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The Lorentz factor, denoted by the Greek letter \( \gamma \), is a crucial element in understanding relativity, especially in special relativity. It's used to determine how much time and space change when you measure them in a moving system compared to a stationary one. The Lorentz factor can be calculated using the formula:

• \( \gamma = \frac{1}{\sqrt{1 - \left(\frac{v}{c}\right)^2}} \)

Here, \( v \) is the speed of the moving object, and \( c \) is the speed of light, approximately \( 3 \times 10^8 \, \mathrm{m/s} \) in a vacuum.

When the speed of our particle is very close to the speed of light, like \( 0.990c \), the Lorentz factor becomes significantly greater than 1. This high value indicates that effects like time dilation and length contraction are pronounced. For our high-energy particle moving at \( 0.990c \), \( \gamma \) is calculated to be approximately 7.088, implying marked changes in perceived measurements of time and distance as viewed from different frames of reference.

Proper distance is a term used in relativity to describe the distance between two points as measured in the rest frame where both points are at rest relative to the observer.

• In simpler terms, it's the distance measured when you're traveling with the moving object and looking at the distance between where the object starts and where it ends.

For the particle in our example, the proper distance is considered to be zero. Why? Because in the rest frame of the particle, it doesn't move relative to itself, which means no distance is covered in its own rest frame.

Think about sitting in a bus: the bus doesn't move relative to you, so you consider the distance from the front to the back of the bus as unchanged. But for an observer outside, watching the bus, the distance the bus covers on the road is different. That's why in the laboratory frame, we measure the distance \( 1.05 \times 10^{-3} \, \mathrm{m} \). From within the particle's frame, due to relativistic effects like length contraction, the distance is much smaller than in the laboratory's frame - calculated by dividing the observed distance by \( \gamma \), resulting in \( 1.48 \times 10^{-4} \, \mathrm{m} \).

The proper lifetime of an object, such as a particle, is the time it 'lives' in its rest frame.

• It's the time experienced by the particle from start (creation) to end (disintegration).

In the context of our swift particle traveling close to light speed, the proper lifetime is different from the time measured by a stationary observer in the laboratory.

To find the proper lifetime, we calculate using the contracted distance in the rest frame of the particle as it zooms along. Using the speed of the particle \( 0.990c \), the proper lifetime \( \tau \) is determined to be approximately \( 5.0 \times 10^{-13} \, \mathrm{s} \). This brief duration is precisely how long the particle exists, as seen from its own perspective - a valuable value for physicists in determining the nature and behavior of unstable particles.

Time dilation is one of the fascinating predictions of special relativity. It shows how time can vary for observers in different inertial frames of motion, particularly when one of them is moving at a significant fraction of the speed of light relative to the other.

• In simpler terms, a moving clock ticks slower compared to a stationary clock as seen by the stationary observer.

This strange effect, while counterintuitive, has been confirmed by many experiments. In our exercise, time dilation means that while the proper lifetime of the particle is \( 5.0 \times 10^{-13} \, \mathrm{s} \), the time observed in the laboratory, or the dilated lifetime, is much longer because the lab frame is stationary relative to the fast-moving particle.

Using the Lorentz factor \( \gamma \), the dilated lifetime \( t \) is calculated as follows: \( t = \gamma \tau \). So, the effective duration measured in the laboratory fame is roughly \( 3.54 \times 10^{-12} \, \mathrm{s} \). This is longer than the proper lifetime, showcasing how speed can stretch time itself as perceived from different frames, a concept that has important implications in particle physics and cosmology.