# Time Dilation, calculate the age of the returning astronaut

apolloxiii

## Homework Statement

"In 2010, a 20-year-old astronaut leaves her twin on Earth and goes on a rocket to explore the galaxy. The rocket moves at 2.7 x 10^8 m/s during the voyage. It returns to earth in the year 2040. Using relativity, calculate the age of the returning astronaut."

## Homework Equations

Δtm = Δts / √(1-((v^2)/(c^2)))

## The Attempt at a Solution

It is a simple calculation, but I am having trouble knowing where to plug in the 30 years of time passed on earth. Subscript "s" is supposed to represent time passed with the observer at rest with respect to an event, and subscript "m" represents time passed with the observer in motion with respect to the event.

Orodruin
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What do you think and why?

time passed with the observer at rest with respect to an event

Note that this statement makes no sense. An event is a single instance in time at a single point in space.

apolloxiii
What do you think and why?

I think that the 30 years is inserted into the equation in the place of Δts, because 30 years has passed from the viewpoint of a stationary observer on earth. It seems to me that solving for Δtm would give you the change in time in the frame of reference of somebody inside the rocket, because that is the frame of reference that is in motion.

Note that this statement makes no sense. An event is a single instance in time at a single point in space.

What I meant to say, and as it says in my textbook, is that the subscript "s" means that the observer is at rest with respect to the event, and that "m" means that the observer is in motion with respect to the event.

Orodruin
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What I meant to say, and as it says in my textbook, is that the subscript "s" means that the observer is at rest with respect to the event, and that "m" means that the observer is in motion with respect to the event.
Again, there is no such thing as being at rest "with respect to an event". An event in itself does not specify a state of motion.

Also note that a reference frame cannot simply "be in motion", motion is always relative to something.

apolloxiii
Again, there is no such thing as being at rest "with respect to an event". An event in itself does not specify a state of motion.

Also note that a reference frame cannot simply "be in motion", motion is always relative to something.

Thanks for the reply. So far studying physics I have not come across the term "event" in a technical sense, so I was not aware of that. I think that the book is using the word event to describe the rocket's trip. An observer on the earth is stationary relative to the event of the rocket's voyage. I thought I should be placing the 30 elapsed years in the place of Δts, because 30 years on earth have passed, and observers on earth are stationary relative to the rocket.

Orodruin
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I think that the book is using the word event to describe the rocket's trip.
If it does that you should immediately throw it away and go to get a better book.

An observer on the earth is stationary relative to the event of the rocket's voyage.
Again, this statement has no meaning. You can only be stationary in a reference frame or relative to another object. It is simply meaningless to call something stationary relative to an event so when you repeat this we have to guess what your intended meaning is and there is no way of really knowing.

observers on earth are stationary relative to the rocket.
This is not true. One thing we can definitely tell is that the Earth and rocket move relative to each other - their relative speed is given to be non zero in the problem.

apolloxiii
This is not true. One thing we can definitely tell is that the Earth and rocket move relative to each other - their relative speed is given to be non zero in the problem.

Ok, so what we are given in the problem is the rocket's speed...relative to the earth? The way I see it, the only information you need to do this problem is one object's speed relative to another, which we have been given. How do I know which variable to solve for? It makes sense to me that I should be solving for the variable of time which is associated with the object that is moving with respect to the other, Δtm, but this book is telling me that Δtm is 30 years, and I need to solve for the change in time experienced by the "stationary" observer.

Orodruin
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the object that is moving with respect to the other
This represents a fundamental misunderstanding of relativity. Both objects are moving relative to the other.
I think it is very bad notation to call an object stationary and the other moving without explicit reference to a given inertial frame, but this might be on the author of your book.

apolloxiii
This represents a fundamental misunderstanding of relativity. Both objects are moving relative to the other.
I think it is very bad notation to call an object stationary and the other moving without explicit reference to a given inertial frame, but this might be on the author of your book.

How would you phrase it, giving explicit reference to a given inertial frame?

apolloxiii
Also, here is a direct quote from the book.
"Up to this point, you have learned that there is no such thing as absolute time or length. Just
by changing from one inertial reference frame to another, the amount of time and length
measured can change. For both of these concepts, you were able to derive equations that
govern how time and length vary from one at-rest reference frame to another. What do you
think happens to the mass of an object if it is viewed from a moving reference frame?"

Is that last sentence ("moving reference frame") another example of the author having a misunderstanding of relativity? Based on what you've said it seems that there is no such thing as a "moving reference frame", just inertial and accelerated.