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Applications

t. t '. S. S '. u. 0. 0 '. x. x '. Applications. Completely symmetrical. It doesn’t matter which observer is assigned to which frame ( S or S ‘ ), as long as you get the signs right. The Lorentz transformation equations reduce to the Galilean transformation for u << c .

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Applications

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  1. t t' S S' u 0 0' x x' Applications • Completely symmetrical. It doesn’t matter which observer is assigned to which frame (S or S‘), as long as you get the signs right. • The Lorentz transformation equations reduce to the Galilean transformation for u << c. • Time dilation and length contraction follow as special cases.

  2. You have just certified my new rocket-speed record of 0.999 c from yourstationary (!) judge’s stand. Unfortunately for me, 40 ns after I pass directly in front of you, there is a malfunction on my ship and an explosion occurs 4 m from my seat toward the front. This triggers a second explosion 40 ns later, 4 m directly behind me. Because disaster is not rare in such events, your judge’s stand records everything (for the coroner’s inquest). What are the spacetime coordinates of the explosion in your reference frame? • Requires full L.T. of spacetime coordinates to your frame (S) since events are separated in space and time in my frame (S‘). • Usual coordinates choices: all motion along x and x‘, with x=x'=0 at t=t'=0.

  3. Comments • The separations in space and time between the events in S are x=x-x=89.3 m andt=t-t=299 ns. • We see that the moving clock is slow, but the simple time dilation calculation would have given the wrong time separation for these two events. • The order of these events, first, then , is preserved in the transformation between frames in this case. • It does not have to be so for a pair of events. The order of two events can be reversed in one frame compared to another, subject to rules that are based on the nature of the “interval” that separates them. In particular, if there is a causal relationship (the occurrence of  is the cause of ), then there can be no frame in which the order is reversed.

  4. How can two observers (Al and Hank) both be at the center of spheres of expanding light fronts from the same event when their positions only coincided instantaneously at t=t'=0? S' (Hank) S (Al) ?? 

  5. Relativistic Invariants Specific combination of spacetime coordinates of an event that has the same value (0) in any IRF. More generally: Very useful for making and interpreting spacetime diagrams.

  6. More on drawing spacetime diagrams: • Plot ct vs. x instead of t vs. x • No distinction between time and space coordinates! • “Light Lines” (two directions) are x= ct (at 45) S

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