Special Relativity: Time Dilation

Introduction

• One of the fascinating consequences of special relativity is the concept of time dilation, which states that time passes differently for objects in motion relative to an observer at rest.
• This effect becomes significant as objects approach speeds close to the speed of light (c).

Two Identical Clocks

• Imagine having two identical and accurate clocks, one stationary and the other moving at a speed close to the speed of light (c).
• According to special relativity, the moving clock measures a shorter time compared to the stationary clock.

Proper Time and Dilated Time

• The time recorded by the moving clock, which is correct for an observer moving with it, is known as the "proper time."
• However, for a stationary observer watching the moving clock, the time interval appears longer, and this is referred to as "time dilation."

The Time Dilation Formula

• Time dilation can be calculated using the following formula:
• $t' = t / \sqrt{(1 - v²/c²)}$
• Where:
  • t' is the dilated time observed by the stationary observer.
  • t is the proper time measured by the moving clock.
  • v is the relative velocity between the moving clock and the stationary observer.
  • c is the speed of light.

Sergei Avdeyev's Example

• Sergei Avdeyev, a Russian cosmonaut, spent 747 days (equivalent to 64,540,800 seconds) aboard the Space Station Mir, which travels at a speed of 7,600 m/s.
• We can calculate how much time Sergei "extended" his life compared to colleagues on Earth using the time dilation formula.

---

Special Relativity: Time Dilation

Calculation

• Plug in the known values into the formula:
• t' = 64,540,800 / $\sqrt{(1 - (7,600^2 / 299,792,458^2))}$
• Calculate this equation step by step:
  • First, calculate the value inside the square root.
  • Then, calculate the square root of that value.
  • Finally, divide the proper time by the square root value.
• After performing the calculation, Sergei experienced only approximately 21 milliseconds less time compared to his colleagues on Earth.

Significance of Significant Figures

• In such examples, where the time difference is extremely small, it's crucial to use a large number of significant figures to ensure accuracy.