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Colorful Road to Relativity Using Occam's Razors and Evenson's Lasers
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Chapter 1(a) Space-Space views: The Lighthouses watch the Ship The animations here show a ship passing two lighthouses at a speed of 0.50c. We postulate that light always travels at a speed of c=299,792,458 meters/second or about c=3.0E8 m/s. There are two lighthouses: the main lighthouse and its partner the north lighthouse located 1 light-sec to the north. They trade light waves back and forth. The distance 1 light-sec=299,792,458m means light is supposed to take exactly one second to travel between the two lighthouses. The two lighthouse were set up as a clock system which 'ticks' every second. Each one has a photo-cell which triggers its light to blink whenever it is hit by a blink wave from the other one. So they claim that they each blink exactly once every second. Events to notice: t=0: The ship going v=-c/2 (right to left) passes the main lighthouse and the zeroth blink is emitted. t=1: The first blink is emitted by the main lighthouse and part of it heads toward the speeding ship. t=2: The first blink wave catches the ship. This is called Event 1. Also at t=2 and the lighthouse emits the second (2nd) blink.This is called Event 2. |
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Chapter 1(b) Space-Time views:
The Ship watches the Lighthouses These animations show 'disagreements' that arise because of the postulate that light always travels at a speed of c=3.0 E8 meters/sec. This postulate demands that light blink waves are perfect circles which expand equally in all directions at speed c around whatever point they are emitted. It does not matter if their source was moving when the blink was emitted, the center of each circle is always fixed once and for all at the time and location of each blink. According to the ship this means that it takes more than a second for a blink wave from one lighthouse to hit the other since the lighthouses move away from their blink centers. This means that the time between blinks is longer than one second since a lighthouse cannot blink until it is hit . Here the blink interval D is about 1.15 seconds. This is the first disagreement: the ship thinks the lighthouse blink rate is slow. Events to notice: t'=0: The lighthouses going v=+c/2 (left to right) pass the ship just as the zeroth blink is emitted. t'=1.15: The first blink is emitted by the speeding main lighthouse and part of it heads back toward the ship. t'=1.75: The first blink wave catches the ship. This is called Event 1. t'=2.30: and the lighthouse emits the second (2nd) blink. This is called Event 2. |
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Chapter 1(c) Dual view Presentations: |
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Chapter 2 Resolving Disagreements
(a)(xy-x'y')coordinate transformation Two mapmakers will give different coordinates for the same object if one of them uses a tipped coordinate axes. One gives (x,y) while the other gives (x',y') for the same point. The space-space demo shows examples of this. The transformation equations are as follows x = x' cos q - y' sin q , y = x' sin q + y' cos q if X' axis is q radians above X, and its slope relative to X is tan q . The slope ratio b:c = sin q : cos q is the tangent tan q = b/c. The other circular trig functions are cos q = 1/Ã(1+b^2/c^2), sin q=(b/c)/Ã(1+b^2/c^2) Note that x^2+y^2=x'^2+y'^2 for all points. This defines the invariant circles which determine the scale markers (tics) of the axes. (b)(xt-x't')coordinate transformation The lighthouse and the ship can resolve their differences by relating their coordinates to tipped space-time coordinate axes. The space-time demo shows examples of this. The transformation equations are as follows x = x' cosh q + y' sinh q , y = x' sinh q + y' cosh q if X' axis has slope v/c = tanh q relative to X. The slope ratio b:c = sinh q : cosh q is the tangent tanh q = b/c. Note that x^2-(ct)^2=x'^2-(ct')'^2 for all points. This defines the invarianthyperbolas which determine the scale markers (tics) of the axes. The other hyperbolic trig functions are coshq = 1/Ã(1-v^2/c^2),sinhq=(v/c)/Ã(1-v^2/c^2) |
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Chapter 3 Space-Time Views (Minkowski Diagrams)
The evolution of time on a Minkowski space-time plot is demonstrated by the animation of ship and lighthouse trajectories. With two observers there are four different ways to display history. First there are two kinds of graphs. The lighthouse would draw a square (x,ct) graph with the ship's (x',ct') graph tipped and squeezed while the ship would like a square (x',ct') graph and let the lighthouse (x,ct) graph be tipped the opposite way. Second, there are two kinds of 'timelines' or ways to separate past, present, and future. The present is shown in each animation as the boundary between the black and white regions. Black and white represent past and future, respectively. In the lighthouse graph the boundary between lighthouse past and future is a line that is horizontial and parallel to the x-axis, while the boundary betaeen ship's past and future is a line that is tipped and parallel to the x' axis. These four animations should help to read any of the possible space time graphs that are possible. However, the most important point is that any one of the graphs is sufficient to describe all the possible histories. They are just four different ways to look at the same thing. |
