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Chapter 33: Problem 30

You're the communications officer on a fast spaceship that takes 50 years in ship time to reach the Andromeda Galaxy, 2 million light years from Earth in the common rest frame of Earth and Andromeda. As soon as you reach Andromeda, your captain orders you to send a radio message to Earth announcing your arrival; he claims the message will reach Earth about a century after you left. You claim it will be much later when the message arrives. Who's right?

Short Answer

Expert verified
The Communications Officer is right. More than 2 million years will have passed on Earth before the radio message reaches there due to the effects of time dilation and the time taken for the radio message to reach Earth.

Step by step solution

01

Understand the problem context

The spaceship is moving close to the speed of light and time on the spaceship is different from time on the Earth due to time dilation effect. The spaceship takes 50 years of ship time to reach Andromeda Galaxy, which is 2 million light years away without accounting time dilation.
02

Calculate the time for the spaceship to reach Andromeda from Earth's perspective

To calculate the time for the spaceship to reach Andromeda from Earth's perspective, it is necessary to take into account time dilation. As the spaceship is traveling at a relativistic speed, time on the spaceship will appear slower as viewed from Earth. The Lorentz factor is given by \(\gamma = 1/\sqrt{1 - v^2/c^2}\), where \(v\) is the velocity of the spaceship, and \(c\) is the speed of light. Since we are not given the spaceship's velocity, we cannot calculate the exact time, but we know that it will be significantly longer than the 50 years of ship time due to time dilation.
03

Establish the time taken for the radio message to reach Earth

Because the radio waves travel at the speed of light, the message will take an additional 2 million years to reach the Earth after it is sent from Andromeda.
04

Compare the Captain's and the Communications officer's claims

The Captain's statement that the message will reach Earth a century after the spaceship’s departure is not correct. Instead, it's in line with the Communications Officer's claim that the arrival of the message on Earth will happen much later, more than 2 million years after the spaceship's departure time.

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Most popular questions from this chapter

You've been named captain of NASA's first interstellar mission since the Voyager robotic spacecraft. You board your spaceship, accelerate quickly to \(0.8 c,\) and cruise at constant speed toward Proxima Centauri, the closest star to our Sun. Proxima Centauri is 4 light-years distant as measured in the two stars' common rest frame. On the way, you conduct various medical experiments to determine the effects of a long space voyage on the human body. Taking your pulse, you find a. it's significantly slower than when you're on Earth. b. it's the same as when you're on Earth. c. it's significantly faster than when you're on Earth.

Use the Lorentz transformations to show that if two events are separated in space and time so that a light signal leaving one event cannot reach the other, then there is an observer for whom the two events are simultaneous. Show that the converse is also true: If a light signal can get from one event to the other, then no observer will find them simultaneous.

Find the speed of a particle whose relativistic kinetic energy is \(50 \%\) greater than the Newtonian value calculated for the same speed.

Two spaceships are racing. The "slower" one passes Earth at \(0.70 c,\) and the "faster" one moves at \(0.40 c\) relative to the slower one. What's the faster ship's speed relative to Earth?

Find the kinetic energy of an electron moving at (a) \(0.0010 c\) (b) \(0.60 c,\) and \((\mathrm{c}) 0.99 c .\) Use suitable approximations where possible.
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