# Find the Proper Time Extrema: Twin Paradox Explained

• Ja4Coltrane
In summary, in the twin paradox, the clock on the geodesic path (the twin who stays at home) will record a longer time interval than the clock on the accelerated path (the twin who goes on a journey and accelerates). This is also true in the thought experiment of throwing the clock up and catching it when it comes back down. The clock on the geodesic path, which experiences no acceleration, will measure a longer proper time than the clock on the distorted path, which experiences acceleration. This is due to the fact that in special relativity, inertial observers always measure the maximum proper time, while in general relativity, inertial observers may measure the maximum, minimum, or even a saddle point of proper time.
Ja4Coltrane
Quick question:
Suppose I hold two initially synchronized clocks on Earth and throw one up and catch it when it comes back down. Now my (small amount of) knowledge of GR tells me that the proper time on the thrown clock should be maximized since it was on a geodesic.

However, this seems like the twin paradox, and in the twin paradox, the thrown clock shows less time than the stationary clock.

Ja4Coltrane said:
Quick question:
Suppose I hold two initially synchronized clocks on Earth and throw one up and catch it when it comes back down. Now my (small amount of) knowledge of GR tells me that the proper time on the thrown clock should be maximized since it was on a geodesic.

However, this seems like the twin paradox, and in the twin paradox, the thrown clock shows less time than the stationary clock.

In the twin paradox, just as in your thought experiment, the clock on geodesic (the twin who stays at home) records a longer time interval than the clock on the distorted path (the twin who flies to a far away star and comes back).

Ja4Coltrane said:
Suppose I hold two initially synchronized clocks on Earth and throw one up and catch it when it comes back down. Now my (small amount of) knowledge of GR tells me that the proper time on the thrown clock should be maximized since it was on a geodesic.

However, this seems like the twin paradox, and in the twin paradox, the thrown clock shows less time than the stationary clock.
You are correct. The thrown clock is on a geodesic (provided we measure from just after it leaves the thrower's hand until just before it returns) so the proper time is maximized. This is exactly analogous to the twin paradox since the thrown clock is the one that goes in a "straight" line. It travels inertially and does not measure any proper acceleration.

Ja4Coltrane said:
Now my (small amount of) knowledge of GR tells me that the proper time on the thrown clock should be maximized since it was on a geodesic.
I think that only holds in SR. In GR geodesic world lines maximize proper-time only locally. You can also have two geodesics meeting twice with different proper-times intervals in between.

Last edited:
Either tonight or tomorrow, I'll post (some of) the fairly simple details of a GR example for which the usual SR result doesn't hold, i.e., the elapsed proper time between meetings for an accelerated clock is greater than for a non-accelerated (geodesic) clock.

The example consists of two clocks that have same $r$, with one clock in geodesic circular orbit (freely falling with no acceleration) and one clock hovering (accelerated).

Ja4Coltrane said:
However, this seems like the twin paradox, and in the twin paradox, the thrown clock shows less time than the stationary clock.

You're misidentifying which clock in the Twin Paradox is the "thrown" clock. In that example, it is the twin that remains on Earth that is analogous to the clock you throw in your own example. In the twin paradox, we are ignoring effects of gravity (which are minimal in any case), so we're really talking about a freely-floating twin (in place of the Earth) and a twin that goes to Vega and accelerates when he gets there. The accelerated twin knows that it was he who accelerated because he feels the acceleration (the acceleration is recorded by an accelerometer, i.e. it causes a mass-on-a-spring to stretch).

In your example, it is the earthbound clock whose accelerometer stretches, while the freely falling clock experiences zero acceleration. Hence the freely falling clock will have the extreme aging.

Note, however, that while in SR you are guaranteed that the inertial observer will have aged by the maximum amount, in GR you are only guaranteed that inertial observers (with no acceleration on their accelerometers) age by extreme amounts. That amount could be maximum, minimum, or even a saddle point. In this example, however, it will indeed be maximum.

## 1. What is the "Twin Paradox"?

The "Twin Paradox" is a thought experiment in special relativity that describes the different aging rates of two identical twins who are separated and then reunited. This paradox highlights the concept of time dilation, where time passes at different rates for observers in different frames of reference.

## 2. How is the "Twin Paradox" explained?

The "Twin Paradox" is explained by considering the different frames of reference of the twins. The twin who stays on Earth is in a relatively stationary frame of reference, while the other twin is in a moving frame of reference due to their travel. This results in the moving twin experiencing time at a slower rate, causing them to age less than the twin on Earth.

## 3. What is the role of velocity in the "Twin Paradox"?

The role of velocity in the "Twin Paradox" is crucial, as it is the factor that causes time dilation to occur. The faster an object moves, the slower time passes for that object from the perspective of a relatively stationary observer. This effect becomes more significant as the velocity approaches the speed of light.

## 4. Are there any real-life examples of the "Twin Paradox"?

While the "Twin Paradox" is a thought experiment, there are real-life examples that demonstrate the concept of time dilation. For instance, astronauts on the International Space Station experience time at a slightly slower rate than those on Earth due to their high orbital velocity. Additionally, particles in particle accelerators can also experience time dilation due to their high speeds.

## 5. How does the "Twin Paradox" impact our understanding of time?

The "Twin Paradox" challenges our traditional understanding of time as a universal constant. It shows that time is not absolute and can vary depending on the frame of reference of the observer. This concept is crucial in the field of physics and has led to the development of theories such as relativity that have greatly impacted our understanding of the universe.

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