I What Is the Geometrical Meaning of Principal Null Directions?

[Umaxo]

Hi,

I know that principal null directions are defined as vector at a point that satisfy the equation $k_{[a}C_{b]cd[e}k_{f]}k^ck^d=0$, where $C_{abcd}$ is weyl tensor. I know they are used for generating solutions to einstein equations and that they are used for algebraic classification of solutions to einstein field equations.

What i cannot find anywhere is some explanation on their geometrical meaning. Do you know what is their geometrical meaning if they have any? - Something one could visualize. Do you know some literature where I could read more about them?

Thanks:)

 

[martinbn]

I am not sure if that's what you are looking for, but when phrased in terms of spinors it seems a clearer. May be not as geometric as you expect but at least not something so difficult to remember as the above.

 

[Umaxo]

I don't know spinors, but what you said seems it is just mathematical convenience.

I am just curious wheter there is something geometricly meaningfull about those vectors.

In kerr spacetime, these principal null directions are used to define new coordinates in which one can extend kerr spacetime over event horizons. When one restricts spacetime to hypersurfaces of constant angular coordinates in these new coordinate system, one gets two dimensional timelike hypersurface of which conformal diagram can be constructed (albeit little wierd).

Now, this is just one choice of hypersurface among many. Nice property of this choice is, that one angular coordinate actually defines coordinate vector fields that is killing. Another nice property is, that these principal null directions actually define null "radial" geodesics (they are radial in infinity) and those lie on these hypersurfaces.
But i am curious, wheter the fact that one of the principal null directions lie in tangent space of hypersurfaces defines some additional "nice" property, perhaps concerning curvature itself, since they are some kind of eigenvectors of weyl tensor?

In literature I checked (Wald - General Relativity, Grifiths/Podolsky -Exact spactimes in Einsteins GR) they are mentioned only in hurry as some mathematically convenient tool. The only physical significance in wald is mentioned:
"Unfortunatelly, since the character of this simplifying assumption is more mathematical than physical, many of the solutions obtained by this approach do not appear of direct physical relevance" which is not very promising.

 

[Ted Jacobson]

The pnd’s are the directions in which a geodesic congruence of light rays experiences no shear, i.e. no astigmatism.