ar X iv : g r - qc / 0 21 10 10 v 1 3 N ov 2 00 2 Can an observer really catch up with light ? ∗

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 will give a time-like curve from p to q. This is a well-known theory proved in the famous book[3]. In the paper we prove that the timelike curves coming from the above-mentioned variation have a proper acceleration which approaches infinity as the time-like curve approaches the null geodesic. This means no observer can be infinitesimally near the light and begin at the same point with the light and finally catch the light. Only separated from the light path finitely, does the observer can begin at the same point with the light and finally catch the light. PACS numbers: 0420 It is well-known that an observer in ”hyperbolic” motion in Minkowski space-time has a constant proper acceleration (the magnitude of the 4-acceleration). The equations of the world line of one of such observers can be expressed concisely as x = x0, y = y0, z = z0, (and the proper acceleration A = x 0 .) where t, x, y, z are the Rindler coordinates, and the line element in these coordinates of the Minkowski metric reads ds = −xdt + dx + dy + dz. (1) (−∞ < t < +∞, x > 0, −∞ < y < +∞, −∞ < z < +∞.) Consider a one-parameter family of hyperbolic observers (x0, being the parameter) with the same y0 and z0, then the proper acceleration, A = x −1 0 , approaches infinity as x0 approaches zero. As a limit case, the curve defined by x = 0, y = y0, z = z0 is a null geodesic. Unlike time-like curves, the concept of 4-acceleration of a null geodesic (or even a null curve), to our knowledge, has not been defined. The fact that A → ∞ as x0 → 0, however, suggests that it seems not unreasonable to define the proper acceleration of a null geodesic (or physically, a photon ) in Minkowski space-time to be infinity. This is indeed what Rindler suggested in his book [1]. The main purpose of the previous paper[2] is to generalize this result to curved space-times, namely, to argue that it is not unreasonable to define the proper acceleration of a null geodesic which is future-complete in curved space-time to be infinity. The present paper continues the main point of the paper [2], and primely concerns the proper acceleration of time-like curve coming from the variation of null geodesic with two end points fixed on the null geodesic, and gives the conclusion that the proper acceleration of this type of time-like curve does approaches infinity as the time-like curve approaches the null geodesic. Because the two end points fixed on the null geodesic, the existence of the time-like curves from variation of γ0 is in question. There are two theorems that concern the existence of the time-like curves: ∗E-mail of Tian:[email protected]. theorem1 [3],[4]. Let γ0(λ) be a smooth causal curve and let p, q ∈ γ0(λ). Then there does not exist a smooth one-parameter family of causal curves σ (u, λ) connecting p and q with σ (0, λ) = γ0(λ) and γu(λ) time-like for all u > 0 (i.e., γ0(λ) cannot be smoothly deformed to a time-like curve) if and only if γ0(λ) is null geodesic with no point conjugate to p along γ0(λ) between p and q. theorem2[3],[4].If there is a point r in (p, q) conjugate to p along γ0(λ), then there will be a variation of γ0(λ) which will give a time-like curve from p to q. We therefore suppose that γ0(λ) is a null geodesic with a point r in (p, q) conjugate to p along γ0(λ) and the existence of the time-like curves connecting p,q obtained from variation is ensured. Precisely ,we have the following definition of the variation of γ0(λ): Let (M,gab) be a 4-dimensional curved space-time and γ0: (0, λq)→ M be a null geodesic, which will later be denoted by γ0(λ) with λ its affine parameter, and with p, q ∈ γ0(λ). We define a variation of γ0 to be a C 1− map [3] σ :[0, ε) × [0, λq] → M such that (1)σ (0, λ) = γ0 (λ), (2)σ (u, 0) = p, σ (u, λq) = q (3)there is a subdivision 0 = λ1 < λ2 < ... < λn = λq of [0, λq] such that σ is C 3 on each [0, ε) × [λi, λi+1]. (4)for each constant u ∈ [0, ε) and u 6= 0, σ (u, λ) is a time-like curve and is represented by γu (λ). Denote by ( ∂ ∂λ )a u ≡ v u the tangent vector to the curve γu(λ), then ( ∂ ∂λ )a 0 ≡ v 0 satisfies the null geodesic equation: ( ∂ ∂λ )b