• dwintz02
In summary, the conversation discusses a scenario where one twin travels in a rocket while the other stays on Earth. The rocket is constructed to have an acceleration of g in its instantaneous rest frame and travels for a total of 20 years (5 years of acceleration, 5 years of deceleration, 5 years of acceleration, and 5 years of deceleration) before landing back on Earth. The questions posed are: a) How old is Pollux according to his watch? b) What year is it on Earth? c) How far away from the Earth did the rocket travel? The conversation also mentions the use of relativistic velocity, time, and acceleration formulas to solve the problem.
dwintz02

## Homework Statement

Assume a rocket ship leaves the Earth in the year 2100. Castor, one of a set of identical twins born in 2080, remains on Earth to work at Mission Control, while the other twin, Pollux, travels in the rocket. Ignore the motion of the Earth relative to the fixed stars. The rocket is constructed so that it has an acceleration g in its instantaneous rest frame (making the astronauts feel at home) Suppose the rocket accelerates in a straight line path for 5 years (by its OWN chronometer), decelerates at the same rate for another 5 years (of its own time), turns around (in negligible time), accelerates back for 5 years, decelerates for 5 years, and then lands on earth.

a) How old is Pollux according to his watch?
b) What year is it on Earth?
c) How far away from the Earth did the rocket travel?

## Homework Equations

Relativistic velocity formulas, time formulas, and acceleration formulas.

## The Attempt at a Solution

So I've taken a time derivative of the velocity formulas and it matches with my book, but this only gives me accelerations in terms of other reference frames. What I'm having trouble with is calculating the equation of motion for the rocket according to the astronauts. How can I continuously add their velocities (because they are accelerating), to say get their velocity as a function of time in their frame? And once I get that, how can I handle the fact that gamma is changing in the time dilation formula?

Daniel

## 1. What is the Twin Paradox with acceleration?

The Twin Paradox with acceleration is a thought experiment in the theory of relativity. It explores the concept of time dilation, where time moves slower for objects in motion compared to those at rest. In this scenario, one twin travels at high speeds while the other stays on Earth. When the traveling twin returns, they will have aged less than the twin on Earth, leading to the paradox.

## 2. What is the difference between the Twin Paradox and the Twin Paradox with acceleration?

The original Twin Paradox only involves one twin traveling at a constant speed, while the Twin Paradox with acceleration involves the traveling twin accelerating and decelerating. This acceleration adds a new component to the thought experiment, as it affects the twin's perception of time and can lead to different outcomes.

## 3. Is the Twin Paradox with acceleration possible in real life?

While the thought experiment is based on the principles of relativity, it is highly unlikely to occur in real life. The speeds and accelerations required for the paradox to be noticeable are not achievable with current technology. Additionally, it is difficult to isolate one twin from all external factors that could affect their aging.

## 4. How is the Twin Paradox with acceleration resolved?

The paradox is resolved by understanding that both twins experience time dilation, but in different ways. The traveling twin experiences time moving slower during acceleration and deceleration, while the twin on Earth experiences time moving slower due to their constant motion. When the traveling twin returns, their experiences of time dilation will result in the paradox being resolved.

## 5. What are the implications of the Twin Paradox with acceleration for space travel?

The Twin Paradox with acceleration highlights the effects of high speeds and acceleration on time. This has implications for space travel, where astronauts traveling at high speeds will experience time dilation. It also raises questions about the possibility of time travel and the need for precise calculations when planning long-distance space missions.

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