Global Coord Atlas Universe: GR Coord Charts, Speed Light

[berra]

I am wondering how big the local coordinate charts in a coordinate atlas of General Relativity can be. Is it just a few nanometres or all the way to the furthest stars? Also I wonder how the speed of light is defined in GR. Is the velocity defined as the differential with respect to coordinate time? Is the coordinate time only defined with respect to an observer, so a coordinate time is always (a diffeomorphism away from) a proper time?

 

[Simon Bridge]

Because of GR we are careful to measure the speed of light (for the purposes of Einstein's postulate) in a local spacetime which is flat ... since the laws of physics must be locally consistent.

I take it you are talking about an "atlas" in the topological sense.
You seem to be asking: how much of space-time do you need to describe in order to get it all?
To quote Deep Thought: "Hmmmm... tricky..."

I don't think anyone knows.

I found a starting point discussion here:
http://physics.stackexchange.com/qu...-about-the-topological-structure-of-spacetime
... starting point only, use the discussion to guide your further inquiries.

I did find: http://at.yorku.ca/t/a/i/c/28.htm
...

But you may have noticed that everyone talks about things being a certain way "locally" and you are wondering how big "locally" actually is... and that "depends". On the local space-time and how you want to organize the charts. Consider the question by analogy with constructing charts for an atlas of the Earth's surface.

 

[WannabeNewton]

I am wondering how big the local coordinate charts in a coordinate atlas of General Relativity can be.

This depends on the characteristic curvature scales of space-time.

Is the velocity defined as the differential with respect to coordinate time?

No it's the change in spatial distance divided by the change in time at a given event on an observer's world-line as measured using the observer's ideal rulers and ideal clock.

Is the coordinate time only defined with respect to an observer, so a coordinate time is always (a diffeomorphism away from) a proper time?

It's the other way around. Coordinate time is always a diffeomorphism away from proper time relative to some observer so it can be defined as the time read by a non-ideal clock carried by an observer (non-ideal meaning the observer has readjusted his clock rate so as to read the coordinate time instead of the proper time).

 

[Dale]

I am wondering how big the local coordinate charts in a coordinate atlas of General Relativity can be. Is it just a few nanometres or all the way to the furthest stars?

This depends on the curvature and the sensitivity. A local chart is the same as an inertial chart to first order, however the curvature causes deviations from flatness to second order. How big depends on how quickly those second-order effects pile up and how sensitive your experiment is to them.

Also I wonder how the speed of light is defined in GR. Is the velocity defined as the differential with respect to coordinate time? Is the coordinate time only defined with respect to an observer, so a coordinate time is always (a diffeomorphism away from) a proper time?

Many coordinate charts do not even have a coordinate time. The geometrical way to understand speed is that it is a function of the "angle" between two worldlines where at least one of the worldlines is timelike. This avoids any confusion about coordinate charts, and using that definition the speed of light is always c wrt any timelike worldline.