# Relativity Error in Hartle's Fig. 7.11?

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1. Jan 1, 2016

### 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.

2. Jan 1, 2016

### George Jones

Staff Emeritus
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.

3. Jan 2, 2016

### 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.