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Chapter 4 Ship, Barn and Butler
Length contraction and past-future paradoxes are shown in this demonstration. We imagine a ship that is too long for its barn or hangar. So it flies in going c/2 so that its contracted length will fit. A butler is provided to open and close the barn doors so that the ship is briefly 'inside'. Unfortunately, reduced budgets force us to make-do with a single butler who must go faster than light in order to do his job. The view of these superluminal shenanigans from the ship is quite remarkable: the butler appears to be a three places at one time, and there are two butlers and an anti-butler which annhilates one of them. The perfect crime? Perhaps, the butler did do it. |
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Chapter 4 (contd) Electric Current
In a current carrying wire electrons of negative (-) charge move relative to nuclei which contain an equal but opposite positive (+) charge. The nuclei are fixed in the (x,ct) frame. In a moving (x',ct') frame there will be more or less (+) than (-) depending on the direction of motion. This yields an electric attraction or repulsion to a charge in the (x',ct') frame. The force is directly proportional to the velocity v of that frame. This force is called magnetism. |
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Chapter 5 Space-Proper-Time Plots
L. C. Epstein has described a view of space-time that is an alternative to the Minkowski plot. In place of the space-time plots he uses space-proper-time or (x,ct) plots. These have the advantage of being much simpler in a number of ways. You hypothesize that everything is moving at exactly the speed of light in (x,ct) space. This is true of so-called 'moving' objects as well as 'fixed' ones. Objects that have velocity in the x-direction must reduce their rate in the ct direction so that the total velocity in (x,ct)-space is still c. This is the well known slowing of proper time for moving observers. The total (vt)^2 + (ct)^2 = (ct)^2 never changes but ct decreases as vt increases. The choice of the ordinary space and time coordinates (x and ct) depends upon the choice of observer. It could be lighthouse (x,ct), ship (x',ct'), or someone else entirely (x'',ct''). But everybody has a proper time and that is what is plotted on the c-tau (ct) axis. |
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Chapter 6 Momentum and Energy
Two-Particle Collisions Two particles constrained to travel along a line are made to collide. The collisions can be viewed as trajectories in space-time (x,ct) or as vectors in momentum and energy (cp,E) space. The rest mass values are set above or in the VaryIt panel. Two reference frames called the laboratory (LAB) and center of momentum (COM) are available in each view afforded by the buttons at the right. In the LAB one of the particles is initially at rest. In the COM the center of gravity is at rest and the total momentum is zero. The COM provides the deepest penetration for a given input energy. It is also provides the simplest visualization and analysis. |
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Chapter 7 Force and Acceleration
Gentlemen start your engines! Two ships constrained to travel along a line are made to undergo what they perceive to be constant acceleration. Their trajectories are hyperbolas in space-time (x,ct), in fact they are invariant hyperbolas. All observers must see the same (x,ct) paths since the acceleration is a constant proper quantity. You might notice that according to the lorentz frame-of-the-moment for Ship-2 the distance to the lighthouse remains frozen at one lite-second during the period of its acceleration away from the lighthouse. Only by turning off the rockets does the Ship begin to move away as seen by this measure of distance. Also, the clocks at the lighthouse (if they could be viewed in this moving frame) are frozen at t=0 during this period. Twin Paradox Twin ships, one with engines off and another blasting madly to reverse its direction, pass each other twice. The proper time between passing for the accelerating ship (Ship-1) is much less than experienced by the fixed ship (Ship-2). So does all this exercise let the occupants of Ship-1 stay younger than the couch potatos in Ship-2? Or, does time just pass more quickly when you're having fun? This is called a 'paradox' since some people thought that a different result would occur in the frame of the moving ship. ( So who is really younger?!) But, that would be an accelerated frame and thereby measurably different . This is not really much of a paradox compared to some of the other crazy things we've seen in non-inertial frames! Relativistic Chicken Rebellious youths of yore would play "Chicken" with often disasterous head-on collisions between speeding cars. Here you can imagine killing yourself a million times deader with a reltivistic version of this time honored sport. Lots of luck! |
Chapter 7 Force and Acceleration |
| Chapter 8 Per-space Per-Time Views (Relawavity Web Site) |