795 lines
45 KiB
Plaintext
795 lines
45 KiB
Plaintext
From: hinson@bohr.physics.purdue.edu (Jason W. Hinson)
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Newsgroups: rec.arts.startrek.tech
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Subject: Relativity and FTL Travel
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Summary: A detailed look at the problem
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Message-ID: <8974@dirac.physics.purdue.edu>
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Date: 18 Dec 92 21:35:54 GMT
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Organization: Purdue University Physics Department
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Lines: 785
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Finally, here it is. It is a length discussion, 794 lines by my count,
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but it is fairly complete for what I intended to do. Also, if you are
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only interested in a particular part, you can just skip the rest.
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What is it about, and who should read it:
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This is a detailed explanation about how relativity and that
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wonderful science fictional invention of faster than light travel do not
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seem to get along with each other. It begins with a simple introduction
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to the ideas of relativity. This section includes some important
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information on space-time diagrams, so if you are not familiar with
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them, I suggest you read it. Then I get into the problems that
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relativity poses for faster than light travel. If you think that there
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are many science fictional ways that we can get around these problems,
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then you probably do not understand the second problem which I discuss
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in the third section, and I strongly recommend that you read it to
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educate yourself. Finally, I introduce my idea (the only one I know of)
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that, if nothing else, gets around this second problem in an interesting
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way.
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The best way to read the article may be to make a hard copy. I
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refer back a few times to a Diagram in the first section, and to have it
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readily available would be nice.
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I hope you can learn a little something from reading this, or at
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least strengthen your understanding of that which you already know.
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Your comments and criticisms are welcome, especially if they indicate
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improvements that can be made for future posts.
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And now, without further delay, here it is.
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Relativity and FTL Travel
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Outline:
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I. An Introduction to Special Relativity
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A. Reasoning for its existence
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B. Time dilation effects
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C. Other effects on observers
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E. Space-Time Diagrams
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D. Experimental support for the theory
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II. The First Problem: The Light Speed Barrier
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A. Effects as one approaches the speed of light
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B. Conceptual ideas around this problem
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III. The Second Problem: FTL Implies The Violation of Causality
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A. What is meant here by causality, and its importance
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B. Why FTL travel of any kind implies violation of causality
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C. A scenario as "proof"
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IV. A Way Around the Second Problem
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A. Warped space as a special frame of reference
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B. How this solves the causality problem
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C. The relativity problem this produces
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D. One way around that relativity problem
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V. Conclusion.
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I. An Introduction to Special Relativity
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The main goal of this introduction is to make relativity and its
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consequences feasible to those who have not seen them before. It should
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also reinforce such ideas for those who are already somewhat familiar
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with them. This introduction will not completely follow the traditional
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way in which relativity came about. It will begin with a pre-Einstein
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view of relativity. It will then give some reasoning for why Einstein's
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view is plausible. This will lead to a discussion of some of the
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consequences this theory has, odd as they may seem. For future
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reference, it will also introduce the reader to the basics of space-time
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diagrams. Finally, I want to mention some experimental evidence that
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supports the theory.
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The idea of relativity was around in Newton's day, but it was
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incomplete. It involved transforming from one frame of reference to
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another frame which is moving with respect to the first. The
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transformation was not completely correct, but it seemed so in the realm
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of small speeds. I give here an example of this to make it clear.
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Consider two observers, you and me, for example. Lets say I am
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on a train which passes you at 30 miles per hour. I through a ball in
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the direction the train is moving, and the ball moves at 10 mph in MY
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point of view. Now consider a mark on the train tracks. You see the
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ball initially moving along at the same speed I am moving (the speed of
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the train). Then I through the ball, and before I can reach the mark on
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the track, the ball is able to reach it. So to you, the ball is moving
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even faster than I (and the train). Obviously, it seems as if the speed
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of the ball with respect to you is just the speed of the ball with
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respect to me plus the speed of me with respect to you. So, the speed
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of the ball with respect to you = 10 mph + 30 mph = 40 mph. This was
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the first, simple idea for transforming velocities from one frame of
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reference to another. In other words, this was part of the first concept
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of relativity.
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Now I introduce you to an important postulate that leads to the
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concept of relativity that we have today. I believe it will seem quite
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reasonable. I state it as it appears in a physics book by Serway: "the
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laws of physics are the same in every inertial frame of reference."
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What it means is that if you observer any physical laws for a given
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situation in your frame of reference, then an observer in a reference
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frame moving with a constant velocity with respect to you should also
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agree that those physical laws apply to that situation.
