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Relativity Example - More Details

If you're reading this page on its own, it won't make much sense. This page is designed to give more detail to the twins paradox example shown on this page.

These extra details are all shown on a single page here to make it easy to print out with the example.


Background on the Twins Paradox

The twin paradox is one of the most often described aspects of special relativity. Unfortunately, it is also one of the most poorly explained. The heart of the paradox is that relativity requires that each twin observe the other's clock as running slower than normal. But since the twins age differently, one of them must have observed the other to have progressed more rapidly than normal through time.

Some texts simply state that the discrepancy is caused because special relativity doesn't handle accelerated frames of reference, and one of the twins is accelerated (3 or 4 times). Others go a little farther with the following explanation:

Each of the twins sees the other's clock as running slow when they are moving apart and fast when they are moving together (due to the Doppler shift). The twin in the rocket sees the transition occurring the instant the rocket turns around and thus sees the Earthbound clock running slow during the first half of the journey and fast during the last half. But the twin on Earth doesn't see the change in direction until well past the halfway point. This is due to the fact that is takes a long time for the turnaround to be seen because it is so far away. Thus the twin on Earth sees the clock on the rocket as running slow for more than half of the duration and fast for less than half. This asymmetry leads to the discrepancy in the twin's ages.

While this is technically accurate, its rather unsatisfying. For one thing, we don't observe similar time discrepancies when dealing with sub-light (e.g. sound) Doppler shifts. For another, it doesn't explain when the twin in the rocket observes the other twin to be moving too fast in time. Relativity indicates that during both the outbound trip and inbound trip, the twin in the rocket must observe Earth's clocks as running slow.

I hope that if you read carefully through the example, and also the extra details on this page, you will have a clear picture of how this all fits together.

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Before Bob Leaves - More Details   top

Here are the tables showing the situation just before Bob leaves. These are trivial, but I've included them for completeness.

Bob's Time 2000 Bob Sees Bob Calculates
Earth Time 2000
advancing at same rate as Bob's
2000
advancing at same rate as Bob's
Star Time 1992
advancing at same rate as Bob's
2000
advancing at same rate as Bob's
Distance to Earth 0 0
not moving
Distance to Star 8 light years 8 light years
not moving

Ann's Time 2000 Ann Sees Ann Calculates
Rocket Time 2000
advancing at same rate as Ann's
2000
advancing at same rate as Ann's
Rocket Distance 0 0
not moving
Star Time 1992 2000, 8 light years away

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When Bob Leaves - More Details   top

Bob's time 2000 Bob sees Bob calculates
Earth time 2000
advancing at 33% of his own
2000
advancing at 60% of his own
Star time 1992
advancing at 300% of his own
2006.4
advancing at 60% of his own
Distance to Earth 0 0
receding at 80% speed of light
Distance to Star 4.8 light years 4.8 light years
approaching at 80% speed of light

Ann's time 2000 Ann sees Ann calculates
Rocket time 2000
advancing at 33% of her own
2000
advancing at 60% of her own
Star time 1992 2000
8 light years away
Distance to Rocket 0 0
receding at 80% speed of light

Here you can see that Bob calculated that the Star clock showed 2000 (same as Earth time) when he left. He can also calculate that because the star is now 3.2 light years closer, that its time has also jumped 6.4 years forward. Of course, he still sees the same time of 2000 (more below).


How can the rate of time change?

It is due to the Doppler Effect. A good analogy is listening to a train whistle change pitch as it passes you. As the train approaches you, the sounds waves between the train and you are compressed, causing you to hear a higher pitch than heard by travellers on the train.

When the train travels away from you, the sound waves are "stretched" and you hear a lower pitch. The train travellers hear a constant pitch, because they are travelling with the train.

In precisely the same way, the time you see on a distant object advances faster than your own as you move towards it, and advances slower than your own as you move away from it. Note that you always see time move forward, it just moves at different rates.


Why doesn't Bob suddenly see the star time showing 1995.2 (4.8 years ago) since the star is now only 4.8 light years away and light always travels at the same speed (regardless of observers' motion)?

Imagine that the clock on the star sends information to you, not in the form of an image of a clock-face, but suppose that it sends out pulses of light in Morse code. This message will not change as pulses travel through space, regardless of the observer'’s frame of reference. The pulses for the time 1992 arrive at Earth when Earth time is 2000.

All observers will see the 1992 pulses hit the Earth when Earth clocks read 2000. Since Bob is still at Earth and the time is still 2000, those 1992 pulses will be hitting his rocket as he departs.

What Bob does see change is the rate of time on the star clock. In this case he sees it advance 3 times faster than his own time.

Bob doesn't see the clock on the star change as he accelerates. He can, however, calculate that it jumps ahead by 6.4 years because it is now closer by 3.2 light-years. Bob sees the Earth clock advancing at less than the 60% he calculates, due to the Doppler effect. The Doppler effect also causes him to see the Star's clock running faster than normal. When he takes the Doppler effect into account, he observes both clocks running slower than normal. Also, Bob's observations of the star's time are extrapolations based on the assumption that the motion of the star hasn't changed.

