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Chapter 33: Problem 39

You're writing a galactic history involving two civilizations that evolve on opposite sides of a \(10^{5}\) -ly-diameter galaxy. In the galaxy's reference frame, civilization B launched its first spacecraft 50,000 years after civilization A. You and your readers, from a more advanced civilization, are traveling through the galaxy at \(0.99 c\) on a line from A to B. Which civilization do you record as having first achieved interstellar travel, and how much in advance of the other?

Short Answer

Expert verified
Civilization A is recorded as the first to achieve interstellar travel, and depending on the calculated time dilation in years, they achieved this ahead of Civilization B.

Step by step solution

01

Calculation of the Lorentz Factor

The Lorentz factor \(\gamma\) accounts for the effects of time dilation in special relativity and is calculated as: \[\gamma = 1/ \sqrt{1 - v^2/c^2} \]where \(v\) is the relative speed of the observer to the stationary frame, which in this case is \(0.99c\). Thus substitute \(v=0.99c\) into the equation to find \(\gamma\).
02

Calculation of Time Dilation

The time dilation formula in special relativity is \[\Delta t' = \gamma \Delta t\]The time period of \(50,000\) years is denoted by \(\Delta t\). Now, calculate for \(\Delta t'\) by substituting the value of \(\Delta t\) and \(\gamma\) from Step 1 into the equation.
03

Comparison of Achieved Space Travel

Now the value of \(\Delta t'\) which should be obtained greater than the initial \(50,000\) years, implies that in the frame of the traveling observer, Civilization B launched its spacecraft \(\Delta t'\) years after Civilization A. Thus, according to the traveler's perspective, it can be inferred that Civilization A first achieved interstellar travel.

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