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## Main Question or Discussion Point

I am considering two vector fields in the spacetime of General Relativity. One is spacelike, the other is timelike, they are normalized and orthogonal:

U.U = -1

V.V = +1

U.V = 0

where dot denotes scalar product.

In addition, it is known the integral curves of U and V always remain in some two dimensional submanifold of the spacetime.

I know for one vector field it's always possible to construct the so called comoving coordinate system in which it has components (1, 0, 0, 0). Is it always possible to find a coordinate system in which the two fields have upper index (contravariant) components:

U -> (1,0,0,0)

V -> (0,1,0,0)

If possible give references where such topics are discussed.

U.U = -1

V.V = +1

U.V = 0

where dot denotes scalar product.

In addition, it is known the integral curves of U and V always remain in some two dimensional submanifold of the spacetime.

I know for one vector field it's always possible to construct the so called comoving coordinate system in which it has components (1, 0, 0, 0). Is it always possible to find a coordinate system in which the two fields have upper index (contravariant) components:

U -> (1,0,0,0)

V -> (0,1,0,0)

If possible give references where such topics are discussed.