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Bob Halfway to the Star - More Details   top

Bob's time 2003 Bob sees Bob calculates
Earth time 2001
advancing at 33% of his own
2001.8
advancing at 60% of his own
Star time 2001
advancing at 300% of his own
2008.2
advancing at 60% of his own
Distance to Earth 2.4 light years 2.4 light years
receding at 80% speed of light
Distance to Star 2.4 light years 2.4 light years
approaching at 80% speed of light

Ann's time 2005 Ann sees Ann calculates
Rocket time 2001.7
advancing at 33% of her own
2003
advancing at 60% of her own
Star time 1997 2005
8 light years away
Distance to Rocket 2.2 light years 4 light years
receding at 80% speed of light

We can now see why it is meaningless to try and compare Ann's and Bob's times. They both know from the relativity formula that each other's time is passing at 60% of their own rate. So in Ann's 2005, she calculates that Bob must be in 2003. In Bob's 2003, he calculates that Ann must be in 2001.8. In Ann's 2001.8,she calculates ... etc!

Just like you can travel to different places at different speeds, you can travel to different times seeing different time rates. Read the facts page if you're interested in seeing how distance and time changes are related.

It looks like Ann can actually calculate Bob's times, however, it might seem that in this case, Ann's calculations are correct, but Bob's are not. It happens to work out this way because this is a special case where Bob has departed Earth, and is travelling in a straight line from, then back to Earth.

In the real world, where small particles really do whiz by the Earth at very high speeds, their speed relative to Earth is constantly changing. As they pass Earth there is a changing component of their speed and direction relative to Earth.

More importantly though, Ann's calculations are based on an assumption that Bob will follow his travel plans exactly and that nothing goes wrong. Ann cannot conclusively say that she knows where Bob is, how fast he is going, or what his time is until she sees where Bob has been, and Ann won't see this until some time in the future. For example, as we'll see later, she won't see Bob turn around at the star until 8 years (her own) after it happened, and yet Bob is due to arrive home in only another 2 years.

We can see how far an object moves, and calculate how much it ages by knowing how much of our own time has passed while it moves. We can also calculate its speed by measuring how far it travels in our own time.

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Bob Arrives at the Star - More Details   top

These figures are at the instant Bob reaches the star, while he is still travelling at 80% of the speed of light towards it.

Bob's Time 2006 Bob Sees Bob Calculates
Earth Time 2002
advancing at 33% of Bob's rate
2003.6
advancing at 60% of Bob's rate
Star Time 2010
advancing at 300% of Bob's rate
2010
advancing at 60% of Bob's rate
Distance to Earth 4.8 light years 4.8 light years
receding at 80% speed of light
Distance to Star 0 0
approaching at 80% speed of light

Ann's Time 2010 Ann Sees Ann Calculates
Rocket Time 2003.3
advancing at 33% of Ann's rate
2006
advancing at 60% of Ann's rate
Rocket Distance 4.4 light years 8 light years
receding at 80% speed of light
Star Time 2002 2010

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Bob Stops at the Star - More Details   top

Bob's Time 2006 Bob Sees Bob Calculates
Earth Time 2002
advancing at same rate as Bob's
2010
advancing at same rate as Bob's rate
Star Time 2010
advancing at same rate as Bob's
2010
advancing at same rate as Bob's
Distance to Earth 8 light years 8 light years
not moving
Distance to Star 0 0
not moving

Ann's Time 2010 Ann Sees Ann Calculates
Rocket Time 2003.3
advancing at 33% of Ann's rate
2006
advancing at same rate as Ann's
Rocket Distance 4.4 light years 8 light years
not moving
Star Time 2002 2010

Bob calculates the clocks at the Star and Earth to be synchronised only when he is stopped with respect to them. When he is moving, he calculates the clock in front of him to be ahead of the clock behind him.

When he accelerates and decelerates, he doesn't see any change in their time and he neither sees nor calculates any change in the time of nearby clocks. He can, however, calculate that distant clocks jump forward or backward in time when he accelerates and decelerates.

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Bob Leaves the Star - More Details   top

Now Bob is still at the Star, but travelling towards Earth at 80% of the speed of light.

Bob's Time 2006 Bob Sees Bob Calculates
Earth Time 2002
advancing at 300% of Bob's rate
2016.4
advancing at 60% of Bob's rate
Star Time 2010
advancing at 33% of Bob's rate
2010
advancing at 60% of Bob's rate
Distance to Earth 4.8 light years 4.8 light years
approaching at 80% speed of light
Distance to Star 0 0
receding at 80% speed of light

Ann's Time 2010 Ann Sees Ann Calculates
Rocket Time 2003.3
advancing at 33% of Ann's rate
2006
advancing at 60% of Ann's rate
Rocket Distance 4.4 light years 8 light years
approaching at 80% speed of light
Star Time 2002 2010

The distance to Earth has jumped closer by 3.2 light years and Earth's time has jumped into the future by 6.4 light years. (Notice that if instead of accelerating back toward Earth, he had accelerated away from it (in the direction he was originally going) he would have observed the same conditions as just before he stopped; which is to say that he would have observed Earth's time to jump back 6.4 years into the past! However, he would only see Earth's clock slowing down, not running backwards.)

