An astronaut is traveling in a space vehicle moving at 0.500...

An astronaut is traveling in a space vehicle moving at 0.500 c relative to the Earth. The astronaut measures her pulse rate at 75.0 beats per minute. Signals generated by the astronaut's pulse are radioed to the Earth when the vehicle is moving in a direction perpendicular to the line that connects the vehicle with an observer on the Earth. (a) What pulse rate does the Earth-based observer measure? (b) What If? 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 framework for understanding how measurements of time and space differ between observers in uniform relative motion. It introduces the idea that the laws of physics are the same in all inertial frames and fundamentally alters classical perceptions of time and space, leading to phenomena such as time dilation and length contraction.

Time Dilation

Time dilation is a consequence of Special Relativity in which a moving clock, as observed from a stationary frame, appears to tick slower than a clock at rest. This concept explains why time intervals measured in one frame (proper time) can differ from those measured in another frame moving at a relative velocity, and it is crucial in understanding how the astronaut's pulse rate is perceived differently by an observer on Earth.

Lorentz Factor

The Lorentz factor, often denoted by ? (gamma), 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 by the equation ? = 1/?(1 - v²/c²), where v is the relative speed and c is the speed of light. This factor is central to calculating how time intervals, such as the spacing of the pulse signals, transform between different inertial frames.

Transverse Doppler Effect

The transverse Doppler effect is a relativistic phenomenon that occurs when a source of periodic signals is moving perpendicular to the line of sight of an observer. Despite the absence of a traditional Doppler shift due to the lack of relative radial motion, time dilation still causes the frequency of the signals (such as the astronaut's pulse rate) to appear reduced when observed from another inertial frame.

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