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?
An orthonormal basis in general relativity is a set of four linearly independent vectors that are perpendicular to each other and have unit length. These vectors are used to describe the geometry of spacetime in general relativity.
In general relativity, an orthonormal basis is used to define the local reference frame at a given point in spacetime. This allows for the measurement of physical quantities, such as the curvature of spacetime, at that point.
An orthonormal basis is significant in general relativity because it provides a way to describe the geometry of spacetime in a way that is independent of any particular coordinate system. This allows for a more precise and consistent understanding of the fundamental laws of physics.
The metric tensor is used to define the inner product between vectors in general relativity. In an orthonormal basis, the metric tensor simplifies to the identity matrix, making mathematical calculations and interpretations of physical quantities more straightforward.
Yes, an orthonormal basis can be used in any coordinate system in general relativity. This is because the basis vectors are defined in terms of the metric tensor, which is a geometric property of spacetime and is independent of any particular coordinate system.