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As an example, consider the conservation of momentum. Say that
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there are two balls coming straight at one another. They collide and go
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off in opposite directions. Conservation of momentum says that if you
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add up the total momentum (mass times velocity) before the collision and
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after the collision, that the two should be identical. Now, let this
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experiment be preformed on a train where the balls are moving along the
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line of the train's motion. An outside observer would say that the
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initial and final velocities of the balls are one thing, while an
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observer on the train would say they were something different. However,
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BOTH observers must agree that the total momentum is the same before and
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after the collision. We should be able to apply this to any physical
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law. If not, (i.e. if physical laws were different for different
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frames of reference) then we could change the laws of physics just by
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traveling in a particular reference frame.
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A very interesting result occurs when you apply this postulate
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to the laws of electrodynamics. What one finds is that in order for the
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laws of electrodynamics to be the same in all inertial reference frames,
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it must be true that the speed of electromagnetic waves (such as light)
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is the same for all inertial observers. Simply stating that may not
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make you think that there is anything that interesting about it, but it
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has amazing consequences. Consider letting a beam of light take the
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place of the ball in the first example given in this introduction. If
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the train is moving at half the velocity of light, wouldn't you expect
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the light beam (which is traveling at the speed of light with respect to
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the train) to look as if it is traveling one and a half that speed with
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respect to an outside observer? Well this is not the case. The old
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ideas of relativity in Newton's day do not apply here. What accounts
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for this peculiarity is time dilation and length contraction.
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Here I give an example of how time dilation can help explain a
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peculiarity that arises from the above concept. Again we consider a
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train, but let's give it a speed of 0.6 c (where c = the speed of light
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which is 3E8 m/s). An occupant of this train shines a beam of light so
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that (to him) the beam goes straight up, hits a mirror at the top of the
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train, and bounces back to the floor of the train where it is detected.
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Now, in my point of view (outside of the train), that beam of light does
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not travel straight up and straight down, but makes an up-side-down "V"
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shape since the train is also moving. Here is a diagram of what I see:
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/|\
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/ | \
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/ | \
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light beam going up->/ | \<-light beam on return trip
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/ | \
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/ | \
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/ | \
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/ | \
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---------|---------->trains motion (v = 0.6 c)
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Lets say that the trip up takes 10 seconds in my point of view. The
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distance the train travels during that time is:
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(0.6 * 3E8 m/s) * 10 s = 18E8 m.
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The distance that the beam travels on the way up (the slanted line to
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the left) must be
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3E8 m/s * 10s = 30E8 m.
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Since the left side of the above figure is a right triangle, and we know
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the length of two of the sides, we can now solve for the height of the
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train:
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Height = [(30E8 m)^2 - (18E8 m)^2]^0.5 = 24E8 m
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(It is a tall train, but this IS just a thought experiment). Now we
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consider the frame of reference of the traveler. The light MUST travel
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at 3E8 m/s for him also, and the height of the train doesn't change
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because only lengths in the direction of motion are contracted.
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Therefore, in his frame the light will reach the top of the train in 24E8 m / 3E8 (m/s) = 8 seconds,
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and there you have it. To me the event takes 10 seconds, while
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according to him it must take only 8 seconds. We each measure time in
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different ways.
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To intensify this oddity, consider the fact that all inertial
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frames are equivalent. That is, from the traveler's point of view he is
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the one who is sitting still, while I zip past him at 0.6 c. So he will
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think that it is MY clock that is running slowly. This lends itself
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over to what seem to be paradoxes which I will not get into here. If
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you have any questions on such things (such as theJ"twin paradox" --
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which can be understood with special relativity, by the way) feel free
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to ask me about them, and I will do the best I can to answer you.
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As I mentioned above, length contraction is another consequence
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of relativity. Consider the same two travelers in our previous example,
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and let each of them hold a meter stick horizontally (so that the length
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of the stick is oriented in the direction of motion of the train). To
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the outside observer, the meter stick of the traveler on the train will
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look as if it is shorter than a meter. Similarly, the observer on the
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train will think that the meter stick of the outside observer is the one
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that is contracted. The closer one gets to the speed of light with
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respect to an observer, the shorter the stick will look to that
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observer. The factor which determines the amount of length contraction
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and time dilation is called gamma.
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Gamma is defined as (1 - v^2/c^2)^(-1/2). For our train (for
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which v = 0.6 c), gamma is 1.25. Lengths will be contracted and time
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dilated (as seen by the outside observer) by a factor of 1/gamma = 0.8,
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which is what we demonstrated with the difference in measured time (8
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seconds compared to 10 seconds). Gamma is obviously an important number
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in relativity, and it will appear as we discuss other consequences of
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the theory.
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Another consequence of relativity is a relationship between
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mass, energy, and momentum. By considering conservation of momentum and
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energy as viewed from two frames of reference, one can find that the
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following relationship must be true for an unbound particle:
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E^2 = p^2 * c^2 + m^2 * c^4
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Where E is energy, m is mass, and p is relativistic momentum which is
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defined as
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p = gamma * m * v (gamma is defined above)
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By manipulating the above equations, one can find another way to express
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the total energy as
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E = gamma * m * c^2
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Even when an object is at rest (gamma = 1) it still has an energy of
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E = m * c^2
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Many of you have seen something like this stated in context with the
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theory of relativity.
