The gravitational light shift and the Sachs-Wolfe effect

Abstract

Using a 3+1 decomposition of spacetime, we derive a new formula to compute the gravitational light shifts as measured by two observers which are normal to the spacelike hypersurfaces defining the foliation. This formula is quite general and is also independent of the existence of Killing fields. Known examples are considered to illustrate the usefulness of the formula. In particular, we focus on the Sachs-Wolfe effect that arises in a perturbed Friedman-Robertson-Walker cosmology.Comment: 8 pages; accepted for publication in General Relativity and Gravitatio

3+1 geodesic equation and images in numerical spacetimes

“…We have re-expressed the geodesic equation within the framework of the 3+1 formalism of general relativity, obtaining equations (25), (28) and (35). Equation (25), ruling the evolution of the particle energy with respect to Eulerian observers, has already been derived (in an equivalent form) by Merlín and Salgado [16]. On the other hand, the system (28) for the position of the geodesic and the redshift formula (35) are novel.…”

“…In this way, the problem considered here generalizes that of Ref. [16], which was limited to Eulerian observers. The redshift factor z is defined by…”

“…This evolution equation for the particle energy relative to the Eulerian observer is equivalent to equation ( 6) of [16]. The latter is actually an evolution equation in term of some affine parameter λ along the geodesic, i.e.…”

“…Note that these observers are not necessarily Eulerian observers. In this way, the problem considered here generalizes that of [16], which was limited to Eulerian observers. The redshift factor z is defined by…”