Proper Time ($t_0$)

Definition of Proper Time

• Proper time ($t_0$) is defined as:

  The time interval between two events measured in a reference frame where the two events occur at the same point in space.

• An event is something that occurs at a particular location and time. Examples include:

  • A rocket landing on a planet
  • A clock hand striking 12
  • A collision between two vehicles
• Proper time is the shortest time interval that any observer can measure between two events.

KEY TAKEAWAY: Proper time is the time measured by an observer who sees the two events happening at the same location.

Understanding the Concept

• Imagine a clock at rest relative to an observer. The time measured by this clock between two events occurring at its location is the proper time.
• If the clock is moving relative to another observer, that observer will measure a longer time interval between the same two events. This longer time interval is called dilated time ($t$).
• Proper time is an invariant quantity, meaning all observers can agree on which observer measures the proper time even if they disagree on the time interval itself.

EXAM TIP: Always identify the reference frame where the two events occur at the same location to determine who measures the proper time.

Example Scenario

Consider a spaceship traveling from Earth to Mars.

• Event 1: The spaceship leaves Earth.

• Event 2: The spaceship arrives at Mars.

• For an astronaut on the spaceship, both events occur at the same location (inside the spaceship). Therefore, the astronaut measures the proper time ($t_0$) for the journey.

• An observer on Earth sees the spaceship move from Earth to Mars. For the Earth observer, the two events occur at different locations in space. The Earth observer measures the dilated time ($t$), which is longer than the proper time.

COMMON MISTAKE: Confusing proper time with dilated time. Remember, proper time is always the shortest time interval measured when the events occur at the same location.

Proper Time vs. Dilated Time

STUDY HINT: Create your own scenarios to practice identifying which observer measures proper time and which measures dilated time.

Formulas and Relationships

• Time Dilation Formula:

  $$t = \gamma t_0$$

  Where:

  • $t$ = Dilated time (time measured by the moving observer)
  • $t_0$ = Proper time (time measured in the rest frame of the event)
  • $\gamma$ = Lorentz factor

• Lorentz Factor:

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

  Where:

  • $v$ = Relative velocity between the two reference frames
  • $c$ = Speed of light in a vacuum (\$2.998 \times 10^8$ m/s)

REMEMBER: The Lorentz factor ($\gamma$) is always greater than or equal to 1. Therefore, dilated time ($t$) is always greater than or equal to proper time ($t_0$).

Implications of Proper Time

• The concept of proper time is fundamental to understanding special relativity.
• It highlights that time is relative and depends on the observer’s frame of reference.
• As an object’s velocity approaches the speed of light, the difference between proper time and dilated time becomes increasingly significant.

APPLICATION: GPS satellites rely on accurate time measurements. Since they are moving relative to observers on Earth, relativistic effects like time dilation must be taken into account to ensure accurate positioning.

Example Problem

A muon (an elementary particle) is created in the upper atmosphere and travels towards the Earth at a speed of $0.99c$. In the muon’s frame of reference, its average lifetime is \$2.2 \times 10^{-6}$ s (proper time). What is the lifetime of the muon as measured by an observer on Earth?

1. Identify the proper time: $t_0 = 2.2 \times 10^{-6}$ s

2. Calculate the Lorentz factor:

   $$\gamma = \frac{1}{\sqrt{1 - \frac{(0.99c)^2}{c^2}}} = \frac{1}{\sqrt{1 - 0.99^2}} \approx 7.09$$
   3. Calculate the dilated time:

   $$t = \gamma t_0 = 7.09 \times 2.2 \times 10^{-6} \text{ s} \approx 1.56 \times 10^{-5} \text{ s}$$

Therefore, the lifetime of the muon as measured by an observer on Earth is approximately \$1.56 \times 10^{-5}$ s.

VCAA FOCUS: VCAA exam questions often involve scenarios where you need to identify the proper time and calculate the dilated time using the time dilation formula.