Is There an Error in Hartle's Fig. 7.11 Caption Regarding Normals and Tangents?

[joneall]

A question concerning FIg. 7.11 on page 173. It seems to me, in light of what is said on the preceding page about null surfaces, that he has interchanged "normal" and "tangent" in the third and fourth sentences of the figure caption. I would say:

"The tangent to the surface l lies in the surface and along the generating light rays. A normal vector t is also shown."

Anyone agree with this? It is not included in the (rather long) list of errata I downloaded. Thanks for you help or corrections.

 

[George Jones]

Unfortunately, I won't be able to look at Hartle until Monday.

Because a null vector is orthogonal to itself, the normal vector to a null surface is also tangent to the null surface. For example, consider a standard orthonormal basis $\left\{ \bf{e}_0 , \bf{e}_1 , \bf{e}_2 , \bf{e}_3 \right\}$ for Minkowski spacetime. A null nsurface that has a tangent space spanned by $\left\{ \bf{e}_0 + \bf{e}_1 , \bf{e}_2 , \bf{e}_3 \right\}$ also has $\bf{e}_0 + \bf{e}_1$ as a normal vector because $\bf{e}_0 + \bf{e}_1$ is orthogonal to each of the vectors that span the tangent space.

 

[joneall]

Thanks for your reply, but I'm not yet sure how to apply that to my question. I see from the math what you say. But I cannot visualize something being both tangent to a surface and normal to it. I presume this is a property of the space or of being at the speed of light, but my poor mind cannot visualize that stuff.

Any helpful hints?

 

[vanhees71]

Draw a Minkowski diagram. There the worldline of a massless particle (a light ray in the sense of the iconal approximation) is the bisector of the coordinate axes. In the sense of the Minkowski fundamental form two vectors are (pseudo-)orthogonal if a lightlike line is its bisector. Thus a lightlike line is pseudo-orthogonal to itself. The problem with Minkowski diagrams is that you must forget about the Euclidean geometry we are used to from childhood on. At the moment were your paper is used as a model of Minkowski space the Euclidean metrics, angles, etc. have no meaning anymore. You have to think of it as a hyperbolic plane!