time to be the second. This means that moving one unit up the
time axis will represent waiting one second of time. We then want to
define the unit for distance to be a light second (the distance light
travels in one second). So if you move one unit to the right on the x
axis, you will be looking at a point in space that is one light second
away from your previous location. In Diagram 2-1, I have marked the
locations of the different space and time units (Note: In my diagrams,
I am using four spaces to be one unit along the x axis and two
character heights to be one unit on the time axis).
With these units, it is interesting to note how a beam of light
is represented in our diagram. Consider a beam of light leaving the
origin and traveling to the right. One second later, it will have
traveled one light second away. Two seconds after it leaves it will
have traveled two light seconds away, and so on. So a beam of light
will always make a line at an angle of 45 degrees to the x and t axes.
I have drawn such a light beam in Diagram 2-2.
Diagram 2-1 Diagram 2-2
t t
^ ^
| | light
+ + /
| | /
+ + /
| | /
-+---+---o---+---+---> x -+---+---o---+---+-> x
| |
+ +
| |
+ +
| |
2.3 Constructing One for a "Stationary" Observer
At this point, we want to decide exactly how to represent events
on this coordinate system for a particular observer. First note that
it is convenient to think of any particular space-time diagram as
being specifically drawn for one particular observer. For Diagram 2-
1, that particular observer (let's call him the O observer) is the one
whose coordinate system has the vertical time axis and horizontal
space axis shown in that diagram. Now, other frames of reference
(which don't follow those axes) can also be represented on this same
diagram (as we will see). However, because we are used to seeing
coordinate systems with horizontal and vertical axes, it is natural to
think of this space-time diagram as being drawn specifically with the
O observer in mind. In fact, we could say that in this space-time
diagram, the O observer is considered to be "at rest".
So if the O observer starts at the origin, then one second later
he is still at x = 0 (because he isn't moving in this coordinate
system). Two seconds later he is still at x = 0, etc. So, he is
always on the time axis in our representation. Similarly, any lines
drawn parallel to the t axis (in this case, vertical lines) will
represent lines of constant position. If a second observer is not
moving with respect to the first, and this second observer starts at a
position two light seconds away to the right of the first, then as
time progresses he will stay on the vertical line that runs through x
= 2.
Next we want to figure out how to represent lines of constant
time. To do this, we should first find a point on our diagram that
represents an event which occurs at the same time as the origin (t =
0). To do this we will use a method that Einstein used. First we
choose a point on the t axis which occurred prior to t = 0. Let's use
an example where this point occurs at t = -3 seconds. At that time we
send out a beam of light in the positive x direction. If the beam
bounces off of a distant mirror at t = 0 and heads back toward the t
axis, then it will come back to the us at t = 3 seconds (because we
know that the light must travel at the same speed going as it does
coming back). So, if we send out a beam at t = -3 seconds and it
returns at t = 3 seconds, then the event of it bouncing off the mirror
occurred simultaneously with the time t = 0 at the origin.
To use this in our diagram, we first pick two points on the t
axis that mark t = -3 and t = 3 (let's call these points A and B
respectively). We then draw one light beam leaving from A in the
positive x direction. Next we draw a light beam coming to B in the
negative x direction. Where these two beams meet (let's call this
point C) marks the point where the original beam bounces off the
mirror. Thus the event marked by C is simultaneous with t = 0 (the
origin). A line drawn through C and o will thus be a line of constant
time. All lines parallel to this line will also be lines of constant
time. So any two events that lie along one of these lines occur at
the same time in this frame of reference. I have drawn this procedure
in Diagram 2-3, and you can see that the x axis is the line through
both o and C which is a line of simultaneity (as one might have
expected).
Now, by constructing a set of simultaneous time lines and
simultaneous position lines we will have a grid on our space-time
diagram. Any event has a specific location on the grid which tells
where and when it occurs. In Diagram 2-4 I have drawn one of these
grids and marked an event (@) that occurred 3 light seconds away to
the left of the origin (x = -3) and 1 second before the origin
(t = -1).
Diagram 2-3 Diagram 2-4
t t
| | | | | | |
B ---+---+---+---+---+---+---
| \ | | | | | |
+ \ ---+---+---+---+---+---+---
| \ | | | | | |
+ \ ---+---+---+---o---+---+--- x
| \ | | | | | |
-+---+---o---+---+---C- x ---@---+---+---+---+---+---
| / | | | | | |
+ / ---+---+---+---+---+---+---
| / | | | | | |
+ /
| /
A
|
2.4 Constructing One for a "Moving" Observer
Now comes an important addition to our discussion of space-time
diagrams. The coordinate system we have drawn will work fine for any
observer who is not moving with respect to the O observer. Now we
want to construct a coordinate system for an observer who IS traveling
with respect to the O observer. The trajectories of two such
observers have been drawn in Diagrams 2-5 and 2-6. Notice that in our
discussion we will always consider moving observers who pass by the O
observer at the time t = 0 and at the position x = 0. Thus, the
origin will mark the event "the two observers pass by one another".
Now, the traveler in Diagram 2-5 is moving slower than the one in
Diagram 2-6. You can see this because in a given amount of time, the
Diagram 2-6 traveler has moved further away from the time axis than
the Diagram 2-5 traveler. So the faster a traveler moves, the more
slanted this line becomes.
Diagram 2-5 Diagram 2-6
t t
| / | /
+ + /
| / | /
+ + /
|` |/
-+---+---o---+---+--- x -+---+---o---+---+- x
,| /|
+ / +
/ | / |
+ / +
/ | / |
What does this line actually represent? Well, consider an object
sitting on this line, right next to our moving observer. If a few
seconds later the object is still sitting on that line (right next to
him), then in his point of view, the object has not moved. The line
is a line of constant ps the same thing for the moving observer as the t axis
represented for the O observer; and in fact, this line becomes the
moving observer's new time axis. We will mark this new time axis as
t' (t-prime). All lines parallel to this slanted line will also be
lines of constant position for our moving observer.
Now, just as we did for the O observer, we want to construct
lines of constant time for our traveling observer. To do this, we
will use the same method that we did for the O observer. The moving
observer will send out a light beam at some time t' = -T, and the beam
will bounce off some mirror so that it returns at time t' = +T. Then
the point at which the beam bounces off the mirror will be
simultaneous with the origin, where t' = t = 0.
There is a very important point to note here. What if instead of
light, we wanted to throw a ball at 0.5 c, have it bounce off some
wall, and then return at the same speed (0.5 c). The problem with
this is that to find a line of constant time for the moving observer,
then the ball must travel at 0.5 c both ways in the reference frame of
the MOVING observer. But we have not yet defined the coordinate
system for the moving observer, so we do not know what a ball moving
at 0.5 c with respect to him will look like on our diagram. However,
because of relativity, we know that the speed of light itself CANNOT
change from one observer to the next. In that case, a beam of light
traveling at c in the frame of the moving observer will also be
traveling at c for the O observer. So, a line which makes a 45 degree
angle with respect to the x and t axes will ALWAYS represent a beam
of t t t'
| / | /
+ B' + /
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