General relativity and accelerated frames

[Nugatory]

it seems from here "even at a point" it's not possible to find such a chart. where am I wrong, please?

"flat at a point" is a statement about the metric tensor at that point, not the Riemann tensor.
I can find coordinates in which the components of the metric tensor at a point are diag(-1,1,1,1) but Riemann still won't vanish.

 

[RiccardoVen]

No this is incorrect. You cannot turn a curved metric into a flat one, ever, not even at a single event. Putting the metric into Minkowski form at a given event, and making the Christoffel symbols vanish there does not mean the metric is flat. The Riemann tensor will still be non-zero there. A handful of GR textbooks will use the terminology "locally flat coordinates" but this is a sacrilegious and downright incorrect term. The equivalence principle only says the gravitational field (= Christoffel symbols) will vanish at the chosen event and that the metric can be put into Minkowski form there. It does not say that the Riemann tensor (= tidal forces) will vanish because they certainly won't-they are a physical characterization of the curvature of space-time. The stipulation of neglecting tidal forces so as to apply the equivalence principle (i.e. the stipulation of a local inertial frame) simply means the Christoffel symbols vanish only at a single event as opposed to on an extended neighborhood of that event and the metric is only Minkowski at this single event. If we start moving away from this event and into an extended neighborhood of it then we will find non-zero terms in the Christoffel symbols and non-trivial terms in the metric that scale with the curvature (i.e. the tidal forces).

As for Gaussian normal coordinates, there is a slight distinction between coordinates and frames but for our purposes here yes a Gaussian normal system is the same thing as a freely falling frame (the local inertial frame has the added property that it is non-rotating which requires the notion of Fermi-Walker transport). One can find explicitly what the Riemann tensor is in this freely falling frame both at its origin (where the metric is Minkowski and the Christoffel symbols = gravitational field vanish) and near the origin up to second order in the characteristic curvature set by the Riemann tensor. See exercise 13.13 in MTW.

OK this clarifies a bit my doubt. It's really rally interesting the distinction you are making betweem gravity = Christoffel symbols and Riemann = tidal forces.
I've not read something like this in any book, and I think it's really clarifying things, at least a bit.
I've to work it out properly, since I think this view can really help me.

I can try to see if MTW has anything on that topic. Chapter 13 would be a good place to start.

OK, I will look at it as well.

In the meantime you can read the following papers:

http://articles.adsabs.harvard.edu//full/1990MNRAS.245..733A/0000735.000.html
http://adsabs.harvard.edu/full/1990MNRAS.245..720A

Thanks for these one as well

 

[RiccardoVen]

"flat at a point" is a statement about the metric tensor at that point, not the Riemann tensor.
I can find coordinates in which the components of the metric tensor at a point are diag(-1,1,1,1) but Riemann still won't vanish.

Yes that's exactly what WBN pointed out in the above post. This was leading confusion to me.
Not it's ok, thanks for your help.