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Relativistic Coordinates-Classic Approach 4.A.1 Appendix 4.A Relativistic Space-time Coordinates The nature of space-time coordinate transformation will be described here using a fictional spaceship traveling at half the speed of light past two lighthouses. In Fig. 4.A.1 the ship is just passing the Main Lighthouse as it blinks in response to a signal from the North lighthouse located at one light second (about 186,000 miles or EXACTLY 299,792,458 meters) above Main. (Such exactitude is the result of 1970-80 work by Ken Evenson's lab at NIST (National Institute of Standards and Technology in Boulder) and adopted by International Standards Committee in 1984.) Now the speed of light c is a constant by civil law as well as physical law! This came about because time and frequency measurement became so much more precise than distance measurement that it was decided to define the meter in terms of c. Fig. 4.A.1 Ship passing Main Lighthouse as it blinks at t=0. This arrangement is a simplified model for a 1Hz laser resonator. The two lighthouses use each other to maintain a strict one-second time period between blinks. And, strict it must be to do relativistic timing. (Even stricter than NIST is the universal agency BIGANN or Bureau of Intergalactic Aids to Navigation at Night.) The simulations shown here are done using RelativIt. Relativistic Coordinates-Classic Approach 4.A.2 Fig. 4.A.2 Main and North Lighthouses blink each other at precisely t=1. At p recisel y t=1 sec. the two lighthouses blink again because that is how long it takes their respective t=0 blink waves to reach each other. This is shown in Fig. 4.A.2. The ship, meanwhile, has only traveled half this far since its speed is c/2. Its velocity is -c/2, that is, negative , since it is going right to left. Next, at p recisel y t=2 sec. the two lighthouses blink again. Also, the first (t=1) blink catches up to the ship and hits it, that is, the ship sees the first blink. This is shown in Fig. 4.A.3. Much of the discussion will center on two happenings or events labeled Happening-1 and Happening-2. ("Event" is accepted physics terminology. "Happening" is oh-so-60's.) The coordinates of Happening-1 are, according to the Lighthouses, (x1=-1, ct1=2) while for Happening-2 they are, according to the Lighthouses, (x2=0, ct2=2). Next, we will see how the ship views all this, that is what are ship coordinates (x1, ct1) and (x2, ct2) for the events. From that we deduce the essential transformation matrix for all events in special relativity. The ship has a very different transcription of these events as shown in the following figures beginning with Fig. 4.A.4. Relativistic Coordinates-Classic Approach 4.A.3 Fig. 4.A.3 Happening 1 (1st blink hits ship) and 2 (2nd blink at Main) both happen at t=2. Relativistic Coordinates-Classic Approach 4.A.4 Fig. 4.A.4 Beginning (t=0) snapshot for ship's view. Now in Fig. 4.A.4, the ship is stationary and that means that the lighthouses are going in the opposite direction with a positive velocity of v=+c/2. Fig. 4.A.4 looks the same as Fig. 4.A.1 except the previous (t=-1) blink wave appears to have been "left behind" by the speeding lighthouses. Therein lies a secret of relativity. Snapshots of light pulses always appear to be circles expanding around the points where they were emitted. This is true no matter how fast you are going, or, more importantly, no matter how fast the emitter is going. You cannot speed up or slow down light by jerking your laser back and forth! So Fig. 4.A.4 and several subsequent figures show previous blink waves expanding around points where the lighthouses were when that light was emitted and all expansions take place at a uniform speed of c. It's the law! And, it's one we can live with. Consequences of this law are quite remarkable. We explore consequences shortly including the fact that the ship sees the North blink-wave tipped by a so-called stellar aberration angle =60° relative to a vertical North-to-South wave ray track seen by the Main Lighthouse. Relativistic Coordinates-Classic Approach 4.A.5 Fig. 4.A.5 Early (t=0.5) snapshot from ship's view. The (t =0.5) view by ship shows the (t =0) blink waves expanded to exactly