A spaceship is traveling away from Earth at 0.87 c . The...

A spaceship is traveling away from Earth at $0.87 c .$ The astronauts report home by radio every 12 h (by their own clocks). (a) At what interval are the reports sent to Earth, according to Earth clocks? (b) At what interval are the reports received by Earth observers, according to their own clocks?

Key Concepts

Lorentz Factor

The Lorentz factor (often denoted as ?) arises in special relativity and quantifies the amount of time dilation (and length contraction) experienced by an object moving at a significant fraction of the speed of light. It is defined as 1/?(1 - v²/c²) and is central to converting proper time intervals to those measured by other inertial observers.

Relativistic Doppler Effect

The relativistic Doppler effect describes the change in the frequency and period of signals, such as electromagnetic radiation, due to the relative motion between the source and the observer. In scenarios where the source is moving away, the signals are received at a lower frequency (or equivalently, at longer intervals) than when they were emitted.

Proper Time

Proper time is the time interval measured by an observer for whom the events occur at the same location, typically the clock carried by that observer. It is an invariant quantity in relativity, serving as the reference time against which time dilation effects are measured.

Time Dilation

Time dilation is the relativistic effect where a clock moving relative to an observer ticks slower than one at rest with respect to that observer. This phenomenon is quantified by the Lorentz factor and accounts for the difference between proper time intervals and those measured in another inertial frame.

Signal Propagation Delay

Signal propagation delay refers to the finite time it takes for a signal, such as a radio wave, to traverse the distance between the source and the observer. In relativistic contexts, this delay must be considered along with time dilation when determining the intervals at which signals are sent and then received by observers in different inertial frames.

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