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CHAPTER 3. LORENTZ TRANSFORMATIONS 18 These are the Lorentz transformation. Note that these two equations reduce to Eq. (3.1) and Eq. (3.2) if c → ∞. One can easily check that the inverse transformation is x ′ + vt ′ x = (3.15) √1 − v 2 /c 2 y = y ′ (3.16) z = z ′ (3.17) t = t′ + vx ′ /c √1 2 − v 2 /c . 2 (3.18) Sometime, we write the Lorentz transformation in matrix form ⎛ ⎛ x ′ ⎞ √ 1 0 0 − v ⎞ √ ⎛ ⎞ 1−β y ′ 2 1−β 2 x ⎜ ⎝ z ′ ⎟ ⎠ = 0 1 0 0 y ⎜ 0 0 1 0 ⎜ ⎟ ⎟ ⎝ z ⎠ t ′ ⎝ 1 ⎠ 0 0 √ t √1−β 2 − v/c2 1−β 2 (3.19) where β = v/c. We illustrate some of the effects that we have discussed by the Lorentz transformation. The time separation in the moving system between two events (x ′ , t ′ ) = (x ′ , t ′ 1) and (x ′ , t ′ ) = (x ′ , t ′ 2) is, of course, t ′ 2 −t ′ 1, if we assume that t ′ 2 > t′ 1 . The time separation between the two events in the stationary system is t 2 − t 1 = t′ 2 + ux ′ /c 2 √ 1 − u 2 /c − t′ 1 + ux ′ /c 2 √ 2 1 − u 2 /c 2 = t ′ 2 − t′ 1 √ 1 − u 2 /c 2 > t ′ 2 − t′ 1 . (3.20) This is time dilation. Similarly, the distance between two ends, (x ′ 1 , t′ ) and (x ′ 2 , t′ ), of a ruler in the moving system is x ′ 2 − x′ 1 . While the coordinates of one end of the ruler in the stationary system is x 1 = x ′ 1 + vt′ √1 − v 2 /c 2 (3.21) t = t′ + vx ′ 1 /c2 √1 − v 2 /c 2 . (3.22) √ Hence, we have t ′ = t 1 − v 2 /c 2 − vx ′ 1 /c2 , and the trajectory of one end is x 1 = 1 √ √ √1 − v 2 /c 2 (x′ 1+vt 1 − v 2 /c 2 −v 2 x ′ 1/c 2 ) = x ′ 1 1 − v 2 /c 2 +vt . (3.23)
CHAPTER 3. LORENTZ TRANSFORMATIONS 19 √ The trajectory of another end is x 2 = x ′ 2 1 − v 2 /c 2 + vt. The distance in the stationary system is √ x 2 − x 1 = (x ′ 2 − x ′ 1) 1 − u 2 /c 2 < x ′ 2 − x′ 1 . (3.24) This is length contraction. In Newtonian mechanics, distance between two points does not change with observers. In special relativity, this is not true anymore. Let two events be labeled by subscripts 1 and 2, and ∆t ′ ≡ t ′ 2 − t′ 1 , etc. Then, c 2 (∆t ′ ) 2 − (∆x ′ ) 2 = c 2 (t ′ 2 − t′ 1 )2 − (x ′ 2 − x′ 1 ⎛ )2 ⎞ = c 2 ⎝ t 2 − t 1 − u/c 2 (x 2 − x 1 ) √ ⎠ 1 − u 2 /c 2 2 − ⎛ ⎞ ⎝ x 2 − x 1 − u(t 2 − t 1 ) √ ⎠ 1 − u 2 /c 2 = c2 (∆t) 2 − 2u∆t∆x + u 2 /c 2 (∆x) 2 − ((∆x) 2 − 2u∆x∆t + u 2 (∆t) 2 ) 1 − u 2 /c 2 = (c2 − u 2 )(∆t) 2 − (1 − u 2 /c 2 )(∆x) 2 1 − u 2 /c 2 = c 2 (∆t) 2 − (∆x) 2 . (3.25) This is called the invariant interval between two events, because the value does not change with observers. • If ∆s 2 ≡ c 2 (∆t) 2 − (∆x) 2 > 0, the two events are time-like related; • if ∆s 2 = 0, they are light-like related; • if ∆s 2 < 0, they are space-like related. (For three dimensional space, ∆s 2 ≡ c 2 (∆t) 2 − (∆x) 2 − (∆y) 2 − (∆z) 2 .) Very often, we compare one event with the origin of the frame. Referring to Fig. 3.1, all events in the upper or lower wedges in the diagram are timelike related to the origin. All events in the left or right wedges are space-like related to the origin. All events on the two diagonal lines are light-like related to the origin. Their significance will be further discussed in the next section. 3.2 Simultaneity How can we say that two events far apart from each other happen at the same time? Since the speed of light is constant, we can send light pulses from the middle point to the two events. If the two light pulses arrive at the same time with the events, they happen simultaneously, Fig. 3.2. 2
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