half the distance between their emission points. Also, the lighthouses have moved half this distance, that is, a quarter of a light- second, and so Main will not be anywhere near the ship at (t =1.0) when the (t =0) blink wave from the North comes down to trigger Main to do its first or (t=1) blink. In fact it’s (t =1.15) before the (t =0) blink from the North finally catches the speeding Main Light to make it blink as shown in the next Fig. 4.A.6. Relativistic Coordinates-Classic Approach 4.A.6 Fig. 4.A.6 Later (t =1.15) snapshot from ship's view finally registers the first lighthouse blinks. Relativistic Coordinates-Classic Approach 4.A.7 Fig. 4.A.7 Later (t =1.75) snapshot from ship's first registers Happening-1. This shows Einstein time dilation. The ship perceives that the lighthouse is running about 15% late at this speed of v=c/2. The next Figs. 4.A.7 and 4.A.8 show something even more surprising to a Newtonian worldview, the relativity of simultaneity where, unlike Fig. 4.A.3, Happening-1 is not simultaneous with Happening-2. Happening-1 (ship hit by 1st blink) happens early at (t =1.75) and before Happening-2 (2nd blink) that occurs at (t =2.30). (Recall Main cannot blink until a blink from the North hits it so Happening-2 doesn't happen until (t =2.30), or twice the time (t =1.15) for the first blink as shown in Fig. 4.A.8.) Relativistic Coordinates-Classic Approach 4.A.8 Fig. 4.A.8 Much later (t =2.30) snapshot from ship's finally registers Happening-2 . Relativistic Coordinates-Classic Approach 4.A.9 The time observed between unit-time blinks by a moving ship is called the Einstein dilation factor . Its classical derivation follows from a simple right triangle whose altitude is c or one light second as shown in Fig. 4.A.9. The triangle base v is the distance traveled by the lighthouse before the North blink wave finally hits it after traveling a distance c along the hypotenuse as seen by ship. This gives the following. 2 2 2 2 2 2 2 2 2 1 c = c + v or: c v = c or: = (4.A.1) () v2 1 c2 Note that the ship or any co-moving ship sees the c hypotenuse ray tipped in the direction of travel by the Stellar aberration angle whose sine is sin = v/c. This is =60° for v/c=1/2 in Fig. 4.A.9 or Fig. 4.A.4. Fig. 4.A.9 Derivation of Einstein time dilation factor or time between blinks . For the above the lighthouse velocity relative to the ship is v=c/2. (4.A.1) gives a time dilation factor of =1/0.75 = 1.1547 very close to the 15% "lateness" in the Fig. 4.A.6 simulation. This lateness grows rapidly and without limit as v approaches c. For v=4c/5, (4.A.1) gives =5/3 = 1.67 that is a 67% lateness or dilation. From this we construct an event table to summarize discrepancies or disagreements between space and time coordinates used by the lighthouses and those used by the ship. This is shown below. Relativistic Coordinates-Classic Approach 4.A.10 Happening 0: Happening 1: Ship gets hit by Happening 2: Main Lighthouse Ship passes Main Lighthouse. first blink from Main Lighthouse. blinks second time. (Lighthouse space) x = 0 x = - 1.00 c x = 0 (Lighthouse time) t = 0 t = 2.00 t = 2.00 (Ship space) x= 0 x= 0 x= c (Ship time) t= 0 t= 1.75 t= 2 = 2.30 One of the most important things to remember about the space coordinate x is that each observer frame carries its own origin (x=0) with it wherever it goes. If a 'Happening' happens to the Lighthouse then it happens at x=0, but if it happens to the ship then it happens at x =0 no matter what the time is. Remembering this saves lots of confusion! Note also: the table above is for a positive lighthouse velocity: v=c/2 relative to the ship. You must always give velocity as one thing relative to another. Absolute velocity is meaningless. We need a table like the one above for the case of a general velocity v of the lighthouse relative to the ship. (Note that if we base ourselves in the frame in which the ship is stationary then the lighthouse moves with a positive velocity v=c/2.) The zero entries stay the same for any value of v. The times for the second blink are t=2 and t = 2 by definition. The ship's reading for the position of the second blink has to be velocity times travel time or v times 2.
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