Use the Lorentz transformation equations in difference form to show that

(A) simultaneity is not an absolute concept and that

(B) a moving clock is measured to run more slowly than a clock that is at rest with respect to an observer.

Question

Use the Lorentz transformation equations in difference form to show that

(A) simultaneity is not an absolute concept and that

(B) a moving clock is measured to run more slowly than a clock that is at rest with respect to an observer.

Accepted Answer

(A) Suppose that two events are simultaneous and separated in space such that [latex] \Delta t^{\prime}=0 ext { and } \Delta x^{\prime} eq 0 [/latex] according to an observer O′ who is moving with speed v relative to O. From the expression for Δt given in Equation 39.14, we see that in this case the time interval Δt measured by observer O is [latex] \Delta t=\gamma v \Delta x^{\prime} / c^{2} [/latex] That is, the time interval for the same two events as measured by O is nonzero, and so the events do not appear to be simultaneous to O.

(B) Suppose that observer O′ carries a clock that he uses to measure a time interval Δt′. He finds that two events occur at the same place in his reference frame [latex] \left(\Delta x^{\prime}=0 ight) [/latex] but at different times [latex]\left(\Delta t^{\prime} eq 0 ight) . ext { Observer } O^{\prime}[/latex] is moving with speed v relative to O, who measures the time interval between the events to be Δt. In this situation, the expression for Δt given in Equation 39.14 becomes [latex]\Delta t=\gamma \Delta t^{\prime}[/latex]. This is the equation for time dilation found earlier (Eq. 39.7), where [latex]\Delta t^{\prime}=\Delta t_{p}[/latex] is the proper time measured by the clock carried by observer O′. Thus, O measures the moving clock to run slow.

[latex] \left.\begin{array}{l} \Delta x=\gamma\left(\Delta x^{\prime}+v \Delta t^{\prime} ight) \ \Delta t=\gamma\left(\Delta t^{\prime}+\frac{v}{c^{2}} \Delta x^{\prime} ight) \end{array} ight\} S ^{\prime} ightarrow S [/latex] (39.14).

[latex]\Delta t=\frac{\Delta t_{p}}{\sqrt{1-\frac{v^{2}}{c^{2}}}}=\gamma \Delta t_{p}[/latex] (39.7).

Question 39.6: Use the Lorentz transformation equations in difference form ...

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