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It is important to note that the mass in the above equations has
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a special definition which we will now discuss. As a traveler approaches
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the speed of light with respect to an observer, the observer sees the
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mass of the traveler increase. (By mass, we mean the property that
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indicates (1) how much force is needed to create a certain acceleration
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and (2) how much gravitational pull you will feel from that object).
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However, the mass in the above equations is defined as the mass measured
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in the rest frame of the object. That mass is always the same. The
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mass seen by the observer (which I will call the observed mass) is given
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by gamma * m. Thus, we could also write the total energy as
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E = (observed mass) * c^2
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That observed mass approaches infinity as the object approaches the
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speed of light with respect to the observer.
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So far we talked about the major consequences of special
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relativity, but now I want to concentrate more specifically on how
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relativity causes a transformation of space and time. Relativity causes
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a little more than can be understood by simple length contraction and
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time dilation. It actually results in two different observers having
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two different space-time coordinate systems. The coordinates transform
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from one frame to the other through what are known as Lorentz
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Transformation. Without getting deep into the math, much can be
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understood about such transforms by considering space-time diagrams.
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A space-time diagram consists of a coordinate system with one
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axis to represent space and another to represent time. Where these two
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principle axes meet is the origin (see Diagram 1 below), and for the
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most part, we consider ourselves to be at that point. Anything above
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the principle space axis is in our future, while anything below that
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axis is in our past. Any event can be described as a point in this axis
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system. For example, consider an event that took place 3 seconds ago
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and was 2 light seconds (the distance light travels in 2 seconds) away
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from you to the left (x = -2 light seconds). This event is marked in
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Diagram 1 as a "*".
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Now consider a traveler going away from the origin to the right.
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As time progresses forward, the traveler gets further and further from
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the time axis. The faster he goes, the more slanted the line he makes
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will be as he is able to get far down the x axis in a short amount of
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time. One important traveler to consider here is light. If we define
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the x axis in light seconds and the time axis in seconds, then light
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will speed away from the origin creating a line at a 45 degree angle to
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the two axes. On diagram 2, I have drawn two lines which represent a
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pulse of light going away from the origin in the plus and minus x
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directions. The two pulses are extended back into the past, as if they
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started from far off, came to the origin, and sped away in the future.
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This figure is known as a light cone.
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A light cone divides a space-time diagram into two major
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sections: the area inside the cone and the area outside the cone. If it
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is impossible for anything to travel faster than light, then the only
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events in the past that you can know about at this moment are those that
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are inside the light cone. Also, the only events that you can influence
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in the future are, again, those inside the light cone.
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Let us now consider (again) an arbitrary traveler who is going
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slower than the speed of light. As a consequence of the Lorentz
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transforms that I have mentioned, the line he makes on the space-time
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diagram becomes his new time line (t'). Because of relativity, his
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space axis will also be transformed. As can be seen in Diagram 3, his
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time axis has been rotated by some angle clockwise, while his space axis
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(x') has been rotated by the same angle counterclockwise. The faster
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the speed, the greater this angle, and as you approach the speed of
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light, the two axis come closer and closer to being the same line (a
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line on the light cone which is at 45 degrees). This gives him a skewed
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set of space-time coordinates that I have tried my best to show on
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Diagram 4 (squint your eyes, and you can see the skewed squares of the
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new coordinate system). It is important to note that in this
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transformation, the position of the light cone does not change. If you
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move one unit down the space axis, and one unit up the time axis, that
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point will still lie somewhere on the light cone. This shows that the
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speed of light has not changed for the moving observer (it still travels
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one light second per second).
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Now let us compare the different ways that each observer views
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space and time. Look at the event marked "*" on Diagram 3. For the
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observer in the x',t' system, the event is in his future (above his
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principle x' axis). For the observer in the x,t system, the event is in
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his past. So how does this make since? Recall two things: (1) you can
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only know about and influence events that are inside the light cone, and
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(2) the light cone does not change for the moving observer. So even if
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an event is in one observers past and in another observers future, it
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will be outside the light cone, and neither observer will be able to
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know about it or influence it. It is the fact that nothing travels
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faster than light that causes this to be true.
