Solving the Mystery of Notation in a Paper - gr-qc/9611042v1

[Mentz114]

Reading this paper

http://lanl.arxiv.org/abs/gr-qc/9611042v1

I'm baffled by this bit on page 3,

The metric of a stationary space-time with time-like Killing vector K = ∂/∂t has the form
ds² = r(dt + ω_idxⁱ)² − r⁻¹dℓ² (i, j, ... = 1, 2, 3), (2.1)
where r = K·K is the length square of the Killing vector, ω_idxⁱ a 1-form, and dℓ² = g_ijdxⁱdx^j the metric in the 3-space S of Killing trajectories.

In a footnote on page 4 we are told

Vectors in the tangent 3-space are set in boldface. A dot notation (.) is used for a scalar product with respect to the 3-metric.

By this, K·K = 0 and equation 2.1 is nonsense. What have I missed ?

 

[martinbn]

The dot notation from the footnote has parentheses as well, and it is just dot, not a cdot.

 

[Mentz114]

The dot notation from the footnote has parentheses as well, and it is just dot, not a cdot.

Yes, thank you. What I missed was that K is not in boldface.

So r = K·K \equiv K^μK_μ ( μ = 0,1,2,3)