8.1 Geometric interpretation of va

Assuming that the derivatives of u and w with respect to time and space tend to zero, the expansions (37View Equation) – (38View Equation) imply (recall that va > 0)
va(𝜃) = lim 𝒦1 ∕2(τ,𝜃), (39 ) τ→ ∞
where the kinetic energy density 𝒦 is given by Equation (28View Equation). In other words, from a geometric point of view, v a is the same object as in the polarized case. Furthermore, it is possible to prove that if expansions of the form of Equations (37View Equation) – (38View Equation) hold with 0 < va(𝜃0) < 1, then the Kretschmann scalar is unbounded along causal curves whose 𝜃-component converges to 𝜃0 in the direction towards the singularity; see Section 7.4. As in the polarized case, we shall refer to va as the velocity. The condition 0 < va < 1 will in the future be referred to as low velocity, whereas va ≥ 1 will be referred to as high velocity.
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