Time dilation

Time Dilation

Overview

• Time dilation refers to the differences in the elapsed time as measured by two observers, due to a relative motion between them.

• This concept emerges from Einstein's theory of relativity, particularly his postulates concerning the constancy of the speed of light in a vacuum for all observers, irrespective of their relative motion.

Einstein’s Postulates

• Different outcomes based on reference frames:

  • When events are viewed from different frames of reference, outcomes may vary significantly.

• Gedanken Experiment:

  • A thought experiment involving a moving train illustrates the differences:

  • Inside the train, observers may perceive two events as simultaneous.

  • Observers outside the train perceive these events occurring at different times.

• Posulates

  • Newtonslawsapplyinanyinertialreferenceframe

  • Thespeadoflightisconstantinallinertialframesofreferance

Observational Differences in Time

• At everyday speeds, the differences in timing are negligible and undetectable without precise instruments.

• At supersonic speeds (e.g., aircraft), timing differences become measurable.

• As speeds approach the speed of light, such as in particle accelerators (e.g., Australian Synchrotron), these time differences become substantial.

• Relativistic effects: Must be considered in calculations involving very high-speed scenarios.

The Light Clock

• A light clock is a thought experiment used to illustrate the concept of time dilation, where a beam of light bounces between two mirrors. As the clock moves at relativistic speeds, an observer at rest would see the light traveling a longer diagonal path, thus taking more time to complete a tick compared to an observer moving with the clock.

  • Working Principle:

  • A light clock consists of mirrors with light reflecting between them. Each reflection counts as 'one tick' of time.

  • Observers perceive light traveling a zigzag path due to the motion of the clock.

Einstein's Equation for Time Dilation

• Equation: $t=t_0y$

  •        $y=\frac{1}{\sqrt{1}-\frac{v^2}{c^2}}$

    • Definitions:

    • t: Time measured by a stationary observer (s).

    • t_0: Proper time, time measured on the moving clock itself.

    • γ (gamma): The Lorentz factor, which adjusts the proper time to the stationary observer's time.

Where:

• v: Speed of the moving observer.

• c: Speed of light in a vacuum, valued at $3.00 imes 10^8 ext{ ms}^{-1}$.

Lorentz Factor  

• Introduced by physicist H.A. Lorentz in relation to the Michelson-Morley experiment.

• Lorentz Factor (γ): $ext{γ} = rac{1}{ rac{1}{ ext{√(1 - (v^2/c^2))}}}$ where:

  • v: Velocity of the object in motion.

  • c: Speed of light.

Timing Observations at Various Speeds

• At low speeds (e.g., 300 ms −1), γ is approximately 1. Thus, the effect of time dilation is minimal.

Example Values for Lorentz Factor  Table 8.2.1

• 
  

• Various values of velocity (v in ms^-1) and corresponding Lorentz factor (γ):
  

  v (m/s)	γ (Lorentz Factor)
  3.00 x 10²	1
  3.00 x 10⁵	1.155
  2.60 x 10⁸	2.00
  2.997 x 10⁸	22.4
  Observations at 99% Speed of Light

  • At 0.99c:

    • A stationary observer sees time on a spaceship slows to about one-seventh of normal time.

    • Example: Clare (stationary) observes one oscillation of her clock correlating to seven oscillations of the spaceship's clock.

  • At the speed of light,

    • Clare perceives the spaceship's clock as completely stopped.

    • Passengers (Amaya and Binh) do not perceive a change in their own clock rate.

  Worked Examples on Time Dilation

  Example 8.2.1: Fast Car

  • Conditions: A car moving at $2.50 imes 10^8 ext{ m/s}$.

  • Observed By: A stationary observer sees 3.00 s pass on the car’s clock.

  • Calculate time observed by the stationary clock during this observation.

  Try Yourself 8.2.1: Fast Scooter

  • Conditions: A scooter moving at $2.98 imes 10^8 ext{ m/s}$.

  • Observed By: Clock on the scooter shows 60.0 s pass.

  • Calculate how much time passes on the observer's clock during this observation.

  Key Questions for Reinforcement

  1. Complete the sentences: Light clock uses the $rac{ ext{speed}}{ ext{oscillation}}$ of light to measure $ext{time}$, as the speed of light is $ext{constant}$ across all frames.

  2. Proper Time (t₀): Refers to the time measured by an observer in the same frame of reference as the event occurring.

  3. Example with an observer on a train platform. Question prompts:

     • a. How far does light travel in one tick in Anna's frame?

     • b. What is the clock tick time in Anna's frame?

     • c. Length of zigzag path for Chloe's view in terms of c and t₀?

  4. Further questions include calculations based on varying speeds and observations from different frames, reinforcing understanding of time dilation.

  Historical and Applied Context

  • GPS and Time Dilation: Satellite clocks run faster than those on Earth due to gravitational time dilation and the relative motion. A total time difference of 0.000038 s over 24 hours observed in GPS systems arises primarily from relativistic effects.

  • Muons and Cosmic Rays: The observation that muons created high above Earth can travel further than expected due to time dilation effects shows real-world implications of Einstein's Theory.

  Conclusion

  • Time dilation remains a fundamental aspect of modern physics, applicable across various fields including astrophysics, GPS technology, and particle physics. All conclusions stem from relativistic effects introduced by Einstein's theory, reshaping our understanding of time and motion in the universe.