Silly time dilation question

[rede96]

I was wondering, imagine if I had twin brothers who each set off in their space ships at the same time, at a speed of 0.5c.

One traveled 20 light years in a straight line, then turned around and traveled 20 light years back to me, the other traveled along a complete circular path, with a circumference of 40 light years.

When the twin brothers retuned to me, which one would have appeared to have aged more?

 

[JesseM]

If they both traveled at the same constant speed they'd both have aged by the same amount, in an inertial frame the rate of a clock depends only on its speed, not on other aspects of its motion.

 

[rede96]

If they both traveled at the same constant speed they'd both have aged by the same amount, in an inertial frame the rate of a clock depends only on its speed, not on other aspects of its motion.

Thanks JesseM.

So acceleration does not play a part? I made an assumption that if the two twins had different rates of accelerations during their trips, then they would age differently.

Is this assumption wrong?

 

[JesseM]

Thanks JesseM.

So acceleration does not play a part? I made an assumption that if the two twins had different rates of accelerations during their trips, then they would age differently.

Is this assumption wrong?

The rate a clock is ticking at any given moment is a function only of its instantaneous speed at that moment, not its acceleration.

 

[jfy4]

The rate a clock is ticking at any given moment is a function only of its instantaneous speed at that moment, not its acceleration.

How does this concept extend to sitting in the gravitational field? Say, I'm sitting on the earth, and someone is 50 ft up in the air. GR says our clocks should run differently. This is a result due to the equivalence between acceleration and uniform gravitational potential is it not? Is this a different matter?

 

[JesseM]

How does this concept extend to sitting in the gravitational field? Say, I'm sitting on the earth, and someone is 50 ft up in the air. GR says our clocks should run differently. This is a result due to the equivalence between acceleration and uniform gravitational potential is it not?

GR time dilation can only be seen as "equivalent" to any form of SR time dilation if you are looking at a small window of space and a brief window of time, such that the effects of spacetime curvature are negligible. I know one can derive the fact that the bottom clock sees the top clock blueshifted and the top clock sees the bottom clock redshifted using the equivalence principle, but I'm not sure if one can actually derive the fact that the bottom clock is actually accumulating less time. If you can it may have something to do with the fact that two objects which are both a constant distance apart in a gravitational field would have to be equivalent to two objects undergoing "Born rigid acceleration" in flat spacetime so that the distance between them as measured in either one's instantaneous inertial rest frame would stay constant from one moment to another...Born rigid acceleration actually requires that the one in front have a higher proper acceleration (see the discussion of Rindler observers, who accelerate in a Born rigid way, here, and also see the Bell spaceship paradox), therefore attains a higher speed more quickly in the inertial frame where they were initially at rest, therefore has greater time dilation in this frame.

 

[jfy4]

Paging through Hartle's Gravity (page 116) I find this relation

$$\Delta\tau_B=\Delta\tau_A\left(1-\frac{gh}{c^2}\right)$$

which is the difference in the rate of reception and emission between two different observers at two different heights in the gravitational field. This appears to constitute time dilation does it not? An observer at high altitude ages more due to acceleration, right?

 

[rede96]

Paging through Hartle's Gravity (page 116) I find this relation

$$\Delta\tau_B=\Delta\tau_A\left(1-\frac{gh}{c^2}\right)$$

which is the difference in the rate of reception and emission between two different observers at two different heights in the gravitational field. This appears to constitute time dilation does it not? An observer at high altitude ages more due to acceleration, right?

Thanks jfy4. (Sorry for the late reply, I've been working away for a few days.) That is part of what I was getting at. After all we adjust the GPS system to account for the effects of GR.

However my original question was born from the Twin Paradox, ie who is moving away from who and thus what determines who ages at a different rate.

Now I am just a novice at this stuff, part time novice at best! So please forgive me if my understanding is a little elementary.

But from reading some of the Twin Paradox threads, it seems that it is the acceleration the twin undergoes that determines that he ages less.

