Problem 7 - Fundamentals of Physics

Question

The premise of the Planet of the Apes movies and book is that hibernating astronauts travel far into Earth's future, to a time when human civilization has been replaced by an ape civilization. Considering only special relativity, determine how far into Earth's future the astronauts would travel if they slept for \(120 \mathrm{y}\) while traveling relative to Earth with a speed of \(0.9990 c\), first outward from Earth and then back again.

Step-by-Step Solution

Answer

The astronauts would travel approximately 53,784 years into the Earth's future.

Step 1: Define the Time Dilation Formula

According to the theory of special relativity, time dilation can be calculated with the formula: \( t' = \frac{t}{\sqrt{1 - \frac{v^2}{c^2}}} \), where \( t' \) is the time experienced by astronauts (Earth time), \( t \) is the proper time (time on spaceship), \( v \) is the velocity of the spaceship, and \( c \) is the speed of light.

Step 2: Plug in the Given Values

We are given \( t = 120 \text{ years} \), \( v = 0.9990c \), and \( c \) is the speed of light. Substitute these values into the time dilation formula: \[ t' = \frac{120}{\sqrt{1 - (0.9990)^2}} \]

Step 3: Calculate the Expression Inside the Square Root

First, calculate \( 0.9990^2 \): \[(0.9990)^2 = 0.998001\]Then, subtract this value from 1:\[1 - 0.998001 = 0.001999\]

Step 4: Calculate the Square Root

Find the square root of \( 0.001999 \): \[\sqrt{0.001999} \approx 0.0447\]

Step 5: Complete the Time Dilation Calculation

Divide the proper time by the calculated value to find the time experienced on Earth:\[ t' = \frac{120}{0.0447} \approx 26892 \text{ years} \]

Step 6: Determine Total Time Traveled into the Future

Since the journey includes both outward and return segments, double the time calculated in Step 5:\[2 \times 26892 \approx 53784 \text{ years} \]

Key Concepts

special relativity

The theory of special relativity, proposed by Albert Einstein in 1905, introduced groundbreaking insights into the nature of space and time. It is based on two postulates: that the laws of physics are the same for all non-accelerating observers, and that the speed of light in vacuum is the same regardless of the motion of the light source or observer. Special relativity fundamentally changes how we perceive time and space, particularly at speeds approaching that of light. A crucial implication of this theory is time dilation, which occurs when an object moves at a significant fraction of the speed of light, causing time to pass at a different rate for the object compared to a stationary observer. Understanding these principles allows us to explore the conditions under which phenomena like time travel become theoretically possible, as illustrated in science fiction scenarios like the Planet of the Apes.

speed of light

The speed of light, denoted by the constant \( c \), is approximately \( 299,792,458 \) meters per second. It is an essential constant in the universe and forms the backbone of the equations in special relativity. One of the profound consequences of the constancy of light's speed is that it leads to the phenomena of time dilation and length contraction.

• It acts as a cosmic speed limit, meaning no object with mass can reach or exceed this speed.
• Whether you are on earth, on a spaceship, or in any frame of reference, light travels at the same speed.

In the context of the exercise, the speed of light provides the benchmark against which the spaceship's velocity is compared. As the spaceship travels at a velocity close to this speed, the relativistic effects become significant, leading to the dramatic difference in the passage of time between the spaceship and Earth.

velocity of spaceship

Velocity is the speed of an object in a specified direction. In scenarios involving special relativity, velocity plays a critical role in determining the effects of time dilation. In the given exercise, our spaceship travels at a velocity of \( 0.9990 c \), meaning it travels at 99.90% of the speed of light.

• At such high velocities, the effects of special relativity become highly pronounced.
• The closer the velocity is to the speed of light, the more pronounced the time dilation effects are.

Calculating time dilation involves substituting the spaceship's velocity into the time dilation formula, allowing us to determine how time differs for the astronauts compared to an observer on Earth.

proper time

Proper time, in the context of relativity, is the time interval measured by an observer who is present at the events being timed. It is considered the true or actual time experienced between two events as measured in the observer's own inertial frame.

• In the exercise, "proper time" refers to the 120 years experienced by the astronauts.
• It is the time that would naturally pass on the spaceship's clocks, as opposed to the time experienced on Earth.

The relativity of experience across different frames of reference underscores how time is not absolute but relative, changing depending on the observer's velocity. Therefore, while the astronauts age 120 years in their reference frame, the calculated Earth time reveals just how long the trip feels here, showing the perplexing yet fascinating nature of time in our universe.