From Special Relativity to Feynman Diagrams.pdf

On the other hand x c has the meaning of the time τ

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On the other hand x / c has the meaning of the time τ AB that a light ray takes to cover the distance x = x B x A ; therefore the condition will be satisfied if: t < τ AB . (1.65) When ( 1.65 ) holds one immediately finds that the velocity V of S with respect to S must satisfy the inequality: V > c t τ AB , (1.66) what is certainly possible with V < c . Having established under what condition it is possible to invert the chronological order of two events by a change of reference frame, let us now assume that the event A sends a physical signal at a velocity c > c and that this signal determines the occurrence of the event B . (For example, with reference to Fig. 1.9, the event A can be the pressing of a switch by the observer S at t A with the emission of a hypothetical signal of velocity c > c whose effect in B is the lighting of a lamp at a later time t B .) In this case the inequality ( 1.65 ) is certainly satisfied since t = x / c and c > c . We then reach the conclusion that, if a signal propagating at velocity c > c existed, that is if it were possible to transmit information at a velocity greater than c , then two causally related events in one reference frame would appear in a different frame in the inverse temporal sequence, thus violating the principle of causality , since the effect would precede its cause. Since we cannot give up the principle of causality, we conclude that: No physical signal can propagate at a velocity greater than the speed of light. Time dilation : Equation 1.60 provides the explicit transformation law for the time intervals. Consider a reference frame S in motion at a constant speed V relative to another frame S (the standard configuration of the two frames is understood) (Fig. 1.9 ). Suppose an observer in S is measuring the time lapse t = t B t A between two events A and B which occur in S at the same place but at different times, so that
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24 1 Special Relativity Fig. 1.9 Event A causing event B t B > t A (for instance two successive positions of the second hand of a clock at rest in S ). If the events occur in S along the x -axis, we then suppose x = x B x A = 0. From ( 1.57 ) it then follows that x = V t . Substituting this relation in ( 1.60 ) we find t = γ ( V ) t V c 2 x = t γ ( V ) 1 V 2 c 2 = t γ ( V ) . (1.67) We conclude that the time lapse measured in S is t = γ ( V ) t > t . (1.68) This means that if an observer at rest in the frame S measures a time interval t , an observer in S will measure a lapse t greater than t by a factor γ ( V ) . As an example, let S be a spacecraft traveling at a high velocity V relative a laboratory frame S on earth, and let time in S and S be measured by two identical clocks. Suppose the observer in S measures the rate at which the clock inside the spacecraft ticks. He notices that the clock in S runs more slowly than his, that is time on the spacecraft flows more slowly than on earth.
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