Question

A space traveler takes off from Earth and moves at speed \(0.988 c\) toward the star Vega, which is \(26.0\) light-years distant. How much time will have elapsed by Earth clocks \((a)\) when the traveler reaches Vega and \((b)\) when the Earth observers receive word from him that he has arrived? ( \(c\) ) How much older will the Earth observers calculate the traveler to be when he reaches Vega than he was when he started the trip?

Short answer

(a) As observed on Earth, the time it will take for the traveler to reach Vega is 26.3 years. (b) The Earth observers will receive word from him 52.3 years after his departure. (c) The Earth observers will calculate the traveler to be 19.32 years older when he reaches Vega than he was when he started the trip.

Step-by-step solution

Calculate the time for the traveler to reach Vega

First, we calculate the time it takes for the traveler to reach Vega as observed on Earth. This can be done by simply dividing the distance to Vega by the speed of the spaceship. \(T = D/v = 26.0 / 0.988 = 26.3\) years

Calculate the time for the message to reach Earth

Secondly, we calculate the time it takes for the message from the traveler to reach Earth. This is simply the distance to Vega divided by the speed of light (since the message travels at the speed of light). \(T_m = D/c = 26.0/1 = 26.0\) years

Calculate the total time for both the traveler to reach Vega and the message to reach Earth

The total time as observed on Earth is the sum of the time it takes for the traveler to reach Vega and the time it takes for the message to reach Earth. \(T_{total} = T + T_m = 26.3 + 26.0 = 52.3\) years

Calculate the time elapsed on the spaceship

The time elapsed on the spaceship (as experienced by the traveler) can be calculated using the time dilation formula: \[ T_0 = T * \sqrt{1 - (v^2/c^2)} = 26.3 * \sqrt{1 - (0.988^2)} = 6.98\] years

Calculate the age difference

The age difference as calculated by the Earth observers is simply the time elapsed on Earth (the time it took for the traveler to reach Vega) minus the time elapsed on the spaceship (as experienced by the traveler). This gives: \[ Age\,Difference = T - T_0 = 26.3 - 6.98 = 19.32\] years.

Key concepts

Time Dilation Explained

Time dilation is a fascinating concept from Einstein's theory of special relativity, describing how time is experienced differently by observers moving relative to each other. When a spaceship travels at a significant fraction of the speed of light, like the 0.988c in this exercise, this effect becomes very noticeable.

• Moving Clocks Run Slower: To someone traveling at high speed, time appears to pass normally. However, compared to an observer at rest, such as those on Earth, the traveler's clock runs more slowly. This means for every year experienced by the traveler, more than a year passes for those on Earth.
• The Lorentz Factor: Time dilation can be quantified using the Lorentz factor, which arises from Lorentz transformation equations. It's calculated as \( \sqrt{1 - (v^2/c^2)} \), where \(v\) is the velocity of the moving object, and \(c\) is the speed of light. The closer \(v\) gets to \(c\), the slower time moves relative to the stationary observer.

In our exercise, by plugging \(v = 0.988c\) into the time dilation formula, we see the significant effect: only about 6.98 years pass on the spaceship when 26.3 years elapse on Earth.

The Fundamental Role of the Speed of Light

The speed of light, represented as \(c\), is a cornerstone of the theory of relativity. It is a constant in all frames of reference, meaning it remains the same irrespective of the motion of the light source or observer.

• Universal Speed Limit: In relativity, nothing can travel faster than light. This universality is why we use the speed of light as a yardstick for cosmic distances (light-years) and for time intervals in relativistic calculations.
• Communication Delays: For messages, as in the spaceship exercise, the time it takes for a signal to travel from Vega to Earth is governed by this speed. After the traveler reaches Vega, Earth's observers wait an additional 26 light-years for the signal to arrive, reinforcing the constraint of \(c\) as the ultimate speed for information transfer.

This exercise shows clearly how the speed of light impacts not only the perception of travel time but also the communication across vast distances in space.

Impacts of Space Travel at High Speeds

Traveling through space at a speed close to that of light, as proposed in the scenario, presents unique challenges and phenomena due to relativistic effects.

• Length Contraction: When moving at near-light speeds, not only time dilation occurs, but also length contraction. To the traveler, distances shrink in the direction of motion, which contributes to shorter perceived travel times when compared to measurements from a stationary reference frame.
• Vivacious Aging Effects: The age difference calculated for the traveler versus those on Earth highlights the curious difference in aging: Earth observers calculate the traveler ages 19.32 years less during the journey. This has profound implications for long-term space travel, where returning astronauts may find their friends and family much older than expected.

Such impacts are crucial considerations in planning future space missions, especially those that might involve speeds approaching the speed of light or travels to distant stars.