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Diagram 1 Diagram 2
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t t
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| | light
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future \ inside /
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| \ cone /
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| \ | /
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| outside \ | / outside
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| cone \ | / cone
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-------------+------------- x -------------+------------- x
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| / | \
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| / | \
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event * | / | \
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| / inside \
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past / cone \
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Diagram 3
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t t'
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| /
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| / ___---> x'
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|/___---'''
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-------------+------------- x
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* ___ ---'''|
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''' / | note: * = event
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/ |
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/ |
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/ |
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Diagram 4 principle t' axis
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/
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+---------------------/-----------+
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|__---/"" / / / / __/--|
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| / / / /__--/""" / |
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| / /___-/-"""/ / / |
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|__/---"/" / / / /__--/|
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| / / / / ___/--""/ / |
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|/ / _/_---/"" / / / | ___--->principle x' axis
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|___-/-"""/ / / / __/---"""
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| / / / /__--/""" / |
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| / / ___O--""/ / / /|
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|_/_---/"" / / / /___-/-| O = Origin
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|/ / / / __/---"/" / |
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| / /__--/""" / / / |
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|___/--""/ / / / _/_---|
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| / / / /___-/-"""/ |
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+---------------------------------+
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These amazing consequences of relativity do have experimental
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foundations. One of these involves the creation of muons by cosmic rays
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in the upper atmosphere. In the rest frame of a muon, its life time is
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only about 2.2E-6 seconds. Even if the muon could traveling at the
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speed of light, it could still only go about 660 meters during its life
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time. Because of that, they should not be able to reach the surface of
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the Earth. However, it has been observed that large numbers of them do
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reach the Earth. From our point of view, time in the muons frame of
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reference is running slow, since the muons are traveling very fast with
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respect to us. So the 2.2E-6 seconds are slowed down, and the muon has
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enough time to reach the earth.
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We must also be able to explain the result from the muons frame
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of reference. So in its point of view, it does only have 2.2E-6 seconds
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to live. However, the muon would say that it is the Earth which is
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speeding toward the muon. Therefore, the distance from the top of the
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atmosphere to the Earth's surface is length contracted. Thus, from its
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point of view, it lives a very small amount of time, but it doesn't have
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that far to go.
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Another verification is found all the time in particle physics.
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The results of having a particle strike a target can only be understood
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if one takes the total energy of the particle to be E = Gamma * m * c^2,
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which was predicted by relativity.
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These are only a few examples that give credibility to the
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theory of relativity. Its predictions have turned out to be true in
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many cases, and to date, no evidence exits that would tend to undermine
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the theory.
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Well, that was a fairly lengthy look at relativity, but how does
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it all apply to faster than light travel? This is what we will look at
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next.
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II. The First Problem: The Light Speed Barrier
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In this section we discuss the first thing (and in some cases
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the only thing) that comes to mind for most people who consider the
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problem of faster than light travel. I call it the light speed barrier.
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As we will see by considering ideas from the previous section, light
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speed seems to be a giant, unreachable wall standing in our way. I also
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introduce a couple of fictional ways to get around this barrier;
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however, part of my reason for introducing these solutions is to show
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that they do not solve the problem discussed in the next section.
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Consider two observers, A and B. Let A be here on Earth and be
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considered at rest for now. B will be speeding past the A at highly
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relativistic speeds. If B's speed is 80% that of light with respect to
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A, then gamma for him (as defined in the previous section) is
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1.6666666... = 1/0.6
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So from A's point of view B's clock is running slow and B's lengths in
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the direction of motion are shorter by a factor of 0.6. If B were
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traveling at 0.9 c, then this factor becomes about 0.436; and at 0.99 c,
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it is about 0.14. As the speed gets closer and closer to the speed of
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light, A will see B's clock slow down infinitesimally slow, and A will
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see B's lengths in the direction of motion becoming infinitesimally
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small.
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In addition, If B's speed is 0.8 c with respect to A, then A
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will see B's observed mass as being larger by a factor of gamma (which
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is 1.666...). At 0.9 c and 0.99 c this factor is about 2.3 and 7.1
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respectively. As the speed gets closer and closer to me speed of light,
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A will see B's observed mass (and thus his energy) get infinitely large.
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Obviously, from A's point of view, B will not be able to reach
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the speed of light without stopping his own time, shrinking to
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nothingness in the direction of motion, and taking on an infinite amount
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of energy.
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Now lets look at the situation from B's point of view, so we
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will consider him be at rest. First, notice that the sun, the other
|
|
planets, the nearby stars, etc. are not moving very relativistically
|
|
with respect to the Earth; so we will consider all of these to be in the
|
|
same frame of reference. Let B be traveling past the earth and toward
|
|
some near by star. In his point of view, the earth, the sun, the other
|
|
star, etc. are the ones traveling at highly relativistic velocities with
|
|
respect to him. So to him the clock on Earth are running slow, the
|
|
energy of all those objects becomes greater, and the distances between
|
|
the objects in the direction of motion become smaller.
|
|
Lets consider the distance between the Earth and the star to
|
|
which B is traveling. From B's point of view, as the speed gets closer
|
|
and closer to that of light, this distance becomes infinitesimally
|
|
small. So from his point of view, he can get to the star in practically
|
|
no time. (This explains how A seems to think that B's clock is
|
|
practically stopped during the whole trip when the velocity is almost
|
|
c.) If B thinks that at the speed of light that distance shrinks to
|
|
zero and that he is able to get there instantaneously, then from his
|
|
point of view, c is the fastest possible speed.
