11.3: First Steps
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invariance off the interval gets us started
Recall that the coordinates y and z transverse to the direction of relative motion between rocket and laboratory have the same values in both frames (Section 3.6):
y=y′z=z′
where primes denote rocket coordinates. A second step makes use of the difference in observed clock rates when the clock is at rest or in motion (Section 1.3 and Box 3-3). Think of a sparkplug at rest at the origin of a rocket frame that moves with speed vrel relative to the laboratory. The sparkplug emits a spark at time t′ as measured in the rocket frame. The sparkplug is at the rocket origin, so the spark occurs at x′=0.
Derive difference in clock rates
Where and when (x and t) does this spark occur in the laboratory? That depends on how fast, vrel , the rocket moves with respect to the laboratory. The spark must occur at the location of the sparkplug, whose position in the laboratory frame is given by
x=vrelt
Now the invariance of the interval gives us a relation between t and t′,
(t′)2−(x′)2=(t′)2−(0)2=t2−x2=t2−(vrelt)2=t2(1−v2rel)
from which
t′=t(1−v2re)1/2
or
t=t′(1−v2rel)1/2 [when x′=0 ]
The awkward expression 1/(1−v2rel )1/2 occurs often in what follows. For simplicity, this expression is given the symbol Greek lower-case gamma: γ.
γ≡1(1−v2re)1/2
Time stretch factor defined
Because it gives the ratio of observed clock rates, γ is sometimes called the time stretch factor (Section 5.8). Strictly speaking, we should use the symbol γrel , since the value of γ is determined by vrel . For simplicity, however, we omit the subscript in the hope that this will cause no confusion. With this substitution, equation 11.3.2 becomes
t=γt′ [when x′=0 ]
Substitute this into the equation x=vrel t above to find laboratory position in terms of rocket measurements:
t=γt′ [when x′=0 ]
Equations 11.3.1, 11.3.3 and 11.3.5 give the first answer to the question, "If we know the space and time coordinates of an event in one free-float frame, what are its space and time coordinates in some other overlapping free-float frame?" These equations are limited, however, since they apply only to a particular situation: one in which both events occur at the same place (x′=0) in the rocket