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Chapter 33: Problem 6

If you're in a spaceship moving at \(0.95 c\) relative to Earth, do you perceive time to be passing more slowly than it would on Earth? Think! Is your answer consistent with the relativity principle?

Short Answer

Expert verified
Yes, the observer on the spaceship would perceive time as passing more slowly compared to how it would pass on Earth, according to an observer on Earth. However, from their own perspective, they would see their time passing normally and see Earth's time running slower. This is consistent with the relativity principle.

Step by step solution

01

Understanding Time Dilation

Time dilation says that the time measured between two events, i.e., the time interval, appears longer for an observer in a relatively moving frame compared to an observer in the rest frame of the events. This effect only becomes noticeable as the relative velocity approaches the speed of light (c). Essentially, time appears to slow down for the moving observer as seen from the observer's rest frame.
02

Application to the Given Problem

In the given scenario, the spaceship is moving at \(0.95 c\) with respect to Earth. Therefore, according to time dilation in special relativity, an observer on Earth would see the clock on the spaceship run slower compared to an identical clock on Earth, i.e., time appears to be passing more slowly for the observer in the spaceship as seen from Earth. However, from the perspective of the person in the spaceship, their clock appears to run normally, and they would perceive the clock on Earth as running slower.
03

Validation with the Relativity Principle

The relativity principle states that the laws of physics have the same form in all admissible frames of reference. For the person in the spaceship, their frame is stationary and Earth鈥檚 frame is moving, and vice versa for someone on Earth. As such, this situation is symmetrical and both observers perceive time to be passing more slowly for the other observer, which is consistent with the relativity principle.

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Most popular questions from this chapter

The Curiosity rover touched down on Mars when Earth and Mars were 14 light- minutes apart. At the instant of touchdown, clocks at Mission Control in Pasadena, California, read 10: 31 PM. As judged by observers on a spacecraft heading along the Earth Mars line at \(0.35 c,\) did touchdown occur before or after the time the clocks in Pasadena read \(10: 31 \mathrm{PM}\), and by how much?

An extraterrestrial spacecraft whizzes through the solar system at \(0.80 c .\) How long does it take to go the 8.3 light-minute-distance from Earth to Sun (a) according to an observer on Earth and (b) according to an alien aboard the ship?

Consider a line of positive charge with line charge density \(\lambda\) as measured in a frame \(S\) at rest with respect to the charges. (a) Show that the electric field a distance \(r\) from this charged line has magnitude \(E=\lambda / 2 \pi \epsilon_{0} r,\) and that there's no magnetic field (no relativity needed here). Now consider the situation in a frame \(S^{\prime}\) moving at speed \(v\) parallel to the line of charge. (b) Show that the charge density measured in \(S^{\prime}\) is given by \(\lambda^{\prime}=\gamma \lambda\), where \(\gamma=1 / \sqrt{1-v^{2} / c^{2}},\) (c) Use the result of (b) to find the electric field in \(S^{\prime}\). since the charge is moving with respect to \(S^{\prime}\), there's a current in \(S^{\prime}\), (d) Find an expression for this current and (e) for the magnetic field it produces. Determine the values of the quantities ( \(f\) ) \(\vec{E} \cdot \vec{B}\) and \((g) E^{2}-c^{2} B^{2}\) in both reference frames, and show that these quantities are invariant. Your result gives a hint at how electric and magnetic fields transform, and demonstrates one instance of the fact that \(\vec{E} \cdot \vec{B}\) and \(E^{2}-c^{2} B^{2}\) are always invariant.

The rest energy of an electron is \(511 \mathrm{keV}\). What's the approximate speed of an electron whose total energy is \(1 \mathrm{GeV} ?\) (Note: No calculations needed!)

Two stars are 50 ly apart, measured in their common rest frame. How far apart are they to a spaceship moving between them at \(0.75 c ?\)
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