Coordinate System Transformation: Lowering/Raising Indices Explained

[GR191511]

In《Introducing Einstein's Relativity Ed 2》on page 106"lowering the first index with the metric,then it is easy to establish,for example by using geodesic coordinates..."
In 《A First Course in General Relativity - 2nd Edition》on page 159 "If we lower the index a,we get(in the locally flat coordinate system at its origin P)..."
What is the relationship between lowering or raising index and coordinate system transformation?

 

[andrewkirk]

The only relationship is that the components of the "lowered" Riemann tensor $R_{abcd}$ have a particularly simple form in that locally flat coordinate system.
Schutz uses that locally flat coordinate system in his presentation because it enables him to develop a fairly simple formula for Riemann tensor components in terms of metric components like $g^{ab}$ and $g_{bu,sv}$, rather than in terms of Christoffel symbols (equation 6.63).
He gets a fairly simple formula for $R^a{}_{bcd}$ in that coordinate system (equation 6.65) and then gets an even simpler formula (equation 6.67) for $R_{abcd}$, ie by lowering the first index.
Most tensor formulas will have coordinate systems in which their component formulas are simple, and in other systems they will be horribly complex. A key challenge in GR is to choose the coordinate system in which the component formulas will be simple and easy(er) to manipulate.

 

[GR191511]

The only relationship is that the components of the "lowered" Riemann tensor $R_{abcd}$ have a particularly simple form in that locally flat coordinate system.
Schutz uses that locally flat coordinate system in his presentation because it enables him to develop a fairly simple formula for Riemann tensor components in terms of metric components like $g^{ab}$ and $g_{bu,sv}$, rather than in terms of Christoffel symbols (equation 6.63).
He gets a fairly simple formula for $R^a{}_{bcd}$ in that coordinate system (equation 6.65) and then gets an even simpler formula (equation 6.67) for $R_{abcd}$, ie by lowering the first index.
Most tensor formulas will have coordinate systems in which their component formulas are simple, and in other systems they will be horribly complex. A key challenge in GR is to choose the coordinate system in which the component formulas will be simple and easy(er) to manipulate.

Thank you！