Problem 4 - Fundamentals of Physics

Question

(Come) back to the future. Suppose that a father is \(20.00 \mathrm{y}\) older than his daughter. He wants to travel outward from Earth for \(2.000 \mathrm{y}\) and then back for another \(2.000 \mathrm{y}\) (both intervals as he measures them ) such that he is then \(20.00 \mathrm{y}\) younger than his daughter. What constant speed parameter \(\beta\) (relative to Earth) is required?

Step-by-Step Solution

Answer

The required speed parameter is \( \beta = 0.705 \).

Step 1: Understanding the Problem

The father wants to travel in such a way that in 4000 years of travel (2000 outward and 2000 back), he ends up being 20 years younger than his daughter, despite initially being 20 years older. This needs to be solved using time dilation, a concept from special relativity, where time runs differently for observers moving at different velocities.

Step 2: Define the Variables and Given Information

Let \( t_f \) be the time elapsed for the father during his travel, and \( t_d \) be the time elapsed for the daughter back on Earth. We know \( t_f = 4000 \text{ years} \). The conditions state that \( t_d - t_f = 40 \text{ years} \), as the father is initially 20 years older and needs to end 20 years younger.

Step 3: Relate Time Dilation and Speed

According to the time dilation formula, the time elapsed in the daughter's frame \( t_d = \frac{t_f}{\sqrt{1-\beta^2}} \), where \( \beta \) is the speed parameter (speed as a fraction of the speed of light). Therefore, we set \( t_d = \frac{4000}{\sqrt{1-\beta^2}} \).

Step 4: Set Up the Equation

From the conditions, \( t_d = t_f + 40 \). Substitute \( t_d = 4000 + 40 \) into the time dilation equation: \[ 4000 + 40 = \frac{4000}{\sqrt{1-\beta^2}} \].

Key Concepts

Special Relativity

Special relativity is a fundamental theory in physics introduced by Albert Einstein in 1905. This theory revolutionized our understanding of how time and space behave, particularly at high velocities, close to the speed of light. A core principle of special relativity is that the laws of physics are the same for all observers, no matter their velocity—provided they're not accelerating. Special relativity has significant implications on our perception of time and space. It tells us that time is not absolute but instead depends on the observer's relative motion.
One of its famous effects is time dilation, which will be further explored in the context of the given problem involving the father and his travel journey. This theory results in time passing at different rates for different observers based on their velocity.

• Important point: Speeding clocks (or fast-moving objects) run more slowly compared to stationary ones as observed from a stationary reference frame.
• This theory is deeply rooted in the cosmic speed limit set by the speed of light, denoted as "c".

The faster you move, the slower time passes in your frame compared to others. This concept is what allows the father to age differently than his daughter on Earth during his space journey.

Velocity in Relativity

In the realm of special relativity, velocity plays a pivotal role in determining time dilation effects. Here, the term "velocity" refers to the speed of an object relative to the speed of light. We use the parameter \( \beta \), defined as \( \beta = \frac{v}{c} \), where \( v \) is the velocity of the object, and \( c \) is the speed of light in a vacuum.
The interesting part about velocities near the speed of light is that they cause significant relativistic effects, one being time dilation, where time observed from different frames doesn't tick at the same rate. For example:

• Low \( \beta \) values (speeds much less than the speed of light) imply negligible differences in time perception across frames.
• High \( \beta \) values (speeds approaching the speed of light) lead to drastic time dilation, meaning a lot more "time" can pass in one frame compared to another.

To solve our problem, the father needs to travel at a particular \( \beta \) such that his time, adjusted for speed, appears slower to match the scenario requirements with his daughter. This precision in velocity aligns the perceived age difference due to special relativity.

Time Perception Difference

Time perception difference is essentially the difference in elapsed time experienced by two observers due to relative motion. In the scenario of the father and daughter, this difference stems from the relativistic effects of the father's high-speed journey through space.
The father's journey, measured in his own frame, takes 4000 years (2000 years each way). However, due to time dilation, more time passes for his daughter left on Earth. This results in the father's experience of time being 'slower.' The calculation involves the time dilation formula:

• Calculate daughter's time \( t_d \) using: \( t_d = \frac{4000}{\sqrt{1-\beta^2}} \).
• Given that when he returns, \( t_d = t_f + 40 \), solve for \( \beta \).

As the problem states, the daughter ages 40 years more than the father during his trip. This difference in aging illustrates the core of time perception difference, showing how speeds approaching light alter time flow with such dramatic impact.