Quick question on where to apply time dilation.

[JJ]

While reading one of Starthrower's threads, I became quite confused.

If an object moves at close to c relative to the earth, but without acceleration, how do we know that time slows down for it instead of our planet? An inertial reference frame is point that doesn't accelerate, correct? Both the planet and the object could then be IRFs. Where does time slow down? What gives?

 

[Hurkyl]

The thing you're missing (and it's a common mistake) is that time dilation is relative.

In the Earth's reference frame, a clock on the object will be observed to run slowly.

In the object's reference frame, clocks on Earth will be observed to run slowly.


(This would, of course, be contradictory in classical mechanics because of absolute simultaneity)

 

[JJ]

That's what I thought, but what happens in the case of a gps satellite? Time dilation is not only observed, but actually felt. It must be due to the acceleration of the satellite, right? The satellite is no longer an inertial reference frame, so time seems to run slower from Earth's point of view, and time speeds up from the satellite's perspective.

 

[Hurkyl]

That's what I thought, but what happens in the case of a gps satellite? Time dilation is not only observed, but actually felt.

What do you mean by "actually felt"?



Anyways, gravitational time dilation is asymmetric; both observers will agree that being deeper in the gravity well tends to cause clocks to run... blarg I can't remember if it's supposed to be faster or slower.

 

[JJ]

By actually felt I mean that if we would take that satellite back to earth, it will be "younger".

A stronger gravity field causes a gravitational time dilation, according to GR. Wouldn't that mean that a satellite's time would run slightly faster than ours?

What else is symmetric? Relativistic mass, length contraction, etc...?

 

[Janus]

By actually felt I mean that if we would take that satellite back to earth, it will be "younger".

A stronger gravity field causes a gravitational time dilation, according to GR. Wouldn't that mean that a satellite's time would run slightly faster than ours?

Okay, I'm going to step in here before you make another common mistake. Gravitational time dilation is not due to local variations in gravitational strength, but due to difference in gravitational potential. For instance, if you had a uniform gravitational field (one which does not change in strength with height), an object higher in the field would run faster, even though it felt the same force of gravity as one lower in the field.

What else is symmetric? Relativistic mass, length contraction, etc...?

Yes and Yes.

 

[JJ]

Yes, I was about to write the gravitational time dilation formula down, but I don't know how to use latex.

I've heard of the twin paradox, where a twin travels in a spaceship and the other stays on earth. About that, I've also read that accelerations are beyond SR and need GR. What gives? What exactly causes the traveling twin to return younger?

I greatly appreciate the help you mentors have given me, thank you very much.

 

[Hurkyl]

The twin paradox is a pseudoparadox; a contradiction only occurs when you make the mistake of assuming the spacebound twin is in one inertial reference frame for the entire trip.

It's another common mistake that you have to invoke GR to handle the twin paradox; the actual computation of the time experienced by each twin is a straightforward application of differential calculus.

 

[JJ]

Then since the Earth is the IRF, the time dialtion would be asymetric?

Edit: GR states that clocks run slower in gravitational fields, and then that gravitational forces and accelerations are equivalent. So do clocks run slower while accelerating? My god, it would all make sense!

 

[Janus]

Then since the Earth is the IRF, the time dialtion would be asymetric?

Edit: GR states that clocks run slower in gravitational fields, and then that gravitational forces and accelerations are equivalent. So do clocks run slower while accelerating? My god, it would all make sense!

No. But what are effected are the measurements made from within the accelerated frame. And those measurements depend on the magnitude of the acceleration, and the distance and direction with respect to the acccleration that the object being measured is.

As to which twin in the twin paradox actually undergoes time dilation, it depends on which twin you are.

The Earth twin sees his brother age slower during almost the entire trip.

The spaceship brother sees the Earth brother age slowly for parts of the trip and age very quicky for other parts of the trip. (the age quickly part is during the turn around when he is both far away and accelerating towards his brother).

Both of these views are equally valid, and neither can be given priority over the other.

 

[JJ]

One last question and I'm done:
Is the formula for time dilation in an accelerated frame
$$t = \frac{t_0}{\sqrt{1 - 2aR/c^2}}$$ ?