In the first paragraph on page 2778, the text immediately following `and we required that γ₀ (λ) be future-complete.' should begin as follows.

We define a variation of γ₀ to be a C^∞ (or at least C^∞) map [2] σ :[0, ε) × (0, ∞) → M such that

(1) σ(0, λ) = γ₀ (λ);
(2) for each constant u [0, ε) and u ≠ 0, σ (u, λ) is a time-like curve and is represented by γ_u (λ);
(3) in the pseudo-orthogonal basis that are parallely transported along the null geodesic γ₀ (λ), which satisfies

the variation vector (see the following for a definition) should not have a component approaching infinity as the parameter λ → ∞;
(4) the first derivative of the variation is not zero (see the following for its meaning).

The text immediately following equation (3) should begin as follows.

Here, we first explain what is meant by the third requirement in the variation map. We give an example: in Minkowski spacetime, in the Cartesian coordinates t, x, y, z, the null geodesic γ₀ (λ) is [λ, λ, 0, 0], and the time-like curve γ_u (λ), u ≠ 0 is a geodesic with [λ, (1-u) λ, 0, 0]; then the variation vector Z ^a is [0, -λ, 0, 0]. In the pseudo-orthogonal basis

has components proportional to the parameter λ. It is not meaningful to concern ouselves with cases like the above example where γ_u (λ), u ≠ 0 is a time-like geodesic, so we exclude them by the third requirement in the definition of the variation map.

We also thank Professor V Perlick for many helpful discussions.