First, recall that a spacetime is a pseudo-Riemmanian manifold $(M,g)$ where $g$ has Lorentzian signature $(-,+,+,+)$.

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. In other papers it is also said that such system can always be built in any spacetime. The question is about how to rigorously construct it.

For that matter, the author picks a function $u \in C^\infty(M)$ satisfying $$g^{ab} (\partial_a u) (\partial_b u)=0$$

Defining $k_a = \partial_a u$ then $k_a$ is a lightlike covector. Thus the level sets $u = \text{const}$ are null hypersurfaces. We furthermore have $k^a \nabla_a k_b=0$ meaning that the integral lines of $k$ are null geodesics - the generators of the null hypersurface. He further supposes $\nabla_a k^a \neq 0$.

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

Let $\theta$ and $\phi$ be any pair of scalar functions that obey the equations $$k^a \nabla_a\theta = k^a\nabla_a \phi = 0 \Longrightarrow \nabla_a \theta \nabla^a\theta \nabla_b \phi\nabla^b \phi - (\nabla_a \phi \nabla^a \theta)^2=D\neq 0.$$

Here the implication sign follows from $\nabla_a k^a \neq 0$, as one can verify by a short calculation. $\theta$ and $\phi$ are constant along each ray; they should be visualized as optical angles.


  1. $u$ is an arbitrary function giving rise to lightlike level sets whose normals satisfy $\nabla_a k ^a =0$.

  2. 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. I mean we cannot just say "these functions exist and are good coordinates", we have to construct these coordinates like one constructs the Riemann normal coordinates, Fermi coordinates, etc.

  3. What he means that the implication sign follows from $\nabla_a k^a \neq 0$? I mean, if I'm not mistaken, considering the inner product among multivectors defined by $$\langle v_1\wedge\cdots \wedge v_k, w_1\wedge \cdots \wedge w_k\rangle = \det (\langle v_i,w_j\rangle)$$ the RHS of the implication sign if actually $\langle \nabla \theta\wedge \nabla \phi,\nabla\theta\wedge \nabla \phi\rangle$. That being nonzero means that $\nabla\phi$ and $\nabla \theta$ are not colinear and hence are linearly independent. But that is a condition on top of $\nabla\phi$ and $\nabla\theta$. I don't see how it follows from $\nabla_a k^a \neq 0$.

So in summary, how to make sense of these angular coordinates and of this short argument from the author of the paper? How do we construct these on any spacetime?


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