Gravitomagnetism and its Measurements with Lunar Laser Ranging, LAGEOS Satellites and Gravity Probe B

It is sometimes claimed that gravitomagnetism, measured by the LAGEOS satellites using the GRACE models [Ciufolini and Pavlis 2004], that might also be detected by refining the Gravity Probe B data analysis [Muhlfelder, Keiser and Turneaure 2007], has already been observed by Lunar Laser Ranging, LLR [Murphy, Nordtvedt and Turyshev 2007]. Here we show that there are two types of gravitomagnetic effects, the first one, measured by LLR, depends on the frame of reference being used in the analysis and can be eliminated by a coordinate transformation (see below), the second effect, measured by LAGEOS with approximately 10 % accuracy and that should be measured by LARES with about 1 % accuracy, is an intrinsic gravitomagnetic effect [Ciufolini 1994, Ciufolini and Wheeler 1995] that cannot be eliminated by means of any coordinate transformation [Ciufolini and Pavlis 2007]. Let us first briefly discuss the interesting formal analogy with electromagnetism. Nevertheless, we have to stress that apart from formal analogies, general relativity, even the linearized theory, and electromagnetism are fundamentally different. The main difference is the equivalence principle: locally (in the spacetime) it is possible to eliminate (in the sense of making arbitrarily small) the effects of the gravitational field in a freely falling frame. This is true for gravity only. In general relativity, the geometry, gαβ , where the various physical phenomena take place, is determined by the energy and by the energy-currents in the universe via the Einstein field equation and since the gravity field, gαβ , has energy and momentum, the gravitational energy contributes itself, in a loop, to the spacetime geometry gαβ . However, in special relativistic electrodynamics, the spacetime geometry ηαβ, where the electromagnetic phenomena take place, is completely unaffected by the electromagnetic phenomena. In electromagnetism, in the frame in which a charge q is at rest, we only have a non-zero electric field E but no magnetic field B. However, if we consider an observer moving with velocity v relative to the charge q,