Question

Two twins, \(A\) and \(B\), are in deep space on similar rockets traveling in opposite directions with a relative speed of \(c / 4\). After a while, twin A turns around and travels back toward twin \(\mathrm{B}\) again, so that their relative speed is \(c / 4\). When they meet again, is one twin younger, and if so which twin is younger? a) Twin A is younger. d) Each twin thinks b) Twin \(B\) is younger. the other is younger. c) The twins are the same age.

Short answer

Based on the calculations and comparison of the proper times experienced by Twin A and B, it can be concluded that Twin A is younger when they meet again. The time dilation caused by their relative speeds in the context of special relativity results in Twin A experiencing less proper time compared to Twin B.

Step-by-step solution

Decode the problem statement

First, we should understand the problem. We have two twins, A and B, in deep space, traveling in opposite directions at a relative speed of c/4. At some point, A turns around and goes back towards B at the same relative speed. We need to find whether one twin is younger when they meet again.

Calculate the proper time for Twin A

In order to find the proper time for Twin A, we need to divide the journey into two parts: the outbound and inbound trips. To calculate the proper time experienced by A during each leg of the trip, we can use the Lorentz transformation formula for time dilation: \(\Delta \tau_A = \Delta t_A \sqrt{1 - \frac{v^2}{c^2}}\) Since their relative speed is c/4, and they are traveling equal distances in both outbound and inbound trips, we can assume that both the outbound and inbound time dilations will be equal. Thus, we just need to find the time dilation for one leg and multiply it by two: \(\Delta \tau_A = 2\Delta t_{A_{leg}} \sqrt{1 - \frac{(c/4)^2}{c^2}}\)

Calculate the proper time for Twin B

Throughout the whole experiment, Twin B is traveling at a relative speed of c/4 with respect to the inertial frame. Thus, we can use the same Lorentz transformation formula for time dilation to find the proper time for Twin B: \(\Delta \tau_B = \Delta t_B \sqrt{1 - \frac{v^2}{c^2}}\) We can substitute the relative speed (c/4) into this formula to find the proper time: \(\Delta \tau_B = \Delta t_B \sqrt{1 - \frac{(c/4)^2}{c^2}}\) Note that \(\Delta t_B\) is equal to twice the time of each leg, the same as \(\Delta t_A\).

Compare Twin A's proper time with Twin B's proper time

Now we can compare the proper times experienced by Twin A and B: Reacting back to the formulas from Step 2 and Step 3 we get: \(\Delta \tau_A = 2\Delta t_{A_{leg}} \sqrt{1 - \frac{(c/4)^2}{c^2}}\) \(\Delta \tau_B = \Delta t_B \sqrt{1 - \frac{(c/4)^2}{c^2}}\) Rewrite the proper time of Twin B: \(\Delta \tau_B = 2\Delta t_{A_{leg}} \sqrt{1 - \frac{(c/4)^2}{c^2}}\) We can see that \(\Delta \tau_A < \Delta \tau_B\), that means Twin A will be younger than Twin B. So, the answer is: a) Twin A is younger.

Key concepts

Special Relativity

Special relativity is a fundamental theory in physics that describes the relationship between space and time. Developed by Albert Einstein in 1905, it revolutionized how we understand motion at speeds close to the speed of light, known as relativistic speeds. At its core, special relativity is based on two postulates: the laws of physics are the same in all inertial frames of reference, and the speed of light in a vacuum is the same for all observers, regardless of their relative motion.

One of the most intriguing aspects of special relativity is the phenomenon of time dilation, which tells us that time can pass at different rates for observers in relative motion. This means that if two individuals are moving at a high velocity relative to each other, each will perceive the other's clock as ticking slower than their own. This is not just a theoretical concept, but an observable reality, confirmed by numerous experiments, such as those involving precise clocks on fast-moving airplanes or satellites in orbit.

Time dilation plays a pivotal role in scenarios involving high-speed travel, such as the thought experiment with twin astronauts, which leads us to consider the consequences of traveling at speeds close to the speed of light.

Lorentz Transformation

The Lorentz transformation equations are the mathematical backbone of Einstein's special relativity theory. They show how measurements of space and time by two observers moving at a constant speed relative to each other are related. The Lorentz transformation is essential for calculating how time and space coordinates change when observed from different inertial frames.

The Lorentz transformation for time shows the interdependence of time and velocity. For time dilation, the equation is expressed as:
\[ \Delta t' = \frac{\Delta t}{\sqrt{1 - \frac{v^2}{c^2}}} \]
where:\

• \(\Delta t'\) is the time interval measured by an observer moving at speed \(v\)
• \(\Delta t\) is the time interval measured by an observer at rest (often referred to as the 'proper time')
• \(v\) is the relative velocity between the observers
• \(c\) is the speed of light in a vacuum

This equation tells us that as an object's velocity approaches the speed of light, the observed time interval dilates, becoming longer for a stationary observer compared to the moving observer's proper time. This explains why Twin A in our exercise will have experienced less time than Twin B upon their reunion.

Proper Time

Proper time, denoted in physics by the symbol \(\tau\), is the actual time measured by a clock following a particular path through spacetime. It represents the time between two events as measured by a clock that is present at both events and moving with the object or observer under consideration. Proper time is an important concept because it is invariant; all observers would agree on the measurement of proper time for a particular event, regardless of their relative motion.

In the context of our twin astronauts, we calculate the proper time for each twin's journey using the Lorentz transformation. The key to understanding proper time is realizing that it's the time experienced by someone (or something) at rest relative to the observed process. For Twin A, who accelerates, turns around, and decelerates, the proper time is the time she would measure on her own clock. Twin B, who remains in consistent motion without turning around, also has his proper time measured from his own perspective.

Without the concept of proper time, we would have no way to make sense of how different observers experience time. It's the cornerstone of analyzing scenarios in special relativity, like our twins' reunification, revealing that movement through space affects the passage of time.