Time Dilation in Special and General Relativity: An Interconnected Relationship?

[bassplayer142]

I see that time is slowed down in a plane that would be moving around the Earth as time dilation. I also see that in a stronger gravitational field slows down time. Are these two values dependant of each other as a connection between special and general relativity? I hope I was clear, thanks.

 

[tiny-tim]

I see that time is slowed down in a plane that would be moving around the Earth as time dilation. I also see that in a stronger gravitational field slows down time. Are these two values dependant of each other as a connection between special and general relativity? I hope I was clear, thanks.

Hi bassplayer142!

No, these two time dilation effects (SR and GR) are completely unconnected (though you can have both together) …

SR time dilation is √(1 - v²/c²), depending on relative speed but not position,

GR time dilation is = √-g₀₀(observee)/√-g₀₀(observer), depending on position but not speed.

 

[Al68]

Hi bassplayer142!

No, these two time dilation effects (SR and GR) are completely unconnected...

Well, they're not completely unconnected, gravitational time dilation is derived directly from SR time dilation. Sure the equations look different, and the effects are calculated independently, but they are very related.

Al

 

[tiny-tim]

Hi Al!

Well, they're not completely unconnected, gravitational time dilation is derived directly from SR time dilation.

I don't think so …

How? ​

(gravitational time dilation is effectively defined as √-g₀₀, isn't it?)

 

[Al68]

Hi Al!


I don't think so …

How? ​

(gravitational time dilation is effectively defined as √-g₀₀, isn't it?)

Gravitational time dilation was predicted/derived by Einstein in 1907? as a result of SR time dilation using accelerated reference frames (a clock "at rest" in a gravitational field is in an accelerated reference frame) and the equivalence principle. Here are a couple of links I could find quickly:

http://en.wikipedia.org/wiki/Gravitational_time_dilation
http://einstein.relativitybook.com/Einstein_gravity.html

Al

 

[yuiop]

There is an interesting connection between gravititational time dilation and SR time dilation.

SR time dialtion for a object moving with velocity (v) in flat space is:

$$t ' = t*\sqrt{1-v^2/c^2}$$

Gravitational time dilation for a stationary object at some radius (r) in a gravitational field is:

$$t ' = t*\sqrt{1-2gm/(rc^2)}$$


Now if we work out the SR time dilation using the velocity that a particle dropped from infinity reaches at radius r, the time dilation works out numerically equivalent to the gravitational time dilatation. This is not how GR time dilation is derived from SR, but it is an interesting coincidence. Of course, if an object is actually dropped from infinity ina gravitational field it has time dilation due to gravitational potentential and due to its falling velocity. I have just posted the equations in another related thread here: https://www.physicsforums.com/showpost.php?p=2009393&postcount=31

 

[bassplayer142]

Thanks everyone. I wish I could take a few courses on more advanced math and relativity to really get a grasp. Bu simple understanding should work for now.

 

[Al68]

There is an interesting connection between gravititational time dilation and SR time dilation.

SR time dialtion for a object moving with velocity (v) in flat space is:

$$t ' = t*\sqrt{1-v^2/c^2}$$

Gravitational time dilation for a stationary object at some radius (r) in a gravitational field is:

$$t ' = t*\sqrt{1-2gm/(rc^2)}$$


Now if we work out the SR time dilation using the velocity that a particle dropped from infinity reaches at radius r, the time dilation works out numerically equivalent to the gravitational time dilatation. This is not how GR time dilation is derived from SR, but it is an interesting coincidence.

I wouldn't call that a coincidence. It's a necessary consequence of SR, since the velocity of an object (rel. to accelerated frame) "dropped" in an accelerated frame (or uniform gravitational field) will be sqrt(2gh). It also has been (correctly) pointed out to me on this forum that gravitational time dilation isn't really GR, since it predates GR. It was derived using SR for accelerated observers, and generalized to a gravitational field using the equivalence principle.

Al

 

[atyy]

I guess there are (at least?) two types of time dilation experiements. In the first type, two atomic clocks start out at the same spacetime location (same event), at which point they are both set to zero and synchronized. After that they follow different paths through spacetime, then meet again at another spacetime location. This experiment can be carried out in special and general relativity, and in both theories, the final reading of each clock is given by the integral of the infinitesimal proper time (spacetime metric) along their spacetime path.

In the second sort of time dilation experiment, the clocks are again synchronized (zero setting is not so important). Then they are moved to different locations in spacetime. Each clock sends a signal to the other clock each time it ticks. We ask, can we permanently make one clock faster so that its ticks exactly coincide with when it receives a signal from the other clock? In some cases, such as two inertial observers in flat spacetime or "stationary" observers in curved spacetime, a one time adjustment is possible. I think this is not possible for arbitary observers, ie. some observers may need multiple adjustments, in which case the idea of relative rate is difficult (?) to define. However, whenever it is possible in SR and GR, the ticking of the clocks is still governed by the infinitesimal proper time (spacetime metric).

 

[yuiop]

...We ask, can we permanently make one clock faster so that its ticks exactly coincide with when it receives a signal from the other clock? In some cases, such as two inertial observers in flat spacetime or "stationary" observers in curved spacetime, a one time adjustment is possible...

As long as the two inertial observers in flat spacetime are at rest with respect to each other then a one time adjustment is sufficient. If the two inertial observers are moving relative to each other, they both see each other's clocks as ticking slower and there is no adjustment that can be made to either clock, that can make them appear to run at the same rate as judged by sending signals to each other at constant intervals. If you meant only one clock is allowed to send signals then I guess that would work.

For two clocks 'stationary' in a gravitational field, the lower clock can be sped up by a one time adjustment and the signals sent to each other would remain synchronised. A falling clock on the other hand can not be synchronised by a one time adjustment with a stationary clock. However, you could work out its 'instantaneous' (infinitesimal) time rate at the time it sent each signal relative to the stationary clock.

 

[atyy]

If you meant only one clock is allowed to send signals then I guess that would work.

Nope, I meant the wrong thing. For what I said to make sense it has to be set up as you described - thanks!

However, you could work out its 'instantaneous' (infinitesimal) time rate at the time it sent each signal relative to the stationary clock.

Yes, I vaguely remember that with rotationally symmetric fields the synchronization always makes some sort of sense along a radius? But does it make sense for arbitrary (radial plus non-radial) motions? In distinguishing between the two sorts of time dilation experiments, I was thinking the first type (comparing total time elapsed) always makes sense, but the second type (asking for a rate adjustment) only makes sense for some observers?