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Chapter 5: Problem 5

To whom does the elapsed time for a process seem to be longer, an observer moving relative to the process or an observer moving with the process? Which observer measures the interval of proper time?

Short Answer

Expert verified
The elapsed time for a process appears to be longer for the observer moving relative to the process (Observer A) than for the observer moving with the process (Observer B), due to the time dilation effect in special relativity. The observer measuring the interval of proper time is the one moving with the process (Observer B).

Step by step solution

01

Understand Elapsed Time and Proper Time

When dealing with special relativity, elapsed time refers to the time interval observed for an event to occur. In the context of this exercise, there are two observers: one moving relative to the process and one moving with the process. Proper time is the time interval measured by an observer who is at rest relative to the event, while the time interval measured by the moving observer is called the relativistic time.
02

Time Dilation in Special Relativity

In special relativity, time dilation is the effect that time appears to pass more slowly for an observer in motion relative to an event than for an observer at rest. This can be illustrated by the time dilation formula: \[ \Delta t = \frac{\Delta t_0}{\sqrt{1 - \frac{v^2}{c^2}}}, \] where \(\Delta t\) is the time interval measured by the moving observer (relativistic time), \(\Delta t_0\) is the proper time measured by the observer at rest relative to the event, \(v\) is the relative velocity between the event and the moving observer, and \(c\) is the speed of light.
03

Compare Elapsed Time for Different Observers

Now, let's compare the elapsed time for the two observers: the one moving relative to the process (Observer A) and the one moving with the process (Observer B). Since Observer A is in motion relative to the process, their measured elapsed time, \(\Delta t_A\), will be greater than the proper time, \(\Delta t_0\), due to time dilation. On the other hand, Observer B is moving with the process, which means they are at rest relative to the event. Therefore, Observer B will measure the proper time \(\Delta t_0\), in which case \(\Delta t_B = \Delta t_0\). Since \(\Delta t_A > \Delta t_0\) and \(\Delta t_B = \Delta t_0\), it follows that: \[ \Delta t_A > \Delta t_B.\]
04

Conclusion

So, the elapsed time for the process appears to be longer for the observer moving relative to the process (Observer A) than for the observer moving with the process (Observer B). The observer measuring the interval of proper time is the one moving with the process (Observer B).

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Most popular questions from this chapter

Consider a thought experiment. You place an expanded balloon of air on weighing scales outside in the early morning. The balloon stays on the scales and you are able to measure changes in its mass. Does the mass of the balloon change as the day progresses? Discuss the difficulties in carrying out this experiment.

(a) Calculate the speed of a 1.00 - \(\mu\) g particle of dust that has the same momentum as a proton moving at \(0.999 c\) (b) What does the small speed tell us about the mass of a proton compared to even a tiny amount of macroscopic matter?

An observer at origin of inertial frame S sees a flashbulb go off at \(x=150 \mathrm{km}, y=15.0 \mathrm{km}, \quad\) and \(z=1.00 \mathrm{km}\) at time \(t=4.5 \times 10^{-4} \mathrm{s} .\) At what time and position in the \(S^{\prime}\) system did the flash occur, if \(S^{\prime}\) is moving along shared \(x\) -direction with \(S\) at a velocity \(v=0.6 c ?\)

To whom does an object seem greater in length, an observer moving with the object or an observer moving relative to the object? Which observer measures the object's proper length?

(a) What is \(\gamma\) if \(v=0.250 c ?\) (b) If \(v=0.500 c ?\)
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