# B Did I use the time dilation formula correctly?

#### Megatronlol

Hi,

We are not learning this in class, but I am giving a presentation on special relativity and as part of my presentation I would like to show that time is not absolute and that if a ship moves away from the Earth for a time t at a speed v then if like 8 years pass on board the ship a greater length of time will pass on Earth due to time slowing down for the people on board the ship.

Here is the work for the calculation. Is it right? I used 8 years ship time for the example (4 years there, 4 years back) and already squared v and c before plugging them in. Any insight is appreciated.

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#### jbriggs444

Homework Helper
due to time slowing down
Time does not slow down. It passes at one second per second for everyone. A better way of looking at it is that more time passed for the people on earth than for those on the space ship. Not that it passed more rapidly or more slowly for either.

#### Megatronlol

Time does not slow down. It passes at one second per second for everyone. A better way of looking at it is that more time passed for the people on earth than for those on the space ship. Not that it passed more rapidly or more slowly for either.
Can you elaborate? I think I worded it poorly. I know that from either frame of reference 1 second is 1 second, but how does one frame experience "more time" than the other?

#### jbriggs444

Homework Helper
The elapsed time on a path through space time depends on the path. In flat space-time (special relativity), it turns out that accelerated paths incur less elapsed time than inertial paths.

#### Dale

Mentor
Here is the work for the calculation. Is it right?
Yes, your calculation is correct. The earth ages 26 years in the time that the traveler ages 8 years traveling at 0.95 c

Can you elaborate? I think I worded it poorly. I know that from either frame of reference 1 second is 1 second, but how does one frame experience "more time" than the other?
Think of it this way. If you have two cars, both drive from Miami to New York, but one drives through Washington DC and the other drives through Chicago then the odometer for the car going through DC will read less than the odometer for the car going through Chicago. This is not because Chicago odometers are shorter than DC odometers. The Chicago miles are the same as DC miles, and the Chicago odometers correctly measure them just like the DC odometers. The difference is simply that the two paths are different and different paths have different lengths. There are simply more miles in the Chicago path.

Similarly for the twins, they take different paths through spacetime so the paths have different lengths.

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#### Nugatory

Mentor
Here is the work for the calculation. Is it right? I used 8 years ship time for the example (4 years there, 4 years back) and already squared v and c before plugging them in. Any insight is appreciated.
The calculation is correct, but it doesn't mean quite what you're thinking it does.

To see the problem, consider that we could just as easily consider the ship to be at rest while the earth is moving away from it... and then the time dilation formula will calculate that it's on earth that time is passing more slowly. So we have the apparent contradiction that the ship clock is slower than the earth clock and also that that the earth clock is slower than the ship clock.

In fact there is no contradiction. Both statements are correct, and the apparent conflict between them goes away when you allow for the relativity of simultaneity - Google for "Einstein train simultaneity" you'll find much good stuff on Einstein's thought experiment demonstrating that concept.

To see how relativity of simultaneity matters, consider exactly what it means to say that one clock is running slower than another: we're sitting at rest on earth and we find that clock A on the spaceship reads 12:00 noon at the same time that clock B on earth and at rest relative to us reads 12:00 noon; a bit later we find that clock A reads 12:30 at the same time that clock B reads 1:00; clearly clock A on the moving spaceship is running slow. That's the time dilation calculation you just did.

But that's the result according to people at rest relative to clock B on earth. Because of relativity of simultaneity, if you're on the moving spaceship you have a different notion of "at the same time". In particular, at the same time that clock A reads 12:30 clock B reads 12:15 and it's clock B that's running slow.

#### Sorcerer

When I read these threads, I really start to understand why educators are starting to lean towards the spacetime diagram explanation of special relativity over just the algebraic one.

For me personally, the analogy of a field that a race is run on (with both moving at the same speed) representing the spacetime interval is the easiest to understand: if you run perpendicular from start to finish, you cover less horizintel ground and it takes less time, but if you choose a crooked or curved path instead, while your journey is longer, you still end up moving the same vertical distance.

Admittedly it’s kind of reversed (the one who changes direction experiences more time), but thinking about that has helped me with spacetime diagrams.

#### sweet springs

I also try to do some explanation.

Say there are three points on a line OAB, OA=AB, staying still with the Earth.

O----A----B

After passing O, the rocket travels 13 years (for the Earth) and passing A with only 4 years of rocket calender consumed.
After passing A, the rocket travels next 13 years and Passing B when rocket calendar shows 8 years passed in total.

When the rocket is passing A, let another rocket #2 is also passing with the same speed but inverse direction with its calendar shows the same time with the original rocket(say #1), i.e. 4 years.
According to symmetric behavior of rocket #1 and #2, when rocket #2 is passing O, the rocket #2 calender shows 8 years passed in total.
The transfer from #1 to #2 at A is equivalent to the return of rocket at A which is our original case.

The earth stays still and the rockets keep moving in the sequence.
We cannot choose rockets to stay still because #1 and #2 have different speed.
We chose the coordinate where the Earth is still.
We can choose the coordinate where #1 is still or #2 is still but cannot choose the coordinate where the both are still. This makes difference.

P.S.
Further question would come as for "Then passing A, the rocket travels next 13 years and Passing B when rocket calender shows 8 years passed in total."
No rocket #2 appears here so we can choose the coordinate where the rocket stays still all the way.
In such a coordinate not rocket but instead the Earth moves so the Earth time should become slow down. Above " " does not contradict with it? I do not explain here now but there is no contradiction.

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"Did I use the time dilation formula correctly?"

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