Visualizing Time Dilation: Exploring the Effects of Relativity on Twin's Journey

[gonegahgah]

Are there any graphs that depict the amount of time dilation that occurs by the fraction of the speed of light for the traveling twin?

 

[JesseM]

You mean, time dilation factor on one axis and velocity on the other? If so just enter "time dilation graph" into a google image search, you'll various examples such as this one.

 

[gonegahgah]

Thanks for that Jesse.

 

[gonegahgah]

I'm not sure what the graph results mean.
The formula give the results of:
0.866c -> 2
0.9428c -> 3
0.9682c -> 4
0.9798c -> 5
etc
Does this mean that the traveling twin will appear to the home twin to take twice as long to travel to and back from their destination when they travel at 0.866c, three times as long when they travel at 0.9428c, four times as long at 0.9682c, etc.?
Can someone please clarify if this is what is meant or something else?

 

[JesseM]

I'm not sure what the graph results mean.
The formula give the results of:
0.866c -> 2
0.9428c -> 3
0.9682c -> 4
0.9798c -> 5
etc
Does this mean that the traveling twin will appear to the home twin to take twice as long to travel to and back from their destination when they travel at 0.866c, three times as long when they travel at 0.9428c, four times as long at 0.9682c, etc.?
Can someone please clarify if this is what is meant or something else?

No, the formula doesn't refer to travel time, it refers to how much the time between ticks of the traveling twin's clock are elongated (dilated) in the Earth twin's frame. For example, if the traveling twin is moving at 0.866c in the Earth's frame, then it takes 2 seconds of time in this frame for the traveling twin's clock to tick forward by 1 second. You can also take the inverse of these numbers to see how much less the traveling twin will have aged in total between leaving Earth and returning, assuming he travels at uniform speed; for example if the traveling twin travels away and back at 0.866c, then if the Earth twin has aged N years during that time, the traveling twin has only aged N/2 years upon return.

The actual travel time in the Earth's frame depends only on the velocity in the Earth's frame, for example if the traveling twin goes to a destination 5 light-years away and returns to Earth, traveling at 0.866c the whole time, the time of this journey in the Earth's frame is (5+5)/0.866 = 11.547 years (which means upon return, the Earth twin will be 11.547 years older while the traveling twin is only 5.7735 years older).

 

[ghwellsjr]

Just in case you were asking for the relative age of the traveling twin compared to the stay-at-home twin, here is a graph that depicts that:

If you complare this graph to the one that JesseM pointed you to, you will see that it is simply the reciprocal of the Y axis. You can also see that the shape of the graph is a simply quarter of a circle.

 

[GrayGhost]

Does this mean that the traveling twin will appear to the home twin to take twice as long to travel to and back from their destination when they travel at 0.866c, three times as long when they travel at 0.9428c, four times as long at 0.9682c, etc.?

gonegahgah,

The gamma factor at 0.866c is 2. This means that the traveling twin ages 1/gamma = 1/2 as much as the inertial twin over the very same interval. As JesseM pointed out, this is the same as saying that the traveling twin experiences a dilated (stretched) duration by a factor of gamma = 2, wrt the traveling twin's aging as the reference.

I recommend you forget the classic twins scenario, and instead focus on all-inertial scenarios first. In the case of the classic twins scenario, one twin (B) undergoes a proper acceleration, and the analysis of the scenario is much more complex. Master the easier all-inertial scenarios first, then go into the more complicated ones. It will save you a great amount of time in the end. Food for thought.

GrayGhost

 

[John232]

It increases exponentially as the object travels closer to the speed of light. Most text that teach the theory should show it...

 

[gonegahgah]

Thank you for that information. That explains it to me a lot better.