A question regarding a specific interval of time and GR

[myoho.renge.kyo]

A. Einstein states in The Principle of Relativity, pp 111 - 112:

"The modification to which the special theory of relativity has subjected the theory of space and time is indeed far-reaching, but one important point has remained unaffected. For the laws of geometry, even according to the special theory of relativity, are to be interpreted directly as laws relating to the possible relative positions of solid bodies at rest; and, in a more general way, the laws of kinematics are to be interpreted as laws which describe the relations of measuring bodies and clocks. To two selected material points of a stationary rigid body there always corresponds a distance of quite definite length, which is independent of the locality and orientation of the body, and is also independent of the time. To two selected positions of the hands of a clock at rest relatively to the privileged system of reference there always corresponds an interval of time of a definite length, which is independent of place and time. We shall soon see that the general theory of relativity cannot adhere to this simple physical interpretation of space and time."

It is now, according to my wrist watch (my wrist watch and the clock of the United States synchronize), 9:11 am, 10/9/2006, here in Burbank, California. Let the two selected positions of the hands of my wrist watch be 6:28 pm, 10/8/2006 thru 6:27 pm, 10/9/2006. Can anyone give me an example of how these two selected positions of the hands of my wrist watch (to which an interval of time of a definite length, 86340 s, corresponds) is not independent of place and time? Or how does this interval of time depend on place and time? Or what is the value of this interval of time if a different place and time is given? Thanks!

 

[cesiumfrog]

I'll assume you were on the ground floor of your building during that time interval. If someone on the second floor had been watching you, they would have thought your clock was running slow - and thus disagreed about the length of the interval. Does that answer your question?

 

[MeJennifer]

...how does this interval of time depend on place and time? Or what is the value of this interval of time if a different place and time is given? Thanks!

This is the case when a distance or duration measurement is done at a location A for a location B and these locations have a different gravitational potential, relative speed or relative acceleration or a combination of those.

Relative speed between A and B contracts lengths and dilates duration.
Relative acceleration between A and B change the amount of contraction and dilation.
Different gravitational potentials between A and B change shape and duration.

 

[pess5]

It is now, according to my wrist watch (my wrist watch and the clock of the United States synchronize), 9:11 am, 10/9/2006, here in Burbank, California. Let the two selected positions of the hands of my wrist watch be 6:28 pm, 10/8/2006 thru 6:27 pm, 10/9/2006.

Can anyone give me an example of how these two selected positions of the hands of my wrist watch (to which an interval of time of a definite length, 86340s, corresponds) is not independent of place and time? Or how does this interval of time depend on place and time? Or what is the value of this interval of time if a different place and time is given? Thanks!

The time interval doesn't depend upon place or time. It depends only on relative velocity. So those in inertial motion at v>0 wrt you will record that your 86340s takes 86340s*gamma per they, where gamma=1/(1-v^2/c^2)^1/2. At say v=0.866c, gamma=2. So your clock runs slower per they, since they see you in motion and consider themselves stationary.

The Lorentz Transformation relates time between 2 frames via this equation …

T = gamma(t-vx/c^2)

X = Beta(x-vt)​

The -vx/c^2 part of the time T eqn produces a temporal offset, and I suspect that this may be related to the question you asked.

You might want to draw yourself some spacetime diagrams, time as a vertical (t) axis, and 1-space as a horizontal (x) axis. If you are unfamiliar with spacetime diagrams, google for "Minkowski worldline diagrams". Draw up a stationary and moving observer on one diagram, plot it out, and you'll see some very interesting things. iT might them become much clearer, much quicker.

pess

 

[myoho.renge.kyo]

wow! thanks to everyone that responded. i have a much better perspective of what is going on. thanks again.