An astronaut is traveling in a space vehicle that has a...

An astronaut is traveling in a space vehicle that has a speed of 0.500$c$ relative to Earth. The astronaut measures his pulse rate at 75.0 beats per minute. Signals generated by the astronaut’s pulse are radioed to Earth when the vehicle is moving perpendicular to a line that connects the vehicle with an Earth observer. (a) What pulse rate does the Earth observer measure? (b) What would be the pulse rate if the speed of the space vehicle were increased to 0.990$c$?

Key Concepts

Special Relativity

Special relativity is the branch of physics developed by Einstein that deals with the physics of objects moving at significant fractions of the speed of light. It introduces concepts such as the constancy of the speed of light in all inertial frames and leads to effects like time dilation and length contraction, which are essential for understanding how measurements differ between relatively moving observers.

Time Dilation

Time dilation is the phenomenon where time is observed to run slower for a moving clock compared to a stationary one. This effect arises from the relative motion between observers and is quantified by the Lorentz factor. It explains why processes, such as an astronaut’s pulse rate, appear to occur at a slower rate when observed from a different inertial frame at high velocities.

Lorentz Factor

The Lorentz factor, denoted as gamma (?), is a critical component in the equations of special relativity, defined as ? = 1/?(1 - v²/c²). It quantifies the degree of time dilation, length contraction, and relativistic mass increase experienced by an object moving at velocity v relative to the speed of light c. It is central to calculating how measurements of time and space differ between reference frames.

Transverse Doppler Effect

The transverse Doppler effect is a relativistic phenomenon observed when a source moves perpendicular to the line of sight of an observer. Unlike the classical Doppler effect, which is due to relative motion along the line of sight, this effect arises purely from time dilation and results in a frequency shift that reflects the slower ticking of the moving clock as seen by the observer.

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