Specifying A Frame
To write down a frame, a coordinate chart on the Lorentzian manifold needs to be chosen. Then, every vector field on the manifold can be written down as a linear combination of the four coordinate basis vector fields:
(Here, the Einstein summation convention is used, and the vector fields are thought of as first order linear differential operators, and the components are often called contravariant components.) In particular, the vector fields in the frame can be expressed this way:
In "designing" a frame, one naturally needs to ensure, using the given metric, that the four vector fields are everywhere orthonormal.
Once a signature is adopted (in the case of a four-dimensional Lorentzian manifold, the signature is −1 + 3), by duality every vector of a basis has a dual covector in the cobasis and conversely. Thus, every frame field is associated with a unique coframe field, and vice versa.
Read more about this topic: Frame Fields In General Relativity
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