|
|
|
|
So from either point of view, it seems that the speed of light
|
|
cannot be reached, much less exceeded. However, through some inventive
|
|
imagination, it is possible to come up with fictional ways around this
|
|
problem. Some of these solutions involve getting from point A to point
|
|
B without traveling through the intermittent space. For example,
|
|
consider a forth dimension that we can use to bend two points in our
|
|
universe closer together (sort of like connecting two points of a "two
|
|
dimensional" piece of paper by bending it through a third dimension and
|
|
touching the two points directly). Then a ship could travel between two
|
|
points without moving through the space in between, thus bypassing the
|
|
light speed barrier.
|
|
Another idea involves bending the space between the points to
|
|
make the distance between them smaller. In a way, this is what highly
|
|
relativistic traveling looks like from the point of view of the
|
|
traveler; however, we don't want the associated time transformation. So
|
|
by fictionally bending the space to cause the space distortion without
|
|
the time distortion, one can imagine getting away from the problem.
|
|
|
|
Again I remind you that these solutions only take care of the
|
|
"light speed barrier" problem. They do not solve the problem discussed
|
|
in the next section, as we shall soon see.
|
|
|
|
|
|
|
|
III. The Second Problem: FTL Implies The Violation of Causality
|
|
|
|
In this section we explore the violation of causality involved
|
|
with faster than light travel. First I will explain what we mean here
|
|
by causality and why it is important that we do not simply throw it
|
|
aside without a second thought. I will then try to explain why
|
|
traveling faster than light by any means (except the one introduced in
|
|
the next section) will produce a violation of causality. Finally,
|
|
attempting to remove any doubt, we will preform a thought experiment to
|
|
show that FTL travel does imply the violation of causality.
|
|
|
|
When I speak of causality, I have the following particular idea
|
|
in mind. Consider an event A which has an effect on another event B.
|
|
Causality would require that event B cannot in turn have an effect on
|
|
event A. For example, let's say that event A is a murderer making a
|
|
decision to shoot and kill his victim. Let's then say that event B is
|
|
the victim being shot and killed by the murderer. Causality says that
|
|
the death of the victim cannot then have any effect on the murderer's
|
|
decision. If the murderer could see his dead victim, go back in time,
|
|
and then decide not to kill him after all, then causality would be
|
|
violated. In time travel "theories," such problems are reasoned with
|
|
the use of multiple time lines and the likes; however, since we do not
|
|
want every excursion to a nearby star to create a new time line, we
|
|
would hope that FTL travel could be done without such causality
|
|
violations. As I shall now show, this is not a simple problem to get
|
|
around.
|
|
|
|
I refer you back to the diagrams in the first section so that I
|
|
can demonstrate the causality problem involved with FTL travel. In
|
|
Diagram 3, two observers are passing by one another. At the moment
|
|
represented by the principle axes shown, the two observers are right
|
|
next to one another an the origin. The x' and t' axes are said to
|
|
represent the K-prime frame of reference (I will call this Kp for
|
|
short). The x and t axes are then the K frame of reference. We define
|
|
the K system to be our rest system, while the Kp observer passes by K at
|
|
a relativistic speed. As you can see, the two observers measure space
|
|
and time in different ways. For example, consider again the event
|
|
marked "*". Cover up the x and t axis and look only at the Kp system.
|
|
In this system, the event is above the x' axis. If the Kp observer at
|
|
the origin could look left and right and see all the way down his space
|
|
axis instantaneously, then he would have to wait a while for the event
|
|
to occur. Now cover up the Kp system and look only at the K system. In
|
|
this system, the event is below the x axis. So to the observer in the K
|
|
system, the event has already occurred.
|
|
Normally, this fact gives us no trouble. If you draw a light
|
|
cone (as discussed in the first section) through the origin, then the
|
|
event will be outside of the light cone. As long as no signal can
|
|
travel faster than the speed of light, then it will be impossible for
|
|
either observer to know about or influence the event. So even though it
|
|
is in one observers past, he cannot know about it, and even though it is
|
|
in the other observers future, he cannot have an effect on it. This is
|
|
how relativity saves its own self from violating causality.
|
|
Now consider what would happen if a signal could be sent
|
|
arbitrarily fast. From K's frame of reference, the event has already
|
|
occurred. For example, say the event occurred a year ago and 5 light
|
|
years away. As long as a signal can be sent at 5 times the speed of
|
|
light, then obviously K can receive a signal from the event. However,
|
|
from Kp's frame of reference, the event is in the future. So as long as
|
|
he can send a signal sufficiently faster than light, he can get a signal
|
|
out to the place where the event will occur before it occurs. So, in
|
|
the point of view of one observer, the event can be know about. This
|
|
observer can then tell the other observer as they pass by each other.
