We explicitly worked out the orbital effects induced on the trajectory of a test particle by the the weak-field approximation of the Kerr-de Sitter metric. It results that the node, the pericentre and the mean anomaly undergo secular precessions proportional to k, which is a measure of the non linearity of the theory. We used such theoretical predictions and the latest observational determinations of the non-standard precessions of the perihelia of the inner planets of the Solar System to put a bound on k getting k <= 10^-29 m^-2. The node rate of the LAGEOS Earth's satellite yields k <= 10^-26 m^-2. The periastron precession of the double pulsar PSR J0737-3039A/B allows to obtain k <= 3 10^-21 m^-2. Interpreting k as a cosmological constant \Lambda, it turns out that such constraints are weaker than those obtained from the Schwarzschild-de Sitter metric.
The General Theory of Relativity (GTR) has passed with excellent results many observational tests, as Solar System and binary pulsars observations show (Ni 2005;Will 2006;Turyshev 2008). As a matter of fact, the current values of the PPN parameters are in agreement with GTR predictions.
However, some observations seem to question the general relativistic model of gravitational interaction on larger scales. On the one hand, the data coming from the galactic rotation curves of spiral galaxies (Binney and Tremaine 1987) cannot be explained on the basis of Newtonian gravity or GTR: the existence of dark matter is postulated to reconcile the theoretical model with observations; furthermore, dark matter can explain the mass discrepancy in galactic clusters (Clowe et al. 2006). On the other hand, a lot of observations, such as the light curves of the type Ia supernovae and the cosmic microwave background (CMB) experiments (Riess et al. 1998;Perlmutter et al. 1999;Bennet et al. 2003), firmly state that our Universe is now undergoing a phase of accelerated expansion.
Actually, the present acceleration of the Universe cannot be explained, within GTR, unless the existence of a cosmic fluid having exotic properties is postulate, i.e. the so called dark energy.
The cosmological constant is one of the candidates to explain (in the GTR framework) what the dark energy is (see e.g. Peebles and Ratra (2003) and references therein). On the other hand, modified gravity models that go beyond GTR have been proposed to try to explain current observations and, among these models, f (R) theories of gravity (Sotiriou and Faraoni 2008) received much attention in recent years. In these theories the gravitational lagrangian depends on an arbitrary function f of the scalar curvature R; they are also referred to as “extended theories of gravity”, since they naturally generalize, on a geometric ground, GTR: namely, when f (R) = R the action reduces to the usual Einstein-Hilbert action, and Einstein’s theory is obtained. It is interesting to point out that the vacuum solutions of GTR with a cosmological constant are also solutions of f (R) gravity vacuum field equations: this is always true in the Palatini formalism, while in metric f (R) gravity this holds for the solutions with constant scalar curvature R (Ferraris et al. 1993;Allemandi et al. 2005;Magnano 1995).
The relevance of the cosmological constant in modern gravitational physics is manifest, and it is interesting to focus on the solutions of Einstein’s field equations with cosmological constant, to investigate its role on different scales. For instance, the Schwarzschild-de Sitter metric, which describes a point-like mass in a space-time with a cosmological constant, has been recently studied by Kagramanova et al. (2006); Sereno and Jetzer (2006); Jetzer and Sereno (2006); Iorio (2006b). In particular, the Schwarzschild-de Sitter metric has been studied to investigate the influence of the cosmological constant on gravitational lensing in Rindler and Ishak (2007); Sereno (2008a); Ruggiero (2007); Sereno (2008b).
In this paper we are concerned with the Kerr-de Sitter metric, which describes a rotating black-hole in a space-time with a cosmological constant (Demianski 1973;Carter 1973;Kerr et al. 2003;Kraniotis 2004Kraniotis , 2005Kraniotis , 2007)). In particular, we want to study the gravito-magnetic (GM) effects in Kerr-de Sitter metric. GM effects are due to the rotation of the sources of the gravitational field: this gives raise to the presence of off-diagonal terms in the metric tensor, which are responsible for a variety of effects concerning orbiting test particles, precessing gyroscopes, moving clocks and atoms and propagating electromagnetic waves (Mashhoon et al. 2001;Ruggiero and Tartaglia 2002;Schäfer 2004;Mashhoon 2007).
They are expected in GTR, but are generally very small and, hence, very difficult to detect (Iorio 2007a). In recent years, there have been some attempts to measure the Lense-Thirring effect (Lense and Thirring 1918) with the LAGEOS and LAGEOS II laser-ranged satellites in the gravitational field of the Earth (Ciufolini and Pavlis 2004); the evaluation of the realistic accuracy reached in such a test and other topics related to it are still matter of debate (Iorio 2006a(Iorio , 2007b)). For other attempts to measure the Lense-Thirring effect in other Solar System scenarios with natural and artificial satellites, see (Iorio 2007a). In April 2004 the Gravity Probe B spacecraft (Everitt et al. 2001) was launched to accurately measure the gravito-magnetic (and geodetic) precession of an orbiting gyroscope (Pugh 1959;Schiff 1960) in the terrestrial space environment: the final results are going to be published. We focus on the GM effects in Kerr-de Sitter metric (the GM precession of an orbiting gyroscope was investigated by Ruggiero (2008)): in particular we work out the GM effects in the weak-field and slow-motion approximation on the orbit of a tes
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