Question

Appearing in the time-dilation and length-contraction formulas, \(\gamma_{y}\) is a reasonable measure of the size of relativistic effects. Roughly speaking, at what speed would observations deviate from classical expectations by \(1 \% ?\)

Short answer

The speed at which observations would deviate from classical expectations by 1% is approximately \(0.141c\), where c is the speed of light.

Step-by-step solution

Understand the relativistic factor formula

Recall the formula for the relativistic factor \(\gamma_{y}\) which is given by \(\gamma_{y} = \frac{1}{\sqrt{1 - v^2/c^2}}\) where v is the velocity of the object and c is the speed of light.

Implement the 1% deviation

According to the problem, the deviation between the speed calculated using the relativistic factor and classical speed is 1%. This means that when \(\gamma_{y}\) calculates a value, it would be 1% different from that if we had calculated it in the classical way. Therefore, \(\gamma_{y} - 1 = 0.01\) (i.e., 1% deviation).

Solve the equation to calculate speed

We can input the equation from Step 2 into the \(\gamma_{y}\) equation to solve for v. From this, we get \(\sqrt{1 - v^2/c^2} = 1 - 0.01 = 0.99\). Squaring both sides gives \(1 - v^2/c^2 = 0.99^2\). Rearranging gives \(v = c \cdot \sqrt{1 - 0.99^2}\).

Compute the speed

Inserting c = 3 * 10^8 m/s into the equation we get \(v = 3 * 10^8 * \sqrt{1 - 0.99^2}\) m/s. Thus, solving gives us v.

Key concepts

Time-Dilation

Time-dilation is a fascinating consequence of Einstein's theory of special relativity, which describes how time can 'slow down' for an object in motion relative to an observer at rest. This effect is especially pronounced at speeds approaching the speed of light. It's explained by the equation \( t' = \gamma t \) where \( t' \) is the time interval measured by a stationary observer, \( t \) is the proper time interval (meaning the time interval as measured in the moving object's own frame), and \( \gamma \) is the relativistic factor.
Imagine a spacecraft travelling close to the speed of light and onboard is a clock. To a stationary observer, the clock on the spacecraft would appear to tick slower than a clock at rest. This implies that less time has passed for the traveling clock compared to one that is stationary. It's a concept that challenges our intuitive understanding of time, but has been validated through numerous experiments, not least by observing the behavior of particles at high speeds in particle accelerators.
For practical interpretations, time-dilation becomes significant at speeds where the relativistic factor changes noticeably. In the exercise, a 1% deviation would mark the speed threshold where classical mechanics starts to disagree with relativistic predictions.

Length-Contraction

Length-contraction is another mind-bending prediction of special relativity. Similar to time-dilation, this phenomenon occurs at very high speeds and tells us that objects physically contract in length in the direction of their motion relative to the observer. The formula to calculate this contraction is \( L = \frac{L_0}{\gamma} \) where \( L_0 \) is the proper length (the length of the object in its rest frame) and \( L \) is the observed length.
Envision a spaceship speeding past Earth; to an observer on earth, the spaceship would seem shorter than its proper length. However, to those within the spaceship, nothing would seem amiss—they would measure the spaceship's length to be the same as when stationary. As with time-dilation, the magnitude of length contraction is determined by the relativistic factor, \( \gamma \) and is only appreciable at velocities close to the speed of light.

Speed of Light

The speed of light, denoted as \( c \) in physics equations, is a fundamental constant that plays a crucial role in the structure of the theories of special and general relativity. Not only is it the speed at which light propagates through a vacuum, it also acts as a universal speed limit according to current scientific understanding.
Einstein's special relativity asserts that nothing can travel faster than light in a vacuum, which is approximately \( 3 \times 10^{8} \) meters per second. This speed isn't just for light; it's the maximum speed for any energy, matter, or information in the universe. The speed of light is central to the equation for the relativistic factor and influences both time-dilation and length-contraction. The invariant speed of light leads to remarkable outcomes such as the relativity of simultaneity and the notion that, for light, time does not pass at all.

Relativistic Factor

The relativistic factor, often represented by the Greek letter \( \gamma \) (gamma), quantifies the amount by which time, length, and relativistic mass change as an object's velocity approaches the speed of light. The formula for the relativistic factor is \( \gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}} \).
This factor is equal to 1 when at rest (since \( v = 0 \) there), meaning there are no relativistic effects and classical physics applies. As the velocity \( v \) increases, \( \gamma \) increases significantly. In the context of the exercise, determining at what speed relativistic effects cause a 1% deviation involves rearranging this formula to solve for \( v \) based on the specified deviation. The resulting speed is a threshold—below it, Newton's classical mechanics is a good approximation, but above it, one must consider relativistic effects to accurately describe physical phenomena.