Question

What is the factor \(\gamma\) for a speed of \(0.99 c\) ? As observed from the ground, by how much would a clock traveling at this speed differ from a ground-based clock after one hour (one hour as measured by the latter, that is)?

Short answer

\(\gamma \approx 7.09\) and the moving clock differs by about 508 seconds.

Step-by-step solution

Understand the Problem

To solve the problem, we need to find the Lorentz factor, \(\gamma\), for a speed of 0.99 times the speed of light, \(c\). Additionally, we need to calculate how much a clock traveling at this speed differs from a clock on the ground after one hour.

Write the Formula for Lorentz Factor

The Lorentz factor \(\gamma\) is given by the formula: \[ \gamma = \frac{1}{\sqrt{1 - \left( \frac{v}{c} \right)^2}} \] where \(v\) is the velocity of the moving object and \(c\) is the speed of light.

Substitute the Given Speed into the Formula

Substitute \(v = 0.99c\) into the formula: \[ \gamma = \frac{1}{\sqrt{1 - (0.99)^2}} \]

Calculate \\(\gamma\\)

Calculate the value: \[ \gamma = \frac{1}{\sqrt{1 - 0.9801}} = \frac{1}{\sqrt{0.0199}} \approx \frac{1}{0.1411} \approx 7.089 \] So, the Lorentz factor is approximately \gamma \approx 7.09\.

Determine the Time Dilation

The time experienced by the moving clock, \(\Delta t'\), is related to the time experienced by the stationary clock, \(\Delta t\), by \[ \Delta t' = \frac{\Delta t}{\gamma} \] where \(\Delta t = 1\ \text{hour} = 3600\ \text{seconds}\).

Calculate the Time Experienced by the Moving Clock

Using the \(\gamma\) calculated, \[ \Delta t' = \frac{3600}{7.089} \approx 508\ \text{seconds} \] This means that the moving clock experiences only about 508 seconds or approximately 8 minutes and 28 seconds when the ground clock measures an hour.

Key concepts

Relativistic Speed

Relativistic speed is a term often used in physics to describe velocities that are a significant fraction of the speed of light, denoted as \( c \). As an object approaches these high speeds, interesting and non-intuitive phenomena predicted by Einstein’s theory of special relativity start to take place. For instance, lengths contract, masses increase, and time dilates.

When we talk about speeds such as 0.99 times the speed of light, we're indeed in the realm of relativistic speeds. At these velocities, conventional Newtonian physics no longer provides accurate predictions. Instead, relativistic mechanics must be used to account for the effects of special relativity.

It's important to note that no object with mass can reach or exceed the speed of light. As an object accelerates closer and closer to this upper limit, its relativistic mass increases, requiring more and more energy for any further acceleration. This is why these velocities are only possible for particles with extremely high energies.

Time Dilation

Time dilation is one of the major consequences of moving at relativistic speeds, as predicted by Einstein's special relativity. It can be counterintuitive. Essentially, time runs slower for an object moving at high speeds compared to a stationary observer.

Consider two identical clocks: one remains stationary on Earth, while the other travels at 0.99 times the speed of light. According to the stationary clock, one hour has passed. But due to time dilation, significantly less time has elapsed on the moving clock. Using the Lorentz factor \( \gamma \), we calculate that only about 508 seconds—or roughly 8 minutes and 28 seconds—have passed on the moving clock. This stark difference is due to the relativity of simultaneity and the effects of traveling at such a high velocity.

Key points about time dilation include:

• The faster you travel, the slower time progresses for you relative to a stationary observer.
• This effect becomes pronounced as the speed approaches the speed of light.
• It has been confirmed by numerous experiments, including those involving particles moving in accelerators.

Special Relativity

Special relativity is a theory formulated by Albert Einstein in 1905. It revolutionized our understanding of space, time, and energy, particularly for objects traveling at high velocities. Special relativity fundamentally differs from Newtonian physics by proposing that the laws of physics are the same for all observers, regardless of their constant velocity relative to each other.

Within this framework, two key principles are:

• The speed of light in vacuum is constant and is the same for all observers, regardless of their motion.
• The physics laws apply equally in all inertial frames, meaning no point of view is special when comparing different uniform-motion states.

These principles lead to remarkable predictions like time dilation and length contraction. For the calculation we discussed, special relativity underpins how the Lorentz factor \( \gamma \) is derived and applied. As objects move faster, they experience time and space differently than stationary objects. This changes how fast clocks tick and how rulers measure lengths while in motion.

Special relativity has profound implications for scientific fields including astrophysics, particle physics, and cosmology, essentially reshaping our comprehension of the universe. Its predictions have been consistently verified by practical experiments, validating it as one of our most reliable scientific theories.