Marissa's spaceship approaches Joseph's at a speed of 0.99...

Marissa's spaceship approaches Joseph's at a speed of 0.99 c. As Marissa passes Joseph, they synchronize their clocks to both read $t=0$ s. When Marissa's clock reads 100 s, she sends a light signal back to Joseph. According to his clock, when does he receive this signal?

Key Concepts

Inertial Reference Frames

In special relativity, different observers who are moving at constant velocities relative to one another constitute inertial reference frames. These frames are central to the analysis of the problem because physical events may be perceived differently (in terms of space, time, and simultaneity) by observers in separate frames, such as Marissa in the moving spaceship and Joseph at rest relative to her passage.

Lorentz Transformation

The Lorentz transformation equations are the mathematical framework that relates the time and space coordinates of events as seen in different inertial reference frames. They are essential for calculating how clocks and distances are observed to change when objects move at speeds close to the speed of light, as in the relative motion between Marissa and Joseph.

Time Dilation

Time dilation is a fundamental phenomenon predicted by special relativity, where a moving clock is observed to tick slower than a stationary one. In this scenario, Marissa’s clock (in her spaceship) runs differently compared to Joseph’s clock, which is critical for analyzing the timing of events such as the emission of the light signal.

Relativity of Simultaneity

The relativity of simultaneity is the concept that events which are simultaneous in one frame of reference may not be simultaneous in another moving frame. This plays a key role in the problem when analyzing how the synchronization of clocks is perceived by Joseph as opposed to Marissa, affecting the interpretation of when the signal is sent and subsequently received.

Invariance of the Speed of Light

A cornerstone of special relativity is that the speed of light in a vacuum is constant (c) in all inertial reference frames regardless of the motion of the source or observer. This principle is crucial in determining the travel time of the light signal between Marissa and Joseph, ensuring that the calculation remains consistent across different frames.

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