Question

Why there are these two angular coordinates in a general spacetime?

Answer 1

In the paper "Gravitational Waves in General Relativity. VIII. Waves in Assymptotically Flat Space-Time" by R. Sachs, the author constructs a system of coordinates in section 2 for a general spacetime.

For that matter, he picks a function u∈C∞(M)u∈C∞(M)satisfying

gab(∂au)(∂bu)=0gab(∂au)(∂bu)=0

Defining ka=∂auka=∂au then kaka is lightlike. Thus the level sets u=constu=const are null hypersurfaces. We furthermore have ka∇akb=0ka∇akb=0 meaning that the integral lines of kk are null geodesics - the generators of the null hypersurface. He further supposes ∇aka≠0∇aka≠0.

The author says such uu can be used to build a coordinate system on its domain. To do it he says:

Let θθ and ϕϕ be any pair of scalar functions that obey the equations

ka∇aθ=ka∇aϕ=0⟹∇aθ∇aθ∇bϕ∇bϕ−(∇aϕ∇aθ)2=D≠0.ka∇aθ=ka∇aϕ=0⟹∇aθ∇aθ∇bϕ∇bϕ−(∇aϕ∇aθ)2=D≠0.

Here the implication sign follows from ∇aka≠0∇aka≠0, as one can verify by a short calculation. θθ and ϕϕ are constant along each ray; they should be visualized as optical angles.

So:

uu is an arbitrary function giving rise to lightlike level sets whose normals satisfy ∇aka=0∇aka=0.

Then he says that there exists these angle functions that are constant on each ray. Well, that is reasonable, but why such angle functions exist? What is their domain of definition, and how they are constructed on an arbitray spacetime? I really don't get how this is done.

What he means that the implication sign follows from ∇aka≠0∇aka≠0? I mean, if I'm not mistaken, considering the inner product among multivectors defined by

⟨v1∧⋯∧vk,w1∧⋯∧wk⟩=det(⟨vi,wj⟩)⟨v1∧⋯∧vk,w1∧⋯∧wk⟩=det(⟨vi,wj⟩)

the RHS of the implication sign if actually ⟨∇θ∧∇ϕ,∇θ∧∇ϕ⟩⟨∇θ∧∇ϕ,∇θ∧∇ϕ⟩. That being nonzero means that ∇ϕ∇ϕ and ∇θ∇θ are not colinear and hence are linearly independent. But that is a condition on top of ∇ϕ∇ϕ and ∇θ∇θ. I don't see how it follows from ∇aka≠0∇aka≠0.

So in summary, how to make sense of these angular coordinates and of this short argument from the author of the paper?

Answer 2

The situation is even more complicated if the two points are separated in time as well as in space. For example, if one observer sees two events occur