B Basic introduction to gravitation as curved spacetime

[cianfa72]

That does define a coordinate chart, provided you also specify how to initially synchronise all the clocks. Unfortunately the clocks won't remain synchronised (by the same criteria), because of "(pseudo-) gravitational time dilation".

Sorry, even if you do not specify how to initially synchronize clocks they still define a coordinate chart -- namely it smoothly maps events of that spacetime region to $\mathbb R^4$ with no singularities.

A better solution is to redefine coordinate time as a constant multiple of proper time (a different constant for each clock) so as to maintain synchronisation.

Do you mean 'adjust' the rate of standard clocks depending on their location in order to maintain synchronization when initially synchronized ?

 

[DrGreg]

Sorry, even if you do not specify how to initially synchronize clocks they still define a coordinate chart -- namely it smoothly maps events of that spacetime region to $\mathbb R^4$ with no singularities.

Well, if you don't specify how to synchronise, you haven't fully defined the chart. But they can be synced any "smooth" way you like. (Any two "nearby" events deemed to be simultaneous must be spacelike-separated.)

Do you mean 'adjust' the rate of standard clocks depending on their location in order to maintain synchronization when initially synchronized ?

That's one way to do it in practice, yes.

 

[cianfa72]

Well, if you don't specify how to synchronize, you haven't fully defined the chart. But they can be synced any "smooth" way you like. (Any two "nearby" events deemed to be simultaneous must be spacelike-separated.)

I assumed the following as reference system (actually the mathematical map it defines from spacetime events to $\mathbb R^4$) from Landau "The Classic Theory of Field - vol 2" sect 82.

In the latter (SR) we meant by a reference system a set of bodies at rest relative to one another in unchanging relative positions. Such systems of bodies do not exist in the presence of a variable gravitational field,
and for the exact determination of the position of a particle in space we must, strictly speaking, have an infinite number of bodies which fill all the space like some sort of "medium". Such a system of bodies with arbitrarily running clocks fixed on them constitutes a reference system in the general theory of relativity.

IMO there is no requirement about clock's rates as well as the procedure employed to synchronize them.

 

[vanhees71]

Of course not, since in a general spacetime you cannot synchronize clocks at different positions.

 

[cianfa72]

Of course not, since in a general spacetime you cannot synchronize clocks at different positions.

Not sure to grasp it: since in GR we cannot synchronize clocks at different positions, there is actually no requirement from the point of view of clock synchronization in order to define a coordinate chart, don't you ?

 

[vanhees71]

Yes, sure. I was confirming what you said in #53.

 

[DrGreg]

IMO there is no requirement about clock's rates as well as the procedure employed to synchronize them.

You are right. The only point I was trying to make was that, to analyse a particular problem, you need to know how one clock relates to another clock, either by specifying what it is or by experimental measurement.

And after initial syncing there's no requirement to remain in sync. In many spacetimes, it's impossible.

But in a particular scenario it may be convenient to choose a coordinate system where there is a specific kind of synchronisation that is maintained.

 

[trainman2001]

That video explaining the concept of gravity to five levels of experience was terrific! Still left with more questions than answers, as were the folks in the video.