I am reading the book 'Gravity' by Hartle and presently I am at the section discussing orthonormal and coordinate bases. I am confused about a few points I had read previously and can't exactly correlate with the above mentioned section.
Here the author says that at every point in spacetime I can define a basis that is orthonormal in the tangent space of that point. This would not necessarily be the coordinate basis which I would get from the line element. This would mean that the metric in the orthonormal basis becomes the flat spacetime metric at the point (from the definition of the components of the metric in terms of the dot product of basis vectors and the requirement of one timelike and three spacelike components). Now, I know that the way to locally transform the metric to the flat spacetime metric is to do a coordinate transformation on the line element that does the above, i.e to go to a local inertial frame for that point.
My question is - Is there any correlation between a coordinate transformation that transforms my line element to that of flat spacetime in the vicinity of a point to that of choosing an orthonormal coordinate system at that point? Is the latter something similar to choosing a coordinate system where my coordinate basis would align with the orthonormal basis at the point? So, I guess my question boils down to the two definitions of the metric - one from the dot product of basis vectors and the other from the line element. Are they completely equivalent?
My last question is - Is the coordinate transformation that takes me to a local inertial frame of a point unique? Can there be more than one coordinate transformations that take me to the local inertial frame of a point? Does this mean that at every point I can choose an orthonormal basis at every point in one and only one unique way?
I am sorry for not writing equations. I am very poor with Tex. I hope I have been able to put my question clearly.