Question

A space explorer \(A\) sets off at a steady \(0.95 c\) to a distant star. After exploring the star for a short time, he returns at the same speed and gets home after a total absence of 80 years (as measured by earth-bound observers). How long do \(A\) 's clocks say that he was gone, and by how much has he aged as compared to his twin \(B\) who stayed behind on earth? [Note: This is the famous "twin paradox." It is fairly easy to get the right answer by judicious insertion of a factor of \(\gamma\) in the right place, but to understand it, you need to recognize that it involves three inertial frames: the earth-bound frame \(\mathcal{S}\), the frame \(\mathcal{S}^{\prime}\) of the outbound rocket, and the frame \(\mathcal{S}^{\prime \prime}\) of the returning rocket. Write down the time dilation formula for the two halves of the journey and then add. Notice that the experiment is not symmetrical between the two twins: \(A\) stays at rest in the single inertial frame \(\delta\), but \(B\) occupies at least two different frames. This is what allows the result to be unsymmetrical.]

Short answer

Twin A's clock says he was gone for about 24.98 years, so A aged 24.98 years compared to B's 80 years.

Step-by-step solution

Understand the Scenario

The problem involves calculating the time experienced by twin A, the space explorer, who travels with a velocity of \(0.95c\) and returns after 80 years according to observers on Earth. Twin B remains on Earth.

Identify Relevant Physics Concepts

This situation involves special relativity and the concept of time dilation. Time dilation is described by the formula:\[ t' = \frac{t}{\gamma} \]where \(t'\) is the time experienced by the moving observer, \(t\) is the time measured by the stationary observer, and \(\gamma\) is the Lorentz factor given by:\[ \gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}} \]

Calculate the Lorentz Factor \(\gamma\)

Given \(v = 0.95c\), calculate \(\gamma\):\[ \gamma = \frac{1}{\sqrt{1 - (0.95)^2}} = \frac{1}{\sqrt{1 - 0.9025}} = \frac{1}{\sqrt{0.0975}} \approx 3.2026 \]

Apply Time Dilation for the Round Trip

The total time for the round trip as measured on Earth is 80 years. Using the time dilation formula:\[ t' = \frac{80}{3.2026} \approx 24.98 \text{ years} \]

Interpretation

The clock on space explorer A's spaceship indicates that only approximately 24.98 years have passed during the 80 years measured on Earth. Twin A will have aged by this amount as compared to their identical twin B.

Key concepts

Twin Paradox

The twin paradox is a thought experiment from Albert Einstein's theory of special relativity. It uses two twins, one traveling on a high-speed journey in space, while the other remains on Earth. Upon returning, the traveling twin ages less than their sibling on Earth. This is because the time experienced by the traveling twin (due to time dilation) is less than that experienced by the stationary twin.
This paradox highlights the non-intuitive nature of special relativity. The key here is that the traveling twin experiences acceleration as they turn around in space and return. Thus, they change inertial frames, which accounts for the age difference.
In this scenario, twin A journeys at 0.95c, experiencing less time as compared to twin B on Earth. The difference isn't symmetrical, meaning twin A ages less due to the journey's specific conditions.

Lorentz Factor

The Lorentz factor, denoted as \(\gamma\), is a crucial component in the equations of special relativity. It accounts for the effects of time dilation and length contraction at relativistic speeds close to the speed of light.
The formula for \(\gamma\) is:

• \[ \gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}} \]

For a velocity of 0.95 times the speed of light (\(v = 0.95c\)), using this formula gives us a Lorentz factor of approximately 3.2026.
This means that time in the moving frame slows down, and lengths contract by a factor of 3.2026 as observed from the stationary frame. It's why, in the exercise, twin A experiences significantly less time than the 80 years measured by observers on Earth.

Special Relativity

Special relativity, formulated by Albert Einstein, revolutionized our understanding of time and space. It asserts that the laws of physics are the same for all observers regardless of their constant velocity relative to each other.
This theory introduces two critical concepts:

• Time Dilation: Moving clocks run slower relative to stationary ones.
• Length Contraction: Moving objects are measured to be shorter in the direction of motion.

In our exercise, time dilation plays a key role. It illustrates how the moving twin's journey results in experiencing less time. The symmetry of Newtonian physics is broken in special relativity, leading to non-trivial conclusions like the twin paradox.