CORRIGENDUM: 'Proper acceleration' of a null geodesic in curved spacetime

Details CORRIGENDUM: 'Proper acceleration' of a null geodesic in curved spacetime CORRIGENDUM: 'Proper acceleration' of a null geodesic in curved spacetime

Oct 2003

adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=2003cqgra..20.4329t&link_type=abstract

Classical and Quantum Gravity, Volume 20, Issue 19, pp. 4329 (2003).

Computer Science

Scientific paper

In the first paragraph on page 2778, the text immediately following `and we required that gamma0 (lambda) be future-complete.' should begin as follows.
We define a variation of gamma0 to be a Cinfty (or at least Cinfty) map [2] sigma :[0, varepsilon) × (0, infty) rightarrow M such that
(1) sigma(0, lambda) = gamma0 (lambda); (2) for each constant u in [0, varepsilon) and u \not= 0, sigma (u, lambda) is a time-like curve and is represented by gammau (lambda); (3) in the pseudo-orthogonal basis ? that are parallely transported along the null geodesic gamma0 (lambda), which satisfies ? the variation vector (see the following for a definition) should not have a component approaching infinity as the parameter lambda rightarrow infty; (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 gamma0 (lambda) is [lambda, lambda, 0, 0], and the time-like curve gammau (lambda), u \not= 0 is a geodesic with [lambda, (1-u) lambda, 0, 0]; then the variation vector Z a is [0, -lambda, 0, 0]. In the pseudo-orthogonal basis ? has components proportional to the parameter lambda. It is not meaningful to concern ouselves with cases like the above example where gammau (lambda), u \not= 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.

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