A line source in Minkowski for the de Sitter spacetime scalar Green's function: Massless minimally coupled case

Abstract

Motivated by the desire to understand the causal structure of physical signals produced in curved spacetimesparticularly around black holes -we show how, for certain classes of geometries, one might obtain its retarded or advanced minimally coupled massless scalar Green's function by using the corresponding Green's functions in the higher dimensional Minkowski spacetime where it is embedded. Analogous statements hold for certain classes of curved Riemannian spaces, with positive definite metrics, which may be embedded in higher dimensional

Gravitational wave memory in dS 4+2 n and 4D cosmology

“…This is, of course, equivalent to the fact that the tail of the de Sitter minimally coupled massless scalar Green's function is a constant in all even dimensions higher than 2 [5]. In even d ≥ 4 dimensions, the tail part of the tensor mode solution in eq.…”

“…On the other hand, that we have been able to successfully utilize Nariai's ansatz [13] to solve Einstein's equations linearized on dS d≥4 suggests it may be worthwhile to try solving them for a background spatially flat FLRW geometry with a more general equation-of-state w. These results could not only provide further insight into potential GW memory effects, they could lead to new perspectives on the study of large scale structure in our universe. Finally, even though [5] The main goal of this section is the derivation of eq. (5).…”

“…Moreover, it is possible to understand why tails exist in odd dimensional flat spacetime by embedding it in one higher dimensional Minkowski [4]. This has prompted further generalizations to de Sitter spacetime [5], which can be viewed as a hyperboloid situated in one higher dimensional flat spacetime. Specifically, the causal structure of the de Sitter scalar Green's function can be related…”

Relation between quantum effects in general relativity and embedding theory

“…In [26,27] a similar result for the relation between two-point Green's functions was obtained, and a relation between the causal structure of the ambient space and that of the embedded Riemannian space was analyzed. The result was also obtained under the assumption that the embedding corresponds to a factorization of the flat metric of the ambient space in form (39).…”

“…This can be done for surfaces corresponding to a so-called warped geometry. The corresponding analyses were performed for the Wightman functions in [25] and for the Green's function in [26,27]. In Sec.…”