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Bob Halfway Back from the Star - More Details   top

Bob's Time 2009 Bob Sees Bob Calculates
Earth Time 2011
advancing at 300% of Bob's rate
2018.2
advancing at 60% of Bob's rate
Star Time 2011
advancing at 33% of Bob's rate
2011.8
advancing at 60% of Bob's rate
Distance to Earth 2.4 light years 2.4 light years
approaching at 80% speed of light
Distance to Star 2.4 light years 2.4 light years
receding at 80% speed of light

Ann's Time 2015 Ann Sees Ann Calculates
Rocket Time 2005
advancing at 33% of Ann's rate
2009
advancing at 60% of Ann's rate
Rocket Distance 6.7 light years 4 light years
approaching at 80% speed of light
Star Time 2007 2015

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Ann Sees Bob Arrive at the Star - More Details   top

Bob's Time 2010.8 Bob Sees Bob Calculates
Earth Time 2016.4
advancing at 300% of Bob's rate
2019.3
advancing at 60% of Bob's rate
Star Time 2011.6
advancing at 33% of Bob's rate
2012.9
advancing at 60% of Bob's rate
Distance to Earth 0.48 light years 0.48 light years
approaching at 80% speed of light
Distance to Star 4.3 light years 4.3 light years
receding at 80% speed of light

Ann's Time 2018 Ann Sees Ann Calculates
Rocket Time 2006
advancing at 33% of Ann's rate
2010.8
advancing at 60% of Ann's rate
Rocket Distance 8 light years 1.6 light years
approaching at 80% speed of light
Star Time 2010 2018

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Ann Sees Bob Stopped at the Star - More Details   top

Bob's Time 2010.8 (same as above)

Ann's Time 2018 Ann Sees Ann Calculates
Rocket Time 2006
advancing at same rate as Ann's
2010.8
advancing at 60% of Ann's rate
Rocket Distance 8 light years 1.6 light years
approaching at 80% speed of light
Star Time 2010 2018

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Ann Sees Bob Leave the Star - More Details   top

Bob's Time 2010.8 (same as above)

Ann's Time 2018 Ann Sees Ann Calculates
Rocket Time 2006
advancing at 300% of Ann's rate
2010.8
advancing at 60% of Ann's rate
Rocket Distance 8 light years 1.6 light years
approaching at 80% speed of light
Star Time 2010 2018

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Ann Sees Bob Halfway Home - More Details   top

After 19 of Ann's years, she sees Bob halfway home. Of course, Bob will actually arrive home in 1 more of Ann's years.

Bob's Time 2011.4 Bob Sees Bob Calculates
Earth Time 2018.2
advancing at 300% of Bob's rate
2019.6
advancing at 60% of Bob's rate
Star Time 2011.8
advancing at 33% of Bob's rate
2013.2
advancing at 60% of Bob's rate
Distance to Earth 0.24 light years 0.24 light years
approaching at 80% speed of light
Distance to Star 4.6 light years 4.6 light years
receding at 80% speed of light

Ann's Time 2019 Ann Sees Ann Calculates
Rocket Time 2009
advancing at 300% of Ann's rate
2011.4
advancing at 60% of Ann's rate
Rocket Distance 4 light years 0.8 light years
approaching at 80% speed of light
Star Time 20112019

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Bob Arrives Home - More Details   top

This is at the instant Bob arrives back at earth, still moving at 240,000 km/sec towards Earth.

Bob's Time 2012 Bob Sees Bob Calculates
Earth Time 2020
advancing at 300% of Bob's rate
2020
advancing at 60% of Bob's rate
Star Time 2012
advancing at 33% of Bob's rate
2013.6
advancing at 60% of Bob's rate
Distance to Earth 0 0
approaching at 80% speed of light
Distance to Star 4.8 light years 4.8 light years
receding at 80% speed of light

Ann's Time 2020 Ann Sees Ann Calculates
Rocket Time 2012
advancing at 300% of Ann's rate
2012
advancing at 60% of Ann's rate
Rocket Distance 0 0
approaching at 80% speed of light
Star Time 2012 2020

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Bob Stops at Home - More Details   top

Bob's Time 2012 Bob Sees Bob Calculates
Earth Time 2020
advancing at same rate as Bob's
2020
advancing at same rate as Bob's
Star Time 2012
advancing at same rate as Bob's
2020
advancing at same rate as Bob's
Distance to Earth 0 0
not moving
Distance to Star 8 light years 8 light years
not moving

Ann's Time 2020 Ann Sees Ann Calculates
Rocket Time 2012
advancing at same rate as Ann's
2012
advancing at same rate as Ann's
Rocket Distance 0 0
not moving
Star Time 2012 2020

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