The Earth is traveling at about 30 km / sec as it orbits the Sun. a. How much more slowly does t

step 1 Question number 10, we know that sun takes 25 days relative to the stars for its rotation. But w

The Earth is traveling at about 30 km / sec as it orbits the...

The Earth is traveling at about $30 \mathrm{km} / \mathrm{sec}$ as it orbits the Sun. a. How much more slowly does time run on the Earth than in the Sun's rest frame? b. Why will the time be measured as slower on the Earth than in the Sun's rest frame? (Why is it not "just relative"?)

The Earth is traveling at about $30 \mathrm{km} / \mathrm{sec}$ as it orbits the Sun.
a. How much more slowly does time run on the Earth than in the Sun's rest frame?
b. Why will the time be measured as slower on the Earth than in the Sun's rest frame? (Why is it not "just relative"?)

Key Concepts

Reference Frame Asymmetry

Even though the principle of relativity suggests that motion is relative, the measurement of time along different world lines is not symmetric when one clock follows a non-inertial or curved path through spacetime, as is the case for an object in orbit. The Earth’s velocity relative to the Sun’s inertial frame causes its clock to accumulate less proper time, making the time dilation effect an absolute comparison of their respective spacetime trajectories rather than a purely relative phenomenon.

Proper Time

Proper time is the time interval measured by a clock that is moving along with the system in its own rest frame. It is the 'true' time that elapses along the world line of the clock. In the context of moving observers, the clock that is in motion records a shorter proper time compared to the coordinate time measured in another frame, such as the Sun’s rest frame in the given problem.

Time Dilation

Time dilation is a fundamental prediction of special relativity, where a clock moving relative to an observer appears to tick slower than a clock at rest with respect to that observer. This effect becomes more pronounced as the relative speed approaches the speed of light; even at much lower speeds, such as the Earth’s orbital velocity, a small time dilation effect occurs, meaning less proper time is accumulated along the moving clock’s path than in the rest frame.

Lorentz Factor

The Lorentz factor, commonly denoted as gamma (?), quantitatively describes time dilation and other relativistic effects. It is defined as 1/?(1 - v²/c²), where v is the relative velocity of the moving frame and c is the speed of light. For velocities much smaller than the speed of light, the deviation from 1 is tiny, but it still provides the mathematical basis for determining how much slower a moving clock ticks compared to a stationary one.

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