# Time Dilation and it's formula

#### RadiantL

Hi, so I think I have a problem with how I am thinking about time dilation anyway here it is

so if two events occur at the same position say light beam going up then back down and this occurs in frame S', this would be the proper time right? the interval between the two events. Now the equation for time dilation is

y = gamma factor
Δt = y Δto

so that means if I was in frame S and watched from S' go by me at very fast speeds... then the the time interval I would measure would increase by the y factor. Now assuming what all I said now is correct. Would that not mean that I am seeing their time go by faster?

Say in S' the proper time between the event be 2...seconds?

now say y = 2...

and now in S, me I would see the whole entire thing happen in

Δt = 4 seconds

So if I see the whole thing go by in 4 seconds doesn't that mean when I look at their clock, their time is going by faster?

Everywhere I read, it says moving clocks run slower... help?

THANKS for any reply!

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#### ghwellsjr

Science Advisor
Gold Member
If you see a clock running slower, then its seconds take longer than your seconds, correct?

#### RadiantL

Right, that's why I'm confused, why is the interval longer? Wouldn't it be less? so
Δt = 1/y * Δto

so that the time you see elapsed from your point of view is smaller?

#### JesseM

Science Advisor
so if two events occur at the same position say light beam going up then back down and this occurs in frame S', this would be the proper time right?
Yes.
the interval between the two events. Now the equation for time dilation is

y = gamma factor
Δt = y Δto
Assuming t0 is the proper time in frame S' and t is the time in some other frame, yes.
so that means if I was in frame S and watched from S' go by me at very fast speeds... then the the time interval I would measure would increase by the y factor. Now assuming what all I said now is correct. Would that not mean that I am seeing their time go by faster?

Say in S' the proper time between the event be 2...seconds?

now say y = 2...

and now in S, me I would see the whole entire thing happen in

Δt = 4 seconds

So if I see the whole thing go by in 4 seconds doesn't that mean when I look at their clock, their time is going by faster?
No, it means you would observe their clock going slower than yours. After all, in 4 seconds of your time, you would see their clock tick forward by only 2 seconds. As an analogy, if you hear me only getting 2 words out for every 4 words you can get out, then you hear me talking more slowly than you, no?

#### RadiantL

Hmm, so

Δt = yΔto

if their proper time in their frame is 3 seconds, and y = 2...then the change in time for me would be 6 seconds...

so from my frame of reference the time that has passed for me would be Δt = 6s, but when I look at them, I see only 3 seconds have gone by for them?

Hmm so how about the other person, in S' would they measure 3 seconds have gone by for themselves and... 3/2 seconds have gone by for me when they are looking at me?

#### Jeronimus

That is because contrary to length contraction, the time-frame (interval) actually expands (dilation means expansion).

If an observer b in S' measures an interval using clocks of let's say 10s, then an observer a in S will measure the interval between the two event points (observer b starts the clock, observer b stops the clock) to be "longer" at 20 seconds if the factor is 2*.

This is because observer a will see the clock of observer b which is at rest in system S' go slower. At half speed at this factor. So rightfully, observer a in his system will measure the time-frame to be twice as long.

The same is true the other way around, since none of the two systems is in any way "better than the other". The laws of physics are the same in both.
If intervals measured by observer a in S are modified by a factor gamma when measured by observer b in S', then the same is true the other way around. Intervals measured by observer b in S' are modified by the SAME factor gamma when measured by observer a in S.

Observer b will measure intervals which are 10 seconds long in system S measured by observer a at rest in S, to take 20 seconds within his system S'. Observer b will see observer's a clocks run slower.

Slower clocks mean LONGER time-intervals.

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#### RadiantL

I think I understand,

Δto = proper time, the interval between two events at same position in say, frame S'
y = gamma
Δt = the time interval between the two events as measured in frame S point of view

because the interval between the two events as measured in frame S point of view is longer.... that means the clock's arrow from S point of view is moving slower, I think I understand, correct me if I'm wrong :P

THANK you everyone that helped me!

oh just one more thing, this delta t's we measure using this equation, mentioned in the above posts, are we allowed to just plug it into the lorentz transformation equations?

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#### ghwellsjr

Science Advisor
Gold Member
I think I understand,

Δto = proper time, the interval between two events at same position in say, frame S'
y = gamma
Δt = the time interval between the two events as measured in frame S point of view

because the interval between the two events as measured in frame S point of view is longer.... that means the clock's arrow from S point of view is moving slower, I think I understand, correct me if I'm wrong :P

THANK you everyone that helped me!

oh just one more thing, this delta t's we measure using this equation, mentioned in the above posts, are we allowed to just plug it into the lorentz transformation equations?
You can as long as you did what you said, "the interval between two events at same position". For example, if you want to see how much proper time has elapsed on a clock that is traveling at 0.6c after 1 second of coordinate S, you need to set the distance to 0.6 in S and LT will give you a time of 0.8 seconds at the same position.

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#### RadiantL

i see... following your example, and using

t' = y((-v/c^2)x + t)

i got the expected answer :)

thanks ghwellsjr for clearing up my question!

#### ghwellsjr

Science Advisor
Gold Member
You're very welcome.

Now let me give you a little suggestion. Instead of using the letter "y" for gamma, you can hit the "Go Advanced" button and you will see a box of Quick Symbols including one for "γ". The only nuisance is that when you click on a Quick Symbol, it inserts it in your post highlighted so you have to click somewhere else to remove the highlight or else your next keystroke will replace your Quick Symbol.

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