Relativity Lite

24 | Relativity Lite on your space too. * In our rocket moving at velocity v = 3 5 c , a watch on our astronaut in figure 1 of chapter 1 would read a time t w w = 5 4 τ to us on Earth, while a clock sitting a dis- tance x away on the floor of the spaceship—that our twin asserts is perfectly synchronized with her watch in her frame of reference—would read to us t x c v c x c c c w = = γ τ τ −     −     5 4 3 5 , a different time, unless the clock is just under the watch on her right hand, at x = 0 . Equally as strange, my measurement of space x ' (this prime is my means to distinguish my space from yours) in the direction of your motion x likewise depends on your time: x x v x c y y z z ' ( ) , , ' = − = −     = γ τ τ 5 4 3 5 '= , but perpendicular distances are unaffected. (For this reason, angles that extend into the x ' direction will be affected.) Not only does relativity force us to give up “absolute time”; it also forces us to give up the idea that “space” (a three-dimensional world that we can group together as { width , length , and height }—or, if you prefer, { north , west , and up }) and “time” (a one-dimensional marker of { duration }) are entirely separate actualities. We have to begin to perceive the universe as a four-dimensional spacetime . To use a word analogy for the math, a spacetime event that you measure as { duration , width , length , and height } may be perceived by someone else moving relative to you as { duration coupled to length , width , length coupled to duration , and height }. Just as space and time constitute a four-dimensional entity of the universe { c t , north , west , up }, so do other common quantities like momentum and energy. I had a fairly petite woman named Catherine Wong in my 2013 Freshman Inquiry class who plays rugby (figure 2). Suppose you are on the field watching her as she barrels into you heading north. What happens? (You get knocked a few feet north.) What else happens? (Some of her motional energy [kinetic energy, or KE] gets deposited into your body, which you experience as pain in your stomach.) Suppose she instead barrels into you heading west while you were still looking south. What happens? (You get knocked few feet west.) What else? (Some of her KE is deposited into your body, which you experience as pain in your side.) Suppose she instead tosses the ball high above your head and you reach up to catch it. What happens? (Your hands recoil a few inches downward.) What else happens? (Some of the ball’s KE gets deposited into your body, which you experience as stinging palms.) * ′ = − ( ) x x vt γ and ′ = − ( ) t t vx c γ 2 for motion in the x direction.