Question

An astronaut, taking his first flight on a space shuttle, brings along his favorite miniature grandfather clock. At launch, the clock and his digital wristwatch are synchronized, and the grandfather clock is pointed in the direction of the shuttle's nose. During the boost phase, the shuttle has an upward acceleration whose magnitude is several times the value of the gravitational acceleration at the Earth's surface. As the shuttle reaches a constant cruising speed after completion of the boost phase, the astronaut compares his grandfather clock's time to that of his watch. Are the timepieces still synchronized? If not, which is ahead of the other? Explain.

Short answer

Answer: No, the digital wristwatch and the miniature grandfather clock are not synchronized after the completion of the boost phase. The grandfather clock is ahead of the astronaut's digital wristwatch due to time dilation caused by the different gravitational forces experienced during the boost phase.

Step-by-step solution

Determine the effect of gravitational acceleration on the two timepieces

During the boost phase, the shuttle experiences an upward acceleration. This will cause a difference in gravitational forces between the two timepieces, with the digital wristwatch experiencing more gravitational force due to the astronaut's body shielding it from the upward acceleration. The miniature grandfather clock experiences less gravitational force.

Analyze the time dilation due to different gravitational forces

The different gravitational accelerations experienced by the wristwatch and the grandfather clock result in a phenomenon known as time dilation. According to Einstein's theory of general relativity, a clock in a stronger gravitational field runs slower compared to a clock in a weaker gravitational field. In this case, the wristwatch is in a stronger gravitational field due to the astronaut's body shielding it from the upward acceleration, causing it to run slower compared to the grandfather clock.

Compare the time shown on both timepieces after the boost phase

When the astronaut compares the time on both timepieces after the boost phase, the wristwatch will show less time than the grandfather clock since it experienced a greater gravitational force during the boost phase.

Conclusion

The digital wristwatch and the miniature grandfather clock are not synchronized after the completion of the boost phase. The grandfather clock is ahead of the astronaut's digital wristwatch. This difference in time is due to the different gravitational forces experienced by the timepieces during the boost phase, causing time dilation.

Key concepts

General Relativity

At the heart of understanding time dilation in gravity is Einstein's theory of General Relativity. Essentially, this theory revolutionized our concept of both space and time by connecting the force of gravity with the curvature of spacetime itself. According to General Relativity, massive objects cause a curvature in spacetime, which in turn determines the motion of smaller objects and light.

This curvature also affects time. The closer you are to a massive object, the more spacetime is curved around it, and the more significant the effects on time. In the context of our exercise, as the shuttle accelerates away from Earth, the difference in the curvature of spacetime experienced by the astronaut and his clocks begins to play a significant role.

Gravitational Acceleration

Gravitational acceleration is the acceleration of an object due to the force of gravity pulling it towards a massive body, like Earth. On Earth's surface, this acceleration has a magnitude of approximately 9.81 meters per second squared (\( g \text{ is about } 9.81 \frac{m}{s^2} \text{ on Earth's surface} \)). In our astronaut's scenario, the upward acceleration of the shuttle is several times that of Earth's gravitational acceleration.

Although both the digital wristwatch and the grandfather clock experience this upward acceleration during the shuttle's boost phase, their positions in the shuttle can lead to them experiencing slightly different effective gravitational accelerations. The grandfather clock, further away from the mass of the astronaut's body, would be less shielded and thus experience a different rate of time compared to the shielded wristwatch.

Time Synchronization

Time synchronization refers to the process of coordinating time measurements between different clocks. In practical terms, it's ensuring all clocks are set to the same time and run at the same rate. Initially, the astronaut's clocks are synchronized, ticking away the seconds identically.

However, when they are subject to varying gravitational forces or accelerations, such as in the shuttle scenario, this synchronization is upset. The differences in gravitational force and acceleration modify how the clocks run: if they're experiencing a stronger gravitational effect, they'll tick slower, and if they're experiencing a lesser effect, they'll tick faster compared to each other.

Affect of Gravity on Clocks

Gravity's influence on timekeeping devices, or clocks, is a direct consequence of general relativity. A clock positioned within a stronger gravitational field will indeed run at a slower pace than a clock in a weaker gravitational field. This principle is vividly demonstrated through the astronaut's wristwatch and grandfather clock.

During the boost phase of the shuttle, the clocks are subjected to different amounts of gravitational force, leading to a noticeable discrepancy in their timekeeping — termed gravitational time dilation. In this specific scenario, since the wristwatch, which is closer to the massive body of the astronaut, encounters a stronger gravitational force, it lags behind the grandfather clock after the acceleration phase is complete.