|
|
Then the second observer can send a signal out that could change that
|
|
event. This is a violation of causality. Basically, when K receives a
|
|
signal from the event, Kp sees the signal as coming from the future.
|
|
Also, when Kp sends a signal to the event, K sees it as a signal being
|
|
sent into the past.
|
|
As a short example of this, consider the following. Instead of
|
|
sending a message out, let's say that Kp sends out a bullet that travels
|
|
faster than the speed of light. This bullet can go out and kill someone
|
|
light-years away in only a few hours (for example) in Kp's frame of
|
|
reference. Now, say he fires this bullet just as he passes by K. Then
|
|
we can call the death of the victim the event (*). Now, in K's frame of
|
|
reference, the victim is already dead when Kp passes by. This means
|
|
that the victim could have sent a signal just after he was shot that
|
|
would reach K before Kp passed by. So K can know that Kp will shoot his
|
|
gun as he passes, and K can stop Kp. But then the victim is never hit,
|
|
and he never sends a message to K. So K doesn't know to stop Kp and Kp
|
|
does shoot the bullet. Obviously, causality is not very happy about
|
|
this logical loop that develops.
|
|
|
|
If this argument hasn't convinced you, then let me try one more
|
|
thought experiment to convince you of the problem. Here, to make
|
|
calculations easy, we assume that a signal can be sent infinitely fast.
|
|
|
|
Person A is on earth, and person B speeds away from earth at a
|
|
velocity v. To make things easy, lets say that v is such that for an
|
|
observer on Earth, person B's clock runs slow by a factor of 2. now,
|
|
person A waits one hour after person B has passed earth. At that time
|
|
person A sends a message to person B which says "I just found a bomb
|
|
under my chair that will take 10 minutes to defuse, but goes off in 10
|
|
seconds ... HELP" He sends it instantaneously from his point of view...
|
|
well, from his point of view, B's clock has only moved half an hour. So
|
|
B receives the message half an hour after passing earth in his frame of
|
|
reference.
|
|
Now we must switch to B's point of view. From his point of
|
|
view, A has been speeding away from him at a velocity v. So, to B, it
|
|
is A's clock that has been running slow. Therefore, when he gets the
|
|
message half an hour after passing earth, then in his frame of
|
|
reference, A's clock has only moved 1/4 an hour. So, B sends a message
|
|
to A that says: "There's a bomb under your chair." It gets to A
|
|
instantaneously, but this time it is sent from B's frame of reference,
|
|
so instantaneously means that A gets the message only 1/4 of an hour
|
|
after B passed Earth. You see that A as received an answer to his
|
|
message before he even sent it. Obviously, there is a causality
|
|
problem, no matter how you get the message there.
|
|
OK, what about speeds grater than c but NOT instantaneous?
|
|
Whether or not you can use the above argument to find a causality
|
|
problem will depend on how fast you have B traveling. If you have a
|
|
communication travel faster than c, then you can always find a velocity
|
|
for B (v < c) such that a causality problem will occur. However, if you
|
|
send the communication at a speed that is less than c, then you cannot
|
|
create a causality problem for any velocity of B (as long as B's
|
|
velocity is also less that c).
|
|
|
|
So, it seems that if you go around traveling faster than the
|
|
speed of light, causality violations are sure to follow you around.
|
|
This causes some very real problems with logic, and I for one would like
|
|
to find a way around such problems. This next section intends to do just
|
|
that.
|
|
|
|
|
|
|
|
|
|
IV. A Way Around the Second Problem
|
|
|
|
Now we can discuss my idea for getting around the causality
|
|
problem produced by FTL travel. I will move through the development of
|
|
the idea step by step so that it is clear to the reader. I will then
|
|
explain how the idea I pose completely gets rid of causality violations.
|
|
Finally, I will discuss the one "bad" side effect of my solution which
|
|
involves the fundamentals of relativity, and I will mention how this
|
|
might not be so bad after all.
|
|
|
|
Join me now on a science fictional journey of the imagination.
|
|
Picture, if you will, a particular area of space about one square light-
|
|
year in size. Filling this area of space is a special field which is
|
|
sitting relatively stationary with respect to the earth, the sun, etc.
|
|
(By stationary, I mean relativistically speaking. That means it could
|
|
still be moving at a few hundreds of thousands of meters per second with
|
|
respect to the earth. Even at that speed, someone could travel for a
|
|
few thousand years and their clock would only be off by a day or two
|
|
from earth's clocks.) So, the field has a frame of reference that is
|
|
basically the same as ours on earth. In our science fictional future, a
|
|
way is found to manipulate the very makeup (fabric, if you will) of this
|
|
field. When this "warping" is done, it is found that the field has a
|
|
very special property. An observer inside the warped area can travel at
|
|
any speed he wishes with respect to the field, and his frame of
|
|
reference will always be the same as that of the field. In our
|
|
discussion of relativity, we saw that in normal space a traveler's frame
|
|
of reference depends on his speed with respect to the things he is
|
|
observing. However, for a traveler in this warped space, this is no
|
|
longer the case.
