# How to Solve the Twin Paradox Problem Using Lorentz Transformations

• GammaLyra
In summary, the problem involves a twin paradox where one twin, Alice, travels at a high speed to a distant star and back while the other twin, Bob, remains on Earth. The problem asks for the coordinates of the jump between the outbound and inbound frames, as well as the age of Bob in the outbound frame before the jump. The Lorentz transformations are used to solve the problem, but there is confusion about the coordinates of Bob in the outbound frame. The error lies in the understanding of proper time and the fact that in the outbound frame, Alice is stationary and her x coordinate is 0. This means that Bob's x coordinate in the outbound frame will also be 0, which may seem counterintuitive but is correct
GammaLyra

## Homework Statement

This is a typical twin paradox problem as laid out in Griffith's Introduction to Electrodynamics, problem 12.16. The problem states that, on their 21st birthday, one of two twins - we'll call her Alice - departs Earth for star X at (4/5)c. Upon arriving at star X, she immediately begins the return journey also at (4/5)c. She returns home at the age of 39 (according to her watch). Her twin Bob remains at Earth during the entirety of the trip.

There are two reference frames associated with Alice: the "outbound" frame in which she travels toward star X away from Earth, and the "inbound" frame in which she returns to Earth. There is also the Earth reference frame.

Part (d) of the problem asks: what are the coordinates (x, t) of the jump (from the outbound frame to the inbound frame) in the outbound frame?

Part (g) of the problem asks: how old does Alice say her brother is right now just before she makes the jump? I.e., how old does Alice think Bob is in the outbound frame right before she jumps to the inbound frame?

## Homework Equations

All that's needed are the usual Lorentz transformations:

talice = γ(tbob - (v/c2)xbob)
tbob = γ(talice + (v/c2)xalice)

## The Attempt at a Solution

I correctly solved part (d) by finding the coordinates of the jump in the Earth frame and transforming them to the outbound frame. This meant transforming the coordinates (12 ly, 15 yrs) to (0, 9 yrs).

For part (g), I want to use the first Lorentz transformation above and solve for tbob:

tbob = talice/γ + (v/c2)xbob

The problem is that I think xbob should be nonzero, when the correct solutions use 0 for this variable. That doesn't make sense to me. As stated above, I think what part (g) is asking is what Bob's coordinates (x, t) are in the outbound frame right before the jump. In the outbound frame, Alice was stationary and her x coordinate in this frame should be 0. Consequently, Bob's x coordinate in the outbound frame should be nonzero, yet it is not. What is the error in my understanding?

Any insight is highly appreciated. Thank you.

Clue: what is the physical interpretation of proper time?

## 1. What is the "twin paradox" problem?

The twin paradox is a thought experiment in Einstein's theory of relativity which examines the concept of time dilation. It involves two twins, one of whom travels at a high speed while the other remains on Earth. When the traveling twin returns, they have aged less than the twin who stayed on Earth, leading to the paradox.

## 2. How does the theory of relativity explain the twin paradox?

According to the theory of relativity, time is relative and can be affected by the speed at which an object is moving. The traveling twin experiences time dilation, meaning that time moves slower for them compared to the stationary twin. This is due to the fact that the faster an object moves, the more it warps the fabric of space-time.

## 3. Is the twin paradox a real phenomenon?

No, the twin paradox is a thought experiment and not a real phenomenon. It is used to illustrate the principles of relativity and does not occur in real life. However, time dilation has been observed and measured in experiments involving high-speed particles, confirming the predictions of the theory of relativity.

## 4. Can the twin paradox be resolved?

Yes, the twin paradox can be resolved by taking into account the acceleration and deceleration of the traveling twin. This is because acceleration affects time dilation, and the traveling twin experiences a change in acceleration when they turn around to return to Earth. When this is taken into consideration, both twins will have experienced the same amount of time when they reunite.

## 5. How does the twin paradox relate to space travel?

The twin paradox is often used to explain the effects of time dilation on space travel. As a space voyager travels at high speeds, time will move slower for them compared to people on Earth, meaning they will age at a slower rate. This has important implications for long-distance space travel and the aging of astronauts compared to people on Earth.

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