Solar system effects in Schwarzschild–de Sitter space–time

Abstract

The Schwarzschild-de Sitter space-time describes the gravitational field of a spherically symmetric mass in a universe with cosmological constant Λ. Based on this space-time we calculate Solar system effects like gravitational redshift, light deflection, gravitational time delay, Perihelion shift, geodetic or de Sitter precession, as well as the influence of Λ on a Doppler measurement, used to determine the velocity of the Pioneer 10 and 11 spacecraft. For Λ = Λ0 ∼ 10 −52 m −2 the cosmological constant plays no role for all of these effects, while a value of Λ ∼ −10 −37 m −2 , if hypothetically held responsible for the Pioneer anomaly, is not compatible with the Perihelion shift.

Solar System planetary orbital motions andf(R) theories of gravity

“…What our results, in agreement with [36,37,38,39], suggest is that, in general, Solar System experiments are not able to constrain k or the other related parameters up to orders of magnitude comparable to the cosmologically relevant values. On the other hand, this fact can be also interpreted by saying that f (R) theories are viable on the Solar System scale, since their predictions are indistinguishable from GR ones, while they are appreciably different only on much larger scales, such as the cosmological one.…”

“…This result is a much smaller than the one obtained in [36], and comparable to the those obtained in [37] and [52], but it is still several orders of magnitude greater than the current value of the cosmological constant Λ 0 ≃ 10 −52 m −2 [53]. As we have seen in Sec.…”

“…The exact solution we found in [26] corresponds to the Schwarzschild-de Sitter metric, which has been studied in the past [33,34,35] and, also, more recently in connection with the problem of the cosmological constant [36,37,38]. The relevance of the Schwarzschild-de Sitter metric on f (R) theories of gravity has also been discussed in [39].…”

Contribution of the cosmological constant to the relativistic bending of light revisited

“…In Schwarzschild spacetime, h is the impact parameter but in Schwarzschild-de Sitter h is the constant of motion J/E where J and E are respectively the momentum and the energy of the photon. The relation between the point of closest approach r0 and the constant of the motion h (or b in other notations) is given by, e.g [12,17] …”

“…The argument for the non-influence of Λ was apparently first made in [9] and has been re-made and reaffirmed by other authors, see for example [10,11,12,13,14]. The common basis of their arguments is that, in the Schwarzschild-de Sitter metric (first derived by Kottler [15]), which applies when Λ is included, Λ nevertheless drops out of the exact r, φ differential equation for a light path (null geodesic).…”