|
|
To help you understand this, lets look at a simple example.
|
|
Consider two ships, A and B, which start out sitting still with respect
|
|
to the special field. They are in regular space, but in the area of
|
|
space where the field exists. At some time, Ship A warps the field
|
|
around him to produce a warped space. He then travels to the edge of
|
|
the warped space at a velocity of 0.999 c with respect to ship B. That
|
|
means that if they started at one end of the field, and A traveled to
|
|
the other end of the field and dropped back into normal space, then B
|
|
says the trip took 1.001001... years. (That's 1 light-year divided by
|
|
0.999 light-years per year.) Now, if A had traveled in normal space,
|
|
then his clock would have been moving slow by a factor of 22.4 with
|
|
respect to B's clock. To observer A, the trip would have only taken
|
|
16.3 days. However, by using the special field, observer A kept the
|
|
field's frame of reference during the whole trip. So he also thinks it
|
|
took 1.001001... years to get there.
|
|
Now, let's change one thing about this field. Let the field
|
|
exist everywhere in space that we have been able to look. We are able
|
|
to detect its motion with respect to us, and have found that it still
|
|
doesn't have a very relativistic speed with respect to our galaxy and
|
|
its stars. With this, warping the field now becomes a means of travel
|
|
within all known space.
|
|
|
|
The most important reason for considering this as a means of
|
|
travel in a science fiction story is that it does preserve causality, as
|
|
I will now attempt to show. Again, I will be referring to Diagram 3 in
|
|
the first section. In order to demonstrate my point, I will be doing
|
|
two things. First, I will assume that the frame of reference of the
|
|
field (let's call it the S frame) is the same as that of the x and t
|
|
system (the K system) shown in Diagram 3. Assuming that, I will show
|
|
that the causality violation discussed in the previous section will not
|
|
occur using the new method of travel. Second, I will show that we can
|
|
instead assume that the S frame is the same as that of the x' and t'
|
|
system (the K-prime--or Kp for short--system), and again causality will
|
|
be preserved.
|
|
Before I do this, let me remind you of how the causality
|
|
violation occurred. The event (*) in the diagram will again be focussed
|
|
on to explore causality. This event is in the past of the K system, but
|
|
it is in the future of the Kp system. Since it is in the past according
|
|
to the K observer, a FTL signal could be sent from the event to the
|
|
origin where K would receive the signal. As the Kp observer passed by,
|
|
K could tell him, "Hay, here is an event that will occur x number of
|
|
light years away and t years in your future." Now we can switch over to
|
|
Kp's frame of reference. He sees a universe in which he now knows that
|
|
at some distant point an event will occur some time in the future. He
|
|
can then send a FTL signal that would get to that distant point before
|
|
the event happens. So he can influence the event, a future that he
|
|
knows must exist. That is a violation of causality. But now we have a
|
|
specific frame of reference in which any FTL travel must be done, and
|
|
this will save causality.
|
|
First, we consider what would happen if the frame of the special
|
|
field was the same as that of the K system. That means that the K
|
|
observer is sitting relatively still with respect to the field. So, in
|
|
the frame of reference of the field, the event "*" IS in the past. That
|
|
means that someone at event "*" can send a message by warping the field,
|
|
and the message will be able to get to origin. Again, the K observer
|
|
has received a signal from the event. So, again he can tell the Kp
|
|
observer about the event as the Kp observer passes by. Again, we switch
|
|
to Kp's frame of reference, and again he is in a universe in which he
|
|
now knows that at some distant point an event will occur some time in
|
|
the future. But here is where the "again's" stop. Before it was
|
|
possible for Kp to then send a signal out that would get to that distant
|
|
point before the event occurs. But NOW, to send a signal faster than
|
|
light, you must do so by warping the field, and the signal will be sent
|
|
in the field's frame of reference. But we have assumed that the field's
|
|
frame of reference is the same as K's frame, and in that frame, the
|
|
event has already occurred. So, as soon as the signal enters the warped
|
|
space, it is in a frame of reference in which the event is over with,
|
|
and it cannot get to the location of the event before it happens. What
|
|
Kp basically sees is that no matter how fast he tries to send the
|
|
signal, he can never get it to go fast enough to reach the event. In
|
|
K's frame, it is theoretically possible to send a signal
|
|
instantaneously; but in Kp's frame, that same signal would have a non-
|
|
infinite speed. So we see that under this first consideration,
|
|
causality is preserved.
|
|
To further convince you of my point, I will now consider what
|
|
would happen if the frame of the special field was the same as that of
|
|
the Kp system instead of the K system. Again, consider an observer at
|
|
the event "*" who wishes to send a signal to K before Kp passes by K.
