Orthonormal Basis in General Relativity - Carroll

[exponent137]

Now, I am reading Carroll's http://arxiv.org/abs/gr-qc/9712019. On a page 88 he defines basic vectors, which are orthonormal (3.114) and basic vectors given by gradients of coordinate functions.
Are these later basis vectors not obviously orthonormal?

 

[Chestermiller]

Now, I am reading Carroll's http://arxiv.org/abs/gr-qc/9712019. On a page 88 he defines basic vectors, which are orthonormal (3.114) and basic vectors given by gradients of coordinate functions.
Are these later basis vectors not obviously orthonormal?

Not if the coordinate system is non-orthogonal.

Chet

 

[exponent137]

I do not imagine enough.
(1) Geometry of Schwarshild. (2) Or desription of bending of a ray because of sun gravity. (3) Or Merkur orbit. (4) And still some typical examples.
Are these examples typicaly use orthonormal basis, or not?

 

[Chestermiller]

I do not imagine enough.
(1) Geometry of Schwarshild. (2) Or desription of bending of a ray because of sun gravity. (3) Or Merkur orbit. (4) And still some typical examples.
Are these examples typicaly use orthonormal basis, or not?

(1) and (2) are orthogonal, but not orthonormal. I'm not familiar with (3). Any time you have a line element with a term involving the product of differentials of two different coordinates, the coordinate system is not orthogonal. Even with simple coordinate systems like cylindrical and spherical, the basis vectors are orthogonal, but not orthonormal (i.e., they are not unit vectors).

Chet