Proper Accelerations of Time-Like Curves near a Null Geodesic

It is well known that when given a null geodesic γ₀(λ) with a point r in (p,q) conjugate to p along γ₀(λ), there will be a variation of γ₀(λ) which can give a time-like curve from p to q. Here we prove that the time-like curves coming from the above-mentioned variation (with the second derivative β₂ ≠ 0) have a proper acceleration A = √A^aA_a which approaches infinity as the time-like curve approaches the null geodesic. Because the curve obtained from variation of the null geodesic must be everywhere time-like, we also discuss the constraint of the‘acceleration’β^a₀ of the variation vector field on the null geodesic γ₀(λ). The acceleration β^a₀ of the variation vector field Z^a on the null geodesic γ₀(λ) cannot be zero.