The discussion revolves around the concept of orthonormal bases in the context of general relativity, specifically referencing Carroll's work. Participants explore the definitions of basic vectors and their properties, particularly in relation to coordinate systems and specific examples from general relativity.
Participants express differing views on the orthonormality of basis vectors in various coordinate systems, indicating that there is no consensus on the use of orthonormal bases in the examples provided.
There is an acknowledgment that the definitions and properties of orthonormal bases depend on the specific coordinate systems used, and the discussion highlights the complexity of these relationships without resolving them.
Not if the coordinate system is non-orthogonal.exponent137 said: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?
(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).exponent137 said: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?