Relative proper times in a compactified universe

• spaghetti3451
In summary, the conversation discusses the concept of a finite box or three-torus in space and how it affects the relative proper times experienced by two observers, one at rest and one moving at a constant velocity. It raises the question of whether different inertial observers in this universe are equivalent and provides links for further reading on the topic.

Homework Statement

Imagine that space (not spacetime) is actually a finite box, or in more sophisticated terms, a three-torus, of size ##L##. By this we mean that there is a coordinate system ##x^{\mu}=(t,x,y,z)## such that every point with coordinates ##(t,x,y,z)## is identified with every point with coordinates ##(t,x+L,y,z)##, ##(t,x,y+L,z)## and ##(t,x,y,z+L)##. Note that the time coordinate is the same. Now consider two observers; observer ##A## is at rest in this coordinate system (constant spatial coordinates), while observer ##B## moves in the ##x##-direction with constant velocity ##v##. ##A## and ##B## begin at the same event, and while ##A## remains still, ##B## moves once around the universe and comes back to intersect the worldline of ##A## without ever having to accelerate (since the universe is periodic). What are the relative proper times experienced in this interval by ##A## and ##B##? Is this consistent with your understanding of Lorentz invariance?

The Attempt at a Solution

Observer ##A## is at rest, so the time elapsed for observer ##A## between observer ##B##'s exit and subsequent return is the proper time for observer ##A##, which is given by ##\tau_{A}=\frac{L}{v}##

The proper time for observer ##B## is given by ##\tau_{B}=\frac{L/\gamma}{v}##.

Are my answers correct? I am a little hesitant about my understanding of what the question actually requires (with regards to the relative proper times), since, as far as I know, there is only one proper time connecting two events, and that's the time elapsed in a frame moving at a velocity ##v## such that it intersects the two events. But, in this problem, I found two proper times.

failexam said:
Observer ##A## is at rest, so the time elapsed for observer ##A## between observer ##B##'s exit and subsequent return is the proper time for observer ##A##, which is given by ##\tau_{A}=\frac{L}{v}##

The proper time for observer ##B## is given by ##\tau_{B}=\frac{L/\gamma}{v}##.

Are my answers correct? I am a little hesitant about my understanding of what the question actually requires (with regards to the relative proper times), since, as far as I know, there is only one proper time connecting two events, and that's the time elapsed in a frame moving at a velocity ##v## such that it intersects the two events. But, in this problem, I found two proper times.
Yes, your answer for the times of A and B are correct. So, in this universe, are different inertial observers equivalent?

There is a lot of discussion that you can find on this topic. For example, here at the forum you can look at https://www.physicsforums.com/threads/the-cosmological-twin-paradox.51197/ which contains some links to papers on the web.

Also,

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1. What is a compactified universe?

A compactified universe is a theoretical concept in physics and cosmology that involves the idea of "wrapping up" or compacting some of the dimensions of our observable universe. This is often used in the study of string theory and other theories of quantum gravity.

2. How does a compactified universe affect the concept of relative proper time?

In a compactified universe, the concept of relative proper time is still applicable. However, due to the compactification of dimensions, there may be variations in the measurement of proper time between different observers, depending on their location and motion within the compactified space.

3. Can relative proper time be measured in a compactified universe?

Yes, relative proper time can still be measured in a compactified universe. However, the measurement may be affected by the compactification of dimensions and may vary for different observers.

4. How does the concept of time dilation apply in a compactified universe?

Time dilation, the phenomenon where time appears to pass slower for an observer in motion, still applies in a compactified universe. However, the compactification of dimensions may cause variations in the amount of time dilation experienced by different observers.

5. Is there any evidence for the existence of a compactified universe?

Currently, there is no direct evidence for the existence of a compactified universe. However, some theories in physics and cosmology, such as string theory, suggest the possibility of a compactified universe, and ongoing research and experiments are being conducted to test these theories.