|
|
The event of K and Kp passing one another has the position of the origin
|
|
in our diagram (as I hope you understand). In order to send this
|
|
signal, the observer at "*" must warp the field and thus enter the
|
|
system of the Kp observer. But in the frame of reference of Kp, when he
|
|
passes by K, the event "*" is in the future. Another way of saying this
|
|
is that in the Kp frame of reference, when the event "*" occurs, Kp will
|
|
have already passed K and gone off on his merry way. So when the signal
|
|
at "*" enters the warped space, it's frame of reference switches to one
|
|
in which K and Kp have already passed by one another. That means that
|
|
it is impossible for "*" to send a signal that would get to K before Kp
|
|
passes by. The possibility of creating a causality violation thus ends
|
|
here.
|
|
Let me summarize the two above scenarios. In the first
|
|
situation, K could know about the event before Kp passes. So Kp can
|
|
know about the event after he passes K, but Kp could not send a signal
|
|
that would then influence the event. In the second situation, Kp can
|
|
send a signal that would influence the event after he passed by K.
|
|
However, K could not know about the event before Kp passed, so Kp cannot
|
|
have previous knowledge of the event before he sends a signal to the
|
|
event. In either case, causality is safe. Also notice that only one
|
|
case can be true. If both cases existed at the same time, then
|
|
causality would be no safer than before. Therefore, only one special
|
|
field can exist, and using it must be the only way that FTL travel can
|
|
be done.
|
|
Many scenarios like the one above can be conceived using
|
|
different events and observers, and (under normal situations) FTL
|
|
travel/communication can be shown to violate causality. However, in all
|
|
such cases, the same types of arguments are used that I have used here,
|
|
and the causality problem is still eliminated by using the special
|
|
field.
|
|
|
|
So, is the the perfect solution where FTL travel exists without
|
|
any side effects that make it logically impossible? Does this mean that
|
|
FTL travel in Star Trek lives, and all we have to do is accept the idea
|
|
that subspace/warped space involves a special frame of reference? Well,
|
|
not quite.
|
|
You see, there is one problem with all of this which involves
|
|
the basic ideas which helped form relativity. We said that an observer
|
|
using our special mode of transportation will always have the frame of
|
|
reference of the field. This means that his frame of reference does not
|
|
change with respect to his speed, and that travel within the warped
|
|
field does not obey Einstein's Relativity. At first glance, this
|
|
doesn't seem too bad, it just sounds like good science fiction. But
|
|
what happens when you observer the outside world while in warp? To
|
|
explore this, let's first look back at why it is necessary for the frame
|
|
of reference to change with respect to speed. We had assumed that the
|
|
laws of physics don't simply change for every different inertial
|
|
observer. It had been found that if the laws of electrodynamics look
|
|
the same to all inertial observers, then the speed of an electromagnetic
|
|
wave such as light must be the same for all observers. This in turn
|
|
made it necessary for different observers to have different frames of
|
|
reference. Now, lets go backwards through this argument. If different
|
|
observers using our special mode of transportation do not have different
|
|
frames of reference, then the speed of light will not look the same to
|
|
all observers. This in turn means that if you are observing an
|
|
electromagnetic occurrence from within the warped space, the laws
|
|
governing that occurrence will look different to you that they would to
|
|
an observer in normal space.
|
|
Perhaps this is not that big of a problem. One could assume
|
|
that what you see from within warped space is not actually occurring in
|
|
real space, but is caused by the interaction between the warped space
|
|
and the real universe. The computer could then compensate for these
|
|
effects and show you on screen what is really happening. I do not,
|
|
however pretend that this is a sound explanation. This is the one part
|
|
of the discussion that I have not delved into very deeply. Perhaps I
|
|
will look further into this in the future, but it seems like science
|
|
fiction could take care of this problem.
|
|
|
|
|
|
|
|
|
|
V. Conclusion.
|
|
|
|
I have presented to you some major concepts of relativity and
|
|
the havoc they play with faster than light travel. I have show you that
|
|
the violation of causality alone is a very powerful deterrent to faster
|
|
than light travel of almost any kind. So powerful are its effects, in
|
|
fact, that I have found only one way to get around them. I hope I have
|
|
convinced you that (1) causality is indeed very hard to get around, and
|
|
(2) my idea for a special field with a particular frame of reference
|
|
does get around it. For the moment, I for one see this as the only way
|
|
I want to consider the possibility of faster than light travel. Though
|
|
I do not expect you to be so adamant about the idea, I do hope that you
|
|
see it as a definite possibility with some desirable outcomes. If
|
|
nothing else, I hope that I have at least educated you to some extent on
|
|
the problems involved when considering the effects of relativity on
|
|
faster than light travel.
|
|
|
|
|
|
|
|
Jason Hinson
|
|
|
|
-Jay
|
|
|