So therefore, my simple logic would say that if it is acceleration that determines which Twin ages, then different rates of acceleration might have different results for how much the twin ages during his trip.

e.g would the twin who went in a straight line and then back again experience different effects (in terms of time dilation) then the other twin going in a giant circle and hence in a constant state of acceleration.

 

[WannabeNewton]

Thanks jfy4. (Sorry for the late reply, I've been working away for a few days.) That is part of what I was getting at. After all we adjust the GPS system to account for the effects of GR.

However my original question was born from the Twin Paradox, ie who is moving away from who and thus what determines who ages at a different rate.

Now I am just a novice at this stuff, part time novice at best! So please forgive me if my understanding is a little elementary.

But from reading some of the Twin Paradox threads, it seems that it is the acceleration the twin undergoes that determines that he ages less.

So therefore, my simple logic would say that if it is acceleration that determines which Twin ages, then different rates of acceleration might have different results for how much the twin ages during his trip.

e.g would the twin who went in a straight line and then back again experience different effects (in terms of time dilation) then the other twin going in a giant circle and hence in a constant state of acceleration.

But it isn't acceleration. It is the relative velocity of the observer in the CMRF.

 

[JesseM]

But it isn't acceleration. It is the relative velocity of the observer in the CMRF.

The relative aging of the twins has nothing to do with the velocity relative to any particular frame (by CMRF do you mean the cosmic microwave background?) The twin that moves inertially between the two meetings will always age more than the twin that accelerates to turn around at some point, regardless of the inertial twin's velocity relative to the CMRF.

 

[JesseM]

So therefore, my simple logic would say that if it is acceleration that determines which Twin ages, then different rates of acceleration might have different results for how much the twin ages during his trip.

No, it's not the rates of acceleration, it's the overall shape of the path through spacetime. Think of it in analogy with paths through 2D space--if you have two points on a plane A and B, and one person travels on a straight line path between A and B while the other travels in a path that looks like two straight segments connected by a short curve (a "V" shape but with the V having a slightly rounded bottom), the one whose path has a non-straight segment will always have a greater path length. But the reason isn't because that person accumulated all the extra distance on the curved part of the path, it's mostly because of the sum of the two segments of the V add up to more than the straight-line path, as in plane geometry a straight line is the shortest distance between two points. I elaborated on this geometric analogy in [post=2972720]this post[/post] if you want to see more details.

Here's a diagram DrGreg posted on another thread a while ago that may help:

Here we have three paths through spacetime which all start from the same location in 2000, and end at the same location in 2020. Path C travels inertially between these two meetings, while the other paths A & B both involve 3 accelerations of exactly the same magnitude and time. But because path B deviates less from the straight path C (since it travels alongside C until accelerating in 2010), its proper time will be closer to that of B (it will have a greater proper time, closer to the maximum proper time of path C), just like how geometrically path B (which includes the straight segment from 2000-2010, not just the "hump" from 2010 to 2020) has a shorter overall length than path A in the diagram.

 

[rede96]

No, it's not the rates of acceleration, it's the overall shape of the path through spacetime.

Hi Jesse, thanks for the reply.

I think I'm getting it, but bear with me a mo :)

Two twins set off from the start point at the same time, leaving me behind at the starting point. (See attached diagram.)

They both travel at the same speed say 0.5c, both travel for the same distance, (eg total distance traveled not distance from the starting point.) but take different paths.

Path A is basically a straight line round trip and Path B is a circular path.

So I guess the questions would be:

a) Would I see them both return to the starting point at the same point in time?

b) Have they both aged the same amount when they returned?

c) Would they have aged at a different rate than me?

 

[JesseM]

So I guess the questions would be:

a) Would I see them both return to the starting point at the same point in time?

b) Have they both aged the same amount when they returned?

c) Would they have aged at a different rate than me?

Yes to all three (and to the last, they have aged less than you when they return).